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ON THE TIME-DEPENDENT SPECIFIC HEAT OF GLASSES AT LOW TEMPERATURES
R. Rammal, R. Maynard
To cite this version:
R. Rammal, R. Maynard. ON THE TIME-DEPENDENT SPECIFIC HEAT OF GLASSES AT LOW TEMPERATURES. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-970-C6-972.
�10.1051/jphyscol:19786430�. �jpa-00217903�
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-970
ON THE TIME-DEPENDENT SPECIFIC HEAT OF GLASSES AT LOW TEMPERATURES
R. Rammal and R. Maynard
Centre de Reohevahes sur les Tris Basses Temperatures, C.N.R.S., B.P. 166X - 38042 Grenoble Cedex, France
Résumé.- Le problème de la chaleur spécifique dépendant du temps a été traité dans le cadre du modèle des défauts "tunnel" couplés aux phonons. La densité spectrale effective des défauts à 2 niveaux a été calculée pour toute fréquence de mesure. Les profils de température ainsi que les constantes de diffusion thermique apparentes mesurés récemment aux très basses températures sont expliqués par ce calcul.
Abstract.- The problem of the time dependent specific heat has been treated in the framework of the model of tunneling defects coupled to the phonons. The effective density of two level defects has been obtained for any measuring frequency. Recent results on the temperature profile as well as apparent thermal diffusion constant at very low temperature have been fitted by this calculation.
The effect of the time dependent measure- ments of disordered systems is of great importance.
First, it gives an apparent character to the proper- ties which must be discriminated from the thermal equilibrium value. For instance, deviations from the linear specific heat observed experimentally in glasses /l/, can be understood in terms of a pro- gressive freezing of the defects contributing to the heat capacity on the time of measurements ins- tead of a true density of states. Secondly, the determination of the distribution of the relaxation times of the defects and its variation in tempera- ture is a reliable objective of this sort of ana- lysis.
The problem of the time dependence of the specific heat of glasses at low temperature was first recognised in the original paper of P.W. An- derson et al. Ill, then studied experimentally
/3/,/4/ and theoretically /5/,/6/. However one of the previous calculation /5/, neglecting the broad distribution of the relaxation times of the de- fects is incorrect.
Let us start from the standard model HI, 111 of two level defects in glasses representing the residual motion of atoms in a double well poten- tial. Since the defects are coupled to phonons, they have relaxation times T., where j labels the
fh
j defect. The coupled Boltzmann equations for phonons and defects have a simple solution within
the approximation of a local temperature :
where a is the time conjugated Laplace variable ("the frequency").
C, , (a) is therefore an "effective" spe- cific heat exhibiting a delay factor (1 + a t . )- 1
characteristic of a relaxation process at frequency 0 and appears as such /8/ in the diffusion equa- tion.
The standard expression for the defects relaxation rate :
T T1 = a T2 E. coth(E./2k^T) involves T. the tun- 3 3 3 J B J neling coupling energy (a is a numerical factor).
Now use the hypothesis fo the original mo- del HI, 111 which assumes an uniform distribution of the tunneling parameter X between 0 and XM A V t
. . / MAX let r „T V = r e MAX and A. = (E.2-r. 2) / z uni-
MIN o J J J form between 0 and Aw.„.
MAX
The effective density of states n(E,a) des- cribing the density of defects of energy E respon- ding at frequency a is obtained from (1) by inte- gration over the tunneling variable r. : ,
and n(E,o) = 0 otherwise,
where x = E/2 k T, x = V'OT*, X = T.J2KT arid
i B o m m m is
T = (2T)3 la is a characteristic relaxation time at temperature T.
The figure 1 shows three distinct regimes : - at E < r . , the density is zero, r appears
m m m therefore as a gap in the density of states.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786430
-
a t x m2
x 5 x a ' t h e e f f e c t i v e d e n s i t y of s t a t e s i s s t r o n g l y reduced from t h e e q u i l i b r i u m v a l u e o = 0, and v a r i e s a s ( X / X ~ ) ~.
-
a t x2
x n(E,o) v a r i e s a s kn(2x/xo).0 '
This p l o t shows d i s t i n c t l y how the time s c a l e i n t r o d u c e s a t y p i c a l f r e e z i n g of t h e d e g r e e s of freedom ;
For x >> xo, t h e r e d u c t i o n of t h e e r f e c - t i v e d e n s i t y of s t a t e s between t h e frequency a and frequency o
-
= O i s independent of E o r x : n(E,O)-
n(E,o) = Rn(l + ( X ~ / X ~ ) ~ ) .The s p e c i f i c h e a t f o l l o w s now d i r e c t l y from ( I ) and (2) :
c ~ ~ ~ ( T , u ) + i g 2 ~ JxxW
A
n ( x , o ) dx (3)m ch2x
An approximate e x p r e s s i o n of Cdef i n t h e interme- d i a t e regime x o > x and kgT m >
rm
i s :s o l u t i o n of t h e d i f f u s i o n e q u a t i o n w i t h two c o n t r i - b u t i o n s C1 and C2 of t h e d e f e c t s s p e c i f i c h e a t : Cdef = C1 + C2 Rn(l +
-
1 ) . CI i sa
independentUT*
and d e s c r i b e e x t r a f a s t r e l a x i n g i m p u r i t i e s pre- s e n t i n t h e sample 131 : C1 = 1.4 ergs/g.K,
C 2 = 0.05 erg1g.k. The d o t s correspond t o measured v a l u e of AT on v i t r e o u s s i l i c a / 3 / . The d o t t e d curve r e p r e s e n t s t h e s o l u t i o n s of t h e d i f f u s i o n e q u a t i o n , b u t f o r C1 = 0.
