Paramagnetism

Paramagnetic materials have atoms with magnetic moments which can be aligned with an applied field so that the induced field will add to the applied field. This results in a larger $B$ -field inside the material than outside and a positive magnetic susceptibility. Paramagnetism is similar to diamagnetism in the sense that the materials will only show magnetization when a magnetic field is applied. Once the applied field is switched off, the moments will randomize again and the magnetization decay exponentially to zero.

The magnetization of a paramagnet is the sum of the aligned magnetic moments per unit volume. The magnetic moment of an atom depends on the total angular momentum quantum number $J$. Since the magnetic quantum numbers are restricted to the values $m_J = (-J,-J+1,\cdots , J-1, J)$, the $z$-component of the magnetic moment can can take the values,

$$\mu_{m_J}=m_J g_J \mu_B,$$

where $g_J$ is the Landé g factor, $\mu_B$ is the Bohr magneton. For a magnetic field applied in the $z$-direction, the energies of the magnetic states will be,

$$E_{m_J} = -\mu_{m_j}B_z = -m_J g_J \mu_B B_z.$$

The occupation probabilty of state $m_J$ is given by a Boltzmann distribution,

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

The $J=\frac{1}{2}$ case
For a spin $\frac{1}{2}$ system where $J = \frac{1}{2}$, there are two states spin-up aligned parallel to the $B$ field and spin-down aligned antiparallel to the $B$ field. The spin up state has a lower energy and will have a higher occupation. Let $N_{\uparrow}$ be the number of spin-up states, $N_{\downarrow}$ be the number of spin-down states, and $N=N_{\uparrow}+N_{\downarrow}$ be the total number of states. The occupations of spin-up and spin-down are given by,

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

Here $\mu =\frac{1}{2} g_{\frac{1}{2}} \mu_B$. The magnetization is magentic moment per unit volume $V$,

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

The occupation probabilities can be used to write this as,

\[ \begin{equation} M=n\mu \tanh\left(\frac{\mu B}{k_B T}\right), \end{equation} \]

where $n=\frac{N}{V}$ is the density of the spins. A plot of the magentization shows that there are two interesting limits, $\mu B >> k_B T$ and $\mu B << k_B T$.

$\large \frac{M}{M_s}$
	$\large \frac{\mu B}{k_BT}$

For $\mu B >> k_B T$, all of the spins align and the magnetization approaches the saturation magnetization $M_s = n\mu$. For small arguments, $\tanh(x)\approx x$ so in the limit $\mu B << k_B T$, the relation between the magnetization an the $B$ field is linear,

\[ \begin{equation} M \approx \frac{n \mu^2 B}{k_B T} = \frac{CB}{T}. \end{equation} \]

The linear magnetic susceptibility $\chi_m =\frac{dM}{dH} = \frac{n\mu_0 g_{1/2}^2 \mu_B^2}{4k_BT}$ describes how the magnetization changes with applied magnetic field near zero field. This has the form $\chi_m= \frac{C}{T}$ which is known as the Curie law where $C$ is called the Curie constant.

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

General $J$
For a general value of the total angular momentum quantum number, the magnetization can be written in terms of the average value of $m_J$,

\[ \begin{equation} M=ng_J \mu_B \langle m_J \rangle \end{equation} \]

The average value of $m_J$ is,

$$\langle m_J \rangle = \sum\limits_{m_J = -J}^{m_J=J}m_Jp_{m_J} = \frac{\sum\limits_{m_J = -J}^{m_J=J}m_j\exp\left(\frac{m_J g_J \mu_B B}{k_{B}T}\right)}{\sum\limits_{m_J = -J}^{m_J=J} \exp\left(\frac{m_J g_J \mu_B B}{k_{B}T}\right)}.$$

Notice that the numerator is the derivative of the denominator so this can be written,

$$\langle m_J \rangle = \frac{1}{Z}\frac{dZ}{dx},$$

Where $Z= \sum\limits_{m_J = -J}^{m_J=J} \exp\left(m_J x\right)$ and $x = \frac{ g_J \mu_B B}{k_{B}T}$. The sum from $-J$ to $J$ can be written as the difference of two infinite sums,

$$Z= \sum\limits_{m_J = -\infty}^{m_J=J} \exp\left(m_J x\right) - \sum\limits_{m_J = -\infty}^{m_J=-J-1} \exp\left(m_J x\right).$$

This is convenient because both of the infinite sums are geometric series that can be summed,

$$Z= \left(e^{Jx} -e^{-(J+1) x}\right)\left(1+e^{-x}+e^{-x^2}++e^{-x^3}+\cdots\right)= \frac{e^{Jx} -e^{-(J+1) x}}{1-e^{-x}}.$$

This can be rearranged to be expressed as, $$Z= \frac{\text{sinh}\left(\left(J+\frac{1}{2}\right)x\right)}{\text{sinh}\left(\frac{x}{2}\right)}.$$ $$ \langle m_{J} \rangle = \frac{1}{Z} \frac{dZ}{dx} = \left ( J + \frac{1}{2} \right ) \coth \left ( \left ( J + \frac{1}{2} \right ) x \right ) - \frac{1}{2} \coth \left ( \frac{1}{2} x \right ).$$
$\langle m_J \rangle = JB_J(Jx)$

$\large x=\frac{g_J\mu_BB}{k_BT}$

Where $B_J(x)$ is the Brillouin function,

$$B_J(x)= \frac{2J+1}{2J}\text{coth}\left(\frac{2J+1}{2J}x\right)-\frac{1}{2J}\text{coth}\left(\frac{x}{2J}\right).$$

The magnetization can then be expressed as,

\[ \begin{equation} M=ng_J \mu_B J \left(\frac{2J+1}{2J} \coth\left( \frac{2J+1}{2J} \frac{g_J\mu_BJB}{k_BT}\right)-\frac{1}{2J} \coth\left(\frac{1}{2J} \frac{g_J\mu_B JB}{k_B T}\right)\right) \end{equation} \]

In the high field limit, $\frac{\mu_BB}{k_BT} >> 1$, the magnetization saturates at $M_s = ng_J\mu_BJ$. In the low field limit, $\frac{\mu_BB}{k_BT} < < 1$, the susceptibility has the form of a Curie law, $$ \chi_m =\frac{dM}{dH} \approx \frac{n\mu_0 g_{J}^2 J(J+1)\mu_B^2}{3k_BT}.$$