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Muffin-tin approximation

January 01, 1970

The muffin-tin approximation is a shape approximation of the potential well in a crystal lattice. It is most commonly employed in quantum mechanical simulations of the electronic band structure in solids. The approximation was proposed by John C. Slater. Augmented plane wave method (APW) is a method which uses muffin-tin approximation. It is a method to approximate the energy states of an electron in a crystal lattice. The basic approximation lies in the potential in which the potential is assumed to be spherically symmetric in the muffin-tin region and constant in the interstitial region. Wave functions (the augmented plane waves) are constructed by matching solutions of the Schrödinger equation within each sphere with plane-wave solutions in the interstitial region, and linear combinations of these wave functions are then determined by the variational method.[1][2] Many modern electronic structure methods employ the approximation.[3][4] Among them APW method, the linear muffin-tin orbital method (LMTO) and various Green's function methods.[5] One application is found in the variational theory developed by Jan Korringa (1947) and by Walter Kohn and N. Rostoker (1954) referred to as the KKR method.[6][7][8] This method has been adapted to treat random materials as well, where it is called the KKR coherent potential approximation.[9]

In its simplest form, non-overlapping spheres are centered on the atomic positions. Within these regions, the screened potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the potential is approximated as a constant. Continuity of the potential between the atom-centered spheres and interstitial region is enforced.

In the interstitial region of constant potential, the single electron wave functions can be expanded in terms of plane waves. In the atom-centered regions, the wave functions can be expanded in terms of spherical harmonics and the eigenfunctions of a radial Schrödinger equation.[2][10] Such use of functions other than plane waves as basis functions is termed the augmented plane-wave approach (of which there are many variations). It allows for an efficient representation of single-particle wave functions in the vicinity of the atomic cores where they can vary rapidly (and where plane waves would be a poor choice on convergence grounds in the absence of a pseudopotential).

See also edit

References edit

  1. ^ Duan, Feng; Guojun, Jin (2005). Introduction to Condensed Matter Physics. Vol. 1. Singapore: World Scientific. ISBN 978-981-238-711-0.
  2. ^ a b Slater, J. C. (1937). "Wave Functions in a Periodic Potential". Physical Review. 51 (10): 846–851. Bibcode:1937PhRv...51..846S. doi:10.1103/PhysRev.51.846.
  3. ^ Kaoru Ohno, Keivan Esfarjani, Yoshiyuki (1999). Computational Materials Science. Springer. p. 52. ISBN 978-3-540-63961-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
  4. ^ Vitos, Levente (2007). Computational Quantum Mechanics for Materials Engineers: The EMTO Method and Applications. Springer-Verlag. p. 7. ISBN 978-1-84628-950-7.
  5. ^ Richard P Martin (2004). Electronic Structure: Basic Theory and Applications. Cambridge University Press. pp. 313 ff. ISBN 978-0-521-78285-2.
  6. ^ U Mizutani (2001). Introduction to the Theory of Metals. Cambridge University Press. p. 211. ISBN 978-0-521-58709-9.
  7. ^ Joginder Singh Galsin (2001). "Appendix C". Impurity Scattering in Metal Alloys. Springer. ISBN 978-0-306-46574-1.
  8. ^ Kuon Inoue; Kazuo Ohtaka (2004). Photonic Crystals. Springer. p. 66. ISBN 978-3-540-20559-3.
  9. ^ I Turek, J Kudrnovsky & V Drchal (2000). "Disordered Alloys and Their Surfaces: The Coherent Potential Approximation". In Hugues Dreyssé (ed.). Electronic Structure and Physical Properties of Solids. Springer. p. 349. ISBN 978-3-540-67238-8. KKR coherent potential approximation.
  10. ^ Slater, J. C. (1937). "An Augmented Plane Wave Method for the Periodic Potential Problem". Physical Review. 92 (3): 603–608. Bibcode:1953PhRv...92..603S. doi:10.1103/PhysRev.92.603.

