The cambridge handbook of physics formulas

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The Cambridge Handbook of Physics Formulas
The Cambridge Handbook of Physics Formulas is a quick-reference aid for students and pro-
fessionals in the physical sciences and engineering. It contains more than 2 000 of the most
useful formulas and equations found in undergraduate physics courses, covering mathematics,
dynamics and mechanics, quantum physics, thermodynamics, solid state physics, electromag-
netism, optics, and astrophysics. An exhaustive index allows the required formulas to be
located swiftly and simply, and the unique tabular format crisply identiﬁes all the variables
involved.
The Cambridge Handbook of Physics Formulas comprehensively covers the major topics
explored in undergraduate physics courses. It is designed to be a compact, portable, reference
book suitable for everyday work, problem solving, or exam revision. All students and
professionals in physics, applied mathematics, engineering, and other physical sciences will
want to have this essential reference book within easy reach.
Graham Woan is a senior lecturer in the Department of Physics and Astronomy at the
University of Glasgow. Prior to this he taught physics at the University of Cambridge
where he also received his degree in Natural Sciences, specialising in physics, and his
PhD, in radio astronomy. His research interests range widely with a special focus on
low-frequency radio astronomy. His publications span journals as diverse as Astronomy
& Astrophysics, Geophysical Research Letters, Advances in Space Science, the Journal of
Navigation and Emergency Prehospital Medicine. He was co-developer of the revolutionary
CURSOR radio positioning system, which uses existing broadcast transmitters to determine
position, and he is the designer of the Glasgow Millennium Sundial.

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The Cambridge Handbook of
Physics Formulas
2003 Edition
GRAHAM WOAN
Department of Physics & Astronomy
University of Glasgow

  
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Cambridge University Press
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2000
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Published in the United States of America by Cambridge University Press, New York
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Contents
Preface page vii
How to use this book 1
1 Units, constants, and conversions 3
1.1 Introduction, 3 • 1.2 SI units, 4 • 1.3 Physical constants, 6
• 1.4 Converting between units, 10 • 1.5 Dimensions, 16
• 1.6 Miscellaneous, 18
2 Mathematics 19
2.1 Notation, 19 • 2.2 Vectors and matrices, 20 • 2.3 Series, summations,
and progressions, 27 • 2.4 Complex variables, 30 • 2.5 Trigonometric and
hyperbolic formulas, 32 • 2.6 Mensuration, 35 • 2.7 Diﬀerentiation, 40
• 2.8 Integration, 44 • 2.9 Special functions and polynomials, 46
• 2.10 Roots of quadratic and cubic equations, 50 • 2.11 Fourier series
and transforms, 52 • 2.12 Laplace transforms, 55 • 2.13 Probability and
statistics, 57 • 2.14 Numerical methods, 60
3 Dynamics and mechanics 63
3.1 Introduction, 63 • 3.2 Frames of reference, 64 • 3.3 Gravitation, 66
• 3.4 Particle motion, 68 • 3.5 Rigid body dynamics, 74 • 3.6 Oscillating
systems, 78 • 3.7 Generalised dynamics, 79 • 3.8 Elasticity, 80 • 3.9 Fluid
dynamics, 84
4 Quantum physics 89
4.1 Introduction, 89 • 4.2 Quantum deﬁnitions, 90 • 4.3 Wave
mechanics, 92 • 4.4 Hydrogenic atoms, 95 • 4.5 Angular momentum, 98
• 4.6 Perturbation theory, 102 • 4.7 High energy and nuclear physics, 103
5 Thermodynamics 105
5.1 Introduction, 105 • 5.2 Classical thermodynamics, 106 • 5.3 Gas
laws, 110 • 5.4 Kinetic theory, 112 • 5.5 Statistical thermodynamics, 114
• 5.6 Fluctuations and noise, 116 • 5.7 Radiation processes, 118

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6 Solid state physics 123
6.1 Introduction, 123 • 6.2 Periodic table, 124 • 6.3 Crystalline
structure, 126 • 6.4 Lattice dynamics, 129 • 6.5 Electrons in solids, 132
7 Electromagnetism 135
7.1 Introduction, 135 • 7.2 Static fields, 136 • 7.3 Electromagnetic fields
(general), 139 • 7.4 Fields associated with media, 142 • 7.5 Force, torque,
and energy, 145 • 7.6 LCR circuits, 147 • 7.7 Transmission lines and
waveguides, 150 • 7.8 Waves in and out of media, 152 • 7.9 Plasma
physics, 156
8 Optics 161
8.1 Introduction, 161 • 8.2 Interference, 162 • 8.3 Fraunhofer diffraction,
164 • 8.4 Fresnel diffraction, 166 • 8.5 Geometrical optics, 168
• 8.6 Polarisation, 170 • 8.7 Coherence (scalar theory), 172 • 8.8 Line
radiation, 173
9 Astrophysics 175
9.1 Introduction, 175 • 9.2 Solar system data, 176 • 9.3 Coordinate
transformations (astronomical), 177 • 9.4 Observational astrophysics, 179
• 9.5 Stellar evolution, 181 • 9.6 Cosmology, 184
Index 187

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Preface
In A Brief History of Time, Stephen Hawking relates that he was warned against including
equations in the book because “each equation... would halve the sales.” Despite this dire
prediction there is, for a scientific audience, some attraction in doing the exact opposite.
The reader should not be misled by this exercise. Although the equations and formulas
contained here underpin a good deal of physical science they are useless unless the reader
understands them. Learning physics is not about remembering equations, it is about appreci-
ating the natural structures they express. Although its format should help make some topics
clearer, this book is not designed to teach new physics; there are many excellent textbooks
to help with that. It is intended to be useful rather than pedagogically complete, so that
students can use it for revision and for structuring their knowledge once they understand
the physics. More advanced users will benefit from having a compact, internally consistent,
source of equations that can quickly deliver the relationship they require in a format that
avoids the need to sift through pages of rubric.
Some difficult decisions have had to be made to achieve this. First, to be short the
book only includes ideas that can be expressed succinctly in equations, without resorting
to lengthy explanation. A small number of important topics are therefore absent. For
example, Liouville’s theorem can be algebraically succinct (˙ = 0) but is meaningless unless ˙
is thoroughly (and carefully) explained. Anyone who already understands what ˙ represents
will probably not need reminding that it equals zero. Second, empirical equations with
numerical coefficients have been largely omitted, as have topics significantly more advanced
than are found at undergraduate level. There are simply too many of these to be sensibly and
confidently edited into a short handbook. Third, physical data are largely absent, although
a periodic table, tables of physical constants, and data on the solar system are all included.
Just a sighting of the marvellous (but dimensionally misnamed) CRC Handbook of Chemistry
and Physics should be enough to convince the reader that a good science data book is thick.
Inevitably there is personal choice in what should or should not be included, and you
may feel that an equation that meets the above criteria is missing. If this is the case, I would
be delighted to hear from you so it can be considered for a subsequent edition. Contact
details are at the end of this preface. Likewise, if you spot an error or an inconsistency then
please let me know and I will post an erratum on the web page.

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Acknowledgments This venture is founded on the generosity of colleagues in Glasgow and
Cambridge whose inputs have strongly influenced the final product. The expertise of Dave
Clarke, Declan Diver, Peter Duffett-Smith, Wolf-Gerrit Früh, Martin Hendry, Rico Ignace,
David Ireland, John Simmons, and Harry Ward have been central to its production, as have
the linguistic skills of Katie Lowe. I would also like to thank Richard Barrett, Matthew
Cartmell, Steve Gull, Martin Hendry, Jim Hough, Darren McDonald, and Ken Riley who
all agreed to field-test the book and gave invaluable feedback.
My greatest thanks though are to John Shakeshaft who, with remarkable knowledge and
skill, worked through the entire manuscript more than once during its production and whose
legendary red pen hovered over (or descended upon) every equation in the book. What errors
remain are, of course, my own, but I take comfort from the fact that without John they
would be much more numerous.
Contact information A website containing up-to-date information on this handbook and
contact details can be found through the Cambridge University Press web pages at
us.cambridge.org (North America) or uk.cambridge.org (United Kingdom), or directly
at radio.astro.gla.ac.uk/hbhome.html.
Production notes This book was typeset by the author in LATEX2ε using the CUP Times fonts.
The software packages used were WinEdt, MiKTEX, Mayura Draw, Gnuplot, Ghostscript,
Ghostview, and Maple V.
Comments on the 2002 edition I am grateful to all those who have suggested improvements,
in particular Martin Hendry, Wolfgang Jitschin, and Joseph Katz. Although this edition
contains only minor revisions to the original its production was also an opportunity to
update the physical constants and periodic table entries and to reflect recent developments
in cosmology.

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How to use this book
The format is largely self-explanatory, but a few comments may be helpful. Although it is
very tempting to flick through the pages to find what you are looking for, the best starting
point is the index. I have tried to make this as extensive as possible, and many equations are
indexed more than once. Equations are listed both with their equation number (in square
brackets) and the page on which they can be found. The equations themselves are grouped
into self-contained and boxed “panels” on the pages. Each panel represents a separate topic,
and you will find descriptions of all the variables used at the right-hand side of the panel,
usually adjacent to the first equation in which they are used. You should therefore not need
to stray outside the panel to understand the notation. Both the panel as a whole and its
individual entries may have footnotes, shown below the panel. Be aware of these, as they
contain important additional information and conditions relevant to the topic.
Although the panels are self-contained they may use concepts defined elsewhere in the
handbook. Often these are cross-referenced, but again the index will help you to locate them
if necessary. Notations and definitions are uniform over subject areas unless stated otherwise.

main January 23, 2006 16:6
1
Chapter 1 Units, constants, and conversions
1.1 Introduction
The determination of physical constants and the definition of the units with which they are
measured is a specialised and, to many, hidden branch of science.
A quantity with dimensions is one whose value must be expressed relative to one or
more standard units. In the spirit of the rest of the book, this section is based around
the International System of units (SI). This system uses seven base units1
(the number is
somewhat arbitrary), such as the kilogram and the second, and defines their magnitudes in
terms of physical laws or, in the case of the kilogram, an object called the “international
prototype of the kilogram” kept in Paris. For convenience there are also a number of derived
standards, such as the volt, which are defined as set combinations of the basic seven. Most of
the physical observables we regard as being in some sense fundamental, such as the charge
on an electron, are now known to a relative standard uncertainty,2
ur, of less than 10−7
.
The least well determined is the Newtonian constant of gravitation, presently standing at a
rather lamentable ur of 1.5 × 10−3
, and the best is the Rydberg constant (ur = 7.6 × 10−12
).
The dimensionless electron g-factor, representing twice the magnetic moment of an electron
measured in Bohr magnetons, is now known to a relative uncertainty of only 4.1 × 10−12
.
No matter which base units are used, physical quantities are expressed as the product of
a numerical value and a unit. These two components have more-or-less equal standing and
can be manipulated by following the usual rules of algebra. So, if 1 · eV = 160.218 × 10−21
· J
then 1 · J = [1/(160.218 × 10−21
)] · eV. A measurement of energy, U, with joule as the unit
has a numerical value of U/ J. The same measurement with electron volt as the unit has a
numerical value of U/ eV = (U/ J) · ( J/ eV) and so on.
1The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.
The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. The second
is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine
levels of the ground state of the caesium 133 atom. The ampere is that constant current which, if maintained in
two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in
vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length. The kelvin,
unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point
of water. The mole is the amount of substance of a system which contains as many elementary entities as there are
atoms in 0.012 kilogram of carbon 12; its symbol is “mol.” When the mole is used, the elementary entities must
be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. The
candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency
540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.
2The relative standard uncertainty in x is defined as the estimated standard deviation in x divided by the modulus
of x (x = 0).

4 Units, constants, and conversions
1.2 SI units
SI base units
physical quantity name symbol
length metrea
m
mass kilogram kg
time interval second s
electric current ampere A
thermodynamic temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
aOr “meter”.
SI derived units
physical quantity name symbol equivalent units
catalytic activity katal kat mol s−1
electric capacitance farad F C V−1
electric charge coulomb C A s
electric conductance siemens S Ω−1
electric potential difference volt V J C−1
electric resistance ohm Ω V A−1
energy, work, heat joule J N m
force newton N m kg s−2
frequency hertz Hz s−1
illuminance lux lx cd sr m−2
inductance henry H V A−1
s
luminous flux lumen lm cd sr
magnetic flux weber Wb V s
magnetic flux density tesla T V s m−2
plane angle radian rad m m−1
power, radiant flux watt W J s−1
pressure, stress pascal Pa N m−2
radiation absorbed dose gray Gy J kg−1
radiation dose equivalenta
sievert Sv [ J kg−1
]
radioactive activity becquerel Bq s−1
solid angle steradian sr m2
m−2
temperatureb
degree Celsius ◦
C K
aTo distinguish it from the gray, units of J kg−1
should not be used for the sievert in practice.
bThe Celsius temperature, TC, is defined from the temperature in kelvin, TK, by TC = TK − 273.15.

1.2 SI units
1
5
SI prefixesa
factor prefix symbol factor prefix symbol
1024
yotta Y 10−24
yocto y
1021
zetta Z 10−21
zepto z
1018
exa E 10−18
atto a
1015
peta P 10−15
femto f
1012
tera T 10−12
pico p
109
giga G 10−9
nano n
106
mega M 10−6
micro µ
103
kilo k 10−3
milli m
102
hecto h 10−2
centi c
101
decab
da 10−1
deci d
aThe kilogram is the only SI unit with a prefix embedded in its
name and symbol. For mass, the unit name “gram” and unit symbol
“g” should be used with these prefixes, hence 10−6 kg can be written
as 1 mg. Otherwise, any prefix can be applied to any SI unit.
bOr “deka”.
Recognised non-SI units
physical quantity name symbol SI value
area barn b 10−28
m2
energy electron volt eV 1.602 18 × 10−19
J
length ˚angström ˚A 10−10
m
fermia
fm 10−15
m
microna
µm 10−6
m
plane angle degree ◦
(π/180) rad
arcminute (π/10 800) rad
arcsecond (π/648 000) rad
pressure bar bar 105
N m−2
time minute min 60 s
hour h 3 600 s
day d 86 400 s
mass unified atomic
mass unit u 1.660 54 × 10−27
kg
tonnea,b
t 103
kg
volume litrec
l, L 10−3
m3
aThese are non-SI names for SI quantities.
bOr “metric ton.”
cOr “liter”. The symbol “l” should be avoided.

1.3 Physical constants
The following 1998 CODATA recommended values for the fundamental physical constants
can also be found on the Web at physics.nist.gov/constants. Detailed background
information is available in Reviews of Modern Physics, Vol. 72, No. 2, pp. 351–495, April
2000.
The digits in parentheses represent the 1σ uncertainty in the previous two quoted digits. For
example, G = (6.673±0.010)×10−11
m3
kg−1
s−2
. It is important to note that the uncertainties
for many of the listed quantities are correlated, so that the uncertainty in any expression
using them in combination cannot necessarily be computed from the data presented. Suitable
covariance values are available in the above references.
Summary of physical constants
speed of light in vacuuma c 2.997 924 58 ×108
m s−1
permeability of vacuumb µ0 4π ×10−7
H m−1
=12.566 370 614 . . . ×10−7
H m−1
permittivity of vacuum 0 1/(µ0c2
) F m−1
=8.854 187 817 . . . ×10−12
F m−1
constant of gravitationc G 6.673(10) ×10−11
m3
kg−1
s−2
Planck constant h 6.626 068 76(52) ×10−34
J s
h/(2π) ¯h 1.054 571 596(82) ×10−34
J s
elementary charge e 1.602 176 462(63) ×10−19
C
magnetic flux quantum, h/(2e) Φ0 2.067 833 636(81) ×10−15
Wb
electron volt eV 1.602 176 462(63) ×10−19
J
electron mass me 9.109 381 88(72) ×10−31
kg
proton mass mp 1.672 621 58(13) ×10−27
kg
proton/electron mass ratio mp/me 1 836.152 667 5(39)
unified atomic mass unit u 1.660 538 73(13) ×10−27
kg
fine-structure constant, µ0ce2
/(2h) α 7.297 352 533(27) ×10−3
inverse 1/α 137.035 999 76(50)
Rydberg constant, mecα2
/(2h) R∞ 1.097 373 156 854 9(83) ×107
m−1
Avogadro constant NA 6.022 141 99(47) ×1023
mol−1
Faraday constant, NAe F 9.648 534 15(39) ×104
C mol−1
molar gas constant R 8.314 472(15) J mol−1
K−1
Boltzmann constant, R/NA k 1.380 650 3(24) ×10−23
J K−1
Stefan–Boltzmann constant,
π2
k4
/(60¯h3
c2
)
σ 5.670 400(40) ×10−8
W m−2
K−4
Bohr magneton, e¯h/(2me) µB 9.274 008 99(37) ×10−24
J T−1
aBy definition, the speed of light is exact.
bAlso exact, by definition. Alternative units are N A−2.
cThe standard acceleration due to gravity, g, is defined as exactly 9.806 65 m s−2.

1
7
General constants
speed of light in vacuum c 2.997 924 58 ×108
m s−1
permeability of vacuum µ0 4π ×10−7
H m−1
=12.566 370 614 . . . ×10−7
H m−1
permittivity of vacuum 0 1/(µ0c2
) F m−1
=8.854 187 817 . . . ×10−12
F m−1
impedance of free space Z0 µ0c Ω
=376.730 313 461 . . . Ω
constant of gravitation G 6.673(10) ×10−11
m3
kg−1
s−2
Planck constant h 6.626 068 76(52) ×10−34
J s
in eV s 4.135 667 27(16) ×10−15
eV s
h/(2π) ¯h 1.054 571 596(82) ×10−34
J s
in eV s 6.582 118 89(26) ×10−16
eV s
Planck mass, (¯hc/G)1/2 mPl 2.176 7(16) ×10−8
kg
Planck length, ¯h/(mPlc) = (¯hG/c3
)1/2 lPl 1.616 0(12) ×10−35
m
Planck time, lPl/c = (¯hG/c5
)1/2 tPl 5.390 6(40) ×10−44
s
elementary charge e 1.602 176 462(63) ×10−19
C
magnetic ﬂux quantum, h/(2e) Φ0 2.067 833 636(81) ×10−15
Wb
Josephson frequency/voltage ratio 2e/h 4.835 978 98(19) ×1014
Hz V−1
Bohr magneton, e¯h/(2me) µB 9.274 008 99(37) ×10−24
J T−1
in eV T−1 5.788 381 749(43) ×10−5
eV T−1
µB/k 0.671 713 1(12) K T−1
nuclear magneton, e¯h/(2mp) µN 5.050 783 17(20) ×10−27
J T−1
in eV T−1 3.152 451 238(24) ×10−8
eV T−1
µN/k 3.658 263 8(64) ×10−4
K T−1
Zeeman splitting constant µB/(hc) 46.686 452 1(19) m−1
T−1
Atomic constantsa
ﬁne-structure constant, µ0ce2
/(2h) α 7.297 352 533(27) ×10−3
inverse 1/α 137.035 999 76(50)
Rydberg constant, mecα2
/(2h) R∞ 1.097 373 156 854 9(83) ×107
m−1
R∞c 3.289 841 960 368(25) ×1015
Hz
R∞hc 2.179 871 90(17) ×10−18
J
R∞hc/e 13.605 691 72(53) eV
Bohr radiusb
, α/(4πR∞) a0 5.291 772 083(19) ×10−11
m
aSee also the Bohr model on page 95.
bFixed nucleus.

Electron constants
electron mass me 9.109 381 88(72) ×10−31
kg
in MeV 0.510 998 902(21) MeV
electron/proton mass ratio me/mp 5.446 170 232(12) ×10−4
electron charge −e −1.602 176 462(63) ×10−19
C
electron speciﬁc charge −e/me −1.758 820 174(71) ×1011
C kg−1
electron molar mass, NAme Me 5.485 799 110(12) ×10−7
kg mol−1
Compton wavelength, h/(mec) λC 2.426 310 215(18) ×10−12
m
classical electron radius, α2
a0 re 2.817 940 285(31) ×10−15
m
Thomson cross section, (8π/3)r2
e σT 6.652 458 54(15) ×10−29
m2
electron magnetic moment µe −9.284 763 62(37) ×10−24
J T−1
in Bohr magnetons, µe/µB −1.001 159 652 186 9(41)
in nuclear magnetons, µe/µN −1 838.281 966 0(39)
electron gyromagnetic ratio, 2|µe|/¯h γe 1.760 859 794(71) ×1011
s−1
T−1
electron g-factor, 2µe/µB ge −2.002 319 304 3737(82)
Proton constants
proton mass mp 1.672 621 58(13) ×10−27
kg
in MeV 938.271 998(38) MeV
proton/electron mass ratio mp/me 1 836.152 667 5(39)
proton charge e 1.602 176 462(63) ×10−19
C
proton speciﬁc charge e/mp 9.578 834 08(38) ×107
C kg−1
proton molar mass, NAmp Mp 1.007 276 466 88(13) ×10−3
kg mol−1
proton Compton wavelength, h/(mpc) λC,p 1.321 409 847(10) ×10−15
m
proton magnetic moment µp 1.410 606 633(58) ×10−26
J T−1
in Bohr magnetons, µp/µB 1.521 032 203(15) ×10−3
in nuclear magnetons, µp/µN 2.792 847 337(29)
proton gyromagnetic ratio, 2µp/¯h γp 2.675 222 12(11) ×108
s−1
T−1
Neutron constants
neutron mass mn 1.674 927 16(13) ×10−27
kg
in MeV 939.565 330(38) MeV
neutron/electron mass ratio mn/me 1 838.683 655 0(40)
neutron/proton mass ratio mn/mp 1.001 378 418 87(58)
neutron molar mass, NAmn Mn 1.008 664 915 78(55) ×10−3
kg mol−1
neutron Compton wavelength, h/(mnc) λC,n 1.319 590 898(10) ×10−15
m
neutron magnetic moment µn −9.662 364 0(23) ×10−27
J T−1
in Bohr magnetons µn/µB −1.041 875 63(25) ×10−3
in nuclear magnetons µn/µN −1.913 042 72(45)
neutron gyromagnetic ratio, 2|µn|/¯h γn 1.832 471 88(44) ×108
s−1
T−1

1
9
Muon and tau constants
muon mass mµ 1.883 531 09(16) ×10−28
kg
in MeV 105.658 356 8(52) MeV
tau mass mτ 3.167 88(52) ×10−27
kg
in MeV 1.777 05(29) ×103
MeV
muon/electron mass ratio mµ/me 206.768 262(30)
muon charge −e −1.602 176 462(63) ×10−19
C
muon magnetic moment µµ −4.490 448 13(22) ×10−26
J T−1
in Bohr magnetons, µµ/µB 4.841 970 85(15) ×10−3
in nuclear magnetons, µµ/µN 8.890 597 70(27)
muon g-factor gµ −2.002 331 832 0(13)
Bulk physical constants
Avogadro constant NA 6.022 141 99(47) ×1023
mol−1
atomic mass constanta mu 1.660 538 73(13) ×10−27
kg
in MeV 931.494 013(37) MeV
Faraday constant F 9.648 534 15(39) ×104
C mol−1
molar gas constant R 8.314 472(15) J mol−1
K−1
Boltzmann constant, R/NA k 1.380 650 3(24) ×10−23
J K−1
in eV K−1 8.617 342(15) ×10−5
eV K−1
molar volume (ideal gas at stp)b Vm 22.413 996(39) ×10−3
m3
mol−1
Stefan–Boltzmann constant, π2
k4
/(60¯h3
c2
) σ 5.670 400(40) ×10−8
W m−2
K−4
Wien’s displacement law constant,c
b = λmT b 2.897 768 6(51) ×10−3
m K
a= mass of 12C/12. Alternative nomenclature for the uniﬁed atomic mass unit, u.
bStandard temperature and pressure (stp) are T = 273.15 K (0◦C) and P = 101 325 Pa (1 standard atmosphere).
cSee also page 121.
Mathematical constants
pi (π) 3.141 592 653 589 793 238 462 643 383 279 . . .
exponential constant (e) 2.718 281 828 459 045 235 360 287 471 352 . . .
Catalan’s constant 0.915 965 594 177 219 015 054 603 514 932 . . .
Euler’s constanta
(γ) 0.577 215 664 901 532 860 606 512 090 082 . . .
Feigenbaum’s constant (α) 2.502 907 875 095 892 822 283 902 873 218 . . .
Feigenbaum’s constant (δ) 4.669 201 609 102 990 671 853 203 820 466 . . .
Gibbs constant 1.851 937 051 982 466 170 361 053 370 157 . . .
golden mean 1.618 033 988 749 894 848 204 586 834 370 . . .
Madelung constantb
1.747 564 594 633 182 190 636 212 035 544 . . .
aSee also Equation (2.119).
bNaCl structure.

1.4 Converting between units
The following table lists common (and not so common) measures of physical quantities.
The numerical values given are the SI equivalent of one unit measure of the non-SI unit.
Hence 1 astronomical unit equals 149.597 9× 109
m. Those entries identiﬁed with a “∗
” in the
second column represent exact conversions; so 1 abampere equals exactly 10.0 A. Note that
individual entries in this list are not recorded in the index, and that values are “international”
unless otherwise stated.
There is a separate section on temperature conversions after this table.
unit name value in SI units
abampere 10.0∗
A
abcoulomb 10.0∗
C
abfarad 1.0∗
×109
F
abhenry 1.0∗
×10−9
H
abmho 1.0∗
×109
S
abohm 1.0∗
×10−9
Ω
abvolt 10.0∗
×10−9
V
acre 4.046 856 ×103
m2
amagat (at stp) 44.614 774 mol m−3
ampere hour 3.6∗
×103
C
˚angstr¨om 100.0∗
×10−12
m
apostilb 1.0∗
lm m−2
arcminute 290.888 2 ×10−6
rad
arcsecond 4.848 137 ×10−6
rad
are 100.0∗
m2
astronomical unit 149.597 9 ×109
m
atmosphere (standard) 101.325 0∗
×103
Pa
atomic mass unit 1.660 540 ×10−27
kg
bar 100.0∗
×103
Pa
barn 100.0∗
×10−30
m2
baromil 750.1 ×10−6
m
barrel (UK) 163.659 2 ×10−3
m3
barrel (US dry) 115.627 1 ×10−3
m3
barrel (US liquid) 119.240 5 ×10−3
m3
barrel (US oil) 158.987 3 ×10−3
m3
baud 1.0∗
s−1
bayre 100.0∗
×10−3
Pa
biot 10.0 A
bolt (US) 36.576∗
m
brewster 1.0∗
×10−12
m2
N−1
British thermal unit 1.055 056 ×103
J
bushel (UK) 36.36 872 ×10−3
m3
bushel (US) 35.23 907 ×10−3
m3
butt (UK) 477.339 4 ×10−3
m3
cable (US) 219.456∗
m
calorie 4.186 8∗
J
continued on next page . . .

1
11
candle power (spherical) 4π lm
carat (metric) 200.0∗
×10−6
kg
cental 45.359 237 kg
centare 1.0∗
m2
centimetre of Hg (0 ◦
C) 1.333 222 ×103
Pa
centimetre of H2O (4 ◦
C) 98.060 616 Pa
chain (engineers’) 30.48∗
m
chain (US) 20.116 8∗
m
Chu 1.899 101 ×103
J
clusec 1.333 224 ×10−6
W
cord 3.624 556 m3
cubit 457.2∗
×10−3
m
cumec 1.0∗
m3
s−1
cup (US) 236.588 2 ×10−6
m3
curie 37.0∗
×109
Bq
darcy 986.923 3 ×10−15
m2
day 86.4∗
×103
s
day (sidereal) 86.164 09 ×103
s
debye 3.335 641 ×10−30
C m
degree (angle) 17.453 29 ×10−3
rad
denier 111.111 1 ×10−9
kg m−1
digit 19.05∗
×10−3
m
dioptre 1.0∗
m−1
Dobson unit 10.0∗
×10−6
m
dram (avoirdupois) 1.771 845 ×10−3
kg
dyne 10.0∗
×10−6
N
dyne centimetres 100.0∗
×10−9
J
electron volt 160.217 7 ×10−21
J
ell 1.143∗
m
em 4.233 333 ×10−3
m
emu of capacitance 1.0∗
×109
F
emu of current 10.0∗
A
emu of electric potential 10.0∗
×10−9
V
emu of inductance 1.0∗
×10−9
H
emu of resistance 1.0∗
×10−9
Ω
Eötvös unit 1.0∗
×10−9
m s−2
m−1
esu of capacitance 1.112 650 ×10−12
F
esu of current 333.564 1 ×10−12
A
esu of electric potential 299.792 5 V
esu of inductance 898.755 2 ×109
H
esu of resistance 898.755 2 ×109
Ω
erg 100.0∗
×10−9
J
faraday 96.485 3 ×103
C
fathom 1.828 804 m
fermi 1.0∗
×10−15
m
Finsen unit 10.0∗
×10−6
W m−2
firkin (UK) 40.914 81 ×10−3
m3

firkin (US) 34.068 71 ×10−3
m3
fluid ounce (UK) 28.413 08 ×10−6
m3
fluid ounce (US) 29.573 53 ×10−6
m3
foot 304.8∗
×10−3
m
foot (US survey) 304.800 6 ×10−3
m
foot of water (4 ◦
C) 2.988 887 ×103
Pa
footcandle 10.763 91 lx
footlambert 3.426 259 cd m−2
footpoundal 42.140 11 ×10−3
J
footpounds (force) 1.355 818 J
fresnel 1.0∗
×1012
Hz
funal 1.0∗
×103
N
furlong 201.168∗
m
g (standard acceleration) 9.806 65∗
m s−2
gal 10.0∗
×10−3
m s−2
gallon (UK) 4.546 09∗
×10−3
m3
gallon (US liquid) 3.785 412 ×10−3
m3
gamma 1.0∗
×10−9
T
gauss 100.0∗
×10−6
T
gilbert 795.774 7 ×10−3
A turn
gill (UK) 142.065 4 ×10−6
m3
gill (US) 118.294 1 ×10−6
m3
gon π/200∗
rad
grade 15.707 96 ×10−3
rad
grain 64.798 91∗
×10−6
kg
gram 1.0∗
×10−3
kg
gram-rad 100.0∗
J kg−1
gray 1.0∗
J kg−1
hand 101.6∗
×10−3
m
hartree 4.359 748 ×10−18
J
hectare 10.0∗
×103
m2
hefner 902 ×10−3
cd
hogshead 238.669 7 ×10−3
m3
horsepower (boiler) 9.809 50 ×103
W
horsepower (electric) 746∗
W
horsepower (metric) 735.498 8 W
horsepower (UK) 745.699 9 W
hour 3.6∗
×103
s
hour (sidereal) 3.590 170 ×103
s
Hubble time 440 ×1015
s
Hubble distance 130 ×1024
m
hundredweight (UK long) 50.802 35 kg
hundredweight (US short) 45.359 24 kg
inch 25.4∗
×10−3
m
inch of mercury (0 ◦
C) 3.386 389 ×103
Pa
inch of water (4 ◦
C) 249.074 0 Pa
jansky 10.0∗
×10−27
W m−2
Hz−1

1
13
jar 10/9∗
×10−9
F
kayser 100.0∗
m−1
kilocalorie 4.186 8∗
×103
J
kilogram-force 9.806 65∗
N
kilowatt hour 3.6∗
×106
J
knot (international) 514.444 4 ×10−3
m s−1
lambert 10/π∗
×103
cd m−2
langley 41.84∗
×103
J m−2
langmuir 133.322 4 ×10−6
Pa s
league (nautical, int.) 5.556∗
×103
m
league (nautical, UK) 5.559 552 ×103
m
league (statute) 4.828 032 ×103
m
light year 9.460 73∗
×1015
m
ligne 2.256∗
×10−3
m
line 2.116 667 ×10−3
m
line (magnetic flux) 10.0∗
×10−9
Wb
link (engineers’) 304.8∗
×10−3
m
link (US) 201.168 0 ×10−3
m
litre 1.0∗
×10−3
m3
lumen (at 555 nm) 1.470 588 ×10−3
W
maxwell 10.0∗
×10−9
Wb
mho 1.0∗
S
micron 1.0∗
×10−6
m
mil (length) 25.4∗
×10−6
m
mil (volume) 1.0∗
×10−6
m3
mile (international) 1.609 344∗
×103
m
mile (nautical, int.) 1.852∗
×103
m
mile (nautical, UK) 1.853 184∗
×103
m
mile per hour 447.04∗
×10−3
m s−1
milliard 1.0∗
×109
m3
millibar 100.0∗
Pa
millimetre of Hg (0 ◦
C) 133.322 4 Pa
minim (UK) 59.193 90 ×10−9
m3
minim (US) 61.611 51 ×10−9
m3
minute (angle) 290.888 2 ×10−6
rad
minute 60.0∗
s
minute (sidereal) 59.836 17 s
month (lunar) 2.551 444 ×106
s
nit 1.0∗
cd m−2
noggin (UK) 142.065 4 ×10−6
m3
oersted 1000/(4π)∗
A m−1
ounce (avoirdupois) 28.349 52 ×10−3
kg
ounce (UK fluid) 28.413 07 ×10−6
m3
ounce (US fluid) 29.573 53 ×10−6
m3
pace 762.0∗
×10−3
m
parsec 30.856 78 ×1015
m

peck (UK) 9.092 18∗
×10−3
m3
peck (US) 8.809 768 ×10−3
m3
pennyweight (troy) 1.555 174 ×10−3
kg
perch 5.029 2∗
m
phot 10.0∗
×103
lx
pica (printers’) 4.217 518 ×10−3
m
pint (UK) 568.261 2 ×10−6
m3
pint (US dry) 550.610 5 ×10−6
m3
pint (US liquid) 473.176 5 ×10−6
m3
point (printers’) 351.459 8∗
×10−6
m
poise 100.0∗
×10−3
Pa s
pole 5.029 2∗
m
poncelot 980.665∗
W
pottle 2.273 045 ×10−3
m3
pound (avoirdupois) 453.592 4 ×10−3
kg
poundal 138.255 0 ×10−3
N
pound-force 4.448 222 N
promaxwell 1.0∗
Wb
psi 6.894 757 ×103
Pa
puncheon (UK) 317.974 6 ×10−3
m3
quad 1.055 056 ×1018
J
quart (UK) 1.136 522 ×10−3
m3
quart (US dry) 1.101 221 ×10−3
m3
quart (US liquid) 946.352 9 ×10−6
m3
quintal (metric) 100.0∗
kg
rad 10.0∗
×10−3
Gy
rayleigh 10/(4π) ×109
s−1
m−2
sr−1
rem 10.0∗
×10−3
Sv
REN 1/4 000∗
S
reyn 689.5 ×103
Pa s
rhe 10.0∗
Pa−1
s−1
rod 5.029 2∗
m
roentgen 258.0 ×10−6
C kg−1
rood (UK) 1.011 714 ×103
m2
rope (UK) 6.096∗
m
rutherford 1.0∗
×106
Bq
rydberg 2.179 874 ×10−18
J
scruple 1.295 978 ×10−3
kg
seam 290.949 8 ×10−3
m3
second (angle) 4.848 137 ×10−6
rad
second (sidereal) 997.269 6 ×10−3
s
shake 100.0∗
×10−10
s
shed 100.0∗
×10−54
m2
slug 14.593 90 kg
square degree (π/180)2∗
sr
statampere 333.564 1 ×10−12
A
statcoulomb 333.564 1 ×10−12
C

1
15
statfarad 1.112 650 ×10−12
F
stathenry 898.755 2 ×109
H
statmho 1.112 650 ×10−12
S
statohm 898.755 2 ×109
Ω
statvolt 299.792 5 V
stere 1.0∗
m3
sth´ene 1.0∗
×103
N
stilb 10.0∗
×103
cd m−2
stokes 100.0∗
×10−6
m2
s−1
stone 6.350 293 kg
tablespoon (UK) 14.206 53 ×10−6
m3
tablespoon (US) 14.786 76 ×10−6
m3
teaspoon (UK) 4.735 513 ×10−6
m3
teaspoon (US) 4.928 922 ×10−6
m3
tex 1.0∗
×10−6
kg m−1
therm (EEC) 105.506∗
×106
J
therm (US) 105.480 4∗
×106
J
thermie 4.185 407 ×106
J
thou 25.4∗
×10−6
m
tog 100.0∗
×10−3
W−1
m2
K
ton (of TNT) 4.184∗
×109
J
ton (UK long) 1.016 047 ×103
kg
ton (US short) 907.184 7 kg
tonne (metric ton) 1.0∗
×103
kg
torr 133.322 4 Pa
townsend 1.0∗
×10−21
V m2
troy dram 3.887 935 ×10−3
kg
troy ounce 31.103 48 ×10−3
kg
troy pound 373.241 7 ×10−3
kg
tun 954.678 9 ×10−3
m3
XU 100.209 ×10−15
m
yard 914.4∗
×10−3
m
year (365 days) 31.536∗
×106
s
year (sidereal) 31.558 15 ×106
s
year (tropical) 31.556 93 ×106
s
Temperature conversions
From degrees
Celsiusa TK = TC + 273.15 (1.1)
TK temperature in
kelvin
TC temperature in
◦Celsius
From degrees
Fahrenheit
TK =
TF − 32
1.8
+ 273.15 (1.2)
TF temperature in
◦Fahrenheit
From degrees
Rankine
TK =
TR
1.8
(1.3) TR temperature in
◦Rankine
aThe term “centigrade” is not used in SI, to avoid confusion with “10−2 of a degree”.

1.5 Dimensions
The following table lists the dimensions of common physical quantities, together with their
conventional symbols and the SI units in which they are usually quoted. The dimensional
basis used is length (L), mass (M), time (T), electric current (I), temperature (Θ), amount of
substance (N), and luminous intensity (J).
physical quantity symbol dimensions SI units
acceleration a L T−2
m s−2
action S L2
M T−1
J s
angular momentum L, J L2
M T−1
m2
kg s−1
angular speed ω T−1
rad s−1
area A, S L2
m2
Avogadro constant NA N−1
mol−1
bending moment Gb L2
M T−2
N m
Bohr magneton µB L2
I J T−1
Boltzmann constant k, kB L2
M T−2
Θ−1
J K−1
bulk modulus K L−1
M T−2
Pa
capacitance C L−2
M−1
T4
I2
F
charge (electric) q T I C
charge density ρ L−3
T I C m−3
conductance G L−2
M−1
T3
I2
S
conductivity σ L−3
M−1
T3
I2
S m−1
couple G, T L2
M T−2
N m
current I, i I A
current density J, j L−2
I A m−2
density ρ L−3
M kg m−3
electric displacement D L−2
T I C m−2
electric field strength E L M T−3
I−1
V m−1
electric polarisability α M−1
T4
I2
C m2
V−1
electric polarisation P L−2
T I C m−2
electric potential difference V L2
M T−3
I−1
V
energy E, U L2
M T−2
J
energy density u L−1
M T−2
J m−3
entropy S L2
M T−2
Θ−1
J K−1
Faraday constant F T I N−1
C mol−1
force F L M T−2
N
frequency ν, f T−1
Hz
gravitational constant G L3
M−1
T−2
m3
kg−1
s−2
Hall coefficient RH L3
T−1
I−1
m3
C−1
Hamiltonian H L2
M T−2
J
heat capacity C L2
M T−2
Θ−1
J K−1
Hubble constant1
H T−1
s−1
illuminance Ev L−2
J lx
impedance Z L2
M T−3
I−2
Ω
1The Hubble constant is almost universally quoted in units of km s−1 Mpc−1
. There are
about 3.1 × 1019 kilometres in a megaparsec.

1.5 Dimensions
1
17
physical quantity symbol dimensions SI units
impulse I L M T−1
N s
inductance L L2
M T−2
I−2
H
irradiance Ee M T−3
W m−2
Lagrangian L L2
M T−2
J
length L, l L m
luminous intensity Iv J cd
magnetic dipole moment m, µ L2
I A m2
magnetic field strength H L−1
I A m−1
magnetic flux Φ L2
M T−2
I−1
Wb
magnetic flux density B M T−2
I−1
T
magnetic vector potential A L M T−2
I−1
Wb m−1
magnetisation M L−1
I A m−1
mass m, M M kg
mobility µ M−1
T2
I m2
V−1
s−1
molar gas constant R L2
M T−2
Θ−1
N−1
J mol−1
K−1
moment of inertia I L2
M kg m2
momentum p L M T−1
kg m s−1
number density n L−3
m−3
permeability µ L M T−2
I−2
H m−1
permittivity L−3
M−1
T4
I2
F m−1
Planck constant h L2
M T−1
J s
power P L2
M T−3
W
Poynting vector S M T−3
W m−2
pressure p, P L−1
M T−2
Pa
radiant intensity Ie L2
M T−3
W sr−1
resistance R L2
M T−3
I−2
Ω
Rydberg constant R∞ L−1
m−1
shear modulus µ, G L−1
M T−2
Pa
specific heat capacity c L2
T−2
Θ−1
J kg−1
K−1
speed u, v, c L T−1
m s−1
Stefan–Boltzmann constant σ M T−3
Θ−4
W m−2
K−4
stress σ, τ L−1
M T−2
Pa
surface tension σ, γ M T−2
N m−1
temperature T Θ K
thermal conductivity λ L M T−3
Θ−1
W m−1
K−1
time t T s
velocity v, u L T−1
m s−1
viscosity (dynamic) η, µ L−1
M T−1
Pa s
viscosity (kinematic) ν L2
T−1
m2
s−1
volume V, v L3
m3
wavevector k L−1
m−1
weight W L M T−2
N
work W L2
M T−2
J
Young modulus E L−1
M T−2
Pa

1.6 Miscellaneous
Greek alphabet
A α alpha N ν nu
B β beta Ξ ξ xi
Γ γ gamma O o omicron
∆ δ delta Π π pi
E ε epsilon P ρ rho
Z ζ zeta Σ σ ς sigma
H η eta T τ tau
Θ θ ϑ theta Υ υ upsilon
I ι iota Φ φ ϕ phi
K κ kappa X χ chi
Λ λ lambda Ψ ψ psi
M µ mu Ω ω omega
Pi (π) to 1 000 decimal places
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679
8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196
4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094
3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912
9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859
5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
e to 1 000 decimal places
2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274
2746639193 2003059921 8174135966 2904357290 0334295260 5956307381 3232862794 3490763233 8298807531 9525101901
1573834187 9307021540 8914993488 4167509244 7614606680 8226480016 8477411853 7423454424 3710753907 7744992069
5517027618 3860626133 1384583000 7520449338 2656029760 6737113200 7093287091 2744374704 7230696977 2093101416
9283681902 5515108657 4637721112 5238978442 5056953696 7707854499 6996794686 4454905987 9316368892 3009879312
7736178215 4249992295 7635148220 8269895193 6680331825 2886939849 6465105820 9392398294 8879332036 2509443117
3012381970 6841614039 7019837679 3206832823 7646480429 5311802328 7825098194 5581530175 6717361332 0698112509
9618188159 3041690351 5988885193 4580727386 6738589422 8792284998 9208680582 5749279610 4841984443 6346324496
8487560233 6248270419 7862320900 2160990235 3043699418 4914631409 3431738143 6405462531 5209618369 0888707016
7683964243 7814059271 4563549061 3031072085 1038375051 0115747704 1718986106 8739696552 1267154688 9570350354

2
Chapter 2 Mathematics
2.1 Notation
Mathematics is, of course, a vast subject, and so here we concentrate on those mathematical
methods and relationships that are most often applied in the physical sciences and engineering.
Although there is a high degree of consistency in accepted mathematical notation, there
is some variation. For example the spherical harmonics, Y m
l , can be written Ylm, and there
is some freedom with their signs. In general, the conventions chosen here follow common
practice as closely as possible, whilst maintaining consistency with the rest of the handbook.
In particular:
scalars a general vectors a
unit vectors ˆa scalar product a·b
vector cross-product a×××b gradient operator ∇
Laplacian operator ∇2
derivative
df
dx
etc.
partial derivatives
∂f
∂x
etc.
derivative of r with
respect to t
˙r
nth derivative
dn
f
dxn
closed loop integral
L
dl
closed surface integral
S
ds matrix A or aij
mean value (of x) x binomial coeﬃcient
n
r
factorial ! unit imaginary (i2
=−1) i
exponential constant e modulus (of x) |x|
natural logarithm ln log to base 10 log10

20 Mathematics
2.2 Vectors and matrices
Vector algebra
Scalar producta a·b=|a||b|cosθ (2.1)
Vector productb
a
a
b
b
c
ˆn (in)
θa×××b=|a||b|sinθ ˆn=
ˆx ˆy ˆz
ax ay az
bx by bz
(2.2)
Product rules
a·b=b·a (2.3)
a×××b=−b×××a (2.4)
a·(b+c)=(a·b)+(a·c) (2.5)
a×××(b+c)=(a×××b)+(a×××c) (2.6)
Lagrange’s
identity
(a×××b)·(c×××d)=(a·c)(b·d)−(a·d)(b·c) (2.7)
Scalar triple
product
(a×××b)·c=
ax ay az
bx by bz
cx cy cz
(2.8)
=(b×××c)·a=(c×××a)·b (2.9)
=volume of parallelepiped (2.10)
Vector triple
product
(a×××b)×××c=(a·c)b−(b·c)a (2.11)
a×××(b×××c)=(a·c)b−(a·b)c (2.12)
Reciprocal vectors
a =(b×××c)/[(a×××b)·c] (2.13)
b =(c×××a)/[(a×××b)·c] (2.14)
c =(a×××b)/[(a×××b)·c] (2.15)
(a ·a)=(b ·b)=(c ·c)=1 (2.16)
Vector a with
respect to a
nonorthogonal
basis {e1,e2,e3}c
a=(e1 ·a)e1 +(e2 ·a)e2 +(e3 ·a)e3 (2.17)
aAlso known as the “dot product” or the “inner product.”
bAlso known as the “cross-product.” ˆn is a unit vector making a right-handed set with a and b.
cThe prime ( ) denotes a reciprocal vector.

2
21
Common three-dimensional coordinate systems
x
y
z
r
θ
φ
ρ
point P
x=ρcosφ=rsinθcosφ (2.18)
y =ρsinφ=rsinθsinφ (2.19)
z =rcosθ (2.20)
ρ=(x2
+y2
)1/2
(2.21)
r =(x2
+y2
+z2
)1/2
(2.22)
θ =arccos(z/r) (2.23)
φ=arctan(y/x) (2.24)
coordinate system: rectangular spherical polar cylindrical polar
coordinates of P: (x,y,z) (r,θ,φ) (ρ,φ,z)
volume element: dx dy dz r2
sinθ dr dθ dφ ρ dρ dz dφ
metric elementsa
(h1,h2,h3): (1,1,1) (1,r,rsinθ) (1,ρ,1)
aIn an orthogonal coordinate system (parameterised by coordinates q1,q2,q3), the diﬀerential line
element dl is obtained from (dl)2 =(h1 dq1)2 +(h2 dq2)2 +(h3 dq3)2.
Gradient
Rectangular
coordinates
∇f =
∂f
∂x
ˆx+
∂f
∂y
ˆy+
∂f
∂z
ˆz (2.25)
f scalar ﬁeld
ˆ unit vector
Cylindrical
coordinates
∇f =
∂f
∂ρ
ˆρ+
1
r
∂f
∂φ
ˆφ+
∂f
∂z
ˆz (2.26)
ρ distance from the
z axis
Spherical polar
coordinates
∇f =
∂f
∂r
ˆr+
1
r
∂f
∂θ
ˆθ+
1
rsinθ
∂f
∂φ
ˆφ (2.27)
General
orthogonal
coordinates
∇f =
ˆq1
h1
∂f
∂q1
+
ˆq2
h2
∂f
∂q2
+
ˆq3
h3
∂f
∂q3
(2.28)
qi basis
hi metric elements

22 Mathematics
Divergence
Rectangular
coordinates
∇·A=
∂Ax
∂x
+
∂Ay
∂y
+
∂Az
∂z
(2.29)
A vector ﬁeld
Ai ith component
of A
Cylindrical
coordinates
∇·A=
1
ρ
∂(ρAρ)
∂ρ
+
1
ρ
∂Aφ
∂φ
+
∂Az
∂z
(2.30)
ρ distance from
the z axis
Spherical polar
coordinates
∇·A=
1
r2
∂(r2
Ar)
∂r
+
1
rsinθ
∂(Aθ sinθ)
∂θ
+
1
rsinθ
∂Aφ
∂φ
(2.31)
General
orthogonal
coordinates
∇·A=
1
h1h2h3
∂
∂q1
(A1h2h3)+
∂
∂q2
(A2h3h1)
+
∂
∂q3
(A3h1h2) (2.32)
qi basis
hi metric
elements
Curl
Rectangular
coordinates ∇×××A=
ˆx ˆy ˆz
∂/∂x ∂/∂y ∂/∂z
Ax Ay Az
(2.33)
ˆ unit vector
A vector ﬁeld
Ai ith component
of A
Cylindrical
ˆρ/ρ ˆφ ˆz/ρ
∂/∂ρ ∂/∂φ ∂/∂z
Aρ ρAφ Az
(2.34)
ρ distance from
the z axis
Spherical polar
ˆr/(r2
sinθ) ˆθ/(rsinθ) ˆφ/r
∂/∂r ∂/∂θ ∂/∂φ
Ar rAθ rAφ sinθ
(2.35)
General
orthogonal
coordinates
∇×××A=
1
h1h2h3
ˆq1h1 ˆq2h2 ˆq3h3
∂/∂q1 ∂/∂q2 ∂/∂q3
h1A1 h2A2 h3A3
(2.36)
qi basis
hi metric
elements
Radial formsa
∇r =
r
r
(2.37)
∇·r =3 (2.38)
∇r2
=2r (2.39)
∇·(rr)=4r (2.40)
∇(1/r)=
−r
r3
(2.41)
∇·(r/r2
)=
1
r2
(2.42)
∇(1/r2
)=
−2r
r4
(2.43)
∇·(r/r3
)=4πδ(r) (2.44)
aNote that the curl of any purely radial function is zero. δ(r) is the Dirac delta function.

2
23
Laplacian (scalar)
Rectangular
coordinates ∇2
f =
∂2
f
∂x2
+
∂2
f
∂y2
+
∂2
f
∂z2
(2.45) f scalar field
Cylindrical
coordinates ∇2
f =
1
ρ
∂
∂ρ
ρ
∂f
∂ρ
+
1
ρ2
∂2
f
∂φ2
+
∂2
f
∂z2
(2.46)
ρ distance
from the
z axis
Spherical
polar
coordinates
∇2
f =
1
r2
∂
∂r
r2 ∂f
∂r
+
1
r2 sinθ
∂
∂θ
sinθ
∂f
∂θ
+
1
r2 sin2
θ
∂2
f
∂φ2
(2.47)
General
orthogonal
coordinates
∇2
f =
1
h1h2h3
∂
∂q1
h2h3
h1
∂f
∂q1
+
∂
∂q2
h3h1
h2
∂f
∂q2
+
∂
∂q3
h1h2
h3
∂f
∂q3
(2.48)
qi basis
hi metric
elements
Differential operator identities
∇(fg)≡f∇g+g∇f (2.49)
∇·(fA)≡f∇·A+A·∇f (2.50)
∇×××(fA)≡f∇×××A+(∇f)×××A (2.51)
∇(A·B)≡A×××(∇×××B)+(A·∇)B +B×××(∇×××A)+(B ·∇)A (2.52)
∇·(A×××B)≡B ·(∇×××A)−A·(∇×××B) (2.53)
∇×××(A×××B)≡A(∇·B)−B(∇·A)+(B ·∇)A−(A·∇)B (2.54)
∇·(∇f)≡∇2
f ≡ f (2.55)
∇×××(∇f)≡0 (2.56)
∇·(∇×××A)≡0 (2.57)
∇×××(∇×××A)≡∇(∇·A)−∇2
A (2.58)
f,g scalar fields
A,B vector fields
Vector integral transformations
Gauss’s
(Divergence)
theorem V
(∇·A) dV =
Sc
A· ds (2.59)
A vector field
dV volume element
Sc closed surface
V volume enclosed
Stokes’s
theorem S
(∇×××A)· ds=
L
A· dl (2.60)
S surface
ds surface element
L loop bounding S
dl line element
Green’s first
theorem
S
(f∇g)· ds=
V
∇·(f∇g) dV (2.61)
=
V
[f∇2
g+(∇f)·(∇g)] dV (2.62)
f,g scalar fields
Green’s second
theorem S
[f(∇g)−g(∇f)]· ds=
V
(f∇2
g−g∇2
f) dV
(2.63)

24 Mathematics
Matrix algebraa
Matrix definition A=





a11 a12 ··· a1n
a21 a22 ··· a2n
...
... ···
...
am1 am2 ··· amn





(2.64)
A m by n matrix
aij matrix elements
Matrix addition C=A+B if cij =aij +bij (2.65)
Matrix
multiplication
C=AB if cij =aikbkj (2.66)
(AB)C=A(BC) (2.67)
A(B+C)=AB+AC (2.68)
Transpose matrixb
ãij =aji (2.69)
(AB...N)= Ñ... ˜BÃ (2.70)
ãij transpose matrix
(sometimes aT
ij, or aij)
Adjoint matrix
(definition 1)c
A†
= Ã∗
(2.71)
(AB...N)†
=N†
...B†
A†
(2.72)
∗ complex conjugate (of
each component)
† adjoint (or Hermitian
conjugate)
Hermitian matrixd H†
=H (2.73) H Hermitian (or
self-adjoint) matrix
examples:
A=



a11 a12 a13
a21 a22 a23
a31 a32 a33


 B=



b11 b12 b13
b21 b22 b23
b31 b32 b33



Ã=



a11 a21 a31
a12 a22 a32
a13 a23 a33


 A+B=



a11 +b11 a12 +b12 a13 +b13
a21 +b21 a22 +b22 a23 +b23
a31 +b31 a32 +b32 a33 +b33



AB=



a11 b11 +a12 b21 +a13 b31 a11 b12 +a12 b22 +a13 b32 a11 b13 +a12 b23 +a13 b33



aTerms are implicitly summed over repeated suffices; hence aikbkj equals k aikbkj.
bSee also Equation (2.85).
cOr “Hermitian conjugate matrix.” The term “adjoint” is used in quantum physics for the transpose conjugate of
a matrix and in linear algebra for the transpose matrix of its cofactors. These definitions are not compatible, but
both are widely used [cf. Equation (2.80)].
dHermitian matrices must also be square (see next table).

2
25
Square matricesa
Trace
trA=aii (2.74)
tr(AB)=tr(BA) (2.75)
A square matrix
aij matrix elements
aii implicitly = i aii
Determinantb
detA= ijk...a1ia2ja3k ... (2.76)
=(−1)i+1
ai1Mi1 (2.77)
=ai1Ci1 (2.78)
det(AB...N)=detAdetB...detN (2.79)
tr trace
det determinant (or |A|)
Mij minor of element aij
Cij cofactor of the
element aij
Adjoint matrix
(definition 2)c adjA= ˜Cij =Cji (2.80)
adj adjoint (sometimes
written Â)
∼ transpose
Inverse matrix
(detA=0)
a−1
ij =
Cji
detA
=
adjA
detA
(2.81)
AA−1
=1 (2.82)
(AB...N)−1
=N−1
...B−1
A−1
(2.83)
1 unit matrix
Orthogonality
condition
aijaik =δjk (2.84)
i.e., Ã=A−1
(2.85)
δjk Kronecker delta (=1
if i=j, =0 otherwise)
Symmetry
If A= Ã, A is symmetric (2.86)
If A=−Ã, A is antisymmetric (2.87)
Unitary matrix U†
=U−1
(2.88) U unitary matrix
† Hermitian conjugate
examples:
A=



a11 a12 a13
a21 a22 a23
a31 a32 a33


 B=
b11 b12
b21 b22
trA=a11 +a22 +a33 trB=b11 +b22
detA=a11 a22 a33 −a11 a23 a32 −a21 a12 a33 +a21 a13 a32 +a31 a12 a23 −a31 a13 a22
detB=b11 b22 −b12 b21
A−1
=
1
detA



a22 a33 −a23 a32 −a12 a33 +a13 a32 a12 a23 −a13 a22
−a21 a33 +a23 a31 a11 a33 −a13 a31 −a11 a23 +a13 a21
a21 a32 −a22 a31 −a11 a32 +a12 a31 a11 a22 −a12 a21



B−1
=
1
detB
b22 −b12
−b21 b11
aTerms are implicitly summed over repeated suffices; hence aikbkj equals k aikbkj.
b
ijk... is defined as the natural extension of Equation (2.443) to n-dimensions (see page 50). Mij is the determinant
of the matrix A with the ith row and the jth column deleted. The cofactor Cij =(−1)i+jMij.
cOr “adjugate matrix.” See the footnote to Equation (2.71) for a discussion of the term “adjoint.”

