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Saturation broadening and narrowing in solids
JOURNAL
OF MAGNETIC
RESONANCE
13,291-298
(1974)
MICHAEL N. ALEXANDER,PAUL L. SAGALYN, AND AMOS J. LEFFLER* Army Materials and Mechanics Research Center, Watertown, Mamachuxtts
02172
Received October 26,1973 The dependence of the NMR line width on the rf field amplitude, Hi, is calculated for an experiment in which the lock-in absorption mode is detected. It is shown that when T,,/7’,, > 6, saturation broadening of a Gaussian line is expected for intermediate rffield intensities (T,, and T,D are, respectively, the spin-lattice relaxation times for the Zeeman and dipolar energy). When the rfheld is very large, saturation narrowing is predicted. For T,,/T,, < 6, only saturation narrowing of a Gaussian line is predicted. The calculated results are in qualitative agreement with lock-in absorption mode data for CaFz and for lithium metal at 77 and 215 K. The importance of careful monitoring of saturation is discussed. INTRODUCTION
The early theory of NMR saturation in solids due to Bloembergen, Purcetl, and Pound (I) predicted that the NMR absorption line would broaden as the intensity of the radio frequency field (rf) was increased. Several years later, Abel1 and Knight (2) reported that saturation narrowing rather than saturation broadening occurred in metallic copper. Subsequently, Redfield (3) performed experiments on copper and aluminum which demonstrated conclusively that the BPP theory does not properly describe NMR saturation in solids. By introducing the concept of spin temperature in the rotating reference frame, he developed a theory of saturation that agrees with experiment at large rf intensity, Provotorov (4) extended the theory to intermediate rf intensities by replacing Redfield’s single spin temperature with separate temperatures appropriate to the Zeeman and dipole-dipole subsystems. In the Provotorov theory the rf mixes the Zeeman and dipolar subsystems, the mixing rate being proportional to the square of the rf amplitude. Thus, at high rf intensity the mixing time is short, a single spin temperature is quickly achieved, and the Provotorov theory goes over to the Redfield theory. The Redfield and Provotorov theories have been shown to predict saturation narrowing, and considerable experimental evidence has been advanced in support of this prediction (2,3,5-11).
In this paper we reconsider the question of saturation broadening in solids. Despite the apparent prevalence of saturation narrowing, broadening has been noted in solids as diverse as metallic lithium at 215 K (II) and solid benzene (12). We have recently observed saturation broadening in crystalline carboranes. In this paper we show that * National Academy of Sciences-National Research Council Senior Research Associate. on leave from Villanova University. Permanent address: Department of Chemistry, Villanova University. Villanova, Pennsylvania 19085. 291 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction Printed in Great Britain
in any form reserved.
292
ALEXANDER,
SAGALYN,
AND
LEFFLER
under suitable conditions the lock-in absorption mode line, calculated by Goldman (.5,6) from the Provotorov theory (4,5), is saturation broadened as well as narrowed. We compare our numerical results with Holcomb’s experimental linewidth data (II) (to the best of our knowledge, only Holcomb has made extensive line width measurements in the low and moderate saturation regime). We also discuss the implications of our results for experiments in which NMR is used as a tool for studying properties of solids. CALCULATION
Goldman (5,6), using the Provotorov theory, has analyzed the signals observed in a cw experiment which employs lock-in detection. In lock-in detection the large magnetic field is modulated at a frequency S2/2n, and the spectrometer detects a signal having the same frequency and phase as the modulation. We analyze this experiment because it is probably the most common scheme employed in cw wide-line NMR. Goldman’s result for the lock-in absorption mode signal, v,(&), is (5, 6) :
Ul Our notation follows that of Goldman: or = yH, = amplitude of rf field in angular frequency units; 7 = nuclear gyromagnetic ratio; do = distance from resonance (either positive or negative- do = 0 at resonance) in angular frequency units (5); D = “local field” in angular frequency units (5); g&J = unsaturated normalized absorption line shape function; r,, = spin-lattice relaxation time for Zeeman energy; T,, = spin-lattice relaxation time for dipole-dipole energy. In Eq. [l], Kis a factor which involves the amplitude of the modulation, the equilibrium magnetization, and ol. As will be seen below, K does not enter into the line width.’ Equation [l] is valid under the following conditions. (i) The modulation amplitude is much smaller than the line width. (ii) 8 Q D : the modulation frequency is much smaller than the line width, expressed in frequency units. (iii) QT,,, QT,, 9 1: the modulation period is much shorter than either the &man or dipolar spin-lattice relaxation times. (iv) rro:g(O) < Sz: the mixing time between the Zeeman and dipolar systems is much longer than one period of the modulation. Note that in the limit of negligible saturation (~1 +Q, VI =&MA,. To obtain the “peak-to-peak” line width observed in the lock-in cw experiments (for negligible saturation this is the width between inflection points of g(A,)), we simply find the A,, which satisfies dvl/dAo = 0. We shall assume that the unsaturated line is Gaussian, a reasonable approximation in many solids : g(Ao) = [l/D(67c)‘/‘]
exp(-Ag/6D2).
