source: sasmodels/sasmodels/models/squarewell.py @ 2f63032

# Note: model title and parameter table are inserted automatically
r"""
This calculates the interparticle structure factor for a square well fluid spherical particles. The mean spherical
approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive
interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing
this calculation to be limited in applicability to well depths |epsilon| < 1.5 kT and volume fractions |phi| < 0.08.

Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential
"shoulder", which may or may not be physically reasonable. The stickyhardsphere model may be a better choice in
some circumstances. Computed values may behave badly at extremely small $qR$.

The well width (|lambda| ) is defined as multiples of the particle diameter (2\*\ *R*\ )

The interaction potential is:

  .. image:: img\squarewell.png

.. math::

    U(r) = \begin{cases}
    \infty & r < 2R \\
    -\epsilon & 2R \leq r < 2R\lambda \\
    0 & r \geq 2R\lambda
    \end{cases}

where $r$ is the distance from the center of the sphere of a radius $R$.

In sasview the effective radius may be calculated from the parameters
used in the form factor $P(q)$ that this $S(q)$ is combined with.

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as

.. math::

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

References
----------

R V Sharma, K C Sharma, *Physica*, 89A (1977) 213.

"""
from numpy import inf

name = "squarewell"
title = "Square well structure factor, with MSA closure"
description = """\
    [Square well structure factor, with MSA closure]
        Interparticle structure factor S(Q)for a hard sphere fluid with
        a narrow attractive well. Fits are prone to deliver non-physical
        parameters, use with care and read the references in the full manual.
        In sasview the effective radius will be calculated from the
        parameters used in P(Q).
"""
category = "structure-factor"
structure_factor = True

#single = False
#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [
    #   [ "name", "units", default, [lower, upper], "type",
    #     "description" ],
    ["radius_effective", "Ang", 50.0, [0, inf], "volume",
     "effective radius of hard sphere"],
    ["volfraction", "", 0.04, [0, 0.08], "",
     "volume fraction of spheres"],
    ["welldepth", "kT", 1.5, [0.0, 1.5], "",
     "depth of well, epsilon"],
    ["wellwidth", "diameters", 1.2, [1.0, inf], "",
     "width of well in diameters (=2R) units, must be > 1"],
    ]

# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
    return 1.0;
    """

Iq = """
// single precision is very poor at extreme small Q, would need a Taylor series
        double req,phis,edibkb,lambda,struc;
        double sigma,eta,eta2,eta3,eta4,etam1,etam14,alpha,beta,gamm;
        double x,sk,sk2,sk3,sk4,t1,t2,t3,t4,ck;
        double S,C,SL,CL;
        x= q;
        
        req = radius_effective;
        phis = volfraction;
        edibkb = welldepth;
        lambda = wellwidth;
        
        sigma = req*2.;
        eta = phis;
        eta2 = eta*eta;
        eta3 = eta*eta2;
        eta4 = eta*eta3;
        etam1 = 1. - eta;
        etam14 = etam1*etam1*etam1*etam1;
        // temp borrow sk for an intermediate calc
        sk = 1.0 +2.0*eta;
        sk *= sk;
        alpha = (  sk + eta3*( eta-4.0 )  )/etam14;
        beta = -(eta/3.0) * ( 18. + 20.*eta - 12.*eta2 + eta4 )/etam14;
        gamm = 0.5*eta*( sk + eta3*(eta-4.) )/etam14;
        
        //  calculate the structure factor
        
        sk = x*sigma;
        sk2 = sk*sk;
        sk3 = sk*sk2;
        sk4 = sk3*sk;
        SINCOS(sk,S,C);
        SINCOS(lambda*sk,SL,CL);
        t1 = alpha * sk3 * ( S - sk * C );
        t2 = beta * sk2 * 2.0*( sk*S - (0.5*sk2 - 1.)*C - 1.0 );
        t3 = gamm*( ( 4.0*sk3 - 24.*sk ) * S - ( sk4 - 12.0*sk2 + 24.0 )*C + 24.0 );
        t4 = -edibkb*sk3*(SL +sk*(C - lambda*CL) - S );
        ck =  -24.0*eta*( t1 + t2 + t3 + t4 )/sk3/sk3;
        struc  = 1./(1.-ck);
        
        return(struc);
"""

Iqxy = """
    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
    """

# ER defaults to 0.0
# VR defaults to 1.0

oldname = 'SquareWellStructure'
oldpars = dict(radius_effective="effect_radius",radius_effective_pd="effect_radius_pd",radius_effective_pd_n="effect_radius_pd_n")
demo = dict(radius_effective=50, volfraction=0.04, welldepth=1.5,
            wellwidth=1.2, radius_effective_pd=0, radius_effective_pd_n=0)
#
tests = [
        [ {'scale': 1.0, 'background' : 0.0, 'radius_effective' : 50.0, 'volfraction' : 0.04,'welldepth' : 1.5, 'wellwidth' : 1.2, 
           'radius_effective_pd' : 0}, [0.001], [0.97665742]]
        ]
# ADDED by: converting from sasview RKH  ON: 16Mar2016