[77c91d0] | 1 | .. fitting_sq.rst |
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| 2 | |
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| 3 | .. Much of the following text was scraped from product.py |
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| 4 | |
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[bc69321] | 5 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[77c91d0] | 6 | |
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[e62c019] | 7 | .. _Product_Models: |
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[77c91d0] | 8 | |
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| 9 | Fitting Models with Structure Factors |
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| 10 | ------------------------------------- |
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| 11 | |
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| 12 | .. note:: |
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| 13 | |
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| 14 | This help document is under development |
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| 15 | |
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[bc69321] | 16 | .. figure:: p_and_s_buttons.png |
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| 17 | |
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| 18 | **Product models**, or $P@S$ models for short, multiply the structure factor |
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| 19 | $S(Q)$ by the form factor $P(Q)$, modulated by the **effective radius** of the |
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| 20 | form factor. |
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[77c91d0] | 21 | |
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| 22 | Many of the parameters in $P@S$ models take on specific meanings so that they |
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| 23 | can be handled correctly inside SasView: |
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| 24 | |
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| 25 | * *scale*: |
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| 26 | |
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[bc69321] | 27 | In simple $P(Q)$ models **scale** often represents the volume fraction of |
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| 28 | material. |
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| 29 | |
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| 30 | In $P@S$ models **scale** should be set to 1.0, as the $P@S$ model contains a |
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| 31 | **volfraction** parameter. |
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[77c91d0] | 32 | |
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| 33 | * *volfraction*: |
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| 34 | |
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[bc69321] | 35 | The volume fraction of material. |
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| 36 | |
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| 37 | For hollow shapes, **volfraction** still represents the volume fraction of |
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| 38 | material but the $S(Q)$ calculation needs the volume fraction *enclosed by* |
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[77c91d0] | 39 | *the shape.* SasView scales the user-specified volume fraction by the ratio |
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| 40 | form:shell computed from the average form volume and average shell volume |
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[bc69321] | 41 | returned from the $P(Q)$ calculation (the original volfraction is divided |
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| 42 | by the shell volume to compute the number density, and then $P@S$ is scaled |
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| 43 | by that to get the absolute scaling on the final $I(Q)$). |
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[77c91d0] | 44 | |
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| 45 | * *radius_effective*: |
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| 46 | |
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[bc69321] | 47 | The radial distance determining the range of the $S(Q)$ interaction. |
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| 48 | |
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| 49 | This may, or may not, be the same as any "size" parameters describing the |
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| 50 | form of the shape. For example, in a system containing freely-rotating |
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| 51 | cylinders, the volume of space each cylinder requires to tumble will be |
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| 52 | much larger than the volume of the cylinder itself. Thus the effective |
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| 53 | radius will be larger than either the radius or half-length of the |
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| 54 | cylinder. It may be sensible to tie or constrain **radius_effective** to one |
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| 55 | or other of these "size" parameters. |
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| 56 | |
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| 57 | If just part of the $S(Q)$ calculation, the value of **radius_effective** may |
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| 58 | be polydisperse. If it is calculated by $P(Q)$, then it will be the weighted |
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[77c91d0] | 59 | average of the effective radii computed for the polydisperse shape |
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| 60 | parameters. |
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| 61 | |
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| 62 | * *structure_factor_mode*: |
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| 63 | |
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[bc69321] | 64 | If the $P@S$ model supports the $\beta(Q)$ *decoupling correction* [1] then |
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| 65 | **structure_factor_mode** will appear in the parameter table after the $S(Q)$ |
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| 66 | parameters. |
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| 67 | |
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| 68 | If **structure_factor_mode = 0** then the *local monodisperse approximation* |
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| 69 | will be used, i.e.: |
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| 70 | |
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| 71 | $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $S(Q)$ + $background$ |
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[77c91d0] | 72 | |
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[bc69321] | 73 | If **structure_factor_mode = 1** then the $\beta(q)$ correction will be |
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| 74 | used, i.e.: |
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[77c91d0] | 75 | |
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[bc69321] | 76 | $I(Q)$ = $(scale$ x $volfraction$ / $volume)$ x $( <F(Q)^2>$ + $<F(Q)>^2$ x $(S(Q)$ - $1) )$ + $background$ |
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[77c91d0] | 77 | |
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[bc69321] | 78 | where $P(Q)$ = $<|F(Q)|^2>$. |
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| 79 | |
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| 80 | This is equivalent to: |
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| 81 | |
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| 82 | $I(Q)$ = $(scale$ / $volume)$ x $P(Q)$ x $( 1$ + $\beta(Q)$ x $(S(Q)$ - $1) )$ + $background$ |
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[77c91d0] | 83 | |
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[bc69321] | 84 | The $\beta(Q)$ decoupling approximation has the effect of damping the |
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| 85 | oscillations in the normal (local monodisperse) $S(Q)$. When $\beta(Q)$ = 1 |
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| 86 | the local monodisperse approximation is recovered. |
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[77c91d0] | 87 | |
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[bc69321] | 88 | More mode options may appear in future as more complicated operations are |
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[77c91d0] | 89 | added. |
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| 90 | |
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| 91 | References |
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| 92 | ^^^^^^^^^^ |
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| 93 | |
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| 94 | .. [#] Kotlarchyk, M.; Chen, S.-H. *J. Chem. Phys.*, 1983, 79, 2461 |
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| 95 | |
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[bc69321] | 96 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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[77c91d0] | 97 | |
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| 98 | *Document History* |
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| 99 | |
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[bc69321] | 100 | | 2019-03-30 Paul Kienzle & Steve King |
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