# Bragg-Williams approximation for the Ising model

Again, we consider a system with the following Hamiltonian:

^{[1]}:

First of all:

Then, for the other term of we have:

The total energy of the system will be:

We can thus see that in the Bragg-Williams approximation also the and exponents are the same of the Weiss mean field theory; in fact, we have seen in Critical exponents of Weiss mean field theory for the Ising model that they come from the expansion of the free energy density for small values of the magnetization when . If we now set into , so that , we get:

^{[2]}in Critical exponents of Weiss mean field theory for the Ising model.

We can also say something^{[3]} in the case . Supposing that our system is uniform, i.e. , we can rewrite as:

In particular if (i.e. ), expanding the right hand side of the self-consistency equation we get a positive linear term (), so that the behaviour of is as shown in the first figure, and the equation has only one solution. On the other hand if then the linear term changes sign and behaves as in the second figure: in this case if is small enough there are three possible solutions, which we have called , and . These are all extrema of , but how can we understand which is a minimum or a maximum? And above all, which of them is the global minimum. If we suppose to be large there will be only one solution, , and as decreases also and will appear; we can therefore argue that for the continuity of the solution is still a minimum also when and are present. Similarly, if we take we can conclude that also is a minimum; therefore will necessarily be a maximum. Now, in order to see which between and is the global minimum of let us take and compute:

This means that as soon as changes sign the global minimum of changes abruptly from to . We are thus obtaining the phenomenology that is indeed observed for a magnet when we change the external field . In other words the sets of points are exactly the graphs of the phase diagram we have seen in Phase transitions and phase diagrams, i.e. the graphs of the magnetization seen as a function of the external field.

- ↑ This form of is very general, and does not depend on the fact that the degrees of freedom of the system can only assume two values: if there is a different number of possible states, say , then can be written in the same form, but will be the probability of one state while the probability of the remaining ones. We will shortly see this when we will apply the Bragg-Williams approximation to the Potts model.
- ↑ Note that also the temperature of the transition is still the same, considering also the factor we have already mentioned.
- ↑ These considerations apply in general also in the other mean field theories considered, but we show them now.