A Step-by-step Picture of Pulsed (Time-Domain) NMR Leslie J. Schwartz St. John Fisher College. Rochester. NY 14618 One of the most challenging aspects of teaching the principles of pulsed NMR a t the undergraduate level is that the rieorous theorv is not easily simplified without distortion. A; explanation in terms of &ant& mechanical mathematics' is not only beyond the scope of the undergraduate (and often eraduate) chemistrv curriculum. hut it also lacks a ''picture" that can be use2 to assimilate quickly new ideas and conceDts. "Arrow diae~ams"'-~are at a more appropri.. . ate level for the beginning student but are sometimes presented yo sketchilv that for the rrallv ~ r o b i n pand thourhtful student they can create more co&sion than they eliminate. The goal of this paper is a description of the pulsed NMR experiment that is as simple and pictorial as possible, hut that stops short of dangerous gen&alizations and oversimplifications that can lead to contradictions. Elements of this description can be found in the reference^'^; this treatment weaves them together in an especially careful way. The description is suitable for the junior or senior undergraduate student and assumes no previous knowledge of quantum mechanics. WO~Y.HO (1) where y is the magnetogyric ratio, which is a constant for a given type of nucleus, and wois called the Larmor frequency. In addition to precession, the net magnetization-vector also slowly approaches the external field direction (the rate of approachis controlled by TI relaxation processes to be discussed later.) The 8 variation is slow compared to the Q variation, however, and can be neglected at this stage of the description. If there is more than one type of spin in the molecule (for example, -OH and -CH2 proton spins), they will experience different local fields, and hence their average spin vectors will precess ahout HOa t slightly different rates. These individual average spin vectors are called spin packets, and the overall net magnetizationvector is well described as asimple 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 maenetic fields with the aim of either relatine" them to the structure of the molecule, or else using them as "handles" with which to monitor a physical process. The local fields can be determined from the slight differences between the precession frequencies of the different spin packets. Unfortunately, a t thermal equilibrium, the precessional motions of the spin packets are undetectable 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 Q direction so that the spins can be considered to he evenly distributed on the surfaces of two cones oriented in the plus and minus z directions in Figure 1. These two conditionsl-5 lead to a net magnetization vector that is stationary along the Ho (8 = 0 ) direction. In order to extract the precession frequencies that are "buried" in the equilibrium net magnetization vector, the vector needs to be perturbed in such a wav that it acauires a nonzero value of 8 . - ~ h pulsed e NMR experiment consists of the application of one or more pulses of electromametic radiatibn, which causes the net magnetization vector i o nu-
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Journal of Chemical Education
Figure 1. (left)The two observable spin orienlations, a and 8, in the presence ola magnetic field hb in the zdirectlon. The polar angie 0 can be 55' or 125'. ma azimuthal angie Q is arbilrary.
Fiaure 2. IrioMI The instantaneous ~recessionaimotion about K of a win v packet Q cnsnges st a rate whlch is propomma to the magnotude of h. 0 slow y decreases lo zero as tnermsi equl brlLm s approacned
tate (tip) away from the z axis. The net magnetization vector is then monitored as it returns to equilibrium via the Q precession and slow 0 decrease of its component spin packets, during which time the precession frequencies are exposed. Theory
The ~hvsicalauantitv underlvine NMR is nuclear spin. Spin is-a fundamental &antity;li& charge and mass, and hence cannot he defined in terms of more intuitively understood quantities. Instead, we "understand" spin analogously to the way in which we understand charge . and mass: through its behavior in a force field. Consider charge. We say that aparticle is charged if, in the nresence of an electric field. the article is accelerated. bharge comes in two varieties:'if the 'particle moves toward a neeative electric field source. we sav that the charee - is DO& . tive; otherwise, it 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 it behaves. A similar description can heeiven for mass in terms of the acceleration of a missive partide in a gravitational field. The relevant field for describing nuclear spin behavior is a magnetic field. This paper will restrict itself to spin-112 nuclei, such as 'H, '3C, 3IP, or '9F. In the presence of a static magnetic field, the particle's spin, which is a vector quantity, will be found to be oriented a t one of two definite 0 angles6 with respect to the field direction as shown in Figure 1.
'
Slichter, C. P. Principles of Magnetic Resonance; Springer-Verlag: New York. 1980. Farrar, T. C.: Becker, E. D. Pulse and Fourier Transform NMRintroduction to Theoiy and Methods: Academic: New York, 1971. Fukushirna. E.; Roeder, S. B. W. Experimental Pulse NMR-A Nuts and Bolts Approach: Addison-Wesley: Reading, MA. 1981. Rabenstein, D. L. J. Chem. Educ. 1984, 61, 909-913. Macomber, R. S. J. Chem. Educ. 1985, 62,213-214. A particle with spin lgeneraily has 21 1 possible spin orientations in a static magnetic field.
