Electron Spin Resonance laboratory guide
PHYS3040 Laboratory – Student Guide
Electron Spin Resonance
1 Introduction
In the original quantum theory proposed by Bohr, only three quantum numbers were required to specify the
state of an electron in an atom: n, l, and m. However, from further studies of the Zeeman effect, including the
anomalous Zeeman effect and the Stern-Gerlach experiment, it became apparent that a fourth quantum number
was required. This additional quantum number is associated with the intrinsic spin angular momentum of the
electron. The interaction of this intrinsic spin and an external applied magnetic field can result in Magnetic
Resonance
Magnetic Resonance spectroscopy is an incredibly broad and active scientific field that continues to underpin
breakthroughs in physics, chemistry, biology, materials science, and medicine. Frequently, these contributions
exploit interactions between the electron (or nuclear) spins with other nearby electrons and nuclei. However,
we are conducting a more foundational electron spin resonance (ESR) experiment exploring the properties of
the relatively free unpaired electrons in a sample of DPPH (diphenyl-picrylhydrazyl).
There exists an unpaired electron in each DPPH molecule whose spin properties correspond closely to those
of a free electron. By subjecting a sample of DPPH to a linear magnetic field and a transverse oscillating
magnetic field, you can observe magnetic resonant absorption by these unpaired electrons and determine several
fundamental spin properties of them.
Learning Goals
• Use classical and quantum concepts to explain magnetic resonance phenomena, and appreciate the diverse
range of applications.
• Derive a simple mathematical model of the dynamics of an electron spin in magnetic field.
• Use the derived model and experimental data to determine the Land´e Splitting Factor and Spin-Spin
relaxation factor in the samples provided.
• Demonstrate laboratory procedures, including the use of lab books and record keeping, proper and justified
procedure, and error control and propagation.
Session 1 Aims
By the end of session 1, you should have
• Collected all data needed to model the relationship between the applied current and resulting magnetic
field.
• Observed and optimised the resonance signal for at least one of the samples provided.
• Explored the effect of different oscillator frequencies.
• Determined the data needed to be collected in session 2 to complete this experiment.
If you are nearing the end of your first session an have not completed these steps, please contact a tutor.
1
2 Experiment
Essential Concepts
Electrons possess a magnetic dipole moment resulting from their angular momentum. Because angular momen-
tum is quantized, the components of the electron magnetic dipole moment are also quantized. If we consider
a free electron in a constant magnetic field, its energy will then be quantized into discrete energy levels. The
energy difference between these levels will depend on several factors such as the strength of the constant mag-
netic field.
The electron can transition between these energy levels by absorbing or emitting electromagnetic radiation with
a frequency corresponding to the energy difference between the levels (∆E = hν). Thus, for a given magnetic
field there will be a frequency, ν, at which the electrons most efficiently absorb electromagnetic radiation. In this
experiment, you will observe the relationship between the magnetic field strength and the resonant frequency
of unpaired electrons in a DPPH (diphenyl-picrylhydrazyl) sample, whose spin properties correspond closely to
those of a free electron.
Figure 1: DPPH Molecule
Equipment
Figure 2 shows an overview of the ESR equipment. You can find an interactive version of this image and an
explanatory video on Blackboard. The different components are:
A) The control Unit
The control unit has a variety of functions:
• Reports the frequency, f, of the RF field in the B)Plug in sample coil that is being driven by the C)amplifier
unit).
• Reports the amplitude of RF field in the B)Plug in sample coil, which will decrease whenever the sample
absorbs energy. A processed version of the field amplitude, the “ESR Signal”, can be displayed on the E)
Oscilloscope.
• Applies a swept magnetic field to the sample by driving a slowly varying current through the Helmholtz
coils. The magnetic field is continuously varied at a frequency of 50 Hz, but the experimenter can set the
average magnitude (A=) and modulation amplitude (A ) of the current. The values of the average current
∼
(A=) and current modulation (A ) can both be displayed on the numeric output, while the instantaneous
∼
current, the “B Signal” can be displayed on the E)Oscilloscope. .