January 01, 1970
muffin, approximation, this, article, technical, most, readers, understand, please, help, improve, make, understandable, experts, without, removing, technical, details, july, 2018, learn, when, remove, this, message, muffin, approximation, shape, approximation. This article may be too technical for most readers to understand Please help improve it to make it understandable to non experts without removing the technical details July 2018 Learn how and when to remove this message The muffin tin approximation is a shape approximation of the potential well in a crystal lattice It is most commonly employed in quantum mechanical simulations of the electronic band structure in solids The approximation was proposed by John C Slater Augmented plane wave method APW is a method which uses muffin tin approximation It is a method to approximate the energy states of an electron in a crystal lattice The basic approximation lies in the potential in which the potential is assumed to be spherically symmetric in the muffin tin region and constant in the interstitial region Wave functions the augmented plane waves are constructed by matching solutions of the Schrodinger equation within each sphere with plane wave solutions in the interstitial region and linear combinations of these wave functions are then determined by the variational method 1 2 Many modern electronic structure methods employ the approximation 3 4 Among them APW method the linear muffin tin orbital method LMTO and various Green s function methods 5 One application is found in the variational theory developed by Jan Korringa 1947 and by Walter Kohn and N Rostoker 1954 referred to as the KKR method 6 7 8 This method has been adapted to treat random materials as well where it is called the KKR coherent potential approximation 9 In its simplest form non overlapping spheres are centered on the atomic positions Within these regions the screened potential experienced by an electron is approximated to be spherically symmetric about the given nucleus In the remaining interstitial region the potential is approximated as a constant Continuity of the potential between the atom centered spheres and interstitial region is enforced In the interstitial region of constant potential the single electron wave functions can be expanded in terms of plane waves In the atom centered regions the wave functions can be expanded in terms of spherical harmonics and the eigenfunctions of a radial Schrodinger equation 2 10 Such use of functions other than plane waves as basis functions is termed the augmented plane wave approach of which there are many variations It allows for an efficient representation of single particle wave functions in the vicinity of the atomic cores where they can vary rapidly and where plane waves would be a poor choice on convergence grounds in the absence of a pseudopotential See also editAnderson s rule Band gap Bloch waves Kohn Sham equations Kronig Penney model Local density approximationReferences edit Duan Feng Guojun Jin 2005 Introduction to Condensed Matter Physics Vol 1 Singapore World Scientific ISBN 978 981 238 711 0 a b Slater J C 1937 Wave Functions in a Periodic Potential Physical Review 51 10 846 851 Bibcode 1937PhRv 51 846S doi 10 1103 PhysRev 51 846 Kaoru Ohno Keivan Esfarjani Yoshiyuki 1999 Computational Materials Science Springer p 52 ISBN 978 3 540 63961 9 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link Vitos Levente 2007 Computational Quantum Mechanics for Materials Engineers The EMTO Method and Applications Springer Verlag p 7 ISBN 978 1 84628 950 7 Richard P Martin 2004 Electronic Structure Basic Theory and Applications Cambridge University Press pp 313 ff ISBN 978 0 521 78285 2 U Mizutani 2001 Introduction to the Theory of Metals Cambridge University Press p 211 ISBN 978 0 521 58709 9 Joginder Singh Galsin 2001 Appendix C Impurity Scattering in Metal Alloys Springer ISBN 978 0 306 46574 1 Kuon Inoue Kazuo Ohtaka 2004 Photonic Crystals Springer p 66 ISBN 978 3 540 20559 3 I Turek J Kudrnovsky amp V Drchal 2000 Disordered Alloys and Their Surfaces The Coherent Potential Approximation In Hugues Dreysse ed Electronic Structure and Physical Properties of Solids Springer p 349 ISBN 978 3 540 67238 8 KKR coherent potential approximation Slater J C 1937 An Augmented Plane Wave Method for the Periodic Potential Problem Physical Review 92 3 603 608 Bibcode 1953PhRv 92 603S doi 10 1103 PhysRev 92 603 Retrieved from https en wikipedia org w index php title Muffin tin approximation amp oldid 1026599243, wikipedia, wiki, book, books, library,

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