Connection between the low temperature acoustic properties and the glass transition temperature of fluoride glasses

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Connection between the low temperature acoustic properties and the glass transition temperature of

fluoride glasses

P. Doussineau, M. Matecki, W. Schön

To cite this version:

P. Doussineau, M. Matecki, W. Schön. Connection between the low temperature acoustic properties and the glass transition temperature of fluoride glasses. Journal de Physique, 1983, 44 (1), pp.101-107.

�10.1051/jphys:01983004401010100�. �jpa-00209564�

Connection between the low temperature ^acoustic properties

and the glass transition temperature ^{of fluoride} glasses

P. Doussineau (*), ^{M. Matecki} (**) ^{and W. Schön} (*)

(*) Laboratoire d’Ultrasons (*), Université Pierre et Marie Curie,

Tour 13, ⁴ place Jussieu, 75230 Paris Cedex 05, ^France

(**) Laboratoire de Chimie Minérale D (+), Université de Rennes Beaulieu, 35042 Rennes Cedex, ^France (Reçu ^{le 22} juin ^1982, révisé le 14 septembre, accepté ^{le 23} septembre 1982)

Résumé.

Les mesures de la variation de vitesse et de l’absorption ^d’ondes acoustiques longitudinales ^{et trans-}

versales de fréquences voisines de 1 GHz dans trois verres fluorés, pour des températures ^entre 0,1 ^{K et} 1,5 K, ^ont permis de déterminer la densité spectrale ^des systèmes à deux niveaux (S2N) qui régissent l’ensemble des propriétés

à basse température, ainsi que les constantes de couplage phonons-S2N. ^{Pour ces trois verres,} la densité spectrale

des S2N varie comme l’inverse de la température de transition vitreuse Tg, ^{tandis que les} constantes de couplage phonons longitudinaux et transversaux avec les S2N varient linéairement avec Tg.

Abstract.

The variations of the phase velocity ^{and the} absorption ^of longitudinal ^and transverse acoustic waves

of frequencies around 1 GHz have been measured in three fluorozirconate glasses ^{in the} temperature range 0.1 K to 1.5 K. The acoustic behaviour at these low temperatures has been ascribed to the existence of two-level systems (TLS). ^{From the} measurements, ^{the TLS} spectral density ^{and the} TLS-phonon coupling ^{constants have been}

determined. The TLS spectral density ^{has been found to be} inversely proportional ^{to the} glass transition temperature

T g. ^The TLS-phonon coupling ^{constants vary} linearly ^with T g.

Classification

Physics ^Abstracts

43.35

62.65

1. Introduction.

It is now well established that all glassy ^or amorphous materials, ^whether insulating, semi-conducting, polymeric ^{or metallic,} have similar low temperature properties [1, 2]. Generally ^{these are} explained by assuming ^{that, in} glass, atoms, ^or groups of atoms, ^can occupy ^two nearly equivalent positions corresponding ^to the minima of asymmetric ^double

well potentials. ^{At low} temperatures ^{the movement of} the atoms mainly ^{occurs via} tunnelling through ^the

energy barrier [3, 4]. Nevertheless this is a phenomeno- logical ^{model and a} microscopic description ^{of the} tunnelling particles ^{is still} lacking. Often this model is referred to as the TLS (two-level system) model because for most of the properties it is sufficient to consider the two lowest _energy states of the tunnelling particle.

Recently several, theoretical and experimental pa- pers appeared, ^{which are} concerned with this problem.

Among ^{them some} papers propose ^a link between the low temperature properties ^{and the} glass ^transition

temperature Tg. ^First Reynolds [5] remarked that the (+) Associated with the Centre National de la Recherche

Scientifique.

phonon ^{mean free} path ^of glasses (deduced ^from

thermal conductivity experiments) ^increases linearly

with Tg. ^{One year later,} Raychaudhuri ^{and Pohl} [6]

measured the heat capacity ^{of water} doped ^K-Ca

nitrates. The coefficient of the quasi ^{linear excess} specific ^{heat at low} temperatures ^scales very well with a

Tg ’ law. Recent measurements of low temperature thermal conductivity ^{in some} K-Ca nitrates show also

a change ^{which can} be connected to the change ^of rj7].

