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Phonon dispersion relations in RbCl and RbF at 80 K
Abstract
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.
... Luckily, several important chlorides have been studied this way. These include halite (NaCl) Schmunk and Winder, 1970;Nilsson, 1979), sylvite (KCl) , RbCl (Raunio and Rolandson, 1970b), MnCl 2 (Escribe et al., 1980), and FeCl 2 (Yelon et al., 1974). Phonon spectra of isotopically substituted chlorides have not been measured, nor have those of chlorine-bearing silicates, such as amphiboles, micas, and clays. ...
... KCl -3 models (Raunio and Rolandson, 1970a) were reoptimized to frequencies reported in ; the Copley et al. (1969) models (II, V, and VI) were reoptimized to the (Copley et al., 1969) measured frequencies. RbCl -3 models (Raunio and Rolandson, 1970a) were reoptimized to frequencies reported by Raunio and Rolandson (1970b). FeCl 2 and MnCl 2 -models (Benedek and Frey, 1980) were not reoptimized. ...
... The frequencies used to constrain the lattice-dynamics calculations for crystals were, for the most part, measured at very low temperatures (77 to 120 K). Measurements made at ϳ300 K in NaCl Schmunk and Winder, 1970), KCl (Tanaka and Hisano, 1989;Wakamura, 1993), and RbCl (Raunio and Rolandson, 1970b) suggest that vibrational frequencies are typically ϳ2 to 5% lower at this temperature than at ϳ120 K. As a consequence, the calculated values of 1000 · ln( 37-35 ) for NaCl, KCl, and RbCl are probably ϳ0.1 to 0.4‰ too high at room temperature. ...
Equilibrium chlorine-isotope (37Cl/35Cl) fractionations have been determined by using published vibrational spectra and force-field modeling to calculate reduced partition function ratios for Cl-isotope exchange. Ab initio force fields calculated at the HF/6-31G(d) level are used to estimate unknown vibrational frequencies of 37Cl-bearing molecules, whereas crystalline phases are modeled by published lattice-dynamics models. Calculated fractionations are principally controlled by the oxidation state of Cl and its bond partners. Molecular mass (or the absence of C-H bonds) also appears to play a role in determining relative fractionations among simple Cl-bearing organic species. Molecules and complexes with oxidized Cl (i.e., Cl0, Cl+, etc.) will concentrate 37Cl relative to chlorides (substances with Cl−). At 298 K, ClO2 (containing Cl4+) and [ClO4]− (containing Cl7+) will concentrate 37Cl relative to chlorides by as much as 27‰ and 73‰, respectively, in rough agreement with earlier calculations. Among chlorides, 37Cl will be concentrated in substances where Cl is bonded to +2 cations (i.e., FeCl2, MnCl2, micas, and amphiboles) relative to substances where Cl is bonded to +1 cations (such as NaCl) by ∼2 to 3‰ at 298 K; organic molecules with C-Cl bonds will be even richer in 37Cl (∼5 to 9‰ at 298 K). Precipitation experiments, in combination with our results, provide an estimate for Cl-isotope partitioning in brines and suggest that silicates (to the extent that their Cl atoms are associated with nearest-neighbor +2 cations analogous with FeCl2 and MnCl2) will have higher 37Cl/35Cl ratios than coexisting brine (by ∼2 to 3‰ at room temperature). Calculated fractionations between HCl and Cl2, and between brines and such alteration minerals, are in qualitative agreement with both experimental results and systematics observed in natural samples. Our results suggest that Cl-bearing organic molecules will have markedly higher 37Cl/35Cl ratios (by 5.8‰ to 8.5‰ at 295 K) than coexisting aqueous solutions at equilibrium. Predicted fractionations are consistent with the presence of an isotopically heavy reservoir of HCl that is in exchange equilibrium with Cl−aq in large marine aerosols.
The present investigation for the alteration of the free-ion wave functions, when the ions are put in a crystal, the effect of the crystal environment is simulated by the Watson potential. The developed perturbation theoretical model for ionic solids by Basu and sengupta has been used for calculation of the different lattice-dynamical properties for the low-polarizability ionic crystal. The feature of used method it requered only input data necessary for the study of (KF) crystal properties are the Hartree-Fock wave functions of the constituent ions. Certain properties like, the dielectric properties and some phonons in symmetry directions, depend on the excited states of the crystal, are rather sensitive to this effect which varies from crystal. A unified study of the phase transition, cohesion, elastic constant, dielectric and vibrational properties of the (KF) crystal is given without using any adjustable free parameter, our approach and overall agreement obtained is well satisfactory. Finally the limitations and reliability of the present model for estimating the effect of the surroundings are critically discussed.
