In 1946, nuclear magnetic resonance (NMR) in condensed matter was discovered simultaneously by Edward Purcell at Harvard and Felix Bloch at Stanford using different techniques. Both groups observed the response of magnetic nuclei, placed in a uniform magnetic field, to a continuous wave radio frequency (RF) magnetic field as the field was tuned through resonance.

In 1950 Ervin Hahn, a young postdoctoral fellow at the University of Illinois, explored the response of magnetic nuclei in condensed matter to pulse bursts of these same RF magnetic fields. Hahn was interested in observing transient effects on the magnetic nuclei after the RF bursts. During these experiments, he observed a spin echo signal after a two-pulse sequence. This discovery, and his brilliant analysis of the experiments, gave birth to a new technique for studying magnetic resonance.

These discoveries and advances have opened up a new form of spectroscopy which has become one of the most important tools in physics, chemistry, geology, biology, and medicine. Magnetic resonance imaging scans (abbreviated MRI – the word “nuclear” was removed to relieve the fears of the scientifically uninformed public) have revolutionized radiology. This imaging technique is completely noninvasive, produces remarkable three-dimensional images, and gives physicians detailed information about the inner working of living systems.

1. PNMR has industrial and agricultural applications in measuring the moisture content of substances

Pulsed nuclear magnetic resonance (PNMR) is an experimental technique used to study the response of magnetic nuclei to an applied magnetic field. In this experiment you will learn the physics of how PNMR works and will make measurements of two characteristic relaxation time constants for protons in a mineral oil sample. These two time constants – and the techniques used to measure them – form the basis of medical MRI imaging.

The physics related to what you are measuring and how the technique works are not complicated; it mostly involves first-year material such as angular momentum, magnetic forces and induced currents. What makes the subject challenging to grasp is that we are dealing with time varying fields and motion in three dimensions which can be difficult to visualize. The pre-lab background research for this experiment is very important, simply because there is a lot to absorb. Before coming to the first day in lab, you and your lab partner should be familiar with the following subjects:

• What is pulsed nuclear magnetic resonance (PNMR) and how is it different from nuclear magnetic resonance (NMR)?
• What is meant by the term “resonance phenomena”?
• What is the physics of single particle spin states for the proton (which is a spin 1/2 particle). Specifically, make sure you understand how a spin 1/2 particle behaves in an external magnetic field in regards to precession, magnetic dipole moment, and energy states.
• What are the macroscopic properties of an ensemble of protons in an external magnetic field? How does the bulk magnetization (the vector sum of the individual proton magnetic moments in the sample) behave when the system is at thermal equilibrium?
• What does it mean to say “when an ensemble of protons is in a non-equilibrium state it relaxes back to its equilibrium state with a characteristic time constant”?
• How do the protons and the bulk magnetization behave when a second magnetic field is applied orthogonal to the original magnetic field?
• What do the terms “90 degree pulse” and “free induction decay (FID)” mean?

Much of the information you need is contained in the Theory and Experimental Technique sections of this wiki. You may find that Google and Wikipedia are also useful.

Here we wish to examine the effect of magnetic fields on protons and other particles having magnetic moment and angular momentum. It is noteworthy that the underlying physics of nuclear magnetic resonance is very similar to that of electron spin resonance and optical pumping.

We will first look at the general case of a spinning magnetic dipole moment in an external magnetic field and then connect that to the behavior of an individual proton.

Single Particle Behavior

The PNMR technique it not sensitive enough to detect the behavior of individual protons, instead we work with the bulk properties of the sample which are the result of the behavior of the ensemble of protons. PNMR is most frequently used to make measurements of the relaxation times which characterize how long it takes for an ensemble of protons to reach equilibrium. These time constants are referred to at T1 and T2.

1. How long does it take a randomly oriented ensemble of magnetic nuclei to become aligned? This time scale is $\mathrm{T}_1$, known as the spin-lattice relaxation time
2. How long does it take for nuclei precessing in phase to get completely out of phase due to nearest neighbor interactions? This is $\mathrm{T}_2$, known as the spin-spin relaxation time

Although we will focus specifically on protons (hydrogen nuclei), everything we discuss is generally applicable to other magnetic nuclei.

