Full Text
STABILITY OF CRYSTAL LATTICES¹
The usual method for investigating the stability of a crystal lattice consists in comparing its energy with the energies of other possible lattices constructed from the same particles. The results obtained² are of relatively little significance because of the complexity and laboriousness of the calculations. On the other hand, it is known that the lattice type was successfully predicted by Goldschmidt³ on the basis of data on atomic and ionic radii. The principle on which such predictions were based generalized the principle of closest packing, valid for a crystal built from atoms of one kind. The generalized principle may be formulated as follows (for a crystal consisting of atoms of two kinds): the most stable configuration of particles is that in which a given particle has the maximum number of neighbors of the other kind situated at the smallest possible distances. Until now this geometrical principle has not been justified by the dynamics of the crystal lattice.
In addition to the above-mentioned method of comparing energies, the method of small oscillations may be applied in studying the stability of a lattice.
Since the number of normal oscillations in a crystal is practically infinite, the application of this method encounters great difficulties⁴.
The method developed by the author of the paper under review is as follows. The crystal is regarded as a thermodynamic system characterized by atomic coordinates \(q_1, q_2, q_3\), etc. (which are subject to a statistical distribution). Denoting by \(a_1, a_2, a_3, \ldots\) the molar parameters describing the macroscopic state of the crystal, then for sufficiently high temperatures (Boltzmann statistics) the state integral is written in the form
\[ Q(a_1, a_2, \ldots, T)=\int \cdots \int e^{-\frac{\varepsilon(q_1,p_1,q_2,\ldots,a_1,a_2,\ldots)}{kT}}\,dq_1dp_1dq_2dp_2\cdots, \]
where \(p\) are the momenta conjugate to \(q\), \(T\) is the temperature, and \(\varepsilon\) is the energy, which must be a function of \(q, p\) and the molar parameters \(a_1, a_2, \ldots\).
Knowing the free energy
\[ A=-kT\lg Q, \]
we find the entropy
\[ S=-\frac{\partial A}{\partial T}, \]
the energy
\[ E=A+TS \]
and the generalized forces
\[ F_r=-\frac{\partial A}{\partial r}. \]
The following six scalar products are adopted as molar coefficients:
\[ \mathbf a_r\cdot \mathbf a_s=a_{rs}\quad (r,s=1,2,3), \]
where \(\mathbf a_1, \mathbf a_2, \mathbf a_3\) are the axial vectors of the elementary cell.
The free energy of the crystal can be represented as a function of \(a_{rs}\) and the temperature in the form
\[ A=U(a_{rs})-kT\lg Q_\nu(a_{rs},T), \]
where \(U\) is the potential energy of the non-oscillating lattice, and \(Q_\nu\) is the part of the state integral that depends on the oscillations.
The generalized forces corresponding to the parameters \(a_{rs}\) are the stress components
\[ A_{rs}=-\frac{\partial A}{\partial a_{rs}}; \]
they are determined by external conditions. For example, for hydrostatic compression \((A_{ii}=-p,\ A_{ik}=0)\).
Solving the last equations gives the equilibrium values for \(a_{rs}\). Stable is that solution for which the quadratic terms in the expansion of \(A\) in small changes \(\delta a_{rs}\) from the equilibrium values \(a^0_{rs}\) are positive. These terms have the form
\[ A-A^0=\frac12 \sum_{\substack{rs\\pq}} A_{rs,pq}(T)\,\delta a_{rs}\delta a_{pq}, \]
where
\[ A_{rs,pq}=\left(\frac{\partial^2 A}{\partial a_{rs}\partial a_{pq}}\right)_0 . \]
The coefficients \(A_{rs,pq}\) are elastic constants, which are expressed here as functions of the temperature.
Thus, the new equilibrium condition is a condition of positivity of the macroscopic deformation energy and can be written in the form of an inequality for the elastic constants.
The equilibrium condition is applied by the author of the article to monatomic cubic lattices. It is well known that monatomic lattices often occur in the form of face-centered lattices, much more rarely in the form of body-centered lattices, and never in the form of a simple cell1. The Goldschmidt principle explains this circumstance as follows: in a face-centered cell the densest packing is possible. The author sets himself the task of substantiating this principle.
The calculation is made under the assumption of central forces acting between atoms. The result of the calculation fully confirms the Goldschmidt principle.
In Born’s textbook[^5] the same calculation is made under concrete assumptions concerning the forces of interaction between atoms, namely, the potential energy is taken in the form
\[ \Phi=u\frac{nm}{n-m}\left(-\frac1m\frac{r_0^m}{r^m}+\frac1n\frac{r_0^n}{r^n}\right), \]
where \(n>m\). (The potential energy is divided into an attractive term, proportional to \(r^{-m}\), and a repulsive one, proportional to \(r^{-n}\).)
A. Kitaigorodskii, Moscow
REFERENCES
- M. Born, Proc. Cambr. Phil. Soc., 36, 160, 1940.
- M. Born and Goeppert-Meyer, Handb. d. Physik, 24/2, 733, 1933.
- V. M. Goldschmidt, Fortschr. d. Mineral, 15, 73, 1931.
- M. Blackman, Proc. Roy. Soc., A 164, 62, 1938; K. Herzfeld and R. H. Lydane, Rhys. Rev., 54, 846, 1938.
- Dhar Misra Rama, Proc. Cambr. Phil. Soc., 36, 173, 1940.
-
Those rare cases in which a substance crystallizes in the form of a simple cubic lattice occur for asymmetric atoms. In this case the assumption of central forces, made by the author, is unjustified. A. K. ↩