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sasmodels/doc/guide/pd/polydispersity.rst
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Polydispersity Distributions
With some models in sasmodels we can calculate the average intensity for a population of particles that exhibit size and/or orientational polydispersity. The resultant intensity is normalized by the average particle volume such that
where $F$ is the scattering amplitude and $langlecdotrangle$ denotes an average over the size distribution.
Each distribution is characterized by its center $bar x$, its width $sigma$, the number of sigmas $N_sigma$ to include from the tails, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. Volume parameters have polydispersity PD (not to be confused with a molecular weight distributions in polymer science) leading to a size distribution of width $text{PD} = sigma / bar x$, but orientation parameters use an angular distributions of width $sigma$. $N_sigma$ determines how far into the tails to evaluate the distribution, with larger values of $N_sigma$ required for heavier tailed distributions. The scattering in general falls rapidly with $qr$ so the usual assumption that $G(r  3sigma_r)$ is tiny and therefore $f(r  3sigma_r)G(r  3sigma_r)$ will not contribute much to the average may not hold when particles are large. This, too, will require increasing $N_sigma$.
Users should note that the averaging computation is very intensive. Applying polydispersion to multiple parameters at the same time or increasing the number of points in the distribution will require patience! However, the calculations are generally more robust with more data points or more angles.
The following six distribution functions are provided:
 Rectangular Distribution
 Uniform Distribution
 Gaussian Distribution
 Lognormal Distribution
 Schulz Distribution
 Array Distribution
 Boltzmann Distribution
These are all implemented as numberaverage distributions.
Rectangular Distribution
The Rectangular Distribution is defined as
where $bar x$ is the mean of the distribution, $w$ is the halfwidth, and Norm is a normalization factor which is determined during the numerical calculation.
Note that the standard deviation and the half width $w$ are different!
The standard deviation is
whilst the polydispersity is
Uniform Distribution
The Uniform Distribution is defined as
f(x) = (1)/( Norm)⎧⎨⎩ 1 for x −  ≤ σ 0 for x −  > σwhere $bar x$ is the mean of the distribution, $sigma$ is the halfwidth, and Norm is a normalization factor which is determined during the numerical calculation.
Note that the polydispersity is given by
PD = σ ⁄
Gaussian Distribution
The Gaussian Distribution is defined as
where $bar x$ is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation.
The polydispersity is
Lognormal Distribution
The Lognormal Distribution is defined as
where $mu=ln(x_text{med})$ when $x_text{med}$ is the median value of the distribution, and Norm is a normalization factor which will be determined during the numerical calculation.
The median value for the distribution will be the value given for the respective size parameter, for example, radius=60.
The polydispersity is given by $sigma$
For the angular distribution
The mean value is given by $bar x = exp(mu+ p^2/2)$. The peak value is given by $max x = exp(mu  p^2)$.
This distribution function spreads more, and the peak shifts to the left, as $p$ increases, so it requires higher values of $N_sigma$ and more points in the distribution.
Schulz Distribution
The Schulz distribution is defined as
where $bar x$ is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation, and $z$ is a measure of the width of the distribution such that
The polydispersity is
Note that larger values of PD might need larger number of points and $N_sigma$. For example, at PD=0.7 and radius=60 Ang, Npts>=160 and Nsigmas>=15 at least.
For further information on the Schulz distribution see: M Kotlarchyk & SH Chen, J Chem Phys, (1983), 79, 2461.
Array Distribution
This userdefinable distribution should be given as a simple ASCII text file where the array is defined by two columns of numbers: $x$ and $f(x)$. The $f(x)$ will be normalized to 1 during the computation.
Example of what an array distribution file should look like:
30  0.1 
32  0.3 
35  0.4 
36  0.5 
37  0.6 
39  0.7 
41  0.9 
Only these array values are used computation, therefore the parameter value given for the model will have no affect, and will be ignored when computing the average. This means that any parameter with an array distribution will not be fitable.
Boltzmann Distribution
The Boltzmann Distribution is defined as
where $bar x$ is the mean of the distribution and Norm is a normalization factor which is determined during the numerical calculation. The width is defined as
which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a characteristic energy per particle.
Note about DLS polydispersity
Many commercial Dynamic Light Scattering (DLS) instruments produce a size polydispersity parameter, sometimes even given the symbol $p$! This parameter is defined as the relative standard deviation coefficient of variation of the size distribution and is NOT the same as the polydispersity parameters in the Lognormal and Schulz distributions above (though they all related) except when the DLS polydispersity parameter is <0.13.
For more information see: S King, C Washington & R Heenan, Phys Chem Chem Phys, (2005), 7, 143
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