Changeset 1423ddb in sasmodels
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 Mar 31, 2019 12:47:58 AM (4 months ago)
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doc/guide/fitting_sq.rst
re62c019 r1423ddb 16 16 .. figure:: p_and_s_buttons.png 17 17 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** 20 of the form factor. 21 22 23 Scattering at vector $\mathbf Q$ for an individual particle with 24 shape parameters $\mathbf\xi$ and contrast $\rho_c(\mathbf r, \mathbf\xi)$ 25 is 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 31 with 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 37 The 1D scattering pattern for monodisperse particles uses the orientation 38 average 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 45 where $F(\mathbf Q,\mathbf\xi)$ uses 46 $\mathbf Q = [Q \sin\theta\cos\phi, Q \sin\theta\sin\phi, Q \cos\theta]^T$. 47 A $u$substitution may be used, with $\alpha = \cos \theta$, 48 $\surd(1  \alpha^2) = \sin \theta$, and 49 $\mathrm d\alpha = \sin\theta\,\mathrm d\theta$. 50 Here, 51 52 .. math:: n = V_f/V(\mathbf\xi) 53 54 is the number density of scatterers estimated from the volume fraction 55 of particles in solution. In this formalism, each incoming 56 wave interacts with exactly one particle before being scattered into the 57 detector. All interference effects are within the particle itself. 58 The detector accumulates counts in proportion to the relative probability 59 at each pixel. The extension to heterogeneous systems is simply a matter of 60 adding the scattering patterns in proportion to the number density of each 61 particle. 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 70 This approximation is valid in the dilute limit, where particles are 71 sufficiently far apart that the interaction between them can be ignored. 72 73 As concentration increases, a structure factor term $S(Q)$ can be included, 74 giving the monodisperse approximation for the interaction between particles, 75 with 76 77 .. math:: I(Q) = n \langle F F^* \rangle S(Q) 78 79 For particles without spherical symmetry, the decoupling approximation (DA) 80 is 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 87 Or equivalently, 88 89 .. math:: I(Q) = P(Q)[1 + \beta\,(S(Q)  1)] 90 91 with form factor $P(Q) = n \langle F F^* \rangle$ and 92 $\beta = \langle F \rangle \langle F \rangle^* \big/ \langle F F^* \rangle$. 93 These approximations can be extended to heterogeneous systems using averages 94 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 95 $n = V_f\big/\langle V \rangle_\mathbf\xi$. 96 Further improvements can be made using the local monodisperse 97 approximation (LMA) or using partial structure factors, as described 98 in \cite{bresler_sasfit:_2015}. 99 100 Many parameters are common amongst $P@S$ models, and take on specific meanings: 24 101 25 102 * *scale*: 26 103 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. 32 124 33 125 * *volfraction*: 34 126 35 The volume fraction of material.36 37 For hollow shapes, **volfraction** still represents the volume fraction of38 material but the $S(Q)$ calculation needs the volume fraction *enclosed by*39 *the shape.* SasView scales the userspecified volume fractionby the ratio40 form:shell computed from the average form volume and average shell volume41 returned from the $P(Q)$ calculation (the original volfraction is divided42 by the shell volume to compute the number density, and then $P@S$ is scaled43 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 userspecified **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 $I(Q)$. 44 136 45 137 * *radius_effective*: 46 138 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 freelyrotating 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 halflength 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 freelyrotating 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 halflength 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. 61 176 62 177 * *structure_factor_mode*: 63 178 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. 90 206 91 207 References
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