J . Phys. Chem. 1993,97,12949-12953
12949
Elastic Behavior of the Carbon Tetrabromide Clathrate of Hexakis(phenylthio)benzene As Determined by Brillouin Spectroscopy Darek Michalski,t Boguslaw Mr62,b.o Harry Kiefte,***Mary Anne Wbite,**tand Maynard J. Cloutert Department of Chemistry, Dalhousie University, Halifax. Nova Scotia B3H 4J3, Canada, and Physics Department, Memorial University of Newfoundland, St. John’s, Newfoundland AI B 3x7, Canada Received: March 23, 1993; In Final Form: September IS, 1993’ Brillouin scattering measurements have been carried out a t room temperature on the CBr4 clathrate of hexakis(pheny1thio)benzene. This material crystallizes in the R j space group (trigonal system), with a host:guest mole ratio of 1:2. The values of all seven adiabatic elastic constants were determined. The material is elastically more anisotropic than the previously investigated ethanol-Dianin’s compound clathrate which has the same space group, yet the bulk modulus and the linear compressibilities of these materials are similar. These investigationsreveal that noncentral forces act in the CBr4-HPTB crystal; these elastic properties may significantly affect other properties such as thermal conductivity.
Introduction Intensive investigations of organic and inorganic host-guest complexes, which belong to the wider family of nonstoichiometric compounds, have been undertaken in recent year~.l-~These materials can have important chemical and physical properties, and their structures and compositions can be used to advantage if the groundwork for the fundamental understanding of their physical properties is sufficiently developed. Clathrates are a subgroup of guest-host complexes, distinguished by the topology of their host-guest aggregate. They are composed of at least two components; the host lattice forms the basic framework that encages the guest molecules. The guest molecules are virtually trapped in the cages formed by the host molecules. There is usually an intermolecularinteraction caused by van der Waals forces between the host and guest, but it is weak in comparison with chemical bonds.415 However, recent studies of clathrate hydrates6 have shown the importance of repulsive guest-host interactions. The unusual architecture of the crystal lattice seems to be the source of many interesting and puzzling physical and chemical properties. It also offers the opportunity to deal with behavior of single molecules or their small clusters in the voids in the host lattice. Elastic constants represent the second derivatives of the energy density with respect to infinitesimal changes in strain tensor components. The stability of a lattice can be said to be prescribed by its elastic properties. For this reason knowledge of elastic properties of matter can make important contributions to understandingprocesses such as structural phase transitions7#* or transitions to superionicstatesgand full understandingof thermal physics. Although knowledge of elastic properties is important to understanding lattice dynamics, there have been very few studies of elasticity of molecular solids. One reason for this is the difficulty in obtaining sufficiently large crystals of good quality; another major problem is that many molecules pack in very low-symmetry crystal structures, making the measurement of the full set of elasticconstants a nontrivial task. Nevertheless, this information is important for full understanding of structural dynamics. Our main interests are in thermal and elastic properties of matter. With regard to the former, we have been investigating the thermal propertiesof clathrates, particularly the importance of individual heat-carrying processes in their thermal conduct Dalhourie University. t Memorial
University of Newfoundland.
#On leave from the IMtitUte of Physics, A. Mickiewicz University, Grunwaldzka 6,60-780 Poznan, Poland. *Aktract published in Advance ACS Abstrocts, November 1, 1993.
