# source:sasmodels/sasmodels/models/spherepy.py@34375ea

core_shell_microgelscostrafo411magnetic_modelrelease_v0.94release_v0.95ticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 34375ea was 34375ea, checked in by Paul Kienzle <pkienzle@…>, 7 years ago

doc build tweaks

• Property mode set to 100644
File size: 4.5 KB
Line
1r"""
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
60import numpy as np
61from numpy import pi, inf, sin, cos, sqrt, log
62
63name = "sphere (python)"
64title = "Spheres with uniform scattering length density"
65description = """\
66P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qR)-qRcos(qR))
67                /(qR)^3]^2 + background
69    V: The volume of the scatter
70    sld: the SLD of the sphere
71    solvent_sld: the SLD of the solvent
72"""
73category = "shape:sphere"
74
75#             ["name", "units", default, [lower, upper], "type","description"],
76parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "",
77               "Layer scattering length density"],
78              ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "",
79               "Solvent scattering length density"],
80              ["radius", "Ang", 50, [0, inf], "volume",
82             ]
83
84
86    return 1.333333333333333 * pi * radius ** 3
87
89    #print "q",q
91    qr = q * radius
92    sn, cn = sin(qr), cos(qr)
93    ## The natural expression for the bessel function is the following:
94    ##     bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1
95    ## however, to support vector q values we need to handle the conditional
96    ## as a vector, which we do by first evaluating the full expression
97    ## everywhere, then fixing it up where it is broken.   We should probably
98    ## set numpy to ignore the 0/0 error before we do though...
99    bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line
100    bes[qr == 0] = 1
101    fq = bes * (sld - solvent_sld) * form_volume(radius)
102    return 1.0e-4 * fq ** 2
103Iq.vectorized = True  # Iq accepts an array of Q values
104
105def Iqxy(qx, qy, sld, solvent_sld, radius):
106    return Iq(sqrt(qx ** 2 + qy ** 2), sld, solvent_sld, radius)
107Iqxy.vectorized = True  # Iqxy accepts arrays of Qx, Qy values
108
110    """
111    Calculate SESANS-correlation function for a solid sphere.
112
113    Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article
114    """
115    d = z / radius
116    g = np.zeros_like(z)
117    g[d == 0] = 1.
118    low = ((d > 0) & (d < 2))
119    dlow = d[low]
120    dlow2 = dlow ** 2
121    g[low] = sqrt(1 - dlow2 / 4.) * (1 + dlow2 / 8.) + dlow2 / 2.*(1 - dlow2 / 16.) * log(dlow / (2. + sqrt(4. - dlow2)))
122    return g
123sesans.vectorized = True  # sesans accepts and array of z values
124