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Far-infrared spectrum of solid methane. Phase II
FAR-INFRARED
22 August 1986
CHEMICAL PHYSICS LETTERS
Volume 129, number 2
SPECTRUM
OF SOLID METHANE.
PHASE II
B.W. BARAN Adaptive
Optics Assocrates, Cambridge, MA 02140, USA
and F.D. MEDINA Department
of Physics, Florida Atlantrc Unrversity, Boca Raton, FL 33431, USA
Received 4 April 1986; in final form 12 June 1986
The volume and temperature dependences of the infrared absorption spectrum of phase II of CH, have been studied. One of the infraredactive translational modes was found to soften on approaching the transition to phase I. Mode of Griineisen parameters and the isothermal compressibility were obtained from the spectra.
1. Introduction Solid methane has been the object of considerable theoretical and experimental inquiry. It exhibits a rich, multi-phase behavior based mainly on differences in orientation of the molecules. Fig. 1 shows a schematic phase diagram. Neutron scattering by Press [ I] on CD, indicates that phase I has an fee structure, with one freely rotating molecule per unit cell. Phase II exhibits an fee structure with space group Fm3c (0:) and eight molecules per unit cell [l] . This phase has partial orientational order, with six of the molecules ordered on sites of D2, symmetry. The other two molecules are orientationally disordered. This lack of order is due to the antiferrorotational arrangement of the molecu1e.s on the D2, site, which acts to cancel the strong molecular field at the 0 sites. There is little agreement as to the orientational order of the molecules in the tetragonal phase III other than the belief that all molecules become ordered [2-51. Work done on methane has been concerned with the accurate determination of the crystal structure, particularly the ordering of the molecules on the lattice sites of each of the phases and the nature of the rotational dynamics. These questions have monopo0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
I 6-
7
TEMPERATURE
( K )
Fig. 1. P- T phase diagram of solid methane.
lized the attentions of neutron scattering, X-ray diffraction and NMR researchers [6-81, which are particularly suited to such studies. Much less work has been done to elucidate the be125
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
havior of the lattice modes. Group theoretical considerations show that there should be three Raman active modes. Two of these are translational and one is librational. Cabana and The [9] saw two weak lines at 52 and 41 cm-l at 9 K. Medina and Daniels [lo] saw three lines at 47,36 and 2 1 cm-l at 8 K. The first value was assigned to the first librational transition (ground state to first excited state) of the molecule on the D,, site, based on good agreement with the frequency and behavior of the mode calculated by Yamamoto et al. [I 1] . The other two modes are translational. Medina [ 121 performed a third study, which measured the lattice frequencies at two different molar volumes. Since rotational and translational modes behave differently under pressure, this becomes a delicate method of probing the environment at the lattice sites. A quantitative measure of these changes is the mode Grtineisen parameter. The Griineisen parameter ri for the ith mode is defined by 7i=-aln+lnI/IT,
(1)
where oi is the frequency of the ith mode and V the molar volume. Translational modes generally have higher Griineisen parameters than librational modes [13] . Medina [12] found that the high-frequency line had a Grtineisen parameter equal to 2.6 at 20 K, compared to 3.2 for the intermediate-frequency line. This was taken as further evidence that the high-frequency line is a librational mode. The low-frequency line was not well enough resolved for one of the samples. Two low-pressure far-infrared studies have been done on phase II by Savoie and Fournier [ 141 and Obriot et al. [ 151. Their results are in basic agreement. Both groups saw two peaks, at 54 and 76 cm-l. These were identified as translational modes based on their isotopic frequency shifts [14]. This work was designed to study the far-infrared active modes of phase II of solid CH, . The experiment was designed so that the temperature dependent effects could be determined separately from the volume, or pressure, dependent effects. In this way, a more complete understanding of the intermolecular potential may be possible.
