--- title: "Why different nuclear potentials can fit the same scattering" description: "Fit a frozen synthetic neutron record, derive Poisson deviance, and use held-out angles and a depth profile to study ambiguity." canonical_url: "https://sandboxphysics.com/stories/nuclear-bridge/inverse-scattering/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nuclear-bridge/inverse-scattering/" last_modified: "2026-10-27" --- [Nuclear physics](https://sandboxphysics.com/stories/nuclear-bridge/) L3 · Chapter 6 # Why different nuclear potentials can fit the same scattering Sandbox Physics · 2026-10-27 Five forward-angle counts can admit both a shallow nuclear well and a deep one. They constrain combinations of outgoing wave amplitudes without necessarily determining depth, radius and absorption separately. The next useful measurement lies where the surviving models make different predictions. ## Before you begin The complex potential, scattering matrix and differential cross section from the preceding chapter. Basic probability is enough; Poisson counts, likelihood and nuisance parameters are introduced here. Every numerical observation is explicitly synthetic. [Review the preceding prerequisite](https://sandboxphysics.com/stories/nuclear-bridge/complex-potential-scattering/) - Construct a potential-parameter likelihood from saved integer counts. - Separate fit agreement, parameter uncertainty and multiple solutions using held-out angles and a depth profile. ## 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. $$ \lambda_i(V,W,R)=\mathcal E\left|f(\theta_i;V,W,R)\right|^2,\qquad n_i\sim\operatorname{Poisson}(\lambda_i) $$ The exposure units cancel those of the differential cross section. The final expected count is dimensionless. 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. $$ \mathcal L=\prod_i\frac{e^{-\lambda_i}\lambda_i^{n_i}}{n_i!},\qquad D=2\sum_i\left[\lambda_i-n_i+n_i\ln\frac{n_i}{\lambda_i}\right] $$ At zero observed count, define the logarithmic term by its limiting value, zero. $$ n_i=0\quad\Longrightarrow\quad D_i=2\lambda_i $$ Seeing no event can strongly disfavor a model that predicts many events at that angle. ## 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. $$ D_{\rm prof}(V)=\min_{W,R}D(V,W,R),\qquad\frac{\mathcal L_{\rm prof}(V)}{\mathcal L_{\max}}=e^{-[D_{\rm prof}(V)-D_{\min}]/2} $$ A profile uses the most favorable nuisance parameters at each depth. The displayed calculation approximates this minimum on a finite grid, with no calibrated confidence threshold. ## Try the reasoning in an experiment ### [Predict first, then challenge a fit](https://sandboxphysics.com/labs/nuclear-optical-potential/) 1. Compare Forward-angle twins, A shallower look-alike and A deeper look-alike. Record their forward deviances and predict where backward curves will separate. 2. Drag the detector backward and include all ten angles. Inspect the old candidates first, then choose Backward angles decide to perform the separate wider-angle fit. 3. Open the depth profile and locate its competing valleys. Export the observations, truth, candidates and profile, and check that the saved counts have stayed fixed. ## Exercises and explanations 1. An angle has zero observed counts. Two models predict $\lambda=0.5$ and $\lambda=8$. What are their deviance contributions? Read the explanation They contribute 1 and 16. A zero-count observation carries information, especially against a model predicting a large mean. 2. The wider angular range rejects the old shallow candidate. Does that reject every shallow potential? Read the explanation No. That candidate is one specific combination of depth, radius and absorption. Reoptimize the other parameters in the shallow region or inspect a profile. This example contains another shallow local fit to the ten-angle data. [Previous: Where the wave goes after a neutron meets a nucleus](https://sandboxphysics.com/stories/nuclear-bridge/complex-potential-scattering/) [Next: Find the collisions that burn deuterium](https://sandboxphysics.com/stories/nuclear-bridge/thermal-rates/) ## Sources and reproduced scope Evaluated level energies and published reaction measurements retain their source versions and uncertainties. Model curves, schematic figures and synthetic scattering records are identified separately. The offline benchmark files state which calculations have been reproduced. 1. [A. M. Moro, FRESCO notes, section 3.1, Optical Model](https://fresco.org.uk/moro/frnotes/node4.html) Equations 1–8: elastic-channel wave, partial-wave cross sections and complex-potential absorption. The local square-well implementation fixes a neutral projectile and centre-of-mass energy. 2. [David Tong, Applications of Quantum Mechanics, scattering theory](https://www.damtp.cam.ac.uk/user/tong/aqm/aqmten.pdf) Partial-wave amplitude, optical theorem, phase shifts and matching. Neutron scattering here fixes a centre-of-mass energy and a spinless central potential.