Ensemble Behavior

The technique of PNMR can be described in general as follows:

1. A sample containing magnetic nuclei is placed in an external magnetic field and allowed to come to equilibrium.

• In our case we will use Glycerin as a sample which provides a large number of protons as our magnetic nuclei.

2. Pulses of radio frequency (RF) oscillating magnetic field are used to reorientate the ensemble of proton spins into some non-equilibrium state.

• For example, we can apply the RF field for a finite time causing an inversion of the Boltzmann distribution as a precursor to measuring the $T_1$ time constant.
• Or, we can rotate the equilibrium bulk magnetization from the $z$-axis into the $xy$-plane where it will precess about the $z$-axis while decaying back to zero for measuring $T_2$.

3. Measurements of the bulk magnetization are performed as it relaxes back to equilibrium.

We will now discuss how our apparatus accomplishes this.

TeachSpin PNMR Technique

Figure 12 shows the components of the apparatus. We will summarize the role of each component here. For more detail on the operation of the electronics, refer to pages 14-25 of the TeachSpin manual in the laboratory.

Figure 12: Components of the PNMR experiment.

[A] Electro-magnet:

The electro-magnetic (Fig. 13) is capable of generating fields up to 10 T between the pole pieces. The strength of the field is determined by the amount of current flowing through the water-cooled coils and magnet power supply. Before turning on the electro-magnet power supply (Fig. 14), you must first turn on the water supply in the next room. You should see a small, steady stream emptying into the sink.

Figure 13: Water-cooled electromagnet

[B] Electro-magnet power supply:

Controls the current flowing through the coils of the electro-magnet, shown in Fig. 14. Only the coarse and fine current controls are used, do not change the settings of the other controls. Before turning on the power supply make sure that both the coarse and fine current controls are set to 0. When turning the power supply off at the end of the day, also make sure that the both current controls are set to zero. Turning the power supply on or off with the current controls not at zero can potentially damage the equipment.

Figure 14: Electro-magnet controls. Current controls are outlined in light blue.

[C] Sample Probe:

The sample probe is a rectangular brass box which slides snugly between the pole pieces of the electro-magnet. The interior of the probe is shown in Fig. 15. There is a hole in the top which accepts a sample vial containing a small amount of material to be studied. When inserted to the proper depth, the sample volume will be in the center of the receiver and Helmholtz coils. The receiver coil is used to detect time varying magnetic fields along the vertical axis (x-axis). The Helmholtz coils are used to create time varying magnetic fields along the horizontal axis (y-axis) which will be used to rotate the nuclei in the sample.

Mineral oil sample	Teflon sample

[D] TeachSpin Electronics Rack

The electronics rack houses three different modules. The power switch is located on the right hand side of the back of the rack. The three modules are as follows:

15 MHz Receiver

A low noise, high gain amplifier connected to the receiver coil in the sample probe. (See Fig. 16a.)

15 MHz Oscillator/Amplifier/Mixer

Sends pulses of 15 MHz ac current to the Helmholtz coils in the sample probe. (See Fig. 16b.) When these pulses turn on, and their duration, is determined by signals from the pulse programmer. Also contains a mixer which is used to compare the frequency of the oscillator signal with the signal induced in the receiver coil.

Pulse Programmer

Allows the user to set up sequences of pulses from the 15 MHz Oscillator. (See Fig. 16c.)

(a)	(b)	(c)
Figure 16: Teachspin electronics rack. (a) The receiver module, which connects to the detector probe and processes the signal. (b) The pulse programmer, which controls the pulses that are sent to the Helmholtz coils. (c) The oscillator, amplifier, and mixer module, which sets the Helmholtz coil's frequency and detects the difference between the applied frequency and the detected frequency (Mixer out)

Make sure the controls of the pulse programmer are set as follows:

• A-width: halfway (This controls how long the applied pulse is)
• Mode: Int (This sets the pulse programmer to use the Internal function generator for timing)
• Repetition time: 100 ms and 10% (pulses should be produced 100 ms * 10% = 10 ms apart)
• Sync: A (sets the Sync out to trigger when pulse A starts.)
• A: On
• B: Off
• Sync Out: connected to oscilloscope external trigger input.

Insert the Glycerin sample. Connect the Detector Out from the 15 MHz receiver to channel 1 of the scope.  his output produces a signal whose amplitude is proportional to the magnitude of the emf in the detector coil, but with the high frequency (~15 MHz) oscillations filtered out. Make sure that the trigger point is centered on the scope display and the trigger is set to External. Set the time base of the scope to about 1 ms.