+
(These are the two"varieties" of spin.) The two orientations represent different energy states of the particle. This is bec a k e 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" cr state in Figure 1 has lower energy than the "partially opposed" 19 state. In an NMR experiment, we are necessarily dealing with many spins simultaneously, and the response that we monitor is thus an average of the behavior of individual spins. The averaging process by which one predicts the behavior of a collection of spins from the behavior of individual spins is not a t all intuitive. A rigorous calculation' shows that depending on the relative number of spins in the two energy states of Figure 1,and depending on the phase distribution of the spiniabout the z axis (i.e., the way that the spins are distributed over the angle @),theaverage spin vector of the svstem can be found a t anv values of and 8. (The average spin vector is the quantum mechanical vector sum [expectationvaluel oftheindividualspinvectors, and will henceforth be referreh to as the net magnetization vector because the magnetic field that i t produces is the induced magnetization of the sample.) T o repeat, while individual spins can be observed at either of only two possible 8 values, the net magnetization \.ertor can he found oriented at 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 Ha, 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 Ho, however, that i t is a good approximation to say that the net magnetization vector precesses about Ho, with precession frequency (in radiansls) given by Plane polarized electroma&etic radiation consists of electric and 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 superposition of two fixed length vectors which rotate in the x-y plane, one clockwise and one counterclockwise, as shown in Figure 3h. The magnitude of each of the two components is one-half of that of the H, vector in Figure 3a, and their rotation frequency is equal to the frequency of the electromagnetic radiation. This latter picture will be seen to be co&enient hecause only one of the two rotating H I romponents has a larae effect on the magnetization vector, and the other will therefore be neglected.. An electromagnetic pulse is created by rapidly turning the source of electromaenetic radiation on and off: the leneth of the pulse is the timFfor which the radiation is bn. he effect of the pulse is most easilv described from the point of view of a reference frame that rotates about the laboratory t axis direction. The laboratorv and r o t ~ t i n areference frames are shown in Figure 4; the &es of the f o k e r are labelled x, y , and z,while those of the latter are x', y', and z. T o illustrate the use of a rotating frame, consider a single nonequilibrium spin packet instantaneously precessing about Ho at a rate determined by 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
Figure 3. (a)The relative directions of the static HOand oscillating H, fieldsin field redrawn as the sum of two fixed-length components that rotate in opposite directions in the x-%plane.
an NMR spectrometer. (b) The osclilating H,
*
Figure 4. Superimposed stationary laboratmy frame (x, y. 2) and precessing rotstlng frame (x', y', I).
stationary in that frame. If the rotating frame precesses a t a slightly different frequency than that of the spin packet, then the packet willcontinue to appear to precess even in the rotating frame, but its precession frequency will he the difference between 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 soin packet amears to Drecess in the rotating frame, as if Ho Ln t c e rotating frame (as been replaced by much weaker "effective Hn". The s recession freauencv . . of a spin packet about the z ax& in the iotating 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 at nonzero 8 i n t h e rotating frame and stili be stationary, as if the precessional effect of the Ho field has been completely nullified.7 Back to the electromagnetic pulse. How do the two rotating HI component vectors of Figure 3b look when viewed 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 appear stationary in the rotating frame, while the counterrotating component will appear to rotate twice as fast in the rotatingframe as in the laboratory frame. Now put the whole picture together: An electromagnetic pulseof frequency woisapplied tlgasingle spin packet that is in thermal equilibrium in a static Hn tield. From the viewpoint of a frame rotating about the 2 axis at frequency wo, there appears to be a static H1 field along the x' axis and a rotating HI field in the xl-y' plane of frequency 2wo that alternately "pushes" and "pulls" the spin packet in opposite directions, thereby having no net effect. If wo is also chosen
a
' A rotating reference frame is a noninertial 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 m/s2, causing people inside to feel as if in free failoravitv "~ . is nullified in that frame. The H, field still has an imoortant effect in the rorating frame n that t s~efifiesthe eqL iibrsum orientaton of the spin packets in the aosence of other fie ds.
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Number 9
Se~tember1988
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Figve 5. (a) The effectof a u / 2 puise. (b) The effect of a r puise
to be eaual to the nrecession freauencv of the soin nacket. then the precessio~aleffect of t6e field in ihe iotatin; frame will be completely nullified, as described earlier. The final result is that the spin packet "sees" only the static HI 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 HI direction is (2) w,-r.H~ Note that H I
