Electron Spin Resonance

The PASCO manual for this experiment can be found here. It is in fact quite good -- it contains both the technical description of the equipment, as well as a brief introduction into the physics of electron spin resonance.
 

Theory behind ESR

When an atom is placed into an external magnetic field, the energy levels split.  This phenomenon is called the Zeeman effect. Electron Spin Resonance involves a resonant coupling of the transition between the Zeeman levels and an external oscillating electromagnetic field (with magnetic component perpendicular to the field causing the Zeeman splitting).

Due to magnetic field B the degenerated S-shell energy level split by , where  is the magnetic moment of the electon. A classical charge with the angular momentum of (1/2)h generates a magnetic moment equal to , the "Bohr magneton." Classically, the magnetic moment of the electron is , but in here we allow an additional scale factor  - the "g-factor" - so that .

When the sample in the magnetic filed B is further exposed to the electromagnetic radiation, the electron in the lower Zeeman level can absorb a quantum of the EM radiation and make a transition to the upper Zeeman level. The total difference between energy levels is then , and thus the condition for the resonance is

Werere h is Planck's constant and  is the frequency of the EM field.

The goal of this experiment is to measure , and to confirm or deny the classical hypothesis.
 

Experimental setup

While the ESR equipment used in the research can be fairly complicated and use strong magnetic fields, for our purposes a relatively week magnetic field is sufficient.

Helmholtz coils

We will use so-called "Helmholtz coils" a pair of coils of radius $R$ which are set appart to the distance equal to $R$. As one can verify from Biot-Savard's law, the magnetic field in the area half-way between the coils and near the axis of the system is nearly homogenous, and equal to

where

• is the magnetic permeability of the vacuum
• N is the number of turns in each coil (which, in our setup, is 320)
• R is the radius of the coils (and the distance between the coils), and
• I is the current passing through them. (The coils are connected in parallel, and since they are identical the current is split in half. The measured current is therefore twice the current in each coil.)

The substance which is used to observe the Electron Spin Resonance is diphenyl-picril-hydrazyl, or DPPH, a organic radical with an unpaired electron appears on one of the nitrogen attoms. That electron does not have any orbital angular momentum (), which greatly simplifies the situation with the Zeeman splitting, since there are only two Zeeman levels and thus only one resonant frequency.
 

Observing the resonant condition

The electron spin resonance is marked by a dramatic increase in absorption of the EM field, and this increase changes the impedance of the RF oscillating circuit. So we can identify the presence of a resonance by observing a drop in the current in the RF circuit.

However, for a constant strength of the external magnetic field B, the frequency of the RF circuit has to be fine-tuned, which is experimentally extremely hard.

Experimentally, this obstacle can be sidestepped by turning the problem on its head: we vary B instead until the resonant condition is reached for a fixed frequency of the RF circuit.  A pulsating B field is created by applying a current to Helmholtz coils that is composed of a DC componet and a (weaker) 60 Hz AC component.  If the value of B causing the electron spin resonance is within the amplitude of the pulsating magnetic field, the resonant condition will be reached twice in each period.

We note that the phase of the current through the coils is offset from the voltage, so we use an additional circuit (here denoted as the "phase shifter") consisting of a variable resistor (of 0--100 k$\Omega$) and a capacitor (100 nF) between the connector to the coils and the oscilloscope input.

The observables then are:

• voltage applied to coils after the phase shifter
• current in the RF circuit
• frequency of the RF circuit

The probe unit

The probe unit is the brain of the experiment: appart from the socket for the coil with the DPPH sample and the output 5-pin DIN connector (connecting it to the ESR adaptor), it has two potentiometers: one controls the frequency of the RF oscillator, and the other the amplitude of the current through the coil with the sample.  The probe unit needs  power supply, and has two outputs:

• "Output Y"  is proportional to the power consumption of the RF circuit, and will exhibit a dip when the resonance is reached
• "f/1000"  is proportional to the AC current through the coil and can be supplied to the scope for a measurement of frequency of the RF circuit.  However note that the probe unit internally contains a frequency divider and thus the observed frequency need to be multiplied by 1000 in order to get the true frequency of the EM field.

Performing the measurement

There are two phases of the ESR measurement:

• identifying the resonance
• measuring 

Usually this is a harder part, as several components need to be adjusted.  There are three DPPH samples distinguished only by the number of turns in the coil.  Take one with the middle number of turns, position both potentiometers to the middle position, and drive a current of about 1-2 A through the coils.  (The voltage on the DC power supply should be about 4-5 V, while the AC voltage of 2-4 V should be sufficient to find the resonance.)

Connect the output of the phase shifter to oscilloscope's Channel 1 (DC coupled), and "Output Y" (AC coupled) to Channel 2.   Adjust the time scale and Channel 1's scale (5 V/div is a good start)  in order to see a sinusoidal signal of the current through the coils.  Adjust Channel 2's scale as well as the DC and AC components of the current in order to see the resonance dips.  Both should be easily visible.

Once the resonance dips have been found, adjust the DC voltage and the phase shifter until the resonant dips are above the points where the AC current is 0.  This is an experimental trick used to simplify the measurement of the magnetic field, as at these points the current through the Helmholtz coils equals the DC current, which is measured directly by the ammeter.  This was the main reason to couple the DC and AC power supplies in parallel -- if we connected them serially (which would have been natural) we could have done away with the 1000 uF capacitor, but then it would have been harder to disentangle the true current at the resonance.)

Now the resonant magnetic field B can be calculated from the DC current and the above formula.  The frequency can be measured directly by hooking "f/1000" to Channel 2 of the digital scope and measuring the frequency of the signal.

Task 1: for one setting of the frequency, measure B for each of the samples (three different coils).

Task 2: for one setting of the frequency, vary the amplitude and verify that B does not change.

Task 3: vary the frequency (take a scan of 5-6 values) with the potentiometer on the probe unit, and repeat the measurement of B and frequency.
 

Measuring  

Take the points from Task 3 and fit them for , and obtain from the coefficients.  (As our model assumes that there is no constant coefficient, use the form y(x) = a x  in the fit.)