Dielectric and anelastic relaxation of crystals containing point defects

A theory is developed which describes the linear, reversible, time-dependent response of a crystal containing point defects to stress or electric fields, respectively known as anelastic and dielectric relaxation. Such relaxation occurs because of the redistribution of the defects among sites which are initially equivalent, but which becomes inequivalent in the presence of the external field. The macroscopic behaviour of such a crystal is found to be describable in terms of the symmetry which can be assigned to the defect. This defect symmetry determines whether or not the crystal will undergo dielectric or anelastic relaxation and, if relaxation can occur, which specific coefficients of elastic compliance or electric susceptibility show the relaxation effect. The latter information, called the ‘selection rules’ tells, in effect, which combination of stress or electric field components is capable of redistributing the defects. Tables are given for these selection rules for all possible defect symmetries in each of the 32 crystal classes. It is also shown that a hitherto unobserved phenomenon of piezoelectric relaxation may occur; the selection rules for this effect are also given.

Aside from its symmetry, the defect can be described as an electric dipole in terms of a suitable dipole moment vector μ, and as an ‘elastic dipole’ in terms of a tensor λ. It is shown that the defect symmetry determines the number of independent components of μ and λ. Finally, a thermodynamic theory is developed which permits calculation of the relaxation strengths for those compliance, susceptibility, and piezoelectric coefficients which undergo relaxation, in terms of the independent components of μ and λ. Applications of the theory to specific cases are then reviewed.     

3-d Model for strain ordering in steel: II Relaxational effects

The spring-defect model developed by us in the accompanying paper I to discuss ferroelasticity, exhibited in the BCT phase of α-iron (BCC metals), is used to analyse anelastic relaxation across the paraelastic to ferroelastic phase transition. The kinetics of the underlying Hamiltonian representing strain-strain interactions is treated within mean-field theory. The relaxation-response relation of the linear response theory is employed to derive explicit expressions for the anelastic strain, the frequency-dependent compliance and the internal friction in terms of the basic parameters of the spring-defect model.     

3-d Model for strain ordering in steel: I Static effects

We present a spring-defect model in 3-dimensions to describe the connection between elastic distortion and interstitial carbon ordering associated with phase transition from a body centred cubic (BCC) to body centered tetragonal (BCT) structure in BCC metals such as α-iron. The presence or the absence of the carbon is modelled in terms of a pseudo spinŝ=+1or -l.An Ising interaction between carbon atoms is recovered after eliminating the lattice degrees of freedom, which is longranged. The coupling between the spin and lattice degrees of freedom allows for a systematic study of ferroelasticity and the variation of the lattice parameter with carbon concentration. The mean field results for the paraelastic to ferroelastic transition, lattice parameter and static compliance are presented. The significant feature of this calculation is not only a derivation of the defect-defect interaction, but also an explicit calculation of the strain dipole tensor associated with each defect, from a microscopic model.     

Phase transitions in 2-dimensional systems of migrating orientable lattice defects

Phase transitions are studied in a system consisting of reorientating and migrating point defects in a two dimensional lattice. Due to the long range (r ^–2) nature of the dominant elastic interaction, surface effects are of central importance and have to be included. After diagonalizing the elastic interaction energy for defects characterized by arbitrary elastic dipole tensors the free energy of the system is minimized with respect to the tensor defect density (which describes the defect distribution in space and over a discrete number of orientations). Different types of phase transitions are obtained depending on the magnitude of the defect anisotropy. The phase below the paraelastic one is characterized for large by an anisotropic but homogeneous distribution, for small by an anisotropic and inhomogeneous distribution with a non linear space dependence. Similarities and differences with 3d results for=0 (or small) are discussed.     

c-director relaxation around a vortex of strength +1 in free-standing smectic-C films

The relaxation of director fields in freely suspended smectic films is studied experimentally by means of polarizing microscopy, and analyzed by solving the torque balance equation under appropriate initial and boundary conditions. We consider in particular the role of anchoring conditions of the c-director at particles and defects in the film. The structure of regular relaxation patterns allows to determine the elastic anisotropy of smectic materials. The splay elastic constant can exceed the bend constant by a factor of two and more. A remarkable consequence of this anisotropy is the stick-slip-like relaxation around a central defect of topological strength s = + 1.     

