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| '''Lattice hard spheres''' (or '''Lattice hard disks''') refers to athermal [[lattice gas|lattice gas]] models, in which pairs of sites separated by less than some (short) distance, <math> \sigma </math>, cannot be simultaneously occupied. | | '''Lattice hard spheres''' refers to athermal [[lattice gas|lattice gas]] models, in which pairs |
| == Brief description of the models ==
| | of sites separated by less than some short distance <math> \sigma </math> cannot be simultaneously occupied. |
| Basically the differences between lattice hard spheres and the standard [[Lattice gas|lattice gas]] model ([[Ising Models|Ising model]]) are the following:
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| *An occupied site excludes the occupation of some of the neighbouring sites.
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| *No energy interactions between pairs of occupied sites -apart of the hard core interactions- are considered.
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| These systems exhibit phase (order-disorder) transitions.
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| == Three-dimensional lattices ==
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| For some results of three-dimensional lattice hard sphere systems see
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| <ref>[http://dx.doi.org/10.1063/1.2008253 A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", Journal of Chemical Physics '''123''' 104504 (2005)]</ref> (on a [[Building up a simple cubic lattice |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.
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| If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]].
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| == Two-dimensional lattices ==
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| === Square lattice ===
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| The model with exclusion of nearest neighbours presents a continuous transition. The critical behaviour at the transition
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| corresponds to the same Universality class of the two-dimensional [[Ising model|Ising Model]], See Ref
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| <ref>[http://dx.doi.org/10.1103/PhysRevB.62.2134 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)] </ref> for a simulation study of this system.
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| For results of two-dimensional systems (lattice hard disks) with different exclusion criteria
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| on a [[building up a square lattice|square lattice]] see <ref>[http://dx.doi.org/10.1063/1.2539141 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)]</ref>.
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| === [[Building up a triangular lattice|Triangular lattice]] ===
| | Basically the differences with the standard [[Lattice gas|lattice gas]] model ([[Ising Models|Ising model]]) are: |
| The [[hard hexagon lattice model|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|hard hexagon lattice model]]).
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| Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]]
| | *An occupied site excludes the occupation of some of the neighboring sites. |
| <ref>[http://dx.doi.org/10.1103/PhysRevB.30.5339 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)]</ref>
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| <ref>[http://dx.doi.org/10.1103/PhysRevB.39.2948 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)]</ref>
| | *No energy interactions between pairs of occupied sites are considered. |
| <ref>[http://dx.doi.org/10.1103/PhysRevE.78.031103 Wei Zhang Youjin Den, ''Monte Carlo study of the triangular lattice gas with first- and second-neighbor exclusions'', Physical Review E '''78''' 031103 (2008)]</ref>.
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| 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.
| | These systems exhibit phase (order-disorder) transitions |
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| | *See Ref. 1 for some results of three-dimensional lattice hard sphere systems (on a [[Building up a simple cubic lattice |simple cubic lattice]]) |
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| | *See Ref 2. for results of two-dimensional systems (lattice hard disks) on a square lattice. |
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| == References == | | == References == |
| <references/>
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| | #[http://dx.doi.org/10.1063/1.2008253 A. Z. Panagiotopoulos, "Thermodynamic properties of lattice hard-sphere models", J. Chem. Phys. 123, 104504 (2005) ] |
| | #[http://dx.doi.org/10.1063/1.2539141 Heitor C. Marques Fernandes, Jeferson J. Arenzon, and Yan Levin "Monte Carlo simulations of two-dimensional hard core lattice gases" J. Chem. Phys. 126, 114508 (2007).] |
| [[category: models]] | | [[category: models]] |