Pressure and temperature dependence of the nuclear quadrupole coupling constant of 23Na in single crystal sodium nitrate

JOURNAL

OF MAGNETIC

RESONANCE

5, 416-428 (1971)

Pressureand Temperature Dependenceof the Nuclear Quadrupole Coupling Constant of 23Na in Single Crystal Sodium Nitrate* G. J. D’ALESSIO AND T. A. SCOTT Department of Physics, University of Florida, Gainesville, Florida 32601 Received, July 6,1971; accepted, August 2,197l NMR of Z3Na in single crystal sodium nitrate has been observed in the temperature interval from 25 to 295°C. The nuclear quadrupole coupling constant vq was measured as a function of temperature at atmospheric pressure and as a function of hydrostatic pressure at eleven fixed temperatures. (&$3P)Ilatm is negative at 25”C, but decreases in magnitude with increasing temperature and is zero at 205”C, and takes on increasingly larger positive values at higher temperatures. At the lambda transition (276°C) this slope abruptly drops to a much smaller value. The unusual negative value of @v&P) at low temperatures is attributed to the anisotropic compressibility of NaNO, and the consequence that this has on the contribution to the electric field gradient that is produced by the six nearest neighbor oxygens. A strong temperature dependence of vg above 150°C is associated with the phase transformation and is viewed as a consequence of increasing dynamic disorder involving hindered reorientation of nitrate groups.

INTRODUCTION

has been extensively studied for well over four decades. In 1931 Kracek (I) first noted the appearance of a gradual phase transformation in the course of a differential heating study. Later specific heat investigation (2-6) confirmed the existence of a characteristic lambda curve with the transformation beginning in the vicinity of 150°C and reaching the lambda point at 276°C. Crystals of sodium nitrate (structure Da,,, 6 RJc) consist of alternate layers of sodium ions and planar nitrate groups. At room temperature the nitrate groups are ordered such that all those in a given layer differ in orientation by 60” (or 180”) from the groups in the two adjacent nitrate layers. The crystal has been the subject of many X-ray studies (7-11). The unit cell, as shown in Fig. 1, contains two molecules and the long symmetry axis of the cell has a height equal to twelve times the spacing between adjacent layers. The triangular nitrate groups lie in planes perpendicular to this axis. The sodium ions have an environmental trigonal symmetry and the electric field gradient (EFG) tensor at the sodium site is axially symmetric. It is the dynamics of the phase transformation that generates most of the interest in sodium nitrate. Although linear thermal expansion measurements (12) associated Sodium

nitrate

* Research Supported by the National

Science Foundation. 416

NQR

OF 23NA

IN

SODIUM

417

NITRATE

ON @

Na

0

0

FIG. 1. The unit cell of sodium nitrate. For clarity only two groups of oxygen atoms are shown.

an anomalous increase in length along the trigonal axis with the transition, the early high-temperature X-ray studies (1346) indicated that in-plane rotation of nitrate groups is responsible for the specific heat anomaly. More recent X-ray studies (I 7-18) favor an order-disorder model over free rotation, but disagree on details of the model. After a number of sometimes contradictory Raman and infrared studies, the freerotation hypothesis was eliminated (19). An informative study of the temperature dependence of the 23Na nuclear quadrupole coupling constant (NQCC) in a single crystal of sodium nitrate has been made by Andrew et al. (20). An unusually rapid decrease of the NQCC occurs with increasing temperature, indicating substantial motional averaging of the EFG at the 23Na site. It is reasonable to assume that motion of the six nearest-neighbor oxygen atoms are particularly influential in producing changes of the EFG at the sodium sites. Andrew et al. hypothesized that angular oscillations of the nitrate groups increase with increasing thermal energy of the solid until reorientation jumps occur. A nitrate group attaining a disordered orientation will reduce the ordering constraints on its neighbors so that as the temperature increases there will be a cooperative decrease in stability of the ordered arrangement until the disorder is complete and the lambda point is reached. The authors further hypothesized that the reorientation of the six nearest-neighbor nitrate groups triggers an anisotropic vibration of the sodium ions in the plane normal to the trigonal axis and that this is the primary cause of the decrease of the NQCC with increasing temperature. In this paper we present the results of a study of the NQCC of 23Na in a single crystal of sodium nitrate. The hydrostatic pressure dependence of the NQCC v, has been measured at eleven fixed temperatures in the range from 25 to 280.5”C. Careful