26 Mathematics
Commutators
Commutator
deﬁnition
[A,B]=AB−BA=−[B,A] (2.89) [·,·] commutator
Adjoint [A,B]†
=[B†
,A†
] (2.90) † adjoint
Distribution [A+B,C]=[A,C]+[B,C] (2.91)
Association [AB,C]=A[B,C]+[A,C]B (2.92)
Jacobi identity [A,[B,C]]=[B,[A,C]]−[C,[A,B]] (2.93)
Pauli matrices
Pauli matrices
σ1 =
0 1
1 0
σ2 =
0 −i
i 0
σ3 =
1 0
0 −1
1=
1 0
0 1
(2.94)
σi Pauli spin matrices
1 2×2 unit matrix
i i2 =−1
Anticommuta-
tion
σiσj +σjσi =2δij1 (2.95) δij Kronecker delta
Cyclic
permutation
σiσj =iσk (2.96)
(σi)2
=1 (2.97)
Rotation matricesa
Rotation
about x1
R1(θ)=


1 0 0
0 cosθ sinθ
0 −sinθ cosθ

 (2.98)
Ri(θ) matrix for rotation
about the ith axis
θ rotation angle
Rotation
about x2
R2(θ)=


cosθ 0 −sinθ
0 1 0
sinθ 0 cosθ

 (2.99)
Rotation
about x3
R3(θ)=


cosθ sinθ 0
−sinθ cosθ 0
0 0 1

 (2.100)
α rotation about x3
β rotation about x2
γ rotation about x3
Euler angles R rotation matrix
R(α,β,γ)=


cosγcosβcosα−sinγsinα cosγcosβsinα+sinγcosα −cosγsinβ
−sinγcosβcosα−cosγsinα −sinγcosβsinα+cosγcosα sinγsinβ
sinβcosα sinβsinα cosβ


(2.101)
aAngles are in the right-handed sense for rotation of axes, or the left-handed sense for rotation of vectors. i.e., a
vector v is given a right-handed rotation of θ about the x3-axis using R3(−θ)v →v . Conventionally, x1 ≡x, x2 ≡y,
and x3 ≡z.

2.3 Series, summations, and progressions
2
27
Progressions and summations
Arithmetic
progression
Sn =a+(a+d)+(a+2d)+···
+[a+(n−1)d] (2.102)
=
n
2
[2a+(n−1)d] (2.103)
=
n
2
(a+l) (2.104)
n number of terms
Sn sum of n successive
terms
a ﬁrst term
d common diﬀerence
l last term
Geometric
progression
Sn =a+ar+ar2
+···+arn−1
(2.105)
=a
1−rn
1−r
(2.106)
S∞ =
a
1−r
(|r|<1) (2.107)
r common ratio
Arithmetic
mean
x a =
1
n
(x1 +x2 +···+xn) (2.108) . a arithmetic mean
Geometric
mean
x g =(x1x2x3 ...xn)1/n
(2.109) . g geometric mean
Harmonic mean x h =n
1
x1
+
1
x2
+···+
1
xn
−1
(2.110) . h harmonic mean
Relative mean
magnitudes
x a ≥ x g ≥ x h if xi >0 for all i (2.111)
Summation
formulas
n
i=1
i=
n
2
(n+1) (2.112)
n
i=1
i2
=
n
6
(n+1)(2n+1) (2.113)
n
i=1
i3
=
n2
4
(n+1)2
(2.114)
n
i=1
i4
=
n
30
(n+1)(2n+1)(3n2
+3n−1) (2.115)
∞
i=1
(−1)i+1
i
=1−
1
2
+
1
3
−
1
4
+...=ln2 (2.116)
∞
i=1
(−1)i+1
2i−1
=1−
1
3
+
1
5
−
1
7
+...=
π
4
(2.117)
∞
i=1
1
i2
=1+
1
4
+
1
9
+
1
16
+...=
π2
6
(2.118)
i dummy integer
Euler’s
constanta γ = lim
n→∞
1+
1
2
+
1
3
+···+
1
n
−lnn (2.119) γ Euler’s constant
aγ 0.577215664...

28 Mathematics
Power series
Binomial
seriesa
(1+x)n
=1+nx+
n(n−1)
2!
x2
+
n(n−1)(n−2)
3!
x3
+··· (2.120)
Binomial
coefficientb
n
Cr ≡
n
r
≡
n!
r!(n−r)!
(2.121)
Binomial
theorem
(a+b)n
=
n
k=0
n
k
an−k
bk
(2.122)
Taylor series
(about a)c
f(a+x)=f(a)+xf(1)
(a)+
x2
2!
f(2)
(a)+···+
xn−1
(n−1)!
f(n−1)
(a)+··· (2.123)
Taylor series
(3-D)
f(a+x)=f(a)+(x·∇)f|a +
(x·∇)2
2!
f|a +
(x·∇)3
3!
f|a +··· (2.124)
Maclaurin
series
f(x)=f(0)+xf(1)
(0)+
x2
2!
f(2)
(0)+···+
xn−1
(n−1)!
f(n−1)
(0)+··· (2.125)
aIf n is a positive integer the series terminates and is valid for all x. Otherwise the (infinite) series is convergent for
|x|<1.
bThe coefficient of xr in the binomial series.
cxf(n)(a) is x times the nth derivative of the function f(x) with respect to x evaluated at a, taken as well behaved
around a. (x·∇)nf|a is its extension to three dimensions.
Limits
nc
xn
→0 as n→∞ if |x|<1 (for any fixed c) (2.126)
xn
/n!→0 as n→∞ (for any fixed x) (2.127)
(1+x/n)n
→ex
as n→∞ (2.128)
xlnx→0 as x→0 (2.129)
sinx
x
→1 as x→0 (2.130)
If f(a)=g(a)=0 or ∞ then lim
x→a
f(x)
g(x)
=
f(1)
(a)
g(1)(a)
(l’Hôpital’s rule) (2.131)

2
29
Series expansions
exp(x) 1+x+
x2
2!
+
x3
3!
+··· (2.132) (for all x)
ln(1+x) x−
x2
2
+
x3
3
−
x4
4
+··· (2.133) (−1<x≤1)
ln
1+x
1−x
2 x+
x3
3
+
x5
5
+
x7
7
+··· (2.134) (|x|<1)
cos(x) 1−
x2
2!
+
x4
4!
−
x6
6!
+··· (2.135) (for all x)
sin(x) x−
x3
3!
+
x5
5!
−
x7
7!
+··· (2.136) (for all x)
tan(x) x+
x3
3
+
2x5
15
+
17x7
315
··· (2.137) (|x|<π/2)
sec(x) 1+
x2
2
+
5x4
24
+
61x6
720
+··· (2.138) (|x|<π/2)
csc(x)
1
x
+
x
6
+
7x3
360
+
31x5
15120
+··· (2.139) (|x|<π)
cot(x)
1
x
−
x
3
−
x3
45
−
2x5
945
−··· (2.140) (|x|<π)
arcsin(x)a
x+
1
2
x3
3
+
1·3
2·4
x5
5
+
1·3·5
2·4·6
x7
7
··· (2.141) (|x|<1)
arctan(x)b



x−
x3
3
+
x5
5
−
x7
7
+···
π
2
−
1
x
+
1
3x3
−
1
5x5
+···
−
π
2
−
1
x
+
1
3x3
−
1
5x5
+···
(2.142)
(|x|≤1)
(x>1)
(x<−1)
cosh(x) 1+
x2
2!
+
x4
4!
+
x6
6!
+··· (2.143) (for all x)
sinh(x) x+
x3
3!
+
x5
5!
+
x7
7!
+··· (2.144) (for all x)
tanh(x) x−
x3
3
+
2x5
15
−
17x7
315
+··· (2.145) (|x|<π/2)
aarccos(x)=π/2−arcsin(x). Note that arcsin(x)≡sin−1
(x) etc.
barccot(x)=π/2−arctan(x).

30 Mathematics
Inequalities
Triangle
inequality
|a1|−|a2|≤|a1 +a2|≤|a1|+|a2|; (2.146)
n
i=1
ai ≤
n
i=1
|ai| (2.147)
Chebyshev
inequality
if a1 ≥a2 ≥a3 ≥...≥an (2.148)
and b1 ≥b2 ≥b3 ≥...≥bn (2.149)
then n
n
i=1
aibi ≥
n
i=1
ai
n
i=1
bi (2.150)
Cauchy
inequality
n
i=1
aibi
2
≤
n
i=1
a2
i
n
i=1
b2
i (2.151)
Schwarz
inequality
b
a
f(x)g(x) dx
2
≤
b
a
[f(x)]2
dx
b
a
[g(x)]2
dx (2.152)
2.4 Complex variables
Complex numbers
Cartesian form z =x+iy (2.153)
z complex variable
i i2 =−1
x,y real variables
Polar form z =reiθ
=r(cosθ+isinθ) (2.154) r amplitude (real)
θ phase (real)
Modulusa
|z|=r =(x2
+y2
)1/2
(2.155)
|z1 ·z2|=|z1|·|z2| (2.156)
|z| modulus of z
Argument
θ =argz =arctan
y
x
(2.157)
arg(z1z2)=argz1 +argz2 (2.158)
argz argument of z
Complex
conjugate
z∗
=x−iy =re−iθ
(2.159)
arg(z∗
)=−argz (2.160)
z ·z∗
=|z|2
(2.161)
z∗ complex conjugate of
z =reiθ
Logarithmb lnz =lnr+i(θ+2πn) (2.162) n integer
aOr “magnitude.”
bThe principal value of lnz is given by n=0 and −π <θ ≤π.

2.4 Complex variables
2
31
Complex analysisa
Cauchy–
Riemann
equationsb
if f(z)=u(x,y)+iv(x,y)
then
∂u
∂x
=
∂v
∂y
(2.163)
∂u
∂y
=−
∂v
∂x
(2.164)
z complex variable
i i2 =−1
x,y real variables
f(z) function of z
u,v real functions
Cauchy–
Goursat
theoremc c
f(z) dz =0 (2.165)
Cauchy
integral
formulad
f(z0)=
1
2πi c
f(z)
z −z0
dz (2.166)
f(n)
(z0)=
n!
2πi c
f(z)
(z −z0)n+1
dz (2.167)
(n) nth derivative
an Laurent coeﬃcients
a−1 residue of f(z) at z0
z dummy variable
Laurent
expansione
x
y
c
c1
c2
z0
f(z)=
∞
n=−∞
an(z −z0)n
(2.168)
where an =
1
2πi c
f(z )
(z −z0)n+1
dz (2.169)
Residue
theorem c
f(z) dz =2πi enclosed residues (2.170)
aClosed contour integrals are taken in the counterclockwise sense, once.
bNecessary condition for f(z) to be analytic at a given point.
cIf f(z) is analytic within and on a simple closed curve c. Sometimes called “Cauchy’s theorem.”
dIf f(z) is analytic within and on a simple closed curve c, encircling z0.
eOf f(z), (analytic) in the annular region between concentric circles, c1 and c2, centred on z0. c is any closed curve
in this region encircling z0.

32 Mathematics
2.5 Trigonometric and hyperbolic formulas
Trigonometric relationships
sin(A±B)=sinAcosB ±cosAsinB (2.171)
cos(A±B)=cosAcosB ∓sinAsinB (2.172)
tan(A±B)=
tanA±tanB
1∓tanAtanB
(2.173)
cosAcosB =
1
2
[cos(A+B)+cos(A−B)] (2.174)
sinAcosB =
1
2
[sin(A+B)+sin(A−B)] (2.175)
sinAsinB =
1
2
[cos(A−B)−cos(A+B)] (2.176)
x
x
−6
−6
−4
−4
−4
−2
−2
−2
−2
0
0
0
0
1
−1
4
4
4
6
6
2
2
2
2
sinx
cosx
tanx
cotx
secx
cscx
cos2
A+sin2
A=1 (2.177)
sec2
A−tan2
A=1 (2.178)
csc2
A−cot2
A=1 (2.179)
sin2A=2sinAcosA (2.180)
cos2A=cos2
A−sin2
A (2.181)
tan2A=
2tanA
1−tan2 A
(2.182)
sin3A=3sinA−4sin3
A (2.183)
cos3A=4cos3
A−3cosA (2.184)
sinA+sinB =2sin
A+B
2
cos
A−B
2
(2.185)
sinA−sinB =2cos
A+B
2
sin
A−B
2
(2.186)
cosA+cosB =2cos
A+B
2
cos
A−B
2
(2.187)
cosA−cosB =−2sin
A+B
2
sin
A−B
2
(2.188)
cos2
A=
1
2
(1+cos2A) (2.189)
sin2
A=
1
2
(1−cos2A) (2.190)
cos3
A=
1
4
(3cosA+cos3A) (2.191)
sin3
A=
1
4
(3sinA−sin3A) (2.192)

2.5 Trigonometric and hyperbolic formulas
2
33
Hyperbolic relationshipsa
sinh(x±y)=sinhxcoshy±coshxsinhy (2.193)
cosh(x±y)=coshxcoshy±sinhxsinhy (2.194)
tanh(x±y)=
tanhx±tanhy
1±tanhxtanhy
(2.195)
coshxcoshy =
1
2
[cosh(x+y)+cosh(x−y)] (2.196)
sinhxcoshy =
1
2
[sinh(x+y)+sinh(x−y)] (2.197)
sinhxsinhy =
1
2
[cosh(x+y)−cosh(x−y)] (2.198)
x
x
−3
−3
−4
−4
−2
−2
−2
−2
0
0
0
0
1
1
−1
−1
4
4
3
3
2
2
2
2
sinhx
coshx
tanhx
tanhx
cothx
sechx
cschx
cosh2
x−sinh2
x=1 (2.199)
sech2
x+tanh2
x=1 (2.200)
coth2
x−csch2
x=1 (2.201)
sinh2x=2sinhxcoshx (2.202)
cosh2x=cosh2
x+sinh2
x (2.203)
tanh2x=
2tanhx
1+tanh2
x
(2.204)
sinh3x=3sinhx+4sinh3
x (2.205)
cosh3x=4cosh3
x−3coshx (2.206)
sinhx+sinhy =2sinh
x+y
2
cosh
x−y
2
(2.207)
sinhx−sinhy =2cosh
x+y
2
sinh
x−y
2
(2.208)
coshx+coshy =2cosh
x+y
2
cosh
x−y
2
(2.209)
coshx−coshy =2sinh
x+y
2
sinh
x−y
2
(2.210)
cosh2
x=
1
2
(cosh2x+1) (2.211)
sinh2
x=
1
2
(cosh2x−1) (2.212)
cosh3
x=
1
4
(3coshx+cosh3x) (2.213)
sinh3
x=
1
4
(sinh3x−3sinhx) (2.214)
aThese can be derived from trigonometric relationships by using the
substitutions cosx→coshx and sinx→isinhx.

34 Mathematics
Trigonometric and hyperbolic deﬁnitions
de Moivre’s theorem (cosx+isinx)n
=einx
=cosnx+isinnx (2.215)
cosx=
1
2
eix
+e−ix
(2.216) coshx=
1
2
ex
+e−x
(2.217)
sinx=
1
2i
eix
−e−ix
(2.218) sinhx=
1
2
ex
−e−x
(2.219)
tanx=
sinx
cosx
(2.220) tanhx=
sinhx
coshx
(2.221)
cosix=coshx (2.222) coshix=cosx (2.223)
sinix=isinhx (2.224) sinhix=isinx (2.225)
cotx=(tanx)−1
(2.226) cothx=(tanhx)−1
(2.227)
secx=(cosx)−1
(2.228) sechx=(coshx)−1
(2.229)
cscx=(sinx)−1
(2.230) cschx=(sinhx)−1
(2.231)
Inverse trigonometric functionsa
x
x
0
0
1
1
1
1
1.6
1.6
2 3 4 5
arcsinx
arccosx
arctanx
arccotx
arccscx
arcsecx
arcsinx=arctan
x
(1−x2)1/2
(2.232)
arccosx=arctan
(1−x2
)1/2
x
(2.233)
arccscx=arctan
1
(x2 −1)1/2
(2.234)
arcsecx=arctan (x2
−1)1/2
(2.235)
arccotx=arctan
1
x
(2.236)
arccosx=
π
2
−arcsinx (2.237)
aValid in the angle range 0≤θ ≤π/2. Note that arcsinx≡sin−1
x etc.

2.6 Mensuration
2
35
Inverse hyperbolic functions
x
x
0
0
0
1
1
1
1
−1
−1
2
2
2
−2
3
4
arsinhx
arcoshx
artanhx
arcothx
arcschxarsechx
arsinhx≡sinh−1
x=ln x+(x2
+1)1/2
(2.238) for all x
arcoshx≡cosh−1
x=ln x+(x2
−1)1/2
(2.239)
x≥1
artanhx≡tanh−1
x=
1
2
ln
1+x
1−x
(2.240) |x|<1
arcothx≡coth−1
x=
1
2
ln
x+1
x−1
(2.241) |x|>1
arsechx≡sech−1
x=ln
1
x
+
(1−x2
)1/2
x
(2.242)
0<x≤1
arcschx≡csch−1
x=ln
1
x
+
(1+x2
)1/2
x
(2.243)
x=0
2.6 Mensuration
Moir´e fringesa
Parallel pattern dM =
1
d1
−
1
d2
−1
(2.244)
dM Moir´e fringe spacing
d1,2 grating spacings
Rotational
patternb dM =
d
2|sin(θ/2)|
(2.245)
d common grating
spacing
θ relative rotation angle
(|θ|≤π/2)
aFrom overlapping linear gratings.
bFrom identical gratings, spacing d, with a relative rotation θ.

36 Mathematics
Plane triangles
Sine formulaa a
sinA
=
b
sinB
=
c
sinC
(2.246)
Cosine
formulas
a
b
c
A
C
B
a2
=b2
+c2
−2bccosA (2.247)
cosA=
b2
+c2
−a2
2bc
(2.248)
a=bcosC +ccosB (2.249)
Tangent
formula
tan
A−B
2
=
a−b
a+b
cot
C
2
(2.250)
Area
area =
1
2
absinC (2.251)
=
a2
2
sinBsinC
sinA
(2.252)
=[s(s−a)(s−b)(s−c)]1/2
(2.253)
where s=
1
2
(a+b+c) (2.254)
aThe diameter of the circumscribed circle equals a/sinA.
Spherical trianglesa
Sine formula
sina
sinA
=
sinb
sinB
=
sinc
sinC
(2.255)
Cosine
formulas
cosa=cosbcosc+sinbsinccosA (2.256)
cosA=−cosBcosC +sinBsinC cosa (2.257)
Analogue
formula
a b
c
A
B
C
sinacosB =cosbsinc−sinbcosccosA (2.258)
Four-parts
formula
cosacosC =sinacotb−sinC cotB (2.259)
Areab E =A+B +C −π (2.260)
aOn a unit sphere.
bAlso called the “spherical excess.”

2.6 Mensuration
2
37
Perimeter, area, and volume
Perimeter of circle P =2πr (2.261)
P perimeter
r radius
Area of circle A=πr2
(2.262) A area
Surface area of spherea A=4πR2
(2.263) R sphere radius
Volume of sphere V =
4
3
πR3
(2.264) V volume
Perimeter of ellipseb
P =4aE(π/2, e) (2.265)
2π
a2
+b2
2
1/2
(2.266)
a semi-major axis
b semi-minor axis
E elliptic integral of the
second kind (p. 45)
e eccentricity
(=1−b2/a2)
Area of ellipse A=πab (2.267)
Volume of ellipsoidc
V =4π
abc
3
(2.268) c third semi-axis
Surface area of
cylinder
A=2πr(h+r) (2.269) h height
Volume of cylinder V =πr2
h (2.270)
Area of circular coned A=πrl (2.271) l slant height
Volume of cone or
pyramid
V =Abh/3 (2.272) Ab base area
Surface area of torus A=π2
(r1 +r2)(r2 −r1) (2.273)
r1 inner radius
r2 outer radius
Volume of torus V =
π2
4
(r2
2 −r2
1)(r2 −r1) (2.274)
Aread
of spherical cap,
depth d
A=2πRd (2.275) d cap depth
Volume of spherical
cap, depth d
V =πd2
R −
d
3
(2.276)
Ω solid angle
z distance from centre
α half-angle subtended
Solid angle of a circle
from a point on its
axis, z from centre
α r
z
Ω=2π 1−
z
(z2 +r2)1/2
(2.277)
=2π(1−cosα) (2.278)
aSphere deﬁned by x2 +y2 +z2 =R2.
bThe approximation is exact when e=0 and e 0.91, giving a maximum error of 11% at e=1.
cEllipsoid deﬁned by x2/a2 +y2/b2 +z2/c2 =1.
dCurved surface only.

38 Mathematics
Conic sections
x xx
y yy
a aa
b
parabola ellipse hyperbola
equation y2
=4ax
x2
a2
+
y2
b2
=1
x2
a2
−
y2
b2
=1
parametric
form
x=t2
/(4a)
y =t
x=acost
y =bsint
x=±acosht
y =bsinht
foci (a,0) (±
√
a2 −b2,0) (±
√
a2 +b2,0)
eccentricity e=1 e=
√
a2 −b2
a
e=
√
a2 +b2
a
directrices x=−a x=±
a
e
x=±
a
e
Platonic solidsa
solid
(faces,edges,vertices)
volume surface area circumradius inradius
tetrahedron
(4,6,4)
a3
√
2
12
a2
√
3
a
√
6
4
a
√
6
12
cube
(6,12,8)
a3
6a2 a
√
3
2
a
2
octahedron
(8,12,6)
a3
√
2
3
2a2
√
3
a
√
2
a
√
6
dodecahedron
(12,30,20)
a3
(15+7
√
5)
4
3a2
5(5+2
√
5)
a
4
√
3(1+
√
5)
a
4
50+22
√
5
5
icosahedron
(20,30,12)
5a3
(3+
√
5)
12
5a2
√
3
a
4
2(5+
√
5)
a
4
√
3+
5
3
aOf side a. Both regular and irregular polyhedra follow the Euler relation, faces−edges+vertices=2.

2.6 Mensuration
2
39
Curve measure
Length of plane
curve l =
b
a
1+
dy
dx
2 1/2
dx (2.279)
a start point
b end point
y(x) plane curve
l length
Surface of
revolution A=2π
b
a
y 1+
dy
dx
2 1/2
dx (2.280) A surface area
Volume of
revolution V =π
b
a
y2
dx (2.281) V volume
Radius of
curvature ρ= 1+
dy
dx
2 3/2
d2
y
dx2
−1
(2.282)
ρ radius of
curvature
Diﬀerential geometrya
Unit tangent ˆτ =
˙r
|˙r|
=
˙r
v
(2.283)
τ tangent
r curve parameterised by r(t)
v |˙r(t)|
Unit principal normal ˆn=
¨r−˙vˆτ
|¨r−˙vˆτ|
(2.284) n principal normal
Unit binormal ˆb= ˆτ×××ˆn (2.285) b binormal
Curvature κ=
|˙r×××¨r|
|˙r|3
(2.286) κ curvature
Radius of curvature ρ=
1
κ
(2.287) ρ radius of curvature
Torsion λ=
˙r·(¨r×××
...
r )
|˙r×××¨r|2
(2.288) λ torsion
Frenet’s formulas
normal plane
osculating
plane
rectifying
plane
origin
ˆn
ˆb
ˆτ
r
˙ˆτ =κvˆn (2.289)
˙ˆn=−κvˆτ +λvˆb (2.290)
˙ˆb=−λvˆn (2.291)
aFor a continuous curve in three dimensions, traced by the position vector r(t).

40 Mathematics
2.7 Differentiation
Derivatives (general)
Power
d
dx
(un
)=nun−1 du
dx
(2.292)
n power
index
Product
d
dx
(uv)=u
dv
dx
+v
du
dx
(2.293)
u,v functions
of x
Quotient
d
dx
u
v
=
1
v
du
dx
−
u
v2
dv
dx
(2.294)
Function of a
functiona
d
dx
[f(u)]=
d
du
[f(u)]·
du
dx
(2.295)
f(u) function of
u(x)
Leibniz theorem
dn
dxn
[uv]=
n
0
v
dn
u
dxn
+
n
1
dv
dx
dn−1
u
dxn−1
+···
+
n
k
dk
v
dxk
dn−k
u
dxn−k
+···+
n
n
u
dn
v
dxn
(2.296)
n
k binomial
coefficient
Differentiation
under the integral
sign
d
dq
q
p
f(x) dx =f(q) (p constant) (2.297)
d
dp
q
p
f(x) dx =−f(p) (q constant) (2.298)
General integral
d
dx
v(x)
u(x)
f(t) dt =f(v)
dv
dx
−f(u)
du
dx
(2.299)
Logarithm
d
dx
(logb |ax|)=(xlnb)−1
(2.300)
b log base
a constant
Exponential
d
dx
(eax
)=aeax
(2.301)
Inverse functions
dx
dy
=
dy
dx
−1
(2.302)
d2
x
dy2
=−
d2
y
dx2
dy
dx
−3
(2.303)
d3
x
dy3
= 3
d2
y
dx2
2
−
dy
dx
d3
y
dx3
dy
dx
−5
(2.304)
aThe “chain rule.”

2
41
Trigonometric derivativesa
d
dx
(sinax)=acosax (2.305)
d
dx
(cosax)=−asinax (2.306)
d
dx
(tanax)=asec2
ax (2.307)
d
dx
(cscax)=−acscax·cotax (2.308)
d
dx
(secax)=asecax·tanax (2.309)
d
dx
(cotax)=−acsc2
ax (2.310)
d
dx
(arcsinax)=a(1−a2
x2
)−1/2
(2.311)
d
dx
(arccosax)=−a(1−a2
x2
)−1/2
(2.312)
d
dx
(arctanax)=a(1+a2
x2
)−1
(2.313)
d
dx
(arccscax)=−
a
|ax|
(a2
x2
−1)−1/2
(2.314)
d
dx
(arcsecax)=
a
|ax|
(a2
x2
−1)−1/2
(2.315)
d
dx
(arccotax)=−a(a2
x2
+1)−1
(2.316)
aa is a constant.
Hyperbolic derivativesa
d
dx
(sinhax)=acoshax (2.317)
d
dx
(coshax)=asinhax (2.318)
d
dx
(tanhax)=asech2
ax (2.319)
d
dx
(cschax)=−acschax·cothax (2.320)
d
dx
(sechax)=−asechax·tanhax (2.321)
d
dx
(cothax)=−acsch2
ax (2.322)
d
dx
(arsinhax)=a(a2
x2
+1)−1/2
(2.323)
d
dx
(arcoshax)=a(a2
x2
−1)−1/2
(2.324)
d
dx
(artanhax)=a(1−a2
x2
)−1
(2.325)
d
dx
(arcschax)=−
a
|ax|
(1+a2
x2
)−1/2
(2.326)
d
dx
(arsechax)=−
a
|ax|
(1−a2
x2
)−1/2
(2.327)
d
dx
(arcothax)=a(1−a2
x2
)−1
(2.328)
aa is a constant.

42 Mathematics
Partial derivatives
Total
differential
df =
∂f
∂x
dx+
∂f
∂y
dy+
∂f
∂z
dz (2.329) f f(x,y,z)
Reciprocity
∂g
∂x y
∂x
∂y g
∂y
∂g x
=−1 (2.330) g g(x,y)
Chain rule
∂f
∂u
=
∂f
∂x
∂x
∂u
+
∂f
∂y
∂y
∂u
+
∂f
∂z
∂z
∂u
(2.331)
Jacobian J =
∂(x,y,z)
∂(u,v,w)
=
∂x
∂u
∂x
∂v
∂x
∂w
∂y
∂u
∂y
∂v
∂y
∂w
∂z
∂u
∂z
∂v
∂z
∂w
(2.332)
J Jacobian
u u(x,y,z)
v v(x,y,z)
w w(x,y,z)
Change of
variable V
f(x,y,z) dxdydz =
V
f(u,v,w)J dudvdw
(2.333)
V volume in (x,y,z)
V volume in (u,v,w)
mapped to by V
Euler–
Lagrange
equation
if I =
b
a
F(x,y,y ) dx
then δI =0 when
∂F
∂y
=
d
dx
∂F
∂y
(2.334)
y dy/dx
a,b fixed end points
Stationary pointsa
maximum minimumsaddle point quartic minimum
Stationary point if
∂f
∂x
=
∂f
∂y
=0 at (x0,y0) (necessary condition) (2.335)
Additional sufficient conditions
for maximum ∂2
f
∂x2
<0, and
∂2
f
∂x2
∂2
f
∂y2
>
∂2
f
∂x∂y
2
(2.336)
for minimum ∂2
f
∂x2
>0, and
∂2
f
∂x2
∂2
f
∂y2
>
∂2
f
∂x∂y
2
(2.337)
for saddle point ∂2
f
∂x2
∂2
f
∂y2
<
∂2
f
∂x∂y
2
(2.338)
aOf a function f(x,y) at the point (x0,y0). Note that at, for example, a quartic minimum ∂2f
∂x2 = ∂2f
∂y2 =0.

2
43
Differential equations
Laplace ∇2
f =0 (2.339) f f(x,y,z)
Diffusiona ∂f
∂t
=D∇2
f (2.340)
D diffusion
coefficient
Helmholtz ∇2
f +α2
f =0 (2.341) α constant
Wave ∇2
f =
1
c2
∂2
f
∂t2
(2.342) c wave speed
Legendre
d
dx
(1−x2
)
dy
dx
+l(l +1)y =0 (2.343) l integer
Associated
Legendre
d
dx
(1−x2
)
dy
dx
+ l(l +1)−
m2
1−x2
y =0 (2.344) m integer
Bessel x2 d2
y
dx2
+x
dy
dx
+(x2
−m2
)y =0 (2.345)
Hermite
d2
y
dx2
−2x
dy
dx
+2αy =0 (2.346)
Laguerre x
d2
y
dx2
+(1−x)
dy
dx
+αy =0 (2.347)
Associated
Laguerre x
d2
y
dx2
+(1+k−x)
dy
dx
+αy =0 (2.348) k integer
Chebyshev (1−x2
)
d2
y
dx2
−x
dy
dx
+n2
y =0 (2.349) n integer
Euler (or
Cauchy) x2 d2
y
dx2
+ax
dy
dx
+by =f(x) (2.350) a,b constants
Bernoulli
dy
dx
+p(x)y =q(x)ya
(2.351) p,q functions of x
Airy
d2
y
dx2
=xy (2.352)
aAlso known as the “conduction equation.” For thermal conduction, f ≡ T and D, the thermal diffusivity,
≡κ≡λ/(ρcp), where T is the temperature distribution, λ the thermal conductivity, ρ the density, and cp the specific
heat capacity of the material.

44 Mathematics
2.8 Integration
Standard formsa
u dv =[uv]− v du (2.353) uv dx=v u dx− u dx
dv
dx
dx (2.354)
xn
dx=
xn+1
n+1
(n=−1) (2.355)
1
x
dx=ln|x| (2.356)
eax
dx=
1
a
eax
(2.357) xeax
dx=eax x
a
−
1
a2
(2.358)
lnax dx=x(lnax−1) (2.359)
f (x)
f(x)
dx=lnf(x) (2.360)
xlnax dx=
x2
2
lnax−
1
2
(2.361) bax
dx=
bax
alnb
(b>0) (2.362)
1
a+bx
dx=
1
b
ln(a+bx) (2.363)
1
x(a+bx)
dx=−
1
a
ln
a+bx
x
(2.364)
1
(a+bx)2
dx=
−1
b(a+bx)
(2.365)
1
a2 +b2x2
dx=
1
ab
arctan
bx
a
(2.366)
1
x(xn +a)
dx=
1
an
ln
xn
xn +a
(2.367)
1
x2 −a2
dx=
1
2a
ln
x−a
x+a
(2.368)
x
x2 ±a2
dx=
1
2
ln|x2
±a2
| (2.369)
x
(x2 ±a2)n
dx=
−1
2(n−1)(x2 ±a2)n−1
(2.370)
1
(a2 −x2)1/2
dx=arcsin
x
a
(2.371)
1
(x2 ±a2)1/2
dx=ln|x+(x2
±a2
)1/2
| (2.372)
x
(x2 ±a2)1/2
dx=(x2
±a2
)1/2
(2.373)
1
x(x2 −a2)1/2
dx=
1
a
arcsec
x
a
(2.374)
aa and b are non-zero constants.

2.8 Integration
2
45
Trigonometric and hyperbolic integrals
sinx dx=−cosx (2.375) sinhx dx=coshx (2.376)
cosx dx=sinx (2.377) coshx dx=sinhx (2.378)
tanx dx=−ln|cosx| (2.379) tanhx dx=ln(coshx) (2.380)
cscx dx=ln tan
x
2
(2.381) cschx dx=ln tanh
x
2
(2.382)
secx dx=ln|secx+tanx| (2.383) sechx dx=2arctan(ex
) (2.384)
cotx dx=ln|sinx| (2.385) cothx dx=ln|sinhx| (2.386)
sinmx·sinnx dx=
sin(m−n)x
2(m−n)
−
sin(m+n)x
2(m+n)
(m2
=n2
) (2.387)
sinmx·cosnx dx=−
cos(m−n)x
2(m−n)
−
cos(m+n)x
2(m+n)
(m2
=n2
) (2.388)
cosmx·cosnx dx=
sin(m−n)x
2(m−n)
+
sin(m+n)x
2(m+n)
(m2
=n2
) (2.389)
Named integrals
Error function erf(x)=
2
π1/2
x
0
exp(−t2
) dt (2.390)
Complementary error
function
erfc(x)=1−erf(x)=
2
π1/2
∞
x
exp(−t2
) dt (2.391)
Fresnel integralsa
C(x)=
x
0
cos
πt2
2
dt; S(x)=
x
0
sin
πt2
2
dt (2.392)
C(x)+i S(x)=
1+i
2
erf
π1/2
2
(1−i)x (2.393)
Exponential integral Ei(x)=
x
−∞
et
t
dt (x>0) (2.394)
Gamma function Γ(x)=
∞
0
tx−1
e−t
dt (x>0) (2.395)
Elliptic integrals
(trigonometric form)
F(φ,k)=
φ
0
1
(1−k2 sin2
θ)1/2
dθ (ﬁrst kind) (2.396)
E(φ,k)=
φ
0
(1−k2
sin2
θ)1/2
dθ (second kind) (2.397)
aSee also page 167.

46 Mathematics
Deﬁnite integrals
∞
0
e−ax2
dx=
1
2
π
a
1/2
(a>0) (2.398)
∞
0
xe−ax2
dx=
1
2a
(a>0) (2.399)
∞
0
xn
e−ax
dx=
n!
an+1
(a>0; n=0,1,2,...) (2.400)
∞
−∞
exp(2bx−ax2
) dx=
π
a
1/2
exp
b2
a
(a>0) (2.401)
∞
0
xn
e−ax2
dx=
1·3·5·...·(n−1)(2a)−(n+1)/2
(π/2)1/2
n>0 and even
2·4·6·...·(n−1)(2a)−(n+1)/2
n>1 and odd
(2.402)
1
0
xp
(1−x)q
dx=
p!q!
(p+q+1)!
(p,q integers >0) (2.403)
∞
0
cos(ax2
) dx=
∞
0
sin(ax2
) dx=
1
2
π
2a
1/2
(a>0) (2.404)
∞
0
sinx
x
dx=
∞
0
sin2
x
x2
dx=
π
2
(2.405)
∞
0
1
(1+x)xa
dx=
π
sinaπ
(0<a<1) (2.406)
2.9 Special functions and polynomials
Gamma function
Deﬁnition Γ(z)=
∞
0
tz−1
e−t
dt [ (z)>0] (2.407)
Relations
n!=Γ(n+1)=nΓ(n) (n=0,1,2,...) (2.408)
Γ(1/2)=π1/2
(2.409)
z
w
=
z!
w!(z −w)!
=
Γ(z +1)
Γ(w+1)Γ(z −w+1)
(2.410)
Stirling’s formulas
(for |z|,n 1)
Γ(z) e−z
zz−(1/2)
(2π)1/2
1+
1
12z
+
1
288z2
−··· (2.411)
n! nn+(1/2)
e−n
(2π)1/2
(2.412)
ln(n!) nlnn−n (2.413)

2
47
Bessel functions
Series
expansion
Jν(x)=
x
2
ν
∞
k=0
(−x2
/4)k
k!Γ(ν +k+1)
(2.414)
Yν(x)=
Jν(x)cos(πν)−J−ν(x)
sin(πν)
(2.415)
Jν(x) Bessel function of the first
kind
Yν(x) Bessel function of the
second kind
Γ(ν) Gamma function
ν order (ν ≥0)
Approximations
x
0
0
2 4 6 8 10
0.5
−0.5
1
−1
J0
J1
Y0
Y1
Jν(x)
1
Γ(ν+1)
x
2
ν
(0≤x ν)
2
πx
1/2
cos x− 1
2 νπ− π
4 (x ν)
(2.416)
Yν(x)
−Γ(ν)
π
x
2
−ν
(0<x ν)
2
πx
1/2
sin x− 1
2 νπ− π
4 (x ν)
(2.417)
Modified Bessel
functions
Iν(x)=(−i)ν
Jν(ix) (2.418)
Kν(x)=
π
2
iν+1
[Jν(ix)+iYν(ix)] (2.419)
Iν(x) modified Bessel function of
the first kind
Kν(x) modified Bessel function of
the second kind
Spherical Bessel
function
jν(x)=
π
2x
1/2
Jν+ 1
2
(x) (2.420)
jν(x) spherical Bessel function
of the first kind [similarly
for yν(x)]
Legendre polynomialsa
Legendre
equation
(1−x2
)
d2
Pl(x)
dx2
−2x
dPl(x)
dx
+l(l +1)Pl(x)=0
(2.421)
Pl Legendre
polynomials
l order (l ≥0)
Rodrigues’
formula Pl(x)=
1
2ll!
dl
dxl
(x2
−1)l
(2.422)
Recurrence
relation
(l +1)Pl+1(x)=(2l +1)xPl(x)−lPl−1(x) (2.423)
Orthogonality
1
−1
Pl(x)Pl (x) dx=
2
2l +1
δll (2.424) δll Kronecker delta
Explicit form Pl(x)=2−l
l/2
m=0
(−1)m l
m
2l −2m
l
xl−2m
(2.425)
l
m binomial coefficients
Expansion of
plane wave
exp(ikz)=exp(ikrcosθ) (2.426)
=
∞
l=0
(2l +1)il
jl(kr)Pl(cosθ) (2.427)
k wavenumber
z propagation axis
z =rcosθ
jl spherical Bessel
function of the first
kind (order l)
P0(x)=1 P2(x)=(3x2
−1)/2 P4(x)=(35x4
−30x2
+3)/8
P1(x)=x P3(x)=(5x3
−3x)/2 P5(x)=(63x5
−70x3
+15x)/8
aOf the first kind.

48 Mathematics
Associated Legendre functionsa
Associated
Legendre
equation
d
dx
(1−x2
)
dPm
l (x)
dx
+ l(l +1)−
m2
1−x2
Pm
l (x)=0
(2.428)
Pm
l associated
Legendre
functions
From
Legendre
polynomials
Pm
l (x)=(1−x2
)m/2 dm
dxm
Pl(x), 0≤m≤l (2.429)
P−m
l (x)=(−1)m (l −m)!
(l +m)!
Pm
l (x) (2.430)
Pl Legendre
polynomials
Recurrence
relations
Pm
m+1(x)=x(2m+1)Pm
m (x) (2.431)
Pm
m (x)=(−1)m
(2m−1)!!(1−x2
)m/2
(2.432)
(l −m+1)Pm
l+1(x)=(2l +1)xPm
l (x)−(l +m)Pm
l−1(x)
(2.433)
!! 5!!=5·3·1 etc.
xx
0.5
0.5
0
0
0
0
1
1
1
1
−1
−1
−1
−1
−0.5
−0.5
2
3
P0
P1
P2
P3 P4
P5
P0
0
P
0
1P 0
2
P
1
1
P 1
2
P
2
2
Legendre polynomials associated Legendre functions
Orthogonality
1
−1
Pm
l (x)Pm
l (x) dx=
(l +m)!
(l −m)!
2
2l +1
δll (2.434)
δll Kronecker
delta
P0
0 (x)=1 P0
1 (x)=x P1
1 (x)=−(1−x2
)1/2
P0
2 (x)=(3x2
−1)/2 P1
2 (x)=−3x(1−x2
)1/2
P2
2 (x)=3(1−x2
)
aOf the ﬁrst kind. Pm
l (x) can be deﬁned with a (−1)m factor in Equation (2.429) as well as Equation (2.430).

2
49
Spherical harmonics
Differential
equation
1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂φ2
Y m
l +l(l +1)Y m
l =0
(2.435)
Y m
l spherical
harmonics
Definitiona Y m
l (θ,φ)=(−1)m 2l +1
4π
(l −m)!
(l +m)!
1/2
Pm
l (cosθ)eimφ
(2.436)
Pm
l associated
Legendre
functions
Orthogonality
2π
φ=0
π
θ=0
Y m∗
l (θ,φ)Y m
l (θ,φ)sinθ dθ dφ=δmm δll (2.437)
Y ∗ complex
conjugate
δll Kronecker
delta
Laplace series
f(θ,φ)=
∞
l=0
l
m=−l
almY m
l (θ,φ) (2.438)
where alm =
2π
φ=0
π
θ=0
Y m∗
l (θ,φ)f(θ,φ)sinθ dθ dφ
(2.439)
f continuous
function
Solution to
Laplace
equation
if ∇2
ψ(r,θ,φ)=0, then
ψ(r,θ,φ)=
∞
l=0
l
m=−l
Y m
l (θ,φ)· almrl
+blmr−(l+1)
(2.440)
ψ continuous
function
a,b constants
Y 0
0 (θ,φ)=
1
4π
Y 0
1 (θ,φ)=
3
4π
cosθ
Y ±1
1 (θ,φ)=∓
3
8π
sinθe±iφ
Y 0
2 (θ,φ)=
5
4π
3
2
cos2
θ−
1
2
Y ±1
2 (θ,φ)=∓
15
8π
sinθcosθe±iφ
Y ±2
2 (θ,φ)=
15
32π
sin2
θe±2iφ
Y 0
3 (θ,φ)=
1
2
7
4π
(5cos2
θ−3)cosθ Y ±1
3 (θ,φ)=∓
1
4
21
4π
sinθ(5cos2
θ−1)e±iφ
Y ±2
3 (θ,φ)=
1
4
105
2π
sin2
θcosθe±2iφ
Y ±3
3 (θ,φ)=∓
1
4
35
4π
sin3
θe±3iφ
aDefined for −l ≤ m ≤ l, using the sign convention of the Condon–Shortley phase. Other sign conventions are
possible.

50 Mathematics
Delta functions
Kronecker delta
δij =
1 if i=j
0 if i=j
(2.441)
δii =3 (2.442)
δij Kronecker delta
i,j,k,... indices (=1,2 or 3)
Three-
dimensional
Levi–Civita
symbol
(permutation
tensor)a
123 = 231 = 312 =1
132 = 213 = 321 =−1 (2.443)
all other ijk =0
ijk klm =δilδjm −δimδjl (2.444)
δij ijk =0 (2.445)
ilm jlm =2δij (2.446)
ijk ijk =6 (2.447)
ijk Levi–Civita symbol
(see also page 25)
Dirac delta
function
b
a
δ(x) dx=
1 if a<0<b
0 otherwise
(2.448)
b
a
f(x)δ(x−x0) dx=f(x0) (2.449)
δ(x−x0)f(x)=δ(x−x0)f(x0) (2.450)
δ(−x)=δ(x) (2.451)
δ(ax)=|a|−1
δ(x) (a=0) (2.452)
δ(x) nπ−1/2
e−n2
x2
(n 1) (2.453)
δ(x) Dirac delta function
f(x) smooth function of x
a,b constants
aThe general symbol ijk... is defined to be +1 for even permutations of the suffices, −1 for odd permutations, and
0 if a suffix is repeated. The sequence (1,2,3,... ,n) is taken to be even. Swapping adjacent suffices an odd (or even)
number of times gives an odd (or even) permutation.
2.10 Roots of quadratic and cubic equations
Quadratic equations
Equation ax2
+bx+c=0 (a=0) (2.454) x variable
a,b,c real constants
Solutions
x1,2 =
−b±
√
b2 −4ac
2a
(2.455)
=
−2c
b±
√
b2 −4ac
(2.456)
x1,x2 quadratic roots
Solution
combinations
x1 +x2 =−b/a (2.457)
x1x2 =c/a (2.458)

2.10 Roots of quadratic and cubic equations
2
51
Cubic equations
Equation ax3
+bx2
+cx+d=0 (a=0) (2.459)
x variable
a,b,c,d real constants
Intermediate
definitions
p=
1
3
3c
a
−
b2
a2
(2.460)
q =
1
27
2b3
a3
−
9bc
a2
+
27d
a
(2.461)
D =
p
3
3
+
q
2
2
(2.462)
D discriminant
If D ≥0, also define: If D <0, also define:
u=
−q
2
+D1/2
1/3
(2.463)
v =
−q
2
−D1/2
1/3
(2.464)
y1 =u+v (2.465)
y2,3 =
−(u+v)
2
± i
u−v
2
31/2
(2.466)
1 real, 2 complex roots
(if D =0: 3 real roots, at least 2 equal)
φ=arccos
−q
2
|p|
3
−3/2
(2.467)
y1 =2
|p|
3
1/2
cos
φ
3
(2.468)
y2,3 =−2
|p|
3
1/2
cos
φ±π
3
(2.469)
3 distinct real roots
Solutionsa
xn =yn −
b
3a
(2.470)
xn cubic roots
(n=1,2,3)
Solution
combinations
x1 +x2 +x3 =−b/a (2.471)
x1x2 +x1x3 +x2x3 =c/a (2.472)
x1x2x3 =−d/a (2.473)
ayn are solutions to the reduced equation y3 +py+q =0.

52 Mathematics
2.11 Fourier series and transforms
Fourier series
Real form
f(x)=
a0
2
+
∞
n=1
an cos
nπx
L
+bn sin
nπx
L
(2.474)
an =
1
L
L
−L
f(x)cos
nπx
L
dx (2.475)
bn =
1
L
L
−L
f(x)sin
nπx
L
dx (2.476)
f(x) periodic
function,
period 2L
an,bn Fourier
coefficients
Complex
form
f(x)=
∞
n=−∞
cn exp
inπx
L
(2.477)
cn =
1
2L
L
−L
f(x)exp
−inπx
L
dx (2.478)
cn complex
Fourier
coefficient
Parseval’s
theorem
1
2L
L
−L
|f(x)|2
dx=
a2
0
4
+
1
2
∞
n=1
a2
n +b2
n (2.479)
=
∞
n=−∞
|cn|2
(2.480)
|| modulus
Fourier transforma
Definition 1
F(s)=
∞
−∞
f(x)e−2πixs
dx (2.481)
f(x)=
∞
−∞
F(s)e2πixs
ds (2.482)
f(x) function of x
F(s) Fourier transform of f(x)
Definition 2
F(s)=
∞
−∞
f(x)e−ixs
dx (2.483)
f(x)=
1
2π
∞
−∞
F(s)eixs
ds (2.484)
Definition 3
F(s)=
1
√
2π
∞
−∞
f(x)e−ixs
dx (2.485)
f(x)=
1
√
2π
∞
−∞
F(s)eixs
ds (2.486)
aAll three (and more) definitions are used, but definition 1 is probably the best.

2.11 Fourier series and transforms
2
53
Fourier transform theoremsa
Convolution f(x)∗g(x)=
∞
−∞
f(u)g(x−u) du (2.487)
f,g general functions
∗ convolution
Convolution
rules
f ∗g =g∗f (2.488)
f ∗(g∗h)=(f ∗g)∗h (2.489)
f f(x) F(s)
g g(x) G(s)
Convolution
theorem
f(x)g(x) F(s)∗G(s) (2.490) Fourier transform
relation
Autocorrela-
tion
f∗
(x) f(x)=
∞
−∞
f∗
(u−x)f(u) du (2.491)
correlation
f∗ complex
conjugate of f
Wiener–
Khintchine
theorem
f∗
(x) f(x) |F(s)|2
(2.492)
Cross-
correlation
f∗
(x) g(x)=
∞
−∞
f∗
(u−x)g(u) du (2.493)
Correlation
theorem
h(x) j(x) H(s)J∗
(s) (2.494)
h,j real functions
H H(s) h(x)
J J(s) j(x)
Parseval’s
relationb
∞
−∞
f(x)g∗
(x) dx=
∞
−∞
F(s)G∗
(s) ds (2.495)
Parseval’s
theoremc
∞
−∞
|f(x)|2
dx=
∞
−∞
|F(s)|2
ds (2.496)
Derivatives
df(x)
dx
2πisF(s) (2.497)
d
dx
[f(x)∗g(x)]=
df(x)
dx
∗g(x)=
dg(x)
dx
∗f(x)
(2.498)
aDeﬁning the Fourier transform as F(s)=
∞
−∞ f(x)e−2πixs dx.
bAlso called the “power theorem.”
cAlso called “Rayleigh’s theorem.”
Fourier symmetry relationships
f(x) F(s) deﬁnitions
even even real: f(x)=f∗
(x)
odd odd imaginary: f(x)=−f∗
(x)
real, even real, even even: f(x)=f(−x)
real, odd imaginary, odd odd: f(x)=−f(−x)
imaginary, even imaginary, even Hermitian: f(x)=f∗
(−x)
complex, even complex, even anti-Hermitian: f(x)=−f∗
(−x)
complex, odd complex, odd
real, asymmetric complex, Hermitian
imaginary, asymmetric complex, anti-Hermitian

54 Mathematics
Fourier transform pairsa
f(x) F(s)=
∞
−∞
f(x)e−2πisx
dx (2.499)
f(ax)
1
|a|
F(s/a) (a=0, real) (2.500)
f(x−a) e−2πias
F(s) (a real) (2.501)
dn
dxn
f(x) (2πis)n
F(s) (2.502)
δ(x) 1 (2.503)
δ(x−a) e−2πias
(2.504)
e−a|x| 2a
a2 +4π2s2
(a>0) (2.505)
xe−a|x| 8iπas
(a2 +4π2s2)2
(a>0) (2.506)
e−x2
/a2
a
√
πe−π2
a2
s2
(2.507)
sinax
1
2i
δ s−
a
2π
−δ s+
a
2π
(2.508)
cosax
1
2
δ s−
a
2π
+δ s+
a
2π
(2.509)
∞
m=−∞
δ(x−ma)
1
a
∞
n=−∞
δ s−
n
a
(2.510)
f(x)=
0 x<0
1 x>0
(“step”)
1
2
δ(s)−
i
2πs
(2.511)
f(x)=
1 |x|≤a
0 |x|>a
(“top hat”)
sin2πas
πs
=2asinc2as (2.512)
f(x)=



1−
|x|
a
|x|≤a
0 |x|>a
(“triangle”)
1
2π2as2
(1−cos2πas)=asinc2
as (2.513)
aEquation (2.499) deﬁnes the Fourier transform used for these pairs. Note that sincx≡(sinπx)/(πx).

2.12 Laplace transforms
2
55
2.12 Laplace transforms
Laplace transform theorems
Definitiona
F(s)=L{f(t)}=
∞
0
f(t)e−st
dt (2.514)
L{} Laplace
transform
Convolutionb
F(s)·G(s)=L
∞
0
f(t−z)g(z) dz (2.515)
=L{f(t)∗g(t)} (2.516)
F(s) L{f(t)}
G(s) L{g(t)}
∗ convolution
Inversec
f(t)=
1
2πi
γ+i∞
γ−i∞
est
F(s) ds (2.517)
= residues (for t>0) (2.518)
γ constant
Transform of
derivative
L
dn
f(t)
dtn
=sn
L{f(t)}−
n−1
r=0
sn−r−1 dr
f(t)
dtr t=0
(2.519)
n integer >0
Derivative of
transform
dn
F(s)
dsn
=L{(−t)n
f(t)} (2.520)
Substitution F(s−a)=L{eat
f(t)} (2.521) a constant
Translation
e−as
F(s)=L{u(t−a)f(t−a)} (2.522)
where u(t)=
0 (t<0)
1 (t>0)
(2.523)
u(t) unit step
function
aIf |e−s0tf(t)| is finite for sufficiently large t, the Laplace transform exists for s>s0.
bAlso known as the “faltung (or folding) theorem.”
cAlso known as the “Bromwich integral.” γ is chosen so that the singularities in F(s) are left of the integral line.

56 Mathematics
Laplace transform pairs
f(t)=⇒F(s)=L{f(t)}=
∞
0
f(t)e−st
dt (2.524)
δ(t)=⇒1 (2.525)
1=⇒1/s (s>0) (2.526)
tn
=⇒
n!
sn+1
(s>0, n>−1) (2.527)
t1/2
=⇒
π
4s3
(2.528)
t−1/2
=⇒
π
s
(2.529)
eat
=⇒
1
s−a
(s>a) (2.530)
teat
=⇒
1
(s−a)2
(s>a) (2.531)
(1−at)e−at
=⇒
s
(s+a)2
(2.532)
t2
e−at
=⇒
2
(s+a)3
(2.533)
sinat=⇒
a
s2 +a2
(s>0) (2.534)
cosat=⇒
s
s2 +a2
(s>0) (2.535)
sinhat=⇒
a
s2 −a2
(s>a) (2.536)
coshat=⇒
s
s2 −a2
(s>a) (2.537)
e−bt
sinat=⇒
a
(s+b)2 +a2
(2.538)
e−bt
cosat=⇒
s+b
(s+b)2 +a2
(2.539)
e−at
f(t)=⇒F(s+a) (2.540)

2.13 Probability and statistics
2
57
Discrete statistics
Mean x =
1
N
N
i=1
xi (2.541)
xi data series
N series length
· mean value
Variancea
var[x]=
1
N −1
N
i=1
(xi − x )2
(2.542)
var[·] unbiased
variance
Standard
deviation
σ[x]=(var[x])1/2
(2.543) σ standard
deviation
Skewness skew[x]=
N
(N −1)(N −2)
N
i=1
xi − x
σ
3
(2.544)
Kurtosis kurt[x]
1
N
N
i=1
xi − x
σ
4
−3 (2.545)
Correlation
coefficientb
r =
N
i=1(xi − x )(yi − y )
N
i=1(xi − x )2 N
i=1(yi − y )2
(2.546)
x,y data series to
correlate
r correlation
coefficient
aIf x is derived from the data, {xi}, the relation is as shown. If x is known independently, then an unbiased
estimate is obtained by dividing the right-hand side by N rather than N −1.
bAlso known as “Pearson’s r.”
Discrete probability distributions
distribution pr(x) mean variance domain
Binomial n
x px
(1−p)n−x
np np(1−p) (x=0,1,... ,n) (2.547)
n
x binomial
coefficient
Geometric (1−p)x−1
p 1/p (1−p)/p2
(x=1,2,3,...) (2.548)
Poisson λx
exp(−λ)/x! λ λ (x=0,1,2,...) (2.549)

58 Mathematics
Continuous probability distributions
distribution pr(x) mean variance domain
Uniform
1
b−a
a+b
2
(b−a)2
12
(a≤x≤b) (2.550)
Exponential λexp(−λx) 1/λ 1/λ2
(x≥0) (2.551)
Normal/
Gaussian
1
σ
√
2π
exp
−(x−µ)2
2σ2
µ σ2
(−∞<x<∞) (2.552)
Chi-squareda e−x/2
x(r/2)−1
2r/2Γ(r/2)
r 2r (x≥0) (2.553)
Rayleigh
x
σ2
exp
−x2
2σ2
σ π/2 2σ2
1−
π
4
(x≥0) (2.554)
Cauchy/
Lorentzian
a
π(a2 +x2)
(none) (none) (−∞<x<∞) (2.555)
aWith r degrees of freedom. Γ is the gamma function.
Multivariate normal distribution
Density function pr(x)=
exp −1
2 (x−µ)C−1
(x−µ)T
(2π)k/2[det(C)]1/2
(2.556)
pr probability density
k number of dimensions
C covariance matrix
x variable (k dimensional)
µ vector of means
Mean µ=(µ1,µ2,... ,µk) (2.557)
T transpose
det determinant
µi mean of ith variable
Covariance C=σij = xixj − xi xj (2.558) σij components of C
Correlation
coeﬃcient
r =
σij
σiσj
(2.559) r correlation coeﬃcient
Box–Muller
transformation
x1 =(−2lny1)1/2
cos2πy2 (2.560)
x2 =(−2lny1)1/2
sin2πy2 (2.561)
xi normally distributed deviates
yi deviates distributed
uniformly between 0 and 1

2
59
Random walk
One-
dimensional
pr(x)=
1
(2πNl2)1/2
exp
−x2
2Nl2
(2.562)
x displacement after N steps
(can be positive or negative)
pr(x) probability density of x
(
∞
−∞ pr(x) dx=1)
N number of steps
l step length (all equal)
rms
displacement
xrms =N1/2
l (2.563) xrms root-mean-squared
displacement from start point
Three-
dimensional
pr(r)=
a
π1/2
3
exp(−a2
r2
) (2.564)
where a=
3
2Nl2
1/2
r radial distance from start
point
pr(r) probability density of r
(
∞
0 4πr2 pr(r) dr =1)
a (most probable distance)−1
Mean distance r =
8
3π
1/2
N1/2
l (2.565)
r mean distance from start
point
rms distance rrms =N1/2
l (2.566) rrms root-mean-squared distance
from start point
Bayesian inference
Conditional
probability
pr(x)= pr(x|y )pr(y ) dy (2.567)
pr(x) probability (density) of x
pr(x|y ) conditional probability of x
given y
Joint
probability
pr(x,y)=pr(x)pr(y|x) (2.568) pr(x,y) joint probability of x and y
Bayes’ theorema
pr(y|x)=
pr(x|y) pr(y)
pr(x)
(2.569)
aIn this expression, pr(y|x) is known as the posterior probability, pr(x|y) the likelihood, and pr(y) the prior
probability.