126
22 August 1986
2. Experimental details The experimental method used to obtain samples of sufficient quality is very similar to that described in ref. [ 16] , and is explained in greater detail there. Briefly, methane gas (minimum purity of 99.99%) was fed into a high-pressure optical cell (designed by FDM, and described in ref. [17]) which is maintained at a constant temperature. The pressure of the fluid was slowly increased until the sample solidified. The melting parameters of methane were determined from Cheng’s [ 181 data. The samples were grown over periods of up to twelve hours and then annealed overnight to remove density gradients. The samples were then cooled nearly isochorically to the temperature of observation. This methodology allows the growth of optically good, nearly stress free samples of known initial volume. Since the molar volume of each sample remains essentially constant over each run, being constrained within the optical cell, it is possible to measure temperature-dependent effects independently of volumedependent effects. A small complication due to changes of the volume of the cell is present as the temperature is lowered. These changes have been estimated using elasticity theory [19] and Bensons’s [20] expressions for the thermal expansion coefficient and modulus of elasticity for the material from which the cell is made. The far-infrared absorption spectra were taken with an RIIC FS-720 Fourier spectrometer using a liquid helium cooled germanium bolometer. The raw spectra were analyzed using a fast Fourier (Cooley-Tukey) transform, based on an analysis by Cannes [21] . The spectra were ratioed against background runs of the empty cell.
3. Results and discussion Three samples, referred to as 1,2, and 3, were grown, respectively, at melting temperatures of 143.6, 169.2, and 228.2K. Their molar volumes remain constant within phase II, and were estimated to be 30.24,29.50, and 27.35 cm3/mole. Spectra were taken at 1 or 2 K intervals from the first appearance of phase II spectra to a low temperature of 10 K, or until phase III was reached. The temperature ranges in phase II were lo-24 K for sampe 1,12-25 K for sample 2, and 28-33 K for sample 3.
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
Two lines were seen in the phase II spectrum, in agreement with Savoie and Fournier [ 141 and Obriot et al. [15] . There appears to be a broad background, independent of the lattice absorption. This seems to be typical of simple molecular crystals [22], but no rigorous explanation has been advanced. It has been suggested [23] that sample imperfections are causing light scattering. Fig. 2 shows the temperature dependence of the observed frequencies for sample 1. The high-frequency mode exhibits a marked decrease in its frequency o2 as the temperature increases. A likely explanation for this behavior is that the mode softens as it nears the I-II transition temperature. Substances which undergo an order-disorder transition, such as NaNO,, NH,Cl and NII,Br [24] are known to have optical soft modes. This softening of the phonon mode is caused by the vanishing of a force constant [25]. In addition, a h-type anomaly appears in the specific
52
F
0
t
0
50
0
u 0
0
,.
0
0
IIO
14 TEMPERATURE
22
I8
2
( K)
big. 2. Temperature dependence of the frequencies w1 (circles) and w2 (squares) of the two infraredactive translational modes in phase II for sample 1.
22 August 1986
heat, since the contribution from the lattice vibrations dominates the thermodynamic functions at low temperatures, and lowering the frequency of the mode increases its contribution to the specific heat of the solid. The II-I transition of solid methane is marked by a A-type anomaly in the specific heat [26]. It is easy to propose a scheme for the II-I phase transition with this interpretation. The unique aspect of the structure of phase II is the ordering of the orientations of the D2, molecules, so as to effectively cancel the molecular field at the 0 sites. This allows the disordered molecules to rotate almost freely, under the influence of only a weak crystalline field, which depends only on the orientation of the disordered molecules. As the temperature rises, the mean oscillation about equilibrium of the ordered molecules must increase. This must affect the molecular field, and due to its randomness, begin to collapse the longrange order. Briganti et al. [27] measured an enormous increase in the population of the librational states, beginning just 3 K below the II-I transition temperature of the crystal at ambient pressure. In addition, the tunnelling data of Press and Kollmar [28] shows that the strength of the molecular field does decrease as the II-I transition is approached, indicating that the orientations of the ordered molecules are becoming more random. Also, the energy of the J = 0 + J = 1 rotational transition of the disordered molecules decrease dramatically just below the transition point. These data, taken together, indicate that phase II gradually begins to destabilize several kelvin below the transition temperature, because of an increase in the librational energy. The temperature dependence of w2 was fitted to the form w2 = A t B exp (-C/T), where T is the absolute temperature. The best fit, shown by the solid curve in fig. 2, was obtained withA = 76.1 cm-l, B = 2200 cm-l, and C = 140 K. A similar behavior was found in sample 2, where the best fit was obtained with.4 = 80.8 cm-l,B= 2010cm-1 and C= 150K. By contrast, the low-frequency mode has an unusual temperature dependence. Its frequency w1 is linear with temperature, being nearly constant in sample 1, while decreasing significantly in sample 2. The data points were fitted to a straight line w1 = A t BT, with A = 49.7 cm-l and B = 0.04 cm-l K-l for sample 1,A = 59.8 cm-l and B = -0.37 cm-l K-l for sample 2. 127
CHEMICAL PHYSICS LETTERS
Volume 129, number 2
22 August 1986
?r 0.2 kbar-1 found by Sprik, Nijmans and Trappeniers [29] from NMR relaxation times, or K, = 3.5 kbar-l
0
x
x
x
x
x
0
0
0
0 12
16 TEMPERATURE
20
24
28
(K)
Fig. 3. Temperature dependence of the mode Griineisen parameters y1 (circles) and 72 (crosses) in phase II.
