Changes in doc/guide/fitting_sq.rst [694c6d0:e62c019] in sasmodels
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doc/guide/fitting_sq.rst
r694c6d0 re62c019 14 14 This help document is under development 15 15 16 **Product models**, or $P@S$ models for short, multiply the form factor 17 $P(Q)$ by the structure factor $S(Q)$, modulated by the **effective radius** 18 of the form factor. For the theory behind this, see :ref:`PStheory` later. 16 .. figure:: p_and_s_buttons.png 19 17 20 **If writing your own** $P@S$ **models, DO NOT give your model parameters** 21 **these names!** 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. 22 21 23 Parameters 24 ^^^^^^^^^^ 25 26 Many parameters are common amongst $P@S$ models, but take on specific meanings: 22 Many of the parameters in $P@S$ models take on specific meanings so that they 23 can be handled correctly inside SasView: 27 24 28 25 * *scale*: 29 26 30 Overall model scale factor. 31 32 To compute number density $n$ the volume fraction $V_f$ (parameterised as 33 **volfraction**) is needed. In most $P(Q)$ models $V_f$ is not defined and 34 **scale** is used instead. Some $P(Q)$ models, such as the *vesicle*, do 35 define **volfraction** and so can leave **scale** at 1.0. 36 37 Structure factor models $S(Q)$ contain **volfraction**. In $P@S$ models 38 this is *also* used as the volume fraction for the form factor model 39 $P(Q)$, *replacing* any **volfraction** parameter in $P(Q)$. This means 40 that $P@S$ models can also leave **scale** at 1.0. 41 42 If the volume fraction required for $S(Q)$ is *not* the volume fraction 43 needed to compute the $n$ for $P(Q)$, then leave **volfraction** as the 44 $V_f$ for $S(Q)$ and use **scale** to define the $V_f$ for $P(Q)$ as 45 $V_f$ = **scale** $\cdot$ **volfraction**. This situation may occur in 46 a mixed phase system where the effective volume fraction needed to compute 47 the structure is much higher than the true volume fraction. 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. 48 32 49 33 * *volfraction*: 50 34 51 The volume fraction of material, $V_f$.35 The volume fraction of material. 52 36 53 For hollow shapes, **volfraction** still represents the volume fraction of 54 material but the $S(Q)$ calculation needs the volume fraction *enclosed by* 55 *the shape.* To remedy this the user-specified **volfraction** is scaled 56 by the ratio form:shell computed from the average form volume and average 57 shell volume returned from the $P(Q)$ calculation when calculating $S(Q)$. 58 The original **volfraction** is divided by the shell volume to compute the 59 number density $n$ used in the $P@S$ model to get the absolute scaling on 60 the final $I(Q)$. 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 $I(Q)$). 61 44 62 45 * *radius_effective*: 63 46 64 The radial distance determining the range of the $S(Q)$ interaction. 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. 65 56 66 This may be estimated from the "size" parameters $\mathbf \xi$ describing 67 the form of the shape. For example, in a system containing freely-rotating 68 cylinders, the volume of space each cylinder requires to tumble will be 69 much larger than the volume of the cylinder itself. Thus the *effective* 70 radius of a cylinder will be larger than either its actual radius or half- 71 length. 72 73 In use, it may be sensible to tie or constrain **radius_effective** 74 to one or other of the "size" parameters describing the form of the shape. 75 76 **radius_effective** may also be specified directly, independent of the 77 estimate from $P(Q)$. 78 79 If **radius_effective** is calculated by $P(Q)$, it will be the 80 weighted average of the effective radii computed for the polydisperse 81 shape parameters, and that average is used to compute $S(Q)$. When 82 specified directly, the value of **radius_effective** may be 83 polydisperse, and $S(Q)$ will be averaged over a range of effective 84 radii. Whether this makes any physical sense will depend on the system. 85 86 .. note:: 87 88 The following additional parameters are only available in SasView 5.0 and 89 later. 90 91 * *radius_effective_mode*: 92 93 Defines how the effective radius (parameter **radius_effective**) should 94 be computed from the parameters of the shape. 95 96 When **radius_effective_mode = 0** then unconstrained **radius_effective** 97 parameter in the $S(Q)$ model is used. *This is the default in SasView* 98 *versions 4.x and earlier*. Otherwise, in SasView 5.0 and later, 99 **radius_effective_mode = k** represents an index in a list of alternative 100 **radius_effective** calculations which will appear in a drop-down box. 101 102 For example, the *ellipsoid* model defines the following 103 **radius_effective_modes**:: 104 105 1 => average curvature 106 2 => equivalent volume sphere 107 3 => min radius 108 4 => max radius 109 110 Note: **radius_effective_mode** will only appear in the parameter table if 111 the model defines the list of modes, otherwise it will be set permanently 112 to 0 for the user-defined effective radius. 113 114 **WARNING! If** $P(Q)$ **is multiplied by** $S(Q)$ **in the FitPage,** 115 **instead of being generated in the Sum|Multi dialog, the** 116 **radius_effective used is constrained (equivalent to** 117 **radius_effective_mode = 1)**. 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. 118 61 119 62 * *structure_factor_mode*: 120 63 121 The type of structure factor calculation to use. 