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Temperature dependence of the A2 vibration modes in α-quartz
Volume 4lA, number 2
PHYSICS LETTERS
11 September 1972
TEMPERATURE DEPENDENCE OF THE A2 VIBRATION MODES IN a-QUARTZ F. GERVAIS, B. PIRIOU and F. CABANNES Centre de Recherches
sur la Physique des Hautes Tempiratures,
C.N.A.S.,
45045
Orlkans, France
Received 13 July 1972 The infrared lattice bands which correspond to the Aa modes of orquartz, are analyzed by means of the classical oscillator model. The observed dependence of the classical parameters on temperature allows to think that non-linear effects occur by approaching the orp phase transition.
Because of its technological and scientific importance, a-quartz has been the subject of many investigations. On the other hand, the lattice force constants of quartz are similar to those which involve the internal vibration modes in silicate crystals, whose anharmonicity is studied elsewhere [ 1,2] . Consequently, the analysis of the temperature dependence of the infrared vibration modes, whose first results are reported in this letter concerning the A2 modes, was undertaken. The experimental procedure and the treatment of the data are those described in [l] . The calculation of the specta (fig. 1) according to the classical dispersion theory, have been carried out by using the oscillator parameters given below for the four A2 normal modes.
The results at room temperature are identical or close to those of Spitzer and Kleinman [3]. As shown by these authors, subsidiary oscillators having a weak oscillator strength, are needed for the calculation of the spectra. Their frequencies are located at 505 532 and 1215 cm-l at room temperature. The A$l) and A$3l modes are forbidden in the /3 phase [4]. Our results show that the oscillator strength of the A$3l mode decreases by approaching the temperature dependence of the A$l) mode frequency is unusual for a normal mode: the frequency shift is positive with increasing temperature, and is especially increased near the Ortp phase transition. Besides, if the contributions to the frequency shifts due to the thermal expan-
8
300
500
700 WAVENUMBER
900
1100
1300
cm-’
Fig. 1. Comparison between experimental reflectivity points and spectra calculated by means of classical oscillator model and by using the parameters given in table 1.
107
Volume
41A, number
PHYSICS
2
Table 1
i
Aq
qT0
lj
T, = 295K
0.67 0.67 0.67
363.5 367.7 379
4.8 9 23
0.65 0.67 0.80
495 487 467
5.2 9.7 21.3
0.11 0.09 0.05
777 776 775
6.7 12 21
0.66 0.65 0.67
1071 1067 1061
7.2 12 28.5
1 T, =495K T3 = 770K
Ti 2
T2 T3
TI 3
T2 i-3
TI 4
T2
r,
E, = 2.383
sion are deduced from the observed frequency values, the anharmonic shifts of frequency for the 1, 3 and 4 modes also appear positive and relatively large at 770K. It has been shown [ 1,5] that the following expression of the damping is convenient at intermediate temperatures for elevated mode frequencies. ~=ai(n,,~t~)+bi[(nb,i+~)2+
l/12]
where the mean number of phonons n = (exp(- hw”/kT) - 1)-l depends on an average w” frequency which is equal to i C$ for a three-phonon process (noted n, ;) and to 5 sli for a four-phonon process (nb, j); ui and bi are cubic and quartic anharmanic contribution parameters respectively. It should be noticed that the contribution to the dependence of
108
LETTERS
11 September
197 2
the dependence of damping on temperature due to the thermal expansion [6] may be neglected in o-quartz for which both volume expansion coefficient and Gruneisen parameter are low. Though only two points I’(T) should be necessary for determining the ai and bi factors, the more extended the temperature range of the data, the more accurate the cubic and quartic anhar manic contributions to the damping. Nevertheless it appears that the damping values observed at 295 and 495K are compatible with low bi/ai ratios (or bi equal to zero), whereas the data at 770K would request a more rapid increasing with temperature. Then it is unlikely that mechanical anharmonicity alone should cause the temperature dependence of the oscillator parameters. The results allows us to think that non-linear terms (non-linear dielectric susceptibility) [7] contribute to the observed values of the frequencies and damping of the A2 modes by approaching the phase transition, in addition to their effect on the oscillator strength Ae3. Additional data concerning the A2 modes and the temperature dependence of the A, and E modes will be presented later. References
[ll F. Gervais, B. Piriou and F. Cabannes,
Phys. St. Solide b 51 (1972) 701. 121 F. Gervais, B. Piriou and F. Cabannes, Phys. St. Solidi b, to be published. [31 W.C. Spitzer and D.A. Kleinman, Phys. Rev. 121 (1961) 1324. 141 J.F. Scott and S.P.S. Porto, Phys. Rev. 161 (1967) 903. (51 T. Sakurai and T. Sato, Phys. Rev. B 4 (197 1) 58’3. [61 J.E. Mooij, Phys. Lett. 29A (1969) 111. [71 R.A. Cowley, Adv. in Phys. 12 (1963) 421.
