Cauchy Relation Research Articles

Crystal dynamics of niobium

The crystal dynamics of niobium is studied employing a two parameter model. The model satisfies the Cauchy relation and the symmetry requirements of the lattice, and the lattice is in equilibrium without recourse to external forces. The agreement between theory and the experiments is fairly good.

Ab Initio Calculations of the Effect of Non-Rigid Ions on the Statics and Dynamics of Oxides

The potential induced breathing of oxygen ions in a crystal, which plays a major role in explaining the violation of the Cauchy relations and the splitting of the longitudinal and transverse optic frequencies in alkaline earth oxides, is analyzed for BaTiO3 in the perovskite structure and the distorted structure corresponding to the tetragonal phase. The results indicate that nonspherical ions are required to describe the ferroelectric transition.

Phonon frequencies and Debye temperature of copper

The phonon frequencies at low temperature and the temperature variation of the Debye temperature of copper have been calculated using a two-parameter model. The model satisfies the Cauchy relation and the symmetry requirements of the lattice, and the lattice is in equilibrium without recourse to external forces. The theoretical phonon frequencies along the principal symmetry directions and the Debye temperature of copper exhibit good agreement with the available neutron scattering and calorimetric measurements.

Bulk elastic properties of metals determined by Brillouin scattering and its application to RCu2Si2(R=rare earth)

Measurements of Brillouin scattering from surface-wave excitations of metals by means of a tandem Fabry-Perot interferometer according to Sandercock allow the reliable determination of bulk elastic moduli and elastic constants, respectively, of polycrystals and single crystals. This determination is independent of Cauchy's relation (cL=3cT, or c12=c44). The accuracy of the method of 5-10% (depending on the penetration depth of the light) has been demonstrated for cubic rare-earth (R) compounds with known elastic constants, such as CePd3 (polycrystal) and CeAl2 (single crystal). The method has been applied to determine the bulk moduli cB of polycrystalline tetragonal RCu2Si2 at 300K with an accuracy of better than 10%. The stable-valent compounds follow cB varies as Q/V (Q=valence of the R ion, V=unit cell volume). With respect to this behaviour, valence instabilities are identified by the softening of cB of EuCu2Si (32%) and YbCu2Si2 (43%). The 'heavy-fermion' superconductor CeCu2Si2 shows a 14% softening of cB with respect to the reference compounds.

Interaction potentials for AgCl and AgBr

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 ${c}_{44}$, 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 ${\mathrm{Ag}}^{+}$ ion.

The low-temperature variation of the elastic constants of lithium hydride and lithium deuteride

The adiabatic elastic constants C11, C12and C44of7LiH and7LiD have been obtained from measurements of ultrasonic wave transit times through single crystals over the range 4.2-300K. The temperature variation follows quite closely the approximation Cij(T)=Cij(0)(1-D epsilon ). The anisotropy factor A=C44/(C11-C12) and the deviation ( Delta =C12-C44) from the Cauchy relation have been used to make comparisons with the alkali halides. It has been found that the Gruneisen parameter gamma is approximately constant at 1.2 for both7LiH and7LiD from 50 to 270K. There are signs of anomalous behaviour of the elastic constants in the region of 11K, where a lambda -type heat capacity anomaly has been reported.

Analysis of a generalized N-particle interaction with application to the elastic constants of BCC crystals

In an earlier work a self-consistency condition for lattice dynamic models was proposed and applied for the purpose of comparing a number of existing models for BCC and FCC crystals. The unphysical result that the Cauchy relation is always satisfied, even in the general tensor force constant model, was obtained. In order to resolve this unsatisfactory situation the previous analysis is re-examined in some detail. The possibility that deficiencies exist in the form assumed previously for the internal potential energy function is checked by starting with a generalized form. That is, the potential energy is written as the sum of arbitrary two-particle and arbitrary three-particle angular contributions. It is then shown that general translational and rotational invariance requires the former to be central whereas the latter satisfy these invariances by definition. It thus follows and is demonstrated that the self-consistency condition was implemented improperly previously and that, in fact, the resultant interparticle forces give rise to pure deformation modes without restriction. This reverses the above unphysical conclusion. As an illustration all of the elastic constants for monatomic BCC systems are determined and those obtained from static deformations are shown to be identical to those obtained from lattice dynamics. The present analysis yields the detailed structure of the standard force constants and demonstrates that it is essential to interpret them as composite quantities. Further, the present analysis is sufficiently complete and general that it gives a basis for the discussion of any particular model.

