Some Problems in the Theory of Dielectric Relaxation at Low Temperatures
Abstract Generated abstract
This paper examines how the classical Debye theory of dielectric relaxation in solid dielectrics should be modified at low temperatures, where lattice behavior becomes quantum mechanical. Using a simple two-position dipole relaxator model, the authors recalculate the jump probability and relaxation time within the Einstein and Debye theories of heat capacity. In both treatments, quantum restrictions on lattice vibrations lead to a more rapid increase of relaxation time with decreasing temperature than predicted by the classical expression, with the Einstein model adding an energy quantum to the activation term and the Debye model introducing a strong temperature-dependent prefactor. The paper suggests that Einstein-type descriptions may apply at moderately low temperatures, Debye-type descriptions at very low temperatures, and mixed spectra may be needed for ionic crystals such as NaCl.
Full Text
SOME PROBLEMS IN THE THEORY OF DIELECTRIC RELAXATION AT LOW TEMPERATURES
Yu. M. Annenkov, E. K. Zavadovskaya, V. A. Shcherbakov
The existing theory of dielectric relaxation in solid dielectrics [1–3] is based on two assumptions:
- The distribution of lattice ions over vibrational energies obeys Boltzmann statistics.
- The vibration frequency of lattice ions does not depend on temperature.
Numerous experimental data show that at medium and high temperatures these assumptions are valid. However, from data on the temperature variation of the thermal properties of solids it follows that below a certain temperature the behavior of solids begins to acquire a quantum character. It is therefore necessary to extend the classical theory of dielectric relaxation to the low-temperature region, refining it with the aid of the quantum theory of heat capacity [4–6].
The discussion will be carried out within the framework of the Einstein and Debye theories of heat capacity for a simple relaxator model: the dipole has two positions of stable equilibrium.
1. Classical theory of relaxation
The theory of dielectric relaxation developed by Debye gives the following temperature dependences of the tangent of the dielectric-loss angle \((\operatorname{tg}\delta)\) and of the relaxation time \((\tau)\):
\[ \operatorname{tg}\delta = \frac{4\pi \chi \cdot \omega \cdot \tau} {4\pi \chi + \varepsilon_{\infty}(1+\omega^2\tau^2)} \approx \frac{4\pi\chi}{\varepsilon_{\infty}}\cdot \frac{\omega\tau}{1+\omega^2\tau^2}, \tag{1} \]
\[ \tau=\frac{1}{2\nu}\cdot e^{U/kT}. \tag{2} \]
Here: \(\omega\) is the frequency of the external field, \(\chi\) is the specific polarizability, \(\nu\) is the averaged vibration frequency of lattice ions, and \(U\) is the magnitude of the potential barrier between localization sites.
The maximum conditions for the frequency and temperature dependences have the form:
\[ \omega_{\max}\approx \frac{1}{\tau}, \tag{3} \]
\[ T_{\max}=\frac{U}{k\ln(2\gamma/\omega)}. \tag{4} \]
If the temperature dependence of \(\chi\) is not taken into account \(\left(\chi\sim \frac{1}{T}\right)\), then the value of \(\operatorname{tg}\delta_{\max}\) does not depend on the temperature or on the frequency of the external field:
\[ \operatorname{tg}\delta_{\max}=\frac{2\pi\chi}{\varepsilon_\infty}. \tag{5} \]
2. Dielectric relaxation within the framework of Einstein’s heat-capacity theory
As is known, the basic propositions of Einstein’s heat-capacity theory are as follows:
-
All atoms of a solid oscillate with the same frequency
\[ \left(\nu_E=\frac{k\Theta_E}{h},\ \text{where } \Theta_E \text{ is the characteristic Einstein temperature}\right) \]
and have energy \(E=n\cdot h\cdot \nu_E\) \((n=0,1,2\ldots)\). -
In thermal equilibrium the ratio of the occupation numbers of two neighboring energy levels \(n\) and \((n+1)\) is equal to \(e^{-h\nu_E/kT}\). In this case the sum of states has the form:
\[ N=\sum_{n=0}^{\infty} e^{-n\nu_E/kT}. \tag{6} \]
If \(U\) is the height of the potential barrier between localization sites, then only ions possessing an energy
\[ E_1=n_1\cdot h\cdot \nu_E \geq U \]
are capable of making a jump.
Therefore the probability of a jump for each weakly bound ion will be proportional to the ratio of the number of states \(N_1\) with energy \(E_1\) to the total number of states \(N\) and to the vibration frequency \(\nu_E\).
