Diffraction of X-rays-electrons and neutrons.ppt

Course Title: Solid State Physics – I
Course Code: PHY-555
Credit Hours: 03
Dr. Khizar Hayat
Assistant Professor, Department of Physics
Abdul Wali Khan University Mardan
12/14/2022
1

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2
Structure of Solids:
Lattices and basis, Symmetry operations, Fundamental types of lattice,
Position and orientation of planes in crystals, Simple crystal structures,
Atomic potential, space groups and binding forces.
Crystal diffraction and reciprocal lattice:
Diffraction of X-rays, Neutrons and electrons from crystals, Bragg’s law, Reciprocal
lattice, Reciprocal lattice to sc, bcc, fcc, orthorhombic and hexagonal crystals, Laue
method, rotating crystal method, Powder methods, Scattered wave amplitude,
Ewald construction and Brillouin zone, Miller Indices, Fourier analysis of the basis.
Phonons and Lattice Vibrations:
Lattice heat capacity, classical model, Einstein model, Enumeration of normal
models, Density of state in one, two and three dimensions, Debye model of heat
capacity, Comparison with experimental results, Thermal conductivity and resistivity,
Umklapp processes.
Free Electron Theory of Solid:
Drude model, Electrical conductivity, Hall effect, Thermal conductivity, The
Sommerfeld theory of electrons, Ground-state energy of electron gas, Thermal
properties of electron gas
Course contents

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Band Theory of Solids:
General theory of electrons in a periodic potential, Bloch’s theorem, Crystal momentum, Density of states,
Electrons in a weak periodic potential, Formation of energy gap, Three schemes to describe energy bands,
Fermi surface.
Transport Properties of Solids:
Motion of electron in bands, Effective mass, Electrical conductivity of metals, electrical Conductivity of localized
electrons, Boltzmann equation
Defects in Crystals:
Crystal imperfections, Thermodynamics of Point defects, Schottky and Frenkel defects, color centres,
Dislocations in Solids, edge dislocation, Screw dislocation Slip and plastic deformation, Stacking faults and
grain Boundaries, Strength of Crystals
Dielectrics and Ferroelectrics:
Maxwell Equations, Polarization, Dielectric Constant and Dielectric Polarizability, Susceptibility, Electronic
Polarizablity, Clausius-Mossotti Relation, Structural Phase Transitions, Ferroelectric crystals, Classification of
Ferroelectric Crystals, Theory of Ferroelectric Displacive Transitions, Thermodynamic theory of Ferroelectric
transition, Ferroelectric Domains, Piezoelectricity.
Diamagnetism and Paramagnetism:
Atomic theory of magnetism, Diamagnetism, Paramagnetism, The quantum numbers, Orbital and spin
magnetic moments of electrons, Langevin theory of Dia and Paramagnetism, Ferromagnetism, Domain theory,
Weiss theory of Ferromagnetism, Magnetic relaxation and resonance phenomena
Semiconductors and Superconductivity:
Intrinsic Semiconductors, Extrinsic semiconductors, Band structure, Energy Gap, Donor and acceptor Level,
Hall Effect, Superconductivity-an introduction, zero resistivity and Meissner effect, Diamagnetism,
susceptibility,Critical field, temperature and current, Type-I and type-II superconductors, BCS theory, electron-
phonon-electron interaction via lattice deformation, ground state of superconductors, Cooper pairs, Coherence
length, the origin of energy gap, London equations (electrodynamics), London penetration depth,
thermodynamics of superconductors, entropy and the Gibbs free energy, Josephson Effect.
Course contents of SSP-II

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Recommended Books
1. N. M. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976.
2. M. A. Omar, Elementary Solid State Physics, Pearson Education 2000.
3. C. Kittle, Introduction to Solid State Physics, 7th Ed. By, Kohn Wiley,
1996.
4. S. O. Pillai, Solids State Physics, New Age International Limited
Publishers, 6th Ed. 2006.
5. M. A. Wahab, Solid State Physics, Narosa Publishing House, 1999.