60 Mathematics
2.14 Numerical methods
Straight-line ﬁttinga
Data {xi},{yi} n points (2.570)
Weightsb {wi} (2.571)
Model y =mx+c (2.572)
Residuals
x
y
y =mx+c
(x,y)
(0,c)
di =yi −mxi −c (2.573)
Weighted
centre
(x,y)=
1
wi
wixi , wiyi
(2.574)
Weighted
moment
D = wi(xi −x)2
(2.575)
Gradient
m=
1
D
wi(xi −x)yi (2.576)
var[m]
1
D
wid2
i
n−2
(2.577)
Intercept
c=y−mx (2.578)
var[c]
1
wi
+
x2
D
wid2
i
n−2
(2.579)
aLeast-squares ﬁt of data to y =mx+c. Errors on y-values only.
bIf the errors on yi are uncorrelated, then wi =1/var[yi].
Time series analysisa
Discrete
convolution (r s)j =
M/2
k=−(M/2)+1
sj−krk (2.580)
ri response function
si time series
M response function duration
Bartlett
(triangular)
window
wj =1−
j −N/2
N/2
(2.581)
wj windowing function
N length of time series
Welch
(quadratic)
window
wj =1−
j −N/2
N/2
2
(2.582)
Hanning
window
w
Welch
Bartlett
Hamming
Hanning
j/N
0
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1
1
wj =
1
2
1−cos
2πj
N
(2.583)
Hamming
window
wj =0.54−0.46cos
2πj
N
(2.584)
aThe time series runs from j =0...(N −1), and the windowing functions peak at j =N/2.

2.14 Numerical methods
2
61
Numerical integration
x
f(x)
h
x0 xN
Trapezoidal rule
xN
x0
f(x) dx
h
2
(f0 +2f1 +2f2 +···
+2fN−1 +fN) (2.585)
h =(xN −x0)/N
(subinterval
width)
fi fi =f(xi)
N number of
subintervals
Simpson’s rulea
xN
x0
f(x) dx
h
3
(f0 +4f1 +2f2 +4f3 +···
+4fN−1 +fN) (2.586)
aN must be even. Simpson’s rule is exact for quadratics and cubics.
Numerical diﬀerentiationa
df
dx
1
12h
[−f(x+2h)+8f(x+h)−8f(x−h)+f(x−2h)] (2.587)
∼
1
2h
[f(x+h)−f(x−h)] (2.588)
d2
f
dx2
1
12h2
[−f(x+2h)+16f(x+h)−30f(x)+16f(x−h)−f(x−2h)] (2.589)
∼
1
h2
[f(x+h)−2f(x)+f(x−h)] (2.590)
d3
f
dx3
∼
1
2h3
[f(x+2h)−2f(x+h)+2f(x−h)−f(x−2h)] (2.591)
aDerivatives of f(x) at x. h is a small interval in x.
Relations containing “ ” are O(h4); those containing “∼” are O(h2).
Numerical solutions to f(x)=0
Secant method xn+1 =xn −
xn −xn−1
f(xn)−f(xn−1)
f(xn) (2.592)
f function of x
xn f(x∞)=0
Newton–Raphson
method
xn+1 =xn −
f(xn)
f (xn)
(2.593) f = df/dx

62 Mathematics
Numerical solutions to ordinary differential equationsa
Euler’s method
if
dy
dx
=f(x,y) (2.594)
and h=xn+1 −xn (2.595)
then yn+1 =yn +hf(xn,yn)+O(h2
) (2.596)
Runge–Kutta
method
(fourth-order)
if
dy
dx
=f(x,y) (2.597)
and h=xn+1 −xn (2.598)
k1 =hf(xn,yn) (2.599)
k2 =hf(xn +h/2,yn +k1/2) (2.600)
k3 =hf(xn +h/2,yn +k2/2) (2.601)
k4 =hf(xn +h,yn +k3) (2.602)
then yn+1 =yn +
k1
6
+
k2
3
+
k3
3
+
k4
6
+O(h5
) (2.603)
aOrdinary differential equations (ODEs) of the form dy
dx =f(x,y). Higher order equations should be
reduced to a set of coupled first-order equations and solved in parallel.

3
Chapter 3 Dynamics and mechanics
3.1 Introduction
Unusually in physics, there is no pithy phrase that sums up the study of dynamics (the way
in which forces produce motion), kinematics (the motion of matter), mechanics (the study of
the forces and the motion they produce), and statics (the way forces combine to produce
equilibrium). We will take the phrase dynamics and mechanics to encompass all the above,
although it clearly does not!
To some extent this is because the equations governing the motion of matter include some
of our oldest insights into the physical world and are consequentially steeped in tradition.
One of the more delightful, or for some annoying, facets of this is the occasional use of
arcane vocabulary in the description of motion. The epitome must be what Goldstein1
calls
“the jabberwockian sounding statement” the polhode rolls without slipping on the herpolhode
lying in the invariable plane, describing “Poinsot’s construction” – a method of visualising the
free motion of a spinning rigid body. Despite this, dynamics and mechanics, including ﬂuid
mechanics, is arguably the most practically applicable of all the branches of physics.
Moreover, and in common with electromagnetism, the study of dynamics and mechanics
has spawned a good deal of mathematical apparatus that has found uses in other ﬁelds. Most
notably, the ideas behind the generalised dynamics of Lagrange and Hamilton lie behind
much of quantum mechanics.
1H. Goldstein, Classical Mechanics, 2nd ed., 1980, Addison-Wesley.

64 Dynamics and mechanics
3.2 Frames of reference
Galilean transformations
Time and
positiona
S
S
r
r
vt
m
r =r +vt (3.1)
t=t (3.2)
r,r position in frames S
and S
v velocity of S in S
t,t time in S and S
Velocity u=u +v (3.3) u,u velocity in frames S
and S
Momentum p =p +mv (3.4)
p,p particle momentum
in frames S and S
m particle mass
Angular
momentum
J =J +mr ×××v+v×××p t (3.5) J,J angular momentum
in frames S and S
Kinetic
energy
T =T +mu ·v+
1
2
mv2
(3.6)
T,T kinetic energy in
frames S and S
aFrames coincide at t=0.
Lorentz (spacetime) transformationsa
Lorentz factor
S S
x x
v
γ = 1−
v2
c2
−1/2
(3.7)
γ Lorentz factor
c speed of light
Time and position
x=γ(x +vt ); x =γ(x−vt) (3.8)
y =y ; y =y (3.9)
z =z ; z =z (3.10)
t=γ t +
v
c2
x ; t =γ t−
v
c2
x (3.11)
x,x x-position in frames
S and S (similarly
for y and z)
t,t time in frames S and
S
Diﬀerential
four-vectorb
dX =(cdt,−dx,−dy,−dz)
(3.12)
X spacetime four-vector
aFor frames S and S coincident at t=0 in relative motion along x. See page 141 for the
transformations of electromagnetic quantities.
bCovariant components, using the (1,−1,−1,−1) signature.
Velocity transformationsa
Velocity
S S
x x
u
v
γ Lorentz factor
=[1−(v/c)2]−1/2
ux =
ux +v
1+uxv/c2
; ux =
ux −v
1−uxv/c2
(3.13)
uy =
uy
γ(1+uxv/c2)
; uy =
uy
γ(1−uxv/c2)
(3.14)
uz =
uz
γ(1+uxv/c2)
; uz =
uz
γ(1−uxv/c2)
(3.15)
c speed of light
ui,ui particle velocity
components in
frames S and S
aFor frames S and S coincident at t=0 in relative motion along x.

3.2 Frames of reference
3
65
Momentum and energy transformationsa
Momentum and energy
S S
x x
v
γ Lorentz factor
=[1−(v/c)2]−1/2
px =γ(px +vE /c2
); px =γ(px −vE/c2
) (3.16)
py =py; py =py (3.17)
pz =pz; pz =pz (3.18)
E =γ(E +vpx); E =γ(E −vpx) (3.19)
c speed of light
px,px x components of
momentum in S and
S (sim. for y and z)
E,E energy in S and S
E2
−p2
c2
=E
2
−p
2
c2
=m2
0c4
(3.20) m0 (rest) mass
p total momentum in S
Four-vectorb P =(E/c,−px,−py,−pz) (3.21) P momentum
four-vector
bCovariant components, using the (1,−1,−1,−1) signature.
Propagation of lighta
Doppler
eﬀect
c
c
α
S
S S
x
x x
y
y y
v
θ
ν
ν
=γ 1+
v
c
cosα (3.22)
ν frequency received in S
ν frequency emitted in S
α arrival angle in S
Aberrationb
cosθ =
cosθ +v/c
1+(v/c)cosθ
(3.23)
cosθ =
cosθ−v/c
1−(v/c)cosθ
(3.24)
γ Lorentz factor
=[1−(v/c)2]−1/2
c speed of light
θ,θ emission angle of light
in S and S
Relativistic
beamingc P(θ)=
sinθ
2γ2[1−(v/c)cosθ]2
(3.25)
P(θ) angular distribution of
photons in S
bLight travelling in the opposite sense has a propagation angle of π+θ radians.
cAngular distribution of photons from a source, isotropic and stationary in S .
π
0 P(θ) dθ =1.
Four-vectorsa
Covariant and
contravariant
components
x0 =x0
x1 =−x1
x2 =−x2
x3 =−x3
(3.26)
xi covariant vector
components
xi contravariant components
Scalar product xi
yi =x0
y0 +x1
y1 +x2
y2 +x3
y3 (3.27)
Lorentz transformations
xi,x i
four-vector components in
frames S and S
x0
=γ[x
0
+(v/c)x
1
]; x
0
=γ[x0
−(v/c)x1
] (3.28)
x1
=γ[x
1
+(v/c)x
0
]; x
1
=γ[x1
−(v/c)x0
] (3.29)
x2
=x
2
; x
3
=x3
(3.30)
γ Lorentz factor
=[1−(v/c)2]−1/2
c speed of light
aFor frames S and S , coincident at t = 0 in relative motion along the (1) direction. Note that the (1,−1,−1,−1)
signature used here is common in special relativity, whereas (−1,1,1,1) is often used in connection with general
relativity (page 67).

Rotating frames
Vector trans-
formation
dA
dt S
=
dA
dt S
+ω×××A (3.31)
A any vector
S stationary frame
S rotating frame
ω angular velocity
of S in S
Acceleration ˙v =˙v +2ω×××v +ω×××(ω×××r ) (3.32)
˙v,˙v accelerations in S
and S
v velocity in S
r position in S
Coriolis force F cor =−2mω×××v (3.33) F cor coriolis force
m particle mass
Centrifugal
force
F cen =−mω×××(ω×××r ) (3.34)
=+mω2
r⊥ (3.35)
F cen centrifugal force
r⊥ perpendicular to
particle from
rotation axis
Motion
relative to
Earth
F cen
r⊥
r
m
ω
ωe
x
y z
λ
m¨x=Fx +2mωe(˙ysinλ−˙zcosλ)
(3.36)
mÿ =Fy −2mωe˙xsinλ (3.37)
m¨z =Fz −mg+2mωe˙xcosλ (3.38)
Fi nongravitational
force
λ latitude
z local vertical axis
y northerly axis
x easterly axis
Foucault’s
penduluma Ωf =−ωe sinλ (3.39)
Ωf pendulum’s rate
of turn
ωe Earth’s spin rate
aThe sign is such as to make the rotation clockwise in the northern hemisphere.
3.3 Gravitation
Newtonian gravitation
Newton’s law of
gravitation
F 1 =
Gm1m2
r2
12
ˆr12 (3.40)
m1,2 masses
F 1 force on m1 (=−F 2)
r12 vector from m1 to m2
ˆ unit vector
Newtonian field
equationsa
g=−∇φ (3.41)
∇2
φ=−∇·g=4πGρ (3.42)
G constant of gravitation
g gravitational field strength
φ gravitational potential
ρ mass density
Fields from an
isolated
uniform sphere,
mass M, r from
the centre a
M r
g(r)=



−
GM
r2
ˆr (r >a)
−
GMr
a3
ˆr (r <a)
(3.43)
φ(r)=



−
GM
r
(r >a)
GM
2a3
(r2
−3a2
) (r <a)
(3.44)
r vector from sphere centre
M mass of sphere
a radius of sphere
aThe gravitational force on a mass m is mg.

3.3 Gravitation
3
67
General relativitya
Line element ds2
=gµν dxµ
dxν
=−dτ2
(3.45)
ds invariant interval
dτ proper time interval
gµν metric tensor
Christoffel
symbols and
covariant
differentiation
Γα
βγ =
1
2
gαδ
(gδβ,γ +gδγ,β −gβγ,δ) (3.46)
φ;γ =φ,γ ≡∂φ/∂xγ
(3.47)
Aα
;γ =Aα
,γ +Γα
βγAβ
(3.48)
Bα;γ =Bα,γ −Γβ
αγBβ (3.49)
dxµ differential of xµ
Γα
βγ Christoffel symbols
,α partial diff. w.r.t. xα
;α covariant diff. w.r.t. xα
φ scalar
Aα contravariant vector
Bα covariant vector
Riemann tensor
Rα
βγδ =Γα
µγΓµ
βδ −Γα
µδΓµ
βγ
+Γα
βδ,γ −Γα
βγ,δ (3.50)
Bµ;α;β −Bµ;β;α =Rγ
µαβBγ (3.51)
Rαβγδ =−Rαβδγ ; Rβαγδ =−Rαβγδ (3.52)
Rαβγδ +Rαδβγ +Rαγδβ =0 (3.53)
Rα
βγδ Riemann tensor
Geodesic
equation
Dvµ
Dλ
=0 (3.54)
where
DAµ
Dλ
≡
dAµ
dλ
+Γµ
αβAα
vβ
(3.55)
vµ tangent vector
(= dxµ/dλ)
λ affine parameter (e.g., τ
for material particles)
Geodesic
deviation
D2
ξµ
Dλ2
=−Rµ
αβγvα
ξβ
vγ
(3.56) ξµ geodesic deviation
Ricci tensor Rαβ ≡Rσ
ασβ =gσδ
Rδασβ =Rβα (3.57) Rαβ Ricci tensor
Einstein tensor Gµν
=Rµν
−
1
2
gµν
R (3.58)
Gµν Einstein tensor
R Ricci scalar (=gµνRµν)
Einstein’s field
equations
Gµν
=8πTµν
(3.59) Tµν stress-energy tensor
p pressure (in rest frame)
Perfect fluid Tµν
=(p+ρ)uµ
uν
+pgµν
(3.60) ρ density (in rest frame)
uν fluid four-velocity
Schwarzschild
solution
(exterior)
ds2
=− 1−
2M
r
dt2
+ 1−
2M
r
−1
dr2
+r2
(dθ2
+sin2
θ dφ2
) (3.61)
M spherically symmetric
mass (see page 183)
(r,θ,φ) spherical polar coords.
t time
Kerr solution (outside a spinning black hole)
ds2
=−
∆−a2
sin2
θ
2
dt2
−2a
2Mrsin2
θ
2
dt dφ
+
(r2
+a2
)2
−a2
∆sin2
θ
2
sin2
θdφ2
+
2
∆
dr2
+ 2
dθ2
(3.62)
J angular momentum
(along z)
a ≡J/M
∆ ≡r2 −2Mr+a2
2 ≡r2 +a2 cos2 θ
aGeneral relativity conventionally uses the (−1,1,1,1) metric signature and “geometrized units” in which G=1 and
c = 1. Thus, 1kg = 7.425×10−28 m etc. Contravariant indices are written as superscripts and covariant indices as
subscripts. Note also that ds2 means (ds)2 etc.

3.4 Particle motion
Dynamics deﬁnitionsa
Newtonian force F =m¨r = ˙p (3.63)
F force
m mass of particle
r particle position vector
Momentum p =m˙r (3.64) p momentum
Kinetic energy T =
1
2
mv2
(3.65)
T kinetic energy
v particle velocity
Angular momentum J =r×××p (3.66) J angular momentum
Couple (or torque) G =r×××F (3.67) G couple
Centre of mass
(ensemble of N
particles)
R0 =
N
i=1 miri
N
i=1 mi
(3.68)
R0 position vector of centre of mass
mi mass of ith particle
ri position vector of ith particle
aIn the Newtonian limit, v c, assuming m is constant.
Relativistic dynamicsa
Lorentz factor γ = 1−
v2
c2
−1/2
(3.69)
γ Lorentz factor
v particle velocity
c speed of light
Momentum p =γm0v (3.70)
p relativistic momentum
m0 particle (rest) mass
Force F =
dp
dt
(3.71)
F force on particle
t time
Rest energy Er =m0c2
(3.72) Er particle rest energy
Kinetic energy T =m0c2
(γ−1) (3.73) T relativistic kinetic energy
Total energy
E =γm0c2
(3.74)
=(p2
c2
+m2
0c4
)1/2
(3.75)
E total energy (=Er +T)
aIt is now common to regard mass as a Lorentz invariant property and to drop the term “rest mass.” The
symbol m0 is used here to avoid confusion with the idea of “relativistic mass” (=γm0) used by some authors.
Constant acceleration
v =u+at (3.76)
v2
=u2
+2as (3.77)
s=ut+
1
2
at2
(3.78)
s=
u+v
2
t (3.79)
u initial velocity
v ﬁnal velocity
t time
s distance travelled
a acceleration

3.4 Particle motion
3
69
Reduced mass (of two interacting bodies)
m1m2
r1r2
r
centre
of
mass
Reduced mass µ=
m1m2
m1 +m2
(3.80)
µ reduced mass
mi interacting masses
Distances from
centre of mass
r1 =
m2
m1 +m2
r (3.81)
r2 =
−m1
m1 +m2
r (3.82)
ri position vectors from centre of
mass
r r =r1 −r2
|r| distance between masses
Moment of
inertia
I =µ|r|2
(3.83) I moment of inertia
Total angular
momentum
J =µr×××˙r (3.84) J angular momentum
Lagrangian L=
1
2
µ|˙r|2
−U(|r|) (3.85)
L Lagrangian
U potential energy of interaction
Ballisticsa
Velocity
ˆx
ˆy
α
v0
h
l
v =v0 cosα ˆx+(v0 sinα−gt) ˆy
(3.86)
v2
=v2
0 −2gy (3.87)
v0 initial velocity
v velocity at t
α elevation angle
g gravitational
acceleration
Trajectory y =xtanα−
gx2
2v2
0 cos2 α
(3.88)
ˆ unit vector
t time
Maximum
height h=
v2
0
2g
sin2
α (3.89)
h maximum
height
Horizontal
range l =
v2
0
g
sin2α (3.90) l range
aIgnoring the curvature and rotation of the Earth and frictional losses. g is assumed
constant.

Rocketry
Escape
velocitya vesc =
2GM
r
1/2
(3.91)
vesc escape velocity
M mass of central body
r central body radius
Specific
impulse
Isp =
u
g
(3.92)
Isp specific impulse
u effective exhaust velocity
g acceleration due to gravity
Exhaust
velocity (into
a vacuum)
u=
2γRTc
(γ−1)µ
1/2
(3.93)
R molar gas constant
γ ratio of heat capacities
Tc combustion temperature
µ effective molecular mass of
exhaust gas
Rocket
equation
(g =0)
∆v =uln
Mi
Mf
≡ulnM (3.94)
∆v rocket velocity increment
Mi pre-burn rocket mass
Mf post-burn rocket mass
M mass ratio
Multistage
rocket ∆v =
N
i=1
ui lnMi (3.95)
N number of stages
Mi mass ratio for ith burn
ui exhaust velocity of ith burn
In a constant
gravitational
field
∆v =ulnM−gtcosθ (3.96)
t burn time
θ rocket zenith angle
Hohmann
cotangential
transferb
a b
transfer ellipse, h
∆vah =
GM
ra
1/2
2rb
ra +rb
1/2
−1
(3.97)
∆vhb =
GM
rb
1/2
1−
2ra
ra +rb
1/2
(3.98)
∆vah velocity increment, a to h
∆vhb velocity increment, h to b
ra radius of inner orbit
rb radius of outer orbit
aFrom the surface of a spherically symmetric, nonrotating body, mass M.
bTransfer between coplanar, circular orbits a and b, via ellipse h with a minimal expenditure of energy.

3.4 Particle motion
3
71
Gravitationally bound orbital motiona
Potential energy
of interaction
U(r)=−
GMm
r
≡−
α
r
(3.99)
U(r) potential energy
M central mass
m orbiting mass ( M)
α GMm (for gravitation)
Total energy E =−
α
r
+
J2
2mr2
=−
α
2a
(3.100)
E total energy (constant)
J total angular momentum
(constant)
Virial theorem
(1/r potential)
E = U /2=− T (3.101)
U =−2 T (3.102)
T kinetic energy
· mean value
Orbital
equation
(Kepler’s 1st
law)
r0
r
=1+ecosφ, or (3.103)
r =
a(1−e2
)
1+ecosφ
(3.104)
r0 semi-latus-rectum
r distance of m from M
e eccentricity
φ phase (true anomaly)
Rate of
sweeping area
(Kepler’s 2nd
law)
dA
dt
=
J
2m
=constant (3.105)
A area swept out by radius
vector (total area =πab)
Semi-major axis a=
r0
1−e2
=
α
2|E|
(3.106)
a semi-major axis
b semi-minor axis
Semi-minor axis b=
r0
(1−e2)1/2
=
J
(2m|E|)1/2
(3.107)
Eccentricityb
m
M
2a
2b ae
A
r0
r
φ
rmax rmin
e= 1+
2EJ2
mα2
1/2
= 1−
b2
a2
1/2
(3.108)
Semi-latus-
rectum r0 =
J2
mα
=
b2
a
=a(1−e2
) (3.109)
Pericentre rmin =
r0
1+e
=a(1−e) (3.110) rmin pericentre distance
Apocentre rmax =
r0
1−e
=a(1+e) (3.111) rmax apocentre distance
Speed v2
=GM
2
r
−
1
a
(3.112) v orbital speed
Period (Kepler’s
3rd law)
P =πα
m
2|E|3
1/2
=2πa3/2 m
α
1/2
(3.113)
P orbital period
aFor an inverse-square law of attraction between two isolated bodies in the nonrelativistic limit. If m is not M,
then the equations are valid with the substitutions m→µ=Mm/(M +m) and M →(M +m) and with r taken as the
body separation. The distance of mass m from the centre of mass is then rµ/m (see earlier table on Reduced mass).
Other orbital dimensions scale similarly, and the two orbits have the same eccentricity.
bNote that if the total energy, E, is < 0 then e < 1 and the orbit is an ellipse (a circle if e = 0). If E = 0, then e = 1
and the orbit is a parabola. If E >0 then e>1 and the orbit becomes a hyperbola (see Rutherford scattering on next
page).

Rutherford scatteringa
b
χ
scattering
centre
trajectory
for α>0
trajectory
for α<0
x
y
rmin (α>0)
rmin (α<0)
aa
Scattering potential
energy
U(r)=−
α
r
(3.114)
α
<0 repulsive
>0 attractive
(3.115)
U(r) potential energy
r particle separation
α constant
Scattering angle tan
χ
2
=
|α|
2Eb
(3.116)
χ scattering angle
E total energy (>0)
b impact parameter
Closest approach
rmin =
|α|
2E
csc
χ
2
−
α
|α|
(3.117)
=a(e±1) (3.118)
rmin closest approach
a hyperbola semi-axis
e eccentricity
Semi-axis a=
|α|
2E
(3.119)
Eccentricity e=
4E2
b2
α2
+1
1/2
=csc
χ
2
(3.120)
Motion trajectoryb 4E2
α2
x2
−
y2
b2
=1 (3.121)
x,y position with respect to
hyperbola centre
Scattering centrec
x=±
α2
4E2
+b2
1/2
(3.122)
Rutherford
scattering formulad
dσ
dΩ
=
1
n
dN
dΩ
(3.123)
=
α
4E
2
csc4 χ
2
(3.124)
dσ
dΩ differential scattering
cross section
n beam flux density
dN number of particles
scattered into dΩ
Ω solid angle
aNonrelativistic treatment for an inverse-square force law and a fixed scattering centre. Similar scattering results
from either an attractive or repulsive force. See also Conic sections on page 38.
bThe correct branch can be chosen by inspection.
cAlso the focal points of the hyperbola.
dn is the number of particles per second passing through unit area perpendicular to the beam.

3.4 Particle motion
3
73
Inelastic collisionsa
m1m1
m2m2v1 v2 v1 v2
Before collision After collision
Coefficient of
restitution
v2 −v1 = (v1 −v2) (3.125)
=1 if perfectly elastic (3.126)
=0 if perfectly inelastic (3.127)
coefficient of restitution
vi pre-collision velocities
vi post-collision velocities
Loss of kinetic
energyb
T −T
T
=1− 2
(3.128)
T,T total KE in zero
momentum frame
before and after
collision
Final velocities
v1 =
m1 − m2
m1 +m2
v1 +
(1+ )m2
m1 +m2
v2 (3.129)
v2 =
m2 − m1
m1 +m2
v2 +
(1+ )m1
m1 +m2
v1 (3.130)
mi particle masses
aAlong the line of centres, v1,v2 c.
bIn zero momentum frame.
Oblique elastic collisionsa
m1
m1
m2m2
v
v1
v2
θ
θ1
θ2
Before collision After collision
Directions of
motion
tanθ1 =
m2 sin2θ
m1 −m2 cos2θ
(3.131)
θ2 =θ (3.132)
θ angle between
centre line and
incident velocity
θi final trajectories
mi sphere masses
Relative
separation angle θ1 +θ2



>π/2 if m1 <m2
=π/2 if m1 =m2
<π/2 if m1 >m2
(3.133)
Final velocities
v1 =
(m2
1 +m2
2 −2m1m2 cos2θ)1/2
m1 +m2
v (3.134)
v2 =
2m1v
m1 +m2
cosθ (3.135)
v incident velocity
of m1
vi final velocities
aCollision between two perfectly elastic spheres: m2 initially at rest, velocities c.

3.5 Rigid body dynamics
Moment of inertia tensor
Moment of
inertia tensora Iij = (r2
δij −xixj) dm (3.136)
r r2 =x2 +y2 +z2
I=



(y2
+z2
) dm − xy dm − xz dm
− xy dm (x2
+z2
) dm − yz dm
− xz dm − yz dm (x2
+y2
) dm



(3.137)
I moment of inertia
tensor
dm mass element
xi position vector of
dm
Iij components of I
Parallel axis
theorem
I12 =I12 −ma1a2 (3.138)
I11 =I11 +m(a2
2 +a2
3) (3.139)
Iij =Iij +m(|a|2
δij −aiaj) (3.140)
Iij tensor with respect
to centre of mass
ai,a position vector of
centre of mass
m mass of body
Angular
momentum
J =Iω (3.141)
J angular momentum
ω angular velocity
Rotational
kinetic energy
T =
1
2
ω·J =
1
2
Iijωiωj (3.142) T kinetic energy
aIii are the moments of inertia of the body. Iij (i = j) are its products of inertia. The integrals are over the body
volume.
Principal axes
Principal
moment of
inertia tensor
I =


I1 0 0
0 I2 0
0 0 I3

 (3.143)
I principal moment of
inertia tensor
Ii principal moments of
inertia
Angular
momentum
J =(I1ω1,I2ω2,I3ω3) (3.144)
J angular momentum
ωi components of ω
along principal axes
Rotational
kinetic energy
T =
1
2
(I1ω2
1 +I2ω2
2 +I3ω2
3) (3.145) T kinetic energy
Moment of
inertia
ellipsoida
T =T(ω1,ω2,ω3) (3.146)
Ji =
∂T
∂ωi
(J is ⊥ ellipsoid surface) (3.147)
Perpendicular
axis theorem
I1 I2
I3
lamina
I1 +I2
≥I3 generally
=I3 ﬂat lamina ⊥ to 3-axis
(3.148)
Symmetries
I1 =I2 =I3 asymmetric top
I1 =I2 =I3 symmetric top (3.149)
I1 =I2 =I3 spherical top
aThe ellipsoid is deﬁned by the surface of constant T.

3
75
Moments of inertiaa
Thin rod, length l
I1
I1
I1
I1
I1
I1
I1
I1
I2
I2
I2
I2
I2
I2
I2
I2
I3
I3
I3
I3
I3
I3
I3
I3
I3
a
a
a
a
b
b
b
b
c
c
c
r
r
r
r
h
l
l
I1 =I2 =
ml2
12
(3.150)
I3 0 (3.151)
Solid sphere, radius r I1 =I2 =I3 =
2
5
mr2
(3.152)
Spherical shell, radius r I1 =I2 =I3 =
2
3
mr2
(3.153)
Solid cylinder, radius r,
length l
I1 =I2 =
m
4
r2
+
l2
3
(3.154)
I3 =
1
2
mr2
(3.155)
Solid cuboid, sides a,b,c
I1 =m(b2
+c2
)/12 (3.156)
I2 =m(c2
+a2
)/12 (3.157)
I3 =m(a2
+b2
)/12 (3.158)
Solid circular cone, base
radius r, height hb
I1 =I2 =
3
20
m r2
+
h2
4
(3.159)
I3 =
3
10
mr2
(3.160)
Solid ellipsoid, semi-axes
a,b,c
I1 =m(b2
+c2
)/5 (3.161)
I2 =m(c2
+a2
)/5 (3.162)
I3 =m(a2
+b2
)/5 (3.163)
Elliptical lamina,
semi-axes a,b
I1 =mb2
/4 (3.164)
I2 =ma2
/4 (3.165)
I3 =m(a2
+b2
)/4 (3.166)
Disk, radius r
I1 =I2 =mr2
/4 (3.167)
I3 =mr2
/2 (3.168)
Triangular platec I3 =
m
36
(a2
+b2
+c2
) (3.169)
aWith respect to principal axes for bodies of mass m and uniform density. The radius of gyration is deﬁned as
k =(I/m)1/2.
bOrigin of axes is at the centre of mass (h/4 above the base).
cAround an axis through the centre of mass and perpendicular to the plane of the plate.

Centres of mass
Solid hemisphere, radius r d=3r/8 from sphere centre (3.170)
Hemispherical shell, radius r d=r/2 from sphere centre (3.171)
Sector of disk, radius r, angle
2θ
d=
2
3
r
sinθ
θ
from disk centre (3.172)
Arc of circle, radius r, angle
2θ
d=r
sinθ
θ
from circle centre (3.173)
Arbitrary triangular lamina,
height ha d=h/3 perpendicular from base (3.174)
Solid cone or pyramid, height
h
d=h/4 perpendicular from base (3.175)
Spherical cap, height h,
sphere radius r
solid: d=
3
4
(2r−h)2
3r−h
from sphere centre (3.176)
shell: d=r−h/2 from sphere centre (3.177)
Semi-elliptical lamina,
height h
d=
4h
3π
from base (3.178)
ah is the perpendicular distance between the base and apex of the triangle.
Pendulums
Simple
pendulum
l
l
l
l
l
θ0
α
I0
a
I1
I2
I3
m
m
m
m
P =2π
l
g
1+
θ2
0
16
+··· (3.179)
P period
g gravitational acceleration
l length
θ0 maximum angular
displacement
Conical
pendulum P =2π
lcosα
g
1/2
(3.180) α cone half-angle
Torsional
penduluma P =2π
lI0
C
1/2
(3.181)
I0 moment of inertia of bob
C torsional rigidity of wire
(see page 81)
Compound
pendulumb
P 2π
1
mga
(ma2
+I1 cos2
γ1
+I2 cos2
γ2 +I3 cos2
γ3)
1/2
(3.182)
a distance of rotation axis
from centre of mass
m mass of body
Ii principal moments of
inertia
γi angles between rotation
axis and principal axes
Equal
double
pendulumc
P 2π
l
(2±
√
2)g
1/2
(3.183)
aAssuming the bob is supported parallel to a principal rotation axis.
bI.e., an arbitrary triaxial rigid body.
cFor very small oscillations (two eigenmodes).

3
77
Tops and gyroscopes
herpolhode
invariable
plane
space
cone
body
conemoment
of inertia
ellipsoid
polhode
θ
2
3J
ω
Ωp
a mg
J3
prolate symmetric top gyroscope
support point
Euler’s equationsa
G1 =I1 ˙ω1 +(I3 −I2)ω2ω3 (3.184)
G2 =I2 ˙ω2 +(I1 −I3)ω3ω1 (3.185)
G3 =I3 ˙ω3 +(I2 −I1)ω1ω2 (3.186)
Gi external couple (=0 for free
rotation)
Ii principal moments of inertia
ωi angular velocity of rotation
Free symmetric
topb
(I3 <I2 =I1)
Ωb =
I1 −I3
I1
ω3 (3.187)
Ωs =
J
I1
(3.188)
Ωb body frequency
Ωs space frequency
Free asymmetric
topc Ω2
b =
(I1 −I3)(I2 −I3)
I1I2
ω2
3 (3.189)
Steady gyroscopic
precession
Ω2
pI1 cosθ−ΩpJ3 +mga=0 (3.190)
Ωp
Mga/J3 (slow)
J3/(I1 cosθ) (fast)
(3.191)
Ωp precession angular velocity
θ angle from vertical
J3 angular momentum around
symmetry axis
m mass
Gyroscopic
stability
J2
3 ≥4I1mgacosθ (3.192)
a distance of centre of mass
from support point
I1 moment of inertia about
support point
Gyroscopic limit
(“sleeping top”)
J2
3 I1mga (3.193)
Nutation rate Ωn =J3/I1 (3.194) Ωn nutation angular velocity
Gyroscope
released from rest
Ωp =
mga
J3
(1−cosΩnt) (3.195) t time
aComponents are with respect to the principal axes, rotating with the body.
bThe body frequency is the angular velocity (with respect to principal axes) of ω around the 3-axis. The space
frequency is the angular velocity of the 3-axis around J, i.e., the angular velocity at which the body cone moves
around the space cone.
cJ close to 3-axis. If Ω2
b <0, the body tumbles.

3.6 Oscillating systems
Free oscillations
Diﬀerential
equation
d2
x
dt2
+2γ
dx
dt
+ω2
0x=0 (3.196)
x oscillating variable
t time
γ damping factor (per unit
mass)
ω0 undamped angular frequency
Underdamped
solution (γ <ω0)
x=Ae−γt
cos(ωt+φ) (3.197)
where ω =(ω2
0 −γ2
)1/2
(3.198)
A amplitude constant
φ phase constant
ω angular eigenfrequency
Critically damped
solution (γ =ω0)
x=e−γt
(A1 +A2t) (3.199) Ai amplitude constants
Overdamped
solution (γ >ω0)
x=e−γt
(A1eqt
+A2e−qt
) (3.200)
where q =(γ2
−ω2
0)1/2
(3.201)
Logarithmic
decrementa ∆=ln
an
an+1
=
2πγ
ω
(3.202)
∆ logarithmic decrement
an nth displacement maximum
Quality factor Q=
ω0
2γ
π
∆
if Q 1 (3.203) Q quality factor
aThe decrement is usually the ratio of successive displacement maxima but is sometimes taken as the ratio of successive
displacement extrema, reducing ∆ by a factor of 2. Logarithms are sometimes taken to base 10, introducing a further
factor of log10 e.
Forced oscillations
Diﬀerential
equation
d2
x
dt2
+2γ
dx
dt
+ω2
0x=F0eiωf t
(3.204)
x oscillating variable
t time
γ damping factor (per unit
mass)
Steady-
state
solutiona
x=Aei(ωf t−φ)
, where (3.205)
A=F0[(ω2
0 −ω2
f )2
+(2γωf)2
]−1/2
(3.206)
F0/(2ω0)
[(ω0 −ωf)2 +γ2]1/2
(γ ωf) (3.207)
tanφ=
2γωf
ω2
0 −ω2
f
(3.208)
ω0 undamped angular frequency
F0 force amplitude (per unit
mass)
ωf forcing angular frequency
A amplitude
φ phase lag of response behind
driving force
Amplitude
resonanceb ω2
ar =ω2
0 −2γ2
(3.209) ωar amplitude resonant forcing
angular frequency
Velocity
resonancec ωvr =ω0 (3.210) ωvr velocity resonant forcing
angular frequency
Quality
factor
Q=
ω0
2γ
(3.211) Q quality factor
Impedance Z =2γ+i
ω2
f −ω2
0
ωf
(3.212) Z impedance (per unit mass)
aExcluding the free oscillation terms.
bForcing frequency for maximum displacement.
cForcing frequency for maximum velocity. Note φ=π/2 at this frequency.

3.7 Generalised dynamics
3
79
3.7 Generalised dynamics
Lagrangian dynamics
Action S =
t2
t1
L(q,˙q,t) dt (3.213)
S action (δS =0 for the motion)
q generalised coordinates
˙q generalised velocities
Euler–Lagrange
equation
d
dt
∂L
∂˙qi
−
∂L
∂qi
=0 (3.214)
L Lagrangian
t time
m mass
Lagrangian of
particle in
external ﬁeld
L=
1
2
mv2
−U(r,t) (3.215)
=T −U (3.216)
v velocity
r position vector
U potential energy
T kinetic energy
Relativistic
Lagrangian of a
charged particle
L=−
m0c2
γ
−e(φ−A·v) (3.217)
m0 (rest) mass
γ Lorentz factor
+e positive charge
φ electric potential
A magnetic vector potential
Generalised
momenta
pi =
∂L
∂˙qi
(3.218) pi generalised momenta
Hamiltonian dynamics
Hamiltonian H =
i
pi˙qi −L (3.219)
L Lagrangian
pi generalised momenta
˙qi generalised velocities
Hamilton’s
equations
˙qi =
∂H
∂pi
; ˙pi =−
∂H
∂qi
(3.220)
H Hamiltonian
qi generalised coordinates
Hamiltonian
of particle in
external ﬁeld
H =
1
2
mv2
+U(r,t) (3.221)
=T +U (3.222)
v particle speed
r position vector
U potential energy
T kinetic energy
Relativistic
Hamiltonian
of a charged
particle
H =(m2
0c4
+|p −eA|2
c2
)1/2
+eφ (3.223)
m0 (rest) mass
c speed of light
+e positive charge
A vector potential
Poisson
brackets
[f,g]=
i
∂f
∂qi
∂g
∂pi
−
∂f
∂pi
∂g
∂qi
(3.224)
[qi,g]=
∂g
∂pi
, [pi,g]=−
∂g
∂qi
(3.225)
[H,g]=0 if
∂g
∂t
=0,
dg
dt
=0 (3.226)
p particle momentum
t time
f,g arbitrary functions
[·,·] Poisson bracket (also see
Commutators on page 26)
Hamilton–
Jacobi
equation
∂S
∂t
+H qi,
∂S
∂qi
,t =0 (3.227) S action

3.8 Elasticity
Elasticity deﬁnitions (simple)a
Stress
F
A
w
l
τ=F/A (3.228)
τ stress
F applied force
A cross-sectional
area
Strain e=δl/l (3.229)
e strain
δl change in length
l length
Young modulus
(Hooke’s law)
E =τ/e=constant (3.230) E Young modulus
Poisson ratiob
σ =−
δw/w
δl/l
(3.231)
σ Poisson ratio
δw change in width
w width
aThese apply to a thin wire under longitudinal stress.
bSolids obeying Hooke’s law are restricted by thermodynamics to −1≤σ ≤1/2, but
none are known with σ <0. Non-Hookean materials can show σ >1/2.
Elasticity deﬁnitions (general)
Stress tensora
τij =
force i direction
area ⊥ j direction
(3.232) τij stress tensor (τij =τji)
Strain tensor ekl =
1
2
∂uk
∂xl
+
∂ul
∂xk
(3.233)
ekl strain tensor (ekl =elk)
uk displacement to xk
xk coordinate system
Elastic modulus τij =λijklekl (3.234) λijkl elastic modulus
Elastic energyb
U =
1
2
λijkleijekl (3.235) U potential energy
Volume strain
(dilatation)
ev =
δV
V
=e11 +e22 +e33 (3.236)
ev volume strain
δV change in volume
V volume
Shear strain
ekl =(ekl −
1
3
evδkl)
pure shear
+
1
3
evδkl
dilatation
(3.237)
δkl Kronecker delta
Hydrostatic
compression
τij =−pδij (3.238) p hydrostatic pressure
aτii are normal stresses, τij (i=j) are torsional stresses.
bAs usual, products are implicitly summed over repeated indices.

3.8 Elasticity
3
81
Isotropic elastic solids
Lamé coefficients
µ=
E
2(1+σ)
(3.239)
λ=
Eσ
(1+σ)(1−2σ)
(3.240)
µ,λ Lamé coefficients
E Young modulus
σ Poisson ratio
Longitudinal
modulusa Ml =
E(1−σ)
(1+σ)(1−2σ)
=λ+2µ (3.241)
Ml longitudinal elastic
modulus
Diagonalised
equationsb
eii =
1
E
[τii −σ(τjj +τkk)] (3.242)
τii =Ml eii +
σ
1−σ
(ejj +ekk) (3.243)
t=2µe+λ1tr(e) (3.244)
eii strain in i direction
τii stress in i direction
e strain tensor
t stress tensor
1 unit matrix
tr(·) trace
Bulk modulus
(compression
modulus)
K =
E
3(1−2σ)
=λ+
2
3
µ (3.245)
1
KT
=−
1
V
∂V
∂p T
(3.246)
−p=Kev (3.247)
K bulk modulus
KT isothermal bulk
modulus
V volume
p pressure
T temperature
Shear modulus
(rigidity modulus)
θsh
τT
µ=
E
2(1+σ)
(3.248)
τT =µθsh (3.249)
ev volume strain
µ shear modulus
τT transverse stress
θsh shear strain
Young modulus E =
9µK
µ+3K
(3.250)
Poisson ratio σ =
3K −2µ
2(3K +µ)
(3.251)
aIn an extended medium.
bAxes aligned along eigenvectors of the stress and strain tensors.
Torsion
Torsional rigidity
(for a
homogeneous
rod) l
a
G
φG=C
φ
l
(3.252)
G twisting couple
C torsional rigidity
l rod length
φ twist angle in
length l
Thin circular
cylinder
C =2πa3
µt (3.253)
a radius
t wall thickness
µ shear modulus
Thick circular
cylinder
C =
1
2
µπ(a4
2 −a4
1) (3.254)
a1 inner radius
a2 outer radius
Arbitrary
thin-walled tube
A
t
t
w
C =
4A2
µt
P
(3.255)
A cross-sectional
area
P perimeter
Long flat ribbon C =
1
3
µwt3
(3.256)
w cross-sectional
width

Bending beamsa
Bending
moment
ds
ξ
(cross section)
neutral surface
x
y
W
FcFc
free
fixed
Gb =
E
Rc
ξ2
ds (3.257)
=
EI
Rc
(3.258)
Gb bending moment
E Young modulus
Rc radius of curvature
ds area element
ξ distance to neutral
surface from ds
I moment of area
Light beam,
horizontal at
x=0, weight
at x=l
y =
W
2EI
l −
x
3
x2
(3.259)
y displacement from
horizontal
W end-weight
l beam length
x distance along beam
Heavy beam EI
d4
y
dx4
=w(x) (3.260)
w beam weight per
unit length
Euler strut
failure
Fc =



π2
EI/l2
(free ends)
4π2
EI/l2
(fixed ends)
π2
EI/(4l2
) (1 free end)
(3.261)
Fc critical compression
force
l strut length
aThe radius of curvature is approximated by 1/Rc d2
y/dx2.
Elastic wave velocitiesa
In an infinite
isotropic solidb
vt =(µ/ρ)1/2
(3.262)
vl =(Ml/ρ)1/2
(3.263)
vl
vt
=
2−2σ
1−2σ
1/2
(3.264)
vt speed of transverse wave
vl speed of longitudinal wave
µ shear modulus
ρ density
Ml longitudinal modulus
= E(1−σ)
(1+σ)(1−2σ)
In a fluid vl =(K/ρ)1/2
(3.265) K bulk modulus
On a thin plate (wave travelling along x, plate thin in z) v(i)
l speed of longitudinal
wave (displacement i)
x
y
z
k
v(x)
l =
E
ρ(1−σ2)
1/2
(3.266)
v(y)
t =(µ/ρ)1/2
(3.267)
v(z)
t =k
Et2
12ρ(1−σ2)
1/2
(3.268)
v(i)
t speed of transverse wave
(displacement i)
E Young modulus
σ Poisson ratio
k wavenumber (=2π/λ)
t plate thickness (in z, t λ)
In a thin circular
rod
vl =(E/ρ)1/2
(3.269)
vφ =(µ/ρ)1/2
(3.270)
vt =
ka
2
E
ρ
1/2
(3.271)
vφ torsional wave velocity
a rod radius ( λ)
aWaves that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout.
bTransverse waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure
waves, or P-waves.

3.8 Elasticity
3
83
Waves in strings and springsa
In a spring vl =(κl/ρl)1/2
(3.272)
vl speed of longitudinal wave
κ spring constantb
l spring length
ρl mass per unit lengthc
On a stretched
string
vt =(T/ρl)1/2
(3.273) vt speed of transverse wave
T tension
On a stretched
sheet
vt =(τ/ρA)1/2
(3.274)
τ tension per unit width
ρA mass per unit area
aWave amplitude assumed wavelength.
bIn the sense κ=force/extension.
cMeasured along the axis of the spring.
Propagation of elastic waves
Acoustic
impedance
Z =
force
response velocity
=
F
˙u
(3.275)
=(E ρ)1/2
(3.276)
Z impedance
F stress force
u strain displacement
Wave velocity/
impedance
relation
if v =
E
ρ
1/2
(3.277)
then Z =(E ρ)1/2
=ρv (3.278)
E elastic modulus
ρ density
v wave phase velocity
Mean energy
density
(nondispersive
waves)
U=
1
2
E k2
u2
0 (3.279)
=
1
2
ρω2
u2
0 (3.280)
P =Uv (3.281)
U energy density
k wavenumber
ω angular frequency
u0 maximum displacement
P mean energy flux
Normal
coefficientsa
r =
ur
ui
=−
τr
τi
=
Z1 −Z2
Z1 +Z2
(3.282)
t=
2Z1
Z1 +Z2
(3.283)
r reflection coefficient
t transmission coefficient
τ stress
Snell’s lawb sinθi
vi
=
sinθr
vr
=
sinθt
vt
(3.284)
θi angle of incidence
θr angle of reflection
θt angle of refraction
aFor stress and strain amplitudes. Because these reflection and transmission coefficients are usually defined in terms
of displacement, u, rather than stress, there are differences between these coefficients and their equivalents defined
in electromagnetism [see Equation (7.179) and page 154].
bAngles defined from the normal to the interface. An incident plane pressure wave will generally excite both shear
and pressure waves in reflection and transmission. Use the velocity appropriate for the wave type.

3.9 Fluid dynamics
Ideal fluidsa
Continuityb ∂ρ
∂t
+∇·(ρv)=0 (3.285)
ρ density
v fluid velocity field
t time
Kelvin circulation
Γ= v· dl =constant (3.286)
=
S
ω· ds (3.287)
Γ circulation
dl loop element
ds element of surface
bounded by loop
ω vorticity (=∇×××v)
Euler’s equationc
∂v
∂t
+(v·∇)v =−
∇p
ρ
+g (3.288)
or
∂
∂t
(∇×××v)=∇×××[v×××(∇×××v)] (3.289)
p pressure
g gravitational field
strength
(v·∇) advective operator
Bernoulli’s equation
(incompressible flow)
1
2
ρv2
+p+ρgz =constant (3.290) z altitude
(compressible
adiabatic flow)d
1
2
v2
+
γ
γ−1
p
ρ
+gz =constant (3.291)
=
1
2
v2
+cpT +gz (3.292)
γ ratio of specific heat
capacities (cp/cV )
cp specific heat capacity
at constant pressure
T temperature
Hydrostatics ∇p=ρg (3.293)
Adiabatic lapse rate
(ideal gas)
dT
dz
=−
g
cp
(3.294)
aNo thermal conductivity or viscosity.
bTrue generally.
cThe second form of Euler’s equation applies to incompressible flow only.
dEquation (3.292) is true only for an ideal gas.
Potential flowa
Velocity potential
v =∇φ (3.295)
∇2
φ=0 (3.296)
v velocity
φ velocity potential
Vorticity condition ω =∇×××v =0 (3.297)
ω vorticity
F drag force on moving
sphere
Drag force on a
sphereb F =−
2
3
πρa3
˙u=−
1
2
Md˙u (3.298)
a sphere radius
˙u sphere acceleration
ρ fluid density
Md displaced fluid mass
aFor incompressible fluids.
bThe effect of this drag force is to give the sphere an additional effective mass equal to half the mass of fluid
displaced.

3.9 Fluid dynamics
3
85
Viscous flow (incompressible)a
Fluid stress τij =−pδij +η
∂vi
∂xj
+
∂vj
∂xi
(3.299)
τij fluid stress tensor
p hydrostatic pressure
η shear viscosity
vi velocity along i axis
Navier–Stokes
equationb
∂v
∂t
+(v·∇)v =−
∇p
ρ
−
η
ρ
∇×××ω+g (3.300)
=−
∇p
ρ
+
η
ρ
∇2
v+g (3.301)
v fluid velocity field
ω vorticity
ρ density
Kinematic
viscosity
ν =η/ρ (3.302) ν kinematic viscosity
aI.e., ∇·v =0, η =0.
bNeglecting bulk (second) viscosity.
Laminar viscous flow
Between
parallel plates h
y
z
r a
a1
a2ω1
ω2
vz(y)=
1
2η
y(h−y)
∂p
∂z
(3.303)
vz flow velocity
z direction of flow
y distance from
plate
η shear viscosity
p pressure
Along a
circular pipea
vz(r)=
1
4η
(a2
−r2
)
∂p
∂z
(3.304)
Q=
dV
dt
=
πa4
8η
∂p
∂z
(3.305)
r distance from
pipe axis
a pipe radius
V volume
Circulating
between
concentric
rotating
cylindersb
Gz =
4πηa2
1a2
2
a2
2 −a2
1
(ω2 −ω1)
(3.306)
Gz axial couple
between cylinders
per unit length
ωi angular velocity
of ith cylinder
Along an
annular pipe
Q=
π
8η
∂p
∂z
a4
2 −a4
1 −
(a2
2 −a2
1)2
ln(a2/a1)
(3.307)
a1 inner radius
a2 outer radius
Q volume discharge
rate
aPoiseuille flow.
bCouette flow.
Draga
On a sphere (Stokes’s law) F =6πaηv (3.308)
F drag force
a radius
On a disk, broadside to flow F =16aηv (3.309) v velocity
η shear viscosity
On a disk, edge on to flow F =32aηv/3 (3.310)
aFor Reynolds numbers 1.

Characteristic numbers
Reynolds
number
Re=
ρUL
η
=
inertial force
viscous force
(3.311)
Re Reynolds number
ρ density
U characteristic velocity
L characteristic scale-length
η shear viscosity
Froude
numbera F=
U2
Lg
=
inertial force
gravitational force
(3.312)
F Froude number
Strouhal
numberb S=
Uτ
L
=
evolution scale
physical scale
(3.313)
S Strouhal number
τ characteristic timescale
Prandtl
number
P=
ηcp
λ
=
momentum transport
heat transport
(3.314)
P Prandtl number
cp Speciﬁc heat capacity at
constant pressure
λ thermal conductivity
Mach
number
M=
U
c
=
speed
sound speed
(3.315)
M Mach number
c sound speed
Rossby
number
Ro=
U
ΩL
=
inertial force
Coriolis force
(3.316)
Ro Rossby number
Ω angular velocity
aSometimes the square root of this expression. L is usually the ﬂuid depth.
bSometimes the reciprocal of this expression.
Fluid waves
Sound waves vp =
K
ρ
1/2
=
dp
dρ
1/2
(3.317)
vp wave (phase) speed
K bulk modulus
p pressure
ρ density
In an ideal gas
(adiabatic
conditions)a
vp =
γRT
µ
1/2
=
γp
ρ
1/2
(3.318)
T (absolute) temperature
µ mean molecular mass
Gravity waves on
a liquid surfaceb
ω2
=gktanhkh (3.319)
vg



1
2
g
k
1/2
(h λ)
(gh)1/2
(h λ)
(3.320)
vg group speed of wave
h liquid depth
λ wavelength
k wavenumber
Capillary waves
(ripples)c ω2
=
σk3
ρ
(3.321) σ surface tension
Capillary–gravity
waves (h λ) ω2
=gk+
σk3
ρ
(3.322)
aIf the waves are isothermal rather than adiabatic then vp =(p/ρ)1/2.
bAmplitude wavelength.
cIn the limit k2 gρ/σ.

3.9 Fluid dynamics
3
87
Doppler effecta
Source at rest,
observer
moving at u k
u
θ
ν
ν
=1−
|u|
vp
cosθ (3.323)
ν ,ν observed frequency
ν emitted frequency
in fluid
Observer at
rest, source
moving at u
ν
ν
=
1
1−
|u|
vp
cosθ
(3.324)
u velocity
θ angle between
wavevector, k, and u
aFor plane waves in a stationary fluid.
Wave speeds
Phase speed vp =
ω
k
=νλ (3.325)
vp phase speed
ν frequency
ω angular frequency (=2πν)
λ wavelength
Group speed
vg =
dω
dk
(3.326)
=vp −λ
dvp
dλ
(3.327)
vg group speed
Shocks
Mach wedgea sinθw =
vp
vb
(3.328)
θw wedge semi-angle
vb body speed
Kelvin
wedgeb
λK =
4πv2
b
3g
(3.329)
θw =arcsin(1/3)=19◦
.5 (3.330)
λK characteristic
wavelength
g gravitational
acceleration
Spherical
adiabatic
shockc
r
Et2
ρ0
1/5
(3.331)
r shock radius
E energy release
t time
ρ0 density of undisturbed
medium
Rankine–
Hugoniot
shock
relationsd
p2
p1
=
2γM2
1 −(γ−1)
γ+1
(3.332)
v1
v2
=
ρ2
ρ1
=
γ+1
(γ−1)+2/M2
1
(3.333)
T2
T1
=
[2γM2
1 −(γ−1)][2+(γ−1)M2
1]
(γ+1)2M2
1
(3.334)
1 upstream values
2 downstream values
p pressure
v velocity
T temperature
ρ density
γ ratio of specific heats
M Mach number
aApproximating the wake generated by supersonic motion of a body in a nondispersive medium.
bFor gravity waves, e.g., in the wake of a boat. Note that the wedge semi-angle is independent of vb.
cSedov–Taylor relation.
dSolutions for a steady, normal shock, in the frame moving with the shock front. If γ =5/3 then v1/v2 ≤4.

