Editing Lattice hard spheres
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These systems exhibit phase (order-disorder) transitions. | These systems exhibit phase (order-disorder) transitions. | ||
== Three-dimensional lattices == | == Three-dimensional lattices == | ||
See Ref. 1 for some results of three-dimensional lattice hard sphere systems (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. | |||
If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]] (See Ref 1). | |||
If next-nearest neighbours are also excluded then the transition becomes [[First-order transitions |first order]]. | |||
== Two-dimensional lattices == | == Two-dimensional lattices == | ||
=== Square lattice === | === Square lattice === | ||
The model with exclusion of nearest neighbours presents a | The model with exclusion of nearest neighbours presents a discontinuous transition. The critical behavior at the transition | ||
corresponds to the same Universality class of the two-dimensional [[Ising model|Ising Model]], See Ref | corresponds to the same Universality class of the two-dimensional [[Ising model|Ising Model]], See Ref | ||
<ref>[http://dx.doi.org/10.1103/PhysRevB.62.2134 Da-Jiang Liu and J. W. Evans | <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'', Phys. Rev. B '''62''', 2134 - 2145 (2000)] </ref> for a simulation study of this system. | ||
on a [[building up a square lattice|square lattice]] | See Ref 2. for results of two-dimensional systems (lattice hard disks) with different exclusion criteria | ||
on a [[building up a square lattice|square lattice]]. | |||
=== [[Building up a triangular lattice|Triangular lattice]] === | === [[Building up a triangular lattice|Triangular lattice]] === | ||
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]]). | 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]]). | ||
Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]] | Other models defined on the triangular lattice (with more excluded positions) have been studied theoretically and by [[Monte Carlo | Monte Carlo simulation]] (Refs 3-5). | ||
It seems (see Ref. 3 and Ref.5) 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 == | == References == | ||
<references/> | <references/> | ||
#[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)] | |||
#[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)] | |||
#[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)] | |||
#[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)] | |||
#[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) (7 pages) ] | |||
[[category: models]] | [[category: models]] |