Three-dimensional Ising Systems Research Articles

Neutron diffraction study of the nondiluted random-field Ising system Fe 1− xCo xCl 2

We report the result of a detailed neutron scattering experiment on a nondiluted three-dimensional random-field Ising system: Fe 1− x Co x Cl 2 in applied fields ( x = 0.255 and 0.275). Systematic peak shape studies were performed as a function of field and temperature with an instrumental resolution of 0.0045 A −1 (half-width at half-maximum). Non-equilibrium behavior was observed and tested to a time scale of 10 hr. Within these limits we find that the low temperature phase of the system has long-range order, even for random fields large enough to reduce T N by 40%.

Phase transitions in diluted magnets: Critical behavior, percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn++ or Mg++ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe++ and Co++ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb2Mn0.5Ni0.5F4 and FecZn1−cF2 which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb2MncMgl−cF4, Rb2CocMgl−cF4, KMncZl-cF3, and MncZnl−cF2 which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.

Random-field effects in Fe1−xMgxCl2

We analyze in detail how random fields are generated in a site-random uniaxial antiferromagnet when a uniform magnet field is applied. Explicit expressions are given for the mean-square random field. We also report on a systematic specific heat study of two three-dimensional Ising systems, pure FeCl2 and site-random Fe0.682Mg0.318Cl2 (MG318), in constant applied fields. The contrasting behavior of the two provide a vivid demonstration of random-field effects. The unusual changes in the shape of the specific heat anomaly observed in MG318 are explained as crossover behavior and not a manifestation of new critical behavior. The crossover exponent φ associated with the random fields is determined from the shape of the magnetic phase boundary. We find φ=1.25±0.10, in excellent agreement with theoretical prediction.

Critical neutron scattering in CeBi

Anisotropic diffuse critical neutron scattering has been observed in the cerium monopnictide CeBi above the antiferromagnetic ordering temperature T N = 25.35 K. The transverse fluctuations were found to be unobservably small. The longitudinal fluctuations diverge at T N and their anisotropy shows that the magnetic interactions within the ferromagnetic sheets are stronger than the interplanar couplings. The data for the longitudinal fluctuations as well as the development of long-range order follow simple power laws with the critical exponents γ = 1.16±0.12, ν = 0.63±0.06, and β = 0.317±0.005, which are in good agreement with those expected from three-dimensional Ising systems.

Critical behavior of the site random Ising antiferromagnetMn1−xZnxF2

Nuclear magnetic resonance has been used to measure $\ensuremath{\beta}$, the critical exponent of the order parameter, in the random Ising antiferromagnet ${\mathrm{Mn}}_{0.864}$${\mathrm{Zn}}_{0.136}$${\mathrm{F}}_{2}$. A single-power-law fit to the measured hyperfine field of the form ${\mathrm{H}}_{\mathrm{hf}}=B{t}^{\ensuremath{\beta}}$ yields $\ensuremath{\beta}=0.349(8)$ over the reduced temperature range of 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ to 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$. This differs from the value $\ensuremath{\beta}=0.335(5)$ obtained by Heller [P. Heller, Phys. Rev. 146, 403 (1966)] for pure Mn${\mathrm{F}}_{2}$ and is consistent with the prediction that random disorder will cause a crossing to a new fixed point in the three-dimensional Ising system. The measured exponent can be explained in terms of recent predictions for the location and extent of the crossover region in reduced temperature.

On the estimation of the equilibrium properties of the kinetic Ising model of ferromagnetism

Using a comparatively small group of spin 1 2 particles set on a simple cubic lattice, an attempt has been made to obtain satisfactory simulation estimates of the equilibrium critical properties of the three-dimensional Ising system. Assuming only nearest-neighbout interactions and regarding a 10×10×10 particle group as a sample from the infinite system, the states of the nearest-neighbour external particles were estimated in a self-consistent manner using probability distribution functions whose moments are equal to the sample moments (particle correlation functions). We obtained in this way a value of 4.45 ± 0.05 (in units of J/ k B) for the critical temperature for the long-range order with exponent 0.31 ± 0.04. These values compare favourably with the expected theoretical values obtained from series expansions. The results for specific heat are less satisfactory. Analytical results have been obtained for the two-particle sample and we find a critical temperature of 4.46, but with a critical exponent of 1.05 for the long-range order.

Phase transition in the two-dimensional Coulomb gas, and the interfacial roughening transition

The behavior of the interface in the three-dimensional (3-D) Ising and related lattice systems is modeled using a 2-D array of columns of varying heights. We show that the "roughening" transition (the transition from a localized to a delocalized interface) is directly related to the metal-insulator transition in a 2-D Coulomb gas. Implications of this relationship are discussed.

Interface sharpness in the Ising system

A simple proof is given for the existence of a sharp interface in three-dimensional Ising systems, at least up to the critical temperature of the corresponding two-dimensional system.

Percolation in a lattice system with particle interaction

The site percolation density in a three-dimensional Ising system with nearest-neighbor interaction is calculated. The interaction lowers the critical particle density, where percolation occurs. A percolation temperature is defined with corresponding critical percolation exponents.

Neutron diffraction determination of the magnetic structure of Dy PO 4

Dy PO 4 is antiferromagnetically ordered below 3.39 K. The spin-structure is described by an antiparallel arrangement of nearest neighbours along the crystallographic c-axis. The magnetic moment at 1.8 K is (9.0±0.3) μ β. These results confirm that Dy PO 4 may be considered as an ideal three-dimensional Ising system.