Study on the dependency of steady state response on the ratio of larmor and rabi frequency for esr using matlab

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 05 Issue: 03 | Mar-2016, Available @ http://www.ijret.org 207
STUDY ON THE DEPENDENCY OF STEADY STATE RESPONSE ON
THE RATIO OF LARMOR AND RABI FREQUENCY FOR ESR USING
MATLAB
Amit Kumar1
, Rajib Chakraborty2
1
Student, Electrical Engineering, Abacus Institute of Engineering and Management, West Bengal, India
2
Assistant Professor, Physics, Abacus Institute of Engineering and Management, West Bengal, India
Abstract
In this project we simulate with very high accuracy specially to study the dependency of the steady state power and dispersion
output on the ratio (r) between Larmor and Rabi frequency for the electron spin resonance experiment by the matlab software
(version 7.9.0.529(R2009b)). Where the sample material (DPPH) has been kept in a strong static magnetic field (B0) and in
orthogonal direction a high frequency electromagnetic field (B1(t)) has been applied. We divide our simulation into two parts. In
the first part we ignore the terms and observe the dependency of the power maximum on the amplitude of the oscillating
e.m. field B1 (for fixed (ωL) Larmor frequency) and on ωL (for fixed B1). Also observe a clear shift (Δω) of the power maxima
(Pmax) from ωL. In our second part we consider the term and the ratio (r) between Larmor and Rabi frequency and
observe the shift (Δω) of the power maxima (Pmax) from ωL and change in peak to peak line width (ΔBPP) with B1 both depends
upon the ratio r. we consider various range of r ([0.83,5], [16,100], [88.3,500], [1000,2000], [833.3,5000]) and observe these
dependency. We observe as the ratio of r increases the output i.e. shift (Δω) and the change in ΔBPP with B1 decreases and
converges to the case of neglecting terms. We also observe the shift (Δω) follows some non linear relationship with B1.
Keywords: E.S.R., Larmor, Rabi, Ratio r, Spin.
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1. INTRODUCTION
In this project we study about the nature of the power
absorption and dispersion curve of electron spin resonance.
We simulate with very high accuracy specially to study the
dependency of the steady state power and dispersion output
on the ratio (r) between Larmor and Rabi frequency for the
electron spin resonance experiment by the matlab software.
Electron paramagnetic resonance (EPR) or electron spin
resonance (ESR) is a spectroscopic technique for studying
mainly paramagnetic materials having unpaired electrons.
The basic concepts of EPR are analogous to those of nuclear
magnetic resonance (NMR), where in NMR spins of atomic
nuclei are excited but in ESR instead of nuclei the electron
spins are excited. EPR spectroscopy is particularly useful for
studying metal complexes or organic radicals. In Kazan
State University first time EPR was observed by Soviet
physicist Yevgeny Zavoisky in 1944, and was developed
independently at the same time by Brebis Bleaney at the
University of Oxford.
2. MAGNETIC MOMENT
Form quantum point of view a free electron (not bounded in
an atom) due to its own intrinsic spin angular momentum
also possesses a Magnetic moment, which is purely a
relativistic effect predicted from relativistic quantum
mechanics. The operator of this spin magnetic moment is:
[25]
The component of the spin operator in any direction has an
eigen value (where ms is = spin quantum
numbers).So the electrons spin magnetic moment (the Eigen
value of the component of spin magnetic moment operator
in that direction) is:
µB=9.2849*10-24
JT-1
3. ZEEMAN EFFECT FOR ELECTRON SPIN
Every electron has a magnetic moment and spin quantum
number s=1/2, with magnetic components ms=+(1/2) and
ms=-(1/2). In the presence of an external magnetic field with
strength B0, the electron's magnetic moment aligns itself
either parallel ms=-(1/2) or anti parallel ms=+(1/2) to the
field, each alignment having a specific energy due to the
Zeeman Effect :
Where
 gs is the electron's so-called g-factor (see also the Landé
g-factor). gs=2.0023 for the free electron
 is the Bohr magneton.
Therefore, the separation between the lower and the upper
state is for unpaired free electrons. This
equation implies that the splitting of the energy levels is

