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### Figure 5.6

dies out with time due to inhomogeneity of the magnetic field and small differences in the local magnetic field experienced by each of the spins, largely due to the presence of nearby nuclear magnets in the tumbling molecule. The individual vectors "fan out" around the cone, and eventually they are randomly oriented again and the net magnetization in the x and j directions is zero (Fig. 5.7, A^B^C^D). The magnitude of this x-y component of the net magnetization vector will decrease exponentially toward zero as the individual magnetic vectors "fan out" over the cone and lose phase coherence (Fig. 5.7, bottom right: Mxy). We can represent this magnetization as an exponential decay with time constant T2 (Fig. 5.7, bottom left: T2 = 0.4 s, Vo = 2 Hz):

My = —Mocos(2nvo t )e-t/T2 Mx = Mo sin(2nVot)e-t/T2

The cosine and sine functions represent the rotation of the net magnetization vector: it starts on the -y' axis (—cos(0) = -1, sin (0) = 0) and moves toward the +x' axis. After 1/4 counterclockwise rotation (vot = 1/4) we have —cos(900) = 0, sin(900) = 1. The decaying exponential function e—t/T2 represents the fanning out of individual vectors and loss of  coherence. We can define a "half-life" for the process just as you might for radioactive decay:

0.5 = e-t/T2; ln(0.5) = ln[e-t/T2] = -t/Tz; ln2 = t/Tr, ti/2 = (ln2) T2 = 0.69372

So after the period of time 0.693 T2 (0.277 s for T2 = 0.4 s) we would see the coherence reduced to 1/2 of its original value; after twice that time (1.386 T2 = 0.554 s) we would see it reduced to 1/4 of its original value, and after three times the half-life (2.079 T2 = 0.832 s) we would see the net magnetization in the x-y plane reduced to one eighth of its original value. T2 (tea-two; the T is always upper case) is called the transverse relaxation time or spin-spin relaxation time, and after this amount of time (T2) the coherence has decayed to 36.8% (e-1) of its original value right after the 90° pulse. After twice T2 we have 13.5% (e-2) of the coherence left, and after three times T2 we have only 5.0% (e-3) left. Sometimes it is more convenient to talk about the rate of loss of coherence, R2 = 1/T2, which is in units of s-1 (Hz) instead of seconds. This is just like a rate constant in chemical kinetics. If you are familiar with exponential decay (e.g., radioactive decay) or, more generally, firstorder processes (e.g., heat flow or first order chemical reactions), you will have no trouble understanding NMR relaxation.

At the same time that the individual spins are losing their phase coherence, some of the spins in the higher energy fi state are "dropping down" to the a state (Fig. 5.8) as the system moves back to thermal equilibrium (the Boltzmann distribution). In Figure 5.8, we start with equal populations (Pa = Pfi = 12) right after the 90° pulse (Fig. 5.8, left). After a time corresponding to one "half-life" (0.693 times T1), 2 spins have dropped down, increasing the population difference to AP = 4 (Pa = 14, Pp = 10). Eventually a total of S spins will move from the fi state to the a state, decreasing the fi state population from N/2 to N/2 - S and increasing the a state population from N/2 to N/2 + S (Fig. 5.8, right: S = 4 and N/2 = 12). The reestablishment of the Boltzmann distribution between the spin states will cause a z component of the net magnetization to appear and grow toward the equilibrium magnitude Mo. This is shown in Fig. 5.7 in the graph at the bottom right for T1 = 0.5 s. Figure 5.8

The z magnetization (Mz) grows from zero to Mo with characteristic time T1 = 0.5 s whereas

the magnetization in the x-y plane (Mxy = [Mx2+Myz] ) decreases from Mo to zero with characteristic time T2 = 0.4 s. Note that in Figure 5.7 the magnitude (length) of the net magnetization vector (Mtot = [Mx2+My2+Mz2]1/2) drops initially because T2 is always shorter than T1; that is, the loss of Mxy is faster than the recovery of Mz. The "regrowth" of Mz is an exponential process, characterized by the function e-t/T1. The mathematical form of Mz is not quite as simple as the loss of transverse (x-y) magnetization because the z component of net magnetization is "growing back" from zero to Mo rather than decaying. What we can say is that the amount of "disequilibrium," defined by the difference between the z magnetization at any point in time, Mz, and the equilibrium value Mo is decaying exponentially:

This is true regardless of the extent or the nature of the perturbation away from the Boltzmann distribution (90o pulse, 180° pulse, saturation, etc.). The z magnetization will always move toward Mo in this way, so that the distance to equilibrium (AMz) is decaying exponentially. For the specific case of a 90° pulse, we can describe the z magnetization by an exponential function that approaches Mo with time constant T1:

AMz(t) = Mz - Mo = -Mo e-t/T1 Mz(t) = Mo - Mo e-t/T1 = Mo(1 - e-t/T1)

since AMz (t = 0) is equal to (0 - Mo) or -Mo. Note that Mz = Mo (1 - 1) = 0 at time zero (e-0 = 1), immediately after the pulse, and after a very long time the exponential term dies away to zero and we have Mz = Mo (Fig. 5.7, bottom right). The populations in the a and ft states as a function of time are shown in Figure 5.9. After the 90° pulse, the z magnetization grows from 0 to 63% of Mo after one T1, to 86% after two times T1, to 95% after three times T1, and to 99% of Mo after five times T1. Often 5T1 is used as a rule of thumb for a complete return to equilibrium—apparently 99% is "good enough for government work." Remember that the z component of net magnetization, Mz, is proportional to the difference in population AP = Pa - Pft, which grows as spins move from the ft state to the a state. The rate of spins "dropping down" from ft to a is driven by the amount of "disequilibrium" or the deviation from the Boltzmann distribution, so we see the rate get slower and slower as the populations get closer and closer to the equilibrium distribution. Figure 5.9

