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Extended Deformable Shell Model Calculation of the Lattice Statics and Dynamics of LiH and LiD Crystals
Abstract
An attempt is made for a unified study of the lattice statics and dynamics of the LiH–LiD crystals. So far it has not been possible to obtain a unified description of the different properties of this crystal on the basis of a single model and with a single set of model parameters. All the previous calculations suffer from this inadequacy and their results show that there are certain problems regarding the stability of the static lattice structure, the cohesive energy, and a consistent description of the dielectric properties and the dispersion of phonons for this crystal. The present calculation based on the extended deformable shell model removes the earlier difficulties and provides a good overall description of the lattice mechanical properties of this crystal with a single set of parameters. Another salient feature of the present investigation is to indicate the effect of the quadrupolar distortion of the charge cloud, in particular, of the hydrogen ion which has been neglected in all the earlier investigations of the lattice mechanics of these crystals. The calculation clearly points out that without including this effect it is not possible to obtain a coherent description of the lattice mechanics in all its totality. Some qualitative justification for this effect being significant for the crystals under consideration is also presented.
It seems likely that isotope effects are most clearly manifested in crystal lattice dynamics, which is evidenced by works in this field that have been published for more than half a century. A great number of stable isotopes and well-developed methods of their separation has made it possible to date to grow crystals of C, LiH, ZnO, ZnSe, CuCl, GaN, GaAs, CdS, Cu2O, Si, Ge and α-Sn with a controllable isotopic composition. The accumulated voluminous theoretical and experimental data suggest that the isotopic composition of a crystal lattice exerts some influence on the thermal, elastic, and vibrational properties of crystals. These effects are quite large and can be readily measured by modern experimental techniques (ultrasound, Brillouin and Raman scattering, and neutron scattering). For example, the change in the lattice constant is Δa/a=10−3 to 10−4, while the change δcik in the elastic constants amounts to several percent. The maximum km (where km is the maximum of thermal conductivity) measured for the most highly enriched (99.99%) sample is 10.5 kW/mK, one order of magnitude higher than for the natural Ge (analogous for C and Si). In addition, crystals of different isotopic compositions possess different Debye temperatures. This difference between a LiH crystal and its deuteride exceeds hundred degrees. Of the same order of magnitude is the difference between Debye temperatures for diamond crystals. Very pronounced and general effects of isotopic substitution are observed in phonon spectra. The scattering lines in isotopically mixed crystals are not only shifted (the shift of LO lines exceeds 100 cm−1) but are also broadened. This broadening is related to the isotopic disorder of a crystal lattice. It is shown in this review that the degree of change in the scattering potential is different for different isotopic mixed crystals. In the case of germanium and diamond crystals, phonon scattering is weak, which allows one to successfully apply the coherent potential approximation (CPA) for describing shift and broadening of scattering lines. In the case of lithium hydride, the change in the scattering potential is so strong that it results in phonon localization, which is directly observed in experiments. Capture the thermal neutrons by isotope nuclei followed by nuclei decay produces new elements in a very large number of possibilities for isotope selective doping of different materials. The review closes with a section describing future developments and applications of isotope technology and engineering.
An adiabatic formulation of the dipolar charge deformation has been found to be quite important in solids. The present formulation is quite general and may be applied to cases where in addition to the dipole deformation the quadrupole is also important. However, as a preliminary application of the formulation we consider homopolar crystals where the symmetry indicates that the dominant multipole deformation of the charge cloud is quadrupolar, dipolar deformation being totally absent. The results obtained give us an idea about the order of magnitude of this effect.
