# Changeset ed5b109 in sasmodels

Ignore:
Timestamp:
Mar 26, 2018 10:43:35 AM (6 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
f4ae8c4
Parents:
5026e05
Message:

docs: reformat dispersity docs to 80 columns

File:
1 edited

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Unmodified
 r5026e05 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. 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. 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. The scattering in general falls rapidly with $qr$ so the usual assumption that $G(r - 3\sigma_r)$ is tiny and therefore $f(r - 3\sigma_r)G(r - 3\sigma_r)$ ^^^^^^^^^^^^^^^^^^^^^^ If applying polydispersion to parameters describing particle sizes, use If applying polydispersion to parameters describing particle sizes, use the Lognormal or Schulz distributions. If applying polydispersion to parameters describing interfacial thicknesses If applying polydispersion to parameters describing interfacial thicknesses or angular orientations, use the Gaussian or Boltzmann distributions. \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. 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 \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. 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! 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. .. 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 \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. 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 \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. 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, which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. ^^^^^^^^^^^^^^^^^^^^^^ 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 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*. 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 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 $N_\sigma$. Note that larger values of PD might need a larger number of points and $N_\sigma$. .. figure:: pd_lognormal.jpg For further information on the Lognormal distribution see: http://en.wikipedia.org/wiki/Log-normal_distribution and http://en.wikipedia.org/wiki/Log-normal_distribution and http://mathworld.wolfram.com/LogNormalDistribution.html ^^^^^^^^^^^^^^^^^^^ 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 (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 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 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. 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 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 M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ p_{DLS} = \sqrt(\nu / \bar x^2) where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. For more information see: