For the classical particle, you will use a spinning billiard ball, containing a magnet embedded at its center. The objectives are to:

• place the ball in a magnetic field and determine the ball's magnetic dipole moment;
• add angular momentum to the ball and observe its motion; and
• determine the relationships among the motion of the ball and its angular momentum and magnetic dipole moment. 

Consider a loop of positive current $I$  whose path encloses area $A$ , as in Fig. 1.

Figure 1: Magnetic dipole moment of a current loop.

The area enclosed by the loop may be considered a vector: the magnitude is the area and the direction is given by the right-hand rule, with the fingers pointing in the direction of positive current flow and the thumb points in the direction of the resulting magnetic dipole moment. Then, the magnetic dipole moment is given by

Consider a magnetic dipole moment placed in an external magnetic field $\overrightarrow B$ , as shown in Fig. 2.

Figure 2: Magnetic dipole in a magnetic field.

The dipole will experience a torque given by

and the energy of interaction between the dipole and the field is given by

Note that the minus sign indicates that the energy is minimum when $\overrightarrow\mu$  and $\overrightarrow B$ are parallel.

CONSIDER: If an object having a magnetic dipole moment (but no angular momentum) is free to move, how will it move in the presence of the magnetic field?

CONSIDER: If the object is given angular momentum, parallel or anti-parallel to its magnetic dipole moment, and is placed it in a magnetic field, how will it move?

To help us answer these questions, consider Fig. 3.

Figure 3: Torque acting on an object with angular momentum $\overrightarrow{L}$  and oppositely directed magnetic dipole moment $\overrightarrow{\mu}$  in a magnetic field (after Eisberg and Resnick, Quantum Physics)

The magnetic field acts on the magnetic dipole moment to produce a torque, given by Eq. (2). This torque gives rise to a change in the angular momentum $d \overrightarrow{L}$  during the time $dt$  such that

The change $d \overrightarrow{L}$  causes $\overrightarrow L$  to precess through an angle $\omega dt$ , where $\omega$  is the precession angular velocity. Note from Fig. 3 that $dL = L \sin \theta$ . Therefore,

$\tau = \mu B \sin \theta = \dfrac{dL}{dt} = \omega L \sin \theta$ and

Eq. (5) is often re-stated as

where $\gamma$  is called the gyromagnetic ratio.

We will use the TeachSpin Magnetic Torque apparatus for this “classical” part of the experiment, shown in Fig. 4. The apparatus consists of the following:

• A control box for supplying current to the magnetic field coils, setting the direction of the fields and controlling the strobe light.
• An external air pump
• A pair of copper wire coils which can produce a magnetic field in the vertical direction. The relation between the current in the coils and the magnetic field produced is

• Cue ball with an embedded magnet and a small black handle for spinning the ball. The magnetic dipole is aligned parallel to the axis containing the black handle.
• Air pump for floating the ball.
• Aluminum rod with a steel tip for holding a sliding plastic mass for changing gravitational torque on the ball.
• Calipers to measure the position of the sliding mass.
• Scale for weighing the sliding mass and the mass of the ball.
• Movable index to indicate a starting and stopping position of precession.
• Saddle to provide a rotating magnetic field, perpendicular to the steady, vertical field.

Figure 4: TeachSpin Magnetic Torque apparatus used in the “classical” part of the experiment.

For this part of the experiment, keep the angular momentum of the ball equal to zero.

Level the air table using the small bubble level mounted at one corner of the base. Use gravitational torque balancing the magnetic torque to determine $\mu$ . To do so:

• Place the aluminum rod in the central hole of the cue ball's black handle, with the rod's magnetic end inserted into the ball.
• Place the sliding plastic mass on the aluminum rod.
• Set the magnetic field to point up, with the field gradient turned off.
• Adjust the current in the magnetic field coils so that the magnetic torque just balances the gravitational torque.

NOTEBOOK: Draw a sketch of the ball showing the magnetic and gravitational torques. Derive the equation relating these torques at equilibrium. 

