| E (MeV) | V0 (MeV) | a (fm) | μ (MeV/c²) | l_max | k (1/fm) | σ_total (barn) |
|---|---|---|---|---|---|---|
| 5 | 50 | 2 | 469 | 6 | 0.245 | ~0.2–1 (depends on δ_l) |
For a spherical potential V(r)=V for r<a and 0 outside, the outside wave number is
k = √(2μE)/ħ, q = √(2μ(E−V))/ħ
Matching the radial solution at r=a gives the phase shift for each angular momentum l:
tan δ_l = [k j'_l(ka) j_l(qa) − q j_l(ka) j'_l(qa)] / [k y'_l(ka) j_l(qa) − q y_l(ka) j'_l(qa)]
Here j_l and y_l are spherical Bessel and Neumann functions. The partial cross section is
σ_l = (4π/k²)(2l+1) sin²(δ_l), σ_total = Σ_l σ_l
The differential cross section uses the scattering amplitude f(θ) built from Legendre polynomials.
- Select a unit system that matches your problem domain.
- Choose an attractive well or repulsive barrier, then set V0.
- Enter energy E, radius a, and reduced mass μ.
- Start with l_max around 6–12, then increase if needed.
- Compute and review δ_l and σ_l for convergence.
- Use the angle table to study anisotropy in dσ/dΩ.
- Export CSV for analysis or PDF for reporting.
1) Why phase shifts matter
In a central potential, each angular-momentum channel is modified by a phase shift δ_l. These δ_l encode the on-shell S-matrix through S_l = exp(2 i δ_l). A rapid change of δ_l with energy often signals strong interaction or a quasi-bound resonance near the scattering threshold.
2) Model parameters and physical scales
The calculator uses a spherical square well or barrier of radius a and strength V0. Two dimensionless scales guide interpretation: ka and qa. For nuclear inputs, k and q are in 1/fm; for SI, they are in 1/m. When ka is small, only low l contribute.
3) Computing k and q from energy
Wave numbers follow nonrelativistic kinetics: k = sqrt(2 μ E)/ħ outside, and q = sqrt(2 μ (E − V))/ħ inside. For a well, V is negative, so q is larger than k at the same E. For a barrier with E below V0, q becomes imaginary.
4) Boundary matching at the radius
Phase shifts are obtained by matching the radial wavefunction and its derivative at r = a. The implementation evaluates spherical Bessel j_l and Neumann y_l at ka, plus j_l at qa, then forms tan δ_l from the standard ratio of matching combinations. The atan2 function preserves the correct quadrant.
5) Resonances and unitarity limits
A practical marker is δ_l near π/2, where sin²δ_l approaches one. In that regime, the l-th partial cross section approaches its unitarity bound, scaling like (4π/k²)(2l+1). Resonance-like behavior typically appears as δ_l crosses π/2 while varying quickly with E, V0, or a.
6) From δ_l to σ_l and σ_total
From δ_l, the code reports sin²δ_l and the partial cross section σ_l, then sums σ_total. In nuclear mode it also converts area to barns using 1 barn = 100 fm², which is convenient for magnitude checks. Convergence is indicated when increasing l_max produces negligible change in σ_total and the dominant σ_l terms.
7) Angular distributions with Legendre sums
Angular structure is computed from f(θ) = (1/k) Σ_l (2l+1) e^{iδ_l} sin(δ_l) P_l(cos θ), followed by dσ/dΩ = |f(θ)|². The table samples θ from 0° to 180° by default, or uses a custom list for targeted angles such as forward and backward scattering.
8) Practical convergence checks
For stable results, begin with l_max between 6 and 12 and increase until both σ_total and the dσ/dΩ pattern stop changing. If ka is large, more partial waves are required because higher l channels remain open. Export CSV to fit δ_l(E) trends and PDF to document model settings and outputs.