# Lattice hard spheres

Lattice hard spheres (or Lattice hard disks) refers to athermal lattice gas models, in which pairs of sites separated by less than some (short) distance, ${\displaystyle \sigma }$, cannot be simultaneously occupied.

## Brief description of the models

Basically the differences between lattice hard spheres and the standard lattice gas model (Ising model) are the following:

• An occupied site excludes the occupation of some of the neighbouring sites.
• No energy interactions between pairs of occupied sites -apart of the hard core interactions- are considered.

These systems exhibit phase (order-disorder) transitions.

## Three-dimensional lattices

For some results of three-dimensional lattice hard sphere systems see [1] (on a simple cubic lattice). The model defined on a simple cubic lattice with exclusion of only the nearest neighbour positions of an occupied site presents a continuous transition. If next-nearest neighbours are also excluded then the transition becomes first order.

## Two-dimensional lattices

### Square lattice

The model with exclusion of nearest neighbours presents a continuous transition. The critical behaviour at the transition corresponds to the same Universality class of the two-dimensional Ising Model, See Ref [2] for a simulation study of this system. For results of two-dimensional systems (lattice hard disks) with different exclusion criteria on a square lattice see [3].

### Triangular lattice

The hard hexagon lattice model belongs to this kind of model. In this model an occupied site excluded the occupation of nearest neighbour positions. This model exhibits a continuous transition, and has been solved exactly (See references in the entry: hard hexagon lattice model). Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by Monte Carlo simulation [4] [5] [6]. It seems that the model with first and second neighbour exclusion presents also a continuous transition, whereas if third neighbours are also excluded the transition becomes first order.