# How to calculate the exchange constant of the Ising model?

The Ising model is a mathematical model of ferro-magnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually, a lattice, allowing each spin to interact with its neighbours. The model allows the identification of phase transitions, as a simplified model of reality [Source]

The algorithm I am using to calculate it is: $$E = -\sum\limits_{\langle i,j \rangle}^{n}J_{ij} s_i s_j - \mu H\sum\limits_{i}^{n} s_i$$

Where $E$ is energy, $J$ is the exchange constant, $s_i$ & $s_j$ represent nearest neighbour pairs (Like $s_{<1,0>} s_{<2,0>}$ or $s_{<5,4>} s_{<5,5>}$). $H$ is the external magnetic field and $\mu$ is the magnetic moment. Usually H is given in units of $\frac{H}{\mu}$ and J is assumed to be isotropic, so the equation simplifies to:

$$E = -J\sum\limits_{\langle i,j \rangle}^{n} s_i s_j - m\sum\limits_{i}^{n} s_i$$ where $m= \frac{H}{\mu}$.

This model can be used to predict the temperature at which a substance looses its ferromagnetism, which for cobalt is 1388K, for iron its 1043K and for nickel its 627K. This change from ferromagnetic to paramagnetic is found by observing a second order phase transition. (A continuous, dramatic change in the slope of the magnetism).

If you use $J=1$, $T_C \approxeq 2.27K$. How could I determine the exchange constants for other substances, if I know their internal molecular lattice structure, or must I determine it numerically, knowing $T_C$?

$J$ can be determined experimentally by knowing, for example, $T_c$. Through a theoretical model including the structure and the species involved you can estimate the value for $J$ through the overlap integrals of the electrons involved.