Ultrasonic attenuation of longitudinal and shear waves in superconducting lead

Longitudinal and shear wave ultrasonic attenuations have been measured in high-purity Pb on two single crystals obtained from the same ingot. The measurements were done at low temperatures, at different frequencies, and in transverse magnetic fields, up to a field of 7.3 kG. The propagation directions in the two crystals were along [100] and [110]. For some propagation and polarization directions the _s / _n ratio is found to be frequency-independent, while for others, large divergences in the _s / _n ratios at different frequencies are observed. A sharp decrease of _s / _n nearT _c is observed for a particular longitudinal wave propagation, but not in any shear wave propagation. In some cases _s / _n is found to be abnormally high and this feature is associated with a peak in attenuation _n and a relatively high _n at 7.2 K. None of the _s / _n curves fits closely to any BCS energy gap. For longitudinal waves the high magnetic field (H) dependence of the normal state attenuation was found to agree qualitatively with the free electron theory for propagation along [100], but not for propagation along [110]. For shear waves the high-field attenuations do not extrapolate to zero asH tends to infinity. For all propagation and polarization directions the high-field attenuations show 1/H ² field dependence.     

Ultrasonic attenuation in normal and superconducting indium

Measurements have been made of the ultrasonic attenuation in the normal and superconducting states of pure In single crystals. The measurements spanned the frequency range of 6–66 MHz. The data, taken in the amplitude-independent regime, displayed the expected deviations from the BCS-type attenuation which are generally attributed to dislocation attenuation as described by Granato and Lücke. Earlier attempts to use this theory to study the amplitude-independent dislocation attenuation of ultrasound in superconductors (e.g., Mason) were generally limited to very narrow frequency ranges. These earlier results were in general agreement with the Granato and Lücke theory. However, the present multiple-frequency measurements are shown to be inconsistent with the Granato and Lücke theory.Supported by the Applied Research Laboratory of The Pennsylvania State University, under contract with the U.S. Naval Sea Systems Command.     

Thermal conductivity of niobium in the purely superconducting and normal states

The thermal conductivity λ of four niobium samples has been measured between 1 and 10 K, both in the superconducting and normal states. The specimens differed in their crystal defect structures due to annealing at different temperatures (dislocations, grain boundaries) and, in one case, to subsequent fast neutron irradiation (dislocation loops). A procedure has been developed with which the electron and phonon contributions to the thermal conductivity can be separated with an accuracy not hitherto obtainable. All the samples proved to have the same energy gap at 0K:δ(0)=(1.95±0.02)kT _c . The phonon conductivity in the superconducting stateλ _p ^s has been compared with the formula of Bardeen, Rickayzen, and Tewordt extended for scattering mechanisms other than phonon-electron interaction. For the unirradiated samples at \({\text{T}} \lesssim 0.15T_{\text{c}} \) , λ _p ^s is proportional toT ², showing that dislocations are mainly responsible for the phonon scattering. The results are qualitatively in agreement with the theory of Klemens, giving a rough indication that the grain boundaries may be considered as arrays of line dislocations. Dislocation loops introduced by the neutron irradiation turn out to behave like clusters of point defects. A second consequence of the irradiation is an enhancement of the original dislocation scattering term.     

Amplitude-dependent ultrasonic attenuation in the normal and superconducting states of indium-doped tin

The amplitude-dependent ultrasonic attenuation was studied at 3.6 and 10 MHz in three annealed single crystals of tin doped with 0.0016, 0.0031, and 0.00928 at% indium. The amplitude of the pulse in each case was varied from 30 to 300 V peak to peak. The results show that the dynamic loss is amplitude dependent, as predicted theoretically by Rogers. The attenuation peaks due to the breakaway of the dislocation from minor pinning points were observed and were used to calculate the electronic damping parameter and the dislocation resonance frequency. The former parameter was calculated from Holstein's theoretical expression and was found to be in agreement with the value determined experimentally.     

Ultrasonic attenuation in the superconducting and intermediate states of pure and doped type I superconductors

The attenuation of longitudinal ultrasonic waves has been measured in single crystals of indium (99.999%), indium doped with 0.003 at % of tin, and indium doped with 0.002 at % of bismuth in the intermediate and superconducting states over the frequency range 10–30 MHz. For the bismuth-doped indium specimen, measurements were taken for three different physical states, i.e., for three different dislocation densities, and for the indium and the tin-doped indium specimens, measurements were for one physical state. For a particular measurement, the same physical state was maintained both in the intermediate and superconducting states. A temperature-dependent oscillatory behavior of the ultrasonic attenuation was observed in the intermediate state in all the three specimens, but in the superconducting state the oscillatory behavior was observed only in the bismuth-doped specimen. Two phases have been identified in the superconducting layers of the intermediate state and there is only one phase in the superconducting state of the bismuth-doped sample. The origin of the two phases in the intermediate state and that of the single phase in the superconducting state of the bismuth-doped sample are discussed. A qualitative explanation is presented for the occurrence of oscillatory attenuation in the intermediate state irrespective of the nature of the dopant and the selective occurrence of oscillatory attenuation in the superconducting state due to the nature of the dopant.     

