2. ELECTRON SPIN RESONANCE

Useful 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.

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.

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 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 semiclassical 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)

(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, 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 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.

Electron spin resonance is also discussed by Melissinos, ("Experiments in Modern Physics," pp. 340-344,

374-377) and by Eisberg and Resnick (p. 369). However, do not concern yourself with the apparatus

discussed by Melissinos.

Equipment

Figure 4: ESR equipment

The equipment used in this experiment is shown in Figure 4. It includes the following: (a) rf (radio

frequency) probes: 3 small plug-in coils (inductors), labeled E, F, and G, (b) rf oscillator (white box on

stand), (c) Frequency adapter (d) LC circuit (coil and white box with adjustable capacitor), (e) Helmholtz

coils: 2 coils of diameter 13.6 cm, (f) DPPH sample (in small test tube). You'll also need a +12V/-12V DC

power supply, frequency meter, oscilloscope, Variac supply, 120-to-25 V transformer, and DMM.

The DPPH contains the electrons whose spins will produce the desired resonance. DPPH is the organic

salt diphenyl-picryl-hydrazil, (C6H5)2N-NC6H2(N02; it contains an electron that is unpaired and has 0

orbital angular momentum (it is in an s orbital) and therefore behaves like an isolated electron. We use

DPPH, in part because it has a very strong and narrow absorption line, with a g=2.0036 factor very close to

the value for free electrons, and also because it’s used in chemistry as a free radical, and ESR actually is

used to study its properties.

Experimental ProcedureTo measure the magnetic moment, e, of the electron, you are to find the resonance condition

described in the introduction and given by 2eB = h. You will need to find resonance for a situation with

known B and . You will create a known uniform B field with a set of Helmholtz coils and create an

oscillating EM field of known frequency by using smaller coils carrying an AC current of known

frequency. Then, by placing the electrons inside the small coils and the small coils inside the Helmholtz

coils, you will provide the electrons with the necessary condition for spin flip transitions. You could either

set a fixed B and vary until absorption from the EM field is seen (i.e. look for an absorption line in a

spectrum) or you can fix and vary B. You will use the latter approach in this experiment. This is easier

because you can put the 60 Hz current from the wall through the Helmholtz coils, which will vary the

magnetic field at a rate much faster than you could vary the frequency of the AC current through the small

coils. Note that the electrons respond to a frequency of tens of MHz, so on the timescale of the electrons,

the magnetic field from the Helmholtz coils is effectively constant. The 60 Hz frequency is also suitable for

monitoring with an inexpensive oscilloscope. When resonance occurs, a drop in the voltage across the

small coils will be seen because the electrons absorb the energy of the electromagnetic field created by the

current, thereby altering the impedance of the coil.

Creating the oscillating electromagnetic field and detecting EM

absorption

The three small plug-in coils are capable of producing EM fields of the following frequencies: E (the

biggest coil): ~ 13-30 MHz; F: ~ 30-75 MHz; G (the smallest): ~ 75-130 MHz. For now, insert coil F

into the oscillator and set up the circuit shown in Figure 5. (Do not insert the DPPH sample yet.) The

adapter ports are labeled and should be connected as follows:

"f/1000" attaches to the frequency meter;

"+12 V", "0", and "-12 V" attach to the DC supply.

Because neither electrode of the RF coil is grounded, you must make a differential measurement of the

voltage across the coil (rather than the more normal type of measurement in which one electrode is

grounded). To do this, first use two 10x probes to connect the top and bottom electrodes of the RF coil to

channels 1 and 2 respectively of your scope. Connect the small alligator clip on one of the 10x probes to the

com connector of your power supply. (Do not connect the other grounding clip.) Your power supply is

“floating” and without this connection, it would not be referenced to ground. Set both channels to the same

sensitivity, e.g. 10 mV/div and then set the scope to subtract channel 2 from channel 1 by pressing the +/-

button.

Figure 5: Oscillator set-up

NOTE: For the sake of the health of the oscillator, please adhere to the following operating rules:

1. Use only a DC power supply.

2. Do not allow the applied voltage to exceed 12 V.

3. Be sure that the voltage is at 0 before turning the oscillator switch on or off.

4. Do not unplug a coil with the oscillator on.

Note that the frequency adapter "mixes down" the frequency by a factor of 1000, as indicated by the

"f/1000" on the port connecting to the frequency meter. (Why do we use the strange term "mixes down"?)