All materials exhibit wave dispersion at 'small' wavelengths leading to non-linearities in experimentally determined dispersion curves. Classical local elasticity fails to predict these non-linearities. Nonlocal continuum mechanics allows for the prediction of the elastic behavior over a considerably wider range of lengthscales. Starting from ab initio lattice dynamics calculations we determine the elastic constants and the phonon dispersion relation for silicon. We verify our results using inelastic neutron scattering data. Next we develop the theoretical and numerical framework to construct nonlocal constitutive equations for longitudinal and transverse acoustic modes.
The composition dependence of the lattice thermal conductivity in NaCl-KCl solid solutions has been measured as a function of composition and temperature. Samples with systematically varied compositions were prepared and the laser flash technique was used to determine the thermal diffusivity from 373 K to 823 K. A theoretical model, based on the Debye approximation of phonon density of state (which contains no adjustable parameters) was used to predict the thermal conductivity of both stoichiometric compounds and fully disordered solid solutions. The predictions obtained with the model agree very well with our measurement. A general method for predicting the thermal conductivity of different halide systems is discussed.
In the present work, the interaction between electrons and long-wavelength transverse optical (TO) vibrations in a polar insulator is considered. The crystal model employed for the theoretical analysis includes classical potentials and electronic polarizabilities. A significant enhancement of the strength of the electron–TO-phonon interaction in ferroelectrics has been found. A microscopic justification of this effect is given. A bridge that relates the interaction of electrons with the polar long-wavelength TO modes of lattice vibrations to the long-range dipole–dipole interaction is established. As an application of our analysis, a novel method for quantitative predictions of the electron–TO–phonon interaction constants in polar crystals via the data of experimental studies is suggested.
A consistent set of shell parameters for alkali halide crystals of both rocksalt and CsCl structures is reported. The fitting procedure, values of the parameters and their sensitivity are discussed. Short range interionic potentials are determined and are shown to give a good description of both static and dynamic properties of the crystals.
Lattice dynamics of rubidium halides have been studied on the basis of a theoretical model recently developed by Verma and Singh (1969) by including the effect of three body forces into the framework of the shell model. The results of the calculation are found to agree fairly well with the available experimental data on dispersion relations, second order Raman spectra and specific heats.
Phonon frequencies in RbBr at 80 K have been obtained by inelastic scattering of thermal neutrons. The measurements were performed in the symmetry directions Delta , Sigma and Lambda and along the line Z. Some zone boundary phonons were also measured at 300 K. Four versions of the shell model were fitted to the experimental frequencies, and the elastic and dielectric constants were calculated from the model parameters. From an eleven parameter model the authors have also calculated the phonon frequency spectrum and the temperature variation of the Debye temperature.
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.
The thermal conductivity of 17 alkali halides has been calculated with a variational method. The calculations confirm previous results from relaxation-time calculations on NaCl and KCl that optic phonons are important in both the heat current and the scattering processes. It is found that the relative contributions to the scattering integral from different combinations of branches are strongly correlated to the mass ratio of the two constituent ions. On the other hand the amount of heat carried by optic phonons is more dependent on the ratio of the static and optic dielectric constants, epsilon 0/ epsilon infinity , than on the mass ratio. The theory of mirror pairs has been applied in an analysis of the thermal conductivity data. It is shown that experiments on Li and LiBr would be a critical test of the variational calculations.
For pt.I see ibid., vol.11, p.1523 (1978). A set of interionic potentials of the Born-Mayer form is derived for the alkali halides. The specification is completely in terms of parameters of two types, (a) parameters which relate to the constituent ions and (b) parameters which apply to all crystals. Thus the parametrisation scheme contains no crystal dependent terms and is of a form which may readily be applied to systems of mixed alkali halides. Each of the nine ions is described by five parameters, its polarisability, its shell charge and three parameters which relate to the ion size. The van der Waals coefficients are calculated consistently from the ion polarisabilities. The model system, which has 47 free parameters, reproduces the lattice constants, high and low frequency dielectric constants and transverse optic frequencies for the 20 crystals to high accuracy. Of crystal properties which are not included in the fitting procedure, the cohesive energies are given in most cases to better than 1%, the NaCl/CsCl structural relative stability is correctly described in all cases except CsF, and reasonable phonon dispersion relations and elastic constants are obtained.
Phonon dispersion relations in RbI have been obtained by inelastic scattering of thermal neutrons. Measurements were performed in the crystallographic symmetry directions Δ, Σ, A, and Z at 80 °K and some selected zone boundary phonons were also measured at 300 °K. A simplified version of the generalized shell model, involving a variable number of parameters was fitted to the experimental dispersion curves. From an eleven parameter model the frequency spectrum, the Debye temperature and other physical quantities have been calculated.Die Phononendispersionsbeziehungen in RbI wurden durch inelastische Streuung thermischer Neutronen erhalten. Die Messungen wurden in den kristallographischen Symmetrierichtungen Δ, Σ, A und Z bei 80 °K durchgeführt. Einige ausgewhlte Zonengrenzen-Phononen wurden auch bei 300 ° K untersucht. Mit einer vereinfachten Version des verallgemeinerten Schalenmodells, das eine variable Anzahl von Parametern enthielt, wurden die experimentellen Dispersionskurven angepaßt. Mittels eines Elf-Parameter-Modells wurden das Frequenzspektrum, die Debye-Temperatur und andere physikalische Größen berechnet.
The polarized second order Raman spectra of three rubidium halide crystals, namely RbCl, RbBr, and RbI are computed on the basis of the theory proposed by Born and Brad-burn, taking into account first as well as second neighbour polarizability coefficients. The three-body force shell model is employed to determine the phonon frequencies, the phonon eigenvectors, and the polarizability tensor. The polarizability coefficients are assumed to be adjustable parameters. It is found that in addition to all the first neighbour coefficients, only eight second neighbour coefficients are sufficient to obtain good agreement between the experimental and the theoretical spectra.
Auf der Grundlage der von Born und Bradburn vorgeschlagenen Theorie wird das polarisierte Ramanspektrum zweiter Ordnung der drei Rubidiumhalogenidkristalle–RbCl, RbBr und RbJ–berechnet, wobei sowohl die Koeffizienten der Polarisierbarkeiten der ersten als auch der zweiten Nachbarn berücksichtigt werden. Für die Bestimmung der Phononenfrequenzen, der Phononeneigenvektoren und der Tensoren der Polarisierbarkeit wird ein Drei-Körperkräfte-Schalenmodell benutzt. Die Koeffizienten der Polarisierbarkeit werden als Anpassungsparameter angesehen. Es wird gefunden, daß zusätzlich zu alien Koeffizienten der ersten Nachbarn nur acht Koeffizienten der zweiten Nachbarn nötig sind, um gute Übereinstimmung zwischen experimentellen und theoretischen Spektren zu erhalten.
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.
Using the breathing shell model with second neighbour interactions taken into consideration, dispersion curves are calculated without any least squares fitting to experimental curves. Accurate calculations are also made of the phonon density of states for all twenty alkali halides. The separate contributions for the positive and negative ions to the phonon density of states are reported.Unter Berücksichtigung von Wechselwirkung mit zweitnächsten Nachbarn werden mit dem „Breathing-Shell-Modell” Dispersionskurven ohne Anpassung an experimentelle Kurven berechnet. Genaue Berechnungen der Phononenzustandsdichte werden für alle zwanzig Alkalihalogenide durchgeführt. Die Beiträge der positiven und negativen Ionen zur Phononenzustandsdichte werden angegeben.
Temporary infrared absorption bands are created in NaCl, KCl, RbCl, and KBr when nominally pure X-ray coloured crystals are irradiated in the F band at low temperatures. This absorption shows a zero-phonon line and a very detailed phonon structure. Some properties such as thermal decay, optical bleaching with polarized light, influence of impurities, and shape of the zero phonon line are studied.
