In the well known Ising model, without any external field (H=0), the energy (E), spins (s) and exchange interaction (J) are related as in the following equation(adsbygoogle = window.adsbygoogle || []).push({});

$$

E = -\sum_{<ij>}J_{ij}s_{i}s_{j}

$$

J_{ij}is site dependent and consists of three components J_{AA}, J_{BB}and J_{AB}where A is say up spin and B is down spin on a lattice (say SL).

For a material of type A_{x}B_{1-x }with system size N = 100 (N is number of spins), I know the values of J_{AA}, J_{BB}and J_{AB}. Here J_{AA}=E(x=1)/n_{AA}and J_{BB}=E(x=0)/n_{BB }are constant for any x. J_{AB}changes with x. n_{AA}is number of AA bonds.

E(x=1) means Energy of the system when x=1 for A_{x}B_{1-x}material, i.e. Energy of the system when the system consists of only up spins (all A). Similarly E(x=0) means energy of the system when there is only down spins (all B).

x is composition, x=0.25 means 25% of N is A spins and rest are B spins.

**Queston 1**: If I increase my system size N to 200, then shall the values of J_{AA}, J_{BB}and J_{AB}change?

**Question 2**: If the values of J_{AA}, J_{BB}and J_{AB}change with N, then with what factor shall I increase it?

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# Dependence of exchange interaction on system size in Ising model

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