.. pd_help.rst .. This is a port of the original SasView html help file to ReSTructured text .. by S King, ISIS, during SasView CodeCamp-III in Feb 2015. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ .. _polydispersityhelp: 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 .. math:: P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an average over the size distribution. Each distribution is characterized by a center value $\bar x$ or $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* the standard deviation, so read the description carefully), the number of sigmas $N_\sigma$ to include from the tails of the distribution, and the number of points used to compute the average. The center of the distribution is set by the value of the model parameter. The meaning of a polydispersity parameter *PD* (not to be confused with a molecular weight distributions in polymer science) in a model depends on the type of parameter it is being applied too. The distribution width applied to *volume* (ie, shape-describing) parameters is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$. However, the distribution width applied to *orientation* (ie, angle-describing) parameters is just $\sigma = \mathrm{PD}$. $N_\sigma$ 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 $G(r - 3\sigma_r)$ is tiny and therefore $f(r - 3\sigma_r)G(r - 3\sigma_r)$ 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 distribution functions are provided: * *Uniform Distribution* * *Rectangular Distribution* * *Gaussian Distribution* * *Boltzmann Distribution* * *Lognormal Distribution* * *Schulz Distribution* * *Array Distribution* * *User-defined Distributions* These are all implemented as *number-average* distributions. Suggested Applications ^^^^^^^^^^^^^^^^^^^^^^ If applying polydispersion to parameters describing particle sizes, use the Lognormal or Schulz distributions. If applying polydispersion to parameters describing interfacial thicknesses or angular orientations, use the Gaussian or Boltzmann distributions. If applying polydispersion to parameters describing angles, use the Uniform distribution. Beware of using distributions that are always positive (eg, the Lognormal) because angles can be negative! The array distribution provides a very simple means of implementing a user- defined distribution, but without any fittable parameters. Greater flexibility is conferred by the user-defined distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Uniform Distribution ^^^^^^^^^^^^^^^^^^^^ The Uniform Distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}} \begin{cases} 1 & \text{for } |x - \bar x| \leq \sigma \\ 0 & \text{for } |x - \bar x| > \sigma \end{cases} where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, $\sigma$ is the half-width, and *Norm* is a normalization factor which is determined during the numerical calculation. The polydispersity in sasmodels is given by .. math:: \text{PD} = \sigma / \bar x .. figure:: pd_uniform.jpg Uniform distribution. The value $N_\sigma$ is ignored for this distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Rectangular Distribution ^^^^^^^^^^^^^^^^^^^^^^^^ The Rectangular Distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}} \begin{cases} 1 & \text{for } |x - \bar x| \leq w \\ 0 & \text{for } |x - \bar x| > w \end{cases} where $\bar x$ ($x_\text{mean}$ in the figure) 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 standard deviation and the half width $w$ are different! The standard deviation is .. math:: \sigma = w / \sqrt{3} whilst the polydispersity in sasmodels is given by .. math:: \text{PD} = \sigma / \bar x .. figure:: pd_rectangular.jpg Rectangular distribution. .. note:: The Rectangular Distribution is deprecated in favour of the Uniform Distribution above and is described here for backwards compatibility with earlier versions of SasView only. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Gaussian Distribution ^^^^^^^^^^^^^^^^^^^^^ The Gaussian Distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}} \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution and *Norm* is a normalization factor which is determined during the numerical calculation. The polydispersity in sasmodels is given by .. math:: \text{PD} = \sigma / \bar x .. figure:: pd_gaussian.jpg Normal distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Boltzmann Distribution ^^^^^^^^^^^^^^^^^^^^^^ The Boltzmann Distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}} \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution and *Norm* is a normalization factor which is determined during the numerical calculation. The width is defined as .. math:: \sigma=\frac{k T}{E} which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. .. figure:: pd_boltzmann.jpg Boltzmann distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Lognormal Distribution ^^^^^^^^^^^^^^^^^^^^^^ The Lognormal Distribution describes a function of $x$ where $\ln (x)$ has a normal distribution. The result is a distribution that is skewed towards larger values of $x$. The Lognormal Distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}}\frac{1}{x\sigma} \exp\left(-\frac{1}{2} \bigg(\frac{\ln(x) - \mu}{\sigma}\bigg)^2\right) where *Norm* is a normalization factor which will be determined during the numerical calculation, $\mu=\ln(x_\text{med})$ and $x_\text{med}$ is the *median* value of the *lognormal* distribution, but $\sigma$ is a parameter describing the width of the underlying *normal* distribution. $x_\text{med}$ will be the value given for the respective size parameter in sasmodels, for example, *radius=60*. The polydispersity in sasmodels is given by .. math:: \text{PD} = \sigma = p / x_\text{med} The mean value of the distribution is given by $\bar x = \exp(\mu+ \sigma^2/2)$ and the peak value by $\max x = \exp(\mu - \sigma^2)$. The variance (the square of the standard deviation) of the *lognormal* distribution is given by .. math:: \nu = [\exp({\sigma}^2) - 1] \exp({2\mu + \sigma^2}) Note that larger values of PD might need a larger number of points and $N_\sigma$. .. figure:: pd_lognormal.jpg Lognormal distribution for PD=0.1. For further information on the Lognormal distribution see: http://en.wikipedia.org/wiki/Log-normal_distribution and http://mathworld.wolfram.com/LogNormalDistribution.html .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Schulz Distribution ^^^^^^^^^^^^^^^^^^^ The Schulz (sometimes written Schultz) distribution is similar to the Lognormal distribution, in that it is also skewed towards larger values of $x$, but which has computational advantages over the Lognormal distribution. The Schulz distribution is defined as .. math:: f(x) = \frac{1}{\text{Norm}} (z+1)^{z+1}(x/\bar x)^z \frac{\exp[-(z+1)x/\bar x]}{\bar x\Gamma(z+1)} where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, *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 .. math:: z = (1-p^2) / p^2 where $p$ is the polydispersity in sasmodels given by .. math:: PD = p = \sigma / \bar x and $\sigma$ is the RMS deviation from $\bar x$. Note that larger values of PD might need a larger number of points and $N_\sigma$. For example, for PD=0.7 with radius=60 |Ang|, at least Npts>=160 and Nsigmas>=15 are required. .. figure:: pd_schulz.jpg Schulz distribution. For further information on the Schulz distribution see: M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461 and M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Array Distribution ^^^^^^^^^^^^^^^^^^ This user-definable 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. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ User-defined Distributions ^^^^^^^^^^^^^^^^^^^^^^^^^^ You can also define your own distribution by creating a python file defining a *Distribution* object with a *_weights* method. The *_weights* method takes *center*, *sigma*, *lb* and *ub* as arguments, and can access *self.npts* and *self.nsigmas* from the distribution. They are interpreted as follows: * *center* the value of the shape parameter (for size dispersity) or zero if it is an angular dispersity. This parameter may be fitted. * *sigma* the width of the distribution, which is the polydispersity parameter times the center for size dispersity, or the polydispersity parameter alone for angular dispersity. This parameter may be fitted. * *lb*, *ub* are the parameter limits (lower & upper bounds) given in the model definition file. For example, a radius parameter has *lb* equal to zero. A volume fraction parameter would have *lb* equal to zero and *ub* equal to one. * *self.nsigmas* the distance to go into the tails when evaluating the distribution. For a two parameter distribution, this value could be co-opted to use for the second parameter, though it will not be available for fitting. * *self.npts* the number of points to use when evaluating the distribution. The user will adjust this to trade calculation time for accuracy, but the distribution code is free to return more or fewer, or use it for the third parameter in a three parameter distribution. As an example, the code following wraps the Laplace distribution from scipy stats:: import numpy as np from scipy.stats import laplace from sasmodels import weights class Dispersion(weights.Dispersion): r""" Laplace distribution .. math:: w(x) = e^{-\sigma |x - \mu|} """ type = "laplace" default = dict(npts=35, width=0, nsigmas=3) # default values def _weights(self, center, sigma, lb, ub): x = self._linspace(center, sigma, lb, ub) wx = laplace.pdf(x, center, sigma) return x, wx You can plot the weights for a given value and width using the following:: from numpy import inf from matplotlib import pyplot as plt from sasmodels import weights # reload the user-defined weights weights.load_weights() x, wx = weights.get_weights('laplace', n=35, width=0.1, nsigmas=3, value=50, limits=[0, inf], relative=True) # plot the weights plt.interactive(True) plt.plot(x, wx, 'x') The *self.nsigmas* and *self.npts* parameters are normally used to control the accuracy of the distribution integral. The *self._linspace* function uses them to define the *x* values (along with the *center*, *sigma*, *lb*, and *ub* which are passed as parameters). If you repurpose npts or nsigmas you will need to generate your own *x*. Be sure to honour the limits *lb* and *ub*, for example to disallow a negative radius or constrain the volume fraction to lie between zero and one. To activate a user-defined distribution, set the following environment variable: SASMODELS_WEIGHTS=path/to/folder/name_of_distribution.py .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 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. .. math:: p_{DLS} = \sqrt(\nu / \bar x^2) where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of $x$. For more information see: S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ *Document History* | 2015-05-01 Steve King | 2017-05-08 Paul Kienzle | 2018-03-20 Steve King | 2018-04-04 Steve King