Modulation-induced sidebands in magnetic resonance

JOURNAL

OF MAGNETIC

RESONANCE

25183495

(1977)

Modulation-inducedSidebandsin Magnetic Resonance* I.

MIYAGAWA,

Y.

HAYASHI,~

AND

Y.

KOTAKE$

Department of Physics and Astronomy, The University of Alabama, University, Alabama 35486 Received July 1,1976 The induced transition probability between Zeeman levels was calculated in the case of a nuclear spin under a modulated magnetic field. It was shown that the intensities of the sidebands can be accurately calculated for a given lineshape of the signal when no significant saturation occurs. This result suggested a technique for accurately measuring the modulation amplitude of magnetic fields. This suggestion was confirmed experimentally in the case of an rf field produced by an ENDOR coil. A quantum-mechanical explanation for the production of sidebands was also found during the course of this calculation. I. INTRODUCTION It is well known in magnetic resonance studies that a magnetic modulation

produces sidebands when the modulation frequency is higher than the signal linewidth (1-5). Several workers (3,4,6, 7) have also shown theoretically that the relative intensity of a sideband is a function of the modulation frequency and amplitude. Thus the modulation amplitude can be determined from the sideband intensity, since one can measure the modulation frequency easily. Actually Burgess and Brown (1) made such a suggestion as early as 1952 (8). A technique such as that suggested by Burgess and Brown recently became important in our laboratory because of the development of a new electron-nuclear-doubleresonance (ENDOR) method (9-13). Of essential importance for this method is an intense radio frequency (rf) field which has to be 10 G or higher at the sample site. The difficult problem is not the production of the intense field but rather the confirmation of its field intensity. The technique suggested by Burgess and Brown should be able to measure the field intensity, if both magnetic field and rf field are applied perpendicular to the ENDOR coil. It is noted, however, that the proposed theories (3, 4, 6, 7) are based on the Bloch equation or Karplus’ theory (14) for microwave spectroscopy. It should be pointed out that the Bloch equation, though a highly useful expression in magnetic resonance, is not quantitatively accurate (25, 16). It should be also pointed out that in the Karplus theory the signal width is assumed to arise from collision broadening. This is not necessarily the case for a magnetic resonance signal ; in many cases, unresolved hyperfine * This work was supported by National Science Foundation Institutional Grant GU 3202. i Permanent address: Department of Physics, Nagoya University, Nagoya, Japan. $ Permanent address: Department of Chemistry, Osaka University, Osaka, Japan. Copyright 0 1977 by Academic Press, Inc. 183

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in any form reserved.

184

MIYAGAWA,

HAYASHI,

AND

KOTAKE

structures contribute to linewidth. Consequently, care must be taken when one tries to use these theories in an accurate determination of a physical quantity such as that of the rf magnetic field. For the reasons given above, in the present work we took a theoretical approach which is entirely different from those of the existing theories for sidebands. We calculated the transition probability between Zeeman levels of a nuclear spin which is under a magnetic field modulation. It was found that when no significant saturation occurs, the intensity of the nth sideband is proportional to [Jn(o)12. In this expression, J, is a Bessel function, and CJ= yHNI/mM, where HM is the modulation amplitude and wM is the modulation frequency. This proportionality was found to be similar to that derived by Haworth and Richards with the Bloch equation. It should be pointed out, however, that the present calculation has shown that the proportionality applied to a signal of any given lineshape, and hence, to a signal from any given sample. It was actually shown in the present work that the assumption for the special case of a Lorentzian lineshape results in the expression for the dc component of the complex magnetic polarization which Haworth and Richards have obtained. This proportionality was examined and confirmed with the use of a DPPH probe which was placed in the center of an ENDOR coil. It was also demonstrated that the distribution of the rf magnetic field can be determined as a function of the position relative to the ENDOR coil. In addition, a quantum-mechanical explanation for sideband production was found during the course of the present calculations. II. THEORY