Temporäre Infrarotabsorptionsbanden werden in NaCl, KCl, RbCl und KBr erzeugt, wenn nominell reine, röntgenverfärbte Kristalle in der F-Bande bei tiefen Temperaturen bestrahlt werden. Diese Absorption zeigt eine Null-Phonon-Linie und eine detailreiche Phononenstruktur. Einige Eigenschaften, wie thermischer Zerfall, optisches Bleichen mit polarisiertem Licht, Einfluß von Störstellen und Form der Null-Phononlinie werden untersucht.
The fundamental role played by lattice vibrations in nearly all fields of solid state physics, be it in connection with absorption of elastic or electromagnetic waves, superconductivity, or ferroelectricity, has stimulated a growing interest in lattice vibrational spectra of pure and perturbed crystal lattices. Such vibrations can be studied directly be means of inelastic neutron scattering which is to be prefered for the study of bulk vibrational modes, and by infrared and Raman spectroscopy which, because of their inherent sensitivity to relatively low defect concentrations, are excellent techniques for investigating the defect induced response of a crystal lattice. Such defects, which may be foreign atoms, ions, or molecules, but which may also be dislocations, voids, and so on, are interesting because they lower the point symmetry of the lattice, thereby relaxing the selection rules for photon-phonon interaction processes, and thus allowing the optical activation of (somewhat perturbed) host lattice vibrations which are inactive in the unperturbed crystal.
Phonon dispersion relations in NaCl at 80°K have been determined in the crystallographic [100], [101], and [111] symmetry directions using inelastic neutron scattering. Some measurements have also been made at 300°K, and at (1.0, 0.5, 0) in reciprocal space. The results are compared with theoretical calculations: those of Karo and Hardy for their "deformed-dipole" model, those of Schröder and Nüsslein for a "breathing-shell" model, and those of Caldwell and Klein for a shell model. Satisfactory agreement is obtained in many places, but significant discrepancies still remain, particularly close to the zone boundaries. Comparison is also made with earlier x-ray work on NaCl. The work was performed on a new crystal spectrometer at the R2 reactor at Studsvik, which is equipped with a two-crystal monochromator.
Phonon dispersion relations in KCl at 80°K have been determined using inelastic neutron scattering. The measurements have been performed mainly in the symmetry directions [100], [110], and [111], but also at some other points in the reciprocal space. More than one third of the phonons in all branches were also studied at 300°K in order to obtain and estimate of the temperature dependence of their frequencies. A comparison is made with dispersion curves calculated by Nüsslein and Schröder for their “breathing shell” variant of the shell model for ionic crystals. The agreement for the dispersion curves is quite satisfactory in the most cases.Die Dispersionskurven der Gitterschwingungen in KCl bei 80°K wurden durch inelastische Neutronenstreuung gemessen. Die Messungen wurden meistens in den kristallographischen Symmetrierichtungen [100], [110] und [111], aber zum Teil auch in anderen Punkten im reziproken Raum durchgeführt. Mehr als ein Drittel der Phononen sind auch bei 300°K untersucht worden, um die Temperaturabhngigkeit ihrer Frequenzen zu erhalten. Die Ergebnisse werden mit dem „breathing shell”-Modell von Nüsslein und Schröder verglichen. Die Übereinstimmung ist in den meisten Fllen sehr befriedigend.
A new feature of the spectrometer is the monochromator unit, which contains two parallel reflecting copper crystals and delivers a beam of monoenergetic neutrons through a fixed collimator which is parallel to the incident beam from the reactor. About half the neutrons reflected by the first crystal are reflected by the second crystal, though the fraction declines at the upper end of our energy scale (about 60 meV). The beam from the monochromator is free from fast neutrons, and the background extremely low. The material, thickness, and mosaic width appropriate to efficient reflection in a monochromator crystal are discussed. The sample and analyzer form a compact separate unit on a stationary base. All the angles of the spectrometer (including those of the monochromator) are continuously variable over wide ranges, with either manual control or automatic control from a programming tape. Two spectrometers which are mirror images of one another are mounted side by side at the same reactor beam hole. They are mainly intended for measurements on phonons with comparatively high resolution. An example of the accuracy attained is given.
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.
