Nuclear Magnetic Resonance
Introduction
Nuclear Magnetic Resonance (NMR) provides an important probe of the microscopic environment in solids and liquids. NMR, as a probe of condensed systems, has become an essential tool in many disciplines, such as physics, chemistry, biology, etc. NMR methods have also become important in areas such as medicine, where they are used as a diagnostic tool. The importance of NMR, both as one of the most fundamental examples of resonance phenomena and as a technologically important tool, makes it an ideal experiment for this laboratory. [1-5]
The nuclei of certain elements possess a spin angular momentum and an associated magnetic moment. When such nuclei are placed in a magnetic field, they can adopt one of a number of quantized orientations, where each orientation corresponds to a particular energy level. The orientation having the lowest energy is the one in which the nuclear magnetic moment is most closely aligned with the direction of an external magnetic field. Nuclear magnetic resonance involves transitions between these energy levels. These transitions may be induced by the absorption of radio frequency (RF) radiation of the correct frequency. The relationship between the electromagnetic frequency n (angular frequency w) and the magnetic field strength Bo is governed by the Larmor equation, n = g Bo / 2p or w = gBo, where g is the gyromagnetic ratio which is a characteristic constant for a particular nucleus.
The local environment of the nuclei in solids and liquids is determined by fluctuations, both static and dynamic, caused by neighboring nuclei and electrons that couple to the nuclear magnetic moment. These fluctuations will shift and broaden the absorption. Measurement of these shifts provides information on the static fluctuations in the systems. Measurement of the width of the line, and the associated relaxation times, provides information on the dynamic fluctuations in the system. The most important contribution arises from fluctuations at the resonance frequency. Thus, NMR provides a method to observe the magnitude of the local fluctuations at a particular frequency and measurements at several frequencies can be used to map the fluctuation spectrum. The NMR experiment utilizes pulsed techniques to measure the position, width, and relaxation time of the resonance. Pulsed techniques, which are now used in most applications, use linear response theory to relate the response after a short perturbation to the more familiar continuous absorption method. This experiment introduces the student to other important techniques such as high frequency signal transmission, pulse manipulation, and signal averaging which is used to recover weak signals from noisy data.
Since the nucleus has spin, an external magnetic field causes it to precess, i.e., it undergoes a circular motion similar to that of a spinning top. The precessional frequency is directly proportional to the applied magnetic field, and is given by the Larmor equation. NMR measurements concern the measurement of this precessional motion. In order to cause transitions between two spin states, a second (radio frequency) field is applied perpendicular to the static field. If the frequency of this field is equal to the precessional frequency of the nucleus, it will induce a transition between the different spin states and a resonance will be observed.
With the absorption of energy, the population of nuclei in the ground state will decrease while the population in the higher energy state will increase until a saturation level is reached. This results in a weakening of the NMR signal. Since there are a number of ways in which the high-spin-state nuclei can relax back into the ground state, the NMR signal is not totally eliminated, and an equilibrium state is reached.
There are two main types of relaxation processes by which the nuclei in the upper state can relax to the lower-energy state. The first is spin-lattice relaxation (interactions of the spins with “the rest of the system”), which has a characteristic time constant T1, the “longitudinal relaxation time.” The spin distribution is out of thermal equilibrium and will lose energy to the other degrees of freedom in the system. The second is spin-spin relaxation, which has a characteristic time constant T2, the “transverse relaxation time.” This redistributes energy between the spins but leaves the total energy of the spin system unchanged, so only the components perpendicular to the field relax. In this experiment, you will attempt to measure T1 and T2 for a given nucleus.
There is another effect which can cause an NMR signal to decay as for a relaxation effect, but it is an experimental artifact, and not an energy loss process. When the total magnetization (by whatever means) is put at some angle with respect to the magnetic field, it starts to precess. If all the spins were to precess at exactly the same frequency, one would observe the transverse component of the magnetization decay with T2. In fact, it seems to decay much faster. Due to variations in its environment (a lack of perfect translational invariance), every spin sees a slightly different field. This means that the local precession frequencies are slightly different, some spins precess faster than others, and when the spins are no longer in sync, the NMR signal decreases. However, this is not a relaxation process since the effect is completely deterministic and reversible. As long as there are no interactions, the spins know where they came from, and if all the spins or all the fields were inverted, the process is perfectly reversible; all the spins precess backward to where they came from, and the NMR signal recovers. This recovery is known as a “spin echo.”
