# Coherent potential approximation > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Coherent_potential_approximation > Markdown URL: https://mediated.wiki/source/Coherent_potential_approximation.md > Source: https://en.wikipedia.org/wiki/Coherent_potential_approximation > Source revision: 1351298164 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) The '''coherent potential approximation''' ('''CPA''') is a method, in [theoretical physics](/source/theoretical_physics), of finding the averaged [Green's function](/source/Green's_function) of an inhomogeneous (or disordered) system. The Green's function obtained via the CPA then describes an effective medium whose scattering properties represent the averaged scattering properties of the disordered system being approximated. It is often described as the 'best' single-site theory for obtaining the averaged Green's function.{{Cite journal |last=Yonezawa |first=Fumiko |last2=Morigaki |first2=Kazuo |date=1973 |title=Coherent Potential Approximation: Basic concepts and applications |url=https://academic.oup.com/ptps/article-lookup/doi/10.1143/PTPS.53.1 |journal=Progress of Theoretical Physics Supplement |language=en |volume=53 |pages=1–76 |doi=10.1143/PTPS.53.1 |issn=0375-9687|url-access=subscription }} It is perhaps most famous for its use in describing the physical properties of [alloys](/source/Alloy) and [disordered magnetic systems](/source/Magnetism), although it is also a useful concept in understanding how [sound waves](/source/sound_waves) scatter in a material which displays spatial inhomogeneity. The coherent potential approximation was first described by [Paul Soven](/source/Paul_Soven),{{Cite journal |last=Soven |first=Paul |date=1967-04-15 |title=Coherent-Potential Model of Substitutional Disordered Alloys |url=https://link.aps.org/doi/10.1103/PhysRev.156.809 |journal=Physical Review |volume=156 |issue=3 |pages=809–813 |doi=10.1103/PhysRev.156.809|url-access=subscription }} and its application in the context of calculations of the [electronic structure](/source/electronic_structure) of materials was pioneered by [Balász Győrffy](/source/Bal%C3%A1zs_Gy%C5%91rffy).{{Cite journal |last=Gyorffy |first=B. L. |date=1972-03-15 |title=Coherent-Potential Approximation for a Nonoverlapping-Muffin-Tin-Potential Model of Random Substitutional Alloys |url=https://link.aps.org/doi/10.1103/PhysRevB.5.2382 |journal=Physical Review B |volume=5 |issue=6 |pages=2382–2384 |doi=10.1103/PhysRevB.5.2382|url-access=subscription }} == Electronic structure (KKR-CPA) == In the context of calculations of the [electronic structure](/source/electronic_structure) of materials, the coherent potential approximation is frequently combined with the [Korringa–Kohn–Rostoker](/source/Korringa%E2%80%93Kohn%E2%80%93Rostoker_method) (KKR) formulation of [density functional theory](/source/density_functional_theory) (DFT) to describe the electronic structure of systems with lattice-based disorder, such as substitutional alloys and magnetic materials at finite temperature.{{Cite journal |last=Ebert |first=H |last2=Ködderitzsch |first2=D |last3=Minár |first3=J |date=2011-09-01 |title=Calculating condensed matter properties using the KKR-Green's function method—recent developments and applications |url=https://iopscience.iop.org/article/10.1088/0034-4885/74/9/096501 |journal=Reports on Progress in Physics |volume=74 |issue=9 |article-number=096501 |doi=10.1088/0034-4885/74/9/096501 |issn=0034-4885|url-access=subscription }}{{Cite book |last=Faulkner, Stocks, Wang |first= |title=Multiple Scattering Theory: Electronic structure of solids |date=December 2018 |publisher=[IoP Publishing](/source/IoP_Publishing) |isbn=978-0-7503-1490-9 |location=Bristol, UK |doi=10.1088/2053-2563/aae7d8}} The KKR formulation of DFT is also sometimes referred to as [multiple scattering theory](/source/multiple_scattering_theory) (MST). When the KKR formulation of DFT is combined with the CPA, it is sometimes referred to as the KKR-CPA. The KKR formulation of DFT rephrases the usual eigenvalue-eigenvector problem (''i.e.'' solving some effective [Schrödinger equation](/source/Schr%C3%B6dinger_equation)) into an electronic scattering problem.