1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | Parameters for this model are the core axial ratio X and a shell thickness, |
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6 | which are more often what we would like to determine and makes the model |
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7 | better behaved, particularly when polydispersity is applied than the four |
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8 | independent radii used in the original parameterization of this model. |
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9 | |
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10 | |
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11 | .. figure:: img/core_shell_ellipsoid_geometry.png |
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12 | |
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13 | The geometric parameters of this model are shown in the diagram above, which |
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14 | shows (a) a cut through at the circular equator and (b) a cross section through |
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15 | the poles, of a prolate ellipsoid. |
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16 | |
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17 | When *X_core < 1* the core is oblate; when *X_core > 1* it is prolate. |
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18 | *X_core = 1* is a spherical core. |
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19 | |
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20 | For a fixed shell thickness *XpolarShell = 1*, to scale the shell thickness |
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21 | pro-rata with the radius set or constrain *XpolarShell = X_core*. |
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22 | |
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23 | When including an $S(q)$, the radius in $S(q)$ is calculated to be that of |
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24 | a sphere with the same 2nd virial coefficient of the outer surface of the |
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25 | ellipsoid. This may have some undesirable effects if the aspect ratio of the |
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26 | ellipsoid is large (ie, if $X << 1$ or $X >> 1$ ), when the $S(q)$ |
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27 | - which assumes spheres - will not in any case be valid. Generating a |
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28 | custom product model will enable separate effective volume fraction and |
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29 | effective radius in the $S(q)$. |
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30 | |
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31 | If SAS data are in absolute units, and the SLDs are correct, then scale should |
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32 | be the total volume fraction of the "outer particle". When $S(q)$ is introduced |
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33 | this moves to the $S(q)$ volume fraction, and scale should then be 1.0, or |
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34 | contain some other units conversion factor (for example, if you have SAXS data). |
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35 | |
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36 | The calculation of intensity follows that for the solid ellipsoid, but |
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37 | with separate terms for the core-shell and shell-solvent boundaries. |
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38 | |
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39 | .. math:: |
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40 | |
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41 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background} |
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42 | |
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43 | where |
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44 | |
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45 | .. math:: |
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46 | :nowrap: |
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47 | |
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48 | \begin{align*} |
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49 | F(q,\alpha) = &f(q,radius\_equat\_core,radius\_equat\_core.x\_core,\alpha) \\ |
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50 | &+ f(q,radius\_equat\_core + thick\_shell, |
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51 | radius\_equat\_core.x\_core + thick\_shell.x\_polar\_shell,\alpha) |
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52 | \end{align*} |
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53 | |
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54 | where |
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55 | |
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56 | .. math:: |
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57 | |
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58 | f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] |
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59 | - \cos[qr(R_p,R_e,\alpha)])} |
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60 | {[qr(R_p,R_e,\alpha)]^3} |
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61 | |
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62 | and |
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63 | |
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64 | .. math:: |
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65 | |
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66 | r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha |
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67 | + R_p^2 \cos^2 \alpha \right]^{1/2} |
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68 | |
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69 | |
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70 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
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71 | $V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the |
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72 | polar radius along the rotational axis of the ellipsoid, $R_e$ is the |
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73 | equatorial radius perpendicular to the rotational axis of the ellipsoid |
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74 | and $\Delta \rho$ (contrast) is the scattering length density difference, |
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75 | either $(sld\_core - sld\_shell)$ or $(sld\_shell - sld\_solvent)$. |
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76 | |
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77 | For randomly oriented particles: |
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78 | |
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79 | .. math:: |
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80 | |
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81 | F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha} |
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82 | |
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83 | For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters |
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84 | will appear when fitting 2D data, see the :ref:`elliptical-cylinder` model |
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85 | for further information. |
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86 | |
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87 | References |
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88 | ---------- |
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89 | see for example: |
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90 | Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461. |
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91 | Berr, S. J. Phys. Chem., 1987, 91, 4760. |
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92 | |
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93 | Authorship and Verification |
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94 | ---------------------------- |
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95 | |
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96 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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97 | * **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015 |
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98 | * **Last Reviewed by:** Richard Heenan **Date:** October 6, 2016 |
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99 | """ |
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100 | |
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101 | import numpy as np |
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102 | from numpy import inf, sin, cos, pi |
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103 | |
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104 | name = "core_shell_ellipsoid" |
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105 | title = "Form factor for an spheroid ellipsoid particle with a core shell structure." |
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106 | description = """ |
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107 | [core_shell_ellipsoid] Calculates the form factor for an spheroid |
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108 | ellipsoid particle with a core_shell structure. |
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109 | The form factor is averaged over all possible |
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110 | orientations of the ellipsoid such that P(q) |
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111 | = scale*<f^2>/Vol + bkg, where f is the |
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112 | single particle scattering amplitude. |
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113 | [Parameters]: |
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114 | radius_equat_core = equatorial radius of core, |
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115 | x_core = ratio of core polar/equatorial radii, |
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116 | thick_shell = equatorial radius of outer surface, |