This is needed because, as indicated above, you're going to need to attain frequencies of tens of MHz,

which is not readable with an inexpensive frequency meter. However, with this adapter you only need a

counter capable of reading tens of kHz. Keep this in mind when setting the scale of the meter.

Plug in and turn on the DC supply, the oscilloscope, and the frequency meter. Turn on the oscillator,

and bring the voltage up to 12 V. Watch the frequency meter and turn up the gain on the oscillator (knob on

the back with curved wedge label indicating strength) until a reasonable frequency reading appears on the

meter. The meter should see frequencies from 30 to 75 kHz (remember that the oscillator signal is mixed

down by a factor of 1000). Ask your instructor for help if, after some effort, you are unable to obtain a

stable display.

You will likely notice that the scope signal is sensitive to the position of the 10X probe cables relative

to your body, and to the position of your hand if it gets close to the RF coil. The human body has a large

effect on RF signals, as you may have noticed when adjusting your TV antenna. Move the 10X probe

cables to a position on the table away from your body.

Watch the oscilloscope and play with the gain on the oscillator. Note the clarity of the sine wave signal

versus the noise as you play with the gain. Set the gain for optimal clarity of the signal. Check that the

frequency reading on the meter agrees with the oscilloscope display. Change the frequency setting by

playing with the knob on top of the oscillator labeled f/MHz. The signal on the oscilloscope you have now

is a display of the voltage across the oscillator versus time. When you achieve resonance, it is this signal

that will show an absorption.

Creating the external magnetic field (Helmholtz coils)

Use the two larger coils to make a set of Helmholtz coils--the requirement for Helmholtz coils is that

the coils be placed along the same axis with a separation equal to 1/2 their diameter. (Solenoids would give

a more uniform field, but they are harder to use in the sample geometry we need to employ here.) The

diameter of these coils is 13.6 cm. Measure the actual spacing between the coils carefully and record it for

later analysis. (You may wish to orient your system to avoid the major effects of the Earth’s gravitational

field; talk with your instructor about this if you are not sure how to proceed.) Connect the coils in series

and to the transformer and Variac as shown in Figure 6. Since you need the current to go in the same

direction in the coils, make sure that the coils face the same way (use the writing on the coils for

orientation). Check with your instructors if you are not sure about this last part, as it can cause confusion

later on!

The magnetic field inside a set of Helmholtz coils is given by

(1)

where R is the coil radius, a is the average coil separation, I the current, and N the number of turns in each

coil. These coils have N = 320. Note that to calculate the magnetic field, you'll need to know the current

through the coils, which you measure indirectly using a DMM as a voltmeter across the 3 Ohm power

resistor. (You should measure the resistance of this combination of power resistors directly, using the 4-

point probe method—see Appendix A.) However, since it is an AC voltage, the DMM (which must be set

on AC) will read the r.m.s. voltage, not the amplitude. So, keep in mind that the amplitude is times

greater. The amplitude can also be read from the oscilloscope display. The scope allows you to

qualitatively monitor the current.

Figure 6: Helmholtz Coils Set-up

The coils are reported to be safe for currents up to 2 Amps RMS. You should find that such high

currents are not required. You also should be careful about leaving smaller currents in the coils for long

periods of time. The coils get quite hot and will slowly degrade over time. So, as a general rule of thumb,

do not leave the coils with more than 0.5 A RMS or so for periods of longer than five minutes.

With the Variac set at 0 Volts, turn on the power. Turn the Variac up until the DMM shows some

reasonable current--e.g. 0.25 A RMS. Set the oscilloscope to trigger off of as well as display channel 2 and

play with it until you see the AC signal of frequency 60 Hz. You should now, effectively, have a display of

magnetic field versus time.

Measuring electron spin resonance

Making sure, first, that the DC power supply is off and the Variac is turned to 0 Volts. Place the DPPH

sample inside coil F (which should still be plugged into the oscillator) and place the coil, with DPPH,

inside the Helmholtz coils set-up, as shown in Figure 7. Make sure that the part of the ampule containing

the black powder is centered in the RF coil. Turn on the DC power supply and the oscillator switch, bring

the voltage up to 12 V, and turn up the gain on the oscillator, as before. Set the frequency to around 40

MHz (40 kHz on meter). Watch the current on the DMM and slowly turn up the Variac. Be sure that you

see the AC signal through the coils appearing on the second oscilloscope. If you don't then stop and adjust

the oscilloscope until you do. Set the current to 0.4 A RMS.

