source: sasmodels/sasmodels/models/flexible_cylinder.py @ f94d8a2

r"""
This model provides the form factor, $P(q)$, for a flexible cylinder where the form factor
is normalized by the volume of the cylinder.
**Inter-cylinder interactions are NOT provided for.**

.. math::

    P(q) = \text{scale} \left<F^2\right>/V + \text{background}

where the averaging $\left<\ldots\right>$ is applied only for the 1D calculation

The 2D scattering intensity is the same as 1D, regardless of the orientation of the q vector which is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}

Definitions
-----------

.. figure:: img/flexible_cylinder_geometry.jpg


The chain of contour length, $L$, (the total length) can be described as a chain of some number of
locally stiff segments of length $l_p$, the persistence length (the length along the cylinder over
which the flexible cylinder can be considered a rigid rod).
The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.

The returned value is in units of $cm^-1$, on absolute scale.

In the parameters, the sldCyl and sldSolv represent the SLD of the chain/cylinder and solvent respectively.


.. figure:: img/flexible_cylinder_1d.jpg

    1D plot using the default values (w/1000 data point).


Our model uses the form factor calculations implemented in a c-library provided by the NIST Center
for Neutron Research (Kline, 2006).


From the reference:

    'Method 3 With Excluded Volume' is used.
    The model is a parametrization of simulations of a discrete representation
    of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous limit.
    See equations (13,26-27) in the original reference for the details.

References
----------

J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and
without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612

Correction of the formula can be found in

W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of
SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548
"""
from numpy import inf

name = "flexible_cylinder"
title = "Flexible cylinder where the form factor is normalized by the volume of the cylinder."
description = """Note : scale and contrast=sld-solvent_sld are both multiplicative factors in the
                model and are perfectly correlated. One or
                both of these parameters must be held fixed
                during model fitting.
              """

category = "shape:cylinder"

#             ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
              ["length",      "Ang",       1000.0, [0, inf],    "volume", "Length of the flexible cylinder"],
              ["kuhn_length", "Ang",        100.0, [0, inf],    "volume", "Kuhn length of the flexible cylinder"],
              ["radius",      "Ang",         20.0, [0, inf],    "volume", "Radius of the flexible cylinder"],
              ["sld",         "1e-6/Ang^2",   1.0, [-inf, inf], "",       "Cylinder scattering length density"],
              ["solvent_sld", "1e-6/Ang^2",   6.3, [-inf, inf], "",       "Solvent scattering length density"],
             ]

source = ["lib/J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"]

demo = dict(scale=1.0, background=0.0001,
            length=1000.0,
            kuhn_length=100.0,
            radius=20.0,
            sld=1.0,
            solvent_sld=6.3)

oldname = 'FlexibleCylinderModel'
oldpars = dict(sld='sldCyl', solvent_sld='sldSolv')


tests = [
         # Accuracy tests based on content in test/utest_other_models.py
         # Currently fails in OCL
         # [{'length':     1000.0,
         #  'kuhn_length': 100.0,
         #  'radius':       20.0,
         #  'sld':           1.0,
         #  'solvent_sld':   6.3,
         #  'background':    0.0001,
         #  }, 0.001, 3509.2187],

         # Additional tests with larger range of parameters
         [{'length':     1000.0,
           'kuhn_length': 100.0,
           'radius':       20.0,
           'sld':           1.0,
           'solvent_sld':   6.3,
           'background':    0.0001,
           }, 1.0, 0.000595345],
        [{'length':        10.0,
           'kuhn_length': 800.0,
           'radius':        2.0,
           'sld':           6.0,
           'solvent_sld':  12.3,
           'background':    0.001,
           }, 0.1, 1.55228],
       [{'length':        100.0,
           'kuhn_length': 800.0,
           'radius':       50.0,
           'sld':           0.1,
           'solvent_sld':   5.1,
           'background':    0.0,
           }, 1.0, 0.000938456]
         ]