Changes in doc/guide/pd/polydispersity.rst [22279a4:eda8b30] in sasmodels
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doc/guide/pd/polydispersity.rst
r22279a4 reda8b30 42 42 calculations are generally more robust with more data points or more angles. 43 43 44 The following distribution functions are provided:44 The following five distribution functions are provided: 45 45 46 46 * *Rectangular Distribution* 47 * *Uniform Distribution*48 47 * *Gaussian Distribution* 49 48 * *Lognormal Distribution* 50 49 * *Schulz Distribution* 51 50 * *Array Distribution* 52 * *Boltzmann Distribution*53 51 54 52 These are all implemented as *number-average* distributions. … … 87 85 Rectangular distribution. 88 86 89 Uniform Distribution90 ^^^^^^^^^^^^^^^^^^^^^^^^91 92 The Uniform Distribution is defined as93 94 .. math::95 96 f(x) = \frac{1}{\text{Norm}}97 \begin{cases}98 1 & \text{for } |x - \bar x| \leq \sigma \\99 0 & \text{for } |x - \bar x| > \sigma100 \end{cases}101 102 where $\bar x$ is the mean of the distribution, $\sigma$ is the half-width, and103 *Norm* is a normalization factor which is determined during the numerical104 calculation.105 106 Note that the polydispersity is given by107 108 .. math:: \text{PD} = \sigma / \bar x109 110 .. figure:: pd_uniform.jpg111 112 Uniform distribution.113 114 87 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 115 88 … … 210 183 ^^^^^^^^^^^^^^^^^^ 211 184 212 This user-definable distribution should be given as a simple ASCII text185 This user-definable distribution should be given as as a simple ASCII text 213 186 file where the array is defined by two columns of numbers: $x$ and $f(x)$. 214 187 The $f(x)$ will be normalized to 1 during the computation. … … 229 202 given for the model will have no affect, and will be ignored when computing 230 203 the average. This means that any parameter with an array distribution will 231 not be fitable. 232 233 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 234 235 Boltzmann Distribution 236 ^^^^^^^^^^^^^^^^^^^^^^ 237 238 The Boltzmann Distribution is defined as 239 240 .. math:: 241 242 f(x) = \frac{1}{\text{Norm}} 243 \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) 244 245 where $\bar x$ is the mean of the distribution and *Norm* is a normalization 246 factor which is determined during the numerical calculation. 247 The width is defined as 248 249 .. math:: \sigma=\frac{k T}{E} 250 251 which is the inverse Boltzmann factor, 252 where $k$ is the Boltzmann constant, $T$ the temperature in Kelvin and $E$ a 253 characteristic energy per particle. 254 255 .. figure:: pd_boltzmann.jpg 256 257 Boltzmann distribution. 204 not be fittable. 258 205 259 206 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
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