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Pulsed Nuclear Magnetic Resonance
John Stoltenberg, David Pengra, Oscar Vilches, and Robert Van Dyck
Department of Physics, University of Washington
3910 15th Ave. NE, Seattle, WA 98195-
1 Background
What we call “nuclear magnetic resonance” (NMR) was developed simultaneously but independently by Edward Purcell and Felix Bloch in 1946. The experimental method and theoretical interpretation they developed is now called “continuous-wave NMR” (CWNMR). A different experimental technique, called “pulsed NMR” (PNMR), was introduced in 1950 by Erwin Hahn. Pulsed NMR is used in magnetic resonance imaging (MRI). Purcell and Bloch won the Nobel Prize in Physics in 1952 for NMR; more recently NMR was the subject of Nobel Prizes in Chemistry in 1991 and 2002. We have both NMR setups in the advanced labs: one is a variation of the CWNMR method, and the other is a pulsed NMR system. The physics underlying NMR is the same for both the continuous and pulsed methods, but the information obtained may be different. Certainly, the words used to describe what is being done in the experiments are different: in the continuous-wave case one tunes a radio-frequency oscillator to “beat” against the resonance of a nuclear magnetic moment in a magnetic field; in the pulsed case, one applies a sequence of RF pulses called π-pulses (180 degree pulses) or π/2-pulses (90 degree pulses) and looks for “free induction decay” and “spin echoes”.
Below, we give a very simplified introduction, based on classical ideas, to the physics of NMR. More thorough discussions, focusing on the CWNMR technique, may be found in in the books by Preston and Dietz [1] and Melissinos [2] (see references). In particular, the chapter in Preston and Dietz gives a nice description of the connection between quantum-mechanical and semi-classical approaches to NMR physics. For a comprehensive treatment of NMR, see the book by Slichter [3].
1.1 Semiclassical ideas
To observe NMR, one needs nuclei with a non-zero angular momentum I and magnetic moment μ. The relationship between these two quantities is
μ = γ I , (1)
where γ is the gyromagnetic ratio. A simple classical calculation would give γ = q/ 2 M , where q is the charge of the nucleus and M is its mass. But quantum mechanics requires this dimensionally correct result to be modified. In practice, we specify γ in units of the nuclear magneton μn ≡ e¯h/ 2 mp times a dimensionless factor g called the “spectroscopic splitting factor” (often just “g factor”):
γ =
gμn ¯h
ge 2 mp
where e is the elementary charge and mp is the mass of one proton. The g factor is on the order of unity. It is positive for some nuclei, and negative for others. For the proton, g = +5.586; a table of various g factors may be found in Preston and Dietz. When a nucleus of moment μ is placed in a magnetic field B 0 it will experience a torque causing a change in the angular momentum following Newton’s second law:
τ = μ × B 0 =
dI dt
γ
dμ dt
Figure 1 shows some of the vectors defined with B 0 lying along the z axis (which is horizontal in most lab setups, including ours). The rate of change of μ is, by Eq. (3), both perpendicular
Figure 1: (a) Vectors defined in semiclassical picture of NMR. (b) Tip of angular momentum vector I, viewed opposite to B 0.
to B 0 and μ itself; hence μ precesses about the direction of B 0. From Fig. 1 it is easy to derive from geometry and Eq. (3) that the magnitudes of the vectors μ × B and dI/dt obey
μB 0 sin θ = μ γ sin θ
dφ dt
thus
ω 0 ≡
dφ dt = γB 0 , (5)
which is the Larmor angular frequency. When we deal with nuclei of spin I = 1/2 (e.g., protons), quantum mechanics tells us that in a magnetic field B 0 the ground state splits into two sublevels, as shown in Fig. 2. The
Figure 2: Splitting of nuclear energy levels due to applied magnetic field.
vary in direction and strength over time, so the net work done by the field would be less, tending to zero as the RF angular frequency moves away from ω 0. A little mathematics shows that this situation follows the standard picture of resonance phenomena—a resonance which involves the interaction between a nucleus with a magnetic field, hence the name nuclear magnetic resonance. In the quantum-mechanical picture for a single nucleus we can say that the probability of a “spin-flip” between the mI = +1/2 and mI = − 1 /2 state is maximized when the RF photon has energy equal to ¯hω 0. One may well wonder whether a complete quantum mechanical approach would give different answers than the classical picture we have presented. It is somewhat surprising to learn that Bloch showed that the classical equations governing NMR can be derived from quantum mechanics. See the text by Slichter for a readable and thorough presentation of this result [3].
