670 BELL SYSTEM TECHNICAL JOURNAL 



sitely charged ions attract each other and draw together until the 

 encroachment repulsion between their closed shells balances the attrac- 

 tion and holds them apart. Conversely if one of two atoms having 

 closed shells normally is converted to an ion, the closed shell arrange- 

 ment will be destroyed and an attraction will result.. For example, 

 He2"'' ions, which may be thought of as formed from an atom and an 

 ion, have been observed in the mass spectrograph.''' The attraction 

 is explained by noting that in this case there are three electrons and 

 the effect of two of them in the lower \s molecular orbital overbalances 

 the one in the upper orbital and gives rise to a net attraction. 



Electrons in Crystals 



We must now investigate the quantum states and their energy 

 levels for electrons in crystals. As in the case of the diatomic molecule 

 we shall study the dependence of the energies upon the distance be- 

 tween atoms, which in the case of a crystal is called the lattice constant. 

 We shall treat the lattice constant as a variable and shall refer to the 

 values for it found experimentally as "observed" or "experimental 

 lattice constants" and indicate them on the figures by the symbol ao. 

 We shall consider the allowed states to be occupied in accordance with 

 Pauli's principle and on this basis find how the energy of the crystal 

 as a whole depends upon the lattice constant. In this section we 

 shall deal with crystals at the absolute zero of temperature and leave 

 the complicating features of thermal effects to a later section. Accord- 

 ing to theory, the equilibrium state of a system at absolute zero is that 

 one which makes the energy least. Hence, a knowledge of the de- 

 pendence of energy upon lattice constant can be used to predict the 

 equilibrium lattice constant — that is, the one which should be found 

 experimentally — for according to the theory quoted above, the equi- 

 librium lattice constant is the one which makes the energy of the 

 crystal least. 



In Fig. 11 we show the potential energy for an electron in a one- 

 dimensional crystal, the distance being measured along a line passing 

 through the atomic nuclei of the constituent atoms. In the interests 

 of simplicity we imagine that high potential walls through which the 

 electron cannot pass bound the crystal at both extremities. These 

 boundary conditions lead to a simpler set of wave functions than would 

 boundary conditions like those discussed for the free atom. The sim- 

 plification of problems by arbitrarily choosing certain boundary con- 

 ditions is a standard device in some branches of quantum mechanics; 

 it introduces an error, but if the crystal is large, the error is negligible; 



7 F. L. Arnot and Marjorie B. M'Ewen, Proc. Roy. Soc, 171, 106, 1939. 