Hellmann-Feynman theorem, elastic moduli, and the Cauchy relations

The Hellmann-Feynman theorem is used to derive a formula for the harmonic elastic moduli of crystals in which every atom is at a center of symmetry in terms of the electron density and its distortion due to the crystal deformation. It is shown that these moduli satisfy the Cauchy relations if the electron density distribution deforms according to the same affine transformation which governs the crystal deformation.

Third-order elastic constants of orthorhombic calcium formate

All 20 independent third-order elastic constants (TOEC) of orthorhombic calcium formate, Ca(HCOO)2, Ca2+. C2H2O2-4, space group Pcab, have been determined with the aid of stress-induced shifts of ultrasonic resonance frequencies. Longitudinal and transverse effects possess negative values indicating the high stability of calcium formate over a wide pressure range. The anisotropy of TOEC, characterized by --c111 > --c333 > --c222, corresponds with the anisotropy of second-order elastic constants (c11 > c33 > c22). The value 5.22 for the pressure derivative of the inverse bulk compressibility coincides with the quasi- invariant value of ca 5 observed in most stable cubic crystals so far investigated. There exists a pronounced departure from third-order Cauchy relations: the 'transverse' constants exceed the corresponding 'shear' components considerably in absolute magnitude in accordance with the behaviour of the second-order Cauchy relations. These effects are explained as originating from the non-centrosymmetric formate ions. A comparison of TOEC of calcium formate and cubic calcium fluoride confirms the typical contribution of the formate ions to the third-order elastic behaviour.

Elastic behaviour of YAG under pressure

Measurements of the effects of hydrostatic and uniaxial pressures on ultrasonic wave velocities are used to obtain the six third-order elastic constants of yttrium aluminium garnet (YAG). Results are used to investigate whether the shear isotropy and the conformity with the Cauchy relation (indicative of central forces), which hold for the second-order elastic constants of YAG, extend into the region of finite strain which determines higher-order elasticity. The compression has been extrapolated up to 150*108 Pa using the Murnaghan equation of state. The vibrational anharmonicity of YAG and yttrium iron garnet (YIG) are compared through the acoustic-mode Gruneisen gammas computed from the pressure dependences of the elastic constants. The attenuation of transverse microwave ultrasonic waves propagated along the (001) direction (the mode used in acoustic devices) due to anharmonic interactions with thermal phonons, calculated at the limits omega tau >>1 and omega tau <1, is shown to be in good agreement with previous experimental attenuation measurements for this mode.

Analysis of the elastic properties of alkali halides

A quantitative explanation for the breakdown of the Cauchy relation for elastic constants of alkali halides has been presented in terms of overlap integrals for neighbor ions using the Lowdin-Lundqvist formulation of the many-body potential. Values of the Cauchy deviation (${c}_{12}\ensuremath{-}{c}_{44}$) are found to present close agreement with experimental data. The analysis has been extended to evaluate the third-order elastic constants and the pressure derivatives of the second-order elastic constants of NaCl-structure alkali halides. An adequate model for the interionic potentials has been developed by considering the short-range repulsive interactions between first and second nearest neighbors and van der Waals dipole-dipole interactions. The results obtained in the present paper compare well with recent experimental data.

