Mean-field Theory

In contrast to dia- and paramagnetism, ferromagnetic materials (e.g. Fe, Ni, Co) have a non-vanishing magnetization $\vec{M} \neq 0$ also at a vanishing external magnetic field $\vec{B}_a = 0$ below a critical temperature $T_c$. In this section the magnetization of the isotropic Heisenberg model in the mean field approximation (german Molekularfeldnäherung or mittlere Feldnäherung) is derived. The Heisenberg model is a very good model for describing the exchange interaction between spins and therefore ideal for modelling ferromagnetism. The Hamiltonian of the isotropic Heisenberg model is given by \[ \begin{equation} H = -J' \sum_{i \: n.n. \: j} \vec{S}_i \cdot \vec{S}_j - \mu B_a \sum_i S^z_i. \nonumber \end{equation} \]

Here, the first sum goes only over the nearest neighbor sites $j$ of lattice site $i$ ($i \: n.n. \: j$). This is because it is (reasonably) assumed that the spin on site $i$ only interacts directly with the neighboring spins but the interaction is vanishing for all lattice sites further away. The exchange coupling constant between the spins on sites $i$ and $j$ is given by $J'$ (in this simple case independent of the sites). In ferromagnetism, parallel spins decrease the energy and $J'<0$. In the case of antiferromagnetism, antiparallel spins decrease the energy and $J'>0$. For paramagnetism $J'=0$. In case of a system of spin $\frac{1}{2}$ particles (for example electrons) the projection of the magnetic moment onto the magnetic field can take two values , $\vec{\mu}\cdot\vec{B}=\pm \frac{g \mu_B}{2}B$ with $g$ being the Landé g factor (for $J=1/2$) and $\mu_B$ the Bohr magneton.

Now mean field approximation is applied: The main idea is that the interaction of two neighboring spins $\vec{S}_i \cdot \vec{S}_j$ can be replaced by the interaction of a single spin of site $i$ with the mean spin value of its surrounding:

$$ \vec{S}_i \cdot \vec{S}_j \rightarrow \vec{S}_i \cdot \langle \vec{S} \rangle .$$

The mean field Hamiltonian can then be written as:

$$ H_{MF} = -\sum_i \vec{S}_i \cdot \left( z J' \langle \vec{S} \rangle + \mu \vec{B}_a \right).$$

In this equation $z$ is the so-called coordination number (number of nearest neighbor sites of spin $i$) and comes from the sum over the nearest neighbor sites. The term in the brackets doesn't depend on any lattice site and acts like a total magnetic field consisting of the external magnetic field, $B_a$ and the mean magnetic field $B_{MF}$:

$$\vec{B}_{MF} = \frac{1}{\mu} \: z \: J' \langle \vec{S} \rangle = \frac{zJ'}{\mu^2 n} \vec{M}. $$

Here $\mu = g\mu_B/2$. In the last expression the average spin has been expressed in terms of the overall magnetization density:

$$ \vec{M} = \underbrace{\frac{N}{V}}_{n} \mu \langle \vec{S} \rangle $$

and with the introduction of the spin density $n$. As defined on the paramagnetism page, the energies of the magnetic states are given by:

$$E_{m_J} =-\vec{\mu}\cdot\vec{B}_{total} = -m_J g\mu_B B_{total},$$

where $m_J = \pm\frac{1}{2}$. The occupation probabilty $p_{m_J}$ of the state $m_J$ is given by a Boltzmann distribution,

$$p_{m_J} = \frac{\exp\left(\frac{-E_{m_J}}{k_{B}T}\right)}{\sum\limits_{m_J} \exp\left(\frac{-E_{m_J}}{k_{B}T}\right)}$$

Magnetization

The derivation for the magnetization follows the same scheme as on the paramagnetism page. Also in this case the two states have different occupation. $N_{\uparrow}$ is the number of spin-up states, $N_{\downarrow}$ is the number of spin-down states and $N=N_{\uparrow}+N_{\downarrow}$ is total number of spins. Like in the case of paramagnetism the spin-up state is more probable. The population probability of both spin states can be written as

\[ \begin{equation} \frac{N_{\uparrow}}{N}=\frac{\exp\left(\frac{\mu B}{k_{B}T}\right)}{\exp\left(\frac{\mu B}{k_{B}T}\right)+\exp\left(\frac{-\mu B}{k_{B}T}\right)} \nonumber \end{equation} \] \[ \begin{equation} \frac{N_{\downarrow}}{N}=\frac{\exp\left(\frac{-\mu B}{k_{B}T}\right)}{\exp\left(\frac{\mu B}{k_{B}T}\right)+\exp\left(\frac{-\mu B}{k_{B}T}\right)}. \nonumber \end{equation} \]

In these equations $\mu=\frac{1}{2} g \mu_B$ and $B=B_{total}$. The magnetization per unit volume is given by

\[ \begin{equation} M = \mu \frac{N_{\uparrow}-N_{\downarrow}}{V} \nonumber \end{equation} \]

By inserting the occupation probabilities this equation can be rewritten in

\[ \begin{equation} M = n \mu \tanh\left(\frac{\mu B_{total}}{k_B T}\right) = M_s \tanh\left(\frac{\mu B_{total}}{k_B T}\right) \nonumber \end{equation} \]

where $M_s$ is the saturation magnetization. The plot of $M/M_s$ shows a similar behavior as in the paramagnetic case.