At the same time Cohen and Grest [8] ^{have inde-} pendently given ^a microscopic description ^{of the} tunnelling ^{centres on} the basis of the free-volume model. In this model TLS originates from the tunnel-

ling ^{of a} particle (atom ^or molecule) surrounding ^voids

frozen at Tg. Moreover with semiquantitative argu- ments they ^{were able to} predict ^{that the} density ^of

states of the tunnelling particles ^{varies as} aTg ^where

a is only weakly dependent ^on the chemical nature of the material.

In order to explore further the connection between the low temperature properties ^of amorphous ^materials

and their glass transition temperature ^{we have} perfor-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004401010100

102

med acoustic propagation measurements in various fluorozirconate glasses with different values of T 9*

Recently ^{it was} shown that in a fluorozirconate glass the usual behaviour of glasses is observed for the ultrasonic properties ^{at low} temperatures [9] : ^the

sound velocity first increases as the logarithm ^{of the}

temperature ^{and the} attenuation at high ^acoustic intensity ^{increases as} the cube of the temperature ^and is frequency independent. ^{It was} also shown from ultrasonic measurements that it is possible ^{to deter-}

mine the TLS spectral density and also the coupling

constants between the TLS and the acoustic wave.

On another hand, ^acoustic propagation is believed to be a good tool for the study ^of glasses because ultra- sonic waves are strongly coupled ^to the TLS and the acoustic properties ^of glasses ^{at low} temperatures have been found to be almost independent ^{of the} impurity ^content of the material [10], contrary ^to dielectric constant [11] ^or specific ^heat [12] ^measure-

ments which are sensitive to other defects such as

aH-.

The fluorozirconate glasses ^were chosen because a

great number of materials can be prepared ^{with the}

same glass ^former ZrF4 and there is a large spread

in their glass transition temperatures.

2. Experimental techniques.

^{In this} paper ^we present the results of ultrasonic measurements in two fluorozirconate glasses : the materials are desi-

gnated by ^{the LAT and BALNA} symbols. ^{We add the}

results obtained previously ^{on another} fluorozirconate

glass : ^V52 [9].

The preparation procedure ^{was the same for the}

three glasses. Details have been given ^elsewhere [13].

For our present purpose ^{we note} that the quenching

rate was about 5 K. s - ’, and all the glasses ^were

annealed near Tg ^{for one hour : in table I are given for}

the three glasses, ^the composition, ^the specific mass,

Table I.

Composition, specific mass, glass, crystalli-

zation and melting temperatures of fluoride glasses.

the glass, crystallization ^and melting temperatures (the last three quantities ^{were measured} by differential thermal analysis). ^The accuracy ^on the glass transition, temperature ^{is about 5 K. The same} glass ^was prepared

with two quenching ^rates (5 ^{K. s - ’ and about}

70 K.s-’). ^In these conditions the glass temperature

was 10 K lower for the fastest quenching ^{rate. Unfor-}

tunately this material was unsuitable for ultrasonic work. The impurities ^{in the} samples ^are paramagnetic

ions (~ 10 ppm Fe), mostly 0H- ions and also oxygen. We ^can bear in mind that in silica glasses ^the

ultrasonic behaviour is not sensitive to the OH con- tent [10].

For each material samples ^{about 5 mm} long,

4 x 4 mm’ section, ^were prepared ^{with two} plane ^and parallel faces. The ultrasonic waves were generated by

resonant quartz ^or LiNbo3 transducers. Standard

pulse ^echo techniques ^{were used. Low} temperatures

were obtained with a dilution refrigerator.

3. Experimental ^results.

The velocities of lon-

gitudinal ^and transverse acoustic waves were first measured at 0.1 K for all the samples. ^The accuracy

was better than 1 %, ^{but it must} be noted that no

correction has been made for the length change ^{of the} samples ^{between room} temperature and 0.1 K. The results are given ^{in table II.}

The attenuation of both longitudinal ^{and transverse}

acoustic waves was measured as a function of the temperature ^{in the} range 0.1 K to about 1.5 K at various frequencies ranging from 130 MHz to 1 900 MHz. The observed behaviour for the LAT and BALNA samples is similar to that previously ^observed

in the V52 glass [9]. The attenuation first increases

as the cube of the temperature ^{and is} frequency ^inde- pendent. ^For each acoustic mode (longitudinal ^and transverse) and for all the samples ^the measurements were performed ^{at least at two different} frequencies ⁱⁿ

Table II.

