Changeset ed5b109 in sasmodels for doc/guide/pd/polydispersity.rst
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 Mar 26, 2018 10:43:35 AM (6 years ago)
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 master, core_shell_microgels, magnetic_model, ticket1257vesicleproduct, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
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doc/guide/pd/polydispersity.rst
r5026e05 red5b109 23 23 average over the size distribution. 24 24 25 Each distribution is characterized by a center value $\bar x$ or $x_\text{med}$, 26 a width parameter $\sigma$ (note this is *not necessarily* the standard deviation, so read 27 the description carefully), the number of sigmas $N_\sigma$ to include from the 28 tails of the distribution, and the number of points used to compute the average. 29 The center of the distribution is set by the value of the model parameter. 30 31 Volume parameters have polydispersity *PD* (not to be confused with a molecular 32 weight distributions in polymer science), but orientation parameters use angular 33 distributions of width $\sigma$. 34 35 $N_\sigma$ determines how far into the tails to evaluate the distribution, with 36 larger values of $N_\sigma$ required for heavier tailed distributions. 25 Each distribution is characterized by a center value $\bar x$ or 26 $x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 27 the standard deviation, so read the description carefully), the number of 28 sigmas $N_\sigma$ to include from the tails of the distribution, and the 29 number of points used to compute the average. The center of the distribution 30 is set by the value of the model parameter. 31 32 Volume parameters have polydispersity *PD* (not to be confused with a 33 molecular weight distributions in polymer science), but orientation parameters 34 use angular distributions of width $\sigma$. 35 36 $N_\sigma$ determines how far into the tails to evaluate the distribution, 37 with larger values of $N_\sigma$ required for heavier tailed distributions. 37 38 The scattering in general falls rapidly with $qr$ so the usual assumption 38 39 that $G(r  3\sigma_r)$ is tiny and therefore $f(r  3\sigma_r)G(r  3\sigma_r)$ … … 62 63 ^^^^^^^^^^^^^^^^^^^^^^ 63 64 64 If applying polydispersion to parameters describing particle sizes, use 65 If applying polydispersion to parameters describing particle sizes, use 65 66 the Lognormal or Schulz distributions. 66 67 67 If applying polydispersion to parameters describing interfacial thicknesses 68 If applying polydispersion to parameters describing interfacial thicknesses 68 69 or angular orientations, use the Gaussian or Boltzmann distributions. 69 70 … … 85 86 \end{cases} 86 87 87 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution,88 $\sigma$ is the halfwidth, and *Norm* is a normalization factor which is89 determined during the numerical calculation.88 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 89 distribution, $\sigma$ is the halfwidth, and *Norm* is a normalization 90 factor which is determined during the numerical calculation. 90 91 91 92 The polydispersity in sasmodels is given by … … 114 115 \end{cases} 115 116 116 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution,117 $w$ is the halfwidth, and *Norm* is a normalization factor which is determined118 during the numerical calculation.117 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 118 distribution, $w$ is the halfwidth, and *Norm* is a normalization 119 factor which is determined during the numerical calculation. 119 120 120 121 Note that the standard deviation and the half width $w$ are different! … … 131 132 132 133 Rectangular distribution. 133 134 .. note:: The Rectangular Distribution is deprecated in favour of the Uniform Distribution 135 above and is described here for backwards compatibility with earlier versions of SasView only. 134 135 .. note:: The Rectangular Distribution is deprecated in favour of the 136 Uniform Distribution above and is described here for backwards 137 compatibility with earlier versions of SasView only. 136 138 137 139 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 147 149 \exp\left(\frac{(x  \bar x)^2}{2\sigma^2}\right) 148 150 149 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution 150 and *Norm* is a normalization factor which is determined during the numerical calculation. 151 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 152 distribution and *Norm* is a normalization factor which is determined 153 during the numerical calculation. 151 154 152 155 The polydispersity in sasmodels is given by … … 170 173 \exp\left(\frac{  x  \bar x  }{\sigma}\right) 171 174 172 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution 173 and *Norm* is a normalization factor which is determined during the numerical calculation. 175 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 176 distribution and *Norm* is a normalization factor which is determined 177 during the numerical calculation. 174 178 175 179 The width is defined as … … 177 181 .. math:: \sigma=\frac{k T}{E} 178 182 179 which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, 183 which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, 180 184 $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. 