The partition function for a two-dimensional binary lattice is evaluated in terms of the eigenvalues of the -dimensional matrix V characteristic for the lattice. Use is made of the properties of the -dimensional "spin"-representation of the group of rotations in -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 -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.
- Received 11 May 1949
- Published in the issue dated October 1949
© 1949 The American Physical Society