A Step-by-step Picture of Pulsed (Time-Domain) NMR Leslie J. Schwarlz St. John Fisher College, Rochester, NY 14618
Oneof themostchallengingaspecrsof teaching the principles of pulsed NMH at the undergraduate level is that the rieorous theow is not easilv without distortion. - simnlified . An explanation in terms of quantum mechanical mathematics' is not onlv bevond the scoDe of the undereraduate (and often graduate) ihemistry cu~riculum,but 2 also lacks a "nicture" that can he used to assimilate quicklv new ideas and concepts. "Arrow diagrams"'" are a t more appropriate level for the beginning student hut are sometimes presented so sketchily that for the really probing and thoughtful student they can create more confusion than they eliminate. The goal of this paper is a description of the pulsed NMR exoeriment that is as simnle and ~ictorialas ~ossihle.but that stops short of dangerous and oveisimplifications that can lead to contradictions. Elements of this description can he found in the references1-=; this treatment weaves them together in an es~eciallvcareful way. The description is suitable for the junior o; senior undergraduate student and assumes no previous knowledge of quantum mechanics
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The physical quantity underlying NMR is nuclear spin. S ~ i is n a fundamental auantitv. like charge and mass. and hence cannot be definedin terms of more ktuitively understood quantities. Instead, we"understand"spin analogouslv to the &ay in which we understand charge andmass: th;ough its behavior in a force field. Consider charge. We say that a particle is charged if, in the presence of an electric field, the particle is accelerated. Charge comes in twovarieties: if the particle moves toward a negative electric field source, we say that the charge is positive; otherwise, i t is negative. We can compute the magnitude of the charge by measuring the acceleration of the particle due to the field. Finally, a charged particle itself creates its own electric field. Notice that this description never says what charge is, hut only how i t behaves. A similar description can be for massin terms of the acceleration of a massive particle in a gravitational field. The relevant field for describine nuclear snin behavior is a magnetic field. This paper will-restrict it'self to spin-112 nuclei. such as 'H.13C.31P.or IgF. In the nresence of a static magnetic field, thkpaiticle's spin, which fs avector quantity, will be found to be oriented at one of two definite 8 angles" with respect to the field direction as shown in Figure 1. (These are the two "varieties" of win.) The two orientations represent different energy states of the particla. This is because a particle with spin creates its own small magnetic field in the direction of its spin vector (again in analogy to the electrical case), which can be acted upon by an external field. The more parallel the two fields are, the lower their interaction energy. Thus, the "partially aligned" a state in Figure 1 has lower energy than the "partially opposed" 0 state. In an NMR exneriment. we are necessarilv dealine with many spins simultaneously, and the response that we monitor is thusanaverageofthe behavior of individual spins.The
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Figure 1. (left)The two observableSpln orlentetions, nand 8, in the presence of amagneticfield/& inthe rdlrection. The polar angle 8 can be 55' or 125'. The azimuthal angle @ is arbitrary. Figure 2. (right) The lnsamaneour precessional motion about hb of a spin packet. @ changss at a rate which is proportional to the magnitude of hb: 0 ~ l o ~decreases ly to zero as thermal equilibrium is approached.