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.: 122 70 123 If the $P@S$ model supports the $\beta(Q)$ *decoupling correction* [1] 124 then **structure_factor_mode** will appear in the parameter table after 125 the $S(Q)$ parameters. 71 $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $S(Q)$ + $background$ 126 72 127 If **structure_factor_mode = 0** then the128 *local monodisperse approximation* will beused, i.e.:73 If **structure_factor_mode = 1** then the $\beta(q)$ correction will be 74 used, i.e.: 129 75 130 .. math:: 131 I(Q) = \text{scale} \frac{V_f}{V} P(Q) S(Q) + \text{background} 76 $I(Q)$ = $(scale$ x $volfraction$ / $volume)$ x $( <F(Q)^2>$ + $<F(Q)>^2$ x $(S(Q)$ - $1) )$ + $background$ 132 77 133 where $P(Q) = \langle F(Q)^2 \rangle$. *This is the default in SasView* 134 *versions 4.x and earlier*. 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 $(S(Q)$ - $1) )$ + $background$ 135 83 136 If **structure_factor_mode = 1** then the $\beta(Q)$ correction will be 137 used, i.e.: 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. 138 87 139 .. math:: 140 I(Q) = \text{scale} \frac{V_f}{V} P(Q) [ 1 + \beta(Q) (S(Q) - 1) ] 141 + \text{background} 142 143 The $\beta(Q)$ decoupling approximation has the effect of damping the 144 oscillations in the normal (local monodisperse) $S(Q)$. When $\beta(Q) = 1$ 145 the local monodisperse approximation is recovered. *This mode is only* 146 *available in SasView 5.0 and later*. 147 148 More mode options may appear in future as more complicated operations are 149 added. 150 151 .. _PStheory: 152 153 Theory 154 ^^^^^^ 155 156 Scattering at vector $\mathbf Q$ for an individual particle with 157 shape parameters $\mathbf\xi$ and contrast $\rho_c(\mathbf r, \mathbf\xi)$ 158 is computed from the square of the amplitude, $F(\mathbf Q, \mathbf\xi)$, as 159 160 .. math:: 161 I(\mathbf Q) = F(\mathbf Q, \mathbf\xi) F^*(\mathbf Q, \mathbf\xi) 162 \big/ V(\mathbf\xi) 163 164 with the particle volume $V(\mathbf \xi)$ and 165 166 .. math:: 167 F(\mathbf Q, \mathbf\xi) = \int_{\mathbb R^3} \rho_c(\mathbf r, \mathbf\xi) 168 e^{i \mathbf Q \cdot \mathbf r} \,\mathrm d \mathbf r = F 169 170 The 1-D scattering pattern for monodisperse particles uses the orientation 171 average in spherical coordinates, 172 173 .. math:: 174 I(Q) = n \langle F F^*\rangle = \frac{n}{4\pi} 175 \int_{\theta=0}^{\pi} \int_{\phi=0}^{2\pi} 176 F F^* \sin(\theta) \,\mathrm d\phi \mathrm d\theta 177 178 where $F(\mathbf Q,\mathbf\xi)$ uses 179 $\mathbf Q = [Q \sin\theta\cos\phi, Q \sin\theta\sin\phi, Q \cos\theta]^T$. 180 A $u$-substitution may be used, with $\alpha = \cos \theta$, 181 $\surd(1 - \alpha^2) = \sin \theta$, and 182 $\mathrm d\alpha = -\sin\theta\,\mathrm d\theta$. 183 Here, 184 185 .. math:: n = V_f/V(\mathbf\xi) 186 187 is the number density of scatterers estimated from the volume fraction $V_f$ 188 of particles in solution. In this formalism, each incoming 189 wave interacts with exactly one particle before being scattered into the 190 detector. All interference effects are within the particle itself. 191 The detector accumulates counts in proportion to the relative probability 192 at each pixel. The extension to heterogeneous systems is simply a matter of 193 adding the scattering patterns in proportion to the number density of each 194 particle. That is, given shape parameters $\mathbf\xi$ with probability 195 $P_\mathbf{\xi}$, 196 197 .. math:: 198 199 I(Q) = \int_\Xi n(\mathbf\xi) \langle F F^* \rangle \,\mathrm d\xi 200 = V_f\frac{\int_\Xi P_\mathbf{\xi} \langle F F^* \rangle 201 \,\mathrm d\mathbf\xi}{\int_\Xi P_\mathbf\xi V(\mathbf\xi)\,\mathrm d\mathbf\xi} 202 203 This approximation is valid in the dilute limit, where particles are 204 sufficiently far apart that the interaction between them can be ignored. 205 206 As concentration increases, a structure factor term $S(Q)$ can be included, 207 giving the monodisperse approximation for the interaction between particles, 208 with 209 210 .. math:: I(Q) = n \langle F F^* \rangle S(Q) 211 212 For particles without spherical symmetry, the decoupling approximation 213 is more accurate, with 214 215 .. math:: 216 217 I(Q) = n [\langle F F^* \rangle 218 + \langle F \rangle \langle F \rangle^* (S(Q) - 1)] 219 220 Or equivalently, 221 222 .. math:: I(Q) = P(Q)[1 + \beta\,(S(Q) - 1)] 223 224 with the form factor $P(Q) = n \langle F F^* \rangle$ and 225 $\beta = \langle F \rangle \langle F \rangle^* \big/ \langle F F^* \rangle$. 226 These approximations can be extended to heterogeneous systems using averages 227 over 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 228 $n = V_f\big/\langle V \rangle_\mathbf\xi$. 229 230 Further improvements can be made using the local monodisperse 231 approximation (LMA) or using partial structure factors [2]. 88 More mode options may appear in future as more complicated operations are 89 added. 232 90 233 91 References … … 236 94 .. [#] Kotlarchyk, M.; Chen, S.-H. *J. Chem. Phys.*, 1983, 79, 2461 237 95 238 .. [#] Bressler I., Kohlbrecher J., Thunemann A.F. *J. Appl. Crystallogr.*239 48 (2015) 1587-1598240 241 96 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 242 97 243 98 *Document History* 244 99 245 | 2019-03-3 1 Paul Kienzle, Steve King & Richard Heenan100 | 2019-03-30 Paul Kienzle & Steve King
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