Isothermal Elastic Constants Research Articles

Temperature Dependence of the Isothermal Elastic Constants of Alkali Metals

AbstractThe temperature dependence of the isothermal elastic constants of alkali metals are calculated using second order perturbation theory and the local Heine‐Abarenkov model potential. The theoretical results are in quantitatively good agreement with available experimental data. The, the qualitative temperature dependence of elastic constants in the dynamical treatment using the expression for the temperature dependent two‐body interactions is obtained.

Nonequilibrium thermodynamics and quasielastic light scattering from crystals. II. Piezoelectric crystals

The methodology described and utilized previously by the author, involving the use of nonequilibrium thermodynamics in the analysis of the spectrum of light scattered quasielastically by crystals, is applied to piezoelectric crystals. An additional thermodynamic variable, the electric polarization, is taken into account and assumed to be a relaxing variable. Solution of the resulting equations in the case of K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ leads, as expected, to the appearance of a polarization relaxation line in the spectrum. The spectrum is calculated for several values of the two relevant parameters: the polarization relaxation frequency and the ratio of ${C}_{66}$ isothermal elastic constants at constant electric field and constant polarization. For a fixed polarization-relaxation frequency, it is found that as the ratio of elastic constants decreases (the characteristic behavior as the ferroelectric phase transition in K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ is approached), the intensity of the relaxation line increases at the expense of the Brillouin lines. In contrast with standard treatments of coupled modes in piezoelectric crystals, no adjustable parameters are involved in determining the spectrum.

Elastic constants of a stressed crystal. III. Static method of defining isothermal elastic constants of third order

A theory is developed which may be the basis for the experimental static determination of the third order elastic constants of crystals of arbitrary symmetry. The application of this theory to cubic symmetry crystals is considered. [Russian Text Ignored].

Elastic constants of a stressed crystal. I. General theory

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Model of the structural phase transformation in sodium azidea)

A microscopic model for the intermolecular interactions is applied to discuss the nature of the α–β structural transformation in sodium azide. NaN3 is an example of an ionic molecular crystal where changes in molecular orientations in the crystal lead to anomalous thermoelastic behavior. Sodium azide differs from other molecular crystals with elastic anomalies such as KCN in that the associated change in orientational order at the transition seems to be quite small. The model, a mean field treatment of the coupling of the rotational motions of the N3− ions to the lattice strains, is shown to lead to indirect N3−–N3− interactions mediated by the lattice. The apparent second-order nature of the transition is examined. Predictions regarding the temperature dependence of the isothermal elastic constants c44 and 1/2(c11−c12) are made.

Temperature dependence of the elastic constants in alkali metals

The isothermal elastic constants Cik(T) in alkali metals are calculated in the pseudopotential model previously described. The standard second approximation in the pseudopotential and the first one in the anharmonicity are used. An expression for Cik(T) convenient for calculations is given in terms of the strain derivatives of the dynamical matrix. The results for the bulk modulus B(T) and shear constant C44(T) agree well with experiment. The shear constants C'=1/2(C11-C12) decrease with increasing T more rapidly than in experiment. For Li this disagreement is large and may again indicate the inadequacy of the local pseudopotential model. For the other alkali metals the discrepancies are much smaller and may be connected, at least partly, with the higher-order anharmonic contributions. The volume dependence of the elastic constants at finite T, the temperature dependence of their pressure derivatives and the sensitivity of the results to the approximations for the dielectric function are also discussed.

Zero and First Sound in NaF and RbI

AbstractZero and first sound as well as the transition region are investigated theoretically in RbI and NaF. The temperature dependence of the first sound, zero sound, and isothermal elastic constants is calculated on the basis of a breathing shell model for the harmonic lattice dynamics and anharmonic parameters from microscopic Grüneisen constants. The theoretical results are compared with ultrasonic data and inelastic neutron scattering results. Own recent neutron experiments in RbI are concerned with the zero sound region at room temperature while the NaF experiments cover also parts of the transition region between first and zero sound at various temperatures between 295 and 700 K. It is shown that the dispersion of sound is satisfactorily described by use of an averaged thermal phonon lifetime. In NaF for each temperature first sound elastic constants and averaged phonon lifetimes are deduced from an analysis of the experimental data.

