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) + \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) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] |
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21 | - \cos[qr(R_p,R_e,\alpha)])} |
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22 | {[qr(R_p,R_e,\alpha)]^3} |
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23 | |
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24 | and |
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25 | |
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26 | .. math:: |
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27 | |
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28 | r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha |
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29 | + R_p^2 \cos^2 \alpha \right]^{1/2} |
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30 | |
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31 | |
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32 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
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33 | $V$ is the volume of the ellipsoid, $R_p$ is the polar radius along the |
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34 | rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular |
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35 | to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the |
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36 | scattering length density difference between the scatterer and the solvent. |
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37 | |
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38 | To provide easy access to the orientation of the ellipsoid, we define |
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39 | the rotation axis of the ellipsoid using two angles $\theta$ and $\phi$. |
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40 | These angles are defined in the |
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41 | :ref:`cylinder orientation figure <cylinder-orientation>`. |
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42 | For the ellipsoid, $\theta$ is the angle between the rotational axis |
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43 | and the $z$-axis. |
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44 | |
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45 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based |
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46 | on the $R_p$ and $R_e$ values, and used as the effective radius for |
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47 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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48 | |
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49 | .. _ellipsoid-1d: |
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50 | |
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51 | .. figure:: img/ellipsoid_1d.jpg |
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52 | |
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53 | The output of the 1D scattering intensity function for randomly oriented |
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54 | ellipsoids given by the equation above. |
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55 | |
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56 | |
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57 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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58 | |
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59 | .. _ellipsoid-geometry: |
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60 | |
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61 | .. figure:: img/ellipsoid_geometry.jpg |
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62 | |
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63 | The angles for oriented ellipsoid. |
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64 | |
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65 | Validation |
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66 | ---------- |
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67 | |
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68 | Validation of our code was done by comparing the output of the 1D model |
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69 | to the output of the software provided by the NIST (Kline, 2006). |
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70 | :num:`Figure ellipsoid-comparison-1d` below shows a comparison of |
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71 | the 1D output of our model and the output of the NIST software. |
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72 | |
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73 | .. _ellipsoid-comparison-1d: |
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74 | |
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75 | .. figure:: img/ellipsoid_comparison_1d.jpg |
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76 | |
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77 | Comparison of the SasView scattering intensity for an ellipsoid |
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78 | with the output of the NIST SANS analysis software. The parameters |
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79 | were set to: *scale* = 1.0, *rpolar* = 20 |Ang|, |
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80 | *requatorial* =400 |Ang|, *contrast* = 3e-6 |Ang^-2|, |
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81 | and *background* = 0.01 |cm^-1|. |
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82 | |
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83 | Averaging over a distribution of orientation is done by evaluating the |
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84 | equation above. Since we have no other software to compare the |
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85 | implementation of the intensity for fully oriented ellipsoids, we can |
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86 | compare the result of averaging our 2D output using a uniform distribution |
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87 | $p(\theta,\phi) = 1.0$. :num:`Figure #ellipsoid-comparison-2d` |
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88 | shows the result of such a cross-check. |
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89 | |
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90 | .. _ellipsoid-comparison-2d: |
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91 | |
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92 | .. figure:: img/ellipsoid_comparison_2d.jpg |
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93 | |
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94 | Comparison of the intensity for uniformly distributed ellipsoids |
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95 | calculated from our 2D model and the intensity from the NIST SANS |
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96 | analysis software. The parameters used were: *scale* = 1.0, |
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97 | *rpolar* = 20 |Ang|, *requatorial* = 400 |Ang|, |
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98 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
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99 | |
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100 | The discrepancy above $q$ = 0.3 |cm^-1| is due to the way the form factors |
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101 | are calculated in the c-library provided by NIST. A numerical integration |
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102 | has to be performed to obtain $P(q)$ for randomly oriented particles. |