{{Cite book |last=Faulkner |first=J S |title=Multiple Scattering Theory |last2=Stocks |first2=G Malcolm |last3=Wang |first3=Yang |date=2018-12-01 |publisher=IOP Publishing |isbn=978-0-7503-1490-9 |doi=10.1088/2053-2563/aae7d8}} It does so by partitioning the one-electron potential of DFT into a collection of spatially-localised potentials around each ionic site, before considering an electron propagating through the system and scattering from these localised potentials. In this manner, the Green's function of the system is obtained. In a system where there is lattice-based disorder (for example, in a substitutional alloy) the CPA provides a means by which to average multiple potentials associated with a single lattice site and obtain an average Green's function (and consequent [electron density](/source/electron_density)) in a physically meaningful way. Although the approach was originally formulated for potentials described within either the muffin tin or atomic sphere approximations (where the spatially localised potential is assumed spherically symmetric) it is now commonplace to use so-called full-potential calculations,{{Cite journal |last=Asato |first=M. |last2=Settels |first2=A. |last3=Hoshino |first3=T. |last4=Asada |first4=T. |last5=Blügel |first5=S. |last6=Zeller |first6=R. |last7=Dederichs |first7=P. H. |date=1999-08-15 |title=Full-potential KKR calculations for metals and semiconductors |url=https://link.aps.org/doi/10.1103/PhysRevB.60.5202 |journal=Physical Review B |volume=60 |issue=8 |pages=5202–5210 |doi=10.1103/PhysRevB.60.5202|url-access=subscription }} where the one-electron potential can have arbitrary spatial dependence. The KKR-CPA has been used with success to study the physics of a variety of alloy systems,{{Cite journal |last=Gonis |first=Antonios |last2=Butler |first2=W. H. |last3=Stocks |first3=G. M. |date=1983-05-09 |title=First-Principles Calculations of Cluster Densities of States and Short-Range Order in ${\mathrm{Ag}}_{c}{\mathrm{Pd}}_{1\ensuremath{-}c}$ Alloys |url=https://link.aps.org/doi/10.1103/PhysRevLett.50.1482 |journal=Physical Review Letters |volume=50 |issue=19 |pages=1482–1485 |doi=10.1103/PhysRevLett.50.1482|url-access=subscription }}{{Cite journal |last=Johnson |first=D. D. |last2=Nicholson |first2=D. M. |last3=Pinski |first3=F. J. |last4=Gyorffy |first4=B. L. |last5=Stocks |first5=G. M. |date=1986-05-12 |title=Density-Functional Theory for Random Alloys: Total Energy within the Coherent-Potential Approximation |url=https://link.aps.org/doi/10.1103/PhysRevLett.56.2088 |journal=Physical Review Letters |volume=56 |issue=19 |pages=2088–2091 |doi=10.1103/PhysRevLett.56.2088|url-access=subscription }}{{Cite journal |last=Stocks |first=G. M. |last2=Butler |first2=W. H. |date=1982-01-04 |title=Mass and Lifetime Enhancement due to Disorder on ${\mathrm{Ag}}_{c}{\mathrm{Pd}}_{1\ensuremath{-}c}$ Alloys |url=https://link.aps.org/doi/10.1103/PhysRevLett.48.55 |journal=Physical Review Letters |volume=48 |issue=1 |pages=55–58 |doi=10.1103/PhysRevLett.48.55|url-access=subscription }}{{Cite journal |last=Gyorffy |first=B. L. |last2=Stocks |first2=G. M. |date=1983-01-31 |title=Concentration Waves and Fermi Surfaces in Random Metallic Alloys |url=https://link.aps.org/doi/10.1103/PhysRevLett.50.374 |journal=Physical Review Letters |volume=50 |issue=5 |pages=374–377 |doi=10.1103/PhysRevLett.50.374|url-access=subscription }} including those where disorder is only present on one sub-lattice{{Cite journal |last=Pindor |first=A J |last2=Temmerman |first2=W M |last3=Gyorffy |first3=B L |date=March 1983 |title=KKR CPA for two atoms per unit cell: application to Pd and PdAg hydrides |url=https://iopscience.iop.org/article/10.1088/0305-4608/13/8/009 |journal=Journal of Physics F: Metal Physics |volume=13 |issue=8 |pages=1627–1644 |doi=10.1088/0305-4608/13/8/009 |issn=0305-4608|url-access=subscription }}{{Cite journal |last=Long |first=N H |last2=Ogura |first2=M |last3=Akai |first3=H |date=2009-02-11 |title=New type of half-metallic antiferromagnet: transition metal pnictides |url=https://iopscience.iop.org/article/10.1088/0953-8984/21/6/064241 |journal=Journal of Physics: Condensed Matter |volume=21 |issue=6 |article-number=064241 |doi=10.1088/0953-8984/21/6/064241 |issn=0953-8984|url-access=subscription }} (the 'inhomogeneous' CPA). In addition, it has been shown that the CPA can very effectively describe magnetism at finite temperature, by considering (weighted) averages taken over all possible spin orientations. This is referred to as the ['disordered local moment' (DLM) picture](/source/Disordered_Local_Moment_(DLM)_Picture){{Cite journal |last=Pindor |first=A J |last2=Staunton |first2=J |last3=Stocks |first3=G M |last4=Winter |first4=H |date=May 1983 |title=Disordered local moment state of magnetic transition metals: a self-consistent KKR CPA calculation |url=https://iopscience.iop.org/article/10.1088/0305-4608/13/5/012 |journal=Journal of Physics F: Metal Physics |volume=13 |issue=5 |pages=979–989 |doi=10.1088/0305-4608/13/5/012 |issn=0305-4608|url-access=subscription }}{{Cite journal |last=Staunton |first=J. |last2=Gyorffy |first2=B.L. |last3=Pindor |first3=A.J. |last4=Stocks |first4=G.M. |last5=Winter |first5=H. |date=November 1984 |title=The "disordered local moment" picture of itinerant magnetism at finite temperatures |journal=Journal of Magnetism and Magnetic Materials |volume=45 |issue=1 |pages=15–22 |doi=10.1016/0304-8853(84)90367-6 |issn=0304-8853}} and can be used to describe the [ferromagnetic](/source/Ferromagnetism) [phase transition](/source/phase_transition) in metals.{{Cite journal |last=Gyorffy |first=B L |last2=Pindor |first2=A J |last3=Staunton |first3=J |last4=Stocks |first4=G M |last5=Winter |first5=H |date=June 1985 |title=A first-principles theory of ferromagnetic phase transitions in metals |url=https://iopscience.iop.org/article/10.1088/0305-4608/15/6/018 |journal=Journal of Physics F: Metal Physics |volume=15 |issue=6 |pages=1337–1386 |doi=10.1088/0305-4608/15/6/018 |issn=0305-4608|url-access=subscription }} === Treatment of systems with short-range order: non-local CPA === The conventional KKR-CPA is a single-site theory and assumes a fully disordered alloy (or equivalent) by considering self-consistently the embedding of a single atom in the uniform [effective medium](/source/Effective_medium_approximations). However, many alloys exhibit at least some level of atomic [short-range order](/source/short-range_order) (ASRO). To account for such systems, the non-local coherent potential approximation (NL-CPA) was developed,{{Cite journal |last=Rowlands |first=D. A. |last2=Staunton |first2=J. B. |last3=Györffy |first3=B. L. |date=2003-03-17 |title=Korringa-Kohn-Rostoker nonlocal coherent-potential approximation |url=https://link.aps.org/doi/10.1103/PhysRevB.67.115109 |journal=Physical Review B |language=en |volume=67 |issue=11 |doi=10.1103/PhysRevB.67.115109 |issn=0163-1829|arxiv=cond-mat/0211328 }} which instead considers the self-consistent embedding of a cluster of atoms in the effective medium. It can be used to describe both the electronic structure{{Cite journal |last=Tulip |first=P. |last2=Staunton |first2=J. |last3=Rowlands |first3=D. |last4=Györffy |first4=B. |last5=Bruno |first5=E. |last6=Ginatempo |first6=B. |date=2006-05-17 |title=Nonsite diagonal properties from the Korringa-Kohn-Rostocker nonlocal coherent-potential approximation |url=https://link.aps.org/doi/10.1103/PhysRevB.73.205109 |journal=Physical Review B |language=en |volume=73 |issue=20 |doi=10.1103/PhysRevB.73.205109 |issn=1098-0121|url-access=subscription }}{{Cite journal |last=Marmodoro |first=Alberto |last2=Ernst |first2=Arthur |last3=Ostanin |first3=Sergei |last4=Sandratskii |first4=Leonid |last5=Trevisanutto |first5=Paolo E. |last6=Lathiotakis |first6=Nektarios N. |last7=Staunton |first7=Julie B. |date=2016-12-21 |title=Short-range ordering effects on the electronic Bloch spectral function of real materials in the nonlocal coherent-potential approximation |url=https://link.aps.org/doi/10.1103/PhysRevB.94.224205 |journal=Physical Review B |language=en |volume=94 |issue=22 |doi=10.1103/PhysRevB.94.224205 |issn=2469-9950|url-access=subscription }} and physical properties, ''e.g.'' transport properties{{Cite journal |last=Tulip |first=P. R. |last2=Staunton |first2=J. B. |last3=Lowitzer |first3=S. |last4=Ködderitzsch |first4=D. |last5=Ebert |first5=H. |date=2008-04-11 |title=Theory of electronic transport in random alloys with short-range order: Korringa-Kohn-Rostoker nonlocal coherent potential approximation |url=https://link.aps.org/doi/10.1103/PhysRevB.77.165116 |journal=Physical Review B |language=en |volume=77 |issue=16 |doi=10.1103/PhysRevB.77.165116 |issn=1098-0121|arxiv=0712.2388 }} of alloys with atomic short-range order. It has also been generalised to arbitrary alloy systems with multiple lattice sites per unit cell.{{Cite journal |last=Marmodoro |first=A |last2=Staunton |first2=J B |date=2011-03-01 |title=Disorder in materials with complex crystal structures: the Non-Local Coherent Potential Approximation for compounds with multiple sublattices |url=https://iopscience.iop.org/article/10.1088/1742-6596/286/1/012033 |journal=Journal of Physics: Conference Series |volume=286 |article-number=012033 |doi=10.1088/1742-6596/286/1/012033 |issn=1742-6596|doi-access=free }} ==References== == Further reading == *{{cite book |author=Ping Sheng |year=1995 |title=Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena |publisher=[Academic Press](/source/Academic_Press) |isbn=978-0-12-639845-8 }} *{{cite journal |author=[Fumiko Yonezawa](/source/Fumiko_Yonezawa) and Kazuo Morigaki |journal=[Progress of Theoretical Physics Supplement](/source/Progress_of_Theoretical_Physics_Supplement) |title=Coherent Potential Approximation: Basic Concepts and Applications |volume=53 |pages=1–76 |year=1973 |doi=10.1143/PTPS.53.1 |bibcode = 1973PThPS..53....1Y |doi-access=free }} *{{cite journal |author=John R. Klauder |title=The modification of electron energy levels by impurity atoms |journal=[Annals of Physics](/source/Annals_of_Physics) |volume=14 |pages=43–76 |year=1961 |doi=10.1016/0003-4916(61)90051-3 |bibcode = 1961AnPhy..14...43K }} Category:Wave mechanics --- Adapted from the Wikipedia article [Coherent potential approximation](https://en.wikipedia.org/wiki/Coherent_potential_approximation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Coherent_potential_approximation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.