On the Stability of Certain Heteropolar Crystals

On the basis of the conventional model of heteropolar crystals it is shown that an irregular crystalline behaviour is to be expected in the region of small values of the exponent, $p$, of the repulsive energy. In the course of this demonstration Madelung's method of obtaining the Coulomb potential for the crystals of the cubical system is justified, and a new, very simple method of calculating the Madelung constant is described. This method is applied to the three crystals of the cubical system and other applications are suggested. The irregular behavior of crystals in the region of small values of the repulsive exponent is shown to be manifested as an instability against various variations by which the geometry of the lattice is altered. In particular, the instability is demonstrated in the case of the "calcite family" of crystals which is evolved by a continuous process from a single parameter $\phi$. The NaCl- and CsCl-types of crystals are members of this family, and a continuous mode of transition from one to the other is thus available. These two types of crystals, in general, are unstable against a variation in which the parameter $\phi$ is changed, ($\phi$-variation), and for small values of the exponent, $p$, all members of the "calcite family" are shown to have a tendency to fall apart into one-dimensional crystals of the Madelung type. Another result of these considerations is that a skew structure, such as that of calcite may be accounted for on the basis of purely central forces. Finally the regions in which respectively one-dimensional and two-dimensional crystals, and the crystals of the cubical system are most stable are calculated. The bearing of these results on the theory of the secondary structure is briefly discussed.