0022-3654/93/2091-12949So4.~0/0
tivity.lOJ1 The importance of clathrates to this research is the potential of singling out the effects of the guest-host interactions, compared with interactions in pure bulk molecular solids. However, the precise phonon dynamicsof most of these materials are unknown. Knowledge of elastic properties is important for the full understanding of dynamical properties, and we have initiated elasticity studies for some clathrates.12.13 The present work extends the earlier work to another inclusion compound system, particularly to investigate the elastic behavior of this system. Many clathrates, such as hydroquinone, Dianin’s compound, and related systems have hexagonal arrangements, through hydrogen bonding of OH groups of six host molecules (Figure 1). The similarity of the structure of the hexa-host system to a hexa-substituted benzene ring was the basis for the proposal and realizationof a new family of inclusion compounds.14J5 Hexakis(phenylthio)benzene,C6(SC6H5)6 (Figure 2; further abbreviated as HFTB), is an archetypal example of these hexa-substituted compounds. A molecule of HFTB is highly symmetrical (Figures 1 and 2). In the solid state, phenyl groups are situated alternatively above and below the plane of thecentral phenyl ring. HPTB crystallizes from appropriate solutions as an inclusion compound.ls All its inclusion compounds appear to have the same R j space group (a trigonal crystallographic system), yet pure HPTB docs not crystallize as a clathrand (unsolvated host lattice of the same structure as the clathrate), although the structurally similar Dianin’s compound d0es.~~J5 This property is due to the peculiar topology of clathrates, in general. They very often owe their existence to the presence of guests, as these secure the stability of the whole lattice. We report here elastic studies of the CBr4 clathrate of H P T B a full structural report will be presented elsewhere.I6 Briefly, there are three host and six guest molecules in the hexagonal unit cell. The cages each contain two guest molecules, with the 3-fold rotation axis collinear with the c axis (Figure 3). The HPTB host molecules, located on points of the 3 symmetry, form the top and the bottom of a cavity. The large thermal parameters of the CBr4 guest molecules in the X-ray study suggest disorder (static or dynamic). CBr4 was chosen as the guest since it was found that more volatile guests (e.g.,CC14)were lost on standing, with disruption of the lattice. The purpose of this work was to determine values of the elastic constants for the CBr4 clathrate of HPTB and thereby determine elastic anisotropy and the anharmonicity in this system. Earlier experiments12 have shown that increasing the guest mass or size in clathrate hydrates stiffens the lattice, As the knowledge of the @ 1993 American Chemical Society
Michalski et al.
12950 The Journal of Physical Chemistry, Vol. 97,No. 49,1993 R
Brillouin scattering provides a noncontactual probe for viscoelastic properties; it has proven to be a serviceable tool in the examination of many types of materials.19 In this technique the inelastic scattering of photons from acoustic lattice vibrations is employed to investigate the elastic properties of crystals. The Brillouin shifts measured in the experiment are directly related to the phonon velocities by the Brillouin equation
R
f
up = (Av/vo)c(n;
a b Figure 1. Close structural analogy between (a) a hydrogen-bonded hexamer unit, e.g., in Dianin’s clathrate and (b) a typical hexasubstituted
benzene analogue, e.g., the hexakis(pheny1thio)benzene molecule. The geometric as well as dimensional (distance d and d’) resemblance can be seen.
v
Figure 2. Molecular structure of hexakis(pheny1thio)benzene (HPTB).
+ n:
- 2nins cos
(1) where up is the phonon velocity of the pth mode, c is the velocity of light in vacuum, vo is the frequency of the laser light, Av is the Brillouin shift, ni and n, are the refractive indexes of the incident and scattered light respectively, and 9 is the scattering angle. These shifts correspond to the slow quasitransverse (TI), fast quasitransverse (T2), and quasilongitudinal (L) acoustical modes. If the Einstein summation scheme is used, Hooke’s law for crystals has the form20 bij = Cijkickl (2) where aijand ckl are the second-rank tensors of stress and strain, respectively, and cijkr is the fourth-rank elastic constant tensor (usually written using Voigt notation in the simplified form as a diagonal matrix with, in the general case, 21 independent components). Symmetry elements of the particular crystallographic system can reduce the number of elastic constants. The form of the elastic constant matrix also depends on the choice of the Cartesian reference coordinates with respect to the crystallographic axes. For all trigonal systems orthogonal coordinates can be chosen in such a way that the matrix has the same topology with six independent nonzero components.20921 Yet, the usual choice for the RII system (R3 and R3) follows “Standards on Piezoelectric Crystals”” withxlla and z((c.This is shown in Figure 3. In this case the elastic stiffness constant matrix is20
‘11
0 0
‘12
‘13
c14
-c2S
‘11
c13
-c14
c2S
c33
0 c44
0 0
0
‘44
‘14
c2s
(c11 - c12)/2 For the RII system, the inverse of cij, i.e., the elastic compliance constant matrix, which is directly related to thermodynamical variables, has the form20 ’11
‘12
‘13
‘14
-‘2S
sll
s13
-s14
s25
0 0
s33
0
0 0
0 2s2,
‘44
2s14
$44
2(Sll - 812) Figure 3. Illustration of the host-guest packing in the CBrr clathrate of HPTB as projected onto the ac plane of the crystallographic axes, or
equivalently onto the xz plane of the orthogonal reference coordinates
in which thestiffness tensor is written. Toshow clearly the guest molecules, two host molecules (one above and one below the cage as seen in this direction) have been excluded.
