Anomalous temperature dependence of third-order elastic constants in NH4Cl and NH4Br near the λ point

Second order elastic constants of NH₄Cl and NH₄Br crystals were measured under hydrostatic and uniaxial pressures near the λ-type phase transitions. Third order elastic constants were calculated from the pressure derivatives of the second order elastic constants. Results show that the temperature dependences of the third order elastic constants of NH₄Cl differ from those of NH₄Br dramatically. In addition, it is found that the coefficients $\text{1}\text{54}\text{(C}{}_{111}\text{+ 6C}{}_{112}\text{+ 2C}{}_{123}\text{)}$ and $\text{(}\text{1}\text{24}\text{√3)(C}{}_{111}\text{? 3C}{}_{112}\text{+ 2C}{}_{123}\text{)}$ of strains in the free energy of NH₄Cl show quite different temperature dependences from those of NH₄Br.     

Elastic anharmonicity of InP: Its relationship to the high pressure transition

Ultrasonic wave transit times have been measured in n-type InP at room temperature using hydrostatic pressures up to 4 kbar. Linear pressure dependences are found for the elastic stiffness moduli implying that at the high pressure structural-electrical transition the shear-to-bulk modulus ratio $\text{(C}{}_{11}\text{?C}{}_{12}\text{)}\text{2B}$ has a (fractional) value which fits the modified Born criterion for stability developed by Demarest $\text{et al}$. The anharmonic force constants and some of the third order elastic constants are found to be smaller the higher the transition pressure for indium III-V compounds.     

Third and fourth order elastic constants of fluoperovskites CsCdF3, TlCdF3, RbCdF3, RbCaF3

The hydrostatic dependence of the second order elastic constants of several fluoperovskites has been measured and studied from a macroscopic point of view and through a force field analysis. The first part of this paper is devoted to the experimental determination of some combinations of the third and fourth order elastic constants. The second part analyses the pressure dependence of the second order elastic constants from a macroscopic point of view. The large values of the second order anharmonicity coefficients $\text{b}{}_{\mathrm{ij}}\text{=}\text{1}\text{2}\text{(}\text{?}{}^2\text{C}{}_{\mathrm{ij}}\text{?P}{}^2\text{)}{}_0\text{C}{}^{\mathrm{(0)}}{}_{\mathrm{ij}}$ are connected to the occurrence of a phase transition in some compounds, whereas the first order coefficients $\text{(}\text{?C}{}_{\mathrm{ij}}\text{?P}\text{)}{}_0$ have an order of magnitude which agrees with a quasiharmonic hypothesis. The third part is devoted to the microscopic interpretation of the third order elastic constants. A rigid ion model is used with either classical Born Mayer potentials or a modification of these potentials with an axial term (Cowley and Rousseau's models).     

The elastic behaviour of the ternary zincblende structure semiconductor CuGe2P3

A single crystal has been grown of CuGe₂P₃, a ternary semiconductor with a zincblende structure in which the copper and germanium atoms are randomly distributed on the cation sites. The second order elastic constants measured by the ultrasonic pulse echo overlap technique (C₁₁ = 13.66, C₁₂ = 6.17, C₄₄ = 6.66 and bulk modulus B = 8.67 in units of 10¹⁰Nm^?2 at 291 K) are closely similar to those of GaP and conform well to Keyes' correlation for zincblende structure crystals. The hydrostatic pressure derivatives of the second order elastic constants ($\text{?C}{}_{11}\text{?P}\text{= 4.40}$, $\text{?C}{}_{12}\text{?P}\text{= 3.9}$, $\text{?C}{}_{44}\text{?P}\text{= 0.93}$ and $\text{?B}\text{?P}\text{= 4.07}$) and the third order elastic constants (C₁₁₁ = ? 8.45, C₁₁₂ = ? 3.49, C₁₂₃ = ? 1.13, C₁₄₄ = ? 2.82, C₁₅₅ = ? 1.44 and C₄₅₆ = ? 0.69 in units of 10¹¹Nm^?2) also resemble those of GaP: even the anharmonicity of the long wavelength acoustic modes of this ternary semiconductor resembles that of its binary “parent” compound. The positive signs of the hydrostatic pressure derivatives and the negative signs of the third order elastic constants show that the acoustic modes at the long wavelength limit stiffen under pressure, an effect which is quantified here by computation of the mode Grüneisen parameters, which are all positive. The harmonic and anharmonic force constants, obtained in the valence force field model, fit well into the pattern shown by related zincblende structure compounds: the ratio ($\text{β}\text{α}$) for bond bending (β) to stretching (α) force constants is greater than 4:1; the dominating anharmonic force constant is γ: most of the anharmonicity is associated with nearest neighbour atoms. Analysis of the temperature dependence of the elastic constants on the basis of a simple anharmonic model again emphasises the similarity between the elastic behaviour of CuGe₂P₃ and GaP. The thermal expansion of CuGe₂P₃ varies almost linearly with temperature between 291 K and 1000 K, the linear coefficient of thermal expansion α being 8.21 ± 0.02 × 10^?6 °C^?1. The similar lattice parameters and elastic behaviour of CuGe₂P₃ and GaP indicate that semiconducting devices made of epitaxial deposits of CuGe₂P₃ on a GaP substrate should prove to be almost strain-free.     

