Vibrational spectrum and temperature behavior of thermal conductivity and specific heat in amorphous dielectrics

JOURNAL OF

ELSEVIER

Journal of Non-Crystalline Solids 202 (1996) 128-136

Vibrational spectrum and temperature behavior of thermal conductivity and specific heat in amorphous dielectrics E.I. Salamatov

*

Physico-Technical Institute, 426001, 132 KiroL' street, Izhecsk, Russia Received 25 April 1995; revised 19 October 1995

Abstract A microscopic model is proposed to describe the temperature behavior of the thermal conductivity and specific heat of amorphous dielectrics. The consideration is carried out in a lattice model of the amorphous state. The main model points are connected with the one-site matrix of phonon scattering on the defects which are impurity oscillators vibrating in a symmetric two-well potential. Depending on the value of the only random parameter of the model, a tunnel state can arise or not on the defect. The mass operator of the system is written in a general form which involves some parameters whose values are unknown. The analytical expressions obtained make it possible to find the model parameter values before numerical calculations. The results demonstrated good agreement with experiments and confirm the validity of the model for describing the properties of amorphous materials over a wide temperature range.

1. Introduction For the past twenty years, many attempts have been made to give a theoretical description of anomalous low-temperature thermal and acoustic properties of amorphous materials. Principal successes in this field are connected with the hypothesis that, in the amorphous matrix, there exist entities which can occupy two stable equilibrium positions separated by a barrier. At low temperatures, the barrier is surmounted via quantum-mechanical tunneling. This two-level system (TLS) model [1,2], generally speaking, is quite satisfactory in describing the properties of dielectric glasses below l K. At higher temperatures, the conventional TLS model

* Tel.: +7-3412 216 977; fax: +7-3412 250 614; e-marl: [email protected].

cannot explain certain universal features of amorphous materials, such as the plateau in the heat conductivity coefficient k(T) and the bump in C / T 3 where C is the specific heat [3,4]. New approaches, based both on modified TLS models [5] and alternative ones [6], are contradictory in nature and applicable, as a rule, only to a particular temperature range. That is why a consistent model is needed which would allow a variety of low-temperature properties of glasses to be described within a unified approach for a wide temperature range. Although exotic quasi-particles, such as fractons [7], solitons [8] etc., are involved in some new approaches, there is good reason to believe phonons to be responsible for anomalous properties of amorphous materials. As far as we know, there are no consistent studies on phonons in disordered systems with due account of various effects - renormaliza-

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E.1. Salamatov / Journal of Non-Cr3'stalline Solids 202 (1996) 128-136

tion of vibrational spectrum, anharmonicity, localization, dynamic interaction between defects, etc. In our previous papers, we studied the influence of these factors on the temperature behavior of the heat conductivity coefficient [9-l 1] and sound velocity [12] in disordered systems. Investigation was pertbrmed by the method of a displacement operator Green's functions [13]. With a knowledge of the Green function of a system one can find all its main properties. So increasing the number of properties considered with the same set of microscopic parameters allows us to obtain additional information about them. Additionally, a judicious simplification of the model and the calculation scheme makes it possible to reduce the number of independent parameters and to obtain analytical expressions relating them to each other. In virtue of the above, the main purposes of this work can be stated as follows: (i) determination of the Green function of the amorphous state within the framework of a simple, physically reasonable model; (ii) derivation of analytical expressions for the specific heat and thermal conductivity coefficients for some characteristic temperatures at which the model parameters can be expressed in terms of experimentally observed quantities; (iii) calculation of these properties over a wide temperature range using the obtained parameter values. From the results of [9-12], it may be deduced that the renormalization of the vibrational spectrum and the quasi-local mode anharmoncity play the leading part in forming the T-dependence of the properties of amorphous dielectrics. Therefore, in this work, the effects of localization and dynamic interaction between defects will be neglected. Such simplification allows us to determine all the model parameters. In Section 2 of this paper, we present the basic equations and discuss the original model suggested in [9,10], which in the next section is somewhat modified by allowing for the quasi-local mode anharmonicity and tunnel states. In Section 4, all the model parameters are determined, while Section 5 is devoted to the discussion of numerical results.