guest-host interaction is very important to the understanding of physical properties of inclusion compounds, the present study is aimed at probing this interaction through Brillouin spectroscopy. This elastic information is also essential for the future understanding of related thermodynamic quantities.
Theory of Experimental Technique We briefly outline the theory of Brillouin spectro~copy~~J8 and the method used here to calculate elastic constants.
The general equations of motion of elastic waves in crystals have nontrivial solutions only if their secular determinant vanishes,lg i.e., if kijkflflk - pU2&id= (3) where u is the appropriate sound velocity which can be measured in the Brillouin scattering experiment, qj and q k are direction cosines, p is the density of the crystal and is the Kronecker delta. The elastic constants can be determined from solution of the Christoffel equation (eq 3), although this is not trivial for low-symmetry systems. In general these equations are highly nonlinear due to coupling through products of elastic constants. One can use particular directions in the crystal for which these equations can be factored to give components of the q matrix or for which they can be simplified to the degree that particular cij values can be found. However, even in this case the phonon
The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12951
Clathrate of Hexakis(phen ylt hio)benzene velocities can lead to ambiguous results due to the existence of quadratic and/or cubic equations. This method is also limited because there are only a few such directions, even if it is possible to cut crystals to these orientations. The low-symmetry RII trigonal system has only a single direction (along the c axis) which allows direct calculation of components of the elastic stiffness constant matrix (c33 and c a in this case). To circumventthese difficulties,we used a proven computerized minimization procedure to calculate the adiabatic elastic constants from the Brillouin shift~~2-24 This procedure does not depend on special directions and is the only tractable method for this crystallographic system. The program, ELCON,24starts from an arbitrary set of elastic constants and fits all elements of the elastic constant tensor to experimental mode stiffnesses, pucxpZ, where u,,, is the experimental acoustic velocity. With every iteration an error vector is computed and its square
R I
(4)
I
I
I 0
I
-10
where ui, the standard deviation error of the experimental value, is minimized by systematically varying the elastic constants until the fit is optimized.
'
,
I
FREQUENCY SHIFT (GHt) Figure4. Representative2-h Brillouinspectrumof [0.470,0.882,-0.039] phonons in CBr4-HPTB. R is of the order lo7 counts. 1/11
Experimental Procedure HPTB was prepared by a literature route25 involving the reaction of c6c16 with C&SNa. Yellow transparent rhombohedral crystals of the CBr4 clathrate of HPTB were grown from chloroform containing excess CBr4.16 At room temperature (1 9 "C) the lattice constants of the CBr4 clathrate of HPTB (hexagonal axes) were found to be a = 14.327 A and c = 20.666
I
I
+ 10
Y
2ooo
2wo X
0
0
A.