In this section we discuss, first, the quantum state of a nuclear spin placed in a magnetic field which consists of a static field and a modulated field. We then calculate the probability of the transition induced between the quantum states by an electromagnetic field at the resonance frequency. When the static magnetic field Ho is applied parallel to the z axis, in the absence of the modulated field, the mth state of the nuclear spin I is given by the wave equation .7ifolm) = kw,lm),

PI

z&‘~= -hy H,,I,,

PI

w, = -yH, m = --coom,

[31

where and where m = (m~I,~m> and yH, = wO. The modulated field Hmod, which is parallel to Ho, may be given by H mod= f& cos o,t,

141

where HM is the modulation amplitude and coNIis the modulation frequency. Then the interaction energy between the modulated field and the spin is given by SF”, = -4 yHMIzcosw,t.

[51

Thus the wave equation of the system is now given by

iha+/at= (x0 + 2eM)$.

Fl

MODULATION-INDUCED

185

SIDEBANDS

The wavefunction $ is easily found to be

lClm = exp[@,sinh-M>-f-6, - aJ)lIm>,

[71

Pm= MM mloM

PI

where and 6, is the phase constant. It is noted that

ixh4,&ol = 0, and hence $, in Eq. [7] is independent of the states other than jm}. The remaining step is an extension of the standard calculation of transition probability. When a radiation field H’ cos ot is applied parallel to the x axis, the new wave equation is given by ih &D/at = (X0 + SPM + SF)@, [91 where 2”’ = -liyH’IXcosot. i-101 The wavefunction

@ may be expanded in terms of $,‘s.

illI where a, is independent of coordinates. Substitution of Eq. [II] into Eq. [9] leads to the relation ri, = iyH’
- Blnk+ q&}]{exp(iwt)

+ exp(-iot)}/2,

WI

where &lk = 4ll - dk,

u31 P41

w mk= co,,,- ok = -q,(m

- k).

[I51

Also in the above calculation, we made use of the following relation at t = 0: a,= 1, a, = 0

form # k.

The selection rule requires m - k = +I, and hence P,,,,c= +yH,lq,,

= I@,

and f&,k = -cm,,.

WI 1171

Now it is noted that exp[ibk

sinb-hd)l

=

2 II=-m

J(Pmk>

exp@%d)y

P31

where n is an integer, and Jn(Pmk)is a Bessel function. Substituting Eq. [IS] into Eq. [12] and then integrating the resulting expression from 0 to t, one obtains am = (W’P)

CmIL IQ exP(-id,,)

2 J, (-Pmk> I191

186

MIYAGAWA,

The transition probability

HAYASHI,

AND

KOTAKE

from the k state to the m state, W,,, is calculated

by

where ~(0,~) is the product of the densities of state at m and k. Substitution of Eq. [19], into Eq. [20] gives wnl, = @% + KLk,

WI

where and WA = WV

I(4~xlk>lzn~,

JI;(-PmkVnt (-Pmk)

x a/at [CO+-

d)o,t/2] .ip sin[(-mu, f 0 + co,,&/21 -cc 1 x (~1~0~IL 0 + o,J1 sin[(n’o, f 0 + q&/2]

[231

It is noted that both Wzk and WA, are independent of phase constant 6,. The first term Wj, is the important part of the transition probability which will be used for determination of the rf field. Now for the allowed transition, p mk= +o(seeEq. [16]), and, since IJn(-o)\2 = ]J,(o)12, W& is given by [241 The positive sign of o corresponds to emission, and the negative sign to absorptionSince the emission probability is the same as the absorption probability, hereafter only the absorption will be considered. W,,O depends on o through p(o - no&. When rz= 0, p(o - noM) becomes p(w), which represents the lineshape whose peak center is at Q, = @Ho. For a given n, p(o - nw,) represents the same lineshape as p(o) but with the peak center at o. + ylwM instead of mo. Thus when the linewidth is less than the modulation frequency, oM, p(o - noM) represents the mth sideband while p(o) represents the central band. The same conclusion is obtained when w is fixed and the magnetic field is swept; the interval of the resulting sideband is given by W&J in this case, since p(Ho + no,/y) should be used instead of p(o, - no&. When the radiation field is weak, and hence the saturation can be ignored, the signal intensity is proportional to the transition probability. Thus Eq. [24] indicates that when the radiation field is weak

