Changeset 9624545 in sasmodels
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 Mar 31, 2019 8:08:06 AM (4 weeks ago)
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doc/guide/fitting_sq.rst
r1423ddb r9624545 14 14 This help document is under development 15 15 16 .. figure:: p_and_s_buttons.png17 18 16 **Product models**, or $P@S$ models for short, multiply the form factor 19 17 $P(Q)$ by the structure factor $S(Q)$, modulated by the **effective radius** 20 of the form factor. 21 18 of the form factor. For the theory behind this, see :ref:`PStheory` later. 19 20 **If writing your own** $P@S$ **models, DO NOT give your model parameters** 21 **these names!** 22 23 Parameters 24 ^^^^^^^^^^ 25 26 Many parameters are common amongst $P@S$ models, but take on specific meanings: 27 28 * *scale*: 29 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. 48 49 * *volfraction*: 50 51 The volume fraction of material, $V_f$. 52 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 userspecified **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 $I(Q)$. 61 62 * *radius_effective*: 63 64 The radial distance determining the range of the $S(Q)$ interaction. 65 66 This may be estimated from the "size" parameters $\mathbf \xi$ describing 67 the form of the shape. For example, in a system containing freelyrotating 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.x and later, 99 **radius_effective_mode = k** represents an index in a list of alternative 100 **radius_effective** calculations which will appear in a dropdown 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 userdefined effective radius. 113 114 **WARNING! If** $P(Q)$ **is multiplied by** $S(Q)$ **in the FitPage,** 115 **instead of being generated in the SumMulti dialog, the** 116 **radius_effective used is constrained (equivalent to** 117 **radius_effective_mode = 1)**. 118 119 * *structure_factor_mode*: 120 121 The type of structure factor calculation to use. 122 123 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. 126 127 If **structure_factor_mode = 0** then the 128 *local monodisperse approximation* will be used, i.e.: 129 130 .. math:: 131 I(Q) = \text{scale} \frac{V_f}{V} P(Q) S(Q) + \text{background} 132 133 where $P(Q) = \langle F(Q)^2 \rangle$. *This is the default in SasView* 134 *versions 4.x and earlier*. 135 136 If **structure_factor_mode = 1** then the $\beta(Q)$ correction will be 137 used, i.e.: 138 139 .. 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.x 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 ^^^^^^ 22 155 23 156 Scattering at vector $\mathbf Q$ for an individual particle with … … 29 162 \big/ V(\mathbf\xi) 30 163 31 with particle volume $V(\mathbf \xi)$ and164 with the particle volume $V(\mathbf \xi)$ and 32 165 33 166 .. math:: 34 167 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 168 e^{i \mathbf Q \cdot \mathbf r} \,\mathrm d \mathbf r = F 36 169 37 170 The 1D scattering pattern for monodisperse particles uses the orientation … … 52 185 .. math:: n = V_f/V(\mathbf\xi) 53 186 54 is the number density of scatterers estimated from the volume fraction 187 is the number density of scatterers estimated from the volume fraction $V_f$ 55 188 of particles in solution. In this formalism, each incoming 56 189 wave interacts with exactly one particle before being scattered into the … … 77 210 .. math:: I(Q) = n \langle F F^* \rangle S(Q) 78 211 79 For particles without spherical symmetry, the decoupling approximation (DA)212 For particles without spherical symmetry, the decoupling approximation 80 213 is more accurate, with 81 214 … … 89 222 .. math:: I(Q) = P(Q)[1 + \beta\,(S(Q)  1)] 90 223 91 with form factor $P(Q) = n \langle F F^* \rangle$ and224 with the form factor $P(Q) = n \langle F F^* \rangle$ and 92 225 $\beta = \langle F \rangle \langle F \rangle^* \big/ \langle F F^* \rangle$. 93 226 These approximations can be extended to heterogeneous systems using averages 94 227 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 95 228 $n = V_f\big/\langle V \rangle_\mathbf\xi$. 229 96 230 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: 101 102 * *scale*: 103 104 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. 124 125 * *volfraction*: 126 127 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 $I(Q)$. 136 137 * *radius_effective*: 138 139 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. 176 177 * *structure_factor_mode*: 178 179 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. 231 approximation (LMA) or using partial structure factors [2]. 206 232 207 233 References … … 210 236 .. [#] Kotlarchyk, M.; Chen, S.H. *J. Chem. Phys.*, 1983, 79, 2461 211 237 238 .. [#] Bressler I., Kohlbrecher J., Thunemann A.F. *J. Appl. Crystallogr.* 239 48 (2015) 15871598 240 212 241 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 213 242 214 243 *Document History* 215 244 216  2019033 0 Paul Kienzle & Steve King245  20190331 Paul Kienzle, Steve King & Richard Heenan
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