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 beta }beta is a large quantity, the integral can be expanded[1][2] in terms of β{displaystyle beta }beta as


∫−∞∞H(ε)eβ(ε−μ)+1dε=∫−∞μH(ε)dε+π26(1β)2H′(μ)+O(1βμ)4{displaystyle int _{-infty }^{infty }{frac {H(varepsilon )}{e^{beta (varepsilon -mu )}+1}},mathrm {d} varepsilon =int _{-infty }^{mu }H(varepsilon ),mathrm {d} varepsilon +{frac {pi ^{2}}{6}}left({frac {1}{beta }}right)^{2}H^{prime }(mu )+Oleft({frac {1}{beta mu }}right)^{4}}int _{{-infty }}^{infty }{frac {H(varepsilon )}{e^{{beta (varepsilon -mu )}}+1}},{mathrm {d}}varepsilon =int _{{-infty }}^{mu }H(varepsilon ),{mathrm {d}}varepsilon +{frac {pi ^{2}}{6}}left({frac {1}{beta }}right)^{2}H^{prime }(mu )+Oleft({frac {1}{beta mu }}right)^{4}

where H′(μ){displaystyle H^{prime }(mu )}H^{prime }(mu ) is used to denote the derivative of H(ε){displaystyle H(varepsilon )}H(varepsilon ) evaluated at ε=μ{displaystyle varepsilon =mu }varepsilon =mu and where the O(xn){displaystyle O(x^{n})}O(x^{n}) notation refers to limiting behavior of order xn{displaystyle x^{n}}x^{n}. The expansion is only valid if H(ε){displaystyle H(varepsilon )}H(varepsilon ) vanishes as ε→−∞{displaystyle varepsilon rightarrow -infty }varepsilon rightarrow -infty and goes no faster than polynomially in ε{displaystyle varepsilon }varepsilon as ε→∞{displaystyle varepsilon rightarrow infty }varepsilon rightarrow 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 }mu and the second term is unchanged.




Contents






  • 1 Application to the free electron model


  • 2 Derivation to second order in temperature


  • 3 Higher order terms and a generating function


  • 4 Notes


  • 5 References





Application to the free electron model


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 )}H(varepsilon ). For these integrals we can then identify β{displaystyle beta }beta as the inverse temperature and μ{displaystyle mu }mu as the chemical potential. Therefore, the Sommerfeld expansion is valid for large β{displaystyle beta }beta (low temperature) systems.



Derivation to second order in temperature


We seek an expansion that is second order in temperature, i.e., to τ2{displaystyle tau ^{2}}tau^2, where β−1=τ=kBT{displaystyle beta ^{-1}=tau =k_{B}T}beta ^{{-1}}=tau =k_{B}T is the product of temperature and Boltzmann's constant. Begin with a change variables to τx=ε−μ{displaystyle tau x=varepsilon -mu }tau x=varepsilon -mu :


I=∫−∞∞H(ε)eβ(ε−μ)+1dε=τ∫−∞∞H(μ+τx)ex+1dx,{displaystyle I=int _{-infty }^{infty }{frac {H(varepsilon )}{e^{beta (varepsilon -mu )}+1}},mathrm {d} varepsilon =tau int _{-infty }^{infty }{frac {H(mu +tau x)}{e^{x}+1}},mathrm {d} x,,}I=int _{{-infty }}^{infty }{frac {H(varepsilon )}{e^{{beta (varepsilon -mu )}}+1}},{mathrm {d}}varepsilon =tau int _{{-infty }}^{infty }{frac {H(mu +tau x)}{e^{{x}}+1}},{mathrm {d}}x,,

Divide the range of integration, I=I1+I2{displaystyle I=I_{1}+I_{2}}I=I_{1}+I_{2}, and rewrite I1{displaystyle I_{1}}I_{1} using the change of variables x→−x{displaystyle xrightarrow -x}xrightarrow -x:


I=τ∫−∞0H(μ+τx)ex+1dx⏟I1+τ∫0∞H(μ+τx)ex+1dx⏟I2.{displaystyle I=underbrace {tau int _{-infty }^{0}{frac {H(mu +tau x)}{e^{x}+1}},mathrm {d} x} _{I_{1}}+underbrace {tau int _{0}^{infty }{frac {H(mu +tau x)}{e^{x}+1}},mathrm {d} x} _{I_{2}},.}I=underbrace {tau int _{{-infty }}^{0}{frac {H(mu +tau x)}{e^{{x}}+1}},{mathrm {d}}x}_{{I_{1}}}+underbrace {tau int _{{0}}^{infty }{frac {H(mu +tau x)}{e^{{x}}+1}},{mathrm {d}}x}_{{I_{2}}},.



