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The Wilson ratio of a metal is the dimensionless ratio of the zero-temperature magnetic susceptibility and the coefficient of the linear temperature term in the electronic specific heat. The relative—value of the Wilson ratio, compared to the Wilson ratio for the non-interacting Fermi-gas, can provide insight into the types of interactions present.

Applications

Fermi liquid theory

The Wilson ratio can be used to characterize strongly correlated Fermi liquids.^[1] The Fermi liquid theory explains the behaviour of metals at very low temperatures. Two important features of a metal which obey this theory are:

At temperatures much below the Fermi temperature the specific heat is proportional to the temperature
The magnetic susceptibility is independent of temperature

Both of these quantities, however, are proportional to the electronic density of states at Fermi energy. Their ratio is a dimensionless quantity called the Wilson (or the Sommerfeld - Wilson) Ratio,^[2] defined as:

R_w = 4π²k_B²Tχ_p/3μ₀(g_eμ_B)²C_elec

After substituting the values of χ_p (Pauli susceptibility) and C_elec (electronic contribution to specific heat), obtained using Sommerfeld theory, the value obtained for R_w in the case of a free electron gas is 1.

In the case of real Fermi-liquid metals, the ratio can differ significantly from 1. The difference arises due to electron - electron interactions within the system. These tend to change the effective electronic mass, which affects both specific heat and magnetic susceptibility. Whether or not this increase in both is given by the same multiplicative factor is shown by the Wilson ratio. In some cases, electron - electron interactions give rise to an additional increase in susceptibility.

The converse is also true, i.e. a deviation of experimental value of R_w from 1 indicates strong electronic correlations.^[3]

References

^ Condensed Concepts (Blogspot) Long live Fermi liquid theory Retrieved March 2015
^ Condensed Concepts (Blogspot) Wilson's ratio for strongly correlated electrons Retrieved March 2015
^ Fundamentals of the Physics of Solids - Volume 2 by Jenö Sólyom

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[1] Condensed Concepts (Blogspot) Long live Fermi liquid theory Retrieved March 2015

[2] Condensed Concepts (Blogspot) Wilson's ratio for strongly correlated electrons Retrieved March 2015

[3] Fundamentals of the Physics of Solids - Volume 2 by Jenö Sólyom

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[2]

[3]

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+{{refimprove|date=March 2015}}
+{{unclear|date=March 2015}}
 The '''Wilson ratio''' of a metal is the dimensionless ratio of the zero-[[temperature]] [[magnetic susceptibility]] and the coefficient of the linear temperature term in the electronic [[specific heat]]. The relative—value of the Wilson ratio, compared to the Wilson ratio for the non-interacting Fermi-gas, can provide insight into the types of interactions present.
+== Applications ==
-To give a specific example, it can be used to characterise [[strongly correlated]] [[Fermi liquids]].<ref>http://condensedconcepts.blogspot.in/2011/04/long-live-fermi-liquid-theory.html</ref>
+=== Fermi liquid theory ===
+The Wilson ratio can be used to characterize [[Strongly correlated material|strongly correlated]] [[Fermi liquid]]s.<ref>Condensed Concepts (Blogspot) [http://condensedconcepts.blogspot.in/2011/04/long-live-fermi-liquid-theory.html ''Long live Fermi liquid theory] Retrieved March 2015</ref> The [[Fermi liquid theory]] explains the behaviour of metals at very low temperatures. Two important features of a metal which obey this theory are:
+# At temperatures much below the [[Fermi temperature]] the specific heat is proportional to the temperature
+# The [[magnetic susceptibility]] is independent of temperature
+Both of these quantities, however, are proportional to the electronic density of states at Fermi energy. Their ratio is a dimensionless quantity called the Wilson (or the Sommerfeld - Wilson) Ratio,<ref>Condensed Concepts (Blogspot) [http://condensedconcepts.blogspot.in/2012/12/wilsons-ratio-for-strongly-correlated.html Wilson's ratio for strongly correlated electrons] Retrieved March 2015</ref> defined as:
-The [[Fermi Liquid Theory]]  is a theory explaining the behaviour of metals at very low temperatures. Two important features of a metal which obeys this theory are:
+{{quote|text = R<sub>w</sub> = 4π<sup>2</sup>k<sub>B</sub><sup>2</sup>Tχ<sub>p</sub>/3μ<sub>0</sub>(g<sub>e</sub>μ<sub>B</sub>)<sup>2</sup>C<sub>elec</sub>}}
-. At temperatures much below the Fermi temperature the specific heat is proportional to the temperature, and
-. The magnetic susceptibility is independent of temperature.
+After substituting the values of χ<sub>p</sub> (Pauli susceptibility) and C<sub>elec</sub> (electronic contribution to specific heat), obtained using Sommerfeld theory, the value obtained for R<sub>w</sub> in the case of a [[free electron gas]] is 1.
-Both of these quantities, however, are proportional to the electronic density of states at Fermi energy. Their ratio is a dimensionless quantity called the Wilson (or the Sommerfeld - Wilson) Ratio.
-<ref>http://condensedconcepts.blogspot.in/2012/12/wilsons-ratio-for-strongly-correlated.html</ref>
+In the case of real Fermi-liquid metals, the ratio can differ significantly from 1. The difference arises due to electron - electron interactions within the system. These tend to change the effective electronic mass, which affects both [[specific heat]] and magnetic susceptibility. Whether or not this increase in both is given by the same multiplicative factor is shown by the Wilson ratio. In some cases, electron - electron interactions give rise to an additional increase in susceptibility.
-R<sub>w</sub> = 4π<sup>2</sup>k<sub>B</sub><sup>2</sup>Tχ<sub>p</sub>/3μ<sub>0</sub>(g<sub>e</sub>μ<sub>B</sub>)<sup>2</sup>C<sub>elec</sub>
+The converse is also true, i.e. a deviation of experimental value of R<sub>w</sub> from 1 indicates strong electronic correlations.<ref>Fundamentals of the Physics of Solids - Volume 2  by Jenö Sólyom</ref>
-After substituting the values of χ<sub>p</sub> (Pauli susceptibility) and C<sub>elec</sub> (electronic contribution to specific heat) obtained using Sommerfeld theory, the obtained value of R<sub>w</sub> for a free electron gas is 1.
-However, the Wilson ratio for real Fermi-liquid metals can differ substantially from 1. (In this regard,even cases like R<sub>w</sub> = 2 are already substantially different from unity).
-The difference from unity arises due to electron - electron interactions in the system. These tend to change the effective electronic mass, which affects both specific heat and magnetic susceptibility. Whether or not this increase in both is given by the same multiplicative factor is shown by the Wilson ratio. In some cases, electron - electron interactions give rise to an additional increase in susceptibility. Thus, in systems where electron - electron correlations are important, the Wilson ratio differs from 1 substantially.
-The converse is also true, i.e. a deviation of experimental value of R<sub>w</sub> from 1 indicates strong electronic correlations.
-<ref>Fundamentals of the Physics of Solids - Volume 2  by Jenö Sólyom</ref>
 ==See also==