686 BELL SYSTEM TECHNICAL JOURNAL 



Closely similar calculations can be carried out for a crystal. One 

 finds the total coulomb energy of all the ions and the total encroach- 

 ment energy; and then one finds the lattice constant that makes the 

 total energy a minimum. The total encroachment energy is easily 

 found; only atoms which are nearest neighbors in the lattice have 

 appreciable overlapping with each other and it is therefore a straight- 

 forward and simple calculation to find the total number of encroach- 

 ments in the crystal. The coulomb energy is not quite so simply 

 found, however, because the electrostatic interaction of a given ion 

 with its nearest neighbors is no more important than its interaction 

 with its vastly larger number of more distant neighbors. The electro- 

 static problem is solved as follows : one considers a NaCl lattice which 

 is perfect except for the absence at one lattice point of a Na+ ion ; one 

 finds by known techniques of electrostatics the value at the vacant 

 lattice point of the electrostatic potential due to the remaining ions; 

 this potential is negative and has a value 



Me SA9e .,. 



where a is the lattice constant and M is a numerical constant known 

 as Madelung's constant, which has a particular value for any special 

 lattice; for the NaCl lattice, M = 13.94. If now a Na+ ion is placed 

 in the vacant lattice point, its electrostatic energy will be — e<i). Sim- 

 ilarly the electrostatic potential at a vacant Cl~ lattice point is +0 and 

 the electrostatic energy of a Cl~ placed there is —€({>. The total 

 electrostatic energy per NaCl molecule in the lattice, however, is not 

 — 2e(f) but only —ecj); the factor 2 does not occur since otherwise the 

 electrostatic interaction between each pair of ions would be included 

 twice. ^^ The total energy per molecule for the crystal can be found 

 by combining the coulomb and the encroachment energies, and the 

 equilibrium lattice constant and binding energy per molecule thence 

 can be derived. 



Using wave functions for Na+ and Cl~ ions obtained by D. R. Har- 

 tree, who has found solutions of Schroedinger's equation numerically, the 

 encroachment energies in NaCl have been evaluated by R. Landshoff.^^ 

 For the lattice constant and binding energy for NaCl he obtains 5.8 8 A 

 and 165 Kg.-cal./gm. atom while experiment gives 5.63A and 183 

 Kg.-cal./gm. atom. 



Some very important theoretical work of a semi-empirical nature has 



1* To see that this is true in a simple case, use the procedure described above to 

 calculate the electrostatic energy of an isolated NaCl molecule. 

 '^Zeits.f. Phys., 102, 201 (1936). 



