## One Spin In A Magnetic Field

The nucleus is viewed as a positively charged sphere that spins on its axis, producing a small magnetic field whose strength (magnetogyric ratio or gamma: y) is characteristic of the particular isotope (e.g., XH, 13C, 31P, etc.).

Throughout this book we will treat y as the "strength of the nuclear magnet," ignoring its units and its definition as a ratio. We will also assume that y is positive.

The spinning is a fundamental property of the nucleus, so it never stops or changes speed and the magnetic field it produces is a constant. The magnetic field provided by the NMR magnet (Bo) is always shown on the vertical axis (z axis) and the spin axis is represented as forming an angle © with the applied magnetic field (Fig. 5.1). Like a compass needle in the earth's magnetic field, the nuclear magnet wants to align with the Bo field, and it experiences

NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology, by Neil E Jacobsen Copyright © 2007 John Wiley & Sons, Inc.

a torque pushing the spin axis toward the z axis. Because it is spinning and possesses angular momentum, however, the torque does not change the angle of the spin axis but instead causes the axis to precess, describing a circular motion in a plane perpendicular to the Bo field, similar to the motion of a gyroscope (a spinning top) in the earth's gravitational field. Although we are always talking about NMR-active nuclei as "spins," it is not the spinning rate we are interested in but rather the precession rate, which is the resonant frequency that forms the basis of nuclear magnetic resonance. This precession rate is proportional to the strength of the nuclear magnet (y), and to the strength of the applied magnetic field Bo:

The precession frequency, also called the Larmor frequency, is represented as vq ("new-zero," in hertz or cycles per second). The division by 2n is sometimes omitted because the units of y can be expressed in hertz per tesla rather than radians per seconds per tesla. When the Larmor frequency is represented in units of radians per second it is called the angular velocity or œQ. These are related by a factor of 2n:

For a typical superconducting magnet with a field strength of 7.05 T, protons (!H nuclei) precess at a rate of 300 million revolutions per second (300 MHz). This frequency is in the radio frequency portion of the electromagnetic spectrum, a bit higher than the frequencies on your FM radio dial (88-108 MHz). The important thing is that vo depends on the nuclear magnet strength y, which is a fundamental property of the type of nucleus we are looking at (e.g., 1H or 13C) and never changes, and the field strength Bo, which depends on the NMR instrument ("200 MHz," "500 MHz," etc.) and can also be changed slightly by NMR hardware (lock system Zo coil, shim coils, or gradient coils) or by the chemical environment of the molecule in which the nucleus finds itself (chemical shift) and the neighboring nuclear magnets (J coupling). Virtually everything we observe in NMR depends on this resonant frequency, and the resonant frequency depends on the effective magnetic field that the nucleus experiences (Beff), which is very close to the NMR magnet's field, Bo. We will return to this fundamental relationship again and again.

In the classical world, all possible angles © can be formed with the magnetic field direction, between 0° (+z axis) and 180° (—z axis). To gain some flavor of the quantum world, however, we will allow only two angles corresponding to the two quantum states of a spin-1/2 nucleus. The lower energy or a state will be represented by an angle of 45°, and the higher energy or ft state will be represented by an angle of 135° to the positive z axis. Thus the axis of the nuclear spin will sweep out a cone as it precesses around the z axis, and this cone will be opening upward (toward the positive z axis) for spins in the a state and downward (toward the negative z axis) for spins in the j state (Fig. 5.1). Our vector model combines these two fundamental aspects of the NMR phenomenon: precession (circular motion in the x-y plane that leads to observable NMR signals) and energy levels (the quantum requirement that each spin be in either the a or aligned energy state or the j or opposed energy state).

5.3 A LARGE POPULATION OF IDENTICAL SPINS: NET MAGNETIZATION

People tend to think of NMR as a quantum phenomenon. But quantum mechanics deals with the options available to a single particle (e.g., two options for a spin-1/2 nucleus) and the probabilities of being in each of the energy states. In NMR we cannot measure something until we have a very large number of spins, like 1020 spins, all behaving statistically in a similar way so their miniscule microscopic magnetic fields add together to generate a macroscopic magnetic signal. Thus NMR is really about statistical mechanics, the sociology of large groups of spins, rather than quantum mechanics, the psychology of individual spins. In order to generate this measurable macroscopic signal the individual spins have to be "organized" so that they behave in a coherent manner, with "teamwork" to make sure that their individual signals do not just cancel each other out. This coherence or organization is provided by the radio frequency (RF) pulse.