2. General formalism and microscopic model The expressions for calculating the lattice heat

129

capacity and the thermal conductivity coefficient of a disordered system are well known [13]:

c ( r ) =/32kBfdo)

g(o)) o)2 cosh2(/3o)/2)/4,

(1) k( T ) = fl z kB f do) gph(to) × o)2D(to) cosh2 ( flo)/2)/4.

(2)

Here g(o)) and gph(o)) are the densities of vibrational and phonon states of the system, D(o)) is the phonon diffusion coefficient, and 13 = h/kBT. All quantities involved in Eqs. (1) and (2) are known to be expressed in terms of the Fourier transform of the one-particle Green function averaged over the impurity configurations:

c(o), k) = ( o)2 _ fio)2(k ) _

o)))

t

Here S(to, k) = P(to, k) + i F(to, k) is the mass operator [ 13]. In this paper, we shall use the lattice model from [9-11] as a microscopic model of a disordered (amorphous) dielectric. This model is based on the assumption that the dynamical properties of the amorphous material qualitatively correspond to those of a crystal lattice which includes specific structural defects. Such defects are much smaller in size than the wave length of low-frequency pbonons and vibrate in an anharmonic potential along one of the generalized coordinates within the defect space. The effective frequency of oscillations along this coordinate is a temperature-dependent quantity discussed in the next section. It is assumed that the phonon scattering at these defects is of resonance character, so that for the one-site scattering matrix one can obtain [9] go)-O)~

t(o), k) --- t ( o ) )

, d - o)2 -

(3)

where o)~ is the effective frequency of the vibrations in an anharmonic potential, e >> 1 is the local perturbation due to the difference of the atomic masses, and r / = 3~rw3e/(2 w D) is the damping at resonance. For an amorphous system, it would appear natural that there exist various types of defects characterized by random parameters described by a distribution

E.I. Salamatoc / Journal of Non-Cr)'stalline Solids 202 (1996) 128-136

130

function. In this paper we assume that the defect type can be determined by the only parameter, for example, cox. Then in an approximation linear in defect concentration (i.e. with neglect of the dynamic interaction between defects), the mass operator becomes:

S(co) = c(t( co)),

(4)

where c is the total defect concentration and (t(co, k)) is the one-site scattering matrix averaged over the defect types [11,14]. The results obtained in [9] give the following relations for the quantities involved in Eqs. (1)-(4): =

,oQ2(l - P(co)/co

),

I/T=

QF( co)/2 co,

D=

gph(co) =

(5)

c2r/3,

2coY'~ImG(k, co)/Tr,

(7)

where Q(co) = (1 - oP/Oco2)-I is the renormalizing factor and the expression for D is written without regard for the quantum corrections. As for the density of vibrational states of a disordered system, it is defined as (8)

where (g~(co)) is the partial density of vibrational states of defect atoms, averaged over the defects types. In the approximation used, the partial density of states is found in solving the single-impurity problem. In the case of resonance scattering, the partial density is described by a Lorentz curve [14]. We should mention some results of our previous works [9,10] that can be useful for understanding of what follows. In Refs. [9,10], it was shown that, in the case of sufficient concentration of heavy impurities, a quasi-gap can arise in the phonon spectrum of the system. In the presence of phonon scattering with the resonance frequency, coo, the quasi-gap lies on the interval (coo, co¢), where co~ = coogl + c e .

k(T) ~ exp( -/3coo).

(10)

Such a behavior of k(T) is characteristic of amorphous materials [15]. Thus, this model is appropriate for describing the properties of amorphous dielectrics at temperatures above 1 K. For lower temperatures, the model should be amplified by the effects connected with the phonon scattering on tunnel states. It can be naturally done within the framework of the local anharmonicity model.