Temperature dependence of the elastic constants of niobium and lead in the normal and superconducting states

The temperature dependence of the elastic constants of niobium and lead was measured in the normal and superconducting states. Both a continuous wave and a coherent gated-carrier technique were used to measure the velocity changes associated with the superconducting transition. The elastic constants were found to exhibit the same temperature dependence as the free energy in either state. The discontinuity in the modulus associated with longitudinal strains and the slope discontinuities at the transition temperature were used to calculate the strain dependence of the transition temperature. The strain dependence in all cases is mainly quadratic.Research supported by a NSF Grant and in part by the Office of Naval Research.     

Dislocation damping and the ultrasonic attenuation in the normal and superconducting states of pure tin

Ultrasonic attenuation studies have been carried out both in the normal and superconducting states in pure (99.9999%) single crystals of tin containing various amounts of dislocations. At low pulse amplitude the normal state attenuation is effectively independent of the dislocation density. The dislocations have been found to introduce significant deviations in the superconducting state attenuations from the BCS values. The extra attenuation Δα_s in the superconducting state due to dislocations has been found to show an inverse fourth power dependence on the temperature. Δα_s has also been shown to exhibit resonance type of behaviour over the frequency range 3 to 50 MHz. This resonance frequency depends on the dislocation density with different power laws in different ranges of the dislocation density.     

Electromagnetic attenuation of shear waves in superconductors

A detailed analysis of the electromagnetic attenuation in superconductors has been carried out, with specific reference to type I materials. The rapid rise atT _c predicted by Cullen and Ferrell is found to be most pronounced in superconductors with low acoustic velocities. Also, the effect occurs at lowest frequencies in these materials. It is demonstrated how the shape of the transition curve reflects anisotropies in electronic and elastic properties. Very close toT _c the attenuation at low frequencies is inversely proportional to the real conductivity function, while at high frequencies it is proportional to that conductivity. Numerical results for Al, In, Pb, and Sn are presented. Although these materials are similar with regard to electronic structure, they are characteristically different with regard to electrodynamic properties as probed by ultrasonic phonons. From the present analysis it is concluded that ultrasonic probing of the electrodynamics of superconductors offers a sensitive and, at the present, unique way of determining the response of a superconductor to a single variable wave number. In addition the wave numbers available in this way are in a range where it has been most difficult to obtain such information by penetration depth studies, namely, in the transition region between pure local and nonlocal behavior.     

The lattice thermal conductivity of normal and superconducting niobium

The lattice thermal conductivity of superconducting and normal Nb as limited by the interaction of phonons with electrons has been deduced from measurements in the superconducting state. The results indicate that the mean free paths of transverse and longitudinal phonons are similar, ～4×10^?5 T ^?1 (cm K), in the normal state. Comparison is made with measurements on other metals. A compilation is included of the ratio of lattice conduction in the normal state to that in the superconducting state, based on the BCS theory of superconductivity.     

Deformation attenuation of ultrasonic shear waves in aluminum

The deformation contribution to the electronic attenuation of ultrasonic shear waves in aluminum is measured to be of the same order of magnitude as for longitudinal waves. This real metal effect is found highly anisotropic, depending both on the directionq/q and the polarization ε of the ultrasound. An anisotropic deformation parameterK _y is therefore suggested as being large whenever ε is close to the edgeWK of the Fermi surface.     

Damping of dislocation and the ultrasonic attenuation in normal and superconducting tin

Ultrasonic attenuation measurements at liquid helium temperatures, 77 K, and 300 K in single crystals of pure tin with frequencies varying from 3 to 50 MHz reveal in a qualitative way the temperature dependence of the electronic damping parameter B _en. An approximate estimate of B _en at 300 K has been made.     

Transmission,attenuation and reflection of shear waves in the human brain

Traumatic brain injuries (TBIs) are caused by acceleration of the skull or exposure to explosive blast, but the processes by which mechanical loads lead to neurological injury remain poorly understood. We adapted motion-sensitive magnetic resonance imaging methods to measure the motion of the human brain in vivo as the skull was exposed to harmonic pressure excitation (45, 60 and 80 Hz). We analysed displacement fields to quantify the transmission, attenuation and reflection of distortional (shear) waves as well as viscoelastic material properties. Results suggest that internal membranes, such as the falx cerebri and the tentorium cerebelli, play a key role in reflecting and focusing shear waves within the brain. The skull acts as a low-pass filter over the range of frequencies studied. Transmissibility of pressure waves through the skull decreases and shear wave attenuation increases with increasing frequency. The skull and brain function mechanically as an integral structure that insulates internal anatomic features; these results are valuable for building and validating mathematical models of this complex and important structural system.     