A large population of identical spins in a sample is called an "ensemble" of spins. This would correspond to a sample with a single compound in solution with only one NMR peak (e.g., the *H spectrum of chloroform, CHCl3). Each individual nucleus in the sample precesses at its resonant frequency around the external magnetic field Bo, which is along the +z axis. Forget about the location of the each molecule within the volume of the sample solution, or the orientation of that molecule relative to the z axis. The nucleus maintains its orientation with respect to the Bo field (a or j state) even as the atom it belongs to is tumbling with the molecule and moving through the solution. The only thing the nucleus can interact with is a magnetic field; it is not "attached" in any way to the molecule so we can think of the spinning nucleus mounted on frictionless bearings ("gimbles" to the nautically inclined) that allow it to stay oriented to the Bo field regardless of the molecule's position or orientation. The orientation of the spin axis of the nucleus can be represented as a magnetic vector, pointing along the spin axis from the South pole of the nuclear magnet to the North pole, with length equal to the magnitude of the spin's magnetic field (y). If all of the vectors representing the magnetic dipoles of the individual spins are lined up in a row, we have a sort of "chorus line" of spins facing us, and all of them will be precessing in the same direction (counterclockwise) and at exactly the same rate, the Larmor frequency vo (Fig. 5.2). But at equilibrium they are all rotating with random phase; that is, at any moment in time if we take a snapshot we see that some are pointing to the right side (y axis), some are pointing to the left side (—y), some to the front (x axis) and some to the back (—x axis), and in fact every possible direction around the cone defined by the 45° angle with the +z axis will be represented (Fig. 5.2(a)). For the moment, we consider only the spins in the lower energy (a) state, which are precessing around a cone which opens upward. This is like a bad ballet company: the dancers are spinning around together, but at any moment some are facing the audience, some have their backs to the audience, some face the right side, and some face the left side in a random fashion. In technical terms we

say that there is no phase coherence in the ensemble. Now we subject the sample to a high power pulse of RF for a precise, short period of time. The effect of the RF pulse is to get the spins "in sync" in terms of phase. After the pulse, all the spins are pointing in the same direction at any point in time (Fig. 5.2(b)). Their precessional motion is now identical and we say that the ensemble has phase coherence. This is like a good ballet company: all the dancers are spinning at the same rate and come around to face the audience at exactly the same time. With this organization of the sample spins extending to the bulk level of, say, 1020 spins, the individual magnetic vectors add together to give a bulk magnetic vector (the "net magnetization") of the sample that is also rotating counterclockwise at the Larmor frequency around the upper cone. The bulk magnetism is large enough to measure, and by placing an electrical coil next to the sample we can detect a weak voltage oscillating at the Larmor frequency. Thus we can detect the signal (the FID) and measure the Larmor frequency very precisely. This concept of organization or phase coherence created in an ensemble of spins by a pulse is fundamental to the understanding of NMR spectroscopy. Without the pulse there is no coherence and the random orientations of the precessing spins cancel out their motions: there is no measurable signal.

Now that the concept of coherence has been introduced, let us make our model of the ensemble of spins a little more accurate. Instead of lining up the spins in a row, we move their magnetic vectors to the same origin, with the South pole of each vector placed at the same point in space (Fig. 5.3(a)). Furthermore, we need to consider both quantum states, the "up" cone (a or lower energy state) and the "down" cone (ft or higher energy state).

For a large population of identical spins, the individual magnetic vectors are all precessing at exactly the same rate (assuming a perfectly homogeneous Bo field) around either the upper cone (a state, "aligned" with Bo) or the lower cone (j state, "opposed" to Bo). The populations will be nearly equal in the two states, with very slightly more spins in the upper cone (lower energy state). In Figure 5.3(a) we have a cartoon representation with 8 spins in the j state and 16 in the a state, so the total population is 24 (N = 24) and an even distribution would be Pa (population in the a state) = 12 and Pp = 12. The energy difference creates a slight preference for the a state, which we have enormously exaggerated in the figure: Pa = n/2 + 8 = 12 + 4 = 16; Pp = N/2 - 8 = 12 - 4 = 8, with 8 = 4. (In the real world 8 is only about 0.001 or 0.0001% of N, not 1/6). Imagine a snapshot at one instant in time: the dipoles are not aligned in any particular direction with respect to the x and y axes, but are spread out evenly over each cone. In other words, the individual spins are all precessing at the same frequency, but their phase is random. Notice that for every vector in the upper cone (a state), there is another vector in the lower cone (j state) exactly opposite that vector. Their magnetic vectors will exactly cancel each other so we can erase them in our picture: they contribute nothing to the net magnetization, which is the only thing we can measure. In the cartoon, we first cancel four pairs of opposing spins (Fig. 5.3(b)), then four more pairs (Fig. 5.3(c)), leaving only 8 spins of the original 24, all of which are in the upper cone (a state). Thus we have wiped out nearly all of the N spins in the ensemble and we are dealing with only the population difference, Pa - Pp = (N/2 + 8) - (N/2 - 8) = 28 = 8. So the vast majority of the spins cancel each other throughout the NMR experiment and we can only detect about one in 105 spins! Remember that in the real world, unlike this example shown in Figure 5.3, 8 is much, much less than N. This gives you some idea why NMR is a relatively insensitive experiment, requiring milligrams (mg) of material rather than micrograms or nanograms.