From the torque diagram, you should see that

where $r$  is the distance from the center of the ball to the center of the sliding mass $m$ , and $g$  is the acceleration due to gravity. Eq. (8) reduces to

As the mass position $r$  is linearly related to the magnetic field $B$ , we can determine the magnitude of the magnetic dipole moment $\mu$  from the slope of a  vs. $r$  plot.

NOTEBOOK: Move the sliding mass to multiple positions along the aluminum rod and, at each position, determine the magnetic field needed to balance the ball. Record the values of current and mass position (with uncertainties) in a table. 
NOTEBOOK and ANALYSIS: Carefully plot magnetic field as a function of position and determine $\mu$  from the slope (roughly for the notebook, and by a fit (with uncertainties) for the report).  

You may provide angular momentum to the ball by spinning it using its black handle. Recall that, for a uniform solid sphere, the angular momentum $L$  is given by

where $M$  is the mass, $R$  is the radius, and $\Omega$  is the spin angular velocity of the sphere. Note that for your sphere, the angular momentum and magnetic dipole moment are either parallel or anti-parallel.

NOTEBOOK: Measure the mass and radius of the ball.

If our ball has angular momentum and we apply an external field, we predict that the ball will precess about the field vector. The precession frequency depends (according to Eq. (5)) on the magnitude of the magnetic dipole moment (measured above), the angular momentum and the magnitude of the magnetic field. We will keep the angular momentum fixed, and measure the precession frequency as a function of field strength.

Angular momentum depends on the spin angular velocity of the ball. In order to keep angular momentum fixed, we must always make our measurements with the ball is spinning at the same angular velocity. In order to achieve this, we will use the following procedure:

1. Turn on the magnetic field and set it to the desired value. (Take your first measurement at an intermediate value.)
2. Turn the magnetic field gradient on. With this setting, the currents in the upper and lower coils are in opposite directions, producing $B=0$  at the center of the apparatus
3. Turn on the strobe light and set its frequency to about 5 Hz. Note that in order to measure this frequency accurately, the frequency counter must count for several seconds. It updates every 10 seconds.
4. Orient the ball so its black handle points toward the strobe light.
5. Spin the ball using the black handle and reduce any wobble with your fingernail or with the tip of a pencil.
6. Wait for the ball to slow until the white dot appears stationary. Lightly touching the end of the black handle can reduce the spin frequency more quickly. If the spin frequency and strobe frequency are equal, the white dot will appear stationary. (Is the converse always true?)
7. Quickly set the position marker as near as possible to the black handle, turn off the field gradient and start the stopwatch to measure the time it takes for the ball to precess one complete cycle.

NOTEBOOK: Repeat steps 1 through 7 at for a range of currents, recording the current and precession period with uncertainties.

NOTEBOOK and ANALYSIS: Calculate the angular momentum, Eq. (10), from your measurements of the mass and radius as well as the known spin frequency. Propagate uncertainties. 

NOTEBOOK and ANALYSIS: Plot $\omega$  vs.$B$  and find the gyromagnetic ratio $\gamma$  from the slope (roughly for the notebook, and by a fit (with uncertainties) for the report).

This part of the experiment provides a qualitative demonstration of how the ball, having both angular momentum and magnetic dipole moment, behaves in a rotating magnetic field, perpendicular to the constant, vertical B field.

Remove the position indicator and install the magnetic field saddle. This saddle provides a field of constant magnitude, which may be rotated in the horizontal plane.

• With the vertical field set to a maximum, start the ball spinning with its black handle midway between the red dots on the saddle. As the ball precesses in one direction, manually rotate the saddle in the other direction. Try to move it smoothly and continuously. What effect does the rotating field have on the precessional motion?
• With the black handle midway between the red dots on the saddle, spin the ball again, but this time try to rotate the saddle in the same direction at a frequency different from the precession frequency. (This is tough!). How does the rotating magnet affect the precession?
• Repeat the experiment with the saddle rotating in the same direction at the same frequency as the precession. This requires some practice! How does the ball move now? Ask the lab staff for help if you have trouble here.

