[5d4777d] | 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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[19dcb933] | 14 | P(Q,\alpha) = {\text{scale} \over V} F^2(Q) + \text{background} |
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[5d4777d] | 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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[19dcb933] | 20 | F(Q) = {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 | \over [Qr(R_p,R_e,\alpha)]^3 } |
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[5d4777d] | 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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[19dcb933] | 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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[5d4777d] | 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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[19dcb933] | 40 | These angles are defined in the |
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| 41 | :ref:`cylinder orientation figure <cylinder-orientation>`. |
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[5d4777d] | 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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[19dcb933] | 47 | $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
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[5d4777d] | 48 | |
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[19dcb933] | 49 | .. _ellipsoid-1d: |
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| 50 | |
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| 51 | .. figure:: img/ellipsoid_1d.JPG |
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[5d4777d] | 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. Our |
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| 58 | implementation of the scattering kernel and the 1D scattering intensity |
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| 59 | use the c-library from NIST. |
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| 60 | |
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[19dcb933] | 61 | .. _ellipsoid-geometry: |
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| 62 | |
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| 63 | .. figure:: img/ellipsoid_geometry.JPG |
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[5d4777d] | 64 | |
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| 65 | The angles for oriented ellipsoid. |
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| 66 | |
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| 67 | Validation |
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| 68 | ---------- |
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| 69 | |
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| 70 | Validation of our code was done by comparing the output of the 1D model |
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| 71 | to the output of the software provided by the NIST (Kline, 2006). |
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[19dcb933] | 72 | :num:`Figure ellipsoid-comparison-1d` below shows a comparison of |
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[5d4777d] | 73 | the 1D output of our model and the output of the NIST software. |
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| 74 | |
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| 75 | .. _ellipsoid-comparison-1d: |
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| 76 | |
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[19dcb933] | 77 | .. figure:: img/ellipsoid_comparison_1d.jpg |
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[5d4777d] | 78 | |
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| 79 | Comparison of the SasView scattering intensity for an ellipsoid |
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| 80 | with the output of the NIST SANS analysis software. The parameters |
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[19dcb933] | 81 | were set to: *scale* = 1.0, *rpolar* = 20 |Ang|, |
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| 82 | *requatorial* =400 |Ang|, *contrast* = 3e-6 |Ang^-2|, |
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| 83 | and *background* = 0.01 |cm^-1|. |
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[5d4777d] | 84 | |
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| 85 | Averaging over a distribution of orientation is done by evaluating the |
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| 86 | equation above. Since we have no other software to compare the |
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| 87 | implementation of the intensity for fully oriented ellipsoids, we can |
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| 88 | compare the result of averaging our 2D output using a uniform distribution |
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[19dcb933] | 89 | $p(\theta,\phi) = 1.0$. :num:`Figure #ellipsoid-comparison-2d` |
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[5d4777d] | 90 | shows the result of such a cross-check. |
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| 91 | |
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| 92 | .. _ellipsoid-comparison-2d: |
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| 93 | |
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[19dcb933] | 94 | .. figure:: img/ellipsoid_comparison_2d.jpg |
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[5d4777d] | 95 | |
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| 96 | Comparison of the intensity for uniformly distributed ellipsoids |
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| 97 | calculated from our 2D model and the intensity from the NIST SANS |
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[19dcb933] | 98 | analysis software. The parameters used were: *scale* = 1.0, |
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| 99 | *rpolar* = 20 |Ang|, *requatorial* = 400 |Ang|, |
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| 100 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
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[5d4777d] | 101 | |
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[19dcb933] | 102 | The discrepancy above *q* = 0.3 |cm^-1| is due to the way the form factors |
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[5d4777d] | 103 | are calculated in the c-library provided by NIST. A numerical integration |
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[19dcb933] | 104 | has to be performed to obtain $P(Q)$ for randomly oriented particles. |
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[5d4777d] | 105 | The NIST software performs that integration with a 76-point Gaussian |