Surface tension
Deﬁnition
σlv =
surface energy
area
(3.335)
=
surface tension
length
(3.336)
σlv surface tension
(liquid/vapour
interface)
Laplace’s
formulaa
R1
R2
surface
h a
θ
θ
σwv
σwl σlv
∆p=σlv
1
R1
+
1
R2
(3.337)
∆p pressure diﬀerence
over surface
Ri principal radii of
curvature
Capillary
constant cc =
2σlv
gρ
1/2
(3.338)
cc capillary constant
ρ liquid density
g gravitational
acceleration
Capillary rise
(circular tube)
h=
2σlv cosθ
ρga
(3.339)
h rise height
θ contact angle
a tube radius
Contact angle cosθ =
σwv −σwl
σlv
(3.340)
σwv wall/vapour surface
tension
σwl wall/liquid surface
tension
aFor a spherical bubble in a liquid ∆p=2σlv/R. For a soap bubble (two surfaces) ∆p=4σlv/R.

4
Chapter 4 Quantum physics
4.1 Introduction
Quantum ideas occupy such a pivotal position in physics that different notations and algebras
appropriate to each field have been developed. In the spirit of this book, only those formulas
that are commonly present in undergraduate courses and that can be simply presented in
tabular form are included here. For example, much of the detail of atomic spectroscopy and of
specific perturbation analyses has been omitted, as have ideas from the somewhat specialised
field of quantum electrodynamics. Traditionally, quantum physics is understood through
standard “toy” problems, such as the potential step and the one-dimensional harmonic
oscillator, and these are reproduced here. Operators are distinguished from observables using
the “hat” notation, so that the momentum observable, px, has the operator ˆpx =−i¯h∂/∂x.
For clarity, many relations that can be generalised to three dimensions in an obvious way
have been stated in their one-dimensional form, and wavefunctions are implicitly taken as
normalised functions of space and time unless otherwise stated. With the exception of the
last panel, all equations should be taken as nonrelativistic, so that “total energy” is the sum
of potential and kinetic energies, excluding the rest mass energy.

90 Quantum physics
4.2 Quantum definitions
Quantum uncertainty relations
De Broglie relation
p=
h
λ
(4.1)
p =¯hk (4.2)
p,p particle momentum
h Planck constant
¯h h/(2π)
λ de Broglie wavelength
Planck–Einstein
relation
E =hν =¯hω (4.3)
k de Broglie wavevector
E energy
ν frequency
Dispersiona
(∆a)2
= (a− a )2
(4.4)
= a2
− a 2
(4.5)
a,b observablesb
· expectation value
(∆a)2 dispersion of a
General uncertainty
relation
(∆a)2
(∆b)2
≥
1
4
i[â, ˆb] 2
(4.6)
â operator for observable a
[·,·] commutator (see page 26)
Momentum–position
uncertainty relationc ∆p∆x≥
¯h
2
(4.7) x particle position
Energy–time
uncertainty relation
∆E∆t≥
¯h
2
(4.8) t time
Number–phase
∆n∆φ≥
1
2
(4.9)
n number of photons
φ wave phase
aDispersion in quantum physics corresponds to variance in statistics.
bAn observable is a directly measurable parameter of a system.
cAlso known as the “Heisenberg uncertainty relation.”
Wavefunctions
Probability
density
pr(x,t) dx=|ψ(x,t)|2
dx (4.10) pr probability density
ψ wavefunction
Probability
density
currenta
j(x)=
¯h
2im
ψ∗ ∂ψ
∂x
−ψ
∂ψ∗
∂x
(4.11)
j =
¯h
2im
ψ∗
(r)∇ψ(r)−ψ(r)∇ψ∗
(r) (4.12)
=
1
m
(ψ∗
ˆpψ) (4.13)
j,j probability density current
¯h (Planck constant)/(2π)
x position coordinate
ˆp momentum operator
m particle mass
real part of
t time
Continuity
equation
∇·j =−
∂
∂t
(ψψ∗
) (4.14)
Schrödinger
equation
ˆHψ =i¯h
∂ψ
∂t
(4.15) H Hamiltonian
Particle
stationary
statesb
−
¯h2
2m
∂2
ψ(x)
∂x2
+V(x)ψ(x)=Eψ(x) (4.16)
V potential energy
E total energy
aFor particles. In three dimensions, suitable units would be particles m−2 s−1.
bTime-independent Schrödinger equation for a particle, in one dimension.

4.2 Quantum definitions
4
91
Operators
Hermitian
conjugate
operator
(âφ)∗
ψ dx= φ∗
âψ dx (4.17)
â Hermitian conjugate
operator
ψ,φ normalisable functions
Position
operator
ˆxn =xn
(4.18)
∗ complex conjugate
x,y position coordinates
Momentum
operator
ˆpn
x =
¯hn
in
∂n
∂xn
(4.19)
n arbitrary integer ≥1
px momentum coordinate
Kinetic energy
operator
ˆT =−
¯h2
2m
∂2
∂x2
(4.20)
T kinetic energy
m particle mass
Hamiltonian
operator
ˆH =−
¯h2
2m
∂2
∂x2
+V(x) (4.21)
H Hamiltonian
V potential energy
Angular
momentum
operators
ˆLz = ˆx ˆpy − ˆy ˆpx (4.22)
ˆL2 = ˆLx
2
+ ˆLy
2
+ ˆLz
2
(4.23)
Lz angular momentum along
z axis (sim. x and y)
L total angular momentum
Parity operator ˆPψ(r)=ψ(−r) (4.24)
ˆP parity operator
r position vector
Expectation value
Expectation
valuea
a = â = Ψ∗
âΨ dx (4.25)
= Ψ|â|Ψ (4.26)
a expectation value of a
â operator for a
Ψ (spatial) wavefunction
x (spatial) coordinate
Time
dependence
d
dt
â =
i
¯h
[ ˆH, â] +
∂â
∂t
(4.27)
t time
Relation to
eigenfunctions
if âψn =anψn and Ψ= cnψn
then a = |cn|2
an (4.28)
ψn eigenfunctions of â
an eigenvalues
n dummy index
cn probability amplitudes
Ehrenfest’s
theorem
m
d
dt
r = p (4.29)
d
dt
p =− ∇V (4.30)
m particle mass
r position vector
p momentum
V potential energy
aEquation (4.26) uses the Dirac “bra-ket” notation for integrals involving operators. The presence of vertical bars
distinguishes this use of angled brackets from that on the left-hand side of the equations. Note that a and â are
taken as equivalent.

92 Quantum physics
Dirac notation
Matrix elementa
anm = ψ∗
n âψm dx (4.31)
= n|â|m (4.32)
n,m eigenvector indices
anm matrix element
ψn basis states
â operator
x spatial coordinate
Bra vector bra state vector = n| (4.33) ·| bra
Ket vector ket state vector =|m (4.34) |· ket
Scalar product n|m = ψ∗
nψm dx (4.35)
Expectation
if Ψ=
n
cnψn (4.36)
then a =
m n
c∗
ncmanm (4.37)
Ψ wavefunction
aThe Dirac bracket, n|â|m , can also be written ψn|â|ψm .
4.3 Wave mechanics
Potential stepa
V(x)
V0
i ii
0 x
incident particle
Potential
function V(x)=
0 (x<0)
V0 (x≥0)
(4.38)
V particle potential energy
V0 step height
Wavenumbers
¯h2
k2
=2mE (x<0) (4.39)
¯h2
q2
=2m(E −V0) (x>0) (4.40)
k,q particle wavenumbers
m particle mass
E total particle energy
Amplitude
reflection
coefficient
r =
k−q
k+q
(4.41)
r amplitude reflection
coefficient
Amplitude
transmission
coefficient
t=
2k
k+q
(4.42)
t amplitude transmission
coefficient
Probability
currentsb
ji =
¯hk
m
(1−|r|2
) (4.43)
jii =
¯hq
m
|t|2
(4.44)
ji particle flux in zone i
jii particle flux in zone ii
aOne-dimensional interaction with an incident particle of total energy E = KE+V. If E < V0 then q is imaginary
and |r|2 =1. 1/|q| is then a measure of the tunnelling depth.
bParticle flux with the sign of increasing x.

4.3 Wave mechanics
4
93
Potential wella
V(x)
−V0
i ii iii
0
xa−a
incident particle
Potential
function V(x)=
0 (|x|>a)
−V0 (|x|≤a)
(4.45)
V0 well depth
2a well width
Wavenumbers
¯h2
k2
=2mE (|x|>a) (4.46)
¯h2
q2
=2m(E +V0) (|x|<a) (4.47)
k,q particle wavenumbers
m particle mass
E total particle energy
Amplitude
reflection
coefficient
r =
ie−2ika
(q2
−k2
)sin2qa
2kqcos2qa−i(q2 +k2)sin2qa
(4.48)
coefficient
Amplitude
transmission
coefficient
t=
2kqe−2ika
2kqcos2qa−i(q2 +k2)sin2qa
(4.49)
coefficient
Probability
currentsb
ji =
¯hk
m
(1−|r|2
) (4.50)
jiii =
¯hk
m
|t|2
(4.51)
jiii particle flux in zone iii
Ramsauer
effectc En =−V0 +
n2
¯h2
π2
8ma2
(4.52)
n integer >0
En Ramsauer energy
Bound states
(V0 <E <0)d
tanqa=
|k|/q even parity
−q/|k| odd parity
(4.53)
q2
−|k|2
=2mV0/¯h2
(4.54)
aOne-dimensional interaction with an incident particle of total energy E =KE+V >0.
bParticle flux in the sense of increasing x.
cIncident energy for which 2qa=nπ, |r|=0, and |t|=1.
dWhen E <0, k is purely imaginary. |k| and q are obtained by solving these implicit equations.

94 Quantum physics
Barrier tunnellinga
V(x)
V0
i ii iii
0 xa−a
incident particle
Potential
function V(x)=
0 (|x|>a)
V0 (|x|≤a)
(4.55)
V0 well depth
2a barrier width
Wavenumber
and tunnelling
constant
¯h2
k2
=2mE (|x|>a) (4.56)
¯h2
κ2
=2m(V0 −E) (|x|<a) (4.57)
k incident wavenumber
κ tunnelling constant
m particle mass
E total energy (<V0)
Amplitude
reflection
coefficient
r =
−ie−2ika
(k2
+κ2
)sinh2κa
2kκcosh2κa−i(k2 −κ2)sinh2κa
(4.58)
coefficient
Amplitude
transmission
coefficient
t=
2kκe−2ika
2kκcosh2κa−i(k2 −κ2)sinh2κa
(4.59)
coefficient
Tunnelling
probability
|t|2
=
4k2
κ2
(k2 +κ2)2 sinh2
2κa+4k2κ2
(4.60)
16k2
κ2
(k2 +κ2)2
exp(−4κa) (|t|2
1)
(4.61)
|t|2 tunnelling probability
Probability
currentsb
ji =
¯hk
m
(1−|r|2
) (4.62)
jiii =
¯hk
m
|t|2
(4.63)
jiii particle flux in zone iii
aBy a particle of total energy E =KE+V, through a one-dimensional rectangular potential barrier height V0 >E.
bParticle flux in the sense of increasing x.
Particle in a rectangular boxa
Eigen-
functions
x
y
z
a
b
c
Ψlmn =
8
abc
1/2
sin
lπx
a
sin
mπy
b
sin
nπz
c
(4.64)
Ψlmn eigenfunctions
a,b,c box dimensions
l,m,n integers ≥1
Energy
levels Elmn =
h2
8M
l2
a2
+
m2
b2
+
n2
c2
(4.65)
Elmn energy
h Planck
constant
M particle mass
Density of
states
ρ(E) dE =
4π
h3
(2M3
E)1/2
dE (4.66)
ρ(E) density of
states (per unit
volume)
aSpinless particle in a rectangular box bounded by the planes x=0, y =0, z =0, x=a, y =b, and
z =c. The potential is zero inside and infinite outside the box.

4.4 Hydrogenic atoms
4
95
Harmonic oscillator
Schr¨odinger
equation −
¯h2
2m
∂2
ψn
∂x2
+
1
2
mω2
x2
ψn =Enψn (4.67)
m mass
ψn nth eigenfunction
x displacement
Energy
levelsa En = n+
1
2
¯hω (4.68)
n integer ≥0
En total energy in nth state
Eigen-
functions
ψn =
Hn(x/a)exp[−x2
/(2a2
)]
(n!2naπ1/2)1/2
(4.69)
where a=
¯h
mω
1/2
Hn Hermite polynomials
Hermite
polynomials
H0(y)=1, H1(y)=2y, H2(y)=4y2
−2
Hn+1(y)=2yHn(y)−2nHn−1(y) (4.70)
y dummy variable
aE0 is the zero-point energy of the oscillator.
Bohr modela
Quantisation
condition
µr2
nΩ=n¯h (4.71)
rn nth orbit radius
Ω orbital angular speed
n principal quantum number
(>0)
Bohr radius a0 =
0h2
πmee2
=
α
4πR∞
52.9pm (4.72)
a0 Bohr radius
µ reduced mass ( me)
−e electronic charge
Orbit radius rn =
n2
Z
a0
me
µ
(4.73)
Z atomic number
h Planck constant
¯h h/(2π)
Total energy En =−
µe4
Z2
8 2
0h2n2
=−R∞hc
µ
me
Z2
n2
(4.74)
En total energy of nth orbit
0 permittivity of free space
me electron mass
Fine structure
constant α=
µ0ce2
2h
=
e2
4π 0¯hc
1
137
(4.75)
α ﬁne structure constant
µ0 permeability of free space
Hartree energy EH =
¯h2
mea2
0
4.36×10−18
J (4.76) EH Hartree energy
Rydberg
constant R∞ =
mecα2
2h
=
mee4
8h3 2
0c
=
EH
2hc
(4.77)
R∞ Rydberg constant
c speed of light
Rydberg’s
formulab
1
λmn
=R∞
µ
me
Z2 1
n2
−
1
m2
(4.78)
λmn photon wavelength
m integer >n
aBecause the Bohr model is strictly a two-body problem, the equations use reduced mass, µ=memnuc/(me+mnuc) me,
where mnuc is the nuclear mass, throughout. The orbit radius is therefore the electron–nucleus distance.
bWavelength of the spectral line corresponding to electron transitions between orbits m and n.

96 Quantum physics
Hydrogenlike atoms – Schrödinger solutiona
Schrödinger equation
−
¯h2
2µ
∇2
Ψnlm −
Ze2
4π 0r
Ψnlm =EnΨnlm with µ=
memnuc
me +mnuc
(4.79)
Eigenfunctions
Ψnlm(r,θ,φ)=
(n−l −1)!
2n(n+l)!
1/2
2
an
3/2
xl
e−x/2
L2l+1
n−l−1(x)Y m
l (θ,φ) (4.80)
with a=
me
µ
a0
Z
, x=
2r
an
, and L2l+1
n−l−1(x)=
n−l−1
k=0
(l +n)!(−x)k
(2l +1+k)!(n−l −1−k)!k!
Total energy En =−
µe4
Z2
8 2
0h2n2
(4.81)
En total energy
Radial
expectation
values
r =
a
2
[3n2
−l(l +1)] (4.82)
r2
=
a2
n2
2
[5n2
+1−3l(l +1)] (4.83)
1/r =
1
an2
(4.84)
1/r2
=
2
(2l +1)n3a2
(4.85)
h Planck constant
me mass of electron
¯h h/2π
µ reduced mass ( me)
mnuc mass of nucleus
Ψnlm eigenfunctions
Ze charge of nucleus
Allowed
quantum
numbers and
selection rulesb
n=1,2,3,... (4.86)
l =0,1,2,... ,(n−1) (4.87)
m=0,±1,±2,... ,±l (4.88)
∆n=0 (4.89)
∆l =±1 (4.90)
∆m=0 or ±1 (4.91)
Lq
p associated Laguerre
polynomialsc
a classical orbit radius, n=1
r electron–nucleus separation
Y m
l spherical harmonics
a0 Bohr radius = 0h2
πmee2
Ψ100 =
a−3/2
π1/2
e−r/a
Ψ200 =
a−3/2
4(2π)1/2
2−
r
a
e−r/2a
Ψ210 =
a−3/2
4(2π)1/2
r
a
e−r/2a
cosθ Ψ21±1 =∓
a−3/2
8π1/2
r
a
e−r/2a
sinθe±iφ
Ψ300 =
a−3/2
81(3π)1/2
27−18
r
a
+2
r2
a2
e−r/3a
Ψ310 =
21/2
a−3/2
81π1/2
6−
r
a
r
a
e−r/3a
cosθ
Ψ31±1 =∓
a−3/2
81π1/2
6−
r
a
r
a
e−r/3a
sinθe±iφ
Ψ320 =
a−3/2
81(6π)1/2
r2
a2
e−r/3a
(3cos2
θ−1)
Ψ32±1 =∓
a−3/2
81π1/2
r2
a2
e−r/3a
sinθcosθe±iφ
Ψ32±2 =
a−3/2
162π1/2
r2
a2
e−r/3a
sin2
θe±2iφ
aFor a single bound electron in a perfect nuclear Coulomb potential (nonrelativistic and spin-free).
bFor dipole transitions between orbitals.
cThe sign and indexing definitions for this function vary. This form is appropriate to Equation (4.80).

4
97
Orbital angular dependence
00
x
y
z
0.20.2
0.2
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
(s)2
(px)2
(py)2
(pz)2
(dx2−y2 )2
(dxz)2
(dz2 )2
(dyz)2
(dxy)2
s orbital
(l =0)
s=Y 0
0 = constant (4.92) Y m
l spherical
harmonicsa
p orbitals
(l =1)
px =
−1
21/2
(Y 1
1 −Y −1
1 )∝cosφsinθ (4.93)
py =
i
21/2
(Y 1
1 +Y −1
1 )∝sinφsinθ (4.94)
pz =Y 0
1 ∝cosθ (4.95)
θ,φ spherical polar
coordinates
d orbitals
(l =2) x
y
z
θ
φ
dx2−y2 =
1
21/2
(Y 2
2 +Y −2
2 )∝sin2
θcos2φ (4.96)
dxz =
−1
21/2
(Y 1
2 −Y −1
2 )∝sinθcosθcosφ (4.97)
dz2 =Y 0
2 ∝(3cos2
θ−1) (4.98)
dyz =
i
21/2
(Y 1
2 +Y −1
2 )∝sinθcosθsinφ (4.99)
dxy =
−i
21/2
(Y 2
2 −Y −2
2 )∝sin2
θsin2φ (4.100)
aSee page 49 for the deﬁnition of spherical harmonics.

98 Quantum physics
4.5 Angular momentum
Orbital angular momentum
Angular
momentum
operators
ˆL=r××× ˆp (4.101)
ˆLz =
¯h
i
x
∂
∂y
−y
∂
∂x
(4.102)
=
¯h
i
∂
∂φ
(4.103)
ˆL2 = ˆLx
2
+ ˆLy
2
+ ˆLz
2
(4.104)
=−¯h2 1
sinθ
∂
∂θ
sinθ
∂
∂θ
+
1
sin2
θ
∂2
∂φ2
(4.105)
L angular
momentum
p linear momentum
r position vector
xyz Cartesian
coordinates
rθφ spherical polar
coordinates
¯h (Planck
constant)/(2π)
Ladder
operators
ˆL± = ˆLx ±i ˆLy (4.106)
=¯he±iφ
icotθ
∂
∂φ
±
∂
∂θ
(4.107)
ˆL±Y ml
l =¯h[l(l +1)−ml(ml ±1)]1/2
Y ml±1
l (4.108)
ˆL± ladder operators
Y
ml
l spherical
harmonics
l,ml integers
Eigen-
functions and
eigenvalues
ˆL2Y ml
l =l(l +1)¯h2
Y ml
l (l ≥0) (4.109)
ˆLzY ml
l =ml¯hY ml
l (|ml|≤l) (4.110)
ˆLz[ ˆL±Y ml
l (θ,φ)]=(ml ±1)¯h ˆL±Y ml
l (θ,φ) (4.111)
l-multiplicity =(2l +1) (4.112)
Angular momentum commutation relationsa
Conservation of angular
momentumb [ ˆH, ˆLz]=0 (4.113)
L angular momentum
p momentum
H Hamiltonian
ˆL± ladder operators
[ ˆLz,x]=i¯hy (4.114)
[ ˆLz,y]=−i¯hx (4.115)
[ ˆLz,z]=0 (4.116)
[ ˆLz, ˆpx]=i¯h ˆpy (4.117)
[ ˆLz, ˆpy]=−i¯h ˆpx (4.118)
[ ˆLz, ˆpz]=0 (4.119)
[ ˆLx, ˆLy]=i¯h ˆLz (4.120)
[ ˆLz, ˆLx]=i¯h ˆLy (4.121)
[ ˆLy, ˆLz]=i¯h ˆLx (4.122)
[ ˆL+, ˆLz]=−¯h ˆL+ (4.123)
[ ˆL−, ˆLz]=¯h ˆL− (4.124)
[ ˆL+, ˆL−]=2¯h ˆLz (4.125)
[ ˆL2, ˆL±]=0 (4.126)
[ ˆL2, ˆLx]=[ ˆL2, ˆLy]=[ ˆL2, ˆLz]=0 (4.127)
aThe commutation of a and b is deﬁned as [a,b]=ab−ba (see page 26). Similar expressions hold for S and J.
bFor motion under a central force.

4
99
Clebsch–Gordan coefficientsa
+1
1/2×1/2 1
+1/2 +1/2 1
0
1 0
+1/2 −1/2 1/2 1/2
−1/2 +1/2 1/2 −1/2
+3/2
1×1/2 3/2
+1 +1/2 1
+1/2
3/2 1/2
+1 −1/2 1/3 2/3
0 +1/2 2/3 −1/3
+2
3/2×1/2 2
+3/2 +1/2 1
+1
2 1
+3/2 −1/2 1/4 3/4
+1/2 +1/2 3/4 −1/4
0
2 1
+1/2 −1/2 1/2 1/2
−1/2 +1/2 1/2 −1/2
+5/2
2×1/2 5/2
+2 +1/2 1
+3/2
5/2 3/2
+2 −1/2 1/5 4/5
+1 +1/2 4/5 −1/5
+1/2
5/2 3/2
+1 −1/2 2/5 3/5
0 +1/2 3/5 −2/5+5/2
3/2×1 5/2
+3/2 +1 1
+3/2
5/2 3/2
+3/2 0 2/5 3/5
+1/2 +1 3/5 −2/5
+1/2
5/2 3/2 1/2
+3/2 −1 1/10 2/5 1/2
1/2 0 3/5 1/15 −1/3
−1/2 +1 3/10 −8/15 1/6
+2
1×1 2
+1 +1 1
+1
2 1
+1 0 1/2 1/2
0 +1 1/2 −1/2
0
2 1 0
+1 −1 1/6 1/2 1/3
0 0 2/3 0 −1/3
−1 +1 1/6 −1/2 1/3
+3
2×1 3
+2 +1 1
+2
3 2
+2 0 1/3 2/3
+1 +1 2/3 −1/3
+1
3 2 1
+2 −1 1/15 1/3 3/5
+1 0 8/15 1/6 −3/10
0 +1 6/15 −1/2 1/10
0
3 2 1
+1 −1 1/5 1/2 3/10
0 0 3/5 0 −2/5
−1 +1 1/5 −1/2 3/10
+3
3/2×3/2 3
+3/2 +3/2 1
+2
3 2
+3/2 +1/2 1/2 1/2
+1/2 +3/2 1/2 −1/2
+1
3 2 1
+3/2 −1/2 1/5 1/2 3/10
+1/2 +1/2 3/5 0 −2/5
−1/2 +3/2 1/5 −1/2 3/10
0
3 2 1 0
+3/2 −3/2 1/20 1/4 9/20 1/4
+1/2 −1/2 9/20 1/4 −1/20 −1/4
−1/2 +1/2 9/20 −1/4 −1/20 1/4
−3/2 +3/2 1/20 −1/4 9/20 −1/4
+7/2
2×3/2 7/2
+2 +3/2 1
+5/2
7/2 5/2
+2 +1/2 3/7 4/7
+1 +3/2 4/7 −3/7
+3/2
7/2 5/2 3/2
+2 −1/2 1/7 16/35 2/5
+1 +1/2 4/7 1/35 −2/5
0 +3/2 2/7 −18/35 1/5
+1/2
7/2 5/2 3/2 1/2
+2 −3/2 1/35 6/35 2/5 2/5
+1 −1/2 12/35 5/14 0 −3/10
0 +1/2 18/35 −3/35 −1/5 1/5
−1 +3/2 4/35 −27/70 2/5 −1/10
+4
2×2 4
+2 +2 1
+3
4 3
+2 +1 1/2 1/2
+1 +2 1/2 −1/2
+2
4 3 2
+2 0 3/14 1/2 2/7
+1 +1 4/7 0 −3/7
0 +2 3/14 −1/2 2/7
+1
4 3 2 1
+2 −1 1/14 3/10 3/7 1/5
+1 0 3/7 1/5 −1/14 −3/10
0 +1 3/7 −1/5 −1/14 3/10
−1 +2 1/14 −3/10 3/7 −1/5
0
4 3 2 1 0
+2 −2 1/70 1/10 2/7 2/5 1/5
+1 −1 8/35 2/5 1/14 −1/10 −1/5
0 0 18/35 0 −2/7 0 1/5
−1 +1 8/35 −2/5 1/14 1/10 −1/5
−2 +2 1/70 −1/10 2/7 −2/5 1/5
mj
l1 ×l2 j j ...
m1 m2 coefficients
m1 m2 j,mj |l1,m1;l2,m2
.
.
.
.
.
.
.
.
.
j,−mj|l1,−m1;l2,−m2 =(−1)l1+l2−j
j,mj|l1,m1;l2,m2
aOr “Wigner coefficients,” using the Condon–Shortley sign convention. Note that a square root is assumed
over all coefficient digits, so that “−3/10” corresponds to − 3/10. Also for clarity, only values of mj ≥ 0 are
listed here. The coefficients for mj < 0 can be obtained from the symmetry relation j,−mj|l1,−m1;l2,−m2 =
(−1)l1+l2−j j,mj|l1,m1;l2,m2 .

100 Quantum physics
Angular momentum additiona
Total angular
momentum
J =L+S (4.128)
ˆJz = ˆLz + ˆSz (4.129)
ˆJ2 = ˆL2 + ˆS2 +2L·S (4.130)
ˆJzψj,mj
=mj¯hψj,mj
(4.131)
ˆJ2ψj,mj
=j(j +1)¯h2
ψj,mj
(4.132)
j-multiplicity=(2l +1)(2s+1) (4.133)
J,J total angular momentum
L,L orbital angular
momentum
S,S spin angular momentum
ψ eigenfunctions
mj magnetic quantum
number |mj|≤j
j (l +s)≥j ≥|l −s|
Mutually
commuting
sets
{L2
,S2
,J2
,Jz,L·S} (4.134)
{L2
,S2
,Lz,Sz,Jz} (4.135)
{} set of mutually
commuting observables
Clebsch–
Gordan
coefficientsb
|j,mj =
ml,ms
ms+ml=mj
j,mj|l,ml;s,ms |l,ml |s,ms
(4.136)
|· eigenstates
·|· Clebsch–Gordan
coefficients
aSumming spin and orbital angular momenta as examples, eigenstates |s,ms and |l,ml .
bOr “Wigner coefficients.” Assuming no L–S interaction.
Magnetic moments
Bohr magneton µB =
e¯h
2me
(4.137)
µB Bohr magneton
me electron mass
Gyromagnetic
ratioa γ =
orbital magnetic moment
orbital angular momentum
(4.138) γ gyromagnetic ratio
Electron orbital
gyromagnetic
ratio
γe =
−µB
¯h
(4.139)
=
−e
2me
(4.140)
γe electron gyromagnetic ratio
Spin magnetic
moment of an
electronb
µe,z =−geµBms (4.141)
=±geγe
¯h
2
(4.142)
=±
gee¯h
4me
(4.143)
µe,z z component of spin
magnetic moment
ge electron g-factor ( 2.002)
ms spin quantum number (±1/2)
Landé g-factorc
µJ =gJ J(J +1)µB (4.144)
µJ,z =−gJµBmJ (4.145)
gJ =1+
J(J +1)+S(S +1)−L(L+1)
2J(J +1)
(4.146)
µJ total magnetic moment
µJ,z z component of µJ
mJ magnetic quantum number
J,L,S total, orbital, and spin
quantum numbers
gJ Landé g-factor
aOr “magnetogyric ratio.”
bThe electron g-factor equals exactly 2 in Dirac theory. The modification ge = 2+α/π +..., where α is the fine
structure constant, comes from quantum electrodynamics.
cRelating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming ge =2.

4
101
Quantum paramagnetism
0
0.2
0.4
0.6
0.8
1
−0.2
−0.4
−0.6
−0.8
−1
5 10−5−10 x
B∞(x)=L(x)
B4(x)
B1(x)
B1/2(x)=tanhx
Brillouin
function
BJ(x)=
2J +1
2J
coth
(2J +1)x
2J
−
1
2J
coth
x
2J
(4.147)
BJ(x)



J +1
3J
x (x 1)
L(x) (J 1)
(4.148)
B1/2(x)=tanhx (4.149)
BJ (x) Brillouin function
quantum number
L(x) Langevin function
=cothx−1/x (see page 144)
M mean magnetisation
n number density of atoms
Mean
magnetisationa M =nµBJgJBJ JgJ
µBB
kT
(4.150)
gJ Land´e g-factor
µB Bohr magneton
B magnetic ﬂux density
M for isolated
spins (J =1/2)
M 1/2 =nµB tanh
µBB
kT
(4.151)
k Boltzmann constant
T temperature
M 1/2 mean magnetisation for
J =1/2 (and gJ =2)
aOf an ensemble of atoms in thermal equilibrium at temperature T, each with total angular momentum quantum
number J.

102 Quantum physics
4.6 Perturbation theory
Time-independent perturbation theory
Unperturbed
states
ˆH0ψn =Enψn (4.152)
(ψn nondegenerate)
ˆH0 unperturbed Hamiltonian
ψn eigenfunctions of ˆH0
En eigenvalues of ˆH0
n integer ≥0
Perturbed
Hamiltonian
ˆH = ˆH0 + ˆH (4.153)
ˆH perturbed Hamiltonian
ˆH perturbation ( ˆH0)
Perturbed
eigenvaluesa
Ek =Ek + ψk| ˆH |ψk
+
n=k
| ψk| ˆH |ψn |2
Ek −En
+... (4.154)
Ek perturbed eigenvalue ( Ek)
|| Dirac bracket
Perturbed
eigen-
functionsb
ψk =ψk +
n=k
ψk| ˆH |ψn
Ek −En
ψn +... (4.155) ψk perturbed eigenfunction
( ψk)
aTo second order.
bTo first order.
Time-dependent perturbation theory
Unperturbed
stationary
states
ˆH0ψn =Enψn (4.156)
ˆH0 unperturbed Hamiltonian
ψn eigenfunctions of ˆH0
En eigenvalues of ˆH0
n integer ≥0
Perturbed
Hamiltonian
ˆH(t)= ˆH0 + ˆH (t) (4.157)
ˆH perturbed Hamiltonian
ˆH (t) perturbation ( ˆH0)
t time
Schrödinger
equation
[ ˆH0 + ˆH (t)]Ψ(t)=i¯h
∂Ψ(t)
∂t
(4.158)
Ψ(t=0)=ψ0 (4.159)
Ψ wavefunction
ψ0 initial state
Perturbed
wave-
functiona
Ψ(t)=
n
cn(t)ψn exp(−iEnt/¯h) (4.160)
where
cn =
−i
¯h
t
0
ψn| ˆH (t )|ψ0 exp[i(En −E0)t /¯h] dt (4.161)
Fermi’s
golden rule
Γi→f =
2π
¯h
| ψf| ˆH |ψi |2
ρ(Ef) (4.162)
Γi→f transition probability per
unit time from state i to
state f
ρ(Ef) density of final states
aTo first order.

4.7 High energy and nuclear physics
4
103
4.7 High energy and nuclear physics
Nuclear decay
Nuclear decay
law
N(t)=N(0)e−λt
(4.163)
N(t) number of nuclei
remaining after time t
t time
Half-life and
mean life
T1/2 =
ln2
λ
(4.164)
T =1/λ (4.165)
λ decay constant
T1/2 half-life
T mean lifetime
Successive decays 1→2→3 (species 3 stable)
N1(t)=N1(0)e−λ1t
(4.166)
N2(t)=N2(0)e−λ2t
+
N1(0)λ1(e−λ1t
−e−λ2t
)
λ2 −λ1
(4.167)
N3(t)=N3(0)+N2(0)(1−e−λ2t
)+N1(0) 1+
λ1e−λ2t
−λ2e−λ1t
λ2 −λ1
(4.168)
N1 population of species 1
λ1 decay constant 1→2
λ2 decay constant 2→3
Geiger’s lawa v3
=a(R −x) (4.169)
v velocity of α particle
x distance from source
a constant
Geiger–Nuttall
rule
logλ=b+clogR (4.170)
R range
b, c constants for each
series α, β, and γ
aFor α particles in air (empirical).
Nuclear binding energy
Liquid drop modela N number of neutrons
A mass number (=N +Z)
B =avA−asA2/3
−ac
Z2
A1/3
−aa
(N −Z)2
A
+δ(A)
(4.171)
δ(A)



+apA−3/4
Z, N both even
−apA−3/4
Z, N both odd
0 otherwise
(4.172)
B semi-empirical binding energy
Z number of protons
av volume term (∼15.8MeV)
as surface term (∼18.0MeV)
ac Coulomb term (∼0.72MeV)
aa asymmetry term (∼23.5MeV)
ap pairing term (∼33.5MeV)
Semi-empirical
mass formula
M(Z,A)=ZMH +Nmn −B (4.173)
M(Z,A) atomic mass
MH mass of hydrogen atom
mn neutron mass
aCoeﬃcient values are empirical and approximate.

104 Quantum physics
Nuclear collisions
Breit–Wigner
formulaa
σ(E)=
π
k2
g
ΓabΓc
(E −E0)2 +Γ2/4
(4.174)
g =
2J +1
(2sa +1)(2sb +1)
(4.175)
σ(E) cross-section for a+b→c
k incoming wavenumber
g spin factor
E total energy (PE + KE)
E0 resonant energy
Total width Γ=Γab +Γc (4.176)
Γ width of resonant state R
Γab partial width into a+b
Γc partial width into c
Resonance
lifetime
τ=
¯h
Γ
(4.177)
τ resonance lifetime
quantum number of R
sa,b spins of a and b
Born scattering
formulab
dσ
dΩ
=
2µ
¯h2
∞
0
sinKr
Kr
V(r)r2
dr
2
(4.178)
dσ
dΩ diﬀerential collision
cross-section
µ reduced mass
K =|kin −kout| (see footnote)
r radial distance
V(r) potential energy of interaction
Mott scattering formulac
dσ
dΩ
=
α
4E
2
csc4 χ
2
+sec4 χ
2
+
Acos α
¯hv lntan2 χ
2
sin2 χ
2 cos χ
2
(4.179)
dσ
dΩ
α
2E
2 4−3sin2
χ
sin4
χ
(A=−1, α v¯h) (4.180)
¯h (Planck constant)/2π
α/r scattering potential energy
χ scattering angle
v closing velocity
A =2 for spin-zero particles, =−1
for spin-half particles
aFor the reaction a+b↔R →c in the centre of mass frame.
bFor a central ﬁeld. The Born approximation holds when the potential energy of scattering, V, is much less than
the total kinetic energy. K is the magnitude of the change in the particle’s wavevector due to scattering.
cFor identical particles undergoing Coulomb scattering in the centre of mass frame. Nonidentical particles obey the
Rutherford scattering formula (page 72).
Relativistic wave equationsa
Klein–Gordon
equation
(massive, spin
zero particles)
(∇2
−m2
)ψ =
∂2
ψ
∂t2
(4.181)
ψ wavefunction
m particle mass
t time
Weyl equations
(massless, spin
1/2 particles)
∂ψ
∂t
=± σx
∂ψ
∂x
+σy
∂ψ
∂y
+σz
∂ψ
∂z
(4.182)
ψ spinor wavefunction
σi Pauli spin matrices
(see page 26)
Dirac equation
(massive, spin
1/2 particles)
(iγµ
∂µ−m)ψ =0 (4.183)
where ∂µ=
∂
∂t
,
∂
∂x
,
∂
∂y
,
∂
∂z
(4.184)
(γ0
)2
=14 ; (γ1
)2
=(γ2
)2
=(γ3
)2
=−14 (4.185)
i i2 =−1
γµ Dirac matrices:
γ0 =
12 0
0 −12
γi =
0 σi
−σi 0
1n n×n unit matrix
aWritten in natural units, with c=¯h=1.

5
Chapter 5 Thermodynamics
5.1 Introduction
The term thermodynamics is used here loosely and includes classical thermodynamics, statis-
tical thermodynamics, thermal physics, and radiation processes. Notation in these subjects
can be confusing and the conventions used here are those found in the majority of modern
treatments. In particular:
• The internal energy of a system is defined in terms of the heat supplied to the system plus
the work done on the system, that is, dU = dQ+ dW.
• The lowercase symbol p is used for pressure. Probability density functions are denoted by
pr(x) and microstate probabilities by pi.
• With the exception of specific intensity, quantities are taken as specific if they refer to unit
mass and are distinguished from the extensive equivalent by using lowercase. Hence specific
volume, v, equals V/m, where V is the volume of gas and m its mass. Also, the specific heat
capacity of a gas at constant pressure is cp =Cp/m, where Cp is the heat capacity of mass m
of gas. Molar values take a subscript “m” (e.g., Vm for molar volume) and remain in upper
case.
• The component held constant during a partial differentiation is shown after a vertical bar;
hence ∂V
∂p T
is the partial differential of volume with respect to pressure, holding temperature
constant.
The thermal properties of solids are dealt with more explicitly in the section on solid state
physics (page 123). Note that in solid state literature specific heat capacity is often taken to
mean heat capacity per unit volume.

106 Thermodynamics
5.2 Classical thermodynamics
Thermodynamic laws
Thermodynamic
temperaturea T ∝ lim
p→0
(pV) (5.1)
T thermodynamic temperature
V volume of a fixed mass of gas
p gas pressure
Kelvin
temperature scale
T/K=273.16
lim
p→0
(pV)T
lim
p→0
(pV)tr
(5.2)
K kelvin unit
tr temperature of the triple point
of water
First lawb dU = dQ+ dW (5.3)
dU change in internal energy
dW work done on system
dQ heat supplied to system
Entropyc
dS =
dQrev
T
≥
dQ
T
(5.4)
S experimental entropy
T temperature
rev reversible change
aAs determined with a gas thermometer. The idea of temperature is associated with the zeroth law of ther-
modynamics: If two systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each
other.
bThe d notation represents a differential change in a quantity that is not a function of state of the system.
cAssociated with the second law of thermodynamics: No process is possible with the sole effect of completely converting
heat into work (Kelvin statement).
Thermodynamic worka
Hydrostatic
pressure
dW =−p dV (5.5) p (hydrostatic) pressure
dV volume change
Surface tension dW =γ dA (5.6)
dW work done on the system
γ surface tension
dA change in area
Electric field dW =E · dp (5.7)
E electric field
dp induced electric dipole moment
Magnetic field dW =B · dm (5.8) B magnetic flux density
dm induced magnetic dipole moment
Electric current dW =∆φ dq (5.9)
∆φ potential difference
dq charge moved
aThe sources of electric and magnetic fields are taken as being outside the thermodynamic system on
which they are working.

5
107
Cycle efficiencies (thermodynamic)a
Heat engine η =
work extracted
heat input
≤
Th −Tl
Th
(5.10)
η efficiency
Th higher temperature
Tl lower temperature
Refrigerator η =
heat extracted
work done
≤
Tl
Th −Tl
(5.11)
Heat pump η =
heat supplied
work done
≤
Th
Th −Tl
(5.12)
Otto cycleb
η =
work extracted
heat input
=1−
V2
V1
γ−1
(5.13)
V1
V2
compression ratio
(assumed constant)
aThe equalities are for reversible cycles, such as Carnot cycles, operating between temperatures Th and Tl.
bIdealised reversible “petrol” (heat) engine.
Heat capacities
Constant
volume
CV =
dQ
dT V
=
∂U
∂T V
=T
∂S
∂T V
(5.14)
CV heat capacity, V constant
Q heat
T temperature
V volume
U internal energy
Constant
pressure
Cp =
dQ
dT p
=
∂H
∂T p
=T
∂S
∂T p
(5.15)
S entropy
Cp heat capacity, p constant
p pressure
H enthalpy
Difference in
heat capacities
Cp −CV =
∂U
∂V T
+p
∂V
∂T p
(5.16)
=
VTβ2
p
κT
(5.17)
βp isobaric expansivity
κT isothermal compressibility
Ratio of heat
capacities
γ =
Cp
CV
=
κT
κS
(5.18)
κS adiabatic compressibility
Thermodynamic coefficients
Isobaric
expansivitya βp =
1
V
∂V
∂T p
(5.19)
V volume
T temperature
Isothermal
compressibility
κT =−
1
V
∂V
∂p T
(5.20)
p pressure
Adiabatic
compressibility
κS =−
1
V
∂V
∂p S
(5.21) κS adiabatic compressibility
Isothermal bulk
modulus
KT =
1
κT
=−V
∂p
∂V T
(5.22) KT isothermal bulk modulus
Adiabatic bulk
modulus
KS =
1
κS
=−V
∂p
∂V S
(5.23) KS adiabatic bulk modulus
aAlso called “cubic expansivity” or “volume expansivity.” The linear expansivity is αp =βp/3.

108 Thermodynamics
Expansion processes
Joule
expansiona
η =
∂T
∂V U
=−
T2
CV
∂(p/T)
∂T V
(5.24)
=−
1
CV
T
∂p
∂T V
−p (5.25)
η Joule coeﬃcient
T temperature
p pressure
U internal energy
Joule–Kelvin
expansionb
µ=
∂T
∂p H
=
T2
Cp
∂(V/T)
∂T p
(5.26)
=
1
Cp
T
∂V
∂T p
−V (5.27)
µ Joule–Kelvin coeﬃcient
V volume
H enthalpy
aExpansion with no change in internal energy.
bExpansion with no change in enthalpy. Also known as a “Joule–Thomson expansion” or “throttling” process.
Thermodynamic potentialsa
Internal energy dU =T dS −pdV +µdN (5.28)
U internal energy
T temperature
S entropy
µ chemical potential
N number of particles
Enthalpy
H =U +pV (5.29)
dH =T dS +V dp+µdN (5.30)
H enthalpy
p pressure
V volume
Helmholtz free
energyb
F =U −TS (5.31)
dF =−S dT −pdV +µdN (5.32)
F Helmholtz free energy
Gibbs free energyc
G=U −TS +pV (5.33)
=F +pV =H −TS (5.34)
dG=−S dT +V dp+µdN (5.35)
G Gibbs free energy
Grand potential
Φ=F −µN (5.36)
dΦ=−S dT −pdV −Ndµ (5.37)
Φ grand potential
Gibbs–Duhem
relation
−S dT +V dp−Ndµ=0 (5.38)
Availability
A=U −T0S +p0V (5.39)
dA=(T −T0)dS −(p−p0)dV (5.40)
A availability
T0 temperature of
surroundings
p0 pressure of surroundings
a dN=0 for a closed system.
bSometimes called the “work function.”
cSometimes called the “thermodynamic potential.”

5
109
Maxwell’s relations
Maxwell 1
∂T
∂V S
=−
∂p
∂S V
=
∂2
U
∂S∂V
(5.41)
U internal energy
T temperature
V volume
Maxwell 2
∂T
∂p S
=
∂V
∂S p
=
∂2
H
∂p∂S
(5.42)
H enthalpy
S entropy
p pressure
Maxwell 3
∂p
∂T V
=
∂S
∂V T
=
∂2
F
∂T∂V
(5.43) F Helmholtz free energy
Maxwell 4
∂V
∂T p
=−
∂S
∂p T
=
∂2
G
∂p∂T
(5.44) G Gibbs free energy
Gibbs–Helmholtz equations
U =−T2 ∂(F/T)
∂T V
(5.45)
G=−V2 ∂(F/V)
∂V T
(5.46)
H =−T2 ∂(G/T)
∂T p
(5.47)
U internal energy
G Gibbs free energy
H enthalpy
T temperature
p pressure
V volume
Phase transitions
Heat absorbed L=T(S2 −S1) (5.48)
L (latent) heat absorbed (1→2)
T temperature of phase change
S entropy
Clausius–Clapeyron
equationa
dp
dT
=
S2 −S1
V2 −V1
(5.49)
=
L
T(V2 −V1)
(5.50)
p pressure
V volume
1,2 phase states
Coexistence curveb
p(T)∝exp
−L
RT
(5.51) R molar gas constant
Ehrenfest’s
equationc
dp
dT
=
βp2 −βp1
κT2 −κT1
(5.52)
=
1
VT
Cp2 −Cp1
βp2 −βp1
(5.53)
Cp heat capacity (p constant)
Gibbs’s phase rule P+F=C+2 (5.54)
P number of phases in equilibrium
F number of degrees of freedom
C number of components
aPhase boundary gradient for a ﬁrst-order transition. Equation (5.50) is sometimes called the “Clapeyron equation.”
bFor V2 V1, e.g., if phase 1 is a liquid and phase 2 a vapour.
cFor a second-order phase transition.

110 Thermodynamics
5.3 Gas laws
Ideal gas
Joule’s law U =U(T) (5.55) U internal energy
T temperature
Boyle’s law pV|T =constant (5.56)
p pressure
V volume
Equation of state
(Ideal gas law)
pV =nRT (5.57) n number of moles
Adiabatic
equations
pVγ
=constant (5.58)
TV(γ−1)
=constant (5.59)
Tγ
p(1−γ)
=constant (5.60)
∆W =
1
γ−1
(p2V2 −p1V1) (5.61)
(Cp/CV )
∆W work done on system
Internal energy U =
nRT
γ−1
(5.62)
Reversible
isothermal
expansion
∆Q=nRT ln(V2/V1) (5.63)
∆Q heat supplied to system
1,2 initial and ﬁnal states
Joule expansiona ∆S =nRln(V2/V1) (5.64) ∆S change in entropy of the
system
aSince ∆Q = 0 for a Joule expansion, ∆S is due entirely to irreversibility. Because entropy is a function of state it
has the same value as for the reversible isothermal expansion, where ∆S =∆Q/T.
Virial expansion
Virial expansion
pV =RT 1+
B2(T)
V
+
B3(T)
V2
+···
(5.65)
p pressure
V volume
T temperature
Bi virial coeﬃcients
Boyle
temperature
B2(TB)=0 (5.66) TB Boyle temperature

5.3 Gas laws
5
111
Van der Waals gas
Equation of state p+
a
V2
m
(Vm −b)=RT (5.67)
p pressure
Vm molar volume
T temperature
a,b van der Waals’ constants
Critical point
Tc =8a/(27Rb) (5.68)
pc =a/(27b2
) (5.69)
Vmc =3b (5.70)
Tc critical temperature
pc critical pressure
Vmc critical molar volume
Reduced equation
of state
pr +
3
V2
r
(3Vr −1)=8Tr (5.71)
pr =p/pc
Vr =Vm/Vmc
Tr =T/Tc
Dieterici gas
Equation of state p=
RT
Vm −b
exp
−a
RTVm
(5.72)
p pressure
Vm molar volume
T temperature
a ,b Dieterici’s constants
Critical point
Tc =a /(4Rb ) (5.73)
pc =a /(4b
2
e2
) (5.74)
Vmc =2b (5.75)
Tc critical temperature
pc critical pressure
Vmc critical molar volume
e =2.71828...
Reduced equation
of state
pr =
Tr
2Vr −1
exp 2−
2
VrTr
(5.76)
pr =p/pc
Vr =Vm/Vmc
Tr =T/Tc
Van der Waals gas Dieterici gas
pr
pr
VrVr
0
0
0
0
33 44 55
0.20.2
0.4
0.4 0.6
0.6 0.8
0.8
1
1
1
1
1.2
1.2
1.4
1.4
1.6
1.8
2
2
2
0.8
0.8
0.9 0.9
1.0
1.0
1.1
1.1
Tr =1.2
Tr =1.2

112 Thermodynamics
5.4 Kinetic theory
Monatomic gas
Pressure p=
1
3
nm c2
(5.77)
p pressure
n number density =N/V
m particle mass
c2 mean squared particle
velocity
Equation of
state of an ideal
gas
pV =NkT (5.78)
V volume
T temperature
Internal energy U =
3
2
NkT =
N
2
m c2
(5.79) U internal energy
Heat capacities
CV =
3
2
Nk (5.80)
Cp =CV +Nk =
5
2
Nk (5.81)
γ =
Cp
CV
=
5
3
(5.82)
CV heat capacity, constant V
Cp heat capacity, constant p
Entropy
(Sackur–
Tetrode
equation)a
S =Nkln
mkT
2π¯h2
3/2
e5/2 V
N
(5.83)
S entropy
¯h =(Planck constant)/(2π)
e =2.71828...
aFor the uncondensed gas. The factor mkT
2π¯h2
3/2
is the quantum concentration of the particles, nQ. Their thermal de
Broglie wavelength, λT , approximately equals n
−1/3
Q .
Maxwell–Boltzmann distributiona
Particle speed
distribution
pr(c) dc=
m
2πkT
3/2
exp
−mc2
2kT
4πc2
dc
(5.84)
m particle mass
T temperature
c particle speed
Particle energy
distribution pr(E) dE =
2E1/2
π1/2(kT)3/2
exp
−E
kT
dE (5.85)
E particle kinetic
energy (=mc2/2)
Mean speed c =
8kT
πm
1/2
(5.86) c mean speed
rms speed crms =
3kT
m
1/2
=
3π
8
1/2
c (5.87)
crms root mean squared
speed
Most probable
speed ˆc=
2kT
m
1/2
=
π
4
1/2
c (5.88) ˆc most probable speed
aProbability density functions normalised so that
∞
0 pr(x) dx=1.

5.4 Kinetic theory
5
113
Transport properties
Mean free patha
l =
1
√
2πd2n
(5.89)
l mean free path
d molecular diameter
n particle number density
Survival
equationb pr(x)=exp(−x/l) (5.90)
pr probability
x linear distance
Flux through a
planec J =
1
4
n c (5.91)
J molecular flux
c mean molecular speed
Self-diffusion
(Fick’s law of
diffusion)d
J =−D∇n (5.92)
where D
2
3
l c (5.93)
D diffusion coefficient
Thermal
conductivityd
H =−λ∇T (5.94)
∇2
T =
1
D
∂T
∂t
(5.95)
for monatomic gas λ
5
4
ρl c cV (5.96)
H heat flux per unit area
T temperature
ρ density
cV specific heat capacity, V
constant
Viscosityd
η
1
2
ρl c (5.97)
η dynamic viscosity
x displacement of sphere in
x direction after time t
Brownian
motion (of a
sphere)
x2
=
kTt
3πηa
(5.98)
t time interval
a sphere radius
Free molecular
flow (Knudsen
flow)e
dM
dt
=
4R3
p
3L
2πm
k
1/2
p1
T
1/2
1
−
p2
T
1/2
2
(5.99)
dM
dt mass flow rate
Rp pipe radius
L pipe length
m particle mass
p pressure
aFor a perfect gas of hard, spherical particles with a Maxwell–Boltzmann speed distribution.
bProbability of travelling distance x without a collision.
cFrom the side where the number density is n, assuming an isotropic velocity distribution. Also known as “collision
number.”
dSimplistic kinetic theory yields numerical coefficients of 1/3 for D, λ and η.
eThrough a pipe from end 1 to end 2, assuming Rp l (i.e., at very low pressure).
Gas equipartition
Classical
equipartitiona Eq =
1
2
kT (5.100)
Eq energy per quadratic degree of
freedom
T temperature
Ideal gas heat
capacities
CV =
1
2
fNk =
1
2
fnR (5.101)
Cp =Nk 1+
f
2
(5.102)
γ =
Cp
CV
=1+
2
f
(5.103)
N number of molecules
f number of degrees of freedom
n number of moles
aSystem in thermal equilibrium at temperature T.

114 Thermodynamics
5.5 Statistical thermodynamics
Statistical entropy
Boltzmann
formulaa
S =klnW (5.104)
klng(E) (5.105)
S entropy
W number of accessible microstates
g(E) density of microstates with energy E
Gibbs entropyb S =−k
i
pi lnpi (5.106) i sum over microstates
pi probability that the system is in microstate i
N two-level
systems
W =
N!
(N −n)!n!
(5.107)
N number of systems
n number in upper state
N harmonic
oscillators
W =
(Q+N −1)!
Q!(N −1)!
(5.108) Q total number of energy quanta available
aSometimes called “configurational entropy.” Equation (5.105) is true only for large systems.
bSometimes called “canonical entropy.”
Ensemble probabilities
Microcanonical
ensemblea pi =
1
W
(5.109)
pi probability that the system is in
microstate i
W number of accessible
microstates
Partition functionb Z =
i
e−βEi
(5.110)
Z partition function
i sum over microstates
β =1/(kT)
Ei energy of microstate i
Canonical ensemble
(Boltzmann
distribution)c
pi =
1
Z
e−βEi
(5.111)
T temperature
Grand partition
function
Ξ=
i
e−β(Ei−µNi)
(5.112)
Ξ grand partition function
Ni number of particles in
microstate i
Grand canonical
ensemble (Gibbs
distribution)d
pi =
1
Ξ
e−β(Ei−µNi)
(5.113)
aEnergy fixed.
bAlso called “sum over states.”
cTemperature fixed.
dTemperature fixed. Exchange of both heat and particles with a reservoir.

5.5 Statistical thermodynamics
5
115
Macroscopic thermodynamic variables
Helmholtz free
energy
F =−kT lnZ (5.114)
T temperature
Z partition function
Grand potential Φ=−kT lnΞ (5.115) Φ grand potential
Ξ grand partition function
Internal energy U =F +TS =−
∂lnZ
∂β V,N
(5.116)
U internal energy
β =1/(kT)
Entropy S =−
∂F
∂T V,N
=
∂(kT lnZ)
∂T V,N
(5.117)
S entropy
Pressure p=−
∂F
∂V T,N
=
∂(kT lnZ)
∂V T,N
(5.118) p pressure
Chemical
potential
µ=
∂F
∂N V,T
=−
∂(kT lnZ)
∂N V,T
(5.119) µ chemical potential
Identical particles
Bose–Einstein distribution Fermi–Dirac distribution
fi
fi
ii
0
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
1.2
1.2
1.2
1.2
1.41.4
1.4
1.61.6
1.6
1.81.8
1.8
22
2
β =1
β =1
5
5
10
10
50
50
(µ=0) (µ=1)
Bose–Einstein
distributiona fi =
1
eβ( i−µ) −1
(5.120)
fi mean occupation number of
ith state
β =1/(kT)
Fermi–Dirac
distributionb fi =
1
eβ( i−µ) +1
(5.121)
i energy quantum for ith state
Fermi energyc
F =
¯h2
2m
6π2
n
g
2/3
(5.122)
F Fermi energy
m particle mass
Bose
condensation
temperature
Tc =
2π¯h2
mk
n
gζ(3/2)
2/3
(5.123)
g spin degeneracy (=2s+1)
ζ Riemann zeta function
ζ(3/2) 2.612
Tc Bose condensation
temperature
aFor bosons. fi ≥0.
bFor fermions. 0≤fi ≤1.
cFor noninteracting particles. At low temperatures, µ F.

