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Pressure and temperature dependences of the dielectric constant, Raman spectra and lattice constant of SnI4†
I. Phys. Chem. Solids, 1975, Vol. 36. pp 1123-l 128. Pergamon Press
Pnnted in Great Bntain
PRESSURE AND TEMPERATURE DEPENDENCES OF THE DIELECTRIC CONSTANT, RAMAN SPECTRA AND LATTICE CONSTANT OF SnI,t P. S. PEERCY, G. A. SAMARA and B. MOROSIN SandiaLaboratories, Albuquerque,NM 87115,U.S.A.
(Received 13January 1975)
Abstract-The pressure and temperature dependences of the Raman frequencies, static dielectric constant and lattice constant of the molecular crystal SnL were investigated. These results are combined to evaluate the pure volume and pure temperature (i.e. volume-independent) dependences of the molecular polarizability and of the Raman-active phonon self-energies. The pressure results also allow the separation of the Raman modes into external and internal modes of the crystal; the external modes exhibit a much stronger pressure dependence, and thereby larger Giineisen parameters, than do the internal modes. The Raman frequencies increase (decrease) with increasing pressure (temperature) whereas the dielectric constant increases with both increasing pressure and temperature, emphasizing the importance of non-volume effects. In fact, it is found that the isobaric temperature dependence of the dielectric constant is dominated by the pure temperature dependence of the molecular polarizability. The pure volume dependence of the polarizability, on the other hand, is relatively small as is perhaps typical of most molecular crystals.
1.
INTRODUCTION
The
molecular crystal SnL has been shown by optical and electrical measurements to undergo an insulator-to-metal transition at high pressure[l]. In dealing with such transitions in molecular crystals, important questions arise as to the exact nature of the transition as well as any precursors to the transition which reflect changes in the chemical bonding and the fundamental physical responses of the material. Toward this end we have investigated the effects of pressure and temperature on the Raman spectra, static dielectric constant and lattice constant of SnL. Although the present measurements extended over a relatively small pressure range (compared to the pressure at which the material becomes a metal) with no indication for any appreciable change in the nature of the chemical bonding, these results show some interesting features which are of fundamental interest in their own right. The purpose of this paper is thus to present and discuss these results. The combination of the pressure and temperature results allowed us to evaluate the pure volume and the pure temperature (i.e. volume-independent) dependences of the molecular polarizability and of the Raman-active phonon self-energies. The pure temperature effects are determined by high order anharmonicities. The pressure dependences of the Raman spectra yield an additional benefit which should be applicable to most molecular crystals. They alow a clean separation of the modes into external and internal modes, the external modes exhibiting much stronger pressure dependences, and thereby larger Griineisen parameters, than do the internal modes. This separability should prove to be an extremely valuable tool for those molecular crystals where there is uncertainty concerning the identification of certain modes tThis work was supported by the U.S. Atomic Energy Commission.
by virtue of the inability to do detailed polarization studies, the existence of near degeneracies, or other factors. ExF’ERMENTAL. RESULTS (a) Raman scatteringresults Figure 1 shows the Raman spectra of SnL at room temperature and pressure. These data, as well as all of the data discussed below, were obtained with the 6471 A line of a krypton laser. Because of the lack of suitable single crystals, detailed polarization studies of the spectra were not performed; however, since Sn14is a relatively simple molecular crystal of cubic symmetry (Pa3), one can readily determine the origins of the various modes. For such a molecular crystal, the optic modes can be separated into internal vibrations of the Snb molecules and external modes which arise from translational and librational motions of the entire SnL molecules. The SnL molecule has Td symmetry and the Ramanactive internal modes have the irreducible representation Thr = A, t E t 2F2. In the free molecule, these vibrational modes occur at [2] 47 (E), 63(F~), 149(A,) and 216 (FJ cm-‘. In crystalline S& the corresponding internal modes occur at 53, 65, 147 and a 209, 215cm-’ pair, respectively, at atmospheric temperature and pressure. The energies of the internal vibrations are shifted slightly by the crystalline environment and the degeneracy of the threefold degenerate F2 modes is reduced by the CJ site symmetry occupied by the SnL units in the crystal. For this lower C3 symmetry, the F2 representations are expected to split into A t E representations. This splitting is clearly evident for the higher frequency F2 mode at 216 cm-’ which splits into a doublet separated by -6 cm-’ at room temperature, whereas the splitting of the lower frequency F2 mode (63 cm-‘) is less than the line-width at room temperature. At low temperature, however, this mode also splits into two modes as
1123
1124
P. S. PEERCY,G. A. SAMARAand B. MOROSIN
L MO
100
FREQUENCY
ml
SHlFlkm-'I
Fig. 1. Raman spectra of SnI, at room temperatureand pressure. The internal modes are labeled with their symmetryin the free molecule.
expected, with a separation of -2 cm-’ at 10°K. The lower frequency modes observed in the spectra consist of two translational modes and one librational mode produced by the relative motions of the tightly bound SnI, molecules. The modes occur at 23, 27 and 36cm-’ at room temperature and pressure. The data are summarized in Table 1. The temperature and pressure dependences of the Raman-active optic modes of SnL are shown in Figs. 2 and 3. (The experimental arrangement used for these measurements has been given previously[4] and will not be repeated here.) All of the modes. display normal temperature dependences with the frequencies decreasing with increasing temperature for T 35O’K. At low temperature the modes are temperature independent, and the frequency decreases essentially linearly with temperature at higher temperature. The largest temperature dependences are observed for the external modes which suggests that the temperature dependences of these modes are dominated by the volume change through thermal expansion of the sample. For the internal modes one can expect contributions to the frequency shifts due to changes in both the intermolecular and intramolecular
Fig. 2. Temperature dependences of the Raman-active modes in SnI..
