# How to choose the values of J and spin parameters in a heterogeneous spin system?

Previous questions (here, here and here) were about modeling magnetic homogeneous systems using Heisenberg and/or Ising hamiltonians:

$$$$\tag{Heisenberg} \hat{H}_H=-\sum_{\langle i j\rangle}J\hat{S}_i\hat{S}_j$$$$

$$$$\tag{Ising} \hat{H}_I=-\sum_{\langle ij\rangle}J\hat{S}_i^z\hat{S}_j^z$$$$

In this work1 (free access), graphene-based systems described by mixed spin-3/2 and spin-5/2 are studied using Ising Hamiltonian. A diagram of the structure is shown bellow (image linked from original address):

The Hamiltonian used is:

$$$$\tag{1} {H_I} = - J\sum\limits_{\left\langle {i,j} \right\rangle } {{S_i}{\sigma _j}} - {J_\sigma }\sum\limits_{\left\langle {i,j} \right\rangle } {{\sigma _i}{\sigma _j}} - {J_S}\sum\limits_{\left\langle {i,j} \right\rangle } {{S_i}{S_j}} - {K_v}\left( {\sum\limits_i {S_i^2} + \sum\limits_j {\sigma _j^2} } \right)$$$$

Here $$\left\langle {i,j} \right\rangle$$ refers to the sum over the nearest neighbors pairs, $$\left[ i,j \right]$$ means sum over the next-nearest neighbors pairs, $$J$$, $$J_\sigma$$ and $$J_S$$ are the exchange interaction constants between sites $$\sigma−S$$, $$\sigma−\sigma$$ and $$S−S$$, respectively (see diagram above), and $$K_v$$ is the crystal field anisotropy constant. The spins moments can take values $$\sigma = ±3/2,±1/2$$ and $$S=±5/2,±3/2,±1/2$$.

My questions are:

• What are the physical justification in setting the values for the exchange interaction constants $$J$$, $$J_\sigma$$ and $$J_S$$?
• What are the physical justifications setting the values of $$\sigma$$ and $$S$$?
1. J.D.Alzate-Cardona, D.Sabogal-Suárez, E.Restrepo-Parra, Critical and compensation behavior of a mixed spin-3/2 and spin-5/2 Ising ferrimagnetic system in a graphene layer. J. Mag. Mag. Matt. 429 34–39 (2017) (DOI: 10.1016/j.jmmm.2017.01.004)