Anomalous spectral density and two-level system-phonon coupling in an insulating magnetic glass

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Anomalous spectral density and two-level

system-phonon coupling in an insulating magnetic glass

P. Doussineau, A. Levelut, M. Matecki, W. Schön, W.D. Wallace

To cite this version:

P. Doussineau, A. Levelut, M. Matecki, W. Schön, W.D. Wallace. Anomalous spectral density and

two-level system-phonon coupling in an insulating magnetic glass. Journal de Physique, 1985, 46 (6),

pp.979-986. �10.1051/jphys:01985004606097900�. �jpa-00210045�

Anomalous spectral density and two-level system-phonon coupling ^{in an} insulating magnetic glass

P. Doussineau (1), A. Levelut (1), ^{M. Matecki} (2), ^{W. Schön} (1) ^{and W. D. Wallace} (1, *)

(1) Laboratoire d’Ultrasons (+), Université Pierre et Marie Curie, ^Tour 13, 4, place Jussieu, 75230 Paris Cedex 05, France

(2) Laboratoire de Chimie Minérale D (+), Université de Rennes Beaulieu. 35042 Rennes Cedex, ^France (Reçu ^le 19 juin 1984, révisé le 21 décembre 1984, accepté ^le 14 février 1985)

Résumé.

Des mesures ultrasonores

basses températures, ^{dans la} gamme de fréquence ^{0,1-1 GHz ont été}

faites

le

magnétique (MnF2)0,65(BaF2)0,2(NaPO3)0,15. ^{La densité} spectrale, ^{P des} systèmes à 2 niveaux

(S2N) ^est deux ordres de grandeur plus petite ^{que dans les autres}

^tandis que les constantes de couplage S2N-phonon 03B31 et 03B3t ^sont anormalement grandes. ^La comparaison

^{les mesures} effectuées

le

non

magnétique (ZnF2)0,6(BaF2)0,2(NaPO3)0,2 ^suggère

^{lien entre ces} valeurs anormales et les propriétés magné- tiques ^du

Abstract.

Low temperature ultrasonic measurements on the magnetic glass (MnF2)0.65(BaF2)0.2(NaPO3)0.15

in the 0.1-1 GHz range give

value for the spectral density ^{P of two level} systems (TLS) ^{two orders of} magnitude

smaller than usual and very large values for the TLS-phonon coupling constants 03B31 and 03B3t. Comparison ^{with the}

case

of the non-magnetic glass (ZnF2)0.6(BaF2)0.2(NaPO3)0.2 ^suggests that these anomalous values

related to the magnetic properties ^{of the} glass.

Classification

Physics ^Abstracts

62.65 - 75.50K - 43.35

1. Introduction.

In this article we report ^on ultrasonic measurements in a magnetic glass which is also a spin-glass. ^Our

purpose ^was originally ^to study the ultrasonic _pro-

perties of this material near the magnetic freezing

temperature Tf

^{3.4 K.} Our results concerning ^this

aspect show that there is no observable transition effect either in the ultrasonic attenuation or in the sound velocity.

However the acoustic behaviour of this material is different from what is usually observed in other

(non-magnetic) insulating glasses. ^We show that this

peculiar ^{behaviour can be} fully explained (for ^{the tem-} perature range from 0.1 K to 5 K) ^{in the framework}

of the two-level system (TLS) theory [ 1-3] and that the difference with standard results is due to very ^anoma- lous values of the fundamental parameters ^{of the} theory, namely ^the spectral density P and the TLS-

(*) Permanent and present ^{address :} Department ^of Physics, ^Oakland University, Rochester, Michigan 48063, U.S.A.

(+) Associated with the Centre National de la Recherche

Scientifique (UA 789).

phonon coupling ^constants y 1 and yt. By way of compa-

rison acoustic measurements in a similar (but ^non- magnetic) glass ^{are shown to} give the ^{usual behaviour}

and therefore the usual values for j5, Yi ^{and yt.}

We first present ^our experimental ^results. They ^are

followed by ^a digest of the TLS theory (with ^a necessary

modification) which is then used for the interpretation

of our data. Finally, ^we discuss the significance ^of

our results on the amorphous ^and magnetic properties

of the magnetic glass.