1.2 Net magnetization and relaxation
Clearly, any physical sample will consist of many atoms and the NMR signal measured by an experiment will be due to the combined effect of the magnetic field on them all. As an example, consider an assembly of N protons, say, in water, glycerin, mineral oil, or animal tissue. The net magnetization M of the sample is the vector sum of all of the individual moments μi. For the example of spin- 12 protons, the net magnetization along the z-axis, Mz would be given by
Mz =
i
γ¯hmI i =
γ¯h (N+ − N−) , (10)
where mI i denotes the state of the ith proton, and N+, N− are the numbers of protons in the
- 12 and −^12 state, respectively. When an RF field B 1 (t) is applied, some protons have their spin flip to align against B 0 ; these absorb energy from the RF field. Other protons experience a spin-flip to align with B 0 ; these give energy back to the RF field. Thus, if the populations of the two sublevels were equal, Mz = 0 and no net energy would be absorbed from the RF field. At room temperature with no RF field but with the steady B 0 field there will be a small difference between the number of protons with spins aligned in the direction of B 0 and the number aligned opposite to it; the magnetic sublevel with mI = +^12 will have a slightly larger population N+ than the population N− in sublevel mI = −^12. The ratio of the populations is proportional to the ratio of the associated Boltzmann factors, according to the principles of statistical mechanics: N+ N−
exp(−U+/kT ) exp(−U−/kT )
exp (+^12 γ¯hB 0 /kT ) exp (−^12 γ¯hB 0 /kT ) = eγ¯hB^0 /kT^. (11)
If one applies the RF B 1 (t) to this equilibrium population, energy goes into the system, since there is an imbalance in the number of up (+^12 ) spins versus down (−^12 ) spins. In a fairly short time, however, this imbalance will vanish, because the continuous transfer of energy involving each moment will cause the populations N+ and N− to become equal. At this point, the net energy absorbed by the system drops to zero, Mz = 0 and the sample is said to be saturated. There is still a tendency for the system to recover its equilibrium configuration, even while it is subject to the oscillating field. The collection of protons could continue to absorb energy
from an oscillating magnetic field B 1 (t) if the equilibrium populations would be restored quickly following a previous absorption event. The decay of Mz to its its equilibrium (nonzero) value is due to energy exchanges between the proton magnetic moments and their local environments.
Let us consider the magnetization a bit more generally. The net magnetization M is a vector, and thus can be broken into components. The component along the direction of the static field Mz is called the longitudinal magnetization, and the components along two orthogonal directions perpendicular to B 0 are called the transverse magnetizations Mx and My. Clearly, each of these components is the sum of the individual component moments μz , μx and μy , respectively, e.g., Mx =
i μxi. Now look again at Fig. 3 which depicts a single moment precessing about B 0. It should be obvious that for a collection of magnetic moments at a particular time, each precessing independently about B 0 , one could have a nonzero Mz but a (possibly) zero Mx or My.
The equilibrium nonzero Mz exists because the moments aligned parallel to B 0 are at a lower energy than those aligned antiparallel. In order to return to equilibrium following absorption from the B 1 (t) field, the moments would need to give energy to their surroundings, conventionally called the “lattice” (even if we are dealing with liquid or gaseous materials). The relaxation of Mz to its equilibrium value is called the “longitudinal” or “spin-lattice” relaxation, and is associated with a characteristic time called T 1. In liquids, T 1 is typically very short but in solids T 1 may be quite long. For example, T 1 is a few milliseconds in water but thousands of seconds in ice. This is one reason why we use liquid samples in the continuous NMR setup, as the short T 1 allows for a continuous input of energy from the oscillator coil.
A nonzero Mx or My is harder to obtain because there is no reason for individual moments to prefer a particular transverse direction. At best Mx and My can oscillate between positive and negative values at the Larmor frequency, if one can first contrive to force the moments to have a net projection onto, say, the x-axis (and indeed one can—we’ll see how). However, the resulting oscillating Mx decays because variations in the local magnetic field cause different moments to precess at different rates. This is called “spin-spin dephasing”, and the characteristic time for this process is called T 2 , the “spin-spin” or “transverse” relaxation time.