Identification of the crystal faces was achieved by goniometry and morphological examination and was confirmed by X-ray diffraction. The c axis (of the hexagonal system) was identified with the slightly longer body diagonal of the crystal samples. The facets of the samples were { 1011). Brillouin scattering measurements were made on four different samples, of average dimensions (4 mm)3. They were cut using a wire saw manufactured by South Bay Technology Inc. with a wire blade (diameter 0.025 mm) and polished using silicon carbide grains of grade 600, 1000, and 1400. This was done to comply with the condition of 90" scattering. The density of CBr4-HPTB a t room temperature (22 "C) was found, by matching thecrystal density withaqueous RbI solutions, to be 1.8630 f 0.0003 g ~ m - which ~ , corresponds to 97% guest molecule occupancy in lacunae in the host lattice. Refractive indexes of CBr4-HPTB were determined, by comparison with standard liquids (Cargille Laboratories) of known refractive indexes, to be n, = n, = 1.803 f 0.003 and n, = 1.803 f 0.003. The Brillouin scattering experiments were carried out using the setup described in detail elsewhere.26 Incident light with wavelength X = 5 14.5 nm was provided by a single-mode argon laser (Spectra Physics 2020-03) and filtered to a power of about 40 mW. The free spectral range utilized in the experiments was 35.57 GHz and was determined by employing a standard block of fused quartz. The finesse was adjusted to 60. The scattered light was analyzed by a triple-pass piezoelectrically scanned Fabry-Perot interferometer (Burleigh R C 110) followed by a cooled photomultiplier connected to the multichannel analyzer of the data acquisition and stabilisation system (Burleigh DAS1). For these experiments, the error in the determination of the frequency shift was about 1%. Results and Discussion The shifts were measured in seven directions giving 16velocities. A representative Brillouin scatteringspectrum is shown in Figure 4. The results of the fitting procedure for the components of the
0
a
m4ooa
b
Figure 5. Sound velocities in the xy plane for (a) the CBr4 clathrate of
HPTB and (b) the ethanol clathrate of Dianin's compound. T1, T2, and L represent the slow quasitransverse, fast quasitransverse, and quasilongitudinal modes, respectively. The anisotropy of the sound velocity for CBr4-HFTB is pronounced for theTl andT2 modes. The anisotropy of the velocities in CBr4-HPTB gives an enhanced view of the rotation of the 6-fold symmetry of the velocity contour away from the Cartesian x, y axes. elastic and stiffness matrices are presented in Table I. Velocities calculated from these values of the cijmatrix were compared with the experimental ones, giving an average error of 4.1% for the slow quasitransverse modes, 3.0% for the fast quasitransverse modes, and 3.3% for the quasilongitudinal modes. For an elastic stiffness matrix to be physically meaningful in terms of crystal stability, certain conditions for the components must be fulfilled.2' For this trigonal system these conditions, Le., (cI1+ c12)c33> 2 c d , (CII- c12)c44 > 2 h 2 + cls2) and ~ 3 3 ~, 4 4 > 0, were met. During the optimization procedure the error in the function value (i.e.,Christoffel equation) was calculated. This indicated how much an elastic constant value could vary before it caused a 1% change in the function. It was expressed to resemble the standard deviation. From this the relative uncertainties in the elastic constants were estimated to be less than 0.04 X 1OloN m-2 (details in Table I). The major source of error is the quality of samples, rather typical for these types of crystals. The optical clarity was relatively poor, and there were cracks which might have been introduced by cutting and polishing during the preparation of these small crystals. The surface and any bulk defects can affect the Brillouin lines and increase the intensity of the central component. (On the other hand, in some cases small crystals have an advantage over big crystals in that they have lower concentrations of innate imperfections.) In addition
12952 The Journal of Physical Chemistry, Vol. 97, No. 49, 1993
Michalski et al.
TABLE I: The Seven Independent Components of the Stiffness and Compliance Matrices for the CBr4 Clathrate of HPTB (cu and su in Units of 1Olo N m-2 and 10-10 m2 N-1, Respectively) ij 12
13
14
25
33
44
0.1 1 0.04 -0.36
0.39 0.04 -0.20
0.14 i 0.02 -0.19
0.21 0.02 -1.16
1.69 i 0.03 0.69
0.29 i 0.02 5.88
11 C
S
1.20 1.24
0.03
*
*
