PX284 - E5c - spin-half paramagnet

a single spin $- 1 / 2$ particle in a magnetic field is a two-state system

\begin{matrix} ↑ & ________ & E_{1} = μ_{B} B \\ ↓ & ________ & E_{1} = - μ_{B} B \end{matrix}

the magnetic moment along $z$ is:

μ = - g μ_{B} m_{s} = \mp μ_{B}

where, $μ_{B}$ is the 'bohr magneton' $= e ℏ / 2 m$ , $g = 2$ , and $m_{s} = \pm 1 / 2$

the energy is:

E = - \vec{μ} \cdot \vec{B}

the single particle partition function is:

Z_{1} = \exp (β μ_{B} B) + \exp (- β μ_{B} B) = 2 \cosh (β μ_{B} B)

a paramagnet is an array of $N$ non-interacting spins
here, the $N$ -particle partition function is:

Z_{N} = (Z_{1})^{N}

for a magnetic system, the first law of thermodynamics can be written as:

d U = T d S - m d B

two possible configurations of the paramagnet could be:
$(A) :$ all moments up, $m = N μ_{B}$
- saves internal energy when $B = 0$
- there is only one microstate
$(B) :$ random orientation with half up and half down
- does not save internal energy
- lots of microstates
- maximum entropy

the hemholtz free energy:

\begin{aligned} F & = U - T d S \\ d F & = S d T - m d B \end{aligned}

where, $m = - {(\frac{\partial F}{\partial B})}_{T}$ , and $M = m / V$ is the magnetization

\begin{aligned} F & = - k_{B} T \ln Z_{n} \\ = - k_{B} T \ln (Z)^{n} \\ = - N k_{B} T \ln [2 \cosh (μ_{B} β B)] \end{aligned}

∴ M = n μ_{B} \tanh (β μ_{B} B)

where, $n = N / V$

$(A)$ is at high $B / T$ , the ground state is favoured $⟹ U$ wins
$(B)$ is at low $B / T$ , all microstates are equiprobable $⟹ S$ wins