-                      red5b109
+                      rf4ae8c4
 The Uniform Distribution is defined as
     .. math::
         f(x) = \frac{1}{\text{Norm}}
         \begin{cases}
 & \text{for } |x - \bar x| \leq \sigma \\
 & \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.
+.. math::
+    f(x) = \frac{1}{\text{Norm}}
+    \begin{cases}
+& \text{for } |x - \bar x| \leq \sigma \\
+& \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.
 …
 The Rectangular Distribution is defined as
     .. math::
         f(x) = \frac{1}{\text{Norm}}
         \begin{cases}
 & \text{for } |x - \bar x| \leq w \\
 & \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.
+.. math::
+    f(x) = \frac{1}{\text{Norm}}
+    \begin{cases}
+& \text{for } |x - \bar x| \leq w \\
+& \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
 …
 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.
+.. 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
 …
 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.
+.. 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
 …
 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} = 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.
+.. 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:
 …
 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.
+.. 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:
 …
 where $\nu$ is the variance of the distribution and $\bar x$ is the mean
 value of x.
+value of $x$.
 For more information see:

Note: See TracChangeset for help on using the changeset viewer.

SasView

Changeset f4ae8c4 in sasmodels

Legend:

doc/guide/pd/polydispersity.rst

Download in other formats: