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Submitted on 1 Jan 1978
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SHORT TIME TRANSIENT TEMPERATURE
PROFILES IN HIGH PURITY VITREOUS SILICA : A
TIME DEPENDENT SPECIFIC HEAT
J. Lewis, J. Lasjaunias, G. Shumacher
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-967
SHORT TIME SCALE TRANSIENT TEMPERATURE PROFILES IN HIGH PURITY VITREOUS SILICA :
A TIME DEPENDENT SPECIFIC HEAT
J.E. Lewis , J.C. Lasjaunias and G. Shumacher
Centre de Recherches sur les Tres Basses Temperatures, CNRS, B.P. 166 X, 38042 Grenoble Cedex, France
Résumé.- On a étudié à très basse température sur de la silice vitreuse de haute pureté (< 1,5 ppm OH, < 0,5 ppm d'ions métalliques) les profils transitoires de température entre 30 ms et 1 s., par une technique répétitive de moyennement de signal. En dessous de 200 mK, les résultats sont ei> accord avec le modèle théorique des systèmes à deux niveaux, pour lequel C(t,T) = AT In t/xx,avec Tx # T- 3. La valeur du coefficient de couplage ngB2 est comparée à celles tirées d'autres expériences. Abstract.- Very low temperature transient temperature profiles in high purity vitreous Si02 (< 1.5 ppm O H , < 0.5 ppm metal ions) have been obtained between 3 0 m s and 1 s., using signal averaging te-chniques. The data below 200 m K are in agreement with a theoretical model for localized two level systems where C(t,T) = A T In t/xx with x^a T ~3. The coupling factor n P is compared to other expe-rimental values.
The excess specific heat in vitreous systems, 1 . 9± .1 and the thermal conductivity that varies as T
below 1 K, can b e understood on the basis of loca-lized two level systems/1/. These states enhance the specific heat above that due to Debye phonons, and resonantly scatter the plane wave phonons so as to produce the observed behaviour. A necessary conse-quence of the tunneling model is that the specific heat should depend on the time scale of measurement, due to the different relaxation rates of excited states, and the time needed for them to equilibrate with the phonons. Until now, the only experimental
test of this prediction, by Goubau and Tait/2/, w a s negative, although Maynard and Rammal / 3 / were able to fit their data by introducing additional excita-tions, presumably linked to the high level of impu-rities in their sample, which effectively masked the intrinsic time-dependent specific heat of the two level systems. A very recent / 4 / test at ultrashort times ( < 1 0- 6s . ) was also negative, though no d e
-tails of the experiment have yet been released. In fact Black/5/, in a recalculation of the tunneling model, claims that both the long time-scale and the
short time-scale specific heats are underestimated by the model, if "accepted" values of the parame-ters are used. In view of its importance, w e have searched for this time dependent specific heat in a very well characterized (< 1.5 p p m O H , <0.5 ppm metal ions) 9.1 cm long, 2.1 cm diameter sample of Sip?. Suprasil W. whose thermal conductivity
Permanent address : Department of Physics, SUNY, Plattsburgh, NY 12901, U.S.A.
and long time specific heat / 6 / , over wide tempera-ture ranges, are well established.
The time dependent effect is predicted to be small, so signal averaging techniques using an 1000 channel analyser were employed to improve signal/ noise ratios. The sample was heated by a brief (0.1 m s ) pulse of energy by an annular heater moun-ted at one end and the resulting temperature tran-sient 9(t) was monitored by an ion-doped silicon thermometer greased to a further annular band s i -tuated in the first half of the sample's length (x/L = 0 . 3 9 ) . The sample w a s linked to the dilution refrigerator by a copper heat leak, whose thermal relaxation time determined the repetition interval
(y 10-30 s.) of heating. The influence of the heat leak on the temperature profiles is negligible for t < 1 s. Short term (y 3 0 s.) temperature stability was excellent ( « 1 m K ) as w a s the long term
('h 0.5 m K ) once dynamical equilibrium had been
achie-ved. Typically, about 1000 repetitions were made over a time of 6-8 h r s , with a 1-2 % relative tem-perature rise. Four profiles were thus obtained, at 2 8 , 5 0 , 113 and 172 m K . The effect w a s masked by the increasing phonon contribution to the specific heat at higher temperatures, i.e. at T > 200 m K the pro-files could be fitted using the long time scale spe-cific heat of the sample to within the experimental uncertainty.