When first trying to find resonance at a new RF frequency, f, it can be helpful to start with a large current
modulation, A , to quickly scan a wide range of magnetic field strengths.
∼
B) The Plug-in RF Coil
Three different plug-in RF coils are provided which each permit oscillations over a different frequency range
(roughly 13 MHz to 30 MHz for the large coil; 30 MHz to 75 MHz for the intermediate coil; 75 MHz to 130 MHz
for the small coil). To cover the full range of available RF frequencies you will need to take several measurements
with each coil. The A) Control unit will report the frequency, f, whenever the amplifier is producing an RF
field. If the control unit reports a frequency of 0 MHz, the amplifier is not currently producing an RF field and
2
Figure 2: ESR equipment overview
to make the amplifier (and you) happy again, you may need to wiggle the connections on the RF plug-in coil,
adjust the frequency control knob, amplifier amplitude knob, or simply turn the amplifier unit off and on.
C) The Amplifier Unit
The amplifier unit holds the B) Plug-in RF coil and the DPPH sample, and generates the oscillating RF
magnetic field. The frequency of the RF field can be changed using the dial on the top of the unit. The
amplitude adjustment knob and power switch are located on the back and do not normally need to be adjusted.
D) Helmholtz Coils
The Helmholtz coils produce the magnetic field the sample is subject to. It is important to position them to
produce the most uniform field, which in theory occurs when the current flows in the same direction around
each coil, the average distance between the coils, d, is equal to the coil radius, r = 6.8 ± 0.5 cm, and the sample
sits at their centre.
E) Oscilloscope
The oscilloscope can visualize the magnetic resonance by graphing the “ESR Signal” versus the “B Signal” from
the A) Control Unit. This requires the oscilloscope to be running in XY mode, with the current strength “B
Signal”) as the X variable (Chan 1) and the “ESR Signal” as the Y variable (Chan 2). It can also be helpful
to set the “B Signal” channel (Chan 1) to AC coupling, so the the oscilloscope subtracts the average value the
current, A=, leaving the sweep centred even as the average current, A=, is adjusted to locate the resonance.
This continuous-wave ESR experiment uses two different sub-systems to control and detect magnetic resonance.
Controlling the resonant frequency: The frequency at which unpaired electrons in the sample can most
efficiently absorb (and emit) radiation, ν, is determined by the relatively strong, spatially uniform magnetic
field generated by the Helmholtz coils. When the magnetic field is stronger, the energy difference between the
spin up and spin down states is greater, and the frequency at which electrons most efficiently (i.e. resonantly)
absorb electromagnetic radiation increases (i.e. ν ∝ “B Signal”).
Detecting resonant absorption: The amplifier continuously drives an oscillating magnetic field of frequency,
f, in the RF plug-in coil. The amplitude of the oscillating field (aka “ESR Signal”) will be large when the sample
does not efficiently absorb the radiation (ν ≪ f or ν ≫ f), decrease when the sample begins to resonantly
absorb radiation (ν ≈ f), and be smallest at resonance (ν = f).
In most ESR experiments, including this one, the magnetic resonance is analysed by scanning the applied
magnetic field while holding the oscillating field frequency, f, constant. To do this, the control unit continuously
sweeps the Helmholtz coil current, IH (and thus magnetic field and spin resonance frequency, ν), over a range,
A= − A ≤ IH ≤ A= + A . When first starting out the tested range of magnetic fields may not overlap with
∼
∼
3
the magnetic resonance (i.e. A= is too small or too large), and the ”ESR Signal” will barely vary over the swept
range. One can then slowly adjust A= while looking for a change in the “ESR Signal” on the oscilloscope. When
the scanned range does overlap with the magnetic resonance, the plot of ”ESR Signal” versus ”B Signal” on the
oscilloscope will show a distinct minimum where the sample most strongly absorbs energy from the oscillating
field.
Data To Collect
1. The signal we can measure using the oscilloscope is the current supplied to the Helmholtz coils, but we
need to know the magnetic field that the DPPH sample is subject to. A magnetometer is provided to help
you experimentally relate these quantities and test the expected theoretical relationship for Helmholtz
coils. Is there another source of magnetic field that should be considered?