5
X-RAY DIFFRACTION,
ELECTRON DIFFRACTION
&
NEUTRON DIFFRACTION
IN CRYSTAL
I. X-Ray
II. Diffraction
III. Diffraction of Waves by Crystals
IV. X-Ray Diffraction
V. Bragg Equation
VI. X-Ray Methods
VII. Neutron & Electron Diffraction
Bertha Röntgen’s
Hand 8 Nov, 1895
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X-RAY CRYSTALLOGRAPHY

6
X-RAY
 X-rays were discovered in 1895 by
the German physicist Wilhelm C.
Röntgen and were so named
because their nature was unknown
at the time.
 He was awarded the Nobel prize
for physics in 1901. Wilhelm Conrad Röntgen
(1845-1923)
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X-RAY PROPERTIES
 X-rays, invisible, highly penetrating electromagnetic
radiation of much shorter wavelength (higher frequency)
than visible light. The wavelength range for X-rays is
from about 10-8 m (100 Å) to about 10-12 m (0.01 Å), the
corresponding frequency range is from about 3 × 1016
Hz to about 3 × 1020 Hz.
 Because, X-rays have wavelengths similar to the size of
atoms, they are useful to explore within crystals.

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The energy of X-rays, like all electromagnetic radiation, is inversely
proportional to their wavelength as given by the Einstein equation:
where E = energy
h = Planck's constant, 6.62517 x 10-27 erg.sec
‫ע‬ = frequency
c = velocity of light = 2.99793 x 1010 cm/sec
λ = wavelength
X-RAY ENERGY

h
E  )
(


c
where 

hc
E 

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Thus, since X-rays have a smaller wavelength than
visible light, they have higher energy. With their higher energy,
X-rays can penetrate matter more easily than can visible light.
Their ability to penetrate matter depends on the density of the
matter, and thus X-rays provide a powerful tool for mapping
internal structures of the human body (bones have higher
density than tissue, and thus are harder for X-rays to penetrate,
fractures in bones have a different density than the bone, thus
fractures can be seen in X-ray pictures).
X-RAY ENERGY

Lattice spacing
typically
o
10
10 m 1

 
o
1A
 
Max von Laue
(1879-1960)
1914 Nobel prize
Laue 1912
X-RAY ENERGY

Energies X-ray, electrons and
neutrons wave-particle
hc
E h
  

hc
E
 
X-ray: o
1A
  
E 12 k eV
Electrons:
h
p k
 

h h
p 2mE
  
-31
e
m 9.1 10 kg

o
1A
 

E 150 eV
Neutrons:
h h
p 2mE
  
o
1A
  -27
n
m 1.6749 10 kg


E 0.08 eV

12
PRODUCTION OF X-RAYS
 Visible light photons and X-ray photons are both produced by
the movement of electrons in atoms. Electrons occupy different
energy levels, or orbitals, around an atom's nucleus.
 When an electron drops to a lower orbital, it needs to release
some energy; it releases the extra energy in the form of a
photon. The energy level of the photon depends on how far the
electron dropped between orbitals.
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Evacuated glass bulb
Anode
Cathode
 X-rays can be produced in a highly evacuated glass bulb, called
an X-ray tube, that contains essentially two electrodes—an anode
made of platinum, tungsten, or another heavy metal of high
melting point, and a cathode. When a high voltage is applied
between the electrodes, streams of electrons (cathode rays) are
accelerated from the cathode to the anode and produce X rays as
they strike the anode.
X-RAY TUBE
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Monochromatic and Broad
Spectrum of X-rays
 X-rays can be created by bombarding a metal target with high
energy (> 104 eV) electrons.
 Some of these electrons excite electrons from core states in the
metal, which then recombine, producing highly monochromatic X-
rays. These are referred to as characteristic X-ray lines.
 Other electrons, which are decelerated by the periodic potential of
the metal, produce a broad spectrum of X-ray frequencies.

