Problem 18 - A Modern Course in Statistical Physics

Question

The molar free energy of a spin system can be written $$ \begin{aligned} \phi(T, H)=\phi_{0}(T) &-\frac{1}{2} J m^{2} \\ &+\frac{1}{2} k_{\mathrm{B}} T[(1+m) \ln (1+m)+(1-m) \ln (1-m)]-m H \end{aligned} $$ where \(J\) is the interaction strength, \(m\) is the net magnetization per mole, \(\phi_{0}(T)\) is the molar free energy in the absence of a net magnetization, \(H\) is an applied magnetic field, \(k_{\mathrm{B}}\) is Boltzmann's constant, and \(T\) is the temperature. (a) Compute the critical temperature (called the Curie temperature). (b) Compute the linear magnetic susceptibility of this system. (Hint: Only consider temperatures in the neighborhood of the critical point where \(m\) is small.)

Step-by-Step Solution

Answer

The Curie temperature is \(T_c = \frac{J}{k_{\mathrm{B}}}\). The linear magnetic susceptibility is \(\chi = \frac{1}{k_{\mathrm{B}} (T - T_c)}\).

Step 1: Understanding the given Free Energy Expression

The given free energy expression for the spin system is: \[ \phi(T, H) = \phi_0(T) - \frac{1}{2} J m^2 + \frac{1}{2} k_{\mathrm{B}} T[(1+m) \ln(1+m) + (1-m) \ln(1-m)] - m H \]

Step 2: Derive equation for critical temperature (Curie temperature)

At the critical temperature, the second derivative of the free energy with respect to magnetization \(m\) should be zero. Start by computing the first derivative of \(\phi(T, H)\) with respect to \(m\): \[ \frac{\partial \phi}{\partial m} = -J m + \frac{1}{2} k_{\mathrm{B}} T \left[ \ln(1+m) + 1 - \ln(1-m) - 1 \right] - H = -J m + \frac{1}{2} k_{\mathrm{B}} T \left[ \ln \left( \frac{1+m}{1-m} \right) \right] - H \]

Step 3: Expand the logarithmic term near the critical point

Using the hint, we consider \(m\) to be small. Use the Taylor expansion for \(\ln(1+m)\) and \(\ln(1-m)\) around \(m = 0\): \[ \ln(1+m) \approx m - \frac{m^2}{2}, \quad \ln(1-m) \approx -m - \frac{m^2}{2} \] Therefore, \[ \ln \left( \frac{1+m}{1-m} \right) \approx 2m \]

Step 4: Simplify the expression for first derivative

Substitute the expansion into the first derivative: \[ \frac{\partial \phi}{\partial m} = -J m + k_{\mathrm{B}} T m - H \] At equilibrium (no external field \(H = 0\)), set this derivative to zero: \[ -J m + k_{\mathrm{B}} T m = 0 \] When \(m eq 0\), this implies: \[ k_{\mathrm{B}} T - J = 0 \] Thus, the Curie temperature \(T_c\) is given by: \[ T_c = \frac{J}{k_{\mathrm{B}}} \]

Step 5: Linear Magnetic Susceptibility

The linear magnetic susceptibility \(\chi\) is determined by: \[ \chi = \left( \frac{\partial m}{\partial H} \right)_{H\to0} \] To compute \(\chi\), perform the second derivative analysis. At \(T \approx T_c\) and for small \(m\): \[ \frac{\partial^2 \phi}{\partial m^2} = -J + k_{\mathrm{B}} T \] At the Curie temperature \(T_c\), this becomes zero. Above \(T_c\), \[ \left( \frac{\partial^2 \phi}{\partial m^2} \right) = k_{\mathrm{B}} (T - T_c) \] Therefore, the magnetic susceptibility: \[ \chi = \frac{1}{k_{\mathrm{B}} (T - T_c)} \]

Key Concepts

molar free energy

The molar free energy, denoted as \(\fhi(T, H)\), represents the energy per mole for a system influenced by temperature and magnetic field. For a spin system, the expression is: \(\fhi(T, H) = \fhi_0(T) - \frac{1}{2} J m^2 + \frac{1}{2} k_B T[(1+m) \text{ln}(1+m) + (1-m) \text{ln}(1-m)] - mH\). Here’s the breakdown:

• \(\fhi_0(T)\) is the intrinsic free energy without magnetization.
• \(J\) indicates the interaction strength within the system.
• \(m\) is the magnetization, representing the net magnetic moment per mole.
• \(H\) is the external magnetic field.
• \(k_B\), Boltzmann's constant, bridges microscopic and macroscopic measurements.
• \(T\) is the temperature of the system.

This expression captures the combined effects of thermal energy, interaction effects, and applied magnetic fields on the system's energy. It plays a crucial role in determining other properties such as the magnetization and susceptibility.

magnetization

Magnetization (\(m\)) refers to the net magnetic moment per mole within the spin system. It's crucial in studying how aligned the magnetic moments are in response to temperature and external fields. In the given expression for molar free energy, magnetization is highlighted as:

• The term \(-\frac{1}{2} J m^2\) covers the interaction energy among magnetic moments.
• The terms \(\frac{1}{2} k_B T [(1+m) \text{ln}(1+m) + (1-m) \text{ln}(1-m)]\) showcase the entropy contribution related to magnetization.
• The term \(-mH\) shows the energy contribution due to an external magnetic field.

Magnetization changes with temperature, reaching zero at high temperatures where the magnet becomes paramagnetic. Understanding magnetization is essential to grasp key thermal and magnetic behaviors like the Curie temperature.

linear magnetic susceptibility

Linear magnetic susceptibility (\(\fchi\)) measures a material's response to an external magnetic field, indicating how easily it can be magnetized. For small magnetization (near the Curie temperature), the formula is: \(\fchi = \frac{\fleft(\fpartial m\fright)}{\fpartial H}\fgmid_{H \rightarrow 0}\).
Here's how to find it:

• From the molar free energy, the first derivative with respect to \(m\) is simplified to \(-J m + k_B T m - H\).
• At equilibrium with \(H=0\), solve \(J m = k_B T m\) to get the Curie temperature: \(T_c = \frac{J}{k_B}\).
• Above \(T_c\), use the second derivative of \(\fleft(\fphi\fright)\) giving \(k_B (\fleft(T - T_c\fright))\) for susceptibility: \(\fchi = \frac{1}{k_B (T - T_c)}\).

This demonstrates how susceptibility approaches infinity as temperature nears and exceeds the Curie point, reflecting heightened magnetic responsiveness.

spin system

A spin system comprises particles with intrinsic spin, influencing their magnetic properties. Spins can align in different ways, impacting the system's free energy and magnetic behavior. For our spin system:

• The expression \(\fphi(T, H)\) accounts for all influences on free energy: interaction energy, entropy effects, and external fields.
• Its magnetization behavior, captured by \(m\), responds to temperature and applied field \(H\).
• The critical behavior around the Curie temperature, defined as \(T_c = \frac{J}{k_B}\), marks the transition from ferromagnetic (ordered spins) to paramagnetic (random spins).

Understanding a spin system helps analyze its magnetic properties and predict behaviors under varying temperature and field conditions, crucial in fields like condensed matter physics and materials engineering.