Changeset 8ed04ff in sasmodels


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Timestamp:
Mar 31, 2019 5:15:02 AM (5 months ago)
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GitHub <noreply@…>
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git-author:
Steve K <smk78@…> (03/31/19 05:15:02)
git-committer:
GitHub <noreply@…> (03/31/19 05:15:02)
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Merge pull request #203 from SasView?/sasview-1164-beta-doc

update beta approx docs. Refs sasview/sasview#1164.

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  • doc/guide/fitting_sq.rst

    re62c019 r1423ddb  
    1616.. figure:: p_and_s_buttons.png 
    1717 
    18 **Product models**, or $P@S$ models for short, multiply the structure factor 
    19 $S(Q)$ by the form factor $P(Q)$, modulated by the **effective radius** of the 
    20 form factor. 
    21  
    22 Many of the parameters in $P@S$ models take on specific meanings so that they 
    23 can be handled correctly inside SasView: 
     18**Product models**, or $P@S$ models for short, multiply the form factor 
     19$P(Q)$ by the structure factor $S(Q)$, modulated by the **effective radius** 
     20of the form factor. 
     21 
     22 
     23Scattering at vector $\mathbf Q$ for an individual particle with 
     24shape parameters $\mathbf\xi$ and contrast $\rho_c(\mathbf r, \mathbf\xi)$ 
     25is computed from the square of the amplitude, $F(\mathbf Q, \mathbf\xi)$, as 
     26 
     27.. math:: 
     28    I(\mathbf Q) = F(\mathbf Q, \mathbf\xi) F^*(\mathbf Q, \mathbf\xi) 
     29        \big/ V(\mathbf\xi) 
     30 
     31with particle volume $V(\mathbf \xi)$ and 
     32 
     33.. math:: 
     34    F(\mathbf Q, \mathbf\xi) = \int_{\mathbb R^3} \rho_c(\mathbf r, \mathbf\xi) 
     35        e^{i \mathbf Q \cdot \mathbf r} \,\mathrm d \mathbf r 
     36 
     37The 1-D scattering pattern for monodisperse particles uses the orientation 
     38average in spherical coordinates, 
     39 
     40.. math:: 
     41    I(Q) = n \langle F F^*\rangle = \frac{n}{4\pi} 
     42    \int_{\theta=0}^{\pi} \int_{\phi=0}^{2\pi} 
     43    F F^* \sin(\theta) \,\mathrm d\phi \mathrm d\theta 
     44 
     45where $F(\mathbf Q,\mathbf\xi)$ uses 
     46$\mathbf Q = [Q \sin\theta\cos\phi, Q \sin\theta\sin\phi, Q \cos\theta]^T$. 
     47A $u$-substitution may be used, with $\alpha = \cos \theta$, 
     48$\surd(1 - \alpha^2) = \sin \theta$, and 
     49$\mathrm d\alpha = -\sin\theta\,\mathrm d\theta$. 
     50Here, 
     51 
     52.. math:: n = V_f/V(\mathbf\xi) 
     53 
     54is the number density of scatterers estimated from the volume fraction 
     55of particles in solution. In this formalism, each incoming 
     56wave interacts with exactly one particle before being scattered into the 
     57detector. All interference effects are within the particle itself. 
     58The detector accumulates counts in proportion to the relative probability 
     59at each pixel. The extension to heterogeneous systems is simply a matter of 
     60adding the scattering patterns in proportion to the number density of each 
     61particle. That is, given shape parameters $\mathbf\xi$ with probability 
     62$P_\mathbf{\xi}$, 
     63 
     64.. math:: 
     65 
     66    I(Q) = \int_\Xi n(\mathbf\xi) \langle F F^* \rangle \,\mathrm d\xi 
     67         = V_f\frac{\int_\Xi P_\mathbf{\xi} \langle F F^* \rangle 
     68         \,\mathrm d\mathbf\xi}{\int_\Xi P_\mathbf\xi V(\mathbf\xi)\,\mathrm d\mathbf\xi} 
     69 
     70This approximation is valid in the dilute limit, where particles are 
     71sufficiently far apart that the interaction between them can be ignored. 
     72 
     73As concentration increases, a structure factor term $S(Q)$ can be included, 
     74giving the monodisperse approximation for the interaction between particles, 
     75with 
     76 
     77.. math:: I(Q) = n \langle F F^* \rangle S(Q) 
     78 
     79For particles without spherical symmetry, the decoupling approximation (DA) 
     80is more accurate, with 
     81 
     82.. math:: 
     83 
     84    I(Q) = n [\langle F F^* \rangle 
     85        + \langle F \rangle \langle F \rangle^* (S(Q) - 1)] 
     86 
     87Or equivalently, 
     88 
     89.. math:: I(Q) = P(Q)[1 + \beta\,(S(Q) - 1)] 
     90 
     91with form factor $P(Q) = n \langle F F^* \rangle$ and 
     92$\beta = \langle F \rangle \langle F \rangle^* \big/ \langle F F^* \rangle$. 