2. You should collect a variety of data points for different EM frequencies and the magnetic field strength
under which they become resonant. You will need enough to perform a linear regression.
3. Finally, you should collect a couple of full spectrum resonance dips, which will be used in the determination
of the spin-spin relaxation (T2) factor. It is sufficient to calculate the (T2) factor for a couple of examples
and average them, you are not expected to linearise this data, or have as many data points as you would
for the calculation of the g factor.
3 Theory and Exercises
Summarise your answers to questions below in your log book.
3.1
Land´e Splitting Factor (g)
Electron spin resonance is a property exclusive to paramagnetic material, due to their non-zero total angular
⃗
momentum. The magnetic moment associated with the total angular momentum J is
µB
⃗
µ⃗ = −gJ
J
J
(1)
¯h
where µB = h¯e/2me is the Bohr magneton (a useful combination of constants), ¯h = h/2π is Planck’s constant,
gJ is the Land´e splitting factor, me is the mass of the electron and e is the fundamental unit of electric charge.
⃗
The energy of a magnetic moment in an external magnetic field B0:
⃗
E = −µ⃗ · B0
(2)
J
Consider only the component of the magnetic moment that is parallel to the magnetic field, as all other
components will be zero. According to quantum mechanics, the allowed values of this component Jz can be
written
Jz = h¯mJ
(3)
where mJ is the angular momentum quantum number, and can only take integer or half integer values (±1/2
in the case of a single spin).
1. Justify the choice of the DPPH sample to approximate a free electron
2. When the spin is in the lower energy level mJ = −1/2, what frequency of electromagnetic radiation is
required to excite it to the upper level? Derive a relationship between frequency ν and the value of the
external magnetic field B0.
Although this quick derivation gives you the expression you will need to determining the Land´e splitting factor,
there is much more to the phenomenon of electron spin resonance. For a more nuanced review of relevant theory,
3.2
Spin-Spin Relaxation
Considering a single electron in isolation might suggest that the resonance condition must be matched exactly.
However for spins embedded in a material, interactions with other spins (of nearby electrons or nuclear spins,
for example) can change the spin energy eigenstates/levels, causing the state of each spin to evolve a little
differently from its neighbours. As a result, even if two electron spins initially start out in the same state,
over time they will evolve into different, uncorrelated states. The timescale over which individual spins become
4
“scrambled” is characterized by the spin-spin relation time, T2, which can be connected to the width of the
magnetic resonance heuristically by considering the time-energy “uncertainty relation”:
¯h
∆ET2 ∼
,
(4)
2
where ∆E is an energy width characterising the spread of relevant eigenstates.
1. Using the expressions above derive the relation for width of the resonance
¯h
∆B0 =
.
(5)
2gJ µBT2
To derive this expression more rigorously, we need to consider the Bloch equations that govern spin dynamics
– see sections 4.3 and 4.4 below), which show that ∆B0 is the half-width of an expected Lorentzian curve. You
may have encountered Lorentian frequency profiles previously in studies of driven classical oscillators.
You will use Eq. 5 to experimentally determine the spin-spin relaxation time in several cases (different oscillator
frequencies and different samples), and seek to explain what this tells you about the material.