116 Thermodynamics
Population densitiesa
Boltzmann
excitation
equation
nmj
nlj
=
gmj
glj
exp
−(χmj −χlj)
kT
(5.124)
=
gmj
glj
exp
−hνlm
kT
(5.125)
nij number density of atoms in
excitation level i of ionisation
state j (j =0 if not ionised)
gij level degeneracy
χij excitation energy relative to
the ground state
Partition
function
Zj(T)=
i
gij exp
−χij
kT
(5.126)
nij
Nj
=
gij
Zj(T)
exp
−χij
kT
(5.127)
νij photon transition frequency
h Planck constant
T temperature
Saha equation (general)
nij =n0,j+1ne
gij
g0,j+1
h3
2
(2πmekT)−3/2
exp
χIj −χij
kT
(5.128)
Zj partition function for
ionisation state j
Nj total number density in
ionisation state j
Saha equation (ion populations)
Nj
Nj+1
=ne
Zj(T)
Zj+1(T)
h3
2
(2πmekT)−3/2
exp
χIj
kT
(5.129)
ne electron number density
me electron mass
χIj ionisation energy of atom in
ionisation state j
aAll equations apply only under conditions of local thermodynamic equilibrium (LTE). In atoms with no magnetic
splitting, the degeneracy of a level with total angular momentum quantum number J is gij =2J +1.
5.6 Fluctuations and noise
Thermodynamic fluctuationsa
Fluctuation
probability
pr(x)∝exp[S(x)/k] (5.130)
∝exp
−A(x)
kT
(5.131)
x unconstrained variable
S entropy
A availability
General
variance var[x]=kT
∂2
A(x)
∂x2
−1
(5.132)
var[·] mean square deviation
T temperature
Temperature
fluctuations var[T]=kT
∂T
∂S V
=
kT2
CV
(5.133)
V volume
Volume
fluctuations
var[V]=−kT
∂V
∂p T
=κT VkT (5.134)
p pressure
Entropy
fluctuations
var[S]=kT
∂S
∂T p
=kCp (5.135) Cp heat capacity, p constant
Pressure
fluctuations
var[p]=−kT
∂p
∂V S
=
KS kT
V
(5.136) KS adiabatic bulk modulus
Density
fluctuations var[n]=
n2
V2
var[V]=
n2
V
κT kT (5.137) n number density
aIn part of a large system, whose mean temperature is fixed. Quantum effects are assumed negligible.

5.6 Fluctuations and noise
5
117
Noise
Nyquist’s noise
theorem
dw =kT ·β (eβ
−1)−1
dν (5.138)
=kTN dν (5.139)
kT dν (hν kT) (5.140)
w exchangeable noise power
T temperature
TN noise temperature
β =hν/(kT)
ν frequency
h Planck constant
Johnson
(thermal) noise
voltagea
vrms =(4kTNR∆ν)1/2
(5.141)
vrms rms noise voltage
R resistance
∆ν bandwidth
Shot noise
(electrical)
Irms =(2eI0∆ν)1/2
(5.142)
Irms rms noise current
I0 mean current
Noise figureb
fdB =10log10 1+
TN
T0
(5.143)
fdB noise figure (decibels)
T0 ambient temperature (usually
taken as 290 K)
Relative power G=10log10
P2
P1
(5.144)
G decibel gain of P2 over P1
P1, P2 power levels
aThermal voltage over an open-circuit resistance.
bNoise figure can also be defined as f =1+TN/T0, when it is also called “noise factor.”

118 Thermodynamics
5.7 Radiation processes
Radiometrya
Radiant energyb
Qe = Le cosθ dA dΩ dt J (5.145)
Qe radiant energy
Le radiance (generally a
function of position
and direction)
θ angle between dir. of
dΩ and normal to dA
Ω solid angle
Radiant flux
(“radiant power”)
Φe =
∂Qe
∂t
W (5.146)
= Le cosθ dA dΩ (5.147)
A area
t time
Φe radiant flux
Radiant energy
densityc We =
∂Qe
∂V
Jm−3
(5.148)
We radiant energy density
dV differential volume of
propagation medium
Radiant exitanced
Me =
∂Φe
∂A
Wm−2
(5.149)
= Le cosθdΩ (5.150)
Me radiant exitance
Irradiancee
Ee =
∂Φe
∂A
Wm−2
(5.151)
= Le cosθdΩ (5.152)
Radiant intensity
x
y
z
θ
φ
dA
dΩ(normal)
Ie =
∂Φe
∂Ω
Wsr−1
(5.153)
= Le cosθ dA (5.154)
Ee irradiance
Ie radiant intensity
Radiance
Le =
1
cosθ
∂2
Φe
dAdΩ
Wm−2
sr−1
(5.155)
=
1
cosθ
∂Ie
∂A
(5.156)
aRadiometry is concerned with the treatment of light as energy.
bSometimes called “total energy.” Note that we assume opaque radiant surfaces, so that 0≤θ ≤π/2.
cThe instantaneous amount of radiant energy contained in a unit volume of propagation medium.
dPower per unit area leaving a surface. For a perfectly diffusing surface, Me =πLe.
ePower per unit area incident on a surface.

5
119
Photometrya
Luminous energy
(“total light”)
Qv = Lv cosθ dA dΩ dt lms (5.157)
Qv luminous energy
Lv luminance (generally a
function of position
and direction)
θ angle between dir. of
dΩ and normal to dA
Ω solid angle
Luminous flux
Φv =
∂Qv
∂t
lumen (lm) (5.158)
= Lv cosθ dA dΩ (5.159)
A area
t time
Φv luminous flux
Luminous
densityb Wv =
∂Qv
∂V
lmsm−3
(5.160)
Wv luminous density
V volume
Luminous
exitancec
Mv =
∂Φv
∂A
lx (lmm−2
) (5.161)
= Lv cosθdΩ (5.162)
Mv luminous exitance
Illuminance
(“illumination”)d
Ev =
∂Φv
∂A
lmm−2
(5.163)
= Lv cosθdΩ (5.164)
Luminous
intensitye
x
y
z
θ
φ
dA
dΩ(normal)
Iv =
∂Φv
∂Ω
cd (5.165)
= Lv cosθ dA (5.166)
Ev illuminance
Iv luminous intensity
Luminance
(“photometric
brightness”)
Lv =
1
cosθ
∂2
Φv
dAdΩ
cdm−2
(5.167)
=
1
cosθ
∂Iv
∂A
(5.168)
Luminous efficacy K =
Φv
Φe
=
Lv
Le
=
Iv
Ie
lmW−1
(5.169)
K luminous efficacy
Le radiance
Φe radiant flux
Ie radiant intensity
Luminous
efficiency
V(λ)=
K(λ)
Kmax
(5.170)
V luminous efficiency
λ wavelength
Kmaxspectral maximum of
K(λ)
aPhotometry is concerned with the treatment of light as seen by the human eye.
bThe instantaneous amount of luminous energy contained in a unit volume of propagating medium.
cLuminous emitted flux per unit area.
dLuminous incident flux per unit area. The derived SI unit is the lux (lx). 1lx=1lmm−2.
eThe SI unit of luminous intensity is the candela (cd). 1cd=1lmsr−1.

120 Thermodynamics
Radiative transfera
Flux density
(through a
plane)
x
y
z
θ
φ
dΩ(normal)
Fν = Iν(θ,φ)cosθ dΩ Wm−2
Hz−1
(5.171)
Mean intensityb
Jν =
1
4π
Iν(θ,φ) dΩ Wm−2
Hz−1
(5.172)
Fν flux density
Iν specific intensity
(Wm−2 Hz−1
sr−1)
Jν mean intensity
Spectral energy
densityc uν =
1
c
Iν(θ,φ) dΩ Jm−3
Hz−1
(5.173)
uν spectral energy density
Ω solid angle
θ angle between normal
and direction of Ω
Specific
emission
coefficient
jν =
ν
ρ
Wkg−1
Hz−1
sr−1
(5.174)
jν specific emission
coefficient
ν emission coefficient
(Wm−3 Hz−1
sr−1)
ρ density
Gas linear
absorption
coefficient
(αν 1)
αν =nσν =
1
lν
m−1
(5.175)
αν linear absorption
coefficient
σν particle cross section
lν mean free path
Opacityd κν =
αν
ρ
kg−1
m2
(5.176) κν opacity
Optical depth τν = κνρ ds (5.177)
τν optical depth, or
optical thickness
ds line element
Transfer
equatione
1
ρ
dIν
ds
=−κνIν +jν (5.178)
or
dIν
ds
=−ανIν + ν (5.179)
Kirchhoff’s lawf
Sν ≡
jν
κν
=
ν
αν
(5.180) Sν source function
Emission from
a homogeneous
medium
Iν =Sν(1−e−τν
) (5.181)
aThe definitions of these quantities vary in the literature. Those presented here are common in meteorology and
astrophysics. Note particularly that the ambiguous term specific is taken to mean “per unit frequency interval” in the
case of specific intensity and “per unit mass per unit frequency interval” in the case of specific emission coefficient.
bIn radio astronomy, flux density is usually taken as S =4πJν.
cAssuming a refractive index of 1.
dOr “mass absorption coefficient.”
eOr “Schwarzschild’s equation.”
fUnder conditions of local thermal equilibrium (LTE), the source function, Sν, equals the Planck function, Bν(T)
[see Equation (5.182)].

5
121
Blackbody radiation
brightness(Bν/Wm−2
Hz−1
sr−1
)
brightness(Bλ/Wm−2
m−1
sr−1
)
frequency (ν/Hz) wavelength (λ/m)
10−20
10−20
10−15
10−10
10−10
10−10
10−5
1
1
1
105
1010
1010
1010
106 108 1012 1014 1016 1018
1020
1020 1022
1030
1040
1050
10−14 10−12 10−8 10−6 10−4 10−2 102
2.7K2.7K
100K100K
103 K103 K
104 K104 K
105 K105 K
106 K
106 K
107 K
107 K
108 K
108 K
109 K
109 K
1010 K
1010 K
νm(T)=c/λm(T) λm(T)
Planck
functiona
Bν(T)=
2hν3
c2
exp
hν
kT
−1
−1
(5.182)
Bλ(T)=Bν(T)
dν
dλ
(5.183)
=
2hc2
λ5
exp
hc
λkT
−1
−1
(5.184)
Bν surface brightness per
unit frequency
(Wm−2 Hz−1
sr−1)
Bλ surface brightness per
unit wavelength
(Wm−2 m−1 sr−1)
h Planck constant
Spectral energy
density
uν(T)=
4π
c
Bν(T) Jm−3
Hz−1
(5.185)
uλ(T)=
4π
c
Bλ(T) Jm−3
m−1
(5.186)
c speed of light
T temperature
uν,λ spectral energy density
Rayleigh–Jeans
law (hν kT)
Bν(T)=
2kT
c2
ν2
=
2kT
λ2
(5.187)
Wien’s law
(hν kT) Bν(T)=
2hν3
c2
exp
−hν
kT
(5.188)
Wien’s
displacement
law
λmT =
5.1×10−3
mK for Bν
2.9×10−3
mK for Bλ
(5.189)
λm wavelength of
maximum brightness
Stefan–
Boltzmann
lawb
M =π
∞
0
Bν(T) dν (5.190)
=
2π5
k4
15c2h3
T4
=σT4
Wm−2
(5.191)
M exitance
σ Stefan–Boltzmann
constant (
5.67×10−8 Wm−2 K−4)
Energy density u(T)=
4
c
σT4
Jm−3
(5.192) u energy density
Greybody M = σT4
=(1−A)σT4
(5.193)
mean emissivity
A albedo
aWith respect to the projected area of the surface. Surface brightness is also known simply as “brightness.” “Speciﬁc
intensity” is used for reception.
bSometimes “Stefan’s law.” Exitance is the total radiated energy from unit area of the body per unit time.

6
Chapter 6 Solid state physics
6.1 Introduction
This section covers a few selected topics in solid state physics. There is no attempt to do
more than scratch the surface of this vast field, although the basics of many undergraduate
texts on the subject are covered. In addition a period table of elements, together with some
of their physical properties, is displayed on the next two pages.
Periodic table (overleaf) Data for the periodic table of elements are taken from Pure Appl. Chem., 71, 1593–1607
(1999), from the 16th edition of Kaye and Laby Tables of Physical and Chemical Constants (Longman, 1995) and
from the 74th edition of the CRC Handbook of Chemistry and Physics (CRC Press, 1993). Note that melting and
boiling points have been converted to kelvins by adding 273.15 to the Celsius values listed in Kaye and Laby. The
standard atomic masses reflect the relative isotopic abundances in samples found naturally on Earth, and the number
of significant figures reflect the variations between samples. Elements with atomic masses shown in square brackets
have no stable nuclides, and the values reflect the mass numbers of the longest-lived isotopes. Crystallographic data
are based on the most common forms of the elements (the α-form, unless stated otherwise) stable under standard
conditions. Densities are for the solid state. For full details and footnotes for each element, the reader is advised to
consult the original texts.
Elements 110, 111, 112 and 114 are known to exist but their names are not yet permanent.

124 Solid state physics
6.2 Periodic table
Titanium
47.867
22 Ti
[Ca]3d2
4 508 295
HEX 1.587
1 943 3 563
name
relative atomic mass (u)
symbol
lattice constant, a (fm)
c/a (angle in RHL,
c/a
b/a in ORC & MCL)
boiling point (K)
density (kgm−3)
crystal type
melting point (K)
electron conﬁguration
atomic number
1
1
Hydrogen
1.007 94
1 H
1s1
89 (β) 378
HEX 1.632
13.80 20.28 2
2
Lithium
6.941
3 Li
[He]2s1
533 (β) 351
BCC
453.65 1 613
Beryllium
9.012 182
4 Be
[He]2s2
1 846 229
HEX 1.568
1 560 2 745
3
Sodium
22.989 770
11 Na
[Ne]3s1
966 429
BCC
370.8 1 153
Magnesium
24.305 0
12 Mg
[Ne]3s2
1 738 321
HEX 1.624
923 1 363 3 4 5 6 7 8 9
4
Potassium
39.098 3
19 K
[Ar]4s1
862 532
BCC
336.5 1 033
Calcium
40.078
20 Ca
[Ar]4s2
1 530 559
FCC
1 113 1 757
Scandium
44.955 910
21 Sc
[Ca]3d1
2 992 331
HEX 1.592
1 813 3 103
Titanium
47.867
22 Ti
[Ca]3d2
4 508 295
HEX 1.587
1 943 3 563
Vanadium
50.941 5
23 V
[Ca]3d3
6 090 302
BCC
2 193 3 673
Chromium
51.996 1
24 Cr
[Ar]3d54s1
7 194 388
BCC
2 180 2 943
Manganese
54.938 049
25 Mn
[Ca]3d5
7 473 891
FCC
1 523 2 333
Iron
55.845
26 Fe
[Ca]3d6
7 873 287
BCC
1 813 3 133
Cobalt
58.933 200
27 Co
[Ca]3d7
8 800 ( ) 251
HEX 1.623
1 768 3 203
5
Rubidium
85.467 8
37 Rb
[Kr]5s1
1 533 571
BCC
312.4 963.1
Strontium
87.62
38 Sr
[Kr]5s2
2 583 608
FCC
1 050 1 653
Yttrium
88.905 85
39 Y
[Sr]4d1
4 475 365
HEX 1.571
1 798 3 613
Zirconium
91.224
40 Zr
[Sr]4d2
6 507 323
HEX 1.593
2 123 4 673
Niobium
92.906 38
41 Nb
[Kr]4d45s1
8 578 330
BCC
2 750 4 973
Molybdenum
95.94
42 Mo
[Kr]4d55s1
10 222 315
BCC
2 896 4 913
Technetium
[98]
43 Tc
[Sr]4d5
11 496 274
HEX 1.604
2 433 4 533
Ruthenium
101.07
44 Ru
[Kr]4d75s1
12 360 270
HEX 1.582
2 603 4 423
Rhodium
102.905 50
45 Rh
[Kr]4d85s1
12 420 380
FCC
2 236 3 973
6
Caesium
132.905 45
55 Cs
[Xe]6s1
1 900 614
BCC
301.6 943.2
Barium
137.327
56 Ba
[Xe]6s2
3 594 502
BCC
1 001 2 173
Lanthanides
57 – 71
Hafnium
178.49
72 Hf
[Yb]5d2
13 276 319
HEX 1.581
2 503 4 873
Tantalum
180.947 9
73 Ta
[Yb]5d3
16 670 330
BCC
3 293 5 833
Tungsten
183.84
74 W
[Yb]5d4
19 254 316
BCC
3 695 5 823
Rhenium
186.207
75 Re
[Yb]5d5
21 023 276
HEX 1.615
3 459 5 873
Osmium
190.23
76 Os
[Yb]5d6
22 580 273
HEX 1.606
3 303 5 273
Iridium
192.217
77 Ir
[Yb]5d7
22 550 384
FCC
2 720 4 703
7
Francium
[223]
87 Fr
[Rn]7s1
300 923
Radium
[226]
88 Ra
[Rn]7s2
5 000 515
BCC
973 1 773
Actinides
89 – 103
Rutherfordium
[261]
104 Rf
[Ra]5f146d2
Dubnium
[262]
105 Db
[Ra]5f146d3?
Seaborgium
[263]
106 Sg
[Ra]5f146d4?
Bohrium
[264]
107 Bh
[Ra]5f146d5?
Hassium
[265]
108 Hs
[Ra]5f146d6?
Meitnerium
[268]
109 Mt
[Ra]5f146d7?
Lanthanides
Lanthanum
138.905 5
57 La
[Ba]5d1
6 174 377
HEX 3.23
1 193 3 733
Cerium
140.116
58 Ce
[Ba]4f15d1
6 711 (γ) 516
FCC
1 073 3 693
Praseodymium
140.907 65
59 Pr
[Ba]4f3
6 779 367
HEX 3.222
1 204 3 783
Neodymium
144.24
60 Nd
[Ba]4f4
7 000 366
HEX 3.225
1 289 3 343
Promethium
[145]
61 Pm
[Ba]4f5
7 220 365
HEX 3.19
1 415 3 573
Samarium
150.36
62 Sm
[Ba]4f6
7 536 363
HEX 7.221
1 443 2 063
Actinides
Actinium
[227]
89 Ac
[Ra]6d1
10 060 531
FCC
1 323 3 473
Thorium
232.038 1
90 Th
[Ra]6d2
11 725 508
FCC
2 023 5 063
Protactinium
231.035 88
91 Pa
[Rn]5f26d17s2
15 370 392
TET 0.825
1 843 4 273
Uranium
238.028 9
92 U
[Rn]5f36d17s2
19 050 285
ORC 1.736
2.056
1 405.3 4 403
Neptunium
[237]
93 Np
[Rn]5f46d17s2
20 450 666
ORC 0.733
0.709
913 4 173
Plutonium
[244]
94 Pu
[Rn]5f67s2
19 816 618
MCL 1.773
0.780
913 3 503

6.2 Periodic table
6
125
BCC body-centred cubic
CUB simple cubic
DIA diamond
FCC face-centred cubic
HEX hexagonal
MCL monoclinic
ORC orthorhombic
RHL rhombohedral
TET tetragonal
(t-pt) triple point
18
13 14 15 16 17
Helium
4.002 602
2 He
1s2
120 356
HEX 1.631
3-5 4.22
Boron
10.811
5 B
[Be]2p1
2 466 1017
RHL 65◦
7
2 348 4 273
Carbon
12.0107
6 C
[Be]2p2
2 266 357
DIA
4 763 (t-pt)
Nitrogen
14.006 74
7 N
[Be]2p3
1 035 (β) 405
HEX 1.631
63 77.35
Oxygen
15.999 4
8 O
[Be]2p4
1 460 (γ) 683
CUB
54.36 90.19
Fluorine
18.998 403 2
9 F
[Be]2p5
1 140 550
MCL 1.32
0.61
53.55 85.05
Neon
20.179 7
10 Ne
[Be]2p6
1 442 446
FCC
24.56 27.07
10 11 12
Aluminium
26.981 538
13 Al
[Mg]3p1
2 698 405
FCC
933.47 2 793
Silicon
28.085 5
14 Si
[Mg]3p2
2 329 543
DIA
1 683 3 533
Phosphorus
30.973 761
15 P
[Mg]3p3
1 820 331
ORC 1.320
3.162
317.3 550
Sulfur
32.066
16 S
[Mg]3p4
2 086 1 046
ORC 2.340
1.229
388.47 717.82
Chlorine
35.452 7
17 Cl
[Mg]3p5
2 030 624
ORC 1.324
0.718
172 239.1
Argon
39.948
18 Ar
[Mg]3p6
1 656 532
FCC
83.81 87.30
Nickel
58.693 4
28 Ni
[Ca]3d8
8 907 352
FCC
1 728 3 263
Copper
63.546
29 Cu
[Ar]3d104s1
8 933 361
FCC
1 357.8 2 833
Zinc
65.39
30 Zn
[Ca]3d10
7 135 266
HEX 1.856
692.68 1 183
Gallium
69.723
31 Ga
[Zn]4p1
5 905 452
ORC 1.001
1.695
302.9 2 473
Germanium
72.61
32 Ge
[Zn]4p2
5 323 566
DIA
1211 3103
Arsenic
74.921 60
33 As
[Zn]4p3
5 776 413
RHL 54◦
7
883 (t-pt)
Selenium
78.96
34 Se
[Zn]4p4
4 808 (γ) 436
HEX 1.135
493 958
Bromine
79.904
35 Br
[Zn]4p5
3 120 668
ORC 1.308
0.672
265.90 332.0
Krypton
83.80
36 Kr
[Zn]4p6
3 000 581
FCC
115.8 119.9
Palladium
106.42
46 Pd
[Kr]4d10
11 995 389
FCC
1 828 3 233
Silver
107.868 2
47 Ag
[Pd]5s1
10 500 409
FCC
1 235 2 433
Cadmium
112.411
48 Cd
[Pd]5s2
8 647 298
HEX 1.886
594.2 1 043
Indium
114.818
49 In
[Cd]5p1
7 290 325
TET 1.521
429.75 2 343
Tin
118.710
50 Sn
[Cd]5p2
7 285 (β) 583
TET 0.546
505.08 2 893
Antimony
121.760
51 Sb
[Cd]5p3
6 692 451
RHL 57◦
7
903.8 1 860
Tellurium
127.60
52 Te
[Cd]5p4
6 247 446
HEX 1.33
723 1 263
Iodine
126.904 47
53 I
[Cd]5p5
4 953 727
ORC 1.347
0.659
386.7 457
Xenon
131.29
54 Xe
[Cd]5p6
3 560 635
FCC
161.3 165.0
Platinum
195.078
78 Pt
[Xe]4f145d96s1
21 450 392
FCC
2 041 4 093
Gold
196.966 55
79 Au
[Xe]4f145d106s1
19 281 408
FCC
1 337.3 3 123
Mercury
200.59
80 Hg
[Yb]5d10
13 546 300
RHL 70◦
32
234.32 629.9
Thallium
204.383 3
81 Tl
[Hg]6p1
11 871 346
HEX 1.598
577 1743
Lead
207.2
82 Pb
[Hg]6p2
11 343 495
FCC
600.7 2 023
Bismuth
208.980 38
83 Bi
[Hg]6p3
9 803 475
RHL 57◦
14
544.59 1 833
Polonium
[209]
84 Po
[Hg]6p4
9 400 337
CUB
527 1 233
Astatine
[210]
85 At
[Hg]6p5
573 623
Radon
[222]
86 Rn
[Hg]6p6
440
202 211
Ununnilium
[271]
110 Uun
Unununium
[272]
111 Uuu
Ununbium
[285]
112 Uub
Ununquadium
[289]
114 Uuq
Europium
151.964
63 Eu
[Ba]4f7
5 248 458
BCC
1 095 1 873
Gadolinium
157.25
64 Gd
[Ba]4f75d1
7 870 363
HEX 1.591
1 587 3 533
Terbium
158.925 34
65 Tb
[Ba]4f9
8 267 361
HEX 1.580
1 633 3 493
Dysprosium
162.50
66 Dy
[Ba]4f10
8 531 359
HEX 1.573
1 683 2 833
Holmium
164.930 32
67 Ho
[Ba]4f11
8 797 358
HEX 1.570
1 743 2 973
Erbium
167.26
68 Er
[Ba]4f12
9 044 356
HEX 1.570
1 803 3 133
Thulium
168.934 21
69 Tm
[Ba]4f13
9 325 354
HEX 1.570
1 823 2 223
Ytterbium
173.04
70 Yb
[Ba]4f14
6 966 (β) 549
FCC
1 097 1 473
Lutetium
174.967
71 Lu
[Yb]5d1
9 842 351
HEX 1.583
1 933 3 663
Americium
[243]
95 Am
[Ra]5f7
13 670 347
HEX 3.24
1 449 2 873
Curium
[247]
96 Cm
[Rn]5f76d17s2
13 510 350
HEX 3.24
1 618 3 383
Berkelium
[247]
97 Bk
[Ra]5f9
14 780 342
HEX 3.24
1 323
Californium
[251]
98 Cf
[Ra]5f10
15 100 338
HEX 3.24
1 173
Einsteinium
[252]
99 Es
[Ra]5f11
HEX
1 133
Fermium
[257]
100 Fm
[Ra]5f12
1 803
Mendelevium
[258]
101 Md
[Ra]5f13
1 103
Nobelium
[259]
102 No
[Ra]5f14
1 103
Lawrencium
[262]
103 Lr
[Ra]5f147p1
1 903

6.3 Crystalline structure
Bravais lattices
Volume of
primitive cell
V =(a×××b)·c (6.1) a,b,c primitive base vectors
V volume of primitive cell
Reciprocal
primitive base
vectorsa
a∗
=2πb×××c/[(a×××b)·c] (6.2)
b∗
=2πc×××a/[(a×××b)·c] (6.3)
c∗
=2πa×××b/[(a×××b)·c] (6.4)
a·a∗
=b·b∗
=c·c∗
=2π (6.5)
a·b∗
=a·c∗
=0 (etc.) (6.6)
a∗,b∗,c∗ reciprocal primitive base
vectors
Lattice vector Ruvw =ua+vb+wc (6.7) Ruvw lattice vector [uvw]
u,v,w integers
Reciprocal lattice
vector
Ghkl =ha∗
+kb∗
+lc∗
(6.8)
exp(iGhkl ·Ruvw)=1 (6.9)
Ghkl reciprocal lattice vector [hkl]
i i2 =−1
Weiss zone
equationb hu+kv+lw =0 (6.10) (hkl) Miller indices of planec
Interplanar
spacing (general)
dhkl =
2π
Ghkl
(6.11)
dhkl distance between (hkl)
planes
Interplanar
spacing
(orthogonal basis)
1
d2
hkl
=
h2
a2
+
k2
b2
+
l2
c2
(6.12)
aNote that this is 2π times the usual deﬁnition of a “reciprocal vector” (see page 20).
bCondition for lattice vector [uvw] to be parallel to lattice plane (hkl) in an arbitrary Bravais lattice.
cMiller indices are deﬁned so that Ghkl is the shortest reciprocal lattice vector normal to the (hkl) planes.
Weber symbols
Converting
[uvw] to
[UVTW]
U =
1
3
(2u−v) (6.13)
V =
1
3
(2v−u) (6.14)
T =−
1
3
(u+v) (6.15)
W =w (6.16)
U,V,T,W Weber indices
u,v,w zone axis indices
[UVTW] Weber symbol
[uvw] zone axis symbol
Converting
[UVTW] to
[uvw]
u=(U −T) (6.17)
v =(V −T) (6.18)
w =W (6.19)
Zone lawa hU +kV +iT +lW =0 (6.20) (hkil) Miller–Bravais indices
aFor trigonal and hexagonal systems.

6.3 Crystalline structure
6
127
Cubic lattices
lattice primitive (P) body-centred (I) face-centred (F)
lattice parameter a a a
volume of conventional cell a3
a3
a3
lattice points per cell 1 2 4
1st nearest neighboursa
6 8 12
1st n.n. distance a a
√
3/2 a/
√
2
2nd nearest neighbours 12 6 6
2nd n.n. distance a
√
2 a a
packing fractionb
π/6
√
3π/8
√
2π/6
reciprocal latticec
P F I
a1 =aˆx a1 = a
2 (ˆy+ ˆz − ˆx) a1 = a
2 (ˆy+ ˆz)
primitive base vectorsd
a2 =aˆy a2 = a
2 (ˆz + ˆx− ˆy) a2 = a
2 (ˆz + ˆx)
a3 =aˆz a3 = a
2 (ˆx+ ˆy− ˆz) a3 = a
2 (ˆx+ ˆy)
aOr “coordination number.”
bFor close-packed spheres. The maximum possible packing fraction for spheres is
√
2π/6.
cThe lattice parameters for the reciprocal lattices of P, I, and F are 2π/a, 4π/a, and 4π/a respectively.
d ˆx, ˆy, and ˆz are unit vectors.
Crystal systemsa
system symmetry unit cellb
latticesc
triclinic none
a=b=c;
α=β =γ =90◦ P
monoclinic one diad [010]
a=b=c;
α=γ =90◦
, β =90◦ P, C
orthorhombic three orthogonal diads
a=b=c;
α=β =γ =90◦ P, C, I, F
tetragonal one tetrad [001]
a=b=c;
α=β =γ =90◦ P, I
trigonald
one triad [111]
a=b=c;
α=β =γ <120◦
=90◦ P, R
hexagonal one hexad [001]
a=b=c;
α=β =90◦
, γ =120◦ P
cubic four triads 111
a=b=c;
α=β =γ =90◦ P, F, I
aThe symbol “=” implies that equality is not required by the symmetry, but neither is it forbidden.
bThe cell axes are a, b, and c with α, β, and γ the angles between b:c, c:a, and a:b respectively.
cThe lattice types are primitive (P), body-centred (I), all face-centred (F), side-centred (C), and
rhombohedral primitive (R).
dA primitive hexagonal unit cell, with a triad [001], is generally preferred over this rhombohedral unit cell.

Dislocations and cracks
Edge
dislocation
ˆl ·b=0 (6.21)
ˆl unit vector line of
dislocation
b,b Burgers vectora
Screw
dislocation
ˆl ·b=b (6.22)
U dislocation energy per
unit length
µ shear modulus
Screw
dislocation
energy per
unit lengthb
b
b
r
L
ˆl
ˆl
U =
µb2
4π
ln
R
r0
(6.23)
∼µb2
(6.24)
R outer cutoff for r
r0 inner cutoff for r
L critical crack length
α surface energy per unit
area
Critical crack
lengthc L=
4αE
π(1−σ2)p2
0
(6.25)
E Young modulus
σ Poisson ratio
p0 applied widening stress
aThe Burgers vector is a Bravais lattice vector characterising the total relative slip
were the dislocation to travel throughout the crystal.
bOr “tension.” The energy per unit length of an edge dislocation is also ∼µb2.
cFor a crack cavity (long ⊥L) within an isotropic medium. Under uniform stress p0,
cracks ≥L will grow and smaller cracks will shrink.
Crystal diffraction
Laue
equations
a(cosα1 −cosα2)=hλ (6.26)
b(cosβ1 −cosβ2)=kλ (6.27)
c(cosγ1 −cosγ2)=lλ (6.28)
a,b,c lattice parameters
α1,β1,γ1 angles between lattice base
vectors and input wavevector
α2,β2,γ2 angles between lattice base
vectors and output wavevector
h,k,l integers (Laue indices)
Bragg’s lawa 2kin.G +|G|2
=0 (6.29)
λ wavelength
kin input wavevector
G reciprocal lattice vector
Atomic form
factor
f(G)=
vol
e−iG·r
ρ(r) d3
r (6.30)
f(G) atomic form factor
r position vector
ρ(r) atomic electron density
Structure
factorb S(G)=
n
j=1
fj(G)e−iG·dj
(6.31)
S(G) structure factor
n number of atoms in basis
dj position of jth atom within basis
Scattered
intensityc I(K)∝N2
|S(K)|2
(6.32)
K change in wavevector
(=kout −kin)
I(K) scattered intensity
N number of lattice points
illuminated
Debye–
Waller
factord
IT =I0 exp −
1
3
u2
|G|2
(6.33)
IT intensity at temperature T
I0 intensity from a lattice with no
motion
u2 mean-squared thermal
displacement of atoms
aAlternatively, see Equation (8.32).
bThe summation is over the atoms in the basis, i.e., the atomic motif repeating with the Bravais lattice.
cThe Bragg condition makes K a reciprocal lattice vector, with |kin|=|kout|.
dEffect of thermal vibrations.

6.4 Lattice dynamics
6
129
Phonon dispersion relationsa
mm m1 m2 (>m1)
a 2a
2(α/m)1/2
ω ω
00 kk−
π
a
π
a
−
π
2a
π
2a
(2α/m2)1/2
(2α/m1)1/2
(2α/µ)1/2
monatomic chain diatomic chain
Monatomic
linear chain
ω2
=4
α
m
sin2 ka
2
(6.34)
vp =
ω
k
=a
α
m
1/2
sinc
a
λ
(6.35)
vg =
∂ω
∂k
=a
α
m
1/2
cos
ka
2
(6.36)
ω phonon angular frequency
α spring constantb
m atomic mass
vp phase speed (sincx≡ sinπx
πx )
vg group speed
λ phonon wavelength
Diatomic
linear chainc ω2
=
α
µ
±α
1
µ2
−
4
m1m2
sin2
(ka)
1/2
(6.37)
a atomic separation
mi atomic masses (m2 >m1)
µ reduced mass
[=m1m2/(m1 +m2)]
Identical
masses,
alternating
spring
constants
mm
a
α1α1 α2
ω2
=
α1 +α2
m
±
1
m
(α2
1 +α2
2 +2α1α2 coska)1/2
(6.38)
=
0, 2(α1 +α2)/m if k =0
2α1/m, 2α2/m if k =π/a
(6.39)
αi alternating spring constants
aAlong infinite linear atomic chains, considering simple harmonic nearest-neighbour interactions only. The shaded
region of the dispersion relation is outside the first Brillouin zone of the reciprocal lattice.
bIn the sense α=restoring force/relative displacement.
cNote that the repeat distance for this chain is 2a, so that the first Brillouin zone extends to |k|<π/(2a). The optic
and acoustic branches are the + and − solutions respectively.

Debye theory
Mean energy
per phonon
modea
E =
1
2
¯hω+
¯hω
exp[¯hω/(kBT)]−1
(6.40)
E mean energy in a mode at ω
ω phonon angular frequency
kB Boltzmann constant
T temperature
Debye
frequency
ωD =vs(6π2
N/V)1/3
(6.41)
where
3
v3
s
=
1
v3
l
+
2
v3
t
(6.42)
ωD Debye (angular) frequency
vs eﬀective sound speed
vl longitudinal phase speed
vt transverse phase speed
Debye
temperature
θD =¯hωD/kB (6.43)
N number of atoms in crystal
V crystal volume
θD Debye temperature
Phonon
density of
states
g(ω) dω =
3Vω2
2π2v3
s
dω (6.44)
(for 0<ω <ωD, g =0 otherwise)
g(ω) density of states at ω
U thermal phonon energy
within crystal
D(x) Debye function
Debye heat
capacity
CV
3NkB
0 1 2
T/θD
CV =9NkB
T3
θ3
D
θD/T
0
x4
ex
(ex −1)2
dx (6.45)
Dulong and
Petit’s law
3NkB (T θD) (6.46)
Debye T3
law
12π4
5
NkB
T3
θ3
D
(T θD) (6.47)
Internal
thermal
energyb
U(T)=
9N
ω3
D
ωD
0
¯hω3
exp[¯hω/(kBT)]−1
dω ≡3NkBT D(θD/T) (6.48)
where D(x)=
3
x3
x
0
y3
ey −1
dy (6.49)
aOr any simple harmonic oscillator in thermal equilibrium at temperature T.
bNeglecting zero-point energy.

6
131
Lattice forces (simple)
Van der Waals
interactiona φ(r)=−
3
4
α2
p¯hω
(4π 0)2r6
(6.50)
φ(r) two-particle potential
energy
r particle separation
αp particle polarisability
Lennard–Jones
6-12 potential
(molecular
crystals)
φ(r)=−
A
r6
+
B
r12
(6.51)
=4
σ
r
12
−
σ
r
6
(6.52)
σ =(B/A)1/6
; =A2
/(4B)
φmin at r =
21/6
σ
(6.53)
ω angular frequency of
polarised orbital
A,B constants
,σ Lennard–Jones
parameters
De Boer
parameter
Λ=
h
σ(m )1/2
(6.54)
Λ de Boer parameter
h Planck constant
m particle mass
Coulomb
interaction
(ionic crystals)
UC =−αM
e2
4π 0r0
(6.55)
UC lattice Coulomb energy
per ion pair
αM Madelung constant
r0 nearest neighbour
separation
aLondon’s formula for fluctuating dipole interactions, neglecting the propagation time between particles.
Lattice thermal expansion and conduction
Grüneisen
parametera γ =−
∂lnω
∂lnV
(6.56)
γ Grüneisen parameter
ω normal mode frequency
V volume
Linear
expansivityb α=
1
3KT
∂p
∂T V
=
γCV
3KT V
(6.57)
α linear expansivity
KT isothermal bulk modulus
p pressure
T temperature
CV lattice heat capacity, constant V
Thermal
conductivity of
a phonon gas
λ=
1
3
CV
V
vsl (6.58)
vs effective sound speed
l phonon mean free path
Umklapp mean
free pathc lu ∝exp(θu/T) (6.59) lu umklapp mean free path
θu umklapp temperature (∼θD/2)
aStrictly, the Grüneisen parameter is the mean of γ over all normal modes, weighted by the mode’s contribution to
CV .
bOr “coefficient of thermal expansion,” for an isotropically expanding crystal.
cMean free path determined solely by “umklapp processes” – the scattering of phonons outside the first Brillouin
zone.

6.5 Electrons in solids
Free electron transport properties
Current density J =−nevd (6.60)
J current density
n free electron number density
vd mean electron drift velocity
Mean electron
drift velocity
vd =−
eτ
me
E (6.61)
τ mean time between collisions (relaxation
time)
me electronic mass
d.c. electrical
conductivity σ0 =
ne2
τ
me
(6.62)
E applied electric field
σ0 d.c. conductivity (J =σE)
a.c. electrical
conductivitya σ(ω)=
σ0
1−iωτ
(6.63) ω a.c. angular frequency
σ(ω) a.c. conductivity
Thermal
conductivity
λ=
1
3
CV
V
c2
τ (6.64)
=
π2
nk2
BτT
3me
(T TF)
(6.65)
CV total electron heat capacity, V constant
V volume
c2 mean square electron speed
T temperature
TF Fermi temperature
Wiedemann–
Franz lawb
λ
σT
=L=
π2
k2
B
3e2
(6.66)
L Lorenz constant ( 2.45×10−8 WΩ K−2)
Hall coefficientc
Jx
Bz
w
Ey
VH
+
RH =−
1
ne
=
Ey
JxBz
(6.67)
RH Hall coefficient
Ey Hall electric field
Jx applied current density
Bz magnetic flux density
Hall voltage
(rectangular
strip)
VH =RH
BzIx
w
(6.68)
VH Hall voltage
Ix applied current (=Jx × cross-sectional area)
w strip thickness in z
aFor an electric field varying as e−iωt.
bHolds for an arbitrary band structure.
cThe charge on an electron is −e, where e is the elementary charge (approximately +1.6×10−19 C). The Hall
coefficient is therefore a negative number when the dominant charge carriers are electrons.

6.5 Electrons in solids
6
133
Fermi gas
Electron density
of statesa
g(E)=
V
2π2
2me
¯h2
3/2
E1/2
(6.69)
g(EF)=
3
2
nV
EF
(6.70)
E electron energy (>0)
g(E) density of states
V “gas” volume
me electronic mass
Fermi
wavenumber
kF =(3π2
n)1/3
(6.71)
kF Fermi wavenumber
n number of electrons per unit
volume
Fermi velocity vF =¯hkF/me (6.72) vF Fermi velocity
Fermi energy
(T =0) EF =
¯h2
k2
F
2me
=
¯h2
2me
(3π2
n)2/3
(6.73) EF Fermi energy
Fermi
temperature
TF =
EF
kB
(6.74)
TF Fermi temperature
Electron heat
capacityb
(T TF)
CVe =
π2
3
g(EF)k2
BT (6.75)
=
π2
k2
B
2EF
T (6.76)
CVe heat capacity per electron
T temperature
Total kinetic
energy (T =0)
U0 =
3
5
nVEF (6.77) U0 total kinetic energy
Pauli
paramagnetism
M =χHPH (6.78)
=
3n
2EF
µ0µ2
BH (6.79)
χHP Pauli magnetic susceptibility
H magnetic field strength
M magnetisation
µB Bohr magneton
Landau
diamagnetism
χHL =−
1
3
χHP (6.80)
χHL Landau magnetic
susceptibility
aThe density of states is often quoted per unit volume in real space (i.e., g(E)/V here).
bEquation (6.75) holds for any density of states.
Thermoelectricity
Thermopowera
E=
J
σ
+ST ∇T (6.81)
E electrochemical fieldb
J current density
σ electrical conductivity
Peltier effect H =ΠJ −λ∇T (6.82)
ST thermopower
T temperature
H heat flux per unit area
Kelvin relation Π=TST (6.83)
Π Peltier coefficient
aOr “absolute thermoelectric power.”
bThe electrochemical field is the gradient of (µ/e)−φ, where µ is the chemical potential, −e the electronic charge,
and φ the electrical potential.

Band theory and semiconductors
Bloch’s theorem Ψ(r+R)=exp(ik·R)Ψ(r) (6.84)
Ψ electron eigenstate
k Bloch wavevector
R lattice vector
r position vector
Electron
velocity
vb(k)=
1
¯h
∇kEb(k) (6.85)
vb electron velocity (for wavevector
k)
¯h (Planck constant)/2π
b band index
Eb(k) energy band
Effective mass
tensor mij =¯h2 ∂2
Eb(k)
∂ki∂kj
−1
(6.86)
mij effective mass tensor
ki components of k
Scalar effective
massa m∗
=¯h2 ∂2
Eb(k)
∂k2
−1
(6.87)
m∗ scalar effective mass
k =|k|
Mobility µ=
|vd|
|E|
=
eD
kBT
(6.88)
µ particle mobility
vd mean drift velocity
E applied electric field
D diffusion coefficient
T temperature
Net current
density
J =(neµe +nhµh)eE (6.89)
J current density
ne,h electron, hole, number densities
µe,h electron, hole, mobilities
Semiconductor
equation
nenh =
(kBT)3
2(π¯h2
)3
(m∗
em∗
h)3/2
e−Eg/(kBT)
(6.90)
Eg band gap
m∗
e,h electron, hole, effective masses
p-n junction
I =I0 exp
eV
kBT
−1 (6.91)
I0 =en2
i A
De
LeNa
+
Dh
LhNd
(6.92)
Le =(Deτe)1/2
(6.93)
Lh =(Dhτh)1/2
(6.94)
I current
I0 saturation current
V bias voltage (+ for forward)
ni intrinsic carrier concentration
A area of junction
De,h electron, hole, diffusion
coefficients
Le,h electron, hole, diffusion lengths
τe,h electron, hole, recombination
times
Na,d acceptor, donor, concentrations
aValid for regions of k-space in which Eb(k) can be taken as independent of the direction of k.

7
Chapter 7 Electromagnetism
7.1 Introduction
The electromagnetic force is central to nearly every physical process around us and is a major
component of classical physics. In fact, the development of electromagnetic theory in the
nineteenth century gave us much mathematical machinery that we now apply quite generally
in other ﬁelds, including potential theory, vector calculus, and the ideas of divergence and
curl.
It is therefore not surprising that this section deals with a large array of physical
quantities and their relationships. As usual, SI units are assumed throughout. In the past
electromagnetism has suﬀered from the use of a variety of systems of units, including the
cgs system in both its electrostatic (esu) and electromagnetic (emu) forms. The fog has now
all but cleared, but some specialised areas of research still cling to these historical measures.
Readers are advised to consult the section on unit conversion if they come across such exotica
in the literature.
Equations cast in the rationalised units of SI can be readily converted to the once common
Gaussian (unrationalised) units by using the following symbol transformations:
Equation conversion: SI to Gaussian units
0 →1/(4π) µ0 →4π/c2
B →B/c
χE →4πχE χH →4πχH H →cH/(4π)
A→A/c M →cM D →D/(4π)
The quantities ρ, J, E, φ, σ, P, r, and µr are all unchanged.

136 Electromagnetism
7.2 Static fields
Electrostatics
Electrostatic
potential
E =−∇φ (7.1)
E electric field
φ electrostatic
potential
Potential
differencea
φa −φb =
b
a
E · dl =−
a
b
E · dl
(7.2)
φa potential at a
φb potential at b
dl line element
Poisson’s Equation
(free space)
∇2
φ=−
ρ
0
(7.3)
ρ charge density
0 permittivity of
free space
Point charge at r
φ(r)=
q
4π 0|r−r |
(7.4)
E(r)=
q(r−r )
4π 0|r−r |3
(7.5)
q point charge
Field from a
charge distribution
(free space)
E(r)=
1
4π 0
volume
ρ(r )(r−r )
|r−r |3
dτ (7.6)
dτ volume element
r position vector
of dτ

-
dτ
r
r
aBetween points a and b along a path l.
Magnetostaticsa
Magnetic scalar
potential
B =−µ0∇φm (7.7)
φm magnetic scalar
potential
B magnetic flux
density
φm in terms of the
solid angle of a
generating current
loop
φm =
IΩ
4π
(7.8)
Ω loop solid angle
I current
Biot–Savart law (the
field from a line
current)
B(r)=
µ0I
4π
line
dl×××(r−r )
|r−r |3
(7.9)
dl line element in
the direction of
the current
r position vector of
dl -
r
r
W
dl
I
s

Ampère’s law
(differential form)
∇×××B =µ0J (7.10)
J current density
µ0 permeability of
free space
Ampère’s law (integral
form)
B · dl =µ0Itot (7.11)
Itot total current
through loop
aIn free space.

7.2 Static ﬁelds
7
137
Capacitancea
Of sphere, radius a C =4π 0 ra (7.12)
Of circular disk, radius a C =8 0 ra (7.13)
Of two spheres, radius a, in
contact
C =8π 0 raln2 (7.14)
Of circular solid cylinder,
radius a, length l
C [8+4.1(l/a)0.76
] 0 ra (7.15)
Of nearly spherical surface,
area S
C 3.139×10−11
rS1/2
(7.16)
Of cube, side a C 7.283×10−11
ra (7.17)
Between concentric spheres,
radii ab
C =4π 0 rab(b−a)−1
(7.18)
Between coaxial cylinders,
radii ab
C =
2π 0 r
ln(b/a)
per unit length (7.19)
Between parallel cylinders,
separation 2d, radii a
C =
π 0 r
arcosh(d/a)
π 0 r
ln(2d/a)
(d a) (7.21)
Between parallel, coaxial
circular disks, separation d,
radii a
C
0 rπa2
d
+ 0 ra[ln(16πa/d)−1] (7.22)
aFor conductors, in an embedding medium of relative permittivity r.
Inductancea
Of N-turn solenoid
(straight or toroidal),
length l, area A ( l2
)
L=µ0N2
A/l (7.23)
Of coaxial cylindrical
tubes, radii a, b (ab)
L=
µ0
2π
ln
b
a
Of parallel wires, radii a,
separation 2d
L
µ0
π
ln
2d
a
per unit length, (2d a) (7.25)
Of wire of radius a bent in
a loop of radius b a
L µ0b ln
8b
a
−2 (7.26)
aFor currents conﬁned to the surfaces of perfect conductors in free space.

Electric fieldsa
Uniformly charged sphere,
radius a, charge q E(r)=



q
4π 0a3
r (r a)
q
4π 0r3
r (r ≥a)
(7.27)
Uniformly charged disk,
radius a, charge q (on axis,
z)
E(z)=
q
2π 0a2
z
1
|z|
−
1
√
z2 +a2
(7.28)
Line charge, charge density
λ per unit length
E(r)=
λ
2π 0r2
r (7.29)
Electric dipole, moment p
(spherical polar
coordinates, θ angle
between p and r)
-

K +−
θ
p
r
Er =
pcosθ
2π 0r3
(7.30)
Eθ =
psinθ
4π 0r3
(7.31)
Charge sheet, surface
density σ
E =
σ
2 0
(7.32)
aFor r =1 in the surrounding medium.
Magnetic fieldsa
Uniform infinite solenoid,
current I, n turns per unit
length
B =
µ0nI inside (axial)
0 outside
(7.33)
Uniform cylinder of
current I, radius a B(r)=
µ0Ir/(2πa2
) r a
µ0I/(2πr) r ≥a
(7.34)
Magnetic dipole, moment
m (θ angle between m and
r)
Br =µ0
mcosθ
2πr3
(7.35)
Bθ =
µ0msinθ
4πr3
(7.36)

K
θ
r
- m
Circular current loop of N
turns, radius a, along axis, z B(z)=
µ0NI
2
a2
(a2 +z2)3/2
(7.37)
The axis, z, of a straight
solenoid, n turns per unit
length, current I
Baxis =
µ0nI
2
(cosα1 −cosα2) (7.38)
-+ Y
z
α1
α2
⊗
aFor µr =1 in the surrounding medium.
Image charges
Real charge, +q, at a distance: image point image charge
b from a conducting plane −b −q
b from a conducting sphere, radius a a2
/b −qa/b
b from a plane dielectric boundary:
seen from free space −b −q( r −1)/( r +1)
seen from the dielectric b +2q/( r +1)

7.3 Electromagnetic fields (general)
7
139
Field relationships
Conservation of
charge
∇·J =−
∂ρ
∂t
(7.39)
J current density
ρ charge density
t time
Magnetic vector
potential
B =∇×××A (7.40) A vector potential
Electric field from
potentials
E =−
∂A
∂t
−∇φ (7.41)
φ electrical
potential
Coulomb gauge
condition
∇·A=0 (7.42)
Lorenz gauge
condition
∇·A+
1
c2
∂φ
∂t
=0 (7.43) c speed of light
Potential field
equationsa
1
c2
∂2
φ
∂t2
−∇2
φ=
ρ
0
(7.44)
1
c2
∂2
A
∂t2
−∇2
A=µ0J (7.45)

-
dτ
r
r
Expression for φ
in terms of ρa
φ(r,t)=
1
4π 0
volume
ρ(r ,t−|r−r |/c)
|r−r |
dτ (7.46)
dτ volume element
r position vector of
dτ
Expression for A
in terms of Ja
A(r,t)=
µ0
4π
volume
J(r ,t−|r−r |/c)
|r−r |
dτ (7.47) µ0 permeability of
free space
aAssumes the Lorenz gauge.
Liénard–Wiechert potentialsa
Electrical potential of
a moving point charge
φ=
q
4π 0(|r|−v·r/c)
(7.48)
q charge
r vector from charge
to point of
observation
v particle velocity
Magnetic vector
potential of a moving
point charge
A=
µ0qv
4π(|r|−v·r/c)
(7.49)
:
j
q
v
r
aIn free space. The right-hand sides of these equations are evaluated at retarded times, i.e., at t =t−|r |/c, where r
is the vector from the charge to the observation point at time t .

Maxwell’s equations
Differential form: Integral form:
∇·E =
ρ
0
(7.50)
closed surface
E · ds=
1
0
volume
ρ dτ (7.51)
∇·B =0 (7.52)
closed surface
B · ds=0 (7.53)
∇×××E =−
∂B
∂t
(7.54)
loop
E · dl =−
dΦ
dt
(7.55)
∇×××B =µ0J +µ0 0
∂E
∂t
(7.56)
loop
B · dl =µ0I +µ0 0
surface
∂E
∂t
· ds (7.57)
Equation (7.51) is “Gauss’s law”
Equation (7.55) is “Faraday’s law”
E electric field
J current density
ρ charge density
ds surface element
dτ volume element
dl line element
Φ linked magnetic flux (= B · ds)
I linked current (= J · ds)
t time
Maxwell’s equations (using D and H)
Differential form: Integral form:
∇·D =ρfree (7.58)
closed surface
D· ds=
volume
ρfree dτ (7.59)
∇·B =0 (7.60)
closed surface
B · ds=0 (7.61)
∇×××E =−
∂B
∂t
(7.62)
loop
E · dl =−
dΦ
dt
(7.63)
∇×××H =Jfree +
∂D
∂t
(7.64)
loop
H · dl =Ifree +
surface
∂D
∂t
· ds (7.65)
D displacement field
ρfree free charge density (in the sense of
ρ=ρinduced +ρfree)
Jfree free current density (in the sense of
J =Jinduced +Jfree)
E electric field
ds surface element
dτ volume element
dl line element
Φ linked magnetic flux (= B · ds)
Ifree linked free current (= Jfree · ds)
t time

7
141
Relativistic electrodynamics
Lorentz
transformation of
electric and
magnetic fields
E =E (7.66)
E⊥ =γ(E +v×××B)⊥ (7.67)
B =B (7.68)
B⊥ =γ(B −v×××E/c2
)⊥ (7.69)
E electric field
measured in frame moving
at relative velocity v
γ Lorentz factor
=[1−(v/c)2]−1/2
parallel to v
⊥ perpendicular to v
Lorentz
transformation of
current and charge
densities
ρ =γ(ρ−vJ /c2
) (7.70)
J⊥ =J⊥ (7.71)
J =γ(J −vρ) (7.72)
J current density
ρ charge density
Lorentz
transformation of
potential fields
φ =γ(φ−vA ) (7.73)
A⊥ =A⊥ (7.74)
A =γ(A −vφ/c2
) (7.75)
Four-vector fieldsa
J
∼
=(ρc,J) (7.76)
A
∼
=
φ
c
,A (7.77)
2
=
1
c2
∂2
∂t2
,−∇2
(7.78)
2
A
∼
=µ0J
∼
(7.79)
J
∼
current density four-vector
A
∼
potential four-vector
2 D’Alembertian operator
aOther sign conventions are common here. See page 65 for a general definition of four-vectors.

7.4 Fields associated with media
Polarisation
Definition of electric
dipole moment
p =qa (7.80)
±q end charges
a charge separation
vector (from − to +)
-
− +p
Generalised electric
dipole moment
p =
volume
r ρ dτ (7.81)
p dipole moment
ρ charge density
dτ volume element
r vector to dτ
Electric dipole
potential
φ(r)=
p ·r
4π 0r3
(7.82)
φ dipole potential
r vector from dipole
0 permittivity of free
space
Dipole moment per
unit volume
(polarisation)a
P =np (7.83)
P polarisation
n number of dipoles per
unit volume
Induced volume
charge density
∇·P =−ρind (7.84) ρind volume charge density
Induced surface
charge density
σind =P ·ˆs (7.85)
σind surface charge density
ˆs unit normal to surface
displacement
D = 0E +P (7.86) D electric displacement
E electric field
susceptibility
P = 0χEE (7.87) χE electrical susceptibility
(may be a tensor)
Definition of relative
permittivityb
r =1+χE (7.88)
D = 0 rE (7.89)
= E (7.90)
r relative permittivity
permittivity
Atomic
polarisabilityc p =αEloc (7.91)
α polarisability
Eloc local electric field
Depolarising fields Ed =−
NdP
0
(7.92)
Ed depolarising field
Nd depolarising factor
=1/3 (sphere)
=1 (thin slab ⊥ to P)
=0 (thin slab to P)
=1/2 (long circular
cylinder, axis ⊥ to P)
Clausius–Mossotti
equationd
nα
3 0
=
r −1
r +2
(7.93)
aAssuming dipoles are parallel. The equivalent of Equation (7.112) holds for a hot gas of electric dipoles.
bRelative permittivity as defined here is for a linear isotropic medium.
cThe polarisability of a conducting sphere radius a is α=4π 0a3. The definition p =α 0Eloc is also used.
dWith the substitution η2 = r [cf. Equation (7.195) with µr =1] this is also known as the “Lorentz–Lorenz formula.”

