Changeset 5026e05 in sasmodels

Timestamp:

Mar 20, 2018 11:19:43 AM (7 years ago)

Author:

smk78

Branches:

master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests

Children:

ed5b109

Parents:

e2f1a41

Message:

Overhaul of polydispersity help for content, readability and accuracy.

File:

• doc/guide/pd/polydispersity.rst (modified) (10 diffs)

doc/guide/pd/polydispersity.rst

@@ -23 +23 @@
 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 $\sigma$.
+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. 
+
+Volume parameters have polydispersity *PD* (not to be confused with a molecular 
+weight distributions in polymer science), but orientation parameters use 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.
@@ -44 +47 @@
 The following distribution functions are provided:
 
+*  *Uniform Distribution*
 *  *Rectangular Distribution*
-*  *Uniform Distribution*
 *  *Gaussian Distribution*
+*  *Boltzmann Distribution*
 *  *Lognormal Distribution*
 *  *Schulz Distribution*
 *  *Array Distribution*
-*  *Boltzmann Distribution*
 
 These are all implemented as *number-average* distributions.
 
-.. 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 $\bar x$ 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 is
-
-.. math:: \text{PD} = \sigma / \bar x
-
-.. figure:: pd_rectangular.jpg
-
-    Rectangular distribution.
-
-
+Additional distributions are under consideration.
+
+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.
+
+The array distribution allows a user-defined distribution to be applied.
+
+.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
 
 Uniform Distribution
-^^^^^^^^^^^^^^^^^^^^^^^^
+^^^^^^^^^^^^^^^^^^^^
 
 The Uniform Distribution is defined as
@@ -102 +85 @@
         \end{cases}
 
-    where $\bar x$ is the mean of the distribution, $\sigma$ is the half-width, and
-    *Norm* is a normalization factor which is determined during the numerical
-    calculation.
-
-    Note that the polydispersity is given by
+    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
@@ -115 +98 @@
 
 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
@@ -123 +142 @@
 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 $\bar x$ is the mean of the distribution and *Norm* is a normalization
-factor which is determined during the numerical calculation.
-
-The polydispersity is
-
-.. math:: \text{PD} = \sigma / \bar x
-
-.. figure:: pd_gaussian.jpg
-
-    Normal distribution.
+    .. 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
@@ -144 +189 @@
 ^^^^^^^^^^^^^^^^^^^^^^
 
+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}{xp}\exp\left(-\frac{(\ln(x) - \mu)^2}{2p^2}\right)
-
-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 $\sigma$
-
-.. math:: \text{PD} = p
-
-For the angular distribution
-
-.. math:: p = \sigma / x_\text{med}
-
-The mean value is given by $\bar x = \exp(\mu+ p^2/2)$. The peak value
-is given by $\max x = \exp(\mu - p^2)$.
-
-.. figure:: pd_lognormal.jpg
-
-    Lognormal distribution.
-
-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.
+    .. 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} = p = \sigma / x_\text{med}
+
+    The mean value of the distribution is given by $\bar x = \exp(\mu+ p^2/2)$ and the peak value 
+    by $\max x = \exp(\mu - p^2)$.
+
+    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 further information on the Lognormal distribution see:
+http://en.wikipedia.org/wiki/Log-normal_distribution and 
+http://mathworld.wolfram.com/LogNormalDistribution.html
 
 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
@@ -182 +234 @@
 ^^^^^^^^^^^^^^^^^^^
 
+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 $\bar x$ 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
-
-.. math:: z = (1-p^2) / p^2
-
-The polydispersity is
-
-.. math:: p = \sigma / \bar x
-
-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.
-
-.. figure:: pd_schulz.jpg
-
-    Schulz distribution.
+    .. 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.
+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
@@ -237 +296 @@
 .. 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 $\bar x$ 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
-
 Note about DLS polydispersity
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -273 +306 @@
 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
@@ -282 +321 @@
 | 2015-05-01 Steve King
 | 2017-05-08 Paul Kienzle
+| 2018-03-20 Steve King