Phonons in sodium fluoride have been studied at room temperature using inelastic neutron scattering techniques. Consistent results were obtained using a cold neutron time-of-flight apparatus and a triple-axis spectrometer. The time-of-flight results were interpolated on to symmetry directions from the observed scattering surfaces. The frequencies in units of 1012 sec-1 of some typical phonons are: TA(0,0,1), ν=4.39±0.04; LO(0,0,0.984), ν=8.52±0.15; LA(0,0,0.972), ν=7.94±0.18; TO(0.488, 0.488, 0.488), ν=6.19±0.07. The optical branches extrapolated to small wave vector are in agreement with the infrared absorption frequency and the Lyddane-Sachs-Teller relation. Hardy and Karo's deformation-dipole model is in agreement with the results to within 6%, but the rigid-ion model differs by as much as 19%. The results are well fitted by a shell model containing nine parameters in which the ionic charge is 0.91.
Measurements of the adiabatic elastic constants from 300°K to 4.2°K have been made on single crystals of lithium chloride and rubidium chloride. The values of the constants at 4.2°K are given in units of 1011 dyn cm2 as c11 = 5.860, c44 = 2.671 and c12 = 2.086 for LiCl and tc11 = 4.499, c44 = 0.497 and c12 = 0.676 for RbCl. The Debye temperatures at 0°K (θ0) as calculated from the elastic constants are 424.7°K for LiCl and 168°K for RbCl. The lattice energies are calculated to be U0 = 196.5 kcal mole for LiCI and U0 = 161.7 kcal mole for RbCl.
The frequency of the TA[100] zone-boundary phonon has been measured in rubidium iodide under hydrostatic pressures of 0 to 3.72 kbar by neutron inelastic scattering. A linear fit to the percentage decrease in frequency gives a value of 13.5+/-0.9% when extrapolated to the NaCl-CsCl polymorphic transition pressure.
A survey is made of those aspects of the theory of lattice vibrations relevant to their spectral behaviour, and the roles of `effective charges', critical points and anharmonicity are considered. The experimental investigations by far infra-red spectroscopy are reviewed, results for ionic, covalent, molecular and monatomic solids being presented. Defect-induced spectra and their use as sources of information on otherwise inactive vibrations are discussed.
Sixteen crystalline ionic halides have been studied in transmission and reflexion experiments in the range 70 to 250 mu at temperatures down to 4^circK. The lattice resonances have been clearly observed, their characteristic frequencies accurately determined and the degree of damping investigated as a function of temperature. The significance of the results is discussed from the point of view of various models of dielectric behaviour-in particular the refinements of the Born-Lorentz model which cover the effect of distortion moments due to the overlap of neighbouring ions. Discrepancies remain between theory and experiment which have a bearing on the calculation of vibration spectra.
The principal purpose of this paper is to discuss the nature of the modes of vibration in a solid. The following is found: (1) Waves in lattices are in general neither transverse nor longitudinal; in particular, they need not be transverse or longitudinal when the propagation vector k is very small. (2) The relationship omegalomegat=(ε0ε∞)12 for "longitudinal" and "transverse" modes in ionic crystals applies, if at all, in a region of small, but nonzero, wave vector k. (3) The derivation of this relationship is based, at least implicitly, on the use of cyclic boundary conditions. (4) The use of cyclic boundary conditions is valid in statistical problems for crystals without long-range forces, but has never been justified for systems with Coulomb forces. (5) If cyclic boundary conditions are nonetheless used, it can be simply shown that for k=0, omegalomegat=1.
The long wave-length, polar lattice vibrations of alkali halide crystals are discussed without making any specific assumptions about the detailed interactions between the ions. This is made possible by the introduction of the effective charge, e*, of an ion defined as follows: All of the positive ions in a crystal slab are displaced by an equal amount in a direction perpendicular to the faces of the slab and all of the negative ions in the opposite direction. Then e* is the ratio of the dipole moment per ion pair induced in the slab by this displacement to the relative displacement of the positive and the negative ions. Expressions are obtained for the frequency, omegal, of the longitudinal vibration and the frequency, omegat, of the transverse vibration in terms of the dielectric constant, k, of the crystal, the dielectric constant, k0, obtained by extrapolating the square of the index of refraction of the crystal from high frequencies to zero frequency, and e*. The ratio of the two frequencies is found to be independent of e* and given by omegalomegat=(kk0)12.