One method which is often employed to study NMR effects is pulse-Fourier transform NMR. In this method a short but intense radio-frequency pulse which extends over the entire bandwidth of frequencies is used to simultaneously excite all of the nuclei. The pattern detected at the receiver is not in the form of peaks, but is a sum of exponentially decaying sine waves, known as a free induction decay (FID). Rapid acquisition and accumulation of these FIDs by a digital scope may be analyzed using Fourier transforms to produce an NMR spectrum. However, in our experiment we shall be using special types of pulses to access directly the time constants T1 and T2 which comprise the FID. The special pulses have a particular duration in time which can cause a spin to rotate by 90° (referred to as a π/2 pulse) or by 180° (referred to as a π pulse). A π/2 pulse results in a FID with a maximum initial amplitude, and a π pulse inverts a spin's direction.
Measurements of T1 are made by using a pulse sequence π - t - π /2, followed by a long delay to allow the spins to fully relax back to the equilibrium state. The peak of the FID signal (immediately after the π /2 pulse) as a function of t is proportional to , and this may be used to determine T1.
The method to measure T2 takes advantage of the spin echo effect to eliminate the artifact caused by field inhomogeneity. The measurement involves the application of a π/2 pulse which causes the magnetic vector to tip away from the z-axis to the x-y plane (say the y-axis). As discussed earlier, the spins fan out into the x-y plane because they have slightly different precession frequencies (due to field inhomogeneities) and spin-spin relaxation. If after a time t a π pulse is applied along the y-axis, all the vectors will flip about the y-axis by 180°. This causes the faster-moving vectors to then be behind the slower-moving ones, and the faster-moving ones will catch up at a time 2t . However, there will not be as many since some spins will be lost to other processes such as spin-spin interactions. The pulse sequence π /2 - t - π results in an echo every 2t seconds. Thus by measuring the pulse height vs delay time t , one can determine the spin-spin relaxation time T2. The peak of the spin echo signal will be proportional to 
and this may be used to determine T2. However, there is an additional relaxation mechanism in a liquid which is related to the motion of its molecules. This introduces yet another exponential decay term with a parameter which depends on the liquid. From your data, you will be able to determine which relaxation effect is most important for your samples.
Water provides an ideal sample for introductory experiments in NMR. The signal from the hydrogen is quite strong and the properties of the system can be varied over a wide range. The effect of fluctuations on the NMR signal will be examined in two ways. First, by adding a magnetic impurity, copper sulfate, to the liquid the strength of the fluctuations can be changed. Secondly, by solidifying the liquid the characteristic frequency of the temporal fluctuations can be changed.
Before undertaking the actual experiment, it is essential that you find the answers to the following questions:
1. What is a complex impedance (as in AC circuit theory)?
2. What is the characteristic impedance r of a transmission line?
3. An important quantity is the impedance 
at the end of a transmission line (having characteristic impedance r), given that the impedance at a length l is 
and that the frequency is such that the wavelength in the transmission line is l. [This is the Transmission Line Formula.] Show that is given by:
4. What is if =r (impedance matching)?
5. What is if l is a half wavelength (l/2)?
6. What isif l is a quarter wavelength (l/4)?
7. For a l/4 line, what is if is open (infinite)?
8. For a l/4 line, what is if is shorted (zero)?
9. What is the behavior of crossed diodes in an NMR circuit?
Additional information regarding this NMR experiment can be found in Elements of Resonance (local copy PDF) and Intro to NMR (local copy PDF)
Experimental Apparatus
Electromagnet and Power Supply
Matec Gated Amplifier and Broadband Receiver (MBR)
Ohaus Explorer analytical balance
Sample solutions of CuSO4
Procedure
Caution: Make sure the cooling water is on before using the electromagnet.
NMR samples are placed in a small test tube inside a small coil, which is part of a resonant circuit, intended to operate around 25MHz.
Connect a coax cable from a function generator to an oscilloscope with a 50 ohm termination. Observe that the received signal is independent of frequency. Add a cable of length about 2m with a tee at the oscilloscope, and the other end of the cable open. Observe that the signal at the oscilloscope is a minimum at a frequency which corresponds to a quarter wavelength in the 2m cable. For the NMR experiment, find three coax cables of the same length, about 2m. Determine the frequency for which these cables are quarter wavelength (around 24.6 MHz). The function generator should remain fixed at this frequency throughout the experiment. The function generator, with an output amplitude of about 0.5V, should be connected to the RF Input of the Matec Pulse Generator (MPG). The RF Input Level meter on the MPG should be just below the red line.
