Statistical Mechanics - Physics at Oregon State University

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106 CHAPTER 5. FERMIONS AND BOSONS Coulomb attraction between the electrons and the ion cores in a solid. A similar large pressure occurs in white dwarf stars. In that case gravity is the force that keeps the particles together. Large temperatures. The other limit of interest is T → ∞. In this case we expect, of course, to find the results for an ideal gas. Like we discussed before, since n = (2S + 1)nQ(T )f 3 (λ) the function f 3 (λ) has to approach zero and hence λ → 0. There- 2 2 fore this function is dominated by the first term in the power series and we find N(T, µ, V ) ≈ (2S + 1)V nQ(T )λ (5.92) Apart from the factor 2S + 1 this is the result we had before. This last factor is a result of the additional degeneracy we introduced by including spin and reduces the chemical potential. In this high temperature limit we find Ω(T, µ, V ) ≈ −(2S + 1)V kBT nQ(T )λ (5.93) Together with the formula for N this yields: and using p = − Ω V : Ω(T, µ, V ) ≈ −NkBT (5.94) p = NkBT (5.95) V Hence the ideal gas law is not influenced by the extra factor 2S+1. The pressure does not change due to the spin degeneracy, unlike the chemical potential, which is now equal to (found by inverting the equation for N): 5.2 Bosons in a box. Integral form. n µ(T, V, N) = kBT log( ) (5.96) (2S + 1)nQ(T ) The discussion in the previous chapter for bosons was again general for all types of bosons, and in order to derive some analytical results we have again to choose a specific model for the orbital energies. Of course we will again take free particles in a box, using ɛo = 2 2M k2 with k = π L (nx, ny, nz). This leads to Ω(T, µ, V ) = −(2S + 1)kBT nx,ny,nz log(Znx,ny,nz(T, µ, V )) (5.97)
5.2. BOSONS IN A BOX. 107 with Znx,ny,nz(T, µ, V ) = 1 − e −1 µ−ɛ(nx,ny ,nz ) kB T (5.98) The same trick applies here as we used for fermions, and we replace the series by an integral: and ˜Ω(T, 1 µ, V ) = V kBT (2S + 1) (2π) 3 d 3 µ− k log(1 − e 2k 2 2M kB T ) (5.99) Ω(T, µ, V ) = ˜ Ω(T, µ, V ) + error(T, µ, V ) (5.100) where the error is defined by making this equation exact. One can again show that also for bosons we have limV →∞ error(T, µ, V ) = 0, but the big difference with a system of fermions is that it is not possible to give an upper-bound to the error which is only a function of V. The value of V we need in order to get for example a 1% error becomes infinitely large when T → 0. Note that µ < 0 for free particles, and hence 0 < λ < 1. Problems also occur in the limit µ → 0, where the integrant diverges at the lower limit of integration. These can be dealt with. The integral is of the form: 1 d 3 k log( 1 − λe − 2 k 2 2Mk B T ) (5.101) and when k is small the argument of the logarithm is approximately log( 2MkBT 2k2 ). Therefore, near the origin we need to evaluate d3k log(k), which behaves well even though the logarithm diverges, since k2 log(k) goes to zero. Hence the function ˜ Ω is well defined for 0 < λ 1, or for µ 0, and there are no problems in this integral with the limit µ → 0. As we will see, this limit corresponds to the limit T → 0. Special functions for bosons. In a manner similar to what we found for fermions, we define a set of special functions by gα(z) = ∞ n=1 zn , |z| < 1 (5.102) nα and use analytic continuation to define these functions everywhere. The formula for ˜ Ω is now manipulated in the same way we did for fermions, and we find ˜Ω(T, µ, V ) = −(2S + 1)V kBT nQ(T )g 5 (λ) (5.103) 2
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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192 CHAPTER 9. GENERAL METHODS: CRI
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230 APPENDIX A. SOLUTIONS TO SELECT
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