A Note on the Eigenvalue Problem in Crystal Statistics

mathematical theory of order-disorder transition. The problem of determining the partition function for a lattice with simple interaction model was found, by severai authors,!) to be reduced to an eigenvalue problem which is easier to handle with,and which, with suitable approximations, would reveal more refined details of the phase transition than the former theories. This method of attack culminated in the work of Onsager,2) who succeeded In giving the rigorous solution of the above mentioned eigenvalue problem for two-dimensional simple square Ising lattice. The mathematical tools involved, however, are so hopelessly complicated that one would quite simply lose sight in the jungle of hypercomplex numbers. A con­ siderable improvement on this mathematical point has been attainp.d by Husimi and Syozi3) who used it to solve the eigenvalue problem for the honeycomb type lattice. The present author, independently of them,' has reached more or less similar ideas and considerations. which it is the purpos~ of the present paper to expose in brief detail. Though as yet no substantial applications has been attempted, nor anything physically new has been derived, it may be hoped that it will do some profit for those who are interested in such problems.