A brief report ^of some aspects of this work was

presented ^elsewhere [4].

2. Experimental ^results,

We have studied the fluorophosphate glasses (MnF2)0.6s(BaF2)0.20(NaP03)0.ls ^and (ZnF2)0.60 (BaF 2)0.20(NaP03)0.20. ^The preparation procedure

and some properties ^{of these} glasses ^{can be found}

elsewhere [5]. Measurements of the magnetic ^sus- ceptibility ^and field-cooled and zero field-cooled

magnetizations have revealed that the Mn glass ^enters

a spin-glass phase ^below T f

^{3.4 K} [6].

In these two glasses ^{we have measured the attenua-}

tion and the relative velocity change ^{for both} longitu-

dinal and transverse ultrasonic waves at frequencies

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004606097900

980

between 100 MHz and 1 GHz and for 0.1 K T

20 K.

Figure ^{1 shows an} example ^{of the} attenuation as a

function of the temperature ^{at the same} frequency ^for

both glasses ^{on a} log-log ^scale. Qualitatively ^{the data}

for both glasses have similar features analogous ^to

those observed in insulating glasses [3] : ^a rapid

increase (as T3) ^at the lowest temperatures ^followed by ^a region where the attenuation varies _very slowly

with the temperature (plateau) ^and finally ^{an increase.}

The differences between the two glasses ^are only ⁱⁿ magnitude. ^{For instance at 1} K, the attenuation in the Mn glass is about four times larger than in the Zn

glass, while the height ^{of the} plateau ^{is two times}

smaller. It is most important ^{to note that the} plateau begins ^{for the Mn} glass ^{at a} temperature ^{much lower} than for the Zn glass (respectively ^{2 K and 6} K). ^We

shall see later that these differences are fully explained by ^{the TLS} theory ^without introducing ^any spin-glass

effect. In particular the lower temperature ^{for the} beginning ^{of the} plateau ^{for the Mn} glass ^indicates

a stronger TLS-phonon coupling.

Figure ^{2 shows an} example ^{of the} temperature ^varia- tion of the velocity ^{at the same} frequency ^{for both} glasses ^{on a} linear scale. The Zn glass ^{shows the usual}

behaviour for ^an insulating glass [3] : ^a rapid ^increase

at low temperatures (which is in fact logarithmic)

followed by ^a broad maximum (between ^{2 and 3 K}

at 260 MHz) ^{and then a} decrease which becomes

quasi-linear ^{above 5} K, up ^to 100 K (not ^{shown in}

the figure). ^{For the Mn} glass ^the temperature ^variation of the velocity ^{shows some} similarities : a logarithmic

increase _up to a maximum and a linear decrease above 5 K. However we wish to emphasize ^two differences : in the Mn glass the maximum occurs at much lower temperature (around ^{0.5 K at this} frequency) ^and beyond its maximum the curve presents ^an upward

curvature below 5 K. Here too, this peculiar ^beha-

viour of the Mn glass ^is explained ^{as an} amorphous

effect only; ^no additional contribution of magnetic

or spin-glass origin ^{is needed. As was the case for the}

attenuation plateau ^{the low} temperature of the maxi-

mum is due to the strong coupling ^{of the} phonons ^with

the TLS of the Mn glass.

The same features are found with transverse ultra- sonic waves as well as longitudinal ^{ones and for all} frequencies used in these experiments (see Figs. 3, ⁴ and 5). ^The slight undulations which may be ^seen in

some figures ^{are artefacts as} they ^{sometimes appear}

in high-frequency ^acoustic experiments.

Now we recall the main predictions ^{of the TLS} theory ^of glasses ^{needed for our} interpretation ^{of the} experimental ^data.

3. Main results of the TLS theory.

The basis of the theory [l, 2] ^{is the} assumption ^{that the}

TLS are the lowest two energy levels of ^a particle tunnelling between the minima of a double well

potential. ^Two parameters characteristic of this poten-

Fig. ^1.

