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Heat propagation in glasses
PHYSICA
Physica B 194-196 (1994) 485-486 North-Holland
Heat Propagation in Glasses P. Strehlowa and W. Dreyerb aphysikalisch-Technische Bundesanstalt, AbbestraBe 2-12, 1000 Berlin 10, Germany bTechnische Universitiit Berlin, FB 9, Miiller-Breslau-StraBe 8, 1000 Berlin 12, Germany In order to describe the phonon-related heat propagation in glasses below 1 K we derive thermodynamic field equations that follow from an underlying kinetic theory including the interaction of phonons and tunneling systems. It is shown that the solution of the derived linearized field equations for given boundary and initial values of heat-pulse experiments are in excellent agreement with the experimental data obtained in vitreous silica. Moreover, a new kind of energy propagation in glasses at low temperatures is predicted. Finally, we calculate the time dependence of heat release due to the relaxation of low-energy excitations, and this requires a numerical solution of the nonlinear field equations. 1. INTRODUCTION It is a fundamental feature of amorphous materials that glasses show a qualitative similarity of their thermal, acoustic and dielectric low-temperature properties which can be explained roughly by the tunneling states model. However, in order to explain the low-temperature behaviour of glasses, which has been observed in ever more complicated heat-pulse experiments and nonlinear heat release measurements, a system of field equations for the variables describing the state of glasses at low temperatures has to be introduced. The interpretation of all these low-temperature experiments in glasses requires that the system of field equations has to be solved for proper boundary and initial values so that the theory may be compared legitimately to the experimental results.
the TS energy density ers and the phonon momentum density Pi have been derived. We find 8teph
+
C2 C?kpk
=
8tp i
+
1/3
=
~ieph
Oters
=
S -(a+l/x) Pi
(1)
-S
where x is the temperature-independent phonon boundary relaxation time. S and a are given by S = 6AzkST5/(rcc3h 3) I
I=
oo I
2
A/T
, with
(2)
x 4 (exp(x) - exp(xT /Ts))dx (exp(x) - 1)(exp(xT / Ts) + 1)
2. THEORY
Based on the coupled Boltzmann equations for phonons and tunneling systems (I'S), field equations for the phonon energy density eph,
a = (15/32)n2AzkT.
(3)
Ts is the temperature of the tunneling states which is equal to T in equilibrium, and the
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E0822-X
486
constants z,A,A and c denote the TS density of states, the TS-phonon coupling, the minimal energy splitting and the Debye velocity, respectively. The energy densities eph and ers are related to T and Ts by eph = ~2k4/(10h3c3) T 4
(4)
(3O
er s = zk2T2AfT Arth~/]-(A/x/T)2xdx exp(xT / Ts) + 1
(5)
3. CALCULATION The solution of the linearized field equations for heat-pulse experiments is in excellent agreement with the experimental data obtained in vitreous silica (Suprasil I) [1] including the overshoot and the non-exponential relaxation behaviour at larger times (Figure 1).
Relaxation on a larger time scale may be observed convenient in heat release measurements If a sample of glass at a temperature T i is suddenly connected to a thermal reservoir at temperatur T O by a thermal link of conductance K, the observed temperature-time curves [2] varies much more slowly with time than an exponential decay. It is important to note that many authors have introduced a specific heat with an explicit time dependence as origin of the nonexponential decay. We have found, however, that the nonlinearity of the field equations is responsible for this, while the specific heat depends on time only via temperature. In Figure 2 the TS part of specific heat, Oers i~q/ST, for Suprasil I is shown. 1'
0.5' ~.~ 0.2
2,. o.1. ~o.o5
E [.tJ z (::3 ra PeN
o.o2 o.o2
0.3. 0.2
o'.1
0".2
Figure 2. Part of the specific heat due to TS in Suprasil I computed by (5).
"
o.1-
¢Y LtJ
I--
o;os
TEMPERATURE, K
0.1
10
102 103 10~ T I H E , /as
105
10
107
Figure 1. Observed heat-pulse shape (circles) in Suprasil I at T = 50 mK. The theoretical temperature profile (solid lines) is computed from Eq. (1) with z = 1.98x 1033 l~tJ-lcm-3, A = 1.02 mK and A = 2.39x 10 -9 cm3s-l. The dashed line corresponds to the Fourier theory. Moreover, as shown in Figure 1, the theory predicts a new pulse at very short times reflecting a wave-like energy transport due to the absorption and emission of phonons by fast relaxing tunnelin states.
