Changeset f4ae8c4 in sasmodels for doc/guide/pd/polydispersity.rst
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 Mar 26, 2018 11:05:02 AM (5 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
red5b109 rf4ae8c4 78 78 The Uniform Distribution is defined as 79 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 80 .. math:: 81 82 f(x) = \frac{1}{\text{Norm}} 83 \begin{cases} 84 1 & \text{for } x  \bar x \leq \sigma \\ 85 0 & \text{for } x  \bar x > \sigma 86 \end{cases} 87 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. 91 92 The polydispersity in sasmodels is given by 93 94 .. math:: \text{PD} = \sigma / \bar x 95 96 .. figure:: pd_uniform.jpg 97 98 Uniform distribution. 99 99 100 100 The value $N_\sigma$ is ignored for this distribution. … … 107 107 The Rectangular Distribution is defined as 108 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 109 .. math:: 110 111 f(x) = \frac{1}{\text{Norm}} 112 \begin{cases} 113 1 & \text{for } x  \bar x \leq w \\ 114 0 & \text{for } x  \bar x > w 115 \end{cases} 116 117 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. 120 121 Note that the standard deviation and the half width $w$ are different! 122 123 The standard deviation is 124 125 .. math:: \sigma = w / \sqrt{3} 126 127 whilst the polydispersity in sasmodels is given by 128 129 .. math:: \text{PD} = \sigma / \bar x 130 131 .. figure:: pd_rectangular.jpg 132 133 Rectangular distribution. 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. 138 138 139 139 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 144 144 The Gaussian Distribution is defined as 145 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 146 .. math:: 147 148 f(x) = \frac{1}{\text{Norm}} 149 \exp\left(\frac{(x  \bar x)^2}{2\sigma^2}\right) 150 151 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. 154 155 The polydispersity in sasmodels is given by 156 157 .. math:: \text{PD} = \sigma / \bar x 158 159 .. figure:: pd_gaussian.jpg 160 161 Normal distribution. 162 162 163 163 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 168 168 The Boltzmann Distribution is defined as 169 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 170 .. math:: 171 172 f(x) = \frac{1}{\text{Norm}} 173 \exp\left(\frac{  x  \bar x  }{\sigma}\right) 174 175 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. 178 179 The width is defined as 180 181 .. math:: \sigma=\frac{k T}{E} 182 183 which is the inverse Boltzmann factor, where $k$ is the Boltzmann constant, 184 $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. 185 186 .. figure:: pd_boltzmann.jpg 187 188 Boltzmann distribution. 189 189 190 190 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 199 199 The Lognormal Distribution is defined as 200 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 201 .. math:: 202 203 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*. 214 215 The polydispersity in sasmodels is given by 216 217 .. math:: \text{PD} = p = \sigma / x_\text{med} 218 219 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 224 225 .. math:: 226 227 \nu = [\exp({\sigma}^2)  1] \exp({2\mu + \sigma^2}) 228 229 Note that larger values of PD might need a larger number of points 230 and $N_\sigma$. 231 232 .. figure:: pd_lognormal.jpg 233 234 Lognormal distribution. 235 235 236 236 For further information on the Lognormal distribution see: … … 249 249 The Schulz distribution is defined as 250 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 251 .. math:: 252 253 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 260 261 .. math:: z = (1p^2) / p^2 262 263 where $p$ is the polydispersity in sasmodels given by 264 265 .. math:: PD = p = \sigma / \bar x 266 267 and $\sigma$ is the RMS deviation from $\bar x$. 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. 272 273 .. figure:: pd_schulz.jpg 274 275 Schulz distribution. 276 276 277 277 For further information on the Schulz distribution see: … … 322 322 323 323 where $\nu$ is the variance of the distribution and $\bar x$ is the mean 324 value of x.324 value of $x$. 325 325 326 326 For more information see:
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