- Email: [email protected]
Far-infrared spectra of solid CH4 under high pressure
CHEMICAL PHYSICS LFXERS
Volume 60, number 1
FAR-INFRARED
15 December 1978
SPECJXA OF SOLID CH4 UNDER HIGH PRESSURE
J. OBRIOT, F. FONDi?RE, Ph. MARTEAU,
H. VU and K. KOBASHI ’
tiboratoire des hteractions MoI&uZzims et des Harites fiessions, cenrre ihiverritaire Punk-iVord* 93430 Viiletaneuse, Jhvzce Received 1 Au,at 1978 Revised manuscript received 7 September 1978 Far-infrared spectra of solid CH4 have been recorded at 4.2 K under pressure, P = 0.4 kbar (phase Ii) and P = 4.5 kbar (phase III). In phase IL two bands have been observed at 53 and 76 cm-‘, while in phase IIi three bands have been seen at 65,70 and 94 cm-‘_ Lattice dynamical calcr?htions for the translational motion in a hznnonic lattice have been cozqared with the observed data_
Introduction
1.
Methane has three phases in the solid state. The phase diagrams of solid CH, and CD, are schematicalIy shown in fig. 1 [l-S] _ Phases I and IX have been extensively studied both experimentally and theoretically [S-7]_ The crystal structure of phase I is fee Fn&n(O~) with a molecule per primitive unit cell [8]_ The molecules have no orientational order. No fundamental op&zal bands have been observed in the * Research Fellow supported by the French Government Permanent address: Department of Chemistry, Faculty of Science, Kyoto University, Kyoto, Japan 606. 6-
1
/ , _* ,
I-
-s _
5
CL
2-
0 0 Fe- 1. PW diagrams CD4 (dashed curve).
90
20
T(K)
‘50
60
of solid CH4 (solid curve) a& solid
lattice viiration region, as might be expected from the crystal symmetry. Tlie low temperature phase 11 is fee Fm3c(O~) with eight molecules per primitive unit cell [S]. Phase II is a phase of partial orientational order in which six of the eight molecules (DW molecules) are ordered and the remaining two molecules (0, molecules) are disordered [S] _ lhe far-infrared spectra of condensed methanes (CH, and CD4) have been observed by Savoie and Fournier 191, an& two absorption bands have been found in phase ll, at 53 and 75 cm-’ for CH4(II) and at 50 and 67 cm- 1 for CD4@)_ ‘Ihe isotopic effect of molecular mass in the observed frequencies indicates that the two bands arise from the transIational lattice vibrations [9]. ?fie Raman spectra of solid CH4(H) have been recorded by Cabana and The [ lo]- In the lattice vibration region, two bands have been found at 41 and 52 cm-‘. Phase III of solid CH, is known to exist under pressure [ 141. A recent NMR study by Nijman and ‘Trappeniers [4] shows that the phase boundary between LI and III lies around 0.4 kbar at low temperatures- On the other hand, phase III of solid CD, exists below 22 K under ordinary pressure [I I-131. The crystal structure is tetragonal but the departure from fee is very small. The orientational ordering in this phase is still unknowns The far-m spectrum of solid CD4(III) has five bands in the lattice vibration i-ogion [VI. In the present paper, we report the far-H2 spectra
Volume 60, number 1
CHEMICAL PHYSICS LETTERS
15 December 1978
of solid CH4 at 4.2 K under pressures, 0.4 kbar and 4.5 kbar. The observed results arc compared with ca?culations of the translational lattice vibrations in a fee harmonic lattice with either 16 or 32 molecules in the primitive unit cell of phase III.
CH4
Solid phase=
2. Experimental
Solid samples were grown from the liquid under a pressure of a few hundred bars, and then cooled down to 4.2 K by immersing the high pressure cell in a
O-I 30 50
liquid helium bath. The sarrpIe thickness was 2 mm.
100
150
200
WAVENUMBER ( cm4 )
Fig- 3- Far-infrared spectrum of solid CH.+ in phase III at 4.2
Soil0
K and 4.5 kbar. Very weak shoulders were observed at 83 and 89 cm-’ in some rnns. Sample thickness is 2 mm.