averaging process by which one predicts the behavior of a collection of s ~ i n from s the behavior of individual soins is not at all intuitive. A rigorous calculation' shows &at depending on the relative number of spins in the two energy states of Figure 1, and depending on the phase distribution of the spins about the z axis (i.e., the way that the spins are distributed over the angle a ) , the auerage spin vector of the system can be found a t any values of and 8. (The averaee spin vector is the quantum~mechanicalvector sum [expecGtion value] of the individual spin vectors, and will henceforth be referred to as the net magnetization vector because the magnetic field that i t produces is the induced magnetization of the sample.) To reneat. . . while individual s ~ i n scan be observed a i either of only two possible 8 valies, the net magnetization vector can be found oriented a t any value of 8. Furthermore, the net magnetization vector is time dependent-it precesses about the external field with frequency determined by the magnitude of the external field (Fig. 2). The external field in an NMR experiment is the vector sum of the static field, called Ho,and small "local" molecular fields due to electrons and nuclei in the vicinity of the nuclear spin of interest. The local fields are so much smaller than HQ, however, that it is a good approximation to say that the net magnetization vector precesses about HQ, with precession frequency (in radiansls) given by wo-r.H, (1) where y is the magnetogyric ratio, which is a constant for a given type of nucleus, and wo is called the Larmor frequency. Editor's Note
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This article was orieinallv in the Seotember is. nuhlished . sue: however,rhetext uas not printed in properorderand the actual srquence could nor br easily deduced. As a servicr to our readers, we are republishing the compi~trarticle in rorrect order and apologize for any confusion that may have
Volume 65
Number 11 November 1988
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In addition to precession, the net magnetization vector also slowly approaches the external field direction (the rate of approach is controlled by TI relaxation processes to be discussed later.) The 0 variation is slow compared to the @ variation, however, and can be neglected a t this stage of the description. If there is more than one type of spin in the molecule (for nroton soins). examnle.. -OH and -CHI-. . .. thev. will exoerience different local fields, and hence their average spin vectors will precess about Ho at slightly different rates. These individual average spin vectors are called spin packets, and the overall net maenetization vector is well described as a simple vector sum of the spin packets. The goal of all NMR experiments, pulsed or continuous wave (cw), is to determine or to monitor the local molecular magnetic fields with the aim of either relating them to the structure of the molecule. or else usine them as "handles" with which tomonitor aphysical process. The local fields can be determined from the slight differences between the precession frequencies of the different spin packets. Unfortunatelv. at thermal eauilihrium, the precessional motions of the spin packets are undetectahle~because(1) there is a slightly larger number of spins in the lower energy orientation than in the higher energy orientation of Figure l,and (2) there is no phase "clustering" of the spins in any particular @ direction so that the soins can be considered to be evenlv distributed on the surfaces of two cones oriented in the plus and minus z directions in Fieure 1.These two conditions1" lead to a net magnetization;ector that is stationary along the Hn (0 = 0)direction. In order to extract the precession frequencies that are "buried" in the equilibrium net magnetization vector, the vector needs to he perturbed in such a way that it acquires a nonzero value of 0. The pulsed NMR experiment consists of the application of o n e o r more pulses of electromagnetic radiation, which causes the net magnetization vector to nutate (tip) away from the z axis. The net magnetizationvector is then monitored as it returns to equilibrium via the @ precession and slow 0 decrease of its component spin packets, during which time the precession frequencies are exposed. Plane polarized electromagnetic radiation consists of electricand magnetic fields that oscillate in fixed planes that are perpendicular to each other and to the direction of propagation of the radiation. Nuclear spins interact with the magnetic field component of the radiation, and so in the following description the electric field is omitted. Figure 3a shows the usual orientations of the static magnetic Ho field and the oscillating magnetic field of the electromagnetic radiation, called HI. As pictured, the H I field is growing and shrinking along the positive and negative x axis. Alternatively, the oscillating HI vector can be pictured as the suoernosition of two fixed length vectors which rotate in the . . X-3 plane, one rlorkwise and one counterclockwise, as shown in Figure 3b. The magnitude of earh of the two components is.one-half . --- ~ ~ of ~that ~ of- the ~ 11, vector in Fieure .la, and their rotation frequency is equal to the frequency of the electromagnetic radiation. This latter picture will be seen to be convenient because only one of the two rotating H I components has a large effect on the magnetization vector, and the other will therefore be neglected. An electromagnetic pulse is created by rapidly turning the source of electromagnetic radiation on and off; the length of the pulse is the time for which the radiation is on. The effect of the pulse is most easily described from the point of view of a reference frame that rotates about the laboratory z axis direction. The laboratory and rotating reference frames are shown in Figure 4; the axes of the former are labelled x , y, and z. while those of the latter are x', Y', and z. T o illustrnte the use of a rotating frame, consider a single nvnequilibrium spin packet instantaneously precessing
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Figure 3. (a) The relative directions of lha static hb and oscillating H,fields in an NMR spectrometer.(b) The oscillating H, fieldredrawn as the sum of two fixad-length components that rotate in opposite directions in lha x-y plane.
