PHYSICAL REVIEW B VOLUME 35, NUMBER 4 1 FEBRUARY 1987 Structure factors associated with the continuous melting of two-dimensional lattice gases: Models with (~3)& ~3)R 30 and p(2X 2) ordered states on triangular nets N. C. Bartelt, T. L. Einstein, and L. D. Roelofs* Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742 (Received 16 December 1985; revised manuscript received 15 May 1986) We study the temperature dependence of the structure factors of two lattice gases which undergo order-disorder phase transitions. Our goal is to determine how much information about the critical behavior of these phase transitions a low-energy electron-diffraction experiment might obtain. We use Monte Carlo simulation to compute the structure factors. Both lattice gases are on triangular nets; one has a (V 3X~3)R30' ordered phase; the other has a p(2X2) ordered phase. The struc- ture factors scale almost halfway from the center of an extra spot to the zone center; for system sizes comparable to those that are physically realizable we see effective critical exponents which are typically within of order 10%%uo of expectations based on universality. Below the transition tempera- ture, nonlinearities in log-log plots are significant, indicating that corrections to scaling cannot be ig- nored. We consider how asymmetries in the structure factor reflect differences between lattice-gas systems and magnetic analogs in the same universality class and also briefly treat the effects of quenched random vacancies and of a fixed concentration of annealed vacancies. I. INTRODUCTION Phase transitions on the surfaces of single crystals are ubiquitous phenomena. ' They are of interest for two reasons. The first is that through analysis of phase dia- grams one hopes to understand microscopic behavior at surfaces. The second is that the understanding of the phase transitions themselves, as problems of statistical physics, is an important test of our basic understanding of two-dimensional critical phenomena. The principal tool for the investigation of surface phases is low-energy elec- tron diffraction (LEED). In this paper we compute, using Monte Carlo simulations, the structure factor for a pair of two-dimensional phase transitions. We then analyze this data straightforwardly — as one might analyze a diffrac- tion experiment — for critical behavior. While our discus- sion is couched in the language of adsorbate phase transi- tions, our work applies to any two-dimensional lattice gas system where the range of long-range order is limited. Perhaps the most fundamental limitations in interpret- ing surface phase transitions in terms of critical phenome- na are finite-size effects. Terraces on surfaces limit the range of long-range order and the spatial scale of fluctua- tions. As single-crystal metal surfaces cannot easily be obtained with more than a hundred atomic spacings be- tween terraces, the correlation length cannot change by much more than an order of magnitude as the surface or- ders. Scaling theories of second-order phase transitions predict the behavior of systems in which the correlation length becomes much greater than microscopic length scales, so it is not clear a priori how well they describe the behavior of typical surface systems. For example, it is difficult to know how close effective exponents measured in LEED experiments will be to exponents computed for the infinite-correlation-length limit. Our direct calcula- tions of these effective exponents for model systems allow some expectation of their meaningfulness to be developed. Of course many of our conclusions can be inferred from the experience gained by the many Monte Carlo simula- tions of finite-size systems ' and various exact results. Most of these works, however, are more interested in ob- taining information about the infinite-system behavior than in the effective exponents themselves, and there are more efficient methods of estimating critical exponents than directly determining effective exponents. Finite-size scaling and Monte Carlo renormalization are examples. These methods of analysis are usually not available to the experimenter; it is difficult to change the system size or to determine many correlation functions. We also note that most of these calculations are for spin systems which have higher symmetries than the lattice gas models appropriate to surface phases, as discussed in more detail below. Analyzing the structure factor also raises points not directly dealt with in most Monte Carlo work. For exam- ple, how small do wave vectors have to be before the structure factors satisfy scaling relations? There are, of course, complications other than finite-size effects in in- terpreting LEED experiments. Two difficulties are finite instrumental resolution and multiple scattering. The former limits the size of correlation lengths which can be measured. This is no difficulty if one desires information about short-range order. (LEED intensity measurements which are sensitive only to short-range correlations will have an energylike singularity at a critical point— allowing the specific-heat exponent a to be measured. ' ) In measuring long-range order one seeks the largest corre- lation lengths possible, requiring deconvolution of an in- strumental response function from the data. " ' Assump- tions made in the deconvolution process (for example as- suming Lorentzian line shapes and circular symmetry) complicate the interpretation of the results. The ultimate resolution one can obtain by deconvolution is limited by 35 1776 1987 The American Physical Society