In this paper some relations between dielectric properties of diagonal cubic ionic crystals are derived on the basis of the shell model for ions of Dick and Overhauser. The relations, which contain no model constants, are in a good agreement with experimental data. Also it is shown that Dick and Overhauser overestimated the number of electrons in the shells of the ions, which accounts for the failure of their quantitative treatment. In the appendix the paper of Hanlon and Lawson on the same subject is 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);
The sound velocities and elastic constants have been measured for pure single crystals of KI and KCl by the ultrasonic pulse technique over the temperature range from 4°K to 300°K. In KI, c11=3.38, c12=0.22, and c44=0.368 in units of 1011 dynes/cm2 at 0°K according to our extrapolations from 80°K. In KCl, measurements of all three constants were made over the entire temperature range down to 4°K and their values at 0°K are c11=4.83, c12=0.54, and c44=0.663 in units of 1011 dynes/cm2. The Debye characteristic temperature at 0°K, theta0, has been calculated from these very low-temperature values of the elastic constants to give 129°+/-2°K for KI and 234°+/-1°K for KCl. The specific heats of KI, KCl, and KBr have been calculated in this paper by using a combination of a Debye and an Einstein term in the expression for the specific heat. These calculations indicate that the agreement between such a simple theory and experiment becomes better as the ratio of ionic radii and masses increases in these three salts. We point out that in the four alkali halides thus far studied in this laboratory, there is evidence that the Cauchy relation, c12=c44, is satisfied at about T=theta0. No reason is offered for this coincidence.
A simple relation is derived between the compressibility and absorption frequency of ionic crystals. This relation takes into account the change which a displacement induces in the electronic configuration, and is independent of the uncertainty concerning the Lorentz force. Comparison with experimental data shows very good agreement in the case of the alkali halides. The changes in the electronic configuration of each ion are described in terms of a dipole moment and in terms of 'local distortions'. The latter are caused by forces not included in the Lorentz force, and it is assumed that they are localized in small regions in the immediate neighbourhood of the neighbouring ions. The local distortions are responsible for the deviation of the polarization from the value given by the Lorentz force; from the sign of this deviation it is concluded that they are due mainly to the non-electrostatic repulsive forces. Although this model allows deviations from spherical charge distribution around the ions, it nevertheless permits the splitting of the energy of certain types of displacements into two-body interactions and leads to the Cauchy relations between the elastic constants.
Phonon frequencies in KC1 and RbC1 have been determined by neutron scattering. The results are used to interpret the second-order Raman spectra in these solids. Assignments of two-phonon processes to the reported Raman peaks are made. It is found that most of these peaks can be accounted for.ZusammenfassungPhononenfrequenzen in KC1 und RbC1 wurden durch inelastische Neutronenstreuung gemessen. Die Ergebnisse werden verwendet um die Ramanspektra von zweiter Ordnung zu erklären. Zweiphononprozessen werden den Ramanspitzen zugeordnet. Fast alle von diesen Spitzen können erklärt werden.
In a systematic study of phonon dispersion relations in NaCl, KCl, and RbCl results were also obtained on phonon widths in some branches, primarily at 80 °K, but to some extent also at 300 °K. The widths for all three chlorides are found to be mostly between 0.04 and 0.10 (in units of 1013 rad s−1). In all cases the increase of the width between 80 and 300 °K is less than the estimated error of the width at 80 °K. The results are discussed and some of the difficulties of such measurements are pointed out.In einem systematischen Studium von Dispersionskurven in NaCl, KCl und RbCl wurden Resultate für Phononenbreiten bei 80, zum Teil auch bei 300 °K erhalten. Die Breiten lagen meistens zwischen 0,04 und 0,10 (Einheit: 1013 rad s−1) für alle drei Chloride. Die Verbreiterung zwischen 80 und 300 °K lag immer innerhalb des berechneten Fehlers der Breite bei 80 °K. Die Ergebnisse werden diskutiert und einige der Schwierigkeiten solcher Messungen hervorgehoben.
The "shell model" of an alkali halide is extended to take into account short-range forces between both first- and second-nearest neighbor atoms in the crystal, the polarizability of both ions, and the possibility that the ionic charge may be less than one electronic charge. The arbitrary parameters of the model have been obtained by means of a least-squares fit to the measured dispersion relations for the lattice vibrations, the dielectric constants, and the elastic constants. For NaI and KBr, models have been found which give excellent agreement both with these measurements and with measurements of the specific heat. This good agreement is, however, obtained only when the simple shell-model concept of the ions is to some extent abandoned. The reasons for this, and the connection with work of other authors, are discussed.