Phonon dispersion in NaF in the region between zero and first sound, measured by inelastic neutron scattering

This is the first inelastic neutron scattering experiment covering the transition region between zero and first sound. Measurements in NaF were performed in a wavevector region 0.06<or=q/qmax<or=0.25 at 295, 500, 600 and 700K. The experimental data cover the zero-sound region at room temperature and the transition region near first sound at 700K. We show that it is possible to describe the frequency dependence of the sound velocities by use of an averaged thermal-photon lifetime. The isothermal elastic constants and the averaged phonon lifetimes are determined at each temperature, and the zero-sound elastic constants at room temperature.

Ultrasonic examination of the elastic characteristics of crystalline β-oxygen

The adiabatic compressibility and the Poisson ratio of solid β-oxygen over the 36–43°K temperature range are determined from measurements of the velocity of longitudinal and transverse ultrasonic waves. Determined also are the isothermal elastic constants, the Gruneisen coefficients, and the Debye temperature. The temperature dependence of the isothermal compressibility is compared with theoretical data.

The elastic constants of thallium chloride in the range 150 to 300K

The adiabatic elastic constants c11, c12 and c44 of TlCl have been obtained from measurements of ultrasonic wave transit times through a specimen of TlCl over the range 150-300K. The isothermal elastic constants, the heat capacity at constant volume and the Gruneisen parameter have been calculated from the elastic constants and associated data. The Gruneisen parameter is found to increase with decreasing temperature. The Debye characteristic temperature of 0K is found to be 141K. The anisotropy factor 2c44T/(c11T-c12T) is found to increase with decreasing temperature in accord with other CsCl-type ionic crystals.

Unpinning of Dislocations within the Framework of the Vibrating‐String Model

AbstractWithin the framework of the vibrating‐string model the frequency of thermally activated unpinning of dislocations from point obstacles is calculated with the aid of a dynamical theory. It is found that the unpinning frequency Γ has the form of an Arrhenius equation, Γ = Γ0e−ΔG/kT, where ΔG is the portion of the free enthalpy of activation which may be obtained by means of elasticity theory using isothermal elastic constants. The pre‐exponential factor Γ0 is shown to decrease with increasing length of the pinned dislocation segments, contrary to the result obtained on the basis of the absolute rate theory.

Adiabatic and isothermal elastic constants for highly anharmonic crystals

The mass operator of the phonon Green function for a highly anharmonic crystal in the ladder approximation is derived. In order to calculate the mass operator in the collision-dominated regime it is necessary to solve the Boltzmann-like equations. These solutions allow us to find the thermal conductivity coefficient and to prove that the thermodynamic relation between adiabatic and isothermal elastic constants is satisfied.

The computation of dynamical moduli of solids under pressure

On the basis of the theory of finite strains, expressions are obtained in general form for the effective adiabatic second order elastic constants of crystals of any symmetry in terms of the isothermal elastic constants of second, third, and higher orders in the free energy decomposition. These expressions are used in the case of crystals of cubic symmetry under hydrostatic conditions to find the elastic wave velocities in mono- and polycrystals, and their pressure dependences. The polycrystal was considered as an isotropic body consisting of a large number of cubic monocrystals. The isotropic elastic constants were calculated from theoretical and experimental results for monocrystals in the Voigt-Reuss-Hill approximation. A method of applying this approximation to thermodynamic effective second order elastic constants is proposed. The results of a computation are compared with data of experiments to measure the sound velocity in polycrystalline NaCl and CsCl specimens under pressures to 100 kbar. The results of this comparison are discussed.

The elastic constants of calcium tungstate, 4·2-300 K

The temperature dependences from 4·2 to 300 K of the seven adiabatic elastic constants of CaWO4 have been determined by measurements of ultrasound wave velocities using the `sing-around' and pulse superposition methods. The adiabatic elastic constants have been calculated from the measured ultrasound wave velocities by a perturbation method using just seven of the eleven measured velocities and by an iterative method using all the measured velocities; agreement has been found between the two sets of values. The complementary data on the thermal expansion and heat capacity at constant pressure of CaWO4 have been used with the present results to find the isothermal elastic constants, the heat capacity at constant volume, the Grüneisen parameters, and the Debye characteristic temperature at 0 K. The Debye characteristic temperature has been found from the heat capacity at constant volume from 10 to 300 K after subtraction of the Einstein heat capacity terms from the total heat capacity.

