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Alternative schemes for double-modulation techniques in magnetic resonance
JOURNAL
OF MAGNETIC
RESONANCE
63,88-94
(1985)
Alternative Schemesfor Double-Modulation Techniques in Magnetic Resonance M. PERI& B. RAKVIN,
AND A. DUL~I~
Ruder Bos’koviC Institute, University of Zagreb, POB 1016, 41001 Zagreb, Croatia, Yugoslavia Received November 2. 1984 It is shown that two double-modulation schemes can be used for the detection of homogeneous spin-packet lines within an inhomogeneously broadened ESR line. 0 1965 Academic Press, Inc.
Homogeneous broadening of spin packets rehects the dynamics of the spin system, and, hence, is often the subject of investigation. In solid samples, the homogeneous broadening is usually unobservable directly from the resonance line since the distribution of static local fields gives rise to a much larger inhomogeneous broadening. The information on homogeneous broadening can be extracted, in such cases, by well known spin-echo techniques, both in NMR (I) and ESR (2). However, while pulsed NMR spectrometers for spin-echo experiments have become very common, pulsed ESR spectrometers are still rare due to a considerable technological sophistication and much larger costs than for a cw instrument. Recently, Rakvin el al. (3) have demonstrated that the cw technique could also be used to extract homogeneous spin-packet lines from an inhomogeneously broadened ESR line. The method was based on the use of two superimposed signals for the modulation of the dc magnetic field. The applicability of this technique has been verified by Rakvin (4) and by Pet-i& et al. (5) in the study of slow motions of nitroxide spin labels. The results obtained have shown that the double-modulation techniques could be considered to be competitive with the electron spin-echo technique employed previously in similar studies 1(2,6, 7). In the present paper, we report on an alternative experimental scheme for the double modulation of the dc magnetic field. We show, theoretically and experimentally, that homogeneous lines can be detected when a modulation signal with a fixed frequency is used for frequency modulation of a second modulation signal, and the frequency of the latter is swept over several multiples of the former. This differs from the formerly used double-modulation scheme which employed the superposition of two modulation signals (j-5). To present the theoretical treatment of the presently proposed scheme, we must give a brief account of the single-modulation scheme using the picture of sideband fields. The equivalence of magnetic field modulation and the frequency modulation of the rf field has been established using classical arguments (8, 9). This equivalence can be proved elegantly using a quantum-mechanical formalism. The Hamiltonian 0022-2364185 $3.00 Copyright Q 1985 by Academic Press. Inc. All rights of reproduction in any form reserved.
88
DOUBLE-MODULATION
89
SCHEMES
for a spin interacting with a modulated magnetic field (HO - H,,,cos w,t) and an rf field Hi rotating at angular frequency o, can be written (in frequency units) % = (cd0- 52 cos o,t)S, + $.q(S+e-iw’ + S-e’“‘)
111
where w. = yHo, !l = yH,,,, and wI = yH,. The Zeeman term can be made constant by performing a time-dependent unitary transformation using the operator U = exp -is, JL sin w,t . :I ( *m The effective Hamiltonian then becomes
PI
Zea = U&“U+ - iU t U’
= woSz + fo,k+exp[-i(wt
+ 2 sin co-t)] + S-exp[i(wt
+ % sin r&)]/
[3]
from which one can see that the rf field has become frequency modulated. The unitary transformation with U given in Eq. [2] can be viewed as a transformation into a frame which oscillates around the z axis with respect to the laboratory frame. The amplitude of the oscillation is Q/w, and the frequency of the oscillation is w,/2a. In such a frame the Zeeman levels are not modulated, but the rf field appears as frequency modulated. For the present purpose it is useful to analyze the interaction of the frequency modulated rf field with an inhomogeneously broadened ESR line. Figure 1 shows a monochromatic field at frequency LO,which, upon frequency modulation, is decomposed into sideband fields according to the well known mathematical identity
FIG. 1. Sideband field pattern of a frequency-modulated rf field. The sideband fields interact with the spin packets in an inhomogeneously broadened ESR line, and give rise to a dc and harmonic components in the signal.
