1 | r""" |
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2 | For information about polarised and magnetic scattering, click here_. |
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3 | |
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4 | .. _here: polar_mag_help.html |
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5 | |
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6 | Definition |
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7 | ---------- |
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8 | |
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9 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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10 | |
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11 | .. math:: |
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12 | |
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13 | I(Q) = \frac{\text{scale}}{V} \cdot \left[ \ |
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14 | 3V(\Delta\rho) \cdot \frac{\sin(QR) - QR\cos(QR))}{(QR)^3} \ |
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15 | \right]^2 + \text{background} |
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16 | |
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17 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
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18 | $R$ is the radius of the sphere, *background* is the background level and |
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19 | *sld* and *solvent_sld* are the scattering length densities (SLDs) of the |
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20 | scatterer and the solvent respectively. |
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21 | |
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22 | Note that if your data is in absolute scale, the *scale* should represent |
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23 | the volume fraction (which is unitless) if you have a good fit. If not, |
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24 | it should represent the volume fraction times a factor (by which your data |
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25 | might need to be rescaled). |
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26 | |
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27 | The 2D scattering intensity is the same as above, regardless of the |
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28 | orientation of $\vec q$. |
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29 | |
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30 | Our model uses the form factor calculations as defined in the IGOR |
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31 | package provided by the NIST Center for Neutron Research (Kline, 2006). |
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32 | |
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33 | Validation |
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34 | ---------- |
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35 | |
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36 | Validation of our code was done by comparing the output of the 1D model |
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37 | to the output of the software provided by the NIST (Kline, 2006). |
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38 | Figure :num:`figure #sphere-comparison` shows a comparison of the output |
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39 | of our model and the output of the NIST software. |
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40 | |
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41 | .. _sphere-comparison: |
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42 | |
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43 | .. figure:: img/sphere_comparison.jpg |
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44 | |
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45 | Comparison of the DANSE scattering intensity for a sphere with the |
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46 | output of the NIST SANS analysis software. The parameters were set to: |
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47 | *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and |
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48 | *background* = 0.01 |cm^-1|. |
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49 | |
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50 | |
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51 | Reference |
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52 | --------- |
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53 | |
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54 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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55 | John Wiley and Sons, New York, (1955) |
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56 | |
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57 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
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58 | """ |
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59 | |
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60 | import numpy as np |
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61 | from numpy import pi, inf, sin, cos, sqrt, log |
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62 | |
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63 | name = "sphere (python)" |
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64 | title = "Spheres with uniform scattering length density" |
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65 | description = """\ |
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66 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qR)-qRcos(qR)) |
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67 | /(qR)^3]^2 + background |
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68 | R: radius of sphere |
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69 | V: The volume of the scatter |
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70 | sld: the SLD of the sphere |
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71 | solvent_sld: the SLD of the solvent |
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72 | """ |
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73 | category = "shape:sphere" |
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74 | |
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75 | # ["name", "units", default, [lower, upper], "type","description"], |
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76 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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77 | "Layer scattering length density"], |
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78 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
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79 | "Solvent scattering length density"], |
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80 | ["radius", "Ang", 50, [0, inf], "volume", |
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81 | "Sphere radius"], |
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82 | ] |
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83 | |
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84 | |
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85 | def form_volume(radius): |
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86 | return 1.333333333333333 * pi * radius ** 3 |
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87 | |
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88 | def Iq(q, sld, solvent_sld, radius): |
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89 | #print "q",q |
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90 | #print "sld,r",sld,solvent_sld,radius |
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91 | qr = q * radius |
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92 | sn, cn = sin(qr), cos(qr) |
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93 | ## The natural expression for the bessel function is the following: |
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94 | ## bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1 |
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95 | ## however, to support vector q values we need to handle the conditional |
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96 | ## as a vector, which we do by first evaluating the full expression |
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97 | ## everywhere, then fixing it up where it is broken. We should probably |
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98 | ## set numpy to ignore the 0/0 error before we do though... |
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99 | bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line |
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100 | bes[qr == 0] = 1 |
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101 | fq = bes * (sld - solvent_sld) * form_volume(radius) |
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102 | return 1.0e-4 * fq ** 2 |
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103 | Iq.vectorized = True # Iq accepts an array of Q values |
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104 | |
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105 | def Iqxy(qx, qy, sld, solvent_sld, radius): |
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106 | return Iq(sqrt(qx ** 2 + qy ** 2), sld, solvent_sld, radius) |
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107 | Iqxy.vectorized = True # Iqxy accepts arrays of Qx, Qy values |
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108 | |
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109 | def sesans(z, sld, solvent_sld, radius): |
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110 | """ |
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111 | Calculate SESANS-correlation function for a solid sphere. |
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112 | |
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113 | Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article |
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114 | """ |
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115 | d = z / radius |
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116 | g = np.zeros_like(z) |
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117 | g[d == 0] = 1. |
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118 | low = ((d > 0) & (d < 2)) |
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119 | dlow = d[low] |
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120 | dlow2 = dlow ** 2 |
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121 | g[low] = sqrt(1 - dlow2 / 4.) * (1 + dlow2 / 8.) + dlow2 / 2.*(1 - dlow2 / 16.) * log(dlow / (2. + sqrt(4. - dlow2))) |
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122 | return g |
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123 | sesans.vectorized = True # sesans accepts and array of z values |
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124 | |
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125 | def ER(radius): |
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126 | return radius |
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127 | |
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128 | # VR defaults to 1.0 |
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129 | |
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130 | demo = dict(scale=1, background=0, |
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131 | sld=6, solvent_sld=1, |
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132 | radius=120, |
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133 | radius_pd=.2, radius_pd_n=45) |
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134 | oldname = "SphereModel" |
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135 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
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