Solutions to Problem 20 in Phys 132: Fermi Energy of a Degenerate Ideal Gas of Electrons, Assignments of Physics

Download Solutions to Problem 20 in Phys 132: Fermi Energy of a Degenerate Ideal Gas of Electrons and more Assignments Physics in PDF only on Docsity! Phys 132 Homework 20 Solutions Professor Crystal Martin TA: Ellie Hadjiyska Problem 20: (Problem 2.1) We are asked to consider an ideal gas of degenerate, non- relativistic electrons with concentration n. (a) We first want to find an expression for the Fermi energy F (the maximum energy of any electron in gas). We can find the Fermi momentum pF (maximum momentum of any electron in gas) by summing over all momentum states within the gas up to this maximum momentum pF (Phillips Eq. (2.26)): N = ∫ pF 0 gs V h3 4πp2 dp = 8πV 3h3 p3F where gs = 2 for the two independent spin states of an electron. This implies that the Fermi momentum is (Phillips Eq. (2.27)) pF = [ 3n 8π ]1/3 h The Fermi energy is therefore F = p2F 2me = h2 2me [ 3n 8π ] 2/3 = ~ 2 2me (3π2n)2/3 where ~ = h/(2π). We now assume that there is a temperature T associated with the gas and the quantum concentration nQ (Phillips Eq. (2.22)) equals the actual concentration n or n = nQ = [ 2πmekT h2 ] 3/2 Combining these expressions gives F = ~ 2 2me [ 3π2 ( 2πmekT h2 ) 3/2 ] 2/3 = (3π2)2/3 4π kT = ( 9π 64 ) 1/3 kT So the ratio of kT to the Fermi energy is kT F = ( 64 9π ) 1/3 ≈ 1.313