Re: [問題] 固態物理
※ 引述《pisces227 (.....)》之銘言：
: (a) show that the chemical potential of a two dimensional “free electron gas”
: may be expressed analytically as
: E /K T
: F B
: μ(T)=k T ln(e^ －1).
: B
: x
: dx e
: Note:∫----- ＝ ln〔--------〕
: x x
: 1＋e 1＋e
∞ 1
E_F = ∫ ---------------------- dε
0 exp{(ε-μ)/K_B*T} + 1
Let x=exp{ε/(K_B*T)}
dε = K_B*T(dx/x) , 1 <= x < ∞
∞ exp{μ/K_B*T}
E_F= K_B*T ∫ --------------------(dx/x)
1 x+exp{μ/K_B*T}
∞ 1 1
=K_B*T * ∫ (--- - -----) dx where a=exp(μ/K_B*T)
1 x x+a
∞ 1 1 ∞
Because ∫ (--- - -----) dx = lnx - ln(x+a) | = ln(1+a)
1 x x+a 1
Thus E_F = K_B*T * ln(1+exp(μ/K_B*T))
=> μ(T)=K_B* T *ln(exp(E_F/(K_B*T)－1)
這比較像是一題微積分
: (b)Calculate the low temperature chemical potential of a two dimensional
: “free electron gas ”by Sommerfeld expansion method
: Observe that the temperature series expansion terminates.
: Compare this result with the exact result of the previous question.
: Discuss the difference between the two results.
: 要請各位高手幫忙解題了，謝謝！
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※ 編輯: mangogogo 來自: 219.84.238.191 (04/06 23:39)
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