90
PER@,
e;(~r
+ f3h
RAKVIN,
wmf) =
AND
5
DltJtiIt
Jn(p)ci(u
+ nwm)f
[41
n=-*
where j3 = Q/w, is the modulation index, and J, is a Bessel function. The lower part of Fig. 1 shows an inhomogeneously broadened ESR line which, according to Portis (IO) and Castner (II), is an envelope of homogeneous lines with a smooth distribution of Larmor frequencies. The tih sideband field gives rise to a dc signal through its interaction with the spin packet of the same Larmor frequency, i.e., (w + nw,). The same spin packet also experiences the interaction with the other sideband fields which are displaced from its resonance by +a,, *2w,, etc. Therefore, the response of the spin packet also contains harmonics of 0,. The total signal from an inhomogeneously broadened ESR hne is a convolution of the signals due to the spin packets at (w + nw,). The spin packets between these frequencies also experience the interaction with the sideband fields, but they do not give rise to a signal with harmonics of 0,. The above picture can be easily extended to the case of double modulation with two superimposed modulation signals (3-5). The Hamiltonian reads 2 = (wg - Q cos w,t - ~cos wkt)S, + $w,(S+P The time-dependent
+ S-e’“‘).
PI
unitary operator U = exp[-G&I
cos w,t + /I’cos w&t)]
WI
transforms the Hamiltonian [S] into an effective Hamiltonian with the constant Zeeman term, while the rf field becomes hequency modulated with two superimposed modulation signals. A repeated use of the mathematical identity [4] leads to
which shows that the doubly modulated rf field is equivalent to a sum of sideband fields whose strengths are given by the product of the Bessel functions, and whose frequencies are at (w + nw, + n’~;). Figure I! illustrates the principle of the decomposition of a monochromatic rf field into sideband fields following the consecutive introduction of the two modulation signals. The principle of the interaction of the sideband fields with the spin packets in an inhomogeneously broadened ESR line is the same as described above for the single-modulation case. The signal contains the dc component, the harmonics of each of the modulation frequencies, and their linear combinations. Each of these components can be detected by the proper choice of the reference signal for the lock-in amplifier. In the previous experiments (3-9, one used the fixed modulation frequency w, as the reference for the lock-in amplifier. Figure 2 shows that, in general, each sideband field is on resonance with one of the spin packets. The detected signal has a certain level. However, as the second modulation frequency oh, is swept, one reaches conditions n&II = n’w& PI when two sideband fields are on resonance with one and the same spin packet. The signal level then changes, and this is detected as the double-modulation spectrum. In particular for .’ = 1, the peaks occur at multiples of the fixed modulation
DOUBLE-MODULATION
SCHEMES
s.*... ..‘....i ..-...., ._.. ..I’.. . .. . I*.. -*
.~“‘i;&, 1
J,(P) !
91
J,(P)““‘.-...., .-.... . ... I 1
FIG.
2. Sideband field pattern of an rf field frequency modulated by two superimposed modulation signals. When one of the modulation frequencies is swept, the hvo indicated sideband fields may approach in frequency so as to interact with one and the same spin packet in the inhomogeneously broadened ESR line.
frequency 0,. The width of the double-modulation resonance is, evidently, given by the homogeneous spin-packet linewidth. The alternative double-modulation scheme, proposed in the present paper, is based on frequency modulation rather than superposition of the two modulation signals. The signal of fixed frequency w, is used for frequency modulation of the second modulation signal, whose frequency wh, is swept in the course of the experiment. The Hamiltonian for this case is given by &” = [w. - SZ’cos(w’,t + 0 sin w,t)]&
+ $Q(S+~+“’ + S-e’“‘).
t91
The modulation of the Zeeman interaction in Eq. [9] is much more complicated than the simple superposition of two modulation signals in Eq. [5]. Again, we look for the time-dependent unitary transformation which Iyields an effective Hamiltonian with the constant Zeeman term. The required transformation operator is U = exp[-iS,
sin(w’m + nw In)t] . 2 ““(‘) n w:, + nw,
The rf field in the transformed frame appears to be frequency modulated by a superposition of a multitude of modulation signals given by I@. [lo]. It can be expressed as exp
Q’JnW
wt + C sin(wh + nw,)t n w& + nw,
. [I l]
92
PERIC, RAKVIN,
AND DIJtiIc
The last line in Eq. [ 1 l] contains a product of sums, which is a very complicated expression. If one wants to obtain the sideband field pattern in analogy to the righthand side of Eq. [7], one has to work out the numerous products. For the present purpose, it is essential to prove that, as the swept frequency WA becomes a multiple of the fixed frequency w,, i.e., , = Cd, kam [=I a number of sideband fields coincide and interact with one and the same spin packet, which is the condition for the observation of a peak in the doublemodulation spectrum. First, one has to note that each sum in Eq. [ 1 l] contributes one of its terms as a factor in the expression which represents a given sideband field. Let us single out two factors in this expression
- + .exp{i[o
+. * 0+ r(oh, + pw,) + * * * -t.+h,
+ qu,) + * * *It}.
