THE QUANTUM PHYSICS OF SOLIDS 711 



configuration of the atom, Fig. 6; we see there that there are unequal 

 numbers of electrons of the two spins. This inequality is produced by 

 the exchange effect which lowers the more occupied set of 3J levels in 

 respect to the less occupied set and produces a stable arrangement with 

 the ?)d levels of one set completely filled. This exchange effect oper- 

 ates in the same way in metallic nickel. In Fig. 28& we show the 

 distribution which results when electrons are shifted from the Zd band 

 of plus spin to that of minus spin until the latter is filled. The ex- 

 change effect produces the displacements of the bands as shown. The 

 arrangement in Fig. 2%h is stable; in order for electrons to be trans- 

 ferred from the filled minus 3d band to the plus band, they would 

 have to increase their energy, a fact which is expressed by drawing 

 the diagram so that the lowest vacant quantum states are appreciably 

 above the highest energy state of the filled d>d band. Thus for nickel 

 an unbalanced distribution of spins prevails both for the free atom 

 and the metal. 



The Energy of Magnetization 



The argument presented above for the stability of the magnetized 

 state shown in Fig. 286 is not really rigorous. We saw that if one 

 electron was transferred from the filled 3J band to one of the vacant 

 states, its energy and, therefore, the energy of the crystal would be 

 raised. In other words, the magnetized state has less energy than a 

 state which is slightly less magnetized. This fact in itself does not 

 prove that the magnetized state is stable; it proves only that it is 

 metastable — i.e., that its energy is less than the energy of other states 

 which differ from it slightly; in order to establish the stability of the 

 magnetized state, it is necessary to prove that its energy is less than 

 the energy of any other state including that of the unmagnetized state 

 shown in Fig. 28a. We may illustrate this necessity by considering 

 the following hypothetical behavior: as the magnetization is reduced 

 from that of Fig. 28Z> to zero (the value for Fig. 28a), the energy might 

 at first increase and then decrease — decreasing so much finally that 

 the energy would be lower for the unmagnetized than for the fully 

 magnetized state. We shall, therefore, discuss the difference in energy 

 between the fully magnetized and unmagnetized states; theory shows 

 that this quantity is the fundamental one whose value determines 

 whether or not ferromagnetism occurs. 



Let us consider the change in energy in going from the unmagnetized 

 state to the magnetized state in Fig. 28. This change in energy can be 

 separated into two contributions, one positive and one negative. The 

 positive contribution comes from an increase in "Fermi energy" or 



