Atomic Force Constants Research Articles

Thermal Conductivity of Paramagnetic Salts at Low Temperatures

We present a simple theory of the effect of impurity-induced changes in atomic force constants and changes in atomic mass on the lifetime of coupled spin-phonon modes in paramagnetic crystals. The result is then employed to study the effect of impurity scattering, boundary scattering, and scattering of the coupled modes by longitudinal fluctuations in spin density on the temperature dependence of the thermal conductivity, and the frequency distribution of the heat flux. The theory can account for the experimental data on MgO doped with ${\mathrm{Cr}}^{2+}$ reported recently by Challis, McConachie, and Williams and offers support for the interpretation of the data presented by these authors.

A lattice theory of morphic effects in crystals of the diamond structure

A microscopic theory of the electric field induced infrared absorption by crystals of the diamond structure is presented in this paper, together with a determination of the modifications in the first-order Raman spectra of such crystals induced by externally applied electric fields and stresses. In connection with this, the shifts and splittings of the triply degenerate k = 0 optical mode frequencies in crystals of the diamond structure in the presence of a static electric field or a strain are determined. It is shown that the shifts in frequencies are of second order in the electric field and are of first order in the strain. In both cases the frequency shifts can be expressed in terms of three parameters. Expressions for these parameters are obtained in terms of atomic force constants and derivatives of the electronic polarizability. The magnitudes and signs of the frequency shifts are estimated on the basis of a simple model of diamond-type crystals.

Contribution to the Spin-Wave Theory of a Dilute Heisenberg Ferromagnet

A spin-wave theory of a dilute Heisenberg ferromagnet is studied using Green's function techniques. The Green's function for one-magnon excitations is calculated in two ways, a decoupling method and a diagram method. An expression for energy is applied to determine the critical concentration for which the ferromagnetic ground state is unstable with respect to the formation of long-wavelength spin waves. For a simple cubic crystal, the result lies close to that obtained using a cluster expansion method. §I. Introduct:i.on The spm-wave theory of a ferromagnetic crystal, contammg some concen­ tration of magnetic atoms of another sort, has been studied by several authors. 1 ) The problem is similar to the lattice vibration problem in which both the mass and the effective atomic force constants are changed. The structure of the energy spectrum which has been obtained within the framework of one-vertex approxi­ mation reflects the existence of localized or quasi-localized levels in the single­ impurity problem.

Phonon Theory of Solid-State Diffusion Including Anharmonic Effects

An anharmonic-lattice-vibration theory of solid-state diffusion, based on classical theory developed previously, is presented in second-quantization form and is evaluated for Cu. Equilibrium statistical mechanics was used, and the Goldstone diagrams were isolated for the interacting-phonon events which contribute to ${D}_{0}$. A displacement transformation suitable for the inhomogeneous strain inherent in the migration mechanism was used. The results of a comprehensive theoretical analysis of the migration portion of ${D}_{0}$ were estimated numerically for Cu, using a nearest-neighbor Debye approximation. The values of ${D}_{0}$ obtained, including an experimental value of the entropy of vacancy formation, were 0.106, 0.091, and 0.078 for temperatures of 293, 793, and 1293\ifmmode^\circ\else\textdegree\fi{}K, respectively. The temperature dependence shown by ${D}_{0}$, in the classical limit, was due to the $T$ dependence of the atomic force constants, thermal expansion, and $T$-dependent anharmonic terms. At lower temperatures, quantum effects introduced terms with inverse powers of $T$ and mass. At 293\ifmmode^\circ\else\textdegree\fi{}K, which is $0.865{\ensuremath{\Theta}}_{D}$ for Cu, there is a 7% decrease of ${D}_{0}$ due to quantum terms. The anharmonic terms introduced no direct effects on the mass dependence of ${D}_{0}$. If the first few anharmonic terms are included in an expansion of the activation energy, the first and most important term is linear in $T$ in the classical limit. Hence, it appears in the experimentally measured ${D}_{0}$, rather than in the activation energy. Therefore, harmonic-lattice-vibration theories or elastic theories of the activation energy should be realistic. A suggestion is made regarding the possibility of controlling the diffusion process by the artificial stimulation of phonons using laser radiation in a selected frequency range.

