Orientational ordering in solid D2 from NMR studies. I. The hcp phase

Using a coherent pulsed NMR spectrometer operating at 5.9 MHz, measurements have been made of the free induction decay and of the solid echo in hcp solid D₂. From these data, and using Fourier transform techniques, the NMR line shapes and longitudinal relaxation times have been found for both o-D₂ (angular momentumJ=0 and spinI=2) and p-D₂ (withJ=1 andI=1) separately. The concentration and temperature extended over the ranges 0.05X(J=1)0.56 and 0.04<T<3 K, respectively. For both o-D₂ and p-D₂, the second moment of the NMR line shape rises smoothly as the temperature decreases, while the relaxation times pass through a minimum. No evidence of a thermal hysteresis could be found in the line shape or relaxation times. The orientational ordering increases continuously as the temperature decreases and the results lead to the conclusion that there is no evidence for a well-defined transition into a phase that has the characteristics of a glassy state, at least above 0.05 K. Calculations of the rms order parameter for p-D₂ as a function ofX andT are presented and the results compared with those for solid H₂. Above 0.5 K, the agreement is very good, while below 0.3 K, (H₂)>(D₂). The orientational polarization of theJ=0 molecules in D₂ by the surroundingJ=1 molecules is measured from the ratio of the lineshape second moments and is found to be in order-of-magnitude agreement with the predictions by A. B. Harris in the high-temperature limit. An analysis and correlation of the various measured relaxation times via energy diffusion models is presented. From the spin-lattice relaxation times of theJ=1 molecules, the orientational fluctuation rates are estimated for various concentrations as a function ofT and compared with the results from H₂. Good agreement is found. Some unusual features in the intensity ratio of theI=1 andI=2 signals that is different from the expected one are described. In an Appendix, corrections made to the observed line shapes to compensate for the instrumental limitations of the pulse spectrometer are outlined.     

Thermal conductivity of solid ortho-para H2 mixtures below 1.6 K

We report thermal conductivity measurements of two hcp H₂ crystals over a temperature range between 0.15 and 1.6 K and an ortho-H₂ concentration range betweenX=0.55 andX=0.01. The orientation of the crystals in the conductivity cell was determined from the anisotropy of the NMR spectrum forX0.01. The conductivity results are analyzed into contributions from (1) phonon scattering by the rotational states of the ortho-H₂ particles and (2) boundary phonon scattering. The results are compared with previous work at higher temperatures. The influence of ortho-H₂ clustering on the thermal conductivity is observed and discussed.     

NMR studies of single crystals of H2. V. NMR anisotropy and order parameter distribution ofX (ortho)≤0.55

We report a study of the NMR line shapes in hcp single crystals of H₂ with ortho concentrationsX0.55 in the regime where there is no longer a transition to a long-range orientationally ordered phase. From the anisotropy of the o-H₂ impurity NMR spectrum at low ortho concentration, reached by ortho-para conversion, the crystal orientation is determined. The second momentM ₂ can be represented by a function of the formM ₂=(X, T)f(cos _Hc ), where _Hc is the angle between the applied magnetic field and the crystalc axis. For a single crystal, the anisotropy functionf(cos _Hc ) is found to be independent of temperature and of ortho concentration within experimental error, and is in very good agreement with predictions based on the first term of the high-temperature expansion ofM ₂ and on other, more general symmetry arguments. An order parameter is defined and the distribution functionP() is calculated from the NMR line shapes under the simplifying assumption that the anisotropy of the order parameter, which gives rise to the observed anisotropy ofM ₂, can be neglected. We giveP() as a function ofX at low temperature, where the line shape is only weakly dependent onT, and as a function ofT at constantX. It is found that the line shapes andP() in both situations evolve continuously and give no hint of a phase transition. These results are discussed in relation to those of magnetic spin-glasses, and it is concluded that the orientational regime in solid H₂, called a quadrupolar glass by previous investigators, cannot be distinguished by symmetry from the orientationally disordered phase that occurs at high temperature.Work supported by NSF grants DMR-81-02993 at Duke University and DMR-79-10153 at the University of Pennsylvania.     

