source: sasmodels/sasmodels/models/sphere.py @ d1fe925

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1r"""
2For information about polarised and magnetic scattering, click here_.
3
4.. _here: polar_mag_help.html
5
6Definition
7----------
8
9The 1D scattering intensity is calculated in the following way (Guinier, 1955)
10
11.. math::
12
13    I(Q) = \frac{\text{scale}}{V} \cdot \left[ \
14        3V(\Delta\rho) \cdot \frac{\sin(QR) - QR\cos(QR))}{(QR)^3} \
15        \right]^2 + \text{background}
16
17where *scale* is a volume fraction, $V$ is the volume of the scatterer,
18$R$ is the radius of the sphere, *background* is the background level and
19*sld* and *solvent_sld* are the scattering length densities (SLDs) of the
20scatterer and the solvent respectively.
21
22Note that if your data is in absolute scale, the *scale* should represent
23the volume fraction (which is unitless) if you have a good fit. If not,
24it should represent the volume fraction times a factor (by which your data
25might need to be rescaled).
26
27The 2D scattering intensity is the same as above, regardless of the
28orientation of $\vec q$.
29
30Our model uses the form factor calculations as defined in the IGOR
31package provided by the NIST Center for Neutron Research (Kline, 2006).
32
33Validation
34----------
35
36Validation of our code was done by comparing the output of the 1D model
37to the output of the software provided by the NIST (Kline, 2006).
38Figure :num:`figure #sphere-comparison` shows a comparison of the output
39of our model and the output of the NIST software.
40
41.. _sphere-comparison:
42
43.. figure:: img/sphere_comparison.jpg
44
45    Comparison of the DANSE scattering intensity for a sphere with the
46    output of the NIST SANS analysis software. The parameters were set to:
47    *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and
48    *background* = 0.01 |cm^-1|.
49
50
51Reference
52---------
53
54A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*,
55John Wiley and Sons, New York, (1955)
56
57*2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.*
58"""
59
60from numpy import inf
61
62name = "sphere"
63title = "Spheres with uniform scattering length density"
64description = """\
65P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qR)-qRcos(qR))
66                /(qR)^3]^2 + background
67    R: radius of sphere
68    V: The volume of the scatter
69    sld: the SLD of the sphere
70    solvent_sld: the SLD of the solvent
71"""
72category = "shape:sphere"
73
74#             ["name", "units", default, [lower, upper], "type","description"],
75parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "",
76               "Layer scattering length density"],
77              ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "",
78               "Solvent scattering length density"],
79              ["radius", "Ang", 50, [0, inf], "volume",
80               "Sphere radius"],
81             ]
82
83
84# No volume normalization despite having a volume parameter
85# This should perhaps be volume normalized?
86form_volume = """
87    return 1.333333333333333*M_PI*radius*radius*radius;
88    """
89
90Iq = """
91    const double qr = q*radius;
92    double sn, cn;
93    SINCOS(qr, sn, cn);
94    const double bes = qr==0.0 ? 1.0 : 3.0*(sn-qr*cn)/(qr*qr*qr);
95    const double fq = bes * (sld - solvent_sld) * form_volume(radius);
96    return 1.0e-4*fq*fq;
97    """
98
99
100Iqxy = """
101    // never called since no orientation or magnetic parameters.
102    //return -1.0;
103    return Iq(sqrt(qx*qx + qy*qy), sld, solvent_sld, radius);
104    """
105
106def ER(radius):
107    return radius
108
109# VR defaults to 1.0
110
111demo = dict(scale=1, background=0,
112            sld=6, solvent_sld=1,
113            radius=120,
114            radius_pd=.2, radius_pd_n=45)
115oldname = "SphereModel"
116oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius')
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