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Three body interaction in ionic crystals
Abstract
An empirical expression for a short range three body interaction in a lattice is suggested which leads to the same force constants as obtained by Basu and Sengupta from the consideration of deformability of the ions. Introducing this additional interaction, the stability of the cesium halide crystals is investigated. The stability is found to be correctly predicted in all cases.RésuméOn suggère une expression empirique représentant l'action réciproque de trois corps à petite distance dans un treillis qui mène aux mêmes constantes de force comme obtenues par Basu et Sengupta en considérant la déformabilité des ions. Comme introduction à cette action réciproque supplémentaire, on examine la stabilité des cristaux d'halite de césium. On s'aperçoit que la stabilité a été correctement prévue dans tous les cas.
... Meanwhile, for particles possessing an internal structure the interaction between a pair of particles is changed due to the presence of a third particle, that can be taken into account by introducing potentials depending on the coordinates of three particles. Such a representation follows from consideration of the exchange and multipole interactions of more than two particles in different orders of the perturbation theory [18][19][20]. Essential is the fact that contribution from threebody forces not only gives quantitative corrections to characteristics of a system calculated with account of only pair interactions, but can prove to be necessary for qualitative understanding of some effects. ...
... Since the quasiparticle energy ε i is the functional of f i , the formula (25) represents a complicated nonlinear equation for the distribution function, being similar to that which takes place in the Landau phenomenological theory of a Fermi liquid [11]. Using (20), (21), we express the energy E 0 through ρ(q, q ′ ): ...
... In particular, the three-body potential can be chosen in the form proposed in [20]: ...
The self-consistent field equations are obtained and the thermodynamics is
built of the many-particle Fermi system at finite temperatures with account of
three-body interactions. It is shown that the delta-like three-body interaction
gives no contribution into the self-consistent field and nonlocality of
three-body interactions should be taken into account for their description
within the framework of the self-consistent field model. The case of a
spatially uniform system is considered and a formula for the fermion's
effective mass with account of contribution from three-body forces is derived.
Dependencies of the effective mass and pressure on density are obtained for the
potential of "semi-transparent sphere" type at zero temperature.
Key words: self-consistent field, three-body interactions, effective mass,
fermion, equation of state
... In particular, the three-body potential can be chosen in the form proposed in Ref. [16]: ...
General formulas are derived for the quasiparticle effective mass and the equation of state of the Fermi system with account of the interparticle attraction at long distances and repulsion at short distances. Calculations are carried out of the equation of state and the effective mass of the Fermi system at zero temperature with the use of the modified Morse potential. It is shown that the pair repulsive forces promote a decrease of the effective mass, and the attractive forces promote its increase. With a certain choice of the parameters of the potential the dependence of the pressure on the density has a nonmonotonic character, which enables to describe the coexistence of the liquid and gaseous phases. The influence of three-body interactions on the equation of state and the effective mass is considered. The calculation results are compared with experimental data concerning the quasiparticle effective mass in the liquid 3He.
Contrary to continuous phase transitions, where renormalization group theory provides a general framework, for discontinuous phase transitions such a framework seems to be absent. Although the thermodynamics of the latter type of transitions is well-known and requires input from two phases, for melting a variety of one-phase theories and models based on solids has been proposed, as a generally accepted theory for liquids is (yet) missing. Each theory or model deals with a specific mechanism using typically one of the various defects (vacancies, interstitials, dislocations, interstitialcies) present in solids. Furthermore, recognizing that surfaces are often present, one distinguishes between mechanical or bulk melting and thermodynamic or surface-mediated melting. After providing the necessary preliminaries, we discuss both types of melting in relation to the various defects. Thereafter we deal with the effect of pressure on the melting process, followed by a discussion along the line of type of materials. Subsequently, some other aspects and approaches are dealt with. An attempt to put melting in perspective concludes this review.
Three-body forces have been used in a study of rare-earth hcp metals, namely Tb and Ho, in a phenomenological way to explain the dispersion results in the ΓKM direction.