Influence of long-range three-center potentials on the lattice dynamics of LiD

The long-range three-center contribution to the dynamical matrix is calculated for LiD from first principles. In addition to nearest-neighbor Li-D overlap, the theory includes also second-nearest-neighbor D-D- overlap of the localized wave functions. We find that the inclusion of the three-center terms improves the agreement of the theoretical and experimental dispersion curves, especially in the TO branch. However, the corrections are rather small. In particular, it is shown that the long-range three-center terms can account only partly for the large violation of the Cauchy relation found in LiD.

Third order elastic constants of cubic K2Zn(CN)4 and K2Hg(CN)4

Abstract The complete third order elasticity tensor has been determined for K 2 Zn(CN) 4 and K 2 Hg(CN) 4 by measuring stress induced shifts of resonance frequencies in thick plates. K 2 Hg(CN) 4 , and in a smaller extent also K 2 Zn(CN) 4 , possess highly anomalous third order elastic constants, not observed until now in any other crystalline material. The most surprising properties of K 2 Hg(CN) 4 are: a negative pressure derivative of the inverse bulk compressibility, an extreme deviation from Cauchy's relation, and a high anisotropy of the pressure derivative of Poisson's ratio under uniaxial stress. The components c 112 and c 123 exhibit large positive values whereas c 111 is much smaller in magnitude than in other stable crystals like alkali halides. The anomalous behaviour still increases at lower temperatures. There exists a relation between the anomalous transversal interactions as expressed in Poisson's ratio and the structural differences in the high pressure (or low temperature) ferroelastic phase and the corresponding cubic prototype. A certain similarity to the third order elastic behaviour of fused quartz is present.

Comparison of the major force constant models for cubic systems using a self-consistency condition

Previously noted similarities between the central pair potential, CPP, the DeLaunay angular force, DAF, and the Lehman, Wolfram, and De Wames axially symmetric, AS, models for lattice dynamics coupled with criticisms of the DAF model, not applicable to the CPP model, has resulted in a puzzling and confusing picture. New physical arguments are used to develop a self-consistency condition, essentially unrecognized previously, which must be included in the use of all these force constant models. In this event, all of the above three models are proven to be identical. We then go on to examine the Clark–Gazis–Wallis, CGW, model. A systematic procedure is presented allowing a practical extension of the expression for the force constants from the current first and second nearest neighbour interactions to include third nearest neighbour interactions. For fcc systems we show that the previous expressions are incomplete and, consequently, overly restrictive and misleading and, hence, causing a current controversy. Also, we point out that the CGW model reproduces the general tensor force, GTF, model, at least to third nearest neighbour interactions. It is important to note that the application of the above mentioned self-consistency condition restores the Cauchy relation in all the models. Finally, numerical results are presented for the phonon spectra of the bcc metals Na and Nb and the fcc metals Cu and Au in order to demonstrate the practical effect of the self-consistency condition. This effect is seen to be larger for metals having larger Cauchy discrepancies. It is inferred that the force constant models, and hence the self-consistency condition, must be modified to yield the Cauchy discrepancy.

Volume forces in simple metals

Within the local pseudopotential model taken to second order, the metallic energy, U, is a sum of a volume dependent and a structure dependent term. Only the second term, expressed as a volume independent, effective ion–ion interaction, is usually retained for lattice dynamic calculations. Here we firstly evaluate the contribution from the volume dependence of U to the longitudinal elastic constants c11and c12 and the bulk modulus B for a variety of models. This contribution, Δbs, which is due to the volume dependence of the screening, was identified by Wallace, Brovman and Kagan, Pethick, and by Finnis but its magnitude is in dispute. We find Δbs is important and very model dependent, ranging from −10 to−50% of B in the alkali metals and Al (but is +50% in Pb for one model) depending on both the pseudopotential and screening employed. Secondly, the usual dynamical matrix is generalized to incorporate this volume dependence. The volume effects are important for long wave phonons only and for most practical purposes may be neglected otherwise. The volume contributions to the pressure and the deviation of the Cauchy relation are also evaluated suggesting that the observed deviation of the Cauchy relation does not provide a reliable estimate of Δbs.