CH4
FHASE I :Ts3o K PHASEH:T&.JK
Pressure was applied
using the metallic
piston
method [14]. Real time spectra were observed by an interferometer, Coderg FS 2000, attached to a computer. l’be experimental
set-up will be described else-
where ClS]. The observed spectra are shown in figs. 2 and 3_ The frequencies of the two bands in phase II at O-4 kbar closely agree with the previous data [9] as seen
30 50
150
loo
200
in table 1. in phase RI at 4.5 kbar, three intense bands have been observed at 6570 and 94 cm-l_
WAVENUMBER (cm-1 )
In
some runs, very weak shoulders appeared at about 83
Fig. 2. Far-infrared spectra of solid C& Solid curve: phase II at 4.2 K and 0.4 kbar. Dashed curve: phase I at 30 R. Sample thickness is 2 mm_
and 89 cm-‘, which are probably the counterpart of similar lines seen in the spectrum of solid CD4 193 _
Table 1 Observed and calculated translational lattice vibration frequencies in phase II Ezrperimental (cm-‘)
Theoretical e,
CH6
CD4
frequency lMJ2)‘”
far-IR a)
3
Ramanb)
mode (activity)
far-IR =)
far-IR ‘)
-7s
76
67
glJ2
53
53
50
2.0
Azg. Trg. TIU C(R) Tr~u(IR), Tzu
2’”
Eg(R), Trg
52 41
From ref. [PI, at 12 K.
e) present work.
b)Fromrcf.
[lO!,atPK.
Cl present work, at 4.2 K and 0.4 kbar.
T2&)
‘1 From ref. [Y] at 2.5K_
91
Volume 60, number 1
CHEMIC_~L PHYSICS LETTERS
3. Theoretical We shah consider the observed result from a stand point of transiational lattice vibrations. In crder to simphfy the calculation, the orientation-dependent potentials are neglected in phase II and are approximately treated in phase III, as will be seen below. This treatment may be justified as a zeroth approximation because the isotropic interaction gives the primary contribution to the second-order force constants in the translational vibrations. The crystaI lattice of phase II is known to be fee [8,12.13] _ Although phase III is tetragonal ~12,13,16,17], the departure from fee is small. The volume change due to the phase transition from II to III amounts to only 0.54% in solid CD4 [13]_ Therefore we use a fee lattice for both phases II and III in the subsequent calculation. The lattice is assumed to be harmonic, and the interactions are counted only between nearest neighbors. At fmt we deal with phase II. Let us take the eight molecules in a primitive unit cell as those numbered from 1 to 8 in fig_ 4. The second-order force constants between molecuiar centers of mass have been supposed to be equal and are denoted by g. The lattice dynamical calculation at k = 0 has been made by the standard method [18]_ Table I gives the calculated frequencies in the unit of cplm)U’, m being the molecular mass,
Fig- 4- fee crystal model as applied to phasesII and III. Each circle represents a methane molecllle. The moIecuIes numbered from 1 to 8 indicate those ir. a primitive unit cell of phase IL Open circks and open circles + solid circles stand for the mokcules in a primitix unit cell of phase III in the case of Z = 16 and Z = 32. respectively-
92
15 December 1975
together with the results of a factor-group analysis. The bigber two branches include the IR-active modes (I,,), and the lowest one includes two Raman-active modes (E, and T.&. The theoretical frequency ratio, 242, of theetwo IR-active branches agrees we11with the observed ratio, 1.42 [9] or 1.40 @resent result) for CH&I), and 1.34 for CD&I) [9] _ Thus, according to our calculated result and the mass isotope effect [9], we conclude that the two bands in the far-IR spectrum in phase II arise from the translational lattice vibrations. In the case of the Raman spectrum of solid CH4(II), a discrepancy exists between theory and experiment, which could not be removed even if the orientational potentials are included. Ran-ranexperiment for solid CD,(n) would help the identification of the 52 cm-l line. Next we deal with phase III Experimental [16,17] and theoretical f 191 investigations suggest that the number of molecules per primitive unit cell (this will be noted by 2, hereafter) equals to either 16 or 32 in phase III Therefore, we take either the 16 molecuies as indicated by the open circles in fig_ 4 [19], or the 32 moIecules as shown by both the open and solid circles in fig_ 4. In order to take into account the tetragonal structure of phase III, we have introduced two kinds of second-order force constants: gab between molecules on the same ab plane, and gc between those on different ab planes. Letfbe the ratio of the the two force constants, f = gc/gab _ In phase III the crystal expands along the c axis and contracts along the a and b axes compared to phase II, so that f 5 1.0. Figs. 5 and 6 show the calculated frequencies at k = 0 as a function off, keepingg& constant. In phase II, the highest frequency branch includes an IR-active mode, as seen in table 1. This holds even in the case of 2 = 16 or 32. Therefore, in the present case of phase III, the highest frequency in the calculated results was adjusted to fit that in the observed bands. Thus the frequencies of the calculated branches were compared with experiment at various f. Good agreement was found in the case of 2 = 32 and f = 0.8 for CH4(III} and f = 0.7 for CD4(III). In table 2, the observed frequencies are compared with the calculated result for 2 = 32, together with the result for Z = 16. A primary difference between Z = I6 and Z = 32 is that, in the case of Z = 16, there is no counterpart of the 6.5 cm-l band of solid CH4 or the 48.0 cm-’ band of solid CD,. Therefore Z = 32 seems to be more likely
15 December 1978
CHEMICAL PHYSICS LETTERS
Volume 60, number 1
Table 2 Observed and caIa&ted translational vibration frequencies
f = 0.7
obs =)
talc b)
obs c,
talc. b)
94
94-o o) * 92.4 * 88.6 *
67
67.0 ‘) * 65.3 * 60.8 *
89 e, 83 e,
829 79.9
78.4 76.8 70.1 66.5 62-7 49.5 47.5 46.8 44-3
70 65
Fig- 5. Translational lattice vibration frequencies at k = 0 as a
function of ffor 2 = 16. The numbers designate the degeneracies See also table 2.