Lowdin's many body effect is accounted to explain the violation of Cauchy's relation (C12=C44) in silver halides. Lundqvists potential model with many body effect has been used to evaluate the third order elastic constants of AgCl and AgBr. A modified Lundqvist potential model is used to calculate the third order elastic constants for the simple AgCl- AgBr mixed system with varying concentration of AgCl and AgBr.
In the present analysis, the interaction system of an fcc d-band metal is considered to be composed of two-body and many-body parts. We use a short-range three-body potential to deduce the contribution of many-body forces to the dynamical matrix of fcc structure. The inclusion of multi-particle forces in the transition metal model potential scheme of Animalu improves, in general, the transverse branches of phonon dispersion curves of Cu, Ag, Au and Pd and gives good agreement with the neutron data for all the metals.
It is shown that the dominant contribution of the breathing degree of freedom of the electron charge cloud is an effective three body interaction and an attractive two body interaction in addition to the usual overlap interaction. The explicit forms for these interactions are obtained and it is found that the form of the three body interaction is identical to that given in the deformable shell model (DSM).
Based on the energetics of bond formation and a statistical model analysis of the electronic charge density deformation in a covalent crystal the existence of an effective three-body interaction is suggested between the particles in a lattice. In addition to describing the different properties of group IV semiconductors the present model together with the idea of the structure dependence of potential parameters provides a satisfactory description of the polymorphic phase transition of these substances.
Auf Grund der Energetik der Bindungsformation und der Analyse eines statistischen Modells der Deformation der elektronischen Ladungsdichte in einem kovalenten Kristall wird die Existenz einer effektiven Dreikörperwechselwirkung zwischen den Teilchen in einem Gitter vorgeschlagen. Zusätzlich zu der Beschreibung verschiedener Eigenschaften von Halbleitern der IV. Gruppe liefert das Modell zusammen mit der Vorstellung der Strukturabhängigkeit der Potentialparameter eine befriedigende Beschreibung des polymorphen Phasenübergangs dieser Substanzen.
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.
The bond-deformation model is developed for compounds having the rocksalt structure—namely, the alkali halides and the alkaline-earth oxides. The full set of nearest-neighbor bond-deformation parameters is presented, and the parameters are related to the Lagrangian and internal strains and to the atomic displacements. The next-nearest-neighbor bond-stretching parameters are shown to be reducible to the nearest-neighbor parameters. A variety of central-force and non-central-force interactions is identified in the expansion of the short-range portion of the strain energy. By a transformation of variables the short-range contributions to the dynamical matrix are obtained. Expressions are derived for the elastic constants and for the force constant associated with the homogeneous polarization of the lattice.
The energy of formation of Schottky defects in LiF, NaCI, NaBr, and NaI is calculated including Van der Waals and the short-range three body potential proposed by Sarkar and Sengupta in the Mott and Littleton scheme. The elastic displacement of the distant neighbours is taken in the form proposed by Boswarva and Lidiard. Due to the uncertainty in the experimental values of the elastic constants, the estimate of the three body contribution is only approximative. Considering the wide range of the experimental values of the energy of formation of Schottky defects the agreement between the calculated and experimental values is satisfactory excepting the case of NaI for which the experimental value seems to be high.
Die Bildungsenergie von Schottkydefekten in LIF, NaCI, NaBr und NaJ wird mit der Methode von Mott und Littleton berechnet, wobei das Van der Waals-Potential und das kurzreichweitige Drei-Körper-Potential benutzt wurcle, das von Sarkar und Sengupta vorgeschlagen wurde. Die elastischen Verschiebungen der entfernten Nachbarn werden in der von Boswara und Lidiard vorgeschlagenen Form beriicksichtigt. Infolge der Unsicherheit in den experimentellen Werten der elastischen Konstanten ist die Berechnung des Drei-Körper-Beitrags nur approximativ. Bei Reriicksichtigung des großen Bereichs der experimentellen Werte der Bildungsenergie von Schottky-Defekten ist die ubereinstimmung zwischen berechneten und experimentellen Werten befriedigend mit Ausnahme von NaJ, fur das die experimentellen Werte zu hoch zu sein scheinen.