418

D’ALESSIO

AND

SCOTT

measurements of the temperature dependence have also been made in order to obtain accurate values of (&,/Z),. The motivation for such a study is that it makes possible the separation of the intrinsic temperature variation of v, from the contributions of volume change to v,. THEORY

The value of the NQCC for 23Na in NaNO, at room temperature was found to be 334_+2 kHz by Pound (21). The magnitude of this quantity makes it advantageous to observe the nuclear quadrupole interaction as a perturbation on the four magnetic energy levels associated with the spin 3/2 23Na nucleus in an external magnetic field. The electric field gradient (EFG) at the sodium site is axially symmetric about the trigonal axis and may be specified (in units of electron charge) by a single parameter 4. In first order the quadrupole interaction produces a spectrum consisting of two satellites symmetrically disposed about the central transition at the Larmor frequency, and the splitting between these satellites is given by (22) v = (e2qQ/2h)(3 cos’ 8’ - 1)

Cl1

where 0’ is the angle between the direction of the external field and the principal 2 axis of the EFG and eQ is the quadrupole moment of the nucleus. For our experiments 8’ is set equal to zero so that v is equal to the quadrupole coupling constant e2qQlh.

The quadrupole coupling constant depends upon volume, and upon temperature due to excitation of lattice vibrations. For an axially symmetric EFG, v. may be expressed as a function of volume through go, the principal value of the EFG, and tp, the amplitudes of the normal modes of vibration; the amplitudes tp are also functions of temperature. Molecular motion at a rate faster than vQ results in averaging of the magnitude of the principal components of the EFG and an oscillation of the principal axes. For an axially symmetric EFG and small amplitude oscillations the measurable maximum principal component of EFG is given by 4 = < 4’(t) > a,(1 - 9 < e”(t) > ..L PI where < q’(t),, > is the time average of the magnitude q for the dominant principal axis and < 02(t) > av is the time average of the square of the inclination of this principal axis with respect to equilibrium. The angular amplitude may be expanded in terms of normal mode amplitudes as < e2 > ay = < C

I

(tP)2Ai

> av)

PI

where the A i are constants with A; ’ having the dimensions of moments of inertia. A theory of the temperature dependence of vp at constant volume has been given in the harmonic approximation by Bayer (23) and later generalized by Kushida (24), who obtains 1 ++v&Y = vg exp (hVi/kT) - 1>I ’ Here v, = e’q,,Q/h is the quadrupole coupling constant for a static molecule in the absence of zero-point vibrations, and vi are the lattice frequencies. At high temperatures

NQR

OF 23NA

IN

SODIUM

419

NITRATE

(kT > hvJ a very accurate limiting form of Eq. [4] is vQ(T) = v,(l+bT+c/T), where b = -$,$A, a$ and

The sum from one to M includes all lattice frequencies that satisfy the inequality, and it is assumed that higher frequencies make a negligible contribution to the temperature dependence. This limit is appropriate for our study of NaNO,. The effects of volume change were incorporated in the theory in a phenomenological manner by Kushida, Benedek, and Bloembergen (KBB) (23, and independently by Gutowsky and Williams (26). The static electric field gradient qo, the lattice frequencies vi, and the parameters A i are all functions of volume; consequently the parameters v,, b, and c depend upon volume. In order to apply Eq. [5] in the analysis of experimental data it is necessary to obtain isometric curves of vQ vs. T. Experimentally vQ is measured as a function of pressure at constant temperature. Pressure vs. volume isotherms are then used to construct vp vs. volume isotherms, from which in turn the isometric vQvs. T curves can be generated. The three parameters v,, b, and c can be deduced from three points on one such isometric curve, and their volume dependence obtained from a family of curves. Full details of this procedure are described in ref. (25) (KBB). Following KBB, it is reasonable to assume for the volume dependence of q. the form 4(V) = 4o(~o’o)wF/,)“(“)~ PI where V/V,, is the volume of the solid at any temperature and pressure relative to the volume at T = 0 K. It follows that dlnv, ~

= dlnqo __~--

= n(V).

