Experimental determination of the phase-transition critical exponents and by integrating methods

Abstract

Integrating methods in low-energy electron diffraction—turning the instrument to low resolution in k∥—can reliably be used to study critical properties of continuous two-dimensional phase transitions and to determine critical exponents α and η. We performed systematic tests of the conditions under which an energylike power dependence of the diffracted intensity of superstructure beams can be observed in order-disorder phase transitions of adsorbed atomic layers belonging to three- and four-state Potts universality classes. As experimental examples, we studied the systems p(2×2)- and (√3×√3)R30°-S/Ru(0001) and (2×2)−2H/Ni(111). We show that above Tc the condition for an energylike singularity, KIξ≫1 (KI: radius of integration in k space, ξ: correlation length) is already reached at KIξ>3, whereas below Tc effective exponents α are already obtained for KIξ<1. Therefore, in the temperature range below Tc this method to determine α turns out to be particularly easy to apply, and we concentrate on the determination of α below Tc in this study. Values of α of 0.67±0.04 and 0.40±0.05 were obtained for four-state Potts and three-state Potts systems, respectively. The latter value indicates small crossover effects to the critical behavior of the peak intensity. These studies were then further extended to the order-disorder transition of p(2×2)−O/Ni(111), which is weakly first order, and to the effects of oxygen impurities in the system (2×2)−2H/Ni(111). We further show that in the limit kξ≫1 the exponent η can be determined from the k∥ dependence of integrated ring intensities around positions of superstructure beams. With the systems mentioned, we demonstrate that the limit of a leading term of critical scattering which is independent of temperature can be reached by carrying out measurements going through the order-disorder phase transitions. Values of η of 0.27±0.10 and of 0.30±0.10 were obtained for the four-state and the three-state Potts systems, in close agreement with theoretical expectations.