THE QUANTUM PHYSICS OF SOLIDS 705 



instead an electrostatic exchange energy like that discussed for atoms 

 in connection with Figs. 4 and 6. 



Paramagnetism and Diamagnetism 

 Let us consider first the so-called "weak spin paramagnetism." 

 This occurs in metals, since they have partially filled bands. In the 

 presence of a magnetic field the spin of the electron is quantized so that 

 the component of its angular momentum in the field direction is either 

 -\-}^h or —}/2^i where h = h/lw {h = Planck's constant) is the quan- 

 tum mechanical unit of angular momentum. The corresponding com- 

 ponents of magnetic moment along the field are — M/s and -f At/s where 

 fi^ is the quantum mechanical unit of magnetic moment known as the 

 Bohr magneton. Letting — e be the charge and m the mass of the 

 electron and c be the speed of light, we have from the quantum theory 



M/3 = ehjlmc. (12) 



The ratio of mechanical moment (i.e. angular momentum) to magnetic 

 moment, taken without regard to sign, is called the "gyro-magnetic 

 ratio." For the spin of the electron its value is mc/e, but for the motion 

 of the electron as a whole, its value is Imcje. Because of the difference 

 between these two values, experimental measurements of the gyro- 

 magnetic effect play a decisive role in the experimental verification of 

 the electron spin theory of ferromagnetism in a way which we shall 

 describe below. 



Half the quantum states in an energy band of a crystal have angular 

 momentum components along the magnetic field of J^i/ and the other 

 half of —]/2^i. In Fig. 26a, we have divided the states in the band into 

 two groups, corresponding to the two spins. We shall refer to one of 

 these as "the band with plus spin" and to the other as "the band with 

 minus spin." When a magnetic field is applied, the energies of the 

 electrons are changed. Thus if an electron in the lowest state of the 

 band with minus spin has an energy £o before the field is applied, it 

 has an energy of Eq — n^H afterwards; the second term represents, 

 of course, the energy of the magnetic dipole jji^ when parallel, as dis- 

 tinguished from anti-parallel, to the field — the situation for minus 

 spin. All the states in the band with minus spin will be thus altered in 

 energy. Similarly all the states in the band with plus spin are dis- 

 placed upwards in energy by jjl^H. This is the situation represented in 

 Fig. 26h. After the displacement we find that some of the electrons 

 in the band with plus spin have higher energies than empty states in 

 the band with minus spin ; such an arrangement is not stable and the 

 electrons will change their quantum states so as to produce the lowest 

 energy possible consistent with the distribution of energy levels shown 



