The stress dependence of superconducting parameters in pure In and Sn and in α-phase In-Sn alloys

We have measured the length change due to the transition from the super-conducting to the normal state in pure In and Sn and in In-Sn alloys. The measurements were made on single-crystal specimens. The alloys had Sn content ranging from zero to 12 at %. Crystals parallel and perpendicular to the tetragonal axis were grown at each alloy content. We find that the relative length changes l/l are very sensitive to alloy content. From l/l we have calculated the stress derivatives H _c/ and T _c / and we find that T _c / changes from +59 to –92 mK/bar for stress along the tetragonal axis, and from –9 to 46 mK/kbar for stress perpendicular to the tetragonal axis. We suggest that these large changes are due to Fermi surface topology changes upon increasing the Sn content in In.     

Integral results for nonlinear diffusion equations

We show how to construct integral results for the multi-dimensional nonlinear diffusion equation c/t=·(D(c)c) and for some generalisations of this. For appropriate boundary conditions these become integral invariants. An application of these results to determining the large-time behaviour of some radially symmetric problems is indicated.     

The volume change at the superconducting transition of lead and aluminum

The volume change at the magnetic field-induced transition from the super-conducting to the normal state has been measured on single crystals of lead and aluminum between 0.3 K andT _c. From these data we have deduced the pressure dependence of the critical fieldH _c, of the critical temperatureT _c, and of the electronic specific heat coefficient . In lead, the results are in good agreement with theoretical calculations by Carbotte, where strong coupling effects are taken into account. We find ln / lnV=3.1±0.8, whereV is the volume. The measurements on aluminum giving ln / lnV=3±4 are consistent with results derived from thermal expansion experiments.This work was in part financially supported by the Schweizerischer Nationalfonds.     

Equation of state of3He near its liquid-vapor critical point

We report high-resolution measurements of the pressure coefficient (P/T) for³He in both the one-phase and two-phase regions close to the critical point. These include data on 40 isochores over the intervals–0.1t+0.1 and–0.2+0.2, wheret=(T–T _c )/T _c and =(– _c )/ _c . We have determined the discontinuity (P/T) of (P/T) between the one-phase and the two-phase regions along the coexistence curve as a function of . The asymptotic behavior of (1/) (P/T) versus near the critical point gives a power law with an exponent (+–1)^–1=1.39±0.02 for0.010.2 or–1×10 ^–2t–10 ^–6 , from which we deduce =1.14±0.01, using =0.361 determined from the shape of the coexistence curve. An analysis of the discontinuity (P/T) with a correction-to-scaling term gives =1.17±0.02. The quoted errors are fromstatistics alone. Furthermore, we combine our data with heat capacity results by Brown and Meyer to calculate (/T) _c as a function oft. In the two-phase region the slope (²/T ²)_c is different from that in the one-phase region. These findings are discussed in the light of the predictions from simple scaling and more refined theories and model calculations. For the isochores 0 we form a scaling plot to test whether the data follow simple scaling, which assumes antisymmetry of – ( _c ,t) as a function of on both sides of the critical isochore. We find that indeed this plot shows that the assumption of simple scaling holds reasonably well for our data over the ranget0.1. A fit of our data to the linear model approximation is obtained for0.10 andt0.02, giving a value of =1.16±0.02. Beyond this range, deviations between the fit and the data are greater than the experimental scatter. Finally we discuss the (P/T) data analysis for ⁴ He by Kierstead. A power law plot of (1/) P/T) versus belowT _c leads to =1.13±0.10. An analysis with a correction-to-scaling term gives =1.06±0.02. In contrast to ³ He, the slopes (²/T ²)_c above and belowT _c are only marginally different.Work supported by a grant from the National Science Foundation.     

Convection in a shallow laterally heated cavity with conducting boundaries

Shallow cavity flows driven by horizontal temperature gradients are analysed over a range of Rayleigh numbersR and Prandtl numbers , whereR is comparable in size to the aspect ratioL(1). Eigenvalue calculations show the existence of a critical Prandtl number R > R _c (), below which the parallel core-flow structure is destroyed for Rayleigh numbersR>R _c(). For other Rayleigh numbers and Prandtl numbers the horizontal scale of influence of the end walls of the cavity is determined.     

