Lattice model

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This article deals with physical calculation models in a general sense. For lattice models in the sense of molecular representations, see rod model .

Lattice models are generally mathematical models in which the degrees of freedom of the system correspond to the elements of a lattice , i.e. H. a countable set of points. This distinguishes them from continuum models, in which degrees of freedom are assigned to each value of an interval . Typical examples are the description of the magnetization of a solid by spins localized at the locations of the atomic nuclei , which are assumed to be fixed and periodic (e.g. Ising model ) or the movement of the conduction electrons by jumping between orbitals localized at the locations of the atomic nuclei ( Hubbard- Model ). The model is used for an approximate description of the physical system.

Lattice models are used, for example, when interactions between bodies are described whose spatial degrees of freedom are limited in such a way that they can only be located at the grid points, or that the remaining variability does not lead to relevant changes in the system to be simulated. The representation as a grid model can lead to a considerable simplification of the necessary calculations.

example

To calculate the band structure of a crystalline solid it is necessary to solve the Schrödinger equation for the ground state. This is approximately possible using the Hohenberg-Kohn theorem with the help of density functional theory . The wave function of a 1-electron state with the energy depending on an effective potential is represented by the following differential equation : ${\displaystyle \varphi _{j}({\vec {r}})}$ ${\displaystyle \epsilon _{j}}$ ${\displaystyle v_{\mathrm {eff} }({\vec {r}})}$

${\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+v_{\mathrm {eff} }({\vec {r}})-\epsilon _{j}\right)\varphi _{j}({\vec {r}})=0}$

This equation contains the position-dependent quantities and . To model the entire band structure, it has to be solved for all electron states. This requires an iterative procedure, since the spin or charge density is a functional , which in turn results from the 1-electron states and their occupation probability, mostly according to the Fermi distribution . ${\displaystyle \varphi _{j}({\vec {r}})}$ ${\displaystyle v_{\mathrm {eff} }({\vec {r}})}$ ${\displaystyle v_{\mathrm {eff} }({\vec {r}})}$ ${\displaystyle \varphi _{j}({\vec {r}})}$

To consider the electronic structure of the same solid in a lattice model, one uses, for example following the tight binding method , a not exactly determined basis of states that are localized around the lattice positions of the atomic cores and formulates the Hamilton operator as a set of parameters , which describe the interaction between the base states and : ${\displaystyle \left|\Psi _{i}\right\rangle }$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle p_{i,j}}$ ${\displaystyle i}$ ${\displaystyle j}$

${\displaystyle {\mathcal {H}}:(M\subset \mathbb {N} )^{2}\mapsto \mathbb {R} ,(i,j)\mapsto p_{i,j}}$.

With this consideration there are no directly location-dependent parameters, but upon closer inspection they depend on the base states and the location-dependent electron density. For certain considerations in the context of perturbation , such as. The calculation of phonon assessing enenergien, the influence of dopants or impurities , the grid model with sufficient accuracy. ${\displaystyle p_{i,j}}$ ${\displaystyle \Psi _{i}({\vec {r}})}$

By further simplifying the lattice model to the relevant parameters or basic conditions and analytical investigations with regard to the parameters, certain properties can then be investigated independently of the exact material.

More grid models

• Lattice-Boltzmann method
• Flory Huggins model
• Ising model
• Potts model
• Lattice theories
• Hubbard model

literature

• H. Römer, T. Filk: Statistical Mechanics . Ed .: University of Freiburg. Freiburg 2012, ISBN 3-527-29228-4 , 8.2 General definitions of grid models, p. 217 ff . (288 p., Uni-freiburg.de [PDF; accessed on January 10, 2020] First edition: VCH, Weinheim 1994, online source contains an edited version).
• D. Vollhardt: Correlated electrons in the solid . In: Physics Journal . tape 9 , no. 8 . Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 2010, p. 31 ff . ( uni-augsburg.de [PDF; accessed on January 10, 2020] Article on the award of the Max Planck Medal ).

Individual evidence

1. ↑ Lexicon of Physics: Lattice Models. Retrieved January 6, 2020 . 
2. ↑ GH Findeegg, T. Hellweg: Grid models of mixtures . In: Statistical Thermodynamics . Springer Spectrum, Berlin, Heidelberg 2015, ISBN 978-3-642-37871-3 , pp. 165–195 , doi : 10.1007 / 978-3-642-37872-0_9 ( springer.com [accessed January 10, 2020]).