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| {{Multiple issues|orphan =February 2009|refimprove =March 2007|context =October 2009}}
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| A '''Kohn anomaly''' is an anomaly in the dispersion relation of a [[phonon]] branch in a metal. For a specific [[wavevector]], the [[frequency]]—and thus the [[energy]]—of the associated phonon is considerably lowered, and there is a discontinuity in its [[derivative]]. They have been first proposed by [[Walter Kohn]] in 1959.<ref>W. Kohn, Image of the [[Fermi surface]] in the vibration spectrum of a metal, ''Phys. Rev. Lett'' '''2''', 393 (1959)</ref> In extreme cases (that can happen in low-dimensional materials), the energy of this phonon is zero, meaning that a static distortion of the lattice appears. This is one explanation for [[Spin density wave|charge density waves]] in solids. The wavevectors at which a Kohn anomaly is possible are the nesting vectors of the Fermi surface, that is vectors that connect a lot of points of the Fermi surface (for a one dimensional chain of atoms this vector would be <math>2k_F</math>).
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| In the phononic spectrum of a metal a Kohn anomaly is a discontinuity in the derivative of the dispersion relation that occurs at certain high symmetry points of the first [[Brillouin zone]], produced by the abrupt change in the screening of lattice vibrations by conduction electrons.
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| Kohn anomalies arise together with [[Friedel oscillations]] when one considers the [[Lindhard approximation]] instead of the [[Thomas-Fermi approximation]] in order to find an expression for the [[dielectric function]] of a homogeneous electron gas. The expression for the [[real part]] <math> \operatorname{Re}(\epsilon (\mathbf{q}, \omega)) </math> of the [[Reciprocal lattice#Reciprocal space|reciprocal space]] [[dielectric function]] obtained following the Lindhard model includes a logarithmic term that is singular at <math> \mathbf{q} = 2\mathbf{k}_F </math>, where <math> \mathbf{k}_F </math> is the [[Fermi energy#Related quantities|Fermi wavevector]]. Although this singularity is quite small in reciprocal space, if one takes the [[Fourier transform]] and passes into real space, the [[Gibbs phenomenon]] causes a strong oscillation of <math> \operatorname{Re}(\epsilon (\mathbf{r}, \omega)) </math> in the proximity of the singularity mentioned above. In the context of phonon [[dispersion relation]]s, these oscillations appear as a vertical [[tangent]] in the plot of <math> \omega ^2(\mathbf{q}) </math>, the so-called Kohn anomalies.
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| Many different systems exhibit Kohn anomalies, including [[graphene]],<ref name="kohngraphene_prl2004">S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Roberston, [http://dx.doi.org/10.1103/PhysRevLett.93.185503 Kohn Anomalies and Electron-Phonon Interactions in Graphite], ''Phys. Rev. Lett.'', '''93''', 185503 (2004)</ref> bulk metals,<ref name="pdkohn">D. A. Stewart, [http://dx.doi.org/10.1088/1367-2630/10/4/043025 Ab initio investigation of phonon dispersion and anomalies in palladium], ''New J. Phys.'', '''10''', 043025 (2008) ''Open Access article''</ref> and many [[low-dimensional systems]] (the reason involves the condition <math> \mathbf{q} = 2 \mathbf{k}_F </math>, which depends on the [[topology]] of the [[Fermi surface]]). However, it is important to emphasize that only materials showing [[metallic]] behaviour can exhibit a Kohn anomaly, as we are dealing with approximations that need a homogeneous electron gas.<ref>'''R. M. Martin''', ''Electronic Structure, Basic Theory and Practical Methods'', Cambridge University Press, 2004, ISBN 0-521-78285-6</ref>
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| ==Notes==
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| <references/>
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| For experimental results, one can turn to
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| [http://prl.aps.org/abstract/PRL/v30/i22/p1144_1 Observation of Giant Kohn Anomaly in the One-Dimensional Conductor K2Pt(CN)4Br0.3· 3H2O, Renker ''et al.'', ''Phys. Rev. Lett.'' 30, 1144]
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| {{DEFAULTSORT:Kohn Anomaly}}
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| [[Category:Condensed matter physics]]
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