Changeset ed5b109 in sasmodels


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Timestamp:
Mar 26, 2018 10:43:35 AM (4 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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f4ae8c4
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5026e05
Message:

docs: reformat dispersity docs to 80 columns

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1 edited

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  • doc/guide/pd/polydispersity.rst

    r5026e05 red5b109  
    2323average over the size distribution. 
    2424 
    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. 
     25Each distribution is characterized by a center value $\bar x$ or 
     26$x_\text{med}$, a width parameter $\sigma$ (note this is *not necessarily* 
     27the standard deviation, so read the description carefully), the number of 
     28sigmas $N_\sigma$ to include from the tails of the distribution, and the 
     29number of points used to compute the average. The center of the distribution 
     30is set by the value of the model parameter. 
     31 
     32Volume parameters have polydispersity *PD* (not to be confused with a 
     33molecular weight distributions in polymer science), but orientation parameters 
     34use angular distributions of width $\sigma$. 
     35 
     36$N_\sigma$ determines how far into the tails to evaluate the distribution, 
     37with larger values of $N_\sigma$ required for heavier tailed distributions. 
    3738The scattering in general falls rapidly with $qr$ so the usual assumption 
    3839that $G(r - 3\sigma_r)$ is tiny and therefore $f(r - 3\sigma_r)G(r - 3\sigma_r)$ 
     
    6263^^^^^^^^^^^^^^^^^^^^^^ 
    6364 
    64 If applying polydispersion to parameters describing particle sizes, use  
     65If applying polydispersion to parameters describing particle sizes, use 
    6566the Lognormal or Schulz distributions. 
    6667 
    67 If applying polydispersion to parameters describing interfacial thicknesses  
     68If applying polydispersion to parameters describing interfacial thicknesses 
    6869or angular orientations, use the Gaussian or Boltzmann distributions. 
    6970 
     
    8586        \end{cases} 
    8687 
    87     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. 
     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. 
    9091 
    9192    The polydispersity in sasmodels is given by 
     
    114115        \end{cases} 
    115116 
    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. 
     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. 
    119120 
    120121    Note that the standard deviation and the half width $w$ are different! 
     
    131132 
    132133        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. 
    136138 
    137139.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    147149               \exp\left(-\frac{(x - \bar x)^2}{2\sigma^2}\right) 
    148150 
    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. 
    151154 
    152155    The polydispersity in sasmodels is given by 
     
    170173               \exp\left(-\frac{ | x - \bar x | }{\sigma}\right) 
    171174 
    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. 
    174178 
    175179    The width is defined as 
     
    177181    .. math:: \sigma=\frac{k T}{E} 
    178182 
    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, 
    180184    $T$ the temperature in Kelvin and $E$ a characteristic energy per particle. 
    181185 
     
    189193^^^^^^^^^^^^^^^^^^^^^^ 
    190194 
    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 $x$. 
     195The Lognormal Distribution describes a function of $x$ where $\ln (x)$ has 
     196a normal distribution. The result is a distribution that is skewed towards 
     197larger values of $x$. 
    193198 
    194199The Lognormal Distribution is defined as 
     
    196201    .. math:: 
    197202 
    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*. 
    207214 
    208215    The polydispersity in sasmodels is given by 
     
    210217    .. math:: \text{PD} = p = \sigma / x_\text{med} 
    211218 
    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 
    216224 
    217225    .. math:: 
     
    219227        \nu = [\exp({\sigma}^2) - 1] \exp({2\mu + \sigma^2}) 
    220228 
    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 
    223232    .. figure:: pd_lognormal.jpg 
    224233 
     
    226235 
    227236For further information on the Lognormal distribution see: 
    228 http://en.wikipedia.org/wiki/Log-normal_distribution and  
     237http://en.wikipedia.org/wiki/Log-normal_distribution and 
    229238http://mathworld.wolfram.com/LogNormalDistribution.html 
    230239 
     
    234243^^^^^^^^^^^^^^^^^^^ 
    235244 
    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. 
     245The Schulz (sometimes written Schultz) distribution is similar to the 
     246Lognormal distribution, in that it is also skewed towards larger values of 
     247$x$, but which has computational advantages over the Lognormal distribution. 
    239248 
    240249The Schulz distribution is defined as 
     
    242251    .. math:: 
    243252 
    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 
    250260 
    251261    .. math:: z = (1-p^2) / p^2 
     
    256266 
    257267    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. 
    261272 
    262273    .. figure:: pd_schulz.jpg 
     
    266277For further information on the Schulz distribution see: 
    267278M 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  
     279M Kotlarchyk, RB Stephens, and JS Huang, *J Phys Chem*, (1988), 92, 1533 
    269280 
    270281.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    310321    p_{DLS} = \sqrt(\nu / \bar x^2) 
    311322 
    312 where $\nu$ is the variance of the distribution and $\bar x$ is the mean value of x. 
     323where $\nu$ is the variance of the distribution and $\bar x$ is the mean 
     324value of x. 
    313325 
    314326For more information see: 
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