1 | r""" |
---|
2 | For information about polarised and magnetic scattering, click here_. |
---|
3 | |
---|
4 | .. _here: polar_mag_help.html |
---|
5 | |
---|
6 | Definition |
---|
7 | ---------- |
---|
8 | |
---|
9 | The 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 | |
---|
17 | where *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 |
---|
20 | scatterer and the solvent respectively. |
---|
21 | |
---|
22 | Note that if your data is in absolute scale, the *scale* should represent |
---|
23 | the volume fraction (which is unitless) if you have a good fit. If not, |
---|
24 | it should represent the volume fraction times a factor (by which your data |
---|
25 | might need to be rescaled). |
---|
26 | |
---|
27 | The 2D scattering intensity is the same as above, regardless of the |
---|
28 | orientation of $\vec q$. |
---|
29 | |
---|
30 | Validation |
---|
31 | ---------- |
---|
32 | |
---|
33 | Validation of our code was done by comparing the output of the 1D model |
---|
34 | to the output of the software provided by the NIST (Kline, 2006). |
---|
35 | Figure :num:`figure #sphere-comparison` shows a comparison of the output |
---|
36 | of our model and the output of the NIST software. |
---|
37 | |
---|
38 | .. _sphere-comparison: |
---|
39 | |
---|
40 | .. figure:: img/sphere_comparison.jpg |
---|
41 | |
---|
42 | Comparison of the DANSE scattering intensity for a sphere with the |
---|
43 | output of the NIST SANS analysis software. The parameters were set to: |
---|
44 | *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and |
---|
45 | *background* = 0.01 |cm^-1|. |
---|
46 | |
---|
47 | |
---|
48 | References |
---|
49 | ---------- |
---|
50 | |
---|
51 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
---|
52 | John Wiley and Sons, New York, (1955) |
---|
53 | |
---|
54 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
---|
55 | """ |
---|
56 | |
---|
57 | from numpy import inf |
---|
58 | |
---|
59 | name = "sphere" |
---|
60 | title = "Spheres with uniform scattering length density" |
---|
61 | description = """\ |
---|
62 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qr)-qr cos(qr)) |
---|
63 | /(qr)^3]^2 + background |
---|
64 | r: radius of sphere |
---|
65 | V: The volume of the scatter |
---|
66 | sld: the SLD of the sphere |
---|
67 | solvent_sld: the SLD of the solvent |
---|
68 | """ |
---|
69 | category = "shape:sphere" |
---|
70 | |
---|
71 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
72 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
---|
73 | "Layer scattering length density"], |
---|
74 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
---|
75 | "Solvent scattering length density"], |
---|
76 | ["radius", "Ang", 50, [0, inf], "volume", |
---|
77 | "Sphere radius"], |
---|
78 | ] |
---|
79 | |
---|
80 | source = ["lib/sph_j1c.c"] |
---|
81 | |
---|
82 | # No volume normalization despite having a volume parameter |
---|
83 | # This should perhaps be volume normalized? |
---|
84 | form_volume = """ |
---|
85 | return 1.333333333333333*M_PI*radius*radius*radius; |
---|
86 | """ |
---|
87 | |
---|
88 | Iq = """ |
---|
89 | const double qr = q*radius; |
---|
90 | const double bes = sph_j1c(qr); |
---|
91 | const double fq = bes * (sld - solvent_sld) * form_volume(radius); |
---|
92 | return 1.0e-4*fq*fq; |
---|
93 | """ |
---|
94 | |
---|
95 | Iqxy = """ |
---|
96 | // never called since no orientation or magnetic parameters. |
---|
97 | //return -1.0; |
---|
98 | return Iq(sqrt(qx*qx + qy*qy), sld, solvent_sld, radius); |
---|
99 | """ |
---|
100 | |
---|
101 | def ER(radius): |
---|
102 | """ |
---|
103 | Return equivalent radius (ER) |
---|
104 | """ |
---|
105 | return radius |
---|
106 | |
---|
107 | # VR defaults to 1.0 |
---|
108 | |
---|
109 | demo = dict(scale=1, background=0, |
---|
110 | sld=6, solvent_sld=1, |
---|
111 | radius=120, |
---|
112 | radius_pd=.2, radius_pd_n=45) |
---|
113 | oldname = "SphereModel" |
---|
114 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
---|