ELECTRON SPIN RESONANCE

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Introduction

One of the intrinsic quantum-mechanical characteristics of fundamental particles

is that they have "spin," or intrinsic angular momentum. As with many quantities in

the quantum mechanics realm, the energy states associated with the spin are

quantized. The spin of a charged particle, like an electron, becomes evident when the

particle is placed in a magnetic field. A charged particle with a "spin" has an

associated magnetic dipole moment and, when placed in an external magnetic field,

the energy of the particle will depend on the orientation of its spin relative to the

magnetic field. In electron spin resonance, electrons in an external magnetic field

absorb energy from an applied oscillating electromagnetic field and change from one

spin orientation to another.

The spins of particles and their interaction with magnetic fields provides a useful

modern-day experimental tool for studying their environment. Electron spin

resonance (ESR), also known as electron paramagnetic resonance, uses the spin of

electrons. The resonance frequency is sensitive to the local environment of the

electrons. In chemistry and medicine the resonance frequency of a proton give

information about the local environment (magnetic field) of the proton. This is the

key to nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).

In this experiment, you will produce electron spin resonance to show that the

electron has two discrete spin energy states and you will measure the value of the

electron's magnetic moment.

ESR in Research

In research, ESR measurements are considerably more complicated that equation 2

would indicate. The electrons and protons in an atom or molecule form a complicated

electromagnetic environment, which is affected by the externally applied magnetic

field. The various energy splittings and shifts that show up in ESR measurements can

therefore provide sensitive information about the internal structure of the atoms and

molecules.

The test sample included with the PASCO ESR Apparatus DPPH, is a particularly

simple substance for ESR measurements. It has an orbital angular momentum of zero,

and only one unpaired electron. Therefore, for a given value of the external magnetic

field, it has only a single resonant frequency. This makes it possible to investigate

some of the basic principles of electron spin resonance, without (or before) getting

into the more complex world of ESR analysis.

Background

An electron bound to an atom can absorb a photon (or packet of electromagnetic

energy) of the right energy and make a transition to a higher energy state, provided

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that the photon energy precisely matches the energy difference between two allowed

states. The energy of a photon depends on its frequency (which can also be

considered in a semi classical picture to be the frequency of oscillation of the electric

and magnetic fields) by E = h. Thus, an electron bound to an atom can absorb or

emit only photons that have specific frequencies (Figure 1).

Figure 1. Electrons bound to atoms can emit or absorb electromagnetic energy

(photons) only in discrete quantities corresponding to the different allowed energy

states in the atom. (a) emission, (b) absorption.

When free electrons (or nearly free electrons in a solid) are placed in a magnetic

field, the z-component of the spin angular momentum (the component parallel to the

magnetic field) can take on two values, and these two spin orientations correspond to

different energies. These two energy states are due to the magnetic dipole moment

of the electron and its orientation relative to the external magnetic field (see Figure 2).

A classical analogue of this phenomenon is the current loop in a magnetic field: To

rotate the magnetic dipole moment of the loop opposite to the applied field, work

must be done. For the spinning electron, there are only two allowed energy states

associated with its interaction with the magnetic field, whereas the energy for a

macroscopic current loop is continuous.

(a)

(b)

Figure 2. (a) Two spin states of electron in a magnetic field (b) This image shows

the energy splitting due to a magnetic field for any spin ½ particle, such as electrons,

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(a) (b)

protons, neutrons or certain nuclei.

(http://www.shu.ac.uk/schools/sci/chem/tutorials/molspec/nmrlev1.gif)

Figure 3. Energy of a classical current loop in a magnetic field

The magnetic moment of the electron differs from the classical analogy of a

current in a loop in one other aspect. The classical magnetic moment is easily

calculated from the circulating current. The electron, on the other hand, is a point

charge so its magnetic dipole moment cannot be viewed simply as a classical

circulating charge. The magnetic moment of an electron is a fundamental, quantum

mechanical property of the electron. As for the current loop, the magnetic moment

and the spin angular momentum of the electron are proportional, but the constant of

proportionality is different.

Prelab question:

The magnetic moment of a current caused by a charge circulating in a loop is given by = IA where A is the area of the loop. Show that in this case = (q/2m)L , where q

is the charge, m the mass of the charges, and L the angular momentum associated with

the circling charges. For electron spins, a correction factor, known as the g-factor, must be included in

the above equation, and we refer to the component of the magnetic moment along the

z axis by substituting h/4 for the magnitude of the angular momentum along this

axis, so that e = ge(q/2m)(h/4). The g-factor for the electron is now known to high

accuracy through both experiment and theory to be 2.002319134.

