Changes in / [052d4c5:0b9c6df] in sasmodels


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  • sasmodels/models/core_shell_parallelepiped.c

    rdbf1a60 re077231  
    5959 
    6060    // outer integral (with gauss points), integration limits = 0, 1 
    61     // substitute d_cos_alpha for sin_alpha d_alpha 
    6261    double outer_sum = 0; //initialize integral 
    6362    for( int i=0; i<GAUSS_N; i++) { 
    6463        const double cos_alpha = 0.5 * ( GAUSS_Z[i] + 1.0 ); 
    6564        const double mu = half_q * sqrt(1.0-cos_alpha*cos_alpha); 
     65 
     66        // inner integral (with gauss points), integration limits = 0, pi/2 
    6667        const double siC = length_c * sas_sinx_x(length_c * cos_alpha * half_q); 
    6768        const double siCt = tC * sas_sinx_x(tC * cos_alpha * half_q); 
    68  
    69         // inner integral (with gauss points), integration limits = 0, 1 
    70         // substitute beta = PI/2 u (so 2/PI * d_(PI/2 * beta) = d_beta) 
    7169        double inner_sum = 0.0; 
    7270        for(int j=0; j<GAUSS_N; j++) { 
    73             const double u = 0.5 * ( GAUSS_Z[j] + 1.0 ); 
     71            const double beta = 0.5 * ( GAUSS_Z[j] + 1.0 ); 
    7472            double sin_beta, cos_beta; 
    75             SINCOS(M_PI_2*u, sin_beta, cos_beta); 
     73            SINCOS(M_PI_2*beta, sin_beta, cos_beta); 
    7674            const double siA = length_a * sas_sinx_x(length_a * mu * sin_beta); 
    7775            const double siB = length_b * sas_sinx_x(length_b * mu * cos_beta); 
    … …  
    9391            inner_sum += GAUSS_W[j] * f * f; 
    9492        } 
    95         // now complete change of inner integration variable (1-0)/(1-(-1))= 0.5 
    9693        inner_sum *= 0.5; 
    9794        // now sum up the outer integral 
    9895        outer_sum += GAUSS_W[i] * inner_sum; 
    9996    } 
    100     // now complete change of outer integration variable (1-0)/(1-(-1))= 0.5 
    10197    outer_sum *= 0.5; 
    10298 

  • sasmodels/models/core_shell_parallelepiped.py

    rf89ec96 r97be877  
    44 
    55Calculates the form factor for a rectangular solid with a core-shell structure. 
    6 The thickness and the scattering length density of the shell or "rim" can be 
    7 different on each (pair) of faces. The three dimensions of the core of the 
    8 parallelepiped (strictly here a cuboid) may be given in *any* size order as 
    9 long as the particles are randomly oriented (i.e. take on all possible 
    10 orientations see notes on 2D below). To avoid multiple fit solutions, 
    11 especially with Monte-Carlo fit methods, it may be advisable to restrict their 
    12 ranges. There may be a number of closely similar "best fits", so some trial and 
    13 error, or fixing of some dimensions at expected values, may help. 
     6The thickness and the scattering length density of the shell or 
     7"rim" can be different on each (pair) of faces. 
    148 
    159The form factor is normalized by the particle volume $V$ such that 
    … …  
    1711.. math:: 
    1812 
    19     I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle 
    20     + \text{background} 
     13    I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} 
    2114 
    2215where $\langle \ldots \rangle$ is an average over all possible orientations 
    23 of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be 
    24 pulled out of the form factor term due to the multiple slds in the model. 
    25  
    26 The core of the solid is defined by the dimensions $A$, $B$, $C$ here shown 
    27 such that $A < B < C$. 
    28  
    29 .. figure:: img/parallelepiped_geometry.jpg 
    30  
    31    Core of the core shell parallelepiped with the corresponding definition 
    32    of sides. 
    33  
     16of the rectangular solid. 
     17 
     18The function calculated is the form factor of the rectangular solid below. 
     19The core of the solid is defined by the dimensions $A$, $B$, $C$ such that 
     20$A < B < C$. 
     21 
     22.. image:: img/core_shell_parallelepiped_geometry.jpg 
    3423 
    3524There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension 
    3625(on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ 
    37 $(=t_C)$ faces. The projection in the $AB$ plane is 
    38  
    39 .. figure:: img/core_shell_parallelepiped_projection.jpg 
    40  
    41    AB cut through the core-shell parallelipiped showing the cross secion of 
    42    four of the six shell slabs. As can be seen, this model leaves **"gaps"** 
    43    at the corners of the solid. 