CD,
*
* * * f
60.5 58.0
51.5 48.4
59.7
56.3 54.7 527 51.4 47.4 43.0 36.3 33.8 322 30.4
*
* * * *
a) Present work, at 4.2 K and 4.5 kbar. b, Present calculated result for 2 = 32. The frequencies desig* correspond to the Z =: 16 case. c, ?zizorn ref. [lo] at 12 K ‘)Tbis frequency is adjisted to f;! the observed value. e, Weak shoulders observed in some ruus in the present work.
neighbors to the frequencies in phases II and RF,- we have made a calculation using an isotropic lknnardJones potential [20] between molecules separated up to 10 A. Such a calculation gives absolute frequencies comparable with the observed data. The lattice distortion from the fee to tetragonal structure in phase III of solid CD4 leads to splittings of the degenerated branches corresponding to an effective value off= 0.9. Therefore we conclude that the difference between gd and zc (f= 0.8 cr 0.7) arises from both 0.U 0.0
02
0.4
f
06
0.8
10
the lattice
distortion
and orientation-dependent
inter-
actions_
Fig. 6. Translational lattice vibration frequencies at k = 0 as a function off for Z = 32. The numbers designate the degeneracics. See also table 2.
4. Discussion
than Z = 16, if the observed spectra are interpreted as the translational bands, like in phase II. However, we cannot rule out 2 = 16, because a tetragonal distortion and ‘he orientation-dependent potentials could lift a degeneracy to reproduce the experimental data. In order to examine the contribution from further
Although the far-IR bands due to molecular rotation have not been observed so far, there is apparent evidence for their existence both experimentally and theoretically. In a fee lattice (phase II and rare gas matrices), the sum of the dipole moments induced by the electric octupole moment of a methane molecule
93
Volume 60, number 1
CHJZMICAL PHYSICS LJZTERS
on the nearest neighbor mo!ecules is non-vanishing. ‘Ihe far-IR spectrum of solid CH4(II)’ due to rotational excitations, has been calcuIated [21] on the basis of the extended James-Keenan model recently estabIished by Yamamoto et aI. [S]. The main result is as follows: A strong absorption line due to 0, molecules exists at 42 cm-’ _ Many lines due to D,, molecules are dispersed around 50 and 100 cm-‘. These Iines have sufficient interzties if the octupole moment does not decrease in the solid phase. The etistence of rotational bands has been proved by Nanba et al_ 1221 in the far-IR spectra of 2.5% methane (CH, and CD,) in a Xe matrix. A sharp band has been found at 47-6 cm-’ and 21.8 cm-’ in the CHe/Xe and CD&Xe samples, respective!y. These data agree well with the theoretical results [21,23,24] _ Accordingly, the reason’ahy rotational bands have not been detected is probably that they become broad and overlap the transIationaI bands_ The dielectric constant E of solid methane closeIy relates to the far-IR spectrum. Constantino and Daniels 13 3 measured E in phases I. II and III of solid CH,_ One of the interesting resuits is that E decreases remarkably in *he phase transition from I to II and increases slightly in phase III. The former fact couid be attributed to a decrease in the number of orientational disordered molecuIes in phase II (0, molecules) to l/4 of those in phase I (all the molecules), since the dipole moments i?duced by the disordered molecules primarily contribute to E [3] _ However, the dielectric constant is related to the far-IR absorption intensity tbroqh the Kramers-Kriinig relation 1251, and the observed far-IR band peaks are mainly the contribution from translational motion. Therefore, we would have to take into account both rotation and
translation in estimating E_ Finally we note the problem to be resoIved in relation to the far-IR spectra of soIid methane, nameIy the direct detection of the rotational bands, the absorption mechanism due to the translational vibrations and the quantitative estimation of the dielectric constant_ Acknowledgement The authors thank Professor R-D. Etters for reading the manuscript_ One of the authors (K-K_) isgrateful to Dr. K. hfaki and Dr. M.L. KIein for valuable suggestions and criticzd discussions. 94
He ako thanks Professor
15 December 1978
H_ Watanabe for kind discussion on the lattice dynamics, and Professors B. Vodar and T_ Yamamoto for kind support. References [ I] R. Stevenson, J_ Chem- Phyr 27 (1957) 656. [2] J-W- Stewart, J_ Phys. Chem. Solids 12 (1959) 122. (31 M-S. Constantine and W-B. Dankis, J_ Chem. Phys. 62 (1975) 764; MS. Constantino, W-B. DanieIs and R.K. Crawford, Phys.
