Statistical Physics

164 10 Dense Gases – Ideal Gases at Low Temperatureandg ′ (µ) = 3 N 1. (10.61)4 E 3/2 µ1/2FUsing these equations, we obtain N(T,V,µ) up to the second order in T :( ) 3/2 µN(T,V,µ) ≃ N + 1 N(πk B T ) 2. (10.62)E F 8 E 3/2Fµ1/2As we expect that µ/E F =1+O(T 2 ), µ in the second term on the right-handside can be replaced by E F .Thenµ/E F can be obtained as follows up to thesecond order in T/T F :[ ( ) ] 2 2/3 ( ) 2µ≃ 1 − π2 T≃ 1 − π2 T. (10.63)E F 8 T F 12 T FIn this equation, we have used the relation that E F = k B T F . This temperaturedependence and the correct temperature dependence of µ up to T =3T F areshown in Fig. 10.5.Fig. 10.5. Temperature dependence of µ/E F as a function of T/T F. The exact resultis shown by the solid line, andthedash-dotted line shows the approximation to thechemical potential (10.63)Internal Energy, Pressure, and Heat CapacityThe internal energy in the absence of a magnetic field isThus, in this caseU(T,V,µ)=∫ ∞0dE 2ED(E)f(E) . (10.64)g(E) =2ED(E) = 3 2 N ( EE F) 3/2, (10.65)G(µ) = 3 5 N µ5/2, (10.66)E 3/2F