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# 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, , cannot be simultaneously occupied.

## Contents

## Brief description of the models[edit]

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[edit]

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[edit]

### Square lattice[edit]

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[edit]

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.

## References[edit]

- ↑ A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", Journal of Chemical Physics
**123**104504 (2005) - ↑ Da-Jiang Liu and J. W. Evans, "Ordering and percolation transitions for hard squares: Equilibrium versus nonequilibrium models for adsorbed layers with c(2×2) superlattice ordering", Physical Review B
**62**, pp 2134 - 2145 (2000) - ↑ Heitor C. Marques Fernandes, Jeferson J. Arenzon, and Yan Levin "Monte Carlo simulations of two-dimensional hard core lattice gases", Journal of Chemical Physics
**126**114508 (2007) - ↑ N. C. Bartelt and T. L. Einstein, "Triangular lattice gas with first- and second-neighbor exclusions: Continuous transition in the four-state Potts universality class", Physical Review B
**30**pp. 5339-5341 (1984) - ↑ Chin-Kun Hu and Kit-Sing Mak, "Percolation and phase transitions of hard-core particles on lattices: Monte Carlo approach", Physical Review B
**39**pp. 2948-2951 (1989) - ↑ Wei Zhang Youjin Den,
*Monte Carlo study of the triangular lattice gas with first- and second-neighbor exclusions*, Physical Review E**78**031103 (2008)