(6)

k

g(co)-=(1-C)gph(co)+c(g~(co)),

not yet excited. At higher temperatures, (9 > T >_ coc (O is the Debye temperature), the contribution of the second-band phonons to the heat conductivity coefficient increases exponentially:

(9)

(If, in the system, there are defects of different types, e signifies the effective mass [14].) When A = co¢ coo is of the order of coo, a plateau in the temperature dependence of k occurs in the region of T>_ co°, where the contribution to the integral (2) from the first-band phonons (below gap) has reached saturation, while the second-band (above gap) phonons are

3. Local anharmonicity and tunnel states In developing the microscopic model of amorphous states, we proceed from the local anharmonicity model [ 11,16]. To take account of the anharmonic effects, we used a pseudoharmonic approximation which enables us to find the T-dependence of the dynamic properties of a disordered system. According to this approximation, the motion of a particle in the anharmonic potential U(r) can be described as a harmonic motion, but in a different (effective) potential U(r): U ( r ) = exp((u 2)/(202/Or2))U(r),

(11)

whose temperature dependence is expressed in terms of the correlator of the root-mean-square shift (u 2) from the equilibrium position, r o. In turn, ro is also a function of temperature and can be found from the condition aU/Or = 0. Having chosen an approximation to find (u 2), we get the self-consistent equations for obtaining the temperature dependence of the effective frequency, w(T). In paper [16], we used an asymmetrical cubic potential to investigate the temperature-activated relaxation processes of amorphous state. In this paper, the two-well symmetric potential [11] for the s-type defect

U~(r) = -Ror2/4 + Q~r4/4,

(12)

with minima at the points r + = +_ V/Ro/(2Q~), the harmonic frequency coo2 = Ro ' and the barrier height Eb~ =Ro/(16Q 2) will be used as a local model

E.L Salamatoc / Journal of Non-Crystalline Solids 202 (1996) 128-136

potential. The parameter a = t o o / E b s ( o ) o = const.) will be used as a measure of anharmonicity. Also, we shall assume this parameter to be the unique independent random parameter which specifies the defect type. The effective frequency of basic oscillators (vibrations near r + ) decreases with increasing temperature, but for each non-zero a there is a temperature T~(ce) at which the basic state goes over into the excited one (vibrations near r = 0), whose effective frequency increases with elevation of temperature. At a > 2.05, the transition temperature Tc = 0 and the oscillator can exist only in an excited state. Allowing for the local anharmonicity, the quasigap in the phonon spectrum becomes a function of temperature, which can lead to a non-trivial behavior of both in the plateau region and at its upper edge. In particular, we considered [11] a maximum appearing in the plateau region that is characteristic of some amorphous materials [17] and high-temperature superconductors [ 18]. An important comment about the pseudoharmonic approximation used in Refs. [11,16] is to be made. It would seem that, in specifying a non-linear system by the effective frequency alone, one can expect to get, at best, a very rough qualitative description of its properties. Actually, however, the situation changes upon introducing the distribution function of a random quantity defining the frequency spectrum of the system at different temperatures. In fact, this spectrum can be regarded now as formed by both the structural disorder and all excited harmonics of the non-linear system. The presence of two-well potentials in the system considered implies that in this system there exist two-level states due to the quantum-mechanical tunneling of particles through the potential barrier. The resonance phonon scattering on such defects is known to be the governing mechanism of scattering in the low-temperature limit. As the exact solution of the quantum-mechanical problem of a non-linear oscillator is unknown, we shall use a quasi-classical approximation at T = 0. In this approximation, the splitting, ~%, of the energy level, E, in a two-well one-dimensional potential, U(r), is defined as [19]

k(T)

tot~=%exp(- fdrv:(U,(r)-E) )/rr,

U,(r)> E.