to errors due to defects, there is uncertainty (few degrees) in the crystal orientation associated with cutting and polishing, and from alignment in the experimental setup. In view of all sources of error, the fit is quite good. The elastic anisotropy of the CBr4 clathrate of HPTB can be assessed from ratios that stem from conditionsfor the appropriate components of the cv matrix for an isotropic material.28 For an elastically isotropic RII system CII= ~33,c12 = c13, 2 c =~ (c11 - c12) and ~ 1 =4 CIS= 0. For CBr4-HPTB, c14 = 0.14 X 1Olo N m-2 and CIS = 4 2 1 X 1O1ON m-2 and the anisotropic factors, C I I / CC~ I~~, / Cand I ~ 2, c ~ / ( c 1 1 c- ~ z )were , found to be 0.71,0.28, and 0.53, respectively. CBr4-HPTB is elasticallymore anisotropic than the ethanol clathrate of Dianin’s compound13 (also RII system, R j structure) for which c15 = 0, c14 = 0.03 X 1OloN m-2 and the respective anisotropic factors were 1.02,0.49, and 0.86. This is seen further in Figure 5 . The R j system can be described as a link between the hexagonal classes and those of trigonal symmetry. It may be characterized by hexagonal axes of an alternating symmetry, or by a 3-fold axis with inversion. To the best of our knowledge,the only R3 trigonal systems for which the full elastic constant tensors have been determined are the ethanol clathrate of Dianin’s compound13and now CBr4-HPTB. The elasticity of the first material was successfully approximated by hexagonal symmetry.’) In contrast, the anisotropy of the elasticity of the CBr4 clathrate of HPTB differs considerablyfrom that represented by systemswith purely hexagonal symmetry. Given that the ethanol adduct of Dianin’s compound is elastically hexagonal, CBr4-HPTB appears to be the lowest symmetry inclusion compound for which elastic information has been obtained. It is instructive to compare some parameters derived from the elastic constants as they very often have more direct physical significance than the tensor itself. The adiabatic bulk modulus, Ba, Le., the ratio of applied hydrostatic pressure to resultant fractional change in volume, which can be expressed as 3
(5) is 6.1 X 109 N m-2 for the CBr4 clathrate of HPTB. This is not greatly different from that of other clathrates, e.g., the ethanol clathrate of Dianin’s compound (B, = 7.8 X lo9 N m-2)13and tetrahydrofuran clathrate hydrate (B, = 8.5 X lo9 N m-2).12 It is also similar to molecular crystals such as stilbene (B, = 6.4 X lo9 N m-2) and naphthalene (B, = 5.3 X lo9 N m-2).29 The values of B, for these inclusion compounds show these systems to be less compressible than some orientationally disordered molecular solids. For example, their values of BEare about twice that of solid CC14 in its orientationally disordered rhombohedral Zbphase(B, = 3.35 X 109Nm-2)mandinitsmetastabledisordered cubic l a phase (B, = 3.30 X 109 N m-33’ and also about twice that of the disordered fcc structure of CBr4 (B, = 3.65 X 109 N m-2).32 However,high compressibilityis not exclusively associated with disorder; it can also be due to other packing and interaction considerations as in crystals of rather flat molecules with delocalized ?r bonds, e.g., anthracene (B, = 3.84 X 109 N m-2),23 toluene (BE= 3.89 X lo9 N m-2),29and biphenyl (B, = 4.46 x lo9 N m-2).29 Experimental studies of clathrate hydrates with different guest species12 show that the guest-host force constant stiffens as the
mass or size of the guest increases. In consideration of results from molecular dynamics simulation^,^^ this can be ascribed to a coupling of the motion of the guest molecules with that of the host lattice. The coupling of the dynamical disorder of the guest molecules with low-frequency vibrations of the host lattice is thought to be responsible for the unusually low thermal conductivity of some inclusion compounds.lOJ1J4 As a result, the temperature profile of their thermoconductivity is more glasslike than that usually associated with crystalline matter. On the basis that the ethanol clathrate of Dianin’s compound and tetrahydrofuran clathrate hydrate have both similar thermal conductivities and similar compressibilities,it would not be unexpected to observein CBr4-HPTB, with its similar B, and similar structural topology, a qualitatively similar thermal conductivity profile. Another physical property, the linear compressibility,8, is the fractional change in length per unit of applied hydrostaticpressure. It varies with direction and in the hexagonal and trigonal systems depends on an angle, a,between the pressure direction and the c axis. It is expressed as
+
,9 = 0, p2 cos2 CY
(6)