The antiferromagnetic oxide crystals behave rather unusually in respect to many physical properties, and the various attempts to understand them in terms of models which satisfactorily describe the simple ionic solids show conspicuous discrepancies between theory and experiment. One of the major difficulties common to all the existing calculations is that it is not possible to give a consistent description of the elastic properties and the dispersion of phonons of such a crystal in the antiferromagnetic phase within the framework of a single model. Moreover, so far, the magnetic and the mechanical properties of these crystals have been studied, one independent of the other. Apart from this difficulty, a simultaneous description of the dielectric properties and the phonon dispersion with the existing models also presents several problems. In a previous work we have suggested a model which is based on a microscopic analysis of the energy expression of an assembly of ions. Assuming a specific form of the superexchange interaction that is suggested by various experimental and theoretical investigations and incorporating the same in the above model, we have been able to reduce significantly the discrepancies mentioned above for the case of NiO crystal. In addition to this, it has also been possible to present a unified description of the magnetic and the mechanical properties of this crystal with a single model. The specific properties that we have attempted to correlate with a single set of model parameters are: the cohesive and the stability properties, the elastic and the dielectric properties, the dispersion relation of phonons and magnons, and the sublattice magnetization properties. We also indicate the rough order of magnitude of the different interactions implied in the model and the relations that they bear to the unusual features of this solid. Some of the limitations of the present model and the directions of further refinement are also discussed.
The copper halide crystals, namely CuI, CuBr, and CuCl, though they have predominantly ionic binding, possess zinc-blende structure, and crystals having this structure have, in general, rather significantly low ionicity. This peculiarity is attendant with a number of unusual properties of this group of solids, and the various attempts to understand them in terms of the existing models show certain problems. Recently we have developed a general energy expression for an assembly of ions occupying an arbitrary configuration from a microscopic analysis of the problem. Inspection of the energy expression shows that within the S2 approximation (up to and including second-order exchange) the following distortions are allowed: the two types of short-range dipolar deformations apart from the long-range dipolar one, scalar, and quadrupolar deformations. A model based on this energy expression has been employed to make a unified study of the different lattice-mechanical properties of the three copper halides. The specific properties that we have tried to correlate are the cohesive energy, the elastic and the dielectric properties, and the dispersion of phonons with the same set of parameters for each crystal. Finally, the results for the different crystals and the possible sources of the remaining discrepancies are discussed.
The large number of available stable isotopes and well developed isotope separation technology have enabled growing crystals of C, LiH, ZnO, CuCl, CuBr, Cu2O, CdS, α-Sn, Ge, Si, etc. with a controlled isotope composition. Experimental and theoretical studies provide evidence that the isotope effect has an influence on the thermal, elastic, and vibrational properties of crystals. In this paper it is shown that in Ge and C crystals isotope effect causes only weak phonon scattering whereas in LiH the scattering potential changes are so strong that they lead to experimentally observable phonon localization. It is emphasized that a systematic description of isotope effects requires that anharmonicity be taken into account.
Theoretical and experimental studies were made of the lattice dynamics ; of alkali halides. A theory of the lattice dynamics of ionic crystals is given ; based on replacement of a polarizable ion by a mcdel in which a rigid shell of ; electrons (taken to have zero mass) can move with respect to the massive ionic ; core. The dipolar approximation then makes the model exactly equivalent to a ; Born-von Karman crystal in which there are two "atoms" of differing charge at ; each lattice point, one of the "atoms" having zero mass. The model was ; specialized to the case of an alkali halide in which only one atom is ; polarizable, and computations of dispersion curves were carried out for Nal. The ; dispersion nu (q) relation of the lattice vibrations in the symmetric STA001!, ; STA110!, and STA111! directions of Nal at 110 deg K were determined by the ; methods of neutron spectrometry. The transverse acoustic, longitudinal ; acoustic, and transverse optic branches were determined completely with a ; probable error of about 3%. -ine dispersion relation for the longitudinal optic ; (LO) branch was determined for the STA001! directions with less accuracy. The ; agreement between the experimental results and the calculations based on the ; shell model, while not complete, is quite satisfactory. The neutron groups ; corresponding to phonons of the LO branch were anomalously energy broadened, ; especially for phonons of long wavelength, suggesting a remarkably short lifetime ; for the phonons of this branch. (auth);
The method of semi-localized crystalline orbital (SLCO) is presented to investigate the influence of homopolar binding on the cohesive energies of ionic crystals. The computations are performed for LiH crystal. As a result, it is shown that the discrepancy between the theoretical value of the cohesive energy of LiH crystal, which was calculated by Lundqvist, and the observed one is removed by taking homopolar binding into account.