1131
The sideband field given by Eq. [ 131 contains the rth term from the pth sum and the sth term from the qth sum. The dots in Eq. [ 131 stand for the terms from the other sums. Leaving the latter unchanged and taking other choices of terms from the pth and qth sums, one obtains different sideband fields. In general, these sideband fields will have different frequencies. However, if condition [ 121 is fulfilled, all the sideband fields with r’ = r + (k + q)Z,
I1
400 I
I
I,
s’ = s - (k + p)l
I1
1
[I41
450
kHz
FIG. 3. Double-modulation spectrum of irradiated quartz. The modulation signal was itself frequency modulated by another modulation signal. The frequency of the former was swept while the second modulation signal had a fixed frequency.
DOUBLE-MODULATION
93
SCHEMES
where 1 = 0, ?I, +2, etc., will have the same frequency. This can easily be verified by replacing r and s in the exponent of Eq. [ 131 by r’ and s’, respectively, and using the condition [ 121. Hence, Eq. [ 121 gives, indeed., the condition for doublemodulation resonance. The diagram of the sideband field pattern is much more complicated than the one presented in Fig. 2 for the superposition of two modulation signals, and we shall not attempt to present it here. Figure 3 shows a double-modulation spectrum obtained using a modulation signal which itself is frequency modulated by another modulation signal. The sample was irradiated quartz, i.e., the same that had been used for the original demonstration of the double-modulation technique (3). Figure 3 shows clearly that the presently proposed double-modulation scheme can be considered as an alternative to the scheme using two superimposed modulation signals. From the experimental point of view, the instrumentation required for the two modulation schemes is the same. The only difference is in the modulation coils. For the first scheme, using two superimposed modulation signals, one may use a single pair of modulation coils or two independent pairs of coils, one for each of the modulation signals. In the latter case, the modulation coils for the fixed frequency signal can be a part of a tuned circuit, while the other pair of coils has to be in a broadbanded circuit to allow for the frequency sweep of the second modulation signal. In the second modulation scheme. Using a modulation signal which itself is frequency modulated by another modulation signal, one has to use a single pair of modulation coils. Obviously, when a single pair of modulation coils is used, one has to have a broadbanded circuit. The block diagram of the experimental setup for the presently proposed double-modulation scheme is shown in Fig. 4. So far, we have not noted significant advantages of one modulation scheme over the other.
reference
signal
af signal
lock - in amplifier
microwave bridge
1 Y-axis (signal) rf
X-Y recorder
X-axis (swept
Ir
sample
frequency) frequency
oscillator (fixed freq.)
signal
rlat
gWa,
oscillator (swept freq .)
FIG. 4. The block diagram of the experimental setup for the double-modulation modulation signal is itself frequency modulated by another modulation signal.
[\Jj modulation coils scheme in which the
94
PER&,
RAKVIN,
AND DUtiIc
REFERENCES 1. T. C. FARRAR AND E. D. BECKER, “Pulse and Fourier Transform NMR,” Academic Press, New York, 1971. 2. L. KEVAN AND R. N. SCHWARTZ (Eds.), “Time Domain Electron Spin Resonance,” WileyInterscience, New York, 1979. 3. B. RAKVIN, T. ISLAM, AND I. MIYAGAWA, Phys. Rev. L&t. 50, 13 13 (1983). 4. B. RAKVIN, Chem. Phys. Z&l. 109, 280 (1984). 5. M. PERK?,B. RAKVIN, AND A. DuL(SIC, J. Chem. Phys. 82, 1079 (1985). 6. L. J. SCHWARTZ,A. E. STILLMAN, AND J. H. FREED, J. CIyem. Phys. 77, 5410 (1982). 7. J. P. HORNAK AND J. H. FREED, Chem. Phys. Z&t. 101, I il5 (1983). 8. 0. HAWORTH AND R. E. RICHARDS, in “Progress in Nuclear Magnetic Resonance Spectroscopy” (J. W. Em&y, J. Freeney, and L. H. SutclilTe, Ed%), Vol. 1, Chap. 1, Pergamon, London, 1966. 9. A. Dutid AND B. RAKVIN, J. Magn. Reson. 52, 323 (1983). 10. A. M. PORTIS, Phys. Rev. 91, 1071 (1953). Il. T. G. CASTNER, Phys. Rev. 115, 1506 (1959).