7.4 Fields associated with media
7
143
Magnetisation
Definition of
magnetic dipole
moment
dm=I ds (7.94)
dm dipole moment
I loop current
ds loop area (right-hand
sense with respect to
loop current)
6
⊗
dm, ds
out in
Generalised
magnetic dipole
moment
m=
1
2
volume
r ×××J dτ (7.95)
m dipole moment
J current density
dτ volume element
r vector to dτ
Magnetic dipole
(scalar) potential
φm(r)=
µ0m·r
4πr3
(7.96)
φm magnetic scalar
potential
µ0 permeability of free
space
Dipole moment per
unit volume
(magnetisation)a
M =nm (7.97)
M magnetisation
n number of dipoles
per unit volume
Induced volume
current density
Jind =∇×××M (7.98) Jind volume current
density (i.e., A m−2)
Induced surface
current density
jind =M×××ˆs (7.99)
jind surface current
density (i.e., A m−1)
ˆs unit normal to
surface
Definition of
magnetic field
strength, H
B =µ0(H +M) (7.100)
H magnetic field
strength
Definition of
magnetic
susceptibility
M =χH H (7.101)
=
χBB
µ0
(7.102)
χB =
χH
1+χH
(7.103)
χH magnetic
susceptibility. χB is
also used (both may
be tensors)
Definition of relative
permeabilityb
B =µ0µrH (7.104)
=µH (7.105)
µr =1+χH (7.106)
=
1
1−χB
(7.107)
µr relative permeability
µ permeability
aAssuming all the dipoles are parallel. See Equation (7.112) for a classical paramagnetic gas and
page 101 for the quantum generalisation.
bRelative permeability as defined here is for a linear isotropic medium.

Paramagnetism and diamagnetism
Diamagnetic
moment of an atom m=−
e2
6me
Z r2
B (7.108)
m magnetic moment
r2 mean squared orbital radius
(of all electrons)
Z atomic number
me electron mass
Intrinsic electron
magnetic momenta m −
e
2me
gJ (7.109)
g Landé g-factor (=2 for spin,
=1 for orbital momentum)
Langevin function
L(x)=cothx−
1
x
(7.110)
x/3 (x
∼ 1) (7.111)
L(x) Langevin function
Classical gas
paramagnetism
(|J| ¯h)
M =nm0L
m0B
kT
(7.112)
M apparent magnetisation
m0 magnitude of magnetic dipole
moment
n dipole number density
Curie’s law χH =
µ0nm2
0
3kT
(7.113)
T temperature
χH magnetic susceptibility
Curie–Weiss law χH =
µ0nm2
0
3k(T −Tc)
(7.114)
Tc Curie temperature
aSee also page 100.
Boundary conditions for E, D, B, and Ha
Parallel
component of the
electric field
E continuous (7.115) component parallel to
interface
Perpendicular
component of the
magnetic flux
density
B⊥ continuous (7.116)
⊥ component
perpendicular to
interface
Electric
displacementb ˆs·(D2 −D1)=σ (7.117)
D1,2 electrical displacements
in media 1 2
ˆs unit normal to surface,
directed 1→2
σ surface density of free
charge
62
1
ˆs
Magnetic field
strengthc ˆs×××(H2 −H1)=js (7.118)
H1,2 magnetic field strengths
in media 1 2
js surface current per unit
width
aAt the plane surface between two uniform media.
bIf σ =0, then D⊥ is continuous.
cIf js =0 then H is continuous.

7.5 Force, torque, and energy
7
145
7.5 Force, torque, and energy
Electromagnetic force and torque
Force between two
static charges:
Coulomb’s law
F 2 =
q1q2
4π 0r2
12
ˆr12 (7.119)
F 2 force on q2
q1,2 charges
r12 vector from 1 to 2
ˆ unit vector
0 permittivity of free
space
-
-
q1 q2r12
F 2
Force between two
current-carrying
elements
dF 2 =
µ0I1I2
4πr2
12
[dl2×××(dl1×××ˆr12)]
(7.120)
dl1,2 line elements
I1,2 currents flowing along
dl1 and dl2
dF 2 force on dl2
µ0 permeability of free
space
*
j
W
dl1
r12
dl2
Force on a
current-carrying
element in a
magnetic field
dF =I dl×××B (7.121)
dl line element
F force
I current flowing along dl
Force on a charge
(Lorentz force)
F =q(E +v×××B) (7.122) E electric field
v charge velocity
Force on an electric
dipolea F =(p ·∇)E (7.123) p electric dipole moment
Force on a magnetic
dipoleb F =(m·∇)B (7.124) m magnetic dipole moment
Torque on an
electric dipole
G =p×××E (7.125) G torque
Torque on a
magnetic dipole
G =m×××B (7.126)
Torque on a
current loop
G =IL
loop
r×××(dlL×××B) (7.127)
dlL line-element (of loop)
r position vector of dlL
IL current around loop
aF simplifies to ∇(p ·E) if p is intrinsic, ∇(pE/2) if p is induced by E and the medium is isotropic.
bF simplifies to ∇(m·B) if m is intrinsic, ∇(mB/2) if m is induced by B and the medium is isotropic.

Electromagnetic energy
Electromagnetic field
energy density (in free
space)
u=
1
2
0E2
+
1
2
B2
µ0
(7.128)
u energy density
E electric field
Energy density in
media
u=
1
2
(D·E +B ·H) (7.129)
D electric displacement
Energy flow (Poynting)
vector
N =E×××H (7.130)
c speed of light
N energy flow rate per unit
area ⊥ to the flow direction
Mean flux density at a
distance r from a short
oscillating dipole
N =
ω4
p2
0 sin2
θ
32π2
0c3r3
r (7.131)
p0 amplitude of dipole moment
( wavelength)
θ angle between p and r
ω oscillation frequency
Total mean power
from oscillating
dipolea
W =
ω4
p2
0/2
6π 0c3
(7.132) W total mean radiated power
Self-energy of a
charge distribution
Utot =
1
2
volume
φ(r)ρ(r) dτ (7.133)
Utot total energy
dτ volume element
r position vector of dτ
φ electrical potential
ρ charge density
Energy of an assembly
of capacitorsb
Utot =
1
2 i j
CijViVj (7.134)
Vi potential of ith capacitor
Cij mutual capacitance between
capacitors i and j
Energy of an assembly
of inductorsc
Utot =
1
2 i j
LijIiIj (7.135) Lij mutual inductance between
inductors i and j
Intrinsic dipole in an
electric field
Udip =−p ·E (7.136)
Udip energy of dipole
p electric dipole moment
Intrinsic dipole in a
magnetic field
Udip =−m·B (7.137) m magnetic dipole moment
Hamiltonian of a
charged particle in an
EM fieldd
H =
|pm −qA|2
2m
+qφ (7.138)
H Hamiltonian
pm particle momentum
q particle charge
m particle mass
aSometimes called “Larmor’s formula.”
bCii is the self-capacitance of the ith body. Note that Cij =Cji.
cLii is the self-inductance of the ith body. Note that Lij =Lji.
dNewtonian limit, i.e., velocity c.

7.6 LCR circuits
7
147
7.6 LCR circuits
LCR definitions
Current I =
dQ
dt
(7.139)
I current
Q charge
Ohm’s law V =IR (7.140)
R resistance
V potential difference
over R
I current through R
Ohm’s law (field
form)
J =σE (7.141)
J current density
E electric field
σ conductivity
Resistivity ρ=
1
σ
=
RA
l
(7.142)
ρ resistivity
A area of face (I is
normal to face)
l length
7
/
A
l
Capacitance C =
Q
V
(7.143)
C capacitance
across C
Current through
capacitor
I =C
dV
dt
(7.144)
I current through C
t time
Self-inductance L=
Φ
I
(7.145)
Φ total linked flux
I current through
inductor
Voltage across
inductor
V =−L
dI
dt
(7.146)
over L
Mutual
inductance
L12 =
Φ1
I2
=L21 (7.147)
Φ1 total flux from loop 2
linked by loop 1
L12 mutual inductance
I2 current through loop 2
Coefficient of
coupling
|L12|=k L1L2 (7.148) k coupling coefficient
between L1 and L2
(≤1)
Linked magnetic
flux through a coil
Φ=Nφ (7.149)
Φ linked flux
N number of turns
around φ
φ flux through area of
turns

Resonant LCR circuits
Phase
resonant
frequencya
ω2
0 =
1/LC (series)
1/LC −R2
/L2
(parallel)
(7.150)
ω0 resonant
angular
frequency
L inductance
C capacitance
R resistance
R L C
series
parallel
Tuningb δω
ω0
=
1
Q
=
R
ω0L
(7.151)
δω half-power
bandwidth
Q quality
factor
Quality
factor
Q=2π
stored energy
energy lost per cycle
(7.152)
aAt which the impedance is purely real.
bAssuming the capacitor is purely reactive. If L and R are parallel, then 1/Q=ω0L/R.
Energy in capacitors, inductors, and resistors
Energy stored in a
capacitor U =
1
2
CV2
=
1
2
QV =
1
2
Q2
C
(7.153)
U stored energy
C capacitance
Q charge
Energy stored in an
inductor U =
1
2
LI2
=
1
2
ΦI =
1
2
Φ2
L
(7.154)
L inductance
Φ linked magnetic ﬂux
I current
Power dissipated in
a resistora
(Joule’s
law)
W =IV =I2
R =
V2
R
(7.155)
W power dissipated
R resistance
Relaxation time τ=
0 r
σ
(7.156)
τ relaxation time
r relative permittivity
σ conductivity
aThis is d.c., or instantaneous a.c., power.
Electrical impedance
Impedances in series Ztot =
n
Zn (7.157)
Impedances in parallel Ztot =
n
Z−1
n
−1
(7.158)
Impedance of capacitance ZC =−
i
ωC
(7.159)
Impedance of inductance ZL =iωL (7.160)
Impedance: Z Capacitance: C
Inductance: L Resistance: R =Re[Z]
Conductance: G=1/R Reactance: X =Im[Z]
Admittance: Y =1/Z Susceptance: S =1/X

7.6 LCR circuits
7
149
Kirchhoﬀ’s laws
Current law
node
Ii =0 (7.161) Ii currents impinging
on node
Voltage law
loop
Vi =0 (7.162) Vi potential diﬀerences
around loop
Transformersa
Z1
Z2

-
y
9
V2
:
z
V1
N1 N2
I2
I1
n turns ratio
N1 number of primary turns
N2 number of secondary turns
V1 primary voltage
V2 secondary voltage
I1 primary current
I2 secondary current
Zout output impedance
Zin input impedance
Z1 source impedance
Z2 load impedance
Turns ratio n=N2/N1 (7.163)
Transformation of voltage and current
V2 =nV1 (7.164)
I2 =I1/n (7.165)
Output impedance (seen by Z2) Zout =n2
Z1 (7.166)
Input impedance (seen by Z1) Zin =Z2/n2
(7.167)
aIdeal, with a coupling constant of 1 between loss-free windings.
Star–delta transformation
‘Star’ ‘Delta’1
1
22 33
Z1
Z2 Z3
Z12
Z23
Z13
i,j,k node indices (1,2, or 3)
Zi impedance on node i
Zij impedance connecting
nodes i and j
Star
impedances
Zi =
ZijZik
Zij +Zik +Zjk
(7.168)
Delta
impedances
Zij =ZiZj
1
Zi
+
1
Zj
+
1
Zk
(7.169)

7.7 Transmission lines and waveguides
Transmission line relations
Loss-free
transmission line
equations
∂V
∂x
=−L
∂I
∂t
(7.170)
∂I
∂x
=−C
∂V
∂t
(7.171)
V potential difference across
line
I current in line
L inductance per unit length
C capacitance per unit length
Wave equation for a
lossless transmission
line
1
LC
∂2
V
∂x2
=
∂2
V
∂t2
(7.172)
1
LC
∂2
I
∂x2
=
∂2
I
∂t2
(7.173)
x distance along line
t time
Characteristic
impedance of
lossless line
Zc =
L
C
(7.174) Zc characteristic impedance
Characteristic
impedance of lossy
line
Zc =
R +iωL
G+iωC
(7.175)
R resistance per unit length
of conductor
G conductance per unit
length of insulator
Wave speed along a
lossless line
vp =vg =
1
√
LC
(7.176)
vp phase speed
vg group speed
Input impedance of
a terminated lossless
line
Zin =Zc
Zt coskl −iZc sinkl
Zc coskl −iZt sinkl
(7.177)
=Z2
c /Zt if l =λ/4 (7.178)
Zin (complex) input impedance
Zt (complex) terminating
impedance
Reflection coefficient
from a terminated
line
r =
Zt −Zc
Zt +Zc
(7.179)
l distance from termination
r (complex) voltage
reflection coefficient
Line voltage
standing wave ratio
vswr=
1+|r|
1−|r|
(7.180)
Transmission line impedancesa
Coaxial line Zc =
µ
4π2
ln
b
a
60
√
r
ln
b
a
(7.181)
Zc characteristic impedance (Ω)
a radius of inner conductor
b radius of outer conductor
permittivity (= 0 r)
Open wire feeder Zc =
µ
π2
ln
l
r
120
√
r
ln
l
r
(7.182)
µ permeability (=µ0µr)
r radius of wires
l distance between wires ( r)
Paired strip Zc =
µ d
w
377
√
r
d
w
(7.183)
d strip separation
w strip width ( d)
Microstrip line Zc
377
√
r[(w/h)+2]
(7.184)
h height above earth plane
( w)
aFor lossless lines.

7.7 Transmission lines and waveguides
7
151
Waveguidesa
Waveguide
equation k2
g =
ω2
c2
−
m2
π2
a2
−
n2
π2
b2
(7.185)
kg wavenumber in guide
a guide height
b guide width
m,n mode indices with respect to
a and b (integers)
c speed of light
Guide cutoff
frequency νc =c
m
2a
2
+
n
2b
2
(7.186)
νc cutoff frequency
ωc 2πνc
Phase velocity
above cutoff
vp =
c
1−(νc/ν)2
(7.187) vp phase velocity
ν frequency
Group velocity
above cutoff
vg =c2
/vp =c 1−(νc/ν)2 (7.188) vg group velocity
Wave
impedancesb
ZTM =Z0 1−(νc/ν)2 (7.189)
ZTE =Z0/ 1−(νc/ν)2 (7.190)
ZTM wave impedance for
transverse magnetic modes
ZTE wave impedance for
transverse electric modes
Z0 impedance of free space
(= µ0/ 0)
Field solutions for TEmn modesc
Bx =
ikgc2
ω2
c
∂Bz
∂x
By =
ikgc2
ω2
c
∂Bz
∂y
Bz =B0 cos
mπx
a
cos
nπy
b
Ex =
iωc2
ω2
c
∂Bz
∂y
Ey =
−iωc2
ω2
c
∂Bz
∂x
Ez =0
(7.191)
a
b
x
y
zField solutions for TMmn modesc
Ex =
ikgc2
ω2
c
∂Ez
∂x
Ey =
ikgc2
ω2
c
∂Ez
∂y
Ez =E0 sin
mπx
a
sin
nπy
b
Bx =
−iω
ω2
c
∂Ez
∂y
By =
iω
ω2
c
∂Ez
∂x
Bz =0
(7.192)
aEquations are for lossless waveguides with rectangular cross sections and no dielectric.
bThe ratio of the electric field to the magnetic field strength in the xy plane.
cBoth TE and TM modes propagate in the z direction with a further factor of exp[i(kgz −ωt)] on all components.
B0 and E0 are the amplitudes of the z components of magnetic flux density and electric field respectively.

7.8 Waves in and out of media
Waves in lossless media
Electric field ∇2
E =µ
∂2
E
∂t2
(7.193)
E electric field
µ permeability (=µ0µr)
permittivity (= 0 r)
Magnetic field ∇2
B =µ
∂2
B
∂t2
(7.194)
t time
Refractive index η =
√
rµr (7.195)
Wave speed v =
1
√
µ
=
c
η
(7.196)
v wave phase speed
η refractive index
c speed of light
Impedance of free space Z0 =
µ0
0
376.7Ω (7.197)
Z0 impedance of free
space
Wave impedance Z =
E
H
=Z0
µr
r
(7.198)
Z wave impedance
Radiation pressurea
Radiation
momentum
density
G =
N
c2
(7.199)
G momentum density
N Poynting vector
c speed of light
Isotropic
radiation
pn =
1
3
u(1+R) (7.200)
pn normal pressure
u incident radiation
energy density
R (power) reflectance
coefficient
Specular
reflection
u
θi
x
y
z
θ
φ
dΩ
(normal)
pn =u(1+R)cos2
θi (7.201)
pt =u(1−R)sinθi cosθi (7.202)
pt tangential pressure
From an
extended
sourceb
pn =
1+R
c
Iν(θ,φ)cos2
θ dΩ dν
(7.203)
Iν specific intensity
ν frequency
Ω solid angle
θ angle between dΩ
and normal to plane
From a point
source,c
luminosity L
pn =
L(1+R)
4πr2c
(7.204)
L source luminosity
(i.e., radiant power)
r distance from source
aOn an opaque surface.
bIn spherical polar coordinates. See page 120 for the meaning of specific intensity.
cNormal to the plane.

7
153
Antennas
Spherical polar geometry:
-
/
6

U
*
z
y
x
r
θ
φ
6p
Field from a short
(l λ) electric
dipole in free
spacea
Er =
1
2π 0
[˙p]
r2c
+
[p]
r3
cosθ (7.205)
Eθ =
1
4π 0
[¨p]
rc2
+
[˙p]
r2c
+
[p]
r3
sinθ (7.206)
Bφ =
µ0
4π
[¨p]
rc
+
[˙p]
r2
sinθ (7.207)
r distance from
dipole
θ angle between r and
p
[p] retarded dipole
moment
[p]=p(t−r/c)
c speed of light
Radiation
resistance of a
short electric
dipole in free
space
R =
ω2
l2
6π 0c3
=
2πZ0
3
l
λ
2
(7.208)
789
l
λ
2
ohm (7.209)
l dipole length ( λ)
λ wavelength
Z0 impedance of free
space
Beam solid angle ΩA =
4π
Pn(θ,φ) dΩ (7.210)
ΩA beam solid angle
Pn normalised antenna
power pattern
Pn(0,0)=1
dΩ differential solid
angle
Forward power
gain
G(0)=
4π
ΩA
(7.211) G antenna gain
Antenna effective
area Ae =
λ2
ΩA
(7.212) Ae effective area
Power gain of a
short dipole
G(θ)=
3
2
sin2
θ (7.213)
Beam efficiency efficiency=
ΩM
ΩA
(7.214)
ΩM main lobe solid
angle
Antenna
temperatureb TA =
1
ΩA 4π
Tb(θ,φ)Pn(θ,φ) dΩ (7.215)
TA antenna
temperature
Tb sky brightness
temperature
aAll field components propagate with a further phase factor equal to expi(kr−ωt), where k =2π/λ.
bThe brightness temperature of a source of specific intensity Iν is Tb =λ2Iν/(2kB).

Reflection, refraction, and transmissiona
w/

θi θr
θt
w/

θi θr
θt
parallel incidence perpendicular incidence

=
U
K
*
Ei Er
BrBi
Bt
Et
Ei Er
BrBi
Et
Bt
ηi
ηt
E electric field
ηi refractive index on
incident side
ηt refractive index on
transmitted side
θr angle of reflection
θt angle of refraction
Law of reflection θi =θr (7.216)
Snell’s lawb ηi sinθi =ηt sinθt (7.217)
Brewster’s law tanθB =ηt/ηi (7.218)
θB Brewster’s angle of
incidence for
plane-polarised
reflection (r =0)
Fresnel equations of reflection and refraction
r =
sin2θi −sin2θt
sin2θi +sin2θt
(7.219)
t =
4cosθi sinθt
sin2θi +sin2θt
(7.220)
R =r2
(7.221)
T =
ηt cosθt
ηi cosθi
t2
(7.222)
r⊥ =−
sin(θi −θt)
sin(θi +θt)
(7.223)
t⊥ =
2cosθi sinθt
sin(θi +θt)
(7.224)
R⊥ =r2
⊥ (7.225)
T⊥ =
ηt cosθt
ηi cosθi
t2
⊥ (7.226)
Coefficients for normal incidencec
R =
(ηi −ηt)2
(ηi +ηt)2
(7.227)
T =
4ηiηt
(ηi +ηt)2
(7.228)
R +T =1 (7.229)
r =
ηi −ηt
ηi +ηt
(7.230)
t=
2ηi
ηi +ηt
(7.231)
t−r =1 (7.232)
electric field parallel to the plane of
incidence
R (power) reflectance coefficient
T (power) transmittance coefficient
⊥ electric field perpendicular to the
plane of incidence
r amplitude reflection coefficient
t amplitude transmission coefficient
aFor the plane boundary between lossless dielectric media. All coefficients refer to the electric field component and
whether it is parallel or perpendicular to the plane of incidence. Perpendicular components are out of the paper.
bThe incident wave suffers total internal reflection if ηi
ηt
sinθi 1.
cI.e., θi =0. Use the diagram labelled “perpendicular incidence” for correct phases.

7
155
Propagation in conducting mediaa
Electrical
conductivity
(B =0)
σ =neeµ=
nee2
me
τc (7.233)
σ electrical conductivity
τc electron relaxation time
µ electron mobility
Refractive index
of an ohmic
conductorb
η =(1+i)
σ
4πν 0
1/2
(7.234)
me electron mass
η refractive index
Skin depth in an
ohmic conductor
δ =(µ0σπν)−1/2
(7.235)
ν frequency
δ skin depth
aAssuming a relative permeability, µr, of 1.
bTaking the wave to have an e−iωt time dependence, and the low-frequency limit (σ 2πν 0).
Electron scattering processesa
Rayleigh
scattering
cross sectionb
σR =
ω4
α2
6π 0c4
(7.236)
σR Rayleigh cross section
ω radiation angular frequency
α particle polarisability
Thomson
scattering
cross sectionc
σT =
8π
3
e2
4π 0mec2
2
(7.237)
=
8π
3
r2
e 6.652×10−29
m2
(7.238)
σT Thomson cross section
me electron (rest) mass
re classical electron radius
c speed of light
Inverse
Compton
scatteringd
Ptot =
4
3
σTcuradγ2 v2
c2
(7.239)
Ptot electron energy loss rate
urad radiation energy density
γ Lorentz factor =[1−(v/c)2]−1/2
v electron speed
Compton
scatteringe
meλ
λ
θ
φ
λ −λ=
h
mec
(1−cosθ) (7.240)
hν =
mec2
1−cosθ+(1/ε)
(7.241)
cotφ=(1+ε)tan
θ
2
(7.242)
λ,λ incident scattered wavelengths
ν,ν incident scattered frequencies
θ photon scattering angle
h
mec
electron Compton wavelength
ε =hν/(mec2)
σKN Klein–Nishina cross section
Klein–Nishina
cross section
(for a free
electron)
σKN =
πr2
e
ε
1−
2(ε+1)
ε2
ln(2ε+1)+
1
2
+
4
ε
−
1
2(2ε+1)2
(7.243)
σT (ε 1) (7.244)
πr2
e
ε
ln2ε+
1
2
(ε 1) (7.245)
aFor Rutherford scattering see page 72.
bScattering by bound electrons.
cScattering from free electrons, ε 1.
dElectron energy loss rate due to photon scattering in the Thomson limit (γhν mec2).
eFrom an electron at rest.

Cherenkov radiation
Cherenkov
cone angle
sinθ =
c
ηv
(7.246)
θ cone semi-angle
c (vacuum) speed of light
η(ω) refractive index
v particle velocity
Radiated
powera
Ptot =
e2
µ0
4π
v
ωc
0
1−
c2
v2η2(ω)
ω dω (7.247)
where η(ω)≥
c
v
for 0ω ωc
Ptot total radiated power
µ0 free space permeability
ωc cutoff frequency
aFrom a point charge, e, travelling at speed v through a medium of refractive index η(ω).
7.9 Plasma physics
Warm plasmas
Landau
length
lL =
e2
4π 0kBTe
(7.248)
1.67×10−5
T−1
e m (7.249)
lL Landau length
Te electron temperature (K)
Electron
Debye length
λDe =
0kBTe
nee2
1/2
(7.250)
69(Te/ne)1/2
m (7.251)
λDe electron Debye length
(m−3)
Debye
screeninga φ(r)=
qexp(−21/2
r/λDe)
4π 0r
(7.252)
φ effective potential
q point charge
r distance from q
Debye
number
NDe =
4
3
πneλ3
De (7.253) NDe electron Debye number
Relaxation
times (B =0)b
τe =3.44×105 T
3/2
e
ne lnΛ
s (7.254)
τi =2.09×107 T
3/2
i
ne lnΛ
mi
mp
1/2
s (7.255)
τe electron relaxation time
τi ion relaxation time
Ti ion temperature (K)
lnΛ Coulomb logarithm
(typically 10 to 20)
Characteristic
electron
thermal
speedc
vte =
2kBTe
me
1/2
(7.256)
5.51×103
T
1/2
e ms−1
(7.257)
vte electron thermal speed
me electron mass
aEffective (Yukawa) potential from a point charge q immersed in a plasma.
bCollision times for electrons and singly ionised ions with Maxwellian speed distributions, Ti

∼ Te. The Spitzer
conductivity can be calculated from Equation (7.233).
cDefined so that the Maxwellian velocity distribution ∝ exp(−v2/v2
te). There are other definitions (see Maxwell–
Boltzmann distribution on page 112).

7.9 Plasma physics
7
157
Electromagnetic propagation in cold plasmasa
Plasma frequency
(2πνp)2
=
nee2
0me
=ω2
p (7.258)
νp 8.98n
1/2
e Hz (7.259)
νp plasma frequency
ωp plasma angular frequency
ne electron number density (m−3)
me electron mass
Plasma refractive
index (B =0)
η = 1−(νp/ν)2 1/2
(7.260)
η refractive index
ν frequency
Plasma dispersion
relation (B =0)
c2
k2
=ω2
−ω2
p (7.261)
ω angular frequency (=2π/ν)
c speed of light
Plasma phase
velocity (B =0)
vφ =c/η (7.262) vφ phase velocity
Plasma group
velocity (B =0)
vg =cη (7.263)
vφvg =c2
(7.264)
vg group velocity
Cyclotron
(Larmor, or gyro-)
frequency
2πνC =
qB
m
=ωC (7.265)
νCe 28×109
B Hz (7.266)
νCp 15.2×106
B Hz (7.267)
νC cyclotron frequency
ωC cyclotron angular frequency
νCe electron νC
νCp proton νC
q particle charge
B magnetic ﬂux density (T)
Larmor
(cyclotron, or
gyro-) radius
rL =
v⊥
ωC
=v⊥
m
qB
(7.268)
rLe =5.69×10−12 v⊥
B
m (7.269)
rLp =10.4×10−9 v⊥
B
m (7.270)
m particle mass (γm if relativistic)
rL Larmor radius
rLe electron rL
rLp proton rL
v⊥ speed ⊥ to B (ms−1)
Mixed propagation modesb
θB angle between wavefront
normal (ˆk) and B
η2
=1−
X(1−X)
(1−X)− 1
2 Y 2 sin2
θB ±S
, (7.271)
where X =(ωp/ω)2
, Y =ωCe/ω,
and S2
=
1
4
Y 4
sin4
θB +Y 2
(1−X)2
cos2
θB
Faraday rotationc
∆ψ =
µ0e3
8π2m2
ec
2.63×10−13
λ2
line
neB · dl (7.272)
=Rλ2
(7.273)
∆ψ rotation angle
λ wavelength (=2π/k)
dl line element in direction of
wave propagation
R rotation measure
aI.e., plasmas in which electromagnetic force terms dominate over thermal pressure terms. Also taking µr =1.
bIn a collisionless electron plasma. The ordinary and extraordinary modes are the + and − roots of S2 when
θB = π/2. When θB = 0, these roots are the right and left circularly polarised modes respectively, using the optical
convention for handedness.
cIn a tenuous plasma, SI units throughout. ∆ψ is taken positive if B is directed towards the observer.

Magnetohydrodynamicsa
Sound speed
vs =
γp
ρ
1/2
=
2γkBT
mp
1/2
(7.274)
166T1/2
ms−1
(7.275)
vs sound (wave) speed
p hydrostatic pressure
ρ plasma mass density
T temperature (K)
Alfvén speed
vA =
B
(µ0ρ)1/2
(7.276)
2.18×1016
Bn
−1/2
e ms−1
(7.277)
mp proton mass
vA Alfvén speed
B magnetic flux density (T)
(m−3)
Plasma beta β =
2µ0p
B2
=
4µ0nekBT
B2
=
2v2
s
γv2
A
(7.278)
β plasma beta (ratio of
hydrostatic to magnetic
pressure)
Direct electrical
conductivity σd =
n2
ee2
σ
n2
ee2 +σ2B2
(7.279)
σd direct conductivity
σ conductivity (B =0)
Hall electrical
conductivity
σH =
σB
nee
σd (7.280) σH Hall conductivity
Generalised
Ohm’s law
J =σd(E +v×××B)+σH
ˆB×××(E +v×××B) (7.281)
J current density
E electric field
v plasma velocity field
ˆB =B/|B|
Resistive MHD equations (single-fluid model)b
∂B
∂t
=∇×××(v×××B)+η∇2
B (7.282)
∂v
∂t
+(v·∇)v =−
∇p
ρ
+
1
µ0ρ
(∇×××B)×××B +ν∇2
v
+
1
3
ν∇(∇·v)+g (7.283)
η magnetic diffusivity
[=1/(µ0σ)]
ν kinematic viscosity
g gravitational field strength
Shear Alfvénic
dispersion
relationc
ω =kvA cosθB (7.284)
k wavevector (k =2π/λ)
θB angle between k and B
Magnetosonic
dispersion
relationd
ω2
k2
(v2
s +v2
A)−ω4
=v2
s v2
Ak4
cos2
θB (7.285)
aFor a warm, fully ionised, electrically neutral p+/e− plasma, µr = 1. Relativistic and displacement current effects
are assumed to be negligible and all oscillations are taken as being well below all resonance frequencies.
bNeglecting bulk (second) viscosity.
cNonresistive, inviscid flow.
dNonresistive, inviscid flow. The greater and lesser solutions for ω2 are the fast and slow magnetosonic waves
respectively.

7.9 Plasma physics
7
159
Synchrotron radiation
Power radiated
by a single
electrona
Ptot =2σTcumagγ2 v
c
2
sin2
θ (7.286)
1.59×10−14
B2
γ2 v
c
2
sin2
θ W
(7.287)
Ptot total radiated power
umag magnetic energy
density =B2/(2µ0)
v electron velocity (∼c)
... averaged
over pitch
angles
Ptot =
4
3
σTcumagγ2 v
c
2
(7.288)
1.06×10−14
B2
γ2 v
c
2
W (7.289)
γ Lorentz factor
=[1−(v/c)2]−1/2
θ pitch angle (angle
between v and B)
c speed of light
Single electron
emission
spectrumb
P(ν)=
31/2
e3
Bsinθ
4π 0cme
F(ν/νch) (7.290)
2.34×10−25
BsinθF(ν/νch) WHz−1
(7.291)
P(ν) emission spectrum
ν frequency
νch characteristic frequency
0 free space permittivity
me electronic (rest) mass
Characteristic
frequency
νch =
3
2
γ2 eB
2πme
sinθ (7.292)
4.2×1010
γ2
Bsinθ Hz (7.293)
F spectral function
K5/3 modiﬁed Bessel fn. of
the 2nd kind, order 5/3
Spectral
function
0
1
1 2 3 4
0.5
F(x)
x
F(x)=x
∞
x
K5/3(y)dy (7.294)
2.15x1/3
(x 1)
1.25x1/2
e−x
(x 1)
(7.295)
aThis expression also holds for cyclotron radiation (v c).
bI.e., total radiated power per unit frequency interval.

Bremsstrahlunga
Single electron and ionb
dW
dω
=
Z2
e6
24π4 3
0c3m2
e
ω2
γ2v4
1
γ2
K2
0
ωb
γv
+K2
1
ωb
γv
(7.296)
Z2
e6
24π4 3
0c3m2
eb2v2
(ωb γv) (7.297)
Thermal bremsstrahlung radiation (v c; Maxwellian distribution)
dP
dV dν
=6.8×10−51
Z2
T−1/2
nineg(ν,T)exp
−hν
kT
Wm−3
Hz−1
(7.298)
where g(ν,T)



0.28[ln(4.4×1016
T3
ν−2
Z−2
)−0.76] (hν kT
∼ 105
kZ2
)
0.55ln(2.1×1010
Tν−1
) (hν 105
kZ2
∼ kT)
(2.1×1010
Tν−1
)−1/2
(hν kT)
(7.299)
dP
dV
1.7×10−40
Z2
T1/2
nine Wm−3
(7.300)
Ze ionic charge
c speed of light
me electronic mass
b collision parameterc
v electron velocity
Ki modiﬁed Bessel functions of
order i (see page 47)
γ Lorentz factor
=[1−(v/c)2]−1/2
P power radiated
V volume
ν frequency (Hz)
W energy radiated
T electron temperature (K)
ni ion number density (m−3)
(m−3)
h Planck constant
g Gaunt factor
aClassical treatment. The ions are at rest, and all frequencies are above the plasma frequency.
bThe spectrum is approximately ﬂat at low frequencies and drops exponentially at frequencies
∼ γv/b.
cDistance of closest approach.

8
Chapter 8 Optics
8.1 Introduction
Any attempt to unify the notations and terminology of optics is doomed to failure. This
is partly due to the long and illustrious history of the subject (a pedigree shared only with
mechanics), which has allowed a variety of approaches to develop, and partly due to the
disparate fields of physics to which its basic principles have been applied. Optical ideas
find their way into most wave-based branches of physics, from quantum mechanics to radio
propagation.
Nowhere is the lack of convention more apparent than in the study of polarisation, and so
a cautionary note follows. The conventions used here can be taken largely from context, but
the reader should be aware that alternative sign and handedness conventions do exist and are
widely used. In particular we will take a circularly polarised wave as being right-handed if,
for an observer looking towards the source, the electric field vector in a plane perpendicular to
the line of sight rotates clockwise. This convention is often used in optics textbooks and has
the conceptual advantage that the electric field orientation describes a right-hand corkscrew
in space, with the direction of energy flow defining the screw direction. It is however opposite
to the system widely used in radio engineering, where the handedness of a helical antenna
generating or receiving the wave defines the handedness and is also in the opposite sense to
the wave’s own angular momentum vector.

162 Optics
8.2 Interference
Newton’s ringsa
nth dark ring
rn
Rr2
n =nRλ0 (8.1)
rn radius of nth ring
n integer (≥0)
R lens radius of curvature
nth bright ring r2
n = n+
1
2
Rλ0 (8.2)
λ0 wavelength in external
medium
aViewed in reflection.
Dielectric layersa
single layer multilayer
1
1R
1−R
RN
1−RN
η1
η1
η2
η3
η3
ηa
ηb
a
N ×{
Quarter-wave
condition
a=
m
η2
λ0
4
(8.3)
a film thickness
m thickness integer
(m≥0)
η2 film refractive index
Single-layer
reflectanceb R =



η1η3 −η2
2
η1η3 +η2
2
2
(m odd)
η1 −η3
η1 +η3
2
(m even)
(8.4)
λ0 free-space wavelength
R power reflectance
coefficient
η1 entry-side refractive
index
η3 exit-side refractive
index
Dependence of
R on layer
thickness, m
max if (−1)m
(η1 −η2)(η2 −η3)0 (8.5)
min if (−1)m
(η1 −η2)(η2 −η3)0 (8.6)
R =0 if η2 =(η1η3)1/2
and m odd (8.7)
Multilayer
reflectancec RN =
η1 −η3(ηa/ηb)2N
η1 +η3(ηa/ηb)2N
2
(8.8)
RN multilayer reflectance
N number of layer pairs
ηa refractive index of top
layer
ηb refractive index of
bottom layer
aFor normal incidence, assuming the quarter-wave condition. The media are also assumed lossless, with µr =1.
bSee page 154 for the definition of R.
cFor a stack of N layer pairs, giving an overall refractive index sequence η1ηa,ηbηa ...ηaηbη3 (see right-hand diagram).
Each layer in the stack meets the quarter-wave condition with m=1.

8.2 Interference
8
163
Fabry-Perot etalona
h
incident rays
θ
θ
η
η
η
∝1
eiφ
e2iφ
e3iφ
Incremental
phase
differenceb
φ=2k0hη cosθ (8.9)
=2k0hη 1−
ηsinθ
η
2 1/2
(8.10)
=2πn for a maximum (8.11)
φ incremental phase difference
k0 free-space wavenumber (=2π/λ0)
h cavity width
θ fringe inclination (usually 1)
θ internal angle of refraction
η cavity refractive index
η external refractive index
n fringe order (integer)
Coefficient of
finesse
F =
4R
(1−R)2
(8.12)
F coefficient of finesse
R interface power reflectance
Finesse
F=
π
2
F1/2
(8.13)
=
λ0
η h
Q (8.14)
F finesse
λ0 free-space wavelength
Q cavity quality factor
Transmitted
intensity
I(θ)=
I0(1−R)2
1+R2 −2Rcosφ
(8.15)
=
I0
1+F sin2
(φ/2)
(8.16)
=I0A(θ) (8.17)
I transmitted intensity
I0 incident intensity
A Airy function
Fringe
intensity
profile
∆φ=2arcsin(F−1/2
) (8.18)
2F−1/2
(8.19)
∆φ phase difference at half intensity
point
Chromatic
resolving
power
λ0
δλ
R1/2
πn
1−R
=nF (8.20)
2Fhη
λ0
(θ 1) (8.21)
δλ minimum resolvable wavelength
difference
Free spectral
rangec
δλf =Fδλ (8.22)
δνf =
c
2η h
(8.23)
δλf wavelength free spectral range
δνf frequency free spectral range
aNeglecting any effects due to surface coatings on the etalon. See also Lasers on page 174.
bBetween adjacent rays. Highest order fringes are near the centre of the pattern.
cAt near-normal incidence (θ 0), the orders of two spectral components separated by δλf will not overlap.

164 Optics
8.3 Fraunhofer diffraction
Gratingsa
coherent plane waves
Young’s
double
slitsb
D
d
N
θi
θi
θn
θn
θn
a
I(s)=I0 cos2 kDs
2
(8.24)
I(s) diffracted intensity
I0 peak intensity
θ diffraction angle
s =sinθ
D slit separation
N equally
spaced
narrow slits
I(s)=I0
sin(Nkds/2)
Nsin(kds/2)
2
(8.25)
λ wavelength
N number of slits
k wavenumber
(=2π/λ)
d slit spacing
Infinite
grating I(s)=I0
∞
n=−∞
δ s−
nλ
d
(8.26)
n diffraction order
δ Dirac delta
function
Normal
incidence
sinθn =
nλ
d
(8.27)
θn angle of diffracted
maximum
Oblique
incidence
sinθn +sinθi =
nλ
d
(8.28)
θi angle of incident
illumination
Reflection
grating
sinθn −sinθi =
nλ
d
(8.29)
Chromatic
resolving
power
λ
δλ
=Nn (8.30)
δλ diffraction peak
width
Grating
dispersion
∂θ
∂λ
=
n
dcosθ
(8.31)
Bragg’s
lawc 2asinθn =nλ (8.32) a atomic plane
spacing
aUnless stated otherwise, the illumination is normal to the grating.
bTwo narrow slits separated by D.
cThe condition is for Bragg reflection, with θn =θi.

8.3 Fraunhofer diffraction
8
165
Aperture diffraction
y
x
z
f(x,y)
sx
sycoherent plane-wave
illumination, normal
to the xy plane
General 1-D
aperturea
ψ(s)∝
∞
−∞
f(x)e−iksx
dx (8.33)
I(s)∝ψψ∗
(s) (8.34)
ψ diffracted wavefunction
I diffracted intensity
θ diffraction angle
s =sinθ
General 2-D
aperture in
(x,y) plane
(small angles)
ψ(sx,sy)∝
∞
f(x,y)e−ik(sxx+syy)
dxdy (8.35)
f aperture amplitude
transmission function
x,y distance across aperture
sx deflection xz plane
sy deflection ⊥ xz plane
Broad 1-D
slitb
I(s)=I0
sin2
(kas/2)
(kas/2)2
(8.36)
≡I0 sinc2
(as/λ) (8.37)
I0 peak intensity
a slit width (in x)
λ wavelength
Sidelobe
intensity
In
I0
=
2
π
2
1
(2n+1)2
(n0) (8.38) In nth sidelobe intensity
Rectangular
aperture
(small angles)
I(sx,sy)=I0 sinc2 asx
λ
sinc2 bsy
λ
(8.39)
a aperture width in x
b aperture width in y
Circular
aperturec I(s)=I0
2J1(kDs/2)
kDs/2
2
(8.40)
J1 first-order Bessel function
D aperture diameter
First
minimumd s=1.22
λ
D
(8.41) λ wavelength
First subsid.
maximum
s=1.64
λ
D
(8.42)
Weak 1-D
phase object
f(x)=exp[iφ(x)] 1+iφ(x) (8.43) φ(x) phase distribution
i i2 =−1
Fraunhofer
limite L
(∆x)2
λ
(8.44)
L distance of aperture from
observation point
∆x aperture size
aThe Fraunhofer integral.
bNote that sincx=(sinπx)/(πx).
cThe central maximum is known as the “Airy disk.”
dThe “Rayleigh resolution criterion” states that two point sources of equal intensity can just be resolved with
diffraction-limited optics if separated in angle by 1.22λ/D.
ePlane-wave illumination.

166 Optics
8.4 Fresnel diffraction
Kirchhoff’s diffraction formulaa
source
ψ0
dA
(source at infinity)
θ
r
r
P
P
ρ
y
x
z
S
dS
ˆs
Source at
infinity
ψP =−
i
λ
ψ0
plane
K(θ)
eikr
r
dA (8.45)
ψP complex amplitude at P
λ wavelength
ψ0 incident amplitude
θ obliquity angle
r distance of dA from P ( λ)where:
Obliquity
factor
(cardioid)
K(θ)=
1
2
(1+cosθ) (8.46)
dA area element on incident
wavefront
K obliquity factor
dS element of closed surface
Source at
finite
distanceb
ψP =−
iE0
λ
closed surface
eik(ρ+r)
2ρr
[cos(ˆs· ˆr)−cos(ˆs· ˆρ)] dS
(8.47)
ˆ unit vector
s vector normal to dS
r vector from P to dS
ρ vector from source to dS
E0 amplitude (see footnote)
aAlso known as the “Fresnel–Kirchhoff formula.” Diffraction by an obstacle coincident with the integration surface
can be approximated by omitting that part of the surface from the integral.
bThe source amplitude at ρ is ψ(ρ)=E0eikρ/ρ. The integral is taken over a surface enclosing the point P.
Fresnel zones
source observerz1 z2
y
Effective aperture
distancea
1
z
=
1
z1
+
1
z2
(8.48)
z effective distance
z1 source–aperture distance
z2 aperture–observer distance
Half-period zone
radius
yn =(nλz)1/2
(8.49)
n half-period zone number
λ wavelength
yn nth half-period zone radius
Axial zeros (circular
aperture) zm =
R2
2mλ
(8.50)
zm distance of mth zero from
aperture
R aperture radius
aI.e., the aperture–observer distance to be employed when the source is not at infinity.

8.4 Fresnel diffraction
8
167
Cornu spiral
S(w)
C(w)
Cornu Spiral
w
w
−∞
∞
1
2
1
√
2
√
3
2
√
5
− 1
2
−1
−
√
2
−
√
3
−2
−
√
5
Edge diffraction
CS(w)+1
2(1+i)
2
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
0.5
1
1.5
2
2
2.5
3
−4 −2 4
intensity
Fresnel
integralsa
C(w)=
w
0
cos
πt2
2
dt (8.51)
S(w)=
w
0
sin
πt2
2
dt (8.52)
C Fresnel cosine integral
S Fresnel sine integral
Cornu spiral
CS(w)=C(w)+iS(w) (8.53)
CS(±∞)=±
1
2
(1+i) (8.54)
CS Cornu spiral
v,w length along spiral
Edge diffraction
ψP =
ψ0
21/2
[CS(w)+
1
2
(1+i)] (8.55)
where w =y
2
λz
1/2
(8.56)
ψP complex amplitude at P
ψ0 unobstructed amplitude
λ wavelength
z distance of P from
aperture plane [see (8.48)]
y position of edge
Diffraction
from a long
slitb
P
coherent
plane waves
y1
y2
z
ψP =
ψ0
21/2
[CS(w2)−CS(w1)] (8.57)
where wi =yi
2
λz
1/2
(8.58)
Diffraction
from a
rectangular
aperture
ψP =
ψ0
2
[CS(v2)−CS(v1)]× (8.59)
[CS(w2)−CS(w1)] (8.60)
where vi =xi
2
λz
1/2
(8.61)
and wi =yi
2
λz
1/2
(8.62)
xi positions of slit sides
yi positions of slit
top/bottom
aSee also Equation (2.393) on page 45.
bSlit long in x.

168 Optics
8.5 Geometrical optics
Lenses and mirrorsa
object
object
image
r2
r1
x2
x1
u
u v
v
f
f
f R
lens mirror
sign convention
+ −
r centred to right centred to left
u real object virtual object
v real image virtual image
f converging lens/
concave mirror
diverging lens/
convex mirror
MT erect image inverted image
Fermat’s principleb
L= η dl is stationary (8.63)
L optical path length
η refractive index
dl ray path element
Gauss’s lens formula
1
u
+
1
v
=
1
f
(8.64)
u object distance
v image distance
f focal length
Newton’s lens
formula
x1x2 =f2
(8.65)
x1 =v−f
x2 =u−f
Lensmaker’s
formula
1
u
+
1
v
=(η−1)
1
r1
−
1
r2
(8.66)
ri radii of curvature of
lens surfaces
Mirror formulac 1
u
+
1
v
=−
2
R
=
1
f
(8.67)
R mirror radius of
curvature
Dioptre number D =
1
f
m−1
(8.68)
D dioptre number (f in
metres)
Focal ratiod
n=
f
d
(8.69)
n focal ratio
d lens or mirror diameter
Transverse linear
magnification
MT =−
v
u
(8.70) MT transverse
magnification
Longitudinal linear
magnification
ML =−M2
T (8.71) ML longitudinal
magnification
aFormulas assume “Gaussian optics,” i.e., all lenses are thin and all angles small. Light enters from the left.
bA stationary optical path length has, to first order, a length identical to that of adjacent paths.
cThe mirror is concave if R 0, convex if R 0.
dOr “f-number,” written f/2 if n=2 etc.

8.5 Geometrical optics
8
169
Prisms (dispersing)
θi θt
δα
prism
Transmission
angle
sinθt =(η2
−sin2
θi)1/2
sinα
−sinθi cosα (8.72)
θt angle of transmission
α apex angle
η refractive index
Deviation δ =θi +θt −α (8.73) δ angle of deviation
Minimum
deviation
condition
sinθi =sinθt =ηsin
α
2
(8.74)
Refractive
index
η =
sin[(δm +α)/2]
sin(α/2)
(8.75) δm minimum deviation
Angular
dispersiona D =
dδ
dλ
=
2sin(α/2)
cos[(δm +α)/2]
dη
dλ
(8.76)
D dispersion
λ wavelength
aAt minimum deviation.
Optical fibres
θm
L
cladding, ηc ηf fibre, ηf
Acceptance angle sinθm =
1
η0
(η2
f −η2
c )1/2
(8.77)
θm maximum angle of incidence
η0 exterior refractive index
ηf fibre refractive index
ηc cladding refractive index
Numerical
aperture
N =η0 sinθm (8.78) N numerical aperture
Multimode
dispersiona
∆t
L
=
ηf
c
ηf
ηc
−1 (8.79)
∆t temporal dispersion
L fibre length
c speed of light
aOf a pulse with a given wavelength, caused by the range of incident angles up to θm. Sometimes called “intermodal
dispersion” or “modal dispersion.”

170 Optics
8.6 Polarisation
Elliptical polarisationa
Elliptical
polarisation
y
x
E0y
E0x
α
a
b
θ
E =(E0x,E0yeiδ
)ei(kz−ωt)
(8.80)
E electric field
k wavevector
z propagation axis
ωt angular frequency ×
time
Polarisation
angleb
tan2α=
2E0xE0y
E2
0x −E2
0y
cosδ
(8.81)
E0x x amplitude of E
E0y y amplitude of E
δ relative phase of Ey
with respect to Ex
α polarisation angle
Ellipticityc
e=
a−b
a
(8.82)
e ellipticity
a semi-major axis
b semi-minor axis
Malus’s lawd I(θ)=I0 cos2
θ (8.83)
I(θ) transmitted intensity
I0 incident intensity
θ polariser–analyser
angle
aSee the introduction (page 161) for a discussion of sign and handedness conventions.
bAngle between ellipse major axis and x axis. Sometimes the polarisation angle is
defined as π/2−α.
cThis is one of several definitions for ellipticity.
dTransmission through skewed polarisers for unpolarised incident light.
Jones vectors and matrices
Normalised
electric fielda E =
Ex
Ey
; |E|=1 (8.84)
E electric field
Ex x component of E
Ey y component of E
Example
vectors:
Ex =
1
0
E45 =
1
√
2
1
1
Er =
1
√
2
1
−i
El =
1
√
2
1
i
E45 45◦ to x axis
Er right-hand circular
El left-hand circular
Jones matrix Et =AEi (8.85)
Et transmitted vector
Ei incident vector
A Jones matrix
Example matrices:
Linear polariser x
1 0
0 0
Linear polariser y
0 0
0 1
Linear polariser at 45◦ 1
2
1 1
1 1
Linear polariser at −45◦ 1
2
1 −1
−1 1
Right circular polariser
1
2
1 i
−i 1
Left circular polariser
1
2
1 −i
i 1
λ/4 plate (fast x) eiπ/4 1 0
0 i
λ/4 plate (fast ⊥x) eiπ/4 1 0
0 −i
aKnown as the “normalised Jones vector.”

8.6 Polarisation
8
171
Stokes parametersa
2b
2a
E0y
E0x
x
y
α
χ
Q
U
V
pI
2α
2χ
Poincaré sphere
Electric fields
Ex =E0xei(kz−ωt)
(8.86)
Ey =E0yei(kz−ωt+δ)
(8.87)
k wavevector
ωt angular frequency × time
δ relative phase of Ey with
respect to Ex
Axial ratiob
tanχ=±r =±
b
a
(8.88)
χ (see diagram)
r axial ratio
Stokes
parameters
I = E2
x + E2
y (8.89)
Q= E2
x − E2
y (8.90)
=pI cos2χcos2α (8.91)
U =2 ExEy cosδ (8.92)
=pI cos2χsin2α (8.93)
V =2 ExEy sinδ (8.94)
=pI sin2χ (8.95)
Ex electric field component x
Ey electric field component y
E0x field amplitude in x direction
E0y field amplitude in y direction
α polarisation angle
p degree of polarisation
· mean over time
Degree of
polarisation p=
(Q2
+U2
+V2
)1/2
I
≤1 (8.96)
Q/I U/I V/I Q/I U/I V/I
left circular 0 0 −1 right circular 0 0 1
linear x 1 0 0 linear y −1 0 0
linear 45◦
to x 0 1 0 linear −45◦
to x 0 −1 0
unpolarised 0 0 0
aUsing the convention that right-handed circular polarisation corresponds to a clockwise rotation of the electric field
in a given plane when looking towards the source. The propagation direction in the diagram is out of the plane.
The parameters I, Q, U, and V are sometimes denoted s0, s1, s2, and s3, and other nomenclatures exist. There is
no generally accepted definition – often the parameters are scaled to be dimensionless, with s0 = 1, or to represent
power flux through a plane ⊥ the beam, i.e., I =( E2
x + E2
y )/Z0 etc., where Z0 is the impedance of free space.
bThe axial ratio is positive for right-handed polarisation and negative for left-handed polarisation using our
definitions.

172 Optics
8.7 Coherence (scalar theory)
Mutual
coherence
function
Γ12(τ)= ψ1(t)ψ∗
2(t+τ) (8.97)
Γij mutual coherence function
τ temporal interval
ψi (complex) wave disturbance
at spatial point i
Complex degree
of coherence
γ12(τ)=
ψ1(t)ψ∗
2(t+τ)
[ |ψ1|2 |ψ2|2 ]1/2
(8.98)
=
Γ12(τ)
[Γ11(0)Γ22(0)]1/2
(8.99)
t time
· mean over time
γij complex degree of coherence
∗ complex conjugate
Combined
intensitya
Itot =I1 +I2 +2(I1I2)1/2
[γ12(τ)]
(8.100)
Itot combined intensity
Ii intensity of disturbance at
point i
real part of
Fringe visibility V(τ)=
2(I1I2)1/2
I1 +I2
|γ12(τ)| (8.101)
if |γ12(τ)| is a
constant:
V =
Imax −Imin
Imax +Imin
(8.102)
Imax max. combined intensity
Imin min. combined intensity
if I1 =I2: V(τ)=|γ12(τ)| (8.103)
Complex degree
of temporal
coherenceb
γ(τ)=
ψ1(t)ψ∗
1(t+τ)
|ψ1(t)2|
(8.104)
=
I(ω)e−iωτ
dω
I(ω) dω
(8.105)
γ(τ) degree of temporal coherence
I(ω) specific intensity
ω radiation angular frequency
c speed of light
Coherence time
and length
∆τc =
∆lc
c
∼
1
∆ν
(8.106)
∆τc coherence time
∆lc coherence length
∆ν spectral bandwidth
Complex degree
of spatial
coherencec
γ(D)=
ψ1ψ∗
2
[ |ψ1|2 |ψ2|2 ]1/2
(8.107)
=
I(ˆs)eikD·ˆs
dΩ
I(ˆs) dΩ
(8.108)
γ(D) degree of spatial coherence
D spatial separation of points 1
and 2
I(ˆs) specific intensity of distant
extended source in direction ˆs
dΩ differential solid angle
Intensity
correlationd
I1I2
[ I1
2 I2
2]1/2
=1+γ2
(D) (8.109)
ˆs unit vector in the direction of
dΩ
k wavenumber
Speckle
intensity
distributione
pr(I)=
1
I
e−I/ I
(8.110) pr probability density
Speckle size
(coherence
width)
∆wc
λ
α
(8.111)
∆wc characteristic speckle size
λ wavelength
α source angular size as seen
from the screen
aFrom interfering the disturbances at points 1 and 2 with a relative delay τ.
bOr “autocorrelation function.”
cBetween two points on a wavefront, separated by D. The integral is over the entire extended source.
dFor wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as
from a thermal source.
eAlso for Gaussian light.

8.8 Line radiation
8
173
8.8 Line radiation
Spectral line broadening
Natural
broadeninga I(ω)=
(2πτ)−1
(2τ)−2 +(ω−ω0)2
(8.112)
I(ω) normalised intensityb
τ lifetime of excited state
Natural
half-width
∆ω =
1
2τ
(8.113)
∆ω half-width at half-power
ω0 centre frequency
Collision
broadening I(ω)=
(πτc)−1
(τc)−2 +(ω−ω0)2
(8.114)
τc mean time between
collisions
p pressure
Collision and
pressure
half-widthc
∆ω =
1
τc
=pπd2 πmkT
16
−1/2
(8.115)
d effective atomic diameter
m gas particle mass
T temperature
c speed of light
Doppler
broadening
ω0
I(ω)
∆ω
I(ω)=
mc2
2kTω2
0π
1/2
exp −
mc2
2kT
(ω−ω0)2
ω2
0
(8.116)
Doppler
half-width ∆ω =ω0
2kT ln2
mc2
1/2
(8.117)
aThe transition probability per unit time for the state is = 1/τ. In the classical limit of a damped oscillator, the
e-folding time of the electric field is 2τ. Both the natural and collision profiles described here are Lorentzian.
bThe intensity spectra are normalised so that I(ω) dω =1, assuming ∆ω/ω0 1.
cThe pressure-broadening relation combines Equations (5.78), (5.86) and (5.89) and assumes an otherwise perfect
gas of finite-sized atoms. More accurate expressions are considerably more complicated.
Einstein coefficientsa
Absorption R12 =B12Iνn1 (8.118)
Rij transition rate, level i→j (m−3 s−1)
Bij Einstein B coefficients
Iν specific intensity of radiation field
Spontaneous
emission
R21 =A21n2 (8.119)
A21 Einstein A coefficient
ni number density of atoms in quantum
level i (m−3)
Stimulated
emission
R21 =B21Iνn2 (8.120)
Coefficient
ratios
A21
B12
=
2hν3
c2
g1
g2
(8.121)
B21
B12
=
g1
g2
(8.122)
h Planck constant
ν frequency
c speed of light
gi degeneracy of ith level
aNote that the coefficients can also be defined in terms of spectral energy density, uν =4πIν/c rather than Iν. In this
case A21
B12
= 8πhν3
c3
g1
g2
. See also Population densities on page 116.