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[19dcb933] | 106 | quadrature rule, which will become imprecise at high $Q$ where the amplitude |
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| 107 | varies quickly as a function of $Q$. The SasView result shown has been |
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[5d4777d] | 108 | obtained by summing over 501 equidistant points. Our result was found |
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[19dcb933] | 109 | to be stable over the range of $Q$ shown for a number of points higher |
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[5d4777d] | 110 | than 500. |
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| 111 | |
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| 112 | REFERENCE |
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| 113 | |
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| 114 | L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
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| 115 | New York, 1987. |
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| 116 | """ |
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| 117 | |
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| 118 | from numpy import pi, inf |
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| 119 | |
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| 120 | name = "ellipsoid" |
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| 121 | title = "Ellipsoid of revolution with uniform scattering length density." |
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| 122 | |
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| 123 | description = """\ |
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| 124 | P(q.alpha)= scale*f(q)^2 + background, where f(q)= 3*(sld |
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| 125 | - solvent_sld)*V*[sin(q*r(Rp,Re,alpha)) |
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| 126 | -q*r*cos(qr(Rp,Re,alpha))] |
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| 127 | /[qr(Rp,Re,alpha)]^3" |
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| 128 | |
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| 129 | r(Rp,Re,alpha)= [Re^(2)*(sin(alpha))^2 |
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| 130 | + Rp^(2)*(cos(alpha))^2]^(1/2) |
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| 131 | |
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| 132 | sld: SLD of the ellipsoid |
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| 133 | solvent_sld: SLD of the solvent |
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| 134 | V: volume of the ellipsoid |
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| 135 | Rp: polar radius of the ellipsoid |
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| 136 | Re: equatorial radius of the ellipsoid |
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| 137 | """ |
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| 138 | |
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| 139 | parameters = [ |
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| 140 | # [ "name", "units", default, [lower, upper], "type", |
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| 141 | # "description" ], |
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| 142 | [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", |
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| 143 | "Ellipsoid scattering length density" ], |
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| 144 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "", |
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| 145 | "Solvent scattering length density" ], |
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| 146 | [ "rpolar", "Ang", 20, [0, inf], "volume", |
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| 147 | "Polar radius" ], |
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| 148 | [ "requatorial", "Ang", 400, [0, inf], "volume", |
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| 149 | "Equatorial radius" ], |
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| 150 | [ "theta", "degrees", 60, [-inf, inf], "orientation", |
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| 151 | "In plane angle" ], |
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| 152 | [ "phi", "degrees", 60, [-inf, inf], "orientation", |
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| 153 | "Out of plane angle" ], |
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| 154 | ] |
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| 155 | |
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| 156 | source = [ "lib/J1.c", "lib/gauss76.c", "ellipsoid.c"] |
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| 157 | |
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| 158 | def ER(rpolar, requatorial): |
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| 159 | import numpy as np |
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| 160 | |
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| 161 | ee = np.empty_like(rpolar) |
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| 162 | idx = rpolar > requatorial |
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| 163 | ee[idx] = (rpolar[idx]**2 - requatorial[idx]**2)/rpolar[idx]**2 |
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| 164 | idx = rpolar < requatorial |
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| 165 | ee[idx] = (requatorial[idx]**2 - rpolar[idx]**2)/requatorial[idx]**2 |
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| 166 | idx = rpolar == requatorial |
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| 167 | ee[idx] = 2*rpolar[idx] |
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| 168 | valid = (rpolar*requatorial != 0) |
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| 169 | bd = 1.0-ee[valid] |
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| 170 | e1 = np.sqrt(ee[valid]) |
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| 171 | b1 = 1.0 + np.arcsin(e1)/(e1*np.sqrt(bd)) |
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| 172 | bL = (1.0+e1)/(1.0-e1) |
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| 173 | b2 = 1.0 + bd/2/e1*np.log(bL) |
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| 174 | delta = 0.75*b1*b2 |
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| 175 | |
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| 176 | ddd = np.zeros_like(rpolar) |
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| 177 | ddd[valid] = 2.0*(delta+1.0)*rpolar*requatorial**2 |
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| 178 | return 0.5*ddd**(1.0/3.0) |
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