Low-energy excitations and hypersonic properties at low temperatures in amorphous media

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Low-energy excitations and hypersonic properties at low temperatures in amorphous media

J. Pelous, R. Vacher

To cite this version:

J. Pelous, R. Vacher. Low-energy excitations and hypersonic properties at low temperatures in amor- phous media. Journal de Physique, 1977, 38 (9), pp.1153-1159. �10.1051/jphys:019770038090115300�.

�jpa-00208681�

LOW-ENERGY EXCITATIONS AND HYPERSONIC ^PROPERTIES

AT LOW TEMPERATURES IN AMORPHOUS MEDIA

J. PELOUS and R. VACHER

Laboratoire de

Spectrométrie Rayleigh-Brillouin (*),

Université des Sciences et

Techniques

^du

Languedoc,

³⁴⁰⁶⁰

Montpellier ^Cedex,

^France

(Reçu

^{le 29 mars}

1977,

^{révisé le 3 mai}

1977, accepte

^{le 12 mai}

1977)

Résumé. ²⁰¹⁴ Nous

présentons

les résultats de mesures par diffusion Brillouin de la vitesse ^et de l’atténuation des ondes acoustiques pour trois verres de silice et deux échantillons de PMMA,

entre 1,7 ^et 15 K. Le maximum de vitesse dû à l’absorption résonnante des phonons ^est observé dans les verres de silice pour les phonons longitudinaux ^de fréquence 35 GHz. Ces résultats expérimentaux,

ainsi que les ^mesures d’atténuation, peuvent être décrits correctement en supposant ^une interaction entre les

phonons

^{et une} distribution de systèmes à deux niveaux.

Dans le PMMA, la vitesse des phonons ^{de 18 GHz devient constante en} dessous de 5 K, ^{et l’atté-} nuation est

proportionnelle

^{à la} température ^{dans tout} l’intervalle de température étudié. Ces résultats peuvent ^être interprétés qualitativement ^en supposant l’existence d’un processus relaxationnel, ^même

si une

description

quantitative n’est pas possible ^à partir ^{de ces seules} valeurs.

Abstract. ²⁰¹⁴ Brillouin scattering measurements of velocity and inverse mean free path ^of phonons

are presented for three silicate

glasses

^{and two} samples of PMMA, ^{in the} temperature range from 1.7 to 15 K. The velocity maximum due to resonant

absorption

^of phonons is observed in the silicate

glasses ^for longitudinal phonons ^of

frequency

35 GHz. These

experimental

results, ^{as well as the}

measurements of the inverse mean free path, ^can be described

satisfactorily

by ^{a model}

assuming

^an

interaction between phonons ^{and two level} systems. ^In PMMA, ^the velocity ^{of 18 GHz}

phonons

^tends

to be constant below 5 K, while the inverse mean free path ^is proportional ^to temperature in the whole temperature range. These results ^can be considered as an evidence of a relaxation process, ^even if a

quantitative

description

^{is not} possible ^on the basis of these data only.

Classification

Physics ^Abstracts

62.65 ^- 78.35

1. Introduction. ^- Several theoretical models have been put ^{forward to}

explain

the anomalous thermal

properties

^of

amorphous

^{media at low}

tempera-

tures

[1].

Those based on the existence of

long-range

disorder

[2]

^{do not}

give

^an

appropriate description

^of

the acoustic

properties [3, 4].

^{On the} contrary, ^the model

assuming

the existence of low _energy excita- tions

[5]

^{with a broad} energy distribution is in _quan- titative agreement with both thermal and acoustic

properties.

These excitations are described in a first

approximation by

^two-level systems (2

LS) interacting

with

phonons

^via

tunnelling.

^{Numerous recent studies}

have demonstrated the

ability

of this model to describe

largely physical properties

^{such as} dielectric relaxa- tion

[6],

^thermal

expansion [7],

^Raman

scattering [8].