    44  
    45  
    46 The total volume of the solid is thus given as 
     26$(=t_C)$ faces. The projection in the $AB$ plane is then 
     27 
     28.. image:: img/core_shell_parallelepiped_projection.jpg 
     29 
     30The volume of the solid is 
    4731 
    4832.. math:: 
    4933 
    5034    V = ABC + 2t_ABC + 2t_BAC + 2t_CAB 
     35 
     36**meaning that there are "gaps" at the corners of the solid.** 
    5137 
    5238The intensity calculated follows the :ref:`parallelepiped` model, with the 
    5339core-shell intensity being calculated as the square of the sum of the 
    54 amplitudes of the core and the slabs on the edges. The scattering amplitude is 
    55 computed for a particular orientation of the core-shell parallelepiped with 
    56 respect to the scattering vector and then averaged over all possible 
    57 orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis 
    58 of the parallelepiped, and $\beta$ is the angle between the projection of the 
    59 particle in the $xy$ detector plane and the $y$ axis. 
    60  
    61 .. math:: 
    62  
    63     P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha 
    64     \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} 
    65  
    66 and 
    67  
    68 .. math:: 
    69  
    70     F(q,\alpha,\beta) 
     40amplitudes of the core and the slabs on the edges. 
     41 
     42the scattering amplitude is computed for a particular orientation of the 
     43core-shell parallelepiped with respect to the scattering vector and then 
     44averaged over all possible orientations, where $\alpha$ is the angle between 
     45the $z$ axis and the $C$ axis of the parallelepiped, $\beta$ is 
     46the angle between projection of the particle in the $xy$ detector plane 
     47and the $y$ axis. 
     48 
     49.. math:: 
     50 
     51    F(Q) 
    7152    &= (\rho_\text{core}-\rho_\text{solvent}) 
    7253       S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ 
    7354    &+ (\rho_\text{A}-\rho_\text{solvent}) 
    74         \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ 
     55        \left[S(Q_A, A+2t_A) - S(Q_A, Q)\right] S(Q_B, B) S(Q_C, C) \\ 
    7556    &+ (\rho_\text{B}-\rho_\text{solvent}) 
    7657        S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ 
    … …  
    8263.. math:: 
    8364 
    84     S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} 
     65    S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L} 
    8566 
    8667and 
    … …  
    8869.. math:: 
    8970 
    90     Q_A &= q \sin\alpha \sin\beta \\ 
    91     Q_B &= q \sin\alpha \cos\beta \\ 
    92     Q_C &= q \cos\alpha 
     71    Q_A &= \sin\alpha \sin\beta \\ 
     72    Q_B &= \sin\alpha \cos\beta \\ 
     73    Q_C &= \cos\alpha 
    9374 
    9475 
    9576where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ 
    96 are the scattering lengths of the parallelepiped core, and the rectangular 
     77are the scattering length of the parallelepiped core, and the rectangular 
    9778slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ 
    9879is the scattering length of the solvent. 
    9980 
    100 .. note::  
    101  
    102    the code actually implements two substitutions: $d(cos\alpha)$ is 
    103    substituted for -$sin\alpha \ d\alpha$ (note that in the 
    104    :ref:`parallelepiped` code this is explicitly implemented with 
    105    $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that 
    106    $du = \pi/2 \ d\beta$.  Thus both integrals go from 0 to 1 rather than 0 
    107    to $\pi/2$. 
    108  
    10981FITTING NOTES 
    11082~~~~~~~~~~~~~ 
    11183 
    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. 
    137  
    138 .. note:: For 2d, constraints must be applied during fitting to ensure that the 
    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. 
     84If the scale is set equal to the particle volume fraction, $\phi$, the returned 
     85value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. However, 
     86**no interparticle interference effects are included in this calculation.** 
     87 
     88There are many parameters in this model. Hold as many fixed as possible with 
     89known values, or you will certainly end up at a solution that is unphysical. 
     90 
     91The returned value is in units of |cm^-1|, on absolute scale. 
     92 
     93NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated 
     94based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ 
     95and length $(C+2t_C)$ values, after appropriately sorting the three dimensions 
     96to give an oblate or prolate particle, to give an effective radius, 
     97for $S(Q)$ when $P(Q) * S(Q)$ is applied. 
