Worked Case: Two-Level Systems and Paramagnets

The Physical Picture

Imagine a collection of \( N \) identical, non-interacting atoms, each possessing a magnetic dipole moment \( \boldsymbol{\mu} \). When placed in a uniform external magnetic field \( \mathbf{B} \), the interaction energy of a single dipole is given by \( E = -\boldsymbol{\mu} \cdot \mathbf{B} \). If we align the field along the \( z \)-axis, the energy depends only on the orientation of the dipole relative to this axis. For a simple spin-1/2 particle, there are only two possible orientations: parallel to the field (lower energy) and anti-parallel (higher energy). This creates a system with exactly two energy levels per particle. Because the particles do not interact, the total energy of the system is simply the sum of the energies of the individual particles. This independence is crucial; it allows us to calculate the partition function for the entire system by simply raising the single-particle partition function to the power of \( N \). This "factorization" is the key mathematical tool that makes the problem solvable and physically transparent.

Single-Particle Partition Function

Let the energy of the lower state be \( E_0 = -\mu B \) and the upper state be \( E_1 = +\mu B \). We define the energy gap as \( \Delta E = E_1 - E_0 = 2\mu B \). The single-particle partition function \( Z_1 \) is the sum of the Boltzmann factors for these two states:

Using the definition of the hyperbolic cosine, \( \cosh(x) = \frac{e^x + e^{-x}}{2} \), we can rewrite this compactly as:

Here, \( \beta = 1/k_B T \). This expression is valid for any temperature \( T \) and field \( B \). Note that \( Z_1 \) is always greater than or equal to 2, reflecting the fact that there are two accessible states.

Thermodynamic Potentials

For \( N \) independent particles, the total partition function is \( Z_N = (Z_1)^N \). The Helmholtz free energy \( F \) is given by \( F = -k_B T \ln Z_N \):

From \( F \), we can derive the internal energy \( U \) and entropy \( S \). The internal energy is the negative derivative of \( \ln Z_N \) with respect to \( \beta \):

Calculating the derivative of \( \ln(2 \cosh(\beta \mu B)) \) with respect to \( \beta \) yields \( \mu B \tanh(\beta \mu B) \). Thus:

The entropy \( S \) is obtained from \( S = -\left( \frac{\partial F}{\partial T} \right)_V \). Alternatively, using \( S = \frac{U - F}{T} \) is often simpler. Substituting the expressions for \( U \) and \( F \):

Dividing by \( N k_B \) gives the entropy per spin in units of \( k_B \), a dimensionless form that is useful for plotting; it rises from 0 at \( T = 0 \) to \( \ln 2 \) at high temperature.

Magnetization and Susceptibility

The magnetization \( M \) is the total magnetic moment per unit volume. For \( N \) particles, the total moment is \( N \langle \mu_z \rangle \). The average moment of a single particle is:

Since \( F_1 = -k_B T \ln(2 \cosh(\beta \mu B)) \), differentiating with respect to \( B \) gives:

This is the Brillouin function for spin-1/2. At low temperatures (\( T \to 0 \)), \( \tanh(\beta \mu B) \to 1 \), so \( \langle \mu_z \rangle \to \mu \). The system is fully magnetized. At high temperatures (\( T \to \infty \)), \( \beta \mu B \ll 1 \), and using the approximation \( \tanh(x) \approx x \), we get:

This linear dependence on \( B \) and inverse dependence on \( T \) is Curie's Law. The magnetic susceptibility \( \chi = \frac{\partial M}{\partial B} \) is therefore \( \chi = \frac{N \mu^2}{V k_B T} \), confirming the \( 1/T \) behaviour characteristic of paramagnets.