Calculator
Formula used
Mean-field theory replaces neighbor spins by their average magnetization m. The effective field becomes J z m + H. The self-consistency equation is:
m = tanh( (J z m + H) / (kB T) )
The mean-field critical temperature for H = 0 is Tc = (J z)/kB. The calculator also reports susceptibility and thermodynamic densities from standard mean-field expressions.
How to use this calculator
- Set J, z, and kB for your units.
- Choose H to break symmetry or keep it zero.
- Select Single T for one state, or Sweep T for a curve.
- Use bisection near the critical region for stability.
- Increase steps in sweeps to resolve rapid changes near Tc.
- Export CSV or PDF for plotting and documentation.
Example data table
These sample inputs help you test ordered and disordered regimes.
| J | z | kB | H | T mode | T or range | Expected behavior |
|---|---|---|---|---|---|---|
| 1.0 | 4 | 1.0 | 0.0 | Sweep | 1.0 to 5.0 | Order below Tc, rapid change near Tc |
| 1.0 | 6 | 1.0 | 0.02 | Sweep | 2.0 to 8.0 | Smoothed transition due to small field |
| 0.8 | 4 | 1.0 | 0.0 | Single | T = 3.5 | Likely weak magnetization above Tc |
Mean-Field Ising Guide
1) What the calculator models
This tool evaluates the mean-field Ising model, a standard approximation for collective behavior in interacting two-state systems. Each spin takes values ±1 and interacts with z neighbors through coupling J, while an external field H biases alignment. Mean-field theory replaces neighbors by their average magnetization m, turning the many-body problem into one self-consistent equation.
2) Core self-consistency equation
The central relation is m = tanh((J z m + H)/(kB T)). Here T controls thermal disorder, and kB sets the energy scale. At high temperature, the hyperbolic tangent becomes nearly linear and m approaches zero. At low temperature, multiple solutions may exist when H = 0.
3) Critical temperature and phase change
For zero field, mean-field theory predicts a critical point at Tc = (J z)/kB. Below Tc, the model supports nonzero magnetization solutions, representing an ordered phase. Above Tc, the stable solution is typically m ≈ 0. In sweeps, you will often see rapid changes near Tc.
4) Susceptibility and response
The calculator reports susceptibility χ = dm/dH, derived by differentiating the self-consistency equation. Near the critical region, χ can become very large, reflecting strong sensitivity to small fields. If the denominator approaches zero numerically, the tool may display INF, indicating a near-divergent response in the approximation.
5) Free energy and internal energy outputs
Thermodynamic densities provide context beyond m. The mean-field free energy per spin combines interaction, field work, and entropic contributions through a ln(2 cosh(·)) term. Internal energy per spin is reported as u = −(J z/2)m² − H m, helping you compare energetic ordering across parameter sets.
6) Heat capacity estimate and data interpretation
Heat capacity is estimated as a numerical derivative cv ≈ du/dT using a small symmetric temperature step. In mean-field theory, cv can show a kink or peak near the transition, depending on chosen parameters and step size. Reduce dT for sharper features, but avoid values so small that rounding noise dominates.
7) Sweep mode for practical analysis
Sweep mode generates a temperature series and solves for m at each point using continuation: the previous solution becomes the next initial guess. This is helpful for mapping curves smoothly. Typical educational settings use J = 1, kB = 1, and z = 4 or 6, giving Tc = z.
8) Solver guidance and stability tips
The bisection solver is robust and recommended near Tc, where fixed-point iteration can slow down or oscillate. If you use fixed-point mode, adjust relaxation (smaller values are steadier). For symmetry-broken solutions at H = 0, provide a nonzero initial guess to select a branch. Export CSV to plot m(T) and χ(T) for reporting.
FAQs
1) What does z represent in this model?
z is the coordination number, the count of nearest neighbors per site. It rescales the effective interaction through Jz and shifts the predicted critical temperature.
2) Why is Tc equal to (J z)/kB?
At H = 0, linearizing tanh near m = 0 gives m ≈ (J z/(kB T)) m. A nontrivial solution appears when the slope reaches one, yielding Tc = J z/kB.
3) Why can there be multiple magnetization solutions?
Below Tc with H = 0, the equation admits symmetric branches ±m because the free energy has two minima. A small field or initial guess selects which branch the solver follows.
4) When should I prefer bisection over fixed-point iteration?
Use bisection near Tc or when results oscillate. Bisection converges reliably if a sign change bracket exists, while fixed-point iteration can slow or fail without enough damping.
5) Why does susceptibility sometimes show INF?
Mean-field susceptibility contains a denominator that can approach zero near the critical region. Numerically, a very small denominator produces extremely large values, displayed as INF.
6) What does the heat capacity estimate represent?
It is a finite-difference approximation of du/dT at fixed H. It highlights where energy changes rapidly with temperature, often near the transition, depending on dT and parameters.
7) How can I reproduce standard textbook curves?
Set kB = 1 and J = 1, choose z = 4 or 6, use H = 0, then run a sweep across Tc. Export CSV and plot m(T) and χ(T).