Attenuation of longitudinal ^{acoustic waves at}

260 MHz

function of temperature ^on

log-log ^{scale for}

the magnetic glass (MnF2)o.6S(BaF2)o.2(NaP03)O.lS ^and

for the non-magnetic glass (ZnF 2)0.6(BaF2)0.2(Napo)3)0.2.

The curves

calculated (see text).

Fig. ^2.

^Relative velocity change ^of longitudinal ^acoustic

waves at 260 MHz

temperature ^on

linear scale for

the two glasses ^of figure ^{1. The two sets of data}

^shifted

arbitrarily ^{relative to} each other. The curves are calculated.

tial are the half-asymmetry ^{A and the} tunnelling 40 energies. ^{However two other} parameters, ^E

d Z + d o ^{and r} = (AO/E)2, ^{are more} often used.

These parameters ^{have a} range of values described

by ^a distribution function P(E, r) ^{for which we} adopt

the following ^form [7, 8] :

where A. ^{is the} (unknown) minimum value of do ^and EM ^{is the upper} cut-off of the TLS spectrum (of ^the

order of kTg). A03BC ^{is a} normalization constant which is only weakly dependent on p (see appendix 1). ^If

It

0 we recover the standard theory [1, 2]. ^Let N ^be

the total density of TLS. The spectral density P03BC ^is

defined as P Jl =. X AJl/2. ^{According to the previous}

remark on A03BC, Pu ^{is weakly} p-dependent ^{and now we}

omit the index _/03BC. The constant P must be not confused with the density ^{of states} n(E) ^which depends ^on E/dm (see appendix 2).

Two different interaction _processes between TLS and acoustic waves are predicted :

i) ^{In the resonant} process, ^an acoustic wave of

frequency ro/2 n ^{induces a resonant} transition between the two levels of the TLS of _energy splitting ^about

Fig ^3.

Attenuation of longitudinal ^{acoustic waves}

function of temperature ^on

log-log ^{scale at} three different

frequencies for the Mn glass. ^The

calculated

Fig. ^4.

^Relative velocity change ^{of acoustic}

^as

function of temperature ^on

semi-log ^{scale at four} frequen-

cies for the Mn glass. L and T refer respectively ^to longitu-

dinal and transverse polarizations. The different sets of data

are shifted arbitrarily ^{relative to} each other. All the

calculated with the

values of K3 and p (see ^{text and}

Table I).

liw. This gives ^{rise to a} velocity change written, ^for hm « ^kT [8] :

where To ^{is an} arbitrary ^reference temperature ^and C1, C, ^{are constants for a given material, related to the}

fundamental parameters ^{of the} theory by the relations :

982

Fig. 5- Attenuation of acoustic waves

function of temperature

linear scales for the Mn glass. ^{Both curves}

are

calculated with the

values of p ^and K3 (see ^text

and Table I).

where vl, vt ^are the velocities of the longitudinal ^or

transverse sound waves and _p the mass density ^{of the}

material. The corresponding attenuation is saturable and is not observed under the experimental ^conditions

of this work.

ii) In the relaxation _process, the acoustic wave

modulates the TLS energy splitting. Because of the relaxation towards the new thermal equilibrium ^with

a characteristic time T1, it results in an attenuation and a dispersion ^{of the wave. Even for TLS} having ^the

same energy splitting ² E, ^{there is a} distribution of the relaxation times T i ^{and we let} Tm(E) ^{be the minimum}

value of T,(E, ^{r). We can calculate} analytically ^the

attenuation and dispersion only ^{for two} limiting ^cases (see appendix 3) :

- At low temperatures (coTl > ¹⁾ the attenuation

depends ^on the mechanism of relaxation. In an insulat-

ing glass, ^{the one} phonon process is dominant. In this

case the relaxation rate is given by :

where K3 ^{is a constant related} to y1, and yt by :

where k is the Boltzmann constant and h is the Planck

constant divided by ^{2 x. The} attenuation is :

The corresponding velocity change ^is negligible ^com- pared ^{to the resonant one.}

-

At high temperatures (wTm1 ¹⁾ the attenuation does not depend ^{on the} temperature (plateau) ^{and has}

the same value independent of the relaxational _pro-

cess :

The corresponding velocity change ^{for an} insulating glass ^is given by

where wo/2 11: ^{is a reference} frequency.