Thermodynamics of a crystal lattice on the basis of a local-equilibrium ensemble

The local-equilibrium statistical ensemble is constructed which describes the thermodynamics of a Bravais lattice taking into account the effects of nonlocality. Correlation-function expressions are obtained for the nonlocal isothermal and adiabatic elastic constants and other quantities generalizing the well-known local expressions 8−12). Crystal dynamics is considered for the case of an inhomogeneous, time-dependent deformation.

Elastic Properties of Metals and Alloys, I. Iron, Nickel, and Iron-Nickel Alloys

A comprehensive compilation is given of elastic properties of iron-nickel alloys. When sufficient data exist, preferred values are recommended. This compilation covers, besides pure iron and pure nickel, the entire binary composition range, both b.c.c. and f.c.c. phases. Elastic constants included are: Young's modulus, shear modulus, bulk modulus (reciprocal compressibility), Poisson's ratio, and single-crystal elastic stiffnesses, both second-order and higher-order. Data are compiled for variation of elastic constants with composition, temperature, pressure, magnetic field, mechanical deformation, annealing, and crystallographic transitions. An overview is given from the vantage points of the electron theory of metals, elasticity theory, and crystallographic theory. Also included are discussions of iso-thermal and adiabatic elastic constants, interrelationships among engineering elastic constants, computation of the latter from single-crystal elastic stiffnesses, and similar topics. Where key data have not been measured, they were generated if possible from existing data using standard formulae. Other gaps, both theoretical and experimental, in the elastic properties of iron-nickel alloys are indicated. A few theoretical results are included where experimental data are nonexistent or scarce. A semantic scheme is proposed for distinguishing elastic constants of solids.

The effect of monovacancies on the elastic properties of rigid disk solids

The isothermal elastic stiffness constants of a system of N rigid disks on a two-dimensional hexagonal lattice confined to an area A are calculated analytically in the high-density limit using one-particle cell cluster theory. The constants are calculated using the following equation: C ijkl = 1 A ( ∂, 2F ∂η ij∂η kl ) η=0 , where C ijkl is the isothermal elastic constant, F is the Helmholtz free energy of the system, and the η ij 's are the strains (η represents all the strains). The isothermal elastic constants are then calculated in the high density limit for the rigid disk system into which a small concentration of monovacancies has been introduced, using the same approach appropriately modified. All the elastic constants are calculated to order 1 t 2 where t = a σ − 1 , a is the smallest distance between lattice sites, and a is the diameter of the disks. All the nonzero isothermal elastic constants are found to be proportional to the temperature. The elastic constant C 1122 = C 2211 which is zero in the perfect system becomes nonzero upon the introduction of monovacancies. However, the fact that C 1122 is found to be zero is an artifact of the one-particle cell cluster approximation.

Self-consistent renormalized phonons in metallic and nonmetallic solids

In this paper we derive expressions for the generalized dynamical matrix (phonon self-energy) of an arbitrary crystal, conducting or insulating, by self-consistently taking into account anharmonicities within the renormalized anharmonic approximation. An integral equation for the generalized Gruneisen function entering into the expression for the phonon self-energy is derived via the dielectric-screening treatment of vibrational excitations. Solution of this equation is shown to be necessary in order to obtain the proper phonon frequencies at all wavelengths. Therefore, our approach automatically provides results for the long-wavelength limit of the static phonon self-energy which are consistent with the isothermal elastic constants derived from the expression of the free energy (elastic sum rule). Experimental implications of the presented theory are discussed.

Adiabatic and isothermal elastic constants from space- and time-dependent response theory

The elastic constants of an anharmonic crystal are studied as an example for discussing the general problem of the static limit in Kubo's linear response theory. By starting from the microscopic theory and by defining wavevector- and frequency-dependent elastic constants it is possible to define adiabatic and isothermal limits. It is shown that phonon transport plays a fundamental role in taking these limits. By summing up the ladder diagrams in the self-energy of the long-wavelength acoustic phonons in a collision-time approximation, the difference between the adiabatic and isothermal elastic constants is obtained in agreement with thermodynamics.

Elasticity of a magnetic lattice near the magnetic critical point

The isothermal elastic constants and the coefficient of anomalous thermal expansion of a magnetic lattice are discussed. The spin system is described by the Ising model with an exchange coupling depending on lattice spacing. A behavior of the elastic constants and the coefficient of thermal expansion is found which is in qualitative agreement with experiments. The isothermal compressibility remains positive nearTc and no thermo-mechanical instability occurs (which would lead to a first-order phase transitions), in contrast to earlier theories.