A Method of Determining the Orderings of the Ising Model with Several Neighbor Interactions under the Magnetic Field and Applications to Hexagonal Lattices

Abstract

435 A method is presented to investigate spin orderings in the ground states of Ising models with several neighbor interactions under the external magnetic field (equivalent to determining ordered structures of binary alloys with arbitrary compositions). \Ve make a configuration polyhedron in the space spanned by r,, r,,-... rn, m, in which linear inequalities are satisfied, where rk is the number of the k-th neighbor down-down spin pairs and m the number of down spins. Spin orderings with values of rk and rn which are coordinates of the vertices of the configuration polyhedron are those in the ground states of Ising models. The method has a transparent outlook and is applied to the hexagonal close-packed (HCP) lattice with up to second neighbor interactions and to the plane hexagonal (PH) lattice with up to third neighbor interactions. Nine spin orderings in the ground state for HCP, which have one-to-one correspondences to those in FCC, and thirteen for PH are found and phase diagrams are obtained. § l. Introduction Studies of the spm orderings in the ground states of Ising models with several neighbor interactions under the magnetic field has become subjects of interest since the state of u = 1/3 of CoCl, · 2H,O at low temperatures found by Kobayashi and Haseda and by Narath 1 J was proved to be the ground state by Kanamori 2 J in terms of the Oguchi-Takano model.'al The problems are equivalent to determining the ordered structures of binary alloys with arbitrary compositions. Developments for several crystal lattices include the following: linear chain with up to second,sl, 4 J third 5 J~sJ and fourth 9 l neighbor interactions, linear chain of S = 1 up to second neighbor interactions, 5 J square lattice up to second 2 J,Jol and third 20 J neighbor interactions , simple cubic lattice up to second'J,Jol and third 1 D. sJ neighbor interactions, body-centered cubic and face-centered cubic lattices up to second neighbor interactions ,')· 15), Jsl, !sal body-centered tetragonal lattice up to second') and third 12 l neighbor interactions, triangular lattice up to second 13), 13 al, 14) and third 20 J neighbor interactions and the spinel lattice.' 1 J The methods of determining the spin orderings in the ground states are classified