In the intermediary range (where w T’ is of the order of 1) the calculations are achieved with a computer (see appendix 3). ^{The total} effect is the sum of the resonant and relaxational contributions.

We have used the form of the distribution function in equation (1) because the standard theory (with

p

0) implies several consequences which ^{are not}

experimentally ^{verified. In} particular, ^the slope ^{of the} logarithmic variation of the velocity ^{at low} tempera-

tures (Eq. (2)) ^{is not} the coefficient Cl,t deduced from the value of the attenuation plateau (Eq. (7)). ^The

introduction of parameter p ^{is a means of} resolving

this difficulty.

4. Application ^{of the TLS} theory ^{to the Mn and Zn} glasses.

Because our samples ^are glasses, ^we begin ^{our inter-} pretation ^{from a} glassy point ^{of view} by applying ^the

TLS theory. ^The differences between the experimental

and the calculated amorphous properties, ^if any, would be attributed to other causes, such as the spin- glass ^{behaviour of the Mn} glass.

-

In figures 1, ^{3 and} 4, ^{one can} observe the expected

low temperature variations : the T3 variation of the attenuation (Eq. (6)) ^{and the} logarithmic increase of the velocity (Eq. (2)). ^{The high} temperature ^attenua- tion plateau (Eq. (7)) ^{is, also present (Figs 1, 3 and 5).}

All these features are observed in both glasses ^{for the}

two polarizations. ^The limiting ^cases give ^{us the values}

of K3, ¹⁴ C1 ^or K3, 03BC, Ct, from which we can calculate

numerically the theoretical values of the attenuation and the velocity ^at any temperature (see appendix 3).

The resulting curves are plotted ^as solid lines in

figures 1, 2, 3, 4 ^{and 5 and.the} corresponding ^{values of} K3, Cl, Ct and p ^{for the two} glasses ^are reported ⁱⁿ

table I. One can see that the fit is _very satisfactory

for both glasses up ^to about 5 K.

Table I.

Numerical results : Column 1 : : a fluoride glass (trade ^mark V52) of composition (ZrF 4)0.575

^

(BaF 2)0.337 5(ThF 4)0.0875’ ^{used as a standard} reference; Column 2 : the Zn glass of composition (ZnF2)o.6(BaF2)o.2 (NaP03)o.2 ; ^{Column 3 : the Mn} glass of composition (MnF2)0.65(BaF2)0.2(NaP03)0.15; Column 4 : ratio of ^the

relevant quantities for ^{Zn and Mn} glasses. ^The upper part of ^{the table} gives ^{the data} concerning the TLS and their

coupling ^to phonons. ^{The lower} part ^{shows some values} useful for ultrasonic experiment interpretation. ^{The values} of ^{column 1 were} calculated from ^{the data} of reference [9] ^{as described in the} present article. The values of ^{columns 2}

and 3 were used for the calculation of ^{the curves in} figures ^1-5.

It is important ^{to note that the same values of} K3

and _p give ^a good fit for both elastic polarizations : indeed, ,u characterizes the spectral distribution of the TLS while K3 ^{measures the} strength ^{of the} one-pho-

non thermal relaxation which is an effect independent

of the perturbation of the TLS population (in particu- lar, independent of the ultrasonic polarization).

For temperatures larger ^{than 5} K, the attenuation still increases and the velocity ^decreases quasi linearly.

This behaviour, ^{which is observed in most} glasses [3]

and is not explained by ^{the TLS} theory, ^{is not} clearly

understood at the present ^time.

Although the ultrasonic properties ^{of the Mn} glass

are quite different from those usually observed in

glasses, ^it appears that they ^are completely explained by amorphous ^{effects, without} introducing any spin- glass ^effect.

5. Discussion of amorphous properties.

From our numerical calculations we have obtained the values of Cl, C,, K3 and p reported ^{in table I. Values}

for a typical glass (V52) [9] ^are included for _compa- rison.

