Curie-Weiss Paramagnetism susceptibility

[unscientific]

Homework Statement

(a) Show the curie-weiss behaviour.
(b) Estimate $\lambda$ and $B_e$ and exchange energy.[/B]

Homework Equations

The Attempt at a Solution

Part(a)
Since even when applied field is zero, $B_{total} \neq 0$ which gives rise to $M\neq 0$. This is a fundamental property of ferromagnetism.
Consider the spin-1/2 ising model:
$$H = \sum\limits_{\langle i,j \rangle} J_{ij} \sigma_i \sigma_j - \sum\limits_i g \mu_B \sigma_i \cdot \vec B$$
$$H_i = \left( g \mu_B B_{applied} - J \sum\limits_j \sigma_j \right)\sigma_i$$
We represent the term in the brackets by an effective field $B_e$:
$$H_i = \left( g \mu_B B_e \right)\sigma_i$$
Average spin at site $i$ is given by
$$\langle \sigma \rangle = -\frac{1}{2} \tanh \left( \frac{\beta g \mu_B B_e}{2}\right)$$
By the mean-field approach, we assume that $\langle \sigma \rangle$ is the same at all sites, giving:
$$g \mu_B B_{app} - J z \sigma_j = g \mu_B B_e$$
where $z$ is the number of nearest neighbours, or coordination number.

This leads to a 'self-consistency' equation
$$\frac{1}{2}\tanh \left[ \frac{\beta}{2} \left( Jz\langle \sigma\rangle - g\mu_B B \right) \right] = \langle \sigma \rangle$$
At zero appleid field, Curie temperature is given when gradient = 1, so
$$\frac{1}{2} \left( \frac{\beta}{2}Jz \right) = 1$$
$$T_c = \frac{Jz}{4 k_B}$$

For the curie-weiss behaviour, we expand for small $\langle \sigma \rangle$:
$$\langle \sigma\rangle = \frac{ \frac{\beta}{4}g \mu_B B_{app} }{frac{\beta}{4} Jz - 1}$$
$$\langle \sigma\rangle = \frac{\frac{1}{4} \frac{g \mu_B B_{app}}{k_B}}{T-T_c}$$
$$M = g n \mu_B \langle \sigma \rangle = \frac{\frac{1}{4} \frac{(g \mu_B)^2 n B_{app}}{k_B}}{T-T_c}$$
$$\chi = \mu_0 \frac{\partial M}{\partial B_{app}} = \frac{1}{4} (g\mu_B)^2 \frac{n}{k_B} \frac{1}{T-T_C}$$

Part(b)
I found the coordination number at $T = 1024 K$ and $J=1$ using
$$T_c = \frac{Jz}{4 k_B}$$
It gave $z = 5.7 \times 10^{-20}$ which seems wrong as the number of neighbours should be a whole number..

 

[unscientific]

bumpp on part (b) - non-integer number of neighbours!

 

[unscientific]

bump on (b) - doesn't make sense to have non-integer number of neighbours

 

[unscientific]

Bump on - (b) please

 

[unscientific]

part (b) - number of neighbours?