PI dln V d In V The relative importance of the volume dependence of q. and of the vibration amplitudes on the volume dependence of vo is given by alnvQ _ dlnvo aln V ---iiln+

db~+

9

dlnV

Wal

or dlnv, aInvQ aln<62>,, Y=dlnV+aln.Y a In V ah v From the functional dependence of v, it follows that ahvQ

and

'

Wbl

420

D’ALESSIO

AND

SCOTT

Combining Eqs. [I l] and [12] gives

11131 where o! is the coefficient of thermal expansion and /3 is the isothermal compressibility. The term on the left is the temperature derivative of the Bayer-Kushida expression and the two derivatives on the right are accessible by standard experiments. EXPERIMENTAL

DETAIL

Excellent single crystals of sodium nitrate are readily produced by growth from a saturated solution. Two such samples were used. The first was cut from a large crystal obtained from Professor L. GuibC. The second was grown over a period of two weeks in our laboratory using a temperature-controlled bath. The samples were oriented with a goniometer and cut with a crystal saw into cylinders that fitted inside the rf coil. The trigonal axis of the crystal was perpendicular to the axis of the cylinder and the samples were oriented in the magnetic field so as to maximize the splitting of the satellite lines (trigonal axis parallel to the magnetic field). A beryllium-copper alloy (Berylco 25) pressure bomb, heat treated to Rockwell C-36 hardness, was used to contain the sample under pressure up to 80 kpsi delivered by an intensifier. Dow 200 silicone fluid was used as hydrostatic pressure medium. Both a manganin cell and a bourdon gauge were employed to measure the pressure, and the measurements are accurate to about one-tenth of a percent. Beryllium-copper bombs proved to be failure-prone when maintained under pressure for prolonged periods at temperatures above 240°C. The pressure bomb was wrapped with noninductive heating tape and surrounded by a thick copper container that acted as an adiabatic shield. A second heater of Nichrome wire was wound on the adiabatic shield. An aluminium radiation shield surrounded the copper container. All components were placed in a closed stainless steel cylinder that was evacuated to IO-’ Torr during a run. Temperature was measured with a platinum resistance thermometer calibrated at the U.S. National Bureau of Standards. This thermometer was set in a cavity drilled in the pressure bomb. The temperature was read on a Mueller bridge. The bridge balance was registered by a Keithley millmicro voltmeter, and automatic temperature control was achieved by sending the unbalance voltage to a programmable power supply that controlled the heater on the bomb. A differential thermocouple was used to detect temperature differences between the bomb and the adiabatic shield, and manual control was exercised on the second heater to minimize this difference. Temperature regulation was $-O.Ol”C near room temperature, declining to +O.O5”C at the highest temperature. The nuclear resonance spectrometer used a Robinson oscillator and was of conventional design. RESULTS

AND

DISCUSSION

Temperature Dependence

The temperature dependence of the NQCC of 23Na in NaNO, in the interval from 25 to 295°C is shown in Fig. 2. The results are in good agreement with previous less detailed studies (20). Of particular interest is the rapid rate of decrease of the NQCC

NQR

OF 23NA

IN SODIUM

421

NITRATE

up to the lambda point as contrasted with the much smaller and approximately constant slope in the high temperature phase. This is shown strikingly in Fig. 3 where the slope (&c/W), is plotted as a function of temperature. The first indication of the phase transformation appears in the neighborhood of 150 to 160X’, and the effect is dramatic above about 250°C. Roughly 50% of the reduction of the NQCC occurs in the last 30” interval before the transition.

I

0

0

I 100

I

I 200

TEMPERATURE

I

t --300

I

(“Cl

FIG. 2. The temperature dependence of the nuclear quadrupole coupling constant vg of 23Na in sodium nitrate at atmospheric pressure. The vertical dashed line indicates the lambda transition.

;

,;,I

0

100

-10 TEMPERATURE

200

300 (“C)

FIG. 3. The slope @v~/W)~ obtained from Fig. 2 as a function of temperature.

422

D'ALESSIO AND SCOTT

Pressure Dependence

The hydrostatic pressure dependence of the NQCC at 25, 90, 140, 175, 190, and 205°C is shown in Fig. 4. In this temperature region the NQCC decreases with increasing pressure (27), which is unusual in that a positive volume dependence is implied. Previous pressure work on NaN03 by Bernheim and Gutowsky had erroneously reported the pressure dependence to be positive at room temperature (28,29). The slope (8vQ/~P), becomes steadily less negative with increasing temperature and

0

-5 N I .x

-10

ho

-15

a -20

-25 0

IO

20

30 PRESSURE

40

50

60

70

(kpsi)

FIG. 4. The change in the nuclear quadrupole coupling constant vg of 23Na in sodium nitrate as a function of hydrostatic pressure for six fixed temperatures between 25 and 205°C.

reaches zero at about 205°C. The behavior above 205°C is shown in Fig. 5. The initial slopes from all measurements, including one above the lambda point, are plotted as a function of temperature in Fig. 6. These data clearly indicate a dynamic instability that is critically dependent upon volume. Analysis

Compressibility data on NaNO, are available from Bridgman’s work (30) only at 30°C and 75°C. Because the uniaxial coefficients of compressibility are anomalous even in this region and unusual behavior is to be expected at higher temperatures near the phase transformation, the volume dependence of the NQCC cannot be calculated reliably above about 100°C at this time. The volume dependence of the NQCC at three temperatures in the range appropriate to the P-V data is shown in Fig. 7. From this graph the isometric temperature dependence of the NQCC was found and is displayed in Fig. 8. These isometric curves are nearly linear in the temperature range covered and it follows from Eq. [S] that the term c/T is relatively negligible compared with the term bT. This is expected because of the high temperature involved in the experiment.

NQROF 23NA IN SODIUM NITRATE I

50

I

I

I

I

I

270’C

40

;

423

-

30

x 20 2 d IO

0 I IO

0

I 20

I 30

I 50

1 40

PRESSURE

I 60

70

(kpsi)

FIG. 5. The change in the nuclear quadrupole coupling constant vg of 23Na in sodium nitrate as a function of hydrostatic pressure for four fixed temperatures between 215 and 270°C.

1.2

-

.* ,”

0.8

-

: I-

0

100 TEMPERATURE

200

300 (“C,

FIO. 6. Initial pressure derivatives of the nuclear quadrupole coupling constant vg plotted vs. temperature. The vertical dashed line indicates the lambda transition temperature.

424 5

0 A

T

=104OC

-5

N I Y

-10

-15 0 ;

-20

-25

-30

V/

V,(T)

FIG. 7. The change in the nuclear quadrupole coupling constant of 23Na in sodium nitrate as a function of volume at three fixed temperatures. VO(T) is the equilibrium volume at zero pressure and specified temperature.

.

V/V(25’C)=

A

v/v

(25OC)=O.997

m

v/v

(25*c)=o.995

1.000

i

320

300

20

40

60

60

TEMPERATURE

FIG. 8. The temperature dependence of the nuclear quadrupole sodium nitrate at constant volume.

