Statistical Mechanics of Paramagnetism

Statistical Mechanics of Paramagnetism

These notes develop the statistical description of an ideal paramagnet starting from microscopic states. Using the Microcanonical Ensemble, we derive the average magnetization, probabilities of spin states, and the statistical definition of temperature.

1. Microscopic Description

The General Quantum Hamiltonian

In a complete quantum mechanical treatment, the Hamiltonian for a charged particle in an electromagnetic field is given by:

\[ H = \frac{1}{2m}\left(\vec{p} - \frac{e}{c}\vec{A}\right)^2 + \mu \vec{S} \cdot \vec{B} \]

Expanding this Hamiltonian reveals two distinct magnetic effects:

• Diamagnetism: Arising from the \( (\vec{p} - \frac{e}{c}\vec{A})^2 \) orbital term.
• Paramagnetism: Arising from the \( \mu \vec{S} \cdot \vec{B} \) spin interaction term.

The Ideal Paramagnet Hamiltonian

For an ideal paramagnet, we assume the orbital contribution is negligible (e.g., \( L=0 \) state) and the entire magnetic moment comes from the spin. The interaction energy of a single spin \( \sigma_i \) with an external field \( H \) is:

\[ \epsilon_i = -\vec{\mu}_i \cdot \vec{H} = -\mu_B H \sigma_i \]

Where \( \sigma_i = \pm 1 \) represents the spin state (up or down).

Macroscopic Observable: Magnetization

The total magnetic moment \( M \) along the axis of the field is the sum of individual moments. This is a function of the microscopic configuration \( \{ \sigma_1, \dots, \sigma_N \} \):

\[ M(\sigma_1, \dots, \sigma_N) = \mu_B \sum_{i=1}^{N} \sigma_i \]

Consequently, the total energy of the system is:

\[ E = -H M = -\mu_B H \sum_{i=1}^{N} \sigma_i \]

2. Ensemble Averages

Following Gibbs' formalism, the average of any observable \( A \) is calculated by summing over all possible configurations \( \sigma \) weighted by their probability \( P(\sigma) \):

\[ \langle A \rangle = \sum_{\sigma} P(\sigma_1, \dots, \sigma_N) A(\sigma_1, \dots, \sigma_N) \]

Specifically for the average magnetization:

\[ \langle M \rangle = \mu_B \sum_{\sigma} \left( \sum_{i=1}^{N} \sigma_i \right) P(\sigma_1, \dots, \sigma_N) \]

Due to the indistinguishability of spins and equal a priori probability, this simplifies to \( N \) times the average of a single spin:

\[ \langle M \rangle = \mu_B N \langle \sigma \rangle \]

3. Isolated Paramagnet (Microcanonical Ensemble)

Consider an isolated system with fixed total energy \( E \) and total number of spins \( N \).

Counting States (\( \Gamma(E) \))

The energy depends on the number of up spins (\( N_+ \)) and down spins (\( N_- \)). Let the net difference be \( q = N_+ - N_- \). The number of distinct configurations corresponding to a specific energy is given by the binomial coefficient:

\[ \Gamma(E) = \frac{N!}{N_+! N_-!} \]

This connects to the energy via \( \langle M \rangle = \mu_B q = \frac{-E}{H} \).

Single Spin Probabilities

The probability of a specific spin (say \( \sigma_1 \)) being +1 is found by summing the probabilities of all configurations where \( \sigma_1 = +1 \). In the microcanonical ensemble, this is the ratio of states with \( \sigma_1=+1 \) to the total states:

\[ P(\sigma_1 = +1) = \frac{(N-1)!}{(N_+-1)! N_-!} \cdot \frac{1}{\Gamma(E)} = \frac{N_+}{N} \]

Expressing this in terms of energy \( E \):

\[ P(\sigma_1 = +1) = \frac{1}{2} - \frac{E/N}{2\mu_B H} \]

Similarly for spin down:

\[ P(\sigma_1 = -1) = \frac{N_-}{N} = \frac{1}{2} + \frac{E/N}{2\mu_B H} \]

Note: The mathematics here is equivalent to the random walk problem.

4. Thermal Equilibrium & Temperature

Subsystem Analysis

Consider the paramagnet divided into two subsystems \( a \) and \( b \) such that \( E_a + E_b = E_{\text{total}} \). The total phase space volume is the product of the individual volumes:

\[ \Gamma_T = \Gamma(E_a, N_a) \cdot \Gamma(E_b, N_b) \]

Equilibrium is defined by maximizing the entropy \( S = k \ln \Gamma \). The condition for equilibrium yields an equality of derivatives:

\[ \frac{\partial S_a}{\partial E_a} = \frac{\partial S_b}{\partial E_b} \equiv k \beta \]

Deriving \( \beta \)

Using Stirling's approximation on the entropy formula derived from \( \Gamma(E) \), the derivative yields:

\[ \beta = \frac{1}{2\mu_B H} \ln \left[ \frac{\frac{1}{2} - \frac{E/N}{2\mu_B H}}{\frac{1}{2} + \frac{E/N}{2\mu_B H}} \right] \]

This leads to the ratio of probabilities for up and down states:

\[ \frac{P(+1)}{P(-1)} = \exp\left( \frac{2\mu_B H}{k_B T} \right) = \exp\left( \frac{\Delta E}{k_B T} \right) \]

This result recovers the standard Boltzmann Factor.

Thermodynamic Identification

We can write the differential of Entropy with respect to magnetization \( M \):

\[ dS = -k_B \beta H dM \]

Comparing this with the Second Law of Thermodynamics \( TdS + HdM = 0 \) (for a magnetic system doing work), we identify the statistical definition of temperature:

\[ \beta = \frac{1}{k_B T} \]