With these two relaxation processes in mind, consider how one would detect the signal of these precessing and relaxing moments. Since they are magnetic, an obvious choice would be to use a pickup coil. The coil also needs to be oriented transverse to the static field in order to sense the precession, since in an orientation parallel to B 0 , the coil would only be sensitive to a slight change in flux from a non-oscillatory Mz ; whereas the transverse component Mx or My will oscillate between positive and negative values at the Larmor frequency ω 0. This arrangement is depicted schematically in Fig. 4, which is taken from the article by Hahn [4]. The top of the figure shows that at t = 0, all of the moments are aligned along the +x direction and the pickup coil has its axis along the y-axis. The static field B 0 points out of the page toward the viewer, thus the moments will precess in the clockwise direction (following the sense of M × B 0 ). The right side of the figure depicts the voltage measured across the pickup coil. Faraday’s law requires that this voltage be proportional to the rate of change of the flux through the coil, and it is a simple matter to prove that the maximum signal from the coil will occur at t = 0 in this setup. (Try it: let the field in the coil be μ 0 M which is rotating about the z-axis at ω 0 , and calculate the rate of change of the flux, dΦ/dt from this.)
1.3 The two experimental methods
In the continuous NMR method, the RF excitation producing the rotating B 1 field is applied all of the time. The resonance is created by having the B 0 field swept slowly through the value which satisfies Eq. (5). A pickup coil surrounding the sample detects the resonance, and this signal is mixed with the fixed RF signal to create “beats” (the term for the modulation of two signals which have nearly the same frequency). The effect of the beats can be seen on an oscilloscope. Once the resonant signal is found, the relationship between the magnetic field and the resonant frequency is fixed, and depending on the givens, the information can be used to extract B 0 itself, γ, or related quantities. One can also crudely measure T 2 ∗ by looking at the decay of the beat signal. In the pulsed NMR method, the RF excitation is applied to the sample in a series of short bursts, or pulses. The application of the RF field for a short time (the “pulse width”) allows the applied torque to rotate the net magnetization M by a specific amount. For example, one can apply a pulse of RF field to rotate all moments by 90◦. If this pulse is applied to a sample initially at equilibrium with a net magnetization M = Mz kˆ, then M will become Mzˆı (i.e., the same magnitude of magnetization now pointing in an orthogonal direction), which will then precess about the z-axis. The pickup coil will see a signal like that shown in Fig. 4 as the magnetization decays back to its equilibrium state. A pulse which accomplishes this trick is called a “π/2 pulse”, and the signal seen as a consequence is called the “free induction decay”. A pulse of a longer duration can flip the net magnetization completely: Mz ˆk → −Mz kˆ; this type of pulse is called a “π pulse”. Interestingly, the free induction decay signal immediately following a π pulse is zero since there is no net transverse component of the magnetization available to induce such a signal. The real utility of the pulse method comes from using a sequence of pulses. By such sequences, one can measure accurately and independently T 1 and T 2 , and also compensate for the effects of field inhomogeneity. The discussion of pulse sequences and their effects are taken up in more detail in the next sections.
2 The pulsed NMR experiment
This experiment is performed with the TeachSpin PS1-A pulsed nuclear magnetic resonance (PNMR) spectrometer. This instrument was designed for use in a teaching laboratory. Its modular design separates the basic functions (oscillator and RF amplifier, RF receiver/detector and pulse programmer) of a PNMR system, allowing the experimenter to investigate and understand the function of each module, and how and why the interconnections between the modules are made. The pulsed NMR technique involves a number of subtle and somewhat complex ideas and methods. One way to learn it is to read through the theory, as laid out here or in the PS1-A manual, and then use the instrument to quickly make the measurements. Another way, perhaps more instructive, is to learn the theory and technique at the same time. In what follows, we will assume that you will proceed along the second way. If, however, you have already read the theory, and would prefer to move directly to the measurement tasks, you can turn to Section 3 (p. 25) for a summary of those. The instruction manual for the PS1-A gives complete details on the instrument and its operation. Our instructions will focus on the particular operations needed to make the measurements. We will refer to relevant passages in the manual by means of a marginal note. Like this. Copies of the manual are available in the lab and online at the course website.