Rev. Letters 29 (1972) 1098. [4] A-J- Nijman and NJ- Trappeniers, Chem. Phys Letters
47 (1977) 188. (51 T. Y-moto, Y_ Kataoka and K- Okada, J_ Chem_ Phys. 66 (1977) 2701; Y. Kataoka, K. Okada and T_ Yamarnoto, Chem. Phys Letters 19 (1973) 365.
[6] hl. Blocm and J-A. Morrison. Surface and defect properties of solids, Vol. 2 (The Chemical Society, London, 1973) p_ 140. [7] A. Cabana, in: Vibrational spectra and structure, Vol. 4, ed. D-R Durig (Elsevier, Amsterdam, 1975) p_ 39. [8] W. Press, J- Chem. Phys. 56 (1972) 2597. [ 91 R. Savoie and R-P. Foumier, Chem. Phys. Letters 7 (1970) 1. [IO]
[ll]
[12]
A. Cabana and N.D. Tl~i, Can. J. Chem. 55 (1977) 3862. J-H. Colwell, E-K. Gilland J-A. Morrison, J. Chem. Phys. 36 (1962) 2223; 39 (1963) 635; 42 (1965) 3144; J-H. Colnell, J_ Chen Phys- 51 (1969) 3820. D-N. Bol’shutkin, V-M_ Gasan. A-L ProkhvatiIov and A-1. Erenburg, J- Struct. Chem. 12 (1971) 313; D-N. Bol’shutkm, V-M. Gasan, A-1. ProIchvatiIov and L-D. Yantsevich, J. Stxuct- Chem. 12 (1971) 1036.
[ 131 D-R Baer, B-k Fraass, D-H_ Riehl and RD. Simmons, J. Chem. Phys 68 (1978) 1411. [ 141 M. Jean-Lous and H. Vu, Rev. Phys AppL 75 (1972) 89. f 15 J J- Obriot, F. Fond&e and Ph. hfarteau, Infrared Phys, to be published. [16] E Ani and E Sandor, Acta Cryst- A31 (1975) Sl88. [ 171 W. Press and A. Hiiller, Proceedings of the NATO Ad-
vanced Study Institute on Anharmonic Lattices, Structural Phase Transitions, and Melting, Geilo, Norway, ed. T_ Riste (Noodhoff, Leyden, 1974) p_ 185. [18] A-k hiaradudin, ElV_ hlontroll, G-H. Weiss and 1-P. Ipatova, Theory of lattice dynamics in the harmonic appro-ximation (Academic Press, New York, 1971). [ 191 K. Maki, Y. Kataoka and T. Yamamoto, to be published. [20] J-0. Hirschfelder,C-F_ Curtiss and R-B_ Bird, Molecular theory of gases and liquids (Wiley, New York, 1954). 1211 K- Kobashi et aL, to be published.