Pulsed NMR studies in solid D2. I. Solid echo

The solid echoes for both I = 1 and I = 2 nuclear spins from a two-pulse sequence are studied forp-D₂ concentrations between 0.06 and 0.5 and for 0.06 < T < 1.5 K. The drop of the I = 1 echo amplitude from Curie's law reported earlier is confirmed and is discussed. The density matrix method, previously developed to describe the I = 1 echo properties in solid H₂ as a function of various parameters, is applied to p-D₂, and similarly accounts for most of the observations. An analysis of the I =2 echoes for o-D₂ is carried out in similar fashion in the absence of a complete density matrix theory. Studies of the echo amplitude for the I = 1 and I = 2 spins versus r.f. pulse widths are reported and discussed. Various measurements associated with the echo decay as a function of the time between the two r.f. pulses are described. They include the I = 2 echo maximum shift respective to that of I = 1, the echo amplitude ratio S (I = 1)/S(I =2), the intermodular nuclear interaction magnitude, the local inhomogeneous field distribution, and the absence of expected satellite echoes for I =2. A comparison is made with the results of a similar study on solid H₂.     

Anisotropic superconductivity in the two-dimensional Hubbard model

We address the problem of anisotropic superconductivity in the two-dimensional Hubbard model. The Eliashberg equations have been generalized to the case which accounts for the anisotropy of the order parameter. Strong local correlations are treated within the mean field slave boson approximation. The superconducting transition temperatureT _c is evaluated as a function of the occupation number. Our results indicate that thed-wave state is the most likely channel for superconductivity for small concentration of holes. We have also derived an approximate analytical formula forT _c valid for any value of the occupation number. In addition, the influence of strong correlations on the electron-phonon coupling function is also discussed.     

Magnetic properties of Sm2Fe14−xCoxAl3 compounds

Compounds in the series Sm₂Fe_14–x Co _x Al₃ (x = 0, 1,2, 3, 4 and 5) have been shown to be of the Th₂Zn₁₇ structure. The Curie temperature is found to increase monotonically from a critical temperature,T _c equal to 471 K for thex=0 sample, toT _c=681 K for thex=5 sample. X-ray diffraction measurements of magnetic field oriented powders showed that all compounds exhibit a room temperature uniaxial anisotropy. Magnetization measurements show that the magnetic anisotropy of Sm₂Fe₁₄Al₃ can be increased substantially by the substitution of even a small quantity of Co for Fe. Results are discussed in terms of possible applications of these compounds as particulate recording media.     

Orientational ordering in solid D2 from NMR studies. II. The cubic ordered phase

An NMR study in the cubic phase of D₂ which exhibits long-range orientational order is reported. The spin systems in both para D₂ (with angular momentum J=1 and spinI=1) and in ortho D₂ (with J=0 andI=2) were investigated. The integrated intensity ratio of theI=1 andI=2 components in the solid echo, the NMR line shapes obtained by Fourier transforming the solid echoes, and the longitudinal relaxation times for theI=1 andI=2 systems are presented. Samples with D₂ concentrationX between 0.69 and 0.62 were studied over the temperature range 0.12<T<4 K. Measurements were carried out before and after repeated thermal cycling through the ordering transition to study the effect of this cycling on the temperature dependence of the relaxation times. The complex behavior of these relaxation times was found to be similar to that in cubic H₂ at a comparable concentration, and the interpretation of this behavior is discussed. Over most of the temperature range, the relaxation time of theI=2 spins is larger than that of theI=1 spins and is believed to be determined by cross-relaxation with the latter, whose intrinsic spin-lattice relaxation is observed. However, at low enough temperatures, the relaxation time for theI=2 spins is found to be shorter than that for theI=1 spins, and this observation cannot be understood on the basis of predictions. Furthermore, theI=2 relaxation time in this temperature region is found to depend on the position within the NMR line. Another observation not yet understood is that the ratio of the integrated intensitiesS(I=1)/S(I=2) in the solid echo is smaller than the theoretically predicted one by about 20%. The NMR line shapes for theI=1 spins in the ordered phase obtained from pulse measurements are compared with those from continuous-wave methods and to those from H₂.     

Preparation and degradation behaviour of Bi-cuprate superconductors

The role of preparation conditions and the effect of addition of Pb in Bi-Sr-Ca-Cu-O (BSCCO) superconductor, on theT _c,J _c and grain orientation have been studied. Calcination at a temperature higher than the melting point of Bi₂O₃ after the prereaction at 800°C leads to formation ofc-axis oriented nearly single phase material. The presence of lead yields a nearly highT _c phase (2223) exhibiting a maximum transition temperatureT _c=110 K and ΔT _c=2 K. The addition of lead and grain orientation together result in an increase inJ _c by nearly two orders of magnitude. The degradation behaviour was studied in terms of changes inT _c,J _c and structure on exposure to atmosphere up to 250 days. Samples with 0.6 Pb were found to be superconducting withT _c=97 K even after 250 days. The highT _c phase was stable against degradation. Our studies indicate that the degradation of leaded BSCCO is a surface phenomenon rather than bulk phenomenon.     