2.1 Pulses and Free Induction Decay
We begin by stating a curious result. Recall that the magnetic moments in our system precess under the influence of the torque from a magnetic field, Eq. 3. This precession occurs with a precession angular frequency ω 0 = γB 0. If we were to transform our coordinate system to one which rotated along with this precession, i.e., we translate coordinates x, y and z, as measured in the fixed lab reference frame to x′, y′^ and z′^ in the rotating frame, where
x′^ = x cos ω 0 t − y sin ω 0 t , y′^ = x sin ω 0 t + y cos ω 0 t , z′^ = z ,
we would find that the magnetic field B 0 vanishes! This must be true, since in this rotating frame (at exactly ω 0 ), there is no precession. PS1-A pp. 5- Now consider what would happen if we apply a time-dependent magnetic field B 1 (t) which rotates at exactly the frequency ω 0 and lies in the x-y plane, e.g.,
B^ 1 (t) = (B 1 cos ω 0 t)ˆı − (B 1 sin ω 0 t)ˆj.
In the rotating frame, B 1 (t) would appear to be a constant. The net effect, as seen from the rotating frame, would be a precession of the moment about an axis lying in the x′-y′^ plane at a frequency ω 1 equal to γB 1. Assume, for an illustration, that we have a moment μ initially pointing along the +z′^ (also the +z) axis, collinear with B 0 , and that we turn on B 1 (t) at exactly t = 0. In the rotating frame, B 1 (t) = B 1 ˆı′, where ˆı′^ is the unit vector along the rotating +x′^ axis. Thus, we would
see μ rotate about the +x′^ axis, initially tilting toward the +y′^ axis (according to μˆk ′ × B 1 ˆı′).
Figure 6: Block diagram of the PS1-A Pulsed NMR apparatus.
negligibly small perturbation to the measured angular rotation frequency—the “Bloch- Siegert shift”.)
- Electronics which can allow one to tune the oscillator to the resonant frequency f 0 : a “mixer” circuit which produces a signal showing the difference between the oscillator and receiver signals.
- Electronics which can act as a switch to turn the RF current on and off in a carefully controlled manner: a “pulse programmer”.
- And, finally, a sample containing moments: the protons (hydrogen nuclei) in water and hydrocarbons.
This apparatus is supplied by the PS1-A system, whose block diagram is shown in Fig. 6. PS1-A p. 15
2.1.1 Operation of the pulse programmer
We will start with learning to operate the pulse programmer module of the apparatus. The pulse programmer creates logic-level pulses (0-4V) which are used by the other electronics to control the application of the RF pulse and detection of the NMR signal. The front panel of the pulse programmer is shown in Fig. 7. The programmer creates two types of pulses, called A and B. To observe free induction decay, you need only use A pulses, but we will use the B pulses later on, so this exercise will look at both types. Details of the controls are given in the PS1-A manual, here we give a brief overview of the PS1-A pulse programmer function. pp. 17-
- Type A pulses are controlled by the A-WIDTH knob—to set the pulse duration—and the REPETITION TIME controls—to set how often the pulses occur. The REPETITION TIME controls are operational only when the MODE switch is set to INT (internal). The other settings of the MODE switch allow a pulse to be initiated by an external signal (into EXT START) or by the press of a button (MAN START).
Figure 7: Diagram of connections needed to examine output of the pulse programmer.
- Type B pulses are coupled to A pulses. They are controlled by the B-WIDTH knob—to set the pulse duration, the NUMBER OF B PULSES thumbwheel switches—to set how many B pulses occur, and the DELAY TIME thumbwheel switches—to set how long after the A pulses B pulses occur and the subsequent time between additional B pulses.
- The SYNC OUT connector and SYNC switch allows a very short (≈ 0. 2 μs) pulse to be used to trigger an oscilloscope at the start of a pulse or pulse sequence. Typically, one connects SYNC OUT to the external trigger input on the scope.
- The BLANKING OUT connector is used to disable the RF detector when the A and B pulses occur in order to prevent overloading the detector amplifier. The M-G OUT con- nector sends a signal to the RF oscillator to enable the “Meiboom-Gill” type of multipulse sequence (see Section 2.2.2).