The mode Griineisen parameters were calculated using the curves fitted to the measured frequencies from samples 1 and 2 between 12 and 24 K. The results are shown in fig. 3. The Grtineisen parameter of the high-frequency mode, r2 , is constant (about 2.4). However, that of the low-frequency mode, 71, drops from about 4 to nearly zero near the II-I transition. This unusual behavior is due to the fact that w1 decreases with temperature for sample 2, but remains nearly constant for sample 1. At 24 K, w1 has nearly the same value in both samples. Unfortunately, sample 3 exists in phase II in too narrow a temperature range to help determine which is the normal behavior for ~1. Thus, at this point, the authors do not have a possible explanation for the temperature dependence ofr1. An equivalent expression for the Griineisen parameter allows the calculation of the isothermal compressibility, K,. This is given by K, = (l/y& @wi/ W)i, , where P is the pressure in kbar. The average value below 23 K is K, = 3.72 kbar-l for both modes, and compares favorably to the value of K, = 3.2 128
which can be extracted from the Raman data of Medina and Daniels [ 10,121. Obriot et al. [ 151 also developed a simple model as an attempt to compare the Raman and infrared data known at the time, by calculating the frequencies of the lattice translations at the zone center. They neglected all orientation-dependent potentials, which was justified as it is the isotropic interaction which has most to do with the arrangement of the lattice [30] and therefore with the lattice translations. They assumed that the seond-order force constants between the molecular centers of mass were all equal. They found that the ratio of frequencies between the two IR active translational modes and the two Raman active translational modes was given by fi : 2 : 4, respectively. Medina has found the frequencies of the Raman translational modes for two samples at pressures near where the two lowest-pressure samples of this study were taken. Comparing the data at 20 K, and correcting for the slight pressure discrepancies by using Medina’s mode Griineisen parameters, the ratios work out tom : 2 :a for the lowest-pressure sample of this study and a : 2 : fl for the nextlowest-pressure sample. This is remarkable for such a simple model although these results indicate that the higher-frequency mode is also dependent on the orientational part of the potential, as we were led to believe from its behavior near the II-I phase transition.
4. Conclusions A number of far-infrared spectra at varying temperatures and molar volumes have been taken on phase II of CH,. Two lines were observed in agreement with previous low-pressure work. Their temperature dependence lends credence to the idea that the phase II transition involves a gradual weakening of the molecular field, as evidenced by the soft-mode like behavior of one of the observed translational modes. Mode Griineisen parameters and, the isothermal compressibility were calculated from the spectra.
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
Acknowledgement This research was carried out at Northeastern University with support from the United States Department of Energy.
References
PI W. Press, J. Chem. Phys. 56 (1972) 2597. PI W. Press, A. Huller, H. Stiller, W. Stirling and R. Carrat, Phys. Rev. Letters 32 (1974) 1354. [31 E. Arzi and E. Sander, Acta Cryst. A31 (1975) 5188. [41 D.R. Baer, B.A. Fraass, D.H. Riehl and R.O. Simmons, J.Chem. Phys. 68 (1978) 1411. [51 A.I. Prokhratilov and A.P. Isakina, Acta Cryst. B36 (1980) 1576. [61 J. Eckert and W. Press, Trans. Am. Cryst. Assoc. 17 (1981) 83. [71 A. Heideman, W. Press, K.J. Lushington and J.A. Morrison, J. Chem. Phys. 75 (1981) 4003. [81 A. Huller and A. Raich, J. Chem. Phys. 71 (1979) 3851. 191 A. Cabana and N.D. The, Can. J. Chem. 55 (1977) 3862. [lOI F.D. Medina and W.B. Daniels, J. Chem. Phys. 70 (1979) 242. 1111 T. Yamamoto, Y. Kataokaand K. Okada, J. Chem. Phys. 66 (1977) 2701.
