Cartan connection and defects in Bravais lattices

The geometrical approach to a field theory of defects in crystalline solids including both dislocations and intrinsic point defects is developed in the framework of Cartan affine differential geometry. A clear distinction between linear and affine connection is made. The discussion is restricted to the basic kinematic level.     

Melting near line and point defects

Power-law exponents are derived for the anomalous thermal properties of solids in which crystal defects act as the centres of nucleation and growth for the transition to the molten or liquid crystalline state. The predictions are borne out by piezothermal or calorimetric data on n-hexadecane, p-azoxyanisole, bismuth and selenium.     

Anelasticity and stress-induced ordering of point defects in crystals

Point defects may exist in a crystal on a set of crystallographically equivalent orientations or sites. The response of a crystal containing point defects to an externally applied stress takes the form of ‘stress-induced ordering’, or preferential alignment of the defects. Recent experiments have used optical and spin resonance techniques to observe this stress-induced ordering directly. This same process also gives rise to anelasticity, e.g. to an internal friction peak. In describing such behaviour, the concept of a point defect as an ‘elastic dipole’, introduced by Kröner and others, is very useful. The elastic dipole is here defined as a second-rank tensor, the ‘λ-tensor’, equal to the average strain per mole fraction of defects all aligned in a particular orientation. This definition is then used to develop a thermodynamic theory of stress-induced ordering, which includes the interaction among defects and the possibility of a ‘reaction’ or interconversion between two species of defects. Applications of the theory are made to defects of various point symmetries in cubic crystals, and a number of specific examples are discussed to illustrate each type of defect symmetry. The usefulness of the theory is that it enables one to calculate the values of the components of the λ-tensor from experimental information and the defect symmetry. Typical values of these λ-parameters range from 10^-2 unity.     

Universal symmetry property of point defects in crystals

Point defects (vacancies, solute atoms, and disorder) are ubiquitous in crystalline solids. With in situ transmission electron microscopy we find clear evidence for the existence of a universal symmetry property of point defects; i.e., the symmetry of short-range order of point defects follows the crystal symmetry when in equilibrium. We further show that this symmetry-conforming property can lead to various interesting effects including "aging-induced microstructure memory" and the associated "time-dependent two-way shape memory."     

Elastic interaction of point defects in cubic and hexagonal crystals

The elastic interaction of two point defects in cubic and hexagonal structures has been considered. On the basis of the exact expression for the tensor Green’s function of the elastic field obtained by the Lifschitz–Rozentsveig for a hexagonal medium, an exact formula for the interaction energy of two point defects has been obtained. The solution is represented as a function of the angle of their relative position on the example of semiconductors such as III-nitrides and α-SiC. For the cubic medium, the solution is found on the basis of the Lifschitz–Rozentsveig Green’s tensors corrected by Ostapchuk, in the weak-anisotropy approximation. It is proven that the calculation of the interaction energy by the original Lifschitz–Rozentsveig Green’s tensor leads to the opposite sign of the energy. On the example of the silicon crystal, the approximate solution is compared with the numerical solution, which is represented as an approximation by a series of spherical harmonics. The range of applicability of the continual approach is estimated by the quantum mechanical calculation of the lattice Green’s function.     

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.     

Luminescence and point defects in cerium-doped lutetium scandium orthoborate crystals

Cerium-doped lutetium scandium orthoborate single crystals as potential scintillation materials were grown by the floating zone (FZ) method for the first time and the Czochralski (Cz) grown crystal was used for comparison. In this paper, the representative composition Lu_0.5Sc_0.5BO₃:0.5at%Ce (abbreviated as LSBO:Ce) was selected as the research target. The phase structure of FZ-grown LSBO:Ce crystal was characterized by X-ray diffraction and its optical properties were investigated using photoluminescence emission (PL), excitation (PLE), radioluminescence (RL) spectra. The defect properties of LSBO:Ce crystals were studied by thermoluminescence (TL) as a function of temperature (300–600 K). The glow curve of FZ-grown LSBO:Ce crystal shows three TSL peaks, at 350, 400, 552 K, corresponding to a trap depth of 0.99, 1.14 and 1.50 eV, respectively. Based on UV-irradiation TL measurements and air-annealing experiments, the former two kinds of traps could be ascribed to the radiation-induced defects and the third kind originated from the oxygen vacancies formed during crystal growth. These defects also existed in the Cz-grown LSBO:Ce crystal.     

A Debye model for point defects in aluminosilicate crystals

Summary In aluminosilicate lattices, exchanges between silicon and aluminum ions are expected to originate charged point defects of opposite signs. Thermodynamic arguments, based on a model analogous to the Debye electrolyte model, allow evaluation of the equilibrium density of defects as a function of temperature. The electric field of point defects is found to be screened by charge distribution even at distances small enough to affect the ground-state energy of electrons trapped by the defect positive charges. Calculations show that large fluctuations of trapped electron energy are expected. These results agree with data from thermoluminescence experiments on different aluminosilicates, which showed broad distributions of electron traps.     