16
 A larger atom is more likely to absorb an X-ray photon in
this way, because larger atoms have greater energy
differences between orbitals -- the energy level more
closely matches the energy of the photon. Smaller atoms,
where the electron orbitals are separated by relatively low
jumps in energy, are less likely to absorb X-ray photons.
 The soft tissue in your body is composed of smaller
atoms, and so does not absorb X-ray photons particularly
well. The calcium atoms that make up your bones are
much larger, so they are better at absorbing X-ray
photons.
Absorption of X-rays
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17
DIFFRACTION
 Diffraction is a wave phenomenon in
which the apparent bending and
spreading of waves occurs when the
waves meet an obstacle.
 Diffraction occurs with electromagnetic
waves, such as light and radio waves,
and also in sound waves and water
waves.
 Simple example of diffraction is double-
slit diffraction.
Width b Variable
(500-1500 nm)
Wavelength Constant
(600 nm)
Distance d = Constant
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18
LIGHT DIFFRACTION
 Light diffraction is caused by light bending around the edge of
an object. The interference pattern of bright and dark lines from
the diffraction experiment can only be explained by the additive
nature of waves:
Brighter light
Darkness
Thus Young’s light interference
experiment proves that light
has wavelike properties.
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LIGHT INTERFERENCE
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Constructive & Destructive Waves
 Constructive interference is
the result of synchronized
light waves that add
together to increase the light
intensity.
 Destructive İnterference .
results when two out-of-phase
light waves cancel each other
out, resulting in darkness.
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Diffraction from a particle and solid
Single particle
 To understand diffraction we also
have to consider what happens when
a wave interacts with a single particle.
The particle scatters the incident
beam uniformly in all directions
Solid material
 What happens if the beam is
incident on solid material? If we
consider a crystalline material, the
scattered beams may add together
in a few directions and reinforce
each other to give diffracted beams
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A crystal is a periodic structure
( unit cells are repeated regularly)
Solid State Physics deals how the waves are propagated
through such periodic structures. Here, we study the crystal
structure through the diffraction of photons (X-ray), nuetrons
and electrons.
Diffraction
X-ray Neutron Electron
Diffraction of Waves by Crystals
The general principles will be the same for each type of waves.
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 The diffraction depends on the crystal structure and on
the wavelength.
 At optical wavelengths such as 5000 angstroms the
superposition of the waves scattered elastically by the
individual atoms of a crystal results in ordinary optical
refraction.
 When the wavelength of the radiation is comparable
with or smaller than the lattice constant, one can find
diffracted beams in directions quite different from the
incident radiation.
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 The structure of a crystal can be determined by
studying the diffraction pattern of a beam of radiation
incident on the crystal.
 Beam diffraction takes place only in certain specific
directions, much as light is diffracted by a grating.
 By measuring the directions of the diffraction and the
corresponding intensities, one obtains information
concerning the crystal structure responsible for
diffraction.
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 X-ray crystallography or X-ray diffraction (XRD) is a
technique in crystallography in which the pattern produced by
the diffraction of x-rays through the closely spaced lattice of
atoms in a crystal is recorded and then analyzed to reveal the
nature of that lattice.
OR
 It is a powerful technique that could "see inside" of crystals and allow
for detailed determination of crystal structures and unit cell size.
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Crystal Structure Determination
A crystal behaves as a 3-D diffraction grating for x-rays
 In a diffraction experiment, the spacing of lines on the grating
can be deduced from the separation of the diffraction maxima
Information about the structure of the lines on the
grating can be obtained by measuring the relative
intensities of different orders
 Similarly, measurement of the separation of the X-ray
diffraction maxima from a crystal allows us to determine
the size of the unit cell and from the intensities of
diffracted beams one can obtain information about the
arrangement of atoms within the cell.
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X-Ray Diffraction Pattern
(110)
(200)
(211)
z
x
y
a b
c
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron (BCC)
Intensity
(relative)
z
x
y
a b
c
z
x
y
a b
c
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Simplified Powder Diffractogram
2q (˚)  * crystal system
* unit cell dimensions
* space group
Intensity  * atomic/ionic positions
* temperature factors
* order/disorder
FWHM  * domain sizes

29
X-Ray Diffraction
W. L. Bragg presented a simple
explanation of the diffracted beams from a
crystal.
The Bragg derivation is simple but is
convincing only since it reproduces the
correct result.
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30
X-Ray Diffraction & Bragg Equation
 English physicists Sir W.H. Bragg and
his son Sir W.L. Bragg developed a
relationship in 1913 to explain why the
cleavage faces of crystals appear to
reflect X-ray beams at certain angles of
incidence (theta, θ).This observation is
an example of X-ray wave
interference. Sir William Henry Bragg (1862-1942),
William Lawrence Bragg (1890-1971)
o 1915, the father and son were awarded the Nobel prize for physics
"for their services in the analysis of crystal structure by means of X-
rays".
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Bragg Equation
 Bragg law identifies the angles of the incident
radiation relative to the lattice planes for which
diffraction peaks occurs.
 Bragg derived the condition for constructive
interference of the X-rays scattered from a set of
parallel lattice planes.
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BRAGG EQUATION
 W.L. Bragg considered crystals to be made up of parallel
planes of atoms. Incident waves are reflected from parallel
planes of atoms in the crystal, with each plane is reflecting
only a very small fraction of the radiation, like a lightly silvered
mirror.
 In mirror like reflection the angle of incidence is equal to the
angle of reflection.
ө
ө
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33
Diffraction Condition
 The diffracted beams are found to occur
when the reflections from planes of atoms
interfere constructively.
 We treat elastic scattering, in which the
energy of X-ray is not changed on reflection.
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34
Bragg Equation
 When the X-rays strike a layer of a crystal, some of them will
be reflected. We are interested in X-rays that are in-phase
with one another. X-rays that add together constructively in x-
ray diffraction analysis in-phase before and after they
reflected.
Incident angle
Reflected angle
Wavelength of X-ray
q
q 
q 
 