After canceling the opposing spins, we see that the x and y components of the individual vectors cancel when they are combined to form the vector sum because all possible directions are equally represented in the population. The motion of precession is thus not detectable in a sample at equilibrium: no voltage will be induced in the probe coil and no NMR signal will be received by the spectrometer. Now consider the z component of the individual vectors remaining in the upper cone (Fig. 5.3(c)). All of these vectors are pointing upwards at a 45° angle to the z axis, so all have the same positive z component. Adding these together we get the vector sum, which is called the net magnetization vector (Fig. 5.3(d)). This vector is a macroscopic property of the sample, and we can view it as a large magnet with potentially measurable properties. At the moment it is stationary because only the x and y components of the individual vectors are moving, and all of these cancel due to their random distribution around the upper cone. We will see, though, how the RF pulse can create from this stationary vector a moving vector that can induce a measurable voltage in the probe coil.

So this net magnetization vector M, pointing along the +z axis and stationary at equilibrium, is the starting material for all NMR experiments. Its magnitude (length of the vector) is called Mo, and it is proportional to the population difference, 28, and to the length of the individual magnetic vectors, the "strength of the nuclear magnet" or y.

Mo = (constant) x (Pa - Pp)eq x y = (constant) x 28 x y

How big is this population difference at equilibrium? The Boltzmann distribution defines the populations of the two states precisely, and it turns out that the equilibrium population difference APeq is proportional to the energy difference between the two states (a and j)

and inversely proportional to the absolute temperature T (in degrees kelvin):

This makes sense because the larger the energy gap between the two states, the greater the preference will be for spins to be in the a state. If we lower the absolute temperature, each individual spin has less thermal energy available so it will be even more likely to prefer the lower energy state. In practice, liquid state NMR cannot benefit much from lowering the absolute temperature because the sample will freeze if we go very far below room temperature, but we can increase the energy gap (AE) by getting stronger and stronger NMR magnets. Because the energy difference is proportional to the Larmor frequency:

AE = hvo = hyBo/2n the energy gap is proportional to both Bo and to y. Thus Mo, our NMR "starting material," depends on Bo and y as follows:

Mo = (constant) x APeq x y = (constant) x (N x AE/T) x y

= (constant) x (N x yBo/T) x y = (constant) x N x y x Bo /T

So the net magnetization at equilibrium is proportional to the number of identical spins in the sample (i.e., the concentration of molecules), the square of the nuclear magnet strength, and the strength of the NMR magnet, and inversely proportional to the absolute temperature. For example, Mo for is 16 times larger than Mo for 13C because yH/yC = 4. This net magnetization vector is the material that we mold, transform and measure in all NMR experiments.

Now consider the effect of a 180° pulse on the ensemble of spins represented in Fig. 5.3. The RF pulse is actually a rotation, and we will see in Chapter 6 that this rotation is exactly analogous to the precession of magnetic vectors around the Bo field. The pulse itself can be viewed as a magnetic field (the "B1" field) oriented in the x-y plane, perpendicular to the Bo field, and for the short period when it is "turned on" it exerts a torque on the individual nuclear magnets that makes them precess counterclockwise around the B1 field. This is shown in Fig. 5.4. Each magnetic vector is rotated by 180°, so the entire structure of two cones is turned upside down, with the upper cone and all its magnetic vectors turned down to become the lower cone, and the lower cone turned up to become the upper cone. This

