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Dependence of exchange interaction on system size in Ising model

by cosmicraga
Tags: ising model
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Mar31-13, 08:47 AM
P: 6
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
E = -\sum_{<ij>}J_{ij}s_{i}s_{j}

Jij is site dependent and consists of three components JAA, JBB and JABwhere A is say up spin and B is down spin on a lattice (say SL).

For a material of type AxB1-x with system size N = 100 (N is number of spins), I know the values of JAA, JBB and JAB. Here JAA=E(x=1)/nAA and JBB=E(x=0)/nBB are constant for any x. JAB changes with x. nAA is number of AA bonds.

E(x=1) means Energy of the system when x=1 for AxB1-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 JAA, JBB and JAB change?

**Question 2**: If the values of JAA, JBB and JAB change with N, then with what factor shall I increase it?
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Mar31-13, 11:28 AM
P: 185
Um this is confusing.
You are describing a system where J is spin dependent. This might be written as
[tex]E = -\sum_{<ij>}J_{ij}(s_i,s_j)s_is_j[/tex]
This is NOT the Ising model. The Ising model has a constant J which is independent of the spin. But you are also describing a system which lacks some fundamental symmetries, so it's very confusing.

It sounds like what you want to describe is an inhomogeneous system, where you have two types of atoms, A and B. The exchange interaction is different between different types of atoms, which is why you have three values for J. This is an Ising model for a binary system.

Usually the answer to your question 1 is no. J is generally regarded as a microscopic interaction parameter which is rather localized and it is not affected by the extent of the system. But you have not specified your boundary conditions. If you have some sort of surface, the value of J might be different at the surface. If you are using periodic boundary conditions then J should not change.
Mar31-13, 11:51 AM
P: 6
>> $$ E = -\sum_{<ij>}J_{ij}(s_i,s_j)s_is_j $$

Yes, this is better representation.

>> This is an Ising model for a binary system.

Yes, you are right.

>> But you have not specified your boundary conditions.

Yes, I am using periodic boundary condition in all directions.

Thanks for your answer. :)

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