First, we will look at type A pulses only. Disconnect all cables from the front panel of the PS1-A PS1-A except the two cables connected to the magnet and sample holder assembly. Turn on the p. 25 power to the PS1-A (the switch is on the back of the case), and turn on the oscilloscope. Connect SYNC OUT to the scope’s EXT TRIG input, and connect A+B OUT to the scope’s CH 1 input. Set the oscilloscope to trigger on “EXT”; set the CH 1 sensitivity to 2V/Div; set the timebase to 20 μs/Div. First, look at A pulses only. Set the pulse programmer controls as follows: A-WIDTH: half-way (12 o’clock) MODE: INT REPETITION TIME: 10 ms, VARIABLE at 10% (≈ 1 ms) SYNC: A A: on B: off If all is working correctly, you should see a trace on the oscilloscope similar to that in the lower trace of Fig. 8. Vary the A-WIDTH knob and note what happens. Then turn the A-WIDTH all the way clockwise, and increase the timebase on the scope to 1 ms/Div. You should see multiple pulses, spaced a couple of milliseconds apart. Note the effects when you change the REPETITION TIME controls.
The modules on the spectrometer are described here. Set the controls as indicated.
15 MHz Receiver This module senses the RF signal produced by the sample-probe coil, PS1-A, amplifies it and rectifies it. The rectified signal is sent to the oscilloscope. The unrectified pp. 22- RF signal is sent to the mixer in order to optimize the oscillator frequency. GAIN: about 30% BLANKING: on TIME CONST: .01 ms TUNING: 12 o’clock
Pulse Programmer You want to set the controls to make only A type pulses. First, calculate the time needed for a π/2 pulse from Eq. (13), given γ = 2. 765 × 104 rad/s-gauss and the magnetic field B 1 ≈ 12 gauss, according to the manufacturer. Then set the programmer PS1-A controls: p. 20 A-WIDTH: Set to calculated tπ/ 2 (use scope) B-WIDTH: Fully CCW NUMBER OF B PULSES: 00 REPETITION TIME: 100 ms, VARIABLE at 100% (≈ 100 ms) SYNC: A A: on B: off
15 MHz OSC/AMP/MIXER This module has three separate circuits in one box. The 15 MHz oscillator is controlled by the FREQUENCY ADJUST knob and the COARSE/FINE switch which changes the significant digit incremented by the knob. The oscillator signal is fed internally to the input of the power amp which is turned on (or “gated”) by the signal going into A+B IN. The output of the power amp, RF OUT is connected to the sample probe excitation coil and delivers an RF power of 150 watts peak power. The M-G PS1-A IN and M-G switch are used to synchronize the phase of the RF signal with the gating p. 20 in order to implement the “Meiboom-Gill” type of multipulse sequence, as described in Section 2.2.2. Set the controls as follows: FREQUENCY: 15. CW-RF: on M-G: on
Now connect the modules together and to the oscilloscope following the diagram shown in Fig. 9. Use the short cables to connect one module to another, and use the long cables to make connections to the oscilloscope. Connect the DETECTOR OUT to CH 1 of the digital oscilloscope, and the MIXER OUT to CH 2. Set the oscilloscope to trigger on “EXT”; set the CH 1 sensitivity to 2V/Div; set the CH 2 sensitivity to 5V/Div; set the timebase to 40 μs/Div. Make sure that the O-ring on the sample vial is 1.5 inches from the bottom and carefully lower the mineral oil sample into the sample holder inside the magnet assembly. After some adjustment of the oscilloscope, you should observe traces that bear some resemblance to those in Fig. 10. If you don’t get anything like Fig. 10 (or anything at all), check the connections on the PS1-A and the switch and knob settings. The CH 1 signal (lower trace) is the free induction decay (FID). This signal is half of the envelope of the RF oscillations generated by the precession of the net magnetization in the x-y plane. It is the
Figure 9: Diagram of connections on the PS1-A spectrometer to make pulsed NMR measure- ments.
Figure 10: Signals of free induction decay. Upper trace: mixer output showing beats indicating the frequency difference between the resonance frequency and the oscillator frequency. Lower trace: detector output showing the FID signal, which is the positive amplitude envelope of the RF signal obtained from the pick-up coil.
signals from the different precession frequencies mix together, and the resulting signal is more complex than a simple dying-off of the transverse magnetization. As you will see, T 2 ∗ is much shorter than either T 2 or T 1 , indicating that magnetic field inhomogeneity is fairly significant in this apparatus.