separations. The measured isobaric temperature dependences of the Raman-active optic modes at 2%“K are given Table 1. Two points should be noted concerning the hydrostatic pressure dependences of the mode frequencies shown in Fig. 3. Fist, the pressure dependences of the external modes are sig&cantly larger than the pressure dependences of the internal modes. This behavior is characteristic of molecular solids and can, in fact, be used to determine the origin of the modes, i.e. whether they are internal or external vibrations [3]. Secondly, the external modes display non-linear pressure dependences at modest pressures whereas the internal modes exhibit linear pressure dependences throughout the 4 kbar range of our measurements. This non-linear behavior for the pressure dependences of the external modes is related to the similar non-linear pressure dependence of the compressibiliby K (Fig. 4). The pressure dependences of these modes are given in Table 1.
Table 1. Room temperature values for the frequencies and their logarithmic temperature and pressure derivatives of the Raman-active modes in SnL. The origins of the various modes are denoted and the isobaric temperature dependences are separated into their pure temperature and pure volume contributions w
ori&in
(a.h/aHm
Y
(alruu/a5$ - - $.umlaP), (10-5/"K)
($/kbr)
(m-l)
(10-5/'K)
+ (ahdar), W5/.K)
23
ext.
6.3 * 0.3
3.1
-50.7
-48.9
-1.8
27 36
ext. ext.
6.5 I 0.3 5.4 * 0;3
3.2 2.7
-47.4 -40.9
-50.4 -41.8
+3.0 +0.9
53
tit. (E)
1.6 f 0.1
0.8
d3.0
-12.5
-0.5
65
irt. (F2)
0.67i 0.1
0;3
-9.2
-5.3
-3.9
int. cq,
0.28t 0.05
Oil
-5.5
-2.2
-3.3
209
jnt.I
0.18t 0.05
Oil
-2.7
-1.4
-1.3
215
wt.
(F2) 0.16+ 0.05
0.1
-3.8
-1.3
-2.5
147
Pressure and temperature dependences of Ihe diefectric constant
1125
are determined from the present measurements and listed in Table 1. The’y, for the external modes are determined from the initial pressure derivatives of 01 and K.These */i are large, ranging from 2.7 to 3.2, whereas p for the internal modes are all less than 1.
Co) X-Ray me~s~r~rne~ts Lattice parameters as a fiction of pressure were determined by least-squares tits to high 28 CuK, reflections obtained with the beryllium high pressure X-ray cell described previously[4]. The results are shown in Fig. 4 in terms of the reduced lattice constant alao. Our room temperature and pressure value of a~ is 12*274(1)&t Our data can be fit to the form alao= 1 t AP t BP2 with A = -6*7(l) x 10e3/kbar and B = 4.3(4) x IO-‘/kba?. Measurements of the lattice constant as a function of temperature yielded a linear thermal expansion coefficient dln a/dT = 5.2 x lo-‘PK over the temperature interval 173-300°K. From these results the volume compressibility is K = 20.1 X 10e31kbar and the volume thermal expansivity is /3 = 15.6 x lo-‘PK. As noted by Wyckoff, an interesting structural feature of Sr& is the larger separation of the iodine atoms within Fig. 3. Hydrostatic pressure dependences of the Raman-active (4.41 A) than between (4*30&/molecules and the contrast modes in SnL at room temperature. of these separations with those in iqdineT [5]. The published structure[6] on Sn14 was determined on the basis of projected (hkO), visually estimated, intensity data which can be divided into strong peaks not very sensitive to the positions of the iodine and tin atoms (h, k divisible by 4), weak, more positional-sensitive data (h even, not divisible by 4, k odd) and a set so weak as to be unobserved (h but not k divisible by 4). Since both the Raman results and the large compressibility suggested that the intermolecul~ rather intr~ole~ul~ I-I separation were becoming shorter with pressure, and since these same separations were already the smaller ones in the published structure[6], the structure of Snb was reexamined using scintillation counter intensity data (363 independent observations using MoKa radiation of which 249 greater than 3u were included in the full matrix least-squares refinement, others only if the observed value was greater than the caiculated value). Our results place the Sn (u, u, U) at u = 0.1278 (5), 6 (u, u, u) at u = O-2533(6) and 1, at {(0*0076(6),-O&l08 (6), O-2550 (5)); the Sn-I separations are 2669(6) and 2.665f8)A; the PRESSIRE II,& int~molec~~ I-I separations are 4.338(g) and 4*368(9)%I, Fig. 4. Pressure dependence of the reduced lattice constant of and the intermoIecu1~ I-I separations are 4,208, 4,402, SnL. 4.414 and 4-486 (9) A. These results do not differ significantly from the previous results 16)when the pooled The mode Gtineisen parameters ~8 defined by the standard deviations are used. Such intermolecular separarelation tions are in general greater than those found in iodine and more closely approach those required for the iodide ion. (1) Furthermore with increasing pressure the separations which probably decrease most will be the longer IThroughout this paper, the value of the error corresuonds to intermolecular I-I separations rather the one shorter the last significant d&k in the function value. separation. We believe that with increasing pressure the SThechemical bond in iodine is 2.678A and the I, molecules lie on mirrors in a structure arranged to yield layers normal to the small rearrangement will be toward the ideal structure, u-axis with four 3~9ANed-ne~bor contacts per moiecuie and with Sn at u = 0.125, I1 at u = 0.25, and I2 at (0,0,0.25), intertayer separations of 4.052,4*353,4-398, and 4~~19[5~. resufting in a more ionic nature to the Sn-I bond.
