[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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[3556ad7] | 4 | The form factor is normalized by the particle volume |
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[5d4777d] | 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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[cade620] | 14 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \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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[cade620] | 20 | F(q,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] |
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[eb69cce] | 21 | - \cos[qr(R_p,R_e,\alpha)])} |
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| 22 | {[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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[3556ad7] | 33 | $V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the polar radius along the |
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[5d4777d] | 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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[416f5c7] | 38 | For randomly oriented particles: |
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| 39 | |
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| 40 | .. math:: |
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| 41 | |
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| 42 | F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha} |
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| 43 | |
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| 44 | |
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[5d4777d] | 45 | To provide easy access to the orientation of the ellipsoid, we define |
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| 46 | the rotation axis of the ellipsoid using two angles $\theta$ and $\phi$. |
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[19dcb933] | 47 | These angles are defined in the |
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[0a7eec11] | 48 | :ref:`cylinder orientation figure <cylinder-angle-definition>`. |
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[5d4777d] | 49 | For the ellipsoid, $\theta$ is the angle between the rotational axis |
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[3556ad7] | 50 | and the $z$ -axis. |
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[5d4777d] | 51 | |
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| 52 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based |
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| 53 | on the $R_p$ and $R_e$ values, and used as the effective radius for |
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[eb69cce] | 54 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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[5d4777d] | 55 | |
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| 56 | |
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[eb69cce] | 57 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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[5d4777d] | 58 | |
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[19dcb933] | 59 | .. _ellipsoid-geometry: |
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| 60 | |
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[2f0c07d] | 61 | .. figure:: img/ellipsoid_angle_projection.jpg |
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[5d4777d] | 62 | |
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[3556ad7] | 63 | The angles for oriented ellipsoid, shown here as oblate, $a$ = $R_p$ and $b$ = $R_e$ |
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[5d4777d] | 64 | |
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| 65 | Validation |
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| 66 | ---------- |
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| 67 | |
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[aa2edb2] | 68 | Validation of the code was done by comparing the output of the 1D model |
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[5d4777d] | 69 | to the output of the software provided by the NIST (Kline, 2006). |
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| 70 | |
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[aa2edb2] | 71 | The implementation of the intensity for fully oriented ellipsoids was |
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| 72 | validated by averaging the 2D output using a uniform distribution |
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| 73 | $p(\theta,\phi) = 1.0$ and comparing with the output of the 1D calculation. |
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[5d4777d] | 74 | |
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| 75 | |
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| 76 | .. _ellipsoid-comparison-2d: |
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| 77 | |
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[19dcb933] | 78 | .. figure:: img/ellipsoid_comparison_2d.jpg |
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[5d4777d] | 79 | |
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| 80 | Comparison of the intensity for uniformly distributed ellipsoids |
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| 81 | calculated from our 2D model and the intensity from the NIST SANS |
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[19dcb933] | 82 | analysis software. The parameters used were: *scale* = 1.0, |
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[a807206] | 83 | *radius_polar* = 20 |Ang|, *radius_equatorial* = 400 |Ang|, |
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[19dcb933] | 84 | *contrast* = 3e-6 |Ang^-2|, and *background* = 0.0 |cm^-1|. |
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[5d4777d] | 85 | |
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[eb69cce] | 86 | The discrepancy above $q$ = 0.3 |cm^-1| is due to the way the form factors |
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[5d4777d] | 87 | are calculated in the c-library provided by NIST. A numerical integration |
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[eb69cce] | 88 | has to be performed to obtain $P(q)$ for randomly oriented particles. |
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[5d4777d] | 89 | The NIST software performs that integration with a 76-point Gaussian |
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[eb69cce] | 90 | quadrature rule, which will become imprecise at high $q$ where the amplitude |
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| 91 | varies quickly as a function of $q$. The SasView result shown has been |
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[5d4777d] | 92 | obtained by summing over 501 equidistant points. Our result was found |