121
1 Revokatov and Lyapukhov have derived a generalization of Eq. [l] in which the line width depends on the modulation amplitude (9). Their equations are very complicated, and are useful only in the limit of extreme saturation.
SATURATION
BROADENING
AND
NARROWING
193
After considerable algebra, and setting A,/ D = x, we obtain (x2 - 3) + (ma;g)(9TlDx2
- 9Tl,, - 3T,,)
+ (no~g)2(3TlzTlD~2
+ 3T;,x4
- 9T,, T,, + 9Tf,x’)
= 0.
PI
Note that because g is a function of x, Eq. [3] is not simply a polynomial equation in s. Equation [3] gives the correct results for low and high saturation. In the limit of negligible saturation, o1 40, Eq. [33 gives x2 = 3, i.e., Ai = 3D*, which corresponds to the width between inflection points of a Gaussian line. In the limit of high saturation we ignore the first two terms on the left side of Eq. [3]. Therefore, to satisfy the equation, we set equal to zero the coefficient of w:, obtaining x4 + x2(8 + 3) - 3s = 0,
Pf
where 6 E T,,/T,,. This is the same result Goldman obtained directly from Eq. [ 1] (Ref. (5), p. 112). We may obtain from Eq. [3] a criterion for saturation broadening of the line. Noting that x2 = 3 for the unsaturated line, we express deviations y from the unsaturated width by defining y = x2 - 3. Substituting into Eq. [3] and requiring y > 0 (line broadening), we find that the first and third terms on the left side of the equation are positive. Since only the term containing o: can be negative, its coefficient must be negative: (2 + y - S/3) < 0.
Therefore, since y > 0,6 > 6. 6 > 6 is a necessary condition for saturation broadening of a Gaussian line. Because Eq. [3] contains both exponential and polynomial terms, we have been unable to derive a tractable analytical expression constituting a sufficient condition for saturation broadening. An alternative approach is numerical evaluation of Eq. [3]. We have not made a detailed numerical search for a sufficient condition. However, with the choice of NMR parameters to be described below, we did find saturation broadening when S = 7; we did not investigate 6 < S < 7 numerically. To find the A0 which satisfy Eqs. [2] and [3], we took T,, = 50 see and took the unsaturated peak-to-peak line width equal to 6.50 Oe, consistent with Holcomb’s data for t9F in CaF, with the large magnetic field parallel to the [l lo] direction (II); ~5was treated as a parameter. The relative linewidth-that is, the lock-in peak-to-peak line width (obtained by solving Eq. [3]) divided by the width of g(A,) (6.50 Ott)---is shown as a function of H1 in Fig. 1. The arrow in Fig. 1 denotes the value of Nj at which the Bloch saturation factor (ZJ), s = nw: g(O),
I4
is unity. It is clear from Fig. 1 that the Goldman-Provotorov theory predicts significant deviations from the unsaturated line width, even for moderate saturation. When ci is large the line broadens markedly as H1 increases, finally narrowing at large H,. Although no single parameter characterizes the saturation behavior in a universal manner, the Bloch saturation factor does appear to retain considerable qualitative significance.
294
ALEXANDER,
SAGALYN,
AND
LEFFLER
Although the values of 6 used in Fig. 1 were chosen for illustrative purposes, they are experimentally reasonable. 6 - lo5 has been observed in some molecular solids (14), and 6 21 1000 has been observed in ferroelectrics (15). In CaF, (for which Fig. 1 presents the results of a model calculation), 6 will depend upon both temperature and the nature and concentration of impurities; we know of values as large as 6 = 80 (12).
10-3
lo-4
101
10-2 HI
tOeI
FIG. 1. Theoretical saturation behavior of relative peak-to-peak NMR line width, in lock-in absorption mode detection. The arrow in the figure designates the value of HI at which the saturation factor (Eq. [5]) is unity. Values of 6 = T1x/T1n used in the calculation appear with each of the curves. The values of other parameters used in the calculation are described in the text.
COMPARISON
WITH
EXPERIMENTAL
DATA
Our calculated CaF, line widths, together with those obtained experimentally by Holcomb (II), are shown in Fig. 2. The theoretical curves in Fig. 2 differ from those in Fig. 1 only in the choice of 6. The fit is qualitative for 6 N 5-12. 81
% G61 .3! L e j
6---------
.x
--__ -.__
P -----p-A---
4 k-
Y, $
r
--6=12
t
821
i
ii lo-4
.lo-3
lo-2
HI (04
FIG. 2. Saturation behavior of lock-in absorption mode line width in CaF,. Large magnetic field is in [ 1lo] direction. Experimental data (II) are shown by circles. The lines are from the present calculation. The arrow denotes the experimental value (II) of Hi at which the saturation factor (Eq. IS]) is unity.
