Statistical Physics

156 10 Dense Gases – Ideal Gases at Low TemperatureVarious thermodynamic variables can be written as a function of T , V , µ,and B in an integral form in terms of this single-particle density of states.We first note that in a magnetic field, a fermion with energy E i has a kineticenergy E K = E i − gµ B sB. Therefore, the density of single-particle states atE i is D(E i − gµ B sB). We also note that the minimum value of E i is gµ B sB.Thus, the total number of particles N, the average energy in the magneticfield Ẽ, the total spin magnetic moment M, and the internal energy U aregiven byandN(T,V,µ,B)= ∑ i= ∑Ẽ(T,V,µ,B)= ∑ i∑s=±1/2∫ ∞s=±1/2= ∑M(T,V,µ,B)= ∑ i∑1e β(Ei−µ) +1gµ BsBs=±1/2∫ ∞s=±1/2= ∑∑1dED(E − gµ B sB)e β(E−µ) +1 , (10.30)E ie β(Ei−µ) +1gµ BsBs=±1/2∫ ∞s=±1/2EdED(E − gµ B sB)e β(E−µ) +1 , (10.31)−gµ B se β(Ei−µ) +1gµ BsB−gµ B sdED(E − gµ B sB)e β(E−µ) +1 , (10.32)U(T,V,µ,M)=Ẽ(T,V,µ,B)+MB= ∑ ∫ ∞dED(E − gµ B sB) E − gµ Bse β(E−µ) +1 . (10.33)s=±1/2gµ BsBFinally, the pressure can be obtained from J, but it can also be written interms of the internal energy U:P (T,V,µ,B)V = −J = k B T ln Ξ(T,V,µ,B)= k B T ∑ ∑lni= k B T ∑s=±1/2s=±1/2∫ ∞gµ BsB[1+e −β(Ei−µ)]dED(E − gµ B sB)ln[1+e −β(E−µ)]