690 BELL SYSTEM TECHNICAL JOURNAL 



According to the quantum mechanical theory of valence, which in 

 itself forms a theory with as many ramifications as the band theory, 

 the electron configuration U'^ls'^lp'^ of carbon is especially suited for 

 forming "electron pair bonds" with other atoms. In forming these 

 bonds the wave functions from one atom and another become dis- 

 torted so as to overlap and form a high electron concentration along 

 the line between the atoms; the energy levels being incompletely filled 

 for the atoms, this overlapping does not produce a repulsion but in- 

 stead a binding together like that produced by the overlapping wave 

 function in Fig. 9b in the hydrogen molecule. The carbon atom is 

 capable of forming four such bonds and forming them most effectively 

 along four lines, making the tetrahedral angles with each other. 



Recently Brill ^^ and his collaborators using x-ray analysis have 

 determined the electron concentration in diamond, in which the car- 

 bon atoms are arranged in a tetrahedral manner. The results of their 

 investigations are shown in Fig. IQA.^^ It is easily seen that the 

 electrons are concentrated in the bonding directions forming homopolar 

 bonds between the atoms. 



The energy band theory, we have said, does not give the clearest 

 picture of the valence crystals; it is, however, especially suited to 

 treatments of the metallic bond. According to the band theory the 

 valence electrons constitute an electron gas — that is, instead of forming 

 electron pair bonds with localized overlapping of the wave functions, 

 they form instead a more or less uniform region of negative charge. 

 In this negative charge the positive ions float. Since the ions repel 

 each other they tend to arrange themselves so as to use their space to 

 best advantage and this requires that they take up one of the "close- 

 packed" arrangements. Let us see why this is true. The close- 

 packed arrangements are those obtained by trying to pack rigid 

 spheres as compactly together as possible. For these arrangements 

 then, the volume per sphere is less than for other arrangements; that 

 is, the close-packed arrangements are the ones which give a minimum 

 volume per sphere for a prescribed value for the distance between 

 sphere centers. Conversely, the close-packed arrangements must be 

 the ones which give a maximum value for the distance between neigh- 

 boring sphere centers for a given value of the volume per sphere. 

 Since the energy of motion, Ep, of the electron gas and, although we 

 have not shown why, the energy Eo, depend for a metal mainly upon 

 the volume, in metals we are interested in cases where the volume per 



" R. Brill, H. G. Grimm, C. Hermann and CI. Peters, /1km. d. Physik, 34, 393 (1939) . 

 ^5 The writer is indebted to Professor Grimm for his permission to reproduce Fig. 

 19 from his article: Naturwissenschaften 27, 1 (1939). 



