Why different nuclear potentials can fit the same scattering

Keep one fixed set of counts

Imagine a detector at several narrow angular acceptances. Combine beam intensity, target thickness, accepted solid angle and efficiency into an effective exposure. A potential then predicts an expected count through its differential cross section.

The example fixes the centre-of-mass energy at $1.2\,\mathrm{MeV}$ and exposure at $\mathcal E=1.5\,\mathrm{sr}\,\mathrm{fm}^{-2}$. At 20, 30, 40, 50 and 60 degrees, the saved synthetic counts are $101,83,68,70,42$. They were generated from the preceding chapter’s potential with seed 3707. Every candidate is compared with these same saved values.

An expected count may be fractional; an observed count is an integer. At twenty degrees the expectation is about 90.6 and the saved outcome is 101. This difference is a sampling fluctuation. Five larger angles are also saved, initially reserved for testing the fit.

Source: A. M. Moro

Turn curve agreement into a likelihood

The Poisson distribution gives the probability of each integer count $n_i$. Assuming independent angles, multiply those probabilities to obtain the likelihood. Fitting means finding potential parameters that assign a larger probability to this fixed record.

Logarithms turn the product into a sum. Compare it with the saturated model, whose expectation at every angle equals the observed count. Twice the log-likelihood difference is the Poisson deviance. Smaller deviance means better agreement for the same selected observations. Its zero-count limit is well defined.

The plotted error bars provide a visual fluctuation scale: the square root of count plus one, divided by exposure. They are not exact Poisson confidence intervals. The fit uses the integer-count likelihood directly.

Worked example: three forward-angle twins

Search over depth, absorption and radius. One local fit gives $(V,W,R)=(36.55,5.00,5.165)$, with depths in MeV and radius in fm. Its five-angle deviance is 2.798. A much shallower well, $(10.75,1.10,5.420)$, gives 2.804. A deeper well, $(77.35,8.00,5.180)$, gives 2.826.

The interior potentials differ substantially, yet their forward fits are almost equally good. Several partial-wave phases and magnitudes combine in a forward amplitude. A depth change can be compensated by radius and absorption changes. Measuring only squared amplitudes at a few angles also loses phase information.

These are local solutions found by a finite grid and coordinate refinement. Their spread demonstrates ambiguity. A parameter standard error or confidence interval requires more information about the likelihood, repeated sampling and model discrepancy.

Source: David Tong

Measure where the predictions separate

The held-out angles are $80^\circ,100^\circ,120^\circ,140^\circ,160^\circ$, with counts $26,5,3,0,0$. Including them raises the three deviances to about 34, 96 and 40. The shallow candidate that described the forward points so well now performs poorly.

A separate ten-angle search finds $V=43.65\,\mathrm{MeV},\ W=3.30\,\mathrm{MeV},\ R=4.350\,\mathrm{fm}$, with deviance about 8.0. Sampling fluctuations shift this optimum from the generating parameters. Wider coverage still does not prove uniqueness: other refitted wells, including a different shallow solution, remain reasonably competitive in this example.

This suggests an experimental design rule. List the surviving predictions, then measure where they disagree and the apparatus works reliably. A new angle can be more informative than many more events at an old angle. A second energy provides another test; energy stays fixed here to isolate angular coverage.

Inspect one parameter without freezing the others

Suppose the quantity of interest is depth $V$. Hold it fixed, then find the best radius and absorption at that depth. Repeating this procedure gives a depth profile. Radius and absorption are nuisance parameters for this question: they affect the prediction but are not the quantity being reported.

The display compares five-angle and ten-angle profiles on the same finite nuisance grid. Each curve subtracts its own lowest grid value. Valleys mark favorable depths; several valleys reveal remaining alternatives. More angular information can change the valleys without making the likelihood single-peaked.

The discrete grid and local optimization limit numerical precision. Zero-count bins also make an unverified large-sample threshold risky. A calibrated interval would require finer optimization and repeated acquisitions with the same sampling process, checking how often the interval contains the generating depth.