22 August 1986
[12] F.D. Medina, J. Chem. Phys. 73 (1980) 77. [13] W.F. Sherman, J. Phys. Cl3 (1980) 4601. [14] R. Savoie and R.P. Fournier, Chem. Phys. Letters 7 (1970) 1. [15] J. Obriot, F. Fond&e, Ph. Marteau, H. Vu and K. Kobashi, Chem. Phys. Letters 60 (1978) 90. [16] F.D. Medina and W.B. Daniels, J. Chem. Phys. 64 (1976) 150. [17] F.D. Medina, Infrared Phys. 20 (1980) 297. [18] V.M. Cheng, W.B. Daniels and R.K. Crawford, Phys. Rev. Bll (1975) 3972. [ 191 R.J. Roark, Formulas for stress and strain (McGraw-Hill, New York, 1954) p. 264. [ZO] D.A. Benson, Ph.D. Thesis, Princeton University (1968). [21] J. Connex, Revue d’optique 40 (1961) 116. 122) L. Ozier and K. Fox, Phys. Letters A27 (1968) 274. [23] W.N. Hardy, I.F. Silvera, K.N. Klump and 0. Schnepp, Phys. Rev. Letters 21 (1968) 291. [24] J.F. Scott, Rev. Mod. Phys. 46 (1974) 82. [25] H.B. Rosenstock, J. Chem. Phys. 35 (1961) 420. [26] K. Clusius, Z. Physik. Chem. 83 (1929) 41. [27] G. Briganti, P. Calvani, F. DeLura and B. Maraviglia, Can. J. Phys. 56 (1978) 1182. [28] W. Press and A. Kollmar, Solid State Commun. 17 (1975) 405. [29] M. Sprik, T. Hijmans and J.J. Trappeniers, Physica 112B (1982) 285. [30] H. Yasuda, Progr. Theoret. Phys. 45 (1971) 1361.
129
22 August 1986
CHEMICAL PHYSICS LETTERS
Volume 129, number 2
SPECTRUM
OF SOLID METHANE.
PHASE II
B.W. BARAN Adaptive
Optics Assocrates, Cambridge, MA 02140, USA
and F.D. MEDINA Department
of Physics, Florida Atlantrc Unrversity, Boca Raton, FL 33431, USA
Received 4 April 1986; in final form 12 June 1986
The volume and temperature dependences of the infrared absorption spectrum of phase II of CH, have been studied. One of the infraredactive translational modes was found to soften on approaching the transition to phase I. Mode of Griineisen parameters and the isothermal compressibility were obtained from the spectra.
1. Introduction Solid methane has been the object of considerable theoretical and experimental inquiry. It exhibits a rich, multi-phase behavior based mainly on differences in orientation of the molecules. Fig. 1 shows a schematic phase diagram. Neutron scattering by Press [ I] on CD, indicates that phase I has an fee structure, with one freely rotating molecule per unit cell. Phase II exhibits an fee structure with space group Fm3c (0:) and eight molecules per unit cell [l] . This phase has partial orientational order, with six of the molecules ordered on sites of D2, symmetry. The other two molecules are orientationally disordered. This lack of order is due to the antiferrorotational arrangement of the molecu1e.s on the D2, site, which acts to cancel the strong molecular field at the 0 sites. There is little agreement as to the orientational order of the molecules in the tetragonal phase III other than the belief that all molecules become ordered [2-51. Work done on methane has been concerned with the accurate determination of the crystal structure, particularly the ordering of the molecules on the lattice sites of each of the phases and the nature of the rotational dynamics. These questions have monopo0 009-2614/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
I 6-
7
TEMPERATURE
( K )
Fig. 1. P- T phase diagram of solid methane.
