14 MAGNETIC PROPERTIES OF NUCLEI magnet tumed off 0.8 0.5 耳04 0.3 色02 0.1 time(units of T1) spin distribution (exaggerated) 5 up, 5 down 6 up, 4 down 5 up, 5 dowt Figure 2.6.(a)Graphical depiction of the approach of a collection of magnetic nuclei toward equilibrium when the external field is turned on(I=0,,.. 3T1), then turned off.(b) Exaggerated pictorial depiction of the spin state distribution at I=0, 3Tlao. resume. This fact presents us with a paradox. The spin system describe how to generate such an oscillating magnetic field. must absorb enough photons for us to be able to detect the but it should not surprise you to learn that it involves electro- signal instrumentally, but not so much as to cause saturation. magnetic radiation of the same frequency At any rate, a This brings us to the question( to be covered more fully in rather strange thing happens when this irradiation by B, Section 3. 1)of how an NMR signal is actually generated. begins: All of the individual nuclear magnetic moments be- Figure 2.7 depicts a collection of /= nuclei at equilibrium, come phase coherent. That is, they focus, tracking the oscil- in a magnetic field aligned with the +z axis.(By convention, lating magnetic field, and form a precessing"bundle"as the static magnetic field, Bo. always defines the z axis. ) Before shown in Figure 2.7c. Provided we have not saturated the irradiation begins(Figure 2. 7a), the nuclei in both spin states system by absorbing too may photons, this phase coherence are precessing with the characteristic frequency, but they are also requires that m tip away from the z axis and begin to z axis. The net nuclear magnetization M is the vector sum of precess around the z axis, again with the characteristic Larmor all the individual nuclear magnetic moments, and its magni- frequency. As such, M now has a component in the x, y plane tude is determined by the excess of up spins over down spins (My) oscillating with the same frequency. The flip angle a Here, M is aligned parallel to Bo; it has noprecessional motion that M makes with the z axis controls the magnitude of M, and no component in the x, y plane. by the relation Now suppose there were a way to produce another mag netic field perpendicular to Bo. However, this new field (B) Mxy=Msin a will be much weaker than Bo, and it will precess in the x,y plane, oscillating at exactly the same frequency as the nuclear The angle a is, in turn, determined by the power and duration magnetic moments(Figure 2.7b).(In Section 3. 1 we will of the irradiation by B. Ultimately, the actual NMR signal
2.4 THE ROTATING FRAME OF REFERENCE 15 Figure 2.7. Precession of a collection of /=i nuclei around external magnetic field Bo. Here. M represents the net nuclear magnetization, the vector sum of all the individual nuclear magnetic moments. (a) Before irradiation by B .(b)The orientation of the rotating magnetic field B. (c)During irradiation by B will be generated from the oscillation of M, v(Section 3. 1), observer in the lab In part(a)of the figure Bo and B, are off, so the maximum signal intensity will occur when a equals so the populations of individual up and down nuclear spins 90°(Why?) are equal, and the magnitude of M is zero. In(b)the equilib There is one other type of relaxation process that must be rium distribution of spins with Bo on has been achieved and mentioned at this point. After irradiation ceases and B, disap- M is aligned along the z axis, even though the individual pears, not only do the populations of the m=+) and m=- nuclear magnetic moments still precess around the z axis Part states revert to the boltzmann distribution but also the indi (c)shows M tipped to an a of 45 through its interaction with vidual nuclear magnetic moments begin to lose their phase B,, and the resulting precession of M describes a cone. Finally coherence and retum to a random arrangement around the z in( d)the flip angle is 90" so the precession of m describes a axis(Figure 2.7a). This latter process, called spin-spin(or disc in the x, y plane transverse)relaxation, causes decay of M, y at a rate control- Instead, suppose the x and y axes were themselves precess led by the spin-spin relaxation time T2. Normally, T2 is ing cl much shorter than T. A little thought should convince you the same frequency the nuclear spins are precessing. Further that if T2<T, then spin-spin(dephasing) relaxation takes suppose we, the observers, were precessing around the z axis place much faster than spin-lattice( Boltzmann distribution) at the same frequency To differentiate this rotating coordinate relaxation system from the fixed (i.e, laboratory frame)system, we will use labels x'., and z to represent the three rotating axes(the 2.4 THE ROTATING FRAME OF REFERENCE a' axis is coincident on and equivalent to the z axis). To us rotating observers, the rotating axes and B, appear stationary. requently in this book we will want to depict nuclear spin and M will rotate in the plane perpendicular to B. These orientations like those shown in Figure 2. 7. More often than relationships are shown in Figure 2.9 not, we will focus our attention on the net nuclear magnetic Clearly, viewing the motion of M in the rotating frame moment(M) rather than on the individual nuclear spins. simplifies our drawings. Still, it is important to remember this Because M will sometimes precess around Bo (i. e, the z about rotating-frame diagrams: Whenever M is anywhere axis), we need a more convenient way than the dashed ellipses except directly along the z axis, it has a component oscillating used so far to depict M as it precesses and changes orientation. in the x,y plane(laboratory frame), and this is what gives rise Henceforth we will use another convention to represent this to an NMR signal precessional motion of M, the rotating frame of reference which is designed to show the effects of B, on M EXAMPLE 2.10 (a)Look back at Figure 2.6. Draw a In Figure 2.8 are shown four representations of M in the laboratory frame diagram that shows M at old"way, the so-called laboratory frame of reference, the [=0, T 2T,..., 6T.()How would the diagram change normal x, y, z coordinate system as viewed by a stationary in the rotating frame?