The deformable-shell model developed by Basu and Sengupta and later substantiated by a potential form by Sarkar and Sengupta has found wide application in describing the different static properties of ionic crystals of both NaCl and CsCl structures. But so far a complete calculation of dynamical properties of ionic crystals has been reported for only one crystal. Further, from a critical comparison of the different lattice-dynamical models which effectively introduce many-body interactions between the ions, we have found that there are certain differences between them, some of which are quite fundamental in nature. Moreover, of the current phenomenological models, the deformable-shell model alone is capable of reasonably treating both the static and the dynamic properties of the crystals. Hence it is important to know the results of the calculation according to different models. In this work we present the lattice-dynamic calculation on the following five crystals, NaCl, NaBr, KI, KCl, and KBr according to the deformable-shell model. In order to obtain the parameters, the well-known macroscopic quantities have been used and no least-square-fitting procedure has been adoped. The parameters obtained from the theory have been used to calculate the phonon dispersion relation in both the symmetry and the off-symmetry directions (where experimental results are available) and the variation of the Debye temperature from the frequency spectra for these crystals. We have consistently used the polarizable negative-ion model for all of them. The results thus obtained agree well with experiment. Other theoretical-model results are also discussed in detail.
The thallium group of halides have certain characteristic properties compared to the other alkali-halide crystals. The peculiarities are manifest in connection with the static lattice structure, the consistent description of the dielectric properties and the phonon dispersion relation, and the unified study of the lattice statics and dynamics. In this report we present a coherent account of the diverse properties of the TlBr crystal from a single model, namely, the deformable shell model with a single set of model parameters. Apart from this we present for the first time the theoretical calculation of the phonon dispersion relation for the NaCl-phase TlBr crystal. This prediction, we feel, will be helpful to the experimentalist in identifying the modes of polarization, the continuity of a particular branch in the overlapping regions and the like, if not quantitatively the frequencies themselves. Moreover, the present investigation tries to indicate the relative importance of the different interatomic interactions in relation to the typicalities. Finally it is seen that an overall description of the lattice mechanical properties of the TlBr crystal in both phases is obtained in the present model. Some of the shortcomings and possible means of improving the present calculation are also pointed out.
A consistent and comprehensive calculation has been performed to describe the dynamics and statics of sodium-halide crystals using a well-known three-body-force shell model (TSM). The computed results on the phonon dispersion, two-phonon Raman and infrared spectra, Debye-temperature variations, dielectric and photoelastic behaviors, harmonic and anharmonic elastic constants, cohesive energy, relative stability, and phase-transition, have shown a reasonably good agreement with their accurately measured data. All these predictions have made use of the same set of TSM parameters throughout. The possible sources of improvements have also been indicated. In view of its overall success, TSM has been regarded as an adequate and appropriate model for the description of lattice-mechanical properties of ionic crystals.
In this report we present a simple statistical-model derivation of the three-body interaction envisaged in the deformable-shell model developed by Basu and Sengupta. Next the model is applied to a unified study of the lattice statics and dynamics of the CsBr crystal, the same set of parameters being used throughout. Among the alkali halide crystals, the cesium group of halides are distinguished from the rest by two characteristic problems of the their own—one related to the stability of the static lattice structure and the other concerning the correlation of the dielectric properties and the dispersion of phonons. The results of the present calculation show that this simple model gives a good description of the lattice mechanics of the CsBr crystal in all its totality and the source of improvement for the remaining discrepancy is suggested.
The static properties of ammonium halides are determined in the Born model of ionic crystals using a modified form for repulsion energy. The repulsion parameters are derived from crystal data. An average value of 0.3933 Å for the hardness parameter, is employed in the evaluation of energy parameter and cohesive energies and other lattice properties of ammonium halides. The present computations are satisfactory when compared with experimental data. An attempt to explain the relative stability of these crystals in hypothetical NaCl-type crystal structure has been made in the present model which gives a correct prediction.