Thermal diffusion in3He-4He mixtures near the tricritical point and λ line

It is shown that in the superfluid phase of helium mixtures, near the tricritical point, the thermal diffusion ratiok _T is positive and, both in the tricritical region and along the coexistence curve, behaves as (c/) _T,p [(T _t –T)/T _t ] ^–1 (same exponent as in the normal phase); and near the line,k _T again is positive and behaves as (c/) _T,p . In both cases, quasielastic light scattering is shown to provide a convenient means of measuring the thermal conductivity at the transition.     

Ultrasonic study of the elastic and nonlinear acoustic properties of ceramic aluminum nitride

Pulse-echo-overlap measurements of ultrasonic wave velocity have been used to determine the elastic stiffness moduli and related elastic properties of aluminum nitride (AlN) ceramic samples as functions of temperature in the range 100–295 K and hydrostatic pressure up to 0.2 GPa at room temperature. Aluminum nitride is an elastically stiff but light ceramic: at 295 K, the longitudinal stiffness (C _L), shear stiffness (), adiabatic bulk modulus (B ^S), Young's modulus (E) and Poisson's ratio () are 373 GPa, 130 GPa, 200 GPa, 320 GPa and 0.234, respectively. The temperature dependences of C _L and B ^S show normal behaviour and can be approximated by the conventional model for vibrational anharmonicity. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to determine the hydrostatic-pressure derivatives of elastic stiffnesses and the acoustic-mode Grüneisen parameters. The values determined at 295 K for the hydrostatic-pressure derivatives (C _L/P) _P=0, (/P) _P=0 and (B ^S/P) _P=0 are 4.7 ± 0.1, 0.22 ± 0.03 and 4.4 ± 0.15, respectively. The adiabatic bulk modulus B ^S and its hydrostatic-pressure derivative (B ^S/P) _P=0 are in good agreement with the results of recent high pressure X-ray diffraction measurements and theoretical calculations. The longitudinal (_L), shear (_S), and mean (^el) acoustic-mode Grüneisen parameters of AlN are positive: the zone-centre acoustic phonons stiffen under pressure. The shear _S (=0.006) is much smaller than the longitudinal _L (=1.09) accounting for the low thermal Gr¨neisen parameter ^th (=0.65) obtained for this ceramic: since the acoustic Debye temperature _D (=980 ± 5 K) is so high, the shear modes play an important role in acoustic phonon population at room temperature. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic AlN.     

Equation of state of a3He-4He mixture near its liquid-vapor critical point

Measurements of the pressure coefficient (P/T)_,x are reported for a ³ He- ⁴ He mixture with a mole fractionX=0.805 of ³ He in the neighborhood of the liquid-vapor critical point. These include data on 16 isochores taken over the density interval–0.50.5 and over the temperature range–0.1 t0.1, where =(– _c )/ _c andt=(T-T _c )/T _c ,with _c andT _c ,respectively, the critical density and temperature of the mixture. From the discontinuity of (P/T)_,x at the boundary between the two-phase and the one-phase regions we determine the dew-bubble curve nearT _c with better precision than was done in recentPVT experiments. From the extrapolation of data not approachingT _c closer than1 mK, (P/T)_,x along the critical isochore appears to be discontinuous atT _c ,while for the isochore / _c 0.92, (P/T)_,x is continuous across the dew curve. It is found that this latter isochore cuts the dew curve at its highest temperature. These observations are discussed in terms of general thermodynamic arguments and theoretical predictions of the asymptotic behavior. We calculate (P/T)_,x from the scaling equation of state proposed by Leung and Griffiths for ³ He- ⁴ He mixtures, using their numerical parameters. In spite of some systematic deviations, especially in the two-phase region, there is in general good agreement with experimental results. In particular, the shape of the measured dew-bubble curve and the apparent discontinuity of (P/T)_,x along the critical isochore show excellent agreement with theory.Work supported by a grant from the National Science Foundation. A report of this work has been presented at the Washington Meeting of the APS [Bull. Am. Phys. Soc. 20, 618 (1975)].     

