1 | # ellipsoid model |
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2 | # Note: model title and parameter table are inserted automatically |
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3 | r""" |
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4 | The form factor is normalized by the particle volume |
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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 output of the 2D scattering intensity function for oriented ellipsoids |
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10 | is given by (Feigin, 1987) |
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11 | |
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12 | .. math:: |
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13 | |
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14 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background} |
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15 | |
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16 | where |
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17 | |
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18 | .. math:: |
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19 | |
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20 | F(q,\alpha) = \Delta \rho V \frac{3(\sin qr - qr \cos qr)}{(qr)^3} |
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21 | |
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22 | for |
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23 | |
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24 | .. math:: |
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25 | |
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26 | r = \left[ R_e^2 \sin^2 \alpha + R_p^2 \cos^2 \alpha \right]^{1/2} |
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27 | |
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28 | |
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29 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
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30 | $V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid, $R_p$ is the polar |
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31 | radius along the rotational axis of the ellipsoid, $R_e$ is the equatorial |
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32 | radius perpendicular to the rotational axis of the ellipsoid and |
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33 | $\Delta \rho$ (contrast) is the scattering length density difference between |
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34 | the scatterer and the solvent. |
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35 | |
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36 | For randomly oriented particles use the orientational average, |
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37 | |
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38 | .. math:: |
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39 | |
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40 | \langle F^2(q) \rangle = \int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)\,d\alpha} |
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41 | |
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42 | |
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43 | computed via substitution of $u=\sin(\alpha)$, $du=\cos(\alpha)\,d\alpha$ as |
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44 | |
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45 | .. math:: |
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46 | |
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47 | \langle F^2(q) \rangle = \int_0^1{F^2(q, u)\,du} |
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48 | |
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49 | with |
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50 | |
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51 | .. math:: |
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52 | |
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53 | r = R_e \left[ 1 + u^2\left(R_p^2/R_e^2 - 1\right)\right]^{1/2} |
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54 | |
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55 | For 2d data from oriented ellipsoids the direction of the rotation axis of |
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56 | the ellipsoid is defined using two angles $\theta$ and $\phi$ as for the |
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57 | :ref:`cylinder orientation figure <cylinder-angle-definition>`. |
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58 | For the ellipsoid, $\theta$ is the angle between the rotational axis |
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59 | and the $z$ -axis in the $xz$ plane followed by a rotation by $\phi$ |
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60 | in the $xy$ plane, for further details of the calculation and angular |
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61 | dispersions see :ref:`orientation` . |
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62 | |
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63 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based |
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64 | on the $R_p$ and $R_e$ values, and used as the effective radius for |
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65 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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66 | |
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67 | |
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68 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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69 | |
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70 | |
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71 | |
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72 | Validation |
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73 | ---------- |
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74 | |
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75 | Validation of the code was done by comparing the output of the 1D model |
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76 | to the output of the software provided by the NIST (Kline, 2006). |
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77 | |
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78 | The implementation of the intensity for fully oriented ellipsoids was |
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79 | validated by averaging the 2D output using a uniform distribution |
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80 | $p(\theta,\phi) = 1.0$ and comparing with the output of the 1D calculation. |
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81 | |
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82 | |
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83 | .. _ellipsoid-comparison-2d: |
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84 | |
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85 | .. figure:: img/ellipsoid_comparison_2d.jpg |
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86 | |
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87 | Comparison of the intensity for uniformly distributed ellipsoids |
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88 | calculated from our 2D model and the intensity from the NIST SANS |
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89 | analysis software. The parameters used were: *scale* = 1.0, |
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90 | *radius_polar* = 20 |Ang|, *radius_equatorial* = 400 |Ang|, |
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91 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
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92 | |
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93 | The discrepancy above $q$ = 0.3 |cm^-1| is due to the way the form factors |
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94 | are calculated in the c-library provided by NIST. A numerical integration |
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95 | has to be performed to obtain $P(q)$ for randomly oriented particles. |
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96 | The NIST software performs that integration with a 76-point Gaussian |
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97 | quadrature rule, which will become imprecise at high $q$ where the amplitude |
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98 | varies quickly as a function of $q$. The SasView result shown has been |
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99 | obtained by summing over 501 equidistant points. Our result was found |