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directly proportional to the magnetic field's strength, as
shown in the diagram below.
Fig 1: Zeeman Effect
When the resonance happens then an unpaired electron
makes a transition between the two energy levels by either
absorbing or emitting a photon of energy hυ. Then we can
write .
4. LARMOR PRECESSION
In physics, Larmor precession (named after Joseph Larmor)
is the precession of the magnetic moments of electrons,
muons, all leptons with magnetic moments, which are
quantum effects of particle spin, atomic nuclei, and atoms
about an external magnetic field. The magnetic field exerts a
torque on the magnetic moment,
where Γ is the torque, µ is the magnetic dipole moment, L is
the angular momentum vector, B is the external magnetic
field, symbolizes the cross product, and ɣ is the
gyromagnetic ratio which gives the proportionality constant
between the magnetic moment and the angular momentum.
The angular momentum vector L precesses about the
external field axis with an angular frequency known as the
Larmor frequency,
Where ω is the angular frequency, is (for a
particle of charge -e) the gyromagnetic ratio, and B is the
magnitude of the magnetic field and g is the g-factor
(normally 1, except in quantum physics).
Simplified, this becomes:
(Energy of the photon hω = )
Where ω is the Larmor frequency, m is mass, -e is the
charge, and B is applied field.
4.1 Rabi Frequency
In case of ESR, EPR or NMR Rabi frequency is the nutation
frequency of the nutation (a periodic motion superimposed
on the precessional circle of the net angular momentum
vector) induced due to applied oscillating radio wave , micro
wave field (B1(t)). This is ω1=ɣB1.
(ɣ= gyromagnetic ratio; B1= amplitude of the oscillating
field)
In quantum mechanical point of view for two level (E1 & E2;
E2 >E1) system the probability of spin flip (i.e. transition
from E1 to E2 level) is a periodic function of time (t);
For any ω the frequency of the applied oscillating
electromagnetic field:
(Detuning factor)
So, angular frequency is called Rabi
frequency in quantum mechanics. At resonance when ω=ω0
then the quantum Rabi frequency became
(i.e. equal to classical Rabi
frequency or nutation frequency).[15,16]
4.2 What is r?
r is the ratio of Larmor frequency(ω0 or ωL) and Rabi
frequency(ω1).
5. MAGNETIC MOMENT DUE TO ELECTRON
SPIN IN DPPH
We know that the electrons spin magnetic moment is
µB=9.2849*10-24
JT-1
Similarly there is nuclear magnetron µN=5.0571*10-27
JT-1
.
It is measured that magnetic moment of the proton
µP=2.7927µN.
µn=-1.9131µN.
µN=eħ/mp
The above numerical value shows that proton and neutron
are having magnetic moment 10-3
time’s .
The nuclei are made up of proton and neutron. The magnetic
moments of the nuclei are much smaller than the atomic
magnetic moments (generally it is in the order of µB.)

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Now we observed for the DPPH (an organic molecule) only
one electron attached with the nitrogen atom is unpaired,
except this electron all electrons are paired. So we can say
the electronic contribution in the molecular magnetic
moment of DPPH must be due to this unpaired electron and
the nuclear contribution in the molecular magnetic moment
caused by all the nucleus (having neutrons and protons) of
DPPH molecule must be negligible compared to this
unpaired electron. Therefore the net resultant vector of this
molecule is obviously in the order of µB. It is measured that
the molecular g factor of DPPH is g=2.0038.The g factor for
the electro spin gs=2.00232. [9]
So we can easily take this molecule as per simulation for
electron spin resonance. Affectively for all practical purpose
we can consider that the net magnetization vector is due to
vector sum of all magnetic dipole of electrons.
Fig 2: Experimental set up for the project.
6. FELIX-BLOCH EQUATION
(Our modeling equation for simulation)
The magnetization vector M of DPPH is vector sum of all
magnetic dipole moment of electrons in unit volume.
(Where i=1 to N; N is number of electrons in unit volume)
In terms of total angular momentum of a sample
Interaction of magnetic moment with magnetic field gives a
torque on the system and changes the angular momentum of
the system
Now electron spins relax to equilibrium value following the
application of r.f. fields. Bloch assumed Mz(t) relax along z
axis with rate 1/T1 and Mx(t), My(t) relax in x-y plane with
different rate 1/T2 respectively , T1 is called spin-lattice
relaxation and T2 is called spin-spin relaxation. Now the
Bloch equation with addition of relaxation becomes:
Where γ is the gyromagnetic ratio (Taken the value of
g=2.002322)
B(t) = (Bx(t), By(t),Bz(t)) is the applied magnetic field.
M(t) = (Mx(t), My(t),Mz(t)) three components of
magnetization vector.
These set of coupled equations are taken as our modeling
equation. The equations are analyzed to simulate the output
of E.S.R. experiment. There are three inputs Bx(t), By(t),
Bz(t) and there are three outputs Mx(t), My(t), Mz(t).
7. STEADY STATE SOLUTION
In our simulation we have taken inputs as:
Bx(t)=B1 cos ωt
By(t)= -B1 sinωt
Bz(t)=B0 (It is a constant field)
And output is M=(Mx, My, Mz). At steady state it is taken
as:
Its matrix representation is given below.
In the above equations,
Complex form of input is given by,
Complex form of magnetization is given by,
Now, we can write susceptibility of the system is given by:-