\[ W_E=\nu_E\cdot \frac{N_1}{N}. \tag{7} \]
Taking into account that
\[ \sum_{n=0}^{\infty} e^{nx}=\frac{1}{1-e^x};\qquad \sum_{n=0}^{n_1} e^{nx}=\frac{1-e^{(n_1+1)x}}{1-e^x}, \]
where
\[ x=-\frac{h\nu_E}{kT}, \]
we obtain
\[ N_1=\sum_{n=0}^{\infty} e^{nx}-\sum_{n=0}^{n_1} e^{nx} =\frac{e^{(n_1+1)x}}{1-e^x}; \tag{8} \]
then
\[ W_E=\nu_E\cdot e^{(n_1+1)x}\simeq \nu_E\cdot e^{-\frac{U+h\nu_E}{kT}} \tag{9} \]
and finally the relaxation time is written in the form
\[ \tau=\frac{1}{2W_E}=\frac{1}{2\nu_E}\cdot e^{\frac{U+h\nu_E}{kT}}. \tag{10} \]
It is not difficult to see that at high temperatures \((h\nu_E \ll kT)\) expression (10) goes over into equation (2). At low temperatures the factor \(e^{h\nu_E/kT}\) substantially increases \(\tau\), i.e., the shift of the frequency peak \(\operatorname{tg}\delta\) with changing temperature occurs faster than in the classical theory. The position of the temperature peak \(\operatorname{tg}\delta\) undergoes practically no additional changes, since usually \(h\nu_E \ll U\):
\[ T_{\max}=\frac{U+h\nu_E}{\kappa \ln(2\nu_E/\omega)}. \tag{11} \]
However, when determining the activation energy from the value of \(T_{\max}\) in the low-temperature region, a correct choice of \(\nu_E\) is necessary.
3. Dielectric relaxation within the framework of the Debye heat-capacity theory
According to Debye’s theory of heat capacity, in a solid consisting of \(N\) atoms, the distribution function of the frequencies of atomic vibrations has the form:
\[ f(\nu)=\frac{12\pi}{c^3}\cdot \nu^2. \tag{12} \]
Here \(c\) is the velocity of sound waves. The number of possible vibrations is \(3N\).
\[ \int_0^{\nu_D} f(\nu)\,d\nu=\frac{4\pi}{c^3}\nu_D^3=3N, \tag{13} \]
\[ \nu_D=\frac{\kappa\Theta_D}{h}, \]
where \(\Theta_D\) is the characteristic Debye temperature.
For \(T>\Theta_D\), all possible vibrations of the lattice are excited in the crystal. For \(T<\Theta_D\), only those vibrations are excited for which the relation
\[ h\nu \leqslant \kappa T \tag{14} \]
is satisfied. The number of excited vibrations in this case is
\[ \int_0^{\nu_T} f(\nu)\,d\nu=\frac{4\pi}{c^3}\nu_T^3;\qquad \nu_T=\frac{\kappa T}{h}. \]
Then the probability of a jump of an ion into a neighboring potential well will be proportional to the density of excited vibrations in the direction of the jump.
\[ W_D\sim \frac{1}{3}\cdot\frac{\nu_T^3}{\nu_D^3} =\frac{T^3}{3\Theta_D^3}. \tag{15} \]
Since the quantum character of low-temperature vibrations is taken into account in condition (14), the further calculation may be carried out in the usual way. It can be shown that for \(T<\Theta_D\) the effective frequency of vibrations is
\[ \nu_{\mathrm{eff}}=\frac{3}{4}\cdot \nu_T =\frac{3}{4}\cdot\frac{\kappa T}{h}. \tag{16} \]
Then the probability of an ion jump per unit time and the relaxation time may be written as
\[ W_D=\frac{T^3}{3\Theta_D^3}\cdot \nu_{\mathrm{eff}}\cdot e^{-U/kT}, \tag{17} \]
\[ \tau=\frac{1}{2W_D}=\frac{2h\Theta_D^3}{\kappa T^4}\cdot e^{U/kT}. \tag{18} \]
Thus, taking into account the temperature dependence of the spectrum of phonon oscillations of a solid leads to a sharp increase in the relaxation time as the temperature is lowered. It can also be shown that above the Debye temperature formula (18) is transformed into expression (2).
4. Conclusion
Thus, the application of the Einstein and Debye theories of heat capacity to the calculation of low-temperature dielectric relaxation leads to a qualitatively identical result: as the temperature is lowered, the relaxation time increases more rapidly than follows from the usual Debye theory of dielectric relaxation. As for the temperature range of applicability of the proposed interpretations, there is as yet an insufficient amount of experimental data for its estimation from studies of the behavior of dielectrics at low temperatures. Analysis of the thermal properties of monatomic solids shows that the Einstein theory describes the behavior of the heat capacity well at not very low temperatures. The Debye theory, on the contrary, describes processes more accurately at very low temperatures. It is reasonable to assume that a similar picture will also be observed for dielectric relaxation. As for crystals of the NaCl type, here a representation of the vibration spectrum as a superposition of Debye and Einstein spectra is more acceptable; moreover, agreement with experiment is improved when there is a large difference in the masses of the anions and cations [7].
REFERENCES
- H. Fröhlich. Theory of Dielectrics. IL, Moscow, 1960.
- A. Lidiard. Ionic Conductivity of Crystals. IL, Moscow, 1962.
- G. I. Skanavi. Physics of Dielectrics (Region of Weak Fields), GTTI, Moscow, 1949.
- F. Seitz. The Modern Theory of Solids. GITTL, Moscow–Leningrad, 1949.
- C. Kittel. Introduction to Solid State Theory. Fizmatgiz, Moscow, 1963.
- T. Leibfried. Microscopic Theory of the Mechanical and Thermal Properties of Crystals. Fizmatgiz, Moscow–Leningrad, 1963.
- Nordwood M. H., Brisco C. V. Phys. Rev., 1958, 112, 1, 45.