181 185 … … 189 193 ^^^^^^^^^^^^^^^^^^^^^^ 190 194 191 The Lognormal Distribution describes a function of $x$ where $\ln (x)$ has a normal distribution. 192 The result is a distribution that is skewed towards larger values of $x$. 195 The Lognormal Distribution describes a function of $x$ where $\ln (x)$ has 196 a normal distribution. The result is a distribution that is skewed towards 197 larger values of $x$. 193 198 194 199 The Lognormal Distribution is defined as … … 196 201 .. math:: 197 202 198 f(x) = \frac{1}{\text{Norm}} 199 \frac{1}{x\sigma}\exp\left(\frac{1}{2}\bigg(\frac{\ln(x)  \mu}{\sigma}\bigg)^2\right) 200 201 where *Norm* is a normalization factor which will be determined during the numerical calculation, 202 $\mu=\ln(x_\text{med})$ and $x_\text{med}$ is the *median* value of the *lognormal* distribution, 203 but $\sigma$ is a parameter describing the width of the underlying *normal* distribution. 204 205 $x_\text{med}$ will be the value given for the respective size parameter in sasmodels, for 206 example, *radius=60*. 203 f(x) = \frac{1}{\text{Norm}}\frac{1}{x\sigma} 204 \exp\left(\frac{1}{2} 205 \bigg(\frac{\ln(x)  \mu}{\sigma}\bigg)^2\right) 206 207 where *Norm* is a normalization factor which will be determined during 208 the numerical calculation, $\mu=\ln(x_\text{med})$ and $x_\text{med}$ 209 is the *median* value of the *lognormal* distribution, but $\sigma$ is 210 a parameter describing the width of the underlying *normal* distribution. 211 212 $x_\text{med}$ will be the value given for the respective size parameter 213 in sasmodels, for example, *radius=60*. 207 214 208 215 The polydispersity in sasmodels is given by … … 210 217 .. math:: \text{PD} = p = \sigma / x_\text{med} 211 218 212 The mean value of the distribution is given by $\bar x = \exp(\mu+ p^2/2)$ and the peak value 213 by $\max x = \exp(\mu  p^2)$. 214 215 The variance (the square of the standard deviation) of the *lognormal* distribution is given by 219 The mean value of the distribution is given by $\bar x = \exp(\mu+ p^2/2)$ 220 and the peak value by $\max x = \exp(\mu  p^2)$. 221 222 The variance (the square of the standard deviation) of the *lognormal* 223 distribution is given by 216 224 217 225 .. math:: … … 219 227 \nu = [\exp({\sigma}^2)  1] \exp({2\mu + \sigma^2}) 220 228 221 Note that larger values of PD might need a larger number of points and $N_\sigma$. 222 229 Note that larger values of PD might need a larger number of points 230 and $N_\sigma$. 231 223 232 .. figure:: pd_lognormal.jpg 224 233 … … 226 235 227 236 For further information on the Lognormal distribution see: 228 http://en.wikipedia.org/wiki/Lognormal_distribution and 237 http://en.wikipedia.org/wiki/Lognormal_distribution and 229 238 http://mathworld.wolfram.com/LogNormalDistribution.html 230 239 … … 234 243 ^^^^^^^^^^^^^^^^^^^ 235 244 236 The Schulz (sometimes written Schultz) distribution is similar to the Lognormal distribution,237 in that it is also skewed towards larger values of $x$, but which has computational advantages 238 over the Lognormal distribution.245 The Schulz (sometimes written Schultz) distribution is similar to the 246 Lognormal distribution, in that it is also skewed towards larger values of 247 $x$, but which has computational advantages over the Lognormal distribution. 239 248 240 249 The Schulz distribution is defined as … … 242 251 .. math:: 243 252 244 f(x) = \frac{1}{\text{Norm}} 245 (z+1)^{z+1}(x/\bar x)^z\frac{\exp[(z+1)x/\bar x]}{\bar x\Gamma(z+1)} 246 247 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, 248 *Norm* is a normalization factor which is determined during the numerical calculation, 249 and $z$ is a measure of the width of the distribution such that 253 f(x) = \frac{1}{\text{Norm}} (z+1)^{z+1}(x/\bar x)^z 254 \frac{\exp[(z+1)x/\bar x]}{\bar x\Gamma(z+1)} 255 256 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 257 distribution, *Norm* is a normalization factor which is determined 258 during the numerical calculation, and $z$ is a measure of the width 259 of the distribution such that 250 260 251 261 .. math:: z = (1p^2) / p^2 … … 256 266 257 267 and $\sigma$ is the RMS deviation from $\bar x$. 258 259 Note that larger values of PD might need a larger number of points and $N_\sigma$. 260 For example, for PD=0.7 with radius=60 Ang, at least Npts>=160 and Nsigmas>=15 are required. 268 269 Note that larger values of PD might need a larger number of points 270 and $N_\sigma$. For example, for PD=0.7 with radius=60 Ang, at least 271 Npts>=160 and Nsigmas>=15 are required. 261 272 262 273 .. figure:: pd_schulz.jpg … … 266 277 For further information on the Schulz distribution see: 267 278 M Kotlarchyk & SH Chen, *J Chem Phys*, (1983), 79, 2461 and 268 M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 279 M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 269 280 270 281 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 310 321 p_{DLS} = \sqrt(\nu / \bar x^2) 311 322 312 where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. 323 where $\nu$ is the variance of the distribution and $\bar x$ is the mean 324 value of x. 313 325 314 326 For more information see:
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