Ising Magnetization Calculator

Formula Used

This calculator supports three commonly used magnetization models for the Ising system. Use consistent units for T, J, h, and kB.

• 2D square exact (zero field):
  For T < Tc, the spontaneous magnetization is
  m(T) = [ 1 − sinh(2J/(kB·T))⁻⁴ ]^1/8
  with Tc = 2J / (kB·ln(1+√2)). For T ≥ Tc, m = 0.
• 1D exact (field allowed):
  m = sinh(βh) / √( sinh²(βh) + exp(−4βJ) )
  where β = 1/(kB·T).
• Mean-field (iterative):
  m = tanh( β( zJm + h ) )
  The solver iterates until the change in m is below the tolerance or the iteration limit is reached.

How to Use This Calculator

1. Choose a Method. Use 2D exact for spontaneous magnetization with zero field.
2. Enter T (temperature) and J (coupling). Use reduced units with kB=1 if desired.
3. If you need a biasing field, enter h and choose 1D exact or mean-field.
4. For mean-field, set z, m0, tolerance, and damping for stable convergence near criticality.
5. Press Calculate. The results appear above the form, and you can export them as CSV or PDF.

1) Magnetization as an order parameter

The Ising model assigns each spin a value of +1 or −1 and couples neighbors through J. The normalized magnetization m is the thermal average spin, so m≈1 signals strong alignment and m≈0 signals disorder. A nonzero field h biases spins and shifts the response curve.

2) Reduced and SI unit choices

For quick studies, reduced units are common: set kB=1 and often J=1. Then temperature is dimensionless and comparisons across runs are direct. If you select SI kB, keep J and h in matching energy units so β=1/(kB·T) remains consistent.

3) Exact 2D square lattice at zero field

This calculator includes the classic closed-form spontaneous magnetization for the 2D square lattice when h=0. For T<Tc, magnetization follows m(T)=[1−sinh(2J/(kB·T))^{−4}]^{1/8}. For T≥Tc, symmetry is restored and m is set to zero.

4) Critical temperature and reduced temperature

The 2D square critical point is Tc=2J/(kB·ln(1+√2)), numerically Tc≈2.269·J/kB. Reporting the reduced temperature T/Tc helps compare different J and kB choices on a single scale. Near T/Tc≈1, m drops rapidly toward zero.

5) Exact 1D response in a field

The one-dimensional model has no finite-temperature transition at h=0, but it has an exact magnetization with a field: m=sinh(βh)/√(sinh²(βh)+exp(−4βJ)). It is a useful baseline for understanding how temperature suppresses alignment and how even small h induces a measurable response.

6) Mean-field model and coordination number

Mean-field replaces neighbors with their average, giving the fixed-point equation m=tanh(β(zJm+h)). It predicts Tc=zJ/kB at zero field. Typical coordination numbers are z=4 (2D square), z=6 (3D simple cubic), and z=8 (bcc). Use z that matches your lattice geometry.

7) Iteration stability near criticality

Close to Tc, fixed-point iterations may oscillate or converge slowly. The damping control blends old and new estimates to stabilize updates. If convergence is difficult, try a smaller damping value (0.2–0.6), increase iterations, and start with a small nonzero m0. The iteration count helps judge numerical reliability.

8) Using exports for analysis

Use CSV when you want to sweep many temperatures, plot m(T), or fit critical behavior in a spreadsheet or script. Use PDF when you need a clean record of a single run for lab notes, homework solutions, or reports. For temperature sweeps, repeat runs and export to build a dataset quickly. Exports are generated from the latest computed values shown above the form.

FAQs

1) What range should magnetization m fall into?

For normalized spins ±1, magnetization is typically between −1 and +1. Values near ±1 indicate strong alignment, while values near 0 indicate disorder or balanced up/down spins.

2) Why does the 2D exact option ignore the field h?

The closed-form 2D spontaneous magnetization implemented here is valid for zero external field. With nonzero field, the exact expression is more complex, so the calculator warns and still computes using h=0 assumptions.

3) What is a good default for reduced units?

Set kB=1 and choose J=1. Then temperatures are dimensionless and the 2D critical point is Tc≈2.269. This is convenient for learning and quick comparisons.

4) How do I pick the coordination number z?

Use the number of nearest neighbors in your lattice. Typical choices are z=4 for a 2D square lattice, z=6 for a 3D simple cubic lattice, and z=2 for a 1D chain.

5) My mean-field result does not converge. What should I change?

Lower the damping (for example 0.3), increase max iterations, and use a small nonzero m0. Near Tc, convergence can be slow, so tighter tolerance may require more iterations.

6) Why can magnetization be nonzero above Tc when h ≠ 0?

An external field biases spins and breaks symmetry, producing a finite response even at high temperatures. The phase transition refers to spontaneous ordering at zero field; with a field, magnetization changes smoothly.

7) Which export should I use for plots and reports?

CSV is best for plotting, fitting, and archiving many runs in spreadsheets. PDF is ideal for a quick, readable snapshot of one run, including inputs, method choice, and computed outputs.

Example Data Table