CONSIDER: What are the conditions that lead to a “spin flip?”  How is this connected to the physics behind the resonance?  

For the quantum system, we will apply a magnetic field and observe the response of free electrons in a sample of DPPH (diphenyl-picri-hydrazyl). The goals are to:

• place the sample in a slowly-varying magnetic field and look for a resonance response with a second, orthogonal radio frequency oscillating magnetic field;
• measure the value of the applied field at different RF values;
• determine the gyromagnetic ratio of the electron; and
• to make the connection between the observed spin-flip of the classical particle and the electron spin resonance seen here.

The electrons in atoms are bound in discrete energy states. Magnetic fields are generated within the atom by the

• orbital motion of the electrons around the atom;
• spin of the electrons; and
• spin of the atomic nucleus.

If atoms are placed in an externally applied magnetic field, the interactions of the applied field with the internal fields listed above cause the energy levels of the atoms to shift. Similarly, if the atoms are placed in a solid, the magnetic fields produced by neighboring atoms will also contribute to energy level shifts.

Electron spin resonance (ESR) is a technique for inducing and detecting transitions among energy levels. Energy level shifts are induced by application of a known magnetic field, while transitions among energy levels is induced by application of electromagnetic radiation of a known frequency. It is found that only for particular combinations of magnetic field and frequency are transitions induced.

Detection is accomplished by measuring the slight decrease in energy in the electromagnetic field which occurs when the energy is absorbed during the transition. A large ensemble of atoms is needed to absorb sufficient energy to be detectable.

It should be noted that the net energy shifts are due to the total field: applied and nearest neighbor. Since we know the value of the applied field, it follows that measuring the frequencies at which resonances occur is a probe into the details of the environment of the solid sample at the atomic scale.

The detailed study of solids using ESR is complex and beyond the scope of this course. Therefore, for simplicity we will study a much simpler system: “free” electrons, not bound to an atom. The sample we will use is the molecule DPPH (diphenyl-picri-hydrazyl), which has one, nearly free, electron per molecule.

The electron is a spin 1/2 particle. Thus, if an electron is placed in a steady magnetic field, the electron can be in only two possible energy states. In the semi-classical model we may think of the magnetic dipole moment (or the angular momentum) of the electron precessing about the applied magnetic field with two possible orientations, one as shown in Fig. 3 and the other with the $\mu$  and $\vec L$  vectors reversed relative to $\overrightarrow B$ .

NOTE: This model is useful, but has some limitations: The presence of a well-defined vector implies knowledge of the x and y-components of the vector, but quantum mechanics tells us that these components are really indeterminate.

In these two states, the magnitudes of the components of spin angular momentum parallel to the field (in the z-direction) are $\pm \hbar / 2$ . The magnetic dipole moments associated with these states are

where  $\mu_B = e\hbar/2m$  is called the “Bohr magneton.” For the free electron – for which all the angular momentum is spin (rather than orbital) angular momentum – we have $g = 2.0023$ . The energy of a magnetic dipole moment in a magnetic field is given by

Thus, the energy difference between the two states is

(Figure is adapted from user manual)

Energy level splitting in an atom with one optically active electron outside a closed subshell-pure spin angular momentum state: $S = 1/2$ , $\ell = 0$ , $j = 1/2$  placed in a magnetic field. 

If we apply electromagnetic radiation with frequency $f$ , such that

where $\Delta E$  is given by Eq. (13), we should induce transitions between the two energy states. Eq. (14), however, can also be written

At resonance, it turns out that $\omega$  of Eq. (15) is the same as that of Fig. 3, i.e., the precession angular velocity of the electron around the magnetic field. Thus, photons of angular frequency $\omega$ carrying angular momentum $\hbar$ , cause the electron's spin to change state. Combining Eqs. (13) and (15) gives

It should be noted that, for a free electron in a magnetic field, the magnitude of the spin magnetic dipole moment is

and the magnitude of the spin angular momentum is

Thus, the ratio of magnetic dipole moment to angular momentum is

which appears also in Eq. (16). 

