Deformation Dipole Research Articles

The displacements and polarization caused by point defects in ionic crystals

Abstract The lattice statics method for the calculation of point defect properties, first proposed by Kanzaki, is here extended to discuss the polarization and displacements caused by charged point defects in ionic crystals. The method allows the determination of the exact displacement and polarization fields for a defect in a given crystal model. Detailed discussion is given for the NaCl structure using the deformation dipole model. It is shown that the long-range displacement field can be rigorously separated into pure strain and pure polarization components, the last being strictly isotropic. For a model without deformation dipoles (Szigeti effective charge e∗=e) the polarization components are exactly as assumed in the method of Mott and Littleton. Different expressions however apply when e∗≠e, as is generally the case. The elastic strengths of the cation vacancies are calculated to be much larger than those of the anion vacancies; this is also an effect of the deformation dipoles, especially of their...

Lattice Response Functions of Imperfect Crystals: Effects Due to a Local Change of Mass and Short-Range Interaction

Lattice response functions, such as the thermal conductivity and dielectric susceptibility of an imperfect crystal with rocksalt structure, are evaluated in terms of the irreducible $T$ matrix accounting for the phonon scattering. It is shown that the effect of defects on thermal conductivity and dielectric susceptibility can be accounted for by expressions which have essentially the same structure. The $T$ matrix for a defect which affects both the mass and the short-range interaction is analyzed according to the irreducible representations of the point group which pertains to the perturbation, and the resonance conditions for ${\ensuremath{\Gamma}}_{1}$, ${\ensuremath{\Gamma}}_{12}$, and ${\ensuremath{\Gamma}}_{15}$ irreducible representations are considered in detail for any positive impurity in KBr crystals. Hardy's deformation-dipole (DD) model is employed for the description of the host-lattice dynamics. A comparison is made with simplified models, such as diatomic linear chains with nearest-neighbor interaction; it is shown that in polar crystals an effective-force constant has to be used in order to give a reliable description of the short-range interaction between the impurity and the host lattice. An attempt is made to define such effective force constants in the framework of the DD model. The numerical calculations concern positive monovalent impurities in KBr crystals. ${\ensuremath{\Gamma}}_{1}$, ${\ensuremath{\Gamma}}_{12}$, and ${\ensuremath{\Gamma}}_{15}$ resonance frequencies are evaluated as a function of the change of mass and nearest-neighbor force constant. For KBr:${\mathrm{Li}}^{+}$ and KBr:${\mathrm{Ag}}^{+}$ we also evaluate the band shape of the absorption spectrum at infrared frequencies; good agreement is found between the theoretical prediction and the experimental data on KBr:${\mathrm{Li}}^{+}$. It is shown that some structures actually observed in the spectrum are due to peaks in the projected density of states of the host lattice, and have nothing to do with resonance scattering. Good agreement is found between the impurity-host-lattice interaction as estimated from a priori calculations and as deduced by fitting the ${\ensuremath{\Gamma}}_{15}$ resonance frequency to the experimental data. A simple explanation of the off-center position of small ions is also suggested. Finally, concentration and stress effects on the absorption coefficient are briefly discussed.

Temperature Dependence of the Width of the Fundamental Lattice-Vibration Absorption Peak in Ionic Crystals. II. Approximate Numerical Results

The results of numerical calculations of the temperature dependence of the width and position of the fundamental lattice-vibration absorption peak in NaCl and LiF are presented. The calculations are carried out in the high-temperature limit, on the basis of the Hardy-Karo deformation dipole model of these crystals. Cubic and quartic anharmonic terms are retained in the crystal Hamiltonian, but the approximation of neglecting the anharmonicity of the Coulomb forces has been made. The expressions for the Fourier-transformed anharmonic force constants have been approximated and simplified by a method suggested by Peierls. The results of the calculations show that the quartic anharmonic terms in the crystal potential energy make a contribution to the width of the fundamental absorption peak which is comparable in magnitude with the contribution from the cubic anharmonic terms, in agreement with the theoretical arguments of Gurevich and Ipatova. Since the quartic anharmonic contribution to the width is proportional to the square of the absolute temperature at high temperatures, these results provide an explanation for the experimental observations that the width increases with a power of the absolute temperature which is intermediate between the first and the second. Quantitatively, the theoretical results are in quite good agreement with the experimental data of Heilmann for the variation with temperature of the width of the fundamental absorption peak in LiF. In the case of NaCl, the agreement between theory and experiment is somewhat poorer, but the theoretical values are still within a factor of about 2 of the experimental values of Hass. The frequency dependence of the imaginary part of the dielectric constant of these two crystals has also been calculated, and is compared with experimental data.

Lattice dynamics and thermal properties of LiH and Lid crystals

Abstract The vibrational spectra and the dispersion curves of LiH and LiD crystals are calculated in the framework of the Hardy deformation dipole model. Some anomalous features of the acoustic dispersion curves, which imply low lattice stability along some directions, are found for both crystals and discussed. The Debye θ's and moment functions are also evaluated.

Determination of the Effective Force Constants between a Substitutional Impurity and Its Nearest Neighbors in an Alkali Halide Crystal

The forces between a substitutional impurity and its nearest neighbors in an alkali halide crystal with rocksalt structure are determined in the harmonic approximation. The deformation dipole model is used for the lattice dynamics of the perfect crystal. Two independent force constants are involved in the problem which is treated in two ways. First, the two force constants are obtained from the equation of motion for the local mode of vibration using experimental values of the local-mode frequencies due to ${\mathrm{H}}^{\ensuremath{-}}$ and ${\mathrm{D}}^{\ensuremath{-}}$ impurities in NaCl and KCl, and assuming that the force constants are the same for different isotopes of the impurity. This method yields no real solutions for the force constants for $U$ centers in KCl. The second method assumes that the force constants for the impurity (say ${\mathrm{H}}^{\ensuremath{-}}$) are the same as those in an alkali hydride crystal having the same lattice spacing as the alkali halide crystal. A Born-Mayer interatomic potential is used for this calculation. The first method gives 45% softening of the force constants for $U$ centers in NaCl; the second method gives 70% softening. The results are compared with those of previous theories and with experimental results on the KBr: ${\mathrm{Li}}^{+}$ system.

Second order dipole absorption in ionic crystals

Abstract A relation is derived for the order of magnitude of the second-order dipole absorption in alkali halides. The relation shows that the second-order dipole has a considerable importance on the magnitude of the infrared absorption in the side-bands. The derivation of the relation is based essentially on two assumptions, both of which are fairly generally accepted for the alkali halides. These are: (i) the electron deformation is governed mainly by overlap repulsion and the internal electric field; (ii) the static charge is approximately equal to the elementary charge, so that the first-order deformation dipole can be obtained from the measured value of the effective charge. Our conclusion concerning the magnitude of the second-order dipole absorption is not contradicted by the fact that the two-phonon absorption in diamond or silicon, which is due to the second-order dipole only, is much weaker than the side-band absorption in ionic crystals. Various considerations indicate that the second-order dipole absorption can be of very different magnitude in different types of solid.