SATURATION
BROADENING
AND
NARROWING
295
Although a Gaussian was used to model the experimental CaF, line shape, the existence of beats on the free induction decay (26) demonstrates that the line is not accurately Gaussian. We have not analyzed the effect on our results of changing details of the line shape. However, we did find that changes in the width of the line did not alter the qualitative nature of the results shown in Fig. 1. In Fig. 2, the increase in the experi,mental line width when HI > 10v2 Oe is anomalous. It is difficult to reconcile such an increase with the Goldman-Provotorov theory. (The line widths measured with the magnetic field parallel to the [lOO] direction are greater than those for the [l lo] direction, but the two appear to converge when H1 is greater than 10M2 Oe (II).)
FIG. 3. Saturation behavior of lock-in absorption mode line width in lithm m&d, at 77 F&@&tern) and 215 K (top). Experimental data (If) are shown by cir&s. Tbt lines give the rewlt of the pewnt c&aUien; there are no adjustable parameters in the caiculatiera,all quantities being ned by ezqmrb#al data (seetext). The (VIOWSdenote the experimentalvalue (II), at each&m&tture, of HI at which the saturation factor (Eq. [5]) is unity.
In Fig. 3 we show experimental data for ‘Li in lithium metal (II), together with curves calculated from Eqs. [2] and [3]. There are no adjustable parameters in the cak~lation. Values of T,, employed are 0.55 set at 77 K and 0.19 see at 215 K (11). The value 6 = 2.0 used for 77 K is an experimental result obtained at that temperature (17); the value 6 = 30 used for 215 K is within a few percent of the experimental result at 218.5 K (18). 6 increases above 180 K because of slow self-diffusion (18). We have taken the peakto-peak width of the unsaturated line to be 6.20 Oe at both 77 and 215 K, following experimental results (II, 19). However, since the square root of the experimental second moment at 77 K is not equal to half the peak-to-peak line width (19), our use of a Gaussian line shape function is an approximation (although line width data are available for 215 K, second moment data have not been published).
296
ALEXANDER,
SAGALYN,
AND
LEFFLER
The increase in lithium line width upon saturation at 215 K is accurately described by the theory (Fig. 3). The saturation behavior at 77 K is not as well represented by the theoretical curve, though the qualitative features are reproduced. The 77 K experimental results themselves are difficult to reconcile with the theory. The experimental line width appears to reach an asymptotic value at large H, which is consistent with 6 N 3. However, since an increase in 6 shifts the onset of narrowing to larger H1 (see Fig. l), 6 = 3 would not fit the experimental data as well as 6 = 2.0 does (see Fig. 3). Redfield and Blume had similar difficulties explaining saturation of the dispersion mode signal in lithium at 77 K (20). DISCUSSION
The principal point we seek to establish is that saturation may in many cases result in significant broadening of the NMR line in solids. We underscore this point because in recent years attention has focused on saturation narrowing. Both saturation broadening and narrowing are of special interest to investigators using line widths and second moments as tools for studying solids. If saturation is not scrupulously avoided the true line width may be either larger or smaller than the linewidth obtained with lock-in detection-and in unfavourable cases the error can be hundreds of percent. Special care is necessary if T,, is long, or if either T,, or T,, is a strong function of temperature. In the former case saturation is difficult to avoid, and in the latter case a spurious temperature dependence of the line width may be recorded. The question of how much saturation maybe tolerated canbeapproached by referring to Fig. 1. Clearly, the extent to which saturation affects the line depends on the parameter 6. The only completely safe procedure is to keep the rf level low enough so that the Bloch saturation factor S (Eq. [5]) is about 0.1. For some values of a-between 10 and 20 for the case considered in Fig. l-the line may be strongly saturated without greatly affecting the lock-in absorption width. However, since the line shape changes upon saturation (5,10, II), second moment data will be incorrect if the line is saturated. At high saturation the line becomes approximately Lorentzian, with rapidly increasing second moment (the second moment of a Lorentzian is infinite). The discussion so far has been confined to lock-in detection. The saturation behavior of the lock-in absorption mode signal, u,(&), is quite different from the saturation of the slow passage absorption mode signal, vO(&), obtained without modulation. This is because modulation effects give rise to the term ~~~g27’,,(d,/D2) in Eq. [l] (5, 6). Thus, c0 cannot in general be obtained by integrating vl, in contrast to what is often assumed. In fact, u0 reaches a maximum (at exact resonance) when S = 1, whereas the maximum amplitude of Q depends on the modulation (see below). The saturation dependence of the line width will also be different for u0 and ul. One can show for a Gaussian line that (using Eq. [4]) u1 is narrowed in the limit of high H1 for all 6 < co, whereas (using formulas Gaines et al. (7) derived from the Provotorov theory) u0 remains broadened in the high-H, limit for all 6 > 9.2 It should be emphasized that the curves in Fig. 1 are in no sense universal curves. .No all-purpose parameter such as the saturation factor characterizes the theory because 2 From Fig. 1 we can see that if 6 is very large, narrowing may require such a large H1 that condition (iv) for the validity of Eq. [l] will be violated. However, in the examples presented in Fig. 1, this validity condition can be satisfied with practical modulation frequencies.
