Muffin-tin approximation

The muffin-tin approximation is a shape approximation of the potential field in an atomistic environment. It is most commonly employed in quantum mechanical simulations of electronic band structure in solids. The approximation was proposed by John C. Slater.^[1][2] Many modern electronic structure methods employ the approximation.^[3][4] Among them are the augmented plane wave (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 Korringa (1947) and by Kohn and Rostocker (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

• Anderson's rule
• Band gap
• Bloch waves
• Kohn–Sham equations
• Kronig–Penney model
• Local-density approximation

References

1. ^ Duan, Feng; Guojun, Jin (2005). Introduction to Condensed Matter Physics. 1. Singapore: World Scientific. ISBN 9789812387110. 
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 3540639616. http://books.google.com/?id=4ErN_IfEN-oC&pg=PA49&dq=%22muffin+tin+approximation%22. 
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. http://books.google.com/?id=B9pVFkCzQasC&dq=intitle:Computational+intitle:Quantum+intitle:Mechanics+intitle:for+intitle:Materials+intitle:Engineers. 
5. ^ Richard P Martin (2004). Electronic Structure: Basic Theory and Applications. Cambridge University Press. pp. 313 ff. ISBN 0521782856. http://books.google.com/?id=dmRTFLpSGNsC&pg=PA316&dq=isbn=0521782856#PPA314,M1. 
6. ^ U Mizutani (2001). Introduction to the Theory of Metals. Cambridge University Press. p. 211. ISBN 0521587093. http://books.google.com/?id=zY5z_UGqAcwC&pg=PA219&dq=%22muffin+tin+approximation%22. 
7. ^ Joginder Singh Galsin (2001). "Appendix C". Impurity Scattering in Metal Alloys. Springer. ISBN 0306465744. http://books.google.com/?id=kmcLT63iX_EC&pg=PA498&dq=KKR+method+band+structure. 
8. ^ Kuon Inoue, Kazuo Ohtaka (2004). Photonic Crystals. Springer. p. 66. ISBN 3540205594. http://books.google.com/?id=GIa3HRgPYhAC&pg=PA66&dq=KKR+method+band+structure. 
9. ^ I Turek, J Kudrnovsky & V Drchal (2000). "Disordered Alloys and Their Surfaces: The Coherent Potential Approximation". In Hugues Dreyssé. Electronic Structure and Physical Properties of Solids. Springer. p. 349. ISBN 3540672389. http://books.google.com/?id=Tb0DW3rSFjEC&pg=PA349&dq=KKR+%22coherent+potential+approximation%22. 
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.