Question

Use a computer to plot $\overline{\mathit{s}}$ as a function of kT/$\epsilon$, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice).

Short answer

Therefore,

$\overline{s}=\tanh \left(\frac{4\overline{s}}{t}\right)$

Step-by-step solution

Given information

Plot $\overline{\mathit{s}}$as a function of kT/$\epsilon$, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice).

Explanation

The average spin alignment is given by:

$\overline{s}=\tanh \left(\beta ϵn\overline{s}\right)$

The number of nearest is given by:

$$\begin{gathered} n=2Inonedimension \\ 4Intwodimensions(squarelattice) \\ 6Inthreedimensions(simplecubiclattice) \\ 8Inthreedimensions(bodycenteredcubiclattice) \\ 12Inthreedimensions(facecenteredcubiclattice) \end{gathered}$$

Consider a two-dimensional lsing model with n = 4, which you can plug into the equation above to get:

$\overline{s}=\tanh \left(4\beta ϵ\overline{s}\right)$

Substitute $\beta =1/k{T}_c$and $t=k{T}_c/ϵ$

$\overline{s}=\tanh \left(\frac{4\overline{s}}{t}\right)$

First, we must solve this equation numerically because it is difficult to solve. To do so, we simply build a list of t values ranging from O01 to 5.0, and then we create a loop to solve this equation numerically for each value of t. I used Python to achieve this, and the code is shown below.

The graph is: