Q016 · Phase / angle / cross section

Partial-Wave Scattering

Send a monoenergetic beam toward a central potential. Move an annular detector, count scattered arrivals, and separate interference in the angle distribution from a change in total scattering.

Recorded outcomes $\text{—}$

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

01 / PREPARE THE QUESTION

Move the detector. Find the missing arrivals.

Scan and save. Increase the beam energy, then repeat. Which valleys move—and which disappear when the detector widens?

Beam travels to the right · annular detector accepts every azimuth · target and detector at schematic scales · direction guides are not trajectories

Detector polar angle

\theta/\mathrm{deg}

60

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

TRY A CONTRAST

Incident energy

E/\mathrm{eV}

0.3

Potential radius

a/\mathrm{nm}

0.35

Attractive depth

V_0/\mathrm{eV}

4

Central potential

Detection & sampling

Detector half width

\beta/\mathrm{deg}

5

Incident fluence per exposure

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

100

Sampling seed

s_0

137

Changing preparation clears active data. Save a sample first. The probe only selects a measurement setting.

Elastic, nonrelativistic scattering with electron mass from an attractive spherical square well or hard sphere. Independent Poisson exposures, uniform incident fluence and ideal annular detection. No Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

02 / RECORD BEFORE EXPLAINING

Count what the detector actually accepts.

$\theta/\mathrm{deg},\quad \overline{d\sigma/d\Omega}\,/\,(\mathrm{nm^2/sr})$ · Blue: active. Amber: most recent other saved sample. Drag to select a probe setting.

Show the model prediction

\text{—}

Each cell is one exposure. Darker cells contain more counts. Whiskers are approximate 95% Poisson score intervals; the detector reports a polar-band average.

03 / CONNECT SHAPE TO THE RESULT

Different waves arrive with different phases.

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

\text{—}

The final three partial waves provide a tail diagnostic. Convergence is also checked by adding eight more waves in scientific tests.

MODEL INSPECTION · SEPARATE FROM COUNTS

Interference changes the angular pattern.

$\theta/\mathrm{deg},\quad |f(\theta)|^2/\mathrm{nm^2}$ · Solid: coherent sum. Dashed: sum of squared components after discarding cross terms. The dashed curve is a diagnostic, not a second physical scattering model.

\text{—}

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

04 / KEEP THE EVIDENCE

Save the settings with the records.

Up to eight frozen samples. Restore to inspect the original settings. New acquisition after saving starts another sample. CSV contains all zero outcomes as well as successful detections.

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. 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: One round target. A pattern full of minima.
One-sentence intuition: Add complex partial-wave amplitudes before taking a squared magnitude. Different angles keep different interference terms.

Core mathematical model

Coherent angular amplitude

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

Polar angle is measured from the incoming positive x direction. The laboratory drawing uses x as the beam axis.

A finite detector

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

An annulus accepts all azimuths and a finite polar band. Recorded counts divide by exposure and solid angle; they estimate a bin average, not a point value.

Independent conservation check

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

Elastic real potentials conserve flux. Angular quadrature and the optical theorem must agree with the partial-wave sum.

Common difficulties

Representation is not a trajectory

Typical misconceptionThe outgoing guide is a calculated classical orbit.

Better mental modelThe guide connects the scatterer to an accepted angle. Complex waves, not classical paths, determine counts.

Model boundaries

Typical misconceptionA numerical pass proves a real apparatus has been reproduced.

Better mental modelElastic, nonrelativistic scattering with electron mass from an attractive spherical square well or hard sphere. Independent Poisson exposures, uniform incident fluence and ideal annular detection. No Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

Run the experiment

01
Predict before counting
Predict whether a larger circular target can produce angular minima. Record an angular scan.
What to observe: Empty charts are empty evidence. Reference curves stay separate until explicitly enabled.
02
Change one physical cause
Compare low energy and multi-wave presets. Broaden the detector and repeat at the same angle.
What to observe: A wide aperture washes out narrow minima; cross section and a conditional angle distribution are different observables.
03
Inspect a failure mode
Remove the potential. Zero counts are still recorded exposures. Compare an impenetrable sphere separately.
What to observe: Export the raw records, including zero outcomes, and reproduce the displayed estimator.