The energy associated with a classical magnetic dipole, of moment , in a

magnetic field B (oriented along the z-axis) is given by E = -•B = -zB. The two

allowed quantum spin energy states of the electron, then, are ±eB, with the lower

state corresponding to “parallel” alignment of the magnetic moment with the B-field,

or when the spin of the negative charge is anti-aligned. (Note that the when there is no

external magnetic field, i.e., B = 0, the two states have the same energy and are

therefore indistinguishable. In quantum mechanical terms, we say that these states are

degenerate.) Since the magnetic dipole moment of an electron along the magnetic

field axis is given by e, as defined above, the energy difference between the two

allowed energy states, and therefore the energy that can be absorbed by the electrons

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is 2eB. The requirement for electrons to absorb a photon and flip its spin, then, is that

h = 2eB. This is considered a resonance situation because the natural response

frequency of the system (the frequency of radiation that the electrons will absorb)

equals the applied frequency (of the oscillating EM field).

In this experiment, the oscillating electromagnetic field is in the radio frequency

range, and is created by an oscillating current in a coil. A quantum of this oscillating

electromagnetic field has energy h, where is the frequency of oscillation. The

electrons will be located inside this coil and inside a larger-scale external magnetic field, created by a set of Helmholtz coils. When h = 2eB the electrons will absorb

energy from the oscillating EM field.

ESR in Theory

The basic setup for electron spin resonance is shown in Fig 1. A test sample is placed in a uniform

magnetic field. The sample is also wrapped within a coil that is connected to an rf oscillator. The

smaller magnetic field induced in the coil by the oscillations of the oscillator is at right angles to the

uniform magnetic field.

Consider, for the moment, a single electron within the test sample. The electron has a magnetic dipole

moment ( ) that is related to its intrinsic angular momentum, or spin, by the vector equation:

(1)

where:

= a constant characteristic of the electron, the g-facotr

= the Bohr magneton =

s = the spin of the electron

= Planck’s constant = .

The magnetic dipole moment of this electron interacts with the uniform magnetic field. Due to its

quantum nature, the electron can orient itself in one of only two ways, with energies equal to

; where is the energy of the electron before the magnetic field was applied. The

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energy difference between these two possible orientations is equal to ; where B is the

magnitude of the magnetic field.

Resonance occurs when the rf oscillator is tuned to a frequency f, such that the energy of the irradiated

photons, , is equal to the difference between the two possible energy states of the electron. Electrons

in the lower energy state can then absorb a photon and jump to the higher energy state. This absorption

on energy effects the permeability of the test sample, which effects the inductance of the coil and

thereby the oscillations of the rf oscillator. The result is an observable change in the current flowing

through the oscillator.

The condition for resonance, therefore, is that the energy of the photons emitted by the oscillator match

the energy difference between the spin states of the electrons in the test sample. Stated mathematically:

(2)

ESR in Practice

For an electron with only two energy states, in a magnetic field of a given magnitude, it would be

necessary to set the rf frequency with considerable accuracy in order to observe resonance. In practice,

this difficulty is solved by varying the magnitude of the magnetic field about some constant value. With

the PASCO ESR Apparatus, this is done by supplying a small ac current, superimposed on a larger dc

current, to a pair of Helmholtz coils. The result is a magnetic field that varies sinusoidally about a

constant value.

If the rf frequency is such that equation 2 is satisfied at some point between the minimum and

maximum values of the sinusoidally varying magnetic field, then resonance will occur twice during

each cycle of the field. Resonance is normally observed using a dual trace oscilloscope. The

oscilloscope traces, during resonance, appear as in Fig. 2. The upper trace is a measure of the current

going to the Helmholtz coils, which is proportional to the magnetic field. The lower trace shows the

envelope of the voltage across the rf oscillator, which dips sharply each time the magnetic field passes

through the resonance point.