lized the attentions of neutron scattering, X-ray diffraction and NMR researchers [6-81, which are particularly suited to such studies. Much less work has been done to elucidate the be125
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
havior of the lattice modes. Group theoretical considerations show that there should be three Raman active modes. Two of these are translational and one is librational. Cabana and The [9] saw two weak lines at 52 and 41 cm-l at 9 K. Medina and Daniels [lo] saw three lines at 47,36 and 2 1 cm-l at 8 K. The first value was assigned to the first librational transition (ground state to first excited state) of the molecule on the D,, site, based on good agreement with the frequency and behavior of the mode calculated by Yamamoto et al. [I 1] . The other two modes are translational. Medina [ 121 performed a third study, which measured the lattice frequencies at two different molar volumes. Since rotational and translational modes behave differently under pressure, this becomes a delicate method of probing the environment at the lattice sites. A quantitative measure of these changes is the mode Grtineisen parameter. The Griineisen parameter ri for the ith mode is defined by 7i=-aln+lnI/IT,
(1)
where oi is the frequency of the ith mode and V the molar volume. Translational modes generally have higher Griineisen parameters than librational modes [13] . Medina [12] found that the high-frequency line had a Grtineisen parameter equal to 2.6 at 20 K, compared to 3.2 for the intermediate-frequency line. This was taken as further evidence that the high-frequency line is a librational mode. The low-frequency line was not well enough resolved for one of the samples. Two low-pressure far-infrared studies have been done on phase II by Savoie and Fournier [ 141 and Obriot et al. [ 151. Their results are in basic agreement. Both groups saw two peaks, at 54 and 76 cm-l. These were identified as translational modes based on their isotopic frequency shifts [14]. This work was designed to study the far-infrared active modes of phase II of solid CH, . The experiment was designed so that the temperature dependent effects could be determined separately from the volume, or pressure, dependent effects. In this way, a more complete understanding of the intermolecular potential may be possible.
126
22 August 1986
2. Experimental details The experimental method used to obtain samples of sufficient quality is very similar to that described in ref. [ 16] , and is explained in greater detail there. Briefly, methane gas (minimum purity of 99.99%) was fed into a high-pressure optical cell (designed by FDM, and described in ref. [17]) which is maintained at a constant temperature. The pressure of the fluid was slowly increased until the sample solidified. The melting parameters of methane were determined from Cheng’s [ 181 data. The samples were grown over periods of up to twelve hours and then annealed overnight to remove density gradients. The samples were then cooled nearly isochorically to the temperature of observation. This methodology allows the growth of optically good, nearly stress free samples of known initial volume. Since the molar volume of each sample remains essentially constant over each run, being constrained within the optical cell, it is possible to measure temperature-dependent effects independently of volumedependent effects. A small complication due to changes of the volume of the cell is present as the temperature is lowered. These changes have been estimated using elasticity theory [19] and Bensons’s [20] expressions for the thermal expansion coefficient and modulus of elasticity for the material from which the cell is made. The far-infrared absorption spectra were taken with an RIIC FS-720 Fourier spectrometer using a liquid helium cooled germanium bolometer. The raw spectra were analyzed using a fast Fourier (Cooley-Tukey) transform, based on an analysis by Cannes [21] . The spectra were ratioed against background runs of the empty cell.
3. Results and discussion Three samples, referred to as 1,2, and 3, were grown, respectively, at melting temperatures of 143.6, 169.2, and 228.2K. Their molar volumes remain constant within phase II, and were estimated to be 30.24,29.50, and 27.35 cm3/mole. Spectra were taken at 1 or 2 K intervals from the first appearance of phase II spectra to a low temperature of 10 K, or until phase III was reached. The temperature ranges in phase II were lo-24 K for sampe 1,12-25 K for sample 2, and 28-33 K for sample 3.
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
Two lines were seen in the phase II spectrum, in agreement with Savoie and Fournier [ 141 and Obriot et al. [15] . There appears to be a broad background, independent of the lattice absorption. This seems to be typical of simple molecular crystals [22], but no rigorous explanation has been advanced. It has been suggested [23] that sample imperfections are causing light scattering. Fig. 2 shows the temperature dependence of the observed frequencies for sample 1. The high-frequency mode exhibits a marked decrease in its frequency o2 as the temperature increases. A likely explanation for this behavior is that the mode softens as it nears the I-II transition temperature. Substances which undergo an order-disorder transition, such as NaNO,, NH,Cl and NII,Br [24] are known to have optical soft modes. This softening of the phonon mode is caused by the vanishing of a force constant [25]. In addition, a h-type anomaly appears in the specific
52
F
0
t
0
50
0
u 0
0
,.
