# Changeset da5536f in sasmodels

Ignore:
Timestamp:
Oct 27, 2017 12:11:24 PM (2 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
eda8b30
Parents:
767dca8
Message:

new chapters in main user and developed docs for Orientation

Location:
doc
Files:
2 edited

### Legend:

Unmodified
 r870a2f4 :ref:Calculator_Interface Orientation and Numerical Integration ------------------------------------- For 2d data from oriented anisotropic particles, the mean particle orientation is defined by angles $\theta$, $\phi$ and $\Psi$, which are not in general the same as similarly named angles in many form factors. The wikipedia page on Euler angles (https://en.wikipedia.org/wiki/Euler_angles) lists the different conventions available. To quote: "Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition." We are using the z-y-z convention with extrinsic rotations $\Psi-\theta-\phi$ for the particle orientation and $x-y-z$ convention with extrinsic rotations $\psi-\theta-\phi$ for jitter, with jitter applied before particle orientation. For numerical integration within form factors etc. sasmodels is mostly using Gaussian quadrature with 20, 76 or 150 points depending on the model.  It also makes use of symmetries such as calculating only over one quadrant rather than the whole sphere.  There is often a U-substitution replacing $\theta$ with $cos(\theta)$ which changes the limits of integration from 0 to $\pi/2$ to 0 to 1 and also conveniently absorbs the $sin(\theta)$ scale factor in the integration.  This can cause confusion if checking equations to say include in a paper or thesis! 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 are also historical quirks such as the parallelepiped model, which has a useless transformation representing j0(a qa) as j0(b qa a/b). Useful testing routines include - :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 (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 from including the U-substitution for theta. :mod:check1d uses sasmodels 1D integration and compares that with a rectangle distribution in $\theta$ and $\phi$, with $\theta$ limits set to $\pm90/\sqrt(3)$ and $\phi$ limits set to $\pm180/\sqrt(3)$ [The rectangle weight function uses the fact that the distribution width column is labelled sigma to decide that the 1-sigma width of a rectangular distribution needs to be multiplied by $\sqrt(3)$ to get the corresponding gaussian equivalent, or similar reasoning.]  This should rotate the sample through the entire $\theta-\phi$ surface according to the pattern that you see in jitter.py when you modify it to use 'rectangle' rather than 'gaussian' for its distribution without changing the viewing angle. When computing the dispersity integral, weights are scaled by abs(cos(dtheta)) to account for the points in phi getting closer together as dtheta increases.  This integrated dispersion is computed at a set of $(qx, qy)$ points $(q cos(\alpha), q sin(\alpha))$ at some angle $\alpha$ (currently angle=0) for each q used in the 1-D integration.  The individual q points should be equivalent to asymint rect-n 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. :mod:sascomp -sphere=n uses the identical rectangular distribution to compute the pattern of the qx-qy grid.  You can see from triaxial_ellipsoid 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). check1d shows that it is different from the sasmodels 1D integral even when at theta=0, psi=0. Cross checking the values with asymint, 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 to compute the high q 1D integral correctly. [Some of that may now be fixed?] 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 these two oriented cylinders here should be equivalent. :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 Testing