On intrinsic and extrinsic effects in the surface impedance of cuprate superconductors

Surface impedance measurements in the normal and superconducting state are an excellent method to study conduction electron dynamics and extended defects. Electron dynamics show up most clearly in the relaxation range, i.e., for distances traveled in one rf periods= _F/ ( _F Fermi velocity) being smaller or of the order of the penetration depth and mean free pathl. For materials with _F10⁷ cm/sec the relaxation range is easily accessible forf0.1 THz. Then, in the normal state, relaxation defines the surface impedance with an intrinsic penetration depth _I approaching the London penetration depth _L andR _{I 0 L}/ 2 as surface resistance, allowing measurement of _L and relaxation time(T, ). In the superconducting state the photon interaction scales with _L/ _L=1/( _F dimension of Cooper pairs forl) and causes at low frequencies an absorption rate growing with, which is decreasing with _F/l. The rate increase proportional to turns to a decrease above 0.1 THz, which is accompanied by a decrease ofA with frequency which is stronger for large and small _F/l. These characteristic dependences allow measurement of material parameters, anisotropy, and dynamics of electrons, especially the relaxation rate. But presently, the rf surface impedanceZ is still shrinking with material improvements, which shows, clearly, that theZ=Z _I+Z _res is still dominated by extrinsic properties summarized inZ _res. We present evidence thatZ _res is due to the large leakage currentj _bl and the smallj _cJ of weak links where the latter destroys the intrinsic shielding from a _I-thin seam _J deep into the bulk. This causes rf residual lossesR _res( ₀)² _J ³ _bl/2.R _res stays finite atT-0 due to _bl(T0) _bl(j _bl) being amplified by ( _J/ _I)³>10³ as a weighting factor. The appropriate measure of weak links are the grain-boundary resistanceR _bn((0)) enhancing _JR _bn andR _resR _bn ² . Thus,Z _res is minimal for minimal extrapolated resistivity(T0).To identify the weak links as a new entity, the H-field dependence is most helpful, because at very low fieldsH _c 1J1/ _J Josephson fluxons penetrate into the weak links. These Josephson fluxons show negligible flux flow or flux creep, and enhanceZ _res by _J(H, T) . The measuredj _cJ(H, T) andj _bl values explainZ _res quantitatively as well as in temperature (a+T ⁿ ) (n1,T<T _c /2) and in field (b+H ⁿ ) (n1,H>H _c1J) dependence. The strength of the field dependencedZ _res/dHZ _res(H _c1J )/H _c2J(T) is not only a measure ofZ _res andH _c2J(T) but is crucial for nonlinear effects and (fluxon) noise also, which limit the performance of rf devices.     

Tilted and Crossing Vortex Chains in Layered Superconductors

No Heading In presence of the Josephson vortex lattice in layered superconductors, small c-axis magnetic field penetrates in the form of vortex chains. In general, structure of a single chain is determined by the ratio of the London [] and Josephson [_J] lengths, = /_J. The chain is composed of tilted vortices at large s (tilted chain) and at small s it consists of crossing array of Josephson vortices and pancake-vortex stacks (crossing chain). We study chain structures at the intermediate s and found two types of phase transitions. For 0.6 the ground state is given by the crossing chain in a wide range of pancake separations a [2–3]_J. However, due to attractive coupling between deformed pancake stacks, the equilibrium separation can not exceed some maximum value depending on the in-plane field and . The first phase transition takes place with decreasing pancake-stack separation a at a = [1 – 2]_J, and rather wide range of the ratio , 0.4 0.65. With decreasing a, the crossing chain goes through intermediate strongly-deformed configurations and smoothly transforms into the tilted chain via the second-order phase transition. Another phase transition occurs at very small densities of pancake vortices, a [20 – 30]_J, and only when exceeds a certain critical value 0.5. In this case small c-axis field penetrates in the form of kinks. However, at very small concentration of kinks, the kinked chains are replaced with strongly deformed crossing chains via the first-order phase transition. This transition is accompanied by a very large jump in the pancake density.PACS numbers: 74.25.Qt, 74.25.Op, 74.20.De     