On the third order elastic constants of AgCl-AgBr mixed system

Lowdin's many body effect is accounted to explain the violation of Cauchy's relation ( C 12= C 44) 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.

Computation of Fourier Interferograms:II. Dispersive two–beam interferogram for Cauchy's dispersion law

An asymmetric dispersive Fourier interferogram is computed for an ideal two-beam interferometer with a dispersive sample in the measuring beam. The assumption that the dispersion of the index of refraction may be described by a Cauchy relation of only two terms leads to a simple, analytically closed expression for the asymmetric model interferogram. The agreement between computed and naturally recorded dispersive Fourier interferograms of nitrogen and methane is, in many points, surprisingly good. The computed model interferogram may therefore lead to a better understanding and an easier interpretation of dispersive Fourier interferograms than has hitherto been possible.

General expressions for inner elastic constants

General expressions are given for the calculation of interlattice tensors and inner elastic constants by summation in real space and reciprocal space in the case of central forces and in reciprocal space for the case of volume-dependent band structure energies. Cauchy relations applicable to two of the tensors have been tabulated.

On Riemann boundary value problems for certain linear elliptic systems in the plane

The partial differential equations which occur in the theory of elastic plates and shells are among those which may be reduced to a first order elliptic system of the type studied by Douglis [3]. Under certain regularity conditions for the coefficients, a Beltrami transformation exists takmg the general first-order, elliptic systems into a normal-Douglis-form. This form can be further simplified, and more concisely represented by utilizing the algebra CY of hypercomplex numbers. The theory of solutions to these linear systems is known as generakzed hyperunai’yticfimction theory (see Gilbert [S, 6, 71 and Gilbert and HiIe [g, 9])* and bears the same relationship to the Douglis theory of hyperanalytic functions as Vekua’s theory to analytic functions. In [I] Hilbert boundary value problems for generalized hyperanalytic functions were studied. Subsequently semilinear Douglis systems were treated by Wendland [13] and Gilbert [7j. Th e p resent work deals with Riemann boundary value problems for linear systems. It is clear that our results may be extended to nonlinear hypercomplex systems resembling the complex cases investigated by Warowna-Dorau [14] and Wolska-Bochenek 1151. A good survey of the methods encountered in the analytic case may be found in the monographs of Gakhov [4J and Muskhelishvili [lo]. The fundamental kernels for the linear system (see [S]) permit the formulation of the Riemann boundary value problem for generalized hyperanalytic functions as a Cauchy type integral relation. In general, there is no similarity principle

Angular forces and normal vibration in nickel

Normal modes of vibration of nuclei in fcc nonmonovalent transition-element nickel have usually been investigated by lattice-dynamical approaches in which the Cauchy discrepancy (${C}_{12}\ensuremath{-}{C}_{44}$) has been attributed to the bulk modulus of the conduction electrons (i.e., ${C}_{12}\ensuremath{-}{C}_{44}={K}_{e}$). The authors consider here an approach in which it is assumed that there are also noncentral-force contributions that combine with the electron gas to break the Cauchy relations of elasticity. For this purpose, the angular interactions are taken from the Clark, Gazis, and Wallis approach and volume forces have been taken from the Krebs scheme, using an appropriate value of the screening parameter taken from the Bohm-Pines plasma theory which takes into account the electron correlations. The frequency versus wave-vector dispersion relations along symmetry directions [$\ensuremath{\zeta}00$], [$\ensuremath{\zeta}\ensuremath{\zeta}0$], and [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$] in the reciprocal space of Ni are computed from the solutions of the secular equation along these directions. The frequency distribution has been calculated with the Blackman's root-sampling technique for a discrete subdivision in wave-vector space. Below $\frac{\ensuremath{\Theta}}{10}$, where the sampling technique gives an inadequate description of the normal modes of vibrations, a modified Houston's spherical six-term integration procedure is employed. Results obtained with the above combination of various interactions are discussed in the light of previous calculations, and the present values are found to yield comparatively better agreement with experiments.