$\large \frac{M}{M_s}$
	$\large \frac{\mu B_{total}}{k_BT}$

Spontaneous magnetization density

The special property of ferromagnetism is the spontaneous magnetization without an external magnetic field $B_a$. To investigate this property the external field is set to zero $(B_a=0)$ so that $B_{total}=B_{MF}$. If this is applied to equation for the magnetization one obtains the so-called self-consistent equation of state of $M$, because $B_{MF}$ depends on $M$ (in the following equations $g_{\frac{1}{2}}$ is abbreviated with $g$):

\[ \begin{equation} M = n \mu \tanh\left(\frac{\mu B_{MF}}{k_B T}\right) = n \mu \tanh\left(\frac{\mu z J' M}{g^2 \mu_B^2 n k_B T}\right) \nonumber \end{equation} \]

By inserting $\mu=\frac{1}{2} g \mu_B$ one gets

\[ \begin{equation} M = \frac{1}{2} n g \mu_B \tanh\left(\frac{z J' M}{2 g \mu_B n k_B T}\right). \nonumber \end{equation} \]

With the definitions $M_s = \frac{1}{2} n g \mu_B$ (saturation magnetization) and $T_c = \frac{z}{4 k_B}J$ (critical temperature) this equation can be simplified to

\[ \begin{equation} M = M_s \tanh\left(\frac{T_c}{T} \frac{M}{M_s}\right). \nonumber \end{equation} \]

The meaning of the two constants will be explained in a few paragraphs. with the parametrization $m=M/M_s$ and $t=T/T_c$ it is possible to simplify this equation even more:

$$m=\tanh\left(\frac{m}{t}\right).$$

A way to solve this equation is a graphical solution: one plots the left side of the equation $(m)$ as straight line and the right side $(\tanh(m/t))$ for different fixed values of $t$. The solution of the equation (except the trivial solution for $m=0$) are the points where the curves for the left and right side intersect. The figure below shows this procedure for some values of $t$:

$\tanh\left(\frac{m}{t}\right)$
	$m$

From these graphical solutions one can see that only curves with $t<1$ have a non-trivial crossing with the left side. This means that only for temperatures $T<T_c$ a spontaneous magnetization can occur and that the magnetization vanishes above this temperature ⇒ $T_c$ is the critical temperature for the phase transition from the ferromagnetic to the paramagnetic phase.
It is possible to solve the transcendental equation of the magnetization numerically for various values of $t$. A possible algorithm to solve this problem is binary search. One obtains the magnetization density as function of temperature. The result is shown in the figure below.

$\large \frac{M}{M_s}$
	$\large \frac{T}{T_c}$

The figure shows the expected behavior of the magnetization density in respect to temperature: the magnetization gets lower for rising temperatures and vanishes at $T_c$.

Susceptibility

In general, the magnetic susceptibility is defined as

$$\chi_m = \left.\frac{\partial M}{\partial H_a}\right\vert_{H_a = 0}.$$

For spin models the relative permeability $\mu_r=1$, because the spins are in vacuum. Therefore $H_a=B_a/\mu_0$ and the equation for the magnetic susceptibility can be written as

$$\chi_m = \left. \mu_0 \frac{\partial M}{\partial B_a}\right\vert_{B_a = 0}.$$

The starting point for the calculation of the susceptibility is the expression for the magnetization:

\[ \begin{equation} M = \frac{1}{2} n g \mu_B \tanh\left(\frac{g \mu_B (B_{MF}+B_a)}{2 k_B T}\right). \nonumber \end{equation} \]

Above $T_c$ the hyperbolic tangent can be expanded with $\tanh(x) \approx x$ (for $x \ll 1$). By applying this expansion on the formula of the magnetization and inserting the expression for $B_{MF}$ one gets:

\[ \begin{equation} M \approx \frac{1}{2} n g \mu_B \left(\frac{g \mu_B (B_{MF}+B_a)}{2 k_B T}\right) = \frac{n g^2 \mu_B^2}{4k_B T} \left(\frac{zJ'}{g^2 \mu_B^2 n} M+B_a\right). \nonumber \end{equation} \]

This equation can be solved for $M$ and simplified with the definition of the critical temperature $(T_c=zJ'/4k_B)$: \[ \begin{equation} M \approx \frac{n g^2 \mu_B^2}{4k_B} \frac{B_a}{T-T_c}. \nonumber \end{equation} \]

Inserting this expression in the equation for the magnetic susceptibility one gets the so-called Curie-Weiss law:

$$\chi_m = \mu_0 \frac{\partial M}{\partial B_a} \approx \frac{\mu_0 n g^2 \mu_B^2}{4k_B} \frac{1}{T-T_c} \approx \frac{C}{T-T_c}.$$

In the figure below the schematic behavior of the magnetic susceptibility is shown:

$\large \chi_m$
	$\large \frac{T}{T_c}$

With this law only the magnetic susceptibility above $T_c$ (paramagnetic behavior) can be modeled. The discussion of the susceptibility below $T_c$ would need different models, but the discussion would exceed the scope of this summary. The same result for the magnetic susceptibility can be obtained with the Landau theory of second order phase transitions.

references

Script Advanced Statistical Physics by Prof. von der Linden
Lectrure slides magnetism by Prof. Hadley

written by Martin Napetschnig, February 2020 edited by Moritz Theissing, May 2020. A discussion including arbitrary values for the angular momentum can be found here.