Eliashberg Spectral Functions for Some High-Tc Superconductors

We used a simplified but more realistic form of a model spectral functions for some high-T _c superconductors (e.g., LSCO, YBCO, BSCCO), avoiding the elaborate process of solving the Eliashberg equations numerically. The resulting spectral functions, constructed using the phonon density of states (PDOS), obtained from INS experiments with the easy use of available expressions, reproduce the observed critical temperature, gap ratio, and several other parameters. T _c is found to have a stronger increase with the electron–phonon coupling constant compared to those obtained from several equations, constructed from numerical solution of the Eliashberg equations for lower values. The obtained functions show widely spreading spectra that inhibit lattice instability by yielding not too large values of (< 2.95).     

Thermal conductivity of solid NaF under high pressure

The thermal conductivity () of solid NaF has been measured over the temperature (T) range 100–350 K and at pressures (P) up to 2.5 GPa, using the transient hot-wire method. Results for (T,P) could be described to a good approximation by the Leibfried-Schlömann formula. It was found that the isochoric temperature derivative of the thermal resistivity W (= ^–1) increased systematically with the mass ratio for the B1-type phases of the sodium and potassium halides.     

Time-dependent behavior of an incompressible elastic fluid. Some homogeneous deformation histories

Summary The theory of the incompressible elastic fluid is designed to describe visco-elastic behavior under finite strain. In terms of this theory it is shown how to calculate the stress response to any strain history. The calculations are carried out and the results displayed for a number of homogeneous strain histories. A knowledge of the response to single step stress relaxation in a given geometry allows one to calculate easily the response to any other history in the same geometry. Furthermore from biaxial stress relaxation data one may calculate the time dependent stress response in any geometry in a manner which resembles the method for predicting the equilibrium response in an isotropic elastic material. An important class of histories comprises the motions following the rest history. To calculate the stress response for such a motion over a given time interval it suffices to have stress relaxation data over the same time interval. Stress response is also calculated for some uniaxial strain histories as well as steady simple shear.

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Zusammenfassung Mit Hilfe der Theorie der inkompressiblen elastischen Flüssigkeit wird das viskoelastische Verhalten bei endlichen Verzerrungen beschrieben. Im Rahmen dieser Theorie wird der Spannungsablauf hinsichtlich einer beliebigen Verzerrungsvorgeschichte berechnet. Die Ergebnisse der Rechnung werden für eine Reihe von Vorgeschichten von homogenen Verzerrungen dargestellt. Die Kenntnis des Relaxationsverhaltens nach einem einzelnen Spannungssprung in einer gegebenen Geometrie gestattet eine einfache Rechnung für jede andere Vorgeschichte in der gleichen Geometrie. Ferner gestattet die Kenntnis der zweiachsigen Spannungs-relaxation die Berechnung des zeitabhängigen Spannungsablaufes bei beliebiger Geometrie in einer Art, die der Ermittlung des Gleichgewichtszustandes in einem isotropen elastischen Material ähnelt. Eine wichtige Gruppe von Vorgeschichten beinhalten die Bewegungen, die auf einen Ruhezustand folgen. Um den Spannungsablauf für eine solche Bewegung über ein vorgegebenes Zeitintervall zu ermitteln, genügt es, Daten für die Spannungsrelaxation in diesem Intervall zu kennen. Für einige einachsige Verzerrungsvorgeschichten und für stationären Schub wird der Spannungsablauf angegeben.

