Sommerfeld expansion

A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution. When the inverse temperature β {displaystyle eta } is a large quantity, the integral can be expanded in terms of β {displaystyle eta } as where H ′ ( μ ) {displaystyle H^{prime }(mu )} is used to denote the derivative of H ( ε ) {displaystyle H(varepsilon )} evaluated at ε = μ {displaystyle varepsilon =mu } and where the O ( x n ) {displaystyle O(x^{n})} notation refers to limiting behavior of order x n {displaystyle x^{n}} . The expansion is only valid if H ( ε ) {displaystyle H(varepsilon )} vanishes as ε → − ∞ {displaystyle varepsilon ightarrow -infty } and goes no faster than polynomially in ε {displaystyle varepsilon } as ε → ∞ {displaystyle varepsilon ightarrow infty } .If the integral is from zero to infinity, then the integral in the first term of the expansion is from zero to μ {displaystyle mu } and the second term is unchanged. Integrals of this type appear frequently when calculating electronic properties, like the heat capacity, in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity H ( ε ) {displaystyle H(varepsilon )} . For these integrals we can then identify β {displaystyle eta } as the inverse temperature and μ {displaystyle mu } as the chemical potential. Therefore, the Sommerfeld expansion is valid for large β {displaystyle eta } (low temperature) systems. We seek an expansion that is second order in temperature, i.e., to τ 2 {displaystyle au ^{2}} , where β − 1 = τ = k B T {displaystyle eta ^{-1}= au =k_{B}T} is the product of temperature and Boltzmann's constant. Begin with a change variables to τ x = ε − μ {displaystyle au x=varepsilon -mu } : Divide the range of integration, I = I 1 + I 2 {displaystyle I=I_{1}+I_{2}} , and rewrite I 1 {displaystyle I_{1}} using the change of variables x → − x {displaystyle x ightarrow -x} : Next, employ an algebraic 'trick' on the denominator of I 1 {displaystyle I_{1}} ,