174 Optics
Lasersa
R1 R2
r1
r2
L
light out
Cavity stability
condition
0≤ 1−
L
r1
1−
L
r2
≤1 (8.123)
r1,2 radii of curvature of end-mirrors
L distance between mirror centres
Longitudinal
cavity modesb νn =
c
2L
n (8.124)
νn mode frequency
n integer
c speed of light
Cavity Q
Q=
2πL(R1R2)1/4
λ[1−(R1R2)1/2]
(8.125)
4πL
λ(1−R1R2)
(8.126)
Q quality factor
R1,2 mirror (power) reﬂectances
λ wavelength
Cavity line
width
∆νc =
νn
Q
=1/(2πτc) (8.127)
∆νc cavity line width (FWHP)
τc cavity photon lifetime
Schawlow–
Townes line
width
∆ν
νn
=
2πh(∆νc)2
P
glNu
glNu −guNl
(8.128)
∆ν line width (FWHP)
P laser power
gu,l degeneracy of upper/lower levels
Nu,l number density of upper/lower
levels
Threshold
lasing condition
R1R2 exp[2(α−β)L]1 (8.129) α gain per unit length of medium
β loss per unit length of medium
aAlso see the Fabry-Perot etalon on page 163. Note that “cavity” refers to the empty cavity, with no lasing medium
present.
bThe mode spacing equals the cavity free spectral range.

9
Chapter 9 Astrophysics
9.1 Introduction
Many of the formulas associated with astronomy and astrophysics are either too specialised
for a general work such as this or are common to other fields and can therefore be found
elsewhere in this book. The following section includes many of the relationships that fall
into neither of these categories, including equations to convert between various astronomical
coordinate systems and some basic formulas associated with cosmology.
Exceptionally, this section also includes data on the Sun, Earth, Moon, and planets.
Observational astrophysics remains a largely inexact science, and parameters of these (and
other) bodies are often used as approximate base units in measurements. For example, the
masses of stars and galaxies are frequently quoted as multiples of the mass of the Sun
(1M = 1.989×1030
kg), extra-solar system planets in terms of the mass of Jupiter, and so
on. Astronomers seem to find it particularly difficult to drop arcane units and conventions,
resulting in a profusion of measures and nomenclatures throughout the subject. However,
the convention of using suitable astronomical objects in this way is both useful and widely
accepted.

176 Astrophysics
9.2 Solar system data
Solar data
equatorial radius R = 6.960×108
m = 109.1R⊕
mass M = 1.9891×1030
kg = 3.32946×105
M⊕
polar moment of inertia I = 5.7×1046
kgm2
= 7.09×108
I⊕
bolometric luminosity L = 3.826×1026
W
effective surface temperature T = 5770K
solar constanta
1.368×103
Wm−2
absolute magnitude MV = +4.83; Mbol = +4.75
apparent magnitude mV = −26.74; mbol = −26.82
aBolometric flux at a distance of 1 astronomical unit (AU).
Earth data
equatorial radius R⊕ = 6.37814×106
m = 9.166×10−3
R
flatteninga
f = 0.00335364 = 1/298.183
mass M⊕ = 5.9742×1024
kg = 3.0035×10−6
M
polar moment of inertia I⊕ = 8.037×1037
kgm2
= 1.41×10−9
I
orbital semi-major axisb
1AU = 1.495979×1011
m = 214.9R
mean orbital velocity 2.979×104
ms−1
equatorial surface gravity ge = 9.780327ms−2
(includes rotation)
polar surface gravity gp = 9.832186ms−2
rotational angular velocity ωe = 7.292115×10−5
rads−1
af equals (R⊕ −Rpolar)/R⊕. The mean radius of the Earth is 6.3710×106 m.
bAbout the Sun.
Moon data
equatorial radius Rm = 1.7374×106
m = 0.27240R⊕
mass Mm = 7.3483×1022
kg = 1.230×10−2
M⊕
mean orbital radiusa
am = 3.84400×108
m = 60.27R⊕
mean orbital velocity 1.03×103
ms−1
orbital period (sidereal) 27.32166d
equatorial surface gravity 1.62ms−2
= 0.166ge
aAbout the Earth.
Planetary dataa
M/M⊕ R/R⊕ T(d) P(yr) a(AU) M mass
Mercury 0.055 274 0.382 51 58.646 0.240 85 0.387 10 R equatorial radius
Venusb
0.815 00 0.948 83 243.018 0.615 228 0.723 35 T rotational period
Earth 1 1 0.997 27 1.000 04 1.000 00 P orbital period
Mars 0.107 45 0.532 60 1.025 96 1.880 93 1.523 71 a mean distance
Jupiter 317.85 11.209 0.413 54 11.861 3 5.202 53 M⊕ 5.9742×1024 kg
Saturn 95.159 9.449 1 0.444 01 29.628 2 9.575 60 R⊕ 6.37814×106 m
Uranusb
14.500 4.007 3 0.718 33 84.746 6 19.293 4 1d 86400s
Neptune 17.204 3.882 6 0.671 25 166.344 30.245 9 1yr 3.15569×107 s
Plutob
0.00251 0.187 36 6.387 2 248.348 39.509 0 1AU 1.495979×1011 m
aUsing the osculating orbital elements for 1998. Note that P is the instantaneous orbital period, calculated from the
planet’s daily motion. The radii of gas giants are taken at 1 atmosphere pressure.
bRetrograde rotation.

9.3 Coordinate transformations (astronomical)
9
177
9.3 Coordinate transformations (astronomical)
Time in astronomy
Julian day numbera
JD =D−32075+1461∗(Y +4800+(M −14)/12)/4
+367∗(M −2−(M −14)/12∗12)/12
−3∗((Y +4900+(M −14)/12)/100)/4 (9.1)
JD Julian day number
D day of month number
Y calendar year, e.g., 1963
M calendar month (Jan=1)
∗ integer multiply
Modified
Julian day
number
MJD =JD−2400000.5 (9.2)
/ integer divide
MJD modified Julian day
number
Day of
week
W =(JD+1) mod 7 (9.3) W day of week (0=Sunday,
1=Monday, ... )
Local civil
time
LCT=UTC+TZC+DSC (9.4)
LCT local civil time
UTC coordinated universal time
TZC time zone correction
DSC daylight saving correction
Julian
centuries
T =
JD−2451545.5
36525
(9.5)
T Julian centuries between
12h UTC 1 Jan 2000 and
0h UTC D/M/Y
Greenwich
sidereal
time
GMST=6h
41m
50s
.54841
+8640184s
.812866T
+0s
.093104T2
−0s
.0000062T3
(9.6)
GMST Greenwich mean sidereal
time at 0h UTC D/M/Y
(for later times use
1s=1.002738 sidereal
seconds)
Local
sidereal
time
LST=GMST+
λ◦
15◦
(9.7)
LST local sidereal time
λ◦ geographic longitude,
degrees east of Greenwich
aFor the Julian day starting at noon on the calendar day in question. The routine is designed around integer
arithmetic with “truncation towards zero” (so that −5/3 = −1) and is valid for dates from the onset of the
Gregorian calendar, 15 October 1582. JD represents the number of days since Greenwich mean noon 1 Jan 4713
BC. For reference, noon, 1 Jan 2000 =JD2451545 and was a Saturday (W =6).
Horizon coordinatesa
Hour angle H =LST−α (9.8)
LST local sidereal time
H (local) hour angle
Equatorial
to horizon
sina=sinδsinφ+cosδcosφcosH (9.9)
tanA≡
−cosδsinH
sinδcosφ−sinφcosδcosH
(9.10)
α right ascension
δ declination
a altitude
A azimuth (E from N)
φ observer’s latitude
Horizon to
equatorial
sinδ =sinasinφ+cosacosφcosA (9.11)
tanH ≡
−cosasinA
sinacosφ−sinφcosacosA
(9.12) +
+
+
+
−
−
−
−
A, H
aConversions between horizon or alt–azimuth coordinates, (a,A), and celestial equatorial coordinates, (δ,α). There
are a number of conventions for defining azimuth. For example, it is sometimes taken as the angle west from south
rather than east from north. The quadrants for A and H can be obtained from the signs of the numerators and
denominators in Equations (9.10) and (9.12) (see diagram).

178 Astrophysics
Ecliptic coordinatesa
Obliquity of
the ecliptic
ε=23◦
26 21 .45−46 .815T
−0 .0006T2
+0 .00181T3
(9.13)
ε mean ecliptic obliquity
T Julian centuries since
J2000.0b
Equatorial to
ecliptic
sinβ =sinδcosε−cosδsinεsinα (9.14)
tanλ≡
sinαcosε+tanδsinε
cosα
(9.15)
α right ascension
δ declination
λ ecliptic longitude
β ecliptic latitude
Ecliptic to
equatorial
sinδ =sinβcosε+cosβsinεsinλ (9.16)
tanα≡
sinλcosε−tanβsinε
cosλ
(9.17) +
+
+
+
−
−
−
−
λ, α
aConversions between ecliptic, (β,λ), and celestial equatorial, (δ,α), coordinates. β is positive above the ecliptic and λ
increases eastwards. The quadrants for λ and α can be obtained from the signs of the numerators and denominators
in Equations (9.15) and (9.17) (see diagram).
bSee Equation (9.5).
Galactic coordinatesa
Galactic
frame
αg =192◦
15 (9.18)
δg =27◦
24 (9.19)
lg =33◦
(9.20)
αg right ascension of
north galactic pole
δg declination of north
galactic pole
Equatorial
to galactic
sinb=cosδcosδg cos(α−αg)+sinδsinδg (9.21)
tan(l −lg)≡
tanδcosδg −cos(α−αg)sinδg
sin(α−αg)
(9.22)
lg ascending node of
galactic plane on
equator
Galactic to
equatorial
sinδ =cosbcosδg sin(l −lg)+sinbsinδg (9.23)
tan(α−αg)≡
cos(l −lg)
tanbcosδg −sinδg sin(l −lg)
(9.24)
δ declination
α right ascension
b galactic latitude
l galactic longitude
aConversions between galactic, (b,l), and celestial equatorial, (δ,α), coordinates. The galactic frame is deﬁned at
epoch B1950.0. The quadrants of l and α can be obtained from the signs of the numerators and denominators in
Equations (9.22) and (9.24).
Precession of equinoxesa
In right
ascension
α α0 +(3s
.075+1s
.336sinα0 tanδ0)N (9.25)
α right ascension of date
α0 right ascension at J2000.0
N number of years since
J2000.0
In
declination
δ δ0 +(20 .043cosα0)N (9.26) δ declination of date
δ0 declination at J2000.0
aRight ascension in hours, minutes, and seconds; declination in degrees, arcminutes, and arcseconds. These equations
are valid for several hundred years each side of J2000.0.

9.4 Observational astrophysics
9
179
9.4 Observational astrophysics
Astronomical magnitudes
Apparent
magnitude
m1 −m2 =−2.5log10
F1
F2
(9.27)
mi apparent magnitude of object i
Fi energy flux from object i
Distance
modulusa
m−M =5log10 D−5 (9.28)
=−5log10 p−5 (9.29)
M absolute magnitude
m−M distance modulus
D distance to object (parsec)
p annual parallax (arcsec)
Luminosity–
magnitude
relation
Mbol =4.75−2.5log10
L
L
(9.30)
L 3.04×10(28−0.4Mbol)
(9.31)
Mbol bolometric absolute magnitude
L luminosity (W)
L solar luminosity (3.826×1026 W)
Flux–
magnitude
relation
Fbol 2.559×10−(8+0.4mbol)
(9.32) Fbol bolometric flux (Wm−2)
mbol bolometric apparent magnitude
Bolometric
correction
BC =mbol −mV (9.33)
=Mbol −MV (9.34)
BC bolometric correction
mV V-band apparent magnitude
MV V-band absolute magnitude
Colour
indexb
B −V =mB −mV (9.35)
U −B =mU −mB (9.36)
B −V observed B −V colour index
U −B observed U −B colour index
Colour
excessc E =(B −V)−(B −V)0 (9.37) E B −V colour excess
(B −V)0 intrinsic B −V colour index
aNeglecting extinction.
bUsing the UBV magnitude system. The bands are centred around 365 nm (U), 440 nm (B), and 550 nm (V).
cThe U −B colour excess is defined similarly.
Photometric wavelengths
Mean
wavelength λ0 =
λR(λ) dλ
R(λ) dλ
(9.38)
λ0 mean wavelength
λ wavelength
R system spectral response
Isophotal
wavelength F(λi)=
F(λ)R(λ) dλ
R(λ) dλ
(9.39)
F(λ) flux density of source (in
terms of wavelength)
λi isophotal wavelength
Effective
wavelength λeff =
λF(λ)R(λ) dλ
F(λ)R(λ) dλ
(9.40) λeff effective wavelength

180 Astrophysics
Planetary bodies
Bode’s lawa
DAU =
4+3×2n
10
(9.41)
DAU planetary orbital radius (AU)
n index: Mercury =−∞, Venus
=0, Earth =1, Mars =2, Ceres
=3, Jupiter=4, ...
Roche limit
R
∼
100M
9πρ
1/3
(9.42)

∼ 2.46R0 (if densities equal) (9.43)
R satellite orbital radius
M central mass
ρ satellite density
R0 central body radius
Synodic
periodb
1
S
=
1
P
−
1
P⊕
(9.44)
S synodic period
P planetary orbital period
P⊕ Earth’s orbital period
aAlso known as the “Titius–Bode rule.” Note that the asteroid Ceres is counted as a planet in this scheme. The
relationship breaks down for Neptune and Pluto.
bOf a planet.
Distance indicators
Hubble law v =H0d (9.45)
v cosmological recession velocity
H0 Hubble parameter (present epoch)
d (proper) distance
Annual
parallax
Dpc =p−1
(9.46)
Dpc distance (parsec)
p annual parallax (±p arcsec from
mean)
Cepheid
variablesa
log10
L
L
1.15log10 Pd +2.47 (9.47)
MV −2.76log10 Pd −1.40 (9.48)
L mean cepheid luminosity
L Solar luminosity
Pd pulsation period (days)
MV absolute visual magnitude
Tully–Fisher
relationb
MI −7.68log10
2vrot
sini
−2.58
(9.49)
MI I-band absolute magnitude
vrot observed maximum rotation
velocity (kms−1)
i galactic inclination (90◦ when
edge-on)
Einstein rings θ2
=
4GM
c2
ds −dl
dsdl
(9.50)
θ ring angular radius
M lens mass
ds distance from observer to source
dl distance from observer to lens
Sunyaev–
Zel’dovich
effectc
∆T
T
=−2
nekTeσT
mec2
dl (9.51)
T apparent CMBR temperature
dl path element through cloud
R cloud radius
... for a
homogeneous
sphere
∆T
T
=−
4RnekTeσT
mec2
(9.52)
Te electron temperature
me electron mass
c speed of light
aPeriod–luminosity relation for classical Cepheids. Uncertainty in MV is ±0.27 (Madore Freedman, 1991,
Publications of the Astronomical Society of the Pacific, 103, 933).
bGalaxy rotation velocity–magnitude relation in the infrared I waveband, centred at 0.90µm. The coefficients
depend on waveband and galaxy type (see Giovanelli et al., 1997, The Astronomical Journal, 113, 1).
cScattering of the cosmic microwave background radiation (CMBR) by a cloud of electrons, seen as a temperature
decrement, ∆T, in the Rayleigh–Jeans limit (λ 1mm).

9.5 Stellar evolution
9
181
Evolutionary timescales
Free-fall
timescalea τﬀ =
3π
32Gρ0
1/2
(9.53)
τﬀ free-fall timescale
ρ0 initial mass density
Kelvin–Helmholtz
timescale
τKH =
−Ug
L
(9.54)
GM2
R0L
(9.55)
τKH Kelvin–Helmholtz timescale
Ug gravitational potential energy
M body’s mass
R0 body’s initial radius
L body’s luminosity
aFor the gravitational collapse of a uniform sphere.
Star formation
Jeans lengtha
λJ =
π
Gρ
dp
dρ
1/2
(9.56)
λJ Jeans length
ρ cloud mass density
p pressure
Jeans mass MJ =
π
6
ρλ3
J (9.57) MJ (spherical) Jeans mass
Eddington
limiting
luminosityb
LE =
4πGMmpc
σT
(9.58)
1.26×1031 M
M
W (9.59)
LE Eddington luminosity
M stellar mass
M solar mass
mp proton mass
c speed of light
aNote that (dp/dρ)1/2 is the sound speed in the cloud.
bAssuming the opacity is mostly from Thomson scattering.
Stellar theorya
Conservation of
mass
dMr
dr
=4πρr2
(9.60)
r radial distance
Mr mass interior to r
ρ mass density
Hydrostatic
equilibrium
dp
dr
=
−GρMr
r2
(9.61)
p pressure
Energy release
dLr
dr
=4πρr2
(9.62)
Lr luminosity interior to r
power generated per unit mass
Radiative
transport
dT
dr
=
−3
16σ
κ ρ
T3
Lr
4πr2
(9.63)
T temperature
σ Stefan–Boltzmann constant
κ mean opacity
Convective
transport
dT
dr
=
γ−1
γ
T
p
dp
dr
(9.64) γ ratio of heat capacities, cp/cV
aFor stars in static equilibrium with adiabatic convection. Note that ρ is a function of r. κ and are functions of
temperature and composition.

182 Astrophysics
Stellar fusion processesa
PP i chain PP ii chain PP iii chain
p+
+p+
→2
1H+e+
+νe
2
1H+p+
→3
2He+γ
3
2He+3
2He→4
2He+2p+
p+
+p+
→2
1H+e+
+νe
2
1H+p+
→3
2He+γ
3
2He+4
2He→7
4Be+γ
7
4Be+e−
→7
3Li+νe
7
3Li+p+
→24
2He
p+
+p+
→2
1H+e+
+νe
2
1H+p+
→3
2He+γ
3
2He+4
2He→7
4Be+γ
7
4Be+p+
→8
5B+γ
8
5B→8
4Be+e+
+νe
8
4Be→24
2He
CNO cycle triple-α process
12
6C+p+
→13
7N+γ
13
7N→13
6C+e+
+νe
13
6C+p+
→14
7N+γ
14
7N+p+
→15
8O+γ
15
8O→15
7N+e+
+νe
15
7N+p+
→12
6C+4
2He
4
2He+4
2He 8
4Be+γ
8
4Be+4
2He 12
6C∗
12
6C∗
→12
6C+γ
γ photon
p+
proton
e+
positron
e−
electron
νe electron neutrino
aAll species are taken as fully ionised.
Pulsars
Braking
index
˙ω ∝−ωn
(9.65)
n=2−
P ¨P
˙P2
(9.66)
ω rotational angular velocity
P rotational period (=2π/ω)
n braking index
Characteristic
agea T =
1
n−1
P
˙P
(9.67)
T characteristic age
L luminosity
c speed of light
Magnetic
dipole
radiation
L=
µ0|¨m|2
sin2
θ
6πc3
(9.68)
=
2πR6
B2
pω4
sin2
θ
3c3µ0
(9.69)
m pulsar magnetic dipole moment
R pulsar radius
Bp magnetic flux density at
magnetic pole
θ angle between magnetic and
rotational axes
Dispersion
measure DM=
D
0
ne dl (9.70)
DM dispersion measure
D path length to pulsar
dl path element
Dispersionb
dτ
dν
=
−e2
4π2
0mecν3
DM (9.71)
∆τ=
e2
8π2
0mec
1
ν2
1
−
1
ν2
2
DM (9.72)
τ pulse arrival time
∆τ difference in pulse arrival time
νi observing frequencies
me electron mass
aAssuming n=1 and that the pulsar has already slowed significantly. Usually n is assumed to be 3 (magnetic dipole
radiation), giving T =P/(2˙P).
bThe pulse arrives first at the higher observing frequency.

9
183
Compact objects and black holes
Schwarzschild
radius
rs =
2GM
c2
3
M
M
km (9.73)
rs Schwarzschild radius
M mass of body
c speed of light
M solar mass
Gravitational
redshift
ν∞
νr
= 1−
2GM
rc2
1/2
(9.74)
r distance from mass centre
ν∞ frequency at inﬁnity
νr frequency at r
Gravitational
wave radiationa Lg =
32
5
G4
c5
m2
1m2
2(m1 +m2)
a5
(9.75)
mi orbiting masses
a mass separation
Lg gravitational luminosity
Rate of change of
orbital period
˙P =−
96
5
(4π2
)4/3 G5/3
c5
m1m2P−5/3
(m1 +m2)1/3
(9.76)
P orbital period
Neutron star
degeneracy
pressure
(nonrelativistic)
p=
(3π2
)2/3
5
¯h2
mn
ρ
mn
5/3
=
2
3
u (9.77)
p pressure
mn neutron mass
ρ density
Relativisticb
p=
¯hc(3π2
)1/3
4
ρ
mn
4/3
=
1
3
u (9.78) u energy density
Chandrasekhar
massc MCh 1.46M (9.79) MCh Chandrasekhar mass
Maximum black
hole angular
momentum
Jm =
GM2
c
(9.80)
Jm maximum angular
momentum
Black hole
evaporation time τe ∼
M3
M3
×1066
yr (9.81) τe evaporation time
Black hole
temperature T =
¯hc3
8πGMk
10−7 M
M
K (9.82)
T temperature
aFrom two bodies, m1 and m2, in circular orbits about their centre of mass. Note that the frequency of the radiation
is twice the orbital frequency.
bParticle velocities ∼c.
cUpper limit to mass of a white dwarf.

184 Astrophysics
9.6 Cosmology
Cosmological model parameters
Hubble law vr =Hd (9.83)
vr radial velocity
H Hubble parameter
d proper distance
Hubble
parametera
H(t)=
˙R(t)
R(t)
(9.84)
H(z)=H0[Ωm0(1+z)3
+ΩΛ0
+(1−Ωm0 −ΩΛ0)(1+z)2
]1/2
(9.85)
0 present epoch
R cosmic scale factor
t cosmic time
z redshift
Redshift z =
λobs −λem
λem
=
R0
R(tem)
−1 (9.86)
λobs observed wavelength
λem emitted wavelength
tem epoch of emission
Robertson–
Walker
metricb
ds2
=c2
dt2
−R2
(t)
dr2
1−kr2
+r2
(dθ2
+sin2
θ dφ2
) (9.87)
ds interval
c speed of light
r,θ,φ comoving spherical polar
coordinates
Friedmann
equationsc
¨R =−
4π
3
GR ρ+3
p
c2
+
ΛR
3
(9.88)
˙R2
=
8π
3
GρR2
−kc2
+
ΛR2
3
(9.89)
k curvature parameter
p pressure
Λ cosmological constant
Critical
density ρcrit =
3H2
8πG
(9.90)
ρ (mass) density
ρcrit critical density
Density
parameters
Ωm =
ρ
ρcrit
=
8πGρ
3H2
(9.91)
ΩΛ =
Λ
3H2
(9.92)
Ωk =−
kc2
R2H2
(9.93)
Ωm +ΩΛ +Ωk =1 (9.94)
Ωm matter density parameter
ΩΛ lambda density parameter
Ωk curvature density parameter
Deceleration
parameter q0 =−
R0
¨R0
˙R2
0
=
Ωm0
2
−ΩΛ0 (9.95) q0 deceleration parameter
aOften called the Hubble “constant.” At the present epoch, 60
∼ H0

∼ 80kms−1 Mpc−1
≡ 100hkms−1 Mpc−1
, where
h is a dimensionless scaling parameter. The Hubble time is tH =1/H0. Equation (9.85) assumes a matter dominated
universe and mass conservation.
bFor a homogeneous, isotropic universe, using the (−1,1,1,1) metric signature. r is scaled so that k = 0,±1. Note
that ds2 ≡(ds)2 etc.
cΛ = 0 in a Friedmann universe. Note that the cosmological constant is sometimes deﬁned as equalling the value
used here divided by c2.

9.6 Cosmology
9
185
Cosmological distance measures
Look-back
time
tlb(z)=t0 −t(z) (9.96)
tlb(z)light travel time from
an object at redshift z
t0 present cosmic time
t(z) cosmic time at z
Proper
distance dp =R0
r
0
dr
(1−kr2)1/2
=cR0
t0
t
dt
R(t)
(9.97)
dp proper distance
R cosmic scale factor
c speed of light
0 present epoch
Luminosity
distancea dL =dp(1+z)=c(1+z)
z
0
dz
H(z)
(9.98)
dL luminosity distance
z redshift
H Hubble parameterb
Flux density–
redshift
relation
F(ν)=
L(ν )
4πd2
L(z)
where ν =(1+z)ν (9.99)
F spectral flux density
ν frequency
L(ν) spectral luminosityc
Angular
diameter
distanced
da =dL(1+z)−2
(9.100)
da angular diameter
distance
k curvature parameter
aAssuming a flat universe (k =0). The apparent flux density of a source varies as d−2
L .
bSee Equation (9.85).
cDefined as the output power of the body per unit frequency interval.
dTrue for all k. The angular diameter of a source varies as d−1
a .
Cosmological modelsa
Einstein – de
Sitter model
(Ωk =0,
Λ=0, p=0
and Ωm0 =1)
dp =
2c
H0
[1−(1+z)−1/2
] (9.101)
H(z)=H0(1+z)3/2
(9.102)
q0 =1/2 (9.103)
t(z)=
2
3H(z)
(9.104)
ρ=(6πGt2
)−1
(9.105)
R(t)=R0(t/t0)2/3
(9.106)
dp proper
distance
H Hubble
parameter
0 present epoch
z redshift
c speed of light
q deceleration
parameter
t(z) time at
redshift z
Concordance
model
(Ωk =0, Λ=
3(1−Ωm0)H2
0 ,
p=0 and
Ωm0 1)
dp =
c
H0
z
0
Ω
−1/2
m0 dz
[(1+z )3 −1+Ω−1
m0]1/2
(9.107)
H(z)=H0[Ωm0(1+z)3
+(1−Ωm0)] (9.108)
q0 =3Ωm0/2−1 (9.109)
t(z)=
2
3H0
(1−Ωm0)−1/2
arsinh
(1−Ωm0)1/2
(1+z)3/2
(9.110)
R cosmic scale
factor
Ωm0 present mass
density
parameter
G constant of
gravitation
ρ mass density
aCurrently popular.

I
Index
Section headings are shown in boldface and panel labels in small caps. Equation numbers
are contained within square brackets.
A
aberration (relativistic) [3.24], 65
absolute magnitude [9.29], 179
absorption (Einstein coefficient) [8.118],
173
absorption coefficient (linear) [5.175], 120
accelerated point charge
bremsstrahlung, 160
Liénard–Wiechert potentials, 139
oscillating [7.132], 146
synchrotron, 159
acceleration
constant, 68
dimensions, 16
due to gravity (value on Earth), 176
in a rotating frame [3.32], 66
acceptance angle (optical fibre) [8.77],
169
acoustic branch (phonon) [6.37], 129
acoustic impedance [3.276], 83
action (definition) [3.213], 79
action (dimensions), 16
addition of velocities
Galilean [3.3], 64
relativistic [3.15], 64
adiabatic
bulk modulus [5.23], 107
compressibility [5.21], 107
expansion (ideal gas) [5.58], 110
lapse rate [3.294], 84
adjoint matrix
definition 1 [2.71], 24
definition 2 [2.80], 25
adjugate matrix [2.80], 25
admittance (definition), 148
advective operator [3.289], 84
Airy
disk [8.40], 165
function [8.17], 163
resolution criterion [8.41], 165
Airy’s differential equation [2.352], 43
albedo [5.193], 121
Alfvén speed [7.277], 158
Alfvén waves [7.284], 158
alt-azimuth coordinates, 177
alternating tensor ( ijk) [2.443], 50
altitude coordinate [9.9], 177
Ampère’s law [7.10], 136
ampere (SI definition), 3
ampere (unit), 4
analogue formula [2.258], 36
angle
aberration [3.24], 65
acceptance [8.77], 169
beam solid [7.210], 153
Brewster’s [7.218], 154
Compton scattering [7.240], 155
contact (surface tension) [3.340], 88
deviation [8.73], 169
Euler [2.101], 26
Faraday rotation [7.273], 157
hour (coordinate) [9.8], 177
Kelvin wedge [3.330], 87
Mach wedge [3.328], 87
polarisation [8.81], 170
principal range (inverse trig.), 34
refraction, 154
rotation, 26
Rutherford scattering [3.116], 72
separation [3.133], 73
spherical excess [2.260], 36
units, 4, 5
˚angström (unit), 5

188 Index
angular diameter distance [9.100], 185
Angular momentum, 98
angular momentum
conservation [4.113], 98
definition [3.66], 68
dimensions, 16
eigenvalues [4.109] [4.109], 98
ladder operators [4.108], 98
operators
and other operators [4.23], 91
definitions [4.105], 98
rigid body [3.141], 74
Angular momentum addition, 100
Angular momentum commutation rela-
tions, 98
angular speed (dimensions), 16
anomaly (true) [3.104], 71
antenna
beam efficiency [7.214], 153
effective area [7.212], 153
power gain [7.211], 153
temperature [7.215], 153
Antennas, 153
anticommutation [2.95], 26
antihermitian symmetry, 53
antisymmetric matrix [2.87], 25
Aperture diffraction, 165
aperture function [8.34], 165
apocentre (of an orbit) [3.111], 71
apparent magnitude [9.27], 179
Appleton-Hartree formula [7.271], 157
arc length [2.279], 39
arccosx
from arctan [2.233], 34
series expansion [2.141], 29
arcoshx (definition) [2.239], 35
arccotx (from arctan) [2.236], 34
arcothx (definition) [2.241], 35
arccscx (from arctan) [2.234], 34
arcschx (definition) [2.243], 35
arcminute (unit), 5
arcsecx (from arctan) [2.235], 34
arsechx (definition) [2.242], 35
arcsecond (unit), 5
arcsinx
from arctan [2.232], 34
arsinhx (definition) [2.238], 35
arctanx (series expansion) [2.142], 29
artanhx (definition) [2.240], 35
area
of circle [2.262], 37
of cone [2.271], 37
of cylinder [2.269], 37
of ellipse [2.267], 37
of plane triangle [2.254], 36
of sphere [2.263], 37
of spherical cap [2.275], 37
of torus [2.273], 37
area (dimensions), 16
argument (of a complex number) [2.157],
30
arithmetic mean [2.108], 27
arithmetic progression [2.104], 27
associated Laguerre equation [2.348], 43
associated Laguerre polynomials, 96
associated Legendre equation
and polynomial solutions [2.428], 48
differential equation [2.344], 43
Associated Legendre functions, 48
astronomical constants, 176
Astronomical magnitudes, 179
Astrophysics, 175–185
asymmetric top [3.189], 77
atomic
form factor [6.30], 128
mass unit, 6, 9
numbers of elements, 124
polarisability [7.91], 142
weights of elements, 124
Atomic constants, 7
atto, 5
autocorrelation (Fourier) [2.491], 53
autocorrelation function [8.104], 172
availability
and fluctuation probability [5.131],
116
Avogadro constant, 6, 9
Avogadro constant (dimensions), 16
azimuth coordinate [9.10], 177
B
Ballistics, 69
band index [6.85], 134
Band theory and semiconductors, 134
bandwidth
and coherence time [8.106], 172

Index
I
189
and Johnson noise [5.141], 117
Doppler [8.117], 173
natural [8.113], 173
of a diffraction grating [8.30], 164
of an LCR circuit [7.151], 148
of laser cavity [8.127], 174
Schawlow-Townes [8.128], 174
bar (unit), 5
barn (unit), 5
Barrier tunnelling, 94
Bartlett window [2.581], 60
base vectors (crystallographic), 126
basis vectors [2.17], 20
Bayes’ theorem [2.569], 59
Bayesian inference, 59
bcc structure, 127
beam bowing under its own weight [3.260],
82
beam efficiency [7.214], 153
beam solid angle [7.210], 153
beam with end-weight [3.259], 82
beaming (relativistic) [3.25], 65
becquerel (unit), 4
Bending beams, 82
bending moment (dimensions), 16
bending moment [3.258], 82
bending waves [3.268], 82
Bernoulli’s differential equation [2.351],
43
compressible flow [3.292], 84
incompressible flow [3.290], 84
Bessel equation [2.345], 43
Bessel functions, 47
beta (in plasmas) [7.278], 158
binomial
coefficient [2.121], 28
distribution [2.547], 57
series [2.120], 28
theorem [2.122], 28
binormal [2.285], 39
Biot–Savart law [7.9], 136
Biot-Fourier equation [5.95], 113
black hole
evaporation time [9.81], 183
Kerr solution [3.62], 67
maximum angular momentum [9.80],
183
Schwarzschild radius [9.73], 183
Schwarzschild solution [3.61], 67
blackbody
energy density [5.192], 121
spectral energy density [5.186], 121
spectrum [5.184], 121
Blackbody radiation, 121
Bloch’s theorem [6.84], 134
Bode’s law [9.41], 180
body cone, 77
body frequency [3.187], 77
body-centred cubic structure, 127
Bohr
energy [4.74], 95
magneton (equation) [4.137], 100
magneton (value), 6, 7
quantisation [4.71], 95
radius (equation) [4.72], 95
radius (value), 7
Bohr magneton (dimensions), 16
Bohr model, 95
boiling points of elements, 124
bolometric correction [9.34], 179
Boltzmann
constant, 6, 9
constant (dimensions), 16
entropy [5.105], 114
excitation equation [5.125], 116
Born collision formula [4.178], 104
Bose condensation [5.123], 115
Bose–Einstein distribution [5.120], 115
boson statistics [5.120], 115
Boundary conditions for E, D, B, and
H, 144
box (particle in a) [4.64], 94
Box Muller transformation [2.561], 58
Boyle temperature [5.66], 110
Boyle’s law [5.56], 110
bra vector [4.33], 92
bra-ket notation, 91, 92
Bragg’s reflection law
in crystals [6.29], 128
in optics [8.32], 164
braking index (pulsar) [9.66], 182
Bravais lattices, 126
Breit-Wigner formula [4.174], 104
Bremsstrahlung, 160
bremsstrahlung

190 Index
single electron and ion [7.297], 160
thermal [7.300], 160
Brewster’s law [7.218], 154
brightness (blackbody) [5.184], 121
Brillouin function [4.147], 101
Bromwich integral [2.518], 55
Brownian motion [5.98], 113
bubbles [3.337], 88
bulk modulus
adiabatic [5.23], 107
general [3.245], 81
isothermal [5.22], 107
bulk modulus (dimensions), 16
Bulk physical constants, 9
Burgers vector [6.21], 128
C
calculus of variations [2.334], 42
candela, 119
candela (SI deﬁnition), 3
candela (unit), 4
canonical
ensemble [5.111], 114
entropy [5.106], 114
equations [3.220], 79
momenta [3.218], 79
cap, see spherical cap
Capacitance, 137
capacitance
current through [7.144], 147
dimensions, 16
energy [7.153], 148
energy of an assembly [7.134], 146
impedance [7.159], 148
mutual [7.134], 146
capacitance of
cube [7.17], 137
cylinder [7.15], 137
cylinders (adjacent) [7.21], 137
cylinders (coaxial) [7.19], 137
disk [7.13], 137
disks (coaxial) [7.22], 137
nearly spherical surface [7.16], 137
sphere [7.12], 137
spheres (adjacent) [7.14], 137
spheres (concentric) [7.18], 137
capacitor, see capacitance
capillary
constant [3.338], 88
contact angle [3.340], 88
rise [3.339], 88
waves [3.321], 86
capillary-gravity waves [3.322], 86
cardioid [8.46], 166
Carnot cycles, 107
Cartesian coordinates, 21
Catalan’s constant (value), 9
Cauchy
inequality [2.151], 30
integral formula [2.167], 31
Cauchy-Goursat theorem [2.165], 31
Cauchy-Riemann conditions [2.164], 31
cavity modes (laser) [8.124], 174
Celsius (unit), 4
Celsius conversion [1.1], 15
centi, 5
centigrade (avoidance of), 15
centre of mass
circular arc [3.173], 76
cone [3.175], 76
disk sector [3.172], 76
hemisphere [3.170], 76
hemispherical shell [3.171], 76
pyramid [3.175], 76
semi-ellipse [3.178], 76
spherical cap [3.177], 76
triangular lamina [3.174], 76
Centres of mass, 76
centrifugal force [3.35], 66
centripetal acceleration [3.32], 66
cepheid variables [9.48], 180
Cerenkov, see Cherenkov
chain rule
function of a function [2.295], 40
partial derivatives [2.331], 42
Chandrasekhar mass [9.79], 183
change of variable [2.333], 42
Characteristic numbers, 86
charge
conservation [7.39], 139
dimensions, 16
elementary, 6, 7
force between two [7.119], 145
Hamiltonian [7.138], 146

Index
I
191
to mass ratio of electron, 8
charge density
dimensions, 16
free [7.57], 140
induced [7.84], 142
Lorentz transformation, 141
charge distribution
electric field from [7.6], 136
energy of [7.133], 146
charge-sheet (electric field) [7.32], 138
Chebyshev equation [2.349], 43
Chebyshev inequality [2.150], 30
chemical potential
from partition function [5.119], 115
Cherenkov cone angle [7.246], 156
Cherenkov radiation, 156
χE (electric susceptibility) [7.87], 142
χH , χB (magnetic susceptibility) [7.103],
143
chi-squared (χ2
) distribution [2.553], 58
Christoffel symbols [3.49], 67
circle
(arc of) centre of mass [3.173], 76
area [2.262], 37
perimeter [2.261], 37
circular aperture
Fraunhofer diffraction [8.40], 165
Fresnel diffraction [8.50], 166
circular polarisation, 170
circulation [3.287], 84
civil time [9.4], 177
Clapeyron equation [5.50], 109
classical electron radius, 8
Classical thermodynamics, 106
Clausius–Mossotti equation [7.93], 142
Clausius-Clapeyron equation [5.49], 109
Clebsch–Gordan coefficients, 99
Clebsch–Gordan coefficients (spin-orbit) [4.136],
100
close-packed spheres, 127
closure density (of the universe) [9.90],
184
CNO cycle, 182
coaxial cable
capacitance [7.19], 137
inductance [7.24], 137
coaxial transmission line [7.181], 150
coefficient of
coupling [7.148], 147
finesse [8.12], 163
reflectance [7.227], 154
reflection [7.230], 154
restitution [3.127], 73
transmission [7.232], 154
transmittance [7.229], 154
coexistence curve [5.51], 109
coherence
length [8.106], 172
mutual [8.97], 172
temporal [8.105], 172
time [8.106], 172
width [8.111], 172
Coherence (scalar theory), 172
cold plasmas, 157
collision
broadening [8.114], 173
elastic, 73
inelastic, 73
number [5.91], 113
time (electron drift) [6.61], 132
colour excess [9.37], 179
colour index [9.36], 179
Common three-dimensional coordinate
systems, 21
commutator (in uncertainty relation) [4.6],
90
Commutators, 26
Compact objects and black holes, 183
complementary error function [2.391], 45
Complex analysis, 31
complex conjugate [2.159], 30
Complex numbers, 30
complex numbers
argument [2.157], 30
cartesian form [2.153], 30
conjugate [2.159], 30
logarithm [2.162], 30
modulus [2.155], 30
polar form [2.154], 30
Complex variables, 30
compound pendulum [3.182], 76
compressibility
adiabatic [5.21], 107
isothermal [5.20], 107
compression modulus, see bulk modulus
compression ratio [5.13], 107
Compton

192 Index
scattering [7.240], 155
wavelength (value), 8
wavelength [7.240], 155
Concordance model, 185
conditional probability [2.567], 59
conductance (definition), 148
conductance (dimensions), 16
conduction equation (and transport) [5.96],
113
conduction equation [2.340], 43
conductivity
and resistivity [7.142], 147
dimensions, 16
direct [7.279], 158
electrical, of a plasma [7.233], 155
free electron a.c. [6.63], 132
free electron d.c. [6.62], 132
Hall [7.280], 158
conductor refractive index [7.234], 155
cone
centre of mass [3.175], 76
moment of inertia [3.160], 75
surface area [2.271], 37
volume [2.272], 37
configurational entropy [5.105], 114
Conic sections, 38
conical pendulum [3.180], 76
conservation of
angular momentum [4.113], 98
charge [7.39], 139
mass [3.285], 84
Constant acceleration, 68
constant of gravitation, 7
contact angle (surface tension) [3.340],
88
continuity equation (quantum physics) [4.14],
90
continuity in fluids [3.285], 84
Continuous probability distributions,
58
contravariant components
in general relativity, 67
in special relativity [3.26], 65
convection (in a star) [9.64], 181
convergence and limits, 28
Conversion factors, 10
Converting between units, 10
convolution
derivative [2.498], 53
discrete [2.580], 60
Laplace transform [2.516], 55
rules [2.489], 53
theorem [2.490], 53
coordinate systems, 21
coordinate transformations
astronomical, 177
Galilean, 64
relativistic, 64
rotating frames [3.31], 66
Coordinate transformations (astronomical),
177
coordinates (generalised ) [3.213], 79
coordination number (cubic lattices), 127
Coriolis force [3.33], 66
Cornu spiral, 167
Cornu spiral and Fresnel integrals [8.54],
167
correlation coefficient
multinormal [2.559], 58
Pearson’s r [2.546], 57
correlation intensity [8.109], 172
correlation theorem [2.494], 53
cosx
and Euler’s formula [2.216], 34
cosec, see csc
cschx [2.231], 34
coshx
cosine formula
planar triangles [2.249], 36
spherical triangles [2.257], 36
cosmic scale factor [9.87], 184
cosmological constant [9.89], 184
Cosmological distance measures, 185
Cosmological model parameters, 184
Cosmological models, 185
Cosmology, 184
cos−1
x, see arccosx
cotx
cothx [2.227], 34
Couette flow [3.306], 85
coulomb (unit), 4
Coulomb gauge condition [7.42], 139

Index
I
193
Coulomb logarithm [7.254], 156
Coulomb’s law [7.119], 145
couple
dimensions, 16
electromagnetic, 145
for Couette flow [3.306], 85
on a current-loop [7.127], 145
on a magnetic dipole [7.126], 145
on a rigid body, 77
on an electric dipole [7.125], 145
twisting [3.252], 81
coupling coefficient [7.148], 147
covariance [2.558], 58
covariant components [3.26], 65
cracks (critical length) [6.25], 128
critical damping [3.199], 78
critical density (of the universe) [9.90],
184
critical frequency (synchrotron) [7.293],
159
critical point
Dieterici gas [5.75], 111
van der Waals gas [5.70], 111
cross section
absorption [5.175], 120
cross-correlation [2.493], 53
cross-product [2.2], 20
cross-section
Breit-Wigner [4.174], 104
Mott scattering [4.180], 104
Rayleigh scattering [7.236], 155
Thomson scattering [7.238], 155
Crystal diffraction, 128
Crystal systems, 127
Crystalline structure, 126
cscx
cschx [2.231], 34
cube
electrical capacitance [7.17], 137
mensuration, 38
Cubic equations, 51
cubic expansivity [5.19], 107
Cubic lattices, 127
cubic system (crystallographic), 127
Curie temperature [7.114], 144
Curie’s law [7.113], 144
Curie–Weiss law [7.114], 144
Curl, 22
curl
cylindrical coordinates [2.34], 22
general coordinates [2.36], 22
of curl [2.57], 23
rectangular coordinates [2.33], 22
spherical coordinates [2.35], 22
current
dimensions, 16
electric [7.139], 147
law (Kirchhoff’s) [7.161], 149
magnetic flux density from [7.11],
136
probability density [4.13], 90
thermodynamic work [5.9], 106
transformation [7.165], 149
current density
dimensions, 16
four-vector [7.76], 141
free [7.63], 140
free electron [6.60], 132
hole [6.89], 134
magnetic flux density [7.10], 136
curvature
in differential geomtry [2.286], 39
parameter (cosmic) [9.87], 184
radius of
and curvature [2.287], 39
plane curve [2.282], 39
curve length (plane curve) [2.279], 39
Curve measure, 39
Cycle efficiencies (thermodynamic), 107
cyclic permutation [2.97], 26
cyclotron frequency [7.265], 157
cylinder
area [2.269], 37
torsional rigidity [3.253], 81
volume [2.270], 37
cylinders (adjacent)
cylinders (coaxial)

194 Index
cylindrical polar coordinates, 21
D
d orbitals [4.100], 97
D’Alembertian [7.78], 141
damped harmonic oscillator [3.196], 78
damping profile [8.112], 173
day (unit), 5
day of week [9.3], 177
daylight saving time [9.4], 177
de Boer parameter [6.54], 131
de Broglie relation [4.2], 90
de Broglie wavelength (thermal) [5.83],
112
de Moivre’s theorem [2.214], 34
Debye
T3
law [6.47], 130
frequency [6.41], 130
function [6.49], 130
heat capacity [6.45], 130
length [7.251], 156
number [7.253], 156
screening [7.252], 156
Debye theory, 130
Debye-Waller factor [6.33], 128
deca, 5
decay constant [4.163], 103
decay law [4.163], 103
deceleration parameter [9.95], 184
deci, 5
decibel [5.144], 117
declination coordinate [9.11], 177
decrement (oscillating systems) [3.202],
78
Definite integrals, 46
degeneracy pressure [9.77], 183
degree (unit), 5
degree Celsius (unit), 4
degree kelvin [5.2], 106
degree of freedom (and equipartition), 113
degree of mutual coherence [8.99], 172
degree of polarisation [8.96], 171
degree of temporal coherence, 172
deka, 5
del operator, 21
del-squared operator, 23
del-squared operator [2.55], 23
Delta functions, 50
delta–star transformation, 149
densities of elements, 124
density (dimensions), 16
density of states
electron [6.70], 133
particle [4.66], 94
phonon [6.44], 130
density parameters [9.94], 184
depolarising factors [7.92], 142
Derivatives (general), 40
determinant [2.79], 25
deviation (of a prism) [8.73], 169
diamagnetic moment (electron) [7.108],
144
diamagnetic susceptibility (Landau) [6.80],
133
Diamagnetism, 144
Dielectric layers, 162
Dieterici gas, 111
Dieterici gas law [5.72], 111
Differential equations, 43
differential equations (numerical solutions),
62
Differential geometry, 39
Differential operator identities, 23
differential scattering cross-section [3.124],
72
Differentiation, 40
differentiation
hyperbolic functions, 41
numerical, 61
of a function of a function [2.295],
40
of a log [2.300], 40
of a power [2.292], 40
of a product [2.293], 40
of a quotient [2.294], 40
of exponential [2.301], 40
of integral [2.299], 40
of inverse functions [2.304], 40
trigonometric functions, 41
under integral sign [2.298], 40
diffraction from
N slits [8.25], 164
1 slit [8.37], 165
2 slits [8.24], 164
circular aperture [8.40], 165
crystals, 128
infinite grating [8.26], 164

Index
I
195
rectangular aperture [8.39], 165
diffraction grating
finite [8.25], 164
general, 164
infinite [8.26], 164
diffusion coefficient (semiconductor) [6.88],
134
diffusion equation
Fick’s first law [5.93], 113
diffusion length (semiconductor) [6.94],
134
diffusivity (magnetic) [7.282], 158
dilatation (volume strain) [3.236], 80
Dimensions, 16
diode (semiconductor) [6.92], 134
dioptre number [8.68], 168
dipole
antenna power
flux [7.131], 146
gain [7.213], 153
total [7.132], 146
electric field [7.31], 138
energy of
electric [7.136], 146
magnetic [7.137], 146
field from
magnetic [7.36], 138
moment (dimensions), 17
moment of
electric [7.80], 142
potential
radiation
field [7.207], 153
radiation resistance [7.209], 153
dipole moment per unit volume
Dirac bracket, 92
Dirac delta function [2.448], 50
Dirac equation [4.183], 104
Dirac matrices [4.185], 104
Dirac notation, 92
direct conductivity [7.279], 158
directrix (of conic section), 38
disc, see disk
discrete convolution, 60
Discrete probability distributions, 57
Discrete statistics, 57
disk
Airy [8.40], 165
centre of mass of sector [3.172], 76
coaxial capacitance [7.22], 137
drag in a fluid, 85
Dislocations and cracks, 128
dispersion
diffraction grating [8.31], 164
in a plasma [7.261], 157
in fluid waves, 86
in quantum physics [4.5], 90
in waveguides [7.188], 151
intermodal (optical fibre) [8.79], 169
measure [9.70], 182
of a prism [8.76], 169
phonon (alternating springs) [6.39],
129
phonon (diatomic chain) [6.37], 129
phonon (monatomic chain) [6.34],
129
pulsar [9.72], 182
displacement, D [7.86], 142
Distance indicators, 180
Divergence, 22
divergence
theorem [2.59], 23
dodecahedron, 38
Doppler
beaming [3.25], 65
effect (non-relativistic), 87
effect (relativistic) [3.22], 65
line broadening [8.116], 173
width [8.117], 173
Doppler effect, 87
dot product [2.1], 20
double factorial, 48
double pendulum [3.183], 76
Drag, 85

196 Index
drag
on a disk to flow [3.310], 85
on a disk ⊥ to flow [3.309], 85
on a sphere [3.308], 85
drift velocity (electron) [6.61], 132
Dulong and Petit’s law [6.46], 130
Dynamics and Mechanics, 63–88
Dynamics definitions, 68
E
e (exponential constant), 9
e to 1 000 decimal places, 18
Earth (motion relative to) [3.38], 66
Earth data, 176
eccentricity
of conic section, 38
of orbit [3.108], 71
of scattering hyperbola [3.120], 72
Ecliptic coordinates, 178
ecliptic latitude [9.14], 178
ecliptic longitude [9.15], 178
Eddington limit [9.59], 181
edge dislocation [6.21], 128
effective
area (antenna) [7.212], 153
distance (Fresnel diffraction) [8.48],
166
mass (in solids) [6.86], 134
wavelength [9.40], 179
efficiency
heat engine [5.10], 107
heat pump [5.12], 107
Otto cycle [5.13], 107
refrigerator [5.11], 107
Ehrenfest’s equations [5.53], 109
Ehrenfest’s theorem [4.30], 91
eigenfunctions (quantum) [4.28], 91
Einstein
A coefficient [8.119], 173
B coefficients [8.118], 173
diffusion equation [5.98], 113
field equation [3.59], 67
lens (rings) [9.50], 180
tensor [3.58], 67
Einstein - de Sitter model, 185
Einstein coefficients, 173
elastic
collisions, 73
media (isotropic), 81
modulus (longitudinal) [3.241], 81
modulus [3.234], 80
potential energy [3.235], 80
elastic scattering, 72
Elastic wave velocities, 82
Elasticity, 80
Elasticity definitions (general), 80
Elasticity definitions (simple), 80
electric current [7.139], 147
electric dipole, see dipole
electric displacement (dimensions), 16
electric displacement, D [7.86], 142
electric field
around objects, 138
static, 136
wave equation [7.193], 152
electric field from
A and φ [7.41], 139
charge distribution [7.6], 136
charge-sheet [7.32], 138
dipole [7.31], 138
disk [7.28], 138
line charge [7.29], 138
point charge [7.5], 136
sphere [7.27], 138
waveguide [7.190], 151
wire [7.29], 138
electric field strength (dimensions), 16
Electric fields, 138
electric polarisability (dimensions), 16
electric polarisation (dimensions), 16
electric potential
from a charge density [7.46], 139
Lorentz transformation [7.75], 141
of a moving charge [7.48], 139
short dipole [7.82], 142
electric potential difference (dimensions),
16
electric susceptibility, χE [7.87], 142
electrical conductivity, see conductivity
Electrical impedance, 148
electrical permittivity, , r [7.90], 142
electromagnet (magnetic flux density) [7.38],
138
electromagnetic
boundary conditions, 144
constants, 7

Index
I
197
fields, 139
wave speed [7.196], 152
waves in media, 152
electromagnetic coupling constant, see fine
structure constant
Electromagnetic energy, 146
Electromagnetic fields (general), 139
Electromagnetic force and torque, 145
Electromagnetic propagation in cold
plasmas, 157
Electromagnetism, 135–160
electron
charge, 6, 7
density of states [6.70], 133
diamagnetic moment [7.108], 144
drift velocity [6.61], 132
g-factor [4.143], 100
gyromagnetic ratio (value), 8
gyromagnetic ratio [4.140], 100
intrinsic magnetic moment [7.109],
144
mass, 6
radius (equation) [7.238], 155
radius (value), 8
scattering cross-section [7.238], 155
spin magnetic moment [4.143], 100
thermal velocity [7.257], 156
velocity in conductors [6.85], 134
Electron constants, 8
Electron scattering processes, 155
electron volt (unit), 5
electron volt (value), 6
Electrons in solids, 132
electrostatic potential [7.1], 136
Electrostatics, 136
elementary charge, 6, 7
elements (periodic table of), 124
ellipse, 38
(semi) centre of mass [3.178], 76
area [2.267], 37
perimeter [2.266], 37
semi-latus-rectum [3.109], 71
semi-major axis [3.106], 71
semi-minor axis [3.107], 71
ellipsoid
moment of inertia of solid [3.163],
75
the moment of inertia [3.147], 74
volume [2.268], 37
elliptic integrals [2.397], 45
elliptical orbit [3.104], 71
Elliptical polarisation, 170
elliptical polarisation [8.80], 170
ellipticity [8.82], 170
E =mc2
[3.72], 68
emission coefficient [5.174], 120
emission spectrum [7.291], 159
emissivity [5.193], 121
energy
density
blackbody [5.192], 121
dimensions, 16
elastic wave [3.281], 83
electromagnetic [7.128], 146
radiant [5.148], 118
spectral [5.173], 120
dimensions, 16
dissipated in resistor [7.155], 148
distribution (Maxwellian) [5.85], 112
elastic [3.235], 80
equipartition [5.100], 113
Fermi [5.122], 115
first law of thermodynamics [5.3],
106
Galilean transformation [3.6], 64
kinetic , see kinetic energy
loss after collision [3.128], 73
mass relation [3.20], 65
of capacitive assembly [7.134], 146
of capacitor [7.153], 148
of charge distribution [7.133], 146
of electric dipole [7.136], 146
of inductive assembly [7.135], 146
of inductor [7.154], 148
of magnetic dipole [7.137], 146
of orbit [3.100], 71
potential , see potential energy
relativistic rest [3.72], 68
rotational kinetic
rigid body [3.142], 74
w.r.t. principal axes [3.145], 74
thermodynamic work, 106
Energy in capacitors, inductors, and
resistors, 148

198 Index
energy-time uncertainty relation [4.8], 90
Ensemble probabilities, 114
enthalpy
Joule-Kelvin expansion [5.27], 108
entropy
Boltzmann formula [5.105], 114
change in Joule expansion [5.64],
110
experimental [5.4], 106
fluctuations [5.135], 116
Gibbs formula [5.106], 114
of a monatomic gas [5.83], 112
entropy (dimensions), 16
, r (electrical permittivity) [7.90], 142
Equation conversion: SI to Gaussian
units, 135
equation of state
Dieterici gas [5.72], 111
ideal gas [5.57], 110
monatomic gas [5.78], 112
van der Waals gas [5.67], 111
equipartition theorem [5.100], 113
error function [2.390], 45
errors, 60
escape velocity [3.91], 70
estimator
kurtosis [2.545], 57
mean [2.541], 57
skewness [2.544], 57
standard deviation [2.543], 57
variance [2.542], 57
Euler
angles [2.101], 26
constant
expression [2.119], 27
value, 9
formula [2.216], 34
relation, 38
strut [3.261], 82
Euler’s equation (fluids) [3.289], 84
Euler’s equations (rigid bodies) [3.186],
77
Euler’s method (for ordinary differential
equations) [2.596], 62
Euler-Lagrange equation
and Lagrangians [3.214], 79
calculus of variations [2.334], 42
even functions, 53
Evolutionary timescales, 181
exa, 5
exhaust velocity (of a rocket) [3.93], 70
exitance
luminous [5.162], 119
radiant [5.150], 118
exp(x) [2.132], 29
expansion coefficient [5.19], 107
Expansion processes, 108
expansivity [5.19], 107
Expectation value, 91
expectation value
Dirac notation [4.37], 92
from a wavefunction [4.25], 91
explosions [3.331], 87
exponential
integral [2.394], 45
exponential constant (e), 9
extraordinary modes [7.271], 157
extrema [2.335], 42
F
f-number [8.69], 168
Fabry-Perot etalon
chromatic resolving power [8.21], 163
free spectral range [8.23], 163
fringe width [8.19], 163
transmitted intensity [8.17], 163
Fabry-Perot etalon, 163
face-centred cubic structure, 127
factorial [2.409], 46
factorial (double), 48
Fahrenheit conversion [1.2], 15
faltung theorem [2.516], 55
farad (unit), 4
Faraday constant, 6, 9
Faraday constant (dimensions), 16
Faraday rotation [7.273], 157
Faraday’s law [7.55], 140
fcc structure, 127
Feigenbaum’s constants, 9
femto, 5
Fermat’s principle [8.63], 168
Fermi