Now explore what happens if you increase the A pulse width beyond tπ/ 2. You should first see the FID signal decrease, nearly to zero, and then increase again. Measure the pulse widths for the pulse that gives a minimum FID signal and the pulse that gives the following maximum. How are these pulses related to tπ/ 2? Discuss the relationship between these pulses, the rotation of M , and the resulting FID signal. Technical note: If the field of the permanent magnet changes in time (due to changes in the temperature of the magnet) the resonant frequency will change, and beats will reappear on PS1-A the mixer output signal. If this happens during the course of your measurements, the p. 16 oscillator frequency should be re-adjusted to minimize the beats.
2.2 Pulse sequences and measurements of T 1 and T 2
Now we will see the real power of the pulsed NMR method in its ability to measure the longitudinal (spin-lattice) relaxation time constant T 1 and the transverse (spin-spin) relaxation time T 2 even when the static magnetic field B 0 is not very uniform.
2.2.1 Measuring T 1 : 2 pulse sequence
The amplitude of the FID signal induced by a π/2 pulse is proportional to the initial longitudinal magnetization Mz. Immediately following a π/2 pulse, Mz ≈ 0, which you could confirm by applying another π/2 pulse at that time and noting that the FID signal was zero. (Indeed, you will have done this already if you think of what a pulse of 2 × tπ/ 2 does!) It takes a while for Mz to recover its equilibrium value, and if you hit the sample with another π/ 2 pulse before equilibrium has been reestablished, then this second FID signal will be weaker than the first one. The longitudinal magnetization relaxes according to an exponential function:
Mz (t) = M 0 − (M 0 − Mi) e−t/T^1 , (14)
where M 0 is the equilibrium value of Mz and Mi is equal to Mz at t = 0. One way of PS1-A estimating T 1 suggested by Eq. (14) is to decrease the time between successive π/2 pulses p. 4 until you see the FID signal drop by about 1/3 (i.e., 1/e) of its “equilibrium” value. You can see this by letting Mi = 0 (what you get after a π/2 pulse) in Eq. (14) and calculating Mz (t = T 1 ). Try this: reduce the REPETITION TIME and see how the FID signal behaves. Can you make an estimate of T 1 by this method? (You may find that you cannot make the FID signal reduce by 1/3 if T 1 is too short. Don’t sweat it; this method is merely qualitative.)
A more elegant measurement of T 1 may be accomplished by a two pulse sequence. If one applies a longer pulse—a π pulse—the net magnetization M can be inverted: M 0 → −M 0. The decay of Mz can be tracked by applying subsequent π/2 pulses at varying intervals after the π pulse, since the FID signal is proportional to Mz at any time. Of particular interest is the point at which Mz momentarily vanishes, or “crosses zero” in its decay from a net negative value to the equilibrium positive value.
Figure 12: Two pulse sequence showing inversion recovery of Mz. The FID signal produced by the π/2 (90◦) pulse has an initial amplitude related to the value of Mz at the end of the interval TI. Taken from reference [5].
This pulse sequence is illustrated in Fig. 12, which is taken from reference [5]. In the figure, we see that a π pulse initially inverts the magnetization, and that a π/2 pulse interrogates the magnetization at a time interval “TI” later. Since the zero crossing point is easily found, we will use this method to measure T 1 for our samples. The spectrometer should already be configured to observe FID pulses (see Section 2.1.2 for settings and Fig. 9 for connections). The strategy is to set up the A pulse so that it rotates the net magnetization by π, and then a single B pulse so that it rotates the net magnetization by π/2. PS1-A pp. 31- Set the REPETITION TIME to allow for a comfortably long recovery time, say 1 sec, ≈ 20% (assuming you are looking at the mineral oil sample). Start with the A pulse only (B off) and adjust the A-WIDTH, oscillator FREQUENCY and receiver TUNING to maximize the FID signal and minimize the beats on the mixer output. Next, increase the A pulse width to minimize the FID signal. This is the pulse width that rotates the net magnetization by π. (As an option you can measure the π/2 and π pulse widths to check the ratio.) Now turn off the A switch, turn on the B switch and set the NUMBER OF B PULSES to 01. Switch SYNC to B. Adjust the B pulse width, starting from fully counterclockwise, to maximize the FID signal (mark on knob should be just below 9 o’clock position). The B pulse is now rotating the net magnetization by π/2. If this B pulse comes close in time (relative to T 1 ) after the π pulse (A), the net magnetization will be rotated by 3π/2 or 270◦. Set the DELAY TIME to 30 μs (0. 03 × 100 ms), turn on both A and B switches, set SYNC to A, and confirm that you get a healthy FID signal.