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Energy stored in the specimen at any instant( t) is given by,
U= -M.B
Power absorbed is,
In the above equation value of is constant. So,
7.1 Output Measurement
For power output v(ω) part is measured and for dispersion
output u(ω) is measured, with the setup shown in figure
So equations for simulations are given below,
For power equation is,
……..(1)
And for dispersion we have taken the value of u (ω).
…………………(2)
7.2 Magnetic Susceptibility of DPPH
This is a dimensionless quantity. We calculate it for DPPH
from the formula:
N is the numbers of dipole in unit volume. N
=21.3811*1026
m-3
.
The total angular momentum quantum number J = ½ (taken
for DPPPH); g=gDPPH=2.0038; T =300K.
After the calculation in S.I. unit
Volume susceptibility in S.I. become,
[Where µ0 is the permeability of the space µ0=
NA-2
in S.I.]
7.3 Relaxation Time (T1 & T2) for DPPH
T1, the longitudinal relaxation time depends on the spin-
lattice coupling, and T2 the transverse relaxation time
depends on spin-spin coupling. For both cases as the
coupling becomes stronger the relaxation time becomes
longer. And longer relaxation time would produce
comparatively low line width ΔBPP (peak to peak distance of
the dispersion curve). In case of DPPH line width (ΔBPP)
strongly depends on the solvent in which the substance has
been crystallized. The lowest observed value of DPPH
crystallized from CS2 is 0.15mT. In general peak to peak
line width (ΔBPP) is reported from 0.15mT to 0.81mT and
relaxation time (T1 & T2) are related with the line width
(ΔBPP) by the following relation :-[10]
[Generally T1>T2]
[For DPPH the value of T2 is very close to T1 ]
[For small value of and very large value of we can
neglect term]
For simulation purpose we have plotted a graph between T2
vs. ΔBPP (0.15mT to 0.81mT) in fig: 34.
For simulation we have taken ΔBPP =0.1732050809 mT. So,
T1=T2=0.378360954*10-7
sec.
7.4 Values Used In Simulation
N=21.38111*1026
number /m3
J= ½
ɣ=0.17619859*1012
radian sec-1
T-1
g= an isotropic g value reported in the literature for DPPH is
2.0036 0.0002 [10]
KB=1.3806488*10-23
J/K
T=300K
µB=9.2849*10-24
JT-1
B0=5.0316*10-4
T, 5.0316*10-2
T, 5.0316*10-1
T, 5.0316T
T1=T2=0.378360954*10-7
sec.
= 4 *10-7
*44.566497