We present interaction potentials for AgCl and AgBr derived from quantum-mechanical approximations and fitted to the experimental lattice constant and energy, elastic and dielectric constants, and special phonon frequencies. The expressions are simple enough for atomistic calculations of crystal defects and molecular dynamics while sufficiently representing the peculiar features of silver halides. The outstanding finding is the dominance of strong van der Waals interaction. Two-body van der Waals forces determine the observed small lattice constant, large lattice energy, and small elastic constant c44, while three-body van der Waals forces contribute mostly to the violation of the Cauchy relation and the large bulk modulus. We could not find convincing indications of partial covalency in silver halides nor of an easy quadrupole deformability of the Ag+ ion.
In the breathing shell model (bsm) the many-body interactions generated by the 'breathing' motion of the ions do not vanish for an alkali halide crystal under homogeneous deformation and hence contribute to many static properties of the lattice. Though this model has been quite successful in the region of lattice dynamics, no attempt has been made to apply them to lattice statics, except in so far as the elastic properties and indirectly the dielectric properties are concerned. It is now well known that no reasonable two-body central interaction is able to predict the correct structure for the caesium chloride structure alkali halides. The authors see whether the many-body interactions bsm purports to describe can yield the correct structure for them and at the same time explain the stability of the other members of the group. The results of the calculation show that within certain limitations bsm correctly predicts the structure of all the alkali halides considered in the present investigation.
Unlike the alkali halides with NaCl structure, the CsCl-structure alkali halides present several problems regarding the cohesive energy, the static lattice structure, the dielectric properties and the phonon dispersion relation. From a critical discussion of the problems it is seen that these differences from the rest of the alkali halides become all the more prominent, when any attempt is made to reproduce the above properties simultaneously in any of the existing models. An attempt is made to calculate simultaneously the lattice statics and dynamics of the CsCl and CsI crystals from a single model with a single set of model parameters. The results of the calculation show that a consistent description for most of the major properties is obtained in the model. The source of the remaining discrepancy is discussed.
The authors discuss the nature of deformation of the electron charge cloud of an ion implied in the deformable shell model and considers an application of the model to the case of LiF crystals. It is found that the properties of the crystal are fairly well reproduced by the present model and certain discrepancies noted by earlier authors are removed. The results of other theoretical investigations are also included for comparison.
Elastic constants have been calculated for the FCC metals Cu, Al and Ni using the Mobius transform and a short-range three-body potential correction. Unlike a previous calculation based only on pair potentials, the present result is in good agreement with experiment since the restriction of the Cauchy relation has been removed. More importantly, it shows the potential application of the Mobius-inversion method for evaluating interatomic potentials from ab initio electronic structure calculations.
A short-range three-body interaction which arises from the deformability of the charge cloud of an atom is included in the interaction between atoms of an inert gas solid and its effect on the dispersion curves and elastic constants is discussed. The result is applied to krypton. Experimental dispersion curves, the bulk modulus at absolute zero, and the Debye temperature are compared with calculated values. It is found that on the whole the inclusion of the three-body potential improves the agreement.Eine Dreikörper-Nahwechselwirkung, die von der Deformierbarkeit der Ladungswolke eines Atoms herrührt, wird in die Wechselwirkungen zwischen den Atomen fester Edelgase einbezogen und ihr Einfluß auf die Dispersionskurven und elastischen Konstanten diskutiert. Die Ergebnisse werden auf Krypton angewendet. Experimentell ermittelte Dispersionskurven, der Volumenmodul am absoluten Nullpunkt und die Debye-Temperatur werden mit berechneten Werten verglichen. Es wird gefunden, daß die Einbeziehung des Dreikörperpotentials die Übereinstimmung im ganzen verbessert.