where j31 = s11 + s12+ SI3 and 8 2 = s13 + s33 - SII- SIZ. For the CBr4 clathrate of HPTB, p1 = 0.68 X 10-10 m2 N-1 and p2 = -0.39 X 10-l0 m2 N-I. For the ethanol clathrate of Dianin’s compound these compressibilities are 0.49 X 10-10 m2 N-1 and -0.22 X 10-l0 m2 N-I, re~pective1y.l~The deviation of the (1,911 + lS21)/I,91lratio from unity is a measure of anisotropy, and these values show again CBr4-HPTB (ratio = 1.57) to be more anisotropicthan the ethanol clathrate of Dianin’scompound (ratio = 1.45). Nevertheless, both crystals present highly anisotropic linear compressibilitiesand in both cases the c/a ratio strongly depends on hydrostatic pressure. As with the ethanol-Dianin clathrate,” the elastic constants of CBr4-HPTB are almost an order of magnitude larger than the very soft hexagonal crystals of &N2 and j3-C0,Z6much smaller (by a factor of 10-50) than harder crystals such as A1203,35but, in fact, very similar to those of ice Zh.’6 The exception in the latter case is c12, which is larger by a factor of 7 in ice Zh (and by 3 in the ethanol clathrate of Dianin’s compound). The elastic constant c12 appears to be low in the present system (uide infra). Although the Cauchy conditions are not fulfilled even for ideal systemsbecauseof thecontributionof the temperaturedependence of the phonons, deviations of Cauchy ratios from unity clearly indicate the character of the potential in a crystal. The geometry of CBr4-HPTB reduces the six general Cauchy ratios to only two, Le., c13/cM and c12/cM(cM = (CII- c12)/2). These ratios are 1.34 and 0.20 respectively for CBr4-HPTB and 1.48 and 0.63 respectively for the ethanol clathrate of Dianin’s compound. The low values of cIzfor ethanol-Dianin and especially for CBr4HPTB are responsible for the significant deviation of the second Cauchy ratio from unity. This indicates the presence of angular forces between atoms and/or torsional interactions between molecules of these clathrates that contribute to the crystal potential. Cauchy C I ~ / C U ratios exceeding unity have been associated with rotational-translational coupling in N2, CO and Ar-02 low values of cgq, are characteristic of soft transverse modes that are susceptible to rotational-translational coupling. Such coupling between the guest and the host lattice of this system is a manifestation of anharmonic behavior that could significantly affect the thermal properties.
Clathrate of Hexakis(pheny1thio)benzene In summary, the completeset of seven elastic stiffnessconstants for the CBr4 clathrate of HPTB has been determined from Brillouin scattering measurements. This seems to be only the second investigationof this kind for this crystallographicsymmetry (RII), preceded by the examination of a clathrate of Dianin's compound.*3 The only other investigation of Brillouin scattering of an inclusion compound appears to be preliminary results for the tetrahydrofuran clathrate hydrate.12 The elastic anisotropies of these clathrates are different, but they all have comparable bulk moduli and linear compressibilities. These detailed studies of elasticity have provided information concerningintermolecular forces that is important to the overall understanding of physical properties of inclusion compounds and other molecular solids.
Acknowledgment. We acknowledge the assistance of J. S. Grossert and K. Wright in synthesisof HPTB and T. S.Cameron in crystallography. We thank K.-H. Brose for providing the ELCON program, The financial support of the Natural Sciences and Engineering Council of Canada (scholarship to D.M. and grantstoH.K., M.J.C.,and M.A.W.) is gratefully acknowledged. D.M. also holds a Killam Memorial Scholarship. References and Notes (1) Davits, J. E. D.; Kemula, W.; Powell, H. M.; Smith, N. 0.J . Inclusion Phenom. 1983,1, 3. (2) Weber, E. Top. Curr. Chem. 1987, 140, 1. (3) Atwood, J. L., Davies, J. E.D., MacNicol, D. D., Eds. Inclusion compounds; Academic Press: London, 1984; Vol. 1, Structural Aspect of Inclusion Compounds Formed by Inorganic and OrganometallicHost lattices; 1984, Vol. 2, Structural Aspect of Inclusion Compounds Formed by Organic Host Lattices; 1984, Vol. 3, Physical Properties and Applications; Oxford UniversityPress,Oxford, 1991,Vol.4,KeyOrganicHost Systems; 1991,Vol. 5, Inorganic and Physical Aspects of Inclusion, Oxford. (4) --In ref - - 3. Vol. 2. I) 1. (5) Belosudov, V. R.: Lavrentiev, M. Yu.; Dyadin, Yu. A. J . Inclusion Phenom. 1991, 10, 399. (6) Rodger, M. P. J . Phys. Chem. 1990, 94, 6080. (7) Askarpour, V.;Kiefte, H. Clouter, M. J. Can. J. Chem. 1988, 66, 541. (8) Mr62, B.; Kiefte, H.; Clouter, M. J.; Tuszynski, J. A. Phys. Rev. B 1991, 43, 641.