1221 T- Nanba, hf. Sagara and hL lkezawa, J. Phys Sot_ Japan 44 (1978) 1755. [23] K. Nishiyama and T. Yamamoto, J. Chem. Phys 58 [24] [25]
(I 973) 1001. K- Nishiyama, J. Chem. Phy% 56 (1972) 5096. H_B. Levine and G_ Birnbaum, Phys_ Rev. 154 (1967) 86_
Volume 60, number 1
FAR-INFRARED
15 December 1978
SPECJXA OF SOLID CH4 UNDER HIGH PRESSURE
J. OBRIOT, F. FONDi?RE, Ph. MARTEAU,
H. VU and K. KOBASHI ’
tiboratoire des hteractions MoI&uZzims et des Harites fiessions, cenrre ihiverritaire Punk-iVord* 93430 Viiletaneuse, Jhvzce Received 1 Au,at 1978 Revised manuscript received 7 September 1978 Far-infrared spectra of solid CH4 have been recorded at 4.2 K under pressure, P = 0.4 kbar (phase Ii) and P = 4.5 kbar (phase III). In phase IL two bands have been observed at 53 and 76 cm-‘, while in phase IIi three bands have been seen at 65,70 and 94 cm-‘_ Lattice dynamical calcr?htions for the translational motion in a hznnonic lattice have been cozqared with the observed data_
Introduction
1.
Methane has three phases in the solid state. The phase diagrams of solid CH, and CD, are schematicalIy shown in fig. 1 [l-S] _ Phases I and IX have been extensively studied both experimentally and theoretically [S-7]_ The crystal structure of phase I is fee Fn&n(O~) with a molecule per primitive unit cell [8]_ The molecules have no orientational order. No fundamental op&zal bands have been observed in the * Research Fellow supported by the French Government Permanent address: Department of Chemistry, Faculty of Science, Kyoto University, Kyoto, Japan 606. 6-
1
/ , _* ,
I-
-s _
5
CL
2-
0 0 Fe- 1. PW diagrams CD4 (dashed curve).
90
20
T(K)
‘50
60
of solid CH4 (solid curve) a& solid
lattice viiration region, as might be expected from the crystal symmetry. Tlie low temperature phase 11 is fee Fm3c(O~) with eight molecules per primitive unit cell [S]. Phase II is a phase of partial orientational order in which six of the eight molecules (DW molecules) are ordered and the remaining two molecules (0, molecules) are disordered [S] _ lhe far-infrared spectra of condensed methanes (CH, and CD4) have been observed by Savoie and Fournier 191, an& two absorption bands have been found in phase ll, at 53 and 75 cm-’ for CH4(II) and at 50 and 67 cm- 1 for CD4@)_ ‘Ihe isotopic effect of molecular mass in the observed frequencies indicates that the two bands arise from the transIational lattice vibrations [9]. ?fie Raman spectra of solid CH4(H) have been recorded by Cabana and The [ lo]- In the lattice vibration region, two bands have been found at 41 and 52 cm-‘. Phase III of solid CH, is known to exist under pressure [ 141. A recent NMR study by Nijman and ‘Trappeniers [4] shows that the phase boundary between LI and III lies around 0.4 kbar at low temperatures- On the other hand, phase III of solid CD, exists below 22 K under ordinary pressure [I I-131. The crystal structure is tetragonal but the departure from fee is very small. The orientational ordering in this phase is still unknowns The far-m spectrum of solid CD4(III) has five bands in the lattice vibration i-ogion [VI. In the present paper, we report the far-H2 spectra
Volume 60, number 1
CHEMICAL PHYSICS LETTERS
15 December 1978
of solid CH4 at 4.2 K under pressures, 0.4 kbar and 4.5 kbar. The observed results arc compared with ca?culations of the translational lattice vibrations in a fee harmonic lattice with either 16 or 32 molecules in the primitive unit cell of phase III.
CH4
Solid phase=
2. Experimental
Solid samples were grown from the liquid under a pressure of a few hundred bars, and then cooled down to 4.2 K by immersing the high pressure cell in a
O-I 30 50
liquid helium bath. The sarrpIe thickness was 2 mm.
100
150
200
WAVENUMBER ( cm4 )
Fig- 3- Far-infrared spectrum of solid CH.+ in phase III at 4.2
Soil0
K and 4.5 kbar. Very weak shoulders were observed at 83 and 89 cm-’ in some rnns. Sample thickness is 2 mm.
