718 BELL SYSTEM TECHNICAL JOURNAL 



If the concentration is decreased below 8.21, electrons will be removed 

 from the upper part with the other spin ; this will result in a decrease in 

 the unbalance and hence in (3, which has for C = 8.21, a value of 2.44 — 

 corresponding to one filled and one empty upper part ; and this decrease 

 will be numerically equal to the decrease in C. Accordingly, the value 

 of iS for iron, C = 8, is 2.44 - 0.21 = 2.23. The numbers 2.44 and 

 2.56 were, of course, chosen so as to obtain this agreement for iron. 

 This theory of Pauling expresses reasonably well the variations in )8 for 

 all the alloys of Fig. 30. 



Criterion for Ferromagnetism 



We must now see how the theory explains the absence of ferro- 

 magnetism for the remaining transition elements. We have seen that 

 the exchange energy lowers and the Fermi energy raises the energy of 

 the magnetized state compared to the unmagnetized state. These two 

 effects very nearly cancel even for the magnetic elements iron, cobalt, 

 and nickel. For the other elements in the transition series, which are 

 not ferromagnetic, the Fermi term apparently exceeds the exchange 

 term. We shall give a theoretical reason for expecting this result. 



In the first place we must indicate how nearly the efifects cancel. 

 Let us take cobalt, which has nine electrons in the ?>d and 4^ bands, as 

 an example. From Fig. 27 we see that for cobalt in the unmagnetized 

 state both 2>d bands are filled to about — 0.46 atomic units. In the 

 magnetized state one band is filled by electrons which have come from 

 levels with less energy of motion in the other band. Since the top 

 of the Zd band comes at about — 0.42 units on Fig. 27, the average 

 gain in energy for each transferred electron is about 0.04 units. Since 

 the number of electrons transferred is 1.7 per atom, the increase in 

 Fermi energy is 0.068 atomic units or 0.9 ev per atom. From an 

 analysis of thermal measurements the value for the actual energy of 

 magnetization is found to be about 0.2 ev per atom; a value which is 

 only about one fourth of the predicted increase in the Fermi energy. 

 Hence the exchange energy exceeds the Fermi energy by only 25 per 

 cent and the two energies nearly cancel. 



The variation in the structure of the Zd band from element to ele- 

 ment was discussed in connection with Fig. 27; we concluded then that 

 the bands become wider as we recede in the periodic table from nickel 

 towards scandium. Greater band width means greater Fermi energy 

 in the magnetized state and this effect opposes the occurrence of ferro- 

 magnetism. The exchange energy can also change. Calculations by 

 Slater,*" which unfortunately are too over-simplified to bear much 



« J. C. Slater, Phys. Rev., 49, 537, 931 (1936). 



