Question

(a) The simplest quantum mechanical model for describing electrical conduction in a metal is the free electron gas in three dimensions. The density of states D(E) is given as: V 2m D(E)- 277 An estimate of the average electron energy can be obtained using the following expression: SEN(E)dE (E) - TH(E)DE where n(E)dE is the number of occupied electron states in an energy interval E to E + dE. Use a suitable expression for n(E) and introduce (and fully rationalise) suitable limits of integration to show that the average electron energy in a metal at T = 0 K is (E) = 0.6E,. Also, briefly comment on why (E) > 0.5E . [5] b) Determine (E) numerically at T = 0 K for Zinc, which has a relative atomic mass of 65.4, a mass density of p = 7.13x10 kgm-, and is assumed to contribute two free electrons per atom for conduction. Give your answer in units of eV. Fundamentally, what is responsible for the fact that the average energy of the quantum free-electron gas is significantly larger than classical predictions?
a) and b)

Answer 1

2 / E82 1.086 X10-25 Fermi Particle Since doctoron is a f (E) TE-EF) (Fermi Dirac Distribution Function) 1+ KT n(E) - D (6) 8 (E) 2(E) 22 (1) E' a 1 1 + expl E-EE KT At T- O f (E) = 1 1 = 1 1+ o 1 + et LED (since EFDE EF EF <E> = SERIE de 2T2 Inte) de 193*** X 12 ( 2 ) 2 E 2. a. de (2) 2 Eh a do <E> EES 3/2 dE E 2/3 EF { F مالم 3 EF = 0.6EF = EFS E²2 de -27 bg 6 m = 3 Ele deron density X = u tom 65. 4 a = 65.4 x 1.66 x 15 kg = S 7.13 x 10 kg m 3 - 6.56 x 1028 m 3 V 1.086 x 1025 5.5.x norge Since EFF +2 372N) 23 2m (1.05 x 10 34) 2 (377 6. 2 x 9.1 x 10-31 23 1.942 x 1030 Es= 1.1 x 10-68 1.82 x 10-30 -39 EF = 6.044 x 10 X 2o 1.56 x 10 9.428x 10-19 J 5- 9 ev