I was reading the following paper by Landau and Lifshitz on ferromagnetism:
https://www.sciencedirect.com/science/article/abs/pii/B9780080363646500089
In the paper, the following expression is used for the density of the energy of the system (inhomogeneity+anisotropy): $$ 1/2 \alpha s'^2+1/2 \beta (s_x^2+s_y^2) $$ Where $s$ is of course the magnetic moment field, $s'$ is its derivative with respect to $x$, and part of the problem is determining $\alpha/\beta$ (since it gives the square of the width of the intermediate region between magnetic domains). I've used here the slightly confusing notation of the paper, where I believe $1/2a$ means $(1/2)a$, on purpose because I don't know if I'm reading something wrong.
Near the end of page 15 (second page of the PDF) there is a derivation I don't understand. It says:
Blockquote The numerical value of the constant α can be obtained approximately in the following way. The energy $1/2\alpha s'^2$ has its maximal possible value, when $s$ changes its direction after every distance equal to the lattice constant $a$ of the crystal, i.e. when $\boldsymbol{s}'^2 \cong s^2/a^2$ This maximum must be of the order $kT_c$ if $T_c$ is the Curie temperature (k is Boltzmann’s constant). Thus, we find approximately $$ \alpha = kT_c/as^2 $$
There's a few things I don't understand. First of all, it seems to me that the maximum of $s'^2$ should be $(2s)^2/a^2$, since it changes direction completely (therefore the total change is twice its magnitude). But I understand it if it's just an order of magnitude calculation. However, I am then unsure exactly where the Curie temperature comes from, and what's more, the final formula does not seem right to me. My calculation would be: $$ \frac{\alpha s^2}{2 a^2} = kT_c$$ therefore: $$ \alpha = \frac{2 a^2 kT_c}{s^2}$$ But this is not at all the formula they find, unless, I've horribly misread something. In fact their formula doesn't even seem dimensionally correct to me.