SATURATION
BROADENING
AND
NARROWING
297
both w:g and o:g’ appear in Eq. [l 1. However, for fixed line width and fixed 8 the data may be characterized by the product of T,,. By relabeling the abscissa so that y2N: T,, remains constant, one may use Fig. 1 for any other Gaussian line having a width of 6.50 Oe between inflection points. The standard operating procedure in magnetic resonance (e.g. Refs. (21, 22)) for avoiding saturation has been either to demonstrate that o1 ocH1, or to find the I$, which maximizes v1 and then operate at a considerably lower rf level. These procedures have their origin in the Bloch and BPP formulations of saturation. Goldman (5) has shou-n that for NMR in solids they yield incorrect results, again because of the term no:p’ T,n(d,/D2) in Eq. [I]. The spin system can be strongly saturated even though ?.I appears to be proportional to HI. Moreover, the value of H, which maximizes I*[ is determined by modulation saturation rather than simply by rf saturation. Therefore. o1 may reach a maximum when S 9 1. Experimental data illustrating these points may be found in Fig. 5.1 of Ref. (5) and in Fig. 5 of Ref. (10). However, a very important result of the theory (5) is that the amplitude at exact resonance of the lock-in dispersion mode signal, ul(0), reaches a maximum when S = i ~ This provides a simple method of measuring S. The old standard operating procedure may, therefore, be used-if it is applied to the center of the lock-in dispersion mode signal. This procedure may also be applied to a1 (0), the on-resonance lock-in absorption mode output at the second harmonic of the modulation frequency (23). If the spectrometer cannot detect either u1 or aZ,then second moments, line widths, and line shapes measured by lock-in methods are open to serious question (the resonance position itself should be unaffected by saturation). It may be possible to choose 11, by calculating S if T,, is known. Another way to determine S is to measure the slow passage absorption signal amplitude without modulation, a,,, as a function of N, (4,5. 7,10). At exact resonance only, U, has maximum amplitude when S = 1. In practice. low frequency noise and inadequate spectrometer response at low frequency may limit use of this technique. Another alternative to lock-in detection is Goldman’s method of slightly saturating single shot passage (Ref. (5), pp. 118-21). The utility of this method is restricted by the requirement that the magnetic field be swept several line widths in a time short compared to T,,. SUMMARY
The dependence of the NMR line width on the rf fieldamplitude HI has beencalculated for an experiment in which the lock-in absorption mode signal is detected. The results, based on theories of Provotorov and Goldman, show that saturation broadening of the line may occur and that it may be pronounced in many cases. A necessary condition for saturation broadening of a Gaussian line is T,,/T,, > 6. At large H1 the line ultimately narrows. -For T,,/T,, < 6, only saturation narrowing occurs for a Gaussian line. The numerical results are qualitatively consistent with experimental data for CaF, and lithium metal. However, the quantitative discrepancies between our numerical results and the experimental data are real, and indicate a need for further experimental tests of the Goldman and Provotorov theories. ACKNOWLEDGMENTS Dr. L. D. Jenningswas of considerableassistancewith the programming. Dr. T. V. made helpful comments upon the manuscript.
Hynes
read and
298
ALEXANDER,
SAGALYN,
AND
LEFFLER
REFERENCES I. N. BL~EMBERGEN, E. M. PURCELL, AND R. V. POUND, Phys. Reo. 73,679 (1948). 2. D. ABELL AND W. KNIGHT, Phys. Rev. 93,940 (1954). 3. A. G. RED~IELD, Whys. Rev. 98,1787 (1955). 4 B. N. PROVOTOROV, Zh. Eksp. Teor. Fiz. 41,1582 (1961); Sou. Phys. JETP 14,1126 (1962). 5. M. GOLDMAN, “Spin Temperature and Nuclear Magnetic Resonance in Solids,” Chapters 4 and
5,
Oxford University Press, Oxford, 1970. 6. M. GOLDMAN, J. Phys. (Paris) 25,843 (1964). 7. J. R. GAINES, E. M. DECASTRO, AND J. G. DAUNT, Phys. Rev. 140, A319 8. 0. P. REVOKATOV AND V. E. LYAPUKHOV, Fiz. Tverd. Tela 13,272 (1971);
(1965). Sov. Phys. SolidState 13,
213 (1971). 9. 0. P. REVOKATOV
AND V. E. LYAPUKHOV,
Fiz. Tverd. Tela 13,488 (1971); Sou. Phys. Solid State 13,
392 (1971). 10. W. R. JANZEN, J. Magn. Resonance 10,263 (1973). 11. D. F. HOLCOMB, Phys. Rev. 112,1599 (1958). 12. J. HAUNT AND R. VAN STEENWINKEL, Z. Naturfosch. 26p, 260 (1971). 13. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Section 3.2.A, Oxford University Press, Oxford, 1961. 14. R. VAN S TEENWWKEL, 2. Naturforsch. 24a, 1526 (1969). 15. R. BLINC, J. PIRS, AND S. ZUMER, Phys. Rev. B 8,15 (1973). 16. I. J. LOWE AND R. E. NORBERG, Phys. Reu. 107,46 (1957). 17. D. P. TUNSTALL, J. Magn. Resonance 6,500 (1972). 18. D. C. AILION AND C. P. SLIGHTER, Phys. Reu. 137, A235 (1965). 19. H. S. GUTOWSKY AND B. R. MCGARVEY, J. Chem. Phys. X&l472 (1952). 20. A. G. RED~IEL.D AND R. J. BLUME, Phys. Rev. 129,1545 (1963). 21. NMR-EPR Staff of Varian Associates, “NMR and EPR Spectroscopy,” Pergamon Press, pp. 9-10,77, New York, 1960. 22. D. J. E. INGRAM, “Spectroscopy at Radio and Microwave Frequencies,” second edition, pp. 237-238, Plenum Press, New York, 1967. 23. Z. TRONTEU, J. L. BIORKSTAM, AND R. JOHNSTON, J. Mug. Resonance 8,35 (1972).
OF MAGNETIC
RESONANCE
13,291-298
(1974)
MICHAEL N. ALEXANDER,PAUL L. SAGALYN, AND AMOS J. LEFFLER* Army Materials and Mechanics Research Center, Watertown, Mamachuxtts
02172
Received October 26,1973 The dependence of the NMR line width on the rf field amplitude, Hi, is calculated for an experiment in which the lock-in absorption mode is detected. It is shown that when T,,/7’,, > 6, saturation broadening of a Gaussian line is expected for intermediate rffield intensities (T,, and T,D are, respectively, the spin-lattice relaxation times for the Zeeman and dipolar energy). When the rfheld is very large, saturation narrowing is predicted. For T,,/T,, < 6, only saturation narrowing of a Gaussian line is predicted. The calculated results are in qualitative agreement with lock-in absorption mode data for CaFz and for lithium metal at 77 and 215 K. The importance of careful monitoring of saturation is discussed. INTRODUCTION
The early theory of NMR saturation in solids due to Bloembergen, Purcetl, and Pound (I) predicted that the NMR absorption line would broaden as the intensity of the radio frequency field (rf) was increased. Several years later, Abel1 and Knight (2) reported that saturation narrowing rather than saturation broadening occurred in metallic copper. Subsequently, Redfield (3) performed experiments on copper and aluminum which demonstrated conclusively that the BPP theory does not properly describe NMR saturation in solids. By introducing the concept of spin temperature in the rotating reference frame, he developed a theory of saturation that agrees with experiment at large rf intensity, Provotorov (4) extended the theory to intermediate rf intensities by replacing Redfield’s single spin temperature with separate temperatures appropriate to the Zeeman and dipole-dipole subsystems. In the Provotorov theory the rf mixes the Zeeman and dipolar subsystems, the mixing rate being proportional to the square of the rf amplitude. Thus, at high rf intensity the mixing time is short, a single spin temperature is quickly achieved, and the Provotorov theory goes over to the Redfield theory. The Redfield and Provotorov theories have been shown to predict saturation narrowing, and considerable experimental evidence has been advanced in support of this prediction (2,3,5-11).
In this paper we reconsider the question of saturation broadening in solids. Despite the apparent prevalence of saturation narrowing, broadening has been noted in solids as diverse as metallic lithium at 215 K (II) and solid benzene (12). We have recently observed saturation broadening in crystalline carboranes. In this paper we show that * National Academy of Sciences-National Research Council Senior Research Associate. on leave from Villanova University. Permanent address: Department of Chemistry, Villanova University. Villanova, Pennsylvania 19085. 291 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction Printed in Great Britain
in any form reserved.
292
ALEXANDER,
SAGALYN,
AND
LEFFLER
under suitable conditions the lock-in absorption mode line, calculated by Goldman (.5,6) from the Provotorov theory (4,5), is saturation broadened as well as narrowed. We compare our numerical results with Holcomb’s experimental linewidth data (II) (to the best of our knowledge, only Holcomb has made extensive line width measurements in the low and moderate saturation regime). We also discuss the implications of our results for experiments in which NMR is used as a tool for studying properties of solids. CALCULATION
Goldman (5,6), using the Provotorov theory, has analyzed the signals observed in a cw experiment which employs lock-in detection. In lock-in detection the large magnetic field is modulated at a frequency S2/2n, and the spectrometer detects a signal having the same frequency and phase as the modulation. We analyze this experiment because it is probably the most common scheme employed in cw wide-line NMR. Goldman’s result for the lock-in absorption mode signal, v,(&), is (5, 6) :
Ul Our notation follows that of Goldman: or = yH, = amplitude of rf field in angular frequency units; 7 = nuclear gyromagnetic ratio; do = distance from resonance (either positive or negative- do = 0 at resonance) in angular frequency units (5); D = “local field” in angular frequency units (5); g&J = unsaturated normalized absorption line shape function; r,, = spin-lattice relaxation time for Zeeman energy; T,, = spin-lattice relaxation time for dipole-dipole energy. In Eq. [l], Kis a factor which involves the amplitude of the modulation, the equilibrium magnetization, and ol. As will be seen below, K does not enter into the line width.’ Equation [l] is valid under the following conditions. (i) The modulation amplitude is much smaller than the line width. (ii) 8 Q D : the modulation frequency is much smaller than the line width, expressed in frequency units. (iii) QT,,, QT,, 9 1: the modulation period is much shorter than either the &man or dipolar spin-lattice relaxation times. (iv) rro:g(O) < Sz: the mixing time between the Zeeman and dipolar systems is much longer than one period of the modulation. Note that in the limit of negligible saturation (~1 +Q, VI =&MA,. To obtain the “peak-to-peak” line width observed in the lock-in cw experiments (for negligible saturation this is the width between inflection points of g(A,)), we simply find the A,, which satisfies dvl/dAo = 0. We shall assume that the unsaturated line is Gaussian, a reasonable approximation in many solids : g(Ao) = [l/D(67c)‘/‘]
exp(-Ag/6D2).
121
1 Revokatov and Lyapukhov have derived a generalization of Eq. [l] in which the line width depends on the modulation amplitude (9). Their equations are very complicated, and are useful only in the limit of extreme saturation.
SATURATION
BROADENING
AND
NARROWING
193
After considerable algebra, and setting A,/ D = x, we obtain (x2 - 3) + (ma;g)(9TlDx2
- 9Tl,, - 3T,,)
+ (no~g)2(3TlzTlD~2
+ 3T;,x4
- 9T,, T,, + 9Tf,x’)
= 0.
PI
Note that because g is a function of x, Eq. [3] is not simply a polynomial equation in s. Equation [3] gives the correct results for low and high saturation. In the limit of negligible saturation, o1 40, Eq. [33 gives x2 = 3, i.e., Ai = 3D*, which corresponds to the width between inflection points of a Gaussian line. In the limit of high saturation we ignore the first two terms on the left side of Eq. [3]. Therefore, to satisfy the equation, we set equal to zero the coefficient of w:, obtaining x4 + x2(8 + 3) - 3s = 0,
Pf
where 6 E T,,/T,,. This is the same result Goldman obtained directly from Eq. [ 1] (Ref. (5), p. 112). We may obtain from Eq. [3] a criterion for saturation broadening of the line. Noting that x2 = 3 for the unsaturated line, we express deviations y from the unsaturated width by defining y = x2 - 3. Substituting into Eq. [3] and requiring y > 0 (line broadening), we find that the first and third terms on the left side of the equation are positive. Since only the term containing o: can be negative, its coefficient must be negative: (2 + y - S/3) < 0.
Therefore, since y > 0,6 > 6. 6 > 6 is a necessary condition for saturation broadening of a Gaussian line. Because Eq. [3] contains both exponential and polynomial terms, we have been unable to derive a tractable analytical expression constituting a sufficient condition for saturation broadening. An alternative approach is numerical evaluation of Eq. [3]. We have not made a detailed numerical search for a sufficient condition. However, with the choice of NMR parameters to be described below, we did find saturation broadening when S = 7; we did not investigate 6 < S < 7 numerically. To find the A0 which satisfy Eqs. [2] and [3], we took T,, = 50 see and took the unsaturated peak-to-peak line width equal to 6.50 Oe, consistent with Holcomb’s data for t9F in CaF, with the large magnetic field parallel to the [l lo] direction (II); ~5was treated as a parameter. The relative linewidth-that is, the lock-in peak-to-peak line width (obtained by solving Eq. [3]) divided by the width of g(A,) (6.50 Ott)---is shown as a function of H1 in Fig. 1. The arrow in Fig. 1 denotes the value of Nj at which the Bloch saturation factor (ZJ), s = nw: g(O),
I4
is unity. It is clear from Fig. 1 that the Goldman-Provotorov theory predicts significant deviations from the unsaturated line width, even for moderate saturation. When ci is large the line broadens markedly as H1 increases, finally narrowing at large H,. Although no single parameter characterizes the saturation behavior in a universal manner, the Bloch saturation factor does appear to retain considerable qualitative significance.
