By L. D. Landau, E. M. Lifshitz

Publish 12 months note: First released 1980 via Butterworth-Heinemann
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A lucid presentation of statistical physics and thermodynamics which develops from the overall ideas to offer quite a few purposes of the idea.

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The result's _ 3(3π2)1/3 *_ (M\m "~Ϊ6 m \Vj ' § fifty eight. the explicit warmth of a degenerate electron gasoline At temperatures that are low in comparison with the degeneracy temperature Tf9 the distribution functionality (57. four) has the shape proven through the damaged line in Fig. 6: it really is substantially various from cohesion or 0 simply in a slender variety of t The degeneracy temperature resembling the electron gasoline density (etmjfPfZ1 is 40Z4'3eV ^ zero. 5X 10*Z** levels. § fifty eight the categorical warmth of a Degenerate Electron fuel 169 values of the strength ε as regards to the proscribing strength ε^. The width of this "tran­ sition sector" of the Fermi distribution is of the order of T. The expressions (57. 6), (57. 7) are the 1st phrases within the expansions of the corresponding amounts in powers of the small ratio T/TF. allow us to now verify the following phrases within the expansions. formulation (56. 6) contains an essential of the shape f /(e)de J <λ—χ)ΐτ+ι zero where/(fi) is a functionality such that the crucial converges; in (56. 6),/(ε) = ε"1· We rework this fundamental through the substitution ε-μ = Tz: = f/(^ T ■J «+l -μΙΤ μ/Τ oo J e~*+] o J e'+l o in thefirstintegral we positioned l/(e~* + \) = 1 - \/(ez + \), acquiring μ zero μΙΤ oo zero zero within the moment of those integrals we exchange the higher restrict by means of infinity, on account that μ/Τ ^> 1 and the essential is speedily convergent. * this provides /=|/(,)d f + rf^ + r ^- : r z ) dz. We now extend the numerator of the second one integrand as a Taylor sequence of powers of z and combine time period via time period: /= f/(e)d |7oode+27y'00 e +27y'oof^o b0 Γ z*dz + j 7 T "'"M (μ I ^ " + Ί ^Η t This quantities to neglecting exponentially small phrases. It needs to be remembered that the growth (58. 1) derived lower than is an asymptotic, now not a convergent, sequence. a hundred and seventy The Fermi and Bose Distributions Substituting the values* of the integrals, we've got eventually 1 = |7(β) άε + ^Τψ(μ) + 1^ΤΤ"(μ)+ . . . . (58. 1) t Integrals of this sort are calculated as follows: oo C oo χ 1 ζ - άζ C _ _ _ = °° °° χχ-ie-* X o ( - ) » e - ' " d z = r U ) ^ (_)« + ! ± (l-2 l -*)7 1 (x) y —, J z * ld e*+l z (ΐ-2*-*)Λ*)ί(*) (jr > o), δ the place ζ(χ) = £ I/Λ* is the Riemann zeta functionality. F o r * = 1, the expression turns into indeterminate; the price of the quintessential is ί = "7ΓΤ *z+l log2 * For A: an excellent integer ( = 2n) the zeta functionality might be expressed when it comes to the Bernoulli numbers B„: i ζ^-^άζ 2 z e +\ 2 Λ -1-1 In „ π*»Βη five the subsequent integrals are calculated equally: J zx~l άζ e*-\ = Γ(χ)ζ(χ) (x> 1). o For or a fair integer ( = In), oo J z2"'1^ (2π)2ηΒη z e -1 4/i o For reference we will supply the values of the 1st few Bernoulli numbers and of a few zeta and gamma services: &l = o» &2 = 30» &3— forty two» &i = 35» C(|) = 2. 612, C(y) = 1. 341, CO) = 1. 202, C(5) = 1. 037, Γ ( | ) = | V^» A4) = TV*· § fifty nine Magnetism of an Electron fuel. vulnerable Fields 171 The 3rd time period within the enlargement is given for reference; it's going to now not wanted right here. placing/ --= ε3/2 in formulation (58. 1) and substituting in (56.

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