where the integral is taken over the region The subscript, ts, denotes the tunnel state of the s-type defect. For the potential Eq. (12), the integral in the exponent is expressed in terms of elliptic functions; however, we confine ourselves to a parabolic approximation of the potential near the point, r = 0. In so doing we get for the splitting of the ground energy level E =
(14)

which relates tots to the main parameter c~ of our model. Since the condition, tot~ << too, must be satisfied, there exists a value of ceq which defines the maximum of tots = toqLet us define the one-site matrix of phonon scattering on TLS at T = 0 in the form

tt~(to) = -Btot~ tooto2/(,o~

- to~ -i,Tt~(to)),

(15) where B is a dimensionless constant related in a certain way to the square of deformation potential [1], and the damping, "% << to, for all to < toq. In the case of a uniform distribution in TLS energies, for the phonon life-time, we get r ~ l / t o , which results in a quadratic dependence of in the low-temperature limit. Below we consider a more general case without assuming a uniform distribution in TLS energies, and we take for finite temperatures.

k(T)

its(to)

=

o-

'~--O"

-Btot~tog

9

to- t a n h ( ~ w / 2 )

- to2_ int (to)).

(16)

The parameter o- is introduced to allow for the possibility that the phonon scattering by the TLS can differ from Eq. (15). The hyperbolic tangent determines the difference in population between the TLS levels. For the TLS contribution to we have

Gt~(w), Gt~(to)=tanh([3to/Z)/(tot2~-to2-i~t~).

(17)

Restricting ourselves to a linear-in-concentration approximation and assuming the function p(c~) to be continuously distributed on the interval (c~l, c~2), we can write for the mass operator of a disordered system [14]: s(to)

(13)

131

=

= c(t

+

(to)) + cgt

(to)),

(is)

132

E.L Salamatoc / Journal of Non-Crystalline Solids 202 (1996) 128-136

where c is the total concentration of two-well anharmonic potentials in the system (structural defects), c' < 1 is a part of them in which the tunnel states are realized, ( ... ) denotes the averaging with the distribution function p(a) in the range from a~ to a 2 for structural defects and from c~ to o% for tunnel states. In this model, the condition c ' < 1 is connected with the presence of defects in which the TLS do not arise (aq < a2). Note also that we do not allow for the asymmetry of the potential destroying the tunnel states. Hence, in what follows, c and ct~ are assumed to be independent of one another. Taking into account the phonon scattering on two level systems results in appearance of an additional term in the expression for g(og) (Eq. (8)), hence the expression for the heat capacity can be represented by three terms:

C(T) = Cts(T) + C~(T) + Cph(T ).

several parameters are interrelated; therefore, it would suffice to consider only the behavior of C(T) and k(T) in order to find all of them. The consideration begins with the low-temperature limit. In this limit, analytical expressions for the heat capacity and conductivity can be obtained, which allows us to substantially reduce the number of fitting parameters of the model. In the present section, we choose wt~ to be an independent random variable and introduce its distribution function in the form, /9(tots)= Rwt~, on the interval (0, ogq). Assuming that the imaginary part of the one-site scattering matrix (16) can be approximated by 6-function, according to Eq. (6), we get for the phonon lifetime 7 = "to(1 -0Pt~ ( o9)/0o92)coth(/3o9/2)/o9 ~+r,

(20)

(19)

Thus, the development of the microscopic model is completed. Let us sum up the above. A lattice model of the amorphous state is used. The defects introduced are impurity oscillators vibrating in a symmetric two-well potential which is characterized by a, the only random parameter of the model. Depending on its value, a tunnel state may or may not arise on the defect. The matrix of phonon scattering on such defects is written in a general form which involves some parameters. In the next section, it will be shown that, for all parameters, we can obtain analytic expressions, relating them to experimentally observed quantities. Note that a similar procedure for describing the low-temperature behavior of k(T) and C(T) was used in Ref. [20].

