Explicit Temperature Dependence Research Articles

The thermodynamics of elastic deformation—I: Equation of state for solids

The isobaric and isothermal volume derivatives of In B, In μ and In μ' are investigated, where B, μ and μ' are the isothermal bulk modulus and the two shear moduli, respectively, of a cubic crystal. In the case of the bulk modulus, the temperature independence of α B (where α is the volume thermal expansion) for a large number of materials, ensures that the derivatives are constant and approximately equal, while for the shear moduli, evidence is advanced that the isothermal derivatives are constant along an isotherm, but not along an isobar except at high temperatures near the melting point. The relationships satisfied by the bulk modulus enable the explicit temperature and pressure dependence of the molar volume, V, thermal expansion, and bulk modulus to be determined. The most adequate representation of the volume dependence of the Grüneisen parameter, γ, appears to be that γ/ V is independent of volume between the Debye and melting temperatures.

Measurement of the upper critical magnetic field of superconductors with magnetic impurities

We have measured the upper critical magnetic field, ${H}_{c2}(T)$, for In-Mn and Pb-Mn films, in which the Mn is a magnetic impurity. The results are compared with the additive-pair-breaking theory of Fulde and Maki, which predicts that the pair-breaking effect of the magnetic impurities is temperature-independent and that the pair-breaking effects of the magnetic impurities and of the applied magnetic field are additive. Furthermore, the theory predicts the explicit temperature dependence of ${H}_{c2}(T)$ The results indicate that the temperature dependence of ${H}_{c2}$ is well described by the Fulde-Maki theory for In-Mn, but not for Pb-Mn, probably because of the strong-electron-phonon coupling in Pb. For both In-Mn and Pb-Mn, the results indicate that the pair-breaking effect of the magnetic impurities is temperature independent, and that the pair-breaking effects of the magnetic impurities and the applied magnetic field are additive, so the concepts inherent in the Fulde-Maki theory are supported. We find no evidence in the data of the Kondo effect or of impurity spin alignment.

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

The pressure derivatives of the elastic moduli of Sr(NO 3) 2, Ba(NO 3) 2 and Pb(NO 3) 2 single crystal, have been measured by the ultrasonic pulse superposition method. The results for Sr(NO 3) 2 and Pb(NO 3) 2 are similar in magnitude and character, while the ones for Ba(NO 3) 2 differ strongly, ∂c 11 ∂P and ∂c 12 ∂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(NO 3) 2 to Pb(NO 3) 2.

The anisotropy dependence of ferromagnetic domain-wall widths in crystalline and amorphous metals

We have used the techniques of Lorentz electron microscopy to study the behavior of ferromagnetic domain-wall widths as a function of the magnetic anisotropy constant K and the foil thickness D. This was done by varying the temperature T in crystalline iron and in amorphous Fe75P15C10 for various D’s. Our results are compared with existing theories. We observed an increase in wall width with temperature in crystalline iron and explained this by assuming the width of the wall to be an explicit function of both the anisotropy constant K (T) and the temperature T. After removing the explicit temperature dependence, the experimental results yielded an expression for the wall width of the form 2.5D/K1/6 within the range of ∼1000 Å foil thickness. An increase in wall width with temperature was also observed for walls in amorphous Fe75P15C10. From the measured wall structures we obtained a value of ∼104 erg/cm3 for the effective anisotropy constant of this amorphous alloy.

Data processing involved in the treatment of the ratio R = TSC/TSL

Starting from the general expressions for TSC and TSL, it is shown how, from the ratio R = TSC/TSL, the explicit temperature dependence ƒ(T) can be separated from the conjucntional (or “functional”) evolution. From a simultaneous recording of TSC and TSL during fractional emptying, an ƒ(T) is deduced which only depends on the properties of the sample. This enables one to find the relative filling of the recombination centres acting in a comparative “single run” experiment. The type of data processing proposed is demonstrated for ZnSe:Al monocrystals.

Temperature dependence of the resistivity in metallic glasses

The temperature coefficient of the resistivity, $\ensuremath{\alpha}$, of metallic glasses is calculated starting from the same formalism which has been used to calculate the resistivity in liquid transition metals. An explicit equation is derived for the temperature dependence which includes both effects due to the decrease in the static structure factor as well as those due to phonons. The magnitude as well as the explicit temperature dependence of $\ensuremath{\alpha}$ is in good agreement with experiment.

Variational calculation of the thermal resistivity of noble metals

An equation is derived which demonstrates the explicit temperature dependence of deviations from Matthiessen's rule in the thermal resistivity of noble metals.