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Symbols used in this work

Symbol x ₁,x ₂,x ₃;x _i position coordinates - X ₁,X ₂,X ₃ particle labels - x (t);x _i (t) position of a tacitly understood particle at timet - (t, ) relative deformation matrix - B (t, ) matric of left relative Cauchy-Green tensor - B _ij (t, ) components ofB - T _e.g.T transpose of a matrix - C (t, ) matrix of right relative Cauchy-Green tensor - I ₁,I ₂,I ₃ principal invariants ofB (orC) - ₁, ₂, ₃ principal extension ratios - extension ratio in pure shear or uniaxial strain - shear measure in simple shear - _ij components of true stress tensor - U potential for incompressible elastic fluid - U _{I 1},U _{I 2} derivatives ofU with respect toI ₁,I ₂ - b _i components of local body force per unit mass - mass density - stress difference in uniaxial strain - a ₁,a ₂,a ₃ functions ofX ₁,X ₂,X ₃ which represent a deformation which is constant over a period of time - W potential for single-step stress relaxation - W _{I 1},W _{I 2} derivatives ofW with respect toI ₁,I ₂ - P (,t) stress difference in pure shear single-step stress relaxation - P _* (,t) time derivative ofP - G (,t) stress difference in uniaxial single step stress relaxation - G _* (,t) time derivative ofG - K (,t) shearing stress in simple shear single step stress relaxation - L ₁ (,t),L ₂ (,t) normal stress differences in simple shear single step stress relaxation - K _* (,t),L _* (,t),L _2* (,t) time derivatives ofK,L ₁,L ₂ - ₁, ₂ forces per unit initial area in biaxial strain following the rest history - a _i components of acceleration - force per unit initial area in uniaxial strain following the rest history - p ₀ ambient pressure - viscosity - v _I,v _II,v _III normal stress measures - rate of shear     

Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem

A two-dimensional axisymmetric mathematical model of electron-beam autocrucible melting is developed and examined. Here, the hypothesis is used that forced convective heat transfer in the melt may be modelled with the help of the coefficient of effective thermal conductivity, _E. A simplified approach is used in which _E is assumed to be known. In another approach the value of _E depends on a prescribed value of a mean melt-stirring velocity and a mean liquid-pool radius which is determined in the course of solving of the problem. With the help of the Kirchoff transformation and a Green function we may reduce the problem to a nonlinear Hammerstein integral equation. Here, a dependence of the thermal-conductivity coefficient on the temperature, _S(T), at the cooled surfaces is disregarded and constant (mean) values of _S are utilized. In order to solve the problem in the case where this dependence _S(T) is taken into account, an axuiliary Green-function method is proposed which also permits to take into account a change of the heat-exchange coefficients on the autocrucible. This reduces the problem to a system of three integral Hammerstein equations. Numerical solutions of the nonlinear integral equations are obtained with the help of a variational (projective-net) method for the case of circular scanning of an electron beam over the heated surface. The computational results are well consistent with experimental data.     

Dynamic Phase Transition in Amorphous YBaCuO Films under Ar Laser Irradiation

The effect of laser irradiation (Ar laser, = 488 nm) on the atomic order and electronic structure of amorphous YBaCuO films is studied. It is shown that the laser irradiation of amorphous YBaCuO films leads to a sequence of processes having an oscillating character: energy accumulation (I _incoh max), accompanied by stronger atomic disordering, and relaxation (J _coh max) due to atomic ordering and a change in the predominant cluster configuration, (11l) – (10l), at incident powers above 2 W. This corresponds to a local topological ordering of the solid solution with the participation of copper–oxygen chains, accompanied by an increase in conductivity by two orders of magnitude.     

Effect of sample density on magnetic penetration depth in YBaCuO ceramic superconductors

We have measured magnetic penetration depth(0) of high-T _c YBCO samples of different density by the cavity resonant frequency shift method at 10, 16.65, and 22.3 GHz microwave frequencies. The value of(0) at 10 GHz is found to decrease from 5850 Å to 2550 Å as the density of the sample increases from 4.4 to 5.3 g/cm³. The results of the frequency response of the penetration depth show a fairly constant value of(0) for all the three samples in the frequency range 10–22.3 GHz. The wide variation observed in the value of(0) for different density samples has been explained in terms of varying Josephson coupling strength in these ceramics.     

