THE QUANTUM PHYSICS OF SOLIDS 707 



minus moment and, since the net moment of Fig. 26a is obviously 

 zero, the net moment produced by the magnetic field is 



8M = 8M- - 8M+ = N{Ei)fxpm. (16) 



The susceptibility of a material, denoted by x> is defined as the mag- 

 netic moment produced per unit volume per unit field: 



The subscript "s" is a reminder that this susceptibility was produced 

 by the spin magnetic moment of the electron. 



Since the moment produced is in the direction of the field, Xs is 

 positive; the susceptibility is of the paramagnetic type. As for its 

 magnitude: in the monovalent metals, as we have said before, the 

 distribution of levels in the bands is well approximated by the free 

 electron formula (4). Using this, we find 



X. =^(2m)3/2(E^ax)^W. (18) 



where £max (= -Ei — -Eo) is the maximum kinetic energy in the band. 

 Before comparing susceptibilities calculated from this expression 

 with experimental values, we must discuss diamagnetism. The elec- 

 trons in the partially filled band of Fig. 26 give formula (18) because of 

 their spin magnetic moments. They give a susceptibility also because 

 of their motion through the crystal. For the case of free electrons, 

 this susceptibility is negative — that is, it is a diamagnetic susceptibility, 

 and, according to a theory we cannot discuss here, in magnitude it is 

 one third of Xs- Denoting it by Xm ("w" for motion of the electron 

 as a whole), we have 



Xm= - (1/3)X.. (19) 



The electrons in the filled bands, corresponding to electrons in closed 

 shells in the ionic cores of the metal, also give rise to diamagnetism. 

 They can give no spin paramagnetism because there is no possibility of 

 transferring electrons from a filled band of one spin to a filled band of 

 the other spin — this would require putting more electrons in the band 

 of one spin than it has quantum states, a violation of Pauli's principle. 

 Denoting by xi the susceptibility of the ionic cores of the metal, we 

 have for the net susceptibility x the equation 



X = Xs + Xm + Xi- (20) 



Specializing this for the case of free electrons in the valence electron 



