source: sasmodels/sasmodels/models/barbell.py @ 3dc41df

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

Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of
the cylinder, *L*, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than
that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell
are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values.

Definition
----------

The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale.

The barbell geometry is defined as

.. image:: img/image105.jpg

where *r* is the radius of the cylinder. All other parameters are as defined in the diagram.

Since the end cap radius
*R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as

*h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`)

The scattered intensity *I(q)* is calculated as

.. math::

    I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right>

where the amplitude *A(q)* is given as

.. math::

    A(Q) =&\ \pi r^2L
        {\sin\left(\tfrac12 QL\cos\theta\right)
            \over \tfrac12 QL\cos\theta}
        {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\
        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
        \cos\left[ Q\cos\theta
            \left(Rt + h + {\tfrac12} L\right)\right]
        \times (1-t^2)
        {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right]
             \over QR\sin\theta \left(1-t^2\right)^{1/2}}

The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form
factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is
the difference of scattering length densities of the cylinder and the surrounding solvent.

The volume of the barbell is

.. math::

    V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right)


and its radius-of-gyration is

.. math::

    R_g^2 =&\ \left[ \tfrac{12}{5}R^5
        + R^4\left(6h+\tfrac32 L\right)
        + R^2\left(4h^2 + L^2 + 4Lh\right)
        + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
        &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
        + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
        \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}

**The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis.

This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, *qmin* = 0.001 |Ang^-1|,
*qmax* = 0.7 |Ang^-1| and the following default values

==============  ========  =============
Parameter name  Units     Default value
==============  ========  =============
scale           None      1.0
len_bar         |Ang|     400.0
rad_bar         |Ang|     20.0
rad_bell        |Ang|     40.0
sld_barbell     |Ang^-2|  1.0e-006
sld_solv        |Ang^-2|  6.3e-006
background      |cm^-1|   0
==============  ========  =============

.. image:: img/image110.jpg

*Figure. 1D plot using the default values (w/256 data point).*

For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for
|theta| = 45 deg and |phi| = 0 deg with default values for other parameters

.. image:: img/image111.jpg

*Figure. 2D plot (w/(256X265) data points).*

.. image:: img/orientation.jpg

Figure. Definition of the angles for oriented 2D barbells.

.. image:: img/orientation2.jpg

*Figure. Examples of the angles for oriented pp against the detector plane.*

REFERENCE

H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230

H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata)

"""
from numpy import pi, inf

name = "barbell"
title = "Cylinder with spherical end caps"
description = """
        Calculates the scattering from a barbell-shaped cylinder. That is a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap
                radius lies outside of the cylinder.
                Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when rad_bar = rad_bell, it is a spherocylinder.
                It must be that rad_bar <(=) rad_bell.
"""

parameters = [
#   [ "name", "units", default, [lower, upper], "type","description" ],
    [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", "Barbell scattering length density" ],
    [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "","Solvent scattering length density" ],
    [ "bell_radius", "Ang",  40, [0, inf], "volume","Spherical bell radius" ],
    [ "radius", "Ang",  20, [0, inf], "volume","Cylindrical bar radius" ],
    [ "length", "Ang",  400, [0, inf], "volume","Cylinder bar length" ],
    [ "theta", "degrees", 60, [-inf, inf], "orientation","In plane angle" ],
    [ "phi", "degrees", 60, [-inf, inf], "orientation","Out of plane angle" ],
    ]

source = [ "lib/J1.c", "lib/gauss76.c", "barbell.c" ]

def ER(radius, length):
    ddd = 0.75*radius*(2*radius*length + (length+radius)*(length+pi*radius))
    return 0.5 * (ddd)**(1./3.)

# parameters for demo
demo = dict(
    scale=1, background=0,
    sld=6, solvent_sld=1,
    bell_radius = 40, radius=20, length=400,
    theta=60, phi=60,
    radius_pd=.2, radius_pd_n=0,
    length_pd=.2,length_pd_n=0,
    theta_pd=15, theta_pd_n=0,
    phi_pd=15, phi_pd_n=0,
    )

# For testing against the old sasview models, include the converted parameter
# names and the target sasview model name.
oldname='BarBellModel'
oldpars=dict(sld='sld_barbell',
             solvent_sld='sld_solv', bell_radius='rad_bell',
             radius='rad_bar',length='len_bar')