2.2.2 Measuring T 2 : spin echoes & multipulse sequence
You may have noticed, while adjusting the controls to set up the π–π/2 pulse sequence in the previous exercise, that you saw an additional bump in the scope trace at twice the delay time. This curious artifact was also noticed by Erwin Hahn in 1949 (when he was still a graduate student). Hahn realized that the signal must be due to a “rephasing” of the transverse components of the magnetic moments in the sample, leading to a nonzero transverse magnetization and subsequent FID signal. He called this process a “spin echo” [6]. In this exercise you will learn how to create a spin echo and how to use it to make a good measurement of the spin-spin relaxation time constant T 2. Although a variety of pulse sequences can produce an echo signal, the strongest echo is produced by a π/2–π pulse sequence. It is also easiest to understand how the transverse rephasing is caused by this particular sequence, which is illustrated in Fig. 14. The pulse
Figure 14: Illustration of how a spin echo is created by a π/2–π pulse sequence. From reference [5].
sequence and resulting detector signal are shown at the top of the figure. A 90◦^ (π/2) pulse tips the net magnetization into the x-y plane, creating the FID signal. This signal decays mainly due to dephasing of the moments as they precess at different rates in a nonuniform B 0 field. The lower part of the figure shows this dephasing from the point of view of the rotating reference frame. In this frame we see that some moments lag behind the group, while others lead. A 180◦^ (π) pulse is applied at time τ after the π/2 pulse. This pulse effectively “flips the pancake” of the collection of dephased magnetic moments about an axis in the x-y plane. The overall precession of the group continues in the same direction, but now those moments which had lagged the group now lead the group, and vice versa. At time 2τ (TE in the figure) all of the moments come back into phase, and we see a recovery of the FID signal. Hahn nicely illustrates the formation of the echo by means of an analogy [4, p. 8]:
Let a team of runners with different but constant running speeds start off at a time t = 0 as they would do at a track meet... At some time T these runners would be distributed around the race track in apparently random positions. The referee fires his gun at a time t = τ > T , and by previous arrangement the racers quickly turn about-face and run in the opposite direction with their original speeds. Obviously, at a time t = 2τ , the runners will return together precisely at the starting line.
Before worrying about T 2 and its relationship to spin echoes, first set up the spectrometer so that you can see an echo. The spectrometer should already be configured to observe FID pulses (see Section 2.1.2 for settings and Fig. 9 for connections). Now you want to set the A pulse so that it rotates the net magnetization by π/2 and then make a single B pulse so that it rotates the net magnetization by π. Start with just the A PS1-A pulse (B pulses turned off) and adjust the electronics (A-WIDTH, TUNING and FREQUENCY) pp. 33 to maximize the FID signal. Set the repetition time to 1 sec, 30% (again, assuming mineral oil as a sample; with other samples you may need a longer time for full recovery of Mz ) and make sure that the M-G switch is on. Next, turn on the B pulses and set the NUMBER OF B PULSES to 01 and the DELAY TIME to 2.5 ms. Finally, adjust the B-WIDTH (starting from fully counterclockwise) to maximize spin echo which occurs at twice the delay time. You should get a scope display similar to that in Fig. 15.
Now note what happens when you increase the delay time. The reduction in the magnitude of the echo signal is due to a loss of “phase memory” among the magnetic moments. This in turn is due to variations in the local magnetic field of each moment over time; it does not depend on the static variation in the applied B 0 field that is responsible for the comparatively short FID decay time. In other words, the reduction in the spin echo amplitude follows T 2 , not T 2 ∗. Hahn explains [4, p. 9]:
The decay of the echo may be understood in terms of the race track analogy if it is assumed now that the runners become fatigued after the start of the race. For this reason they may change their speeds erratically or even drop out of the race completely. Consequently, following the second gun shot (the second pulse) some of the racers may return together at the starting line, but not all of them.
In terms of the analogy, T 2 ∗ depends on the inherent differences in base speeds among the runners; in terms of our collection of moments this is like the different precession speeds of the