     98 
     99For 2d data the orientation of the particle is required, described using 
     100angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further 
     101details of the calculation and angular dispersions see :ref:`orientation`. 
     102The angle $\Psi$ is the rotational angle around the *long_c* axis. For example, 
     103$\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector. 
     104 
     105For 2d, constraints must be applied during fitting to ensure that the 
     106inequality $A < B < C$ is not violated, and hence the correct definition 
     107of angles is preserved. The calculation will not report an error, 
     108but the results may be not correct. 
    143109 
    144110.. figure:: img/parallelepiped_angle_definition.png 
    145111 
    146112    Definition of the angles for oriented core-shell parallelepipeds. 
    147     Note that rotation $\theta$, initially in the $x-z$ plane, is carried 
     113    Note that rotation $\theta$, initially in the $xz$ plane, is carried 
    148114    out first, then rotation $\phi$ about the $z$ axis, finally rotation 
    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. 
     115    $\Psi$ is now around the axis of the cylinder. The neutron or X-ray 
     116    beam is along the $z$ axis. 
    151117 
    152118.. figure:: img/parallelepiped_angle_projection.png 
    … …  
    154120    Examples of the angles for oriented core-shell parallelepipeds against the 
    155121    detector plane. 
    156  
    157  
    158 Validation 
    159 ---------- 
    160  
    161 Cross-checked against hollow rectangular prism and rectangular prism for equal 
    162 thickness overlapping sides, and by Monte Carlo sampling of points within the 
    163 shape for non-uniform, non-overlapping sides. 
    164  
    165122 
    166123References 
    … …  
    178135 
    179136* **Author:** NIST IGOR/DANSE **Date:** pre 2010 
    180 * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 
     137* **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016 
    181138* **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 
    182 * **Last Reviewed by:** Paul Butler **Date:** May 24, 2018 - documentation 
    183   updated 
     139* Cross-checked against hollow rectangular prism and rectangular prism for 
     140  equal thickness overlapping sides, and by Monte Carlo sampling of points 
     141  within the shape for non-uniform, non-overlapping sides. 
    184142""" 
    185143 

  • sasmodels/models/parallelepiped.c

    rdbf1a60 r108e70e  
    3838            inner_total += GAUSS_W[j] * square(si1 * si2); 
    3939        } 
    40         // now complete change of inner integration variable (1-0)/(1-(-1))= 0.5 
    4140        inner_total *= 0.5; 
    4241 
    … …  
    4443        outer_total += GAUSS_W[i] * inner_total * si * si; 
    4544    } 
    46     // now complete change of outer integration variable (1-0)/(1-(-1))= 0.5 
    4745    outer_total *= 0.5; 
    4846 

  • sasmodels/models/parallelepiped.py

    rf89ec96 ref07e95  
    22# Note: model title and parameter table are inserted automatically 
    33r""" 
     4The form factor is normalized by the particle volume. 
     5For information about polarised and magnetic scattering, see 
     6the :ref:`magnetism` documentation. 
     7 
    48Definition 
    59---------- 
    610 
    7 This model calculates the scattering from a rectangular solid 
    8 (:numref:`parallelepiped-image`). 
    9 If you need to apply polydispersity, see also :ref:`rectangular-prism`. For 
    10 information about polarised and magnetic scattering, see 
    11 the :ref:`magnetism` documentation. 
     11 This model calculates the scattering from a rectangular parallelepiped 
     12 (\:numref:`parallelepiped-image`\). 
     13 If you need to apply polydispersity, see also :ref:`rectangular-prism`. 
    1214 
    1315.. _parallelepiped-image: 
    … …  
    1921 
    2022The three dimensions of the parallelepiped (strictly here a cuboid) may be 
    21 given in *any* size order as long as the particles are randomly oriented (i.e. 
    22 take on all possible orientations see notes on 2D below). To avoid multiple fit 
    23 solutions, especially with Monte-Carlo fit methods, it may be advisable to 
    24 restrict their ranges. There may be a number of closely similar "best fits", so 
    25 some trial and error, or fixing of some dimensions at expected values, may 
    26 help. 
    27  
    28 The form factor is normalized by the particle volume and the 1D scattering 
    29 intensity $I(q)$ is then calculated as: 
     23given in *any* size order. To avoid multiple fit solutions, especially 
     24with Monte-Carlo fit methods, it may be advisable to restrict their ranges. 
     25There may be a number of closely similar "best fits", so some trial and 
     26error, or fixing of some dimensions at expected values, may help. 