Dispersion curves for the lattice vibrations propagating in the [00zeta], [zetazeta0], and [zetazetazeta] directions in NaI at 100°K and in KBr at 90°K have been measured using neutron spectrometry, and the results compared with calculations based on a simple shell model. Both substances obey the Lyddane-Sachs-Teller relation. In addition, some measurements were made on KBr at 400°K most frequencies showed a decrease of a few percent. At this temperature, the acoustical modes show no significant energy broadening, the transverse optical modes show slight broadening, and the longitudinal optical modes show very considerable broadening. The anomalous broadening of the LO modes is not yet understood and requires further study. It appears to be specimen-dependent as well as temperature-dependent.
The frequency/wave vector dispersion relation nuj(q) for the normal modes of vibration of pure potassium iodide at 90°K has been measured by inelastic neutron scattering techniques. The experiments were performed with both a time-of-flight machine and a triple-axis crystal spectrometer. The results obtained from the latter spectrometer have been analyzed in terms of various interionic-force models. A satisfactory fit to the results is provided by an eleven-parameter dipole approximation model, which allows for the polarizability of both ions. Comparison of normal-mode frequencies calculated from this model with those observed in the time-of-flight experiments shows generally good agreement. The frequency distribution function for the normal modes, computed from the best-fit model, displays a well-defined energy gap, from 8.65 to 11.85 meV, separating the acoustic and optic modes. The moments of this function are in excellent agreement with values derived from heat-capacity data. By assuming an exponential form for the nearest-neighbor short-range force constants, the temperature variation of the thermal expansion, and the frequency shift and inverse lifetime of the transverse optic mode of very long wavelength, have been calculated and compared with the available experimental results. The occurrence of localized impurity modes in the energy gap in the frequency distribution for potassium iodide doped with potassium nitrite is briefly discussed.
Long-wavelength acoustic phonons have been studied for each of the [00zeta]T, [00zeta]L, [0zetazeta]T1, and [0zetazeta]L branches in KBr at 95 and 463°K by means of inelastic neutron scattering. This study has shown that the elastic constants of KBr determined by inelastic neutron scattering (zero-sound regime) exhibit a different temperature dependence from those determined by ultrasonic techniques (first-sound regime). For example, the change in the zero-sound elastic constant for the [0zetazeta]T1 branch between 95 and 463°K is (23+/-4)%, while the change in the first-sound elastic constant determined by Haussühl using ultrasonic techniques is (35.1+/-1.2)%. Cowley's theory for first and zero sound shows reasonable agreement with experiment.
Vibrational distribution functions are derived for the sodium chloride lattice for a series of increasingly dense, evenly distributed points in the Brillouin zone. The rigid-ion and the deformable-ion models are compared, and the effect of including next-nearest-neighbor repulsive interactions is considered. Calculation of the Debye characteristic temperature {\Theta{}}_{D}(T) and the moments of the distribution {\mu{}}_{m} as functions of sample density indicates that for the densest sample used the {\Theta{}}_{D}(T) curve is established down to 2\ifmmode^\circ\else\textdegree\fi{}K and the errors in the moments are insignificant for m\ge-. Singularities in the density-of-states and the combined density-of-states distributions are related to critical points and crossover features in the dispersion curves, which are also evaluated for high densities of points along certain symmetry directions. Comparison is made with the experimental second-order Raman spectrum of rocksalt, and it is shown how these measurements provide extensive indirect tests of the details of the theoretical lattice frequency spectra.
A set of electronic polarizabilities has been obtained from a least-squares fit of experimental refraction data using simple additivity and a Lorentz factor of 4π/3. Except for the fluorides, the electronic polarizability values of the alkali-halide crystals calculated from this set agree with the experimental data within 3 percent. Similar least-squares fits were attempted with various values of the Lorentz factor, the best fit being obtained for 4π/3. On the basis of 4π/3, the additivity assumption and the alkali-halide set, polarizabilities have been obtained for other ions. The best values for the sodium D line in A3 are Li+ 0.03, Na+ 0.41, K+ 1.33, Rb+ 1.98, Cs+ 3.34, F- 0.64, Cl- 2.96, Br- 4.16, I- 6.43, Ca++ 1.1, Sr++ 1.6, Ba++ 2.5, O-- 0.5-3.2, S-- 4.8-5.9, Se-- 6.0-7.5, Te-- 8.3-10.2, Ag+ 2.4, Cu+ 1.6, Cu++ 0.2, Zn++ 0.8, Cd++ 1.8, Ge4+ 1, Sn4+ 3.4, Pb++ 4.9. Values represented by a spread indicate ions that cannot be treated additively.
