6.8 Nonlinear fit to a resonance

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §6.8. Rewritten in original wording. Continues 6.6b Least-squares fitting labs. Labs: 6.8b Nonlinear least-squares labs.

When least squares stops being linear algebra

Back in 6.5 the neutron cross-section table (Table 6.1) was interpolated without ever using the theory behind it — the Breit–Wigner resonance formula

$$f(E)=\frac{f_r}{(E-E_r)^2+\Gamma^2/4}.$$

Now the goal is to actually fit this formula to the data: find the values of $E_r$ (resonance energy), $f_r$ (peak height), and $\Gamma$ (width) that minimize $\chi^2$. Because $f(E)$ is not linear in these three parameters, the $\chi^2$-minimization conditions $\partial\chi^2/\partial a_m=0$ from 6.6 become nonlinear simultaneous equations — the closed-form linear-regression formulas no longer apply.

Rewriting the theory in fit-parameter notation

Relabel $a_1=f_r$, $a_2=E_r$, $a_3=\Gamma^{2}/4$ and $x=E$. The model, its three partials $\partial g/\partial a_m$, and the nonlinear residuals $f_1=f_2=f_3=0$ follow — pick each equation among lookalikes.

Stationarity of $\chi^{2}$ still means $\partial\chi^{2}/\partial a_m=0$; for Breit–Wigner that is exactly the residual system above. Walk the stationarity logic once more if you want the abstract form.

Newton-Raphson in three dimensions

Linearize the three residuals about a guess $\mathbf a$, build the Jacobian by forward differences, and update $\mathbf a\leftarrow\mathbf a+\Delta\mathbf a$. Pick each equation among lookalikes.

Practical notes on the search

• Starting guesses matter. Because $f_r,E_r,\Gamma$ each have direct physical meaning (peak height, peak location, full width at half maximum), a rough graph of the data gives good starting values — much better than an arbitrary guess.
• Local vs. global minima. Newton-Raphson only guarantees convergence to a root, not necessarily the best one. Repeat the search from a grid of different starting guesses; if different searches converge to different minima, keep the one with the lowest $\chi^2$.
• Ill-conditioning is common. Parameters that affect the fit weakly (or that are strongly correlated with another parameter) make the Jacobian nearly singular, producing wild, unstable correction steps. This is a real hazard of nonlinear fitting, not just a bug — backtracking (accepting only a fraction of $\Delta\mathbf a$ per step) and better initial guesses both help.

Quiz

Nonlinear least-squares fit

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Why can the Breit-Wigner fit not use the closed-form linear-regression formulas from 6.6?

Pick an answer to see if you are right.

Next: 6.8b Nonlinear least-squares labs.