No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Deformable shell model and the properties of LiF crystals
Abstract
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.
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.
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 The methods of the harmonic theory are no longer applicable here, because the anharmonic terms cause a coupling between the different modes of oscillations.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.
The frequency wave-vector dispersion relation for the normal modes of vibration of Li7F at 298°K has been measured by means of slow-neutron inelastic scattering techniques. Triple-axis crystal spectrometers at two reactor facilities at the Oak Ridge National Laboratory were employed, mostly in the "constant-Q" mode of operation. The results can be satisfactorily fitted by a seven-parameter dipole approximation model involving nearest-neighbor and second-nearest-neighbor F--F- short-range forces, a variable ionic charge, and the polarizability of the F- ions only. The applicability of other force models to Li7F is also discussed. Certain inconsistencies have been found between the slopes of the acoustic branches near q=0 and the appropriate velocities of sound as measured by ultrasonic techniques. It is believed that these arise from anharmonic effects. The frequency distribution of the normal modes has been computed with high precision from the best-fit dipole approximation model, together with related quantities such as the lattice heat capacity, entropy, and Debye-Waller factors for each ion. Combined (two-phonon) density-of-states functions have also been calculated; comparisons are made between these one- and two-phonon distribution functions and certain features observed in optical and infrared absorption experiments on LiF.
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.
The specific heat of iron-vanadium alloys with 8–45 at.% iron was measured between 125°K and 625°K in an adiabatic calorimeter. The total specific heat was separated into a lattice contribution, a dilatation correction, and a term linear in temperature. The coefficient of the linear term was associated with the electronic specific heat coefficient, γ, and compared with results of specific heat measurements at liquid helium temperatures. These agree for the alloys with up to 33 at.% iron, but differ considerably for the alloys with higher iron concentration.A similar analysis of enthalpy measurements made on these alloys from 800 to 1200°K yields γ values which agree with the liquid helium results for all alloy compositions.It is concluded that the interpretation of the liquid helium measurements is essentially correct and that between 100°K and 700°K there are magnetic contributions to the specific heat of the alloys of 35 and 45 at.% iron.
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 values of the elastic constants of LiF at 4.2°K are found to be c11=12.46×1011, c44=6.49×1011, and c12=4.24×1011 dynes/cm2. At 300°K c11=11.12×1011, c44=6.28×1011, and c12=4.20×1011 dynes/cm2. A Debye characteristic temperature, thetaD, has been calculated from these values by using the Houston method of averaging the velocity over different directions in the crystal. At 4.2°K, thetaD=734°K in good agreement with the specific heat work of Martin. At 300°K, thetaD=708°K which is not in good agreement with values derived from specific heat measurements. A model has been suggested in which a combination of Debye and Einstein terms is used in the specific heat theory.
Houston's formula first derived for an approximate determination of the frequency spectrum of cubic crystals, essentially approximates the integral over the surface of a unit sphere of any function I(theta, varphi) which is invariant under the operations of the cubic symmetry group in terms of the values of I(theta, varphi) along the three directions (100), (110), and (111). In this paper approximate formulas are given for the integral in terms of the values of I(theta, varphi) along the above three and any or all of the (210), (211), and (221) directions. As an application, Debye temperatures Theta are calculated for nine cubic crystals. It appears that the Theta values calculated from the formula containing the values of I along all the above six directions may be expected to be correct to about 1% for crystals for which 0.25
The solution of the generalized master equation of Zwanzig for a macroscopic system, is approximated by the solution of a markoffian master equation. The reliability of this approximation is studied at arbitrary times.
We have measured the Brillouin spectrum of LiF at hypersound frequencies in the [100] and [110] planes. In addition to the longitudinal mode we have observed the pure transverse modes in these planes. Both the polarization and the relevant intensities confirm the assignment. The measured elastic constants are compared with ultrasonic data.
Dielectric and spectroscopic measurements have been made on the alkali halides and several thallium and silver halides, at 2 and 290 K, in order to determine, for each, values for the parameters which characterize the dispersion of the dielectric constant due to lattice vibrations. These have then been used to assess critically the current theoretical treatments and interpretations of the electronic structures and interionic forces in ionic crystals. The measurements were of audio and radio-frequency dielectric constants, of far infrared transmission spectra and of refractive indices through the visible spectrum.
Vibrational frequency spectra of sodium chloride derived from two theoretical models which allow for ionic deformability, instead of treating the ions as rigid point charges, have been used to calculate effective Debye temperatures and distribution function moments.The results are compared with those deduced from specific heat measurements. For the second, more refined, model the agreement with experiment is considerably better than that achieved when the ions are treated as rigid point charges as in the simple Born theory.
Phonon dispersion relations have been obtained in the Delta , Sigma and Lambda symmetry directions using inelastic scattering of thermal neutrons. The results are used to calculate effective charges and effective compressibilities from the relations derived by Szigeti (1949 and 1950). Elastic constants are calculated from the slopes of the dispersion curves for small wave vectors and compared to those obtained in ultrasonic work. The dispersion curves are compared with calculations by Karo using the deformed-dipole-next-nearest- neighbours model. The Lyddane-Sachs-Teller (1941) relation is found to be satisfied in all instances except in RbCl at 80 K. Results from earlier work on NaCl and KCl are included for comparison.
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.
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 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.