# P(r) Calculation

## Description

This tool calculates a real-space distance distribution function, P(r), using the inversion approach (Moore, 1980).

P(r) is set to be equal to an expansion of base functions of the type

$\Phi_{n(r)} = 2 r sin(\frac{\pi n r}{D_{max}})$

The coefficient of each base function in the expansion is found by performing a least square fit with the following fit function

$\chi^2=\frac{\sum_i (I_{meas}(Q_i)-I_{th}(Q_i))^2}{error^2}+Reg\_term$

where $$I_{meas}(Q_i)$$ is the measured scattering intensity and $$I_{th}(Q_i)$$ is the prediction from the Fourier transform of the P(r) expansion.

The $$Reg\_term$$ term is a regularization term set to the second derivative $$d^2P(r)/d^2r$$ integrated over $$r$$. It is used to produce a smooth P(r) output.

## Using P(r) inversion

The user must enter

• Number of terms: the number of base functions in the P(r) expansion.
• Regularization constant: a multiplicative constant to set the size of the regularization term.
• Maximum distance: the maximum distance between any two points in the system.

P(r) inversion requires that the background be perfectly subtracted. This is often difficult to do well and thus many data sets will include a background. For those cases, the user should check the “Estimate background level” option and the module will do its best to estimate it. If you know the background value for your data, select the “Input manual background level” option. Note that this value will be treated as having 0 error.

The P(r) module is constantly computing in the background what the optimum number of terms should be as well as the optimum regularization constant. These are constantly updated in the buttons next to the entry boxes on the GUI. These are almost always close and unless the user has a good reason to choose differently they should just click on the buttons to accept both. {D_max} must still be set by the user. However, besides looking at the output, the user can click the explore button which will bring up a graph of chi^2 vs Dmax over a range around the current Dmax. The user can change the range and the number of points to explore in that range. They can also choose to plot several other parameters as a function of Dmax including: I0, Rg, Oscillation parameter, background, positive fraction, and 1-sigma positive fraction.

The p(r) calculator accepts any number of data sets in the p(r) data set combo box. Switching between data sets in the combo box allows for individual fits and review of fit parameters. The displayed plots will also update to include the latest fit results for the selected data set.

The ‘Calculate All’ button will run the p(r) calculation for each file, sequentially. The calculator will estimate the number of terms, background, and regularization constant for each file prior to running the p(r) calculation. The final results for all data sets are then presented in a savable data table.

## Reference

P.B. Moore J. Appl. Cryst., 13 (1980) 168-175

Note

This help document was last modified by Steve King, 26 Jan 2021