RELATIONSHIP BETWEEN THE SHORT-TIME SPECIFIC HEAT AND THE ULTRASONIC PROPERTIES OF GLASSES

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Submitted on 1 Jan 1978

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RELATIONSHIP BETWEEN THE SHORT-TIME

SPECIFIC HEAT AND THE ULTRASONIC

PROPERTIES OF GLASSES

James Black

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-963

RELATIONSHIP BETWEEN THE SHORT-TIME SPECIFIC HEAT

AND THE ULTRASONIC PROPERTIES OF GLASSES'

"

James L. Black

Dept. of Physics, Brookhaven National Laboratory, Upton, New York 11973, U.S.A.

Résumé.- Onaétabli des prédictions théoriques pour des expériences d'impulsion de chaleur réalisées dans des verres à des températures inférieures à 1 K. Ces calculs sont basés sur le "modèle de tun-nelling", à l'aide de paramètres déterminés à partir d'expériences d'atténuation ultrasonore et d'écho de phonon dans de la silice vitreuse. Les résultats théoriques sont comparés aux expériences d'impulsion de chaleur de Goubau et de Tait /11/. La valeur théorique de la chaleur spécifique à court terme est d'un ordre de grandeur environ inférieure à la valeur expérimentale. Cet écart im-portant est discuté en fonction des suppositions fondamentales du "modèle de tunnelling".

Abstract.- Theoretical predictions are made for heat-pulse experiments in glasses at temperatures below 1 K. These calculations are based upon the tunnelling model, using parameters determined from ultrasonic attenuation and phonon-echo experiments in vitreous silica. The theoretical results are compared with the heat-pulse experiments of Goubau and Tait /ll/. The theoretical value of the short-time specific heat is found to be roughly an order of magnitude smaller than the experimental value. This important discrepancy is discussed in terms of the underlying assumptions of the tunnelling model.

The tunnelling model /1,2/ of amorphous so-lids has had considerable success in explaining the remarkable low-temperature properties of glasses /3/. According to this model, glasses contain groups of atoms which can quantum-mechanically tunnel between two local minima in the potential energy. A reasonable distribution of energy-splittings for these levels leads to the observed linear specific heat 11*1. Furthermore the coupling of these tunnel-ling levels to long-wavelength phonons leads to observable effects in the thermal conductivity /5/, the ultrasonic attenuation /6/, and the velocity of sound HI. Most dramatically, these levels cause nonlinear phonon effects /3/, including phonon echoes /8,9/.

This work explores the issue of consistency between the specific heat and the ultrasonic experi-ments in glasses. Within the tunnelling model, the

specific heat contains contributions not only from the strongly coupled levels which are observed in phonon scattering experiments but also from more weakly coupled levels. This happens whenever there

is a broad distribution of phonon/tunnelling-level couplings /1,2,10/. By reducing the time-scale of the specific heat experiment, some weakly-coupled levels are excluded, leading to a predicted reduc-tion in the specific heat /l/. The heat-pulse experiments of Goubau and Tait /ll/, however, show

only a small reduction in the specific heat. In fact, the short-time specific heat observed in /ll/ is too large to be consistent with phonon-scattering expe-riments, as is seen in detailed calculations repor-ted here /12/.

The primary difficulty in computing the heat-pulse behaviour in fused silica is the fact that phonon diffusion and tunnelling-level equilibration occur on roughly the same time scale (100 us). This difficulty is overcome by using the linearized cou-pled-Boltzmann-equation formalism /12,13/ within the local temperature approximation :

where AT is the Laplace transform of the local tem-perature rise, K is the thermal conductivity, and Q is the heat-pulse input energy density. The va-riables E, n (E), and R,-(E) are respectively the energy splitting, the density of states, and the "spin-lattice" relaxation rate for tunnelling levels. The label E, allow a distinction to be made among tunnelling levels with different coupling strengths.

T Work supported at Harvard University by NSF Grant DMR 72-02977-A03 and at Brookhaven by U.S. Depart-ment of Energy Contract EY-76-C-02-0016.

In order to complete the calculation, assump-

approach is currently under investigation.

tions about the distribution of coupling strengths

must be made. Generally, these coupling strengths

vary as ye

,

where

depends on the tunnelling

barrier height and

is a deformation potential

/

In the "standard" model

the dis-

tribution function

is assumed to be constant.

By taking values of

P

and y from ultrasonic atte-

nuation

and phonon-echo experiments

the

detector-face temperature rise shown in figure

is

Goubau and Tait 0.0

'0 4 0 80 120 160 2 0 0 2 4 0

Fig.

Heat-pulse predictions based on standard

levels alone. Temperature rise is normalized by the

t

"phonon specific heat only" value

obtained. Deviations from this "standard" model are

accounted for by introducing "anomalous" levels

which are assumed to not contribute to

P

or

because they are weakley coupled to phonons.

Heatpulse predictions with anomalous levels of va-

rious coupling strengths are shown in figure

The

values

P

and y are the same as in figure

but the

anomalous levels cause additional (unobserved) pho-

non scattering (especially Fig.

From figures I and

it is apparent that

the experimental results of reference

exhibit

a

short-time specific heat which greatly exceeds

the theoretical predictions derived from ultrasonic

experiments. This result appears to be quite general

in that it does not depend strongly on the tunnel-

ling model assumptions. ~esolution

of this diffi-

culty may lie in questioning the concept of a local

temperature. That is, it may be more appropriate to

consider the separate propagation of each phonon

T = 1 6 0 m K

P = 0.2 1 lo3' erg" cm-3 With Anomalous Levels

Gwbau and Tait

TIME (microseconds)

Fig.

Heat-pulse predictions based on standard

and anomalous levels. Anomalous levels lead to pho-

non-scattering rates which exceed the observed

amounts

by

c)

and d)

I wish to thank B.I. Halperin for many

helpful discussions.

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