Pressure dependence of elastic behaviour and force constants of GaP

The six third order elastic constants of GaP have been obtained from measurements of the effect of hydrostatic and uniaxial pressures on ultrasonic wave velocities. The valence force field model has been used to obtain the bond-bending and stretching force constants. Confirmatory evidence is added to the hypotheses that zinc blende structure crystals undergo the densification transition under pressure when (1) a modified Born stability criterion that $\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}\text{B}$ equals about 0.2 is reached at the phase transition pressure P_t, and (2) the ratio $\text{β}{}_{\mathrm{t}}\text{α}{}_{\mathrm{t}}$ of the second order bond-bending β_t to bond-stretching α_t, force constants is about 0.1. These findings suggest that a macroscopic shear takes place at the transition     

Elastic moduli and mode gammas of GaP: Their relationship to those of other isomorphic crystals and the high pressure structural-electrical transition

The transit times of ultrasonic waves associated with piezoelectrically inactive modes have been measured in sulfur-dope, n-type GaP using hydrostatic pressures up to 5 kbar. The velocities and elastic constants of most modes increase linearly with pressure but $\text{C}{}_{\mathrm{S}}\text{=}\text{1}\text{2}\text{(}\text{C}{}_{11}\text{?}\text{C}{}_{12}\text{)}$ decreases slightly. The behavior of the elastic constants and their associated mode gammas are compared to those of other III–V compounds and of Si and Ge. We suggest that the behavior of C_S is related to the occurrence of the structural-electrical transition which occurs at high pressure. A modified Born criterion for lattice stability developed by Demarest $\text{et}$$\text{al}$. is used to explain this relationship. The contribution of thermal dilatation to the temperature dependence of each elastic constant is deduced. Using it and data from the literature we find that the explicit thermal contribution to each elastic constant is significant and in the case of C₁₂ has an anomalous sign.     

The elastic constants for single-crystal lead and indium from room temperature to the melting point

The adiabatic elastic stiffness constants of indium and lead were measured from room temperature to their respective melting points using the ultrasonic pulse-echo method. The average temperature derivatives at constant pressure, $\text{dC}{}_{\mathrm{ij}}\text{dT}\text{= C'}{}_{\mathrm{ij}}$, for the indium constants are:

$\text{C'}{}_{11}\text{= ?0.032 C'}{}_{12}\text{= ?0.0055 C'}{}_{13}\text{= ?0.011 C'}{}_{33}\text{= ?0.032 C'}{}_{44}\text{= ?0.0059 C'}{}_{66}\text{= ?0.022}$

in units of 10¹⁰dyne cm^?2deg^?1, and are referred to axes coincident with those of the face-centered tetragonal unit cell. The average temperature derivatives at constant pressure for the lead elastic constants are, in the same units,

$\text{C'}{}_{11}\text{= ?0.0264 C'}{}_{12}\text{= ?0.0156 C'}{}_{44}\text{= ?0.0155.}$

The elastic anisotropies of both lead and indium increase markedly with increasing temperature.     

Third order elastic constants of rubidium chloride

The third order elastic constants of RbCl, determined by measurements of the static stress dependence of ultrasonic waves, are found to be (in units of 10¹²$\text{dynes}\text{cm}{}^2$)

$\text{C}{}_{111}\text{= ?6.71 ± 0.1 C}{}_{123}\text{= 0.05 ± 0.07}$

$\text{C}{}_{112}\text{=?0.18 ± 0.04 C}{}_{144}\text{= 0.11 ± 0.02}$

$\text{C}{}_{166}\text{=?0.17 ± 0.01 C}{}_{456}\text{= 0.4 ± 0.01}$

. The calculation of third order constants using a rigid ion Born model is briefly discussed, and results are compared to the measurements. The comparison qualitatively supports the model, but no quantitative evaluation of the repulsive interaction is possible.     