I1=τ∫−∞0H(μ+τx)ex+1dx=τ∫0∞H(μ−τx)e−x+1dx{displaystyle I_{1}=tau int _{-infty }^{0}{frac {H(mu +tau x)}{e^{x}+1}},mathrm {d} x=tau int _{0}^{infty }{frac {H(mu -tau x)}{e^{-x}+1}},mathrm {d} x,}I_{1}=tau int _{{-infty }}^{0}{frac {H(mu +tau x)}{e^{{x}}+1}},{mathrm {d}}x=tau int _{0}^{infty }{frac {H(mu -tau x)}{e^{{-x}}+1}},{mathrm {d}}x,

Next, employ an algebraic 'trick' on the denominator of I1{displaystyle I_{1}}I_{1},


1e−x+1=1−1ex+1,{displaystyle {frac {1}{e^{-x}+1}}=1-{frac {1}{e^{x}+1}},,}{frac {1}{e^{{-x}}+1}}=1-{frac {1}{e^{x}+1}},,

to obtain:


I1=τ∫0∞H(μ−τx)dx−τ∫0∞H(μ−τx)ex+1dx{displaystyle I_{1}=tau int _{0}^{infty }H(mu -tau x),mathrm {d} x-tau int _{0}^{infty }{frac {H(mu -tau x)}{e^{x}+1}},mathrm {d} x,}I_{1}=tau int _{{0}}^{infty }H(mu -tau x),{mathrm {d}}x-tau int _{0}^{{infty }}{frac {H(mu -tau x)}{e^{{x}}+1}},{mathrm {d}}x,

Return to the original variables with −τdx=dε{displaystyle -tau mathrm {d} x=mathrm {d} varepsilon }-tau {mathrm {d}}x={mathrm {d}}varepsilon in the first term of I1{displaystyle I_{1}}I_{1}. Combine I=I1+I2{displaystyle I=I_{1}+I_{2}}I=I_{1}+I_{2} to obtain:


I=∫−∞μH(ε)dε+τ∫0∞H(μ+τx)−H(μ−τx)ex+1dx{displaystyle I=int _{-infty }^{mu }H(varepsilon ),mathrm {d} varepsilon +tau int _{0}^{infty }{frac {H(mu +tau x)-H(mu -tau x)}{e^{x}+1}},mathrm {d} x,}I=int _{{-infty }}^{mu }H(varepsilon ),{mathrm {d}}varepsilon +tau int _{0}^{{infty }}{frac {H(mu +tau x)-H(mu -tau x)}{e^{{x}}+1}},{mathrm {d}}x,

The numerator in the second term can be expressed as an approximation to the first derivative, provided τ{displaystyle tau }tau is sufficiently small and H(ε){displaystyle H(varepsilon )}H(varepsilon ) is sufficiently smooth:


ΔH=H(μ+τx)−H(μ−τx)≈2τxH′(μ)+⋯,{displaystyle Delta H=H(mu +tau x)-H(mu -tau x)approx 2tau xH'(mu )+cdots ,,}Delta H=H(mu +tau x)-H(mu -tau x)approx 2tau xH'(mu )+cdots ,,

to obtain,


I=∫−∞μH(ε)dε+2τ2H′(μ)∫0∞xdxex+1{displaystyle I=int _{-infty }^{mu }H(varepsilon ),mathrm {d} varepsilon +2tau ^{2}H'(mu )int _{0}^{infty }{frac {xmathrm {d} x}{e^{x}+1}},}I=int _{{-infty }}^{mu }H(varepsilon ),{mathrm {d}}varepsilon +2tau ^{2}H'(mu )int _{0}^{{infty }}{frac {x{mathrm {d}}x}{e^{{x}}+1}},

The definite integral is known[3] to be:



∫0∞xdxex+1=π212{displaystyle int _{0}^{infty }{frac {xmathrm {d} x}{e^{x}+1}}={frac {pi ^{2}}{12}}}int _{0}^{{infty }}{frac {x{mathrm {d}}x}{e^{{x}}+1}}={frac {pi ^{2}}{12}}.