Equilibrium configurations and energetics of point defects in two-dimensional colloidal crystals

We demonstrate a novel method of introducing point defects (mono- and divacancies) in a confined monolayer colloidal crystal by manipulating individual particles with optical tweezers. Digital video microscopy is used to study defect dynamics in real space and time. We verify the numerical predictions that the stable configurations of the defects have reduced symmetry compared to the triangular lattice and discover that in addition they are characterized by distinct topological arrangements of the particles in the defect core. Surprisingly, point defects are thermally excited into separated dislocations, from which we extract the dislocation pair potential.     

Interactions of point defects with dislocations in silver chloride crystals

Internal friction techniques at 35 kHz were used to study the interactions between point defects and dislocations in crystals of silver chloride. The Granato-Lücke plots of the amplitude-dependent decrement were not linear, but were concave upward. Various analyses of these data all gave pin-dislocation binding energies of approximately 0·2 eV. An extensive study was made of the decay of the amplitude-independent decrement after “excitation” by high-amplitude oscillation; the Granato-Lücke L⁴ law was found valid, but the arrival of pins proceeded as t^1/3, rather than the usual t^2/3 found in deformed specimens. A first-order model of the response of a slightly perturbed Maxwell atmosphere about the dislocation is presented. The model quantitatively describes the experimental results. The activation energy of the rate constant was found to be 0·42 eV, consistent with data on solute diffusion and strain-aging.     

Elastic displacement field of point defects in anisotropic cubic crystals

The elastic displacement field of point defects in cubic crystals is calculated for weak anisotropy by second order perturbation theory and by a variational procedure. The results are compared with numerical calculations for Cu. Further analytical approximations are given for the volume change in an infinite crystal and for the interaction energy of two point defects.     

Electron paramagnetic resonance study of impurities and point defects in oxide crystals

In this topic review the results of the X-band electron paramagnetic resonance (EPR) measurements of Mn, Co, Cr, Fe ions in YAlO₃ (YAP) crystals and Fe ions in LiNbO₃ (LNO) crystals and of chromium doped Bi₁₂GeO₂₀ (BGO) and Ca₄GdO(BO₃)₃ single crystals, are presented. It is well known that the oxide crystals (for example:YAP, LNO, BGO) are one of the most widely used host materials for different optoelectronic applications. The nature of point defect of impurities and produced in the oxide crystal after irradiation by bismuth ions and after irradiation by the ²³⁵U ions with energy 9.47 MeV/u and fluency 5?×?10¹¹?cm^?1 is discussed. The latter is important for applications of these oxide crystal as laser materials.     

Screening of the electric field in covalent crystals containing point defects

An expression is derived within the framework of the Debye-Hückel approximation for the screening length of a field in a p-type semiconductor taking into account the energy spread of immobile acceptor levels and the density of states tail of the valence band. It is shown that the screening length depends additively on the product of the carrier density and their diffusion coefficient ratio (for free holes and holes hopping via acceptors).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 41–43, November, 1984.     

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.     

Formation of elementary radiation defects in halide crystals with different types of crystal lattice

Pulsed spectrometry with time resolution has been used to study processes for creation of self-localized excitons and F centers under the action of a pulsed electron flux of nanosecond duration at temperatures in the region from 80 to 600 K in CaF₂ and LiF crystals. It is shown that an alternative to creation of self-localized excitons in the triplet state with increased crystal temperature during irradiation is formation of an F-H pair. A comparative analysis of the effectiveness of the processes for creation of elementary defects in the crystals studied is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 13–19, January, 1995.     

Localized optical modes induced by point and extended defects in ionic crystals

It is shown that the vector character of the relative displacement of atoms in ionic crystals causes a nonexponential (power) law decay of localized optical mode amplitudes at large distances from point or linear defects.     

Generation and accumulation of point defects in FCC single crystals upon plastic strain

A model of point defect generation and accumulation in a FCC crystal upon strain with a constant rate at room temperature is suggested in the present work. The model results are in agreement with the available experimental data. The influence of point defects on the formation of fragmented substructures is analyzed. The model is verified using experimental data on the stress‒strain curve, dislocation density, vacancy concentration, and misorientation angle.     

The distribution of the valence electron density in predominantly ionic crystals with different Bravais sublattices

Based on the theory of the local-density functional, self-consistent valence electron densities are calculated for CaO with a rock-salt lattice, CaF₂ with a fluorite lattice, K₂O with an antifluorite lattice, and for their constituent sublattices. It is shown that in the crystals with different Bravais sublattices, the anionic sublattice is a framework with covalent bonds containing a metal sublattice inside of them. The coupling between the sublattices is characterized by the density difference, which is deflned as the difference between the total electron density and the densities of the individual sublattices. The density difference is found to be an order of magnitude smaller than the crystal and sublattice densities, which is evidence of weak hybridization of the sublattices and of the predominately ionic character of the bonding between them.