4 Further Background and Theory
4.1
Electron paramagnetism
We are considering an almost free electron. This is an unpaired electron in the DPPH (diphenyl-picryl-hydrazyl)
molecule whose spin properties correspond closely to those of a free electron. Classically a particle which has an
angular momentum rotates, and if it is charged will have also a magnetic moment, proportional to the angular
moment. The proportionality factor is called the magnetomechanical factor γ. This factor should be negative
for electrons because they have a negative electrical charge but it is conventional to take γ as the magnitude of
ˆ
the magnetomechanical factor and include the negative sign in the equation µˆ = −γS, where the spin angular
ˆ
momentum operator is S. We ignore the kinetic energy of the electron and therefore when the electron is placed
in a static magnetic field B, its energy operator (Hamiltonian) reads
ˆ
ˆ
ˆ
ˆ
ˆ
H = −µˆ · B = γS · B ≡ γ(SxBx + SyBy + SzBz)
(6)
First, we consider a static magnetic field B = B0 and find the eigenvalues of this Hamiltonian. Of course, the
eigenvalues of the Hamiltonian do not depend on the choice of the coordinate system but a wise choice of the
reference frame makes this step simpler. We chose the direction of z-axis to be parallel to the direction of the
magnetic field B0. The Hamiltonian then simplifies to
ˆ
ˆ
H = γSzB0
(7)
1
The eigenvalues of this Hamiltonian (eq. (6)) are ±2 γ¯hB0, where the lower energy state corresponds to spin
h¯
projection −2 . The energy difference between these levels is ∆E = γ¯hB0. The time-dependent wave-function
includes an oscillating phase factor exp(−iEh¯ t). The absolute value of the phase of a quantum state does not
have a physical meaning but the relative phase between the two states does. Relatively to the lower eigenstate
of the electron, the upper eigenstate will have a time dependent phase factor of exp(−i∆E t) ≡ exp(−iω0t),
h¯
where ω0 ≡ γB0.
A transition between the two energy levels can be induced by applying a radiation field of frequency ω such that
ω ≈ ω0. At thermal equilibrium, the lower energy state has a greater probability of occupation, leading to an
excess of magnetic dipoles aligned with the applied magnetic field. Therefore some energy will be transferred
from the field to the electrons. Note that photons, representing the quanta of energy of the electromagnetic
field oscillating at frequency ω have energy ¯hω. The energy of the whole system will be conserved upon such a
transition (one photon is absorbed and the electron is promoted to the higher energy state). Additionally to the
energy conservation, the angular moment should be also preserved upon the transition. Fortunately, circular
polarised photons carry an angular moment of exactly h¯ and therefor the moment conservation is automatically
satisfied for circular polarised magnetic field.
The resonance condition ω ≈ γB0 can be detected by observing the absorption of radio-frequency power on
resonance while little power will be absorbed off resonance.
5
Figure 3: Electron magnetic dipole in static B-field showing Larmor precession and correct orientation of B1.
Such a measurement lies at the core of the ESR technique and enables determination of the magnetomechanical
factor
ω0
γ =
(8)
B0
once the values of B0 and the corresponding ω0 are measured.
Classical electrodynamics (which by definition does not take electron spin into account) predicts a magnitude
e
2me
of γ0 =
for the electron, a value which remains the same whatever the electron’s environment. Here, e and
me are the charge and rest mass of an electron, respectively. Quantum mechanics agrees with this value of γ
only when the magnetic dipole arises from purely orbital angular momentum. If the angular momentum is due
to the electron’s intrinsic spin or to some combination of spin and orbital angular momentum, the quantum
mechanical result differs from the classical prediction. (And it is the quantum mechanical value which agrees
with experiment.) We use the Lande g factor to express the deviation of γ from the classical value by
γ
g =
(9)
γ0
4.2
Orienting the magnetic field and electromagnetic radiation to produce ESR
We now consider an electron magnetic dipole in its lower energy state as a magnetic ‘top’ in order to predict
how to apply the radiation field to induce transitions to the higher energy level. Figure 3 shows the relative
directions of B0, the electron spin S, Sz, and the magnetic dipole moment µ. (Don’t worry for the moment
about the rest of the diagram; it will become clear as you read this section.) First, let us examine the behaviour
with time of the expectation value of the magnetic dipole operator in the static field B0. As you will see in
lectures, for an operator F which does not depend on time, the time rate of change of its expectation value is
given by the general result
d
i
⟨F⟩ =
⟨[H, F]⟩
(10)
dt
¯h
where H is the Hamiltonian for the system. We thus obtain
d
⟨µ⟩ = γB × ⟨µ⟩
(11)
dt
Question 1 Derive eqn 11 by applying the general result of eqn 10 to the magnetic dipole operator. [Hint:
you will need to use the commutation laws for the spin angular momentum operators.]
