# Changeset a5cb9bc in sasmodels

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Timestamp:
Sep 12, 2018 5:19:05 PM (10 days ago)
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ticket-608-user-defined-weights
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35d2300
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01dba26
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 r01dba26 -------------------------------------------- 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 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 The resultant intensity is then normalized by the average particle volume such that P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an average over the distribution $f(x; \bar x, \sigma)$, giving .. math:: P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} f(x; \bar x, \sigma) F^2(q, x)\, dx + \text{background} Each distribution is characterized by a center value $\bar x$ or $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* <<<<<<< HEAD the standard deviation, so read the description carefully), the number of sigmas $N_\sigma$ to include from the tails of the distribution, and the However, the distribution width applied to *orientation* (ie, angle-describing) parameters is just $\sigma = \mathrm{PD}$. ======= 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* parameters is just $\sigma = \mathrm{PD}$. >>>>>>> master $N_\sigma$ determines how far into the tails to evaluate the distribution, Users should note that the averaging computation is very intensive. Applying 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 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. **This may not be suitable. See Suggested Applications below.** .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace the term 'polydispersity' (see Pure Appl. Chem., (2009), 81(2), 351-353 _ 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 .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace the term 'polydispersity' (see Pure Appl. Chem., (2009), 81(2), 351-353 _ 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. 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. is conferred by the user-defined distribution. .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 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 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 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 PI = \mu_{2} / \bar \Gamma^2 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the intensity-weighted average value, of the distribution of decay rates. PI = \sigma^2 / 2\bar \Gamma^2 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** to be defined as RP = \sigma / \bar \Gamma = \sqrt{2 \cdot PI} PI values smaller than 0.05 indicate a highly monodisperse system. Values 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 The **size polydispersity P-parameter** is defined as the relative standard deviation coefficient of variation .. math:: 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 value of $R$. Here, the product $P \bar R$ is *equal* to the standard deviation of the Lognormal distribution.