Eq. (16) is often re-written as

where $\gamma$  is called the gyromagnetic ratio. Eq. (20) is analogous to Eq. (6) for the classical case. In both cases $\gamma$  is the ratio of magnetic dipole moment to angular momentum. The factor $g$  is required by quantum mechanics.

Above, we make the argument that a spin flip occurs when the system absorbs a photon of energy $E = \hbar \omega$  which carries with it one unit of angular momentum $\hbar$  (in the correct direction). In practice, however, we do not use photons, but instead a radio frequency (RF) oscillating magnetic field

which possess intrinsic energy and angular momentum. This oscillating field can be decomposed into two counter-rotating fields in the x-y plane as 

In the classical case, we saw that a rotating magnetic field in the direction of precession at the correct frequency could induce a spin flip, whereas a rotating magnetic field in the wrong direction (even with the correct frequency) had no effect; thus, we see that our oscillating field has a resonance-inducing part and an ignorable part.

We use the Daedalon ESR apparatus (shown in Fig. 5), consisting of the following:

• a 60 Hz AC power supply for Helmholtz coils
• a tunable radio frequency oscillator with frequency and feedback controls
• Helmholtz coils, connected in parallel such that the magnetic field at the center is given by

*The sample we will use is the molecule DPPH (diphenyl-picri-hydrazyl), which has one, nearly free, electron per molecule.

Figure 5: The Daedelon ESR apparatus used in the “quantum” part of the experiment.

The user's manual for the apparatus can be found here

Figure 6: ESR electrical connections

Make the electrical connections as shown in Fig. 6:

• Adjust the height of the probe to be the same as the center of the Helmholtz coils.
• Slide the probe through the side slot in the Helmholtz coil support, so that the axes of the probe's RF coil and Helmholtz coils are perpendicular.
• Set the Helmholtz coil current its maximum value.
• Set the scope to display in voltage vs. time mode. Ground both channels 1 and 2 and move their traces to the center (zero volt) line. Then, DC-couple both channels.
• Channel 1 (the signal from the “sense” output) should be a sine wave of frequency 60 Hz and amplitude several volts.
• Channel 2 (the signal from the “video” output) should show a noisy, roughly flat signal, possibly with periodic spikes on it.

We will now try to find an electron spin resonance signal:

• The “tuning” knob on the ESR head controls a tunable capacitor in an amplifier circuit, and consequently the frequency of the oscillator. As you turn the knob, you will see the frequency reading on the power supply change, possibly jumping discontinuously or at times even going to zero; this is normal.
• The “feedback” knob on the ESR head controls a tunable resistor in the same circuit. As you turn the knob, you adjust the “Q factor“ of the amplifier (and, to a lesser extent, the resonant frequency). You will find that setting this knob too low causes the frequency reader to go to zero.
• Our oscillator is most sensitive when it is “marginal”, i.e. when the amplification gain is only slightly more than the loss in the circuit. When the feedback is properly set, the frequency reading should be stable across a continuous range of frequencies (about 20-30 MHz). You should also find large “spikes” on the signal on channel 2 (the “video” signal) that rise up out of an otherwise noisy baseline. (See Fig. 7).
• Note that if the RF oscillator stops oscillating, the frequency display will read zero and no resonance spikes will appear on the scope. If that happens, readjust the frequency and/or feedback controls until you find a new stable point.

Figure 7: An example of the scope trace as viewed in “time versus voltage” mode with resonance “spikes” visible.

Once you've found a clear signal, we can make a measurement of the magnetic field at the time of the resonance so that we can pair this value of $B$  with the supplied value of $f$ :

• Measure the voltage to the Helmholtz coil (the sine-wave voltage) at which resonance occurs in the oscillator signal. Since the modulating (60 Hz) field is periodic, you should see these resonances repeat at periodic points across the scope screen. Record the voltage value for several of these resonances and estimate the uncertainty in each. How many resonances occur in one cycle?
• Note that the “sense” signal (channel 1) is such that 1 volt is produced by 1 amp flowing from the power supply (1/2 amp flowing through each coil). With this current, calculate the magnetic field from Eq. (21).
• Increase the frequency selected by the sensor in the ESR head via the tuning knob and re-adjust the $B$  field to find a new resonance.