Figure 18: An example of the difference between the Detector Out (Ch 1) versus the RF Out (Ch 2). While it cannot be seen on this scale, the RF out signal has a ~15 MHz component that makes it appear quite noisy.

Now watch the signal on channel 1 as you slowly turn up the field on the electromagnet. You should find resonance (a peak of several volts) somewhere around a setting of 100 on the coarse current control of the electromagnet power supply. The signal you are looking at is the free induction decay (FID) signal. Over how wide of a range of coarse current settings can you observe resonance?  

The signal detected by the receiver coil will pick up the resonant frequency at which the proton magnetic moments precess. However, since this frequency is in the MHz range, it is difficult to make a precise measurement with just the oscilloscope. Instead, we will use a device called a frequency mixer, which takes two frequencies as input and produces a signal whose frequency is either the sum or difference of the two input frequencies. If we compare our measured resonance frequency to an internal reference frequency that we know well, we can tell how close we are to the reference frequency by looking at the beat frequency produced by the mixer output.

In our case, we can easily distinguish beats in the kHz range as we get close to matching the resonance and reference frequencies. (See Fig. 19.)

(A)

(B)

(C)

Figure 19: Examples of different mixer outputs: (A) Many beats, far from resonance (B) Some visible beats, still not at resonance (C) Three visible beats, nearly at resonance. Note that the scale on all three images are the same, but that (B) and (C) include the detector output signal.

Adjust either the magnetic field strength or the reference RF frequency until the beat frequency drops (close) to zero. At this point, we know that the protons are precessing at a frequency equal to the reference frequency.

Vary the A pulse width over its full range and observe the effect on the amplitude of the FID. Adjust the gain on the Receiver to nearly maximize the FID amplitude, with no clipping of the FID.

Adjust the receiver gain so the FID amplitude is about 10 to 10.5 V (with no clipping) and the mixer output shows zero beat frequency. Experiment with the height of the sample in its holder. Iterate as needed to find the strongest signal.

No magnet produces a perfectly uniform field. However, it is possible to find a region of maximum uniformity (the sweet spot). To do so, move the sample probe around in the magnet gap while observing the changing shape of the FID. A uniform field is indicated by a long, smooth exponential decay of the FID. In a perfectly uniform field the exponential decay constant of the FID would be $T_2$, the spin-spin relaxation time constant, and would be on the order of 10 ms. Field inhomogeneities can dominate the relaxation time with a time constant of order 0.1 ms, and thus we cannot reliably measure $T_2$ with this method. 

While at resonance (zero-beat condition) remove the sample tube containing the Glycerin sample and measure the magnetic field using the Hall effect gaussmeter. Calibrate your gaussmeter using the calibration magnets provided (see Fig. 20). Place the tip of the gaussmeter probe in the same region of the magnet where the sample sat. Note that the flat face of the Hall effect crystal must be perpendicular to the direction of the magnetic field being measured. Make a careful estimate of the uncertainty of this measurement.

From your measured values of resonant frequency and magnetic field, calculate γ for protons. Because of the imprecise manner in which we are measuring the magnetic field at resonance, by using the hall effect gaussmeter, this is not going to be a highly precise measurement.

Repeat the measurement for Florine atoms in the teflon sample.

Also solid materials tend to produce smaller amplitude signals with much faster decay times. So the FID can be difficult to find, you can make your life easier by estimating how much the magnetic field needs to increase relative to protons.

(A)	(B)
Figure 20. (A) DC Gaussmeter (B) calibration magnet set

We now know how to do the following:

• measure the magnitude of a bulk magnetization in the $xy$-plane which precesses about the $z$-axis, and
• reorient the bulk magnetizations in the sample.

The inversion recovery method uses a series of RF pulse pairs to measure the rate at which the magnetization along the $z$-axis relaxes to equilibrium. The method is as follows:

• Allow the sample to come to equilibrium in $B_0 \hat{\bf z}$.
• Apply a $180^\circ$ pulse to rotate the bulk magnetization from the +$z$-axis to the -$z$-axis. This is equivalent to inverting the Boltzmann distribution of proton spin states.
• After the $180^\circ$ pulse, the sample is allowed to relax back to equilibrium for  $\tau$ seconds.
• A $90^\circ$ pulse is applied to rotate the magnetization on the $z$-axis, which will have relaxed part way back to equilibrium, into the $xy$-plane where it will begin to precess and induce a current in the receiver coil. The amplitude of this FID signal is proportional to the magnitude of the magnetization which existed on the z-axis immediately before the 90º pulse was applied.

