Description

This tool calculates a real-space distance distribution function, P(r), using the inversion approach (Moore, 1908).

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

|bigphi|_n(r) = 2.r.sin(|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 = |bigsigma|_i [ I_meas(Q_i) - I_th(Q_i) ] ² / (Error) ² + Reg_term

where I_meas(Q) is the measured scattering intensity and I_th(Q) 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 d²P(r) / dr² 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.

Reference

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