Experimental confirmation of the isotopic volume effect in superconducting molybdenum by means of energy-dispersive x-ray diffration at low temperatures

Differences in the lattice constants of Mo-100 and Mo-92 have been measured by x-ray diffraction, in order to search directly for the volume effect of isotopes in a superconductor. No significant difference in the lattice constanta(Mo-100)–a(Mo-92) could be detected at 290 K, while the differences –0.0014±0.0008 and –0.0029±0.0009 Å were detected at 85.3 and 4.31 K, respectively. These values, and their temperature dependence, are considered to be theoretically reasonable. The exponent in the isotope effect defined by T_cM^- is represented thermodynamically by =–( lnT _c / lnM)–( lnT _c / lnV)(d lnV/d lnM). From the results, d lnV/d lnM is found to be –0.033±0.009 at 4.31 K. Then, the second term representing the isotopic volume effect is estimated to be about 0.09, with lnT _c / lnV2.81. The observed value of is 0.33, so that the contribution of the second term, 0.09, is 27% of the value of . It becomes quite clear that the isotopic volume effect in superconducting Mo should not be neglected.     

Pressure dependence of T c for lead

The superconducting transition temperature T _c for Pb has been determined as a function of hydrostatic pressure up to 26 kbar. T _c is found to more closely follow a linear pressure dependence, with T _c/P = –(3.61 ± 0.05)× 10^–5K/bar, than a linear volume dependence, as previously reported. The volume dependence of the electron-phonon parameter derived from these measurements, d(ln )/d(ln V) = 4.2 ± 0.1, is in good agreement with theoretical calculations by Ott and Sorbello and by Carbotte and Vashishta. The variation of for larger volume changes is discussed.Supported by the Australian Research Grant Committee.     

Interaction between a dislocation and an impurity in KCl: Mg2+ single crystals

The interaction between a dislocation and the impurity in KCl: Mg²⁺ (0.035 mol% in the melt) was investigated at 77–178 K with respect to the two models: one is the Fleischer's model and the other the Fleischer's model taking account of the Friedel relation. The latter is termed the F-F. The dependence of strain-rate sensitivity due to the impurities on temperature for the specimen was appropriate to the Fleischer's model than the F-F. Furthermore, the activation enthalpy, H, for the Fleischer's model appeared to be nearly proportional to the temperature in comparison with the F-F. The Friedel relation between effective stress and average length of the dislocation segments is exact for most weak obstacles to dislocation motion. However, above-mentioned results mean that the Friedel relation is not suitable for the interaction between a dislocation and the impurity in the specimen. Then, the value of H(T _c) at the Fleischer's model was found to be 0.61 eV. H(T _c) corresponds to the activation enthalpy for overcoming of the strain field around the impurity by a dislocation at 0 K. In addition, the Gibbs free energy, G ₀, concerning the dislocation motion was determined to be between 0.42 and 0.48 eV on the basis of the following equation ln / = G ₀/(kT_p0)1 – (T/T _c)^{1/2 –1}(T/T _c)^1/2 + ln ₀/where k is the Boltzmann's constant, T the temperature, T _c the critical temperature at which the effective stress due to the impurities is zero, _p0 the effective shear stress without thermal activation, and ₀ the frequency factor.     

