Formal solutions for response functions of superconductors and 3He(B): II. Applications to acoustic and quasihomogeneous regime

Formal solutions for the autocorrelation functions of density and the transversal current are discussed in the acoustic and quasihomogeneous regime. The poles of these functions are obtained without any restrictions imposed on Landau parameters. The formula for sound dispersion at T = 0 is generalized by the inclusion of terms of the relative order of $\text{(}\text{kv}\text{2Δ}\text{)}{}^2$, (k is the wave vector, v is the Fermi velocity, Δ is the energy gap, $\text{h}\text{?}\text{≡ 1}$). The dispersion formulae for transversal and longitudinal excitations with a gap for ³He(B) are also given, with an accuracy up to the terms of the order of $\text{(}\text{kv}\text{2Δ}\text{)}{}^2$, for 0 ? T ? T_c, and without any restrictions imposed on Landau parameters. Under these last assumptions our autocorrelation functions are calculated in the polar as well as non-polar regions. It is shown that if T > 0, the transversal function vanishes at some ω, such that $\text{0 ? ω}{}^2\text{? (}\text{12}\text{5}\text{)Δ}{}^2$. Moreover,the zero of the density autocorrelation function is distanced from its pole by an amount of the order of $\text{(}\text{kv}\text{2Δ}\text{)}{}^2$.     

Anomalies in the elastic constants of antiferromagnetic uranium mononitride

Measured anomalies in the elastic constants of a single crystal of cubic anti-ferromagnetic uranium mononitride are reported. A dip of about 10% occurs in C₁₁ at 6 K below the Néel point $\text{T}{}_{\mathrm{<mtext>N</mtext>}}\text{?53}\text{K}$. Starting at T_N a renormalization in C₄₄ which is proportional to the square of the sublattice ,magnetization also occurs. The results are in general agreement with model calculations of the sound velocity renormalization due to spin-phonon interactions.     

Ground state of ferromagnet with antiferromagnetic impurity

The Heisenberg ferromagnet with an antiferromagnetic impurity and arbitrary spin is considered. The method is suggested for constructing the ground state of such a system, the method using the Jacobi matrix technique. As an example, there has been investigated the ferromagnet with a simple cubic lattice and the matrix spin $\text{S =}\text{1}\text{2}$, and the impurity spin S′ = 1. The energy and wave function of the ground state are found as dependent on the system parameters.     

The Faddeev equations for the Heisenberg ferromagnet

The Faddeev equations for the three-magnon T-matrix of the Heisenberg ferromagnet with nearest neighbour interactions are derived for the cubic lattice in arbitrary dimensions. The extreme case of spin $\text{1}\text{2}$ is considered and the kinematical restriction, that only one spin deviation per site is possible, has been taken into account rigorously. Hence the T-matrix is unitary and suited for the study of bound state as well as scattering state properties. The analytic solution of the homogeneous Faddeev equations in one dimension is given.     

Excitations in the antiferromagnetic Heisenberg chain

Non-singlet excitations of the antiferromagnetic Heisenberg chain of N atoms with spin $\text{1}\text{2}$ are examined. It is found that the energy of the lowest lying excitations for total spin S and wave number q converges to $\text{(}\text{π}\text{2}\text{)|}\text{sin}\text{q |}\text{as}\text{N → ∞,}\text{if only}\text{S?}\text{ln}\text{N}$.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

Acoustic soft modes of Nb3Sn in high magnetic fields

In single crystal Nb₃Sn the effect of a magnetic field upon the soft sound velocity v_s (110, 1$\text{1}$0) was measured at different temperatures above the structural phase transition point. The observed quadratic variation with field is in accordance with theory. But the magnitude of the field dependence is much smaller than expected and can only be explained by existing theories if the magnetic moment of the electron would vary by as much as a factor of two within an energy interval of some 100 K.     