100 (“C)

coupling

constant of a3Na in

NQR

OF 23NA

IN

SODIUM

425

NITRATE

From the numerical data in Fig. 7 and Eqs. [9], [lOa], and [lob],

we obtain

n(V) = 3.48 + 0.20

and aInvQ

81n ---2

= 0.07

t-141 aln<02>.V alnv for the values of these parameters at 25°C and 1 atm. The important conclusion to be drawn is that the change in the volume of the unit cell at this temperature is itself much more effective in altering the NQCC through a strong volume dependence of qO than is the variation in the lattice amplitudes due to the same volume change. It should be mentioned that an earlier estimate of n(V) = 5.57 (31) was based on the method of Ref. (26), where the observed temperature dependence of those lattice frequencies which are assumed to be important in altering the EFG was incorporated into the calculation. The KBB analysis does not require such an assumption. The positive volume dependence of qOis unusual but may be understood by taking note of the anisotropic properties of the crystal, particularly the compressibility. Assuming that an ionic model is appropriate for calculation, the EFG at the sodium site is proportional to

7 ‘ep,

Cl51

where yi is the distance from the sodium ion to the ith charge in the lattice and 0, is the angle between the trigonal axis and Ti. The angular factor in the numerator will change due to anisotropic compression. A significant part of the total EFG may be produced by the six nearest neighbor oxygens but this is not established. However, the experimental evidence is strong that it is these oxygens that are responsible for most of the variation observed in the EFG due to anisotropic volume changes and molecular motion. The angles ei are the same for all six oxygens and the crucial observation is that the angle at 25°C is 54.298”, which is very close to the “magic angle” of 54.733” at which 3 cos’ 8 - 1 = 0. Furthermore, under hydrostatic pressure this angle increases due to the anisotropic compressibility. For a pressure of 1 kbar the angle becomes 54.345”. The resulting decrease of the numerator of Eq. [15] as 8i approaches the magic angle dominates the increase of EFG that comes from compression through the factor r; 3. This provides a consistent explanation of the unusual negative pressure dependence of vo shown in Fig. 4. It has been pointed out by Betsuyaku (32) among others that one must be cautious in using a point-charge model to calculate the EFG even in ionic crystals because charge is distributed in the bonds and may not be sufficiently localized. An a priori point-charge calculation of the EFG in NaNO, by Bernheim and Gutowsky (28) in fact does not provide a definitive explanation of the experimental data. However, the above explanation of the positive volume dependence of vg, although argued from Eq. [15], should be valid because it relies not on the accuracy of a point charge model in explaining the correct magnitude of the EFG but rather the explanation of the sense of the effect of small changes in the lattice parameters on the dominant terms of Eq. [15]. From Eq. [ 131 and experimental data one may obtain the temperature derivative of vo at constant volume. The lack of information on compressibility again limits the range of analysis; however, the fact that (ava/aP), = 0 at 205°C permits one to apply

426

D’ALESSIO

AND

SCOTT

Eq. [ 131 at this point without knowledge of /I. Data for the three temperatures 25,90, and 205°C are given in Table 1. For 205°C < T < T, the term (a/,0)(&o/P), is positive and increases in magnitude with increasing temperature. Conversely, the term (lJvQ/~T), is negative and increases greatly in magnitude as T, is approached. The behavior of (dvQ/8T),, ti’ is determined by the balance between these competing terms. Accurate measurements of compressibility are very desirable so that the constant TABLE

1

VALUES OF THE TERMS IN &. 8” (cm%g)

($1)

25 90 205 a From b From

1.39 x 10-4 1.51 x W4 Ref. Ref.