These steps are repeated, for different time intervals $\tau$. Using Eq. (15) (reproduced below), the time constant $T_1$ can be found by plotting FID amplitude versus delay time, $\tau$:

Procedure:

1. Establish resonance with the A pulse and set the width to 90º. Adjust (if necessary) so that the sample is again in the sweet spot of the magnet and set the gain so that the voltage is between 10 V and 10.5 V.
2. Set the A pulse to $180^\circ$, the first minimum.
3. Turn off the A pulse, turn on the B pulse and trigger the scope on the B pulse*.
4. Set the B width to give a $90^\circ$ pulse by maximizing the first FID signal.
5. Turn both A and B pulses on and set the scope to trigger on the B pulse.
6. Optimize the output signal by checking the magnet sweet spot and small tuning of the A and B pulse widths.
7. Adjust the delay time settings.  What effect does this have on your output signal?

In principle $T_2$ can be extracted from the decay of the FID following a $90^\circ$ pulse. In reality, however, the situation is not so easy because protons in the sample experience different net magnetic fields due to two effects. One effect is the nearest neighbor spin-spin interaction which is a characteristic of the chemical environment that the protons are in. This is the quantity of physical interest. The second effect is the inhomogeneity of the applied magnetic field. No matter how well constructed, no real magnetic will produce a perfectly uniform magnetic field between its poles. Protons in different regions of the electromagnetic field experience different magnetic field strengths leading to different precession frequencies. As you should know by now, nature likes to arrange things so as to make it difficult for physicists to make measurements. So, it should not be surprising that the systematic effect of the electromagnet field inhomogeneity on the spread of precession frequencies in the sample dominates over the effect of the spin-spin interactions. Therefore, simply measuring the decay constant of the FID does not give $T_2$, but instead provides a measure of the field gradient of the electromagnet. In order to measure $T_2$ we need to use a very clever, but subtle technique developed by Erwin Hahn known as spin echo. Here we explain the logic of the spin echo measurement.

The spin echo method proceeds as follows:

• The sample is allowed to come to thermal equilibrium.
• A $90^\circ$ is applied to the sample to rotate $M_z$ into the $xy$-plane where it begins precessing and decaying due to the effects of both the spin-spin interactions and the field inhomogeneities.
• The sample is allowed to relax partway back to equilibrium for $\tau$ seconds. During this time protons which are in a stronger region of the electromagnet field will precess ahead of protons which happen to be in a weaker field region.
• A $180^\circ$ pulse is applied to the sample. This pulse causes any remaining magnetization in the $xy$-plane, $M_{xy}$, to precess $180^\circ$ about the $y^*$ axis (in the rotating frame of the proton). 
• After the $180^\circ$ $M_{xy}$ remains in the $xy$-plane but now the protons in the stronger field regions, which were getting ahead of protons in the weaker field regions, are actually behind the slower precessing protons in the weaker field regions.
• After  $\tau$ seconds, all of the proton dipole moments which  diverged  for time $\tau$ before the $180^\circ$ pulse will have reconverged. This reconvergence induces a current in the receiver coil which is called the echo pulse.

So what does the above sequence accomplish in terms of separating $T_2$ from the effects of the electromagnet? The key is that the 5th and 6th bullet points above only apply to precession frequency differences caused by the electromagnet, which are assumed to be constant over the time it takes to make the measurement. Thus, protons in a strong region of the electromagnet field which were precessing ahead of protons in weaker regions before the $180^\circ$ pulse, remain in that same strong region after the 180º pulse and as a result catch up to the slower precessing protons at the same rate as before the 180º pulse. This means that the effect of the electromagnet field inhomogeneities is reversible. This assumption however is NOT true for the nearest neighbor spin-spin interactions which fluctuate randomly on time scales small compared to the duration of a single measurement. So, while the $180^\circ$ pulse has the effect of reversing the magnetization loss due to the electromagnet, it has no effect on the loss of magnetization due to spin-spin interaction. Thus the amplitude of the echo is proportional to  $M_{xy}$ after it has been allowed to relax for $2\tau$ seconds under the effects of the spin-spin interaction only. Figure 21 illustrates the spin echo signal.

