Physics · Molecular physics and thermodynamics · Year 4

Phase transition and fluctuations

Why does a heated magnet suddenly stop attracting, and what trembles at that point?

The situation

We hang several paper clips in a chain from a black ceramic (ferrite) magnet taken from a refrigerator door, and heat it on an electric hotplate. At about 450 °C the clips fall off one after another — the magnet loses its strength. And a glass ampoule of carbon dioxide near its critical point turns milky white. Why does order vanish suddenly at a certain temperature, and what exactly “trembles” at that point?

Topic

Phase transitions and fluctuations. Heating a magnet: magnetisation vanishes at the Curie point (Ising model)

Key ideas

Ising model (Metropolis)
Curie point, Landau theory
Fluctuations, Rayleigh scattering

Formulas

E = - J \sum_{⟨ i j ⟩} s_{i} s_{j} - h \sum_{i} s_{i}, s_{i} = \pm 1

The Ising model: if neighbouring spins point the same way, the energy is lower ( $J > 0$ — ferromagnetic).

P = \min (1, e^{- Δ E / k T})

The Metropolis algorithm: the computer flips a spin with a Boltzmann probability.

k T_{C} = \frac{2 J}{\ln (1 + \sqrt{2})} \approx 2.269 J, m = {(1 - \sinh^{- 4} \frac{2 J}{k T})}^{1 / 8}

Onsager's exact solution (1944): the Curie point; the formula for spontaneous magnetisation (announced by Onsager, proved by Yang in 1952).

F (m) = F_{0} + a (T - T_{C}) m^{2} + b m^{4} - h m

Landau theory: a series in the order parameter $m$ . For $T < T_{C}$ there are two minima.

χ = \frac{N (⟨ m^{2} ⟩ - {⟨ m ⟩}^{2})}{k T}, C = \frac{⟨ E^{2} ⟩ - {⟨ E ⟩}^{2}}{k T^{2}}, \frac{σ}{⟨ X ⟩} \propto \frac{1}{\sqrt{N}}

Susceptibility and heat capacity come from fluctuations; the relative fluctuation falls as $1 / \sqrt{N}$ .

Theory

Ferromagnetism and the Ising model

In ferromagnets such as iron and nickel, the spins of neighbouring atoms tend to point the same way because of the exchange interaction (a quantum effect) — macroscopic regions, domains, form (in ferrites the moments of the two sublattices are opposite but unequal — ferrimagnetism, and the net magnetisation is not zero). Thermal motion destroys this order. The Ising model is the simplest picture: at the lattice sites there are only “up” or “down” spins. The lattice on the screen is a “magnified” view of a small region inside a magnet; we calculate it with the Metropolis method (Monte Carlo). Ferrites (iron oxides) are the material of ceramic magnets and transformer cores. The laboratory on hysteresis and domains is in the “Magnetic properties of matter” experiment.

Determining the Curie point (lab)

The order parameter is the magnetisation $m$ : it falls to zero at $T_{C}$ . At the Curie point the susceptibility $χ$ and the heat capacity $C$ reach a maximum — this is how $T_{C}$ is found. Change the temperature step by step and record $m$ and $χ$ at each point (or switch on automatic measurement). On a finite lattice the peak lies slightly above $T_{C}$ : for L = 64 at $\approx 2.30$ – $2.35 J / k$ . The temperature scale is fitted to the chosen material: $T = (T_{I s i n g} / 2.269) \cdot T_{C}$ ; for the ferrite $T_{C} \approx 450$ °C, for iron 770 °C, for gadolinium ≈ 20 °C.

First- and second-order phase transitions

The Ehrenfest classification: in a first-order transition the first derivatives of the free energy (density, magnetisation, entropy) jump — latent heat, hysteresis, metastable states (boiling, melting). For $T < T_{C}$ , if you sweep the field $h$ from positive to negative, $m$ jumps and a loop forms. In a second-order transition $m$ falls to zero continuously, and the second derivatives — $C$ , $χ$ — jump or tend to infinity (the Curie point, the liquid–vapour critical point). Along the line of second-order transitions these jumps are linked by the Ehrenfest equations.

Landau theory

Landau expanded $F$ as a series in the order parameter: $F = F_{0} + a (T - T_{C}) m^{2} + b m^{4} - h m$ . For $T > T_{C}$ there is one minimum ( $m = 0$ ); for $T < T_{C}$ there are two symmetric minima — the system chooses one of them (spontaneous symmetry breaking; the ball on the track shows this). The theory gives $m_{0} \propto {(T_{C} - T)}^{1 / 2}$ and a jump in $C$ . In the exact 2D solution, however, $m \propto {(T_{C} - T)}^{1 / 8}$ : near $T_{C}$ the fluctuations are so large that the mean-field approach breaks down (the Ginzburg criterion) — compare them on the graph.

Fluctuations: the Gaussian distribution and 1/√N

The deviation of a quantity from its mean value is a fluctuation, and its measure is the variance $σ^{2} = ⟨ x^{2} ⟩ - {⟨ x ⟩}^{2}$ . In thermodynamic theory the probability of a deviation is $\propto e^{Δ S / k}$ , which leads to the Gaussian distribution. The relative fluctuation of an additive quantity made up of N independent parts is $\propto 1 / \sqrt{N}$ — unnoticeable in a macroscopic body. Fluctuations are linked to response functions: $⟨ Δ E^{2} ⟩ = k T^{2} C$ , $⟨ Δ M^{2} ⟩ = k T χ$ — this is exactly how $C$ and $χ$ are calculated in the experiment. In small systems even the work done fluctuates. Thermal fluctuations (the trembling of a galvanometer mirror, noise in a resistor, fluctuations in a photon gas) set the sensitivity limit of measuring instruments.

Correlations and critical opalescence

The pair correlation function $G (r) = ⟨ s_{0} s_{r} ⟩ - {⟨ s ⟩}^{2}$ shows how much neighbouring regions fluctuate “in agreement”: $G \propto e^{- r / ξ}$ , and the correlation length $ξ$ tends to infinity at $T_{C}$ . For many-particle distribution functions, in the Bogoliubov chain of equations each equation depends on the next; in practice the chain is cut off by an approximation. If we treat the lattice as a lattice gas (spin “up” = a molecule), density fluctuations scatter light: the Rayleigh law $I \propto 1 / λ^{4}$ — the sky is blue. At the critical point the compressibility and the fluctuations are enormous — the substance turns milky white (critical opalescence). Light scattered at 90° is polarised.

Did you know?

In 1944 Lars Onsager exactly solved the two-dimensional Ising model (without a field). The critical temperature $k T_{C} = 2 J / \ln (1 + \sqrt{2})$ had been found before him, in 1941, by H. Kramers and G. Wannier using symmetry (duality). The model is named after Ernst Ising: in 1925 he showed that there is no phase transition in a one-dimensional chain.
M. Smoluchowski explained critical opalescence in 1908 by density fluctuations, and in 1910 A. Einstein derived the formula for the scattering intensity. The critical point of carbon dioxide is about 31 °C and 7.4 MPa.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Refrigerator and heat pump — Grade 10
Real gases — Grade 10
Maxwell–Boltzmann distribution — Grade 10
Transport phenomena — Year 1
Quantum statistics — Year 4