Inputs
Use Joules for strict SI. eV is also supported and converted internally.
Example Data
These sample rows illustrate typical inputs and the resulting equilibrium magnetization.
| z | T (K) | J (J) | H (J) | Expected trend |
|---|---|---|---|---|
| 6 | 300 | 1.0e-21 | 0 | Often disordered if T > Tc |
| 6 | 30 | 1.0e-21 | 0 | More ordering as T decreases |
| 8 | 300 | 3.0e-21 | 0 | Higher Tc can allow ordering |
| 6 | 300 | 1.0e-21 | 2.0e-22 | Field biases magnetization sign |
Formula Used
The Bragg-Williams mean-field condition treats spins as uncorrelated and replaces neighbors by an average magnetization. The equilibrium magnetization m satisfies the self-consistent equation:
A convenient free-energy density per spin (up to an additive constant) is:
The calculator reports the equilibrium value m_eq, the critical temperature Tc = zJ/kB, the reduced temperature T/Tc, and the corresponding F, U, and S per spin.
How to Use This Calculator
- Enter z, T, and the coupling J in J or eV.
- Set H to zero for spontaneous ordering, or use a small bias.
- Choose an initial guess m0 if you expect a specific branch.
- Adjust max iterations and tolerance near criticality.
- Press Calculate and export results using CSV or PDF.
Bragg-Williams Mean-Field Modeling for Ordering
1) What the model represents
The Bragg-Williams approach is a mean-field description of cooperative ordering where each site interacts with an averaged neighborhood. In magnetic language, the order parameter is the magnetization per spin, m, constrained to the range −1 to +1. This calculator solves the self-consistent condition and reports thermodynamic quantities per spin to support quick comparison across materials, lattices, and temperature sweeps.
2) Core equation and parameters
The equilibrium condition uses m = tanh((zJm + H)/(kB T)), where z is the coordination number, J is the coupling energy, and H is the field-like bias term. The constant kB = 1.380649×10⁻²³ J/K anchors the temperature scale. Entering J in eV is supported using 1 eV = 1.602176634×10⁻¹⁹ J.
3) Critical temperature and lattice data
In this mean-field theory, the critical temperature is Tc = zJ/kB. Typical coordination numbers include simple cubic z = 6, body-centered cubic z = 8, and face-centered cubic z = 12. Increasing z or J raises Tc, making ordering possible at higher temperatures within the model assumptions.
4) Free energy, energy, and entropy outputs
The calculator evaluates a standard mean-field free-energy density per spin, combining an interaction term −(zJ/2)m², a bias contribution −Hm, and a mixing entropy term based on probabilities (1±m)/2. It also reports internal energy U and entropy S = (U − F)/T, helping you track energetic stabilization versus entropic disorder.
5) Reduced temperature for comparisons
The reduced temperature T/Tc enables data collapse across different lattices or couplings. Values above 1 typically yield small m when H = 0, while values below 1 can support nonzero solutions. For sensitivity studies, sweep T and keep z fixed, then compare the resulting m_eq and F.
6) Numerical stability near Tc
Close to Tc, convergence slows because the slope of the fixed-point map approaches unity. The solver uses damping to improve stability. If results appear branch-dependent, try a smaller nonzero H, increase max iterations, or provide a physically motivated initial guess m0, such as 0.01 for weak ordering or 0.8 for strongly ordered states.
7) Interpreting the field term
The parameter H biases the sign and magnitude of m. With H > 0, solutions tend toward positive magnetization, and with H < 0 they tend negative. Using a small bias is a practical way to select a stable branch when multiple solutions exist below Tc.
8) Practical workflow and exports
Start with lattice-appropriate z, select a physically plausible J, and compute Tc to plan your temperature range. Then run a series of temperatures and export each result to CSV for plotting m_eq versus T/Tc, or to PDF for documentation. This supports quick lab-style reporting and consistent classroom demonstrations.
FAQs
1) What does Bragg-Williams assume?
It assumes each site feels an average neighborhood, so correlations are neglected. This simplifies interactions into a mean-field term using the order parameter m.
2) Why can there be multiple solutions below Tc?
Below Tc the free-energy landscape can develop two symmetry-related minima. Small changes in m0 or H may select different stable branches with similar magnitude but opposite sign.
3) How should I choose z?
Use the coordination number for your lattice: simple cubic 6, body-centered cubic 8, and face-centered cubic 12 are common. If unknown, start with 6 and test sensitivity.
4) What units should J and H use?
Use Joules for strict SI. You may also enter eV, which is converted internally using 1 eV = 1.602176634×10⁻¹⁹ J.
5) Why does convergence slow near Tc?
Near criticality, the self-consistency map becomes nearly tangent to the identity line. Increasing iterations, tightening tolerance carefully, or adding a tiny bias H improves stability.
6) Are F, U, and S absolute values?
They are per-spin quantities within a chosen reference, so an additive constant may differ between conventions. Trends versus temperature and parameter changes are the most meaningful outputs.
7) What should I export for plotting?
Export m_eq and T/Tc to build a reduced curve, and include F to compare stability. CSV is convenient for spreadsheets, while PDF is useful for reports and sharing.