应变晶体的弹性偶极子模型

将物体视为一弹性连续介质,其中任一处在应变状态的体积元以一弹性偶极子模拟之,本文给出了它的等效偶极矩的表示式,并将其划分为与此体积元及其周围基体的固有性质有关的永久偶极矩和决定于介质所处的应变状态的感生偶极矩两部分,它们分别使弹性介质具有顺弹性和介弹性。本文给出了在各向同性的弹性连续介质中弹性偶极子所产生的位移场的表示式,以及在考虑到其他应力源的强度和偶极子矩间存在着相互弛豫作用的情况下,弹性偶极子与其他应力场间相互作用的式子,它是Kroner公式加上高级修正项,将此模型用于讨论熟知的Cottrell气团和两对称中心间的相互作用时,得到与前人一致且更为细致的结果。 关键词：     

Partial relaxation magnitudes for anelastic and dielectric relaxation due to point defects

This paper continues the development of a general theory of anelastic and dielectric relaxation of crystals containing point defects, when such defects can react to form different species each of which may occur in several crystallographically equivalent orientations. The present objective is to calculate the partial relaxation magnitudes when such defects give rise to more than one relaxation (e.g. more than one Debye peak) of a given elastic or dielectric constant. A general calculation is presented which incorporates both the thermodynamic and kinetic, aspects of the relaxation process. In particular, each partial magnitude is associated with one relaxational normal coordinate, rather than with a particular defect species. The results are then reviewed in a form suitable for the reader who wishes to use them without going through the detailed derivation. Finally, the theory is applied to some specific cases, in particular, to the substitutional-interstitial pair in b.c.c. metals.     

Defect equilibria and conduction mechanisms in ice

Dielectric and elastic relaxation processes in ice may be explained by means of two types of point defects. At the crossover they interchange their roles as majority and minority mechanism. From conductivity measurements at the crossover it is possible to determine the defect mobilities, without knowing the effective charges. Measurements from different laboratories lead to the same results and to the conclusion that in contrast to earlier theoretieal ideas the ion states as such do not play an essential role in the Debye relaxation. The two mechanisms responsible for this process are Bjerrum defects as envisaged in earlier concepts and Bjerrum-ion aggregates. These defects allow an explanation of the temperature dependence of the conductivity in pure and doped ice.     

A model for the propagation of nonlinear dispersive strain waves in a solid with defect clusters

A model for the propagation of nonlinear dispersive one-dimensional longitudinal strain waves in an isotropic solid with quadratic nonlinearity of elastic continuum is developed with taking into account the interaction with atomic defect clusters. The governing nonlinear dispersive-dissipative equation describing the evolution of longitudinal strain waves is derived. An approximate solution of the model equation was derived which describes asymmetrical distortion of geometry of the solitary strain wave due to the interaction between the strain field and the field of clusters. The contributions of the finiteness of the relaxation times of cluster-induced atomic defects to the linear elastic modulus and the lattice dissipation and dispersion parameters are determined. The amplitudes and width of the nonlinear waves depend on the elastic constants and on the properties of the defect subsystem (atomic defects, clusters) in the medium. The explicit expression is received for the sound velocity dependence upon the fractional cluster volume, which is in good agreement with experiment. The critical value of cluster volume fraction for the influence on the strain wave propagation is determined.     

Density matrix formalism for anelastic relaxation

A density matrix formalism is developed for anelastic (mechanical) relaxation in crystalline materials with point defects characterized by elastic dipoles. The time-dependent approach to equilibrium of the strain response under the action of a constant applied stress is deduced. The formalism parallels the one used in nuclear magnetic relaxation. The anelastic relaxation time is determined as a function of the parameters occurring in the defect hopping term in the Hamiltonian. This term is responsible for the dissipation of the anelastic ‘potential’ energy into the host lattice. In a lengthy concluding section, the following aspects are discussed point by point: the advantages of the formalism presented, its scope and special cases; the physical implications of the expression obtained for the relaxation time; the similarities and differences between magnetic relaxation and anelastic relaxation, etc.     

Paraelastic behavior of potassium cyanide

The low temperature Curie-type paraelastic behavior of low concentration anionic impurities dissolved in simple cubic alkali halide crystals has already been analysed for a number of cases. The CN⁻ radical in particular can substitute the halide ion in any concentration (up to 100% forming pure cyanide crystals), producing room temperature alkali halide-like crystals that above about 85% concentration show low temperature ferroelastic phases of crystallization. This paper presents the experimental evidence of room temperature paraelastic alignment exploring some analogies between elastic and magnetic (or electric) properties in the high concentration impurity range and above the phase transition (Curie) temperature.     

Defect stability in thorium monocarbide:An ab initio study

The elastic properties and point defects of thorium monocarbide(ThC) have been studied by means of density functional theory based on the projector-augmented-wave method. The calculated electronic and elastic properties of ThC are in good agreement with experimental data and previous theoretical results. Five types of point defects have been considered in our study, including the vacancy defect, interstitial defect, antisite defect, schottky defect, and composition-conserving defect. Among these defects, the carbon vacancy defect has the lowest formation energy of 0.29 eV. The second most stable defect(0.49 eV) is one of composition-conserving defects in which one carbon is removed to another carbon site forming a C2 dimer. In addition, we also discuss several kinds of carbon interstitial defects, and predict that the carbon trimer configuration may be a transition state for a carbon dimer diffusion in Th C.     