Pressure derivatives of the elastic moduli of strontium,barium and lead nitrate

The pressure derivatives of the elastic moduli of Sr(NO₃)₂, Ba(NO₃)₂ and Pb(NO₃)₂ single crystal, have been measured by the ultrasonic pulse superposition method. The results for Sr(NO₃)₂ and Pb(NO₃)₂ are similar in magnitude and character, while the ones for Ba(NO₃)₂ differ strongly, $\text{?c}{}_{11}\text{?P}$ and $\text{?c}{}_{12}\text{?P}$ even From the measured pressure derivatives, the mode Grüneisen gammas and the Grüneisen constant as a function of temperature have been determined, and the latter is correlated with the experimental values, deduced from thermal expansion. The explicit temperature dependence of the elastic moduli is calculated, and found to be always negative, increasing in absolute value from Sr(NO₃)₂ to Pb(NO₃)₂.     

On the interlattice displacement of rhombohedral structure

The expressions for the components of the interlattice displacement to the first order in strain and hence the expressions for the SOE constants are obtained for the rhombohedral lattice using the method of homogeneous deformation. Two and three body interactions are considered up to fourth neighbours in the calculations. The special strains that are generally chosen in explaining the temperature variation of thermal expansion of a uniaxial crystal, which make the interlattice displacement zero, are shown to be dependent [$\text{η}{}_{\mathrm{zz}}\text{= ?}\text{3}\text{2p}{}^2\text{C}\text{D}\text{(η}{}_{\mathrm{xx}}\text{+ η}{}_{\mathrm{yy}}\text{)}$]. The internal displacement factors A, B, C and D are calculated for the metals As, Sb and Bi to get an idea on the magnitude of C/D.     

Elastic constant pressure derivatives and stability of the bcc phase of InTl alloys

Hydrostatic-pressure derivatives of the elastic stiffness constants have been measured ultrasonically in a bcc InTl alloy of composition near the eutectoid. The shear modulus $\text{1}\text{2}\text{(C}{}_{11}\text{?C}{}_{12}\text{)}$ is small, but does not soften under pressure: the eutectoid reaction is not directly associated with a mode softening shear instability. Anharmonic contributions to the excess vibrational entropy from the q 〈110〉, e 〈1$\text{1}$0〉 modes which stabilize the bcc structure are small compared with the harmonic contributions.     

Magnetostrictive strain and thermal expansion in NdCo5 in the spin orientation temperature region

Magnetostrictive strain and thermal expansion measurements in an oriented assembly of NdCo₅ particles along the c-axis were carried out between 77 and 330 K. The measurement strains in the c-derection and transverse to the c-direction exhibited anomalous behavior at 285 and 245 K, which are assoiciated with the spin reorientation process in this temperature range. The estimated magnetostrictive energy $\text{1}\text{2}\text{E(λ}{}_{\mathrm{c}}\text{- λ⊥}{}_{\mathrm{c}}\text{)}{}^2$ is comparable to the anisotropy energy at 285 K. There is also a pronouced anomaly in ther thermal expansion in the spin reorientation temperature region.     

Theoretical analysis of the centrifugal distortion contributions to the rotational spectra of 2Π diatomics

The centrifugal distortion contributions to the rotational energies of diatomic molecules are derived from the resolution of the vibration-rotation wave equation. The unknown radial dependence of the fine structure constants is taken into account by means of a Taylor expansion around the equilibrium distance. Hence, one obtains the expressions of the centrifugal corrections associated with each fine structure constant in terms of the equilibrium values of its radial derivatives. The case of ²Π states is examined in detail. The dependence of the centrifugal distortion effects upon the choice of the coupling scheme representation is exhibited and a ²Π energy matrix containing the centrifugal constants of any order is proposed. Such a matrix is appropriate to fit the data for any value of the rotational quantum number. The theoretical expressions of the energy levels are related to the experimental data and the correlations between the spin-orbit centrifugal and spin-rotation contributions are put in evidence. It is shown that very compact formulas can be derived allowing a straightforward evaluation of the successive radial derivatives of the spin-orbit function in terms of the spectroscopic data $\text{A}{}^{\mathrm{(1)}}\text{? ?α}{}_{\mathrm{A}}\text{(}\text{w}{}_{\mathrm{e}}\text{B}{}_{\mathrm{e}}\text{)}$; $\text{A}{}^{\mathrm{(2)}}\text{? ?(1 +}\text{α}{}_{\mathrm{B}}\text{w}{}_{\mathrm{e}}\text{2B}{}_{\mathrm{e}}{}^2\text{)A}{}^{\mathrm{(1)}}$; …. Application of these results to the case of several molecules is considered and discussed.     

