720 BELL SYSTEM TECHNICAL JOURNAL 



and when Eu — Ej^ is negative, the reverse is true. Hence a positive 

 value for Ev — Ej^ is a necessary and sufficient condition for ferro- 

 magnetism. The variable on the horizontal scale is ra (the distance be- 

 tween nearest neighboring atoms in the crystal) divided by Td (the aver- 

 age radius for the ^d wave function). Small values of rajra mean 

 crowding together of the atoms, large values of the Fermi energy, and 

 no ferromagnetism. Certain values of rajrd, such as are found for iron, 

 cobalt, and nickel, favor ferromagnetism. Very large values of Valrd 

 mean widely separated atoms and low Fermi energy and, conse- 

 quently, ferromagnetism; however, for very widely separated atoms, 

 the energy of interaction between them is small and so is the energy of 

 magnetization. The curve shown in Fig. 32 is only qualitative. The 

 theory that the curve should have this form was first worked out by 

 Bethe using the "atomic" rather than the band theory of magnetism; 

 for the reasons discussed above, however, no quantitative theoretical 

 curve is available. Ratios of raird have been calculated by Slater * 

 and occur as indicated for several elements. This curve can be con- 

 sidered from either of two viewpoints. We may imagine that ta re- 

 mains constant, as it does approximately for the transition elements, 

 and that ra varies from element to element; we then get the result 

 shown in Fig. 32. On the other hand we may consider a definite 

 chemical element thus fixing r^; then Fig. 32 tells us how the energy 

 of magnetization depends on the lattice constant or volume of the 

 sample. We shall use this in the following paragraphs to explain the 

 effects of magnetism upon thermal expansion. 



Magnetism and Thermal Expansion 

 In Fig. 33a we show a solid curve which represents for iron in the 

 magnetized state the dependence of the energy Ej^j upon the lattice 

 constant a. In Fig. Z2>h is shown, on a relatively enlarged energy 

 scale, the value of Ev — Em as taken from Fig. 32 with rd thought of 

 as fixed, and a the lattice constant in place of Ta- The position of 

 curve {b) has been adjusted so that the point marked O , corresponding 

 to iron in Fig. 32, comes at the equilibrium distance or minimum of 

 the Em curve. Adding the solid curves of (a) and (h) (adjusting the 

 energy scales, of course) gives the dashed curve representing the energy 

 Ev of the unmagnetized state shown in Fig. 33a. We are now in a 

 position to make predictions about the thermal expansion of iron. 



Let us imagine that the iron is somehow made to stay in the mag- 

 netized state. Then its expansion curve, lattice constant versus 

 temperature, will be shown as in Fig. Zic by the solid heavy line. Next 



* J. C. Slater, Phys. Rev., 36, 57 (1930). 



