Analyze partial-wave scattering with flexible physics inputs. Switch methods, inspect outputs, and compare energy trends. Export clean reports, tables, and charts for study today.
Large screens show 3 columns, smaller screens show 2, and mobile shows 1.
The chart compares phase shift and total partial cross section across the selected energy range.
These rows are illustrative sample scenarios for checking the interface and output style.
| Method | Energy (MeV) | l | Key inputs | Approx. phase shift | Use case |
|---|---|---|---|---|---|
| Hard-sphere | 5.0 | 0 | a = 3.5 fm, μ = 469.5 MeV/c² | Moderate positive shift | Simple elastic reference model |
| Effective-range | 2.0 | 0 | as = 5.2 fm, re = 1.8 fm | Large low-energy s-wave shift | Threshold behavior study |
| Direct S-matrix | 8.0 | 1 | Sl = 0.54 + 0.72i | Complex elastic and reaction output | Inelasticity inspection |
| Coefficients | 6.0 | 2 | A = 1.0, B = 0.25 | Small positive phase | Wave matching exercises |
The calculator first converts energy into a wave number using
k = √(2μE) / ħc
For an impenetrable sphere, the boundary condition gives
tan(δl) = jl(ka) / nl(ka)
Near threshold, the s-wave obeys
k cot(δ0) = -1/as + (rek²)/2
If Sl = ηe2iδl, then
δl = ½ arg(Sl) and η = |Sl|
For elastic channels, the calculator uses
σl = (4π/k²)(2l+1) sin²(δl)
For direct S-matrix input, the elastic and reaction contributions are separated using η.
A scattering phase shift measures how an interaction changes the phase of an outgoing partial wave relative to free motion. It is a compact way to describe elastic scattering dynamics at each angular momentum.
The reduced mass sets the momentum scale for two-body scattering. It appears in the wave-number relation and therefore affects the phase shift, cross sections, and the energy trend shown in the graph.
Use the hard-sphere option when you want a simple elastic benchmark with an excluded radius. It is useful for intuition, testing, and rough comparisons against more realistic interaction models.
It approximates low-energy s-wave scattering with the scattering length and effective range. This is especially helpful near threshold, where detailed potential information is often replaced by compact parameters.
When |Sl| is below 1, part of the incoming flux leaves the elastic channel. The calculator interprets that loss through η and reports a reaction contribution alongside the elastic partial cross section.
A phase shift near 90° can suggest resonance-like behavior, especially when it changes rapidly with energy. The calculator adds a resonance note, but you should still confirm the interpretation with a fuller physical model.
That convention matches a common asymptotic wave decomposition where coefficient ratios encode the phase offset. If your source uses a different sign or normalization, transform the coefficients before entering them.
No. The results are only as accurate as the selected model and your inputs. The tool is designed for study, estimation, and structured comparisons rather than replacing a full numerical scattering solver.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.