1 | r""" |
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2 | For information about polarised and magnetic scattering, see |
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3 | the :ref:`magnetism` documentation. |
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4 | |
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5 | Definition |
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6 | ---------- |
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7 | |
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8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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9 | |
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10 | .. math:: |
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11 | |
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12 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
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13 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
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14 | \right]^2 + \text{background} |
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15 | |
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16 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
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17 | $r$ is the radius of the sphere and *background* is the background level. |
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18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
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19 | scatterer and the solvent respectively, whose difference is $\Delta\rho$. |
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20 | |
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21 | Note that if your data is in absolute scale, the *scale* should represent |
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22 | the volume fraction (which is unitless) if you have a good fit. If not, |
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23 | it should represent the volume fraction times a factor (by which your data |
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24 | might need to be rescaled). |
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25 | |
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26 | The 2D scattering intensity is the same as above, regardless of the |
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27 | orientation of $\vec q$. |
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28 | |
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29 | Validation |
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30 | ---------- |
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31 | |
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32 | Validation of our code was done by comparing the output of the 1D model |
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33 | to the output of the software provided by the NIST (Kline, 2006). |
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34 | |
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35 | |
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36 | References |
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37 | ---------- |
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38 | |
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39 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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40 | John Wiley and Sons, New York, (1955) |
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41 | |
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42 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
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43 | """ |
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44 | |
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45 | import numpy as np |
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46 | from numpy import inf |
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47 | |
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48 | name = "sphere" |
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49 | title = "Spheres with uniform scattering length density" |
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50 | description = """\ |
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51 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
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52 | /(qr)^3]^2 + background |
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53 | r: radius of sphere |
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54 | V: The volume of the scatter |
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55 | sld: the SLD of the sphere |
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56 | sld_solvent: the SLD of the solvent |
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57 | """ |
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58 | category = "shape:sphere" |
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59 | |
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60 | # ["name", "units", default, [lower, upper], "type","description"], |
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61 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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62 | "Layer scattering length density"], |
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63 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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64 | "Solvent scattering length density"], |
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65 | ["radius", "Ang", 50, [0, inf], "volume", |
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66 | "Sphere radius"], |
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67 | ] |
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68 | |
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69 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c"] |
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70 | |
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71 | c_code = """ |
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72 | static double form_volume(double radius) |
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73 | { |
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74 | return sphere_volume(radius); |
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75 | } |
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76 | |
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77 | static void Fq(double q, double *F1,double *F2, double sld, double solvent_sld, double radius) |
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78 | { |
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79 | const double fq = sas_3j1x_x(q*radius); |
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80 | const double contrast = (sld - solvent_sld); |
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81 | const double form = 1e-2 * contrast * sphere_volume(radius) * fq; |
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82 | *F1 = form; |
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83 | *F2 = form*form; |
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84 | } |
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85 | """ |
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86 | |
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87 | # TODO: figure this out by inspection |
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88 | have_Fq = True |
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89 | |
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90 | def ER(radius): |
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91 | """ |
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92 | Return equivalent radius (ER) |
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93 | """ |
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94 | return radius |
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95 | |
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96 | # VR defaults to 1.0 |
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97 | |
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98 | def random(): |
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99 | radius = 10**np.random.uniform(1.3, 4) |
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100 | pars = dict( |
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101 | radius=radius, |
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102 | ) |
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103 | return pars |
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104 | |
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105 | tests = [ |
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106 | [{}, 0.2, 0.726362], |
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107 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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108 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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109 | 0.2, 0.228843], |
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110 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "ER", 120.], |
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111 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.], |
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112 | ] |
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