y=(ggr=-;(y),
1126
P. S. PEEKY, G. A. SAMARA and 3. hk3ROSiN
(c) Dielectric measurements Static dielectric constant measurements were made at a frequency of 10 kHz using a General Radio Model 1615A transformer ratio arm capacitance bridge on disks of fused Sn14 powder prepared in a 0.63 cm diameter pellet maker at an applied pressure of -10 kbar. The pressure measurements were made in a piston-cylinder apparatus, consisting of a 0.95 cm i.d. x 1.91 cm o.d. x 6.35 cm long tungsten carbide chamber fitted with hardened steel binding rings and mylar insulated, tungsten carbide pistons. The measurements extended to -30 kbar. The sample disks with lead foil electrodes attached to the large faces were encapsulated in a nylon jacket and loaded into the pressure cylinder. The compressibility of nylon is close to that of SnL. The pressure dependence of the static dielectric constant E is shown in Fig. 5. Below 2 kbar sample compaction effects were noted, hence, the response below this pressure was obtained by extrapolating back to zero pressure the smooth, reproducible and reversible higher
Fig. 6. Temperature dependence of the dielectric constant of SnL at 6 kbar. The circles represent the changes in measured sample capacitance, whereas the solid line represents the data corrected for the dimensional changes of the sample.
In Figs. 5 and 6, E is found to increase with both increasing pressure and temperature, suggesting nonvolume effects must be important. DISCUSSION
(a) Phonon anhamonicities
The isobaric temperature dependence of a thermodynamic quantity I++(+= w or E for the present discussion) for cubic, or isotropic, symmetry is composed of two contributions: (i) changes which arise solely from the change in volume with temperature and (ii) the explicit temperature dependence at constant volume. Treating 4 as an explicit function of volume V and temperature T yields I
I
I
10
2Q
M
J
PRESSURElk$rl
Fig. 5. Pressure dependence of the reduced dielectric constant of SnI,. The circles represent the changes in measured sample capacitance, whereas the solid line represents the data corrected for the dimensional changes of the sample. pressure response. The data points in Fig. 5 represent the
measured fractional changes in sample capacitance C whereas the solid line represents the fractional changes in E obtained from the capacitance data by taking account of the dimensional changes of the sample due to compression, recalling that E = Ct/roA, where t and A are sample thickness and area and By= 8.85 x lo-‘* f/m is the permittivity of free space. At 1 bar E = 6~3(5) and c increases quite markedly with pressure, the initial slope being (a In E/BP)T= 3.60 x 10~*/kbar. The temperature measurements extend from 77 to 300°K and were also taken in the piston cylinder apparatus contained within a high pressure, low temperature cryostat similar to that described by Lyon et al.[71. The temperature dependence of Q, taken at 6 kbar, is shown in Fig. 6. The symbols represent actual data, and the solid line represents I after correcting the data for dimensional changes. A dashed line is used below 200°K in the region where thermal expansion data are not available.
(2) The first term on the right-hand side of eqn (2) is the pure-volume contribution to the isobaric temperature dependence of + which results from thermal expansion and the second term is the pure-temperature contribution which arises from cubic and quartic (“higher-order”) anharmonicities. Thus, measurements of S(T) and +(P) allows one to separately determine the contributions from thermal expansion and higher-order anharmonicities to the isobaric temperature dependence of $. In certain cases, one can further identify the higher-order anharmonicities as being dominated by either cubic or quartic contributions[8] and compare the results with calculation[9, lo]. To investigate the origins of the isobaric temperature dependences of the Raman-active phonons, we have evaluated the pure-volume and pure-temperature contributions to the frequency shifts via eqn (2). Results based on initial room temperature and pressure derivatives are given in Table 1. This analysis demonstrates that the temperature dependences of the phonon frequencies of the external modes are dominated by the pure-volume
Pressure and temperature dependences of fhe dielecbic constant
(thermal expansivity) contributions. In fact, for these modes the pure-volume contribution to the frequency shifts is an order of magnitude larger than the puretemperature contribution. For the case of the internal modes, the pure-volume and pure-temperature contributions are comparable (with the exception of the 53 cm-’ mode where the pure-volume contribution dominates). For these modes the frequency shifts and attendant phonon anh~monicities are small. Fu~hermore, the con~butions from higher-order anharmonicities are negative which prevents further identification of the origin of the anharmonicities. Cubic anharmonicities result in negative definite contributions to the phonon self-energies whereas the contributions from quartic anharmonicities can be either negative or positive[S]. Therefore one cannot identify whether the higher-order contributions to the phonon self-energies result from cubic or quartic anharmonicities.
1127
second contribution may be obtained from eqn (3) since
All quantities in eqns (4a) and (Sa) are thus known except (3 In a/U)“, which can then be determined by substitution. The various quantities evaluated at 293°K are given in eqns (4b) and (5b), where there is a one-to-one co~es~ndence between the terms in eqns (4a) and (4b) and in eqns (Sa) and (!%). l-84 = -1.56 t 0.37 t 3_03(inunits of lO+PK),
(4b)
1.55 = 2,0110.46 (in units of IO-‘/kbar).
