Changeset 97d172c in sasmodels
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 Aug 9, 2018 10:17:24 AM (2 months ago)
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 master, F1F2models_grethe, beta_approx, cudatest, py3, ticket1074gammainc, ticket1084, ticket1102pinhole, ticket1142pluginreload, ticket1155BE_PolyElectrolyte, ticket1157, ticket608userdefinedweights, ticket_1156
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doc/guide/pd/polydispersity.rst
rd712a0f r97d172c 8 8 .. _polydispersityhelp: 9 9 10 Polydispersity Distributions 11  12 13 With some models in sasmodels we can calculate the average intensity for a 14 population of particles that exhibit size and/or orientational 15 polydispersity. The resultant intensity is normalized by the average 16 particle volume such that 10 Polydispersity & Orientational Distributions 11  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 17 the actual parameter it is being applied too. 18 19 The resultant intensity is then normalized by the average particle volume such 20 that 17 21 18 22 .. math:: … … 21 25 22 26 where $F$ is the scattering amplitude and $\langle\cdot\rangle$ denotes an 23 average over the sizedistribution $f(x; \bar x, \sigma)$, giving27 average over the distribution $f(x; \bar x, \sigma)$, giving 24 28 25 29 .. math:: … … 30 34 Each distribution is characterized by a center value $\bar x$ or 31 35 $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 32 the standard deviation, so read the description carefully), the number of 33 sigmas $N_\sigma$ to include from the tails of the distribution, and the 34 number of points used to compute the average. The center of the distribution 35 is set by the value of the model parameter. The meaning of a polydispersity 36 parameter *PD* (not to be confused with a molecular weight distributions 37 in polymer science) in a model depends on the type of parameter it is being 38 applied too. 36 the standard deviation, so read the description of the distribution carefully), 37 the number of sigmas $N_\sigma$ to include from the tails of the distribution, 38 and the number of points used to compute the average. The center of the 39 distribution is set by the value of the model parameter. 39 40 40 41 The distribution width applied to *volume* (ie, shapedescribing) parameters 41 42 is relative to the center value such that $\sigma = \mathrm{PD} \cdot \bar x$. 42 However, the distribution width applied to *orientation* (ie, angledescribing)43 parameters is just$\sigma = \mathrm{PD}$.43 However, the distribution width applied to *orientation* parameters is just 44 $\sigma = \mathrm{PD}$. 44 45 45 46 $N_\sigma$ determines how far into the tails to evaluate the distribution, … … 51 52 52 53 Users should note that the averaging computation is very intensive. Applying 53 polydispersion to multiple parameters at the same time or increasing the 54 number of points in the distribution will require patience! However, the 55 calculations are generally more robust with more data points or more angles. 54 polydispersion and/or orientational distributions to multiple parameters at 55 the same time, or increasing the number of points in the distribution, will 56 require patience! However, the calculations are generally more robust with 57 more data points or more angles. 56 58 57 59 The following distribution functions are provided: … … 72 74 the term 'polydispersity' (see `Pure Appl. Chem., (2009), 81(2), 73 75 351353 <http://media.iupac.org/publications/pac/2009/pdf/8102x0351.pdf>`_ 74 in order to make the terminology describing distributions of properties 75 unambiguous. Throughout the SasView documentation we continue to use the 76 term polydispersity because one of the consequences of the IUPAC change is 77 that orientational polydispersity would not meet their new criteria (which 78 requires dispersity to be dimensionless). 76 in order to make the terminology describing distributions of chemical 77 properties unambiguous. However, these terms are unrelated to the 78 proportional size distributions and orientational distributions used in 79 SasView models. 79 80 80 81 Suggested Applications 81 82 ^^^^^^^^^^^^^^^^^^^^^^ 82 83 83 If applying polydispersion to parameters describing particle sizes, use84 If applying polydispersion to parameters describing particle sizes, consider using 84 85 the Lognormal or Schulz distributions. 85 86 86 87 If applying polydispersion to parameters describing interfacial thicknesses 87 or angular orientations, usethe Gaussian or Boltzmann distributions.88 or angular orientations, consider using the Gaussian or Boltzmann distributions. 88 89 89 90 If applying polydispersion to parameters describing angles, use the Uniform … … 332 333 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 333 334 334 Many commercial Dynamic Light Scattering (DLS) instruments produce a size 335 polydispersity parameter, sometimes even given the symbol $p$\ ! This 336 parameter is defined as the relative standard deviation coefficient of 337 variation of the size distribution and is NOT the same as the polydispersity 338 parameters in the Lognormal and Schulz distributions above (though they all 339 related) except when the DLS polydispersity parameter is <0.13. 340 341 .. math:: 342 343 p_{DLS} = \sqrt(\nu / \bar x^2) 344 345 where $\nu$ is the variance of the distribution and $\bar x$ is the mean 346 value of $x$. 335 Several measures of polydispersity abound in Dynamic Light Scattering (DLS) and 336 it should not be assumed that any of the following can be simply equated with 337 the polydispersity *PD* parameter used in SasView. 338 339 The dimensionless *Polydispersity Index (PI)* is a measure of the width of the 340 distribution of autocorrelation function decay rates (*not* the distribution of 341 particle sizes itself, though the two are inversely related) and is defined by 342 ISO 22412:2017 as 343 344 .. math:: 345 346 PI = \mu_{2} / \bar \Gamma^2 347 348 where $\mu_\text{2}$ is the second cumulant, and $\bar \Gamma^2$ is the 349 intensityweighted average value, of the distribution of decay rates. 350 351 *If the distribution of decay rates is Gaussian* then 352 353 .. math:: 354 355 PI = \sigma^2 / 2\bar \Gamma^2 356 357 where $\sigma$ is the standard deviation, allowing a *Relative Polydispersity (RP)* 358 to be defined as 359 360 .. math:: 361 362 RP = \sigma / \bar \Gamma = \sqrt{2.PI} 363 364 PI values smaller than 0.05 indicate a highly monodisperse system. Values 365 greater than 0.7 indicate significant polydispersity. 366 367 The *size polydispersity Pparameter* is defined as the relative standard 368 deviation coefficient of variation 369 370 .. math:: 371 372 P = \sqrt\nu / \bar R 373 374 where $\nu$ is the variance of the distribution and $\bar R$ is the mean 375 value of $R$. Here, the product $P \bar R$ is *equal* to the standard 376 deviation of the Lognormal distribution. 377 378 P values smaller than 0.13 indicate a monodisperse system. 347 379 348 380 For more information see: 349 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 381 `ISO 22412:2017, International Standards Organisation (2017) <https://www.iso.org/standard/65410.html>`_. 382 `Polydispersity: What does it mean for DLS and Chromatography <http://www.materialstalks.com/blog/2014/10/23/polydispersitywhatdoesitmeanfordlsandchromatography/>`_. 383 `Dynamic Light Scattering: Common Terms Defined, Whitepaper WP111214. Malvern Instruments (2011) <http://www.biophysics.bioc.cam.ac.uk/wpcontent/uploads/2011/02/DLS_Terms_defined_Malvern.pdf>`_. 384 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143. 385 T Allen, in *Particle Size Measurement*, 4th Edition, Chapman & Hall, London (1990). 350 386 351 387 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 357 393  20180320 Steve King 358 394  20180404 Steve King 395  20180809 Steve King
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