Transverse and longitudinal ultrasonic attenuation measurements of the superconducting energy gap in high-purity rhenium

The attenuation of transverse and longitudinal waves propagating along thec axis in high-purity rhenium has been measured in the normal and superconducting states. In the superconducting state the attenuation was compared with the simple BCS theory using as adjustable parameters the gap parameter and the residual attenuation atT _c (for transverse waves). Using transverse-wave attenuation a gap parameter of 3.50±0.10 was found, while the longitudinal attenuation data yielded a gap parameter of 2.90±0.10. At present, the only apparent explanation for this difference is that the integrals for transverse-and longitudinal wave energy loss in an ultrasonic wave contain different azimuthal angle dependencies and thus average an anisotropic energy gap differently.Research sponsored by the Air Force Office of Scientific Research under AFOSR Grant No. 71-2079.     

Longitudinal ultrasonic attenuation in normal and superconducting lead at low temperatures

We have measured longitudinal ultrasonic attenuation along the [110] direction in normal and superconducting states in two single crystals of lead, one made from high-purity lead and the other made with high-purity lead doped with 0.1 at % gold. In both specimens an amplitude-dependent effect in the superconducting state has been observed. The data have been taken in the frequency range from 12 to 108 MHz. In high-purity lead the amplitude-independent ratio α_s/α_n shows the frequency dependence observed by Randorff and Marshall, whereas in the doped specimen this ratio shows a very small spread with frequency. In both specimens deformation does not change the α_s/α_n ratio appreciably.     

Thermodynamic properties of superconducting niobium

The thermodynamic properties of superconducting Nb are calculated numerically from the solution of the Eliashberg equations on the imaginary axis for several possible electron-phonon spectral densities ²()F(). Comparison with experiment is made in order to see which spectrum gives the best agreement, and functional derivatives with respect to ²()F() are used to estimate how this agreement might be improved by small changes in ²()F(). Possible gap anisotropy effects are also considered with the help of a simple model anisotropy for the interaction.Research supported by Natural Science and Engineering Research Council of Canada.     

The plastic flow stress of metals and alloys in the superconducting and normal states

The observed difference in plasticity in the normal and superconducting states in metals and alloys has been attributed by most authors to an electron-dislocation interaction. Granato has associated this interaction with a dislocation damping property. We suggest a different model, in which the dislocation is thermally activated over a barrier whose effective temperature is raised by dislocation kinetic energy and which cools at a rate dependent on the thermal conductivity. State differences in flow stress are then attributed to differences in conductivity. While previous models have been used to describe the temperature dependence of the flow stress difference alone, this paper describes several other aspects of the phenomenon which demonstrate the validity of our model.     

Plasticity of lead single crystals in the superconducting and normal states at 4.2K

The flow stress increment _SN associated with a superconducting-normal phase transition has been studied as a function of deformation and strain rate for several lead single crystals at 4.2 K. In the linear work hardening range, the total applied stress _S determines the size of _SN. The dislocation inertial model recently proposed by Granato for the enhanced plasticity of the superconducting state is evaluated in detail with respect to the dependence of _SN on the distance between obstacles to dislocation movement. Reasonable agreement between theory and experimental results is obtained for the linear work hardening range, if it is assumed that the obstacles are forest dislocations. For the normal-state electronic drag coefficient a valueB _eN1.1×10^–4 dyn · sec/cm² is estimated.The experimental part of this work was performed at Materials Science Division, Argonne National Laboratory, Argonne, Illinois, under the auspices of the U.S. Atomic Energy Commission.     

Magnetic and superconducting properties of niobium oxides

Low temperature magnetic susceptibilities of niobium oxides have been measured. A homogeneous sample of NbO prepared by arc melting and checked by using X-ray and metallographic techniques exhibits no ferromagnetism but becomes superconducting at 1.20°K.     

Electrical resistivity and peak effect in superconducting niobium

The flux flow resistivity _f has been measured in foils of high purity niobium as a function of magnetic field and temperature. The magnetic field dependence of _f just belowH _C2 is much stronger than predicted by the recent theory of Maki for the pure limit. NearH _C2 a peak in the critical current is observed that is independent of any voltage criterion.Based on work performed partly under the auspices of the U.S. Atomic Energy Commission.     

Thermal conductivity of Ta-Nb alloys in superconducting,mixed, and normal states

The thermal conductivity of the type II superconductors Ta₉₅Nb₅, Ta₈₀Nb₂₀, Ta₆₀Nb₄₀, and Ta₂₀Nb₈₀ has been measured as a function of magnetic field up to 14 kOe and of temperature between 0.5 and 4.5 K. The temperature dependence of the thermal conductivity in the superconducting state is well accounted for on the basis of the Bardeen, Rickayzen, and Tewordt theory above 0.4T _c . The lattice thermal conductivity limited by dislocation scattering is observed below 0.3T _c . The thermal conductivity near the upper critical fieldH _c2 shows a linear dependence on magnetic field as predicted theoretically by Caroli and Cyrot. After the correction of the phonon contribution, the experimental results for the dirtiest sample, Ta₆₀Nb₄₀, are found in good quantitative agreement with the theory. Deviations from the theory for less dirty alloys depend on the electron mean free path.