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Connection between the low temperature acousticproperties and the glass transition temperature of

fluoride glassesP. Doussineau, M. Matecki, W. Schön

To cite this version:P. Doussineau, M. Matecki, W. Schön. Connection between the low temperature acoustic propertiesand the glass transition temperature of fluoride glasses. Journal de Physique, 1983, 44 (1), pp.101-107.�10.1051/jphys:01983004401010100�. �jpa-00209564�

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Connection between the low temperature acoustic propertiesand 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. 2014 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, ontpermis de déterminer la densité spectrale des systèmes à deux niveaux (S2N) qui régissent l’ensemble des propriétésà basse température, ainsi que les constantes de couplage phonons-S2N. Pour ces trois verres, la densité spectraledes S2N varie comme l’inverse de la température de transition vitreuse Tg, tandis que les constantes de couplagephonons longitudinaux et transversaux avec les S2N varient linéairement avec Tg.

Abstract. 2014 The variations of the phase velocity and the absorption of longitudinal and transverse acoustic wavesof frequencies around 1 GHz have been measured in three fluorozirconate glasses in the temperature range 0.1 Kto 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 beendetermined. The TLS spectral density has been found to be inversely proportional to the glass transition temperatureT g. The TLS-phonon coupling constants vary linearly with T g.

J. Physique 44 (1983) 101-107 JANVIER 1983,

Classification

Physics Abstracts43.35 - 62.65

1. Introduction. - It is now well established thatall glassy or amorphous materials, whether insulating,semi-conducting, polymeric or metallic, have similarlow temperature properties [1, 2]. Generally these areexplained by assuming that, in glass, atoms, or groupsof atoms, can occupy two nearly equivalent positionscorresponding to the minima of asymmetric doublewell potentials. At low temperatures the movement ofthe atoms mainly occurs via tunnelling through theenergy barrier [3, 4]. Nevertheless this is a phenomeno-logical model and a microscopic description of thetunnelling particles is still lacking. Often this model isreferred to as the TLS (two-level system) model becausefor most of the properties it is sufficient to considerthe 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 thelow temperature properties and the glass transitiontemperature Tg. First Reynolds [5] remarked that the

(+) Associated with the Centre National de la RechercheScientifique.

phonon mean free path of glasses (deduced fromthermal conductivity experiments) increases linearlywith Tg. One year later, Raychaudhuri and Pohl [6]measured the heat capacity of water doped K-Canitrates. The coefficient of the quasi linear excess

specific heat at low temperatures scales very well with aTg ’ law. Recent measurements of low temperaturethermal conductivity in some K-Ca nitrates show alsoa change which can be connected to the change ofrj7].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-volumemodel. In this model TLS originates from the tunnel-ling of a particle (atom or molecule) surrounding voidsfrozen at Tg. Moreover with semiquantitative argu-ments they were able to predict that the density ofstates of the tunnelling particles varies as aTg wherea is only weakly dependent on the chemical nature ofthe material.

In order to explore further the connection betweenthe low temperature properties of amorphous materialsand their glass transition temperature we have perfor-

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

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

Recently it was shown that in a fluorozirconate

glass the usual behaviour of glasses is observed for theultrasonic properties at low temperatures [9] : thesound velocity first increases as the logarithm of thetemperature and the attenuation at high acoustic

intensity increases as the cube of the temperature andis frequency independent. It was also shown fromultrasonic measurements that it is possible to deter-mine the TLS spectral density and also the couplingconstants between the TLS and the acoustic wave.On another hand, acoustic propagation is believed tobe a good tool for the study of glasses because ultra-sonic waves are strongly coupled to the TLS and theacoustic properties of glasses at low temperatureshave been found to be almost independent of theimpurity content of the material [10], contrary to

dielectric constant [11] or specific heat [12] measure-ments which are sensitive to other defects such asaH-.The fluorozirconate glasses were chosen because a

great number of materials can be prepared with thesame glass former ZrF4 and there is a large spreadin their glass transition temperatures.

2. Experimental techniques. - In this paper we

present the results of ultrasonic measurements intwo fluorozirconate glasses : the materials are desi-gnated by the LAT and BALNA symbols. We add theresults obtained previously on another fluorozirconateglass : 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 quenchingrate was about 5 K. s - ’, and all the glasses wereannealed near Tg for one hour : in table I are given forthe 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 differentialthermal analysis). The accuracy on the glass transition,temperature is about 5 K. The same glass was preparedwith two quenching rates (5 K. s - ’ and about70 K.s-’). In these conditions the glass temperaturewas 10 K lower for the fastest quenching rate. Unfor-tunately this material was unsuitable for ultrasonicwork. The impurities in the samples are paramagneticions (~ 10 ppm Fe), mostly 0H- ions and also

oxygen. We can bear in mind that in silica glasses theultrasonic 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 andparallel faces. The ultrasonic waves were generated byresonant quartz or LiNbo3 transducers. Standard

pulse echo techniques were used. Low temperatureswere obtained with a dilution refrigerator.