Optical and ultrasonic properties of europium phosphate glasses

To provide an overall picture of the vibrational properties of phosphate glasses containing high concentrations of europium, measurements have been made of their ultrasonic wave velocities and attenuation, optical absorption spectra and laser-induced fluorescence. To examine their valence state, the fluorescence of the glasses has been examined. The spectra do not show any obvious sign of divalent europium ions, only trivalent europium ion fluorescence being observed. Room-temperature absorption spectra of these glasses also provide evidence of only the absorption bands of trivalent europium. The elastic stiffnessesC ₁₁ andC ₄₄ continue to increase down to low temperatures and the ultrasonic attenuation is characterized by a broad peak, properties which are consistent with thermally activated relaxations of two-level systems. The longitudinal and shear ultrasonic wave velocities decrease under pressure; the hydrostatic pressure derivatives (C ₁₁/P) _{T, P=0} and (C ₄₄/P) _{T, P=0} of the elastic stiffness tensor componentsC _IJ and (B/P) _{T, P=0} of the bulk modulus,B ₀, are negative. When compressed, the europium phosphate glasses, like their samarium analogues, show the interesting property of becoming easier to squeeze. Measurements, using a pulse superposition technique, of the effect of uniaxial stress on ultrasonic wave velocities have been used to determine the temperature dependences of the third-order elastic stiffness tensor components of (Eu₂O₃)_0.186(P₂O₅)_0.814 and (Eu₂O₃)_0.20(P₂O₅)_0.80 glasses between 77 and 400 K. The uniaxial and hydrostatic pressure dependences of the elastic constants quantify the cubic term in the strain Hamiltonian and the vibrational anharmonicity of the long-wavelength phonons of these glasses. The acoustic mode Grüneisen parameters are negative: application of pressure induces a decrease in the long-wavelength acoustic phonon mode frequencies. As the temperature is reduced the pressure-induced mode softening becomes enhanced. The hydrostatic pressure derivative (C ₁₁/P) _{T, P=0} is larger than (C ₄₄/P) _{T, P=0} over the whole temperature range, and the longitudinal acoustic mode Grüneisen parameter _L is larger than that _S of the shear wave: the longitudinal mode softens more with pressure than the shear mode. Murnaghan's equation-of-state is used to determine the compressionV(P)/V ₀.     

Thermal pressure coefficients of ethanenitrile,propanenitrile, and butanenitrile in the region 295–395 K

The thermal pressure coefficient (p/T) _v has been measured for ethanenitrile from 299 to 364 K, for propanenitrile from 295 to 377 K, and for butanenitrile from 297 to 398 K. The results are discussed in terms of the diminishing role of polarity in the alkanenitrile series and of a corresponding-states approach using gas-liquid critical properties as reduction factors. Although (p/T) _v varies unevenly with chain length, the reduced quantity shows a more regular behavior similar to that of the related quantity the cohesive energy density.     

Specific heat capacity of solids under pressure from measurements of (∂T/∂P)s

A procedure is described for calculating specific heat capacity under pressure, c _p (T, P), from data for c _p (T, 0) and adiabatic (T/P) _s. The main advantage is that (T/P)_s can be readily measured under high-pressure conditions.     

Diffusive relaxation processes in liquid3He-4He mixtures. I. Normal phase

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients T and c are reached via a diffusive transient process described by a series of terms modes involving characteristic times _n . These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=k _T ² (/c) _T,P /TC _P,c coupling the mass and thermal diffusion. Herek _T is the thermal diffusion ratio, (/c) _T,P ^–1 is the concentration susceptibility, is the chemical potential difference between the components, andC _P,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times (T) and (c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different³He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure³He (i.e., in the absence of mass diffusion) we find the observed (T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated³He-⁴He mixtures, we observe (c) and (T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(³He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of (c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (T _t,X _t) is approached for the mixtureX=X _t0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toT andT _t, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.Work supported by grants from the National Science Foundation and the Research Corporation and by an A. P. Sloan fellowship to one of the authors (RPB).     

Thermal transport properties in the normal phase of dilute3He-4He mixtures

We present steady-state measurements of the thermal diffusion ratiok _T and of the heat conductivity for three dilute mixtures of³He in⁴He with concentrations 9×10^–3X(³He)5×10^–2 at saturated vapor pressure in the normal phase close to the superfluid transition. The data are compared with predictions by Dohm and Folk from the renormalization group (RG) theory. From auxiliary determinations of thermodynamic derivatives for these mixtures, we obtain the separation factor =–(k _T /T)×(/X) _T,P /(/T) _X,P above T over the range wherek _T is positive. Here is the mass density. From the transients of X(t) as a function of time, we obtain an estimation for the mass diffusion coefficientD and compare the results with predictions by Dohm and Folk and with results from other experiments.     

