710 



BELL SYSTEM TECHNICAL JOURNAL 



Fig. 16. The 3d levels give a band capable of containing ten electrons 

 per atom, five with each spin ; and the 45 band can hold two electrons 

 per atom, one with each spin. Curves representing N{E) for these 

 bands, calculated for the case of copper by Slater and Krutter, are 

 shown in Fig. 27. We see that the 45 band is much wider in energy 

 than the M and that it contains only one-fifth as many electronic states. 

 The band structure will be similar for all the transition elements; the 

 energy scales, however, will be different. As is shown in Fig. 6 the Zd 

 electrons are more tightly bound for copper than for nickel or chro- 

 mium. Corresponding to this tighter binding, the d>d wave functions 

 of copper extend less in space than those of nickel and chromium and 

 consequently they overlap less between atoms and the Zd band is 

 narrower for copper. Progressing towards decreasing atomic number 

 in the sequence of elements from copper to scandium, the ?>d band will 

 continually widen; and this widening, as we shall see later, can help 

 account for the absence of ferromagnetism for the elements before 

 iron in the periodic table. 



(a) 



+ SPIN 



(b) 



-SPIN + SPIN 



ELECTRONS 



PER ATOM 



= 5 



UNMAGNETIZED STATE MAGNETIZED STATE 



NUMBER OF QUANTUM STATES AND ELECTRONS PER UNIT ENERGY 



Fig. 28 — The ferromagnetism of nickel. 



In Fig. 28, we give a simplified representation of the 45 and M bands 

 split into two sets according to the spin. (We may, if we wish, sup- 

 pose that a magnetic field is applied along which the spin is quantized, 

 but that the field is so weak that the displacement of the energy bands 

 produced by it is negligible; this supposition is not necessary, however, 

 for regarding the spin we shall need only the fact that all the electrons 

 in the + spin band have parallel spins which are anti-parallel to those 

 in the — spin band.) For the element nickel there are 28 electrons, 10 

 of which are in the M and 45 bands. They can fill the bands as indi- 

 cated in Fig. 28a. Let us compare this distribution with the electron 