B) Recent experiments done by Lewis and L a s j a u n i a s
141,
on a very pure s i l i c a and f o r a d i f f e r e n t ran- g e of temperature and time a r e r e p o r t e d i n t h i s i s s u e . On f i g u r e 1 of r e f e r e n c e 141, t h e d a t a s e x h i b i t d e p a r t u r e from t h e s t a n d a r d s o l u t i o n of t h e d i f f u s i o n s e q u a t i o n s (with a time independent s p e c i f i c h e a t l a t s h o r t time. This d e p a r t u r e h a s been f i t t e d by t h e s o l u t i o n of t h e d i f f u s i o n equa- t i o n w i t h t h e frequency dependent s p e c i f i c h e a t / 4 / . The c h a r a c t e r i s t i c r e l a x a t i o n time -rL of e x p r e s s i o n ( 4 ) h a s been found t o b e -cf = 1.3 x~ o - ~ ~ T - ~ s . This v a l u e i s s m a l l e r t h a n t h e r e l a x a - t i o n time deduced from u l t i - a s o n i c experiment/&/ and corresponds t o t h e f a s t e s t time of t h e d i s t r i b u t i o n f o r d e f e c t s t h e energy s p l i t t i n g of which i s kgT.
Fig. 1 : The e f f e c t i v e energy d e n s i t y of two l e v e l d e f e c t s a t f i n i t e frequency o and f o r o = 0 v e r s u s energy s p l i t t i n g .
This e x ~ r e s s i o n shows f o r or*>> 1 (low temperature of h i g h frequency) t h e a p p a r e n t speci- f i c h e a t v a r i e s a s T ~ / U
,
w h i l e f o r o r X << 1 (high temperatures o r lower frequency) i t i s p r o p o r t i o - n a l t o T % ~ T ~ / ( J . For o going t o z e r o , t h e s p e c i f i c h e a t r e a c h e s a f i n i t e v a l u e Cdef(o = 0 ) correspon- d i n g t o t h e s t a t i c d e n s i t y of s t a t e s of F i g u r e 1 .The problem of t h e Laplace transform of t h e d i f f u s i o n e q u a t i o n w i t h t h e s p e c i f i c h e a t Cdef (0) h a s been performed /8/ f o r any time and s p a c e v a r i s b l e f o r an i n i t i a l p u l s e a t one end of t h e
s l a b . S i n c e t h e d e t a i l s 181 cannot b e r e p o r t e d h e r e indeed we p r e s e n t two f i t s w i t h t h e experi- ments :
A ) On f i g u r e 2, t h e temperature p r o f i l e i s f i t t e d a s a f u n c t i o n of time a t x = L , i n a s l a b of t h i c - kness L, s u b j e c t e d t o a h e a t p u l s e a t x = 0 a t t = 0. The continuous c u r v e corresponds t o t h e
F i g . 2 : T r a n s i e n t temperature p r o f i l e a s a func- t i o n of time d e l a y .
The b a s i c time dependent e f f e c t i n g l a s s e s would b e b e t t e r i n v e s t i g a t e d by measuring t h e re- l a x a t i o n of t h e e n t h a l p y H ( t ) which canbe o b t a i n e d
from t h e h e a t flow Q = a H / a t needed, t o m a i n t a i n t h e temperature c o n s t a n t a f t e r quenching. S i n c e t h e time e v o l u t i o n of e n t h a l p y i s c o n t r o l e d by t h e d i s t r i b u t i o n of t h e r e l a x a t i o n time, one c a n ex- p e c t a v a r i a t i o n i n bt f o r l a r g e time o r r a t h e r
6
i n l / t . T h i s behaviour which h a s been observed r e c e n t l y i n s p i n g l a s s e s191,
shows t h a t t h e d i s - t r i b u t i o n of t h e r e l a x a t i o n times i s probably si- m i l a r t o t h a t of g l a s s e s .References
/l/Lasjaunias, J.C., Ravex, A., Thoulouze, D. and Vandorpe, M., Interna- tional Conference on Phonon Scattering in Solids, p. 138, University of Nottingham, Plenum Press (1975).
/2/ Anderson, P.W., Halperin, B.I. and Varma, C.M., Phil. Mag. 2
1(1972).
/3/ Goubau, W.M. and Tait, R.A., Phys. Rev. Lett. 2 1220 (1975).
/ 4 /
Lewis, J.E., and Lasjaunias, J.C., "Short time scale transient tempera-
ture profiles in high purity vitreous silica
:a time dependent specific heat", LT 15, this issue.
/ 5 /
Heinrichs, J. and Kumar, N., Phys. Rev. Lett. 36 1406 (1976).
161
Black, J.L., to be published.
/7/ Phillips,W.A., J. Low Temp. Phys. 7 351 (1972).
/8/