CH4
FHASE I :Ts3o K PHASEH:T&.JK
Pressure was applied
using the metallic
piston
method [14]. Real time spectra were observed by an interferometer, Coderg FS 2000, attached to a computer. l’be experimental
set-up will be described else-
where ClS]. The observed spectra are shown in figs. 2 and 3_ The frequencies of the two bands in phase II at O-4 kbar closely agree with the previous data [9] as seen
30 50
150
loo
200
in table 1. in phase RI at 4.5 kbar, three intense bands have been observed at 6570 and 94 cm-l_
WAVENUMBER (cm-1 )
In
some runs, very weak shoulders appeared at about 83
Fig. 2. Far-infrared spectra of solid C& Solid curve: phase II at 4.2 K and 0.4 kbar. Dashed curve: phase I at 30 R. Sample thickness is 2 mm_
and 89 cm-‘, which are probably the counterpart of similar lines seen in the spectrum of solid CD4 193 _
Table 1 Observed and calculated translational lattice vibration frequencies in phase II Ezrperimental (cm-‘)
Theoretical e,
CH6
CD4
frequency lMJ2)‘”
far-IR a)
3
Ramanb)
mode (activity)
far-IR =)
far-IR ‘)
-7s
76
67
glJ2
53
53
50
2.0
Azg. Trg. TIU C(R) Tr~u(IR), Tzu
2’”
Eg(R), Trg
52 41
From ref. [PI, at 12 K.
e) present work.
b)Fromrcf.
[lO!,atPK.
Cl present work, at 4.2 K and 0.4 kbar.
T2&)
‘1 From ref. [Y] at 2.5K_
91
Volume 60, number 1
CHEMIC_~L PHYSICS LETTERS
3. Theoretical We shah consider the observed result from a stand point of transiational lattice vibrations. In crder to simphfy the calculation, the orientation-dependent potentials are neglected in phase II and are approximately treated in phase III, as will be seen below. This treatment may be justified as a zeroth approximation because the isotropic interaction gives the primary contribution to the second-order force constants in the translational vibrations. The crystaI lattice of phase II is known to be fee [8,12.13] _ Although phase III is tetragonal ~12,13,16,17], the departure from fee is small. The volume change due to the phase transition from II to III amounts to only 0.54% in solid CD4 [13]_ Therefore we use a fee lattice for both phases II and III in the subsequent calculation. The lattice is assumed to be harmonic, and the interactions are counted only between nearest neighbors. At fmt we deal with phase II. Let us take the eight molecules in a primitive unit cell as those numbered from 1 to 8 in fig_ 4. The second-order force constants between molecuiar centers of mass have been supposed to be equal and are denoted by g. The lattice dynamical calculation at k = 0 has been made by the standard method [18]_ Table I gives the calculated frequencies in the unit of cplm)U’, m being the molecular mass,
Fig- 4- fee crystal model as applied to phasesII and III. Each circle represents a methane molecllle. The moIecuIes numbered from 1 to 8 indicate those ir. a primitive unit cell of phase IL Open circks and open circles + solid circles stand for the mokcules in a primitix unit cell of phase III in the case of Z = 16 and Z = 32. respectively-
92
15 December 1975
together with the results of a factor-group analysis. The bigber two branches include the IR-active modes (I,,), and the lowest one includes two Raman-active modes (E, and T.&. The theoretical frequency ratio, 242, of theetwo IR-active branches agrees we11with the observed ratio, 1.42 [9] or 1.40 @resent result) for CH&I), and 1.34 for CD&I) [9] _ Thus, according to our calculated result and the mass isotope effect [9], we conclude that the two bands in the far-IR spectrum in phase II arise from the translational lattice vibrations. In the case of the Raman spectrum of solid CH4(II), a discrepancy exists between theory and experiment, which could not be removed even if the orientational potentials are included. Ran-ranexperiment for solid CD,(n) would help the identification of the 52 cm-l line. Next we deal with phase III Experimental [16,17] and theoretical f 191 investigations suggest that the number of molecules per primitive unit cell (this will be noted by 2, hereafter) equals to either 16 or 32 in phase III Therefore, we take either the 16 molecuies as indicated by the open circles in fig_ 4 [19], or the 32 moIecules as shown by both the open and solid circles in fig_ 4. In order to take into account the tetragonal structure of phase III, we have introduced two kinds of second-order force constants: gab between molecules on the same ab plane, and gc between those on different ab planes. Letfbe the ratio of the the two force constants, f = gc/gab _ In phase III the crystal expands along the c axis and contracts along the a and b axes compared to phase II, so that f 5 1.0. Figs. 5 and 6 show the calculated frequencies at k = 0 as a function off, keepingg& constant. In phase II, the highest frequency branch includes an IR-active mode, as seen in table 1. This holds even in the case of 2 = 16 or 32. Therefore, in the present case of phase III, the highest frequency in the calculated results was adjusted to fit that in the observed bands. Thus the frequencies of the calculated branches were compared with experiment at various f. Good agreement was found in the case of 2 = 32 and f = 0.8 for CH4(III} and f = 0.7 for CD4(III). In table 2, the observed frequencies are compared with the calculated result for 2 = 32, together with the result for Z = 16. A primary difference between Z = I6 and Z = 32 is that, in the case of Z = 16, there is no counterpart of the 6.5 cm-l band of solid CH4 or the 48.0 cm-’ band of solid CD,. Therefore Z = 32 seems to be more likely
15 December 1978
CHEMICAL PHYSICS LETTERS
Volume 60, number 1
Table 2 Observed and caIa&ted translational vibration frequencies
f = 0.7
obs =)
talc b)
obs c,
talc. b)
94
94-o o) * 92.4 * 88.6 *
67
67.0 ‘) * 65.3 * 60.8 *
89 e, 83 e,
829 79.9
78.4 76.8 70.1 66.5 62-7 49.5 47.5 46.8 44-3
70 65
Fig- 5. Translational lattice vibration frequencies at k = 0 as a
function of ffor 2 = 16. The numbers designate the degeneracies See also table 2.
