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doc/guide/pd/polydispersity.rst (modified) (10 diffs)
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doc/guide/pd/polydispersity.rst
r92d330fd r5026e05 23 23 average over the size distribution. 24 24 25 Each distribution is characterized by its center $\bar x$, its width $\sigma$, 26 the number of sigmas $N_\sigma$ to include from the tails, and the number of 27 points used to compute the average. The center of the distribution is set by the 28 value of the model parameter. Volume parameters have polydispersity *PD* 29 (not to be confused with a molecular weight distributions in polymer science) 30 leading to a size distribution of width $\text{PD} = \sigma / \bar x$, but 31 orientation parameters use an angular distributions of width $\sigma$. 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 32 35 $N_\sigma$ determines how far into the tails to evaluate the distribution, with 33 36 larger values of $N_\sigma$ required for heavier tailed distributions. … … 44 47 The following distribution functions are provided: 45 48 49 * *Uniform Distribution* 46 50 * *Rectangular Distribution* 47 * *Uniform Distribution*48 51 * *Gaussian Distribution* 52 * *Boltzmann Distribution* 49 53 * *Lognormal Distribution* 50 54 * *Schulz Distribution* 51 55 * *Array Distribution* 52 * *Boltzmann Distribution*53 56 54 57 These are all implemented as *number-average* distributions. 55 58 56 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 57 58 Rectangular Distribution 59 ^^^^^^^^^^^^^^^^^^^^^^^^ 60 61 The Rectangular Distribution is defined as 62 63 .. math:: 64 65 f(x) = \frac{1}{\text{Norm}} 66 \begin{cases} 67 1 & \text{for } |x - \bar x| \leq w \\ 68 0 & \text{for } |x - \bar x| > w 69 \end{cases} 70 71 where $\bar x$ is the mean of the distribution, $w$ is the half-width, and 72 *Norm* is a normalization factor which is determined during the numerical 73 calculation. 74 75 Note that the standard deviation and the half width $w$ are different! 76 77 The standard deviation is 78 79 .. math:: \sigma = w / \sqrt{3} 80 81 whilst the polydispersity is 82 83 .. math:: \text{PD} = \sigma / \bar x 84 85 .. figure:: pd_rectangular.jpg 86 87 Rectangular distribution. 88 89 59 Additional distributions are under consideration. 60 61 Suggested Applications 62 ^^^^^^^^^^^^^^^^^^^^^^ 63 64 If applying polydispersion to parameters describing particle sizes, use 65 the Lognormal or Schulz distributions. 66 67 If applying polydispersion to parameters describing interfacial thicknesses 68 or angular orientations, use the Gaussian or Boltzmann distributions. 69 70 The array distribution allows a user-defined distribution to be applied. 71 72 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 90 73 91 74 Uniform Distribution 92 ^^^^^^^^^^^^^^^^^^^^ ^^^^75 ^^^^^^^^^^^^^^^^^^^^ 93 76 94 77 The Uniform Distribution is defined as … … 102 85 \end{cases} 103 86 104 where $\bar x$ is the mean of the distribution, $\sigma$ is the half-width, and105 *Norm* is a normalization factor which is determined during the numerical106 calculation.107 108 Note that the polydispersityis given by87 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, 88 $\sigma$ is the half-width, and *Norm* is a normalization factor which is 89 determined during the numerical calculation. 90 91 The polydispersity in sasmodels is given by 109 92 110 93 .. math:: \text{PD} = \sigma / \bar x … … 115 98 116 99 The value $N_\sigma$ is ignored for this distribution. 100 101 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 102 103 Rectangular Distribution 104 ^^^^^^^^^^^^^^^^^^^^^^^^ 105 106 The Rectangular Distribution is defined as 107 108 .. math:: 109 110 f(x) = \frac{1}{\text{Norm}} 111 \begin{cases} 112 1 & \text{for } |x - \bar x| \leq w \\ 113 0 & \text{for } |x - \bar x| > w 114 \end{cases} 115 116 where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the distribution, 117 $w$ is the half-width, and *Norm* is a normalization factor which is determined 118 during the numerical calculation. 119 120 Note that the standard deviation and the half width $w$ are different! 121 122 The standard deviation is 123 124 .. math:: \sigma = w / \sqrt{3} 125 126 whilst the polydispersity in sasmodels is given by 127 128 .. math:: \text{PD} = \sigma / \bar x 129 130 .. figure:: pd_rectangular.jpg 131 132 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. 117 136 118 137 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 123 142 The Gaussian Distribution is defined as 124 143 125 .. math:: 126 127 f(x) = \frac{1}{\text{Norm}} 128 \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) 129 130 where $\bar x$ is the mean of the distribution and *Norm* is a normalization 131 factor which is determined during the numerical calculation. 132 133 The polydispersity is 134 135 .. math:: \text{PD} = \sigma / \bar x 136 137 .. figure:: pd_gaussian.jpg 138 139 Normal distribution. 144 .. math:: 145 146 f(x) = \frac{1}{\text{Norm}} 147 \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) 148 149 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. 151 152 The polydispersity in sasmodels is given by 153 154 .. math:: \text{PD} = \sigma / \bar x 155 156 .. figure:: pd_gaussian.jpg 157 158 Normal distribution. 