712 BELL SYSTEM TECHNICAL JOURNAL 



"energy of motion," which was discussed in connection with the bind- 

 ing energy of metals. This energy is positive because after the shift 

 to the magnetized state, electrons have moved from states in the band 

 of plus spin to states which lie higher — in respect to the bottom of the 

 bands in both cases — in the band of the minus spin ; that is, the elec- 

 trons which have moved from one band to the other have all gained 

 "energy of motion." The negative contribution to the energy comes 

 from the exchange effect. This causes the lowering of the filled band 

 and the raising of the unfilled band; since there are more electrons in 

 the lowered band than in the raised band, there is a net decrease in 

 energy due to this exchange effect. Thus we have a positive change in 

 Fermi energy and a negative change in exchange energy in going from 

 the unmagnetized to the magnetized state. If the exchange energy 

 has a greater change than the Fermi energy, the energy of the mag- 

 netized state is lower and the metal is ferromagnetic. 



No satisfactory calculations have as yet been made for these energy 

 differences. In order to calculate them, accurate values for the dis- 

 tribution of states in the M band are needed, and the mathematical 

 methods available for computing this distribution are not as yet very 

 satisfactory. Next the exchange effect energy must be found; this is 

 also difficult to calculate accurately. Finally, the description given 

 here is over-simplified; in particular another energy term, known as the 

 correlation energy, must be included; this energy acts somewhat like 

 an exchange energy but between the bands of different spins and it 

 tends to cancel out the exchange energy. Although these difficulties 

 greatly mar the usefulness of the theory of ferromagnetism repre- 

 sented in Fig. 28, this theory is able to correlate a large amount of 

 experimental material in a very natural way; and since it is the theory 

 based on the concepts of energy bands, it is the one that we shall dis- 

 cuss in this paper. In passing, however, we must state that there are 

 other theories of ferromagnetism which in some ways are more suc- 

 cessful and in other ways less successful than the band theory. Some 

 of these are atomic rather than band theories. An example of this type 

 of difference in method of attack was given in the discussion of the 

 binding energy of sodium chloride; two treatments were given: for one 

 the basis being the ions and for the other the energy bands. In the 

 case of sodium chloride, however, the theoretical equivalence of the 

 two methods is easily demonstrated. In the case of ferromagnetism, 

 the two theories are not equivalent and are both simplifications of a 

 more complex and as yet unsatisfactorily explored intermediate case. 



Although no satisfactory calculations of the energy difference be- 

 tween the magnetized and unmagnetized states of metals exist, the 



