source: sasmodels/sasmodels/models/hollow_rectangular_prism.py @ 71b751d

# rectangular_prism model
# Note: model title and parameter table are inserted automatically
r"""

This model provides the form factor, $P(q)$, for a hollow rectangular
parallelepiped with a wall of thickness $\Delta$.


Definition
----------

The 1D scattering intensity for this model is calculated by forming
the difference of the amplitudes of two massive parallelepipeds
differing in their outermost dimensions in each direction by the
same length increment $2\Delta$ (Nayuk, 2012).

As in the case of the massive parallelepiped model (:ref:`rectangular-prism`),
the scattering amplitude is computed for a particular orientation of the
parallelepiped with respect to the scattering vector and then averaged over all
possible orientations, giving

.. math::
  P(q) =  \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \,
  \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi

where $\theta$ is the angle between the $z$ axis and the longest axis
of the parallelepiped, $\phi$ is the angle between the scattering vector
(lying in the $xy$ plane) and the $y$ axis, and

.. math::
  :nowrap:

  \begin{align*}
  A_{P\Delta}(q) & =  A B C
    \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)}
    {\left( q \frac{C}{2} \cos\theta \right)} \right]
    \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)}
    {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right]
    \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)}
    {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\
    & - 8
    \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right)
    \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]}
    {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right]
    \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]}
    {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right]
    \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]}
    {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right]
  \end{align*}

where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling
$A \le B \le C$, and the volume $V$ of the parallelepiped is

.. math::
  V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta)

The 1D scattering intensity is then calculated as

.. math::
  I(q) = \text{scale} \times V \times (\rho_\text{p} -
  \rho_\text{solvent})^2 \times P(q) + \text{background}

where $\rho_\text{p}$ is the scattering length of the parallelepiped,
$\rho_\text{solvent}$ is the scattering length of the solvent,
and (if the data are in absolute units) *scale* represents the volume fraction
(which is unitless).

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 long *C* axis. For example,
$\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector.

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.

.. figure:: img/parallelepiped_angle_definition.png

    Definition of the angles for oriented hollow rectangular prism.
    Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then
    rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the prism.
    The neutron or X-ray beam is along the $z$ axis.

.. figure:: img/parallelepiped_angle_projection.png

    Examples of the angles for oriented hollow rectangular prisms against the
    detector plane.


Validation
----------

Validation of the code was conducted by qualitatively comparing the output
of the 1D model to the curves shown in (Nayuk, 2012).


References
----------

R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854
"""

import numpy as np
from numpy import pi, inf, sqrt

name = "hollow_rectangular_prism"
title = "Hollow rectangular parallelepiped with uniform scattering length density."
description = """
    I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background
        P(q,theta,phi) = (2/pi/V^2) * double integral from 0 to pi/2 of ...
           (AP1-AP2)^2(q)*sin(theta)*dtheta*dphi
        AP1 = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2)
        AP2 = S(q*C'*cos(theta)) * S(q*A'*sin(theta)*sin(phi)) * S(q*B'*sin(theta)*cos(phi))
        C' = (C/2-thickness)
        B' = (B/2-thickness)
        A' = (A/2-thickness)
        S(x) = sin(x)/x
"""
category = "shape:parallelepiped"

#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld",
               "Parallelepiped scattering length density"],
              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
               "Solvent scattering length density"],
              ["length_a", "Ang", 35, [0, inf], "volume",
               "Shorter side of the parallelepiped"],
              ["b2a_ratio", "Ang", 1, [0, inf], "volume",
               "Ratio sides b/a"],
              ["c2a_ratio", "Ang", 1, [0, inf], "volume",
               "Ratio sides c/a"],
              ["thickness", "Ang", 1, [0, inf], "volume",
               "Thickness of parallelepiped"],
              ["theta", "degrees", 0, [-360, 360], "orientation",
               "c axis to beam angle"],
              ["phi", "degrees", 0, [-360, 360], "orientation",
               "rotation about beam"],
              ["psi", "degrees", 0, [-360, 360], "orientation",
               "rotation about c axis"],
             ]

source = ["lib/gauss76.c", "hollow_rectangular_prism.c"]
have_Fq = True

def ER(length_a, b2a_ratio, c2a_ratio, thickness):
    """
    Return equivalent radius (ER)
    thickness parameter not used
    """
    b_side = length_a * b2a_ratio
    c_side = length_a * c2a_ratio

    # surface average radius (rough approximation)
    surf_rad = sqrt(length_a * b_side / pi)

    ddd = 0.75 * surf_rad * (2 * surf_rad * c_side + (c_side + surf_rad) * (c_side + pi * surf_rad))
    return 0.5 * (ddd) ** (1. / 3.)

def VR(length_a, b2a_ratio, c2a_ratio, thickness):
    """
    Return shell volume and total volume
    """
    b_side = length_a * b2a_ratio
    c_side = length_a * c2a_ratio
    a_core = length_a - 2.0*thickness
    b_core = b_side - 2.0*thickness
    c_core = c_side - 2.0*thickness
    vol_core = a_core * b_core * c_core
    vol_total = length_a * b_side * c_side
    vol_shell = vol_total - vol_core
    return vol_total, vol_shell


def random():
    a, b, c = 10**np.random.uniform(1, 4.7, size=3)
    # Thickness is limited to 1/2 the smallest dimension
    # Use a distribution with a preference for thin shell or thin core
    # Avoid core,shell radii < 1
    min_dim = 0.5*min(a, b, c)
    thickness = np.random.beta(0.5, 0.5)*(min_dim-2) + 1
    #print(a, b, c, thickness, thickness/min_dim)
    pars = dict(
        length_a=a,
        b2a_ratio=b/a,
        c2a_ratio=c/a,
        thickness=thickness,
    )
    return pars


# parameters for demo
demo = dict(scale=1, background=0,
            sld=6.3, sld_solvent=1.0,
            length_a=35, b2a_ratio=1, c2a_ratio=1, thickness=1,
            length_a_pd=0.1, length_a_pd_n=10,
            b2a_ratio_pd=0.1, b2a_ratio_pd_n=1,
            c2a_ratio_pd=0.1, c2a_ratio_pd_n=1)

tests = [[{}, 0.2, 0.76687283098],
         [{}, [0.2], [0.76687283098]],
        ]