q 2q
Total Diffracted
Angle q
2

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35
The line CE is equivalent
to the distance between
the two layers (d)
Bragg Equation
 These two x-ray beams travel slightly different distances. The
difference in the distances traveled is related to the distance
between the adjacent layers.
 Connecting the two beams with perpendicular lines shows the
difference between the top and the bottom beams.
sin
DE d q

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36
Bragg Law
 The length DE is the same as EF, so the total distance
traveled by the bottom wave is expressed by:
 Constructive interference of the radiation from successive
planes occurs when the path difference is an integral
number of wavelenghts. This is the Bragg Law.
sin
EF d q

sin
DE d q

2 sin
DE EF d q
 
2 sin
n d
 q

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Bragg Equation
where, d is the spacing of the planes and n is the order of diffraction.
 Bragg reflection can only occur for wavelength
 This is why we cannot use visible light. No diffraction occurs when
the above condition is not satisfied.
 The diffracted beams (reflections) from any set of lattice planes
can only occur at particular angles pradicted by the Bragg law.

q n
d 
sin
2
d
n 2



38
Bragg Equation
Since Bragg's Law applies to all sets of crystal planes, the lattice
can be deduced from the diffraction pattern, making use of general
expressions for the spacing of the planes in terms of their Miller
indices. For cubic structures
Note that the smaller the spacing the higher the angle of diffraction.
The diffraction pattern will reflect the symmetry properties of the
lattice.
2 sin
d n
q 

2 2 2
a
d
h k l

 
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Spacing dhkl between successive (hkl) planes
In cubic systems: 2
2
2
l
k
h
a
dhkl



x
y
2
2
110
2 a
d 
a d110
2
110
a
d 
Top view

40
Types of X-ray camera
There are many types of X-ray camera to
sort out reflections from different crystal
planes. We will study only three types of X-ray
photograph that are widely used for the simple
structures.
1. Laue photograph
2. Rotating crystal method
3. Powder photograph
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41
X-RAY DIFFRACTION METHODS
X-Ray Diffraction Method
Laue Rotating Crystal Powder
Orientation
Single Crystal
Polychromatic Beam
Fixed Angle
Lattice constant
Single Crystal
Monochromatic Beam
Variable Angle
Lattice Parameters
Polycrystal (powdered)
Monochromatic Beam
Variable Angle
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42
LAUE METHOD
 The Laue method is mainly used to determine the
orientation of large single crystals while radiation is
reflected from, or transmitted through a fixed crystal.
 The diffracted beams form arrays of spots, that lie on curves
on the film.
 The Bragg angle is fixed for every set of planes in the crystal.
Each set of planes picks out and diffracts the particular
wavelength from the white radiation that satisfies the Bragg law
for the values of d and θ involved.
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43
Back-reflection Laue Method
 In the back-reflection method, the film is placed between the
x-ray source and the crystal. The beams which are diffracted
in a backward direction are recorded.
X-Ray Film
Single
Crystal
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44
Transmission Laue Method
 In the transmission Laue method, the film is placed behind
the crystal to record beams which are transmitted through
the crystal.
X-Ray
Film
Single
Crystal
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45
Laue Pattern
The symmetry of the spot
pattern reflects the
symmetry of the crystal
when viewed along the
direction of the incident
beam.
Laue method is often used to
determine the orientation of
single crystals by means of
illuminating the crystal with
a continuos spectrum of X-
rays;
Single crystal
Continous spectrum of x-
rays
Symmetry of the crystal;
orientation
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46
ROTATING CRYSTAL METHOD
 In the rotating crystal method, a
single crystal is mounted with
an axis normal to a
monochromatic x-ray beam.
A cylindrical film is placed
around it and the crystal is
rotated about the chosen axis.
 As the crystal rotates, sets of lattice planes will at some
point make the correct Bragg angle for the monochromatic
incident beam, and at that point a diffracted beam will be
formed.
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47
ROTATING CRYSTAL
METHOD
Lattice constant of the crystal can be
determined by means of this method; for a
given wavelength if the angle at which a
reflection occurs is known, can be
determined.
hkl
d
q
2 2 2
a
d
h k l