This model has also been used for the

analysis

^of

nuclear

spin

relaxation

[9]

^{and IR}

absorption experi-

ments

[10].

^Some

departures

of the thermal

properties

from the theoretical

predictions [11]

may be accounted

for

through

refinements of the model. Some

light

^has

been shed

recently

^on the relation between excitations

responsible

^for

transport properties

^{and for}

specific

heat

[12]

^{as well as on} the connection between elastic and dielectric behaviour

[13]. However,

^{the micro-}

scopic

^nature of the defects and the

universality

^{of this}

description

^are still unclear.

The acoustic

properties

^of

amorphous

^{media at low}

temperatures have been studied _up to now for ultra- sonic

frequencies

^{in the} range 10 MHz-10 GHz

[13,14, 15, 16].

^Brillouin

scattering

allows the

study

^of

higher frequency

^{elastic waves, with} very low acoustic _powers.

In a recent Brillouin

scattering experiment

^{we have}

probed

^{the resonant} interaction between 2 LS and

higher

energy thermal

phonons

^{of 35 GHz}

frequency

in vitreous silica

[17].

In this _paper, we

present

measurements of

hyper-

sonic

velocity

^and

absorption

^{between 1.8 and 15 K}

for three silica or silicate

glasses

of various

composi-

tions as well as for two

samples

^of

polymethylmetha- crylate (PMMA)

of different

purities.

These results

76

(*) Equipe de Recherche Associee au C.N.R.S. no 460.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019770038090115300

1154

allow us to test the

validity

^{of the 2} LS model for

high

energy excitations and to look at the variations of the parameters of this model. The

qualitative

differencies between

inorganic glasses

^and

organic polymers

^are

also discussed.

2.

Experimental.

^{- The} conditions of the Brillouin

scattering experiments

^have been described in refe-

rence

[18].

^The

resolving

power of the

spectrometer

was

improved by decreasing

^the

frequency jitter

^{of the}

laser due to the

cooling

^water flow. This was obtained

by separating

^the

cavity

mirrors from the tube. The instrumental width was then

decreased by

^{a factor of 3.}

Within the _accuracy of

frequency

shift measurements,

changes

^{in sound}

velocity

^{of about 5 x}

^10-5

^{can be}

observed. The inverse mean free

path

is obtained from the values of the linewidth with an accuracy of 50

cm-1.

For the

experiments

^{above 4.2}

^K,

^the

samples

^were

cooled

by

convection of helium _gas. The

sample

^was

enclosed in a copper

cell,

whose lower

part

^was immersed in

liquid

helium for measurements between 4.2 and 1.8 K. For this latter temperature, ^the

sample

was

entirely

immersed in

liquid

^helium

during

^the

experiments.

^The measurement of the

intensity

^ratio

of the anti-Stokes and Stokes Brillouin components

was used as a sensitive check of the temperature ^{of the}

scattering

^volume

[19].

The

samples investigated

^were

synthetic

^silica

« Suprasil

^{II » (1 200} ppm OH-

ions, Heraeus-Schott, Germany),

^fused quartz ^«

Puropsil

^{» . (OH - content}

lower than 20 _ppm,

Electroquartz, France),

^{« BK7 »}

sodium borosilicate

glass (Heraeus-Schott).

^{We also}

studied two

samples

of PMMA : the first was commer-

cial

Plexiglass,

the second a

high-purity sample (kindly supplied by

^S.

Hunklinger).

3. Silicate

glasses.

^- 3 .1 RESULTS AND DISCUSSION.

- The measurements of

velocity

and attenuation of 33 GHz

hypersonic

^{waves in the two}

samples

^of

vitreous silica are shown in

figures

^{1 and 2.} The results for BK7 are

plotted

ⁱⁿ

figures

^{3 and 4}

(frequency :

37.5

GHz).