We now evaluate eqn 11 explicitly, recalling that B0 = (0, 0, B0). The three component equations are
6
dµx
=
=
=
−ω0µy
ω0µx
0
(12)
(13)
(14)
dt
dµy
dt
dµz
dt
From the last equation it is evident that the component of the dipole moment parallel to B0 (i.e. µz) is constant
with time; the dipole remains aligned with the field. The equations for µx and µy, on the other hand, indicate
precession about the direction of B0 at a constant rate and angle. The sense of precession is marked on Figure
3; note that the dipole precesses in the opposite sense to the direction of ⟨µ⟩ × B0, as the electron is negatively
charged. To determine the angular rate of precession, we differentiate eqn 12 or 13 and substitute Eqn. (13) or
d2µx
dt2
d2µy
=
=
−ω2µx
(15)
(16)
0
−ω2µy
0
dt2
We can choose the orientation of the x and y axis so that at the time zero the y-component of µ⃗ is zero while
its x-component is µ . Then the solution of these equations is
⊥
µx
µy
=
=
µ cos(ω0t)
(17)
(18)
⊥
µ sin(ω0t)
⊥
The component of the magnetic moment in the xy-plane rotates with angular frequency ω0 = γB0. This is
known as the Larmor precession frequency, as the precession of the magnetic dipole moment is termed Larmor
precession.
What is significant, is that the Larmor frequency is equal to the transition frequency between paramagnetic
energy levels found in the previous section. This suggests that not only should the applied high frequency
radiation field have frequency ω0 (= ωL) in order to ‘flip’ the dipole into the higher energy state (the energy
condition); the magnetic part of this field should ‘follow the precessing dipole around’. In other words, the
applied radiation field must be circularly polarised, with its magnetic part B1 rotating in the same sense as the
precession of µ. In order to ‘follow µ around’ as it precesses, the field must further be phased so that B1 is
We can see how B1 applied in the appropriate way acts to ‘flip’ the magnetic dipole into its higher energy state
by looking at the ‘torque’ exerted by B1 on µ. (Thinking about the ‘torque’ is a simplified way of looking
at the change of µ with time in the presence of B1, but without determining the exact form of equations
d
describing dt ⟨µ⟩.) The direction of the torque is that of −µ × B1, which is illustrated in Figure 3. As shown,
the torque is in the same plane as µ and B0 and acts to increase the inclination angle θ between these two
vectors. In a classical description, θ would increase uniformly with time, gradually increasing the potential
energy associated with µ. In our (quantised) case, θ can take on one of only two values, corresponding to
the two paramagnetic energy states. Hence if the ‘torque’ applied by B1 meets the conditions of our quantised
system, as set out earlier, applying the radiation field (of which B1 is the magnetic part) will induce a transition
into the higher-energy paramagnetic state, as desired.
One of the conditions on B1 was that it should take the same sense as the precession of µ. Let us consider what
would happen if we applied B1 with the correct frequency, but the wrong sense, that is, counterpropagating
with respect to the precession. In this case, the direction of the ‘torque’ of B1 on µ would change as µ precessed
around B0, sometimes acting to increase θ and sometimes acting to decrease it. The tendency would then be
to set θ oscillating, not to ‘flip’ it to its higher-energy value. We thus realise the importance of applying B1 in
the correct sense.
4.3
Spin-lattice and spin-spin relaxation
We now know how to induce an ESR transition in electrons from their lower to higher paramagnetic states.
What happens, however, when we apply the radiation field (B1) over an extended time? To answer this question,
we must now consider an ensemble of electron spins (as found in our macroscopically sized sample of DPPH,
for example), rather than a single electron spin dipole. Before turning on the radiation field, the spins will be
7
in thermal equilibrium, which means there will be a greater population density of electrons in the lower state
than in the upper state. When the radiation field is applied, energy will be absorbed from the field and induce
transitions from the lower to the upper paramagnetic state. The population density in the upper state will
gradually increase and that in the lower state gradually decrease, until the two densities equalise. Once this
occurs, the rates of absorption from the lower state and emission from the upper state will become equal, and
there will be no further nett absorption of energy from the radiation field. This effect is called saturation.