     93These approximations can be extended to heterogeneous systems using averages 
     94over size, $\langle \cdot \rangle_\mathbf\xi = \int_\Xi P_\mathbf\xi \langle\cdot\rangle\,\mathrm d\mathbf\xi \big/ \int_\Xi P_\mathbf\xi \,\mathrm d\mathbf\xi$ and setting 
     95$n = V_f\big/\langle V \rangle_\mathbf\xi$. 
     96Further improvements can be made using the local monodisperse 
     97approximation (LMA) or using partial structure factors, as described 
     98in \cite{bresler_sasfit:_2015}. 
     99 
     100Many parameters are common amongst $P@S$ models, and take on specific meanings: 
    24101 
    25102* *scale*: 
    26103 
    27   In simple $P(Q)$ models **scale** often represents the volume fraction of 
    28   material. 
    29    
    30   In $P@S$ models **scale** should be set to 1.0, as the $P@S$ model contains a 
    31   **volfraction** parameter. 
     104    Overall model scale factor. 
     105 
     106    To compute number density $n$ the volume fraction $V_f$ is needed.  In 
     107    most $P(Q)$ models $V_f$ is not defined and **scale** is used instead. 
     108    Some $P(Q)$ models, such as *vesicle*, do define **volfraction** and so 
     109    can leave **scale** at 1.0. 
     110 
     111    The structure factor model $S(Q)$ has **volfraction**.  This is also used 
     112    as the volume fraction for the form factor model $P(Q)$, replacing the 
     113    **volfraction** parameter if it exists in $P$. This means that 
     114    $P@S$ models can leave **scale** at 1.0. 
     115 
     116    If the volume fraction required for $S(Q)$ is *not* the volume fraction 
     117    needed to compute the number density for $P(Q)$, then leave 
     118    **volfraction** as the volume fraction for $S(Q)$ and use 
     119    **scale** to define the volume fraction for $P(Q)$ as 
     120    $V_f$ = **scale**  $\cdot$  **volfraction**.  This situation may 
     121    occur in a mixed phase system where the effective volume 
     122    fraction needed to compute the structure is much higher than the 
     123    true volume fraction. 
    32124 
    33125* *volfraction*: 
    34126 
    35   The volume fraction of material. 
    36  
    37   For hollow shapes, **volfraction** still represents the volume fraction of 
    38   material but the $S(Q)$ calculation needs the volume fraction *enclosed by* 
    39   *the shape.* SasView scales the user-specified volume fraction by the ratio 
    40   form:shell computed from the average form volume and average shell volume 
    41   returned from the $P(Q)$ calculation (the original volfraction is divided 
    42   by the shell volume to compute the number density, and then $P@S$ is scaled 
    43   by that to get the absolute scaling on the final $I(Q)$). 
     127    The volume fraction of material. 
     128 
     129    For hollow shapes, **volfraction** still represents the volume fraction of 
     130    material but the $S(Q)$ calculation needs the volume fraction *enclosed by* 
     131    *the shape.*  Thus the user-specified **volfraction** is scaled by the ratio 
     132    form:shell computed from the average form volume and average shell volume 
     133    returned from the $P(Q)$ calculation when calculating $S(Q)$.  The original 
     134    **volfraction** is divided by the shell volume to compute the number 
     135    density $n$ used in $P@S$ to get the absolute scaling on the final $I(Q)$. 
    44136 
    45137* *radius_effective*: 
    46138 
    47   The radial distance determining the range of the $S(Q)$ interaction. 
    48    
    49   This may, or may not, be the same as any "size" parameters describing the 
    50   form of the shape. For example, in a system containing freely-rotating 
    51   cylinders, the volume of space each cylinder requires to tumble will be 
    52   much larger than the volume of the cylinder itself. Thus the effective 
    53   radius will be larger than either the radius or half-length of the 
    54   cylinder. It may be sensible to tie or constrain **radius_effective** to one 
    55   or other of these "size" parameters. 
    56  
    57   If just part of the $S(Q)$ calculation, the value of **radius_effective** may 
    58   be polydisperse. If it is calculated by $P(Q)$, then it will be the weighted 
    59   average of the effective radii computed for the polydisperse shape 
    60   parameters. 