Parameters used to describe the _propaga- tion of ^{acoustic waves in three} fluoride glasses ^{at low}

temperature. ^{All these} parameters ^{are either} directly

measured in this work or deduced using equations ( 1) ^to

(4). ^Their meanings ^are given ^{in the text.}

order to check the frequency independence. ^Moreover,

in order ^to improve ^the accuracy ^we measured the attenuation at the highest possible frequency ^because

the T3 law is obeyed up ^{to a} higher temperature ^when the frequency increases. Some curves are shown in

figures ^{1 and} 2, using logarithmic coordinates. In fact the attenuation is the change ^{as a} function of the

Fig. ^1.

Attenuation of longitudinal (left) ^{and transverse} (right) ultrasonic waves at various frequencies ^{as a function}

of the temperature in the BALNA glass.

Fig. ^2.

Attenuation of longitudinal (left) ^{and transverse} (right) ultrasonic waves at various frequencies ^{as a function}

of the temperature in the LAT glass.

temperature ^{and the} procedure ^is equivalent ^{to the}

subtraction of a temperature independent background.

Taking ^{into account all the errors} (signal ^{to noise} ratio,

accuracy of the variable attenuators, ...) ^{we have}

estimated our accuracy better than 0.4 dB. cm - 1. In

one experiment ^the accuracy ^{was a} factor of four better

by using ^a longer ^acoustic path.

In this _paper we are concerned only ^{with the} high amplitude attenuation. In the previous paper ^on the V52 glass [9] it has been shown that the usual satura- tion effects are observed in this class of materials.

The relative change ^{of the} phase velocity ^{of the}

acoustic wave was measured in the same temperature range 0.1 K to 1.5 K and at various frequencies

between 250 MHz and 1 000 MHz. Under these

conditions, ^as it is well known for glasses, ^{the sound} velocity ^{increases as the} logarithm ^{of the} tempe-

rature with a slope ^{which is} frequency independent.

With our experimental set-up ^{we were able to detect} relative velocity changes smaller than 5 x 10- 6.

Some curves are shown in figures ^{3 and 4} using ^semi- logarithmic coordinates.

In all the measurements we have checked that there

was no heating ^{effect at} the lowest temperatures by varying ^the intensity ^of the ultrasonic wave by ^more

than 20 dB.

4. Theory.

^{In order} to extract the parameters ^of the TLS theory ^{from our acoustic} measurements we

need first to recall ^some results of this theory [1].

An important quantity ^{for our} purpose must be

introduced now. In the standard TLS theory [3, 4] P is

the product of the number of TLS _per unit volume

by ^a normalization factor of the distribution function of the double-well parameters. ^{P is not the} density

Fig. ^3.

^Phase velocity change ^of longitudinal (bottom)

and transverse (top) ultrasonic waves as a function of the

logarithm ^{of the} temperature in the BALNA glass.

104

Fig. ^4.

^Phase velocity change ^of longitudinal (bottom)

and transverse (top) ultrasonic wave as a function of the

logarithm ^{of the} temperature in the LAT glass.

of states n(E), ^{as measured} by specific ^heat experiments,

but the two quantities ^are related. ^{In order to avoid}

confusion we propose ^to call P the spectral density

of the TLS. The density ^{of states} n(E) ^can be calculated

knowing ^the distribution function. In the standard

theory n(E) ^is given by :

where Am ^is the minimum value for the tunnelling

energy. The second expression is valid when E > Am.

It is believed that this condition is generally ^fulfilled.

Therefore n is ^{taken as a constant. The knowledge}

of the limits o , the distribution function allows the

density ^of states to be calculated exactly. ^{In this case}

the proportionality factor between P and the density

of states is known. Unfortunately ^no experimental

determination has been made of these limits.

The interaction of an ultrasonic wave of frequency M/2 ^{n with a TLS of} energy E ^occurs by ^{two different}

processes. In the resonant process ^a phonon ^of energy hco is absorbed by the TLS of the same energy. The

resulting attenuation is power dependent ^{and obser-}

vable at low intensity of the acoustic wave. In our

experiments ^we have worked at high ^acoustic power and therefore the resonant attenuation is saturated.

The corresponding velocity change ^is given ^{in the}

case hcv « ^kT by

with

where To ^{is an} arbitrary ^reference temperature, P is the

constant spectral density ^{of the} TLS, 1B ^{the TLS-} phonon coupling constant, p the specific mass, vT the sound velocity

^{L or T stands for the} polariza-

tion of the acoustic wave.

