A spectral line is modelled as a Gaussian,

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

I(\lambda) = I_0 \exp\!\left[-\dfrac{(\lambda-\lambda_0)^2}{2w^2}\right]

, with the width fixed at

w = 2.0\,\mathrm{nm}

(known from the instrument) and the amplitude

I_0

and centre

\lambda_0

to be fitted. Three intensities are measured, each with the same uncertainty:

$\lambda$ (nm)	498	500	502
$I$ (counts)	40	90	55

Starting from the guess

I_0 = 80\,\mathrm{counts}

\lambda_0 = 499\,\mathrm{nm}

, carry out one Gauss-Newton step. (a) Evaluate the residuals and the two partial derivatives

\partial I/\partial I_0

and

\partial I/\partial \lambda_0

at the starting guess, for the three wavelengths. (b) Assemble the

2\times2

normal equations

J^{\mathsf T}J\,\Delta\mathbf{p}=J^{\mathsf T}\mathbf{r}

(the common weight cancels as in a fit with equal uncertainties). (c) Solve for

\Delta I_0

and

\Delta \lambda_0

and report the updated

I_0

and

\lambda_0

to three significant figures.

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