FIG. 1. ^- Variation of the velocity ^{of 33 GHz} longitudinal hyper-

sounds vs. temperature ^{in vitreous silica :} (+) suprasil II, (D) puropsil. ^The straight ^{lines are} the best fits from _eq. (2) ^for (- - -)

suprasil ^and (...) puropsil.

FIG. 2. ^- Variation of the inverse mean free path ^{of 33 GHz} longitudinal phonons ^vs. temperature in vitreous silica : (+) suprasil ^II, (~) puropsil. ^The straight ^{lines are} the values of the resonant absorption ^from eq. (1) calculated with the values of

nMl2

obtained from velocity measurements for (- - -) suprasil ^and (...) puropsil.

FIG. 3. ^- Variation of the velocity ^of longitudinal hypersounds (37 GHz) ^vs. temperature ^{in BK7.}

FIG. 4. ^- Variation of the inverse mean free path ^of longitudinal phonons (37 GHz) ^vs. temperature ^{in BK7.}

For the three

samples,

^the

velocity

goes

through

^a

maximum in the _range from 6 to 9 K. This maximum is similar to that observed for ultrasonic

frequencies

at lower

temperature.

The maximum can be

explained by assuming

^an

interaction of

phonons

^{with 2 LS via}

tunnelling.

^At

very low temperature, ^{the resonant}

absorption

^of

phonons

^{is the}

predominant

process

[20].

^{The cor-}

responding

^sound

absorption

^is

given by :

where n is the

density

^{of 2 L} states, p the

density

^{of the}

glass, v

^the

velocity

^of

longitudinal phonons, M1

^the

energy

corresponding

^{to the}

coupling

^between

pho-

nons and 2 LS

by

^direct process,

w/2 n

^the

frequency

of the

hypersonic

^{waves and x =}

hw/(2 kB T).

^This

expression

^{is valid under the}

condition, always

fulfilled in

spontaneous

^Brillouin

scattering,

^{that 2 LS}

are unsaturated.

The related variation

dvres

^{of the}

velocity

^with

temperature

is obtained

by using

^{the Kramers-}

Kronig

relation between

dvres

^and ares,

assuming

^that

tanh

(x) ~

^{x :}

At

higher

temperatures ^the

hypersonic absorption

arel is

mainly

^{due to the} relaxation of 2 LS

[20].

arel is

given by

E is the level

splitting

^{of the 2}

LS,

^D describes the energy shift of the level

splitting,

ⁱ the relaxation time

given by :

The indices I and t refer to the

longitudinal

^and

transverse

polarization

^of

phonons.

The related

change

^of

velocity Ave,

^{is : :}

Below 5

K,

the relaxational contribution is

negli- gible.

Within the _accuracy of our

experiments,

^a

logarithmic

increase of the

velocity

is observed in this range. The

comparison

of the results with _eq.

(1) gives

The values must be

compared

^to those deduced in reference

[14]

from ultrasonic measurements :

nMl2 =

^{2.6 x}

¹⁰⁸

erg.

cm- 3

for

Suprasil II ;

nMl2

^{= 2.4 x}

¹⁰⁸

erg.

cm- 3

for

Suprasil

^W

(a high purity synthetic

silica with a very low OH-

content).

A similar behaviour was observed for

BK7,

^{with a} maximum near 9 K and a value of

nMi equal

^to

(2.1

±

0.3)

^x

¹⁰⁸ erg . cm- 3,

^near the ultrasonic value (2.6 ^x

108 erg. cm-3) reported

in reference

[14].

The

comparison

^{of the}

nMl

^{values in} ultrasonic and Brillouin

scattering experiments

shows that this

product

^is

nearly

^{constant in a}

large frequency

range.

Variations of OH - content can be invoked to

explain

the small difference of

nM,2

^between

Suprasil

^{W and}

Suprasil

^II in ultrasonic measurements. On the other

hand,

strong differences of thermal

conductivity

^have

been observed between

samples

^{of fused} quartz ^and

synthetic

^silica

[21].