Figure 1 shows the profile at 113 m K plotted as In 6/tT vs 1/t and is clearly nonlinear. The d a -shed line is the behaviour expected using a
Fig. 1 : Transient temperature profile at T = 113mK.
tant specific heat equal to the long time scale va- lue (t > several seconds). At short time scales the actual profile rises faster than predicted on clas- sical diffusion theory, indicating a time varying specific heat. The smooth curve is the computer fit via an inverse Laplace transform of the time depen- dent specific heat prediction of the generalized two level tunneling model/3/, where the contribu- tion of the Debye phonons has also been included, estimated from values of the sound velocity/6/. It can be shown/l,3/ that, at intermediate times T*<< t<<?
,
the time dependent specific heat ofthe two level systems is C(t,T) = CI lnt/~" where C1 = nok2~/12 with no the constant value for the
distribution assumed in the original model, and
T~ = a/233k3~3= is the shortest relaxation
time of the defects. ?, the longest relaxation ti- me of the defects, is related to a cut-off in the marimum barrier height between levels i.e. to a gap in the low energy density of states.
-
(a
'
= (B~/v: + ~B:/v:)/~T~M~=~P(v;~+~v;~)
/2npji4where is the average phonon-defect coupling, V the
sound velocity, p the mass density and 1 and t re- fer ta the phonon polarization.
X The result of the fit, with C1 and T as ad- justable parameters is shown in figure 2. The strin- gent theoretical temperature dependences of C1 and T~ are-clearly met. Here C1 = 0.42 T and T" = 1.3
x IO-'~T-~. These lead to a value no = 2.64 x
~ m - ~ e r ~ - l a n d = 5.4 eV and hence n o 9 = 2 x lolo erg/cm3.
n is in good agreement with values dedaced from long time scale specific heat measurements/6,7/ but
5
is larger than previously reported from dif-ferent experiments, although a wide-spread of values exists : from 0.3 eV (sound velocity variation/8/), 0.6 eV (thermal conductivity /7/), to a few eV @ho- non echoes 191, direct measurement of T1/10/). This
results in an excessively large scattering strength ' Y no?, which from a variety of measurements, is
%lo8
erg/cm3 / 1 1 1.
The discrepancy is due to the ultrashort re-
X
laxation time T of the study (% n s. at 0.1 K ra- ther than % ps. expected from theory). It does not seem possible, given the experimental profiles, to
have both n and in good agreement with published
values, although this point has not pursued, as the fit is done essentially on a trial and error basis due to the complicated form of the heat equation in the presence of a time varying specific heat.
If the effect is genuine, the most probable inference because of the excellent temperature va-
x
riations of C1 and T
,
we must conclude that it is necessary to introduce a distribution of strength coupling B2 to take in account different experiments in vitreous silica.References
/I/ Anderson,P.W., Halperin,B.I. and Varma,C.M., Phil. Mag. 25 (1972) 1
-
Phillips,W.A., J. Low Temp. Phys. L(1972) 351 121 Goubau,W.M. and Tait,R.A., Phys. Rev. Lett.
2
(1975)1220
/ 3 / Rammal,R., Thesis Grenoble (1 977) (unpublished) Maynard,R. and Rammal,R., this issued (LT 15)
/ 4 / Kummer,R.B., Navayanamurti,V. and Dynes,R.E., Bull. A.P.S. 23 (1978) 336
-
/5/ Black,J.L., to be published
161 Lasjaunias,J.C., Ravex,A. and Vandorpe,M. and Hunklinger,S., Sol. State Commun.
17
(1975) 1045/7/ Vandorpe,M., Lasjaunias,J.C. and Maynard,R., this issue (LT15) 181 Piche,L., Maynard,R., Hunklinger,S., and Jzckle,J., Phys. Rev.
Lett.
32
(1974) 1426/9/ Golding,B. and Graebner,J.E., Phys. Rev. Lett.