We first note that the two glasses ^{of this} experiment

have values of C1 ^and Ct ^{which are} comparable ^and

close to those of V52. On the other hand, ^{the two values} of K3 ^{are very} different : for the Zn glass this value is in the usual _range while it is two orders of magnitude greater ^{for the Mn} glass. ^We observe in this latter

case the logarithmic ^decrease (sum ^{of the resonant} logarithmic ^increase (Eq. (2)) and the relaxational

logarithmic ^decrease (Eq. (8))) ^{of the} velocity ^{in the} high temperature regime mT§q « ^{1 above the maxi-}

mum (see Fig. 4) precisely ^{because of this} large ^value.

Indeed, ^{in most} glasses ^the unexplained quasi ^linear

decrease of the velocity (see the data in Fig. 2) ^domi-

nates the logarithmic contributions given by equa- tions (8) ^and (2). ^{In order that the} regime change ^can

be clearly identified it is necessary that the tempera-

ture T. ^{defined by} wTT(TR) - ¹ be much smaller than about 5 K. It is _easy to show that TR - Wl/3 K3-1/3. ^{For our Mn glass the} regime change

takes place ^{at a} temperature TR = ^{0.5 K, which is}

lower than usual because of the anomalously large

value of K3. ^{Such a regime} change ^{has been observed}

at very low frequency (in the kilohertz range) ^{for the}

dielectric constant [ 10] and for the sound velocity [ 1] ]

of a borosilicate. In these _cases, this was due to the

frequency dependence ^of TR.

With the help ^of equations (3) ^and (5) ^{we are able}

to deduce the fundamental parameters ^{of the} theory :

the spectral density P ^{and the} coupling ^constants yl, yt.

We emphasize that, contrary ^{to a} widely ^{held belief, the}

acoustic method is the only ^{one able} to provide separately the values of the spectral density P ^{and the} coupling ^constant _yl,, (and ^not only ^the product

Py 2 ). The values obtained are reported ^{in table I. As}

it could be guessed ^{after our} discussion about K3 (see Eq. (5)) the values of yl and yt ^are in the normal range for the ^Zn glass ^and quite ^{a bit} larger (by ^a

factor about 10) ^{for the} Mn glass. On another hand,

while the spectral density P is normal for the Zn glass,

it is smaller by ^{about two orders of} magnitude ^for

the Mn glass.

984

Finally, ^{this last} feature leads us to discuss our

results in connection with the free volume theory ^of glasses. At the end of the theory, ^too long ^{to be} report- ed here, ^{Cohen and Grest} [12] ^have suggested ^{that the}

TLS density ^of states must be proportional ^{to the}

inverse of the glass transition temperature Tg. ^A

similar law has been checked in a series of fluoride compounds [13] ^{where it was} found that PT9 -

(2.6 ± 0.5) ^{1048 j-1 m- 3 K. In} _our magnetic ^mate-

rial where Tg

^{570 K we find} FT9 - ^{0.02 x 1048}

J-1 m- 3 K. These values, differing by ^{two orders of} magnitude, ^{do not} support ^the universality ^{of the}

free volume theory.

In summary, ^we emphasize ^the following surprising

result : the Mn glass contains fewer TLS but they ^are

more strongly coupled ^to phonons ^{than in most} glasses.

6. Discussion of magnetic properties.

In the previous ^{section we} have discussed our experi-

mental results from an amorphous point ^{of view.}

However our Mn glass ^{is also a} spin glass [6] ^and

we discuss now how our results _may be related to the

magnetic properties. Several ultrasonic experiments

have already ^been performed ⁱⁿ magnetic glasses.

We first review them briefly.

