Ignore:
Timestamp:
May 26, 2018 5:29:07 PM (4 years ago)
Author:
butler
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

addresses #896

File:
1 edited

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  • sasmodels/models/core_shell_parallelepiped.py

    r96153e4 rfc7bcd5  
    88parallelepiped (strictly here a cuboid) may be given in *any* size order as 
    99long as the particles are randomly oriented (i.e. take on all possible 
    10 orientations see notes on 2D below). To avoid multiple fit solutions, e 
    11 specially with Monte-Carlo fit methods, it may be advisable to restrict their 
     10orientations see notes on 2D below). To avoid multiple fit solutions, 
     11especially with Monte-Carlo fit methods, it may be advisable to restrict their 
    1212ranges. There may be a number of closely similar "best fits", so some trial and 
    1313error, or fixing of some dimensions at expected values, may help. 
     
    1717.. math:: 
    1818 
    19     I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle  
     19    I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle 
    2020    + \text{background} 
    2121 
     
    110110~~~~~~~~~~~~~ 
    111111 
    112 There are many parameters in this model. Hold as many fixed as possible with 
    113 known values, or you will certainly end up at a solution that is unphysical. 
    114  
    115  
    116 NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated 
    117 based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ 
    118 and length $(C+2t_C)$ values, after appropriately sorting the three dimensions 
    119 to give an oblate or prolate particle, to give an effective radius 
    120 for $S(q)$ when $P(q) * S(q)$ is applied. 
    121  
    122 For 2d data the orientation of the particle is required, described using 
    123 angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details 
    124 of the calculation and angular dispersions see :ref:`orientation` . 
    125  
    126 The angle $\Psi$ is the rotational angle around the $C$ axis. 
    127 For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis 
    128 oriented parallel to the y-axis of the detector with $A$ along the x-axis. 
    129 For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated 
    130 $\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, 
    131 before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis 
    132 of the particle to obtain the final orientation of the parallelepiped. 
     112#. There are many parameters in this model. Hold as many fixed as possible with 
     113   known values, or you will certainly end up at a solution that is unphysical. 
     114 
     115#. The 2nd virial coefficient of the core_shell_parallelepiped is calculated 
     116   based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ 
     117   and length $(C+2t_C)$ values, after appropriately sorting the three 
     118   dimensions to give an oblate or prolate particle, to give an effective radius 
     119   for $S(q)$ when $P(q) * S(q)$ is applied. 
     120 
     121#. For 2d data the orientation of the particle is required, described using 
     122   angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ 
     123   and $\phi$ define the orientation of the director in the laboratry reference 
     124   frame of the beam direction ($z$) and detector plane ($x-y$ plane), while 
     125   the angle $\Psi$ is effectively the rotational angle around the particle 
     126   $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the 
     127   $B$ axis oriented parallel to the y-axis of the detector with $A$ along 
     128   the x-axis. For other $\theta$, $\phi$ values, the order of rotations 
     129   matters. In particular, the parallelepiped must first be rotated $\theta$ 
     130   degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ 
     131   axis (in the $x-y$ plane). Applying orientational distribution to the 
     132   particle orientation (i.e  `jitter` to one or more of these angles) can get 
     133   more confusing as `jitter` is defined **NOT** with respect to the laboratory 
     134   frame but the particle reference frame. It is thus highly recmmended to 
     135   read :ref:`orientation` for further details of the calculation and angular 
     136   dispersions. 
    133137 
    134138.. note:: For 2d, constraints must be applied during fitting to ensure that the 
    135    inequality $A < B < C$ is not violated, and hence the correct definition 
    136    of angles is preserved. The calculation will not report an error, 
    137    but the results may be not correct. 
     139   order of sides chosen is not altered, and hence that the correct definition 
     140   of angles is preserved. For the default choice shown here, that means 
     141   ensuring that the inequality $A < B < C$ is not violated,  The calculation 
     142   will not report an error, but the results may be not correct. 
    138143 
    139144.. figure:: img/parallelepiped_angle_definition.png 
    140145 
    141146    Definition of the angles for oriented core-shell parallelepipeds. 
    142     Note that rotation $\theta$, initially in the $xz$ plane, is carried 
     147    Note that rotation $\theta$, initially in the $x-z$ plane, is carried 
    143148    out first, then rotation $\phi$ about the $z$ axis, finally rotation 
    144     $\Psi$ is now around the axis of the particle. The neutron or X-ray 
    145     beam is along the $z$ axis. 
     149    $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray 
     150    beam is along the $z$ axis and the detecotr defines the $x-y$ plane. 
    146151 
    147152.. figure:: img/parallelepiped_angle_projection.png 
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