H. Yoshizawa

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 the three dimensional Ising magnet: Fe x Zn1?x F2

We have carried out a comprehensive neutron scattering study of random field effects in the diluted three dimensional Ising antiferromagnet FexZn1−xF2 withx=0.35 and 0.5. Emphasis is on the global trands from the small to the large random field regimes. It is found, as in previous experiments, that when the system is cooled in a field it evolves from the high temperature paramagnetic state to a low temperature domain wall state. The low temperature peaks are well-described by Lorentzian squared profiles although for thex=0.5 sample extinction made the measurements difficult. In both samples, the results show that in the field-cooled state the correlation length varies asH−v withv=2.2±0.1. In thex=0.35 sample this power law holds over a length scale varying from 2 to 1500 lattice constants. At low fields pretransitional behavior similar to that observed previously in Fe0.6Zn0.4F2 is found. AtTN (H=0) it is found that the correlation length also scales algebraically withH but withv=0.86±0.04. Pronounced history-dependent effects are observed below the phase boundary determined by the peak in the critical scattering. For example, on cooling in zero-field, raising the field and then warming, long range order survives up to the phase boundary; at this point it appears to convert abruptly into the finite correlation length field cooled state although elucidation of the explicit nature of this transition is complicated by rounding due to a concentration gradient. These results are discussed in the context of recent theories incorporating metastability effects as well as recent experiments.

The effects of random fields on the critical and bicritical behavior of Mn x Zn1−xF2

We have carried out a comprehensive study of the static and dynamic spin-spin correlations of MnxZn1−xF2 in a magnetic field. Samples withx=0.75 andx=0.5 have been studied. This system exhibits behavior closely related, if not identical, to that of the Random Field Ising Model (RFIM). An additional feature of MnxZn1−xF2 is that it exhibits an easily accessible bicritical point; thus one can study the changeover from the RFIM to the uniformXY model with a transverse random field. Quite generally, the instantaneous spin-spin correlations in a field are described by a combination of Lorentzian, Lorentzian-squared and delta function terms the latter corresponds to the long range order (LRO) component. In the Ising phase one finds history dependent behavior as discussed previously. In theXY phase, except very near the spin-flop boundary, one finds ergodic behavior withXY LRO and Lorentzian squared Ising fluctuations. Rather complicated instability effects are found all along the spin-flop boundary. Further, when one establishes LRO in theXY phase and lowers the field through the spin-flop value, one obtains a LRO Ising state in thex=0.75 sample whereas one obtains the field-cooled domain state in thex=0.50 sample. This dramatic difference in behavior is not understood. Our results on the RFIM aspects of the problem are consistent with our previous studies. The transition is dominated by the metastability effects with an underlying equilibrium transition which is either first order or weakly second order (β≈0). The underlying transition manifests itself directly in measurements of the dynamic response nearTN(H). From the data above the metastability boundary we deduce for the static correlation length exponentv=1.4±0.3 in good agreement with theory. We find for the RFIM crossover exponent φRF=1.5±0.2 where the errors represent the spread in values obtained from different techniques. Finally, we have determined in detail the field-temperature phase diagram of thex=0.5 sample including the critical behavior along the spin-flop line; the latter transition appears to be second order for an extended region.

Disorder, random fields, and competing interactions in antiferromagnets

Neutron scattering studies of disordered antiferromagnets have proved to be a very profitable way of studying random systems. Several recent examples are selected and include a detailed study of the phase transition of a d = 3 Ising system showing a well defined transition with properties different from these of a pure d = 3 Ising system. Much of the article is then concerned with the effect of a random field on the ordering and phase transitions. It is shown that random fields do have a large effect on the critical properties and in practice destroy the long range order in the good Ising systems with d = 2 and d = 3, although not in a nearly Heisenberg-like system. These results are compared with current theories and the discrepancies discussed. Finally measurements on a system with competing interactions are discussed and shown to be strongly influenced by the random fields produced in that system.

Metastability and a Temporal Phase Transition in the Random Field Ising Model

At the preceding school in this series, we reviewed1 the then current experimental understanding of the properties of an Ising model in a random field. Since then there have been a large number of both theoretical2 and experimental studies3,4,5,6but as we describe below the problem is still not completely understood. As reviewed in this school by Villain7, it is now generally accepted that the lower critical dimension in equilibrium, dl, is 2. Experiments performed by cooling random antiferromagnets in a uniform field to produce a random staggered field, have shown that long range order is not achieved in these experiments and that at low temperatures the properties are history dependent. Recently theories8 have been developed to reconcile these experimental results with the theoretical prediction that dl = 2, by considering the energy barriers to domain wall motion in the presence of random fields. In order to examine this problem in more detail and to test these theories in detail we have performed new measurements on the random d = 3 antiferromagnet Mno.75Zn0.25F2 and Belanger et al. have performed measurements of the d = 2.antiferromagnet Rb2Co0. 85Mg0.15F4. We choose Mn0.75Zn0.25F2 because the MnxZn1−xF2 system is well understood. The spins interact with Heisenberg interactions and weaker dipolar interactions, which are responsible for the uniaxial symmetry. The system therefore belongs to the Ising universality class but there are very many low energy spin waves which might be expected to relax the system towards thermodynamic equilibrium. At low temperatures and applied field H = 0, the spin wave gap is about 8 K and many of our measurements are at about 40 K.

A Weakly Anisotropic Ising Model in a Random Field

The application of a uniform field to a random antiferromagnet produces a random staggered field [1], and so enables the effect of applying a random field to be studied experimentally in a convenient and controllable manner. Detailed experiments have been performed on the good Ising models Rb2CoxMg1−x F2 [2], CoxZn1−xF2 [3], and FexZn1−xF2 and as reviewed in detail elsewhere [4], the results reveal a number of unexpected features which are still only partially understood. In particular when the samples are cooled from the paramagnetic phase in an applied field, the system shows unusual behaviour, characterised by a Lorentzian squared profile in the quasi-elastic scattering cross-section, and a lack of long range order. In contrast if the samples are cooled in zero applied field, and the field applied at low temperatures, the long range antiferromagnetic state persists. On heating, the long range order persists, in the case of the three-dimensional systems, up to the phase transition temperature.