Electrostatic Potentials and Their Spatial Derivatives About Point Defects in Ionic Crystals.

Methods which have been developed and employed for the calculation of point defects in ionic crystals are described. These methods involve a detailed Hartree-Fock-like treatment by way of angular-momentum-dependent exchange potentials of the ions within an inner region ${\ensuremath{\tau}}_{1}$ centered about the defect. In the outer region ${\ensuremath{\tau}}_{2}$, composed of the rest of the crystal, the ions are treated in two different approximations, the most successful of which involves an effective-mass formalism. Polarization effects are included by expressions of the form resulting from the work of Toyozawa, Haken, and Schottky. Although the main purpose of the paper is to describe the methods, applications are made to the calculation of the electronic energy levels in the optical absorption of the $F$ center in KCl and NaCl. Good agreement with experiment is found only if polarization effects, which are important even in the ground state, are included. Additional applications will be given in subsequent papers.

The energy, the elastic dipole tensor and the entropy of point defects in ionic crystals are usually calculated by the Mott-Littleton approach, which treats a single defect in an infinite crystal. The authors suggest that there may be advantages, particularly for the dipole tensor and the entropy, in performing the calculations in periodically repeating geometry. They examine how this 'supercell' method can be used for the energy and the dipole tensor, paying attention to the problem that for charged defects the repeating unit carries a net charge, so that the Coulomb energy is divergent. They test the supercell method on a number of both charged and uncharged defects, and show that the results for the energy and dipole tensor are in close agreement with those obtained by the conventional approach.

Any deviation from a perfect periodic lattice or structure in a crystal is an imperfection. There are three main types of imperfections, point, line and plane imperfections1, of which we shall deal only with the point defects in this chapter. The point imperfections encountered in ionic solids are: (i) chemical impurities (ii) vacant lattice sites and (iii) interstitial ions. Real crystals are always imperfect in some respect. In thermal equilibrium in an otherwise perfect crystal a certain number of lattice vacancies or interstitials are always present, because the entropy is increased by the presence of disorder in the structure and at a finite temperature, the equilibrium condition of a crystal happens to be a state of minimum free energy.

The authors present and describe a new embedded-cluster approach for calculating both the electronic and the spatial structure of point defects in ionic crystals with self-consistent incorporation of the polarisation effects into the eigenvalue spectrum and the total energy of a crystal. For the first time both the electronic structure and the ionic displacements within a large cluster containing a defect are obtained in the framework of the same calculating scheme. It permits one to describe correctly back-coupling between redistribution of the electron density in a defective region and polarisation of the rest of the crystal outside the cluster, which can play a vital role for the study of small polaron effects. Making use of a 45-atom cluster with 175 electrons and the semi-empirical INDO method, the authors have simulated the impurity-induced trapping of a hole in the (Li)0 centre in an MgO crystal. The calculations do not use a priori assumptions and do confirm that one-centre localisation of a hole in the ground state is energetically favourable. They show that polarisation energies can differ considerably for the ground and charge-transfer excited states of the defect. It is also demonstrated that the generally accepted model of the O- hole polaron is oversimplified: a considerable charge redistribution between Li+ and another O- along the (100) direction takes place.

Various mechanisms for the formation of point defects in ionic crystals are considered. Their probable importance as a function of the energy of the ionizing radiation and the temperature at which the irradiation takes place is discussed. Complex centers formed from aggregates of simple point defects are reviewed and their occurrence is considered in terms of the expected spatial distribution of the point defects. The effect of high-energy X-rays and protons on the ionic conductivity is dis- cussed in terms of a new interpretation of the conductivity curve as a function of temperature, and it is suggested that radiation-enhanced diffusion may be important in explaining the experimental data.