Acoustic velocity measurements were done in

amorphous ferromagnetic ^CoP [14] ^{at low} tempera-

tures. The slope ^{of the} logarithmic ^{increase was} found

to be _very small, but from this single datum, P and Yh Yt ^cannot be separately ^{deduced. No} comparison

is therefore possible ^{with our results.}

Acoustic velocity experiments ^{down to 1.3 K in the}

100 _{MHz range} by Lin and Thomas [15] ^{in various}

aluminosilicate glasses ^{have shown that} glasses ^with magnetic ^ions (Mn+ + ^and Co+ +) ^{have a different}

behaviour than glasses ^without magnetic ^{ions. For}

the non-magnetic glasses the maximum of the sound

velocity ^{occurs at about 2 K} while for the magnetic

ones no maximum was observed in the explored temperature range. Analogous results in amorphous aluminosilicate spin glasses ^{were obtained} by ^Moran

et al. [16] ^{and were} explained ^{as a} negative ^contribu-

tion in the vicinity ^{of the} freezing temperature ^{of the}

spins. All these results look similar to those ^we have obtained in our Mn glass. ^{However since we were able}

to perform experiments ^{down to 0.1 K and at fre-} quencies ^{above 100 MHz we observe the} velocity

maximum. This allows us to interpret ^{our results in}

terms of amorphous properties ^{alone and we see no}

evidence of magnetic ^{effects at} Tf. ^In particular ^the

low temperature of the maximum and the upward

curvature mentioned above are explained by ^the large

value of the constant K3 ^{and not} by ^{some additional} magnetic ^{effect We} believe that the results of Lin and Thomas [15] and Moran et al. [16] may be interpreted

in the same manner. The case of metallic spin glasses (such ^as Cu-Mn) [17, 18] is outside the _scope of this discussion because of their crystalline structure, ^but

results similar to Moran et ale are reported, i.e., ^a decrease in the velocity ^{near the} spin-glass tempera- ture Tf.

The magnetic ^state of this material might give ^{rise to}

the existence of magnetic ^two-level systems (MTLS).

These entities, ^first proposed by Anderson et al. [1]

have been invoked to explain ^thermal conductivity

measurements on some spin glasses [19, 20]. ^{In our}

case the spectral density (of TLS) ^{we measure is lower}

than usually ^{found in a} non-magnetic glass. Therefore,

it is absolutely unnecessary ^to introduce supplemen-

tary ^{MTLS in the} interpretation ^{of our} experiments.

Coming ^{back to our essential} result, ^{it can be} pushed

forward as a tempting hypothesis ^{that the} spectral density of the TLS and their coupling ^with phonons

are strongly dependent ^on the presence of magnetic

ions. Up ^{to now these} quantities ^{were believed to be}

the same for all glasses whatever their chemical com-

position ^or microscopic structure. The possibility

of modifying drastically ^{the TLS} spectral density by introducing magnetic ions could be an interesting

way ^to gain ^an understanding ^{of the} microscopic origin of the TLS.

7. Conclusion

The acoustic attenuation and velocity change ^mea-

surements in the amorphous spin-glass (MnF2)o. 6 s (Ba.F2)o.2(NaPO3)0.15 ^have given ^the following ^re-

sults : i) within the _accuracy of our data there is no

anomaly ^{observed at the} spin-glass transition; ii) ^the

acoustic behaviour calculated within the framework of the TLS theory ^is sufficient to explain ^the experi-

mental results below 5 K. But surprisingly ^{it appears}

that in this magnetic glass ^{the TLS} spectral density

and the TLS-phonon coupling ^{constants are sub-} stantially different from those observed in ^non-

magnetic glasses. ^The comparison with another glass

of the same series (but non-magnetic) indicates that the presence of magnetic ions may be ^at the origin

of this difference. There is presently ^no microscopic explanation ^of this feature. In order to clarify ^this point, ^more experimental ^work is necessary, especially by varying ^the concentration and the nature of the

magnetic ^{ions and} performing ^acoustic experiments

with an applied magnetic ^field Appendices,

In the present appendices ^we gather ^{some useful}

results of the TLS theory.

1.

The distribution ^function P(E, r) ^{defined in} equation (1) ^{contains a} normalization constant A.

which a priori depends ^{on the value of} Il. We show here

that A ^is only weakly ,u-dependent. ^The normalization

condition reads :

or

It is _easy to show that in the domain _p > - 2 ^where

the integral ^{over r} converges it is a decreasing ^function of Il. Therefore the normalization constant A. ^increases

when _p increases. In the case of the standard theory (p

0) ^the integral ^is easily calculated and gives

Practically, the values of 03BC ^are found in the domain

spreading ^{from 0 to} -1 ^{We have} computed ^{the ratio}

Ap/Ao in the interval 0 2 ^for (EM/dm)

^106.