The effect of the many body interaction proposed by Sarkar and Sengupta is considered in detail. The method of homogeneous deformation is extended to include many body interactions in which the potential depends on several scaler distances. The general expressions for the elastic constants obtained by this method are compared with those obtained from the long wave theory. Including the various central interactions the model is applied to calculate the elastic constants, the cohesive energy, and the phase transition pressure for the six halides of potassium and rubidium. It is found that the inclusion of the three body interaction considerably improves the agreement.
Der Effekt der Vielteilchenwechselwirkung, auf den von Sarkar und Sengupta hingewiesen wurde, wird im Einzelnen betrachtet. Die Methode homogener Deformation wird erweitert unter Einbeziehung der Vielteilchenwechselwirkung, bei der das Potential von einigen Maßstabslängen abhängt. Die allgemeinen Ausdrücke für die elastischen Konstanten, die nach dieser Methode erhalten werden, werden mit den nach der Langwellentheorie erhaltenen Werten verglichen. Unter Einschluß der verschiedenen Zentralwechselwirkungen wird das Modell verwendet, um die elastischen Konstanten, die Kohäsionsenergie und den Phasenübergangsdruck für die sechs Halogenide des Kalium und des Rubidium zu berechnen. Die Berücksichtigung der Dreikörperwechselwirkung verbessert die Übereinstimmung beträchtlich.
Using the Heitler-London approach a simple expression for the first-order exchange energy between a system of interacting atoms or ions is obtained in the S2 approximation. It is shown that this expression leads to two-body, three-body, and four-body interactions only. A specific expression for the three-body interaction is evaluated using several different approximations. Assumptions which lead to the expression previously obtained by Lundqvist are discussed. It is found that these expressions vanish for neutral particles. Some modification in the assumption is made and an expression for three-body interaction for neutral particles is obtained. Numerical calculations are made for argon, xenon, and helium and the results compared with those obtained by others. The three-body interaction is also applied to study the relative stability of different alkali halide structures. It is found that this three-body potential has a strong preference for CsCl structure.
Mit der Heitler-London-Methode wird ein einfacher Ausdruck für die Austauschenergie erster Ordnung zwischen einem System wechselwirkender Atome oder Ionen in der S2-Näherung erhalten. Es wird gezeigt, daß dieser Ausdruck nur zu Zwei-Teilchen-, Drei-Teilchen- und Vier-Teilchen-Wechselwirkungen führt. Für Drei-Teilchen-Wechselwirkung wird mit verschiedenen Näherungen ein spezifischer Ausdruck abgeleitet. Annahmen, die zu dem kurzlich von Lundquist erhaltenen Ausdruck führen, werden diskutiert. Es wird gefunden, daß diese Ausdrücke für neutrale Teilchen verschwinden. Numerische Berechnungen werden fur Argon, Xenon und Helium dürchgeführt und die Ergebnisse mit denen anderer Autoren verglichen. Die Drei-Teilchen-Wechselwirkung wird auch zur Untersuchung der relativen Stabilität verschiedener Alkalihalogenidstrukturen benutzt. Es wird gefunden, daß dieses Drei-Teilchen-Potential eine starke Bevorzugung der CsCl-Struktur ergibt.
The model potential including a short range three-body interaction which was used by Sarkar and Sengupta [1,2] to calculate various static properties of some alkali halide crystals has been extended to include overlap interaction between next nearest neighbours. This has been used to calculate the cohesive energy, harmonic elastic constants, phase transition pressure, and transitionvolume for transition from sodium chloride to the cesium chloride structure in the case of potassium and rubidium halides. The results have been compared with experimental values and with the theoretical calculations done on other models. It is found that the inclusion of overlap interaction between next nearest neighbours considerably improves the overall agreement between theoretical and experimental results.