The subsurface electric potential in oriented single crystals of silver chloride has been directly mapped out by a technique based on the equilibrium distribution of a very small concentration of a charged radiotracer. The potential was found to vary with depth according to the Gouy-Chapman solution of Poisson's equation, with screening lengths in the range 10\char21{}20 nm. The surfaces were all negative, relative to the deep interior, by 0.10\char21{}0.30 V, depending on temperature and surface orientation. The present results on the subsurface potential difference can be employed to resolve the Frenkel-pair formation enthalpy and entropy, as determined from transport measurements, into the formation enthalpies and entropies of the individual point defects comprising the pair. For the silver chloride (110) surface, for example, ${G}_{v}$=0.74 eV and ${G}_{i}$=0.37 eV at 150 \ifmmode^\circ\else\textdegree\fi{}C; the best values obtained to date for ${H}_{v}$, ${H}_{i}$, ${S}_{v}$, and ${S}_{i}$, are 0.77 eV, 0.70 eV, 0.79k, and 9.0k, respectively.

Investigation of the electronic structure of point defects in ionic crystals by the cluster scattered wave method with the self-consistent calculation of the lattice distortion and long-range polarization

The Stark effect of localized electronic centres is reviewed by using a few selected examples of impurity centres or simple defects. Important information on the symmetry of centres and on the position of energy levels is obtained through the electric-field-induced absorption changes. Typical results are summarized for the Stark effect on the zerophonon lines and on the broad absorption band of point defects in ionic crystals and of impurities in semiconductors.

Excitation of Plasmons and Interband Transitions by Electrons. By H. Raether. (Berlin, Heidelberg, New York: Springer-Verlag, 1980.) [Pp. 192+subject index.] Price DM 59, – –/US $32.50 hardback. Physics of Superionic Conductors. Edited by M. B. Salamon. (Berlin, Heidelberg, New York: Springer-Verlag, 1979.) [Pp. xii+248+subject index.] Price DM 59, – –/US $32.50 hardback. Electronic and Vibrational Properties of Point Defects in Ionic Crystals. By Y. Farces and M. P. Fontana. (Amsterdam: North Holland, 1979.) [Pp. xix+247+author and subject index.] Price Dfl. 100.00/US $48.75 hardback. Defects and Diffusion in Solids. An Introduction. By S. Mrowec. (Amsterdam, Oxford, New York: Elsevier, 1980.) [Pp. 464+subject index.] Price Dfl. 170.00/US $83.00 hardback. Fluctuation Phenomena. Edited by E. W. Montroll and J. L. Lebowitz. (Amsterdam: North-Holland, 1979.) [Pp. 340+author and subject index.] Price Dfl. 120.00/US $58.50 hardback. Crystal Growth: A Tutorial Approach. Edited by W. Bardsley, D. T. J. Hurle and J. B. Mullin. (Amsterdam: North-Holland, 1979.) [Pp. vi+401+subject index.] Price Dfl. 110.00/US $53.75 hardback. Liquid Crystals. By S. Chandrasekhar. (Cambridge University Press, 1980.) [Pp. x+336+subject index.] Price £8.95 paperback. Light Scattering in Solids. Edited by J. L. Birman, H. Z. Cummins and K. K. Rebane. (New York: Plenum Press, 1979.) [Pp. xxi+523+author and subject index.] Price US $55.00 hardback (camera-ready typescript). Electrostatics 1979. Edited by J. Lowell. (London, Bristol: The Institute of Physics, 1979.) [Pp. 349+author index.] Price £32.00 hardback. Auger Electron Spectroscopy Reference Manual. By G. E. Mcguire. (New York: Plenum Press, 1979.) [Pp. ix+129.] Price US $39.50 paperback. Laser Handbook, Volume 3. Edited by M. L. Stitch. (Amsterdam: North-Holland, 1979.) [Pp. ix+846+author and subject index.] Price Dfl. 250.00/ US $122.00 hardback.

A mechanism of interaction of dislocations with point defects is suggestedon the basis of a study of deformation-stimulated luminescence (DL)from irradiated KCl. DL is considered to be a result of destruction ofhole colour centres by dislocations. The centres destroyed by dislocationsare responsible for irradiation-induced strengthening of alkali halides.