Furthermore, the relation between the energy band structure and SLCO is discussed and it is shown that SLCO corresponds to Wannier's functions of the valence and conduction bands for the case of LiH crystal where the valence and conduction bands are non-degenerate.
A microscopic theory of lattice dynamics of ionic crystals is developed in terms of non-orthogonal highly localized wave functions. It is shown that this approach is in close correspondence to some shell models and leads to a microscopic understanding of the model parameters. The phonon dispersion of LiD has been calculated without using any adjustable parameters and it agrees with experiment to within 10% on the average.
An attempt is made to incorporate the effect of the quadrupolar distortion of the charge cloud in the framework of the deformable shell model. All the relevant equations and lattice sums are evaluated for the CsCl structure and a preliminary application of the complete model which takes into account simultaneously the scalar, the dipolar and the quadrupolar deformation considered for a CsCl structure crystal. So far there is no estimate of the effect of the quadrupolar distortion on the different properties of a crystal belonging to CsCl structure. The calculation indicates that the effect is quite considerable for certain properties and its inclusion improves the overall agreement with experiment.
Recently one of us (S.S.J.) showed that the deformation dipole model and the shell model are two special forms of a more general dipole model of lattice dynamics that follows rigorously from Zeyher's microscopic theory. Here we use a phenomenological approach to generalize the new model into a deformable-ion model (DIM) which is also consistent with Zeyher's microscopic theory. The DIM contains the long-range Coulomb interactions in the dipole approximation and it allows the rigid-ion short-range interactions to be modified by self-energy corrections due to symmetry-allowed deformations of the ions. We study the room-temperature phonon dispersion curves of 7LiD to compare different forms of the DIM and in the process arrive at a simple form of DIM which is in very good overall agreement with dielectric, elastic, and neutron scattering data.
In previous papers by Misra(1) and by Born and Misra(2) lattice sums of the type required in discussing the stability of a cubic crystal of the Bravais type in which the forces are central have been calculated. In the investigation of the thermodynamic properties of crystals a more general type of lattice sum occurs, which involves the phases of the waves. In the present paper a method of calculating these sums is developed and tables are computed.(Received November 14 1942)
The model for the lattice dynamics of alkali halides as given in a previous paper is generalized to include the case in which both ions are polarizable. Dispersion curves and density of states are calculated for LiF and NaCl at 0 °K. The Debye characteristic temperatures ΘD(T) for 0 °K and 300 °K are computed for these crystals, and good agreement with experimental values is obtained.In einer vorhergehenden Arbeit wurde ein neues Modell für die Gitterdynamik der Alkalihalogenide vorgeschlagen. Dieses Modell wird verallgemeinert für den Fall, daß beide Ionen polarisierbar sind. Anschließend werden die Dispersionskurven und Zustandsdichten für NaCl und LiF bei 0 °K angegeben. Die ebenfalls berechneten Debye-Temperaturen ΘD(T) dieser Kristalle für 0 und 300 °K stimmen gut mit experimentellen Ergebnissen überein.
The lattice vibrations of the silver halides AgCl and AgBr have been investigated using a shell model with deformable ions. The deformability of the d-shell of the Ag+-ion and its covalent coupling to the neighbours have been considered by introducing two new degrees of freedom in the electronic shell of the Ag+-ion. They are of quadrupolar (Γ12+) and of rotational (Γ15+) symmetry character. The calculations have been confirmed by comparison with the unpolarized Raman spectra of second order.
