Q016 · Phase shifts, angles and cross sections

Partial-Wave Scattering

Why does scattering off a single round target show so many minima? Because the partial waves interfere. Each one scatters with its own phase shift, and their complex amplitudes add before the sum is squared, so they reinforce at some angles and cancel at others. The total cross section is a separate quantity: one number for all angles together.

Send a beam of a single energy at a central potential. Move a ring-shaped detector, count the scattered particles, and tell interference in the angular pattern apart from a change in the total amount of scattering.

Recorded outcomes $\text{—}$

Aperture-averaged cross section $\text{—}$

01 / Choose what to test

Move the detector to find where arrivals go missing.

Raise the energy and scan again. Which valleys disappear when the detector gets wider?

Beam to the right; schematic scales; lines mark directions, not trajectories.

Detector polar angle

\theta/\mathrm{deg}

60

Each angular point is an independent exposure. Counts include zero outcomes.

Try a comparison

Incident energy

E/\mathrm{eV}

1.5

Potential radius

a/\mathrm{nm}

0.55

Attractive depth

V_0/\mathrm{eV}

4

Central potential

Detection & sampling

Changing the preparation clears the current data. Moving the detector only selects a measurement angle, so the existing scan remains available.

Detector half width

\beta/\mathrm{deg}

5

Incident fluence per exposure

\mathcal F/\mathrm{nm}^{-2}

100

Sampling seed

s_0

137

Save before changing the preparation.

Model assumptions

Particles with the electron mass scatter elastically and nonrelativistically from an attractive spherical square well or a hard sphere. Each exposure is an independent Poisson count, the incident fluence is uniform, and the ring detector is ideal. The model has no Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

Simulated detector exposures

Count what the detector actually accepts.

$\theta/\mathrm{deg},\quad \overline{d\sigma/d\Omega}\,/\,(\mathrm{nm^2/sr})$ · Blue: current scan. Amber: saved comparison. Drag to choose the angle.

Show the model prediction

\text{—}

Counts and uncertainty

Each cell is one simulated exposure. Darker cells contain more counts. The amber curve uses the most recent saved sample other than the active scan. Whiskers are approximate 95% Poisson score intervals; the detector reports a polar-band average.

A conserved probability current

Every partial wave sits on the same circle.

$\operatorname{Re}t_l,\quad\operatorname{Im}t_l$ · Gold: largest partial cross section. Cyan: other waves.

Change the energy and follow the gold point around the circle.

1

Why probability flow gives a circle

\begin{gathered}S_l=e^{2i\delta_l}=1+2it_l\\t_l=e^{i\delta_l}\sin\delta_l\\(\operatorname{Re}t_l)^2+\left(\operatorname{Im}t_l-\tfrac12\right)^2=\tfrac14\end{gathered}

A real potential returns all incoming flux to the elastic channel. Changing the energy moves each amplitude around the circle. The top of the circle gives the largest possible partial cross section at that energy.

This is a nonrelativistic potential model. The dimensionless circle relation also occurs in relativistic partial waves once their phase-space normalization is included.

Continue from probability flow to scattering amplitudes

Partial cross sections, angular interference and saved samples

03 / Connect the shape to the result

Different partial waves carry different phases.

$l,\quad \sigma_l/\mathrm{nm^2}$ · This decomposition comes from the model. It is not a measurement of individual partial waves.

\text{—}

The last three partial waves show how much the tail contributes. The scientific tests also check convergence by adding eight more waves.

Model view, separate from the counts

Interference changes the angular pattern.

$\theta/\mathrm{deg},\quad |f(\theta)|^2/\mathrm{nm^2}$ · Solid: coherent sum. Dashed: the squared components added without their cross terms. The dashed curve is only a diagnostic; it is not a second physical scattering model.

\text{—}

$r/\mathrm{nm},\quad E/\mathrm{eV}$

04 / Save the evidence

Save the settings with the records.

You can keep up to eight frozen samples. Restore one to see its original settings. Acquiring after a save starts a new sample. The CSV contains every zero outcome as well as every successful detection.

Model limits, units & sources

A real, finite-range central potential and elastic electron-mass scattering. The incident plane wave travels along positive x. The annular detector accepts every azimuth and a polar interval excluding two-degree beam ports at each pole. Radius and depth define a hypothetical target; no specific material is implied. Target and detector use schematic scales; the guide lines indicate directions rather than trajectories. The hard-sphere option replaces the interior by a zero-wavefunction boundary.

The partial-wave cutoff increases with incident momentum and radius. Detector integration uses Gaussian quadrature; independent angular exposures follow Poisson counting. The displayed dot set is at most 128 conditionally sampled arrivals from the last exposure. The raw CSV retains each full exposure count and fluence. Target recoil, absorption, inelastic channels, multiple scattering and Coulomb tails are omitted.

Fitzpatrick · Partial waves · Phase shifts · Resonant scattering

Start with the essentials

Focus question: Why does scattering off a single round target show so many minima?
One-sentence intuition: Add the complex partial-wave amplitudes first and square the magnitude afterward. That is why each angle keeps its own interference terms.

Core mathematical model

Scattering amplitude as a sum of partial waves

f(\theta)=\frac1k\sum_{l=0}^{l_{\max}}(2l+1)e^{i\delta_l}\sin\delta_l P_l(\cos\theta)

The polar angle is measured from the incoming direction, positive x, which is also the beam axis in the drawing.

Counts in a detector of finite size

\langle N\rangle=\mathcal F\int_{\Delta\Omega}|f(\theta)|^2\,d\Omega

The ring accepts every azimuth over a finite band of polar angles. Dividing the counts by exposure and solid angle estimates the average over that band, not the value at a single angle.

Total cross section, computed independently

\sigma=\frac{4\pi}{k^2}\sum_l(2l+1)\sin^2\delta_l=\frac{4\pi}{k}\operatorname{Im} f(0)

A real potential with only elastic scattering conserves flux, so integrating over angle and using the optical theorem must both agree with the partial-wave sum.

Common difficulties

A picture is not a trajectory

Typical misconceptionThe outgoing guide line is a computed classical path.

Better mental modelThe line only connects the scatterer to the angle the detector accepts. The counts come from complex wave amplitudes, not from classical paths.

What the model leaves out

Typical misconceptionIf the numerical checks pass, a real apparatus has been reproduced.

Better mental modelParticles with the electron mass scatter elastically and nonrelativistically from an attractive spherical square well or a hard sphere. Each exposure is an independent Poisson count, the incident fluence is uniform, and the ring detector is ideal. The model has no Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

Run the experiment