In the second _process, the acoustic wave modulates the TLS splitting. ^{The return to} the thermal equilibrium

takes place ^{with a} characteristic relaxation time Tl.

This leads to a relaxational attenuation which is

given by

This equation ^{is valid} only when the dominant relaxa- tional _process is the direct or one phonon process and if the condition wT 1 > 1 (low temperature regime) is fulfilled. In the above equation K3 ^is given by

We have used the notation of reference [ 14J. ^{We are not}

concerned here with the high temperature regime.

5. Interpretation.

Clearly ^{our results on the LAT}

and BALNA samples ^are qualitatively ^{similar to those}

obtained previously ^{on the V52} glass ^{and are well} explained in the framework of the tunnelling ^model.

The logarithmic ^increase of the sound velocity ^between

0.1 and about 1 K is ascribed to the resonant interac- tion between the TLS and the elastic wave. With

equation (1) ^{and from our results we} determined CL

and CT for the LAT and the BALNA samples. ^The

values are given ^{in table II. For the BALNA} glass ^the experimental points ^{for the} temperatures ^{around 1 K} (as ^{shown in} figure 3) ^are systematically above the In T

curve as extrapolated from the lowest temperatures.

This effect was observed for the longitudinal ^{and the}

transverse waves and at all the frequencies. ^{Such an}

effect has already been observed in silica glasses ^and

has been interpreted ^{as the} consequence of a slight

increase of the TLS density ^{of states with the} energy

[ 15J. ^{A similar} explanation ^can be used for the BALNA

glass. ^{It must be} emphasized that this effect was not observed in the LAT and V52 glasses.

From the attenuation curves and with equation (3)

we calculated the products CL K3 ^and CT K3 ^{from the} longitudinal ^and transverse measurements respectively.

Knowing CL ^and CT ^{from the} velocity measurements

we obtain two values for K3. ^For every material the two values agree very well within the experimental ^accu-

racy. In table II, K3 ^is given for the three glasses ^with

the estimated uncertainty.

Now from CL, CT ^and K3, ^{and with} equations (2)

and (4) we ^can obtain values for the TLS spectral density ^{P and the} longitudinal ^and transverse coupling constants TL ^and yT. The calculated values are given

in table II.

_

The accuracy ^on the determinations of P, 7L and _7T

reflects mainly the accuracy ^on the value of K3’ ^{In the}

uncertainty ^we have included the _accuracy of one measurement and the scattering of the results when the

frequency ^{or the} polarization ^{of the wave is} changed.

In order to test the prediction of Cohen and Grest [8]

and to check the ^{result of} Raychaudhuri ^{and Pohl} [6]

the values of P are plotted ^{as a} function of Tg ^{1 in} figure ^5. Although ^{the error bars are rather} large, ^our results agree very well with a variation of the form

aT-’.

6. Discussion and comments. - The question ^arises

of the validity ^{and the} generality ^{of the P -} Tg ’ ^law.

In this respect ^{we discuss some} points ^{and make some}

comments. Some of these are strongly in favour of the

T-’ law. Some others raise serious difficulties about the above interpretation.

1. Particularly serious is the problem ^to know if the result of figure ^{5 is not an} effect of changes ^{in the} composition ^{of the} samples. ^The ZrF4 ^{content of our}

three samples ^is almost identical (between ^{52 and}

60 %) but the other constituents are in part ^different.

Unfortunately ^{it is difficult to answer this} question.

The only way ^to prepare various samples ^of exactly

the same glass ^{with different} Tg ^{values is} by varying ^the quenching ^{rate. We saw} above that only ^small changes

in Tg ^are obtained in that _{way. As} a consequence only experiments ^{on the same series of} glasses (for example

silica based glasses, fluorozirconate glasses, ^K-Ca

nitrate glasses, ...) ^are possible. ^We would also mention that Raychaudhuri ^{and Pohl} [6] have found that the

excess specific ^{heat of a} 50-50 K-Ca nitrate sample

scales _very well with those of water-doped ^{40-60 K-Ca}

nitrate samples ^{on a} Tg ^{law. They have also observed}

the same behaviour on a series of various silica-based

glasses.

2. Cohen and Grest [8] ^have predicted ^not only ^a

Tg dependence for the TLS density ^{of states but also}

that the coefficient must be only weakly ^material dependent. ^{Of course much more work is} necessary

Fig. ^5.