Index
I
199
energy [6.73], 133
velocity [6.72], 133
wavenumber [6.71], 133
fermi (unit), 5
Fermi energy [5.122], 115
Fermi gas, 133
Fermi’s golden rule [4.162], 102
Fermi–Dirac distribution [5.121], 115
fermion statistics [5.121], 115
fibre optic
acceptance angle [8.77], 169
dispersion [8.79], 169
numerical aperture [8.78], 169
Fick’s first law [5.92], 113
Fick’s second law [5.95], 113
field equations (gravitational) [3.42], 66
Field relationships, 139
fields
depolarising [7.92], 142
electrochemical [6.81], 133
gravitational, 66
static E and B, 136
Fields associated with media, 142
film reflectance [8.4], 162
fine-structure constant
expression [4.75], 95
value, 6, 7
finesse (coefficient of) [8.12], 163
finesse (Fabry-Perot etalon) [8.14], 163
first law of thermodynamics [5.3], 106
fitting straight-lines, 60
fluctuating dipole interaction [6.50], 131
fluctuation
of density [5.137], 116
of entropy [5.135], 116
of pressure [5.136], 116
of temperature [5.133], 116
of volume [5.134], 116
probability (thermodynamic) [5.131],
116
variance (general) [5.132], 116
Fluctuations and noise, 116
Fluid dynamics, 84
fluid stress [3.299], 85
Fluid waves, 86
flux density [5.171], 120
flux density–redshift relation [9.99], 185
flux linked [7.149], 147
flux of molecules through a plane [5.91],
113
flux–magnitude relation [9.32], 179
focal length [8.64], 168
focus (of conic section), 38
force
and acoustic impedance [3.276], 83
and stress [3.228], 80
between two charges [7.119], 145
between two currents [7.120], 145
between two masses [3.40], 66
central [4.113], 98
centrifugal [3.35], 66
Coriolis [3.33], 66
critical compression [3.261], 82
dimensions, 16
Newtonian [3.63], 68
on
charge in a field [7.122], 145
current in a field [7.121], 145
electric dipole [7.123], 145
magnetic dipole [7.124], 145
sphere (potential flow) [3.298], 84
sphere (viscous drag) [3.308], 85
unit, 4
Force, torque, and energy, 145
Forced oscillations, 78
form factor [6.30], 128
formula (the) [2.455], 50
Foucault’s pendulum [3.39], 66
four-parts formula [2.259], 36
four-scalar product [3.27], 65
four-vector
momentum [3.21], 65
spacetime [3.12], 64
Four-vectors, 65
Fourier series
complex form [2.478], 52
real form [2.476], 52
Fourier series, 52
Fourier series and transforms, 52
Fourier symmetry relationships, 53
Fourier transform

200 Index
cosine [2.509], 54
derivatives
and inverse [2.502], 54
general [2.498], 53
Gaussian [2.507], 54
Lorentzian [2.505], 54
shah function [2.510], 54
shift theorem [2.501], 54
similarity theorem [2.500], 54
sine [2.508], 54
step [2.511], 54
top hat [2.512], 54
triangle function [2.513], 54
Fourier transform, 52
Fourier transform pairs, 54
Fourier transform theorems, 53
Fourier’s law [5.94], 113
Frames of reference, 64
Fraunhofer diffraction, 164
Fraunhofer integral [8.34], 165
Fraunhofer limit [8.44], 165
free charge density [7.57], 140
free current density [7.63], 140
Free electron transport properties, 132
free energy [5.32], 108
free molecular flow [5.99], 113
Free oscillations, 78
free space impedance [7.197], 152
free spectral range
Fabry Perot etalon [8.23], 163
laser cavity [8.124], 174
free-fall timescale [9.53], 181
Frenet’s formulas [2.291], 39
frequency (dimensions), 16
Fresnel diffraction
Cornu spiral [8.54], 167
edge [8.56], 167
long slit [8.58], 167
rectangular aperture [8.62], 167
Fresnel diffraction, 166
Fresnel Equations, 154
Fresnel half-period zones [8.49], 166
Fresnel integrals
and the Cornu spiral [8.52], 167
in diffraction [8.54], 167
Fresnel zones, 166
Fresnel-Kirchhoff formula
plane waves [8.45], 166
spherical waves [8.47], 166
Friedmann equations [9.89], 184
fringe visibility [8.101], 172
fringes (Moiré), 35
Froude number [3.312], 86
G
g-factor
electron, 8
Landé [4.146], 100
muon, 9
gain in decibels [5.144], 117
galactic
coordinates [9.20], 178
latitude [9.21], 178
longitude [9.22], 178
Galactic coordinates, 178
Galilean transformation
of angular momentum [3.5], 64
of kinetic energy [3.6], 64
of momentum [3.4], 64
of time and position [3.2], 64
of velocity [3.3], 64
Galilean transformations, 64
Gamma function, 46
gamma function
and other integrals [2.395], 45
gas
adiabatic expansion [5.58], 110
adiabatic lapse rate [3.294], 84
constant, 6, 9, 86, 110
Dieterici, 111
Doppler broadened [8.116], 173
flow [3.292], 84
giant (astronomical data), 176
ideal equation of state [5.57], 110
ideal heat capacities, 113
ideal, or perfect, 110
internal energy (ideal) [5.62], 110
isothermal expansion [5.63], 110
linear absorption coefficient [5.175],
120
molecular flow [5.99], 113
monatomic, 112
paramagnetism [7.112], 144
pressure broadened [8.115], 173
speed of sound [3.318], 86

Index
I
201
temperature scale [5.1], 106
Van der Waals, 111
Gas equipartition, 113
Gas laws, 110
gauge condition
Coulomb [7.42], 139
Lorenz [7.43], 139
Gaunt factor [7.299], 160
Gauss’s
law [7.51], 140
lens formula [8.64], 168
theorem [2.59], 23
Gaussian
electromagnetism, 135
Fourier transform of [2.507], 54
integral [2.398], 46
light [8.110], 172
optics, 168
probability distribution
k-dimensional [2.556], 58
1-dimensional [2.552], 58
Geiger’s law [4.169], 103
Geiger-Nuttall rule [4.170], 103
General constants, 7
General relativity, 67
generalised coordinates [3.213], 79
Generalised dynamics, 79
generalised momentum [3.218], 79
geodesic deviation [3.56], 67
geodesic equation [3.54], 67
geometric
mean [2.109], 27
progression [2.107], 27
Geometrical optics, 168
Gibbs
constant (value), 9
entropy [5.106], 114
free energy [5.35], 108
Gibbs’s phase rule [5.54], 109
Gibbs–Helmholtz equations, 109
Gibbs-Duhem relation [5.38], 108
giga, 5
golden mean (value), 9
golden rule (Fermi’s) [4.162], 102
Gradient, 21
gradient
gram (use in SI), 5
grand canonical ensemble [5.113], 114
grand partition function [5.112], 114
grand potential
from grand partition function [5.115],
115
grating
formula [8.27], 164
resolving power [8.30], 164
Gratings, 164
Gravitation, 66
gravitation
field from a sphere [3.44], 66
general relativity, 67
Newton’s law [3.40], 66
Newtonian, 71
Newtonian field equations [3.42], 66
gravitational
collapse [9.53], 181
constant, 6, 7, 16
lens [9.50], 180
potential [3.42], 66
redshift [9.74], 183
wave radiation [9.75], 183
Gravitationally bound orbital motion,
71
gravity
and motion on Earth [3.38], 66
waves (on a fluid surface) [3.320],
86
gray (unit), 4
Greek alphabet, 18
Green’s first theorem [2.62], 23
Green’s second theorem [2.63], 23
Greenwich sidereal time [9.6], 177
Gregory’s series [2.141], 29
greybody [5.193], 121
group speed (wave) [3.327], 87
Grüneisen parameter [6.56], 131
gyro-frequency [7.265], 157
gyro-radius [7.268], 157
gyromagnetic ratio
electron [4.140], 100

202 Index
proton (value), 8
gyroscopes, 77
gyroscopic
limit [3.193], 77
nutation [3.194], 77
precession [3.191], 77
stability [3.192], 77
H
H (magnetic field strength) [7.100], 143
half-life (nuclear decay) [4.164], 103
half-period zones (Fresnel) [8.49], 166
Hall
coefficient (dimensions), 16
conductivity [7.280], 158
effect and coefficient [6.67], 132
voltage [6.68], 132
Hamilton’s equations [3.220], 79
Hamilton’s principal function [3.213], 79
Hamilton-Jacobi equation [3.227], 79
Hamiltonian
charged particle (Newtonian) [7.138],
146
charged particle [3.223], 79
of a particle [3.222], 79
quantum mechanical [4.21], 91
Hamiltonian (dimensions), 16
Hamiltonian dynamics, 79
Hamming window [2.584], 60
Hanbury Brown and Twiss interferometry,
172
Hanning window [2.583], 60
harmonic mean [2.110], 27
Harmonic oscillator, 95
harmonic oscillator
damped [3.196], 78
energy levels [4.68], 95
entropy [5.108], 114
forced [3.204], 78
mean energy [6.40], 130
Hartree energy [4.76], 95
Heat capacities, 107
heat capacity (dimensions), 16
heat capacity in solids
Debye [6.45], 130
heat capacity of a gas
Cp −CV [5.17], 107
constant pressure [5.15], 107
constant volume [5.14], 107
for f degrees of freedom, 113
ratio (γ) [5.18], 107
heat conduction/diffusion equation
Fick’s second law [5.96], 113
heat engine efficiency [5.10], 107
heat pump efficiency [5.12], 107
heavy beam [3.260], 82
hectare, 12
hecto, 5
Heisenberg uncertainty relation [4.7], 90
Helmholtz equation [2.341], 43
Helmholtz free energy
hemisphere (centre of mass) [3.170], 76
hemispherical shell (centre of mass) [3.171],
76
henry (unit), 4
Hermite equation [2.346], 43
Hermite polynomials [4.70], 95
Hermitian
conjugate operator [4.17], 91
matrix [2.73], 24
symmetry, 53
Heron’s formula [2.253], 36
herpolhode, 63, 77
hertz (unit), 4
Hertzian dipole [7.207], 153
hexagonal system (crystallographic), 127
High energy and nuclear physics, 103
Hohmann cotangential transfer [3.98],
70
hole current density [6.89], 134
Hooke’s law [3.230], 80
l’Hôpital’s rule [2.131], 28
Horizon coordinates, 177
hour (unit), 5
hour angle [9.8], 177
Hubble constant (dimensions), 16
Hubble constant [9.85], 184
Hubble law
as a distance indicator [9.45], 180
in cosmology [9.83], 184
hydrogen atom
eigenfunctions [4.80], 96
energy [4.81], 96

Index
I
203
Schrödinger equation [4.79], 96
Hydrogenic atoms, 95
Hydrogenlike atoms – Schrödinger so-
lution, 96
hydrostatic
compression [3.238], 80
condition [3.293], 84
equilibrium (of a star) [9.61], 181
hyperbola, 38
Hyperbolic derivatives, 41
hyperbolic motion, 72
Hyperbolic relationships, 33
I
I (Stokes parameter) [8.89], 171
icosahedron, 38
Ideal fluids, 84
Ideal gas, 110
ideal gas
adiabatic equations [5.58], 110
internal energy [5.62], 110
isothermal reversible expansion [5.63],
110
law [5.57], 110
Identical particles, 115
illuminance (definition) [5.164], 119
illuminance (dimensions), 16
Image charges, 138
impedance
acoustic [3.276], 83
dimensions, 17
electrical, 148
impedance of
capacitor [7.159], 148
coaxial transmission line [7.181], 150
electromagnetic wave [7.198], 152
forced harmonic oscillator [3.212],
78
free space
value, 7
inductor [7.160], 148
lossless transmission line [7.174], 150
lossy transmission line [7.175], 150
microstrip line [7.184], 150
open-wire transmission line [7.182],
150
paired strip transmission line [7.183],
150
terminated transmission line [7.178],
150
waveguide
TE modes [7.189], 151
TM modes [7.188], 151
impedances
in parallel [7.158], 148
in series [7.157], 148
impulse (dimensions), 17
impulse (specific) [3.92], 70
incompressible flow, 84, 85
indefinite integrals, 44
induced charge density [7.84], 142
Inductance, 137
inductance
dimensions, 17
energy [7.154], 148
energy of an assembly [7.135], 146
impedance [7.160], 148
mutual
energy [7.135], 146
self [7.145], 147
voltage across [7.146], 147
inductance of
cylinders (coaxial) [7.24], 137
solenoid [7.23], 137
wire loop [7.26], 137
wires (parallel) [7.25], 137
induction equation (MHD) [7.282], 158
inductor, see inductance
Inelastic collisions, 73
Inequalities, 30
inertia tensor [3.136], 74
inner product [2.1], 20
Integration, 44
integration (numerical), 61
integration by parts [2.354], 44
intensity
correlation [8.109], 172
luminous [5.166], 119
of interfering beams [8.100], 172
radiant [5.154], 118
specific [5.171], 120
Interference, 162
interference and coherence [8.100], 172
intermodal dispersion (optical fibre) [8.79],

204 Index
169
internal energy
ideal gas [5.62], 110
Joule’s law [5.55], 110
monatomic gas [5.79], 112
interval (in general relativity) [3.45], 67
invariable plane, 63, 77
inverse Compton scattering [7.239], 155
Inverse hyperbolic functions, 35
inverse Laplace transform [2.518], 55
inverse matrix [2.83], 25
inverse square law [3.99], 71
Inverse trigonometric functions, 34
ionic bonding [6.55], 131
irradiance (definition) [5.152], 118
irradiance (dimensions), 17
isobaric expansivity [5.19], 107
isophotal wavelength [9.39], 179
isothermal bulk modulus [5.22], 107
isothermal compressibility [5.20], 107
Isotropic elastic solids, 81
J
Jacobi identity [2.93], 26
Jacobian
in change of variable [2.333], 42
Jeans length [9.56], 181
Jeans mass [9.57], 181
Johnson noise [5.141], 117
joint probability [2.568], 59
Jones matrix [8.85], 170
Jones vectors
examples [8.84], 170
Jones vectors and matrices, 170
Josephson frequency-voltage ratio, 7
joule (unit), 4
Joule expansion (and Joule coefficient) [5.25],
108
Joule expansion (entropy change) [5.64],
110
Joule’s law (of internal energy) [5.55],
110
Joule’s law (of power dissipation) [7.155],
148
Joule-Kelvin coefficient [5.27], 108
Julian centuries [9.5], 177
Julian day number [9.1], 177
Jupiter data, 176
K
katal (unit), 4
Kelvin
circulation theorem [3.287], 84
relation [6.83], 133
temperature conversion, 15
temperature scale [5.2], 106
wedge [3.330], 87
kelvin (SI definition), 3
kelvin (unit), 4
Kelvin-Helmholtz timescale [9.55], 181
Kepler’s laws, 71
Kepler’s problem, 71
Kerr solution (in general relativity) [3.62],
67
ket vector [4.34], 92
kilo, 5
kilogram (SI definition), 3
kilogram (unit), 4
kinematic viscosity [3.302], 85
kinematics, 63
kinetic energy
for a rotating body [3.142], 74
Galilean transformation [3.6], 64
in the virial theorem [3.102], 71
loss after collision [3.128], 73
of a particle [3.216], 79
of monatomic gas [5.79], 112
operator (quantum) [4.20], 91
w.r.t. principal axes [3.145], 74
Kinetic theory, 112
Kirchhoff’s (radiation) law [5.180], 120
Kirchhoff’s diffraction formula, 166
Kirchhoff’s laws, 149
Klein–Nishina cross section [7.243], 155
Klein-Gordon equation [4.181], 104
Knudsen flow [5.99], 113
Kronecker delta [2.442], 50
kurtosis estimator [2.545], 57
L
ladder operators (angular momentum) [4.108],
98

Index
I
205
Lagrange’s identity [2.7], 20
Lagrangian (dimensions), 17
Lagrangian dynamics, 79
Lagrangian of
charged particle [3.217], 79
particle [3.216], 79
two mutually attracting bodies [3.85],
69
Laguerre equation [2.347], 43
Laguerre polynomials (associated), 96
Lamé coefficients [3.240], 81
Laminar viscous flow, 85
Landé g-factor [4.146], 100
Landau diamagnetic susceptibility [6.80],
133
Landau length [7.249], 156
Langevin function (from Brillouin fn) [4.147],
101
Langevin function [7.111], 144
Laplace equation
solution in spherical harmonics [2.440],
49
Laplace series [2.439], 49
Laplace transform
convolution [2.516], 55
derivative of transform [2.520], 55
inverse [2.518], 55
of derivative [2.519], 55
substitution [2.521], 55
translation [2.523], 55
Laplace transform pairs, 56
Laplace transform theorems, 55
Laplace transforms, 55
Laplace’s formula (surface tension) [3.337],
88
Laplacian
Laplacian (scalar), 23
lapse rate (adiabatic) [3.294], 84
Larmor frequency [7.265], 157
Larmor radius [7.268], 157
Larmor’s formula [7.132], 146
laser
cavity Q [8.126], 174
cavity line width [8.127], 174
cavity modes [8.124], 174
cavity stability [8.123], 174
threshold condition [8.129], 174
Lasers, 174
latent heat [5.48], 109
lattice constants of elements, 124
Lattice dynamics, 129
Lattice forces (simple), 131
lattice plane spacing [6.11], 126
Lattice thermal expansion and conduc-
tion, 131
lattice vector [6.7], 126
latus-rectum [3.109], 71
Laue equations [6.28], 128
Laurent series [2.168], 31
LCR circuits, 147
LCR definitions, 147
least-squares fitting, 60
Legendre equation
and polynomials [2.421], 47
Legendre polynomials, 47
Leibniz theorem [2.296], 40
length (dimensions), 17
Lennard-Jones 6-12 potential [6.52], 131
lens blooming [8.7], 162
Lenses and mirrors, 168
lensmaker’s formula [8.66], 168
Levi-Civita symbol (3-D) [2.443], 50
l’Hôpital’s rule [2.131], 28
Liénard–Wiechert potentials, 139
light (speed of), 6, 7
Limits, 28
line charge (electric field from) [7.29], 138
line fitting, 60
Line radiation, 173
line shape
collisional [8.114], 173
Doppler [8.116], 173
natural [8.112], 173
line width
collisional/pressure [8.115], 173
Doppler broadened [8.117], 173
natural [8.113], 173
Schawlow-Townes [8.128], 174
linear absorption coefficient [5.175], 120
linear expansivity (definition) [5.19], 107

206 Index
linear expansivity (of a crystal) [6.57],
131
linear regression, 60
linked flux [7.149], 147
liquid drop model [4.172], 103
litre (unit), 5
local civil time [9.4], 177
local sidereal time [9.7], 177
local thermodynamic equilibrium (LTE),
116, 120
ln(1+x) (series expansion) [2.133], 29
logarithm of complex numbers [2.162],
30
logarithmic decrement [3.202], 78
London’s formula (interacting dipoles) [6.50],
131
longitudinal elastic modulus [3.241], 81
look-back time [9.96], 185
Lorentz
contraction [3.8], 64
factor (γ) [3.7], 64
force [7.122], 145
Lorentz (spacetime) transformations, 64
Lorentz factor (dynamical) [3.69], 68
Lorentz transformation
in electrodynamics, 141
of four-vectors, 65
of momentum and energy, 65
of time and position, 64
of velocity, 64
Lorentz-Lorenz formula [7.93], 142
Lorentzian distribution [2.555], 58
Lorentzian (Fourier transform of) [2.505],
54
Lorenz
constant [6.66], 132
gauge condition [7.43], 139
lumen (unit), 4
luminance [5.168], 119
luminosity distance [9.98], 185
luminosity–magnitude relation [9.31], 179
luminous
density [5.160], 119
efficacy [5.169], 119
efficiency [5.170], 119
energy [5.157], 119
exitance [5.162], 119
flux [5.159], 119
intensity (dimensions), 17
intensity [5.166], 119
lux (unit), 4
M
Mach number [3.315], 86
Mach wedge [3.328], 87
Maclaurin series [2.125], 28
Macroscopic thermodynamic variables,
115
Madelung constant (value), 9
Madelung constant [6.55], 131
magnetic
diffusivity [7.282], 158
flux quantum, 6, 7
monopoles (none) [7.52], 140
permeability, µ, µr [7.107], 143
quantum number [4.131], 100
scalar potential [7.7], 136
susceptibility, χH , χB [7.103], 143
vector potential
from J [7.47], 139
of a moving charge [7.49], 139
magnetic dipole, see dipole
magnetic field
around objects, 138
dimensions, 17
static, 136
strength (H) [7.100], 143
Magnetic fields, 138
magnetic flux (dimensions), 17
magnetic flux density (dimensions), 17
magnetic flux density from
current [7.11], 136
current density [7.10], 136
dipole [7.36], 138
electromagnet [7.38], 138
line current (Biot–Savart law) [7.9],
136
solenoid (finite) [7.38], 138
solenoid (infinite) [7.33], 138
uniform cylindrical current [7.34],
138
waveguide [7.190], 151

Index
I
207
wire [7.34], 138
wire loop [7.37], 138
Magnetic moments, 100
magnetic vector potential (dimensions), 17
Magnetisation, 143
magnetisation
dimensions, 17
isolated spins [4.151], 101
quantum paramagnetic [4.150], 101
magnetogyric ratio [4.138], 100
Magnetohydrodynamics, 158
magnetosonic waves [7.285], 158
Magnetostatics, 136
magnification (longitudinal) [8.71], 168
magnification (transverse) [8.70], 168
magnitude (astronomical)
–flux relation [9.32], 179
–luminosity relation [9.31], 179
absolute [9.29], 179
apparent [9.27], 179
major axis [3.106], 71
Malus’s law [8.83], 170
Mars data, 176
mass (dimensions), 17
mass absorption coefficient [5.176], 120
mass ratio (of a rocket) [3.94], 70
Mathematical constants, 9
Mathematics, 19–62
matrices (square), 25
Matrix algebra, 24
matrix element (quantum) [4.32], 92
maxima [2.336], 42
Maxwell’s equations, 140
Maxwell’s equations (using D and H),
140
Maxwell’s relations, 109
Maxwell–Boltzmann distribution, 112
Maxwell-Boltzmann distribution
mean speed [5.86], 112
most probable speed [5.88], 112
rms speed [5.87], 112
speed distribution [5.84], 112
mean
arithmetic [2.108], 27
geometric [2.109], 27
harmonic [2.110], 27
mean estimator [2.541], 57
mean free path
and absorption coefficient [5.175],
120
Maxwell-Boltzmann [5.89], 113
mean intensity [5.172], 120
mean-life (nuclear decay) [4.165], 103
mega, 5
melting points of elements, 124
meniscus [3.339], 88
Mensuration, 35
Mercury data, 176
method of images, 138
metre (SI definition), 3
metre (unit), 4
metric elements and coordinate systems,
21
MHD equations [7.283], 158
micro, 5
microcanonical ensemble [5.109], 114
micron (unit), 5
microstrip line (impedance) [7.184], 150
Miller-Bravais indices [6.20], 126
milli, 5
minima [2.337], 42
minimum deviation (of a prism) [8.74],
169
minor axis [3.107], 71
minute (unit), 5
mirror formula [8.67], 168
Miscellaneous, 18
mobility (dimensions), 17
mobility (in conductors) [6.88], 134
modal dispersion (optical fibre) [8.79],
169
modified Bessel functions [2.419], 47
modified Julian day number [9.2], 177
modulus (of a complex number) [2.155],
30
Moiré fringes, 35
molar gas constant (dimensions), 17
molar volume, 9
mole (SI definition), 3
mole (unit), 4
molecular flow [5.99], 113
moment
electric dipole [7.81], 142
moment of area [3.258], 82
moment of inertia

208 Index
cone [3.160], 75
cylinder [3.155], 75
dimensions, 17
disk [3.168], 75
ellipsoid [3.163], 75
elliptical lamina [3.166], 75
rectangular cuboid [3.158], 75
sphere [3.152], 75
spherical shell [3.153], 75
thin rod [3.150], 75
triangular plate [3.169], 75
two-body system [3.83], 69
moment of inertia ellipsoid [3.147], 74
Moment of inertia tensor, 74
moment of inertia tensor [3.136], 74
Moments of inertia, 75
momentum
dimensions, 17
generalised [3.218], 79
Momentum and energy transformations,
65
Monatomic gas, 112
monatomic gas
entropy [5.83], 112
equation of state [5.78], 112
internal energy [5.79], 112
pressure [5.77], 112
monoclinic system (crystallographic), 127
Moon data, 176
motif [6.31], 128
motion under constant acceleration, 68
Mott scattering formula [4.180], 104
µ, µr (magnetic permeability) [7.107], 143
multilayer films (in optics) [8.8], 162
multimode dispersion (optical fibre) [8.79],
169
multiplicity (quantum)
j [4.133], 100
l [4.112], 98
multistage rocket [3.95], 70
Multivariate normal distribution, 58
Muon and tau constants, 9
muon physical constants, 9
mutual
inductance (definition) [7.147], 147
inductance (energy) [7.135], 146
mutual coherence function [8.97], 172
N
nabla, 21
Named integrals, 45
nano, 5
natural broadening profile [8.112], 173
natural line width [8.113], 173
Navier-Stokes equation [3.301], 85
nearest neighbour distances, 127
Neptune data, 176
neutron
Compton wavelength, 8
gyromagnetic ratio, 8
magnetic moment, 8
mass, 8
molar mass, 8
Neutron constants, 8
neutron star degeneracy pressure [9.77],
183
newton (unit), 4
Newton’s law of Gravitation [3.40], 66
Newton’s lens formula [8.65], 168
Newton’s rings, 162
Newton’s rings [8.1], 162
Newton-Raphson method [2.593], 61
Newtonian gravitation, 66
noggin, 13
Noise, 117
noise
figure [5.143], 117
Johnson [5.141], 117
Nyquist’s theorem [5.140], 117
shot [5.142], 117
normal (unit principal) [2.284], 39
normal distribution [2.552], 58
normal plane, 39
Nuclear binding energy, 103
Nuclear collisions, 104
Nuclear decay, 103
nuclear decay law [4.163], 103
nuclear magneton, 7
number density (dimensions), 17
numerical aperture (optical fibre) [8.78],
169
Numerical differentiation, 61
Numerical integration, 61

Index
I
209
Numerical methods, 60
Numerical solutions to f(x)=0, 61
Numerical solutions to ordinary dif-
ferential equations, 62
nutation [3.194], 77
Nyquist’s theorem [5.140], 117
O
Oblique elastic collisions, 73
obliquity factor (diffraction) [8.46], 166
obliquity of the ecliptic [9.13], 178
observable (quantum physics) [4.5], 90
Observational astrophysics, 179
octahedron, 38
odd functions, 53
ODEs (numerical solutions), 62
ohm (unit), 4
Ohm’s law (in MHD) [7.281], 158
Ohm’s law [7.140], 147
opacity [5.176], 120
open-wire transmission line [7.182], 150
operator
angular momentum
and other operators [4.23], 91
definitions [4.105], 98
Hamiltonian [4.21], 91
kinetic energy [4.20], 91
momentum [4.19], 91
parity [4.24], 91
position [4.18], 91
time dependence [4.27], 91
Operators, 91
optic branch (phonon) [6.37], 129
optical coating [8.8], 162
optical depth [5.177], 120
Optical fibres, 169
optical path length [8.63], 168
Optics, 161–174
Orbital angular dependence, 97
Orbital angular momentum, 98
orbital motion, 71
orbital radius (Bohr atom) [4.73], 95
order (in diffraction) [8.26], 164
ordinary modes [7.271], 157
orthogonal matrix [2.85], 25
orthogonality
associated Legendre functions [2.434],
48
Legendre polynomials [2.424], 47
orthorhombic system (crystallographic), 127
Oscillating systems, 78
osculating plane, 39
Otto cycle efficiency [5.13], 107
overdamping [3.201], 78
P
p orbitals [4.95], 97
P-waves [3.263], 82
packing fraction (of spheres), 127
paired strip (impedance of) [7.183], 150
parabola, 38
parabolic motion [3.88], 69
parallax (astronomical) [9.46], 180
parallel axis theorem [3.140], 74
parallel impedances [7.158], 148
parallel wire feeder (inductance) [7.25],
137
paramagnetic susceptibility (Pauli) [6.79],
133
paramagnetism (quantum), 101
Paramagnetism and diamagnetism, 144
parity operator [4.24], 91
Parseval’s relation [2.495], 53
Parseval’s theorem
integral form [2.496], 53
series form [2.480], 52
Partial derivatives, 42
partial widths (and total width) [4.176],
104
Particle in a rectangular box, 94
Particle motion, 68
partition function
atomic [5.126], 116
macroscopic variables from, 115
pascal (unit), 4
Pauli matrices, 26
Pauli matrices [2.94], 26
Pauli paramagnetic susceptibility [6.79],
133
Pauli spin matrices (and Weyl eqn.) [4.182],
104
Pearson’s r [2.546], 57
Peltier effect [6.82], 133
pendulum
compound [3.182], 76
conical [3.180], 76
double [3.183], 76

210 Index
simple [3.179], 76
torsional [3.181], 76
Pendulums, 76
perfect gas, 110
pericentre (of an orbit) [3.110], 71
perimeter
of circle [2.261], 37
of ellipse [2.266], 37
Perimeter, area, and volume, 37
period (of an orbit) [3.113], 71
Periodic table, 124
permeability
dimensions, 17
magnetic [7.107], 143
of vacuum, 6, 7
permittivity
dimensions, 17
electrical [7.90], 142
of vacuum, 6, 7
permutation tensor ( ijk) [2.443], 50
perpendicular axis theorem [3.148], 74
Perturbation theory, 102
peta, 5
petrol engine efficiency [5.13], 107
phase object (diffraction by weak) [8.43],
165
phase rule (Gibbs’s) [5.54], 109
phase speed (wave) [3.325], 87
Phase transitions, 109
Phonon dispersion relations, 129
phonon modes (mean energy) [6.40], 130
Photometric wavelengths, 179
Photometry, 119
photon energy [4.3], 90
Physical constants, 6
Pi (π) to 1 000 decimal places, 18
Pi (π), 9
pico, 5
pipe (flow of fluid along) [3.305], 85
pipe (twisting of) [3.255], 81
pitch angle, 159
Planck
constant, 6, 7
constant (dimensions), 17
function [5.184], 121
length, 7
mass, 7
time, 7
Planck-Einstein relation [4.3], 90
plane polarisation, 170
Plane triangles, 36
plane wave expansion [2.427], 47
Planetary bodies, 180
Planetary data, 176
plasma
beta [7.278], 158
dispersion relation [7.261], 157
frequency [7.259], 157
group velocity [7.264], 157
phase velocity [7.262], 157
refractive index [7.260], 157
Plasma physics, 156
Platonic solids, 38
Pluto data, 176
p-n junction [6.92], 134
Poincaré sphere, 171
point charge (electric field from) [7.5],
136
Poiseuille flow [3.305], 85
Poisson brackets [3.224], 79
Poisson distribution [2.549], 57
Poisson ratio
and elastic constants [3.251], 81
simple definition [3.231], 80
Poisson’s equation [7.3], 136
polarisability [7.91], 142
Polarisation, 170
Polarisation, 142
polarisation (electrical, per unit volume)
[7.83], 142
polarisation (of radiation)
angle [8.81], 170
axial ratio [8.88], 171
degree of [8.96], 171
elliptical [8.80], 170
ellipticity [8.82], 170
reflection law [7.218], 154
polarisers [8.85], 170
polhode, 63, 77
Population densities, 116
potential
chemical [5.28], 108
difference (and work) [5.9], 106
difference (between points) [7.2], 136
electrical [7.46], 139
electrostatic [7.1], 136
energy (elastic) [3.235], 80
energy in Hamiltonian [3.222], 79

Index
I
211
energy in Lagrangian [3.216], 79
field equations [7.45], 139
four-vector [7.77], 141
grand [5.37], 108
Liénard–Wiechert, 139
magnetic scalar [7.7], 136
magnetic vector [7.40], 139
thermodynamic [5.35], 108
Potential flow, 84
Potential step, 92
Potential well, 93
power (dimensions), 17
power gain
antenna [7.211], 153
short dipole [7.213], 153
Power series, 28
Power theorem [2.495], 53
Poynting vector (dimensions), 17
Poynting vector [7.130], 146
pp (proton-proton) chain, 182
Prandtl number [3.314], 86
precession (gyroscopic) [3.191], 77
Precession of equinoxes, 178
pressure
critical [5.75], 111
degeneracy [9.77], 183
dimensions, 17
fluctuations [5.136], 116
hydrostatic [3.238], 80
in a monatomic gas [5.77], 112
radiation, 152
waves [3.263], 82
primitive cell [6.1], 126
primitive vectors (and lattice vectors) [6.7],
126
primitive vectors (of cubic lattices), 127
Principal axes, 74
principal moments of inertia [3.143], 74
principal quantum number [4.71], 95
principle of least action [3.213], 79
prism
determining refractive index [8.75],
169
deviation [8.73], 169
minimum deviation [8.74], 169
transmission angle [8.72], 169
Prisms (dispersing), 169
probability
conditional [2.567], 59
density current [4.13], 90
distributions
continuous, 58
discrete, 57
joint [2.568], 59
Probability and statistics, 57
product (derivative of) [2.293], 40
product (integral of) [2.354], 44
product of inertia [3.136], 74
progression (arithmetic) [2.104], 27
progression (geometric) [2.107], 27
Progressions and summations, 27
projectiles, 69
propagation in cold plasmas, 157
Propagation in conducting media, 155
Propagation of elastic waves, 83
Propagation of light, 65
proper distance [9.97], 185
Proton constants, 8
proton mass, 6
proton-proton chain, 182
pulsar
braking index [9.66], 182
characteristic age [9.67], 182
magnetic dipole radiation [9.69], 182
Pulsars, 182
pyramid (centre of mass) [3.175], 76
pyramid (volume) [2.272], 37
Q
Q, see quality factor
Q (Stokes parameter) [8.90], 171
Quadratic equations, 50
quadrature, 61
quadrature (integration), 44
quality factor
Fabry-Perot etalon [8.14], 163
forced harmonic oscillator [3.211],
78
free harmonic oscillator [3.203], 78

212 Index
LCR circuits [7.152], 148
quantum concentration [5.83], 112
Quantum definitions, 90
Quantum paramagnetism, 101
Quantum physics, 89–104
Quantum uncertainty relations, 90
quarter-wave condition [8.3], 162
quarter-wave plate [8.85], 170
quartic minimum, 42
R
Radial forms, 22
radian (unit), 4
radiance [5.156], 118
radiant
energy [5.145], 118
exitance [5.150], 118
flux [5.147], 118
intensity (dimensions), 17
radiation
bremsstrahlung [7.297], 160
Cherenkov [7.247], 156
field of a dipole [7.207], 153
flux from dipole [7.131], 146
resistance [7.209], 153
synchrotron [7.287], 159
Radiation pressure, 152
radiation pressure
extended source [7.203], 152
isotropic [7.200], 152
momentum density [7.199], 152
point source [7.204], 152
specular reflection [7.202], 152
Radiation processes, 118
Radiative transfer, 120
radiative transfer equation [5.179], 120
radiative transport (in stars) [9.63], 181
radioactivity, 103
Radiometry, 118
radius of curvature
in bending [3.258], 82
relation to curvature [2.287], 39
radius of gyration (see footnote), 75
Ramsauer effect [4.52], 93
Random walk, 59
random walk
one-dimensional [2.562], 59
three-dimensional [2.564], 59
range (of projectile) [3.90], 69
Rankine conversion [1.3], 15
Rankine-Hugoniot shock relations [3.334],
87
Rayleigh
resolution criterion [8.41], 165
scattering [7.236], 155
theorem [2.496], 53
Rayleigh-Jeans law [5.187], 121
reactance (definition), 148
reciprocal
lattice vector [6.8], 126
matrix [2.83], 25
vectors [2.16], 20
reciprocity [2.330], 42
Recognised non-SI units, 5
rectangular aperture diffraction [8.39],
165
rectangular coordinates, 21
rectangular cuboid moment of inertia [3.158],
75
rectifying plane, 39
recurrence relation
associated Legendre functions [2.433],
48
Legendre polynomials [2.423], 47
redshift
–flux density relation [9.99], 185
cosmological [9.86], 184
gravitational [9.74], 183
Reduced mass (of two interacting bod-
ies), 69
reduced units (thermodynamics) [5.71],
111
reflectance coefficient
and Fresnel equations [7.227], 154
dielectric film [8.4], 162
dielectric multilayer [8.8], 162
reflection coefficient
dielectric boundary [7.230], 154
potential barrier [4.58], 94
potential step [4.41], 92
potential well [4.48], 93

Index
I
213
transmission line [7.179], 150
reflection grating [8.29], 164
reflection law [7.216], 154
Reflection, refraction, and transmis-
sion, 154
refraction law (Snell’s) [7.217], 154
refractive index of
dielectric medium [7.195], 152
ohmic conductor [7.234], 155
plasma [7.260], 157
refrigerator efficiency [5.11], 107
regression (linear), 60
relativistic beaming [3.25], 65
relativistic doppler effect [3.22], 65
Relativistic dynamics, 68
Relativistic electrodynamics, 141
Relativistic wave equations, 104
relativity (general), 67
relativity (special), 64
relaxation time
and electron drift [6.61], 132
in a conductor [7.156], 148
in plasmas, 156
residuals [2.572], 60
Residue theorem [2.170], 31
residues (in complex analysis), 31
resistance
and impedance, 148
dimensions, 17
energy dissipated in [7.155], 148
radiation [7.209], 153
resistivity [7.142], 147
resistor, see resistance
resolving power
chromatic (of an etalon) [8.21], 163
of a diffraction grating [8.30], 164
Rayleigh resolution criterion [8.41],
165
resonance
forced oscillator [3.209], 78
resonance lifetime [4.177], 104
resonant frequency (LCR) [7.150], 148
Resonant LCR circuits, 148
restitution (coefficient of) [3.127], 73
retarded time, 139
revolution (volume and surface of), 39
Reynolds number [3.311], 86
ribbon (twisting of) [3.256], 81
Ricci tensor [3.57], 67
Riemann tensor [3.50], 67
right ascension [9.8], 177
rigid body
angular momentum [3.141], 74
kinetic energy [3.142], 74
Rigid body dynamics, 74
rigidity modulus [3.249], 81
ripples [3.321], 86
rms (standard deviation) [2.543], 57
Robertson-Walker metric [9.87], 184
Roche limit [9.43], 180
rocket equation [3.94], 70
Rocketry, 70
rod
bending, 82
stretching [3.230], 80
waves in [3.271], 82
Rodrigues’ formula [2.422], 47
Roots of quadratic and cubic equations, 50
Rossby number [3.316], 86
rot (curl), 22
Rotating frames, 66
Rotation matrices, 26
rotation measure [7.273], 157
Runge Kutta method [2.603], 62
Rutherford scattering, 72
Rutherford scattering formula [3.124], 72
Rydberg constant, 6, 7
and Bohr atom [4.77], 95
dimensions, 17
Rydberg’s formula [4.78], 95
S
s orbitals [4.92], 97
S-waves [3.262], 82
Sackur-Tetrode equation [5.83], 112
saddle point [2.338], 42
Saha equation (general) [5.128], 116
Saha equation (ionisation) [5.129], 116
Saturn data, 176
scalar effective mass [6.87], 134
scalar product [2.1], 20
scalar triple product [2.10], 20
scale factor (cosmic) [9.87], 184
scattering
angle (Rutherford) [3.116], 72
Born approximation [4.178], 104
Compton [7.240], 155

214 Index
crystal [6.32], 128
inverse Compton [7.239], 155
Klein-Nishina [7.243], 155
Mott (identical particles) [4.180], 104
potential (Rutherford) [3.114], 72
processes (electron), 155
Rayleigh [7.236], 155
Rutherford [3.124], 72
Thomson [7.238], 155
scattering cross-section, see cross-section
Schawlow-Townes line width [8.128], 174
Schrödinger equation [4.15], 90
Schwarz inequality [2.152], 30
Schwarzschild geometry (in GR) [3.61],
67
Schwarzschild radius [9.73], 183
Schwarzschild’s equation [5.179], 120
screw dislocation [6.22], 128
secx
secant method (of root-finding) [2.592],
61
sechx [2.229], 34
second (SI definition), 3
second (time interval), 4
second moment of area [3.258], 82
Sedov-Taylor shock relation [3.331], 87
selection rules (dipole transition) [4.91],
96
self-diffusion [5.93], 113
self-inductance [7.145], 147
semi-ellipse (centre of mass) [3.178], 76
semi-empirical mass formula [4.173], 103
semi-latus-rectum [3.109], 71
semi-major axis [3.106], 71
semi-minor axis [3.107], 71
semiconductor diode [6.92], 134
semiconductor equation [6.90], 134
Series expansions, 29
series impedances [7.157], 148
Series, summations, and progressions, 27
shah function (Fourier transform of) [2.510],
54
shear
modulus [3.249], 81
strain [3.237], 80
viscosity [3.299], 85
waves [3.262], 82
shear modulus (dimensions), 17
sheet of charge (electric field) [7.32], 138
shift theorem (Fourier transform) [2.501],
54
shock
Rankine-Hugoniot conditions [3.334],
87
spherical [3.331], 87
Shocks, 87
shot noise [5.142], 117
SI base unit definitions, 3
SI base units, 4
SI derived units, 4
SI prefixes, 5
SI units, 4
sidelobes (diffraction by 1-D slit) [8.38],
165
sidereal time [9.7], 177
siemens (unit), 4
sievert (unit), 4
similarity theorem (Fourier transform) [2.500],
54
simple cubic structure, 127
simple harmonic oscillator, see harmonic
oscillator
simple pendulum [3.179], 76
Simpson’s rule [2.586], 61
sinx
and Euler’s formula [2.218], 34
sinc function [2.512], 54
sine formula
planar triangles [2.246], 36
spherical triangles [2.255], 36
sinhx
sin−1
x, see arccosx
skew-symmetric matrix [2.87], 25
skewness estimator [2.544], 57
skin depth [7.235], 155
slit diffraction (broad slit) [8.37], 165
slit diffraction (Young’s) [8.24], 164
Snell’s law (acoustics) [3.284], 83
Snell’s law (electromagnetism) [7.217], 154
soap bubbles [3.337], 88
solar constant, 176
Solar data, 176
Solar system data, 176

Index
I
215
solenoid
finite [7.38], 138
infinite [7.33], 138
self inductance [7.23], 137
solid angle (subtended by a circle) [2.278],
37
Solid state physics, 123–134
sound speed (in a plasma) [7.275], 158
sound, speed of [3.317], 86
space cone, 77
space frequency [3.188], 77
space impedance [7.197], 152
spatial coherence [8.108], 172
Special functions and polynomials, 46
special relativity, 64
specific
charge on electron, 8
emission coefficient [5.174], 120
heat capacity, see heat capacity
definition, 105
dimensions, 17
intensity (blackbody) [5.184], 121
specific impulse [3.92], 70
speckle intensity distribution [8.110], 172
speckle size [8.111], 172
spectral energy density
spectral function (synchrotron) [7.295],
159
Spectral line broadening, 173
speed (dimensions), 17
speed distribution (Maxwell-Boltzmann) [5.84],
112
speed of light (equation) [7.196], 152
speed of light (value), 6
sphere
area [2.263], 37
capacitance of adjacent [7.14], 137
capacitance of concentric [7.18], 137
close-packed, 127
collisions of, 73
geometry on a, 36
gravitation field from a [3.44], 66
in a viscous fluid [3.308], 85
in potential flow [3.298], 84
Poincaré, 171
polarisability, 142
volume [2.264], 37
spherical Bessel function [2.420], 47
spherical cap
area [2.275], 37
volume [2.276], 37
spherical excess [2.260], 36
Spherical harmonics, 49
spherical harmonics
Laplace equation [2.440], 49
orthogonality [2.437], 49
spherical polar coordinates, 21
spherical shell (moment of inertia) [3.153],
75
spherical surface (capacitance of near) [7.16],
137
Spherical triangles, 36
spin
and total angular momentum [4.128],
100
degeneracy, 115
electron magnetic moment [4.141],
100
Pauli matrices, 26
spinning bodies, 77
spinors [4.182], 104
Spitzer conductivity [7.254], 156
spontaneous emission [8.119], 173
spring constant and wave velocity [3.272],
83
Square matrices, 25
standard deviation estimator [2.543], 57
Standard forms, 44
Star formation, 181
Star–delta transformation, 149
Static fields, 136
statics, 63
Stationary points, 42
Statistical entropy, 114
Statistical thermodynamics, 114
Stefan–Boltzmann constant, 9
Stefan–Boltzmann constant (dimensions),
17

216 Index
Stefan-Boltzmann constant, 121
Stefan-Boltzmann law [5.191], 121
stellar aberration [3.24], 65
Stellar evolution, 181
Stellar fusion processes, 182
Stellar theory, 181
step function (Fourier transform of) [2.511],
54
steradian (unit), 4
stimulated emission [8.120], 173
Stirling’s formula [2.411], 46
Stokes parameters, 171
Stokes parameters [8.95], 171
Stokes’s law [3.308], 85
Stokes’s theorem [2.60], 23
Straight-line fitting, 60
strain
simple [3.229], 80
tensor [3.233], 80
volume [3.236], 80
stress
dimensions, 17
in fluids [3.299], 85
simple [3.228], 80
tensor [3.232], 80
stress-energy tensor
and field equations [3.59], 67
perfect fluid [3.60], 67
string (waves along a stretched) [3.273],
83
Strouhal number [3.313], 86
structure factor [6.31], 128
sum over states [5.110], 114
Summary of physical constants, 6
summation formulas [2.118], 27
Sun data, 176
Sunyaev-Zel’dovich effect [9.51], 180
surface brightness (blackbody) [5.184],
121
surface of revolution [2.280], 39
Surface tension, 88
surface tension
Laplace’s formula [3.337], 88
work done [5.6], 106
surface tension (dimensions), 17
surface waves [3.320], 86
survival equation (for mean free path) [5.90],
113
susceptance (definition), 148
susceptibility
Landau diamagnetic [6.80], 133
magnetic [7.103], 143
Pauli paramagnetic [6.79], 133
symmetric matrix [2.86], 25
symmetric top [3.188], 77
Synchrotron radiation, 159
synodic period [9.44], 180
T
tanx
tangent [2.283], 39
tangent formula [2.250], 36
tanhx
tan−1
x, see arctanx
tau physical constants, 9
Taylor series
one-dimensional [2.123], 28
three-dimensional [2.124], 28
telegraphist’s equations [7.171], 150
temperature
antenna [7.215], 153
Celsius, 4
dimensions, 17
Kelvin scale [5.2], 106
thermodynamic [5.1], 106
Temperature conversions, 15
temporal coherence [8.105], 172
tensor
Einstein [3.58], 67
electric susceptibility [7.87], 142
ijk [2.443], 50
fluid stress [3.299], 85
magnetic susceptibility [7.103], 143
Ricci [3.57], 67
Riemann [3.50], 67
strain [3.233], 80
stress [3.232], 80
tera, 5
tesla (unit), 4
tetragonal system (crystallographic), 127
tetrahedron, 38
thermal conductivity

Index
I
217
diffusion equation [2.340], 43
dimensions, 17
phonon gas [6.58], 131
transport property [5.96], 113
thermal de Broglie wavelength [5.83], 112
thermal diffusion [5.93], 113
thermal diffusivity [2.340], 43
thermal noise [5.141], 117
thermal velocity (electron) [7.257], 156
Thermodynamic coefficients, 107
Thermodynamic fluctuations, 116
Thermodynamic laws, 106
Thermodynamic potentials, 108
thermodynamic temperature [5.1], 106
Thermodynamic work, 106
Thermodynamics, 105–121
Thermoelectricity, 133
thermopower [6.81], 133
Thomson cross section, 8
Thomson scattering [7.238], 155
throttling process [5.27], 108
time (dimensions), 17
time dilation [3.11], 64
Time in astronomy, 177
Time series analysis, 60
Time-dependent perturbation theory, 102
Time-independent perturbation theory,
102
timescale
free-fall [9.53], 181
Kelvin-Helmholtz [9.55], 181
Titius-Bode rule [9.41], 180
tonne (unit), 5
top
asymmetric [3.189], 77
symmetric [3.188], 77
symmetries [3.149], 74
top hat function (Fourier transform of)
[2.512], 54
Tops and gyroscopes, 77
torque, see couple
Torsion, 81
torsion
in a thick cylinder [3.254], 81
in a thin cylinder [3.253], 81
in an arbitrary ribbon [3.256], 81
in an arbitrary tube [3.255], 81
in differential geometry [2.288], 39
torsional pendulum [3.181], 76
torsional rigidity [3.252], 81
torus (surface area) [2.273], 37
torus (volume) [2.274], 37
total differential [2.329], 42
total internal reflection [7.217], 154
total width (and partial widths) [4.176],
104
trace [2.75], 25
trajectory (of projectile) [3.88], 69
transfer equation [5.179], 120
Transformers, 149
transmission coefficient
Fresnel [7.232], 154
potential barrier [4.59], 94
potential step [4.42], 92
potential well [4.49], 93
transmission grating [8.27], 164
transmission line, 150
coaxial [7.181], 150
equations [7.171], 150
impedance
lossless [7.174], 150
lossy [7.175], 150
input impedance [7.178], 150
open-wire [7.182], 150
paired strip [7.183], 150
reflection coefficient [7.179], 150
vswr [7.180], 150
wave speed [7.176], 150
waves [7.173], 150
Transmission line impedances, 150
Transmission line relations, 150
Transmission lines and waveguides, 150
transmittance coefficient [7.229], 154
Transport properties, 113
transpose matrix [2.70], 24
trapezoidal rule [2.585], 61
triangle
area [2.254], 36
inequality [2.147], 30
plane, 36
spherical, 36
triangle function (Fourier transform of)
[2.513], 54
triclinic system (crystallographic), 127
trigonal system (crystallographic), 127
Trigonometric and hyperbolic defini-

218 Index
tions, 34
Trigonometric and hyperbolic formulas, 32
Trigonometric and hyperbolic integrals,
45
Trigonometric derivatives, 41
Trigonometric relationships, 32
triple-α process, 182
true anomaly [3.104], 71
tube, see pipe
Tully-Fisher relation [9.49], 180
tunnelling (quantum mechanical), 94
tunnelling probability [4.61], 94
turns ratio (of transformer) [7.163], 149
two-level system (microstates of) [5.107],
114
U
U (Stokes parameter) [8.92], 171
UBV magnitude system [9.36], 179
umklapp processes [6.59], 131
energy-time [4.8], 90
general [4.6], 90
momentum-position [4.7], 90
number-phase [4.9], 90
underdamping [3.198], 78
unified atomic mass unit, 5, 6
uniform distribution [2.550], 58
uniform to normal distribution transfor-
mation, 58
unitary matrix [2.88], 25
units (conversion of SI to Gaussian), 135
Units, constants and conversions, 3–18
universal time [9.4], 177
Uranus data, 176
UTC [9.4], 177
V
V (Stokes parameter) [8.94], 171
van der Waals equation [5.67], 111
Van der Waals gas, 111
van der Waals interaction [6.50], 131
Van-Cittert Zernicke theorem [8.108], 172
variance estimator [2.542], 57
variations, calculus of [2.334], 42
Vector algebra, 20
Vector integral transformations, 23
vector product [2.2], 20
vector triple product [2.12], 20
Vectors and matrices, 20
velocity (dimensions), 17
velocity distribution (Maxwell-Boltzmann)
[5.84], 112
velocity potential [3.296], 84
Velocity transformations, 64
Venus data, 176
virial coefficients [5.65], 110
Virial expansion, 110
virial theorem [3.102], 71
vis-viva equation [3.112], 71
viscosity
dimensions, 17
from kinetic theory [5.97], 113
kinematic [3.302], 85
shear [3.299], 85
viscous flow
between cylinders [3.306], 85
between plates [3.303], 85
through a circular pipe [3.305], 85
through an annular pipe [3.307], 85
Viscous flow (incompressible), 85
volt (unit), 4
voltage
across an inductor [7.146], 147
bias [6.92], 134
Hall [6.68], 132
law (Kirchhoff’s) [7.162], 149
standing wave ratio [7.180], 150
thermal noise [5.141], 117
volume
dimensions, 17
of cone [2.272], 37
of cube, 38
of cylinder [2.270], 37
of dodecahedron, 38
of ellipsoid [2.268], 37
of icosahedron, 38
of octahedron, 38
of parallelepiped [2.10], 20
of pyramid [2.272], 37
of revolution [2.281], 39
of sphere [2.264], 37
of spherical cap [2.276], 37
of tetrahedron, 38
of torus [2.274], 37
volume expansivity [5.19], 107
volume strain [3.236], 80
vorticity and Kelvin circulation [3.287],

Index
I
219
84
vorticity and potential flow [3.297], 84
vswr [7.180], 150
W
wakes [3.330], 87
Warm plasmas, 156
watt (unit), 4
wave impedance
in a waveguide [7.189], 151
Wave mechanics, 92
Wave speeds, 87
wavefunction
and expectation value [4.25], 91
and probability density [4.10], 90
diffracted in 1-D [8.34], 165
hydrogenic atom [4.91], 96
perturbed [4.160], 102
Wavefunctions, 90
waveguide
cut-off frequency [7.186], 151
equation [7.185], 151
impedance
TE modes [7.189], 151
TM modes [7.188], 151
TEmn modes [7.190], 151
TMmn modes [7.192], 151
velocity
group [7.188], 151
phase [7.187], 151
Waveguides, 151
wavelength
Compton [7.240], 155
de Broglie [4.2], 90
photometric, 179
redshift [9.86], 184
thermal de Broglie [5.83], 112
waves
capillary [3.321], 86
in a spring [3.272], 83
in a thin rod [3.271], 82
in bulk fluids [3.265], 82
in fluids, 86
in infinite isotropic solids [3.264], 82
magnetosonic [7.285], 158
on a stretched sheet [3.274], 83
on a stretched string [3.273], 83
on a thin plate [3.268], 82
sound [3.317], 86
surface (gravity) [3.320], 86
transverse (shear) Alfvén [7.284], 158
Waves in and out of media, 152
Waves in lossless media, 152
Waves in strings and springs, 83
wavevector (dimensions), 17
weber (unit), 4
Weber symbols, 126
weight (dimensions), 17
Weiss constant [7.114], 144
Weiss zone equation [6.10], 126
Welch window [2.582], 60
Weyl equation [4.182], 104
Wiedemann-Franz law [6.66], 132
Wien’s displacement law [5.189], 121
Wien’s displacement law constant, 9
Wien’s radiation law [5.188], 121
Wiener-Khintchine theorem
in Fourier transforms [2.492], 53
in temporal coherence [8.105], 172
Wigner coefficients (spin-orbit) [4.136],
100
Wigner coefficients (table of), 99
windowing
Bartlett [2.581], 60
Hamming [2.584], 60
Hanning [2.583], 60
Welch [2.582], 60
wire
magnetic flux density [7.34], 138
wire loop (inductance) [7.26], 137
wire loop (magnetic flux density) [7.37],
138
wires (inductance of parallel) [7.25], 137
work (dimensions), 17
X
X-ray diffraction, 128
Y
yocto, 5
yotta, 5
Young modulus
and Lamé coefficients [3.240], 81

220 Index
and other elastic constants [3.250],
81
Hooke’s law [3.230], 80
Young modulus (dimensions), 17
Young’s slits [8.24], 164
Yukawa potential [7.252], 156
Z
Zeeman splitting constant, 7
zepto, 5
zero-point energy [4.68], 95
zetta, 5
zone law [6.20], 126

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