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The Journal of Physical Chemistry, Vol. 97, No. 49, 1993 12953 (9) Ngoepe, P. E.; Comins, J. D. Phys. Rev. Lett. 1988, 61, 978. (10) Tse, J.; White, M. A. J . Phys. Chem. 1988, 92, 5006. (11) Zakrzewski, M.; White, M. A. Phys. Rev. B 1992,45,2809. (12) Kiefte, H.; Clouter, M. J.; Gagnon, R.E. J. Phys. Chem. 1985,89, 3103. (13) Zakrzewski, M.; Mroz, B.; Kiefte, H.; White, M. A.; Clouter, M. J. J. Phys. Chem. 1991,95,1783. The value of c1) in Table I1 of this reference is listed in error as 0.3 X 1010N m-2; it should have k e n listed as C I , = 0.03 X 10'0 N m-2. (14) MacNicol, D. D.; Wilson, D. R. J . Chem. Soc., Chem. Commun. 1976, 494. (15) Hardy, A. D. U.; MacNicol, D. D.; Wilson, D. R. J . Chem. Soc., Perkin Tram. 2 1979, 1011. (16) Michalski, D.; et a/., to be published. (17) Vacher, R.; Boyer, L. Phys. Rev. B 1972, 6, 639. (18) Cummins, H. Z.; Schoen, P. E.; Arecchi, F. T. Loser Handbook; Schulz-Dubois, E. 0.. Ed.; North-Holland Amsterdam, 1972;Vol. 2, p 1029. (19) Dil, J. G. Rep. Prog. Phys. 1982,45,285. (20) Ney, J. F. Physical Properties of Crystals; Oxford Clarendon Press: New York, 1957. (21) Auld, B. A. Acousric Fields and Waves in Solids; John Wiley and Sons: New York, 1973; Vol. 1. (22) Brose, K.-H.; Eckhardt, C. J. Chem. Phys. Lett. 1986, 125, 235. (23) Dye, R. C.; Eckhardt, C. J. J . Chem. Phys. 1989, 90,2090. (24) Brose, K.-H.; ELCON. A Computer Program for Fitting Elastic Constants to Phonon Velocities; Department of Chemistry, Wayne State University, Detroit, 1989. (25) MacNicol, D. D.; Mallinson, P. R.; Murphy, A,; Sym, G. J. Tetrahedron Lett. 1982, 23, 4131. (26) Ahmad, S. F.; Kiefte, H.; Clouter, M. J.; Whitmore, M. D. Phys. Rev. B 1982, 26,4239. (27) Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Oxford Clarendon Press: London, 1954. (28) Farnell, G. W. Can. J. Phys. 1961, 39, 65. (29) Kitaigorodsky, A. I. Molecular Crystals and Molecules; Academic Press: New York, 1973. (30) Zuk, J.; Kiefte, H.; Clouter, M. J. J . Chem. Phys. 1991, 95, 1950. (31) Zuk, J.; Kiefte, H.; Clouter, M. J. J . Chem. Phys. 1990, 92, 917. (32) Zuk, J.; Brake, D. M.; Kiefte, H.; Clouter, M. J. J. Chem. Phys. 1989, 91, 5285. (33) Tse,J. S.;Klein, M. L.; McDonald, I. R. J. Chem. Phys. 1984,81, 6 146. (34) Ross, R. G. J. Inclusion Phenom. Mol. Recog. Chem. 1990,8,227. (35) Watchman, J. B.; Tefft, W. E.; Lam, D. G., Jr.; Stinchfield, R. P. J. Natl. Bur. Srd. 1960,64A, 213. (36) Gammon, P. H.; Kiefte, H.; Clouter, M. J. J. Phys. Chem. 1983,87, 4025.