Die Gitterschwingungen der Silberhalogenide AgCl und AgBr wurden mit Hilfe eines erweiterten Schalenmodells mit deformierbaren Ionen untersucht. Die Deformierbarkeit der d-Schale des Ag+-Ions sowie die kovalente Bindung dieses Ions an seine nächsten Nachbarn wurden durch Einführen zweier elektronischer Freiheitsgrade am Ag+-Ion mit quadrupolartiger (Γ12+) und rotationsartiger (Γ10+) Symmetrie berucksichtigt. Die Modellrechnungen wurden durch Vergleich mit den unpolarisierten Ramanspektren zweiter Ordnung bestätigt.
The simple shell model which takes account of the polarizability of the ions is improved by including the effect of deformations of the electron shell during the vibrations of the ions. The deformability is introduced in a phenomenological way and it is shown that this ultimately leads to three body interactions between like particles. Because of the inclusion of many body effects the elastic constants do not satisfy in general the Cauchy relation C12 = C44. The theory is applied to the calculation of the dispersion curves for NaI and the result is compared with the simple shell model curves and with the curves obtained by neutron diffraction experiments. Definite improvement of the agreement for the deformable shell model is noticed.
Das einfache Schalenmodell, das die Polarisierbarkeit der Ionen berücksichtigt, wird durch Einbeziehung der Verformungen der Elektronenschalen während der Schwingungen der Ionen verbessert. Die Deformierbarkeit wird phänomenologisch eingeführt und es wird gezeigt, daß dies zu einer Dreikörperwechselwirkung zwischen gleichen Teilchen führt. Wegen der Einbeziehung der Vielkörpereffekte erfüllen die elastischen Konstanten im allgemeinen nicht die Cauchy-Relation C12 = C44. Die Theorie wird angewendet zur Berechnung der Dispersionskurven für NaJ und das Ergebnis wird mit Kurven nach dem einfachen Schalenmodell und mit Kurven, die durch Neutronenbeugungsexperimente erhalten werden, verglichen. Das Modell mit deformierbaren Schalen zeigt deutlich bessere Übereinstimmung.
Twenty-one independent structure factors of LiH have been measured with X-rays to an absolute accuracy of ∼ per cent, and fifteen with neutrons to an absolute accuracy of per cent. The electron density in the hydrogen ion at room temperature has been determined to within 0·004 electrons Å−3, and it is shown that the hydrogen ion is contracted in the crystalline environment to approximately the degree calculated by WALLER and LUNDQVIST and by HURST. The X-ray data suggest that the ionic charge is between 0·8 and 1 electron, while a discussion of the lattice dynamics shows that the infra-red reflectivity and the dielectric constants indicate an ionic charge in the same range.
Crystalline lithium hydride, with four electrons per primitive unit cell, has the simplest electronic configuration of all ionic crystals with the sodium-chloride-type structure, and thus represents an excellent test case for existing theories of lattice dynamics of these substances. Lattice vibrations in this system have been studied by measuring the phonon dispersion curves along the three high-symmetry directions in a single crystal of LiD7, using the techniques of coherent inelastic scattering of thermal neutrons. The isotopes Li7 and D were chosen over normal Li and H because of their more favorable neutron cross sections. All measurements were taken at room temperature, using a triple-axis neutron spectrometer operating in the constant-Q mode. Uncertainties in the measured frequencies are estimated to be no greater than 3-5%. The transverse zone center frequency is in excellent agreement with infrared absorption data. The experimental dispersion curves have been least-squares-fitted to several versions of the rigid-ion and shell models of lattice dynamics. A seven-parameter shell model fits the data to within the experimental accuracy. In this model, the ionic charge was variable and only the hydride ions were polarizable. Short-range forces were assumed between nearest neighbors (Li+ and H−) and between negative second neighbors (H− and H−). The ionic charge obtained from the fit indicates that the bonding in LiD7 is about 88% ionic, in good agreement with estimates obtained from electronegativity considerations. The second-neighbor bond-stretching force constant was found to be about 7% of that obtained for the nearest neighbors. The model parameters obtained for LiD7 were used to calculate dispersion curves for LiH7, and frequency distribution functions for both substances. The frequency distribution for LiH7 has a gap between the acoustic and optic modes, whereas that for LiD7 does not.