The Gated RF Output of the MPG should be set up as follows: A l/4 cable should go from the Gated RF Output to a set of crossed diodes in series, followed by a tee. From one part of the tee the second l/4 cable should go to the NMR cell. From the other part of the tee the third l/4 cable should lead to the input of the Matec Broadband Receiver (MBR), which should be shunted to ground with a second set of crossed diodes.
The NMR setup operates in two modes: the transmit mode, when a large amplitude pulse (a few hundred volts) is present and the crossed diodes act as closed switches, and the receive mode, when the small NMR signal (less than 0.5V) is present and the crossed diodes act as open switches. Consider the behavior of the NMR setup with the three l/4 cables, etc., during the transmit mode and the receive mode.
Set the Matec as follows: Sequence Interval, coarse .01-.11, vernier 3.00 (approximately a 30Hz repetition rate); Pulse Separation, M=10, A=0.10, B=2.00 (approximately 200 ms separation); Pulse Width #1, coarse .5-10, fine 2.50 (~3 ms width); Pulse Width #2, coarse .5-10, fine 6.3 (~6 ms width); RF Power Low, level 60; Pulse Selector #1 switch up, #2 switch down, Ext Pulse down; RF Cycle Increment switches on Continuous; Power switch on, High Voltage switch on.
Place a small probe coil (inside a test tube) into the NMR coil, and monitor it with an oscilloscope; use a 6 dB, 50 ohm attenuator at the scope, with 10V/div, 1ms/div settings. Adjust the MPG Impedance Matching and RF Tuning controls, and the two variable capacitors in the NMR cell to produce the largest pulse with smooth exponential initiation and decay. The two variable capacitors are connected in series and parallel with the NMR coil; the capacitor on the left (looking from the capacitors toward the coil) is in parallel with the coil, and the capacitor on the right is connected in series between the coil and the coax. The series capacitor is more important when driving the coil, since it can set the LC series to resonate at the drive frequency. The parallel capacitor is more important for impedance matching the coil circuit with the MBR during the receive mode; the proper way to adjust the parallel capacitor is to connect the probe coil to the signal generator and adjust the parallel capacitor for maximum signal out of the MBR.
Place a sample of water with high concentration of copper sulfate in the NMR coil. Turn on the magnet cooling water, and slowly increase the current to around 6.05A. Set the Gain of the Matec Broadband Receiver (MBR) to 9.00, with the Filter at 5MHz and Multiplier of 0.8. Trigger the scope with MPG Monitor Pulse #1, and monitor the MBR output with the scope at 2V/div and 50ms/div. Increase the current very slowly (to about 6.1A) until a clean exponential Free Induction Decay (FID) is observed on the scope.
Turn Pulse Selector #2 switch on, and observe the spin echo. Adjust the pulse widths to get the best echo. To find T2, use the p/2 - t - p pulse sequence, and measure the height of the spin echo as a function of the pulse separation t. To find T1, use a p - t - p/2 pulse sequence, and measure the top of the FID following the second pulse as a function of the pulse separation t.
Determine T1 and T2 as functions of copper sulfate concentration, from about 0.15 down to 0.01 molar concentrations.
Other liquid solutions to try include: copper nitrate (Cu(NO3)2) in water, copper nitrate in methanol, and Zeolite in methanol.
References
[1] C.P. Slichter, Principles of Magnetic Resonance, Harper & Row, New York (1963).
[2] E. Fukushima and S. B. W. Roeder, Experimental Pulse NMR: A Nuts and Bolts Approach, Addison-Wesley, Redwood City (1981).
[3] Thomas C. Farrar, Pulse Nuclear Magnetic Resonance Spectroscopy, Farragut Press,
Chicago (1989).
[4] D. C. Ailion, Methods of Experimental Physics, Academic Press, New York, Vol. 21, pp. 439-482 (1983).
[5] R. J. Abraham, J. Fisher, and P. Loftus, Introduction to NMR Spectroscopy, John Wiley & Sons, New York (1988).
[6] D. W. Preston and E. R. Dietz, The Art of Experimental Physics, Wiley, New York (1985).
[7] C. P. Poole Jr. and H. A. Farach, Relaxation in Magnetic Resonance, London (1971).
[8] A. Abragam, The Principles of Nuclear Magetism, Oxford (1961).
[9] C. P. Slichter, Principles of Magnetic Resonance, Springer-Verlag, Berlin, (1978).
[10] R. T. Schumacher, Introduction to Magnetic Resonance, Principles and Applications, New York (1970).
[11] E. D. Bedcer, High Resolution NMR, New York (1980).