L OcWd12>

[251

where 1, is the intensity of the nth sideband. This result is similar to that obtained by Haworth and Richards (4). It should be emphasized, however, that the present result

MODULATION-INDUCED

187

SIDEBANDS

applies to the case of any lineshape, and hence to signals from any real samples in general, as long as the radiation field is weak. In addition, since

one obtains F I,” = IO, n--m

1271

where Z” is the signal intensity under no modulation. Equation [27] indicates that the sum is independent of 0, and hence of NM, under a given weak radiation field. For signals of magnetic resonance in general, it is far easier to detect the first or second derivative of an absorption signal under a low-frequency modulation than the signal itself. In such cases, one can replace p in Eq. [24] by 8p/8Ho or Pp/azHo for the following reason. In integration in Eq. [20], the time t is required to be much longer than 271/o, that is, one period of the electromagnetic radiation. Hence it is noted that t can be chosen in such a way that it is much shorter than the period of the audiofrequency modulation. Consequently, in calculation of the transition probability, one can ignore the time-dependent change in Ho resulting from the audiofrequency modulation. For this reason, when an audiofrequency modulation is used, the resulting signal should be given by derivatives of Eq. [24]. The second term is not important for the present purpose. It can be shown that this term consists of harmonic terms of o,t which do not contribute to the dc component of transition probability (17). When the modulated field is perpendicular to Ho, its contribution to the sideband intensity should be of the order of (HM/Ho)2 times that given by Eq. [22] or Eq. [24] when Ho 9 HM. Hence, such a contribution can be ignored unless, for example, for an ESR experiment at 9 GHz, HM > 300 G. Thus when the rf field is not parallel to Ho, it is expected that only the z component of HM will contribute significantly to the sidebands if Ho $ HM. The present calculation does not include interaction terms such as hyperfine interaction and zero-field splitting. It should be noted, however, that even when these terms are included, the present calculations should be fairly accurate if the dc magnetic field is sufficiently high and hence [I=, A?] N 0 (Paschen-Back case). III.

SPECIAL

CASE:

LORENTZIAN

LINESHAPE

It is of interest to consider the special case of a Lorentzian lineshape since a stationary solution of the Bloch equation is a Lorentzian. The case of I= 4 will be considered. The lineshape function may be given by P(O) = T2/{1 + TZco - wo)2)n,

Lw

where l/T2 is the linewidth parameter and o. is the center of the absorption. Thus, p(o - noM) = T,/(l + T:(o - o. - nc~~)~}n.

Lw

Substitution of Eq. [29] into Eq. [24] in the case of I = 4 gives the transition probability W%L, I/Z =

WY,,

-1/2

= (y2H”

T2/@

;

JnZ(M1

+ T;(w

-

00 - fimd2).

i30E

188

MIYAGAWA,

HAYASHI,

AND

KOTAKE

It is noted that p(o - nwM) given by Eq. [29] has a peak at o = w0 + IZO~. Thus when l/T, c ~MM,Eq. [30] represents a set of the signals: the nth signal appears at w = o0 f nmhl and its intensity is proportional to ,Z,~(CT). Furthermore, it would be of interest to compute the complex susceptibility for the Lorentzian lineshape. The power P which is absorbed by a sample of unit volume is related to the imaginary part x”(o) of the complex susceptibility and the strength of the resonance field H’ by (18) P = W(H’)2 II” (w)/2.

The absorbed power is also related to the transition probability tion difference AN as

[311 IV0 and the popula-

P=hwW”AN,

~321

and hence one obtains x” (co) = 2h W” AN/(H’)2.

