Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis

The partition function for a two-dimensional binary lattice is evaluated in terms of the eigenvalues of the $2^n$-dimensional matrix V characteristic for the lattice. Use is made of the properties of the $2^n$-dimensional "spin"-representation of the group of rotations in $2n$-dimensions. In consequence of these properties, it is shown that the eigenvalues of V are known as soon as one knows the angles of the $2n$-dimensional rotation represented by V.

Together with the eigenvalues of V, the matrix $\Psi$ which diagonalizes V is obtained as a spin-representation of a known rotation. The determination of $\Psi$ is needed for the calculation of the degree of order.

The approximation, in which all the eigenvalues of V but the largest are neglected, is discussed, and it is shown that the exact partition function does not differ much from the approximate result.