Thermal expansion of zirconium—II

The generalised Gruneisen parameters γ′ = ?? log ω/??′ and γ″ = ?? log ω/??″ have been calculated for various normal mode frequencies using the force model for Zirconium obtained on Keating's approach. There is a general agreement between the normalised frequency distribution curve of Zirconium obtained on the present model and that of Bezdek et al. The temperature dependence of the effective Gruneisen functions $\text{γ}\text{?}{}_{\mathrm{\perp }}\text{(T)}$ and $\text{γ}\text{?}{}_{\mathrm{|}}\text{(T)}$ has been calculated using the procedure of Blackman. The high temperature limits of $\text{γ}\text{?}{}_{\mathrm{\perp }}\text{(T)}$, $\text{γ}\text{?}{}_{\mathrm{|}}\text{(T)}$ and $\text{γ}\text{?}{}_{\mathrm{v}}$ are in good agreement with those obtained by Goldak et al. from an analysis of the thermal expansion data of Zirconium.     

The Hilbert-Schmidt method in the three-particle problem and determination of the vertex parts tdn and td1n

Vertex constants for virtual decays $\text{t→d+n}\text{and}\text{t→d}{}^1\text{+n}$ are calculated using the resonance expansion technique for the three-particle amplitude suggested recently in ref. [1]. In the model with a spin-dependent separable NN-potential values $\text{G}{}_{\mathrm{<mtext>tdn</mtext>}}{}^2\text{=1.92}\text{fm}\text{, G}{}_{\mathrm{<mtext>td1n</mtext>}}{}^2\text{=?0.38}\text{fm}$ are obtained.     

Renormalization group improved predictions for quarkonium decay

A method recently suggested to resolve the ambiguity related to scheme dependence in perturbative QCD is applied to calculations of radiative corrections to quarkonium decay. Renormalization group improved predictions are given for the ratio $\text{R=Г(}{}^1\text{S}{}_0\text{→had)/Г(}{}^3\text{S}{}_1\text{→had)}$, as a function of $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$. The values obtained are substantially larger than in conventional schemes. One finds in particular that R>182 for charmonium, and R>174 for bottomium, independently of the value of the quark mass and $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$. It follows that $\text{Г(}\text{n}{}_{\mathrm{c}}\text{→}\text{had}\text{)>6.9}\text{MeV}$ and $\text{Г(}\text{n}{}_{\mathrm{b}}\text{→}\text{had}\text{)>3.6}\text{MeV}$. Values of the wave function at the origin as a function of $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$ are also extracted from the measured gluonic width of the ³S₁ state. An upper bound for $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$ is obtained under some assumptions.     

Application of finite strain theory to non-cubic crystals

The application of finite strain theory to a crystal of orthorhombic or higher elastic symmetry, which is subjected to a purely extensional deformation parallel to the crystallographic axes, yields stress-strain relations and effective elastic constants for hydrostatic stresses. Alternative formulations of the theory are obtained when the free energy is written as Taylor series in E, the invariant analogue of the Eulerian strain tensor, or η, the Lagrangian strain tensor. In most cases, the formulation in terms of E provides a better approximation when the Taylor series are truncated at the second or third order. In the case of cubic crystals, the stress-strain relations reduce to the Birch equation. For non-cubic crystals, the P-V relations calculated using the non-cubic and Birch equations will differ due to the effects of elastic anisotropy. A comparison between the second-order approximations of the non-cubic stress-strain relations and the cubic Birch equation suggests that the difference in volume will be less than 1% for most materials. The difference in volume is reduced when the third-order approximations are used. When the third-order terms are retained, the stress-strain relations calculated using the Eulerian formulation agree with measured linear compression data for quartz to 150 kbar $\text{(}\text{P}\text{K}{}_0\text{= 0.4).}$ For zinc, the calculated pressure-volume relation agrees with the shock-wave Hugoniot up to 750 kbar $\text{(}\text{P}\text{K}{}_0\text{= 1.2),}$ although both calculated and Hugoniot P-V relations disagree with X-ray compression data. At pressures greater than 300 kbar, the calculated axial ratio of zinc approaches that for other hexagonal metals $\text{(}\text{c}\text{a}\text{? 1.63).}$     

Dispersion relations and lattice thermal expansion of dysprosium

The lattice dynamics, second and third order elastic constants and the lattice thermal expansion of dysprosium have been calculated using Keating's approach. The ten third order elastic constants are calculated using four anharmonic parameters. The present model reproduces the measured pressure derivatives of the second order elastic constants of dysprosium well. The low and high temperature limits γ?_I and γ?_H of the lattice thermal expansion are evaluated and the agreement between the calculated γ?_H and that obtained from the thermal expansion and specific heat data is satisfactory.     

Upper limit on the proton decay lifetime in SU(5) theory

Both diagrams which contribute to proton decays are calculated in the MIT bag model for all two-body decay modes. Their relative phases are model-determined. All results are presented in such a way that they can be easily deduced for any $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}$. In minimal SU(5), with $\text{λ}{}_{\mathrm{<mtext>MS</mtext>}}\text{= 160}\text{MeV}$, a theoretical upper limit seems to be 10³⁰ y.