[331

In the case of a nucleus of Z = f, the equilibrium polarization MO is given by MO = -yfi AN/2,

L341

and hence Eq. [33] is rewritten by f (co) = -4 w:

112,112

Mo/WW.

E351

Substitution of Eq. [30] into Eq. 1351gives x” (co) = -3 yMo T, 2 J;(D)/{ 1 + T:(o - coo- y10~)~).

[361 n One obtains the real component x’(o) from X”(W) with the aid of the Kramers-Kronig equations (18) f(o)

= x’(m) + (I/n)

i f(cu’)do’/(o’

- Co) 1371

= &MO T, 2 J$) n

{-T, (co - 00 - nu&&/{l

+ Ti(w - 00 - ncoh.1)2).

If m1 is defined by m, z u - iv = H’f

- iH’f,

it follows that [l - iT2 (o - 00 + nwM)]/[l + T$(co - o. + IZOM)‘]. [381 n=-ml This is identical with the expression which Haworth and Richards obtained for the dc component of the signal with the use of the Bloch equation when saturation is ignored (see Ref. (4, first equation on p. 8); note that HI = H’/2). In the case of a nucleus of Z # 3, a simple relation such as that given by Eq. [34] does not hold in general between MO and AN. Under the high-temperature approximation, however, the population difference between the m - 1 and m states is approximately given by [N/(2Z+ l)]hyH,/(kT), and hence one obtains Eq. [38]. Thus in the case of a nucleus with Z other than +, Eq. [38] is accurate only when the high-temperature approximation applies. ml = 3 2 miyH’MoTJ:(~)

MODULATION-INDUCED IV.

189

SIDEBANDS

QUANTUM-MECHANICAL SIDEBAND

EXPLANATION PRODUCTION

OF

THE

Substitution of a relation similar to Eq. [18] into Eq. [7] gives *, = s bnGw~-@h

L39.I

+ nwdtllm)l,

where 6, = Jn(-Pm) exN4J. It is noted that the inside of the bracket of Eq. [39] is a wave function with energy A(o, + no,). Thus an effect of the modulated field is such that a state with energy tie, = firno, is converted into a combination of states with energies fi(mo, + no,); the latter states have the same time-independent part as the original one. Consequently under a modulated field, the resonance frequencies will be w0 + (n - n’)wM instead of the single frequency oO. Thus this discussion provides a quantum-mechanical explanation of the appearance of side bands under a modulated magnetic field. Furthermore, the present discussion can be generalized easily. Instead of Eq. [4], in general, the modulated field may be given by H rncd

WI

=f@>z

which, instead of Eq. [7], leads to a wavefunction ym j f(f)

dt’ + 6, - w,t

0

Now

in Eq. [41], /f(t’)dt’

II .

II411

may be expanded in a Fourier series, and hence exp{iym ;

0

n

f(t’)dt’} may be given by a product of the series, each of which resembles the right side of Eq. [18]. Consequently on the basis of the preceding discussion, one expects the appearance of many sidebands. In most cases of spectroscopic investigations, splitting of a spectrum is known to. occur as a result of mixing two or more states by perturbation. According to the present calculation, the mechanism of splitting into sidebands in magnetic resonance is rather unusual, since no mixing of the Zeeman states is involved. As is shown by Eq. [39], the wavefunction which is transformed by modulation has no contribution from states other than the original mth state. V. EXPERIMENTAL

The ENDOR spectrometer employed for previous work (9-13) was used for the present observation. In addition to the rf modulation, the magnetic field was modulated at 16 kHz, and the second harmonic of the resulting signal was detected. The microwave frequency was 9 GHz. Signals from a tiny DPPH crystal, which is less than 0.1 mm on a side, were observed at room temperature. This crystal was prepared from a benzene solution by Dr. A. S. Jones, to whom we are grateful. VI.