0
0
IIO
14 TEMPERATURE
22
I8
2
( K)
big. 2. Temperature dependence of the frequencies w1 (circles) and w2 (squares) of the two infraredactive translational modes in phase II for sample 1.
22 August 1986
heat, since the contribution from the lattice vibrations dominates the thermodynamic functions at low temperatures, and lowering the frequency of the mode increases its contribution to the specific heat of the solid. The II-I transition of solid methane is marked by a A-type anomaly in the specific heat [26]. It is easy to propose a scheme for the II-I phase transition with this interpretation. The unique aspect of the structure of phase II is the ordering of the orientations of the D2, molecules, so as to effectively cancel the molecular field at the 0 sites. This allows the disordered molecules to rotate almost freely, under the influence of only a weak crystalline field, which depends only on the orientation of the disordered molecules. As the temperature rises, the mean oscillation about equilibrium of the ordered molecules must increase. This must affect the molecular field, and due to its randomness, begin to collapse the longrange order. Briganti et al. [27] measured an enormous increase in the population of the librational states, beginning just 3 K below the II-I transition temperature of the crystal at ambient pressure. In addition, the tunnelling data of Press and Kollmar [28] shows that the strength of the molecular field does decrease as the II-I transition is approached, indicating that the orientations of the ordered molecules are becoming more random. Also, the energy of the J = 0 + J = 1 rotational transition of the disordered molecules decrease dramatically just below the transition point. These data, taken together, indicate that phase II gradually begins to destabilize several kelvin below the transition temperature, because of an increase in the librational energy. The temperature dependence of w2 was fitted to the form w2 = A t B exp (-C/T), where T is the absolute temperature. The best fit, shown by the solid curve in fig. 2, was obtained withA = 76.1 cm-l, B = 2200 cm-l, and C = 140 K. A similar behavior was found in sample 2, where the best fit was obtained with.4 = 80.8 cm-l,B= 2010cm-1 and C= 150K. By contrast, the low-frequency mode has an unusual temperature dependence. Its frequency w1 is linear with temperature, being nearly constant in sample 1, while decreasing significantly in sample 2. The data points were fitted to a straight line w1 = A t BT, with A = 49.7 cm-l and B = 0.04 cm-l K-l for sample 1,A = 59.8 cm-l and B = -0.37 cm-l K-l for sample 2. 127
CHEMICAL PHYSICS LETTERS
Volume 129, number 2
22 August 1986
?r 0.2 kbar-1 found by Sprik, Nijmans and Trappeniers [29] from NMR relaxation times, or K, = 3.5 kbar-l
0
x
x
x
x
x
0
0
0
0 12
16 TEMPERATURE
20
24
28
(K)
Fig. 3. Temperature dependence of the mode Griineisen parameters y1 (circles) and 72 (crosses) in phase II.
The mode Griineisen parameters were calculated using the curves fitted to the measured frequencies from samples 1 and 2 between 12 and 24 K. The results are shown in fig. 3. The Grtineisen parameter of the high-frequency mode, r2 , is constant (about 2.4). However, that of the low-frequency mode, 71, drops from about 4 to nearly zero near the II-I transition. This unusual behavior is due to the fact that w1 decreases with temperature for sample 2, but remains nearly constant for sample 1. At 24 K, w1 has nearly the same value in both samples. Unfortunately, sample 3 exists in phase II in too narrow a temperature range to help determine which is the normal behavior for ~1. Thus, at this point, the authors do not have a possible explanation for the temperature dependence ofr1. An equivalent expression for the Griineisen parameter allows the calculation of the isothermal compressibility, K,. This is given by K, = (l/y& @wi/ W)i, , where P is the pressure in kbar. The average value below 23 K is K, = 3.72 kbar-l for both modes, and compares favorably to the value of K, = 3.2 128
which can be extracted from the Raman data of Medina and Daniels [ 10,121. Obriot et al. [ 151 also developed a simple model as an attempt to compare the Raman and infrared data known at the time, by calculating the frequencies of the lattice translations at the zone center. They neglected all orientation-dependent potentials, which was justified as it is the isotropic interaction which has most to do with the arrangement of the lattice [30] and therefore with the lattice translations. They assumed that the seond-order force constants between the molecular centers of mass were all equal. They found that the ratio of frequencies between the two IR active translational modes and the two Raman active translational modes was given by fi : 2 : 4, respectively. Medina has found the frequencies of the Raman translational modes for two samples at pressures near where the two lowest-pressure samples of this study were taken. Comparing the data at 20 K, and correcting for the slight pressure discrepancies by using Medina’s mode Griineisen parameters, the ratios work out tom : 2 :a for the lowest-pressure sample of this study and a : 2 : fl for the nextlowest-pressure sample. This is remarkable for such a simple model although these results indicate that the higher-frequency mode is also dependent on the orientational part of the potential, as we were led to believe from its behavior near the II-I phase transition.
