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.     

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 shear elastic mode near structural transition in the band Jahn-teller model

The temperature dependence of the pressure derivative of the shear elastic constant $\text{dC}{}_{\mathrm{<mtext>S</mtext>}}\text{dp}$ for degenerate electronic system is worked out in the band Jahn Teller model and is shown to follow C_S linearly as the temperature decreases and approaches the transition temperature T_M. The constant of proportionality gives useful informations on the characteristics of the band Jahn-Teller transition. This analysis can explain well the recently observed experimental results on Nb₃Sn.     

Third order elastic constants and thermal expansion of terbium

The third-order elastic constants and the temperature variation of the effective Grüneisen functions of terbium have been calculated on the basis of Keating's method. The pressure derivatives of the second order elastic constants of terbium have been obtained by interpolation of the experimental pressure derivatives of gadolinium and dysprosium. The ten third order elastic constants of terbium are calculated using four third order anharmonic parameters obtained from its interpolated pressure derivatives. The low and high temperature limits $\text{λ}{}_{\mathrm{L}}$ and $\text{λ}{}_{\mathrm{H}}$ of the lattice thermal expansion are evaluated. The agreement between the calculated $\text{λ}{}_{\mathrm{H}}$ and that obtained from thermal expansion and specific heat data of terbium is good.     

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 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.     

Hole contribution to the elastic constants of SnTe

Ultrasonic wave velocities have been measured in SnTe single crystals with hole concentrations of 1.0 and 4.5 × 10²⁰/cm³. The shear elastic stiffness constant C₄₄ is sensitive to the hole concentration but $\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}$ is not, a result which is consistent with the valence band pockets being sited at the L points. The non-ellipsoidal, non-parabolic multivalley band model has been used to calculate the hole contribution to the elastic constants. The calculated difference between the shear constant C₄₄ (2.78 × 10¹⁰ dyne cm^-2) for the two crystals is in agreement with that measured experimentally (2.67 × 10¹⁰ dyne cm^-2). The shear deformation potential constant E_u for the SnTe valence band is 7.8 eV at 293°K.     

The pressure dependence of elastic moduli of NiF2 to 10 kbar

At room temperature, the pressure dependence of the elastic stiffness moduli of NiF₂ have been measured by pulse superposition method to 10 kbar. The observed strong decreasing behavior in the modulus $\text{C}{}_{\mathrm{<mtext>s</mtext>}}\text{=}\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}$ is discussed.     

Light scattering study of dispersion and attenuation of hypersound in adamantane

The Brillouin scattering techniques have been used to measure the velocity dispersion of hypersonic acoustic waves in the “high temperature” disordered cubic phase of adamantane. Shear waves, characteristic of the C₄₄ elastic constant, show no significant dispersion. Longitudinal waves propagating in the (001) plane show strong velocity dispersion. The measures have been performed at the same temperature T = 295.7 K. Using a classical single relaxation time model for the dispersion as a function of frequency at temperature T, the L-mode data have been correctly fitted.The importance of the dispersion $\text{(C}{}_{\mathrm{\infty }}\text{? C}{}_{\mathrm{o}}\text{)}\text{C}{}_0$ for the elastic constants is 20% for C₁₁, 51% for C₁₂ #1% for C₄₄ and ?2.8% for $\text{(C}{}_{11}\text{? C}{}_{12}\text{)}\text{2}$. The fitted relaxation time is τ ? 9 × 10^?11 sec.     

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.     

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 influence of excess oxygen on the elastic properties of non-stoichiometric SrO crystals

Measurements have been carried out of the elastic constants of SrO in the virgin undoped state and of the changes produced in them by equilibrium doping with oxygen at ? 1200°C and oxygen partial pressure of 0.95 atm. The method used was Papadakis' pulse-echo overlap technique in conjunction with thermogravimetric analysis (T.G.A.) to determine mass and density changes due to oxygen doping.The values obtained for C₁₁, C₁₂ and C₄₄ of the virgin crystal at 23°C are

$\text{C}{}_{11}\text{= 17.60 ± 0.03 × 10}{}^{11}\text{dynes/cm}{}^2$

;

$\text{C}{}_{12}\text{= 4.808 ± 0.007 × 10}{}^{11}\text{dynes/cm}{}^2$

;

$\text{C}{}_{44}\text{= 5.577 ± 0.008 × 10}{}^{11}\text{dynes/cm}{}^2$

.(These values are in very good agreement with those of Son and Bartels [2].)Values for $\text{δC}{}_{11}\text{C}{}_{11}$ and $\text{δC}{}_{44}\text{C}{}_{44}$ were found to be ?1.74% and ?0.86% respectively. Accurate valu $\text{δC}{}_{12}\text{C}{}_{12}$ could not be obtained because of sample size limitations after quenching. However, C₁₂ was shown to definitely increase due to doping.Analysis of the results indicate that the elastic modulus changes can only be attributed to the formation of cation vacancies during doping. Analysis of the T.G.A. behavior indicates that this cation vacancy formation is probably associated with the presence of various tripositive cation and uninegative anion species depending upon the impurity concentrations of the sample. This implied impurity-controlled cation vacancy concentration is consistent with the earlier observed extrinsic nature of cation diffusion in SrO at 1200°C.     