EXPERIMENTAL

CONFIRMATION

The present calculation may be confirmed by examining the two relations given by Eqs. [25] and [27] experimentally. A probe of DPPH was used for this purpose, since the

190

MIYAGAWA,

HAYASHI,

AND

KOTAKE

signal is strong and is not saturated at room temperature unless a very high microwave power is applied. In the experiments given in this section, the probe was placed at the center of the ENDOR coil. The first relation was confirmed as follows. Sideband signals were observed for an rf modulation field of a given frequency and a given amplitude under a given microwave power level. Figure 1 shows examples of such observations when 0,/27c = 17.0 MHz.

(C)

13.2

---It+ 6.1 G FIG. 1. Sidebands of the ESR absorption from a tiny DPPH sample placed at the center of the ENDOR coil carrying an rf current at 17 MHz. In addition to the rf field the magnetic field was modulated at 16 MHz, and the second harmonic of the resulting signal was detected. The separation of the sideband peaks is 6.1 G, which is equivalent to the rf frequency, 17 MHz. The number to the right of each curve shows the corresponding rf field amplitude (NM) in gauss. The microwave frequency was 9 GHz, and the observation was made at room temperature. The static magnetic field was applied perpendicular to the ENDOR coil. The relative gain of the amplifier for the upper two curves was 5, and that for the lower two was 1.

Now the proportionality given by Eqs. [25] indicates that in each absorption curve the observed value of I,/&, should result in the same 0, and hence in the same H, (seeEq. [ 16]), regardless ofn. Table 1 lists the values of HM which were calculated from intensity ratios I,,/&, in the case of the sidebands given in Fig. 1. One notices that the value of HM is the same within experimental error under a given microwave level. Thus this fact confirms the first relation. The same experiment confirmed the second relation. Table 2 lists 2 I,/IO, where I0 is n

MODULATION-INDUCED

SIDEBANDS

191

the intensity of the signal which is obtained when no modulation is applied. One finds that this value remains unity within experimental error, although I&,, increases from 6.0 to 16.8 G. Note that in Fig. 1 the relative gain of the amplifier for the upper two curves was 3, whereas that for the lower two was 1. TABLE RADIO

FREQUENCY

1

AMPLITUDES (HM) WHICH SIDEBAND SIGNALS SHOWN

HM (G)

WERE ESTIMATED IN FIG. 2”

FROM THE

at radio frequency level :b

A

B

C

D

9.21 9.21 9.15 9.15 -

13.1 13.1 13.1 13.1 13.3 13.3 -

I-4

6.00 5.96 -

16.8 16.8 16.9 16.9 16.8 16.8 16.7 16.7

Experimental error

0.07

0.1

0.3

0.3

Signal

11 I-1 12 Z-2 13 I-3 14

a The measurement was made for w,/2n = 17.0 MHz under a given microwave power for the unsaturated range. HM was estimated from the intensity ratio of the signal and the central signal in the band, that is, Ij/Io (j # 0). The signals for a given rf level give the same H,, within the experimental error. b See Fig. 2. TABLE SUMSOF

2

THE PEAK INTENSITIES

Radio frequency levels* A Average H, 1 LIZ” II Experimental

(G)

error

B

C

D

5.98

9.18

0.98

0.99

13.2 1.01

16.8 0.97

k-o.03

kO.03

kO.03

io.03

a The table lists 2 Z,/Z”for the rf levels given in Fig. 2, where I” is the intensity of the nonmodulated

sigial under the same microwave power level. The value of 2 In/Z0 is ” independent of HM and remains 1.00 within the experimental error. The microwave power level and will were similar to those for Table 1. b See Fig. 2. 7

192

MIYAGAWA,

HAYASHI,

AND

KOTAKE

In the present experiment, the applied rf field was less than 20 G, and Ho N 3400 G. Thus, when the rf field is not parallel to H,, the effect of the modulation should essentially arise from the z component of the rf field, as was discussed in Section II. This prediction was also confirmed. Table 3 lists the observed value of HNI as a function of a, TABLE ORIENTATION