固/固型二维正方晶格声子晶体缺陷态研究*

固/固型声子晶体在抑制噪音和隔离振动等工程领域有着潜在的应用。利用有限元方法对存在缺陷的二维正方晶格金/环氧树脂声子晶体进行了研究,数值计算结果表明弹性波在点缺陷处局域,带隙中出现多条缺陷模,点缺陷数目的增加对声子晶体局域特性产生显著影响;弹性波沿着线缺陷传播形成波导,改变线缺陷结构可以改变弹性波传播方向和实现信号的分离。对声子晶体缺陷特性的研究可为声学滤波器和波导等器件的设计提供参考。     

Williams-watts dielectric relaxation: A fractal time stochastic process

Dielectric relaxation in amorphous materials is treated in a defect-diffusion model where relaxation occurs when a mobile defect, such as a vacancy, reaches a frozen-in dipole. The random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite. When there are many more defects than dipoles, the Williams-Watts decaying fractional exponential relaxation law is derived. The argument of the exponential is related to the number of distinct sites visited by the random walk of the defect. For the same reaction dynamics but with more traps than walkers, an algebraically decaying relaxation is found.     

Low temperature thermal conductivity of OH− and OD−-doped LIF single crystals

Thermal conductivity measurements between 0.1 and 70K of OH^? and OD^? doped LiF indicates resonant phonon scattering at 0.86cm^? for OH^? and 0.43cm^? for OD^?. Hydroxyl impurities trapped at divalent impurity sites do not scatter phonons resonantly and it is shown that inhomogenous distribution of strong scattering centers modify thermal conductivity curves considerably. A simple tunneling model for the isotope shift suggests an effect similar to the “polaron” model of paraelastic centers. A relaxation time for phonon scattering by OH^? is derived, showing that the ground state may be understood in terms of a direct process, underlining that this system provides a connection between the theory of paraelectric/paraelastic impurities and Jahn-Teller systems.     

Nonlinear Longitudinal Strain Waves in a Solid Exposed to Pulsed Laser Radiation

The propagation of longitudinal strain waves in a solid with quadratic nonlinearity of elastic continuum was studied in the context of a model that takes into account the joint dynamics of elastic displacements in the medium and the concentration of the laser-induced point defects. The input equations of the problem are reformulated in terms of only the total displacements of the medium points. In this case, the presence of structural defects manifests itself in the emergence of a delayed response of the system to the propagation of the strain-related perturbations, which is characteristic of media with relaxation or memory. The model equations describing the nonlinear displacement wave were derived with allowance made for the values of the relaxation parameter. The influence of the generation, relaxation, and the strain-induced drift of defects and the flexoelectricity on the propagation of this wave was analyzed. It is shown that, for short relaxation times of defects, the strain can propagate in the form of both shock fronts and solitary waves (solitons). Exact solutions depending on the type of relation between the coefficients in the equation and describing both the shock-wave structures and the evolution of solitary waves are presented. In the case of longer relaxation times, shock waves do not form and the strain wave propagates only in the form of solitary waves or a train of solitons. The contributions of the finiteness of the defect-recombination rate and the flexoelectricity to linear elastic moduli and spatial dispersion are determined.     

A neutron diffraction study of the glass transition in (KBr)0.47(KCN)0.53

The molecular crystal (KBr)_0.47 (KCN)_0.53 has been investigated by elastic neutron diffraction at the transition from the paraelastic to the orientational glass state. The freezing temperature is characterized by the onset of a momentum transfer dependent broadening of the diffraction lines indicating the transition from a crystalline to an amorphous state.     

Dielectric and anelastic relaxation of crystals containing point defects. II

In part I (1965, Adv. Phys., 14, 101), a theory was developed which treated the thermodynamics of dielectric and anelastic relaxation due to point defects in crystals from the viewpoint of the point symmetry of the defect as well as of the crystal. In the present paper this theory is extended to treat the kinetics of relaxation. Equations are derived which express the relaxation times of electrically and stress active modes of relaxation in terms of the rates of reorientation between one particular defect orientation and all of the other equivalent configurations. Explicit expressions are then given for these relaxation times for commonly occurring crystal and defect symmetries. The reorientation frequencies which appear in these expressions may be converted into the appropriate atom or ion jump rates; this final step can generally be carried out merely by inspection of the crystal model. The possibility that two or more relaxations due to a given point defect may be widely separated on a frequency or temperature scale (a situation which is called a ‘frozen-free split’), and the anomalies connected with such behaviour, are discussed. Finally, various examples which have been studied in the literature, of relaxations due to point defects, are reviewed in the framework of the present theory.     

Effect of poisson ratio on cellular structure formation

Mechanically active cells in soft media act as force dipoles. The resulting elastic interactions are long ranged and favor the formation of strings. We show analytically that due to screening, the effective interaction between strings decays exponentially, with a decay length determined only by geometry. Both for disordered and ordered arrangements of cells, we predict novel phase transitions from paraelastic to ferroelastic and antiferroelastic phases as a function of the Poisson ratio.