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*, John Wiley and Sons, New York, (1955) |
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40 | |
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41 | Source |
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42 | ------ |
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43 | |
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44 | `sphere.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/sphere.py>`_ |
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45 | |
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46 | `sphere.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/sphere.c>`_ |
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47 | |
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48 | Authorship and Verification |
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49 | ---------------------------- |
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50 | |
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51 | * **Author:** |
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52 | * **Last Modified by:** |
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53 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
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54 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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55 | """ |
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56 | |
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57 | import numpy as np |
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58 | from numpy import inf |
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59 | |
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60 | name = "sphere" |
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61 | title = "Spheres with uniform scattering length density" |
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62 | description = """\ |
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63 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
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64 | /(qr)^3]^2 + background |
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65 | r: radius of sphere |
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66 | V: The volume of the scatter |
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67 | sld: the SLD of the sphere |
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68 | sld_solvent: the SLD of the solvent |
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69 | """ |
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70 | category = "shape:sphere" |
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71 | |
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72 | # ["name", "units", default, [lower, upper], "type","description"], |
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73 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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74 | "Layer scattering length density"], |
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75 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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76 | "Solvent scattering length density"], |
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77 | ["radius", "Ang", 50, [0, inf], "volume", |
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78 | "Sphere radius"], |
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79 | ] |
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80 | |
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81 | source = ["lib/sas_3j1x_x.c", "sphere.c"] |
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82 | have_Fq = True |
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83 | radius_effective_modes = ["radius"] |
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84 | |
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85 | def random(): |
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86 | """Return a random parameter set for the model.""" |
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87 | radius = 10**np.random.uniform(1.3, 4) |
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88 | pars = dict( |
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89 | radius=radius, |
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90 | ) |
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91 | return pars |
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92 | |
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93 | tests = [ |
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94 | [{}, 0.2, 0.726362], # each test starts with default parameter values inside { }, unless modified. Then Q and expected value of I(Q) |
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95 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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96 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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97 | 0.2, 0.2288431], |
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98 | [{"radius": 120., "radius_pd": 0.02, "radius_pd_n":45}, |
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99 | 0.2, 792.0646662454202, 1166737.0473152, 120.0, 7246723.820358589, 1.0], # the longer list here checks F1, F2, R_eff, volume, volume_ratio = call_Fq(kernel, pars) |
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100 | # But note P(Q) = F2/volume |
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101 | # F and F^2 are "unscaled", with for n <F F*>S(q) or for beta approx I(q) = n [<F F*> + <F><F*> (S(q) - 1)] |
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102 | # for n the number density and <.> the orientation average, and F = integral rho(r) exp(i q . r) dr. |
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103 | # The number density is volume fraction divided by particle volume. |
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104 | # Effectively, this leaves F = V drho form, where form is the usual 3 j1(qr)/(qr) or whatever depending on the shape. |
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105 | # [{"@S": "hardsphere"}, |
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106 | # 0.01, 55.881884232102124], # this is current value, not verified elsewhere yet |
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107 | # [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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108 | # 0.2, 1.233304061, [1850806.119736], 120.0, 8087664.1226, 1.0], # the longer list here checks F1, F2, R_eff, volume, volume_ratio = call_Fq(kernel, pars) |
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109 | # [{"@S": "hardsphere"}, |
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110 | # 0.2, 0.14730859242492958], # this is current value, not verified elsewhere yet |
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111 | # [{"@S": "hardsphere"}, |
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112 | # 0.1, 0.7940350343811906], # this is current value, not verified elsewhere yet |
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113 | [{"@S": "hardsphere", |
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114 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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115 | "volfraction":0.2, |
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116 | "radius_effective":45.0, # uses this (check) |
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117 | "structure_factor_mode": 1, # 0 = normal decoupling approximation, 1 = beta(Q) approx |
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118 | "radius_effective_mode": 0 # equivalent sphere, there is only one valid mode for sphere. says -this used r_eff =0 or default 50? |
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119 | }, 0.01, 1316.2990966463444 ], |
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120 | [{"@S": "hardsphere", |
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121 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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122 | "volfraction":0.2, |
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123 | "radius_effective":50.0, # hard sphere structure factor |
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124 | "structure_factor_mode": 1, # 0 = normal decoupling approximation, 1 = beta(Q) approx |
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125 | "radius_effective_mode": 0 # this used r_eff =0 or default 50? |
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126 | }, 0.01, 1324.7375636587356 ], |
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127 | [{"@S": "hardsphere", |
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128 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45, |
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129 | "volfraction":0.2, |
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130 | "radius_effective":50.0, # hard sphere structure factor |
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131 | "structure_factor_mode": 1, # 0 = normal decoupling approximation, 1 = beta(Q) approx |
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132 | "radius_effective_mode": 1 # this used 120 ??? |
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133 | }, 0.01, 1579.2858949296565 ] |
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134 | ] |
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135 | # putting None for expected result will pass the test if there are no errors from the routine, but without any check on the value of the result |
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