Eur. Phys. J. B 32, 527–535 (2003) DOI: 10.1140/epjb/e2003-00118-3 T HE EUROPEAN P HYSICAL JOURNAL B SCDA for 3D lattice gases with repulsive interaction Ya.G. Groda 1 , P. Argyrakis 2 , G.S. Bokun 1 , and V.S. Vikhrenko 1, a 1 Belarussian State Technological University, 13a Sverdlova Str., Minsk 220 050, Belarus 2 Department of Physics, University of Thessaloniki, 54006 Thessaloniki, Greece Received 7 August 2002 / Received in final form 22 January 2003 Published online 24 April 2003 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2003 Abstract. The selfconsistent diagram approximation (SCDA) is generalized for three-dimensional lattice gases with nearest neighbor repulsive interactions. The free energy is represented in a closed form through elementary functions. Thermodynamical (phase diagrams, chemical potential and mean square fluctua- tions), structural (order parameter, distribution functions) as well as diffusional characteristics are inves- tigated. The calculation results are compared with the Monte Carlo simulation data to demonstrate high precision of the SCDA in reproducing the equilibrium lattice gas characteristics. It is shown that simi- larly to two-dimensional systems the specific statistical memory effects strongly influence the lattice gas diffusion in the ordered states. PACS. 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) – 05.60.-k Transport processes – 05.70.Ce Thermodynamic functions and equations of state 1 Introduction Extensive applications of lattice models [1–7] for the de- scription of a variety of physical systems and processes make it topical to improve statistical-mechanical meth- ods of their investigation. The approaches developed ear- lier are characterized either insufficient precision (different mean field approximations [8,9]) or require tedious calcu- lations (series expansions [4,10]) and cannot be used for most practical calculations. Although lattice models are described by a discrete vector of states and the Marko- vian master equation the strong interparticle interactions being accounted of lead to their complex properties and give rise to considerable difficulties well known in many body theories. Properties of lattice systems crucially depend on the type of interparticle interactions. For example, attractive interactions lead to first order phase transitions while or- der – disorder phase transitions are ordinary observed for repulsive interactions. Therefore, many approaches appro- priate for the former cannot be used for the later or at least must be considerably modified because the ordering in a lattice system lowers its symmetry and requires to divide the lattice into sublattices. Recently [11] the selfconsistent diagram approxima- tion (SCDA) was suggested and applied to lattice gases with attractive nearest neighbor interactions [11,12]. It was shown that SCDA well reproduces the thermody- namic (chemical potential versus concentration, its con- a e-mail: vvikhre@mail.ru centration derivative, phase transition curves) and struc- tural (probabilities for a pair of lattice sites to be occupied by particles or vacancies) properties for two- (2D) as well as three-dimensional (3D) systems. Moreover, it was suc- cessfully used for describing the diffusional properties of lattice gases [12,13]. Here we generalize SCDA for nearest neighbor repulsive interactions on 3D low packed (simple cubic (SC) and body centered cubic (BCC)) lattices. The paper is organized as follows. In the next section the necessary definitions are introduced and the free en- ergy is represented as a diagram expansion in renormal- ized Mayer functions. Section 3 is devoted to evaluation of the mean potentials that renormalize the Mayer func- tions on the basis of the concept of minimal susceptibility. In Section 4 statistical mechanical calculations are com- pared with Monte Carlo simulation results. The last sec- tion concludes. 2 The free energy and the order parameter For lattice gases with repulsive interparticle interactions at sufficiently low temperatures different ordered states can exist. The sublattice decomposition of the lattice is used for the analysis whether the system is in an ordered or disordered state [14]. We consider the simplest case when the lattice is represented by two sublattices (A and B). Lattice gases with repulsive nearest neighbor (NN) interactions on honeycomb, square, simple cubic and body centered cubic lattices are examples of such systems. The potential energy of the system of n particles on N lattice