Quasi-stationary criticality of the Order-Parameter of the d=3 Random-Field Ising Antiferromagnet Fe(0.85)Zn(0.15)F2: A Synchrotron X-ray Scattering Study

The critical exponent beta =0.17(1) for the three-dimensional random-field Ising model (RFIM) order parameter upon zero-field cooling (ZFC) has been determined using extinction-free magnetic x-ray scattering techniques for Fe(0.85)Zn(0.15)F2. This result is consistent with other exponents determined for the RFIM in that Rushbrooke scaling is satisfied. Nevertheless, there is poor agreement with equilibrium computer simulations, and the ZFC results do not agree with field-cooling (FC) results. We present details of hysteresis in Bragg scattering amplitudes and line shapes that help elucidate the effects of thermal cycling in the RFIM, as realized in dilute antiferromagnets in an applied field. We show that the ZFC critical-like behavior is consistent with a second-order phase transitions, albeit quasi-stationary rather than truly equilibrium in nature, as evident from the large thermal hysteresis observed near the transition.