(5b)
and
Note from eqn (5b) that the two volume-dependent contributions K and -~(a in a/a In V)T have opposite signs. The K contribution is positive and its origin is clear: it is a density term which accounts for the increase in the The pressure and temperature measurements of Qgiven density of pol~~ble molecules. The K(a In a/a In V)T above are sufficient to identify the origins of the isobaric contribution on the other hand is negative and much temperatnre dependence of the dielectric constant. At smaller than the K contribution, and the latter thus room temperature, these data and eqn (2) yield the dominates in determining the relatively large observed pure-volume contribution [-/3l~(a In e/J& = -27-9x increase in l with pressure. The quantity (a In a/t? In V)T 10mSloK and the pure temperature contribution is relatively small and positive, being = 0.23. Its positive (a In r/U)” = 69.2 x IO-‘PK. These two contributions sign can be qualitatively understood simply on the basis of the fact that the polarizability of a molecule or group of have different signs, and the isobaric temperature dependence of E is dominated by the higher-order molecules in an electric field increases as the available anharmonic (pure-temperature) contribution. This qualitavolume for molecular distortion increases-i.e. the tive behavior was evident from the data since t increased restoring forces between molecules decreases, and with both increasing temperature and pressure. thereby a increases, with increasing volume. The small Because SnL has cubic symmetry, we can examine the mag~~de of (a In crla In V), is probably characteristic tem~ra~re and pressure dependences of E from a more of molecular crystals. For normal ionic crystals such as fundamental point of view. For a cubic or isotropic the alkaii halides (a ln cu/In V)T = l-2. As can be seen from eqn (4b) the two volume material, the total polarizability a of a macroscopic small sphere of volume V is related to the dielectric constant by contributions, -p and @(a In a/a In V), to the temperathe macroscopic Clausius-Mossotti relation[ll, 121, ture dependence of e also have opposite signs, with the former dominating so that the sum, i.e. the pure volume c-l 47ra effect, is negative. Very clearly the observed isobaric -=__, 1-l-2 3v increase of E with temperature arises from the dominance of pure temperature effect (a In a/H)“. The origin of this Combining eqns (2) and (3) allows one to evaluate the effect can be understood in terms of a model of particle or isobaric temperature dependence of e in terms of the molecule in a potential well. For a parabolic well the force pure-volume and pure-temperat~e con~butions to the constant k is independent of amplitude and the polariza~l~zab~ity a as bility, which is proportional to l/k, is thus independent of temperat~e. Any deviations from a parabolic potential causes k, and hence a, to be amplitude- or temperaturedependent. A detailed understanding of this effect would require knowledge of the electronic and lattice contribu@a) tions to a as well as the electronic structure and lattice dynamics of SnI,. It is emphasized here that the above a The right-hand side of eqn (4a) consists of three is the total macroscopic polarizability of SnL and consists contributions. The first, ,9, represents the change in e due of the sum of the electronic polarizability arkc and the to change in density. The second, @(a In a/a In V), is the lattice polarizability aI. To separate the two effects and change in e due to the change in the polarizability of a extend the above analysis further, measurements of the fixed number of molecules with changing volume. In this refractive index, and its pressure and temperature case, these two cont~~utions together constitute the pure dependences are also needed. Such meas~ements yield volume effect. The third ~n~bution, (a In a/a ln T)“, is information on mG which undoubtedIy plays an importhe voIume-independent, or pure tempera~re, effect. The tant role in the case of molecular crystals like SnL.
P. S. PEERCY, G. A. SAMARA and B. MOROSIN
1128 suMiuARY
We have reexamined the structure of the molecular crystal Snb and measured the thermal expansivity and volume compressibility using X-ray techniques. These results were combined with measurements of the pressure and temperature dependences of the static dielectric constant and Raman-active phonons to evaluate the anharmonic contributions to the phonon self-energies and to the isobaric temperature dependence of the dielectric constant. Although this material is very compressible, we found that the isobaric temperature dependence of the polarizability is dominated by higher-order anharmonicities rather than by the volume contribution. The external Raman-active phonons displayed large temperature and pressure dependences which are dominated by the changes in volume with temperature and pressure; contrary to this behavior, the internal molecular vibrations were much less sensitive to temperature and pressure and their isobaric temperature dependences consist of comparable contributions from higher-order anharmonicities and thermal expansion.
Acknowledgements-Tlte expert tecbniial assistance of J. D. Rluck, B. E. Hammons, and R. A. Trudo in making the measurements is gratefully acknowledged.
REFERJtNCES
1. Riggleman B. M. and Drickamer H. G., J. Chem. Phys. 38, 2721 (1%3). 2. Stammreich H., Fomeris R. and Tavares Y., J. Chem. Phys. 25, 1278(1956). 3. Peercy P. S. and Samara G. A., Phys. Rev. B8, 2033 (1973). 4. Morosin B. and Schirber J. E., J. Appl. Cryst. 7,295 (1974). 5. Wyckoff R. W. G., Crystal Structures Vol. 1, p. 53, Vol. 2, p. 131. Interacience, New York (1%5). 6. Mellor F. and Fankucken I., Acta Cryst. 8, 343 (1955). 7. Lyon D. N., McWban D. B. and Stevens A. L., Reo. Sci. Inst. 38, 1234(1967). 8. See, e.g. Samara G. A. and Peercy P. S., Phys. Rev. B7, 1131 (1973). 9. Maradudin A. A. and Fien A. E., Phys. Rev. 128,2589(1962). 10. Cowley R. A., Adu. Phys. 12, 421 (1963);Phil. Msg. 11, 673 (1965). 11. FrBhlich, Theory ofDielectrics Appendix 3, p. 169.Clarendon Press, Oxford (1949). 12. Bosman A. J. and Havinga E. E., Phys. Rev. 129,1593(1963).