Antiplane strain problem of diamond inclusion

In this paper, the singular stress fields created by an antiplane deformation at an inclusion corner are studied. It is shown that these singular stress fields can be separated into two independent types: a symmetric type with the stress singularity of 1/ ^1–1 and a skew-symmetric type with the stress singularity of 1/ ^1–2. These two types of the singular stress field can not occur simultaneously at the corner. If G ₂<G ₁, there exists only the singularity of the skew-symmetric type, and if G ₂>G ₁, there exists only the singularity of the symmetric type. A general expression of stress fields in the vicinity of the corner is presented. In the expression the singular stress fields for the symmetric type and the skew-symmetric type are defined in terms of the constants K _III, ₁ and K _III, ₂, respectively. K _III, ₂ and K _III, ₂ have to be determined from the complete boundary conditions of the given problem. For the problem of an infinite plate containing a diamond inclusion and subjected to a uniform longitudinal shear stress at infinity, the values of K _III, ₁ and K _III, ₂ are obtained by body force method. In the body force method, the investigated stresses are simulated by the superposition of the fundamental stress fields due to point forces. In order to obtain accurate solutions, the basic density functions of the distributed point forces are used, so that the stress singularities at the corner tip can be simulated by the point forces.     

Effect of growth rate and lamellar spacing on microhardness in the directionally solidified Pb-Cd, Sn-Zn and Bi-Cd eutectic alloys

Lead-cadmium, Zinc-tin and Bismuth-cadmium of (99.99%) high purity eutectic alloys were melted in a graphite crucible under vacuum atmosphere. These eutectic alloys were directionally solidified upward with a constant temperature gradient G and different growth rates V in the Bridgman type directional solidification furnace. The lamellar spacings and microhardness H _V were measured from both transverse section and longitudinal section of the specimen. The variations of H _V with respect to V and have been determined by using the linear regression analysis method. H _V values increase with the increasing values of V and decrease with the increasing values. The Hall-petch type relationships obtained in this work have been compared with the previous works.     

The stress dependence of the crack growth rate during creep

An equation is derived for the crack growth rate under creep conditions. In the model, the propagation of a grain boundary crack is controlled by the plastic growth of cavities located in the grain boundaries ahead of the crack. It is assumed that the cavities grow by power law creep in the elastic crack tip stress field. Hence, the stress dependence of the crack velocity is provided through the elastic stress intensity factor, i.e., dC/dt=BK _I ^p .The cavity spacing, , appears as an important factor in the coefficient,B ^–(p–2)/2. At large values of , corresponding to less severe creep damage in the grain boundaries, the above equation would predict very low values for the crack velocity. Under such conditions, we suggest that another mechanism, whose stress dependence is provided through the net section stress, becomes active, i.e., dC/dt=B _net ^p . Since increases with decreasing applied stress, one should observe the _net correlation at low stresses. The results of recent creep crack growth experiments which tend to support this hypothesis are presented.

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Résumé On dérive une équation décrivant la vitesse de propagation d'une fissure dans les conditions de fluage. Dans ce modèle, la propagation d'une fissure aux frontières de grains est contrôlée par la croissance dans le domaine plastique de cavités situées aux frontières de grains en avant de la fissure.On suppose que les cavités s'étendent dans le champ de contraintes élastiques situées à l'extrémité de la fissure en suivant une loi de fluage parabolique. Dès lors, la dépendance de la vitesse de la fissuration en fonction de la contrainte est fournie par un facteur d'intensité de contrainte élastique, c'est-à-dire dC/dt=BK _I ^p .L'espace entre les cavités, , apparaît être un facteur important dans les coefficientsB. Pour de grandes valeurs de , qui correspondent à un dommage moins sévère par fluage aux frontières des grains, l'équation ci-dessus permettrait de prédire des valeurs très faibles de la vitesse de fissuration.Sous ces conditions, il est suggèré qu'un autre mécanisme, dont la dépendance de la contrainte est fournie par la contrainte agissant sur la section droite, devient plus actif; on a alors dC/dt=B _nette ^p .Comme augmente lorsque la contrainte appliquée diminue, on devrait observer une corrélation de nette à basses contraintes.Les résultats d'essais de croissance de fissure sous des conditions de fluage effectués récemment tendent à supporter cette hypothèse et sont présentés.

