THE QUANTUM PHYSICS OF SOLIDS 



683 



we should expect encroachment repulsions between these Ions when 

 their wave functions begin to overlap. In the band picture this repul- 

 sion results from the spreading of the 3d band; since the band spreads 

 more to higher energies than to lower energies and since it is full, the 

 average energy of an electron in it increases as the lattice constant 

 decreases. Thus the same result, repulsion between closed shells, is 



3d I 2 



-0.1 

 12-0.2 - 



O -0.3 



5 



,-0.6 

 -0.7 - 

 -0,8 - 

 -0.9 



I 2 3ao4 5 6 7 8 9 10 II 



LATTICE CONSTANT IN ANGSTROMS 



Fig. 16 — Energy bands for copper versus lattice constant. 



found for the ions in a metal as for the rare gas atoms. For elements 

 whose atoms have partially filled 3d levels the situation is quite differ- 

 ent. For them only part of the levels of the 3d band will be filled 

 and there will be a decrease in the energy of the 3d electrons in the 

 metal as compared to the atom. This has been proposed by Seitz and 

 Johnson as an explanation of the fact that the highest binding energies 

 for the metals of a transition series occur for those that have approxi- 

 mately half-filled 3d bands and for which consequently nearly all of 

 the 3d electrons have lower energies than in the atomic state. ^^ The 

 very high melting point metals — columbium, molybdenum, tantalum, 

 and tungsten — come approximately at the middle of their transition 

 series. In Table II we give the binding energies for a number of the 

 transition elements. 



15 F. Seitz and R. P. Johnson, Jour. App. Phys., 8, 84, 186, 246 (1937). 



