Polydisperisty and Angular Distributions

Calculates the form factor for a polydisperse and/or angular population of particles with uniform scattering length density. The resultant form factor is normalized by the average particle volume such that P(q) = scale*<F*F>/Vol + bkg, where F is the scattering amplitude and the < > denote an average over the size distribution.  Users should use PD for a size distribution and Sigma for an angular distribution.

Note that this computation is very time intensive thus applying polydispersion/angular distrubtion for more than one paramters or increasing Npts values might need extensive patience to complete the computation.

The following five distribution functions are provided;

• Rectangular distribution
• Array distribution
• Gaussian distribution
• Lognormal distribution
• Schulz distribution

Rectangular distribution

The x_mean is the mean of the distribution, w is the half-width, and Norm is a normalization factor which is determined during the numerical calculation. Note that the Sigma and the half width w are different.

The standard deviation is

.

The PD (polydispersity) is

.

Array distribution

This distribution is to be given by users as a txt file where the array should be defined by two columns in the order of x and f(x) values. The f(x) will be normalized by SansView during the computation.

Example of an array in the file;

30        0.1

32        0.3

35        0.4

36        0.5

37        0.6

39        0.7

41        0.9

We use only these array values in the computation, therefore the mean value given in the control panel, for example ‘radius = 60’, will be ignored.

Gaussian distribution

The x_mean is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.

The PD (polydispersity) is

.

Lognormal distribution

The mu = ln(x_med),  x_med is the median value of the distribution, and Norm is a normalization factor which will be determined during the numerical calculation. The median value is the value given in the size parameter in the control panel, for example, “radius = 60”.

The PD (polydispersity) is given by sigma,

.

For the angular distribution,

The mean value is given by x_mean =exp(mu+p^2/2).

The peak value is given by x_peak=exp(mu-p^2).

This distribution function spreads more and the peak shifts to the left as the p increases, requiring higher values of Nsigmas and Npts.

Schulz distribution

The x_mean is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.

The z = 1/p^2 – 1.

The PD (polydispersity) is

.