source: sasmodels/sasmodels/models/squarewell.py @ 56b2687

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Last change on this file since 56b2687 was 56b2687, checked in by Paul Kienzle <pkienzle@…>, 8 years ago

Merge branch 'master' into polydisp

Conflicts:

README.rst
sasmodels/core.py
sasmodels/data.py
sasmodels/generate.py
sasmodels/kernelcl.py
sasmodels/kerneldll.py
sasmodels/sasview_model.py

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1# Note: model title and parameter table are inserted automatically
2r"""
3This calculates the interparticle structure factor for a square well fluid spherical particles. The mean spherical
4approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive
5interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing
6this calculation to be limited in applicability to well depths |epsilon| < 1.5 kT and volume fractions |phi| < 0.08.
7
8Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential
9"shoulder", which may or may not be physically reasonable. The stickyhardsphere model may be a better choice in
10some circumstances. Computed values may behave badly at extremely small $qR$.
11
12The well width (|lambda| ) is defined as multiples of the particle diameter (2\*\ *R*\ )
13
14The interaction potential is:
15
16  .. image:: img/squarewell.png
17
18.. math::
19
20    U(r) = \begin{cases}
21    \infty & r < 2R \\
22    -\epsilon & 2R \leq r < 2R\lambda \\
23    0 & r \geq 2R\lambda
24    \end{cases}
25
26where $r$ is the distance from the center of the sphere of a radius $R$.
27
28In sasview the effective radius may be calculated from the parameters
29used in the form factor $P(q)$ that this $S(q)$ is combined with.
30
31For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as
32
33.. math::
34
35    q = \sqrt{q_x^2 + q_y^2}
36
37References
38----------
39
40R V Sharma, K C Sharma, *Physica*, 89A (1977) 213.
41
42"""
43from numpy import inf
44
45name = "squarewell"
46title = "Square well structure factor, with MSA closure"
47description = """\
48    [Square well structure factor, with MSA closure]
49        Interparticle structure factor S(Q)for a hard sphere fluid with
50        a narrow attractive well. Fits are prone to deliver non-physical
51        parameters, use with care and read the references in the full manual.
52        In sasview the effective radius will be calculated from the
53        parameters used in P(Q).
54"""
55category = "structure-factor"
56structure_factor = True
57single = False
58
59#single = False
60#             ["name", "units", default, [lower, upper], "type","description"],
61parameters = [
62    #   [ "name", "units", default, [lower, upper], "type",
63    #     "description" ],
64    ["radius_effective", "Ang", 50.0, [0, inf], "volume",
65     "effective radius of hard sphere"],
66    ["volfraction", "", 0.04, [0, 0.08], "",
67     "volume fraction of spheres"],
68    ["welldepth", "kT", 1.5, [0.0, 1.5], "",
69     "depth of well, epsilon"],
70    ["wellwidth", "diameters", 1.2, [1.0, inf], "",
71     "width of well in diameters (=2R) units, must be > 1"],
72    ]
73
74# No volume normalization despite having a volume parameter
75# This should perhaps be volume normalized?
76form_volume = """
77    return 1.0;
78    """
79
80Iq = """
81// single precision is very poor at extreme small Q, would need a Taylor series
82        double req,phis,edibkb,lambda,struc;
83        double sigma,eta,eta2,eta3,eta4,etam1,etam14,alpha,beta,gamm;
84        double x,sk,sk2,sk3,sk4,t1,t2,t3,t4,ck;
85        double S,C,SL,CL;
86        x= q;
87       
88        req = radius_effective;
89        phis = volfraction;
90        edibkb = welldepth;
91        lambda = wellwidth;
92       
93        sigma = req*2.;
94        eta = phis;
95        eta2 = eta*eta;
96        eta3 = eta*eta2;
97        eta4 = eta*eta3;
98        etam1 = 1. - eta;
99        etam14 = etam1*etam1*etam1*etam1;
100        // temp borrow sk for an intermediate calc
101        sk = 1.0 +2.0*eta;
102        sk *= sk;
103        alpha = (  sk + eta3*( eta-4.0 )  )/etam14;
104        beta = -(eta/3.0) * ( 18. + 20.*eta - 12.*eta2 + eta4 )/etam14;
105        gamm = 0.5*eta*( sk + eta3*(eta-4.) )/etam14;
106       
107        //  calculate the structure factor
108       
109        sk = x*sigma;
110        sk2 = sk*sk;
111        sk3 = sk*sk2;
112        sk4 = sk3*sk;
113        SINCOS(sk,S,C);
114        SINCOS(lambda*sk,SL,CL);
115        t1 = alpha * sk3 * ( S - sk * C );
116        t2 = beta * sk2 * 2.0*( sk*S - (0.5*sk2 - 1.)*C - 1.0 );
117        t3 = gamm*( ( 4.0*sk3 - 24.*sk ) * S - ( sk4 - 12.0*sk2 + 24.0 )*C + 24.0 );
118        t4 = -edibkb*sk3*(SL +sk*(C - lambda*CL) - S );
119        ck =  -24.0*eta*( t1 + t2 + t3 + t4 )/sk3/sk3;
120        struc  = 1./(1.-ck);
121       
122        return(struc);
123"""
124
125# ER defaults to 0.0
126# VR defaults to 1.0
127
128demo = dict(radius_effective=50, volfraction=0.04, welldepth=1.5,
129            wellwidth=1.2, radius_effective_pd=0, radius_effective_pd_n=0)
130#
131tests = [
132        [ {'scale': 1.0, 'background' : 0.0, 'radius_effective' : 50.0, 'volfraction' : 0.04,'welldepth' : 1.5, 'wellwidth' : 1.2, 
133           'radius_effective_pd' : 0}, [0.001], [0.97665742]]
134        ]
135# ADDED by: converting from sasview RKH  ON: 16Mar2016
136
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