3.85 x 1O-6 3.86 x 1O-6 -

[ 131

(wanp Wz/“C) -281f15 -464i15 - 1208 f 25

- 4.76 zt 0.20 - 3.21 i. 0.20 0

-172 -128 0

-453i25 -592xt25 - 1208 + 25

(12). (30).

volume term may be calculated. This would permit a detailed comparison with the theoretical formula in Eq. [4]. The data at 205°C show that the Bayer-Kushida term is remarkably strong and are indicative therefore of a strong motional averaging process quite independent of effects of volume change. The Phase Transition

It has been argued on the basis of X-ray data that in the high temperature phase of a nitrate group with the surrounding six nearest-neighbor ions is either calcite-like (18) or aragonite-like (17). These two situations are illustrated in Fig. 9a,b. A definite conclusion could not be reached from the X-ray work due to the difficulty of incorporating oscillatory motion and correlation among nitrate groups into the proper scattering model and also because of experimental difficulties with the observations at high temperatures. An important fact deduced from the X-ray work between 25 and 200°C is that the direction of the maximum amplitude of vibration of an oxygen atom is inclined at roughly a 45” angle to the equilibrium plane of the nitrate group of which it is a constituent (10). This motion reflects the symmetry of the crystal in that its direction is approximately perpendicular to the plane formed by the two nearest-neighbor sodium ions and the nitrogen atom to which the oxygen is bound. Thus hindered rotations of the NO, groups do not take place in the plane of the NO,. The oxygen atoms will pass through points of aragonite-type coordination as they move out of this plane and then back into it on completion of the rotation. Aragonite positions have the symmetry feature that they allow an oxygen atom to be approximately equidistant from three sodium ions. Stromme (33) contends that with increasing temperature the aragonite positions become allowed equilibrium positions. He subscribes to a four-point disorder model as shown in Fig. 9c for the high temperature phase. His calculation of the configuration (T > 7” the coordination

NQR

OF 23NA

IN

SODIUM

NITRATE

0

0 (a)

421

(b)Araganite

Calcite 0 0

0

0

0 0

(cl

Four

-point

FIG. 9. A projection view of calcite and aragonite configurations of nitrate groups as seen looking down the trigonal axis. Oxygen atoms are shown by large circles, nitrogen atoms by small circles. The dashed lines in each case illustrate the alternative ordering position reached by 60” rotations. The four-point disorder model obtained from a combination of all ordered and disordered orientations for (a) and (b) is illustrated in (c).

entropy of a four-point disorder model agrees with values obtained from specific heat data (3,4). Recent neutron scattering experiments by Powell and Martel (34) indicate the existence of aragonite-type positions at high temperature. The authors also suggest that the in-plane rotation normal mode of the nitrate groups is damped or becomes soft at the lambda point. This is compatible with the strong temperature dependence of the NQCC. The gradual phase transformation and the accompanying reduction of the EFG at the sodium site begins with the onset of hindered rotation of the nitrate groups. As nitrate groups occupy disordered orientations, neighboring groups are influenced to assume disordered orientations also, the vibration force constants are reduced so that the oxygen amplitude increases. Thermal expansion permits aragonite-type orientations to become oxygen equilibrium positions. Because these positions are out of plane, vibrational components parallel to the trigonal axis develop in amplitude and the thermal expansion along that axis becomes progressively greater as more nitrate groups attain aragonite-type orientations. The increasingly large magnitude assumed by the quantity (&@P), as the temperature approaches T, is clear evidence of the role of volume in the transition. Compression increases vp by decreasing the population of aragonite-type positions and restoring a small proportion of the low-temperature