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8. EXPLANATION OF ALL THE SIMULATED
OUTPUT GRAPH IN MATLAB
(In this simulation we have used equation (1), (2) for power
absorption and dispersion respectively.)
8.1 Simulation After Neglecting ω1
2
T1T2 Terms
Fig 3:- b1=10.^-4; b2=2*10.^-4; b3=3*10.^-4; b4=4*10.^-4;
b5=5*10.^-4. (b1, b2, b3, b4, and b5 are different magnetic
field).In this graph for each curve magnetic field along z
axis (B0) is constant but rf field is time varying along x axis
also (B1) is different for each and every curve. We have
observed that with the increase in the magnetic field the
maximum power absorption point is also increases. There is
also a slight shift of the median of maximal’s from Larmor
frequency (ex: for b1=10-4
T median of maximal’s is at
89.16265 MHz but ωL=88. 5985628 MHz).
Fig 4:- b1=10.^-4; b2=2*10.^-4; b3=3*10.^-4; b4=4*10.^-
4; b5=5*10.^-4. (b1, b2, b3, b4, and b5 are different
magnetic field). Frequency is in MHz range (radio
frequency). In this graph the condition is same as fig 3.
Fig 5:- w1=81.5985628; w2=84.6285628; w3=87.6485628;
w4=90.6685628; w5=93.7085628 (w1, w2, w3, w4, and w5
are Larmor frequency in MHz range). In this graph for each
curve magnetic field (B1=10-4
.) is constant but (B0) is
different for each and every curve. For the variation in (B0)
we have taken different values of frequency (ω0) is denoted
by w1, w2.etc.By changing the value of Larmor frequency
we can sifts the point of the power absorption. In this case
the frequency is in the range of MHz, so the bandwidth is
large and accuracy is less. Also the maximum points are
increasing w.r.t. ω0.
Fig 6:- w1=81.5985628; w2=84.6285628; w3=87.6485628;
w4=90.6685628; w5=93.7085628 (w1, w2, w3, w4, and w5
are Larmor frequency in MHz range). In this graph for each
curve magnetic field (B1= 5*10-4
Tesla) is constant.

_______________________________________________________________________________________
Fig 7:- b1=10.^-4; b2=2*10.^-4; b3=3*10.^-4; b4=4*10.^-4;
b5=5*10.^-4. (b1, b2, b3, b4, and b5 are different magnetic
field).In this graph for each curve magnetic field along z
axis (B0) is constant but micro wave field is time varying
along x axis also (B1) is different for each and every curve.
We have observed that with the increase in the magnetic
field the maximum power absorption point is also increases.
For the micro wave frequency the shift of maximal’s are not
observed up to four decimal place but we will observe if we
able to calculate the values for more than four decimal
place.
Fig 8:- b1=10.^-4; b2=2*10.^-4; b3=3*10.^-4; b4=4*10.^-
4; b5=5*10.^-4. (b1, b2, b3, b4, and b5 are different
magnetic field). Frequency is in GHz range (microwave
frequency). In this graph for each curve magnetic field along
z axis (B0) is constant but micro wave field is time varying
along x axis also (B1) is different for each and every curve.
The accuracy in the case of microwave frequency is very
high.
Fig 9:- w1=88.5985628; w2=88.6285628; w3=88.6485628;
w4=88.6685628; w5=88.7085628 (w1, w2, w3, w4, and w5
are Larmor frequency in GHz range).In this graph for each
curve magnetic field (B1) is constant but micro wave field is
time varying along x axis also (B0) is different for each and
every curve. For the variation in (B0) we have taken
different values of frequency (ω0). By changing the value of
Larmor frequency we can sifts the point of the power
absorption. In this case the frequency is in the range of GHz,
so the bandwidth is very small and accuracy is very high.
For w1= 88.5985628GHz the median of the maximal’s is
sifts at 88.5987GHz.
Fig 10:- w1=81.5985628; w2=84.6285628;
w3=87.6485628; w4=90.6685628; w5=93.7085628 (w1,
w2, w3, w4, and w5 are Larmor frequency in GHz range). In
this graph conditions are same as fig 17. Here changes of the
absorption maxima with the Larmor frequency are
prominent. For w3= 87.6485628GHz the median of the
maximal’s is sifts at 87.6488GHz.