Critical behaviour of a uniaxial ferromagnet,ErCl3·6H2O with predominant dipolar interactions

The parallel magnetic susceptibility χ of a uniaxial ferromagnet ErCl₃·6H₂O has been measured between 0.3 and 4.2K and specially near T_c = 0.353 K. The predominant contribution to the Curie-Weiss temperature is due to the dipolar interactions. χ is proportional to ?^?γ with $\text{? =}\text{T}\text{T}{}_{\mathrm{c}}\text{?1}$ in the range 10^?3 < ? < 5 × 10^?2. The γ value, γ = 1.01 ±0.03 is consistent with the theoretical prediction for a uniaxial dipolar ferromagnet.     

Stress-induced crossover phenomena in nickel above its Curie point

The temperature dependence of the anisotropic critical scattering of neutrons from a nickel single crystal was observed under the influence of elastic uniaxial mechanical stress of 140 g/mm². The temperature was varied between 10^-4 < τ = [T ? T_c(Δ)]/T_c(0) < 10^-2, and the momentum transfer between 1.4 × 10^-3 < $\text{q = (2π/λ) sin θ < 6.6 × 10}{}^{-3}\text{A}\text{?}{}^{\mathrm{-I}}$. The expected crossover from isotropic (Heisenberg) to anisotropic (XY) behaviour should occur between 10^-5 < τ < 10^-3, in good agreement with the experimental result.     

Pairing-isopairing competition in odd-odd nuclei

We discuss the systematics of 0⁺, v = 0, T = |T_Z| + 1 levels in odd-odd nuclei and conclude about the ground state isospin and seniority of these nuclei. For N = Z nuclei starting from the $\text{1f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ shell v_g.s = 0 |T_g.s = |T_z + 1. Otherwise v_g.s = 2, T_g.s = |T_z|.     

Effects of electron interaction on the change in sound velocity under de Haas-van Alphen conditions

Based on a temperature propagator technique in the grand ensemble of an interacting electron gas, the oscillatory sound velocity is examined under the de Haas-van Alphen conditions. In consideration of the oscillation of the Fermi energy (chemical potential) and the first order exchange effects, the dHvA oscillations of the sound velocity are shown to have the same one phase as in the case of an ideal electron gas, in agreement with experimental results. For large electron density, that is, for very small r_s, and by a proper renormalization of the Fermi energy, we have succeeded in eliminating one of the two oscillatory functions which have a phase difference of $\text{π}\text{2}$.     

Das kritische magnetische Verhalten von EuO

Mössbauer effect and magnetization measurements were employed in order to study the static and especially the dynamic magnetic properties of the nearly Heisenberg ferromagnet EuO near its Curie temperature,T _c=69.2 °K. The critical exponent β of the spontaneous magnetization was determined to be β=0.34±0.02. It was shown that critical slowing down of spin fluctuations takes place nearT _c with spin relaxation times between 7×10^?11 sec (T=1.01T _c) and 1.5×10^?1 sec (T=1.03T _c). The experimental values of the relaxation time were found to be in satisfactory agreement with theoretically computed ones. Just belowT _c the Mössbauer spectra exhibit relaxation effects, which are characteristic for the occurence of critical super-paramagnetism. Investigations of several samples indicated quantitatively, that critical superparamagnetism has its origin in the non ideal composition of the real crystal.     

Solitary wave in a magnetoelastic easy plane Heisenberg ferromagnet

The spin fluctuations of an easy plane Heisenberg ferromagnet are coupled to the transverse acoustical vibrations. In the long wavelength limit, the possibility of a solitary wave with a bell shape is analysed. It is observed that the velocity v of the solitary wave is bounded by the higher and lower velocities of the acousto-magnetic polaritons for this system. The behaviour of the solitary wave is analysed as a function of this velocity where a splitting of the solitary wave occurs at a critical velocity.     

Nuclear relaxation studies of the spin-Peierls transition

Nuclear relaxation studies of the spin-Peierls transition in TTF-BDT are presented. At low temperature, but above T_c, the results agree in magnitude, frequency dependence and temperature dependence with calculations based on the pseudofermion treatment of the $\text{S =}\text{1}\text{2}$, 1-D Heisenberg antiferromagnet. The sharp decrease in the rate below T_c reflects the continuous opening of a magnetic energy gap at the spin-Peierls transition.     