Magnetically and Thermally Modulated Microwave Absorption in Y1Ba2Cu3O7–x Single Crystal near Tc

The microwave absorption R in the Y ₁ Ba ₂ Cu ₃ O _7–x single crystals was investigated near T _c 92K and in the external magnetic field 0 < H 9kOe. A modified ESR spectrometer was used in the experiment. The method of temperature modulation, along with the usual method of magnetic-field modulation, was first applied in studying of the microwave response of these crystals. Peaks in the temperature dependencies of the signals R/H and R/T observed in the vicinity of T _c were differently shaped and shifted one with respect to another. The evolution of the peaks with variation of the magnetic field and angle between H and the c-axis was traced. It has been shown that the observed difference of the temperature dependencies of the derivatives R/H and R/T occurs due to the field-induced broadening of the superconducting transition, which is inherent in the high-T _c superconductors.     

Cross-Correlated Dynamic Resistance of a Direct Current Superconducting Quantum Interference Device

This paper presents experimental data on a comparative study of a dc SQUID with voltage and current bias. We introduce a cross-correlated dynamic resistance of the device defined as a ratio R _dCV = _VC/_IV, where _VC = (V/) _A is the slope of the voltage-to-flux characteristics measured in the current bias mode and _IV = (I/) _A is the slope of the current-to-flux characteristics measured with voltage bias. It has been found that R _dCV may deviate strongly from the dynamic resistance observed in the current bias mode of operation. The intrinsic energy resolution of the SQUID remains unchanged for both modes of operation, but the current noise of the voltage biased device scales with the cross-correlated dynamic resistance. In our SQUID with the loop inductance L = 105 pH, is equal to 37 h in the white noise region at a temperature of 4.2 K.     

Ultrasonic study of the temperature and pressure dependences of the elastic properties of β-silicon nitride ceramic

Ultrasonic wave velocity measurements have been used to determine the elastic stiffness moduli and related elastic properties of high-purity, dense -Si₃N₄ ceramic samples as functions of temperature in the range 150–295 K and hydrostatic pressure up to 0.2 GPa at room temperature. Due to its covalently bonded, rigid structural framework -Si₃N₄ is an elastically stiff material; the elastic stiffness moduli of the ceramic at 295 K are: C _L = 396 GPa, = 119 GPa, B ^S = 238 GPa, E = 306 GPa, Poisson's ratio = 0.285. The longitudinal elastic stiffness C _L increases with decreasing temperature and shows a knee at about 235 K; the decrease in slope below the knee indicates mode softening. The shear elastic stiffness shows mode softening which results in a plateau centred at about 235 K and an anomalous decrease with further reduction in temperature. The hydrostatic-pressure derivatives of elastic stiffnesses at 295 K are (C _L/P) _P=0 = 4.5 ± 0.1, (B ^S/P) _P=0 = 4.3 ± 0.1 and (/P) _P=0 = 0.17 ± 0.02 (pressure < 0.12 GPa). An interesting feature of the nonlinear acoustic behaviour of this ceramic is that, in the pressure range above 0.12 GPa, the values obtained for (/P) _P=0 and the shear mode Grüneisen parameter (_S) are small and negative, indicating acoustic-mode softening under these higher pressures. Both the anomalous temperature and pressure dependences of the shear elastic stiffness indicate incipient lattice shear instability. The shear _S(=0.005) is much smaller than the longitudinal _L(=1.18) accounting for the thermal Grüneisen parameter ^th(=1.09): since the acoustic Debye temperature _D(=923 ± 5 K) is so high, the shear modes play an important role in acoustic phonon population at room temperature. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic -Si₃N₄.