source: sasmodels/sasmodels/models/core_shell_cylinder.py @ 6cefbc9

# core shell cylinder model
# Note: model title and parameter table are inserted automatically
r"""
The form factor is normalized by the particle volume.

Definition
----------

The output of the 2D scattering intensity function for oriented core-shell
cylinders is given by (Kline, 2006)

.. math::

    I(q,\alpha) = \frac{\text{scale}}{V_s} F^2(q) + \text{background}

where

.. math::

    F(q) = &\ (\rho_c - \rho_s) V_c
           \frac{\sin \left( q \tfrac12 L\cos\alpha \right)}
                {q \tfrac12 L\cos\alpha}
           \frac{2 J_1 \left( qR\sin\alpha \right)}
                {qR\sin\alpha} \\
         &\ + (\rho_s - \rho_\text{solv}) V_s
           \frac{\sin \left( q \left(\tfrac12 L+T\right) \cos\alpha \right)}
                {q \left(\tfrac12 L +T \right) \cos\alpha}
           \frac{ 2 J_1 \left( q(R+T)\sin\alpha \right)}
                {q(R+T)\sin\alpha}

and

.. math::

    V_s = \pi (R + T)^2 (L + 2T)

and $\alpha$ is the angle between the axis of the cylinder and $\vec q$,
$V_s$ is the volume of the outer shell (i.e. the total volume, including
the shell), $V_c$ is the volume of the core, $L$ is the length of the core,
$R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$
is the scattering length density of the core, $\rho_s$ is the scattering
length density of the shell, $\rho_\text{solv}$ is the scattering length
density of the solvent, and *background* is the background level.  The outer
radius of the shell is given by $R+T$ and the total length of the outer
shell is given by $L+2T$. $J1$ is the first order Bessel function.

.. _core-shell-cylinder-geometry:

.. figure:: img/core_shell_cylinder_geometry.jpg

    Core shell cylinder schematic.

To provide easy access to the orientation of the core-shell cylinder, we
define the axis of the cylinder using two angles $\theta$ and $\phi$.
(see :ref:`cylinder model <cylinder-angle-definition>`)

NB: The 2nd virial coefficient of the cylinder is calculated based on
the radius and 2 length values, and used as the effective radius for
$S(q)$ when $P(q) \cdot S(q)$ is applied.

The $\theta$ and $\phi$ parameters are not used for the 1D output.

Validation
----------

Validation of our code was done by comparing the output of the 1D model to
the output of the software provided by the NIST (Kline, 2006).

Averaging over a distribution of orientation is done by evaluating the
equation above. Since we have no other software to compare the
implementation of the intensity for fully oriented cylinders, we
compared the result of averaging our 2D output using a uniform
distribution $p(\theta,\phi) = 1.0$.

Reference
---------
see, for example, Ian Livsey  J. Chem. Soc., Faraday Trans. 2, 1987,83, 1445-1452

2016/03/18 - Description reviewed by RKH
"""

from numpy import pi, inf

name = "core_shell_cylinder"
title = "Right circular cylinder with a core-shell scattering length density profile."
description = """
P(q,alpha)= scale/Vs*f(q)^(2) + background,
      where: f(q)= 2(sld_core - solvant_sld)
        * Vc*sin[qLcos(alpha/2)]
        /[qLcos(alpha/2)]*J1(qRsin(alpha))
        /[qRsin(alpha)]+2(sld_shell-sld_solvent)
        *Vs*sin[q(L+T)cos(alpha/2)][[q(L+T)
        *cos(alpha/2)]*J1(q(R+T)sin(alpha))
        /q(R+T)sin(alpha)]

    alpha:is the angle between the axis of
        the cylinder and the q-vector
    Vs: the volume of the outer shell
    Vc: the volume of the core
    L: the length of the core
        sld_shell: the scattering length density of the shell
    sld_solvent: the scattering length density of the solvent
    background: the background
    T: the thickness
        R+T: is the outer radius
     L+2T: The total length of the outershell
    J1: the first order Bessel function
     theta: axis_theta of the cylinder
     phi: the axis_phi of the cylinder
"""
category = "shape:cylinder"

#             ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld",
               "Cylinder core scattering length density"],
              ["sld_shell", "1e-6/Ang^2", 4, [-inf, inf], "sld",
               "Cylinder shell scattering length density"],
              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
               "Solvent scattering length density"],
              ["radius", "Ang", 20, [0, inf], "volume",
               "Cylinder core radius"],
              ["thickness", "Ang", 20, [0, inf], "volume",
               "Cylinder shell thickness"],
              ["length", "Ang", 400, [0, inf], "volume",
               "Cylinder length"],
              ["theta", "degrees", 60, [-inf, inf], "orientation",
               "In plane angle"],
              ["phi", "degrees", 60, [-inf, inf], "orientation",
               "Out of plane angle"],
             ]

source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "core_shell_cylinder.c"]

def ER(radius, thickness, length):
    """
    Returns the effective radius used in the S*P calculation
    """
    radius = radius + thickness
    length = length + 2 * thickness
    ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius))
    return 0.5 * (ddd) ** (1. / 3.)

def VR(radius, thickness, length):
    """
    Returns volume ratio
    """
    whole = pi * (radius + thickness) ** 2 * (length + 2 * thickness)
    core = pi * radius ** 2 * length
    return whole, whole - core

demo = dict(scale=1, background=0,
            sld_core=6, sld_shell=8, sld_solvent=1,
            radius=45, thickness=25, length=340,
            theta=30, phi=15,
            radius_pd=.2, radius_pd_n=1,
            length_pd=.2, length_pd_n=10,
            thickness_pd=.2, thickness_pd_n=10,
            theta_pd=15, theta_pd_n=45,
            phi_pd=15, phi_pd_n=1)
# ADDED by:  RKH  ON: 18Mar2016 renamed sld's etc