16 MAGNETIC PROPERTIES OF NUCLEI Figure 28. Depiction of M in the laboratory frame: see text D Solution: (a)See Figure 2. 10.(b)Since M is at all time The result shown in Example 2. I I is very important. The aligned with the z axis. the diagram would look exactly rotating magnetic field B,(depicted initially in Figure 2. 7b) the same in the rotating frame 0 now appears stationary(here, along the x axis)in the rotating frame. The significance of this fact is shown in Figure 2.12 While the external field Bo tends to keep M aligned with the a EXAMPLE 2.11 Look back at Figure 2.7 Draw a ro- z(2)axis, B, causes M to precess around the x' axis, tilting tating frame diagram that shows how M changes orientation Min they, plane (The overall motion of M in the laboratory when B, is turned on long enough to give an a of 90, then frame would describe a complex spiral, but the component of turned off at t=0. Show the orientation of M initially, then at precession around Bo does not show up in the rotating frame. a=90, then at t values of T2, 272, 372, and oo. Note that BI Since B, is much weaker than Bo, the precession of M around is positioned along the x axis. (You may neglect the effects B, is much slower than its precession around B.The stronger of spin flips and longitudinal relaxation for the purposes of B, is and the longer it is on, the more M will precess around it, increasing the flip angle a. When B, is turned off. Relaxes exponentially back toward the z axis at a rate governed by O Solution: See Figure 2. 11. Note that M remains in the y, z plane. z z Figure 2.9. Axes of the rotating frame of reference, as viewed (a)by a stationary observer in the laboratory frame and(b)by an observer precessing in the rotating frame
4 THE ROTATING FRAME OF REFERENCE 17 Figure 2.10. Pictorial solution to Example 2. 10. Compare with Figure 2.6a B T2 2T B B, off Figure 2.11. Pictorial solution to Example 2.11 B a=180° Figure 2. 12. Precession of M around B, in the rotating frame
18 MAGNETIC PROPERTIES OF NUCLEI 2.5 RELAXATION MECHANISMS AND CORRELATION TIMES the nuclear spins once B, is turned off. This mechanism for spin-spin relaxation is usually the dominant one and gives rise to an effective spin-spin relaxation time known as Ti The complete microscopic details of how longitudinal(spin where T%<T2(the"natural"spin-spin relaxation time). All lattice, TD and transverse(spin-spin, T2) relaxation occur is three types of spin-spin relaxation are driven by the second beyond the scope of this book. But a little further discussion law of thermodynamics: In the absence of other forces, a might be profitable in order to provide us with at least a system will tend spontaneously to attain that arrangement qualitative understanding of the subject with maximum entropy( disorder) Look again at Figure 2.7c. What causes the individual up Thermodynamics also tells us that systems tend spontane and down nuclear spins in a"bundled "set of identical target ously toward equilibrium, which is characterized by a mint (observed) nuclei to randomize their phasing(defocus)after mum energy(or free energ, to be exact). One component of B, is turned off? One mechanism for spin-spin relaxation can free energy is the entropy mentioned above. Normally the be pictured as follows. Suppose that one of the up spins and dominant component of free energy is enthalpy (heat con one of the down spins instantaneously exchange energy In tent). Within a magnetic field, the equilibrium( Boltzmann this way, the up spin is converted to a higher energy down distribution of nuclear spins is the one with minimum en spin, and vice versa, with no net change in energy. However, thalpy(and maximum entropy ). Any other distribution will although the orientations(up or down) have exchanged, the have higher enthalpy(and free energy). For such a higher exact phasing has not, as shown in Figure 2. 13. Repetition of enthalpy distribution to relax back to equilibrium, it mus this process with other pairs of up/down spins will have the ssipate its excess energy to the surroundings. In the context ultimate effect of randomizing their phasing, driving Mx,y to of NMR, these surroundings(the lattice)comprise other zero and with it the NMR signal nearby nonidentical magnetic nuclei that can, but need not Spin-spin relaxation can also occur when other nearby necessarily, be part of the same molecule as the nuclei of oscillating magnetic or electric fields interfere with the exter- interest The lattice can also be regarded as an infinite heat nal field Bo, causing some of the nuclei to experience a (energy) sink to or from which energy can be transferred slightly augmented magnetic field while others experience a without changing its temperature htly diminished one. Those nuclei in the region of the The most important mechanism for spin-lattice relaxation augmented field will precess slightly faster, while those ex- involves a direct(through space)interaction between the periencing the diminished field will precess slightly slower magnetic dipole of a target nucleus and that of lattice nuclei ( Figure 2.14). This will result in the "fanning out of individ- Since lattice nuclei are undergoing constant periodic motion ual spin vectors again with no net energy change (e.g, rotation and translation), the local magnetic fields due As we will see in Section 3. 2, there is also a limit to the to their magnetic moments will also be oscillating at the same homogeneity of Bo itself. Even the finest magnets produce a frequencies. When the frequency of this motion is comparable field strength that varies ever so slightly around the region to the frequency of precession of the target nucleus(. g, 250 containing the sample. And this small range in field strength MHz), there can be a mutual spin flip But since these nuclei causes nuclei in one part of the sample to precess at very are nonidentical, there will be a net change in energy accom- slightly different frequencies, again leading to dephasing of panying the exchange. That is, energy will either be passed to Figure 2. 13. Result of one spin-spin exchange, shown in the laboratory frame. Compare with Figure 2.7c. Here. M is not shown