159 160 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 161 162 Boltzmann Distribution 163 ^^^^^^^^^^^^^^^^^^^^^^ 164 165 The Boltzmann Distribution is defined as 166 167 .. math:: 168 169 f(x) = \frac{1}{\text{Norm}} 170 \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) 171 172 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. 174 175 The width is defined as 176 177 .. math:: \sigma=\frac{k T}{E} 178 179 which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, 180 $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. 181 182 .. figure:: pd_boltzmann.jpg 183 184 Boltzmann distribution. 140 185 141 186 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 144 189 ^^^^^^^^^^^^^^^^^^^^^^ 145 190 191 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 $x$. 193 146 194 The Lognormal Distribution is defined as 147 195 148 .. math:: 149 150 f(x) = \frac{1}{\text{Norm}} 151 \frac{1}{xp}\exp\left(-\frac{(\ln(x) - \mu)^2}{2p^2}\right) 152 153 where $\mu=\ln(x_\text{med})$ when $x_\text{med}$ is the median value of the 154 distribution, and *Norm* is a normalization factor which will be determined 155 during the numerical calculation. 156 157 The median value for the distribution will be the value given for the 158 respective size parameter, for example, *radius=60*. 159 160 The polydispersity is given by $\sigma$ 161 162 .. math:: \text{PD} = p 163 164 For the angular distribution 165 166 .. math:: p = \sigma / x_\text{med} 167 168 The mean value is given by $\bar x = \exp(\mu+ p^2/2)$. The peak value 169 is given by $\max x = \exp(\mu - p^2)$. 170 171 .. figure:: pd_lognormal.jpg 172 173 Lognormal distribution. 174 175 This distribution function spreads more, and the peak shifts to the left, as 176 $p$ increases, so it requires higher values of $N_\sigma$ and more points 177 in the distribution. 196 .. math:: 197 198 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*. 207 208 The polydispersity in sasmodels is given by 209 210 .. math:: \text{PD} = p = \sigma / x_\text{med} 211 212 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 216 217 .. math:: 218 219 \nu = [\exp({\sigma}^2) - 1] \exp({2\mu + \sigma^2}) 220 221 Note that larger values of PD might need a larger number of points and $N_\sigma$. 222 223 .. figure:: pd_lognormal.jpg 224 225 Lognormal distribution. 226 227 For further information on the Lognormal distribution see: 228 http://en.wikipedia.org/wiki/Log-normal_distribution and 229 http://mathworld.wolfram.com/LogNormalDistribution.html 178 230 179 231 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 182 234 ^^^^^^^^^^^^^^^^^^^ 183 235 236 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. 239 184 240 The Schulz distribution is defined as 185 241 186 .. math:: 187 188 f(x) = \frac{1}{\text{Norm}} 189 (z+1)^{z+1}(x/\bar x)^z\frac{\exp[-(z+1)x/\bar x]}{\bar x\Gamma(z+1)} 190 191 where $\bar x$ is the mean of the distribution and *Norm* is a normalization 192 factor which is determined during the numerical calculation, and $z$ is a 193 measure of the width of the distribution such that 194 195 .. math:: z = (1-p^2) / p^2 196 197 The polydispersity is 198 199 .. math:: p = \sigma / \bar x 200 201 Note that larger values of PD might need larger number of points and $N_\sigma$. 202 For example, at PD=0.7 and radius=60 |Ang|, Npts>=160 and Nsigmas>=15 at least. 203 204 .. figure:: pd_schulz.jpg 205 206 Schulz distribution. 242 .. math:: 243 244 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 250 251 .. math:: z = (1-p^2) / p^2 252 253 where $p$ is the polydispersity in sasmodels given by 254 255 .. math:: PD = p = \sigma / \bar x 256 257 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. 261 262 .. figure:: pd_schulz.jpg 263 264 Schulz distribution. 207 265 208 266 For further information on the Schulz distribution see: 209 M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461. 267 M Kotlarchyk & S-H Chen, *J Chem Phys*, (1983), 79, 2461 and 268 M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 210 269 211 270 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 237 296 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 238 297 239 Boltzmann Distribution240 ^^^^^^^^^^^^^^^^^^^^^^241 242 The Boltzmann Distribution is defined as243 244 .. math::245 246 f(x) = \frac{1}{\text{Norm}}247 \exp\left(-\frac{ | x - \bar x | }{\sigma}\right)248 249 where $\bar x$ is the mean of the distribution and *Norm* is a normalization250 factor which is determined during the numerical calculation.251 The width is defined as252 253 .. math:: \sigma=\frac{k T}{E}254 255 which is the inverse Boltzmann factor,256 where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a257 characteristic energy per particle.258 259 .. figure:: pd_boltzmann.jpg260 261 Boltzmann distribution.262 263 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ264 265 298 Note about DLS polydispersity 266 299 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ … … 273 306 related) except when the DLS polydispersity parameter is <0.13. 274 307 308 .. math:: 309 310 p_{DLS} = \sqrt(\nu / \bar x^2) 311 312 where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. 313 275 314 For more information see: 276 315 S King, C Washington & R Heenan, *Phys Chem Chem Phys*, (2005), 7, 143 … … 282 321 | 2015-05-01 Steve King 283 322 | 2017-05-08 Paul Kienzle 323 | 2018-03-20 Steve King
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