-                      rd712a0f
+                      r97d172c
 .. _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
+Polydispersity & Orientational Distributions
+--------------------------------------------
+For some models we can calculate the average intensity for a population of
+particles that possess size and/or orientational (ie, angular) distributions.
+In SasView we call the former *polydispersity* but use the parameter *PD* to
+parameterise both. In other words, the meaning of *PD* in a model depends on
+the actual parameter it is being applied too.
+The resultant intensity is then normalized by the average particle volume such
+that
 .. math::
 …
 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an
 average over the size distribution $f(x; \bar x, \sigma)$, giving
+average over the distribution $f(x; \bar x, \sigma)$, giving
 .. math::
 …
 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 standard deviation, so read the description of the distribution 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 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 $\sigma = \mathrm{PD}$.
+However, the distribution width applied to *orientation* parameters is just
+$\sigma = \mathrm{PD}$.
 $N_\sigma$ determines how far into the tails to evaluate the distribution,
 …
 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.
+polydispersion and/or orientational distributions 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:
 …
            the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2),
 -353 <http://media.iupac.org/publications/pac/2009/pdf/8102x0351.pdf>`_
+           in order to make the terminology describing distributions of properties
+           unambiguous. Throughout the SasView documentation we continue to use the
+           term polydispersity because one of the consequences of the IUPAC change is
+           that orientational polydispersity would not meet their new criteria (which
+           requires dispersity to be dimensionless).
+           in order to make the terminology describing distributions of chemical
+           properties unambiguous. However, these terms are unrelated to the
+           proportional size distributions and orientational distributions used in
+           SasView models.
 Suggested Applications
 ^^^^^^^^^^^^^^^^^^^^^^
 If applying polydispersion to parameters describing particle sizes, use
+If applying polydispersion to parameters describing particle sizes, consider using
 the Lognormal or Schulz distributions.
 If applying polydispersion to parameters describing interfacial thicknesses
 or angular orientations, use the Gaussian or Boltzmann distributions.
+or angular orientations, consider using the Gaussian or Boltzmann distributions.
 If applying polydispersion to parameters describing angles, use the Uniform
 …
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+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 $x$.
+Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and
+it should not be assumed that any of the following can be simply equated with
+the polydispersity *PD* parameter used in SasView.
+The dimensionless *Polydispersity Index (PI)* is a measure of the width of the
+distribution of autocorrelation function decay rates (*not* the distribution of
+particle sizes itself, though the two are inversely related) and is defined by
+ISO 22412:2017 as
+.. math::
+    PI = \mu_{2} / \bar \Gamma^2
+where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the
+intensity-weighted average value, of the distribution of decay rates.
+*If the distribution of decay rates is Gaussian* then
+.. math::
+    PI = \sigma^2 / 2\bar \Gamma^2
+where $\sigma$ is the standard deviation, allowing a *Relative Polydispersity (RP)*
+to be defined as
+.. math::
+    RP = \sigma / \bar \Gamma = \sqrt{2.PI}
+PI values smaller than 0.05 indicate a highly monodisperse system. Values
+greater than 0.7 indicate significant polydispersity.
+The *size polydispersity P-parameter* is defined as the relative standard
+deviation coefficient of variation
+.. math::
+    P = \sqrt\nu / \bar R
+where $\nu$ is the variance of the distribution and $\bar R$ is the mean
+value of $R$. Here, the product $P \bar R$ is *equal* to the standard
+deviation of the Lognormal distribution.
+P values smaller than 0.13 indicate a monodisperse system.
 For more information see:
+S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143
+`ISO 22412:2017, International Standards Organisation (2017) <https://www.iso.org/standard/65410.html>`_.
+`Polydispersity: What does it mean for DLS and Chromatography <http://www.materials-talks.com/blog/2014/10/23/polydispersity-what-does-it-mean-for-dls-and-chromatography/>`_.
+`Dynamic Light Scattering: Common Terms Defined, Whitepaper WP111214. Malvern Instruments (2011) <http://www.biophysics.bioc.cam.ac.uk/wp-content/uploads/2011/02/DLS_Terms_defined_Malvern.pdf>`_.
+S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143.
+T Allen, in *Particle Size Measurement*, 4th Edition, Chapman & Hall, London (1990).
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 …
 | 2018-03-20 Steve King
 | 2018-04-04 Steve King
+| 2018-08-09 Steve King

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doc/guide/pd/polydispersity.rst

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