Changeset da5536f in sasmodels
 Timestamp:
 Oct 27, 2017 12:11:24 PM (21 months ago)
 Branches:
 master, core_shell_microgels, magnetic_model, ticket1257vesicleproduct, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
 Children:
 eda8b30
 Parents:
 767dca8
 Location:
 doc
 Files:

 3 added
 2 edited
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doc/developer/overview.rst
r870a2f4 rda5536f 166 166 :ref:`Calculator_Interface` 167 167 168 Orientation and Numerical Integration 169  170 171 For 2d data from oriented anisotropic particles, the mean particle orientation is defined by angles $\theta$, $\phi$ and $\Psi$, which 172 are not in general the same as similarly named angles in many form factors. The wikipedia page on Euler angles 173 (https://en.wikipedia.org/wiki/Euler_angles) lists the different conventions available. To quote: "Different authors may use different 174 sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles 175 should always be preceded by their definition." 176 177 We are using the zyz convention with extrinsic rotations $\Psi\theta\phi$ for the particle orientation and $xyz$ convention with 178 extrinsic rotations $\psi\theta\phi$ for jitter, with jitter applied before particle orientation. 179 180 For numerical integration within form factors etc. sasmodels is mostly using Gaussian quadrature with 20, 76 or 150 points depending on 181 the model. It also makes use of symmetries such as calculating only over one quadrant rather than the whole sphere. There is often a 182 Usubstitution replacing $\theta$ with $cos(\theta)$ which changes the limits of integration from 0 to $\pi/2$ to 0 to 1 and also conveniently 183 absorbs the $sin(\theta)$ scale factor in the integration. This can cause confusion if checking equations to say include in a paper or thesis! 184 Most models use the same core kernel code expressed in terms of the rotated view (qa, qb, qc) for both the 1D and the 2D models, but there 185 are also historical quirks such as the parallelepiped model, which has a useless transformation representing j0(a qa) as j0(b qa a/b). 186 187 Useful testing routines include  188 189 :mod:`asymint` a direct implementation of the surface integral for certain models to get a more trusted value for the 1D integral using a reimplementation of the 2D kernel in python and mpmath 190 (which computes math functions to arbitrary precision). It uses $\theta$ ranging from 0 to $\pi$ and $\phi$ ranging from 0 to $2\pi$. It perhaps would benefit 191 from including the Usubstitution for theta. 192 193 :mod:`check1d` uses sasmodels 1D integration and compares that with a rectangle distribution in $\theta$ and $\phi$, with $\theta$ limits set to 194 $\pm90/\sqrt(3)$ and $\phi$ limits set to $\pm180/\sqrt(3)$ 195 [The rectangle weight function uses the fact that the distribution width column is labelled sigma to decide 196 that the 1sigma width of a rectangular distribution needs to be multiplied by $\sqrt(3)$ to get the corresponding gaussian equivalent, 197 or similar reasoning.] This should rotate the sample through the entire $\theta\phi$ 198 surface according to the pattern that you see in jitter.py when you modify it to use 'rectangle' rather than 'gaussian' for its distribution 199 without changing the viewing angle. When computing the dispersity integral, weights are scaled by abs(cos(dtheta)) to account for the points in 200 phi getting closer together as dtheta increases. This integrated dispersion is computed at a set of $(qx, qy)$ points $(q cos(\alpha), q sin(\alpha))$ 201 at some angle $\alpha$ (currently angle=0) for each q used in the 1D integration. The individual q points should be equivalent to asymint rectn 202 when the viewing angle is set to (theta,phi,psi) = (90, 0, 0). Such tests can help to validate that 2d intensity is consistent with 1d models. 203 204 :mod:`sascomp sphere=n` uses the identical rectangular distribution to compute the pattern of the qxqy grid. You can see from triaxial_ellipsoid 205 that there may be something wrong conceptually since the pattern is no longer circular when the view (theta,phi,psi) is not (90, phi, 0). 206 check1d shows that it is different from the sasmodels 1D integral even when at theta=0, psi=0. Cross checking the values with asymint, 207 the sasmodels 1D integral is better at low q, though for very large structures there are not enough points in the integration for sasmodels 1D 208 to compute the high q 1D integral correctly. [Some of that may now be fixed?] 209 210 The :mod:`sascomp` utility can be used for 2d as well as 1d calculations to compare results for two sets of parameters or processor types, for example 211 these two oriented cylinders here should be equivalent. 212 213 :mod:`\./sascomp 2d cylinder theta=0 phi=0,90 theta_pd_type=rectangle phi_pd_type=rectangle phi_pd=10,1 theta_pd=1,10 length=500 radius=10` 214 168 215 169 216 Testing 
doc/guide/index.rst
rc0d7ab3 rda5536f 13 13 resolution.rst 14 14 magnetism/magnetism.rst 15 orientation/orientation.rst 15 16 sesans/sans_to_sesans.rst 16 17 sesans/sesans_fitting.rst
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