Das Modellpotential mit kurzreichweitiger Dreiteilchen-Wechselwirkung, das von Sarkar und Sengupta [1, 2] benutzt wurde, um verschiedene statische Eigenschaften einiger Alkalihalogenidkristalle zu berechnen, wird erweitert und schließt Überlappungswechselwirkung zwischen übernächsten Nachbarn ein. Es wird benutzt, um die Kohäsionsenergie, die harmonischen elastischen Konstanten, Phasenübergangsdruck und Übergangsvolumen für den Übergang von der Natriumchloridstruktur zur Cäsiumchloridstruktur für den Fall von Kalium- und Rubidiumhalogenid zu berechnen. Die Ergebnisse werden mit experimentellen Werten und mit theoretischen Berechnungen an anderen Modellen verglichen. Es wird gefunden, daß die Einbeziehung der Überlappungswechselwirkung zwischen übernächsten Nachbarn beträchtlich die Gesamtübereinstimmung zwischen theoretischen und experimentellen Ergebnissen verbessert.
Many theoretical models have so far been suggested to study the lattice-dynamical properties of ionic crystals. One difficulty, common to all of them is that they do not explain the Cauchy relation breakdown. A critical comparison is presented of some improved models that do explain the breakdown of the Cauchy relation. Most applications of these models have been made in the field of lattice-dynamical properties, such as dispersion curves, theta -T curves, etc. Few attempts have so far been made to apply these models to the study of static properties as cohesive energy, relative stability, phase transition pressure, or anharmonic properties as variation of bulk modulus with temperature and pressure, thermal expansion of solids, etc.
The Lundqvist three-body interaction which was originally deduced for rigid ions is rederived in the framework of a shell model. Some discrepancies in a previous interpretation are pointed out. Application is considered for NaCl crystal. If the parameter of the three-body interaction is determined from the overlap integrals it is found that the calculated dispersion curves, in general, deviate more from the experimental ones. Some static properties are also calculated.
Im Rahmen des Schalenmodells wird die Lundqvist-Drei-Körper-Wechselwirkung, die ursprünglich für starre Ionen erhalten wurde, erneut abgeleitet. Auf einige Widersprüche der früheren Interpretation wird hingewiesen. Für NaCl-Kristalle werden Anwendungen betrachtet. Wenn der Parameter der Drei-Körper-Wechselwirkung aus dem Überlappungsintegral bestimmt wird, zeigt es sich, daß die berechneten Dispersionskurven im allgemeinen mehr von den experimentellen abweichen. Einige statische Eigenschaften werden ebenfalls berechnet.
The phase transition data at 298 °K of potassium chloride crystals are analysed on the basis of two alternative models, (a) the Tosi-Fumi prescription (TFP) in which overlap parameters are taken to be different in the two phases and (b) the three-body potential model (TBPM) used by Sarkar and Sengupta [1]. The effect of finite temperature and that of zero point energy is included using the Einstein approximation. It is found that the transition data are well reproduced in both the models with TFP giving slightly better values. The temperature coefficient of transition pressure comes out with the right sign but has too high a magnitude in both the models. In elastic data, however, TFP gives poor agreement while for TBPM the agreement is much better.Die Phasenübergangswerte bei 298 °K von Kaliumchloridkristallen werden auf der Grundlage zweier Alternativmodelle analysiert: a) der Tosi-Fumi-Vorschrift (TFP), bei der die Überlappungsparameter in den beiden Phasen als verschieden angesehen werden, und b) dem Drei-Körper-Potential-Modell (TBPM) von Sarkar und Sengupta [1]. Der Einfluß der endlichen Temperatur und der Nullpunktsenergie wird mit der Einstein-Nherung berück-sichtigt. Es wird gefunden, daß die Übergangswerte in beiden Modellen gut wiedergegeben werden, wobei TFP leicht bessere Werte liefert. Der Temperaturkoeffizient des Übergangsdruckes ergibt sich mit dem richtigen Vorzeichen, jedoch mit zu hohen Absolutwerten in beiden Modellen. Bei den elastischen Werten Mefert jedoch TFP schlechtere Übereinstimmung, whrend die übereinstimmung für TBPM viel besser ist.