The ESR Equipment

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Included Equipment

The ESR Apparatus is available in three separate packages (see Fig. 3):

The ESR Probe Unit (SE-9634) includes:

The probe Unit with base

Three rf Probes and a DPPH sample in a vial

The passive Resonant Circuit

The current Measuring Lead for the Probe Unit

The ESR Basic System (SE-9635) includes:

The ESR Probe Unit (se-9634)

A pair of Helmholtz Coils with bases

The ESR Adapter (SE-9637)

The Complete ESR System (SE-9636) includes:

The ESR Probe Unit (SE-9634)

A pair of Helmholtz Coils with bases

The Control Unit

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Control Unit Technical Data:

Power Requirement: 120 or 240 VAC; 60Hz

Fuse: 1.6A, 220V (sil-bio) for 120VAC; 0.8A, 220V (sil-bio) for 120VAC

Magnetic Field Supply: 0-10 VDC; 0-5 VAC; maximum current 3A (not overload protected)

Phase Shifter: 0-90˚ Digital Frequency Display: 4 digits

ESR Adapter

If you are not using the Control Unit, the ESR Adapter can bs used to connect the probe Unit to the

necessary power supply, frequency meter, and oscilloscope. See the section, Setup with the Basic ESR

System, for details of the connections. See the Appendix if you would like details for building your

own adapter.

Helmholtz Coils

The Helmholtz coils provide a highly uniform magnetic field in which to place the sample material for

the ESR measurement. They should be connected in parallel and placed so that the separation between

them is equal to the radius (see Fig. 7). When this is the case, the magnetic field in the central area

between the two ciols is highly uniform, and is equal to the value shown in Fig. 7.

The Passive Resonant Circuit can be used to demonstrate resonant energy absorption in a non-quantum

system. It is just an LC circuit with an adjustable capacitance. It replaces the test sample and the

Helmholtz coils in the ESR experiment.

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Required Equipment

To perform ESR experiments with the Complete ESR System, you’ll need the following additional

equipment:

A DC ammeter capable of measuring up to 3A

A dual trace oscilloscope

Connecting wires with banana plug connectors

Setup

Connect the Helmholtz coils to the Control Unit, as shown in Fig. 9. (The coils should be connected in

parallel-terminal A to terminal A, and Z to Z.) Connect an ammeter in series, as shown, to monitor the

current to the coils.

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Important: the DC current supply in the Control Unit is not overload protected. Do not let the

current to the Helmholtz coils exceed 3A.

Position the Helmholtz coils so that they are parallel and facing in the same direction, and their

separation is equal to approximately one half their diameter.

Connect the X output of the Control Unit to channel 1 of a dual trace oscilloscope. Set the oscilloscope

controls as follws:

Sensitivity: 1 or 2 V/div

Sweep Rate: 2 or 5 ms/div

Coupling: DC

Set the center knob on the Control Unit, to zero, then slowly vary , the left knob, from 0 to

10V and observe the trace on the oscilloscope. It should be a clean, straight line, showing that the DC

component of the current to the Helmholtz coils is constant. ( controls the DC current going to the

Helmholtz coils.)

Note: If the oscilloscope trace is not straight your Control Unit is probably not set for the

proper line voltage.

Set at approximately midscale, then turn clockwise, to increase the AC component of the

current to the Helmholtz coils. The trace on the oscilloscope should now show a smooth sine wave, as

in Fig. 10, corresponding to an AC magnetic field that is superimposed upon a constant DC field.

Fig 10 Oscilloscope – current to the Helmholtz coils

Connect the Y output of the Control Unit to channel 2 of the oscilloscope. Set the oscilloscope controls

for channel 2 as follow:

Sensitivity: 0.5 or 1 V/div

Coupling: DC

Connect the Probe Unit to the Control Unit, as shown in Fig. 9.

Plug the medium sized rf Probe into the Probe Unit, and insert the sample of DPPH into the coil of the

probe.

Turn on the Probe Unit by flipping the On/Off switch to the up (I) position. Then turn the Amplitude

knob on the Probe Unit to a medium setting.

The Frequency meter on the Control Unit should now indicate the frequency of the rf oscillations.

Adjust the Frequency control knob on the Probe Unit to produce an output of approximately 50MHz.

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Set to about the 4th position above zero (at about the 11 o’clock position).

Increase from zero to a medium value, so the Helmholtz coil current is about 1.0 A. The

oscilloscope traces should now look as in Fig. 11. Then channel 1 trace shows the current to the

Helmholtz coils, which is proportional to the magnetic field produced by the coils. The channel 2 trace

shows the envelope of the voltage across the rf oscillator, with the pulses showing the points of

resonance absorption. If you see no resonance pulses, slowly vary or the rf frequency, until you

do.