OF MAGNETIC
RESONANCE
63,88-94
(1985)
Alternative Schemesfor Double-Modulation Techniques in Magnetic Resonance M. PERI& B. RAKVIN,
AND A. DUL~I~
Ruder Bos’koviC Institute, University of Zagreb, POB 1016, 41001 Zagreb, Croatia, Yugoslavia Received November 2. 1984 It is shown that two double-modulation schemes can be used for the detection of homogeneous spin-packet lines within an inhomogeneously broadened ESR line. 0 1965 Academic Press, Inc.
Homogeneous broadening of spin packets rehects the dynamics of the spin system, and, hence, is often the subject of investigation. In solid samples, the homogeneous broadening is usually unobservable directly from the resonance line since the distribution of static local fields gives rise to a much larger inhomogeneous broadening. The information on homogeneous broadening can be extracted, in such cases, by well known spin-echo techniques, both in NMR (I) and ESR (2). However, while pulsed NMR spectrometers for spin-echo experiments have become very common, pulsed ESR spectrometers are still rare due to a considerable technological sophistication and much larger costs than for a cw instrument. Recently, Rakvin el al. (3) have demonstrated that the cw technique could also be used to extract homogeneous spin-packet lines from an inhomogeneously broadened ESR line. The method was based on the use of two superimposed signals for the modulation of the dc magnetic field. The applicability of this technique has been verified by Rakvin (4) and by Pet-i& et al. (5) in the study of slow motions of nitroxide spin labels. The results obtained have shown that the double-modulation techniques could be considered to be competitive with the electron spin-echo technique employed previously in similar studies 1(2,6, 7). In the present paper, we report on an alternative experimental scheme for the double modulation of the dc magnetic field. We show, theoretically and experimentally, that homogeneous lines can be detected when a modulation signal with a fixed frequency is used for frequency modulation of a second modulation signal, and the frequency of the latter is swept over several multiples of the former. This differs from the formerly used double-modulation scheme which employed the superposition of two modulation signals (j-5). To present the theoretical treatment of the presently proposed scheme, we must give a brief account of the single-modulation scheme using the picture of sideband fields. The equivalence of magnetic field modulation and the frequency modulation of the rf field has been established using classical arguments (8, 9). This equivalence can be proved elegantly using a quantum-mechanical formalism. The Hamiltonian 0022-2364185 $3.00 Copyright Q 1985 by Academic Press. Inc. All rights of reproduction in any form reserved.
88
DOUBLE-MODULATION
89
SCHEMES
for a spin interacting with a modulated magnetic field (HO - H,,,cos w,t) and an rf field Hi rotating at angular frequency o, can be written (in frequency units) % = (cd0- 52 cos o,t)S, + $.q(S+e-iw’ + S-e’“‘)
111
where w. = yHo, !l = yH,,,, and wI = yH,. The Zeeman term can be made constant by performing a time-dependent unitary transformation using the operator U = exp -is, JL sin w,t . :I ( *m The effective Hamiltonian then becomes
PI
Zea = U&“U+ - iU t U’
= woSz + fo,k+exp[-i(wt
+ 2 sin co-t)] + S-exp[i(wt
+ % sin r&)]/
[3]
from which one can see that the rf field has become frequency modulated. The unitary transformation with U given in Eq. [2] can be viewed as a transformation into a frame which oscillates around the z axis with respect to the laboratory frame. The amplitude of the oscillation is Q/w, and the frequency of the oscillation is w,/2a. In such a frame the Zeeman levels are not modulated, but the rf field appears as frequency modulated. For the present purpose it is useful to analyze the interaction of the frequency modulated rf field with an inhomogeneously broadened ESR line. Figure 1 shows a monochromatic field at frequency LO,which, upon frequency modulation, is decomposed into sideband fields according to the well known mathematical identity
FIG. 1. Sideband field pattern of a frequency-modulated rf field. The sideband fields interact with the spin packets in an inhomogeneously broadened ESR line, and give rise to a dc and harmonic components in the signal.