294
ALEXANDER,
SAGALYN,
AND
LEFFLER
Although the values of 6 used in Fig. 1 were chosen for illustrative purposes, they are experimentally reasonable. 6 - lo5 has been observed in some molecular solids (14), and 6 21 1000 has been observed in ferroelectrics (15). In CaF, (for which Fig. 1 presents the results of a model calculation), 6 will depend upon both temperature and the nature and concentration of impurities; we know of values as large as 6 = 80 (12).
10-3
lo-4
101
10-2 HI
tOeI
FIG. 1. Theoretical saturation behavior of relative peak-to-peak NMR line width, in lock-in absorption mode detection. The arrow in the figure designates the value of HI at which the saturation factor (Eq. [5]) is unity. Values of 6 = T1x/T1n used in the calculation appear with each of the curves. The values of other parameters used in the calculation are described in the text.
COMPARISON
WITH
EXPERIMENTAL
DATA
Our calculated CaF, line widths, together with those obtained experimentally by Holcomb (II), are shown in Fig. 2. The theoretical curves in Fig. 2 differ from those in Fig. 1 only in the choice of 6. The fit is qualitative for 6 N 5-12. 81
% G61 .3! L e j
6---------
.x
--__ -.__
P -----p-A---
4 k-
Y, $
r
--6=12
t
821
i
ii lo-4
.lo-3
lo-2
HI (04
FIG. 2. Saturation behavior of lock-in absorption mode line width in CaF,. Large magnetic field is in [ 1lo] direction. Experimental data (II) are shown by circles. The lines are from the present calculation. The arrow denotes the experimental value (II) of Hi at which the saturation factor (Eq. IS]) is unity.
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Although a Gaussian was used to model the experimental CaF, line shape, the existence of beats on the free induction decay (26) demonstrates that the line is not accurately Gaussian. We have not analyzed the effect on our results of changing details of the line shape. However, we did find that changes in the width of the line did not alter the qualitative nature of the results shown in Fig. 1. In Fig. 2, the increase in the experi,mental line width when HI > 10v2 Oe is anomalous. It is difficult to reconcile such an increase with the Goldman-Provotorov theory. (The line widths measured with the magnetic field parallel to the [lOO] direction are greater than those for the [l lo] direction, but the two appear to converge when H1 is greater than 10M2 Oe (II).)
FIG. 3. Saturation behavior of lock-in absorption mode line width in lithm m&d, at 77 F&@&tern) and 215 K (top). Experimental data (If) are shown by cir&s. Tbt lines give the rewlt of the pewnt c&aUien; there are no adjustable parameters in the caiculatiera,all quantities being ned by ezqmrb#al data (seetext). The (VIOWSdenote the experimentalvalue (II), at each&m&tture, of HI at which the saturation factor (Eq. [5]) is unity.
In Fig. 3 we show experimental data for ‘Li in lithium metal (II), together with curves calculated from Eqs. [2] and [3]. There are no adjustable parameters in the cak~lation. Values of T,, employed are 0.55 set at 77 K and 0.19 see at 215 K (11). The value 6 = 2.0 used for 77 K is an experimental result obtained at that temperature (17); the value 6 = 30 used for 215 K is within a few percent of the experimental result at 218.5 K (18). 6 increases above 180 K because of slow self-diffusion (18). We have taken the peakto-peak width of the unsaturated line to be 6.20 Oe at both 77 and 215 K, following experimental results (II, 19). However, since the square root of the experimental second moment at 77 K is not equal to half the peak-to-peak line width (19), our use of a Gaussian line shape function is an approximation (although line width data are available for 215 K, second moment data have not been published).
