---
title: "How a circle limits the strength of scattering"
description: "Decompose an angular pattern into partial waves, derive the unitarity circle and calculate a resonance cross section."
canonical_url: "https://sandboxphysics.com/stories/scattering-amplitudes/partial-wave-unitarity/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/scattering-amplitudes/partial-wave-unitarity/"
last_modified: "2026-10-27"
---

[Scattering amplitudes](https://sandboxphysics.com/stories/scattering-amplitudes/)

L1 · Chapter 2

# How a circle limits the strength of scattering

Sandbox Physics · 2026-10-27

Even strong scattering has only a finite incoming probability in each partial wave. Plot its complex amplitude and elastic scattering traces a circle. This geometry turns conservation of probability into a precise restriction on an amplitude.

## Before you begin

Probability current, complex amplitudes and cross sections from the preceding chapter. You need modulus and phase. Angular-momentum partial waves, phase shifts and unitarity are introduced here; only the first two Legendre polynomials are needed initially.

[Review the preceding prerequisite](https://sandboxphysics.com/stories/scattering-amplitudes/flux-and-amplitude/)

- Derive the elastic unitarity circle and calculate a resonant cross section.
- Separate the shared probability argument from the different normalizations of potential and relativistic scattering.

## Build an angular pattern from simpler shapes

A central potential preserves angular momentum. Decompose its wave into sectors numbered $\ell=0,1,2,\ldots$, called partial waves. Their angular dependence is described by Legendre polynomials $P_\ell(\cos\vartheta)$. The first is a constant; the second changes sign with the direction cosine.

Far from the target, each radial wave contains incoming and outgoing spherical waves. A potential changes their relative phase. For a real central potential without absorption, outgoing and incoming currents are equal in each sector. Only a phase remains.

$$
f(\vartheta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)t_\ell P_\ell(\cos\vartheta),\qquad P_0(x)=1,\quad P_1(x)=x
$$

This defines a dimensionless partial-wave amplitude. The wave number still has units of inverse length.

$$
S_\ell=e^{2i\delta_\ell},\qquad t_\ell=\frac{S_\ell-1}{2i}=e^{i\delta_\ell}\sin\delta_\ell
$$

The phase shift specifies the change in the incoming–outgoing relation. The factor of two follows from the usual sinusoidal radial-wave convention.

Source: David Tong

## Draw probability conservation

Substitute $t_\ell=x+iy$ into $|S_\ell|=1$. Expand the modulus and complete the square. The resulting circle is centered halfway up the imaginary axis and has radius one half. Varying an elastic phase shift moves a point around its boundary.

If other final states can carry probability, the elastic outgoing current can be smaller. The allowed point moves into the disk. A point outside would require more elastic outgoing probability than entered. Unitarity preserves probability for the complete scattering operator; restricting attention to one channel gives an inequality.

$$
|1+2it_\ell|^2\leq1\quad\Longleftrightarrow\quad\operatorname{Im}t_\ell\geq|t_\ell|^2
$$

Equality holds when this partial wave has only elastic scattering.

$$
x^2+\left(y-\frac12\right)^2\leq\frac14
$$

The interior allows probability to leave the elastic channel. The exterior is forbidden.

Partial-wave amplitude

$t=e^{i\delta}\sin\delta$

Gold point: elastic phase shift of sixty degrees. The circle center allows elastic probability to leave the channel. $(\operatorname{Re}t,\operatorname{Im}t)=(\sqrt3/4,3/4)$

## Worked example with one surviving partial wave

When the wavelength greatly exceeds the potential range, centrifugal barriers usually suppress higher angular momenta. Keep only $\ell=0$, whose angular shape is constant. Integrating the differential cross section over the sphere gives the expression below.

Set $k=2\,\mathrm{nm}^{-1}$ and $\delta_0=\pi/6$. First calculate $t_0=\sqrt3/4+i/4$, whose squared modulus is one quarter. Then $\sigma=\pi/4\,\mathrm{nm}^2$. At phase shift $\pi/2$ the point reaches the top of the circle and $\sigma=\pi\,\mathrm{nm}^2$, the elastic upper limit for this wave number and partial wave.