The elastic constants and their temperature and pressure derivatives of AgBr-AgCl mixed crystals

The three independent second-order elastic constants and their temperature and pressure derivatives have been measured for four AgBr-AgCl mixed crystals, with 19.5, 39.1, 56.6 and 78.7 mole % AgCl, using the ultrasonic pulse-echo technique at room temperature. The explicit temperature dependence of the elastic constants is calculated and is found to be much larger than that of other NaCl structure crystals. The violation of the Cauchy relation C 12 = C 44 is found to be significant and increases between AgBr and AgCl. The high temperature limit of the Gruneisen parameter is calculated from the elastic data. A comparison is made between the elastic properties of the silver halides and the alkali halides.

Effect of sharp structure in the electronic density of states on the electron-phonon coupling in highTc, superconductors

Abstract The evidence for sharp structure in the electronic density of states N(e) of certain classes of high T c superconductors is based on the normal state temperature dependence of the spin susceptibility x, electronic specific-heat coefficient γ, electrical resistivity p, and nuclear spin-lattice relaxation rate 1/T1. Model calculations of x, γ, p, and T1 in the rigid N(ε) approximation and of x and γ for a temperature dependent N(ε) are presented for V3Ga and V3Si. The model calculations have been used to calculate the explicit temperature dependence of the electron mass enhancement and the renormalized electronic specific heat coefficient. The temperature dependence of the dielectric screening, which can have an important effect on renormalizing the phonon frequencies at low temperatures, is also discussed.

One-dimensional conductivity for strongly scattered electrons — application to CdS

The d.c. conductivity of a one-dimensional system containing strong scatters is computed. The explicit temperature dependence for the conductivity is obtained. The model is applied to electrons localized on screw dislocations in strained CdS. The calculation yields excellent agreement with experimental results.

On the NMR properties near the 3He A-normal transition

The NMR properties of the A phase of liquid 3He very close to the transition temperature are investigated on the basis of the Leggett theory and of the assumption that the state is of the Anderson-Brinkman-Morel type. The explicit temperature dependence of the NMR frequencies is obtained and is shown to be quite different from that expected from a simple extrapolation of the lower temperature data.

EFFECT OF HYDROSTATIC PRESSURE ON THE ISOMER SHIFTS OF THE 6.2 keV RESONANCE OF181Ta IN TRANSITION METALS

Pressure dependence of the isomer shifts has been studied for 181Ta impurities in lattice hosts of Ta, W and Pt. The results are used to analyse earlier results of temperature dependence of the isomer shift. Values for the volume and explicit temperature dependences are derived.

Thermal variation of the spin relaxation rate in nearly magnetic metals

Considering either the nuclear spin resonance on the nucleus of a nearly magnetic atom or the electron spin resonance of a magnetic impurity in a nearly magnetic host, it is shown that taking into account an explicit temperature dependence of the Stoner susceptibility of the interacting electrons yields an important thermal decrease of 1/T1T around the spin-fluctuation temperature. The physical case of the nuclear spin relaxation rate of platinum is discussed.

Theory of the Temperature Derivative of the Refractive Index in Transparent Crystals

A simplified model is introduced to study the temperature dependence of the index of refraction $n$ in the transparent regime of crystals. The dielectric susceptibility is written as a sum of an electronic contribution due to band-to-band transitions and a lattice contribution, viewed as due to a single mode ${\ensuremath{\omega}}_{0}$ Each of the latter undergoes a temperature variation consisting of contributions due to thermal expansion, as well as contributions due to the explicit temperature dependence at constant volume. The temperature derivative $\frac{\mathrm{dn}}{\mathrm{dT}}$ is investigated for various materials. It is found that for zinc-blende- and diamond-type semiconductors, electronic effects, in particular the temperature variation of the band gap at constant volume, yield the dominant contribution. Theoretical calculations of the latter are carried out employing a temperature-dependent-pseudopotential band-structure model; the resulting values for $\frac{\mathrm{dn}}{\mathrm{dT}}$ are in good agreement with experiment. For ionic materials, it is found that both lattice and electronic contributions may be important, as are both explicit temperature variation and thermal-expansion effects. Experimental data on the temperature dependence of ${\ensuremath{\omega}}_{0}$ and the band gap are employed to obtain good agreement with the frequency variation of $\frac{\mathrm{dn}}{\mathrm{dT}}$ for a variety of ionic crystals. The results demonstrate that the major physical mechanisms responsible for $\frac{\mathrm{dn}}{\mathrm{dT}}$ can be understood within the present simplified model, and that the model is useful in predicting the magnitude and frequency dependence of $\frac{\mathrm{dn}}{\mathrm{dT}}$ for a wide variety of crystals of interest.