Spectral density of two-level systems ^{for three} fluoride glasses ^{as a} function of. the inverse of their glass

transition temperatures. The values corresponding ^{to two}

silica-based glasses (pure Si02 ^{and a} borosilicate BK7)

are plotted ^{on the same} diagram.

to test this prediction. ^{But in a first} attempt, ^{we have} examined the literature ^{to see} if the same set of acoustic

experiments ^{as ours in} fluorozirconate glasses ^have

been reported ^{for some other} glasses. ^{We found two} compounds : ^a borosilicate glass ^{where all the} experi-

ments were performed by ^{the same} group [ 16, 17],

and silica (unfortunately for this material the data

were obtained by ^{two different} [15, 17, 18] groups).

We have analysed ^the experimental ^{data on attenua-}

tion and velocity changes ^{as above} for the fluorozir- conate glasses. ^The corresponding ^values of CL, CT, K 3

and P, yL and _yT, are given ^{in table III. For this set of}

values it is difficult to estimate the error bars and

consequently ^{we did not} give them. The values dis-

played ^{in table III are} somewhat different from

previously published estimations for P, yL and _TT [19].

But we must bear in mind that in present ^{work we} have taken into account the distribution of the coupling

constants. Therefore the meaning ^{of the} coupling

constants is not exactly ^{the same in the various} papers.

Moreover, ^echo experiments ^can provide ^{values of}

the coupling ^constants y without the knowledge ^{of the} density ^{of states. This was done for} amorphous S’02

where the longitudinal coupling ^constant yL ^was measured [20]. ^{It was} found 1’L ^{= (1-5 ±} 0.4) ^eV

which is compatible ^with the value 1.04 eV quoted here,

if the errors ^{bars are} taken into account. If we plot ^the

values of P for silica and the borosilicate glasses ^{on the}

same diagram ^{as our} values for the fluozirconate

glasses ^{as a} function of T 9’ ^{we see that} surprisingly they agree with the same P = aTg ’ ^{law, as shown in}

figure 5. It appears that the P = aTg 1 ^{law has some}

kind of generality. ^More experimental work in various

glassy materials is needed in order to confirm this assertion.

3. The low temperature specific ^{heat has been}

measured in a V52 sample and in another fluorozir- conate glass ^not investigated ^here [21]. Although

Table III.

Parameters describing ^the propagation of ^{acoustic waves} in vitreous Si02 ^{and the} borosilicate

glass ^{BK7 and their} interaction with two-level systems.

All these parameters ^are deduced from ^the experiments reported ⁱⁿ references [15] ^to [18] using equations (I)

to (4).

106

a large ^excess specific ^heat (about ^ten times larger ^than

in silica) has been detected this specific ^{heat cannot be}

fitted by ^{a linear T law or a T° law with v around 1.2}

as in silica ^glasses. Therefore it is difficult to compare

our P values with the density ^{of states} determined from

specific ^heat measurements. On another hand, ^{as has}

been discussed ^above, it is worth mentioning ^{that the}

value of P deduced from acoustic experiments ^is

different from, but related to, ^the density ^{of states} no

given by the linear temperature part ^{of the} specific

heat below 1 K.

4. Our treatment of the experimental ^data (and consequently ^our conclusion) ^{assumes the} validity ^of

the TLS theory in its standard version [3, 4]. ^But

because this theory, although phenomenological, ^has

been demonstrated to be _very powerful for the inter-

pretation of the low temperature properties ^of glasses,

we think this point ^{is not} controversial. Probably ^a

more or less slight modification of the distribution function different from one material to the other could give ^another way ^to explain ^the differences observed in the acoustic behaviour of the various fluorozirconate glasses.

5. Finally ^{we turn to} the values of the coupling constants yL ^and YT’ Usually ^{it is} thought ^{that these} quantities ^are of the order of 1 eV and almost inde-

pendent of the material [22]. Apparently ^{this is true for}

the fluorozirconate glasses. YL is between 0.71 eV and 1.13 eV and yT between 0.43 eV and 0.65 eV for the three glasses studied here. But a more careful exami- nation of the results shows that the values of _yL and yT scales _very well with a linear dependence ^on Tg

as shown in figure 6. This result is quite surprising. ^To

our knowledge ^{there is no} prediction about such a

variation. We ^must note that the YL ^and yT values for the borosilicate and silica glasses ^{do not} agree with a

Fig. ^6.

TLS-phonon coupling ^{constants for three fluoride} glasses ^{as a function of their} glass transition temperatures.