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117 | x_polar_shell = ratio of polar shell thickness to equatorial shell thickness, |
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118 | sld_core = SLD_core |
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119 | sld_shell = SLD_shell |
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120 | sld_solvent = SLD_solvent |
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121 | background = Incoherent bkg |
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122 | scale =scale |
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123 | Note:It is the users' responsibility to ensure |
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124 | that shell radii are larger than core radii. |
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125 | oblate: polar radius < equatorial radius |
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126 | prolate : polar radius > equatorial radius - this new model will make this easier |
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127 | and polydispersity integrals more logical (as previously the shell could disappear). |
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128 | """ |
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129 | category = "shape:ellipsoid" |
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130 | |
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131 | # pylint: disable=bad-whitespace, line-too-long |
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132 | # ["name", "units", default, [lower, upper], "type", "description"], |
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133 | parameters = [ |
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134 | ["radius_equat_core","Ang", 20, [0, inf], "volume", "Equatorial radius of core"], |
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135 | ["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"], |
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136 | ["thick_shell", "Ang", 30, [0, inf], "volume", "thickness of shell at equator"], |
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137 | ["x_polar_shell", "", 1, [0, inf], "volume", "ratio of thickness of shell at pole to that at equator"], |
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138 | ["sld_core", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Core scattering length density"], |
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139 | ["sld_shell", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Shell scattering length density"], |
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140 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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141 | ["theta", "degrees", 0, [-360, 360], "orientation", "elipsoid axis to beam angle"], |
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142 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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143 | ] |
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144 | # pylint: enable=bad-whitespace, line-too-long |
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145 | |
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146 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"] |
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147 | have_Fq = True |
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148 | |
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149 | def ER(radius_equat_core, x_core, thick_shell, x_polar_shell): |
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150 | """ |
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151 | Returns the effective radius used in the S*P calculation |
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152 | """ |
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153 | from .ellipsoid import ER as ellipsoid_ER |
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154 | polar_outer = radius_equat_core*x_core + thick_shell*x_polar_shell |
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155 | equat_outer = radius_equat_core + thick_shell |
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156 | return ellipsoid_ER(polar_outer, equat_outer) |
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157 | |
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158 | def random(): |
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159 | volume = 10**np.random.uniform(5, 12) |
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160 | outer_polar = 10**np.random.uniform(1.3, 4) |
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161 | outer_equatorial = np.sqrt(volume/outer_polar) # ignore 4/3 pi |
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162 | # Use a distribution with a preference for thin shell or thin core |
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163 | # Avoid core,shell radii < 1 |
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164 | thickness_polar = np.random.beta(0.5, 0.5)*(outer_polar-2) + 1 |
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165 | thickness_equatorial = np.random.beta(0.5, 0.5)*(outer_equatorial-2) + 1 |
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166 | radius_polar = outer_polar - thickness_polar |
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167 | radius_equatorial = outer_equatorial - thickness_equatorial |
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168 | x_core = radius_polar/radius_equatorial |
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169 | x_polar_shell = thickness_polar/thickness_equatorial |
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170 | pars = dict( |
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171 | #background=0, sld=0, sld_solvent=1, |
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172 | radius_equat_core=radius_equatorial, |
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173 | x_core=x_core, |
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174 | thick_shell=thickness_equatorial, |
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175 | x_polar_shell=x_polar_shell, |
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176 | ) |
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177 | return pars |
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178 | |
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179 | q = 0.1 |
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180 | # tests had in old coords theta=0, phi=0; new coords theta=90, phi=0 |
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181 | qx = q*cos(pi/6.0) |
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182 | qy = q*sin(pi/6.0) |
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183 | # 11Jan2017 RKH sorted tests after redefinition of angles |
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184 | tests = [ |
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185 | # Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py |
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186 | [{'radius_equat_core': 200.0, |
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187 | 'x_core': 0.1, |
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188 | 'thick_shell': 50.0, |
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189 | 'x_polar_shell': 0.2, |
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190 | 'sld_core': 2.0, |
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191 | 'sld_shell': 1.0, |
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192 | 'sld_solvent': 6.3, |
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193 | 'background': 0.001, |
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194 | 'scale': 1.0, |
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195 | }, 1.0, 0.00189402], |
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196 | |
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197 | # Additional tests with larger range of parameters |
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198 | [{'background': 0.01}, 0.1, 11.6915], |
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199 | |
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200 | [{'radius_equat_core': 20.0, |
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201 | 'x_core': 200.0, |
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202 | 'thick_shell': 54.0, |
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203 | 'x_polar_shell': 3.0, |
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204 | 'sld_core': 20.0, |
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205 | 'sld_shell': 10.0, |
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206 | 'sld_solvent': 6.0, |
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207 | 'background': 0.0, |
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208 | 'scale': 1.0, |
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209 | }, 0.01, 8688.53], |
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210 | |
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211 | # 2D tests |
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212 | [{'background': 0.001, |
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213 | 'theta': 90.0, |
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214 | 'phi': 0.0, |
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215 | }, (0.4, 0.5), 0.00690673], |
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216 | |
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217 | [{'radius_equat_core': 20.0, |
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218 | 'x_core': 200.0, |
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219 | 'thick_shell': 54.0, |
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220 | 'x_polar_shell': 3.0, |
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221 | 'sld_core': 20.0, |
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222 | 'sld_shell': 10.0, |
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223 | 'sld_solvent': 6.0, |
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224 | 'background': 0.01, |
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225 | 'scale': 0.01, |
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226 | 'theta': 90.0, |
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227 | 'phi': 0.0, |
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228 | }, (qx, qy), 0.01000025], |
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229 | ] |
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