Compressibility of eleven alkali halides and its variations with pressure and temperature have been determined by measurements by Bridgman's new method, up to 12,000 atm. for both 30° C and 75° C. The samples were all single simple cubic crystals each grown from the melt in a new way, described elsewhere. The error in the values of the compressibility at zero pressure, κ0, is probably less than one per cent; in the values for variation with pressure, ψ0, and temperature the error may be 5 and 20 per cent respectively. By extrapolation approximate values of κ0 for absolute zero are found. Periodic relations. Both κ0 and ψ0 when plotted against the alkali ion for a series of salts of the same halogen ion, or vice-versa, show similar behavior. The curves break sharply at the ion similar to argon (K or Cl) the rate of increase suddenly decreasing. This behavior is also shown by the grating space as measured by Davey, and tends to corroborate Bohr's theory of atomic structure according to which there is a discontinuity in atomic formation at argon, additional electrons going into inner shells.
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.
Measured values of the static dielectric constant between 1.5°K and 290°K are reported for seventeen metal halides together with calculated values of the effective charge parameter and the L0 mode frequencies at 4.2°K.
The paper comprises theoretical and experimental studies 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 model 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 Kármán crystal in which there are two "atoms" of differing charge at each lattice point, one of the "atoms" having zero mass. The model has been specialized to the case of an alkali halide in which only one atom is polarizable, and computations of dispersion curves have been carried out for sodium iodide. We have determined the dispersion ν(q) relation of the lattice vibrations in the symmetric [001], [110], and [111] directions of sodium iodide at 110°K 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%. The dispersion relation for the longitudinal optic (LO) branch was determined for the [001] directions with less accuracy. Frequencies of some important phonons with their errors (units 1012 cps) are: TA[0,0,1]1.22±0.04, LA[0,0,1] 1.82±0.06, TA[½,½,½]1.52±0.05, LA[½,½,½]2.32±0.06, TO[0,0,0] 3.60±0.1, TO[0,0,1]3.80±0.1, TO[½,½,½]3.50±0.1. 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.
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 adiabatic elastic constants of NaCl have been measured at 77.3°K and 4.2°K using the ultrasonic pulse echo technique. The results are as follows (in units of 1011 dyne cm−2: View Within Article where . These values have been used to calculate the adiabatic bulk modulus and to re-evaluate the low temperature value of the Gruneisen parameter as determined from thermal expansion data and microscopic mode gammas.
Thermo-elastic constants have been measured on 15 alkali halides of the NaCl type with ultrasonic methods. The constants show only small variations within the crystal group and may therefore be regarded as typical for NaCl type crystals.
In elastic scattering experiments, it is appropriate to consider the scattering pattern in the space of momentum and energy transfers, and a proper treatment of resolution takes account of momentum and energy together, both as regards the correlation of momentum and energy increments in the beams of incident and scattered particles and as regards the structure of the cross section under observation. A description is given of the calculation of resolution widths in measurements on a phonon dispersion surface with a three‐axis crystal spectrometer, and of the manner in which the two main contributions to the energy resolution width may be minimized by focusing— which involves a suitable choice of the monochromator and analyser crystals, the beam path, and the incident and scattered energies. Focusing usually improves resolution markedly without loss of intensity, and often with a gain in the peak intensity of a recorded resonance. An experimental illustration is given. Cases where focusing entails loss of intensity are discussed in connection with intensity formulas.
The infrared lattice reflection spectra of LiCl, LiBr, KF, RbF and CsF have been measured in the spectral region from 95 to 1500 cm−1 and have been analysed by the Kramers-Krönig method. The following fundamental dispersion frequencies have been obtained: LiCl, 191 ± 2 cm−1; LiBr, 159 ± 4 cm−1; KF, 190 ± 2 cm−1; RbF, 156 ± 2 cm−1; CsF, 115(?) cm−1. The effective charges and compressibilities have been computed with the aid of the Szigeti relations.
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.
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