     

Sound propagation in 3He-4He mixtures near the superfluid transition

Measurements of the acoustic attenuation and dispersion in liquid ³He-⁴He mixtures near the superfluid transition T (x) are reported. The frequency range is /2gp=1–45 MHz and the ³He mole fraction X of the mixtures is 0.007, 0.05, 0.15, and 0.36. Comparisons are made with the measurements of Buchal and Pobell for similar mixtures obtained in the kHz region, and on the whole, the consistency between the two experiments is very satisfactory. An analysis is then performed using both the kHz and MHz data. In the normal phase, where the energy dissipation is caused by order parameter fluctuations having a lifetime _F , the attenuation data can all be scaled according to the expression = (T )f(_F. Here (T )^1+y, with y being a function of the mole fraction X and _F(T–T )^–x, with x increasing weakly with X. In the superfluid phase, we attempt a similar scaling representation, which is found to be fairly successful, but where x(T\s-T ) is roughly 15% larger than x(T>T ). In the superfluid phase we also analyze the attenuation data, assuming the additivity of relaxation and fluctuation-dissipation mechanism, and discuss the relaxation times so derived. In contrast to the attenuation, the dispersion data cannot be brought satisfactorily into a scaling representation. However, at T , we find U()-U(0)^y as predicted by Kawasaki, where y is in good agreement with the values from attenuation experiments.Supported by a grant from the National Science Foundation.     

The libron spectra and conversion as a function of ortho-para concentration in solid H2 and D2

A study of the Raman scattering spectra of orientationally ordered ortho-H₂ and para -D₂ as a function of J = 0 impurities is presented. Libron peak frequencies over the concentration region 0.75 <c ₁< 0.98 are determined and show a linear concentration dependence that can be represented by the relation (1)[1 – K(1 – c₁)]. A least squares fit of our data gives values of K = 1.27 ± 0.05, 1.28 ± 0.05, and 0.96 ± 0.05 in D₂ where = E _g, T _g ⁽¹⁾ , and T _g ⁽²⁾ , respectively, and K 1.32 ± 0.05 and 0.97 ± 0.05 for the E _g and T _g ⁽²⁾ modes in H₂. Results are compared to the values calculated by Diehl et al. using the coherent potential approximation for the J = 0 impurity problem. Extrapolated values for (1) were obtained that are slightly higher than those given by Hardy, Silvera, and McTague, resulting in an increased value for the effective interaction parameter _eff. Conversion rates for H₂ and D₂ are determined, resulting in k _H2 = 0.0193 ± 0.0007h^–1 and k _D2 = 0.00063 ± 0.0001 h^–1.Supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).     

Longitudinal critical current in type II superconductors. I. Helical vortex instability in the bulk

The vortex lattice in type II superconductors is unstable against the growth of helical perturbations if the current along the vortices exceeds a critical value. The longitudinal critical current, the pitch, and the spatially varying amplitude of the elliptically polarized helices are calculated from the London theory at the onset of instability in planar current distributions far from the surface. For weak pinning (_L² c ₆₆) the wavelength and width of the mode extend over the entire specimen, and the critical current is 2H(c ₆₆/c ₁₁)^1/4. For moderate pinning (c _{66 L}² c ₁₁) the wavelength and width are close to Campbell's pinning length (c ₁₁/_L)^1/2, and the critical current times its mean density is 2H ²(_L/c ₁₁)^1/2. For strong pinning (_L² c ₁₁) helical instability occurs at pin-free vortex sections, the helix wavelength is 2.2d, and the critical current density is 0.47Hd/² (H, d, c ₁₁ and c ₆₆), and _L are the magnetic field, spacing, elastic moduli, and pinning parameter of the vortex lattice, and is the magnetic penetration depth).     

Anomalous hysteresis of the critical current in long proximity-effect bridges

The critical current hysteresis of long Ag-Sn proximity-effect bridges, observed in the current-voltage characteristics obtained at helium bath temperatures T above and below the point of helium T , shows a jump discontinuity at T = T . The relevance of the observed phenomenon to existing theories of superconductors driven into resistive states by direct currents is discussed.Research carried out at the Tata Institute of Fundamental Research.     