     27 
     28The 1D scattering intensity $I(q)$ is calculated as: 
    3029 
    3130.. Comment by Miguel Gonzalez: 
    … …  
    4039 
    4140    I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 
    42            \left< P(q, \alpha, \beta) \right> + \text{background} 
     41           \left< P(q, \alpha) \right> + \text{background} 
    4342 
    4443where the volume $V = A B C$, the contrast is defined as 
    45 $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ 
    46 is the form factor corresponding to a parallelepiped oriented 
    47 at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ 
    48 (the angle between the projection of the particle in the $xy$ detector plane 
    49 and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all 
    50 orientations. 
     44$\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, 
     45$P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented 
     46at an angle $\alpha$ (angle between the long axis C and $\vec q$), 
     47and the averaging $\left<\ldots\right>$ is applied over all orientations. 
    5148 
    5249Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the 
    53 form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) 
     50form factor is given by (Mittelbach and Porod, 1961) 
    5451 
    5552.. math:: 
    … …  
    6966    \mu &= qB 
    7067 
    71 where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been 
    72 applied. 
    73  
    74 For **oriented** particles, the 2D scattering intensity, $I(q_x, q_y)$, is 
    75 given as: 
    76  
    77 .. math:: 
    78  
    79     I(q_x, q_y) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 P(q_x, q_y) 
     68The scattering intensity per unit volume is returned in units of |cm^-1|. 
     69 
     70NB: The 2nd virial coefficient of the parallelepiped is calculated based on 
     71the averaged effective radius, after appropriately sorting the three 
     72dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and 
     73length $(= C)$ values, and used as the effective radius for 
     74$S(q)$ when $P(q) \cdot S(q)$ is applied. 
     75 
     76For 2d data the orientation of the particle is required, described using 
     77angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details 
     78of the calculation and angular dispersions see :ref:`orientation` . 
     79 
     80.. Comment by Miguel Gonzalez: 
     81   The following text has been commented because I think there are two 
     82   mistakes. Psi is the rotational angle around C (but I cannot understand 
     83   what it means against the q plane) and psi=0 corresponds to a||x and b||y. 
     84 
     85   The angle $\Psi$ is the rotational angle around the $C$ axis against 
     86   the $q$ plane. For example, $\Psi = 0$ when the $B$ axis is parallel 
     87   to the $x$-axis of the detector. 
     88 
     89The angle $\Psi$ is the rotational angle around the $C$ axis. 
     90For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis 
     91oriented parallel to the y-axis of the detector with $A$ along the x-axis. 
     92For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated 
     93$\theta$ degrees in the $z-x$ plane and then $\phi$ degrees around the $z$ axis, 
     94before doing a final rotation of $\Psi$ degrees around the resulting $C$ axis 
     95of the particle to obtain the final orientation of the parallelepiped. 
     96 
     97.. _parallelepiped-orientation: 
     98 
     99.. figure:: img/parallelepiped_angle_definition.png 
     100 
     101    Definition of the angles for oriented parallelepiped, shown with $A<B<C$. 
     102 
     103.. figure:: img/parallelepiped_angle_projection.png 
     104 
     105    Examples of the angles for an oriented parallelepiped against the 
     106    detector plane. 
     107 
     108On introducing "Orientational Distribution" in the angles, "distribution of 
     109theta" and "distribution of phi" parameters will appear. These are actually 
     110rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, 
     111perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ 
     112these are parallel to the $Y$ and $X$ axes of the instrument.) The third 
     113orientation distribution, in $\psi$, is about the $c$ axis of the particle, 
     114perpendicular to the $a$ x $b$ face. Some experimentation may be required to 
     115understand the 2d patterns fully as discussed in :ref:`orientation` . 
     116 
     117For a given orientation of the parallelepiped, the 2D form factor is 
     118calculated as 
     119 
     120.. math:: 
     121 
     122    P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1}{2}qA\cos\alpha)}\right]^2 
     123                  \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1}{2}qB\cos\beta)}\right]^2 
     124                  \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1}{2}qC\cos\gamma)}\right]^2 
     125 
     126with 
     127 
     128.. math:: 
     129 
     130    \cos\alpha &= \hat A \cdot \hat q, \\ 
     131    \cos\beta  &= \hat B \cdot \hat q, \\ 
     132    \cos\gamma &= \hat C \cdot \hat q 
     133 
     134and the scattering intensity as: 
     135 
     136.. math:: 
     137 
     138    I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) 
    80139            + \text{background} 
    81140 
    … …  
    89148   with scale being the volume fraction. 