NOTEBOOK: For one or two frequencies, record the voltages (four per cycle) that correspond to resonance. Calculate the corresponding $B$-field values.( You will measure these values (and more) again using a different procedure below.)

There is an alternate way to measure the voltages at which resonance occur. If we place the oscilloscope into “XY mode” (selectable in the “Display” menu on the scope under the “Format” option), then the trace which appears will be a “loop” such that channel 1 will serve as the (time-dependent) x-coordinate and channel 2 will serve as the (time-dependent) y-coordinate. Since both signals are periodic with the same frequency (60 Hz), the loop traces the same pattern each cycle and the image is stable.

Let us follow the trace for one of these cycles. (See Fig. 8). Beginning at the leftmost point, our Helmholtz voltage (x-coordinate) is at the minimum value. As this voltage increases (following the trace to the right), we pass through a resonance point and the voltage on the oscillator signal (y-axis) rises. Continuing, we go off resonance (y falls), we pass through zero field, and then we come to the corresponding resonance point for positive Helmholtz voltage (y rises again). Once we reach the maximum voltage in the Helmholtz coils (maximum x-value), the voltage decreases and the the trace reverses direction. Moving now from right to left, we see the resonance on the positive half, pass through zero, then see the resonance on the negative half before we finally return to where we started.

Figure 8: Examples of the scope trace as viewed in “XY mode”. The image on the left is out-of-phase (note the four distinct peaks), whereas the image on the right has been adjusted so that the corresponding pairs of resonance peaks overlap and are in-phase.

Likely, you will initially see a trace which looks like the image on the left in Fig. 8 with four distinct peaks. This means that the four resonances are appearing on the scope at four different voltages within the cycle. We know that these resonances in fact should occur only at two different voltages ( $+V_0$  and $-V_0$ ). Therefore, the oscillator voltage signal must be out of sync with the Helmholtz voltage signal. To adjust the phase relationship between the two signals, adjust the “Phase Null” knob on the front of the power supply until the pairs of peaks overlap (as seen in the right image of Fig. 8). In XY-mode, the voltage measured along the x-axis from the center of the loop to each peak corresponds to the voltage in the Helmholtz coil at the time of resonance. However, since there is some ambiguity in the “center” of the loop, a more accurate measurement would be to measure the voltage along the x-axis from peak to peak (which corresponds to twice the resonant voltage) and divide by two.

NOTEBOOK: Using either the “time vs. voltage” or “XY” mode, record values of the voltage in the coils at each resonance for fixed frequency. Collect data over the full range of frequencies at as many values as you can.

NOTEBOOK and ANALYSIS: Convert from $V$  to$I$  to $B$ , and from $f$ to $\omega$. Plot $\omega$ vs. $B$ and find the gyromagnetic ratio $\gamma$  from the slope (roughly for the notebook, and by a fit (with uncertainties) for the report). Is this value consistent with Eq. (16)? Note that if your dominant uncertainties are in the magnetic field, you may with to invert the axes and fit to find $1/\gamma$ .

We now want to test the relation between the direction of the RF field and the applied Helmholtz field:

• With a resonance signal visible, turn the Helmholtz coil so that it applies a field parallel to the axis of the RF coil.
• Observe the effect on the resonance signal seen on the scope.

CONSIDER: What do you observe? Does this make sense with your understanding of the physics of the spin flip resonance? In particular, what effect does rotating the field have on angular momentum in our system?

As the TA looks over your notebook, they will be looking for general good practices and for specific things. Be prepared to point to (and explain) the appropriate places in your lab notebook where the following information is contained.

Electron Spin Resonance Analysis Rubric