 
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48
Rotating Crystal Method
The reflected beams are located on the surface of
imaginary cones. By recording the diffraction patterns (both
angles and intensities) for various crystal orientations, one
can determine the shape and size of unit cell as well as
arrangement of atoms inside the cell.
Film
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49
THE POWDER METHOD
If a powdered specimen is used, instead of a
single crystal, then there is no need to rotate
the specimen, because there will always be
some crystals at an orientation for which
diffraction is permitted. Here a monochromatic
X-ray beam is incident on a powdered or
polycrystalline sample.
This method is useful for samples that are
difficult to obtain in single crystal form.
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50
THE POWDER METHOD
The powder method is used to determine the value
of the lattice parameters accurately. Lattice parameters
are the magnitudes of the unit vectors a, b and c which
define the unit cell for the crystal.
For every set of crystal planes, by chance, one or
more crystals will be in the correct orientation to give
the correct Bragg angle to satisfy Bragg's equation.
Every crystal plane is thus capable of diffraction. Each
diffraction line is made up of a large number of small
spots, each from a separate crystal. Each spot is so
small as to give the appearance of a continuous line.
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52
Application of XRD
1. Differentiation between crystalline and amorphous
materials;
2. Determination of the structure of crystalline materials;
3. Determination of electron distribution within the atoms, and
throughout the unit cell;
4. Determination of the orientation of single crystals;
5. Determination of the texture of polygrained materials;
6. Measurement of strain and small grain size…..etc
XRD is a nondestructive technique. Some of the uses of
x-ray diffraction are;
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53
Advantages and
disadvantages of X-rays
Advantages;
 X-ray is the cheapest, the most convenient and
widely used method.
 X-rays are not absorbed very much by air, so
the specimen need not be in an evacuated
chamber.
Disadvantage;
 They do not interact very strongly with lighter
elements.
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54
Difraction Methods
Diffraction
X-ray Neutron Electron
Different radiation source of neutron or
electron can also be used in diffraction
experiments.
The physical basis for the diffraction of
electron and neutron beams is the same as that
for the diffraction of X rays, the only difference
being in the mechanism of scattering. 12/14/2022

55
Neutron Diffraction
 Neutrons were discovered in 1932 and their wave
properties was shown in 1936.
 λ ~1A°; Energy E~0.08 eV. This energy is of the same
order of magnitude as the thermal energy kT at room
temperature, 0.025 eV, and for this reason we speak of
thermal neutrons.
E = p2/2m p = h/λ
E=Energy λ=Wavelength
p=Momentum
mn=Mass of neutron = 1,67.10-27kg
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56
Neutron Diffraction
 Neutron does not interact with electrons in the crystal.
Thus, unlike the x-ray, which is scattered entirely by
electrons, the neutron is scattered entirely by nuclei
 Although uncharged, neutron has an intrinsic magnetic
moment, so it will interact strongly with atoms and ions
in the crystal which also have magnetic moments.
 Neutrons are more useful than X-rays for determining
the crystal structures of solids containing light
elements.
 Neutron sources in the world are limited so neutron
diffraction is a very special tool.
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57
Neutron Diffraction
Neutron diffraction has several advantages over its x-
ray counterpart;
 Neutron diffraction is an important tool in the investigation
of magnetic ordering that occur in some materials.
 Light atoms such as H are better resolved in a neutron
pattern because, having only a few electrons to scatter
the X ray beam, they do not contribute significantly to the
X ray diffracted pattern.
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58
Electron Diffraction
Electron diffraction has also been used in the analysis of
crystal structure. The electron, like the neutron, possesses wave
properties;
Electrons are charged particles and interact strongly with
all atoms. So electrons with an energy of a few eV would be
completely absorbed by the specimen. In order that an
electron beam can penetrate into a specimen , it necessitas a
beam of very high energy (50 keV to 1MeV) as well as the
specimen must be thin (100-1000 nm)
0
2A


eV
m
h
m
k
E
e
e
40
2
2 2
2
2
2





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59
Electron Diffraction
If low electron energies are used, the penetration depth
will be very small (only about 50 A°), and the beam will be
reflected from the surface. Consequently, electron diffraction is
a useful technique for surface structure studies.
Electrons are scattered strongly in air, so diffraction
experiment must be carried out in a high vacuum. This brings
complication and it is expensive as well.
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Diffraction of X-rays-electrons and neutrons.ppt

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