The saturation is reduced due to spin-lattice relaxation. On absorbing energy from the radiation field, the
electrons are no longer in thermal equilibrium with the lattice, but effectively at higher temperature. Any inter-
action between the electrons and the lattice will tend to restore thermal equilibrium; energy will be transferred
on average from the electrons to the lattice. As the electrons lose energy, they return to the lower paramagnetic
state. This relaxation process means that the population density of the lower state remains greater than that
of the upper state and permits continuous absorption of energy from the radiation field by the sample, making
the ESR phenomenon more readily observable. The characteristic time T1 taken to restore thermal equilibrium
is known as the spin-lattice relaxation time. (In DPPH, T1 is of the order of 10−5 s.)
A second relaxation process is spin-spin relaxation, which comprises interactions between electrons in the sample,
without nett transfer of energy out of the ensemble. For instance, two electrons could swap the state of their
spins (a process known as Heisenberg exchange); this doesn’t change the total energy of the ensemble, but it
does reduce the lifetime of an individual spin [1]. The characteristic spin-spin relaxation time is designated T2.
It is of the order of 10−8 s in DPPH. We can qualitatively explain line broadening due to spin-spin interactions
through the notion that each magnetic dipole sees a magnetic field consisting of B0 and a fluctuating component
δB due to the effects of neighbouring dipoles. As a result, the resonant frequency varies somewhat, such that
absorption occurs (and is observed) over a range of frequencies.
The theory which takes into account the relaxation processes is outlined in the next section.
4.4
The Bloch equations and line shape function
The macroscopic magnetisation M is a sum (over all electrons) of the electron magnetic moments µ and its
d
change with time is governed by a relation equivalent to that of Eq. 11, that is, dt M = γB × M, where B is
the applied magnetic field which now has two parts, a static field B0 and and oscillating field B1
B = B0 + B1 = B1 cos(ωt)i + B1 sin(ωt)j + B0k
(19)
Substituting and evaluating the cross-product, we obtain the component equations
dMx
=
=
=
−ω0My + γB1 sin(ωt)Mz
(20)
(21)
(22)
dt
dMy
ω0Mx − γB1 cos(ωt)Mz
dt
dMz
−γB1 sin(ωt)Mx + γB1 cos(ωt)My
dt
These are known as the Bloch equations.
The Bloch equations model the behaviour of the macroscopic magnetic dipole moment in the applied static and
radiation fields; however, we have yet to add terms to take relaxation effects into account. We do this by noting
the expected values of the components of M a long time after the field is turned on, and adding additional
terms to the equations to ensure they approach these equilibrium values. This is achieved most easily if we
look at what happens in the absence of the radiation field B1, i.e. set B1 = 0. In this case, at equilibrium,
only Mz should be non-zero, that is, the macroscopic dipole moment will be aligned with the static field B0,
as expected for a paramagnetic material. We write the dipole moment at equilibrium as M = (0, 0, M0), where
χ
µ0
M0 is related to the strength of the applied (static) field B0 through the magnetic susceptibility: M0 =
B0.
If, however, we put B1 = 0 in equations 20-22, they take on the form of eqns 12-14 (with the µ’s replaced
dMz
dt
by M’s). In that case,
= 0, so Mz would always remain at its initial value, rather than approaching the
equilibrium value M0. It is spin-lattice relaxation which tends to restore Mz to M0, on a timescale given by the
M0−Mz
spin-lattice relaxation time T1. Adding a term
accounts for the effects of spin-lattice relaxation.
T1
As noted above, we expect the transverse (i.e. x and y) components of M to be zero at equilibrium, while eqns
12 and 13 give oscillatory Mx and My. The second relaxation process, spin-spin relaxation, is relevant here, as
8
−My
to
−Mx
it tends to destroy Mx and My. This is built into the equations by adding a term
T2
T2
dMx
dt
Mx
T2
=
=
=
−ω0My + γB1 sin(ωt)Mz −
(23)
(24)
(25)
dMy
dt
My
ω0Mx − γB1 cos(ωt)Mz −
T2
dMz
dt
Mz − M0
−γB1 sin(ωt)Mx + γB1 cos(ωt)My −
T1
We are interested in the steady-state solution to these equations.
