Cross-Section Computation

A scattering computation becomes physically useful only after its amplitudes are converted to flux-normalized observables. Cross-section evaluation adds angular-momentum truncation, angular resolution, and cancellation errors to the radial solver’s uncertainty. The formulas here apply to a spinless distinguishable-particle central-potential problem with a short-range exterior convention.

Required background. Cross sections and partial-wave cross sections define the physics. Phase extraction supplies numerical $S_\ell$.

Amplitudes from the scattering matrix

For incident wave number $k>0$,

$$f(\theta)=\frac1{2ik} \sum_{\ell=0}^{L}(2\ell+1)(S_\ell-1) P_\ell(\cos\theta), \qquad \frac{d\sigma}{d\Omega}=|f(\theta)|^2.$$

For elastic scattering, $S_\ell=e^{2i\delta_\ell}$. If phases are small, evaluate $S_\ell-1$ with a stable exponential-minus-one operation or the equivalent $2ie^{i\delta_\ell}\sin\delta_\ell$; direct subtraction can lose relative accuracy.

Compute Legendre polynomials by a stable recurrence in the required angular range. Sum complex amplitudes before taking their modulus. Summing $|f_\ell|^2$ at each angle removes interference and gives a wrong differential cross section.

Integral and optical checks

Orthogonality gives

$$\sigma_{\rm el} =\frac{\pi}{k^2}\sum_{\ell=0}^{L} (2\ell+1)|S_\ell-1|^2.$$

For elastic unitary scattering this reduces to $4\pi k^{-2}\sum_\ell(2\ell+1)\sin^2\delta_\ell$. If flux is lost from the retained elastic channel,

$$\sigma_{\rm reaction} =\frac{\pi}{k^2}\sum_\ell(2\ell+1)(1-|S_\ell|^2).$$

For passive absorption it is nonnegative. The optical theorem uses $\sigma_{\rm total}=\sigma_{\rm el}+\sigma_{\rm reaction}$, giving $4\pi\,\operatorname{Im}f(0)/k$. Comparing only the elastic sum with the optical total in an absorbing problem would falsely diagnose a numerical failure.

For a truncation at $L$, $|f|^2$ is a polynomial of degree at most $2L$ in $x=\cos\theta$. Thus Gauss–Legendre quadrature with at least $L+1$ points integrates that truncated expression exactly in exact arithmetic. This is an independent implementation check, not a guarantee that the omitted partial waves are negligible.

An isotropic benchmark

If only the $s$ wave is nonzero,

$$f=\frac{e^{i\delta_0}\sin\delta_0}{k}, \qquad \sigma_{\rm el}=\frac{4\pi}{k^2}\sin^2\delta_0.$$

At $\delta_0=\pi/2$, $f=i/k$ and $\sigma_{\rm el}=4\pi/k^2$. Integrating the angle-independent differential result and applying the optical theorem must agree with that number. This tests the $2\ell+1$, $2ik$, and solid-angle factors simultaneously.

For a purely absorbing $s$ wave with $S_0=0$, elastic and reaction cross sections each equal $\pi/k^2$, while the total is $2\pi/k^2$. Absorption does not mean zero elastic scattering.

Error budgets at difficult angles

A phase error can matter strongly near an interference minimum, where the exact amplitude is nearly zero. Use absolute error there, and avoid plotting unstable relative errors. The amplitude perturbation obeys

$$|\Delta f(\theta)| \leq\frac1{2k} \sum_\ell(2\ell+1)|\Delta S_\ell|$$

because $|P_\ell(x)|\leq1$ on $[-1,1]$. Forward scattering adds partial waves coherently and often needs a larger cutoff than an integrated observable.

Increase $L$ and angular resolution independently. For Coulomb tails, separate the known long-range amplitude and use the appropriate Coulomb-modified phases; the short-range partial-wave sum is not convergent in the same naive sense. Identical particles and spin channels require symmetry and statistical factors defined in their domain treatment.

Exercises

For $L=3$, how many Gauss–Legendre points suffice to integrate the truncated differential cross section exactly?

Solution

The polynomial degree is at most six. Four-point Gauss–Legendre quadrature is exact through degree seven, so four points suffice. This tests quadrature of the truncated amplitude; the physical cutoff may still need to increase beyond three.

References

• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.