^{In our}

experiments, nM12

^{is found}

to be smaller in

Puropsil

^{than in}

Suprasil

^{II. The}

difference, slightly higher

^than that observed in ultra- sonic

experiments,

^{could be}

mainly assigned

^{to the}

variation of OH- content. Some small contribution from the thermal treatment could also be invoked.

Our

velocity

measurements agree with the _assump- tion of a resonant interaction between

phonons

^and

2 LS _up to 8 K. In this _range the excess heat

capacity, mainly governed by Debye phonons,

^{does not}

give

any information about the distribution of the excitations.

Our results

give

first evidence of 2 L excitations

having

energy levels as

high

^{as 5 x}

^10-4

^eV.

Above 10-15

K,

^a contribution to sound

absorption

can occur from

many-phonons

processes, and in the

following

^we will restrict our

analysis

^{to the} tempera-

tures below 10 K where the

one-phonon

process is dominant. The similarities between the

experimental

curves confirm that

transport properties

^are

weakly

sensitive to the

impurity

^content. The inverse mean

free

path ^{l -1}

^of

phonons

decreases with

temperature

down to 5 K as well for the two

samples

^{of silica}

(Fig. 2)

^{as for BK7}

(Fig. 4).

^The

T3-dependence expected

^{from eq.}

(3),

^for arel when Q)’t > 1 is observed in a very small

temperature

range and under

the

condition of

substracting

^a residual contribution of about 50

cm-1

for vitreous silica.

Between 2 and 5

K,

^the results show that l -1 is lower than 100

cm-1

in accordance with the measure- ments in

Suprasil

^I

by

^{H. E. Jackson} and coworkers

[22]

and our

previous independent

measurement in

Supra-

sil II

[17].

This is also in

agreement

^{with an}

extrapola-

tion in our

frequency-temperature

range of the ultra- sonic measurements of

Golding

and coworkers

[16],

which

gives

⁷⁰

^cm-’

^{for T = 2 K and v = 33 GHz.}

Within the _accuracy of our

experiments (about

50

cm-1)

the increase of

l-1

with

decreasing tempe- ’

rature due to the resonant interaction is not observed.

The calculation of

lresl

^from eq.

(1)

^{and the}

experi-

mental value of

nM,2

deduced from

velocity

^measure-

ment

give

^{a value}

significantly higher

^{than the}

experi-

1156

ment for the three

samples.

This confirms our

preceed- ing

result and has

already

been noted

by Golding

et al.

[16].

^A

possible explanation

^{of this}

disagreement

is that

assuming

^a

logarithmic dependence

^{of the}

velocity,

^{i.e. a constant value for}

nM12,

^{leads to an}

overestimate

ofnM/

^from

velocity

measurements.

3.2 FIT OF THE THEORETICAL PARAMETERS. ^- Three parameters, n, M and D are involved in the theoretical calculation of l -1 and

w/v.

^These parameters ^must be taken as functions of the _{energy E.}

The excess

density

^{of states}

n(E)

determined from

specific

^heat measurements must be viewed with caution when

applied

^{to the}

coupling

^{with 33 GHz}

phonons. Furthermore,

it is known

[12]

^{that not all}

the excess states are

equally important

^{for the trans-}

port

properties. Therefore,

^if

n(E)

is assumed

unknown,

the number of

parameters

^{is too}

high

^{to be}

deduced from a

fitting

of acoustic wave

velocity

^and

attenuation measurements. It must also be noted that the relative

uncertainty

^{of our} measurements of l-1 below 5 K is _very

high,

^so that the fit of the resonant

part of the sound

absorption (eq. (1))

^{cannot be made}

with sufficient _accuracy.