Mutual friction and the thermal conductivity of superfluid helium near Tλ

Temperature gradients in superfluid helium carrying a heat current in a 13.8-mm-wide tube or in a 0.1-mm-wide slit have been measured at 0.09 m°K–T<16 m°K. The results are interpreted in terms of a mutual friction force between the normal component of He II and vortices in its counterflowing superfluid component. This force is found to diverge near T with a mutual friction constant proportional to (T–T)^T 0.35±0.06.     

Conventional Electronic Structure of MgB2 and ZrB2: LDA vs. de Haas-v. Alphen & ARPES Data

We compare full potential LDA band calculations of the Fermi surfaces areas and band masses of MgB₂ and ZrB₂ previously reported and new dHvA data. Discrepancies in areas in MgB₂ can be removed by a small shift of bands relative to bands. Comparison of effective masses lead to orbit averaged el-ph coupling constants =1.3 and =0.5, whereas for ZrB₂ only weak el-ph coupling with <0.3 is found. The ARPES data can be also well described by the LDA showing the presence of surface states.     

Heat Transfer in a Packed Bed at Moderate Values of the Reynolds Number

Results are given of an experimental investigation of the coefficient of wall heat transfer of a round tube filled with a packed medium formed by monodisperse glass spheres of different diameters (d _p = 0.9, 3.2, 8.9 mm) in a stabilized region of heat transfer under conditions of filtering of water and aqueous solution of glycerin. A two-layer model of heat transfer is used to calculate the contribution made by the heat resistance of the flow core and of the wall zone using the measured coefficients of heat transfer and temperature profiles across the packed bed. The form of dependence for the effective coefficient of thermal conductivity is determined. Data are given of the measurement of the coefficient of wall heat transfer of annular channel filled both with a single layer of spheres with packing of two types (cubic and rhombohedral) and with several layers of spheres with random packing in a stabilized region of heat transfer under conditions of filtering of water. It is demonstrated that, in the case of inertial mode of filtering of liquid through the packed bed, the values of the Nusselt number both in the tube and in the annular channel correspond to the relation Nu _e (d _e/D)Pe^1/2 _e . A semi-empirical correlation is suggested, which generalizes well our experimental data (and the data of other authors) on heat transfer in the tube and in the annular channel. A theoretical model is suggested, according to which the variation of heat transfer is defined by the behavior of the effective coefficient of thermal conductivity _ef/ _f Pe^1/2 _d associated with the predominant contribution made to convective heat transfer by the transport processes in vortex cells with closed lines of flow.     

Effect of phase fluctuations upon penetration depth

The temperature dependence of the static penetration (T) has been used as a guide to the nature of the superconducting state in high-T _c materials. It has been argued that an algebraic temperature dependence in the ratio (T)/(0) [(T) — (0)]/(0) at low temperature is evidence for d-wave pairing. This paper examines the effect of superconducting phase fluctuations upon (T) and finds an algebraic dependence over a broad range of temperature.     

A new method for the evaluation of thermal conductivity and thermal diffusivity from transient hot strip measurements

The standard straight-line fit to data of a transient hot strip (THS) experiment to determine the thermal conductivity and thermal diffusivitya suffers from two major drawbacks: First, due to the statistical nature of the estimation procedure, there is no relation between the uncertainty of the measured value on one hand and the transport properties obtained on the other. Second, in order to account for he heat capacity of the strip and outer boundary conditions, two intervals of the plot must he rejected before analyzing it. So far, these intervals are selected arbitrarily. We now treat the THS working equation as a function of the four parameters concerned. a.U ₀ (initial voltage), andt ₀ (time delay). Chi-square fittings. following the Levenberg-Marquardt algorithm. are performed separately for several overlapping time intervals of the entire plot to find and a with minimal standard deviation. In the course of subsequent iterations an individual weighting factor is applied to each point to account for systematic errors. This procedure yields the "best" values of anda along with their individual errors. comprising the systematic and the statistical errors. Experimental results on Pyrex glass 7740 were taken to verify the new procedure.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder. Colorado, U.S.A.