Changeset f4ae8c4 in sasmodels


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Timestamp:
Mar 26, 2018 11:05:02 AM (5 years ago)
Author:
Paul Kienzle <pkienzle@…>
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
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d86f0fc
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ed5b109
Message:

doc: remove extra indentation on dispersion distribution descriptions

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1 edited

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  • doc/guide/pd/polydispersity.rst

    red5b109 rf4ae8c4  
    7878The Uniform Distribution is defined as 
    7979 
    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 half-width, 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. 
     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 
     88where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 
     89distribution, $\sigma$ is the half-width, and *Norm* is a normalization 
     90factor which is determined during the numerical calculation. 
     91 
     92The polydispersity in sasmodels is given by 
     93 
     94.. math:: \text{PD} = \sigma / \bar x 
     95 
     96.. figure:: pd_uniform.jpg 
     97 
     98    Uniform distribution. 
    9999 
    100100The value $N_\sigma$ is ignored for this distribution. 
     
    107107The Rectangular Distribution is defined as 
    108108 
    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 half-width, 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. 
     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 
     117where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 
     118distribution, $w$ is the half-width, and *Norm* is a normalization 
     119factor which is determined during the numerical calculation. 
     120 
     121Note that the standard deviation and the half width $w$ are different! 
     122 
     123The standard deviation is 
     124 
     125.. math:: \sigma = w / \sqrt{3} 
     126 
     127whilst 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. 
    138138 
    139139.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    144144The Gaussian Distribution is defined as 
    145145 
    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. 
     146.. math:: 
     147 
     148    f(x) = \frac{1}{\text{Norm}} 
     149            \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) 
     150 
     151where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 
     152distribution and *Norm* is a normalization factor which is determined 
     153during the numerical calculation. 
     154 
     155The polydispersity in sasmodels is given by 
     156 
     157.. math:: \text{PD} = \sigma / \bar x 
     158 
     159.. figure:: pd_gaussian.jpg 
     160 
     161    Normal distribution. 
    162162 
    163163.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    168168The Boltzmann Distribution is defined as 
    169169 
    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. 
     170.. math:: 
     171 
     172    f(x) = \frac{1}{\text{Norm}} 
     173            \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) 
     174 
     175where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 
     176distribution and *Norm* is a normalization factor which is determined 
     177during the numerical calculation. 
     178 
     179The width is defined as 
     180 
     181.. math:: \sigma=\frac{k T}{E} 
     182 
     183which 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. 
    189189 
    190190.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    199199The Lognormal Distribution is defined as 
    200200 
    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. 
     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 
     207where *Norm* is a normalization factor which will be determined during 
     208the numerical calculation, $\mu=\ln(x_\text{med})$ and $x_\text{med}$ 
     209is the *median* value of the *lognormal* distribution, but $\sigma$ is 
     210a parameter describing the width of the underlying *normal* distribution. 
     211 
     212$x_\text{med}$ will be the value given for the respective size parameter 
     213in sasmodels, for example, *radius=60*. 
     214 
     215The polydispersity in sasmodels is given by 
     216 
     217.. math:: \text{PD} = p = \sigma / x_\text{med} 
     218 
     219The mean value of the distribution is given by $\bar x = \exp(\mu+ p^2/2)$ 
     220and the peak value by $\max x = \exp(\mu - p^2)$. 
     221 
     222The variance (the square of the standard deviation) of the *lognormal* 
     223distribution is given by 
     224 
     225.. math:: 
     226 
     227    \nu = [\exp({\sigma}^2) - 1] \exp({2\mu + \sigma^2}) 
     228 
     229Note that larger values of PD might need a larger number of points 
     230and $N_\sigma$. 
     231 
     232.. figure:: pd_lognormal.jpg 
     233 
     234    Lognormal distribution. 
    235235 
    236236For further information on the Lognormal distribution see: 
     
    249249The Schulz distribution is defined as 
    250250 
    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 = (1-p^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. 
     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 
     256where $\bar x$ ($x_\text{mean}$ in the figure) is the mean of the 
     257distribution, *Norm* is a normalization factor which is determined 
     258during the numerical calculation, and $z$ is a measure of the width 
     259of the distribution such that 
     260 
     261.. math:: z = (1-p^2) / p^2 
     262 
     263where $p$ is the polydispersity in sasmodels given by 
     264 
     265.. math:: PD = p = \sigma / \bar x 
     266 
     267and $\sigma$ is the RMS deviation from $\bar x$. 
     268 
     269Note that larger values of PD might need a larger number of points 
     270and $N_\sigma$. For example, for PD=0.7 with radius=60 |Ang|, at least 
     271Npts>=160 and Nsigmas>=15 are required. 
     272 
     273.. figure:: pd_schulz.jpg 
     274 
     275    Schulz distribution. 
    276276 
    277277For further information on the Schulz distribution see: 
     
    322322 
    323323where $\nu$ is the variance of the distribution and $\bar x$ is the mean 
    324 value of x. 
     324value of $x$. 
    325325 
    326326For more information see: 
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