1 | r""" |
---|
2 | For information about polarised and magnetic scattering, see |
---|
3 | the :ref:`magnetism` documentation. |
---|
4 | |
---|
5 | Definition |
---|
6 | ---------- |
---|
7 | |
---|
8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
9 | |
---|
10 | .. math:: |
---|
11 | |
---|
12 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
---|
13 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
---|
14 | \right]^2 + \text{background} |
---|
15 | |
---|
16 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
---|
17 | $r$ is the radius of the sphere and *background* is the background level. |
---|
18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
---|
19 | scatterer and the solvent respectively, whose difference is $\Delta\rho$. |
---|
20 | |
---|
21 | Note that if your data is in absolute scale, the *scale* should represent |
---|
22 | the volume fraction (which is unitless) if you have a good fit. If not, |
---|
23 | it should represent the volume fraction times a factor (by which your data |
---|
24 | might need to be rescaled). |
---|
25 | |
---|
26 | The 2D scattering intensity is the same as above, regardless of the |
---|
27 | orientation of $\vec q$. |
---|
28 | |
---|
29 | Validation |
---|
30 | ---------- |
---|
31 | |
---|
32 | Validation of our code was done by comparing the output of the 1D model |
---|
33 | to the output of the software provided by the NIST (Kline, 2006). |
---|
34 | |
---|
35 | |
---|
36 | References |
---|
37 | ---------- |
---|
38 | |
---|
39 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
---|
40 | John Wiley and Sons, New York, (1955) |
---|
41 | |
---|
42 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
---|
43 | """ |
---|
44 | |
---|
45 | import numpy as np |
---|
46 | from numpy import inf |
---|
47 | |
---|
48 | name = "sphere" |
---|
49 | title = "Spheres with uniform scattering length density" |
---|
50 | description = """\ |
---|
51 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
---|
52 | /(qr)^3]^2 + background |
---|
53 | r: radius of sphere |
---|
54 | V: The volume of the scatter |
---|
55 | sld: the SLD of the sphere |
---|
56 | sld_solvent: the SLD of the solvent |
---|
57 | """ |
---|
58 | category = "shape:sphere" |
---|
59 | |
---|
60 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
61 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
62 | "Layer scattering length density"], |
---|
63 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
---|
64 | "Solvent scattering length density"], |
---|
65 | ["radius", "Ang", 50, [0, inf], "volume", |
---|
66 | "Sphere radius"], |
---|
67 | ] |
---|
68 | |
---|
69 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c"] |
---|
70 | |
---|
71 | # No volume normalization despite having a volume parameter |
---|
72 | # This should perhaps be volume normalized? |
---|
73 | form_volume = """ |
---|
74 | return sphere_volume(radius); |
---|
75 | """ |
---|
76 | |
---|
77 | Iq = """ |
---|
78 | return sphere_form(q, radius, sld, sld_solvent); |
---|
79 | """ |
---|
80 | |
---|
81 | def ER(radius): |
---|
82 | """ |
---|
83 | Return equivalent radius (ER) |
---|
84 | """ |
---|
85 | return radius |
---|
86 | |
---|
87 | # VR defaults to 1.0 |
---|
88 | |
---|
89 | def random(): |
---|
90 | radius = 10**np.random.uniform(1.3, 4) |
---|
91 | pars = dict( |
---|
92 | radius=radius, |
---|
93 | ) |
---|
94 | return pars |
---|
95 | |
---|
96 | tests = [ |
---|
97 | [{}, 0.2, 0.726362], |
---|
98 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
---|
99 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
---|
100 | 0.2, 0.228843], |
---|
101 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "ER", 120.], |
---|
102 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.], |
---|
103 | ] |
---|