The temperature dependence of the dielectric constant in solid H2, D2,4He,and Ne

The dielectric constant ε of solid hexagonal close-packed H₂ (X?0), H₂(X?0.7), D₂(X=0.33), and He ⁴ at constant volume and of H₂(X=0.75) and Ne at saturated vapor pressure is studied as a function of temperature between 1.17 and 10 K. HereX is the fraction of molecules with angular rotational momentumJ=1, the fraction 1?X havingJ=0. These are the first reported measurements on van der Waals solids at constant volume. The experiments were carried out using a thick-walled capacitance cell, the measurement frequency being 14 MHz. The observed changes of ε withT are expressed in terms of the effective polarizability α_CM defined by the Clausius-Mosotti relation. A reduced polarizability α*=α_CM (T)/α_CM (T=1.17 K) is presented as a function of temperature. The initial purpose of the measurements was to observe the effect of rotational ordering of the(J=1) molecules in H ₂ , to be detected by a change in polarizability, as suggested by A. B. Harris. With increasing temperature it was found that α_CM decreased of the order of 0.1% for H ₂ and D ₂ . Also, differences were found between H₂(X=0) and H₂(X=0.75), which are briefly discussed. For He ₄ , α_CM decreased by about 3×10 ^?5 between 1.17 K and the melting point, 3.8 K. An analysis of the results showed that the temperature variation of α_CM could not be caused by the effect of lattice vibrations. Also, a simple calculation shows that the rearrangement of molecular positions in H ₂ and D ₂ at constant volume would have to be quite substantial to account for the observed temperature change in α_CM, which change is therefore not understood. The dielectric constant of Ne was measured at saturated vapor pressure and corrections were made to obtain the change of α_CM at constant volume. Rather different results were obtained for Ne than for the other solids: α_CM rose sharply with temperature, and this behavior also is not understood. Measurements of the diectric constant of H₂(X=0) and H₂(X=0.73) near the melting curve showed that the polarizability α_CM is the same within 0.1% in both the liquid and the solid phases. In the solid and liquid phases the density differences between H₂(X=0) and H₂(X=0.75) and also the differences in the respective melting pressures are in good agreement with the previous published work.     

Influence of sintering time and quenching on the formation of highT c phase

Influence of sintering time and quenching in Bi_{2 −x} Pb _x Ca₂Sr₂Cu₃O _y (x=0.0, 0.1, 0.2, 0.25, 0.3 and 0.4) samples have been studied by resistance and XRD measurements. In samples sintered at 850°C for 4 days,T _c(0) increases with Pb concentration.T _c(0) increased from 81 K forx=0.0 to 109 K inx=0.30 sample and then decreased. Increasing the sintering time to 10 days decreased theT _c Quenching further decreased theT _c(0). From X-ray diffraction patterns, the intensity peaks of low and highT _c phases have been measured. The addition of Pb promotes highT _c-phase. Sintering time, slow cooling and rapid quenching studies show that there is an optimum sintering time and cooling rate to produce a highT _c-phase.     

Pulsed NMR in solid D2. II. Stimulated echo

A study of the NMR stimulated echo resulting from an r.f. three-pulse sequence is presented for hcp D₂ with para-D₂ concentrations X between 0.06 and 0.49 and over the temperature range 0.07 < T < 1.5 K. The echo has two components, a sharp one on top of a broad one, which are the signals of the I = I (p-D₂) and the I= 2 (o-D₂) spins respectively. The echo widths reflect the different degrees of molecular orientational ordering. The echo amplitudes are studied as a function of the experimental parameters in the r.f. pulse sequence. The temperature dependence of the amplitude shows the expected Curie law behavior for the I = 2 spins, but a strong deviation from this law for the I = 1 spins. This anomalous behavior is consistent with that found for the I = 1 solid echo signal. The echo decay time _E is found to be very different for both spin systems, both in magnitude and in temperature dependence. The comparison of the I=1 spin results in D₂ with those in H₂, where the molecular orientational ordering is also from quadrupolar interaction, shows substantial differences. These are discussed in terms of the spin diffusion within each NMR line, and of the cross-relaxation between the two I = 1 and I = 2 spin systems.     