Mode propagation in superconductors near Tc

The linearized Eliashberg theory for dirty superconductors in the vicinity of T_c is used to generate the dynamic Ginzburg-Landau (GL) equation for the order parameter and the coupled Boltzmann equation for the quasi-particle distribution function, with and without magnetic impurities and inelastic phonon scattering. The Schmid and Schön (SS) separation of the order parameter into longitudinal (real) and transverse (imaginary) components is shown to be exact to linear order. The former does not couple with the wave vector (k) and frequency (ω) dependent external field, while the latter does. Explicit expressions are exhibited for the current and charge densities to leading order [$\text{Δ}\text{T}$ in the gap regime, in which Δτ_s > 1 where τ_s is the conduction electron spin flip lifetime; and $\text{Δ}{}^2\text{T(ω + iDk}{}^2\text{)}$ in the gapless regime, Δτ_s < 1. D is the diffusion constant]. The continuity equation is shown to be obeyed to this order.Use of the Poisson equation in relation to the transverse component leads to a propagating mode, which, in the absence of magnetic impurities, reduces to the SS result. We find that the mode propagates only in the gap regime, whereas it is purely diffusive in the gapless regime and for gapless superconductors. In the presence of magnetic impurities, the solution reduces to the SS limit, but with reduced mode velocity and damping. The charge density appropriate to the propagating mode is shown to be zero to order $\text{ω}\text{ω}{}_{\mathrm{p}}{}^2\text{τ}{}_1$, where τ₁ is the impurity scattering time and θ_p the plasma frequency. The characteristic penetration length of an electric field into a superconductor is calculated, taking into account the phonon inelastic scattering time. The result reproduces the Waldram and SS expressions. Careful examination is made of the conditions under which these results pertain. The characteristic penetration length is also obtained in the gapless regime and for gapless superconductors for which T_cτ_s < 1.Finally, the longitudinal and transverse pair susceptibilities are calculated with and without magnetic impurities and in the gap and gapless regimes.     

Effects of dipolar forces on critical sound attenuation in uniaxial systems

Critical sound attenuation in uniaxial dipolar systems is investigated above and below T_c by using the dynamic renormalization-group analysis for an elastically coupled Ising-type model at d = 4. We predict a logarithmic crossover of the divergence of sound damping as $\text{t =}\text{|T ? T}{}_{\mathrm{c}}\text{|}\text{T}{}_{\mathrm{c}}\text{→ 0}$ and derive a universal, crossover-independent ratio R = 12.984|lnt| between the relaxation and the fluctuation damping below T_c. The effects of elastic anisotropy on the phase transition are discussed.     

Localized versus itinerant electrons in hexagonal FeGe

Hexagonal FeGe is a metallic antiferromagnet with T_N = 411K. At higher temperatures FeGe is a uniaxial antiferromagnet with spins parallel to the c-axis and ferromagnetic alignment within each c-plane. Susceptibility and magnetization measurements show that the spins leave the c-axis below 30K. From the molecular field model (MF) we propose that the spins then form a double cone structure. The spin components parallel to the c-axis retain the uniaxial structure while the c-plane components have a turn angle in the vicinity of 180° between neighbour planes, which implies absence of long range order. The MF results are in good agreement with Fermi surface calculations, which show the existence of several spin density wave vectors in the vicinity of $\text{Q =}\text{π}\text{c}$, thus explaining the indeterminacy of the turn angle in the MF model.     

Spin-pair correlations of the classical Heisenberg ferromagnet above and below the critical point: Results of self-consistent Monte Carlo calculations

The spin-pair correlation functions of the classical Heisenberg ferromagnet with simple cubic lattice have been calculated in the paramagnetic and ferromagnetic temperature range using the self-consistent Monte Carlo method. The results agree with high-temperature series expansions aboveT _c , for low temperatures with spin-wave theory. By two different approaches the divergence of the ferromagnetic homogeneous susceptibility in zero field throughout the ferromagnetic temperature range could be verified. The functional dependence of the static susceptibilityχ _T (k) upon the inverse correlation lengthκ ₁ is discussed above and belowT _c and a Fourier transform for the explicit dependence of the spin correlations upon correlation length belowT _c is given. According to these results the scaling assumptionv=v′ for the exponents of the correlation length in the critical region is consistent with a divergent ferromagnetic susceptibility.