COMPRESSIBILITY OF METALS. 227 



Substituting above, 



d~L = }3v + t 



Now in a solid under normal conditions we may suppose the kinetic 

 energy of motion of centers of mass of the atoms to be a function of 

 temperature only, so that at constant temperature this part of L is 

 constant. But the equation above gives at moderate pressures an 

 increase of L with pressure. As Schottky points out, this means an 

 increase of internal kinetic energy of rotation of the electrons in the 

 atoms about the nuclei, which again means a decrease of atomic 

 radius. In other words, the atoms shrink in size as pressure increases. 

 This reminds one strongly of the compressible atom of Richards. 22 

 What is more, the change of size is very considerable, so that when a 

 substance is compressed the force between the atoms changes for two 

 reasons, both of the same order of magnitude, one reason being the 

 change in the distance between atomic centers, and the other the 

 change in the dimensions of the atom. This will evidently introduce 

 another term into Born's equation, and will essentially modify his 

 results. The modified analysis is perhaps worth giving. While we 

 are giving the analysis, we may as well carry the work one step further 

 than Born has in his published papers, because we can thereby get the 

 variation of compressibility with pressure. 



If the atom is deformable, the coefficient of the repulsive force can 

 no longer be regarded as a constant, but will depend on the distance 

 of separation of the atoms. Born's analysis for the forces exerted by 

 a cube suggests that this coefficient is proportional to the fourth power 

 of the atomic radius. 



We assume as the starting point, as does Born, that the potential 

 energy per unit cell of the lattice is of the form 



a b 



$ = — - -4- — 



8 is the lattice constant, and may be computed in terms of the lattice 

 structure and its elementary charges. The coefficient b should be 

 directly computable in terms of the details of the structure, but it must 

 also satisfy a condition imposed by the stability of the lattice, n 

 should also be computable in terms of the details of the structure, but 

 again may also be found in another way, namely, from the compressi- 

 bility. Thus we have here the possibility of two checks, one on n and 