This ratio _may correspond ^to Emlk

^{1 000 K and} d m/k

^{I mK.} A AIAO ^{varies by} less than _{5 per} cent over the calculated range. Moreover this ratio de-

creases when EM/dm increases. A value of dm ^much

smaller than 1 mK, ^{as is} probable, ^would give ^{a still}

weaker variation. Therefore, ^{in most cases we can} take Ap ^as independent of Il.

In the article we compare the numerical values of the

spectral density Pp ^{obtained in glasses} with different p and we conclude at the smaller P. ^is, the smaller is the total density ^{of TLS. We have to} justify ^this point by justifying ^the underlying hypotheses. ^We

first point ^out that the dominant term of the equation giving Au ^is

where it is assumed that d m EM. Only the corrective terms are p-dependent. ^Therefore All ^is only weakly (in ^fact logarithmically) dependent of dm. ^{On the other}

hand we adopt ^the generally accepted hypothesis

that the upper cut-off EM is related to the glassy ^tem-

perature Tg. Therefore when we compare ^two glasses

with nearly equal Tg ^(as is done in this article) ^we

can take the two values of EM ^as nearly equal.

We can conclude, then, that the values of A,, ^{for two}

glasses ^{with the same} Tg ^are nearly equal, ^{even if we}

have no information ^about Am. Therefore the ratio of the two Pu ^{also gives} the ratio of the two densities JV of TLS.

2.

The density ^{of states} n(E) is defined as

Consequently, ^{we have}

n,,(E) ^{appears as the} product ^{of the constant} Pu ^(which depends very weakly on p ^{as seen} above) by ^{a function} of E/dm, A. being unknown, ^the relationship ^between

PI1 ^and nu(E) ^{even for a given E is unknown too.}

Consequently ^{these two} quantities ^{must be} carefully distinguished.

In order to illustrate the physical meaning ^{of the}

parameter p ^{we can} give ^the explicit ^{forms of} nu(E)

in two cases where the calculations are easy (p = ⁰ and It

2)

The integrals ^{of these two functions over the} energy range ^must be equal (the normalization condition).

Consequently ^{these two} positive ^{functions must cross}

at some value of the _energy as can be seen in figure ^6.

(In ^{this case we} must take into ^{account the} slight difference between Po ^and _PI/2). ^The physical ^mean- ing ^{is now clear :} the increase of p ^is equivalent ^{to a}

transfer of low-energy ^{TLS to} higher energies.

3.

The relative variation of the complex ^elastic

constant bc(w, T) ^{due to} the relaxational _process is

given by [8] :

where v

cvT’1 (E, T) and fl

l/kT. ^{The other} symbols ^are defined in the article. Generally speaking

the inner integral ^cannot be calculated analytically

for arbitrary values of the parameter ,u. However, ^it

Fig. ^6.

Density ^{of states} n(E) (in arbitrary units)

function of the reduced energy E/dm for p

^{0 and} p = 2

for

hypothetical glass ^with EM/dm ^{= 10.}

986

can be done (at ^{least in} principle) for any rational value of _{p. We} have performed the calculation of this

integral ^{over r for the} following values : p

0, i, i, i and 2 I ^which adequately ^{covers the} range (0, 2) ^of experimentally observed values. The results are com-

plicated functions of E. Finally, ^{the outer} integral ^is

evaluated with a computer. The best fit to the experi-

mental data determines the values of the free

(unknown) parameters (Ct,t, ^{K3, p).} The results do not

depend ^{on the value of} dm ^as long ^{as it is} negligible compared ^{to kT.}

Note added in proof :

Saturation effects have been observed on the attenuation of 963 MHz transverse acoustic waves

propagating ^{in the} MnF2 glass ^at 0.165 K The

magnitude of the critical acoustic flux _necessary to saturate the attenuation is much higher ^{than in a non-} magnetic glass (V52 ^or silica for example). ^{This means}

that the relaxation times are shorter (and consequently

the phonon-TLS coupling larger) in this kind of material as compared ^{to a} non-magnetic glass.

References

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