6.4 Magnetization search

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §6.4. Rewritten in original wording. Continues 6.3 Newton–Raphson search. Labs: 6.4b Magnetization labs.

Problem: $M(T)$ for a mean-field magnet

$N$ spin-$1/2$ moments $\mu$ sit in an effective field and thermalize at temperature $T$. Boltzmann weights favor alignment with the field, so more spins occupy the lower level $N_L$ than the upper level $N_U$. The net magnetization is

$$M=\mu\,(N_L-N_U).$$

In the Weiss molecular-field picture the dominant field is proportional to $M$ itself ($B=\lambda M$), which closes into a transcendental relation with $M$ on both sides. Strip it to reduced variables and a residual — pick each equation among lookalikes.

As $T\to 0$, $\tanh\to 1$ and $M\to N\mu$ (full alignment). $m=0$ always solves the reduced equation; spontaneous order $0<m(t)\le 1$ exists only for $t<1$. Sweeping $t$ and root-finding $f(m,t_i)=0$ builds $m(t)$.

Searching strategy

Reuse the tools from §6.2–6.3:

1. At fixed $t=0.5$, bracket the nontrivial zero (e.g. $[0.5,1]$ — $f$ changes sign) and bisect.
2. Start Newton near that root (e.g. $m_0=0.9$) with a forward-difference $f'$.
3. Compare iteration counts; then tabulate $m(t)$ on a grid $t\in(0,1]$.

Six significant figures for $t=0.5$ give

$$m(0.5)\approx 0.957504.$$

Quiz

Magnetization search

1 / 4

The reduced equation we root-find is…

Pick an answer to see if you are right.

Next: 6.4b Magnetization labs — bisect and Newton at $t=0.5$, then build $m(t)$. Fitting continues at 6.5 Lagrange interpolation.