In summary, we have demonstrated a consistent description of both heat-pulse and heat release experiments in glasses by solving the field equations for the variables ers,eph and Pi. REFERENCES
1. P.Strehlow, W.Dreyer and M.Meissner, Phonon Scattering in Condensed Matter VII, edited by M.Meissner and R.O. Pohl, Springer Series in Solid State Sciences, Vol. 112 (1993) 2. J. Zimmermann and G. Weber, Phys. Rev.Lett. 46, 661 (1981)
Physica B 194-196 (1994) 485-486 North-Holland
Heat Propagation in Glasses P. Strehlowa and W. Dreyerb aphysikalisch-Technische Bundesanstalt, AbbestraBe 2-12, 1000 Berlin 10, Germany bTechnische Universitiit Berlin, FB 9, Miiller-Breslau-StraBe 8, 1000 Berlin 12, Germany In order to describe the phonon-related heat propagation in glasses below 1 K we derive thermodynamic field equations that follow from an underlying kinetic theory including the interaction of phonons and tunneling systems. It is shown that the solution of the derived linearized field equations for given boundary and initial values of heat-pulse experiments are in excellent agreement with the experimental data obtained in vitreous silica. Moreover, a new kind of energy propagation in glasses at low temperatures is predicted. Finally, we calculate the time dependence of heat release due to the relaxation of low-energy excitations, and this requires a numerical solution of the nonlinear field equations. 1. INTRODUCTION It is a fundamental feature of amorphous materials that glasses show a qualitative similarity of their thermal, acoustic and dielectric low-temperature properties which can be explained roughly by the tunneling states model. However, in order to explain the low-temperature behaviour of glasses, which has been observed in ever more complicated heat-pulse experiments and nonlinear heat release measurements, a system of field equations for the variables describing the state of glasses at low temperatures has to be introduced. The interpretation of all these low-temperature experiments in glasses requires that the system of field equations has to be solved for proper boundary and initial values so that the theory may be compared legitimately to the experimental results.
the TS energy density ers and the phonon momentum density Pi have been derived. We find 8teph
+
C2 C?kpk
=
8tp i
+
1/3
=
~ieph
Oters
=
S -(a+l/x) Pi
(1)
-S
where x is the temperature-independent phonon boundary relaxation time. S and a are given by S = 6AzkST5/(rcc3h 3) I
I=
oo I
2
A/T
, with
(2)
x 4 (exp(x) - exp(xT /Ts))dx (exp(x) - 1)(exp(xT / Ts) + 1)
2. THEORY
Based on the coupled Boltzmann equations for phonons and tunneling systems (I'S), field equations for the phonon energy density eph,
a = (15/32)n2AzkT.
(3)
Ts is the temperature of the tunneling states which is equal to T in equilibrium, and the
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved S S D I 0921-4526(93)E0822-X
486
constants z,A,A and c denote the TS density of states, the TS-phonon coupling, the minimal energy splitting and the Debye velocity, respectively. The energy densities eph and ers are related to T and Ts by eph = ~2k4/(10h3c3) T 4
(4)
(3O
er s = zk2T2AfT Arth~/]-(A/x/T)2xdx exp(xT / Ts) + 1
(5)
3. CALCULATION The solution of the linearized field equations for heat-pulse experiments is in excellent agreement with the experimental data obtained in vitreous silica (Suprasil I) [1] including the overshoot and the non-exponential relaxation behaviour at larger times (Figure 1).
Relaxation on a larger time scale may be observed convenient in heat release measurements If a sample of glass at a temperature T i is suddenly connected to a thermal reservoir at temperatur T O by a thermal link of conductance K, the observed temperature-time curves [2] varies much more slowly with time than an exponential decay. It is important to note that many authors have introduced a specific heat with an explicit time dependence as origin of the nonexponential decay. We have found, however, that the nonlinearity of the field equations is responsible for this, while the specific heat depends on time only via temperature. In Figure 2 the TS part of specific heat, Oers i~q/ST, for Suprasil I is shown. 1'
0.5' ~.~ 0.2
2,. o.1. ~o.o5
E [.tJ z (::3 ra PeN
o.o2 o.o2
0.3. 0.2
o'.1
0".2
Figure 2. Part of the specific heat due to TS in Suprasil I computed by (5).
"
o.1-
¢Y LtJ
I--
o;os
TEMPERATURE, K
0.1
10
102 103 10~ T I H E , /as
105
10
107
Figure 1. Observed heat-pulse shape (circles) in Suprasil I at T = 50 mK. The theoretical temperature profile (solid lines) is computed from Eq. (1) with z = 1.98x 1033 l~tJ-lcm-3, A = 1.02 mK and A = 2.39x 10 -9 cm3s-l. The dashed line corresponds to the Fourier theory. Moreover, as shown in Figure 1, the theory predicts a new pulse at very short times reflecting a wave-like energy transport due to the absorption and emission of phonons by fast relaxing tunnelin states.
In summary, we have demonstrated a consistent description of both heat-pulse and heat release experiments in glasses by solving the field equations for the variables ers,eph and Pi. REFERENCES
1. P.Strehlow, W.Dreyer and M.Meissner, Phonon Scattering in Condensed Matter VII, edited by M.Meissner and R.O. Pohl, Springer Series in Solid State Sciences, Vol. 112 (1993) 2. J. Zimmermann and G. Weber, Phys. Rev.Lett. 46, 661 (1981)