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The increase in lithium line width upon saturation at 215 K is accurately described by the theory (Fig. 3). The saturation behavior at 77 K is not as well represented by the theoretical curve, though the qualitative features are reproduced. The 77 K experimental results themselves are difficult to reconcile with the theory. The experimental line width appears to reach an asymptotic value at large H, which is consistent with 6 N 3. However, since an increase in 6 shifts the onset of narrowing to larger H1 (see Fig. l), 6 = 3 would not fit the experimental data as well as 6 = 2.0 does (see Fig. 3). Redfield and Blume had similar difficulties explaining saturation of the dispersion mode signal in lithium at 77 K (20). DISCUSSION
The principal point we seek to establish is that saturation may in many cases result in significant broadening of the NMR line in solids. We underscore this point because in recent years attention has focused on saturation narrowing. Both saturation broadening and narrowing are of special interest to investigators using line widths and second moments as tools for studying solids. If saturation is not scrupulously avoided the true line width may be either larger or smaller than the linewidth obtained with lock-in detection-and in unfavourable cases the error can be hundreds of percent. Special care is necessary if T,, is long, or if either T,, or T,, is a strong function of temperature. In the former case saturation is difficult to avoid, and in the latter case a spurious temperature dependence of the line width may be recorded. The question of how much saturation maybe tolerated canbeapproached by referring to Fig. 1. Clearly, the extent to which saturation affects the line depends on the parameter 6. The only completely safe procedure is to keep the rf level low enough so that the Bloch saturation factor S (Eq. [5]) is about 0.1. For some values of a-between 10 and 20 for the case considered in Fig. l-the line may be strongly saturated without greatly affecting the lock-in absorption width. However, since the line shape changes upon saturation (5,10, II), second moment data will be incorrect if the line is saturated. At high saturation the line becomes approximately Lorentzian, with rapidly increasing second moment (the second moment of a Lorentzian is infinite). The discussion so far has been confined to lock-in detection. The saturation behavior of the lock-in absorption mode signal, u,(&), is quite different from the saturation of the slow passage absorption mode signal, vO(&), obtained without modulation. This is because modulation effects give rise to the term ~~~g27’,,(d,/D2) in Eq. [l] (5, 6). Thus, c0 cannot in general be obtained by integrating vl, in contrast to what is often assumed. In fact, u0 reaches a maximum (at exact resonance) when S = 1, whereas the maximum amplitude of Q depends on the modulation (see below). The saturation dependence of the line width will also be different for u0 and ul. One can show for a Gaussian line that (using Eq. [4]) u1 is narrowed in the limit of high H1 for all 6 < co, whereas (using formulas Gaines et al. (7) derived from the Provotorov theory) u0 remains broadened in the high-H, limit for all 6 > 9.2 It should be emphasized that the curves in Fig. 1 are in no sense universal curves. .No all-purpose parameter such as the saturation factor characterizes the theory because 2 From Fig. 1 we can see that if 6 is very large, narrowing may require such a large H1 that condition (iv) for the validity of Eq. [l] will be violated. However, in the examples presented in Fig. 1, this validity condition can be satisfied with practical modulation frequencies.
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both w:g and o:g’ appear in Eq. [l 1. However, for fixed line width and fixed 8 the data may be characterized by the product of T,,. By relabeling the abscissa so that y2N: T,, remains constant, one may use Fig. 1 for any other Gaussian line having a width of 6.50 Oe between inflection points. The standard operating procedure in magnetic resonance (e.g. Refs. (21, 22)) for avoiding saturation has been either to demonstrate that o1 ocH1, or to find the I$, which maximizes v1 and then operate at a considerably lower rf level. These procedures have their origin in the Bloch and BPP formulations of saturation. Goldman (5) has shou-n that for NMR in solids they yield incorrect results, again because of the term no:p’ T,n(d,/D2) in Eq. [I]. The spin system can be strongly saturated even though ?.I appears to be proportional to HI. Moreover, the value of H, which maximizes I*[ is determined by modulation saturation rather than simply by rf saturation. Therefore. o1 may reach a maximum when S 9 1. Experimental data illustrating these points may be found in Fig. 5.1 of Ref. (5) and in Fig. 5 of Ref. (10). However, a very important result of the theory (5) is that the amplitude at exact resonance of the lock-in dispersion mode signal, ul(0), reaches a maximum when S = i ~ This provides a simple method of measuring S. The old standard operating procedure may, therefore, be used-if it is applied to the center of the lock-in dispersion mode signal. This procedure may also be applied to a1 (0), the on-resonance lock-in absorption mode output at the second harmonic of the modulation frequency (23). If the spectrometer cannot detect either u1 or aZ,then second moments, line widths, and line shapes measured by lock-in methods are open to serious question (the resonance position itself should be unaffected by saturation). It may be possible to choose 11, by calculating S if T,, is known. Another way to determine S is to measure the slow passage absorption signal amplitude without modulation, a,,, as a function of N, (4,5. 7,10). At exact resonance only, U, has maximum amplitude when S = 1. In practice. low frequency noise and inadequate spectrometer response at low frequency may limit use of this technique. Another alternative to lock-in detection is Goldman’s method of slightly saturating single shot passage (Ref. (5), pp. 118-21). The utility of this method is restricted by the requirement that the magnetic field be swept several line widths in a time short compared to T,,. SUMMARY
The dependence of the NMR line width on the rf fieldamplitude HI has beencalculated for an experiment in which the lock-in absorption mode signal is detected. The results, based on theories of Provotorov and Goldman, show that saturation broadening of the line may occur and that it may be pronounced in many cases. A necessary condition for saturation broadening of a Gaussian line is T,,/T,, > 6. At large H1 the line ultimately narrows. -For T,,/T,, < 6, only saturation narrowing occurs for a Gaussian line. The numerical results are qualitatively consistent with experimental data for CaF, and lithium metal. However, the quantitative discrepancies between our numerical results and the experimental data are real, and indicate a need for further experimental tests of the Goldman and Provotorov theories. ACKNOWLEDGMENTS Dr. L. D. Jenningswas of considerableassistancewith the programming. Dr. T. V. made helpful comments upon the manuscript.
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