5.5.2 Relaxation After a 180° Pulse

At the end of a 180° pulse, the populations are inverted so that there is a slight excess (N/2 + 8) in the upper energy (ft) state and a slight deficit (N/2 — 8) in the lower energy (a) state (Fig. 5.10, Ti = 0.5 s). This is twice as far from equilibrium as the situation immediately after a 90° pulse (AP = —28 after a 180° pulse, 0 after a 90° pulse, and 28 at equilibrium). Spins drop down from the ft state to the a state, reducing the population in the ft state and increasing the population in the a state. After 8 spins have dropped down (time = 11/2 = 0.693 Ti = 0.35 s), we have reached the point where populations are equal in the two states (Mz = 0). This is half of the way to equilibrium. Spins continue to drop down until in all 28 spins have dropped down from ft to a, leaving a population of N/2 + 8 in the a state and N/2 - 8 in the ft state. In mathematical terms, Mz = —Mo at the end of the 180° pulse, so

AMz (t = 0) = Mz (t = 0) — Mo = —Mo — Mo = —2Mo

AMZ (t) = Mz (t) — Mo = AMz (t = 0)e—t/T1 = —2Moe—t/T1

Thus Mz starts at — Mo and after 0.693 T1 it equals zero (halfway to equilibrium from the starting point: Fig. 5.10). After two half-lives it equals 1/2 M o (3/4 of the way to equilibrium) and after 3 x 0.693 x T1 it equals 3/4 Mo. After a long time it equals Mo. It is important to recognize that this return to equilibrium, which moves the net magnetization vector from the —z axis to the +z axis after a 180° pulse, is not a rotation. This process of longitudinal or spin-lattice relaxation involves only the process of spins dropping down from the higher energy state to the lower energy state and therefore cannot create magnetization in the x'-y' plane (coherence). After a 180° pulse there is no coherence, so the net magnetization vector simply shrinks in magnitude along the — z axis until it reaches zero magnitude, and then grows along the +z axis until it reaches the magnitude Mo defined by thermal equilibrium. Time Figure 5.10

5.5.3 Spin Temperature

It is sometimes convenient to compare longitudinal (T1) relaxation to the process of heat flow (also a first-order process). When we apply a pulse to the spins, we "heat them up," and as they return to equilibrium they "cool down" again to the temperature of their surroundings. We can even define a "spin temperature" (Ts) as the temperature corresponding to a given population difference between the a and ft states:

where AE is the energy gap between the a and ft states. This is the same as the Boltzmann relationship, except that the population ratio can assume any value and not just the equilibrium ratio. In Figure 5.8, the equilibrium ratio at room temperature (300 K) was 8/16 = 0.5 or e-0 693. If we start to saturate the spins and the ratio increases to 9/15 = 0.714 or e-0 511, the spin temperature has increased to 407 K (407 = 300 x 0.693/0.511). Full saturation (Pft/Pa = 1) corresponds to an infinite spin temperature (e-AE^ = e-0 = 1) and inversion (180o pulse) corresponds to a negative spin temperature of -300 K (Pft/Pa = e-AE/(-T) = e+AE/T). These are not physically reasonable temperatures, but the concept is still useful that we are "heating up" the spins when we promote spins from the a state to the ft state. After we heat up the spins, T1 relaxation can be viewed as a flow of heat from the spin "container" to the "outside world" of the sample solution (sometimes called the "lattice," a term from solid-state NMR). The amount of energy is very small compared to average thermal energy of the molecules, so the sample temperature increases only very slightly, but the spin temperature goes down and approaches the sample temperature in an exponential manner.

### 5.5.4 T1 versus T2

Because T2 is always smaller than T1, the loss of magnetization in the x-y plane will always be faster than the re-establishment of magnetization along the z axis (Fig. 5.7, lower right). The equation for Mz is independent of the equations for Mx and My. This means we can deal with transverse relaxation (the T2 process) and longitudinal relaxation (the T1 process) as completely separate phenomena. Any transverse (x-y plane) component of the magnetization will undergo exponential decay with time constant T2 and any nonequilibrium longitudinal component (z axis) will approach Mo exponentially with time constant T1. It is Figure 5.11

important to realize that relaxation is not a rotation back to the z axis: the net magnetization vector only rotates in a plane perpendicular to the x-y plane during an RF pulse. In the absence of pulses, the x and y components decay toward zero with time constant T2 and the z component recovers toward Mo with the longer time constant T1. The combination of precession, loss of coherence (T2 relaxation) and the slower recovery of z magnetization (Ti relaxation) after a 90o pulse is illustrated in Figure 5.11. The tip of the net magnetization vector describes an inward spiral over the surface of a circular "tent" with a single pole in the center (+z axis). In the extreme case where T2 is much less than T1, the x and y components would decay to zero first, and then the z magnetization would "grow back" along the +z axis to the Mo value. Relaxation may seem like a troublesome side issue, sort of like friction in classical mechanics, but we will see later that cross relaxation, the process by which the relaxation of one spin influences the relaxation of a nearby spin, leads to the NOE, which is an important method for measuring distances within a molecule.

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