The magnetic field is now being swept up and down at 60 Hz with a high enough amplitude to meet

the resonance condition at some point (or points) in the 60 Hz cycle. However, if we want to have a prayer

of observing the resonance (which we hope to do by observing the changes in the amplitude of the voltage

across the RF coil), we must readjust the time scale of the first scope so as to match that of the second.

Also, to make sure the two scopes are in sync, set the "Trigger Source" on both to "Line". This causes them

to trigger off the 60 Hz of the incoming AC power, so that the two scopes will now be in sync. Since the

RF frequency is so much higher than 60 Hz, the first scope should now be displaying a fat band. You won't

be able to discern the sinusoidal nature of the signal, but you know it's still there, and you can easily see

variations in the amplitude. Can you see any evidence of resonance (which would be indicated by decreases

in the amplitude in synchronization with the 60 Hz signal of the magnetic field)?

Most likely, you won't see any changes in amplitude at all. To accentuate the changes, the

manufacturer has included a special circuit. First, it measures the amplitude of the RF voltage applied to the

RF coil (see Figure 7). This amplitude is a "slowly" varying signal -- in our case it varies at 60 Hz (which is

slow indeed compared to RF frequencies). Next, this amplitude is passed through a high-pass filter, which

blocks the DC level, so that only changes in the amplitude get through. Finally, it amplifies this change-in-

amplitude signal by a factor of 56,000. The resulting signal, which comes out of the "Y" connector, is

extremely sensitive to changes in the amplitude of the RF signal.

Dial the Variac back to zero, then connect the Y output to channel 2 of the second scope. Now increase

the Variac slowly. You will notice that some of the 60 Hz signal gets through to the Y output. Why?

Continue dialing the Variac up until you see sharp dips in the Y output. You have found electron spin

resonance! Again, you will likely not be able to see any changes on the first scope (which is just measuring

the voltage across the RF coil directly). Now that you have seen that this measurement is not helpful for the

DPPH experiment, you may disconnect the 10X probes from the RF coil, and not use the first scope for the

rest of the experiment.

Figure 7: DPPH sample and oscillator inside Helmholtz coils

Raise the Variac voltage a little higher and note that you see four dips per cycle of the magnetic field.

Why does this happen? (Check with your instructor at this point to ensure you understand the signal.)

Record the oscillator frequency and the current through the Helmholtz coils for the resonance.

Note: There is a phase shift between the two signals, so you can't simply read the current that appears

to be in line with the dips. You need to use your knowledge of a sine wave. For example, you should be

able to figure out the voltage for resonance when you have exactly two dips (at what point in the sine wave

must resonance occur). Since there is a range over which two dips occur, a more precise method would be

to set the voltage so that you have four equally spaced dips. At what points in the sine wave must these dips

occur now? When you go to higher oscillator frequencies (see below) , you’ll probably find that you can’t

go to high enough voltages to get four equally spaced dips. In these cases, the "two dips" approach will

suffice.

Note now that the dips are not narrow spikes, but have significant widths. This is an artifact of the

circuit that creates the y output. Measure the widths of the dips. Consider other physical factors that would

contribute to the broadening of the dips and make some calculations to estimate their quantitative effect.

You should find that the effect of these other physical factors is not sufficient to explain the observed

widths.

Find electron spin resonance for about 9 to 10 different frequencies, using about 3 frequency settings

per plug-in coil, and measure the current and frequency of each resonance. Measure the widths of the dips

for several cases to determine if they depend on any measurable parameter (see Analysis).

WARNING! Be careful when changing coils. Remember to turn off the oscillator before changing coils or

you might damage the oscillator.

Analysis

Calculate the magnetic field for each ESR case you observe and make a plot of vs.. B. Compute

the theoretical slope of this curve from the discussion above, being careful about units. From the slope of

the best-fit line to your actual data, determine the magnetic moment of the electron, and its uncertainty.

Show that the classical relation between angular momentum and magnetic moment of a dipole disagrees

with your results for the electron. Calculate the g factor of the electron, including error bars. In your report,

discuss the effect of the Earth’s magnetic field (about 10-4 T) and any other source of experimental

uncertainty, on your experiment.

Consider what might determine the observed widths of the resonant dips. Is it the homogeneity of the

magnetic field, or the lifetime of the excited state, or something else? Try to make estimates to test your

ideas.