CLL = (L,L; 0, 0) measurements and a structure due to a P partial wave in the pp system

Measurements of C_LL of pp elastic scattering near θ_c.m. = 90° at thirteen energies between 300 and 800 MeV are reported. These, together with previous values of C_NN, are used to extract values of two quantities, $\text{?}{}_{\mathrm{<mtext>s</mtext>}}$ and $\text{?}{}_{\mathrm{<mtext>t</mtext>}}$, which contain only spin-singlet and only coupled spin-triplet partial waves, respectively. The $\text{?}{}_{\mathrm{<mtext>s</mtext>}}$ curve, which is not dependent on C_LL, exhibits the behavior expected for the previously conjectured ¹D₂ resonance. The $\text{?}{}_{\mathrm{<mtext>t</mtext>}}$ curve also exhibits a resonance-like behavior, which could be due either to the ³P₀ or the ³P₂ partial wave.     

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.     

Attenuation of longitudinal ultrasonic wave near the diffuse phase transition in CdF2

The attenuation of a longitudinal ultrasonic wave propagating in the [111] direction in CdF₂ is studied as function of temperature from 300K to 1030K. An ultrasonic attenuation peak has been observed for the first time near 983K. This peak is used to define the diffuse transition temperature (T_c = 983K) in CdF₂ which is well below its melting temperature of 1372K. The Arrhenius activation energy of anion motion above T_c was obtained from the temperature dependence of the attenuation and the theory of the dynamics of the coupled crystalline-cage-charged-liquid fluctuations. The elastic constant, $\text{C}{}_{11}\text{+2}\text{C}{}_{12}\text{+4}\text{C}{}_{44}\text{)}\text{3}$, measured simulataneously with the ultrasonic attenuation displays a large decrease near 983K in addition to the nearly linear decrease in the elastic constant with 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.     

Temperature derivatives at constant volume of the elastic constants of the alkali halides

Values of dC/dT)_V have been computed from experimental dC/dT)_P and dC/dP)_T for the three elastic constants (expressed as $\text{B}{}_{\mathrm{T}}\text{, C}{}_{44}\text{and}\text{(C}{}_{11}\text{?C}{}_{12}\text{)}\text{2)}$ of each of the 16 Li, Na, K, and Rb halides. The dC/dT)_V measure the explicit dependence of C on T, the effect of thermal expansion having been removed. The dC/dT)_V are all small (compared with dC/dT)_P), are all negative, and vary quite systematically and smoothly with anion, with cation and with the three elastic constants. Negative dB_T/dT)_V is accounted for both thermodynamically and by available lattice dynamical calculations. dC/dT)_V for shear constant is expressed in terms of two vibrational mode parameters, whose values can then be estimated from the observed value, and the trend of dC/dT)_V with Debye temperature.     

Information on the dynamics of the reaction KN → K1N from KLN interactions

If $\text{K}{}_{\mathrm{S}}{}^1\text{,}{}_{\mathrm{L}}\text{is a K}{}^1$ resonance decaying into K_S,L (the short- and long-lived kaon) and a neutral system S^o of pions, one can isolate the C-even and C-odd, crossed-channel contributions to $\text{KN → K}{}^1\text{N}$ by using the reactions $\text{K}{}_{\mathrm{L}}\text{N → K}{}_{\mathrm{S}}{}^1\text{,}{}_{\mathrm{L}}\text{N}$ whether S^o is a C-eigenstate, or a mixture of C-even and C-odd states. Applications to the production of K¹(890) and the Q-meson are discussed, and simple numerical predictions made for Q_S,L production. Q-production data indicate approximate t-channel helicity conservation for the ω and P' exchanges at vertices involving a spin change, in similarity to the belief for the pomeron. Q_S,L production data can give information also on Q-decays.     

Excited vibrational state microwave spectra,structure, and force-field of trifluorosilane

The J = 2?1 microwave spectrum of six isotopic species of HSiF₃ has been observed and assigned in excited states of five of the six fundamental vibrations. The assignment is based on relative intensities, double resonance experiments, and trial anharmonic force constant calculations. Analysis of the spectra leads to experimental values for five of the $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ constants, all three l-doubling constants q_t, one Fermi resonance constant φ₂₃₃, and one zeta constant $\text{ζ}{}_{\mathrm{6,\; 6}}{}^{\mathrm{(z)}}$.The harmonic force field has been refined to all the available data on vibration wavenumbers, centrifugal distortion constants, and zeta constants. The cubic anharmonic force field has been refined to the data on $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ and q_t constants, using two models: a valence force model with two cubic force constants for SiH and SiF stretching, and a more sophisticated model. With the help of these calculations, the following equilibrium structure has been determined: $\text{r}{}_{\mathrm{e}}\text{(}\text{SiH}\text{) = 1.4468(±5)}\text{A}\text{?}$, $\text{r}{}_{\mathrm{e}}\text{(}\text{SiF}\text{) = 1.5624(±1)}\text{A}\text{?}$, ∠HSiF = 110.64(±3)°,     

Relative sticking coefficients of H2 and D2 on tungsten

The relative sticking coefficients of H₂ and D₂ on the (100) and (110) planes of W have been measured. It is found that on (100) W, in agreement with a previous determination by Tamm and Schmidt, but that on the close packed (110) plane . This result suggests that qualitatively different mechanisms probably control condensation rates on the two planes. It is also shown that the coverage dependence of S for the tightly bound β₂ state on (100) W is constant rather than decreasing as 1?θ as had been suggested previously by Madey.