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ULTRASONIC RELAXATION IN α-SULFUR
M. Boissier, R. Vacher, Bernard Perrin
To cite this version:
M. Boissier, R. Vacher, Bernard Perrin. ULTRASONIC RELAXATION IN α-SULFUR. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-611-C6-613. �10.1051/jphyscol:19816179�. �jpa-00221261�
JOURNAL D E PHYSIQUE
ColZoque C6, suppzgment au n012, Tome 42, de'cembre 1981 page C6-611
ULTRASONIC RELAXATION IN a-SULFUR
M . B o i s s i e r , R . Vacher and B . erri in*
Laboratoire de Spectrome'trie RayZeigh BriZlouin (ERA 460), Universite' des Sciences et Techniques du Languedoc, Place E. BataiZZon, 34060 MontpeZZier Cedex, France
* Dgpartement de Recherches Physiques (LA 711, Tour 22, Universitl P. e t M.
Curie, 75230 Paris Cedex 05, France
Abstract.- We present new measurements of the temperature dependence of acoustic damping in a-sulfur. The anomalous elastic properties observed in this crystal are described in terms of a "molecular relaxation".
Previous studies of the longitudinal acoustic wave propagation in a-sulfur have revealed an unusual behaviour. The sound absorption is very high for a dielec- tric crystal. Furthermore, Bril louin scattering measurements have given evidence of a large departure from the usual w2-dependence of the attenuation and shown the re- lated velocity dispersion in the frequency range 10-25 G H Z . ~ A phenomenological des- cription of these properties has already been given in tern of a relaxational pro- cess involving a single relaxation time.3
We want to discuss here several arguments either qualitative or quantitative allowing to interpret the relaxational process as a "molecular relaxation". We also present recent Brillouin scattering measurements of the acoustic absorption a as a function of temperature. The results, shown in Fig.1, indicate a temperature depen- dence of a very different from that observed in others dielectric crystals. This confirms that the Akhieser damping mechanism4 (which, in most cases, accounts well
TEMPERATURE ( K )
for the acoustic damping in dielectric crystals for the frequency-temperature range
laxation, will be called "molecular re-
4
-
3 -'E
B L!
-9 - 2 -
z Q
t 3
z w 1
z t-
( l , 0 , 0 ) longitud8nol waves
+ 25 GHz x 2 2 GHZ o 16 GHz
/
/x F1g.1
o 0XuQ u x
considered here) is insufficient to ex- plain the results in a-sulfur.
In molecular crystals, the internal vi- brations of molecules are slightly affec- ted by the lattice vibrations. Therefore, internal and external vibrations of mole- cules are weakly coupled in such crystals.
If an external perturbation is applied to the crystal, this weak coupling corres- ponds to long relaxation times of the phonon populations during the thermaliza- tion of the phonon gas. This effect,
TJO M O 300 which can give rise to an ultrasonic re-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816179
C6-6 12 JOURNAL DE PHYSIQUE
laxation" in the following. The existence of such a phenomenon in solids has first been suggested by Liebermann.5 Yore re~ently,~ it has been shown that, for this re- laxation mechanism, the amplitude of the relaxational term reflects the anisotropy of the solid. In contrast, the frequency dependence is not affected. Furthermore, a new expression for the relaxation amplitude has been proposed ; when the relaxa- tion spectrum reduces to a single time, the following expression for a is obtained
where s and w are the velocity and frequency of the elastic wave, respectively,
p the mass density, T the absolute temperature, CV the specific heat at constant volume, CI the specific heat associated with internal vibration modes, cijM the elastic constants, Bke the components of the thermal expansion tensor, ei and k the components of the polarization and normalizedwave vector of the wave, and j T
the relaxation time.
On the basis of the following arguments, we propose the "molecular relaxa- tion" for explaining the anomalous acoustic properties of a-sulfur :
(i) According to (I], the transverse waves propagating along the symmetry axes of the (orthorhombic) crystal do not give rise to "molecular relaxation", in agreement with experiment.
(L) As the coefficient of the relaxation in (I) varies strongly with tempe- rature, it can be shown that this variation is compatible with the experimental re- sults for a plotted in Fig.1.
(LLL) The analysis of the frequency dependence of the attenuation shows that the expression a = (AT/(I+w~T~) + c)w2 ( 2
can be fitted to the experimental results. The coefficients A, related to the ampli- tude of the relaxation, C, describing that of the Akhieser damping and -c are given in Table 1. In this Table, -r appears nearly insensitive to the anisotropy, in agree- ment with an interpretation in terms of a "molecular relaxation".
(iv) By comparing Eqs. (I] and ( 2 ) , an estimate of CI can be deduced for each of the propagation directions studied, the other quantities appearing in ( 1) being known. This value has, obviously, to be the same for all directions. The results, shown in Table 2, indicate a (statistical) dispersion smaller than 4 % for CI. The- refore, the anisotropy of the relaxation amplitude is well described by the "molecu- lar relaxation" model.
(v] Finally, CI was calculated directly from the frequencies of the internal modes measured in Ramn scattering and infrared absorption experiments. In this cal- culation, we neglected the spatial dispersion of internal modes. Such an estimate gives 0.98 x 106 J.m-3 . K - l , which is close enouph from the average value of CI in Table 2 (0.81 x lo6 J.~-~.K-').
In conclusion, we suggest that the relaxational behaviour of the acoustic properties of a-sulfur would be the first evidence of "molecular relaxation" in an
Table 1 -
inorganic molecular crystal. It seems therefore that this phenomenon is a characte- ristic property of what we can term, in a broad sense, the "molecular",crystals.
Propagation direction
A m-' x s)
C m-')
T
(PSI
J . S a p r i e l , L. RivoaZZan and J.L. R i b s t , J . Phys. ( P a r i s ) 33 ( 1 9 7 2 ) C6-150.
R. Vacher, J . SaprieZ and M. B o i s s i e r , J . AppZ. Phys. 45 ( 1 9 7 4 ) 2855.
R. Vacher, M. B o i s s i e r and F. Michard, i n " L i g h t S c a t t e r i n g i n S o l i d s " , ( F Z m a r i o n , P a r i s , 1975) p. 651-5.
A. A k h i e s e r , J . Phys. (U.S.S.R. ) 2 ( 1 9 3 9 ) 277.
L.N. Liebermann, Phys. Rev. 113 ( 1 9 5 9 ) 1052.
B. P e r r i n , Phys. Rev. ( t o b e p u b l i s h e d ) .
(100) (010) (001) (110) (101) (011)
5.6 9.0 3.2 8.2 6.2 4.1
5.7 2.6 6.4 3.3 1 . 5 3 . 9
10.3 10.3 10.5 10.5 10.5 10.5
Table 2 - Experimental estimate of CI at 293 K from relation ( 7 ) ( p = 2.07 x l o 3 Kg.m-3 ; qr = 1.39 x l o 6 J.~-~.K-').
