P(r) Inversion Perspective

The inversion approach is based on Moore, J. Appl. Cryst., (1980) 13, 168-175.

P(r) is set to be equal to an expansion of base functions of the type phi_n(r) = 2 * r * sin(π * 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:

χ ^2 = sum_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 d^2 P(r) / dr^2 integrated over r. It is used to produce a smooth P(r) output.

The following are user inputs:

• 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.