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103 | The NIST software performs that integration with a 76-point Gaussian |
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104 | quadrature rule, which will become imprecise at high $q$ where the amplitude |
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105 | varies quickly as a function of $q$. The SasView result shown has been |
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106 | obtained by summing over 501 equidistant points. Our result was found |
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107 | to be stable over the range of $q$ shown for a number of points higher |
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108 | than 500. |
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109 | |
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110 | References |
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111 | ---------- |
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112 | |
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113 | L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
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114 | New York, 1987. |
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115 | """ |
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116 | |
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117 | from numpy import inf |
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118 | |
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119 | name = "ellipsoid" |
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120 | title = "Ellipsoid of revolution with uniform scattering length density." |
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121 | |
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122 | description = """\ |
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123 | P(q.alpha)= scale*f(q)^2 + background, where f(q)= 3*(sld |
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124 | - solvent_sld)*V*[sin(q*r(Rp,Re,alpha)) |
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125 | -q*r*cos(qr(Rp,Re,alpha))] |
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126 | /[qr(Rp,Re,alpha)]^3" |
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127 | |
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128 | r(Rp,Re,alpha)= [Re^(2)*(sin(alpha))^2 |
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129 | + Rp^(2)*(cos(alpha))^2]^(1/2) |
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130 | |
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131 | sld: SLD of the ellipsoid |
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132 | solvent_sld: SLD of the solvent |
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133 | V: volume of the ellipsoid |
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134 | Rp: polar radius of the ellipsoid |
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135 | Re: equatorial radius of the ellipsoid |
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136 | """ |
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137 | category = "shape:ellipsoid" |
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138 | |
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139 | # ["name", "units", default, [lower, upper], "type","description"], |
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140 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
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141 | "Ellipsoid scattering length density"], |
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142 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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143 | "Solvent scattering length density"], |
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144 | ["rpolar", "Ang", 20, [0, inf], "volume", |
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145 | "Polar radius"], |
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146 | ["requatorial", "Ang", 400, [0, inf], "volume", |
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147 | "Equatorial radius"], |
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148 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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149 | "In plane angle"], |
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150 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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151 | "Out of plane angle"], |
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152 | ] |
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153 | |
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154 | source = ["lib/J1.c", "lib/gauss76.c", "ellipsoid.c"] |
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155 | |
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156 | def ER(rpolar, requatorial): |
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157 | import numpy as np |
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158 | |
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159 | ee = np.empty_like(rpolar) |
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160 | idx = rpolar > requatorial |
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161 | ee[idx] = (rpolar[idx] ** 2 - requatorial[idx] ** 2) / rpolar[idx] ** 2 |
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162 | idx = rpolar < requatorial |
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163 | ee[idx] = (requatorial[idx] ** 2 - rpolar[idx] ** 2) / requatorial[idx] ** 2 |
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164 | idx = rpolar == requatorial |
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165 | ee[idx] = 2 * rpolar[idx] |
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166 | valid = (rpolar * requatorial != 0) |
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167 | bd = 1.0 - ee[valid] |
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168 | e1 = np.sqrt(ee[valid]) |
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169 | b1 = 1.0 + np.arcsin(e1) / (e1 * np.sqrt(bd)) |
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170 | bL = (1.0 + e1) / (1.0 - e1) |
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171 | b2 = 1.0 + bd / 2 / e1 * np.log(bL) |
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172 | delta = 0.75 * b1 * b2 |
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173 | |
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174 | ddd = np.zeros_like(rpolar) |
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175 | ddd[valid] = 2.0 * (delta + 1.0) * rpolar * requatorial ** 2 |
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176 | return 0.5 * ddd ** (1.0 / 3.0) |
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177 | |
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178 | |
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179 | demo = dict(scale=1, background=0, |
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180 | sld=6, solvent_sld=1, |
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181 | rpolar=50, requatorial=30, |
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182 | theta=30, phi=15, |
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183 | rpolar_pd=.2, rpolar_pd_n=15, |
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184 | requatorial_pd=.2, requatorial_pd_n=15, |
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185 | theta_pd=15, theta_pd_n=45, |
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186 | phi_pd=15, phi_pd_n=1) |
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187 | oldname = 'EllipsoidModel' |
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188 | oldpars = dict(theta='axis_theta', phi='axis_phi', |
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189 | sld='sldEll', solvent_sld='sldSolv', |
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190 | rpolar='radius_a', requatorial='radius_b') |
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