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 | Our model uses the form factor calculations as defined in the IGOR |
---|
31 | package provided by the NIST Center for Neutron Research (Kline, 2006). |
---|
32 | |
---|
33 | Validation |
---|
34 | ---------- |
---|
35 | |
---|
36 | Validation of our code was done by comparing the output of the 1D model |
---|
37 | to the output of the software provided by the NIST (Kline, 2006). |
---|
38 | Figure :num:`figure #sphere-comparison` shows a comparison of the output |
---|
39 | of 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 | |
---|
51 | Reference |
---|
52 | --------- |
---|
53 | |
---|
54 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
---|
55 | John 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 | |
---|
60 | import numpy as np |
---|
61 | from numpy import pi, inf, sin, cos, sqrt, log |
---|
62 | |
---|
63 | name = "sphere" |
---|
64 | title = "Spheres with uniform scattering length density" |
---|
65 | description = """\ |
---|
66 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qR)-qRcos(qR)) |
---|
67 | /(qR)^3]^2 + background |
---|
68 | R: radius of sphere |
---|
69 | V: The volume of the scatter |
---|
70 | sld: the SLD of the sphere |
---|
71 | solvent_sld: the SLD of the solvent |
---|
72 | """ |
---|
73 | category = "shape:sphere" |
---|
74 | |
---|
75 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
76 | parameters = [["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", |
---|
81 | "Sphere radius"], |
---|
82 | ] |
---|
83 | |
---|
84 | |
---|
85 | def form_volume(radius): |
---|
86 | return 1.333333333333333 * pi * radius ** 3 |
---|
87 | |
---|
88 | def Iq(q, sld, solvent_sld, radius): |
---|
89 | #print "q",q |
---|
90 | #print "sld,r",sld,solvent_sld,radius |
---|
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 |
---|
103 | Iq.vectorized = True # Iq accepts an array of Q values |
---|
104 | |
---|
105 | def Iqxy(qx, qy, sld, solvent_sld, radius): |
---|
106 | return Iq(sqrt(qx ** 2 + qy ** 2), sld, solvent_sld, radius) |
---|
107 | Iqxy.vectorized = True # Iqxy accepts arrays of Qx, Qy values |
---|
108 | |
---|
109 | def sesans(z, sld, solvent_sld, radius): |
---|
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 |
---|
123 | sesans.vectorized = True # sesans accepts and array of z values |
---|
124 | |
---|
125 | def ER(radius): |
---|
126 | return radius |
---|
127 | |
---|
128 | # VR defaults to 1.0 |
---|
129 | |
---|
130 | demo = dict(scale=1, background=0, |
---|
131 | sld=6, solvent_sld=1, |
---|
132 | radius=120, |
---|
133 | radius_pd=.2, radius_pd_n=45) |
---|
134 | oldname = "SphereModel" |
---|
135 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
---|