680 



BELL SYSTEM TECHNICAL JOURNAL 



be obtained. In Table I, we compare theoretical and experimental 

 values of lattice constant, binding energy, and compressibility calcu- 

 lated by the method described above. The theoretical values were 

 computed by Bardeen ^^ who has added some refinements and correc- 

 tions to the original calculations. 



TABLE I 



Although the theory can give quite satisfactory values for the 

 various physical quantities shown in Table I, it cannot as yet predict 

 precisely what crystalline form a metal like sodium will take. In 

 carrying out the calculations discussed above, it was assumed that the 

 atoms were arranged in a body-centered cubic lattice. Now the cor- 

 rect theoretical procedure would be to calculate the energy for all 

 conceivable arrangements of the atoms and then to select that arrange- 

 ment giving the least energy of all as the theoretically predicted 

 equilibrium arrangement. This program is, of course, too laborious 

 to be practical^ — furthermore experience shows that metals, with but 

 few exceptions, crystallize in one of three forms : body-centered cubic, 

 face-centered cubic, and hexagonal close-packed. For this reason it 

 might be regarded as sufficient to calculate the energies for the face- 

 centered cubic and hexagonal close-packed and to compare these with 

 that for the body-centered cubic. When such calculations are carried 

 out, however, it is found that the minimum energies calculated for 

 the three forms differ among themselves by amounts which are negli- 

 gible in view of approximations necessary in making the calculations. 

 Hence the theory cannot predict with any certainty which form really 

 has the lowest energy; it does predict, however, that all three forms 

 do have nearly the same energy and gives a value for this energy. 

 Actually the binding energy of sodium must be greatest for the body- 

 centered cubic lattice because this form is the one that occurs in nature 

 and so must be the form of lowest energy. However, it is probable 

 that the diff'erence in energy between the various possible allotropic 



12 J. Bardeen, Jour. Cheni. Phys., 6, 367, 372 (1938). 