Dispersion Relations for Hexagonal Close-Packed Crystal Lattices

The dispersion relations for waves propagating in the [0001] and [$01\overline{1}0$] directions were calculated, using a Born---von K\'arm\'an model of the hexagonal close-packed crystal lattice. The interactions between nearest neighbors in the plane were considered to be central forces, while the interactions between nearest and next-nearest neighbors out of the plane were considered to be noncentral forces, i.e., the interactions involved both central and angular forces. The atomic force constants were evaluated (1) from the elastic constants by the method of long waves, (2) from a least-squares analysis of the eigenvalues of the secular equation for elastic waves propagating in certain symmetric crystallographic directions at critical points in the Brillouin zone, i.e., from a least-squares analysis of the dispersion relations, and (3) from a least-squares analysis of the elastic constants and the dispersion relations. The correlation between theoretical and experimental dispersion relations from the noncentral-force model is good for magnesium and zinc and is limited for beryllium. If the atomic force constants have been calculated by a least-squares analysis of certain points in the dispersion relations with or without the elastic constants, these atomic force constants greatly enhance the agreement between theory and experiment. As a result of this work it seems that better agreement between theory and experiment is obtained if angular forces are included in a model of the hcp crystal lattice rather than simply extending the sphere of interaction to more and more neighbors.

Noncentral force model for hexagonal close‐packed crystal lattices (ii)

AbstractA Born‐von Karman model of the hexagonal close‐packed crystal lattice was considered using a combination of central and angular forces between various sets of neighbors. The interactions between nearest neighbors in the plane were considered to be central forces while the interaction between both nearest and next‐nearest neighbors out of the plane were considered to be noncentral forces, i.e. the interactions involved both central and angular forces. The equations of motion were evolved and the atomic force constants were evaluated by three different methods. The atomic force constants were evaluated 1. from the elastic constants by the method of long waves, 2. from a least squares analysis of the eigenvalues of the secular equation for elastic waves propagating in certain symmetric crystallographic directions at critical points in the Brillouin zone, i.e. from a least squares analysis of the dispersion relations, and 3. from a least squares analysis of the elastic constants and the dispersion relations. The eigenvalues of the secular equation were calculated for a mesh of seven hundred thirty‐five (735) points in that portion (one twenty‐fourth) of the Brillouin zone irreducible under symmetry operations. The frequency spectrum was obtained from these eigenvalues. The lattice specific heats as a function of temperature were calculated using the appropriate frequency distribution function. The Debye and Einstein characteristic temperatures were also calculated as a function of temperature and compared to experimental values. The insensitivity of the calculated thermodynamic properties of the hexagonal close‐packed crystal lattice on the frequency spectrum is demonstrated for zinc by calculating three different frequency spectra, one for each method of evaluating the atomic force constants, and from these spectra computing the lattice specific heats as a function of temperature. The Debye characteristic temperatures do not change significantly although the details of the frequency spectra differ appreciably. The conclusion has been drawn that better agreement between theory and experiment is reached if angular forces are included in a model of the hexagonal close‐packed crystal lattice rather than simply extending the sphere of interaction to more and more neighboring atoms.

The influence of the electron - phonon - interaction on the interatomic force constants

A procedure following the ideas of Fröhlich and Nakajima is developed to determine not only the frequency but also the polarization vectors of the lattice waves and the Fourier transforms of the general atomic force constants in metals.

Crystal Dynamics of Copper

The frequency/wave-vector dispersion relation for the symmetry directions [$00\ensuremath{\zeta}$], [$0\ensuremath{\zeta}\ensuremath{\zeta}$], [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$], and [$0\ensuremath{\zeta}1$] in copper at room temperature has been determined by inelastic neutron scattering. Interplanar and atomic force constants have been obtained. First-neighbor interactions are dominant, but longer range forces, extending to at least sixth-nearest neighbors, are also present. Physically realistic errors have been assigned to many of the force constants; to our knowledge this has not been done previously. The analysis of the data, and the excellent agreement between measurements carried out at the Chalk River and McMaster University reactors, indicate that, on the average, the errors conventionally assigned to frequencies in neutron-scattering studies are too large by about a factor of 2 to be standard deviations. Frequency distributions have been obtained and used to calculate various thermodynamic quantities which agree quite well with experiment. (Discrepancies of a few percent appear to be accounted for by anharmonic effects.) Our results are in rather good agreement with those of Sinha obtained by quite different experimental methods.

Concentration Dependences of Elastic Constants and Debye Temperatures for Binary Alloys

Concentration dependence of elastic constants for binary alloys with cubic symmetry are calculated by the lattice theory of Born and von Karman. Central forces between nearest are next-nearest neighbors and angular forces only between nearest neighbors are taken account. Atomic force constants for A-B alloys are given by the average of the atomic force constants for A-A, B-B and A-B pairs weighted by the probability of appearance for these pairs. The elastic constants are expressed by the mean atomic force constants and their concentration dependence is given by a quadratic equation of concentration, which is consistent with the experiments for Ag-Au and Cu-Ni alloys. The variation of the Debye temperatures with concentration is calculated by using the calculated elastic constants and its result is compared with the experiments.

