1 | .. _ellipsoid: |
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2 | |
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3 | Ellipsoid |
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4 | ======================================================= |
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5 | |
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6 | Ellipsoid of revolution with uniform scattering length density. |
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7 | |
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8 | =========== =================================== ============ ============= |
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9 | Parameter Description Units Default value |
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10 | =========== =================================== ============ ============= |
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11 | scale Source intensity None 1 |
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12 | background Source background |cm^-1| 0 |
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13 | sld Ellipsoid scattering length density |1e-6Ang^-2| 4 |
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14 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 1 |
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15 | rpolar Polar radius |Ang| 20 |
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16 | requatorial Equatorial radius |Ang| 400 |
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17 | theta In plane angle degree 60 |
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18 | phi Out of plane angle degree 60 |
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19 | =========== =================================== ============ ============= |
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20 | |
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21 | The returned value is scaled to units of |cm^-1|. |
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22 | |
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23 | |
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24 | The form factor is normalized by the particle volume. |
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25 | |
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26 | Definition |
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27 | ---------- |
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28 | |
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29 | The output of the 2D scattering intensity function for oriented ellipsoids |
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30 | is given by (Feigin, 1987) |
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31 | |
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32 | .. math:: |
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33 | |
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34 | P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background} |
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35 | |
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36 | where |
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37 | |
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38 | .. math:: |
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39 | |
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40 | F(Q) = {3 (\Delta rho)) V (\sin[Qr(R_p,R_e,\alpha)] |
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41 | - \cos[Qr(R_p,R_e,\alpha)]) |
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42 | \over [Qr(R_p,R_e,\alpha)]^3 } |
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43 | |
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44 | and |
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45 | |
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46 | .. math:: |
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47 | |
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48 | r(R_p,R_e,\alpha) = \left[ R_e^2 \sin^2 \alpha |
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49 | + R_p^2 \cos^2 \alpha \right]^{1/2} |
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50 | |
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51 | |
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52 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
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53 | $V$ is the volume of the ellipsoid, $R_p$ is the polar radius along the |
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54 | rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular |
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55 | to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the |
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56 | scattering length density difference between the scatterer and the solvent. |
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57 | |
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58 | To provide easy access to the orientation of the ellipsoid, we define |
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59 | the rotation axis of the ellipsoid using two angles $\theta$ and $\phi$. |
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60 | These angles are defined in the |
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61 | :ref:`cylinder orientation figure <cylinder-orientation>`. |
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62 | For the ellipsoid, $\theta$ is the angle between the rotational axis |
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63 | and the $z$-axis. |
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64 | |
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65 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based |
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66 | on the $R_p$ and $R_e$ values, and used as the effective radius for |
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67 | $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
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68 | |
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69 | .. _ellipsoid-1d: |
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70 | |
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71 | .. figure:: img/ellipsoid_1d.JPG |
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72 | |
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73 | The output of the 1D scattering intensity function for randomly oriented |
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74 | ellipsoids given by the equation above. |
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75 | |
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76 | |
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77 | The $\theta$ and $\phi$ parameters are not used for the 1D output. Our |
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78 | implementation of the scattering kernel and the 1D scattering intensity |
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79 | use the c-library from NIST. |
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80 | |
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81 | .. _ellipsoid-geometry: |
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82 | |
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83 | .. figure:: img/ellipsoid_geometry.JPG |
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84 | |
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85 | The angles for oriented ellipsoid. |
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86 | |
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87 | Validation |
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88 | ---------- |
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89 | |
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90 | Validation of our code was done by comparing the output of the 1D model |
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91 | to the output of the software provided by the NIST (Kline, 2006). |
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92 | :num:`Figure ellipsoid-comparison-1d` below shows a comparison of |
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93 | the 1D output of our model and the output of the NIST software. |
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94 | |
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95 | .. _ellipsoid-comparison-1d: |
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96 | |
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97 | .. figure:: img/ellipsoid_comparison_1d.jpg |
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98 | |
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99 | Comparison of the SasView scattering intensity for an ellipsoid |
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100 | with the output of the NIST SANS analysis software. The parameters |
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101 | were set to: *scale* = 1.0, *rpolar* = 20 |Ang|, |
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102 | *requatorial* =400 |Ang|, *contrast* = 3e-6 |Ang^-2|, |
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103 | and *background* = 0.01 |cm^-1|. |
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104 | |
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105 | Averaging over a distribution of orientation is done by evaluating the |
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106 | equation above. Since we have no other software to compare the |
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107 | implementation of the intensity for fully oriented ellipsoids, we can |
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108 | compare the result of averaging our 2D output using a uniform distribution |
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109 | $p(\theta,\phi) = 1.0$. :num:`Figure #ellipsoid-comparison-2d` |
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110 | shows the result of such a cross-check. |
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111 | |
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112 | .. _ellipsoid-comparison-2d: |
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113 | |
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114 | .. figure:: img/ellipsoid_comparison_2d.jpg |
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115 | |
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116 | Comparison of the intensity for uniformly distributed ellipsoids |
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117 | calculated from our 2D model and the intensity from the NIST SANS |
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118 | analysis software. The parameters used were: *scale* = 1.0, |
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119 | *rpolar* = 20 |Ang|, *requatorial* = 400 |Ang|, |
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120 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
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121 | |
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122 | The discrepancy above *q* = 0.3 |cm^-1| is due to the way the form factors |
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123 | are calculated in the c-library provided by NIST. A numerical integration |
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124 | has to be performed to obtain $P(Q)$ for randomly oriented particles. |
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125 | The NIST software performs that integration with a 76-point Gaussian |
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126 | quadrature rule, which will become imprecise at high $Q$ where the amplitude |
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127 | varies quickly as a function of $Q$. The SasView result shown has been |
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128 | obtained by summing over 501 equidistant points. Our result was found |
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129 | to be stable over the range of $Q$ shown for a number of points higher |
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130 | than 500. |
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131 | |
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132 | REFERENCE |
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133 | |
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134 | L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
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135 | New York, 1987. |
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136 | |
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