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8.2. Simulation Without Neglecting ω1
2
T1T2 Terms
(Study the dependency of steady state power absorption
and dispersion output on the ratio r)
We have also simulated power absorption and power
dispersion curve without neglecting the value of ω1
2
T1T2.
Results of simulations are given below. The value of ω1 is
taken in the MHz range in all simulations
Fig 11:- b1=10.^-4; b2=2*10.^-4; b3=4*10.^-4; b4=6*10.^-
4. (b1, b2, b3, b4, and b5 are different magnetic field).In this
graph for each curve magnetic field along z axis (B0) is
constant but rf field is time varying along x axis also (B1) is
different for each and every curve. (In this graph B0 is
constant even though the maximum power absorption point
is at different frequency for the curves having different
value of magnitude of magnetic field B1.).In fig 11, 12 we
have taken two values of ω0 (ω1 =88. 5985628MHz and ω2
=150 MHz as indicated in graphs)
Fig 12:- b1=10.^-4; b2=2*10.^-4; b3=4*10.^-4; b4=6*10.^-
4. (b1, b2, b3, b4, and b5 are different magnetic field). (In
this graph we can see that as the magnitude of the magnetic
field B1 increases bandwidth is also increases.)
Fig 13:-After calculation we have got
Fig 14

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Fig 15: Graph of shift of maxima from Larmor frequency vs
magnitude of oscillating magnetic field. In this graph we can
see that the shift is non linear.
Fig 16
Fig 17
Fig 18
Fig 19
Fig 20

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Fig 21
Fig 22
Fig 23
Fig 24:- graph for the relaxation time vs magnetic field peak
to peak (ΔBPP).
9. CONCLUSIONS
In the first part of the simulation after ignoring the
terms we have observed the dependency of the power
maximum on the amplitude of the oscillating e.m. field B1
(for fixed (ωL) Larmor frequency) and on ωL (for fixed B1).
Also a clear shift (Δω) of the power maxima (Pmax) from ωL
is observed.
In our second part of the simulation we have considered the
term and we have observed as the ratio r increases
the output i.e. shift (Δω) and the change in ΔBPP with B1
decreases and converges to the case of neglecting
terms. We also observe the shift (Δω) follows some non
linear relationship with B1.
It would be very interesting to simulate the response of the
system if applied micro wave field is not simply cosine
function but a different periodic function of time and carry
some information (as an amplitude modulated micro wave
field or frequency modulated micro wave field.).
REFERENCES
[1]. https://en.wikipedia.org/wiki/Bloch_equations.
[2].
https://en.wikipedia.org/wiki/Electron_paramagnetic_resona
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[3]. https://en.wikipedia.org/wiki/Zeeman_effect
[4]. https://en.wikipedia.org/wiki/Land%C3%A9_g-factor
[5]. https://en.wikipedia.org/wiki/Gyromagnetic_ratio
[6]. https://en.wikipedia.org/wiki/Larmor_precession
[7].http://www.unistuttgart.de/gkmr/lectures/lectures_WS_0
203/ magnetisation_ blochequ.PDF
[8]. Book Eighth edition introduction to Solid State Physics
by CHARLES KITTEL
[9]. http://www.its.caltech.edu/~derose/labs/exp6.html
[10]. N. D. Yordanov, Appl. Magn. Reson. 10 (1996) 339–
350.

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Electron Spin Resonance. PhDthesis, ETH Z¨urich, No.
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[19]. C. POOLE and H. FARACH, Relaxation in Magnetic
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Relaxation Phenomena in Solids.Hilger, London, 1969.
[21]. R.Brandle; G.Kruger and W.Muller warmuth;
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spectroscopy J.Magn. reson. 131; 86-91
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[24]. Line space of electron paramagnetic resonance signals
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BIOGRAPHIES
Mr.Amit kumar is Pursing B.Tech Fourth
year in Electrical Engineering from Abacus
Institute of Engineering and Managament
(A.I.E.M.) Natungram, Mogra, Dist.
Hooghly W.B., INDIA, Affiliated to West
Bengal University of Technology
(W.B.U.T.). braj.amit777@gmail.com
Rajib Chakraborty Assistant professor of
physics in the department of physics of
A.I.E.M. (Affiliated Tech college of West
Bengal University of Technology).
Completed B.Sc (Physics hons) from
IGNOU and M.Sc physics from Manipal
University (Sikkim). Also qualified in
CSIR-UGC-NET in physical Sc. Have Research interests
mainly on Quantum information Science, Quantum
Entanglement and it’s influence on quantum Game theory.
(rajib_30@yahoo.co.in)

Study on the dependency of steady state response on the ratio of larmor and rabi frequency for esr using matlab

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