Figure 21: Expected signal from a spin-echo measurement.

By plotting the amplitude of the echo as a function of time $(t=2\tau)$, the time constant $T_2$ can be found from

The spin echo method due to Hahn uses a pulse sequence $90^\circ \rightarrow \tau \rightarrow 180^\circ \rightarrow \tau \rightarrow \textrm{"echo"}$ . Our starting point is the setup we have for the first FID experiment, with a $90^\circ$ A pulse, which we will use together with a $180^\circ$ B pulse. The $90^\circ$ pulse rotates the net magnetization away from the $z$-axis into the $x$-$y$ plane where it precesses at the Larmor frequency. We allow the magnetization to decay for a time $\tau$. The magnetization decays due to the differences in precession frequency caused by both the reversible effects of field inhomogeneities and the irreversible effects of nearest neighbor interactions with the reversible effects dominating. We now apply a $180^\circ$ pulse to reverse magnetization within the $x$-$y$ plane. At a time $\tau$ after the $180^\circ$ pulse all of the spins that were out of phase due to field inhomogeneities will be back in phase and an echo of the magnetization will appear in the detector where any loss of amplitude (relative to the amplitude of the FID following the initial $90^\circ$ pulse) is due only to the irreversible nearest neighbor interactions.

1. Establish resonance.
2. Make sure that $M_0$ is between 10 V and 10.5 V
3. Turn off the B pulse and trigger the scope on the A pulse.
4. Set the A Width to give a $90^\circ$ pulse.
5. Maximize the FID signal and find the magnet sweet spot.
6. Turn off the A pulse temporarily, turn on the B pulse and trigger the scope on the B pulse.
7. Set the B Width to give a $180^\circ$ pulse. Turn up the B width through its first maximum ($90^\circ$ pulse) to its first minimum which is a $180^\circ$ pulse.
8. Turn both A and B pulses on and set the scope to trigger on the B pulse.
9. For a short delay time $\tau$, view the echo and tune the B width to maximize the echo

Part II of the experiment is intentionally left open-ended. You may need to consult outside resources for more theory information or equipment manuals for details about the apparatus. Given your experience collecting data in Part I, use your judgement to determine effective collection and analysis strategies, and budget your remaining time in lab appropriately.

The information given below is only a suggestion for how to proceed. Based on your interests and the quality of your work after Part I, you and the faculty instructor may discuss alternate goals. You are encouraged to bring ideas and propose ideas at your meeting, but you should also be prepared to defend approaches which stray far from the outline below.

For the second part of the experiment you will use the techniques learned above in Part I to measure $T_1$ and $T_2$ (of protons) as a function of Cu$^{2+}$ ion concentration and compare with accepted results.

You will prepare samples of Cu$^{2+}$ in doubly deionized (DDI) water. Use molar concentrations of $10^{-2}$, $10^{-3}$, $10^{-4}$, $10^{-5}$, and $10^{-6}$, where CuSO$_4$ has a molecular weight of 159.6 g/mol and H$_2$O has a molecular weight of 18.0 g/mol. The lab staff can show you how to use the necessary equipment for preparing such samples in the chemical preparation room.

[1] Pulsed NMR Apparatus Manual, Teach Spin Inc.

[2] C. P. Slichter, Principles of Magnetic Resonance, Springer, New York, 1996.

[3] H. Carr and E. Purcell, "Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments", Phys. Rev. 94(3), 630 (1954).

[4] N. Bloembergen, E.M. Purcell, and R.V. Pound, "Nuclear Magnetic Resonance", Nature 160( 4066), 475 (1947).

[5] E. L. Hahn, "Spin Echoes", Physical Review 80(4), 580 (1950).

[6] N. Bloembergen, "Encounters in Magnetic Resonances", World Scientific Series in 20th Century Physics 15, World Scientific,1996.

[7] R. R. Ernst and W. A. Anderson, "Application of Fourier Transform Spectroscopy to Magnetic Resonance", The Review of Scientific Instruments 37(1), 1966.