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100 | to be stable over the range of $q$ shown for a number of points higher |
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101 | than 500. |
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102 | |
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103 | Model was also tested against the triaxial ellipsoid model with equal major |
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104 | and minor equatorial radii. It is also consistent with the cyclinder model |
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105 | with polar radius equal to length and equatorial radius equal to radius. |
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106 | |
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107 | References |
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108 | ---------- |
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109 | |
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110 | L A Feigin and D I Svergun. |
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111 | *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, |
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112 | Plenum Press, New York, 1987. |
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113 | |
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114 | A. Isihara. J. Chem. Phys. 18(1950) 1446-1449 |
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115 | |
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116 | Authorship and Verification |
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117 | ---------------------------- |
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118 | |
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119 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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120 | * **Converted to sasmodels by:** Helen Park **Date:** July 9, 2014 |
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121 | * **Last Modified by:** Paul Kienzle **Date:** March 22, 2017 |
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122 | """ |
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123 | from __future__ import division |
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124 | |
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125 | import numpy as np |
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126 | from numpy import inf, sin, cos, pi |
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127 | |
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128 | name = "ellipsoid" |
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129 | title = "Ellipsoid of revolution with uniform scattering length density." |
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130 | |
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131 | description = """\ |
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132 | P(q.alpha)= scale*f(q)^2 + background, where f(q)= 3*(sld |
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133 | - sld_solvent)*V*[sin(q*r(Rp,Re,alpha)) |
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134 | -q*r*cos(qr(Rp,Re,alpha))] |
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135 | /[qr(Rp,Re,alpha)]^3" |
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136 | |
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137 | r(Rp,Re,alpha)= [Re^(2)*(sin(alpha))^2 |
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138 | + Rp^(2)*(cos(alpha))^2]^(1/2) |
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139 | |
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140 | sld: SLD of the ellipsoid |
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141 | sld_solvent: SLD of the solvent |
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142 | V: volume of the ellipsoid |
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143 | Rp: polar radius of the ellipsoid |
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144 | Re: equatorial radius of the ellipsoid |
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145 | """ |
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146 | category = "shape:ellipsoid" |
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147 | |
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148 | # ["name", "units", default, [lower, upper], "type","description"], |
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149 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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150 | "Ellipsoid scattering length density"], |
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151 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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152 | "Solvent scattering length density"], |
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153 | ["radius_polar", "Ang", 20, [0, inf], "volume", |
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154 | "Polar radius"], |
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155 | ["radius_equatorial", "Ang", 400, [0, inf], "volume", |
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156 | "Equatorial radius"], |
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157 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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158 | "ellipsoid axis to beam angle"], |
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159 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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160 | "rotation about beam"], |
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161 | ] |
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162 | |
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163 | |
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164 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "ellipsoid.c"] |
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165 | have_Fq = True |
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166 | effective_radius_type = [ |
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167 | "average curvature", "equivalent volume sphere", "min radius", "max radius", |
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168 | ] |
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169 | |
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170 | def random(): |
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171 | """Return a random parameter set for the model.""" |
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172 | volume = 10**np.random.uniform(5, 12) |
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173 | radius_polar = 10**np.random.uniform(1.3, 4) |
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174 | radius_equatorial = np.sqrt(volume/radius_polar) # ignore 4/3 pi |
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175 | pars = dict( |
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176 | #background=0, sld=0, sld_solvent=1, |
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177 | radius_polar=radius_polar, |
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178 | radius_equatorial=radius_equatorial, |
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179 | ) |
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180 | return pars |
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181 | |
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182 | demo = dict(scale=1, background=0, |
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183 | sld=6, sld_solvent=1, |
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184 | radius_polar=50, radius_equatorial=30, |
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185 | theta=30, phi=15, |
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186 | radius_polar_pd=.2, radius_polar_pd_n=15, |
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187 | radius_equatorial_pd=.2, radius_equatorial_pd_n=15, |
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188 | theta_pd=15, theta_pd_n=45, |
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189 | phi_pd=15, phi_pd_n=1) |
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190 | q = 0.1 |
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191 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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192 | qx = q*cos(pi/6.0) |
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193 | qy = q*sin(pi/6.0) |
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194 | tests = [ |
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195 | [{}, 0.05, 54.8525847025], |
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196 | [{'theta':80., 'phi':10.}, (qx, qy), 1.74134670026], |
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197 | ] |
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198 | del qx, qy # not necessary to delete, but cleaner |
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