428

D’ALESSIO AND SCOTT

ordered structure. At the lambda point the distribution of nitrate groups between calcite-type and aragonite-type positions reaches equilibrium and the phase transformation is complete. It is agreed with Andrew et al. (20) that the reduction of the NQCC is due to the relative motion of the sodium ions and their neighbors, particularly the six nearest neighbor oxygen atoms. However, the motion of the sodium ions is believed not to be so important as is the appearance of new oxygen equilibrium positions and the large vibrational amplitudes associated with the oxygen atoms at elevated temperatures. ACKNOWLEDGMENTS We wish to thank P. C. Canepa for substantial assistance in the setup of the experimental apparatus. Informative conversations and correspondence with Professor E. R. Andrew, Professor D. G. Hughes, and Professor R. A. Bernheim are gratefully acknowledged. REFERENCES 1. F. C. KRACEK, J. Amer. Chem. Sot. 53, 2609 (1931). 2. H. MIEKK-OJA, Ann. Acad. Sci. Fenn. Ser. AZ7, 1 (1941). 3. V. A. ~OKOLOV AND N. E. SMIDT, Zzv. Sekt. Fiz-Khim. Anal. Inst. Obxhch. Neorg. Khim. Akad. Nauk SSSR 26, 123 (1955). 4. A. MUSTAJOKI, Ann. Acad. Sci. Fenn. Ser. AVI 6, 1 (1957). 5. V. C. REIN~B~ROUGH and F. E. W. WETMORE, Aust. J. Chem. 20, 1 (1967). 6. L. KUBICAR, Fyz. Cas. 18, 58 (1968). 7. W. L. BRAGG, Proc. Roy. Sot. London A 89,468 (1914). 8. R. W. G. WYCKOFF, Phys. Rev. 16, 149 (1920). 9. M. KANTOLA AND E. VILHONEN, Ann. Acad. Sci. Fenn. Ser. A VI 54, 1 (1960). IO. P. CHERIN, W. C. HAMILTON, AND B. POST, Acta Crystaffogr. 23,455 (1967). Il. K. V. K. RAO and K. S. MURTHY. J. Phys. Chem. Solids 31, 887 (1969). 12. J. B. AUSTIN AND R. H. H. PIERCE, JR., J. Amer. Chem. Sot. 55, 661 (1933). 13. F. C. KRACEK, E. POSNJAK, AND S. B. HENDRICKS,J. Amer. Chem. Sot. 53,3339 (1931). 14. J. M. BLJ~OET and J. A. A. KETELAAR, J. Amer. Chem. Sot. 54,625 (1932). IS. J. A. A. KETELAAR and B. STRIJIK, Rec. Trav. Chim. 64, 174 (1945). 16. P. E. TAHVONEN, Ann. Acad. Sci. Fenn. Ser. AZ 42, 1 (1947). 17. L. A. SIEGEL,J. Chem. Phys. 17, 1146 (1949). 18. Y. SHINNAKA, J. Phys. Sot. Jap. 19, 1281 (1964). 19. R. M. HEXTIZR,Spectrochim. Acta 10, 291 (1958). 20. E. R. ANDREW, R. G. EADES, J. W. HENNEL, AND D. G. HUGHES, Proc. Phys. Sac. London 79,954 (1962). 21. R. V. POUND, Phys. Rev. 79, 685 (1950). 22. M. H. COHEN and F. REIF, “Solid State Physics” (F. Seitz and D. Turnbull, Eds.), Vol. 5, p. 337, Academic Press, New York, 1957. 23. H. BAYER. 2. Phys. 130,227 (1951). 24. T. KUSHWA, J. Sci. Hiroshima Univ. Ser. A 19, 327 (1955). 25. T. KUSHIDA, G. B. BENEDEK, AND N. BLOEMBERGEN,Phys. Rev. 104, 1364 (1956). 26. H. S. GIJTOWSKY AND G. A. WILLIAMS, Phys. Rev. 105,464 (1957). 27. G. J. D’ALESSM) AND T. A. SCO?T, Bull. Amer. Phys. Sot. 15,214 (1970). 28. R. A. B~RNHEIM AND H. S. GIJT~WSKY, J. Chem. Phys. 32, 1072 (1960). 29. R. A. BERNHEIMAND H. S. GU?OWSKY, J. Chem. Phys. 54, 1431 (1970). 30. P. W. BRWGMAN, Proc. Amer. Acad. Arts Sci. 64, 51 (1925). 31. G. J. D’ALJNIO AND T. A. Scorr, Bull. Amer. Phys. Sot. 15, 1323 (1970). 32. H. BIXSUYAKU, J. Chem. Phys. 51,2546 (1969). 33. K. 0. S~~OMME, Actu Chem. Stand, 23, 1616 (1969). 34. B. M. POWELL and P. MARTEL, Bull. Amer. Phys. Sot. 16, 311 (1971).