Scattering of Phonons by Dislocations

The theories of Carruthers and Klemens of the scattering of phonons by dislocations are compared, and it is shown that Carruthers' approach yields larger cross sections in several ionic crystals: LiF, NaF, NaCl, KCl, KBr, KI, NaI, and MgO. Klemens employs a Grüneisen model for anharmonic coupling, while Carruthers deals directly with the atomic force constants. In the limited number of cases where experiments reveal the expected τ−1 ∝ ω (K ∝ T2) dependence, Carruthers' theory agrees better with the observed results, although both approaches underestimate the strength of the scattering. It is suggested that dislocations acting cooperatively might be the cause of the discrepancies.

Analysis of Debye-Waller-Factor and Mössbauer-Thermal-Shift Measurements. I. General Theory

We give a simple theoretical analysis of the dynamics of an arbitrary atom in a general harmonic solid. The atom under discussion may be an impurity. Several general results are found which limit the possible temperature dependences of the mean-square displacement and mean-square velocity in a way which is described. These results are expected to be most useful in analyzing experiments involving Debye-Waller-factor and M\ossbauer-thermal-shift measurements. As an illustration the allowed range of mean-square displacements at $T=0$ and $T=80\ifmmode^\circ\else\textdegree\fi{}$K corresponding to a measured value at $T=298\ifmmode^\circ\else\textdegree\fi{}$K is given. These results also provide strong consistency relations that experimental data or numerical calculations should satisfy. One especially interesting result indicates a possible method for determining a simple sum over atomic force constants from Debye-Waller-factor measurements. This sum, which in general is not obtainable from any other type of measurements, could be used as a convenient check on atomic-force-constant models. The dependence of the mean-square displacement and mean-square velocity on the various masses and force constants in the lattice is described. The relation between our results derived in the harmonic approximation and experimentally measurable quantities is discussed. Finally, several experiments which appear to be interesting are mentioned.

Lattice Vibrations and Debye Temperatures of Copper and Aluminum

Dynamical matrix elements have been presented for the axially symmetric model for cubic crystals, when interactions out to any neighbor are considered. The model has been applied to obtain the vibration spectra and Debye temperatures of copper and aluminum. In the case of copper, results have also been presented for the phonon-dispersion curves, vibration spectrum and Debye temperatures as obtained from the tensor-force model using White's atomic force constants. Comparison of the results with experimental data shows that the axially symmetric model gives rather poor results for Debye temperatures of Cu and Al.

Mathematical Techniques in the Study of Coriolis Coupling in Tetrahedral XY4 Molecules

AbstractRelations between Coriolis coupling coefficients in tetrahedral XY4 molecules are evaluated, using the method of Cα and related matrices. It appears that the Coriolis coupling may be studied in terms of two independent Coriolis coefficients, e.g. ζ24 and ζ44. For these two quantities explicit equations are deduced, containing atomic masses, vibrational frequencies (in terms of λ values), and force constants of the molecule.

Temperature Dependence of the Debye-Waller Factor for Copper and Aluminum

The Debye-Waller factor for copper has recently been determined by Flinn et al. by making x-ray intensity measurements from 4-500\ifmmode^\circ\else\textdegree\fi{}K. Flinn et al. were able to explain their results using a central force model for the copper lattice. However, it is well known that this model is inadequate in explaining the vibrational properties of the copper lattice over the entire wavelength region. Since various sets of force constants have been proposed for copper, we calculated the Debye-Waller factor for these sets. It is found that with the present experimental accuracy it is impossible to select between them. Calculations are also given for aluminum using Walker's atomic force constants.

On the relation between grey and white tin (α-Sn and β-Sn)

A model capable of representing the structure of either grey or white tin and susceptible to lattice dynamical analysis has been studied. Values of atomic force constants for the models have been estimated and used in the appropriate secular equations to calculate frequency distributions of lattice modes for the two phases. From these distributions, thermodynamic quantities (in particular the specific heat and the vibrational contribution to the free energy) have been computed and the results are compared with observed data; fair agreement is found in view of the assumptions involved and the lack of any observed elastic constants for grey tin. The values of the force constants in relation to the phase change are discussed and a suggestion about the interpretation of the angular stiffnesses in terms of the interaction of the Fermi surface with Brillouin polyhedra is offered.