Pnnted in Great Bntain
PRESSURE AND TEMPERATURE DEPENDENCES OF THE DIELECTRIC CONSTANT, RAMAN SPECTRA AND LATTICE CONSTANT OF SnI,t P. S. PEERCY, G. A. SAMARA and B. MOROSIN SandiaLaboratories, Albuquerque,NM 87115,U.S.A.
(Received 13January 1975)
Abstract-The pressure and temperature dependences of the Raman frequencies, static dielectric constant and lattice constant of the molecular crystal SnL were investigated. These results are combined to evaluate the pure volume and pure temperature (i.e. volume-independent) dependences of the molecular polarizability and of the Raman-active phonon self-energies. The pressure results also allow the separation of the Raman modes into external and internal modes of the crystal; the external modes exhibit a much stronger pressure dependence, and thereby larger Giineisen parameters, than do the internal modes. The Raman frequencies increase (decrease) with increasing pressure (temperature) whereas the dielectric constant increases with both increasing pressure and temperature, emphasizing the importance of non-volume effects. In fact, it is found that the isobaric temperature dependence of the dielectric constant is dominated by the pure temperature dependence of the molecular polarizability. The pure volume dependence of the polarizability, on the other hand, is relatively small as is perhaps typical of most molecular crystals.
1.
INTRODUCTION
The
molecular crystal SnL has been shown by optical and electrical measurements to undergo an insulator-to-metal transition at high pressure[l]. In dealing with such transitions in molecular crystals, important questions arise as to the exact nature of the transition as well as any precursors to the transition which reflect changes in the chemical bonding and the fundamental physical responses of the material. Toward this end we have investigated the effects of pressure and temperature on the Raman spectra, static dielectric constant and lattice constant of SnL. Although the present measurements extended over a relatively small pressure range (compared to the pressure at which the material becomes a metal) with no indication for any appreciable change in the nature of the chemical bonding, these results show some interesting features which are of fundamental interest in their own right. The purpose of this paper is thus to present and discuss these results. The combination of the pressure and temperature results allowed us to evaluate the pure volume and the pure temperature (i.e. volume-independent) dependences of the molecular polarizability and of the Raman-active phonon self-energies. The pure temperature effects are determined by high order anharmonicities. The pressure dependences of the Raman spectra yield an additional benefit which should be applicable to most molecular crystals. They alow a clean separation of the modes into external and internal modes, the external modes exhibiting much stronger pressure dependences, and thereby larger Griineisen parameters, than do the internal modes. This separability should prove to be an extremely valuable tool for those molecular crystals where there is uncertainty concerning the identification of certain modes tThis work was supported by the U.S. Atomic Energy Commission.
by virtue of the inability to do detailed polarization studies, the existence of near degeneracies, or other factors. ExF’ERMENTAL. RESULTS (a) Raman scatteringresults Figure 1 shows the Raman spectra of SnL at room temperature and pressure. These data, as well as all of the data discussed below, were obtained with the 6471 A line of a krypton laser. Because of the lack of suitable single crystals, detailed polarization studies of the spectra were not performed; however, since Sn14is a relatively simple molecular crystal of cubic symmetry (Pa3), one can readily determine the origins of the various modes. For such a molecular crystal, the optic modes can be separated into internal vibrations of the Snb molecules and external modes which arise from translational and librational motions of the entire SnL molecules. The SnL molecule has Td symmetry and the Ramanactive internal modes have the irreducible representation Thr = A, t E t 2F2. In the free molecule, these vibrational modes occur at [2] 47 (E), 63(F~), 149(A,) and 216 (FJ cm-‘. In crystalline S& the corresponding internal modes occur at 53, 65, 147 and a 209, 215cm-’ pair, respectively, at atmospheric temperature and pressure. The energies of the internal vibrations are shifted slightly by the crystalline environment and the degeneracy of the threefold degenerate F2 modes is reduced by the CJ site symmetry occupied by the SnL units in the crystal. For this lower C3 symmetry, the F2 representations are expected to split into A t E representations. This splitting is clearly evident for the higher frequency F2 mode at 216 cm-’ which splits into a doublet separated by -6 cm-’ at room temperature, whereas the splitting of the lower frequency F2 mode (63 cm-‘) is less than the line-width at room temperature. At low temperature, however, this mode also splits into two modes as
1123
1124
P. S. PEERCY,G. A. SAMARAand B. MOROSIN
L MO
100
FREQUENCY
ml
SHlFlkm-'I
Fig. 1. Raman spectra of SnI, at room temperatureand pressure. The internal modes are labeled with their symmetryin the free molecule.
expected, with a separation of -2 cm-’ at 10°K. The lower frequency modes observed in the spectra consist of two translational modes and one librational mode produced by the relative motions of the tightly bound SnI, molecules. The modes occur at 23, 27 and 36cm-’ at room temperature and pressure. The data are summarized in Table 1. The temperature and pressure dependences of the Raman-active optic modes of SnL are shown in Figs. 2 and 3. (The experimental arrangement used for these measurements has been given previously[4] and will not be repeated here.) All of the modes. display normal temperature dependences with the frequencies decreasing with increasing temperature for T 35O’K. At low temperature the modes are temperature independent, and the frequency decreases essentially linearly with temperature at higher temperature. The largest temperature dependences are observed for the external modes which suggests that the temperature dependences of these modes are dominated by the volume change through thermal expansion of the sample. For the internal modes one can expect contributions to the frequency shifts due to changes in both the intermolecular and intramolecular
Fig. 2. Temperature dependences of the Raman-active modes in SnI..
