FR07-B · Different wells behind the same pattern

Nuclear Optical Potential

Can one scattering pattern determine a nuclear potential? Several nuclear potentials can fit the same finite set of scattering counts. Their predictions often separate at angles that were left out of the fit, so a wider angular range can distinguish the competing explanations.

Send a neutron wave toward a model calcium nucleus. Compare potentials that fit the same forward-angle counts, then use backward angles to separate them.

Elastic cross section $2.32\,\mathrm b$

Loss from elastic channel $1.10\,\mathrm b$

Poisson deviance $4.95$

At the detector $39.4\,\mathrm{fm}^{2}\,\mathrm{sr}^{-1}$

One saved scattering record

Which potential fits?

Drag the gold detector to compare directions.

Attractive depth

V/\mathrm{MeV}

42

Absorptive depth

W/\mathrm{MeV}

4

Well radius

R/\mathrm{fm}

4.5

Detector angle

\theta/{}^\circ

50

Use all ten angles in the comparison

Model and keyboard controls

A neutral, spinless neutron–calcium-40 square-well model at a fixed centre-of-mass energy of 1.2 MeV. The complex well describes elastic-channel absorption. Spin, Coulomb forces, surface diffuseness, individual resonances and coupled channels are omitted. Counts are reproducible synthetic Poisson observations. Target size and wave amplitude use schematic display scales.

The same scattering solution sets the wave pattern and the angular prediction. The scene shows the real part of the exact stationary exterior partial-wave solution. Colour uses one fixed compressed amplitude scale. The animation slows its common phase for display; it does not simulate individual neutron paths. The masked interior is an illustration of the potential region.

Focus the scene and press Left or Right to move the detector. Up and Down change the attractive depth. Reset restores the original potential and five-angle comparison.

Model assumptions

Saved synthetic Poisson record · seed 3707

Similar forward, different backward

Five forward angles enter the fit.

Gold shows your potential; violet, blue, coral and green show candidates. Pale points are fitted, gray points held out; bars approximate counting errors.

$\frac{d\sigma}{d\Omega}\ [\mathrm{fm}^{2}\,\mathrm{sr}^{-1}],\quad 1\,\mathrm b=100\,\mathrm{fm}^{2}$

Upper bars use ten angles; lower bars use five. Shorter bars mean smaller deviance.

Inspect the depth profile

At each depth, radius and absorption are searched on the same finite grid. Each curve subtracts its own lowest grid value. Gold uses five angles; blue uses all ten. Multiple valleys show the remaining ambiguity. The grid is coarse and these curves do not define calibrated confidence intervals.

Scattering normalization and sources

Vertical counting-error bars use the square root of count plus one divided by exposure. They are an approximate visual scale; the fit uses the Poisson count likelihood directly.

\Delta\!\left(\frac{d\sigma}{d\Omega}\right)=\frac{\sqrt{n+1}}{\mathcal E}

The residual panel automatically rescales symmetrically to include every unmodified value, and its ticks show the current range. Each count residual is divided by the square root of count plus one.

Candidate numbers follow the exported array. These local fits illustrate ambiguity; their spread does not constitute a confidence interval.

U(r)=\begin{cases}-V-iW&r<R\\0&r\geq R\end{cases},\quad f(\theta)=\frac1{2ik}\sum_{\ell=0}^{12}(2\ell+1)(S_\ell-1)P_\ell(\cos\theta)

\sigma_{\rm el}=\frac\pi{k^2}\sum_\ell(2\ell+1)|S_\ell-1|^2,\quad\sigma_{\rm reac}=\frac\pi{k^2}\sum_\ell(2\ell+1)(1-|S_\ell|^2)

The entrance wave travels to the right; detector angles are measured from that forward direction in the centre-of-mass frame. The reduced mass uses the neutron mass and an explicitly approximate calcium mass of forty atomic mass units. Twelve orbital partial waves are retained beyond the s wave. The record stays at one energy when a potential changes.

The grid search and local coordinate refinement use the Poisson count likelihood. A second search fits all ten angles. The omitted channels are represented only by lost elastic flux; absorption does not specify the probabilities of particular reactions.

Moro · FRESCO optical-model equations (1)–(8)Tong · Scattering Theory, §10.2 Frozen record and fit candidates

Study nuclear structure, scattering and stellar rates

Start with the essentials

Focus question: Can one scattering pattern determine a nuclear potential?
One-sentence intuition: Good agreement at the measured angles can leave competing potentials that disagree sharply at unmeasured angles.

Core mathematical model

A potential with attraction and absorption

U(r)=\begin{cases}-V-iW&r<R\\0&r\geq R\end{cases},\quad V,W\geq0

The real depth changes the elastic wave phase. The negative imaginary part reduces the outgoing elastic flux. Radius is measured in femtometres; the energy and both depths are in MeV.

From partial waves to an angular distribution

\begin{aligned}f(\theta)&=\frac1{2ik}\sum_{\ell=0}^{12}(2\ell+1)(S_\ell-1)P_\ell(\cos\theta)\\\frac{d\sigma}{d\Omega}&=|f(\theta)|^2\end{aligned}

Each orbital angular momentum contributes a complex scattering amplitude. Their interference makes the angle dependence. The same matrices determine the exterior wave shown on the stage.

Compare predictions with counts

D=2\sum_i\left[\lambda_i-n_i+n_i\ln\frac{n_i}{\lambda_i}\right],\quad\lambda_i=\mathcal E\,|f(\theta_i)|^2

The saved count is n; its predicted Poisson mean is lambda. Exposure combines beam, target, accepted solid angle and efficiency. A zero observed count contributes twice the predicted mean. A smaller deviance indicates a better fit to the selected angles.

Common difficulties

Absorption refers to one channel

Typical misconceptionAn absorbed neutron has vanished from the full physical system.

Better mental modelThe imaginary potential represents probability flowing into omitted reaction and excitation channels. This model reports their combined loss from the elastic channel.

A fitted curve can have several explanations

Typical misconceptionA small fit statistic proves the potential parameters are uniquely known.

Better mental modelThe forward counts constrain combinations of phases. Several potentials can reproduce those combinations. Additional angles or energies test the alternatives.

Run the experiment

01
Compare the forward-angle twins
Predict whether a much shallower well must spoil the fit. Compare Forward-angle twins with A shallower look-alike.
What to observe: The potential profiles change substantially. The forward points and their fitted curves stay close, and the forward deviance hardly changes.
02
Measure where predictions separate
Drag the detector toward a backward angle. Then turn on Use all ten angles in the comparison.
What to observe: The held-out counts penalize the shallow candidate much more strongly. These new angles carry information missing from the forward pattern.
03
Fit the wider angular range
Choose Backward angles decide, then export the record and fits. Compare the included angles and the unchanged saved counts.
What to observe: A separate search fits all ten saved angles. Candidate spread describes alternatives found by this finite search; it does not supply a statistical confidence interval.

Nuclear Optical Potential

Similar forward, different backward