However,

this simultaneous fit is a very sensitive criterion to test the values of these parameters. ^The

comparison

has been effected for

temperatures

^below

10 K,

where the

one-phonon

process is dominant. As the results are very similar for the three

samples

of silicate

glasses studied,

^{our above}

analysis

is restricted to vitreous silica. Without direct information on the

coupling

^constant

D,

^this para- meter is

generally

^assumed

energy-independent.

^This

assumption

^{is in}

agreement

with the fit of ultrasonic

experiments [20].

^Different

expressions

^for

n(E)

^and

M(E)

^{have been}

proposed by

several authors from both theoretical considerations and

experimental,

determination. In the

following,

^{we will} try ^{to fit our} results from an

extrapolation

^{of these}

expressions

^to

high energies.

The first set of parameters ^{used for}

comparison

^was

that used

by

Jackle and coworkers for ultrasonic

frequencies :

To a first

approximation, n(E)

^{was assumed to be}

constant and

equal

^to no ⁼ 7.8 x

10" erg-’ ^{cm- 3.}

The

corresponding

values for v and

l-1

are shown in

figures

^{5 and 6}

(dashed-dotted lines).

^We also tried the

expression given by Stephens [23]:

where a ⁼ 0.078

(K-2), already

^used

by

Piche et al.

[14]

to

analyse

^their measurements. The

results, given

ⁱⁿ

our

preceding

paper, ^{are seen to} be inconsistent for

hypersonic frequencies (dashed

^{lines in}

figures

⁵

and

6).

While the increase with temperature ^{of the} calculated values for the inverse mean free

path

^{is too}

FIG. 6. ^- Comparison ^{of our inverse mean free} path measurements

(+) ⁱⁿ suprasil ^{II with} various theoretical curves (See text).

small when n is taken

equal

^to no, this increase is over-

estimated if

Stephen’s expression

is used. This result suggests ^{that the} increase of

n(E)

^for

high

energy excitations is smaller than that deduced from heat

capacity

measurements.

Dependencies

^of

n(E)

^smaller

than

E2

have

already

^been

proposed by

^other

authors

[7, 24],

^as for instance :

where

Eo

^{is an} energy gap. This

expression,

^deduced

from theoretical considerations must be associated with an energy

dependence

^{of the}

coupling

energy M

given by

leading

^{to a constant value for}

^{nM 2.}

^{The calculated}

values

of l-1

and

w/v

obtained from these

expressions

were much lower than the

experimental

^values.

The form n ⁼ _no EV was used

by La.sjaunias

^{et al.}

[11] ]

to describe the low

temperature

^thermal

properties

^of

vitreous silica. The values ^{v ~} 0.22 was obtained

by

these authors from

specific

^heat measurements, ^while thermal

conductivity

measurements led to v ~ 0.05.

Keeping

^the

values

of reference

[20]

^{for the}

coupling parameters M,

^{M and}

D,

^we tried this

expression

ⁱⁿ

our calculation of

v(T)

^and

l-1(T).

^{The results were}

still lower than the

experimental

^values.

Finally

^the

best fit was obtained

by taking

the form of _eq.

(6)

for

n(E),

with 0.010 a 0.015 (solid ^{lines in}

figures

^{5 and}

6).

^{It must} be noted that the result is _very sensitive to the form taken for

n(E).

However,

it is obvious that the solution is not

unique,

^{due to the}

high

number of

parameters.

Therefore,

^we have tried to fit the

experimental

^results

by varying

^the

coupling energies, keeping

^{the form}

of

n(E)

^{to the above}

expression.

^{In the} temperature range under

consideration,

the result is

weakly

^sensi-

tive to this variation and no conclusion can be drawn

on this

point.

A new set of

parameters

has been determined

recently by Golding et

^al.

[16],

^{from the}

analysis

^of very low temperature ultrasonic measurements. These authors used _eq.

