Magnetization steps and relevant cluster statistics for a diluted Heisenberg layer: Nearest-neighbor cluster model on the square lattice

Abstract

A theory for magnetization steps (MST's) from a strongly diluted Heisenberg antiferromagnet on the square lattice will be presented in several papers. In this first paper, general results for cluster models are reviewed, including an updated in-depth discussion of cluster types. A detailed equilibrium theory for the nearest-neighbor (NN) cluster model on the square lattice is then presented. The fraction $x$ of cations that are magnetic is assumed to be well below the site percolation concentration, ${x}_{c}=0.593$ for this model. Isotropic exchange interactions (Heisenberg exchange) are the only intracluster interactions. Neighbors are classified by symmetry, instead of the traditional classification by the distance $r$. Neighbors in the same symmetry class have the same isotropic exchange constant $J$, so that the $J$'s too can be classified by these symmetry classes. For the square lattice, twelve of the neighbors' symmetry classes correspond to the twelve shortest $r$'s. The corresponding $J$'s are as follows: ${J}_{1}$ for NN's, ${J}_{2}$ for second neighbors, and so on up to ${J}_{12}$. Clusters are divided into ``types.'' Each type, $c$, is specified by the cluster size ${n}_{c}$ (the number of spins), and by a ``bond list'' that specifies the $J$'s for all spin pairs in the cluster. The bond list determines the cluster's exchange Hamiltonian. The relevant statistics is therefore the statistics of cluster types, not of cluster sizes. The main assumption in the statistics is that the magnetic ions are randomly distributed over all cation sites. For the NN cluster model on the square lattice, there are 3290 cluster types with sizes ${n}_{c}\ensuremath{\leqslant}12$. Perimeter polynomials (PP's) for cluster types, analogous to the usual PP's for cluster sizes, are given for all 3290 cluster types. The energy eigenvalues for the 10 cluster types with sizes ${n}_{c}\ensuremath{\leqslant}5$ were determined for magnetic ions with spin $S=5∕2$, such as ${\mathrm{Mn}}^{2+}$ and ${\mathrm{Fe}}^{3+}$. The average magnetic moment ${\ensuremath{\mu}}_{c}(T,B)$ per cluster, for these 10 cluster types, was then obtained at temperature $T$ and magnetic field $B$. At low $T$, all cluster types with sizes ${n}_{c}>1$ give rise to MST's. The total magnetization $M$ is proportional to a statistically-weighted average of ${\ensuremath{\mu}}_{c}(T,B)$ over all cluster types. The contribution from cluster types with ${n}_{c}\ensuremath{\leqslant}5$ is dominant when $x\ensuremath{\leqslant}0.25$. This contribution is evaluated exactly. Two alternative approximations are used to evaluate the contribution from the larger clusters. Examples of calculated magnetization curves are given. In addition to the MST's, the magnetization exhibits a fast rise at low $B$. This rise ends in a plateau (the plateau of ``apparent saturation''). The apparent saturation value ${M}_{s}$ is calculated for $x$ up to 0.25. This paper is accompanied by electronically accessible tables (EPAPS) of numerical results for all 3290 cluster types.