The two sets of points ^{refer to} longitudinal (top) ^{and trans-}

verse (bottom) polarizations.

linear Tg dependence including the values of the fluorozirconate glasses. Therefore the Tg ^variation

of _YL and yT ^{is not so} general ^{(if it exists) as the} Tg

variation of P.

7. Conclusioa

We have presented ^{in this} paper

experimental results which give ^a strong indication for

a connection between the TLS spectral density ^of

fluorozirconate glasses and their glass ^transition

temperature. Our results confirm a theoretical ana-

lysis ^which predicts ^a Tg ^{1 law and} previous specific

heat measurements in another kind of glass. ^{Of course}

the generality ^{of this} behaviour has ^to be checked further by ^{much more} experimental work in other

glasses.

References

[1] ^{See collected} papers in Amorphous Solids, ^edited by

W. A. Phillips (Springer Verlag) ^1981.

[2] BLACK, ^J. L., in Metallic Glasses, ^edited by ^{H. J. Gün-}

therodt and H. Beck (Springer Verlag) 1981, p.167.

[3] ANDERSON, ^P. W., HALPERIN, ^{B. I. and} VARMA, ^C. M.,

Philos. Mag. ²⁵ (1972) ^1.

[4] PHILLIPS, ^W. A., ^{J. Low} Temp. Phys. ⁷ (1972) ^351.

[5] ^{REYNOLDS Jr., C. L., J.} Non-Cryst. ^{Solids 30} (1978) ^371.

[6] RAYCHAUDHURI, ^{A. K. and} POHL, ^R. O., Solid State Commun. 37 (1980) 105 ; Phys. ^{Rev. B 25} (1982)

1310.

[7] KLITSNER, T., RAYCHAUDHURI, ^{A. K. and} POHL, ^R. O.,

J. Physique Colloq. ⁴² (1981) ^C6-66.

[8] COHEN, ^{M. H. and} GREST, ^G. S., Phys. Rev. Lett. 45

(1980) 1271 ; Solid State Commun. 39 (1981) ^143.

[9] DOUSSINEAU, P. and MATECKI, M., ^J. Physique-Lett.

42 (1981) ^L-267.

[10] HUNKLINGER, S., PICHE, L., LASJAUNIAS, ^{J. C. and} DRANSFELD, K., ^J. Phys. ^{C 8} (1975) ^L-423.

[11] ^VON SCHICKFUS, ^{M. and} HUNKLINGER, S., ^J. Phys. ^C

9 (1976) ^L-439.

[12] LASJAUNIAS, ^J. C., RAVEX, A., ^{VANDORPE, M. and} HUNKLINGER, S., Solid State Commun. 17 (1975)

1045.

[13] POULAIN, ^{M. and} LUCAS, J., Verres Refract. ³² (1978) 505 ;

LECOQ, A. and POULAIN, M., ^Verres Refract. ³⁴ (1980)

333 and references therein.

[14] DOUSSINEAU, P., FRENOIS, Ch., LEISURE, ^R. G., ^LEVE-

LUT, A. and PRIEUR, J.-Y., ^J. Physique ⁴¹ (1980)

1193.

[15] PICHE, L., MAYNARD, R., HUNKLINGER, ^{S. and} JÄCKLE, J., Phys. Rev. Lett. 32 (1974) ^1426.

[16] HUNKLINGER, ^{S. and} PICHE, L., Solid State Commun.

17 (1975) ^1189.

[17] JÄCKLE, J., PICHE, L., ARNOLD, ^{W. and} HUNKLINGER, S., ^J. Non-Cryst. ^{Solids 20} (1976) ^365.

[18] GOLDING, B., GRAEBNER, ^{J. E. and} KANE, ^A. B., Phys. Rev. Lett. 37 (1976) ^1248.

[19] HUNKLINGER, ^{S. and} ARNOLD, W., ⁱⁿ Physical ^Acous-

tics, ^edited by R. N. Thurston and W. P. Mason

(Academic Press, ^New York) 1976, Vol. 12, p.155.

[20] GRAEBNER, J. E. and GOLDING, B., Phys. ^{Rev. B 19} (1979) 964.

[21] LASJAUNIAS, ^J. C., ^{to be} published.

[22] ^{See for} example, PHILLIPS, ^W. A., ^{J. Low} Temp. Phys.

11 (1973) 757, ^or

JOFFRIN, ^{J. and} LEVELUT, A., ^J. Physique ³⁶ (1975) ^811.