(6)

^for

n(E),

^{with a = 0.1}

(K2).

no ^was taken

equal

^{to 3.5 x}

¹⁰³¹ erg-1 ^{cm- 3}

^{in order}

to account for the fact that all the excitations involved in heat

capacity

^{do not take}

part

ⁱⁿ

transport

pro-

perties. They

also chose M ⁼ 1.6 eV. No information

on the relaxation _process can be obtained from these very low temperature ^results.

Therefore,

^we

adapted

the parameters ^{in order to} obtain the best fit of our

velocity

^maximum

(M

^{= 1.6}

eV,

^{D = 2.35}

eV).

^With

these

values,

^a

plateau

is obtained for

l -1

above 6

K,

in

disagreement

^with

experiment

^{(dotted lines in}

figures

^{5 and}

6).

Our results seem to rule out the

assumption

^{of a}

constant or

only slightly

energy

dependent density

^of

states, ^{and are more} consistent with a

quadratic

variation of

n(E), subject

^to the condition of

taking

for the a-coefficient a value smaller than that obtained from heat

capacity

measurements and ultrasonic

experiments. However,

^{for a} best fit the

frequency dependence

^{of the}

coupling energies

should also be taken into account, but this

frequency dependence

cannot be deduced from

only hypersonic

^measu-

rements.

4.

Polymethylmethacrylate.

^{- 4.} .1 RESULTS. ^- The

velocity

measurements for the two

samples

^{are shown}

in

figure

^{7. Within} the accuracy of our

experiments,

the

logarithmic temperature dependence

^{of the}

velocity

at low temperature ^{is not} observed :

instead,

^the

velocity

^{tends to be constant below 5 K both for}

commercial

plexiglass

and for the

high purity sample.

The decrease of

velocity

^with

increasing

temperature

FIG. 7. - Variation of the velocity ^of longitudinal hypersound (18 GHz) ^vs. temperature in PMMA : (+) high purity sample;

(0) commercial sample.

due to relaxation is

higher

^{in PMMA} than in silicate

glasses.

^A

large

difference between the two

samples

must also be noted.

PMMA is a strong ^{scatterer of}

light :

^the

intensity

of the Brillouin lines is _very

high.

^This material is also a

strong ^{scatterer of}

phonons,

and the Brillouin line- width is

larger

than in usual

inorganic glasses.

^For

these _reasons, the inverse mean free

path

^{is easier to}

measure and the results are more accurate than those in vitreous silica. The inverse mean free

path,

pro-

portional

^{to T between 1.7} and 10 K for the two

samples,

^is

higher by

^a factor of 2.5 in

high-purity

PMMA than in commercial

plexiglass. However,

^we

cannot rule out the

possibility

that this

surprising

difference is due to some

partial crystallinity

^{of the}

commercial

sample,

and further

experiments

^{will be}

necessary ^on this

point.

FIG. 8. ^- Variation of the inverse mean free path ^of longitudinal hypersound (18 GHz) ^vs. temperature ^{in PMMA :} (+) high purity

sample; (0) commercial sample.

1158

4.2 DISCUSSION. ^- The similarities between the thermal

properties

^{of PMMA} and those of

inorganic glasses suggest

^{that the 2} LS model could also be sufficient to describe the low temperature

physical properties

^of

organic polymers.

This is in

agreement

with the observation of ^a minimum of the dielectric constant in PMMA. Ultrasonic studies in PMMA have not been

performed

up ^{to now} in the _very low temperature range

(below

¹

K)

^{where the resonant}

interaction between

phonons

^{and 2 LS can be}

expected

to dominate. Without an ultrasonic determination of the parameters ^{of the}

model,

^{it is not}

possible

^to try ^a numerical

comparison

^{of our} results with theoretical

values,

^{so that our}

analysis

below will be restricted to

qualitative

considerations.

It must be noted that a maximum of ultrasonic

velocity

^has

recently

been observed in

AS2S3 [25],

ⁱⁿ

which the sound

velocity

^{has a} value similar to that in PMMA. This maximum is less

pronounced

^and

occurs at lower

temperature

than in vitreous silica.

