Partial Waves and Threshold Scattering

For a central potential, rotational symmetry separates scattering into angular-momentum channels called partial waves. Each channel carries a radial phase shift. At low energy, only a few channels may matter, and a small number of threshold parameters can describe an interaction even when the potential is too strong for Born theory.

Helpful background. The scattering amplitude fixes the outgoing-wave convention; spherical harmonics supply the angular basis. Operators and unitarity explains the probability-conservation constraints used below.

Replace an angular pattern by radial channels

Begin with the partial-wave expansion. For a scalar particle in a central potential, each angular momentum $\ell$ satisfies a radial equation with a centrifugal term. Outside a short-range interaction, the radial wave can be compared with its free counterpart. Phase shifts defines the resulting $\delta_\ell(k)$ and explains why it is defined modulo $\pi$ unless a continuous branch is chosen.

For a single elastic channel, $S_\ell=e^{2i\delta_\ell}$ with real $\delta_\ell$. The partial-wave cross sections add the contributions to the angle-integrated observable. In the distinguishable-particle convention,

$$\sigma_{\mathrm{el}}=\frac{4\pi}{k^2} \sum_{\ell=0}^{\infty}(2\ell+1)\sin^2\delta_\ell.$$

The angular-momentum separation is exact. Estimating the phase shifts or stopping the sum at finite $\ell$ introduces an approximation. For an interaction of range $R$, $kR$ gives a useful first estimate of how many angular-momentum channels may contribute, but convergence of the observable should decide the cutoff.

Understand what low energy simplifies

Low-energy scattering explains the suppression of higher partial waves for suitably short-range interactions. The $s$-wave often dominates when $kR\ll1$, provided exchange symmetry permits it and there is no exceptional higher-wave resonance at threshold.

The scattering length introduces the leading $s$-wave parameter $a$. For finite $a$ at sufficiently low momentum,

$$f_0(k)\approx\frac1{-a^{-1}-ik}, \qquad \sigma_0\approx\frac{4\pi a^2}{1+k^2a^2}.$$

This retains the imaginary term required by elastic unitarity. The simpler plateau $\sigma_0\approx4\pi a^2$ needs $k|a|\ll1$ as well. If $|a|\gg R$, the plateau and the range scale are widely separated; low energy alone does not mean that $ka$ is small.

The effective-range expansion adds the next term,

$$k\cot\delta_0=-\frac1a+\frac12r_e k^2+\cdots.$$

Use it in $f_0=(k\cot\delta_0-ik)^{-1}$ rather than expanding away the unitarity term. The range of validity depends on the potential’s tail and nearby singularities. Narrow resonances can introduce a large effective range, making a scattering-length-only description insufficient even when $kR$ is small.

Three settings that need a different treatment

Long-range forces. Coulomb scattering has a logarithmic asymptotic phase. Ordinary short-range boundary conditions and threshold formulas cannot simply be reused for an unscreened $1/r$ potential.

Indistinguishable particles. Identical-particle scattering combines direct and exchange amplitudes and selects allowed spatial symmetries. For identical spin-polarized fermions, the symmetric spatial $s$-wave is forbidden. Spin preparation and final-state counting therefore change the cross section.

One spatial dimension. One-dimensional scattering revisited organizes left/right leads and, for a symmetric potential, parity channels. Three-dimensional angular-momentum cross sections do not directly apply to reflection and transmission probabilities.

The hard-sphere example and low-energy square-well comparison are useful next calculations. Continue to poles and resonances to understand why threshold parameters can become large.

Exercises

The largest elastic s-wave contribution. Use the partial-wave formula to find the maximum $\sigma_0$ at fixed $k>0$.

Solution

Since $\sin^2\delta_0\leq1$, $\sigma_0\leq4\pi/k^2$. Equality occurs at $\delta_0=\pi/2$ modulo $\pi$. This is the distinguishable-particle elastic bound; exchange restrictions and multiplicities must be treated separately for identical particles.

Does a positive scattering length prove a bound state? A hard sphere of radius $R$ has $a=R>0$. Can one infer a bound-state energy $-\hbar^2/(2\mu a^2)$, where $\mu$ is the reduced mass?

Solution

No. A purely impenetrable repulsive sphere has no bound state. The shallow-bound-state relation requires the relevant pole to lie in the domain where the low-energy continuation is accurate, typically with $a$ much larger than the range and controlled range corrections. For the hard sphere, $a=R$, so extending the scattering-length approximation to $k=i/a$ reaches the range scale and is unjustified.

References

• Newton, R. G. Scattering Theory of Waves and Particles. 2nd ed., Dover (2002; reprint of the 1982 edition). Partial waves, threshold behavior, and long-range scattering.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Phase shifts, cross sections, and channel conventions.