Pressure Dependence of Mossbauer Fraction and Energy Shift for Fe6' in Cu Lattice at 298 and 94 °K

AbstractThe pressure variations of the Mössbauer fraction and the second order Doppler shift for Fe57 in Cn lattice at 298 and 94 °K have been calculated taking into account the effects of change of mass and force constant due to impurity atoms. Due to the large error in the experimental value of Williamson and Ingalls no conclusion regarding the change of force constant with temperature has been obtained. The pressure dependence of centre shift at the two temperatures enables one to calculate the explicit temperature dependence of isomer shift with pressure. It is shown that at 82 kbar the isomer shift at 298 °K becomes equal to that at 94 °K. This is explained using the band model of lattice.

Differences in the Temperature Dependence of the Hyperfine Field and Magnetization of Ferromagnetic Metals

The temperature dependences of the magnetization and of the hyperfine field in ferromagnetic metals such as Fe or Ni are different even after transformation to constant volume. It is suggested that the effect of lattice vibrations can explain the observed explicit temperature dependence of the hyperfine field at constant volume and that the deviation of the latter from the temperature dependence of the magnetization at constant pressure is well described by considering the thermal expansion also. Some applications of this model to related problems are discussed.

NMR study of solid and liquid aluminium

Knight shift and spin-lattice relaxation were accurately measured in solid and liquid aluminium from room temperature to 1100°C. The Knight shift in the solid exhibits a strong explicit temperature dependence. K( a), the reciprocal enhancement factor of the Korringa relation, is found to be temperature dependent, an effect which is unaccounted for by the present methods of calculating the susceptibility.

Temperature Dependence of the Damping of Nonlinear Lattice Resonance in Ionic Crystals

A general expression for the $n\mathrm{th}$-order nonlinear complex-susceptibility tensor is derived by use of many-time temperature-dependent Green's functions. The damping constant for nonlinear absorption is obtained from the susceptibility tensor. It is shown that the $n\mathrm{th}$-order nonlinear susceptibility tensor is generated from lower-order Green's functions and that the explicit temperature dependence of the damping constant is not affected by increasing the strength of the applied field. It is found, by observing the frequency dependence of the susceptibility, that the frequency $\ensuremath{\omega}$ of the applied field is converted into $n\mathrm{th}$-harmonic waves of frequency $n\ensuremath{\omega}$.

Pressure Dependence of the Elastic Constants of SrTiO3

The effects of hydrostatic pressure between 0 and 22 kbar on the elastic constants of SrTiO3 were measured at 295°K. All constants increased linearly with pressure in this range. The pressure derivatives of the adiabatic constants dCij/dP are: 10.21±0.15, 3.51±0.15, and 1.24±0.05 for C11, C12, and C44, respectively. These pressure derivatives, in conjunction with the recent second-harmonic generation results of Mackey and Arnold, allow us to determine the complete set of third-order elastic constants. The results, in units of 1012 dyn/cm2, are: C111=−49.6±4.3, C112=−7.7±1.6, C123=+0.2±4.3, C144=−8.1±2.4, C166=−3.0±1.2, and C456=0.9±2.7. Combined with the temperature derivatives of the (second-order) elastic constants at constant pressure, the pressure results allow a determination of the explicit volume and temperature dependences of these constants. In addition, the present results allow us to calculate the acoustic mode Grüneisen parameters, an approximate value of the limiting Grüneisen constant, γ0=1.52, and the coefficients of the polynominal and logarithmic (Murnaghan) forms of the high-pressure equation of state of SrTiO3.

Temperature Coefficient of the Refractive Index of Diamond- and Zinc-Blende-Type Semiconductors

We have calculated the temperature coefficient of the long-wavelength refractive index of several group-IV and III-V semiconductors, using the Penn model for the electronic contribution to the dielectric constant. The isotropic band gap of this model is identified with the band gap at the $X$ point of the Brillouin zone, which can be simply expressed in terms of pseudopotential coefficients. The explicit temperature dependence of this gap is calculated by applying to these pseudopotential coefficients the appropriate Debye-Waller factors. The thermal expansion effect is obtained in the manner suggested recently by Van Vechten. Good agreement between the calculated and the observed temperature dependence of the long-wavelength refractive index is found.