This page will briefly describe how to use this tool to fit a physical model to scattering data. Using the obtained fitting parameters, we can gain some insight into the physics of the scattering process. The data is a set of differential cross section measurements made at different scattering angles (θ) in the range of approximately [0,π]. The experiment has been carried out at some fixed particle energy E=C k2, where C is a constant and k is the magnitude of the incident wave vector. The method of partial waves and the Born approximation give us a simple model to fit our scattering data.
The actual data is measured in k2 times the differential cross section. The uncertainties in these measurements are 10%, regardless of angle. In the next section, we will upload the scattering data into the curve fitting tool. If you have not already done so, launch the Curve Fitting Tool.
This tutorial assumes some familiarity with the Curve Fitting Tool. There is a beginner tutorial available for first time users.
Upload Data
The data for our scattering experiment should be uploaded to the tool from the user's computer. First download the scattering data to your system and click on "Launch Upload Tool" to upload the XML file. If all is well, the "View Data" and "Input Parameters" frames should "unfreeze" and are available for use. Click "Next" to view the scattering data.
View Data
You may like to see what this scattering data looks like. To do this, in the "View Data" frame, click on "Plot Data". First notice that our cross section values are represented by y and our scattering angle θ is represented by x. It is these variables we will use in our model equation. Notice also that because our error is 10% regardless of scattering angle, the error bars get smaller for smaller cross section or larger angle. The fitting routine will therefore try to fit the larger angle points more closely to the data than the smaller angles. Click "Next" to provide a fitting model to the data.
Input Parameters
We would like to fit this data to a partial wave model using the Born approximation. Given this simplification, we can deduce that the wavefunction of the scattered beam (see, for example, Modern Quantum Mechanics by Sakurai) at large distances can be represented by a sum of spherically outgoing and spherically incoming waves. The incoming or incident waves are not affected by the scatterer, but the outgoing waves are modulated by a coefficient. If the distance from scatterer to detector is large, the requirements of probability conservation (unitarity) suggest that only the phase of the outgoing wave (denoted by δl) is changed when the incident beam interacts with the scatterer, that is, the modulation coefficient mentioned above is simply a phase factor: e2iδl.
To see how the above discussion relates to the data we would like to fit, first note that the differential cross section of some scattering process as a function of scattering angle θ is given by the absolute square of a scattering "amplitude" f(θ). This factor is a sum of l partial spherical waves, where l is the orbital angular momentum quantum number. This quantum number arises because we assume that the scattering potential is spherically symmetric or invariant to rotations in three dimensions. A consequence of this rotational invariance is that the transition operator in the Born approximation is a scalar whose matrix elements (by the Wigner-Eckart theorem) in the l,m (where m is the quantum number representing the z component of the orbital angular momentum) basis have a simple mathematical relation. Working in this basis and using the assumption that our scattering coefficient is merely a phase factor gives us a functional form of the scattering amplitude:
where Pl is the lth Legendre polynomial.
Truncating the series above to include only s and p-waves (l=0 and l=1, respectively), we obtain a functional form for y, the fitting model for our data:
where δs=δ0, δp=δ1 and θ is the scattering angle or x in our model. We can further separate this model into an s-wave term (which is the l=0 contribution):
![y_{s} = P[2]=sin^{2}\delta_{s}](../../../images/s_wave.jpg "y_{s} = P[2]=sin^{2}\delta_{s}")
a p-wave term (which is the l=1 contribution):
![y_{p} = P[0]\cos^{2}\theta = 9 \sin^{2}\delta_{p}\cos^{2}\theta](../../../images/p_wave.jpg "y_{p} = P[0]\cos^{2}\theta = 9 \sin^{2}\delta_{p}\cos^{2}\theta")
and an interference term (a combination of s and p wave contributions):
![y_{int} = P[1]\cos\theta =6\cos(\delta_{s}-\delta_{p})\sin\delta_{s}\sin\delta_{p}\cos\theta](../../../images/int_wave.jpg "y_{int} = P[1]\cos\theta =6\cos(\delta_{s}-\delta_{p})\sin\delta_{s}\sin\delta_{p}\cos\theta")
where y=ys + yp + yint. Notice the above equations have been written in terms of three fitting parameters P[0], P[1] and P[2]. These are the fitting parameters we will use for our model. Once the fitting parameters are obtained, we can use the equations above to find the phase shifts for s and p-wave scattering. In the "Input Parameter" panel, please enter the information as it appears in the image below:
Notice that initial guesses for the fitting parameters are all zero. Click on "Fit Curve" to obtain your fitting data:
Using the values for the fitting parameters and the equations above we find that |δs|=1.03 rad and |δp|=0.2482 rad. Click "Next" to see how well your fit looks.
View Fit
Click "View Plot" to see the fitted model against the scattering data:
Notice that the fit (the solid curve) is a pretty good match to the data.
Challenge
Can you devise a method of determining the sign of δs or δp? Could you include higher order waves (d-waves, for example) and produce a better fit? What is the value of δd?