Hence,


I=∫−∞∞H(ε)eβ(ε−μ)+1dε≈∫−∞μH(ε)dε+π26β2H′(μ){displaystyle I=int _{-infty }^{infty }{frac {H(varepsilon )}{e^{beta (varepsilon -mu )}+1}},mathrm {d} varepsilon approx int _{-infty }^{mu }H(varepsilon ),mathrm {d} varepsilon +{frac {pi ^{2}}{6beta ^{2}}}H'(mu ),}I=int _{{-infty }}^{infty }{frac {H(varepsilon )}{e^{{beta (varepsilon -mu )}}+1}},{mathrm {d}}varepsilon approx int _{{-infty }}^{mu }H(varepsilon ),{mathrm {d}}varepsilon +{frac {pi ^{2}}{6beta ^{2}}}H'(mu ),


Higher order terms and a generating function


We can obtain higher order terms in the Sommerfeld expanion by use of a
generating function for moments of the Fermi distribution. This is given by


∫−∞∞dϵ2πeτϵ/2π{11+eβ(ϵ−μ)−θ(−ϵ)}=1τ{(τT2)sin⁡(τT2)eτμ/2π−1},0<τT/2π<1.{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}e^{tau epsilon /2pi }left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{tau }}left{{frac {({frac {tau T}{2}})}{sin({frac {tau T}{2}})}}e^{tau mu /2pi }-1right},quad 0<tau T/2pi <1.}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}e^{tau epsilon /2pi }left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{tau }}left{{frac {({frac {tau T}{2}})}{sin({frac {tau T}{2}})}}e^{tau mu /2pi }-1right},quad 0<tau T/2pi <1.}

Here kBT=β−1{displaystyle k_{rm {B}}T=beta ^{-1}}{displaystyle k_{rm {B}}T=beta ^{-1}} and Heaviside step function −θ(−ϵ){displaystyle -theta (-epsilon )}{displaystyle -theta (-epsilon )} subtracts the divergent zero-temperature contribution.
Expanding in powers of τ{displaystyle tau }tau gives, for example [4]



∫−∞∞dϵ2π{11+eβ(ϵ−μ)−θ(−ϵ)}=(μ2π),{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}=left({frac {mu }{2pi }}right),}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}=left({frac {mu }{2pi }}right),}

∫−∞∞dϵ2π(ϵ2π){11+eβ(ϵ−μ)−θ(−ϵ)}=12!(μ2π)2+T24!,{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}left({frac {epsilon }{2pi }}right)left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}+{frac {T^{2}}{4!}},}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}left({frac {epsilon }{2pi }}right)left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}+{frac {T^{2}}{4!}},}

∫−∞∞dϵ2π12!(ϵ2π)2{11+eβ(ϵ−μ)−θ(−ϵ)}=13!(μ2π)3+(μ2π)T24!,{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{2!}}left({frac {epsilon }{2pi }}right)^{2}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{3!}}left({frac {mu }{2pi }}right)^{3}+left({frac {mu }{2pi }}right){frac {T^{2}}{4!}},}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{2!}}left({frac {epsilon }{2pi }}right)^{2}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{3!}}left({frac {mu }{2pi }}right)^{3}+left({frac {mu }{2pi }}right){frac {T^{2}}{4!}},}

∫−∞∞dϵ2π13!(ϵ2π)3{11+eβ(ϵ−μ)−θ(−ϵ)}=14!(μ2π)4+12!(μ2π)2T24!+78T46!,{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{3!}}left({frac {epsilon }{2pi }}right)^{3}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{4!}}left({frac {mu }{2pi }}right)^{4}+{frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}{frac {T^{2}}{4!}}+{frac {7}{8}}{frac {T^{4}}{6!}},}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{3!}}left({frac {epsilon }{2pi }}right)^{3}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{4!}}left({frac {mu }{2pi }}right)^{4}+{frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}{frac {T^{2}}{4!}}+{frac {7}{8}}{frac {T^{4}}{6!}},}