3. Experimental results. - The velocities of lon-

gitudinal and transverse acoustic waves were firstmeasured at 0.1 K for all the samples. The accuracywas better than 1 %, but it must be noted that nocorrection has been made for the length change of thesamples between room temperature and 0.1 K. Theresults 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 andBALNA samples is similar to that previously observedin the V52 glass [9]. The attenuation first increasesas the cube of the temperature and is frequency inde-pendent. For each acoustic mode (longitudinal andtransverse) and for all the samples the measurementswere 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 lowtemperature. All these parameters are either directlymeasured in this work or deduced using equations ( 1) to(4). Their meanings are given in the text.

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order to check the frequency independence. Moreover,in order to improve the accuracy we measured theattenuation at the highest possible frequency becausethe T3 law is obeyed up to a higher temperature whenthe frequency increases. Some curves are shown infigures 1 and 2, using logarithmic coordinates. In factthe 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 functionof the temperature in the BALNA glass.

Fig. 2. - Attenuation of longitudinal (left) and transverse(right) ultrasonic waves at various frequencies as a functionof the temperature in the LAT glass.

temperature and the procedure is equivalent to thesubtraction of a temperature independent background.Taking into account all the errors (signal to noise ratio,accuracy of the variable attenuators, ...) we haveestimated our accuracy better than 0.4 dB. cm - 1. Inone experiment the accuracy was a factor of four betterby using a longer acoustic path.

In this paper we are concerned only with the highamplitude attenuation. In the previous paper on theV52 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 temperaturerange 0.1 K to 1.5 K and at various frequenciesbetween 250 MHz and 1 000 MHz. Under these

conditions, as it is well known for glasses, the soundvelocity increases as the logarithm of the tempe-rature with a slope which is frequency independent.With our experimental set-up we were able to detectrelative 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 therewas no heating effect at the lowest temperatures byvarying the intensity of the ultrasonic wave by morethan 20 dB.

4. Theory. - In order to extract the parameters ofthe TLS theory from our acoustic measurements weneed 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 isthe product of the number of TLS per unit volumeby a normalization factor of the distribution functionof 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 thelogarithm of the temperature in the BALNA glass.

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Fig. 4. - Phase velocity change of longitudinal (bottom)and transverse (top) ultrasonic wave as a function of thelogarithm 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 avoidconfusion we propose to call P the spectral densityof the TLS. The density of states n(E) can be calculatedknowing the distribution function. In the standard

theory n(E) is given by :

where Am is the minimum value for the tunnellingenergy. 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 knowledgeof the limits o , the distribution function allows thedensity of states to be calculated exactly. In this casethe proportionality factor between P and the densityof states is known. Unfortunately no experimentaldetermination 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 differentprocesses. In the resonant process a phonon of energyhco is absorbed by the TLS of the same energy. Theresulting attenuation is power dependent and obser-vable at low intensity of the acoustic wave. In ourexperiments we have worked at high acoustic powerand therefore the resonant attenuation is saturated.The corresponding velocity change is given in thecase 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, vTthe sound velocity = L or T stands for the polariza-tion of the acoustic wave.

In the second process, the acoustic wave modulatesthe TLS splitting. The return to the thermal equilibriumtakes 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 processand if the condition wT 1 > 1 (low temperatureregime) is fulfilled. In the above equation K3 is given by

We have used the notation of reference [ 14J. We are notconcerned here with the high temperature regime.

5. Interpretation. - Clearly our results on the LATand BALNA samples are qualitatively similar to thoseobtained previously on the V52 glass and are wellexplained in the framework of the tunnelling model.The logarithmic increase of the sound velocity between0.1 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 CLand CT for the LAT and the BALNA samples. Thevalues are given in table II. For the BALNA glass theexperimental points for the temperatures around 1 K

(as shown in figure 3) are systematically above the In Tcurve as extrapolated from the lowest temperatures.This effect was observed for the longitudinal and thetransverse waves and at all the frequencies. Such aneffect has already been observed in silica glasses andhas been interpreted as the consequence of a slightincrease of the TLS density of states with the energy[ 15J. A similar explanation can be used for the BALNAglass. It must be emphasized that this effect was notobserved in the LAT and V52 glasses.From the attenuation curves and with equation (3)

we calculated the products CL K3 and CT K3 from thelongitudinal and transverse measurements respectively.Knowing CL and CT from the velocity measurements

we obtain two values for K3. For every material the twovalues agree very well within the experimental accu-racy. In table II, K3 is given for the three glasses withthe estimated uncertainty.Now from CL, CT and K3, and with equations (2)

and (4) we can obtain values for the TLS spectraldensity P and the longitudinal and transverse couplingconstants TL and yT. The calculated values are givenin table II.

_

The accuracy on the determinations of P, 7L and 7Treflects mainly the accuracy on the value of K3’ In the

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uncertainty we have included the accuracy of onemeasurement and the scattering of the results when thefrequency 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, ourresults agree very well with a variation of the formaT-’.

6. Discussion and comments. - The question arisesof the validity and the generality of the P - Tg ’ law.In this respect we discuss some points and make somecomments. Some of these are strongly in favour of theT-’ law. Some others raise serious difficulties aboutthe above interpretation.

1. Particularly serious is the problem to know if theresult of figure 5 is not an effect of changes in thecomposition of the samples. The ZrF4 content of ourthree samples is almost identical (between 52 and60 %) but the other constituents are in part different.Unfortunately it is difficult to answer this question.The only way to prepare various samples of exactlythe same glass with different Tg values is by varying thequenching rate. We saw above that only small changesin Tg are obtained in that way. As a consequence onlyexperiments on the same series of glasses (for examplesilica based glasses, fluorozirconate glasses, K-Canitrate glasses, ...) are possible. We would also mentionthat Raychaudhuri and Pohl [6] have found that theexcess specific heat of a 50-50 K-Ca nitrate samplescales very well with those of water-doped 40-60 K-Canitrate samples on a Tg law. They have also observedthe 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 alsothat the coefficient must be only weakly materialdependent. Of course much more work is necessary

Fig. 5. - Spectral density of two-level systems for threefluoride glasses as a function of. the inverse of their glasstransition temperatures. The values corresponding to twosilica-based glasses (pure Si02 and a borosilicate BK7)are plotted on the same diagram.

to test this prediction. But in a first attempt, we haveexamined the literature to see if the same set of acoustic

experiments as ours in fluorozirconate glasses havebeen reported for some other glasses. We found twocompounds : a borosilicate glass where all the experi-ments were performed by the same group [ 16, 17],and silica (unfortunately for this material the datawere 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 3and P, yL and yT, are given in table III. For this set ofvalues 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 wehave taken into account the distribution of the couplingconstants. Therefore the meaning of the couplingconstants is not exactly the same in the various papers.Moreover, echo experiments can provide values ofthe coupling constants y without the knowledge of thedensity of states. This was done for amorphous S’02where the longitudinal coupling constant yL was

measured [20]. It was found 1’L = (1-5 ± 0.4) eVwhich is compatible with the value 1.04 eV quoted here,if the errors bars are taken into account. If we plot thevalues of P for silica and the borosilicate glasses on thesame diagram as our values for the fluozirconate

glasses as a function of T 9’ we see that surprisinglythey agree with the same P = aTg ’ law, as shown infigure 5. It appears that the P = aTg 1 law has somekind of generality. More experimental work in variousglassy materials is needed in order to confirm thisassertion.

3. The low temperature specific heat has beenmeasured in a V52 sample and in another fluorozir-conate glass not investigated here [21]. Although

Table III. - Parameters describing the propagationof acoustic waves in vitreous Si02 and the borosilicateglass BK7 and their interaction with two-level systems.All these parameters are deduced from the experimentsreported in references [15] to [18] using equations (I)to (4).

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a large excess specific heat (about ten times larger thanin silica) has been detected this specific heat cannot befitted by a linear T law or a T° law with v around 1.2as in silica glasses. Therefore it is difficult to compareour P values with the density of states determined fromspecific heat measurements. On another hand, as hasbeen discussed above, it is worth mentioning that thevalue of P deduced from acoustic experiments isdifferent from, but related to, the density of states nogiven by the linear temperature part of the specificheat below 1 K.

4. Our treatment of the experimental data (andconsequently our conclusion) assumes the validity ofthe TLS theory in its standard version [3, 4]. Butbecause this theory, although phenomenological, hasbeen demonstrated to be very powerful for the inter-pretation of the low temperature properties of glasses,we think this point is not controversial. Probably amore or less slight modification of the distributionfunction different from one material to the othercould give another way to explain the differencesobserved in the acoustic behaviour of the variousfluorozirconate glasses.

5. Finally we turn to the values of the couplingconstants yL and YT’ Usually it is thought that thesequantities are of the order of 1 eV and almost inde-

pendent of the material [22]. Apparently this is true forthe fluorozirconate glasses. YL is between 0.71 eV and1.13 eV and yT between 0.43 eV and 0.65 eV for thethree 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 Tgas shown in figure 6. This result is quite surprising. Toour knowledge there is no prediction about such avariation. We must note that the YL and yT values forthe borosilicate and silica glasses do not agree with a

Fig. 6. - TLS-phonon coupling constants for three fluorideglasses 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 thefluorozirconate glasses. Therefore the Tg variationof YL and yT is not so general (if it exists) as the Tgvariation of P.

7. Conclusioa - We have presented in this paperexperimental results which give a strong indication fora connection between the TLS spectral density offluorozirconate glasses and their glass transition

temperature. Our results confirm a theoretical ana-lysis which predicts a Tg 1 law and previous specificheat measurements in another kind of glass. Of coursethe generality of this behaviour has to be checkedfurther by much more experimental work in otherglasses.

References

[1] See collected papers in Amorphous Solids, edited byW. 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.

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[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, orJOFFRIN, J. and LEVELUT, A., J. Physique 36 (1975) 811.