4. Estimation of model parameter values In view of the universal behavior of amorphous material properties, in this paper, we do not match the model parameters to a particular material but find only typical values. It is convenient to classify the parameters introduced in the microscopic model under three types: the energy ones (ogq, o9o, w~), the structure parameters (c, c~) and those describing a particular defect (B, e, ~r). Additionally, we must determine the distribution function, p ( a ) . It should be noted that

where

ro=4/(ct~TrRBo9b

9--

o-

),

and Pt~(og) is the real part of ct~(tt~(og)). In the limits o9 << ogq, T ~ 0 we have for the phonon density [9] gph(o9) = 3og"/ogDV ~ 3 ,~1 -- Pt~(og)/o9 z .

(21)

In view of Eqs. (3) and (4), the expression for the phonon diffusion coefficient takes the form D ~ L,~"o coth(~o9/2)/(3o9'~+~).

(22)

The approximate equality results from the fact that the dependence of Pt~(og) on o9 logarithmically differs from the quadratic one, hence, to a good accuracy, we can put ( 1 - OPt~(og)/Oo92) = ( 1 Pt~(og)/o92). Again, with neglect of the logarithmic shift from the quadratic dependence in gph(W), after substituting Eqs. (21) and (22) into the integral (2), we get for the heat conductivity coefficient

k(T) = koT 3 ,T ~,.

(23)

In this frequency range, the main contribution to the density of vibrational states comes from the two level systems. We have from Eq. (17) gt~(w) = tanh(~o9/2) Ro9;',

(24)

which, after substitution into Eq. (1) yields

C(T) = CoTv+'.

(25)

E.L Salamator/ Journal of Non-Cr3,stalline Solids 202 (1996) 128-136 As the power behavior of k(T) and C(T) in the limit T ~ 0 is a well experimentally determined quantity, we can find the values of ",/ and o-, which can differ from the standard ones: o-= 1 and y = 0 [1]. In this paper, we shall not consider the experimentally observed shifts from the linear (for C(T)) and quadratic (for k(T)) dependencies; therefore, we obtain o- = 1 and ",/= 0 (R = 1 / w / ) . In this case, we have

C,, = ( 2 . 1 k n / 1 9 t x ) W ,

(26)

k,, = 4 k ~ / ( 3 6h19 ) W / B ,

(27)

where W = (c t w D / w ~) is a new independent parameter involved in the expressions for both the mass operator and the partial density of states, 6 3 = (41T/3)(//,/N A p) [15], /z is the molecular weight, p is the mass density, and NA is the Avogadro number. To obtain Eq. (27) we used the relation of the Debye temperature to the sound velocity [15]:

kB19 = hL.o27r/6. Bearing in mind that, in amorphous dielectrics, the standard value of C o amounts to 5 - 5 0 e r g / g / K 2, the molecular weight is of the order of 50 a.u., the mass density p ~ 2 - 1 0 g / c m 2 and 19-- 300 K [15], we can take W ~ 10 -3. The quantity k o is also universal enough (k o ~ 10 -5 W / K / c m ) , which, in view of the found value of W, yields B ~ l0 2. Thus, consideration of the low-temperature limit allows us to define the model parameters that describe the phonon - TLS interaction. Moreover, cots being related to a by Eq. (14), we can find the distribution function for the quantity ce. Since p(wt~) = const., we have from Eq. (14) p(a)

= R' exp( - 2 ' r r / a ) / o ~ 2.

(28)

Thus, consideration of the low-temperature limit allows us to find the following four model parameters toq, ct~, B, o- and to define the distribution function p(c~). Now we pass to the remaining model parameters. It should be noted that, according to Section 2, the energy characteristics Wo, w c determine the plateau region on the k(T) curve. From comparison with experimental data [15], we can put oJo ~ 0.1 w D and w~ ~ 0.5w D on the energy scale. Then from Eq. (9), we have ce ~ 30. Assuming the maximum of the function C ( T ) / T 3 in the region T ~ 10 K to be due