∫−∞∞dϵ2π14!(ϵ2π)4{11+eβ(ϵ−μ)−θ(−ϵ)}=15!(μ2π)5+13!(μ2π)3T24!+(μ2π)78T46!,{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{4!}}left({frac {epsilon }{2pi }}right)^{4}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{5!}}left({frac {mu }{2pi }}right)^{5}+{frac {1}{3!}}left({frac {mu }{2pi }}right)^{3}{frac {T^{2}}{4!}}+left({frac {mu }{2pi }}right){frac {7}{8}}{frac {T^{4}}{6!}},}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{4!}}left({frac {epsilon }{2pi }}right)^{4}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{5!}}left({frac {mu }{2pi }}right)^{5}+{frac {1}{3!}}left({frac {mu }{2pi }}right)^{3}{frac {T^{2}}{4!}}+left({frac {mu }{2pi }}right){frac {7}{8}}{frac {T^{4}}{6!}},}

∫−∞∞dϵ2π15!(ϵ2π)5{11+eβ(ϵ−μ)−θ(−ϵ)}=16!(μ2π)6+14!(μ2π)4T24!+12!(μ2π)278T46!+3124T68!.{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{5!}}left({frac {epsilon }{2pi }}right)^{5}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{6!}}left({frac {mu }{2pi }}right)^{6}+{frac {1}{4!}}left({frac {mu }{2pi }}right)^{4}{frac {T^{2}}{4!}}+{frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}{frac {7}{8}}{frac {T^{4}}{6!}}+{frac {31}{24}}{frac {T^{6}}{8!}}.}{displaystyle int _{-infty }^{infty }{frac {depsilon }{2pi }}{frac {1}{5!}}left({frac {epsilon }{2pi }}right)^{5}left{{frac {1}{1+e^{beta (epsilon -mu )}}}-theta (-epsilon )right}={frac {1}{6!}}left({frac {mu }{2pi }}right)^{6}+{frac {1}{4!}}left({frac {mu }{2pi }}right)^{4}{frac {T^{2}}{4!}}+{frac {1}{2!}}left({frac {mu }{2pi }}right)^{2}{frac {7}{8}}{frac {T^{4}}{6!}}+{frac {31}{24}}{frac {T^{6}}{8!}}.}


A similar generating function for the odd moments of the Bose function is
∫0∞dϵ2πsinh⁡(ϵτ/π)1eβϵ−1=14τ{1−τTtan⁡τT},0<τT<π{displaystyle int _{0}^{infty }{frac {depsilon }{2pi }}sinh(epsilon tau /pi ){frac {1}{e^{beta epsilon }-1}}={frac {1}{4tau }}left{1-{frac {tau T}{tan tau T}}right},quad 0<tau T<pi }{displaystyle int _{0}^{infty }{frac {depsilon }{2pi }}sinh(epsilon tau /pi ){frac {1}{e^{beta epsilon }-1}}={frac {1}{4tau }}left{1-{frac {tau T}{tan tau T}}right},quad 0<tau T<pi }



Notes




  1. ^ Ashcroft & Mermin 1976, p. 760.


  2. ^ Fabian, J. "Sommerfeld's expansion" (PDF). Universitaet Regensburg. Retrieved 2016-02-08..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  3. ^ "Definite integrals containing exponential functions". SOS Math. Retrieved 2016-02-08.


  4. ^ R. Loganayagam, P. Surówka (2012). "Anomaly/Transport in an Ideal Weyl gas". JHEP. 04 (4): 2012:97. arXiv:1201.2812. Bibcode:2012JHEP...04..097L. CiteSeerX 10.1.1.761.5605. doi:10.1007/JHEP04(2012)097.



References




  • Sommerfeld, A. (1928). "Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik". Zeitschrift für Physik. 47: 1–3. Bibcode:1928ZPhy...47....1S. doi:10.1007/BF01391052.


  • Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics. Thomson Learning. p. 760. ISBN 978-0-03-083993-1.




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