A resonance in a real central potential often accompanies a phase shift passing through a right angle. Rapid variation within a narrow energy interval produces a peak. Background, higher partial waves and possible inelastic channels determine how well a single resonance describes a particular scattering process.

$$
\sigma_{\rm el}=\frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell\quad\Longrightarrow\quad\sigma_{\ell=0}\leq\frac{4\pi}{k^2}
$$

Orthogonality removes cross terms after full angular integration. The differential cross section at a fixed angle still includes interference.

## Use the circle with a stated normalization

Relativistic partial waves often retain a two-body phase-space factor $\rho(s)$ explicitly and use $S_\ell=1+2i\rho a_\ell$. The quantity on our circle is then $t_\ell=\rho a_\ell$. One cannot identify the two amplitudes by renaming them.

Potential scattering offers a concrete way to explore phase shifts. The later gluon amplitudes describe massless gauge fields at fixed color order and have additional spin and channel structure. The last two chapters move to one spatial dimension. Conservation of probability connects these examples, but each needs its own objects and normalization.

Source: M. F. Paulos

## Try the reasoning in an experiment

### [Follow one wave around its unitarity circle](https://sandboxphysics.com/labs/partial-wave-scattering/)

1. Choose Almost isotropic and identify the dominant gold partial-wave point. Predict how deepening the attraction moves it.
2. Vary depth or energy continuously. Find a cross-section peak and a gold point near the circle top; compare the circle with the angular pattern.
3. Raise the energy or increase the well radius and watch higher partial waves participate. Explain why integrated cross sections add by sector but fixed-angle amplitudes must be summed first.

## Exercises and explanations

1. Is $t=0.4+0.1i$ allowed? Show the calculation.

   Read the explanation

   Its squared modulus is $0.17$ and its imaginary part is $0.1$, so the inequality fails. It lies outside the circle and gives an elastic $|S|^2=1.28$, exceeding one.

2. Where does $S_\ell=0$ lie? Does its elastic scattering cross section vanish?

   Read the explanation

   The definition gives $t_\ell=i/2$, the disk center. The outgoing elastic radial wave vanishes, but the scattering amplitude measures the change from free propagation. Its squared modulus remains $1/4$. The remaining probability enters other channels.

3. Double the wave number at fixed phase shift. How does the lowest-wave cross section change?

   Read the explanation

   It falls by four. The dimensionless circle point stays fixed; restoring area units retains the wavelength scale.

[Previous: How a scattering amplitude becomes a detector count](https://sandboxphysics.com/stories/scattering-amplitudes/flux-and-amplitude/) [Next: Why split a momentum into two spinors?](https://sandboxphysics.com/stories/scattering-amplitudes/helicity-and-color/)

## Sources and reproduced scope

Worked examples and schematic figures are authored for this course. Numerical values are calculated model references. The published comparison recalculates a specified analytic CDD curve; it does not copy experimental data, source artwork or the paper’s numerical optimization points. Source locations and reproduced scope are listed below.

1. [David Tong, Topics in Quantum Mechanics, Chapter 10, Scattering Theory](https://www.damtp.cam.ac.uk/user/tong/aqm/aqmten.pdf)

   Sections 10.2.1–10.2.3: three-dimensional asymptotic waves, cross sections and partial waves. We derive the current-to-area relation and unitarity-circle examples for the course; the linked square-well experiment retains its own model and units.

2. [M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap II: Two Dimensional Amplitudes, arXiv:1607.06110v2; JHEP 11 (2017) 143](https://arxiv.org/html/1607.06110v2)

   Section 2 equations (1)–(3) fix crossing, elastic unitarity and coupling normalization. Equations (9), (13)–(14), Figures 5a and 6 give the sine-Gordon reference, CDD factor and physical strip. We recalculate the analytic Figure 5a reference at external mass one and exchanged mass square root of three, its pole residue and the complex value at energy squared six. We do not reproduce the dispersive optimization of section 2.1/Appendix A. The rational three-parameter family is our extension.