Question 2 For the steady-state solution to the Bloch equations, assume that M can be expressed as follows
ꢀ
ꢁ
ꢀ
ꢁ
χ′
χ′′
µ0
χ′
χ′′
µ0
M =
B1 cos(ωt) +
B1 sin(ωt) i +
B1 sin(ωt) −
B1 cos(ωt) j + Mzk
(26)
µ0
µ0
where Mz, χ′, andχ′′ do not change with time. Solve these equations to derive the expression for χ′′ given in
eqn 27. [Hint: if A cos(ωt) + B sin(ωt) = 0 is to hold for all times t, both coefficients A and B must be zero.]
ω0χ0T2
1 + γ2B21T1T2 + (ω0 − ω)2 T22
χ′′ =
(27)
(28)
The energy loss rate due to the changing magnetisation is derived in classical electromagnetic field theory
dM
˙
W = B ·
dt
Using the expression for M above and expression for the total applied field
Question 3 Show that the mean rate of energy dissipation W, where
2π
ω
Z
ω
˙
W =
Wdt
(29)
(30)
2π
0
reads
ωχ′′B21
W =
µ0
This expression for W indicates that the amount of energy absorbed depends on the frequency ω and magnitude
B1 of the applied radiation field, and the quadrature (out-of-phase) component of the magnetic susceptibility,
that is χ′′. Due to the resonance effect, χ′′ and the absorption strongly depend on the difference ω − ω0. This
dependence defines the line shape function
1
L(ω − ω0) =
(31)
1 + γ2B12T1T2 + (ω0 − ω)2 T22
If the condition γ2B12T1 << 1 is fulfilled (with a small magnitude B1 one can always achieve it), then
1
L(ω − ω0) =
(32)
1 + (ω0 − ω)2 T22
As we can see, the interaction with the environment (represented here by phenomenological constants T1 and
T2) tend to broaden the ESR spectral line, which would be a sharp spike if the dipoles were interacting only
with B0 and B1, as seen in Figure 4. The shape of the curve in Fig. 4 is very common in different fields of
physics and is called Lorentzian function or simply Lorentzian.
In the experiment, the detected signal is proportional to the power loss and the overall vertical scaling depends
on the details of the experimental setup. What you will see on the screen of the oscilloscope is the dependence
of the RF power PRF on the external field B0. That is
1
PRF(B0) = A0 − A1
(33)
1 + (γB0 − ω)2 T22
where A0 and A1 are two unessential constants, and we have used the relation ω0 = γB0. You have to read the
experimental section to understand how to relate the horizontal coordinate on the screen to the values of B0.
9
Figure 4: Broadened ESR line shape.
Magnetic Field B0 (T)
RF power (arb. units)
B0=ω/γ
Figure 5: Theoretical dependence of the RF power on the magnetic field B0.).
References
[1] Gerson, F. and W. Huber, Electron Spin Resonance Spectroscopy of Organic Radicals. 2003: Wiley-VCH.
QD 96.E4 G47.
[2] Ikeya, M., New Applications of Electron Spin Resonance: Dating, Dosimetry and Microscopy. 1993: World
Scientific. QC 763.I38.
[3] Servant, R. and E. Palangi´e, Comptes rendus hebdomadaires de l’Acad´emie des Sciences s´erie B, 1975.
280(8): p. 239-42.
[4] Crane, H.R., The g Factor of the Electron. Scientific American, 1968. 218: p. 72-85.
[5] Marcley, R.G., Apparatus for Electron Paramagnetic Resonance at Low Fields. American Journal of Physics,
1961. 29(8): p. 492-97.
[6] Pake, G.E., Fundamentals of Nuclear Magnetic Resonance Absorption I and II. American Journal of Physics,
1950. 18: p. 438-452, 473-486.
[7] Pake, G.E., Paramagnetic Resonance. 1962: W. A. Benjamin. QC 762.P3.
[8] Poole, C.P., Electron Spin Resonance. 1967: Interscience Publishers. QC 762.P6.
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