Lattice Vibrations of Zincblende Structure Crystals

In this article we present expressions, based on the shell model of lattice dynamics, for all important long-wavelength properties of the lattice vibration spectrum of a zincblende structure crystal, and for the vibration frequencies having wave vectors at the Brillouin zone boundary in the [100] direction. General first and second neighbor short-range force constants are used. The formulas are presented in the distortion dipole form of Mashkevich and Tolpygo, which uses one constant, $\ensuremath{\alpha}$, to describe the electronic polarizability of each atom, rather than the two redundant constants $Y$ and $k$, the shell charge and polarization spring, respectively, used in the shell model. Some of the force constants in our expression are evaluated for GaAs, InSb, AlSb, and ZnS with the aid of available experimental data. We were unable to fit all of the data for any substance with only first neighbor force constants. When small, realistic values of second neighbor constants were used, a multitude of fits resulted. The absence of a criterion determining the best fit results in a spread of possible ionic charge values of about one electronic charge for the 3-5 compounds, and an inability to decide whether ZnS is very ionic or rather covalent. Our results point up the necessity of calculations of atomic force constants from fundamental quantum mechanics.

Lattice Dynamics of White Tin

The theoretical lattice vibrational spectrum for white tin (body-centered tetragonal) is examined on the basis of the axially symmetric lattice dynamics model including fourth-nearest-neighbor interactions. The atomic force constants are derived using experimental data for the elastic constants, specific heat, Debye temperature, and the Debye-Waller factor at 100\ifmmode^\circ\else\textdegree\fi{}K. The 6\ifmmode\times\else\texttimes\fi{}6 dynamical matrix is diagonalized along the principal symmetry directions and the resulting dispersion curves are presented. The spectrum is characterized by low-lying optical modes which interact strongly with the acoustic modes, particularly near the surface of the first Brillouin zone. These optical modes give rise to a large density of states at intermediate acoustical frequencies and contribute significantly to both the specific heat and Debye-Waller factor at low temperature.

The determination of the vibrational entropies of solute atoms in dilute copper solid solutions by the measurement of vapour pressures

An effusion method has been used to investigate the equilibrium between dilute copper based solid solutions and their vapours over a large range of temperature. Neutron irradiated alloys were used and a gamma-ray spectrometer enabled measurements on very dilute alloys to be carried out. The change of vibrational entropy S + M − S 0 v occurring when a solvent atom is replaced by a solute atom and the corresponding enthalpy change H + u − H 0 v were derived from the experimental data. The results are given in the table together with the values of H + u , the energy of a solute atom in the solution with respect to a solute atom at rest in a vacuum. System H u + − H v 0 kcal mole H u + kcal mole S u + − S v 0 k Cu-0.1 At. %Ag 26.6 ± 0.69 −45.9 ± 0.69 4.71 ± 0.29 Cu-0.5 At. %Au 2.01 ± 0.29 −70.5 ± 0.29 7.07 ± 0.12 Cu-2.5 At. %Au 1.84 ± 0.26 −70.7 ± 0.26 6.85 ± 0.11 To explain the very large increases in vibrational entropy two models have been discussed. In the first, atomic force constants were used to estimate the change in vibrational frequency with solute diameter for a cluster of 13 atoms. In the second model elastic theory was used to calculate the change in entropy due to the shear and dilational strains around a solute atom. The experimental values of S + u - S 0 v lie between those derived from the two models. Experimental values of the vibrational entropies deduced by other workers from solvus curves and e.m.f. measurements are also compared with the calculated values.

Lattice Dynamics of Alpha Uranium

The method developed by Begbie and Born has been applied to alpha uranium, where equations are developed which give the macroscopic elastic constants in terms of the microscopic force constants. Interactions of an atom with its first through fourth nearest neighbors, which involve twelve atoms, are considered. Through symmetry considerations, nineteen atomic force constants enter into this force system. An independent determination of the force constants is required before a valid verification of the solutions can be made. However, using measured values of the nine elastic constants, two sets of force constants are evaluated, one based upon quasi-central forces and the other upon neglect of fourth nearest neighbors. (auth)

A method of evaluating diffusion coefficients in crystals∗

The mechanism of diffusion proposed by Rice is reconsidered with the aid of an extension of Kac's theorem. The analysis yields an activation energy which, as predicted by Zener, is simply related to the minimal local deformation energy. It is also found that in contrast with Rice's treatment, the activation energy may be calculated directly from the atomic force constants without resorting to normal mode analysis.