133

to the second term in Eq. (19) (the assumption was verified by numerical calculations), we can easily obtain Cm~x = cks/l~. Since the experimental values of ( C ( T ) / T 3 ) r = lOK ~ 0.1 ( e r g / g / K ) , we get c ~ 0.1. At such magnitudes of c the parameter e should take on large values ( ~ 100), which can be a consequence of using the lattice model of the amorphous state. However, this result does not contradict Ref. [21] in which the possibility that the effective detect mass assumes large values in amorphous materials was discussed. Thus, the suggested model enables us to define the characteristic values of all parameters before performing numerical calculations. The results of calculations were found to vary only slightly with variation of the microscopic model parameters in the vicinity of these characteristic values. So, in this paper, we present the results for only one set of parameters: c = 0.1, ~ = 300, B = 250, Wo = 0.1 w D, W = 10 -3. Let us dwell on p(c~). Since the parameters, c and c~t are believed to be independent (see Section 3), some other distribution functions are used in calculations along with Eq. (28). It was found that, contrary to the sound velocity [12], the integral properties C and k depend only slightly on the form of p ( a ) . Hence, in the paper, we present but the results obtained with p(ot) = 6(oz),

5. Discussion of results Fig. 1 shows the density of vibrational states of the system under consideration in the limit T ~ 0, the contributions to g ( w ) from particular terms being designated by dashed lines. When the local anharmonicity is not taken into account, only the low-frequency range of the spectrum will change with elevation of temperature as a result of equalization of level population in the TLS. The T-dependence of the system heat capacity and the contributions to this from particular terms in Eq. (19) are shown in Fig. 2. Three temperature ranges, in which the dependence C(T) is governed by a specific mechanism, can be identified. At T < 0.01 19 the main contribution to C(T) comes from the TLS, at 0.01 19 < T < 0.1 ~9 from the vibrations localized on structural defects, while at T > 0.119 the phonon contribution becomes dominant. The connection be-

E.I. Salamatot, / Journal of Non-Crystalline Solids 202 (1996) 128-136

134

10

1o -,

1o -,

1o

i0-~.

,o -,

I~

\

/ COq,

/I' t

I0

-~

/ /

/

~Io-' 0

~ / t/

5 / 10

/,,

' ' ' '"']10 -2 . . . . . . . .

10' -, . . . . . . .

~1

........

10 ~

/

10 -~

10 -~

lines).

(10) (curve 3), which does not contradict the results of Ref. [15]. The slow increase of k(T) in the plateau region is due to equalization of level popula-

10 -~ 10 ":

E lO-..

I

I

Iltll,I

10 -~

I

I

II1[11[

10 "'

4

I

t

".

1

IIIII1~

I

I

IILIII

110

t/

/,

/3

10

-7

10"8

-e.

,

,,,,,,,i

. . . . . . .

10 -~

u

. . . . .

1 0 -a

Fig. 2. The temperature dependence of particular terms according to Fig. 1.

C(T)

,,,i

10 -~

,

,

,,,,,u

1

T/O

and contributions of

J

........

10 "4

10 -"

10 -s

x

-5.

I0 -7

- -

,/

-i

o

I 0

I'

Fig. 3. The functions C(T)/T s for amorphous (1) and ideal (2) systems and the corresponding densities of vibrational states

(..9

10

/

X

T,/@

(dashed

tween the T-dependence of the heat capacity and the form of g(to) is clearly seen in Fig. 3 in which the function C ( T ) / T s is presented along with g(to). The maximum observed is certain to be due to the vibrations localized on defects, and the width of the peak is a function of the size of the quasi-gap in the phonon spectrum. The role of the gap in formation of the plateau on the k(T) curve is shown in Fig. 4. Note that the high-temperature behavior of k is described by Eq.

10

/

I'I

2/ /

101

(.0 / (-,{)O

~

..~.~

/'~L-"// / / ,,

2 /

Fig. I. The density of vibrational states at T = 0 in arbitraryunits. The particular contributions are shown by dashed lines: (1) TLS, (2) structural defects, (3) density of phonon states.

10 -z

/

/

1 0 -7 1Q -3 '

~-

j

,

2

/

10

\

1o-'-

r.o

iI

/~11

10-'"

\

I

10

' -3

'

......

r 10

........ -Z

........

