Yukawa Scattering Lab

Choose E and the asymptotic impact parameter b: the single path updates immediately. Sweep b explicitly compares40 paths and can be cancelled. Changing m, g or E removes the old sweep; changing b updates its current marker. Potential probes update directly. Only input values and the selected view are retained, including unfinished probe text; paths, sweeps, animation position and results are regenerated, never saved. Reset restores the live defaults.

The attractive model is V(r) = −g² exp(−mr)/r and Fᵣ = −g² exp(−mr)(1/r² + m/r). Units are dimensionless with projectile mass μ = ħ = c = 1; m sets inverse range, not a real particle mass. Examples are illustrative. For m = 0 the exact Coulomb model is used; for g = 0 the path is straight. The origin is singular for nonzero coupling.

Screened paths use the Binet equation for w = b/r with incoming conditions at infinity, adaptive RK4 and an energy-invariant check. Inadequate convergence is reported instead of a finite angle. Near orbiting, narrow structures may be missed by the 40-point sweep. χ is the signed outgoing direction in [−180°, 180°]; multiple turns can produce a positive principal angle even with attraction. Gaps denote unresolved samples.

The x/y window is ±8 with equal scales. The dotted circle marks 1/m when it fits; a range outside this window is not drawn. Blue is Yukawa; red dashed is the optional exact Coulomb comparison. The optional trace reveals visible arclength uniformly, not physical time; it pauses when hidden and respects reduced motion.

Potential displays |V| or |Fᵣ| with logarithmic axes. Readings retain the negative sign of V and Fᵣ. Born displays fB(q) = 2g²/(q² + m²) for μ = ħ = 1, also on logarithmic axes. Zero coupling gives zero on a linear vertical scale. The q plot starts at 0.01; the numeric probe also accepts q = 0, where m = 0 and g > 0 diverges. The first Born approximation assumes sufficiently weak scattering; it is not an exact quantum solution or a cross section. Classical trajectories and the quantum approximation are separate views.

Reference: Fitzpatrick: Born approximation.