CHAPTER 6 h/ 2 2 1. The energy eigenvalues are k = k . The mean value over the volume of a sphere in k space is 2m <> = h/ 2 k 2 dk k 2 3 h/ 2 2 3 = k F = F . 2m k 2 dk 5 2m 5 The total energy of N electrons is 3 U0 = N F . 5 2a. In general p = -U/V at constant entropy. At absolute zero all processes are at constant entropy (the 23 3 h/ 2 32 N = N , whence 5 2m V 3 Third Law), so that p = dU 0 dV, where U 0 = N F 5 p= 2 U0 . (b) Bulk modulus 3 V 2 dp 2 dU 0 2 U 0 2 U 0 10 U 0 2 U0 . B = V = V + + = = 2 dV 9 V 3 V 3V dV 3 V 3 V (c) For Li, U0 3 = (4.7 1022 cm 3 ) (4.7 eV) (1.6 1012 erg eV) V 5 = 2.1 1011 erg cm 3 = 2.1 1011 dyne cm 2 , whence B = 2.3 1011 dyne cm-2. By experiment (Table 3.3), B = 1.2 1011 dyne cm-2. 3. The number of electrons is, per unit volume, n = 0 d D() 1 e ( ) +1 , where D() is the density of orbitals. In two dimensions m 1 d () 2 0 h/ +1 e m = 2 ( + log (1 + e )), h/ n= where the definite integral is evaluated with the help of Dwight [569.1]. 2 1033 \u0011 1057 nucleons, and roughly an equal number of electrons. In a 4a. In the sun there are 24 1.7 10 white dwarf star of volume 6-1