I 10

-p

T/O

Fig. 4. The r-dependence of k(T). The phonon diffusion coefficient (l) and density of phonon states (2) are shown by dashed lines. Curve 3 denotes the high-temperature behavior of k(T) according to ( k ( r ) ~ exp(-0.35T/O)). The centered curve is calculated without allowance for equalization of level population in T L S .

E.L Salamator / Journal of Non-Co'stalline Solids 202 (1996) 128-136

tion in the TLS. This effect can be seen on the centered curve, calculated after substitution of unity for th(/3w/2). One can obtain a flatter plateau in a natural way, by introducing into the model a restriction on the phonon free path length imposed via the specimen size L. In this case, the effective number of TLS contributing to phonon scattering diminishes, which leads to a smaller increase in k(T) in the plateau region. The results of calculation of k(T) are presented in Fig. 5 for three different values of L. One can see that, at L = 2 X 10 -4 m (curve 2), the quadratic dependence of k(T) is conserved at T ~ 0, but the TLS contribution in the plateau region becomes small. In the case of small value of L = 10 -5 m (curve 3) as T ~ 0, we get k(T) ~ T 3, and the TLS can be neglected. Note that such a behavior of k(T) is observed in porous glasses [22].

6. Summary The approach proposed for calculating the heat capacity and thermal conductivity of amorphous system is distinguished by the fact that the main model points are associated with the one-site matrix of i 10 ~ 10 -1=

10 -2 i iJllrll i

10 -~ , IIiflt[ I

(,c) /C,c] D10 1

i f[rl[i I

i

i J l l l l l i-

1

10 (.3 10 - '

10 -5

30"s

' ' ,",u

I 0 -'

........

I 0 -=

F i g . 5. T h e T - d e p e n d e n c e o f

~ 10 -z

k(T)

' ....... 1 I 0 -'

.....

,,,

T/o

for three soeciment sizes: (1)

L ~ 0, ( 2 ) L = 2 × 10 . 4 m , ( 3 ) L = 10 . 4 m. T h e b e h a v i o r o f th e p h o n o n d i f f u s i o n c o e f f i c i e n t is s h o w n by d a s h e d l i n e s at L ~ 0

(1) and L= 10-~ (3).

135

10 ":

2

10 -"

I 0 -~"

/ , ~

~

~

10 -6.

I 0 -~'"

10 -"

10 -~

10 -=

I D -~

I

10

T/0

Fig. 6. The temperaturebehaviorof k(r), C(T)/T 3 and ~:(r)= (v(T)- r(0))/l,(0)+ ~:o (see Ref. [12]) in arbitrary units.

phonon scattering on the defects which are impurity oscillators vibrating in a symmetric two-well potential. Depending on the value of the only random parameter of the model, a tunnel state may or may riot arise on the defect. The physical quantities involved in calculations: the densities of vibrational and phonon states, the group velocity and lifetime of phonons are obtained on the basis of the assumptions made. This process makes it possible to find the temperature dependence of several experimentally observed parameters within the framework of the one and the same calculation procedure, defining simultaneously the relation between the model parameters. Along with the quantities k(T) and C(T) discussed above, the T-dependence of the adiabatic sound velocity was calculated earlier within the same model. The weak dependence of k(T) and C(T) on the local anharmonicity parameters allows them to be presented together with the T-dependence of the adiabatic sound velocity calculated in paper [12], without any matching of parameters. The temperature dependence of the heat capacity, thermal conductivity coefficient and adiabatic sound velocity are shown in Fig. 6 which demonstrates good agreement with the experimental data available, confirming the validity of the model suggested for describing the properties of amorphous dielectrics over a wide temperature range.

136

E.L Salamatoc / Journal of Non-Crystalline Solids 202 (1996) 128-136

Acknowledgements The author is grateful to E.P. Chulkin and A.P. Zhernov for several illuminating discussions.

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