PHYSICS LETTERS
11 September 1972
TEMPERATURE DEPENDENCE OF THE A2 VIBRATION MODES IN a-QUARTZ F. GERVAIS, B. PIRIOU and F. CABANNES Centre de Recherches
sur la Physique des Hautes Tempiratures,
C.N.A.S.,
45045
Orlkans, France
Received 13 July 1972 The infrared lattice bands which correspond to the Aa modes of orquartz, are analyzed by means of the classical oscillator model. The observed dependence of the classical parameters on temperature allows to think that non-linear effects occur by approaching the orp phase transition.
Because of its technological and scientific importance, a-quartz has been the subject of many investigations. On the other hand, the lattice force constants of quartz are similar to those which involve the internal vibration modes in silicate crystals, whose anharmonicity is studied elsewhere [ 1,2] . Consequently, the analysis of the temperature dependence of the infrared vibration modes, whose first results are reported in this letter concerning the A2 modes, was undertaken. The experimental procedure and the treatment of the data are those described in [l] . The calculation of the specta (fig. 1) according to the classical dispersion theory, have been carried out by using the oscillator parameters given below for the four A2 normal modes.
The results at room temperature are identical or close to those of Spitzer and Kleinman [3]. As shown by these authors, subsidiary oscillators having a weak oscillator strength, are needed for the calculation of the spectra. Their frequencies are located at 505 532 and 1215 cm-l at room temperature. The A$l) and A$3l modes are forbidden in the /3 phase [4]. Our results show that the oscillator strength of the A$3l mode decreases by approaching the temperature dependence of the A$l) mode frequency is unusual for a normal mode: the frequency shift is positive with increasing temperature, and is especially increased near the Ortp phase transition. Besides, if the contributions to the frequency shifts due to the thermal expan-
8
300
500
700 WAVENUMBER
900
1100
1300
cm-’
Fig. 1. Comparison between experimental reflectivity points and spectra calculated by means of classical oscillator model and by using the parameters given in table 1.
107
Volume
41A, number
PHYSICS
2
Table 1
i
Aq
qT0
lj
T, = 295K
0.67 0.67 0.67
363.5 367.7 379
4.8 9 23
0.65 0.67 0.80
495 487 467
5.2 9.7 21.3
0.11 0.09 0.05
777 776 775
6.7 12 21
0.66 0.65 0.67
1071 1067 1061
7.2 12 28.5
1 T, =495K T3 = 770K
Ti 2
T2 T3
TI 3
T2 i-3
TI 4
T2
r,
E, = 2.383
sion are deduced from the observed frequency values, the anharmonic shifts of frequency for the 1, 3 and 4 modes also appear positive and relatively large at 770K. It has been shown [ 1,5] that the following expression of the damping is convenient at intermediate temperatures for elevated mode frequencies. ~=ai(n,,~t~)+bi[(nb,i+~)2+
l/12]
where the mean number of phonons n = (exp(- hw”/kT) - 1)-l depends on an average w” frequency which is equal to i C$ for a three-phonon process (noted n, ;) and to 5 sli for a four-phonon process (nb, j); ui and bi are cubic and quartic anharmanic contribution parameters respectively. It should be noticed that the contribution to the dependence of
108
LETTERS
11 September
197 2
the dependence of damping on temperature due to the thermal expansion [6] may be neglected in o-quartz for which both volume expansion coefficient and Gruneisen parameter are low. Though only two points I’(T) should be necessary for determining the ai and bi factors, the more extended the temperature range of the data, the more accurate the cubic and quartic anhar manic contributions to the damping. Nevertheless it appears that the damping values observed at 295 and 495K are compatible with low bi/ai ratios (or bi equal to zero), whereas the data at 770K would request a more rapid increasing with temperature. Then it is unlikely that mechanical anharmonicity alone should cause the temperature dependence of the oscillator parameters. The results allows us to think that non-linear terms (non-linear dielectric susceptibility) [7] contribute to the observed values of the frequencies and damping of the A2 modes by approaching the phase transition, in addition to their effect on the oscillator strength Ae3. Additional data concerning the A2 modes and the temperature dependence of the A, and E modes will be presented later. References
[ll F. Gervais, B. Piriou and F. Cabannes,
Phys. St. Solide b 51 (1972) 701. 121 F. Gervais, B. Piriou and F. Cabannes, Phys. St. Solidi b, to be published. [31 W.C. Spitzer and D.A. Kleinman, Phys. Rev. 121 (1961) 1324. 141 J.F. Scott and S.P.S. Porto, Phys. Rev. 161 (1967) 903. (51 T. Sakurai and T. Sato, Phys. Rev. B 4 (197 1) 58’3. [61 J.E. Mooij, Phys. Lett. 29A (1969) 111. [71 R.A. Cowley, Adv. in Phys. 12 (1963) 421.