separations. The measured isobaric temperature dependences of the Raman-active optic modes at 2%“K are given Table 1. Two points should be noted concerning the hydrostatic pressure dependences of the mode frequencies shown in Fig. 3. Fist, the pressure dependences of the external modes are sig&cantly larger than the pressure dependences of the internal modes. This behavior is characteristic of molecular solids and can, in fact, be used to determine the origin of the modes, i.e. whether they are internal or external vibrations [3]. Secondly, the external modes display non-linear pressure dependences at modest pressures whereas the internal modes exhibit linear pressure dependences throughout the 4 kbar range of our measurements. This non-linear behavior for the pressure dependences of the external modes is related to the similar non-linear pressure dependence of the compressibiliby K (Fig. 4). The pressure dependences of these modes are given in Table 1.
Table 1. Room temperature values for the frequencies and their logarithmic temperature and pressure derivatives of the Raman-active modes in SnL. The origins of the various modes are denoted and the isobaric temperature dependences are separated into their pure temperature and pure volume contributions w
ori&in
(a.h/aHm
Y
(alruu/a5$ - - $.umlaP), (10-5/"K)
($/kbr)
(m-l)
(10-5/'K)
+ (ahdar), W5/.K)
23
ext.
6.3 * 0.3
3.1
-50.7
-48.9
-1.8
27 36
ext. ext.
6.5 I 0.3 5.4 * 0;3
3.2 2.7
-47.4 -40.9
-50.4 -41.8
+3.0 +0.9
53
tit. (E)
1.6 f 0.1
0.8
d3.0
-12.5
-0.5
65
irt. (F2)
0.67i 0.1
0;3
-9.2
-5.3
-3.9
int. cq,
0.28t 0.05
Oil
-5.5
-2.2
-3.3
209
jnt.I
0.18t 0.05
Oil
-2.7
-1.4
-1.3
215
wt.
(F2) 0.16+ 0.05
0.1
-3.8
-1.3
-2.5
147
Pressure and temperature dependences of Ihe diefectric constant
1125
are determined from the present measurements and listed in Table 1. The’y, for the external modes are determined from the initial pressure derivatives of 01 and K.These */i are large, ranging from 2.7 to 3.2, whereas p for the internal modes are all less than 1.
Co) X-Ray me~s~r~rne~ts Lattice parameters as a fiction of pressure were determined by least-squares tits to high 28 CuK, reflections obtained with the beryllium high pressure X-ray cell described previously[4]. The results are shown in Fig. 4 in terms of the reduced lattice constant alao. Our room temperature and pressure value of a~ is 12*274(1)&t Our data can be fit to the form alao= 1 t AP t BP2 with A = -6*7(l) x 10e3/kbar and B = 4.3(4) x IO-‘/kba?. Measurements of the lattice constant as a function of temperature yielded a linear thermal expansion coefficient dln a/dT = 5.2 x lo-‘PK over the temperature interval 173-300°K. From these results the volume compressibility is K = 20.1 X 10e31kbar and the volume thermal expansivity is /3 = 15.6 x lo-‘PK. As noted by Wyckoff, an interesting structural feature of Sr& is the larger separation of the iodine atoms within Fig. 3. Hydrostatic pressure dependences of the Raman-active (4.41 A) than between (4*30&/molecules and the contrast modes in SnL at room temperature. of these separations with those in iqdineT [5]. The published structure[6] on Sn14 was determined on the basis of projected (hkO), visually estimated, intensity data which can be divided into strong peaks not very sensitive to the positions of the iodine and tin atoms (h, k divisible by 4), weak, more positional-sensitive data (h even, not divisible by 4, k odd) and a set so weak as to be unobserved (h but not k divisible by 4). Since both the Raman results and the large compressibility suggested that the intermolecul~ rather intr~ole~ul~ I-I separation were becoming shorter with pressure, and since these same separations were already the smaller ones in the published structure[6], the structure of Snb was reexamined using scintillation counter intensity data (363 independent observations using MoKa radiation of which 249 greater than 3u were included in the full matrix least-squares refinement, others only if the observed value was greater than the caiculated value). Our results place the Sn (u, u, U) at u = 0.1278 (5), 6 (u, u, u) at u = O-2533(6) and 1, at {(0*0076(6),-O&l08 (6), O-2550 (5)); the Sn-I separations are 2669(6) and 2.665f8)A; the PRESSIRE II,& int~molec~~ I-I separations are 4.338(g) and 4*368(9)%I, Fig. 4. Pressure dependence of the reduced lattice constant of and the intermoIecu1~ I-I separations are 4,208, 4,402, SnL. 4.414 and 4-486 (9) A. These results do not differ significantly from the previous results 16)when the pooled The mode Gtineisen parameters ~8 defined by the standard deviations are used. Such intermolecular separarelation tions are in general greater than those found in iodine and more closely approach those required for the iodide ion. (1) Furthermore with increasing pressure the separations which probably decrease most will be the longer IThroughout this paper, the value of the error corresuonds to intermolecular I-I separations rather the one shorter the last significant d&k in the function value. separation. We believe that with increasing pressure the SThechemical bond in iodine is 2.678A and the I, molecules lie on mirrors in a structure arranged to yield layers normal to the small rearrangement will be toward the ideal structure, u-axis with four 3~9ANed-ne~bor contacts per moiecuie and with Sn at u = 0.125, I1 at u = 0.25, and I2 at (0,0,0.25), intertayer separations of 4.052,4*353,4-398, and 4~~19[5~. resufting in a more ionic nature to the Sn-I bond.