     139    The radial distance determining the range of the $S(Q)$ interaction. 
     140 
     141    This may be estimated from the "size" parameters $\mathbf \xi$ describing 
     142    the form of the shape.  For example, in a system containing freely-rotating 
     143    cylinders, the volume of space each cylinder requires to tumble will be 
     144    much larger than the volume of the cylinder itself.  Thus the effective 
     145    radius will be larger than either the radius or the half-length of the 
     146    cylinder.  It may be sensible to tie or constrain **radius_effective** 
     147    to one or other of these "size" parameters. **radius_effective** may 
     148    also be specified directly, independent of the estimate from $P(Q)$. 
     149 
     150    If it is calculated by $P(Q)$, **radius_effective** will be the 
     151    weighted average of the effective radii computed for the polydisperse 
     152    shape parameters, and that average used to compute $S(Q)$.  When 
     153    specified directly, the value of **radius_effective** may be 
     154    polydisperse, and $S(Q)$ will be averaged over a range of effective 
     155    radii.  Whether this makes any physical sense will depend on the system. 
     156 
     157* *radius_effective_mode*: 
     158 
     159    Selects the **radius_effective** value to use. 
     160 
     161    When **radius_effective_mode = 0** then the **radius_effective** 
     162    parameter in the $P@S$ model is used.  Otherwise 
     163    **radius_effective_mode = k** is the index into the list of 
     164    **radius_effective_modes** defined by the model indicating how the 
     165    effective radius should be computed from the parameters of the shape. 
     166    For example, the *ellipsoid* model defines the following:: 
     167 
     168        1 => average curvature 
     169        2 => equivalent volume sphere 
     170        3 => min radius 
     171        4 => max radius 
     172 
     173    **radius_effective_mode** will only appear in the parameter table if 
     174    the model defines the list of modes, otherwise it will be set permanently 
     175    to 0 for user defined effective radius. 
    61176 
    62177* *structure_factor_mode*: 
    63178 
    64   If the $P@S$ model supports the $\beta(Q)$ *decoupling correction* [1] then 
    65   **structure_factor_mode** will appear in the parameter table after the $S(Q)$ 
    66   parameters. 
    67    
    68   If **structure_factor_mode = 0** then the *local monodisperse approximation* 
    69   will be used, i.e.: 
    70  
    71     $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $S(Q)$ + $background$ 
    72  
    73   If **structure_factor_mode = 1** then the $\beta(q)$ correction will be 
    74   used, i.e.: 
    75  
    76     $I(Q)$ = $(scale$ x $volfraction$ / $volume)$ x $( <F(Q)^2>$ + $<F(Q)>^2$ x $(S(Q)$ - $1) )$ + $background$ 
    77  
    78     where $P(Q)$ = $<|F(Q)|^2>$. 
    79      
    80   This is equivalent to: 
    81    
    82     $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $( 1$ + $\beta(Q)$ x $(S(Q)$ - $1) )$ + $background$ 
    83  
    84   The $\beta(Q)$ decoupling approximation has the effect of damping the 
    85   oscillations in the normal (local monodisperse) $S(Q)$. When $\beta(Q)$ = 1 
    86   the local monodisperse approximation is recovered. 
    87  
    88   More mode options may appear in future as more complicated operations are 
    89   added. 
     179    The type of structure factor calculation to use. 
     180 
     181    If the $P@S$ model supports the $\beta(Q)$ *decoupling correction* [1] 
     182    then **structure_factor_mode** will appear in the parameter table after 
     183    the $S(Q)$ parameters. 
     184 
     185    If **structure_factor_mode = 0** then the 
     186    *local monodisperse approximation* will be used, i.e.: 
     187 
     188    .. math:: 
     189        I(Q) = \text{scale} \frac{V_f}{V} P(Q) S(Q) + \text{background} 
     190 
     191    where $P(Q) = \langle F(Q)^2 \rangle$. 
     192 
     193    If **structure_factor_mode = 1** then the $\beta(Q)$ correction will be 
     194    used, i.e.: 
     195 
     196    .. math:: 
     197        I(Q) = \text{scale} \frac{V_f}{V} P(Q) [ 1 + \beta(Q) (S(Q) - 1) ] 
     198        + \text{background} 
     199 
     200    The $\beta(Q)$ decoupling approximation has the effect of damping the 
     201    oscillations in the normal (local monodisperse) $S(Q)$. When $\beta(Q) = 1$ 
     202    the local monodisperse approximation is recovered. 
     203 
     204    More mode options may appear in future as more complicated operations are 
     205    added. 
    90206 
    91207References 
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