90
PER@,
e;(~r
+ f3h
RAKVIN,
wmf) =
AND
5
DltJtiIt
Jn(p)ci(u
+ nwm)f
[41
n=-*
where j3 = Q/w, is the modulation index, and J, is a Bessel function. The lower part of Fig. 1 shows an inhomogeneously broadened ESR line which, according to Portis (IO) and Castner (II), is an envelope of homogeneous lines with a smooth distribution of Larmor frequencies. The tih sideband field gives rise to a dc signal through its interaction with the spin packet of the same Larmor frequency, i.e., (w + nw,). The same spin packet also experiences the interaction with the other sideband fields which are displaced from its resonance by +a,, *2w,, etc. Therefore, the response of the spin packet also contains harmonics of 0,. The total signal from an inhomogeneously broadened ESR hne is a convolution of the signals due to the spin packets at (w + nw,). The spin packets between these frequencies also experience the interaction with the sideband fields, but they do not give rise to a signal with harmonics of 0,. The above picture can be easily extended to the case of double modulation with two superimposed modulation signals (3-5). The Hamiltonian reads 2 = (wg - Q cos w,t - ~cos wkt)S, + $w,(S+P The time-dependent
+ S-e’“‘).
PI
unitary operator U = exp[-G&I
cos w,t + /I’cos w&t)]
WI
transforms the Hamiltonian [S] into an effective Hamiltonian with the constant Zeeman term, while the rf field becomes hequency modulated with two superimposed modulation signals. A repeated use of the mathematical identity [4] leads to
which shows that the doubly modulated rf field is equivalent to a sum of sideband fields whose strengths are given by the product of the Bessel functions, and whose frequencies are at (w + nw, + n’~;). Figure I! illustrates the principle of the decomposition of a monochromatic rf field into sideband fields following the consecutive introduction of the two modulation signals. The principle of the interaction of the sideband fields with the spin packets in an inhomogeneously broadened ESR line is the same as described above for the single-modulation case. The signal contains the dc component, the harmonics of each of the modulation frequencies, and their linear combinations. Each of these components can be detected by the proper choice of the reference signal for the lock-in amplifier. In the previous experiments (3-9, one used the fixed modulation frequency w, as the reference for the lock-in amplifier. Figure 2 shows that, in general, each sideband field is on resonance with one of the spin packets. The detected signal has a certain level. However, as the second modulation frequency oh, is swept, one reaches conditions n&II = n’w& PI when two sideband fields are on resonance with one and the same spin packet. The signal level then changes, and this is detected as the double-modulation spectrum. In particular for .’ = 1, the peaks occur at multiples of the fixed modulation
DOUBLE-MODULATION
SCHEMES
s.*... ..‘....i ..-...., ._.. ..I’.. . .. . I*.. -*
.~“‘i;&, 1
J,(P) !
91
J,(P)““‘.-...., .-.... . ... I 1
FIG.
2. Sideband field pattern of an rf field frequency modulated by two superimposed modulation signals. When one of the modulation frequencies is swept, the hvo indicated sideband fields may approach in frequency so as to interact with one and the same spin packet in the inhomogeneously broadened ESR line.
frequency 0,. The width of the double-modulation resonance is, evidently, given by the homogeneous spin-packet linewidth. The alternative double-modulation scheme, proposed in the present paper, is based on frequency modulation rather than superposition of the two modulation signals. The signal of fixed frequency w, is used for frequency modulation of the second modulation signal, whose frequency wh, is swept in the course of the experiment. The Hamiltonian for this case is given by &” = [w. - SZ’cos(w’,t + 0 sin w,t)]&
+ $Q(S+~+“’ + S-e’“‘).
t91
The modulation of the Zeeman interaction in Eq. [9] is much more complicated than the simple superposition of two modulation signals in Eq. [5]. Again, we look for the time-dependent unitary transformation which Iyields an effective Hamiltonian with the constant Zeeman term. The required transformation operator is U = exp[-iS,
sin(w’m + nw In)t] . 2 ““(‘) n w:, + nw,
The rf field in the transformed frame appears to be frequency modulated by a superposition of a multitude of modulation signals given by I@. [lo]. It can be expressed as exp
Q’JnW
wt + C sin(wh + nw,)t n w& + nw,
. [I l]
92
PERIC, RAKVIN,
AND DIJtiIc
The last line in Eq. [ 1 l] contains a product of sums, which is a very complicated expression. If one wants to obtain the sideband field pattern in analogy to the righthand side of Eq. [7], one has to work out the numerous products. For the present purpose, it is essential to prove that, as the swept frequency WA becomes a multiple of the fixed frequency w,, i.e., , = Cd, kam [=I a number of sideband fields coincide and interact with one and the same spin packet, which is the condition for the observation of a peak in the doublemodulation spectrum. First, one has to note that each sum in Eq. [ 1 l] contributes one of its terms as a factor in the expression which represents a given sideband field. Let us single out two factors in this expression
- + .exp{i[o
+. * 0+ r(oh, + pw,) + * * * -t.+h,
+ qu,) + * * *It}.