y=(ggr=-;(y),
1126
P. S. PEEKY, G. A. SAMARA and 3. hk3ROSiN
(c) Dielectric measurements Static dielectric constant measurements were made at a frequency of 10 kHz using a General Radio Model 1615A transformer ratio arm capacitance bridge on disks of fused Sn14 powder prepared in a 0.63 cm diameter pellet maker at an applied pressure of -10 kbar. The pressure measurements were made in a piston-cylinder apparatus, consisting of a 0.95 cm i.d. x 1.91 cm o.d. x 6.35 cm long tungsten carbide chamber fitted with hardened steel binding rings and mylar insulated, tungsten carbide pistons. The measurements extended to -30 kbar. The sample disks with lead foil electrodes attached to the large faces were encapsulated in a nylon jacket and loaded into the pressure cylinder. The compressibility of nylon is close to that of SnL. The pressure dependence of the static dielectric constant E is shown in Fig. 5. Below 2 kbar sample compaction effects were noted, hence, the response below this pressure was obtained by extrapolating back to zero pressure the smooth, reproducible and reversible higher
Fig. 6. Temperature dependence of the dielectric constant of SnL at 6 kbar. The circles represent the changes in measured sample capacitance, whereas the solid line represents the data corrected for the dimensional changes of the sample.
In Figs. 5 and 6, E is found to increase with both increasing pressure and temperature, suggesting nonvolume effects must be important. DISCUSSION
(a) Phonon anhamonicities
The isobaric temperature dependence of a thermodynamic quantity I++(+= w or E for the present discussion) for cubic, or isotropic, symmetry is composed of two contributions: (i) changes which arise solely from the change in volume with temperature and (ii) the explicit temperature dependence at constant volume. Treating 4 as an explicit function of volume V and temperature T yields I
I
I
10
2Q
M
J
PRESSURElk$rl
Fig. 5. Pressure dependence of the reduced dielectric constant of SnI,. The circles represent the changes in measured sample capacitance, whereas the solid line represents the data corrected for the dimensional changes of the sample. pressure response. The data points in Fig. 5 represent the
measured fractional changes in sample capacitance C whereas the solid line represents the fractional changes in E obtained from the capacitance data by taking account of the dimensional changes of the sample due to compression, recalling that E = Ct/roA, where t and A are sample thickness and area and By= 8.85 x lo-‘* f/m is the permittivity of free space. At 1 bar E = 6~3(5) and c increases quite markedly with pressure, the initial slope being (a In E/BP)T= 3.60 x 10~*/kbar. The temperature measurements extend from 77 to 300°K and were also taken in the piston cylinder apparatus contained within a high pressure, low temperature cryostat similar to that described by Lyon et al.[71. The temperature dependence of Q, taken at 6 kbar, is shown in Fig. 6. The symbols represent actual data, and the solid line represents I after correcting the data for dimensional changes. A dashed line is used below 200°K in the region where thermal expansion data are not available.
(2) The first term on the right-hand side of eqn (2) is the pure-volume contribution to the isobaric temperature dependence of + which results from thermal expansion and the second term is the pure-temperature contribution which arises from cubic and quartic (“higher-order”) anharmonicities. Thus, measurements of S(T) and +(P) allows one to separately determine the contributions from thermal expansion and higher-order anharmonicities to the isobaric temperature dependence of $. In certain cases, one can further identify the higher-order anharmonicities as being dominated by either cubic or quartic contributions[8] and compare the results with calculation[9, lo]. To investigate the origins of the isobaric temperature dependences of the Raman-active phonons, we have evaluated the pure-volume and pure-temperature contributions to the frequency shifts via eqn (2). Results based on initial room temperature and pressure derivatives are given in Table 1. This analysis demonstrates that the temperature dependences of the phonon frequencies of the external modes are dominated by the pure-volume
Pressure and temperature dependences of fhe dielecbic constant
(thermal expansivity) contributions. In fact, for these modes the pure-volume contribution to the frequency shifts is an order of magnitude larger than the puretemperature contribution. For the case of the internal modes, the pure-volume and pure-temperature contributions are comparable (with the exception of the 53 cm-’ mode where the pure-volume contribution dominates). For these modes the frequency shifts and attendant phonon anh~monicities are small. Fu~hermore, the con~butions from higher-order anharmonicities are negative which prevents further identification of the origin of the anharmonicities. Cubic anharmonicities result in negative definite contributions to the phonon self-energies whereas the contributions from quartic anharmonicities can be either negative or positive[S]. Therefore one cannot identify whether the higher-order contributions to the phonon self-energies result from cubic or quartic anharmonicities.
1127
second contribution may be obtained from eqn (3) since
All quantities in eqns (4a) and (Sa) are thus known except (3 In a/U)“, which can then be determined by substitution. The various quantities evaluated at 293°K are given in eqns (4b) and (5b), where there is a one-to-one co~es~ndence between the terms in eqns (4a) and (4b) and in eqns (Sa) and (!%). l-84 = -1.56 t 0.37 t 3_03(inunits of lO+PK),
(4b)
1.55 = 2,0110.46 (in units of IO-‘/kbar).
