# Changeset 7e6bc45e in sasmodels

Ignore:
Timestamp:
Dec 20, 2017 2:33:39 PM (3 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
06ee63c
Parents:
93fbc34
Message:

update orientation docs

Location:
doc/guide
Files:
2 edited

### Legend:

Unmodified
 r82592da ================== With two dimensional small angle diffraction data SasView will calculate With two dimensional small angle diffraction data sasmodels will calculate scattering from oriented particles, applicable for example to shear flow or orientation in a magnetic field. In general we first need to define the reference orientation of the particles with respect to the incoming neutron or X-ray beam. This is done using three angles: $\theta$ and $\phi$ define the orientation of the axis of the particle, angle $\Psi$ is defined as the orientation of the major axis of the particle cross section with respect to its starting position along the beam direction. The figures below are for an elliptical cross section cylinder, but may be applied analogously to other shapes of particle. of the particle's $a$-$b$-$c$ axes with respect to the incoming neutron or X-ray beam. This is done using three angles: $\theta$ and $\phi$ define the orientation of the $c$-axis of the particle, and angle $\Psi$ is defined as the orientation of the major axis of the particle cross section with respect to its starting position along the beam direction (or equivalently, as rotation about the $c$ axis). There is an unavoidable ambiguity when $c$ is aligned with $z$ in that $\phi$ and $\Psi$ both serve to rotate the particle about $c$, but this symmetry is destroyed when $\theta$ is not a multiple of 180. The figures below are for an elliptical cross section cylinder, but may be applied analogously to other shapes of particle. .. note:: Definition of angles for oriented elliptical cylinder, where axis_ratio b/a is shown >1, Note that rotation $\theta$, initially in the $x$-$z$ b/a is shown >1. Note that rotation $\theta$, initially in the $x$-$z$ plane, is carried out first, then rotation $\phi$ about the $z$-axis, finally rotation $\Psi$ is around the axis of the cylinder. The neutron or X-ray beam is along the $z$ axis. or X-ray beam is along the $-z$ axis. .. figure:: with $\Psi$ = 0. Having established the mean direction of the particle we can then apply angular orientation distributions. This is done by a numerical integration over a range of angles in a similar way to particle size dispersity. In the current version of sasview the orientational dispersity is defined with respect to the axes of the particle. Having established the mean direction of the particle (the view) we can then apply angular orientation distributions (jitter). This is done by a numerical integration over a range of angles in a similar way to particle size dispersity. The orientation dispersity is defined with respect to the $a$-$b$-$c$ axes of the particle, with roll angle $\Psi$ about the $c$-axis, yaw angle $\theta$ about the $b$-axis and pitch angle $\phi$ about the $a$-axis. More formally, starting with axes $a$-$b$-$c$ of the particle aligned with axes $x$-$y$-$z$ of the laboratory frame, the orientation dispersity is applied first, using the Tait-Bryan _ $x$-$y'$-$z''$ convention with angles $\Delta\phi$-$\Delta\theta$-$\Delta\Psi$. The reference orientation then follows, using the Euler angles _ $z$-$y'$-$z''$ with angles $\phi$-$\theta$-$\Psi$.  This is implemented using rotation matrices as .. math:: R = R_z(\phi)\, R_y(\theta)\, R_z(\Psi)\, R_x(\Delta\phi)\, R_y(\Delta\theta)\, R_z(\Delta\Psi) To transform detector $(q_x, q_y)$ values into $(q_a, q_b, q_c)$ for the shape in its canonical orientation, use .. math:: [q_a, q_b, q_c]^T = R^{-1} \, [q_x, q_y, 0]^T The inverse rotation is easily calculated by rotating the opposite directions in the reverse order, so .. math:: R^{-1} = R_z(-\Delta\Psi)\, R_y(-\Delta\theta)\, R_x(-\Delta\phi)\, R_z(-\Psi)\, R_y(-\theta)\, R_z(-\phi) The $\theta$ and $\phi$ orientation parameters for the cylinder only appear when fitting 2d data. On introducing "Orientational Distribution" in the angles, "distribution of theta" and "distribution of phi" parameters will when fitting 2d data. On introducing "Orientation Distribution" in the angles, "distribution of theta" and "distribution of phi" parameters will appear. These are actually rotations about the axes $\delta_1$ and $\delta_2$ of the cylinder, the $b$ and $a$ axes of the cylinder cross section. (When $\theta = \phi = 0$ these are parallel to the $Y$ and $X$ axes of the instrument.) The third orientation distribution, in $\Psi$, is about the $c$ axis of the particle. Some experimentation may be required to understand the 2d patterns fully. A number of different shapes of distribution are available, as described for polydispersity, see :ref:polydispersityhelp . of the cylinder, which correspond to the $b$ and $a$ axes of the cylinder cross section. (When $\theta = \phi = 0$ these are parallel to the $Y$ and $X$ axes of the instrument.) The third orientation distribution, in $\Psi$, is about the $c$ axis of the particle. Some experimentation may be required to understand the 2d patterns fully. A number of different shapes of distribution are available, as described for size dispersity, see :ref:polydispersityhelp. Earlier versions of SasView had numerical integration issues in some circumstances when distributions passed through 90 degrees. The distributions in particle coordinates are more robust, but should still be approached with care for large ranges of angle. Given that the angular dispersion distribution is defined in cartesian space, over a cube defined by .. math:: [-\Delta \theta, \Delta \theta] \times [-\Delta \phi, \Delta \phi] \times [-\Delta \Psi, \Delta \Psi] but the orientation is defined over a sphere, we are left with a map projection _ problem, with different tradeoffs depending on how values in $\Delta\theta$ and $\Delta\phi$ are translated into latitude/longitude on the sphere. Sasmodels is using the *equirectangular* projection. In this projection, square patches in angular dispersity become wedge-shaped patches on the sphere. To correct for the changing point density, there is a scale factor of $\sin(\Delta\theta)$ that applies to each point in the integral. This is not enough, though. Consider a shape which is tumbling freely around the $b$ axis, with $\Delta\theta$ uniform in $[-180, 180]$. At $\pm 90$, all points in $\Delta\phi$ map to the pole, so the jitter will have a distinct angular preference. If the spin axis is normal to the beam (which will be the case for $\theta=90$ and $\Psi=90$), the scattering pattern should be circularly symmetric, but it will go to zero at $\pm 90$ due to the $\sin(\Delta\theta)$ correction. This problem does not appear for a shape that is tumbling freely around the $a$ axis, with $\Delta\phi$ uniform in $[-180, 180]$, so swap the $a$ and $b$ axes so $\Delta\theta < \Delta\phi$ and adjust $\Psi$ by 90. This works with the existing sasmodels shapes due to symmetry. There are alternative projections. The *sinusoidal* projection works by scaling $\Delta\phi$ as $\Delta\theta$ increases, and dropping those points outside $[-180, 180]$. The distortions are a little less for middle ranges of $\Delta\theta$, but they are still severe for large $\Delta\theta$ and the model is much harder to explain. The *Guyou* projection has an excellent balance with reasonable distortion in both $\Delta\theta$ and $\Delta\phi$, as well as preserving small patches. However, it is considerably more expensive to implement, and we have not yet computed the distortion correction, measuring the degree of stretch at the point $(\Delta\theta, \Delta\phi)$ in the correction. .. note:: Note that the form factors for oriented particles are also performing numerical integrations over one or more variables, so care should be taken, especially with very large particles or more extreme aspect ratios. In such cases results may not be accurate, particularly at very high Q, unless the model has been specifically coded to use limiting forms of the scattering equations. For best numerical results keep the $\theta$ distribution narrower than the $\phi$ distribution. Thus for asymmetric particles, such as elliptical_cylinder, you may need to reorder the sizes of the three axes to acheive the desired result. This is due to the issues of mapping a rectangular distribution onto the surface of a sphere. Note that the form factors for oriented particles are performing numerical integrations over one or more variables, so care should be taken, especially with very large particles or more extreme aspect ratios. In such cases results may not be accurate, particularly at very high Q, unless the model has been specifically coded to use limiting forms of the scattering equations. Users can experiment with the values of *Npts* and *Nsigs*, the number of steps used in the integration and the range spanned in number of standard deviations. The standard deviation is entered in units of degrees. For a "rectangular" distribution the full width should be $\pm \sqrt(3)$ ~ 1.73 standard deviations. The new "uniform" distribution avoids this by letting you directly specify the For best numerical results keep the $\theta$ distribution narrower than the $\phi$ distribution. Thus for asymmetric particles, such as elliptical_cylinder, you may need to reorder the sizes of the three axes to acheive the desired result. This is due to the issues of mapping a rectanglar distribution onto the surface of a sphere. Users can experiment with the values of *Npts* and *Nsigs*, the number of steps used in the integration and the range spanned in number of standard deviations. The standard deviation is entered in units of degrees. For a "rectangular" distribution the full width should be $\pm \sqrt(3)$ ~ 1.73 standard deviations. The new "uniform" distribution avoids this by letting you directly specify the half width. The angular distributions will be truncated outside of the range -180 to +180 degrees, so beware of using saying a broad Gaussian distribution with large value of *Nsigs*, as the array of *Npts* may be truncated to many fewer points than would give a good integration,as well as becoming rather meaningless. (At some point in the future the actual polydispersity arrays may be made available to the user for inspection.) The angular distributions may be truncated outside of the range -180 to +180 degrees, so beware of using saying a broad Gaussian distribution with large value of *Nsigs*, as the array of *Npts* may be truncated to many fewer points than would give a good integration,as well as becoming rather meaningless. (At some point in the future the actual dispersion arrays may be made available to the user for inspection.) Some more detailed technical notes are provided in the developer section of this manual :ref:orientation_developer . This definition of orientation is new to SasView 4.2.  In earlier versions, the orientation distribution appeared as a distribution of view angles. This led to strange effects when $c$ was aligned with $z$, where changes to the $\phi$ angle served only to rotate the shape about $c$, rather than having a consistent interpretation as the pitch of the shape relative to the flow field defining the reference orientation.  Prior to SasView 4.1, the reference orientation was defined using a Tait-Bryan convention, making it difficult to control.  Now, rotation in $\theta$ modifies the spacings in the refraction pattern, and rotation in $\phi$ rotates it in the detector plane. *Document History* | 2017-11-06 Richard Heenan | 2017-12-20 Paul Kienzle
 rc654160 If the scattering is dependent on the orientation of the shape, then you will need to include *orientation* parameters *theta*, *phi* and *psi* at the end of the parameter table.  Shape orientation uses *a*, *b* and *c* axes, corresponding to the *x*, *y* and *z* axes in the laboratory coordinate system, with *z* along the beam and *x*-*y* in the detector plane, with *x* horizontal and *y* vertical.  The *psi* parameter rotates the shape about its *c* axis, the *theta* parameter then rotates the *c* axis toward the *x* axis of the detector, then *phi* rotates the shape in the detector plane.  (Prior to these rotations, orientation dispersity will be applied as roll-pitch-yaw, rotating *c*, then *b* then *a* in the shape coordinate system.)  A particular *qx*, *qy* point on the detector, then corresponds to *qa*, *qb*, *qc* with respect to the shape. The oriented C model is called as *Iqabc(qa, qb, qc, par1, par2, ...)* where at the end of the parameter table.  As described in the section :ref:orientation, the individual $(q_x, q_y)$ points on the detector will be rotated into $(q_a, q_b, q_c)$ points relative to the sample in its canonical orientation with $a$-$b$-$c$ aligned with $x$-$y$-$z$ in the laboratory frame and beam travelling along $-z$. The oriented C model is called using *Iqabc(qa, qb, qc, par1, par2, ...)* where *par1*, etc. are the parameters to the model.  If the shape is rotationally symmetric about *c* then *psi* is not needed, and the model is called