Changeset a5cb9bc in sasmodels


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Timestamp:
Sep 12, 2018 5:19:05 PM (6 years ago)
Author:
Paul Kienzle <pkienzle@…>
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  • doc/guide/pd/polydispersity.rst

    r01dba26 ra5cb9bc  
    1111-------------------------------------------- 
    1212 
    13 For some models we can calculate the average intensity for a population of  
    14 particles that possess size and/or orientational (ie, angular) distributions.  
    15 In SasView we call the former *polydispersity* but use the parameter *PD* to  
    16 parameterise both. In other words, the meaning of *PD* in a model depends on  
     13For some models we can calculate the average intensity for a population of 
     14particles that possess size and/or orientational (ie, angular) distributions. 
     15In SasView we call the former *polydispersity* but use the parameter *PD* to 
     16parameterise both. In other words, the meaning of *PD* in a model depends on 
    1717the actual parameter it is being applied too. 
    1818 
    19 The resultant intensity is then normalized by the average particle volume such  
     19The resultant intensity is then normalized by the average particle volume such 
    2020that 
    2121 
    … …  
    2424  P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} 
    2525 
    26 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an  
     26where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an 
    2727average over the distribution $f(x; \bar x, \sigma)$, giving 
    2828 
    2929.. math:: 
    3030 
    31   P(q) = \frac{\text{scale}}{V} \int_\mathbb{R}  
     31  P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} 
    3232  f(x; \bar x, \sigma) F^2(q, x)\, dx + \text{background} 
    3333 
    3434Each distribution is characterized by a center value $\bar x$ or 
    3535$x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 
    36 <<<<<<< HEAD 
    3736the standard deviation, so read the description carefully), the number of 
    3837sigmas $N_\sigma$ to include from the tails of the distribution, and the 
    … …  
    4746However, the distribution width applied to *orientation* (ie, angle-describing) 
    4847parameters is just $\sigma = \mathrm{PD}$. 
    49 ======= 
    50 the standard deviation, so read the description of the distribution carefully),  
    51 the number of sigmas $N_\sigma$ to include from the tails of the distribution,  
    52 and the number of points used to compute the average. The center of the  
    53 distribution is set by the value of the model parameter. 
    54  
    55 The distribution width applied to *volume* (ie, shape-describing) parameters  
    56 is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$.  
    57 However, the distribution width applied to *orientation* parameters is just  
    58 $\sigma = \mathrm{PD}$. 
    59 >>>>>>> master 
    6048 
    6149$N_\sigma$ determines how far into the tails to evaluate the distribution, 
    … …  
    6755 
    6856Users should note that the averaging computation is very intensive. Applying 
    69 polydispersion and/or orientational distributions to multiple parameters at  
    70 the same time, or increasing the number of points in the distribution, will  
    71 require patience! However, the calculations are generally more robust with  
     57polydispersion and/or orientational distributions to multiple parameters at 
     58the same time, or increasing the number of points in the distribution, will 
     59require patience! However, the calculations are generally more robust with 
    7260more data points or more angles. 
    7361 
    … …  
    9078**This may not be suitable. See Suggested Applications below.** 
    9179 
    92 .. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace  
    93            the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2),  
    94            351-353 <http://media.iupac.org/publications/pac/2009/pdf/8102x0351.pdf>`_  
    95            in order to make the terminology describing distributions of chemical  
    96            properties unambiguous. However, these terms are unrelated to the  
    97            proportional size distributions and orientational distributions used in  
     80.. note:: In 2009 IUPAC decided to introduce the new term 'dispersity' to replace 
     81           the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2), 
     82           351-353 <http://media.iupac.org/publications/pac/2009/pdf/8102x0351.pdf>`_ 
     83           in order to make the terminology describing distributions of chemical 
     84           properties unambiguous. However, these terms are unrelated to the 
     85           proportional size distributions and orientational distributions used in 
    9886           SasView models. 
    9987 
    … …  
    113101The array distribution provides a very simple means of implementing a user- 
    114102defined distribution, but without any fittable parameters. Greater flexibility 
    115 is conferred by the user-defined distribution.  
     103is conferred by the user-defined distribution. 
    116104 
    117105.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
    … …  
    442430^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 
    443431 
    444 Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and  
    445 it should not be assumed that any of the following can be simply equated with  
     432Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and 
     433it should not be assumed that any of the following can be simply equated with 
    446434the polydispersity *PD* parameter used in SasView. 
    447435 
    448 The dimensionless **Polydispersity Index (PI)** is a measure of the width of the  
    449 distribution of autocorrelation function decay rates (*not* the distribution of  
    450 particle sizes itself, though the two are inversely related) and is defined by  
     436The dimensionless **Polydispersity Index (PI)** is a measure of the width of the 
     437distribution of autocorrelation function decay rates (*not* the distribution of 
     438particle sizes itself, though the two are inversely related) and is defined by 
    451439ISO 22412:2017 as 
    452440 
    … …  
    455443    PI = \mu_{2} / \bar \Gamma^2 
    456444 
    457 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the  
     445where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the 
    458446intensity-weighted average value, of the distribution of decay rates. 
    459447 
    … …  
    464452    PI = \sigma^2 / 2\bar \Gamma^2 
    465453 
    466 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)**  
     454where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** 
    467455to be defined as 
    468456 
    … …  
    471459    RP = \sigma / \bar \Gamma = \sqrt{2 \cdot PI} 
    472460 
    473 PI values smaller than 0.05 indicate a highly monodisperse system. Values  
     461PI values smaller than 0.05 indicate a highly monodisperse system. Values 
    474462greater than 0.7 indicate significant polydispersity. 
    475463 
    476 The **size polydispersity P-parameter** is defined as the relative standard  
    477 deviation coefficient of variation   
     464The **size polydispersity P-parameter** is defined as the relative standard 
     465deviation coefficient of variation 
    478466 
    479467.. math:: 
    … …  
    482470 
    483471where $\nu$ is the variance of the distribution and $\bar R$ is the mean 
    484 value of $R$. Here, the product $P \bar R$ is *equal* to the standard  
     472value of $R$. Here, the product $P \bar R$ is *equal* to the standard 
    485473deviation of the Lognormal distribution. 
    486474 
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