Changeset a5cb9bc in sasmodels
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 Sep 12, 2018 5:19:05 PM (10 months ago)
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doc/guide/pd/polydispersity.rst
r01dba26 ra5cb9bc 11 11  12 12 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 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 17 17 the actual parameter it is being applied too. 18 18 19 The resultant intensity is then normalized by the average particle volume such 19 The resultant intensity is then normalized by the average particle volume such 20 20 that 21 21 … … 24 24 P(q) = \text{scale} \langle F^* F \rangle / V + \text{background} 25 25 26 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an 26 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an 27 27 average over the distribution $f(x; \bar x, \sigma)$, giving 28 28 29 29 .. math:: 30 30 31 P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} 31 P(q) = \frac{\text{scale}}{V} \int_\mathbb{R} 32 32 f(x; \bar x, \sigma) F^2(q, x)\, dx + \text{background} 33 33 34 34 Each distribution is characterized by a center value $\bar x$ or 35 35 $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 36 <<<<<<< HEAD37 36 the standard deviation, so read the description carefully), the number of 38 37 sigmas $N_\sigma$ to include from the tails of the distribution, and the … … 47 46 However, the distribution width applied to *orientation* (ie, angledescribing) 48 47 parameters 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 the53 distribution is set by the value of the model parameter.54 55 The distribution width applied to *volume* (ie, shapedescribing) parameters56 is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$.57 However, the distribution width applied to *orientation* parameters is just58 $\sigma = \mathrm{PD}$.59 >>>>>>> master60 48 61 49 $N_\sigma$ determines how far into the tails to evaluate the distribution, … … 67 55 68 56 Users 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 57 polydispersion and/or orientational distributions to multiple parameters at 58 the same time, or increasing the number of points in the distribution, will 59 require patience! However, the calculations are generally more robust with 72 60 more data points or more angles. 73 61 … … 90 78 **This may not be suitable. See Suggested Applications below.** 91 79 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 351353 <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 351353 <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 98 86 SasView models. 99 87 … … 113 101 The array distribution provides a very simple means of implementing a user 114 102 defined distribution, but without any fittable parameters. Greater flexibility 115 is conferred by the userdefined distribution. 103 is conferred by the userdefined distribution. 116 104 117 105 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 442 430 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 443 431 444 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 432 Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and 433 it should not be assumed that any of the following can be simply equated with 446 434 the polydispersity *PD* parameter used in SasView. 447 435 448 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 436 The dimensionless **Polydispersity Index (PI)** is a measure of the width of the 437 distribution of autocorrelation function decay rates (*not* the distribution of 438 particle sizes itself, though the two are inversely related) and is defined by 451 439 ISO 22412:2017 as 452 440 … … 455 443 PI = \mu_{2} / \bar \Gamma^2 456 444 457 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the 445 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the 458 446 intensityweighted average value, of the distribution of decay rates. 459 447 … … 464 452 PI = \sigma^2 / 2\bar \Gamma^2 465 453 466 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** 454 where $\sigma$ is the standard deviation, allowing a **Relative Polydispersity (RP)** 467 455 to be defined as 468 456 … … 471 459 RP = \sigma / \bar \Gamma = \sqrt{2 \cdot PI} 472 460 473 PI values smaller than 0.05 indicate a highly monodisperse system. Values 461 PI values smaller than 0.05 indicate a highly monodisperse system. Values 474 462 greater than 0.7 indicate significant polydispersity. 475 463 476 The **size polydispersity Pparameter** is defined as the relative standard 477 deviation coefficient of variation 464 The **size polydispersity Pparameter** is defined as the relative standard 465 deviation coefficient of variation 478 466 479 467 .. math:: … … 482 470 483 471 where $\nu$ 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 472 value of $R$. Here, the product $P \bar R$ is *equal* to the standard 485 473 deviation of the Lognormal distribution. 486 474
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