3

DEPENDENCE

OF HMa

HM ((3 47 0

15 30 45 60 15 90

Observed

Calculated

12.9 12.4 11.1 9.16 6.55 3.24 0

12.9 12.4 11.2 9.12 6.53 3.24 0

a In the case of a given applied field, HM was measured as a function of the angle (a) between Ho and the line perpendicular to the plane of the ENDOR coil. The calculated value is Hz cos GC,where Hi is the value of HM for CI= 0. The microwave power level and wy were similar to those for Table 1.

the angle between Ho and the rf field, when the rf field is 12.9 G. As the table shows, the observed value was found to be identical with HL cos a, where H& is the value of HM for CI= 0. The proportionality given by Eq. [25] does not hold when saturation of the signal occurs. Table 4 lists the values of HM which were estimated for several microwave power levels. As one can see, under an intense microwave field such as the case of 0 db, the estimated value of HM significantly differs from one sideband to another. TABLE MEASURED

VALUES

4

OF HM AS A FUNCTION

OF MICROWAVE

POWERS

Microwave power -20 db -15 db -10 db 0 db

18.0 18.0 18.0 18.1

18.2 18.0 18.2 18.4

18.3 18.2 18.3 18.3

18.2 18.1 18.1 17.8

LIThe measurement was made for several sideband peaks under a given rf field, and the value was estimated from Zj/Io as in the case for Table 1. 0~/2z = 17.0 MHz.

MODULATION-INDUCED VII. MEASUREMENT

SIDEBANDS

193

OF rf FIELD

The preceding experiment confirmed the theory described in Section II. Thus it is possible to measure the rf field by measuring the intensity ratio of the sidebands under a microwave power at which no significant saturation occurs. In formulating the theory no approximation was made, and hence the proportionality given by Eq. [25] is accurate. Consequently, in principle, an rf field of any magnitude can be determined with the use of the proposed technique. However, in practice, measurement of a small rf field was found to be difficult because of the weak sideband signal. For example, in the range of 16 to 30 MHz, it was found difficult to measure a field weaker than 0. I G. However, it appeared that there is no upper bound for measurable field intensity.

TEFLON -ENDOR

FRAME Ho

coi

5 mm FIG. 2. Cross section of the ENDOR coil at its center. The cross section plane is perpendicularto the cavity axis. The ENDOR coil, which is supported by a Teflon frame, is a ten-turn loop made of plastic coated No. 38 copper wire. The arrow marked by & indicatesthe direction of the rf field at the position marked by the circle (senseof the direction is not important). The arrow marked by I& indicatesth applied dc magnetic field. When the rf field at the center was 12.1 G, the rf field at the position on a surfaceof the Teflon frame wasfound to be 7.6 G, and its direction to be 30” in terms of 0.

Since the DPPH probe was less than 0.1 mm on a side, it was possible to measure fairly accurately the distribution of the rf field as a function of position relative to the ENDOR coil. It was also possible to measure the direction of the rf field by observing the sidebands when the ENDOR coil is perpendicular to Ho and those when these two are parallel. Figure 2 illustrates the result of such a measurement whencoh1/2n = 17 MHz: when the rf field at the center was 12.1 G, the rf field at the position marked by an arrow which is on a surface of the Teflon frame was found to be 7.6 G, and its direction (0) to be 30.0” from Ho. In principle, the proposed technique should apply to any nuclear magnetic resonance using any sample. Thus we conducted a rather extensive examination of proton magnetic resonance in the case of water samples containing various concentrations of copper sulfate. In this examination, the resonance frequency was 14 MHz, and the modulation frequency ranged from 1 to 14 kHz. It was found difficult to suppress saturation effectively even for a high concentration of copper sulfate under a weak radiation field. Consequently, a different sideband peak resulted in a different vaJue of NM for a given modulation field. It was also found that when the modulation intensity varied, the sum of the sideband intensities varied. Thus this technique could not determine the modulation field accurately in the case of proton resonance.

194

MIYAGAWA,

HAYASHI,

AND

KOTAKE

For the reason given above, we feel that usefulness of the proposed technique is rather limited practically, although in principle its accuracy should be high in any case. Measurement of the rf field from an ENDOR coil is a rather special case to which the proposed technique applies effectively. VIII. CONCLUSIONS

It was shown that the induced transition probability between pure Zeeman levels can be calculated accurately in the case of a nuclear spin under a modulated magnetic field. The result applies to a signal of any lineshape under a modulation frequency which can be higher than the Zeeman frequency. On the basis of this result, it was also shown that as long as no significant saturation occurs, the intensity relation for sidebands, I,, cc IJ,( 2, which has been pointed out by other workers, is accurate for a signal of any lineshape and for any modulation frequency. The present results should still be fairly accurate when interactions such as hyperfine interaction are included, if the dc magnetic field is high enough (Paschen-Back case). It was shown that, in the special case of a Lorentzian lineshape, a classical limit of the present calculation results in the complex susceptibility which has been obtained with the use of the Bloch equation. A quantum-mechanical explanation was given for the production of sidebands in magnetic resonance. It was pointed out that no mixing of the Zeeman states occurs as a result of modulation, and hence the mechanism is rather unusual compared to most cases of splittings of signals in spectrascopy. It should be mentioned that the corresponding case of Stark modulation has also been treated quantum mechanically (19). However, this case involves a mixing of the original quantum states, and hence the mechanism of sideband production is different from the present case of magnetic modulation. The proportional relation, 1, CC1Jnj2,was examined experimentally in the case of the rf field from an ENDOR coil. The relation was found to be accurate within the experimental error, which was only a few percentage points in terms of the estimated rf field. On the basis of the present investigation, an experimental method for an accurate measurement of rf magnetic field was suggested and actually demonstrated.

1.

2. 3.

4. 5.

6. 7.

8.

REFERENCES J. H. BURGESS AND R. M. BROWN, Rev. Sci. Instrum. 23, 334 (1952). N. S. GARIF’ YANOV, Zh. Eksperinz. i Teor. Fiz. 32,609 (1957); Soviet Phys. JETP 32, 503 (1957). R. GABILLARD AND R. PONCHEL, Proc. Colloq. A 11,749 (1962). 0. HAWORTH AND R. E. RICHARDS, Progr. NMR Spectrosc. 1,1 (1966). C. P. POOLE, JR., “Electron Spin Resonance,” Chap. 10, Interscience, New York, 1967. B. A. JACOBSOHN AND R. K. WANGSNESS, Phys. Rev. 13,942 (1948). J. D. MACOMBER AND J. S. WAUGH, Phys. Rev. A 140,1419 (1965). See also M. T. JONES,J. Magn. Resonance 11, 207 (1973). This paper lists more recent works on magnetic modulation.

9. I. MIYAGAWA,

R. B. DAVIDSON,

H. A. HELMS,

JR., AND B. A. WILKINSON,

JR., J. Magn.

10,156 (1973). 10. H. A. HELMS, JR., Il. I. SUZUKI, J. Phys. 12. I. MIYAGAWA AND 23. Y. KOTAKE AND I.

I. SUZUKI, AND I. MIYAGAWA, J. Chem. Phys. 59,5055 Sot. Japan 37,1379 (1974). T. S. CHEN, Bull. Amer. Phys. Sot. 20,424 (1975). MIYAGAWA, J. Chem. Phys. 64,3169 (1976).

(1973).

Resonance

MODULATION-INDUCED

SIDEBANDS

195

14. R. KARPLUS, Phys. Rev. 73,1027 (1948). 15. A. M. PORTIS,Phys. Rev. 91,107l (1953). Id. A. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 520, Oxford University Press, London, 1961. 17. I. MIYAGAWA, to be published. 18. See, for example, C. P. SLICHTER,“Principles of Magnetic Resonance,” Chap. 2, Harper and Row, New York, 1963. 19. See, for example, C. H. TOWNESAND A. L. SCHAW~OW, “Microwave Spectroscopy,” pp. 273-283, McGraw-Hill, New York, 1955.