1131
The sideband field given by Eq. [ 131 contains the rth term from the pth sum and the sth term from the qth sum. The dots in Eq. [ 131 stand for the terms from the other sums. Leaving the latter unchanged and taking other choices of terms from the pth and qth sums, one obtains different sideband fields. In general, these sideband fields will have different frequencies. However, if condition [ 121 is fulfilled, all the sideband fields with r’ = r + (k + q)Z,
I1
400 I
I
I,
s’ = s - (k + p)l
I1
1
[I41
450
kHz
FIG. 3. Double-modulation spectrum of irradiated quartz. The modulation signal was itself frequency modulated by another modulation signal. The frequency of the former was swept while the second modulation signal had a fixed frequency.
DOUBLE-MODULATION
93
SCHEMES
where 1 = 0, ?I, +2, etc., will have the same frequency. This can easily be verified by replacing r and s in the exponent of Eq. [ 131 by r’ and s’, respectively, and using the condition [ 121. Hence, Eq. [ 121 gives, indeed., the condition for doublemodulation resonance. The diagram of the sideband field pattern is much more complicated than the one presented in Fig. 2 for the superposition of two modulation signals, and we shall not attempt to present it here. Figure 3 shows a double-modulation spectrum obtained using a modulation signal which itself is frequency modulated by another modulation signal. The sample was irradiated quartz, i.e., the same that had been used for the original demonstration of the double-modulation technique (3). Figure 3 shows clearly that the presently proposed double-modulation scheme can be considered as an alternative to the scheme using two superimposed modulation signals. From the experimental point of view, the instrumentation required for the two modulation schemes is the same. The only difference is in the modulation coils. For the first scheme, using two superimposed modulation signals, one may use a single pair of modulation coils or two independent pairs of coils, one for each of the modulation signals. In the latter case, the modulation coils for the fixed frequency signal can be a part of a tuned circuit, while the other pair of coils has to be in a broadbanded circuit to allow for the frequency sweep of the second modulation signal. In the second modulation scheme. Using a modulation signal which itself is frequency modulated by another modulation signal, one has to use a single pair of modulation coils. Obviously, when a single pair of modulation coils is used, one has to have a broadbanded circuit. The block diagram of the experimental setup for the presently proposed double-modulation scheme is shown in Fig. 4. So far, we have not noted significant advantages of one modulation scheme over the other.
reference
signal
af signal
lock - in amplifier
microwave bridge
1 Y-axis (signal) rf
X-Y recorder
X-axis (swept
Ir
sample
frequency) frequency
oscillator (fixed freq.)
signal
rlat
gWa,
oscillator (swept freq .)
FIG. 4. The block diagram of the experimental setup for the double-modulation modulation signal is itself frequency modulated by another modulation signal.
[\Jj modulation coils scheme in which the
94
PER&,
RAKVIN,
AND DUtiIc
REFERENCES 1. T. C. FARRAR AND E. D. BECKER, “Pulse and Fourier Transform NMR,” Academic Press, New York, 1971. 2. L. KEVAN AND R. N. SCHWARTZ (Eds.), “Time Domain Electron Spin Resonance,” WileyInterscience, New York, 1979. 3. B. RAKVIN, T. ISLAM, AND I. MIYAGAWA, Phys. Rev. L&t. 50, 13 13 (1983). 4. B. RAKVIN, Chem. Phys. Z&l. 109, 280 (1984). 5. M. PERK?,B. RAKVIN, AND A. DuL(SIC, J. Chem. Phys. 82, 1079 (1985). 6. L. J. SCHWARTZ,A. E. STILLMAN, AND J. H. FREED, J. CIyem. Phys. 77, 5410 (1982). 7. J. P. HORNAK AND J. H. FREED, Chem. Phys. Z&t. 101, I il5 (1983). 8. 0. HAWORTH AND R. E. RICHARDS, in “Progress in Nuclear Magnetic Resonance Spectroscopy” (J. W. Em&y, J. Freeney, and L. H. SutclilTe, Ed%), Vol. 1, Chap. 1, Pergamon, London, 1966. 9. A. Dutid AND B. RAKVIN, J. Magn. Reson. 52, 323 (1983). 10. A. M. PORTIS, Phys. Rev. 91, 1071 (1953). Il. T. G. CASTNER, Phys. Rev. 115, 1506 (1959).