Figure 4. Superimposed stationary laboratw frame (x. y, zl rotating frame (x', y'. 4.
and precessing
about Ho a t a rate determined hy the magnitude of Ho (and that of local fields, which are neglected in the following description). If the rotating frame has the same rotation frequency as the spin packet, then the packet will appear stationary in that frame. If the rotating frame precesses at a slightly different frequency than that of the spin parket, then the parket will rontinue toappear to precesseven in the rotating frame, hut its precession frequency will be the difference hetween its laboratory precession frequency and the frequency of the rotating frame. The closer the rotating frame frequency is to the spin packet precession frequency, the slower the spin packet appears to precess in the rotating frame, as if Ho in the rotating frame has been replaced by a much weaker "effective Ho". The precession frequency of a spin packet about the z axis in the rotating frame is thus not simply determined by the magnitude of the laboratory Ho field (plus any local fields) but, in addition, is dependent on the rotating frame frequency; also, it is possible for spin packets to be a t nonzero 0 in the rotating frame and still be stationary, as if the precessional effect of the Ha field has been comnletelv n ~ l l i f i e d . ~ Back td the eiectromagnetic pulse. How do the two rotating HI component vectors of Figure 3b look when viewed - -
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' Slichter. C. P. Principles of Magnetic Resonance; Springer-Ver-
lag: New York. 1980. Farrar, T. C.: Becker, E. D. Pulse and Fourier Transform NMRIntroduction to TheoryandMethods; Academic: New York, 1971. Fukushima, E.; Roeder, S. 6. W. Experimental Pulse NMR-A Nuts and BolfsApproach; Addison-Wesley: Reading. MA, 1981. 4Rabenstein,D. L. J. Chem. Educ. 1984, 61,909-913. 5Macomber, R. S. J. Chem. Educ. 1985, 62, 213-214. A particle with spin I generally has 21 1 possible spin orientations in a static magnetic field. ' A rotating reference frame is a noninertiai frame of reference; therefore, it makes sense that laws of physics are altered within it. (For example, the precessional effect of H, can be nullified.)A more familiar noninertial reference frame is an elevator accelerating downwards at 9.8 mls2, causing people inside to feel as if in free faligravity is nullified in that frame. The H, field still has an important effectin the rotating frame in that it specifies the equilibrium orientation of the spin packets in the absence of other fields.
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Figure 5. (a) The effect of a *I2 pulse. (b) The effect of a s pulse.
from a rotating reference frame whose rotation frequency is equal to the radiation frequency? The H I component that in the laboratory frame rotates in the same direction as the rotating frame will amear stationarv in the rotatine frame. while tLe c o ~ n t e r r o i a i i ncomponent ~ will appear rotate twice as fast in the rotatinn frame as in the laboratory frame. Now put the whole picture together: An electromagnetic pulse of frequency wois applied to a single spin packet that is in thermal equilihrium in a static Ho field. From the viewpoint of a frame rotating about the z axis a t frequency wo, there appears to he a static H I field along the x' axis and a rotating HI field in the x'-y' plane of frequency 2w0 that alternately "pushes" and "pulls" the spin packet in opposite directions, thereby having no net effect. If wo is also chosen to he equal to the precession frequency of the spin packet, then the precessional effect of the Ho field in the rotating frame will he completely nullified, as descrihed earlier. The final result is that the spin packet "sees" only the static H I field along the x' axis and thus begins to precess about that axis at a frequency determined by the magnitude of HI (plus any local fields), as shown in Figure 5a. In analogy with eq 1, the precession frequency of the spin packet about the H I direction is w,ay.H,
(2)
Note that HI