4. Conclusions A number of far-infrared spectra at varying temperatures and molar volumes have been taken on phase II of CH,. Two lines were observed in agreement with previous low-pressure work. Their temperature dependence lends credence to the idea that the phase II transition involves a gradual weakening of the molecular field, as evidenced by the soft-mode like behavior of one of the observed translational modes. Mode Griineisen parameters and, the isothermal compressibility were calculated from the spectra.
Volume 129, number 2
CHEMICAL PHYSICS LETTERS
Acknowledgement This research was carried out at Northeastern University with support from the United States Department of Energy.
References
PI W. Press, J. Chem. Phys. 56 (1972) 2597. PI W. Press, A. Huller, H. Stiller, W. Stirling and R. Carrat, Phys. Rev. Letters 32 (1974) 1354. [31 E. Arzi and E. Sander, Acta Cryst. A31 (1975) 5188. [41 D.R. Baer, B.A. Fraass, D.H. Riehl and R.O. Simmons, J.Chem. Phys. 68 (1978) 1411. [51 A.I. Prokhratilov and A.P. Isakina, Acta Cryst. B36 (1980) 1576. [61 J. Eckert and W. Press, Trans. Am. Cryst. Assoc. 17 (1981) 83. [71 A. Heideman, W. Press, K.J. Lushington and J.A. Morrison, J. Chem. Phys. 75 (1981) 4003. [81 A. Huller and A. Raich, J. Chem. Phys. 71 (1979) 3851. 191 A. Cabana and N.D. The, Can. J. Chem. 55 (1977) 3862. [lOI F.D. Medina and W.B. Daniels, J. Chem. Phys. 70 (1979) 242. 1111 T. Yamamoto, Y. Kataokaand K. Okada, J. Chem. Phys. 66 (1977) 2701.
22 August 1986
[12] F.D. Medina, J. Chem. Phys. 73 (1980) 77. [13] W.F. Sherman, J. Phys. Cl3 (1980) 4601. [14] R. Savoie and R.P. Fournier, Chem. Phys. Letters 7 (1970) 1. [15] J. Obriot, F. Fond&e, Ph. Marteau, H. Vu and K. Kobashi, Chem. Phys. Letters 60 (1978) 90. [16] F.D. Medina and W.B. Daniels, J. Chem. Phys. 64 (1976) 150. [17] F.D. Medina, Infrared Phys. 20 (1980) 297. [18] V.M. Cheng, W.B. Daniels and R.K. Crawford, Phys. Rev. Bll (1975) 3972. [ 191 R.J. Roark, Formulas for stress and strain (McGraw-Hill, New York, 1954) p. 264. [ZO] D.A. Benson, Ph.D. Thesis, Princeton University (1968). [21] J. Connex, Revue d’optique 40 (1961) 116. 122) L. Ozier and K. Fox, Phys. Letters A27 (1968) 274. [23] W.N. Hardy, I.F. Silvera, K.N. Klump and 0. Schnepp, Phys. Rev. Letters 21 (1968) 291. [24] J.F. Scott, Rev. Mod. Phys. 46 (1974) 82. [25] H.B. Rosenstock, J. Chem. Phys. 35 (1961) 420. [26] K. Clusius, Z. Physik. Chem. 83 (1929) 41. [27] G. Briganti, P. Calvani, F. DeLura and B. Maraviglia, Can. J. Phys. 56 (1978) 1182. [28] W. Press and A. Kollmar, Solid State Commun. 17 (1975) 405. [29] M. Sprik, T. Hijmans and J.J. Trappeniers, Physica 112B (1982) 285. [30] H. Yasuda, Progr. Theoret. Phys. 45 (1971) 1361.
129