CD,
*
* * * f
60.5 58.0
51.5 48.4
59.7
56.3 54.7 527 51.4 47.4 43.0 36.3 33.8 322 30.4
*
* * * *
a) Present work, at 4.2 K and 4.5 kbar. b, Present calculated result for 2 = 32. The frequencies desig* correspond to the Z =: 16 case. c, ?zizorn ref. [lo] at 12 K ‘)Tbis frequency is adjisted to f;! the observed value. e, Weak shoulders observed in some ruus in the present work.
neighbors to the frequencies in phases II and RF,- we have made a calculation using an isotropic lknnardJones potential [20] between molecules separated up to 10 A. Such a calculation gives absolute frequencies comparable with the observed data. The lattice distortion from the fee to tetragonal structure in phase III of solid CD4 leads to splittings of the degenerated branches corresponding to an effective value off= 0.9. Therefore we conclude that the difference between gd and zc (f= 0.8 cr 0.7) arises from both 0.U 0.0
02
0.4
f
06
0.8
10
the lattice
distortion
and orientation-dependent
inter-
actions_
Fig. 6. Translational lattice vibration frequencies at k = 0 as a function off for Z = 32. The numbers designate the degeneracics. See also table 2.
4. Discussion
than Z = 16, if the observed spectra are interpreted as the translational bands, like in phase II. However, we cannot rule out 2 = 16, because a tetragonal distortion and ‘he orientation-dependent potentials could lift a degeneracy to reproduce the experimental data. In order to examine the contribution from further
Although the far-IR bands due to molecular rotation have not been observed so far, there is apparent evidence for their existence both experimentally and theoretically. In a fee lattice (phase II and rare gas matrices), the sum of the dipole moments induced by the electric octupole moment of a methane molecule
93
Volume 60, number 1
CHJZMICAL PHYSICS LJZTERS
on the nearest neighbor mo!ecules is non-vanishing. ‘Ihe far-IR spectrum of solid CH4(II)’ due to rotational excitations, has been calcuIated [21] on the basis of the extended James-Keenan model recently estabIished by Yamamoto et aI. [S]. The main result is as follows: A strong absorption line due to 0, molecules exists at 42 cm-’ _ Many lines due to D,, molecules are dispersed around 50 and 100 cm-‘. These Iines have sufficient interzties if the octupole moment does not decrease in the solid phase. The etistence of rotational bands has been proved by Nanba et al_ 1221 in the far-IR spectra of 2.5% methane (CH, and CD,) in a Xe matrix. A sharp band has been found at 47-6 cm-’ and 21.8 cm-’ in the CHe/Xe and CD&Xe samples, respective!y. These data agree well with the theoretical results [21,23,24] _ Accordingly, the reason’ahy rotational bands have not been detected is probably that they become broad and overlap the transIationaI bands_ The dielectric constant E of solid methane closeIy relates to the far-IR spectrum. Constantino and Daniels 13 3 measured E in phases I. II and III of solid CH,_ One of the interesting resuits is that E decreases remarkably in *he phase transition from I to II and increases slightly in phase III. The former fact couid be attributed to a decrease in the number of orientational disordered molecuIes in phase II (0, molecules) to l/4 of those in phase I (all the molecules), since the dipole moments i?duced by the disordered molecules primarily contribute to E [3] _ However, the dielectric constant is related to the far-IR absorption intensity tbroqh the Kramers-Kriinig relation 1251, and the observed far-IR band peaks are mainly the contribution from translational motion. Therefore, we would have to take into account both rotation and
translation in estimating E_ Finally we note the problem to be resoIved in relation to the far-IR spectra of soIid methane, nameIy the direct detection of the rotational bands, the absorption mechanism due to the translational vibrations and the quantitative estimation of the dielectric constant_ Acknowledgement The authors thank Professor R-D. Etters for reading the manuscript_ One of the authors (K-K_) isgrateful to Dr. K. hfaki and Dr. M.L. KIein for valuable suggestions and criticzd discussions. 94
He ako thanks Professor
15 December 1978
H_ Watanabe for kind discussion on the lattice dynamics, and Professors B. Vodar and T_ Yamamoto for kind support. References [ I] R. Stevenson, J_ Chem- Phyr 27 (1957) 656. [2] J-W- Stewart, J_ Phys. Chem. Solids 12 (1959) 122. (31 M-S. Constantine and W-B. Dankis, J_ Chem. Phys. 62 (1975) 764; MS. Constantino, W-B. DanieIs and R.K. Crawford, Phys.