The Orientational Ordering of Molecular Hydrogen in 2D

The orientational ordering of ortho-hydrogen molecules on a triangular lattice is calculated as a function of ortho concentration using the method of restricted traces. The calculated temperatures for the transitions from the disordered phase to the long range pinwheel ordered phase show a strong concentration dependence with a critical concentration X _c=73% below which the long range ordering is not realized in this model. The results are consistent with recent NMR studies of ortho-para hydrogen mixtures adsorbed as a commensurate × structure on hexagonal boron nitride for which X _c=69±4%. The values of the observed transition temperatures can only be understood if there is a large renormalization of the effective quadrupolar interaction constant in 2D.     

The T-criterion for engineering materials based on strength anisotropy

Summary In most engineering materials the yield limits in tension and compression are in general different. Their ratio,R= _0C / _0T , characterizing the strength anisotropy of the material, was found to influence significantly the modes of both plane stress and plane strain fractures. The theoretical analysis presented in this paper introduced the necessary modifications of theT-criterion of fracture in order to cover the effect of the strength anisotropy, called the strength-differential effect (SDE). According to the statement of the proposed fracture criterion, the maximum value of the ratio of strain energy density components,T _R =T _V /T _D , when calculated along the elastic-plastic boundary around the tip of a crack indicates the angle of initial crack path. Crack onset is characterized by a critical value of the SED. For the determination of the elastic-plastic boundary the most general form of a failure criterion, that is the paraboloid failure condition was used.With 12 Figures     

Intrinsic intergranular transport current density in high-Tc superconductors

The intrinsic, intergranular, critical current densityJ _c(H=0,TT _c) in sintered untextured YBCO specimens of different grain size is self-consistently determined using a recently proposed critical state model [1] on ZFC magnetizationM(H, T) and on pulse transportJ _ct(T) measurements. Flux-creep effects, which are significant in these materials, are treated and theJ _c(T) results are corrected to their intrinsic values.     

3He diffusion in liquid isotopic mixtures

Spin echo techniques have been used to study the diffusion of³He in liquid isotopic mixtures with molar concentrations in the range fromX ₃=0.137toX ₃=0.850 over a temperature range from 3.0 to 0.4 K. AboveT the diffusive behavior is very similar to that of pure³He, although there is an increased scattering associated with⁴He. BelowT the diffusion coefficient increases with decreasing temperature and shows a dependence on the³He number density much like that of a classical gas. The excitations of⁴He have little influence on the³He diffusion coefficient, even at temperatures well above 1 K.The Rhodes Trust and the National Research Council of Canada supported one of us (R.B.H.) during the course of this research.     

Upper critical field anisotropy in the Chevrel phase compound PbMo6S8

The anisotropy of the upper critical field,H _c2, was measured on deoxygenated single crystals of the Chevrel phase compound PbMo₆S₈. The superconducting transition temperature,T _c, was determined with the applied field pointing in various directions in the hexagonal plane. At a fixed value of the external field,T _c is an oscillating function of direction. Since theH _c2 versusT curve is linear in the field region we probed, the shape of the anisotropy inT _c at constantH is the same as the anisotropy ofH _c2 at constantT. The data reflect the low-temperature symmetry of the crystal structure. Two-fold symmetry was observed in the plane. The data were fitted to the effective mass model with uniaxial symmetry. The anisotropy ofH _c2 is more complex than that previously observed by us in Cu_1.8Mo₆S₈. We relate this observation to the structural differences of these compounds.     

Collective modes and sound propagation in a magnetic field in superfluid3He-B

A high-resolution, ultrasonic (12–89 MHz) acoustic impedance technique has been used to investigate the order parameter collective modes in superfluid³He-B over a pressure range of 0–15 bar and in magnetic fields up to 180 mT. In agreement with earlier experiments, theJ=2 real squashing mode has been observed to split into five components in small magnetic fields. However, contrary to earlier theoretical estimates, the Zeeman shifts have been found to become extremely nonlinear as the magnetic field is increased. The extent of this nonlinearity is largest at low pressures and at temperatures close toT _c. In comparison with recent theoretical work, the nonlinear Zeeman shifts may be explained as a result of two effects. First, there is a significant distortion of the B-phase energy gap in large magnetic fields. Second, there is an important coupling between the sameJ _zsubstates of the differentJ modes. In this sense the nonlinear evolution of the real squashing mode constitutes the observation of the Paschen-Back effect in³He-B. A comparison of the observed Zeeman shifts with theoretical expressions has yielded information about particle-particle and particle-hole interaction effects in the superfluid. In the limitT 0 and above a threshold field, the real squashing mode has been found to possess additional structure. TheJ _z=0 substate has been observed to split into a doublet. The separation between the two components of the doublet is of the order of 100–200 kHz and remains independent of the magnetic field. The origin of the doublet may be understood in terms of a recent theory which postulates a texture-dependent collective mode frequency. Further, at extremely small fields the effects due to dispersion of the real squashing modes have been found to be important. The magnitude of the dispersion-induced mode splitting in zero field is found to be consistent with theoretical predictions. TheJ=2 squashing mode has also been studied in the presence of a magnetic field. TheJ _z=0 state of the squashing mode is observed to shift to lower temperatures in a magnetic field. An additional field dependence of the observed acoustic impedance is interpreted as the evolution of theJ _z=–1, –2 states, but appears to be inconsistent with theoretical predictions.     