Fig. 11 the Oscilloscope Display

Your traces may not be symmetrical, as they are in Fig. 11. This is because of the inductance of the

Helmholtz coils, which causes the current through them, and therefore the magnetic field they produce,

to lag the voltage that drives them. You can compensate for this delay by adjusting , the Phase Shifter

control knob, until the traces are symmetrical. When symmetrical, the traces properly reflect the

relationship between the modulating magnetic field and the resonant pulses.

ESR in the X-Y Mode

ESR is often observed with the oscilloscope in the X-Y mode. For this mode of observation, connect

the X and Y outputs of the Control Unit to the X and Y inputs of the oscilloscope, respectively. In this

mode, the horizontal displacement of the trace indicates the magnitude of the magnetic field between

the Helmholtz coils. The vertical displacement indicates the signal from the Prob Unit. As before, two

resonance pulses can be observed since the magnetic field passes through the correct value twice each

cycle. By adjusting the Phase Shifter, the two peaks can be brought into coincidence. The resulting

trace will appear as in Fig. 12.

Fig. 12 the Oscilloscope Display

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Fig. 13 Using the Control Unit

Making ESR Measurements

Whether you are using the Complete ESR System of the ESR Basic System, the same technique is used

for making measurements.

Setup the apparatus as described in the

appropriate section.

Adjust the rf frequency and the DC current to the Helmholtz coils until you locate the resonance pulses.

Adjust the phase shifter so that the resonance pulses are symmetric with respect to the oscilloscope

trace that shows the current to the Helmholtz coils.

Refine the adjustment of the DC current until the resonance pulses occur when the AC component of

the current to the Helmholtz coils is zero.

To do this:

Making sure that channel 1 of the oscilloscope (the trace showing the current to the Helmholtz coils) is

in the AC coupled mode.

Using the oscilloscope controls, ground the input to channel 1, zero the trace, and then un-ground the

input.

Adjust the DC current. As you do, notice how the resonance pulses move closer together or farther

apart. Adjust the DC current, and the phase shifter if necessary, until the pulses occur just when the AC

current to the Helmholtz coils is zero. (This is most accurately accomplished if you adjust the vertical

position of the channel 2 trace so that the bottom of the resonance pulses are just at the zero level of the

channel 1 trace.)

With these adjustments, the oscilloscope traces should appear as in Figure 17. Everything is set for

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making ESR measurements. Since the current has been adjusted so that the resonance pulses occur

when the AC current to the coils is zero, the current to the Helmholtz coils at resonance is just the DC

value indicated by the ammeter. The resonant frequency is equal to the value on the Control Unit

display (or the value indicated by your frequency meter multiplied by 1,000).

Fig. 17 Scope Display

Measure the rf frequency and the DC current. Then vary the current and find the new resonance

frequency. Do this for several values of the frequency.

The magnitude of the magnetic field between the Helmholtz coils is directly proportional to the current

supplied to the coils. You can determine the magnitude of the field using the following equation (easily

derived from the Biot-Savart Law):

where:

=

N = number of turns in each coil

R = the radius of the Helmholtz coils (which is equal to their separation when they are properly

arranged)

I = current passing through the coil

If you are using the test sample of DPPH, you can now determine the g-factor for the electron using the

equation .

you might damage the oscillator.

CURRENT(A) MAGNETIC FIELD(mT) FREQUENCY(MHz)

.26 .55 15.3

.35 .74 20.6

.44 .93 25.2

.51 1.O8 30.1

.60. 1.27 35.7

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.69 1.46 40.4

.77 1.63 45.1

.86 .1.82 50.5

.94 1.99 55.7

.1.0 2.12 60.6

1.10 2.33 65.2

1.20 2.54 70.8

1.30 2.75 75.0

1.35 2.85 80.7

1.45 3.07 85.3

1.55 3.28 90.5

1.60 3.38 95.1

1.70 3.60 100.3

1.80 3.81 105.0

1.90 3.02 110.7

1.95 4..12 115.3

2.00 4.23 120.8

2.10 4.44 125.2

2.20 4.65 130.0

Reference:

Griffiths, Introduction to Quantum Mechanics, 4.4 (pp. 171 on),

Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, Sec. 8.1 to

8.3.

www.wikipedia.com

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