On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics

The first six coefficients in the expansion of the susceptibility χ, and its inverse, χ ^?1, in ascending powers of the reciprocal temperature, have been determined for the Heisenberg model of a ferromagnetic, for any spin value, S, and any lattice. The first five coefficients appropriate to the magnetic specific heat, C, have also been found. For the body-centred and face-centred cubic lattices, the χ and C coefficients are tabulated for half-integral S from 1/2 to 3.

From these coefficients estimates have been made of the reduced Curie temperatures, θs _c= k T _c/J. It is found that for the simple, body-centred and face-centred cubic lattices the formula reproduces the estimated Curie temperatures fairly accurately. Here X=S(S+1) and z is the lattice coordination-number.

It is found that, suitably scaled, the theoretical curves for inverse susceptibility against temperature above the Curie point are rather insensitive to the spin value and to the precise lattice structure. The ratio of their initial to their final gradients is approximately 0·3. A comparison is made with the experimental values of χ ^?1 for both iron and nickel. If iron is represented by the Heisenberg model with S=1, then the observed Curie temperature corresponds to a J value of 1·19×10^?2 ev.

Brief consideration is given to the use of the tabulated coefficients for antiferromagnetic problems.     

Ultrasonic investigation of phase transitions in K2SnCl6

The velocities and the attenuation of ultrasonic waves have been investigated as a function of temperature (255KT350K) for the cubic phase of K₂SnCl₆ which undergoes a structural transition atT _c1263K. An anomalous decrease of the shear stiffness constantc=1/2(c ₁₁ –c ₁₂) of about 30% between room temperature andT _c1 is found in this cubic high temperature phase whenT _c1 is approached from above. Whereas the softening ofc extends over a substantial temperature range (about 50 K), the other shear constantc ₄₄ shows only a weak decrease, which is an order of magnitude smaller and restricted toT–T _c110 K. NearT _c1, a strong increase occurs in the attenuation of the shear acoustic wave propagated along the [111] direction, while such an anomalous attenuation is not observed for the transverse acoustic wave propagated along [100]. Furthermore, atT _c1 a hysteresis is detected for the longitudinal sound wave velocityv _L[100], which is restricted to the temperature region 262KT263.5 K. In addition, the room-temperature elastic constants of (NH₄)₂SiF₆ and the hydrostatic pressure derivatives of the elastic constants of K₂SnCl₆ and (NH₄)₂SiF₆ at room-temperature are presented and discussed in terms of mode softening behaviour.     

On the kinetics of a two-level system with a diffusion-modulated interaction

For a two-level system corresponding to a particle of spin ½ in a random field in the Z direction, the relaxation function

has been estimated, the magnitude Ω(t) being the sum of the isotropic interactions of the particle in question with particles j executing diffusional motion. Specifically Ω(t)=ω₀ + Σ ω(Mj, rj), where ω₀ = constant, M_j is a random time-independent parameter, ω(M, r) decreases with r faster than r ^-3 and r _j = r _j (t) is a diffusional, random function. From the expression for <σ⁺(t)>, we establish general features of the relaxation phenomenon for diffusional processes, and calculate the relaxation rate 1/T ₂ and relaxation shift Δω to be 1/T ₂-iΔω = 4π CAv_M λ _M , where C is the concentration of particles and λ _M is the scattering length for an equation of the Schrödinger type with an imaginary potential -iω(M, r) instead of U/?, and diffusion coefficient instead of ?/2m. We also found that for the case of ‘external’ relaxation, the Redfield approach proved valid only under the simultaneous restrictions of low concentration and weak interaction.