Ultrasonic propagation in3He-4He mixtures near the tricritical point

We present measurements of the singular sound attenuation _sing in liquid mixtures of³He and⁴He near the superfluid transition and at temperatures above the phase separation curve. The mole fractionX of³He ranged from 0.55 to 0.73 and the frequency /2 varied between 1 and 45 MHz. The temperature range was 0.75–1.4 K, with the greatest emphasis on the tricritical region nearX _t=0.67 andT _t=0.87 K. From the change in slopedU/dT of the sound velocityU, we present a new determination of the phase separation curve, which is compared with previous measurements. The sound attenuation peak along the superfluid transition becomes broader in temperature asT is decreased. In addition, there is also an increase in sound attenuation as the phase separation temperatureT is approached. ForX<X _t these two peaks merge into one asX _t is approached. For a given frequency, the attenuation has a maximum value at the tricritical point. Estimates of the contribution _D of mass diffusion to the attenuation for³He-⁴He mixtures with 0<X<0.55 and comparison with experimental values show that _D becomes relatively more important asX increases, and that atX=0.55 it effectively accounts for all of the observed singular attenuation, at least at megahertz frequencies. Hence we assume that for mixtures withX>0.55 the observed attenuation can be analyzed solely in terms of the diffusive relaxation mechanism. The mass diffusion parameterD is then determined from the data. AtX=0.55,D diverges asT is approached, which is consistent with theoretical expectations and experimental results. NearT _t, there is a crossover to a tricritical regime, and it is found that approximatelyD(T–T _t) ^Z withZ=0.32±0.1. Mode coupling predictions are thatZ=1/2 while recent renormalization group calculations giveZ=1/3. The attenuation curves in the tricritical region at the various frequencies can be represented satisfactorily but not perfectly by a scaling function with a characteristic relaxation time (T–T _t) ^–x withx=1.7±0.15. This time corresponds to order-parameter fluctuations. Its temperature dependence is in excellent agreement with renormalization-group calculations that givex5/3, while expectations from dynamic scaling are thatx=3/2. Our analysis also gives the variation of the amplitudes of andD with the direction of approach toT _t. A comprehensive theory for interpreting all the data, in the normal as well as in the superfluid phase, is lacking at this time.Supported by a grant from the National Science Foundation. A preliminary account of this work was presented inBull. Am. Phys. Soc. 21, 229 (1976).     

Optical study of charge dynamics along thec-axis in high-Tc superconductors

Comparing the optical spectra withEc for typical high-T _c superconducting cuprates, we discuss the charge dynamics along thec-axis. The plasma energy or the mass anisotropic factorm _* ^c //m _* ^b is one of the key parameters determining the spectrum forEc. In Bi₂Sr₂CaCu₂O₈ and La_2–x Sr _x CuO₄ with a largem _* ^c /m _* ^b , the plasma energy is smaller than the superconducting gap energy, and thus the supercurrent along thec-axis is a kind of Josephson current flowing through insulating layers such as the BiO layers. On the other hand, in YBa₂Cu₃O₇ with a smallm _* ^c /m _* ^a , it seems that the coherent supercurrent flows along thec-axis. Although the spectrum forEc strongly depends on samples, presumably due to the difference in the hole concentration, a clear anisotropy between theab- and thec-directions is observed in the characteristics energy scales such as the plasma energy as well as the reflectivity knee energy.     

Upper critical field of aD-wave superconductor stabilized by antiferromagnetic fluctuations

We have derived microscopic equations for the upper critical magnetic field of aD-wave superconductor stabilized by antiferromagnetic spin fluctuations. We present numerical results for the reduced fieldh _c2(t) as a function of reduced temperaturet when the magnetic field is along thez axis and also zero-order results when it is along thex ory axis. Two differentD-wave models are considered. The angular dependence ofH _c2 forT nearT _c is given as a function of polar angleϑ in thez−x orz−y plane.