Methods described in a previous paper are applied to the calculation of the electronic structure of the $F$ center in KCl, KBr, KI, and NaCl. Structure of the ions is explicitly taken into account in a region close to the defect site, and an effective-mass method is used in the more distant regions. Dielectric polarization effects are included on the basis of the formulation due to Toyozawa, Haken, and Schottky, but results in best agreement with experiment are obtained by a procedure which deviates somewhat from that formulation, as might be expected also on purely theoretical grounds. It does not appear to be possible to obtain agreement with experiment even for absorption energies if polarization effects are neglected. The calculated energy differences of the $1s$, $2p$, $3p$, and $2s$ states in absorption agree very well with presently available experimental data. Agreement with experiment is fairly good for the $1s$, $2s$, and $2p$ energies in emission. The calculated optical transition matrix elements are very sensitive to the details of the model; those obtained from our best model and wave functions are compatible with experimental results. The results for the spin density at the nearest-neighbor ions are reasonably close to the experimental values.

ABSTRACTWe propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.

Valence stability of aliovalent ions is mostly correlated with lattice site potential in ionic crystals. Madelung electrostatic potential is obtained by adding all the lattice site potentials for all the ions present in a crystal structure. Therefore, valence stability and the stability of a crystal structure can be better understood with consideration of both the lattice site potential and Madelung electrostatic potential. This was first demonstrated more than four decades ago by one of the present authors. We revisit this situation by using re-calculated lattice site potential and Madelung electrostatic potential for perovskite structure type ABO3 compounds using a new computer program VESTA. We show that the formation of a perovskite structure type compound with the general formula ABO3 (where A and B are cations and O is an oxide ion) becomes energetically favorable when it has a higher Madelung electrostatic potential than the combined Madelung electrostatic potential of parent binary compounds AO and B2O3 or BO2. It is further shown that strong lattice site potential results in stability of high valence or high valence ions can be stabilized in a lattice site with strong lattice-site potential. It further follows that certain ions experience maximum lattice site potential at the B ion lattice site of the perovskite structure when compared to other structures such as fluorite BO2, rutile BO2 and corundum B2O3. Therefore, (i) the stability of an ion with a high (and uncommon) valence state at the B site being higher than that at the A site, (ii) occurrence of point defects at A or O sites with weak lattice site potentials, respectively and (iii) instability of perovskite A4+B2+O3, and A5+B1+O3 compounds, respectively can be rationalized by lattice site potential and Madelung electrostatic potential analysis.

The paper briefly reviews the development of the atomistic method of calculating the properties of defects in ionic crystals, as it derives from the early work of Mott and Littleton (1938). The physical principles and assumptions of the method are described, although the main emphasis is on the development of the technique. It is pointed out that appeal to the quasi-harmonic approximation has made it possible to extend the technique so that the characteristics of defects can now be successfully obtained as functions of temperature. The insights into atomic transport phenomena which these calculations provide are illustrated by three examples.

We propose an analytic approach for calculating the electrostatic energy of proteins or protein complexes in aqueous solution. This method, termed CVCEL (Circular Variance Continuum ELectrostatics), is fitted to Poisson calculations and is able to reproduce the corresponding energies for different choices of solute dielectric constant. CVCEL thus treats both solute charge interactions and charge self-energies, and it can also deal with salt solutions. Electrostatic damping notably depends on the degree of solvent exposure of the charges, quantified here in terms of circular variance, a measure that reflects the vectorial distribution of the neighbors around a given center. CVCEL energies can be calculated rapidly and have simple analytical derivatives. This approach avoids the need for calculating effective atomic volumes or Born radii. After describing how the method was developed, we present test results for coarse-grain proteins of different shapes and sizes, using different internal dielectric constants and different salt concentrations and also compare the results with those from simple distance-dependent models. We also show that the CVCEL approach can be used successfully to calculate the changes in electrostatic energy associated with changes in protein conformation or with protein-protein binding.