(5b)
and
Note from eqn (5b) that the two volume-dependent contributions K and -~(a in a/a In V)T have opposite signs. The K contribution is positive and its origin is clear: it is a density term which accounts for the increase in the The pressure and temperature measurements of Qgiven density of pol~~ble molecules. The K(a In a/a In V)T above are sufficient to identify the origins of the isobaric contribution on the other hand is negative and much temperatnre dependence of the dielectric constant. At smaller than the K contribution, and the latter thus room temperature, these data and eqn (2) yield the dominates in determining the relatively large observed pure-volume contribution [-/3l~(a In e/J& = -27-9x increase in l with pressure. The quantity (a In a/t? In V)T 10mSloK and the pure temperature contribution is relatively small and positive, being = 0.23. Its positive (a In r/U)” = 69.2 x IO-‘PK. These two contributions sign can be qualitatively understood simply on the basis of the fact that the polarizability of a molecule or group of have different signs, and the isobaric temperature dependence of E is dominated by the higher-order molecules in an electric field increases as the available anharmonic (pure-temperature) contribution. This qualitavolume for molecular distortion increases-i.e. the tive behavior was evident from the data since t increased restoring forces between molecules decreases, and with both increasing temperature and pressure. thereby a increases, with increasing volume. The small Because SnL has cubic symmetry, we can examine the mag~~de of (a In crla In V), is probably characteristic tem~ra~re and pressure dependences of E from a more of molecular crystals. For normal ionic crystals such as fundamental point of view. For a cubic or isotropic the alkaii halides (a ln cu/In V)T = l-2. As can be seen from eqn (4b) the two volume material, the total polarizability a of a macroscopic small sphere of volume V is related to the dielectric constant by contributions, -p and @(a In a/a In V), to the temperathe macroscopic Clausius-Mossotti relation[ll, 121, ture dependence of e also have opposite signs, with the former dominating so that the sum, i.e. the pure volume c-l 47ra effect, is negative. Very clearly the observed isobaric -=__, 1-l-2 3v increase of E with temperature arises from the dominance of pure temperature effect (a In a/H)“. The origin of this Combining eqns (2) and (3) allows one to evaluate the effect can be understood in terms of a model of particle or isobaric temperature dependence of e in terms of the molecule in a potential well. For a parabolic well the force pure-volume and pure-temperat~e con~butions to the constant k is independent of amplitude and the polariza~l~zab~ity a as bility, which is proportional to l/k, is thus independent of temperat~e. Any deviations from a parabolic potential causes k, and hence a, to be amplitude- or temperaturedependent. A detailed understanding of this effect would require knowledge of the electronic and lattice contribu@a) tions to a as well as the electronic structure and lattice dynamics of SnI,. It is emphasized here that the above a The right-hand side of eqn (4a) consists of three is the total macroscopic polarizability of SnL and consists contributions. The first, ,9, represents the change in e due of the sum of the electronic polarizability arkc and the to change in density. The second, @(a In a/a In V), is the lattice polarizability aI. To separate the two effects and change in e due to the change in the polarizability of a extend the above analysis further, measurements of the fixed number of molecules with changing volume. In this refractive index, and its pressure and temperature case, these two cont~~utions together constitute the pure dependences are also needed. Such meas~ements yield volume effect. The third ~n~bution, (a In a/a ln T)“, is information on mG which undoubtedIy plays an importhe voIume-independent, or pure tempera~re, effect. The tant role in the case of molecular crystals like SnL.
P. S. PEERCY, G. A. SAMARA and B. MOROSIN
1128 suMiuARY
We have reexamined the structure of the molecular crystal Snb and measured the thermal expansivity and volume compressibility using X-ray techniques. These results were combined with measurements of the pressure and temperature dependences of the static dielectric constant and Raman-active phonons to evaluate the anharmonic contributions to the phonon self-energies and to the isobaric temperature dependence of the dielectric constant. Although this material is very compressible, we found that the isobaric temperature dependence of the polarizability is dominated by higher-order anharmonicities rather than by the volume contribution. The external Raman-active phonons displayed large temperature and pressure dependences which are dominated by the changes in volume with temperature and pressure; contrary to this behavior, the internal molecular vibrations were much less sensitive to temperature and pressure and their isobaric temperature dependences consist of comparable contributions from higher-order anharmonicities and thermal expansion.
Acknowledgements-Tlte expert tecbniial assistance of J. D. Rluck, B. E. Hammons, and R. A. Trudo in making the measurements is gratefully acknowledged.
REFERJtNCES
1. Riggleman B. M. and Drickamer H. G., J. Chem. Phys. 38, 2721 (1%3). 2. Stammreich H., Fomeris R. and Tavares Y., J. Chem. Phys. 25, 1278(1956). 3. Peercy P. S. and Samara G. A., Phys. Rev. B8, 2033 (1973). 4. Morosin B. and Schirber J. E., J. Appl. Cryst. 7,295 (1974). 5. Wyckoff R. W. G., Crystal Structures Vol. 1, p. 53, Vol. 2, p. 131. Interacience, New York (1%5). 6. Mellor F. and Fankucken I., Acta Cryst. 8, 343 (1955). 7. Lyon D. N., McWban D. B. and Stevens A. L., Reo. Sci. Inst. 38, 1234(1967). 8. See, e.g. Samara G. A. and Peercy P. S., Phys. Rev. B7, 1131 (1973). 9. Maradudin A. A. and Fien A. E., Phys. Rev. 128,2589(1962). 10. Cowley R. A., Adu. Phys. 12, 421 (1963);Phil. Msg. 11, 673 (1965). 11. FrBhlich, Theory ofDielectrics Appendix 3, p. 169.Clarendon Press, Oxford (1949). 12. Bosman A. J. and Havinga E. E., Phys. Rev. 129,1593(1963).