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 which diagonalizes V is obtained as a spin-representation of a known rotation. The determination of 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.