# Changeset fc7bcd5 in sasmodels

Ignore:
Timestamp:
May 26, 2018 5:29:07 PM (4 years ago)
Branches:
master, core_shell_microgels, magnetic_model, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
Children:
f89ec96
Parents:
96153e4
Message:

Final? edits to Parallelepiped (and core shell version) documentation

Location:
sasmodels/models
Files:
2 edited

### Legend:

Unmodified
 r96153e4 parallelepiped (strictly here a cuboid) may be given in *any* size order as long as the particles are randomly oriented (i.e. take on all possible orientations see notes on 2D below). To avoid multiple fit solutions, e specially with Monte-Carlo fit methods, it may be advisable to restrict their orientations see notes on 2D below). To avoid multiple fit solutions, especially with Monte-Carlo fit methods, it may be advisable to restrict their ranges. There may be a number of closely similar "best fits", so some trial and error, or fixing of some dimensions at expected values, may help. .. math:: I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle + \text{background} ~~~~~~~~~~~~~ There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical. NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ and length $(C+2t_C)$ values, after appropriately sorting the three dimensions to give an oblate or prolate particle, to give an effective radius for $S(q)$ when $P(q) * S(q)$ is applied. For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details of the calculation and angular dispersions see :ref:orientation . The angle $\Psi$ is the rotational angle around the $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated $\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis of the particle to obtain the final orientation of the parallelepiped. #. There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical. #. The 2nd virial coefficient of the core_shell_parallelepiped is calculated based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ and length $(C+2t_C)$ values, after appropriately sorting the three dimensions to give an oblate or prolate particle, to give an effective radius for $S(q)$ when $P(q) * S(q)$ is applied. #. For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ and $\phi$ define the orientation of the director in the laboratry reference frame of the beam direction ($z$) and detector plane ($x-y$ plane), while the angle $\Psi$ is effectively the rotational angle around the particle $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the order of rotations matters. In particular, the parallelepiped must first be rotated $\theta$ degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ axis (in the $x-y$ plane). Applying orientational distribution to the particle orientation (i.e  jitter to one or more of these angles) can get more confusing as jitter is defined **NOT** with respect to the laboratory frame but the particle reference frame. It is thus highly recmmended to read :ref:orientation for further details of the calculation and angular dispersions. .. note:: For 2d, constraints must be applied during fitting to ensure that the inequality $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, but the results may be not correct. order of sides chosen is not altered, and hence that the correct definition of angles is preserved. For the default choice shown here, that means ensuring that the inequality $A < B < C$ is not violated,  The calculation will not report an error, but the results may be not correct. .. figure:: img/parallelepiped_angle_definition.png Definition of the angles for oriented core-shell parallelepipeds. Note that rotation $\theta$, initially in the $xz$ plane, is carried 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 now around the axis of the particle. The neutron or X-ray beam is along the $z$ axis. $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray beam is along the $z$ axis and the detecotr defines the $x-y$ plane. .. figure:: img/parallelepiped_angle_projection.png
 r96153e4 ---------- This model calculates the scattering from a rectangular parallelepiped This model calculates the scattering from a rectangular solid (:numref:parallelepiped-image). If you need to apply polydispersity, see also :ref:rectangular-prism. For applied. NB: The 2nd virial coefficient of the parallelepiped is calculated based on the averaged effective radius, after appropriately sorting the three dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and length $(= C)$ values, and used as the effective radius for $S(q)$ when $P(q) \cdot S(q)$ is applied. For 2d data the orientation of the particle is required, described using angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details of the calculation and angular dispersions see :ref:orientation . The angle $\Psi$ is the rotational angle around the $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis oriented parallel to the y-axis of the detector with $A$ along the x-axis. For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated $\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis of the particle to obtain the final orientation of the parallelepiped. .. note:: For 2d, constraints must be applied during fitting to ensure that the inequality $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, but the results may be not correct. .. _parallelepiped-orientation: .. figure:: img/parallelepiped_angle_definition.png Definition of the angles for oriented parallelepiped, shown with \$A