Rev. Letters 29 (1972) 1098. [4] A-J- Nijman and NJ- Trappeniers, Chem. Phys Letters
47 (1977) 188. (51 T. Y-moto, Y_ Kataoka and K- Okada, J_ Chem_ Phys. 66 (1977) 2701; Y. Kataoka, K. Okada and T_ Yamarnoto, Chem. Phys Letters 19 (1973) 365.
[6] hl. Blocm and J-A. Morrison. Surface and defect properties of solids, Vol. 2 (The Chemical Society, London, 1973) p_ 140. [7] A. Cabana, in: Vibrational spectra and structure, Vol. 4, ed. D-R Durig (Elsevier, Amsterdam, 1975) p_ 39. [8] W. Press, J- Chem. Phys. 56 (1972) 2597. [ 91 R. Savoie and R-P. Foumier, Chem. Phys. Letters 7 (1970) 1. [IO]
[ll]
[12]
A. Cabana and N.D. Tl~i, Can. J. Chem. 55 (1977) 3862. J-H. Colwell, E-K. Gilland J-A. Morrison, J. Chem. Phys. 36 (1962) 2223; 39 (1963) 635; 42 (1965) 3144; J-H. Colnell, J_ Chen Phys- 51 (1969) 3820. D-N. Bol’shutkin, V-M_ Gasan. A-L ProkhvatiIov and A-1. Erenburg, J- Struct. Chem. 12 (1971) 313; D-N. Bol’shutkm, V-M. Gasan, A-1. ProIchvatiIov and L-D. Yantsevich, J. Stxuct- Chem. 12 (1971) 1036.
[ 131 D-R Baer, B-k Fraass, D-H_ Riehl and RD. Simmons, J. Chem. Phys 68 (1978) 1411. [ 141 M. Jean-Lous and H. Vu, Rev. Phys AppL 75 (1972) 89. f 15 J J- Obriot, F. Fond&e and Ph. hfarteau, Infrared Phys, to be published. [16] E Ani and E Sandor, Acta Cryst- A31 (1975) Sl88. [ 171 W. Press and A. Hiiller, Proceedings of the NATO Ad-
vanced Study Institute on Anharmonic Lattices, Structural Phase Transitions, and Melting, Geilo, Norway, ed. T_ Riste (Noodhoff, Leyden, 1974) p_ 185. [18] A-k hiaradudin, ElV_ hlontroll, G-H. Weiss and 1-P. Ipatova, Theory of lattice dynamics in the harmonic appro-ximation (Academic Press, New York, 1971). [ 191 K. Maki, Y. Kataoka and T. Yamamoto, to be published. [20] J-0. Hirschfelder,C-F_ Curtiss and R-B_ Bird, Molecular theory of gases and liquids (Wiley, New York, 1954). 1211 K- Kobashi et aL, to be published.
1221 T- Nanba, hf. Sagara and hL lkezawa, J. Phys Sot_ Japan 44 (1978) 1755. [23] K. Nishiyama and T. Yamamoto, J. Chem. Phys 58 [24] [25]
(I 973) 1001. K- Nishiyama, J. Chem. Phy% 56 (1972) 5096. H_B. Levine and G_ Birnbaum, Phys_ Rev. 154 (1967) 86_