A similar behaviour in PMMA would

explain

^that

this maximum is not observed in ^our

experiments.

The acoustic

properties

^{of PMMA down to 1.7 K}

for

hypersonic frequencies

appear ^to be dominated

by

a relaxation _process. This can result from an increase with _energy of

n(E)

^{faster in PMMA than in vitreous}

silica,

^{as indicated}

by

^the

analysis

^of

specific

^heat

results

[23],

^{or from a}

high

^value of D in PMMA. These

hypotheses

^{lead to a} shift of the

velocity

^{maximum to}

lower

temperatures,

^{and to a} strong increase of

lrell,

in

qualitative agreement

^{with our}

experiments.

The contribution of the resonant interaction between

phonons

^{and 2 LS seems to be masked}

by

the relaxa- tion. This can be

supported by

^a

rough

calculation of

lresl

^{in the dominant}

phonon approximation.

^The

specific

^{heat C} and thermal

conductivity

^{K in PMMA}

are known

[1].

^{Their values are near} those in vitreous silica. 18 GHz

phonons

^are dominant for heat trans-

port properties

^{near 0.2}

^K,

^{and the mean free}

path

^/can

be calculated at this

temperature

^{from K =}

(1 J3)

^Cvl.

Assuming

^that

lresl

^is

proportional

^to

T-1,

^{the value}

l- 1 =

50

cm-1

is found for PMMA at 2 K

showing

that this contribution is

negligible

ⁱⁿ

comparison

^to

the

experimental

^result.

In

conclusion,

^the

qualitative

differences between the acoustic

properties

of PMMA and those of silicate

glasses

^at very low temperatures ^{cannot be invoked to} rule out the 2 LS model in

organic glasses. However,

it seems difficult to

predict

^{the T}

dependence

^{of 1- 1 in}

a

large

temperature range from _eq.

(3).

Lastly,

the decrease with

temperature

^of

l -1

shows

that,

ⁱⁿ

PMMA,

^the

geometrical scattering

^{of elastic}

waves

by

^frozen-in fluctuations do not contribute

appreciably

^{to sound}

absorption,

^{as we have}

already

noted for some silicate

glasses [26].

5. Conclusion. ^- Our measurements of

velocity

and attenuation of

longitudinal hypersonic

^waves ( ~ 35

GHz)

^{at low}

temperatures

in three silicate

glasses

^can be accounted for

by assuming

^{a resonant}

interaction between thermal

phonons

^and

low-energy

excitations _up to 10 K.

By using

^the

simplified

^des-

cription

of these excitations

by

²

LS,

^{our results can be}

compared

^to calculations. The excess

density

^{of states}

is unknown in the energy range under consideration and the

extrapolation

^{of the}

expressions

^of

n(E)

obtained

previously

^{for lower}

energies

^{does not lead}

to a

satisfactory

calculation. We obtained a

good

^fit

by assuming

^a

parabolic

increase of

n(E),

^{with a coeffi-}

cient smaller than that obtained from

specific

^heat

measurements. This suggests ^{that the} energy

depen-

dence of the

density

of the excitations

responsible

^for

the

transport properties

^can differ from that of the total

density

^{of states.}

Furthermore,

^the

coupling

coefficients

M1

^{and D seem to be}

nearly

^{constant with}

, energy.

The

study

^{of two PMMA}

samples

^{has shown that}

the resonant process has ^a

negligible

effect in the _range from 2 to 10 K. On the other

hand,

the relaxational contribution is strong ^{and leads to an}

unexplained T-dependence

^{of the}

absorption.

^Further

experiments by

ultrasonic

propagation

^{or Brillouin}

scattering

^at

lower

temperatures

^are necessary ^to

explain

^this

point.

Acknowledgments.

^- The authors are very

grateful

to the referees for

helpful

^comments.

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