Phase Transitions

Phase transitions occur when a system undergoes a qualitative change in its macroscopic properties. First-order transitions involve latent heat; second-order (continuous) transitions are characterized by order parameters and critical exponents near the critical point.

Key Concepts

Order parameter: quantifies degree of order (e.g., magnetization M)
First-order: discontinuous order parameter (e.g., liquid-gas)
Second-order: continuous order parameter, diverging correlation length
Ising model: spins ±1 with nearest-neighbor coupling J
Mean-field critical temperature: T_c = zJ/k_B (z = coordination number)

Key Equations

Ising Hamiltonian

H = -J\sum_{\langle ij\rangle} s_i s_j - h\sum_i s_i

Spontaneous magnetization (MF)

M = \tanh(zJM/k_BT)

Curie-Weiss susceptibility

\chi = \frac{C}{T - T_c}

Critical exponent (MF)

M \propto |T - T_c|^{1/2}\text{ as }T\to T_c^-

Worked Example

Example Problem

Problem

A 2D square Ising model has J=0.1 eV, coordination z=4. Estimate the mean-field T_c in K.

Solution

T_c = zJ/k_B = 4×0.1×1.6×10⁻¹⁹/1.38×10⁻²³ = 4640 K. (Exact 2D Ising: T_c = 2J/(k_B ln(1+√2)) ≈ 2.27J/k_B = 2630 K.)

Practice

Exercises

7 problems

1 of 7

A 1D Ising chain has J=0.02 eV per bond. What is the mean-field T_c in K for z=2?

Your answer

2 of 7

The Curie-Weiss susceptibility has C=1.0 (SI) and T_c=300 K. Find χ at T=350 K.

Your answer

3 of 7

Near T_c, the order parameter scales as M ∝ (T_c-T)^β with β=1/2 (mean field). If M=0.5 at T=T_c-4 K, find M at T=T_c-1 K.

Unlock Exercise 3