means that every spin that was in the a state is now in the p state and every spin that was in the p state is now in the a state. Each individual magnetic vector is still precessing at the Larmor frequency counterclockwise around the z axis (Bo field), and they are still randomly distributed around each of the cones. The only obvious difference is that the population of the upper cone (a or lower energy state) is now N/2 — 8, or a tiny bit less than half of the spins in the ensemble. This is because every one of those spins started out in the lower cone (higher energy state) at equilibrium, before the pulse. The population in the lower cone (P or higher energy state) is now N/2 + or a tiny bit more than half of the spins. We have inverted the population distribution, and Herr Dr. Prof. Boltzmann is turning over in his grave, because you are not supposed to have more spins in the higher energy state than in the lower energy state! As before, we cancel the exactly opposing pairs of spins until we have only 28 spins in the lower cone (P state) and these vectors are then combined by vector addition to give the net magnetization vector, M (Fig. 5.4, right). This vector is now pointing along the — z axis, with the same magnitude, Mo, that it had at equilibrium. We see a general pattern here: rotating the ensemble of magnetic vectors by an angle © has the effect of rotating the net magnetization vector M by the same angle ©. This is not surprising because the net magnetization is just the vector sum of all the individual magnetic vectors.

5.4 COHERENCE: NET MAGNETIZATION IN THE X-Y PLANE

The effect of a 90o pulse (i.e., turning on the B\ field for half of the time that we used for the 180o pulse) is more interesting (Fig. 5.5). Again the entire double-cone structure is rotated counterclockwise by the pulse, this time stopping with the (formerly) upper cone at the left-hand side and the (formerly) lower cone at the right-hand side. Each individual magnetic vector has experienced a 90o rotation around the B1 vector, which is extending out toward us. As before, we can cancel all of the exactly opposing magnetic vectors, leaving just 28 spins in the left-hand side cone. Now we have to wave the magic quantum wand because most of the magnetic vectors are violating the rule that they must choose either the upper cone or the lower cone, as defined by the Bo field direction. In other words, they must be either "aligned" (45o angle to Bo) with or "opposed" (135o angle to Bo) to the Bo field in our model. We sort these out by rotating them around the left-hand side cone to the nearest point where they are in a "proper" cone pointing either up or down. Now we have 8 spins in the upper cone and 8 spins in the lower cone, with a population difference AP of zero. The 90o pulse has destroyed the equilibrium population difference. More importantly, all of these magnetic vectors are pointing to the left-hand extreme of their respective cones at the moment the pulse stops. We have created phase coherence because at this instant

in time all of the spins point to the left side and will start their precession around Bo in unison. The net magnetization is the vector sum of all 25 magnetic vectors, and it points to the left side (Fig. 5.5, right) and has the same magnitude (Mo) as the vector sum formed from the 25 surviving magnetic vectors in the upper cone at equilibrium. As we saw with the 180° pulse, the effect of the 90° pulse on the net magnetization vector is to rotate it in a counterclockwise direction around the B1 field (this time by 90°), just as it does to each of the individual magnetic vectors. Hopefully this exercise will give you some confidence in the validity of the vector model, which ignores the individual nuclear spins and deals only with the net magnetization vector M. Pulse rotations are applied to the net magnetization vector and we do not need to worry about the complex behavior of the individual spins any more. But the model of the individual nuclear magnetic vectors combining to form the net magnetization is very important to keep in your head because it gives us an understanding of the nature of phase coherence. We will see when we look at NMR relaxation how the individual behavior of spins contributes to the loss of coherence over time.

The simplest NMR experiment is just to apply a 90° pulse to the sample and then record the FID signal. A pulse of high power and short duration radio frequency energy at the Larmor frequency will have the effect of organizing the individual precessing spins into a coherent, in-phase motion so that at any instant in time all 25 of the "net" spins are oriented in the same direction with respect to the x and y axes. Now, instead of canceling out (random phase), the x and y components of the individual spins add together (phase coherence) to form a net magnetization vector in the x-y plane that rotates at the Larmor frequency (Fig. 5.6). Because at equilibrium each of the N/2 - 5 spins in the higher energy state was directly opposed to one of the spins in the lower energy state, only the 25 spins representing the equilibrium population difference, AP, contribute to this vector. This rotating magnetic vector induces a sinusoidal voltage in the probe coil of the spectrometer that can be amplified and detected to give a free induction decay (FID). Fourier transformation of the FID gives a frequency domain spectrum with a single peak at the Larmor frequency.

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