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== | {{Electronic structure methods}} | ||
The '''Korringa–Kohn–Rostoker approximation''' or '''KKR method'''<ref name=Korringa> | |||
{{cite journal | |||
| author = J. Korringa | year = 1947 | |||
| title = On the calculation of the energy of a Bloch wave in a metal | |||
| journal = Physica | volume = XIII | pages = 392–400 }} | |||
</ref><ref name=KohnRostoker> | |||
{{cite journal | |||
| author = W. Kohn, N. Rostoker | year = 1954 | |||
| title = Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium | |||
| journal = Phys.Rev. | volume = 94 | pages = 1111–1120 }} | |||
</ref><ref name=JonesMarch> | |||
{{cite book |author= W. Jones, N. H. March |title=Theoretical Solid State Physics |year=1973 |publisher= Wiley and Sons - Dover Publications | |||
|url=http://books.google.com/books?id=7eTq_ndhFpgC |isbn=0-486-65015-4 }} | |||
</ref> is used to calculate the [[electronic band structure]] of [[solid]]s. It is a [[Green's function]] method which matches the different types of one-electron wave functions and their derivatives which are used in a [[muffin-tin approximation]]. | |||
==Introduction== | |||
In [[Solid-state physics|solid state physics]] the properties of electrons and potentials are determined by the spherical symmetry of the interior of the atoms and the point group symmetry of the crystal lattice. The [[wave function]] of an electron and the potential in the KKR method are both composed of two parts. A linear combination of spherical harmonics, which is multiplied by a radial wave function, is used for the wave function on the inside of the atoms where the potential is the potential of the atom. Linear combinations of plane waves are used for the wave function in the regions between the atoms where the potential is constant. | |||
== | ==Mathematical formulation== | ||
The [[Schrödinger equation]] can be expressed like | |||
:<math>[\nabla^2 + E]\psi(\bold{r}) = V(\bold{r})\psi(\bold{r})</math> | |||
where <math>V(\bold{r})</math> is the potential of the solid and <math>\psi(\bold{r})</math> is the wave function of the electron that has to be calculated. | |||
The unperturbed [[Green's function]] is defined as the solution of | |||
:<math>[\nabla^2 + E]G(\bold{r},\bold{r}') = \delta(\bold{r}-\bold{r}')</math> | |||
A plane wave can be expanded as | |||
:<math>e^{i\bold{k}\cdot \bold{r}} = \sum_l (2 l + 1) i^l j_l(kr)P_l(\cos\theta)</math> | |||
where <math>j_l(kr)</math> are [[Bessel_function#Spherical_Bessel_functions:_jn.2C_yn|spherical Bessel functions]] and <math>P_l(\cos\theta)</math> are [[Legendre polynomials]]. | |||
==References== | |||
{{reflist|2}} | |||
{{DEFAULTSORT:Korringa-Kohn-Rostoker approximation}} | |||
[[Category:Electronic structure methods]] | |||
Revision as of 11:44, 22 December 2013
Template:Electronic structure methods The Korringa–Kohn–Rostoker approximation or KKR method[1][2][3] is used to calculate the electronic band structure of solids. It is a Green's function method which matches the different types of one-electron wave functions and their derivatives which are used in a muffin-tin approximation.
Introduction
In solid state physics the properties of electrons and potentials are determined by the spherical symmetry of the interior of the atoms and the point group symmetry of the crystal lattice. The wave function of an electron and the potential in the KKR method are both composed of two parts. A linear combination of spherical harmonics, which is multiplied by a radial wave function, is used for the wave function on the inside of the atoms where the potential is the potential of the atom. Linear combinations of plane waves are used for the wave function in the regions between the atoms where the potential is constant.
Mathematical formulation
The Schrödinger equation can be expressed like
where is the potential of the solid and is the wave function of the electron that has to be calculated.
The unperturbed Green's function is defined as the solution of
A plane wave can be expanded as
where are spherical Bessel functions and are Legendre polynomials.
References
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
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