The consideration of paired interionic forces cannot explain the lattice dynamics of Be, Tb and Ho. In the present work, a phenomenological force constant model has been developed which considers three-body forces in addition to the usual forces of paired character. This approach explains the correct ordering of dispersion branches and the experimental degeneracy of the K-point in Be, Tb and Ho. Further good agreement is found between the theoretical curves and the neutron data of three high symmetry directions for all the three hcp metals.
The Coulomb, van der Waals and repulsive lattice sums occurring in the higher order elastic constants up to sixth order have been calculated for the rocksalt and cesium chloride structures. Numerical values of the static elastic constants up to sixth order based on a rigid ion model with van der Waals and Born-Mayer type central force interaction between first and second nearest neighbors are calculated for several alkali halides representing both structure types. Fair agreement with the available experimental third and fourth order elastic constant data is found.
The long wavelength limit of the optical frequencies and of the microscopic Grueneisen parameters, the elastic constants and their first and second pressure derivatives are calculated for the spinel structure from a modified rigid ion model. The model includes Coulomb, first and second neighbor two-body central force interaction, and the three-body central force interaction of Basu and Sengupta for the tetrahedral and octahedral cation-anion units. Application to Al2MgO4 is made on the basis of (a) a rigid ion model and (b) the above modified rigid ion model with the three-body forces included for the MgO4 tetrahedra. In addition to two parameters which are determined by the equilibrium conditions the models contain three and four parameters, respectively, which are fitted to experimental values of one Raman and two i.r. frequencies and, for model (b), to one elastic constant. Good to fair agreement with experimental data of the remaining optical frequencies, elastic constants and their first pressure derivatives and the macroscopic Grüneisen parameter is obtained. Both models results in new assignments for the Raman and i.r. active modes.
There are presently experimental data on FCC metals, concerning fundamentally harmonic properties (dispersion curves, second order elastic constants) and essentially anharmonic ones (third order elastic constants, thermal expansion). However, at present, there is not a simple interatomic potential that can reproduce adequately all this experimental information simultaneously. In this work we develop several programs to calculate the second order elastic constants and the dispersion curves using a general central two body interaction up to 19th neighbors and common nearest neighbor three body interactions. The difference from previous works is that here the three body potential is a purely angular one, uncoupled from the two body forces, allowing a quantitative evaluation of its relative importance.
The extended x-ray-absorption fine structure (EXAFS) of three alkali-metal halides (NaBr, KBr, and RbCl) is investigated at pressures ranging from 0 to 8 GPa. Parameter fitting is used to extract the change of the nearest-neighbor distance and thus the pressure, as well as the absolute values of the second cumulant as a function of pressure, σ2(p), and the third cumulant at zero pressure, σ(3)(0). For all three materials, a sharp reduction with pressure of σ2(p) is found. Classical statistical mechanics is then invoked to calculate the first, second, and third moments of the nearest-neighbor distance and thus their second and third cumulants. The integration is done with the Monte Carlo technique. For the potential energy U in the Boltzmann factor, the generalized Huggins-Mayer pair potential is used along with a three-body potential term due to charge transfer. Generally, model calculations and EXAFS data match well. High-pressure EXAFS data can therefore be regarded as a help to assess, and even exclude, potential parameters from the literature.
This chapter describes the lattice theory of the anharmonic effects. This theory contains an extension of the generally used harmonic approximation. The chapter discusses the calculation the thermal and caloric data of an ideal single crystal and to indicate the relationships among them. The methods of the harmonic theory are no longer applicable here, because the anharmonic terms cause a coupling between the different modes of oscillations. In the case of sound waves, there is a superposition of mechanically excited vibrations and thermal ones, which are already present. The anharmonicity causes an exchange of energy between thermal and mechanical vibrations. The mechanical sound waves therefore also loose energy to the thermal vibrations. Macroscopically, this leads to damping effects. The chapter describes the theory of the anharmonic effects that is as comprehensive as possible. The symmetry and invariance properties of the potential energy are discussed.
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);