    90149 
    91 Where $P(q_x, q_y)$ for a given orientation of the form factor is calculated as 
    92  
    93 .. math:: 
    94  
    95     P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} 
    96                    {2}qA\cos\alpha)}\right]^2 
    97                   \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} 
    98                    {2}qB\cos\beta)}\right]^2 
    99                   \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} 
    100                    {2}qC\cos\gamma)}\right]^2 
    101  
    102 with 
    103  
    104 .. math:: 
    105  
    106     \cos\alpha &= \hat A \cdot \hat q, \\ 
    107     \cos\beta  &= \hat B \cdot \hat q, \\ 
    108     \cos\gamma &= \hat C \cdot \hat q 
    109  
    110  
    111 FITTING NOTES 
    112 ~~~~~~~~~~~~~ 
    113  
    114 #. The 2nd virial coefficient of the parallelepiped is calculated based on 
    115    the averaged effective radius, after appropriately sorting the three 
    116    dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and 
    117    length $(= C)$ values, and used as the effective radius for 
    118    $S(q)$ when $P(q) \cdot S(q)$ is applied. 
    119  
    120 #. For 2d data the orientation of the particle is required, described using 
    121    angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ 
    122    and $\phi$ define the orientation of the director in the laboratry reference 
    123    frame of the beam direction ($z$) and detector plane ($x-y$ plane), while 
    124    the angle $\Psi$ is effectively the rotational angle around the particle 
    125    $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the 
    126    $B$ axis oriented parallel to the y-axis of the detector with $A$ along 
    127    the x-axis. For other $\theta$, $\phi$ values, the order of rotations 
    128    matters. In particular, the parallelepiped must first be rotated $\theta$ 
    129    degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ 
    130    axis (in the $x-y$ plane). Applying orientational distribution to the 
    131    particle orientation (i.e  `jitter` to one or more of these angles) can get 
    132    more confusing as `jitter` is defined **NOT** with respect to the laboratory 
    133    frame but the particle reference frame. It is thus highly recmmended to 
    134    read :ref:`orientation` for further details of the calculation and angular 
    135    dispersions. 
    136  
    137 .. note:: For 2d, constraints must be applied during fitting to ensure that the 
    138    order of sides chosen is not altered, and hence that the correct definition 
    139    of angles is preserved. For the default choice shown here, that means 
    140    ensuring that the inequality $A < B < C$ is not violated,  The calculation 
    141    will not report an error, but the results may be not correct. 
    142     
    143 .. _parallelepiped-orientation: 
    144  
    145 .. figure:: img/parallelepiped_angle_definition.png 
    146  
    147     Definition of the angles for oriented parallelepiped, shown with $A<B<C$. 
    148  
    149 .. figure:: img/parallelepiped_angle_projection.png 
    150  
    151     Examples of the angles for an oriented parallelepiped against the 
    152     detector plane. 
    153  
    154 .. Comment by Paul Butler 
    155    I am commenting this section out as we are trying to minimize the amount of 
    156    oritentational detail here and encourage the user to go to the full 
    157    orientation documentation so that changes can be made in just one place. 
    158    below is the commented paragrah: 
    159    On introducing "Orientational Distribution" in the angles, "distribution of 
    160    theta" and "distribution of phi" parameters will appear. These are actually 
    161    rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, 
    162    perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ 
    163    these are parallel to the $Y$ and $X$ axes of the instrument.) The third 
    164    orientation distribution, in $\psi$, is about the $c$ axis of the particle, 
    165    perpendicular to the $a$ x $b$ face. Some experimentation may be required to 
    166    understand the 2d patterns fully as discussed in :ref:`orientation` . 
    167  
    168150 
    169151Validation 
    … …  
    174156angles. 
    175157 
     158 
    176159References 
    177160---------- 
    178161 
    179 .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, 
    180    14 (1961) 185-211 
    181 .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 
     162P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 
     163 
     164R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 
    182165 
    183166Authorship and Verification 
    … …  
    186169* **Author:** NIST IGOR/DANSE **Date:** pre 2010 
    187170* **Last Modified by:**  Paul Kienzle **Date:** April 05, 2017 
    188 * **Last Reviewed by:**  Miguel Gonzales and Paul Butler **Date:** May 24, 
    189   2018 - documentation updated 
     171* **Last Reviewed by:**  Richard Heenan **Date:** April 06, 2017 
    190172""" 
    191173 
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