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[eb69cce] | 93 | to be stable over the range of $q$ shown for a number of points higher |
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[5d4777d] | 94 | than 500. |
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| 95 | |
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[eb69cce] | 96 | References |
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| 97 | ---------- |
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[5d4777d] | 98 | |
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[431caae] | 99 | L A Feigin and D I Svergun. |
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| 100 | *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, |
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| 101 | Plenum Press, New York, 1987. |
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[5d4777d] | 102 | """ |
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| 103 | |
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[3c56da87] | 104 | from numpy import inf |
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[5d4777d] | 105 | |
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| 106 | name = "ellipsoid" |
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| 107 | title = "Ellipsoid of revolution with uniform scattering length density." |
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| 108 | |
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| 109 | description = """\ |
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| 110 | P(q.alpha)= scale*f(q)^2 + background, where f(q)= 3*(sld |
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[3556ad7] | 111 | - sld_solvent)*V*[sin(q*r(Rp,Re,alpha)) |
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[3e428ec] | 112 | -q*r*cos(qr(Rp,Re,alpha))] |
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| 113 | /[qr(Rp,Re,alpha)]^3" |
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[5d4777d] | 114 | |
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| 115 | r(Rp,Re,alpha)= [Re^(2)*(sin(alpha))^2 |
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[3e428ec] | 116 | + Rp^(2)*(cos(alpha))^2]^(1/2) |
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[5d4777d] | 117 | |
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[3e428ec] | 118 | sld: SLD of the ellipsoid |
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[3556ad7] | 119 | sld_solvent: SLD of the solvent |
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[3e428ec] | 120 | V: volume of the ellipsoid |
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| 121 | Rp: polar radius of the ellipsoid |
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| 122 | Re: equatorial radius of the ellipsoid |
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[5d4777d] | 123 | """ |
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[a5d0d00] | 124 | category = "shape:ellipsoid" |
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[5d4777d] | 125 | |
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[3e428ec] | 126 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 127 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[3e428ec] | 128 | "Ellipsoid scattering length density"], |
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[42356c8] | 129 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[3e428ec] | 130 | "Solvent scattering length density"], |
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[a807206] | 131 | ["radius_polar", "Ang", 20, [0, inf], "volume", |
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[3e428ec] | 132 | "Polar radius"], |
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[a807206] | 133 | ["radius_equatorial", "Ang", 400, [0, inf], "volume", |
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[3e428ec] | 134 | "Equatorial radius"], |
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| 135 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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| 136 | "In plane angle"], |
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| 137 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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| 138 | "Out of plane angle"], |
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| 139 | ] |
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| 140 | |
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[43b7eea] | 141 | source = ["lib/sph_j1c.c", "lib/gauss76.c", "ellipsoid.c"] |
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[5d4777d] | 142 | |
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[a807206] | 143 | def ER(radius_polar, radius_equatorial): |
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[5d4777d] | 144 | import numpy as np |
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| 145 | |
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[a807206] | 146 | ee = np.empty_like(radius_polar) |
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| 147 | idx = radius_polar > radius_equatorial |
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| 148 | ee[idx] = (radius_polar[idx] ** 2 - radius_equatorial[idx] ** 2) / radius_polar[idx] ** 2 |
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| 149 | idx = radius_polar < radius_equatorial |
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| 150 | ee[idx] = (radius_equatorial[idx] ** 2 - radius_polar[idx] ** 2) / radius_equatorial[idx] ** 2 |
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| 151 | idx = radius_polar == radius_equatorial |
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| 152 | ee[idx] = 2 * radius_polar[idx] |
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| 153 | valid = (radius_polar * radius_equatorial != 0) |
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[3e428ec] | 154 | bd = 1.0 - ee[valid] |
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[5d4777d] | 155 | e1 = np.sqrt(ee[valid]) |
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[3e428ec] | 156 | b1 = 1.0 + np.arcsin(e1) / (e1 * np.sqrt(bd)) |
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| 157 | bL = (1.0 + e1) / (1.0 - e1) |
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| 158 | b2 = 1.0 + bd / 2 / e1 * np.log(bL) |
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| 159 | delta = 0.75 * b1 * b2 |
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[5d4777d] | 160 | |
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[a807206] | 161 | ddd = np.zeros_like(radius_polar) |
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| 162 | ddd[valid] = 2.0 * (delta + 1.0) * radius_polar * radius_equatorial ** 2 |
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[3e428ec] | 163 | return 0.5 * ddd ** (1.0 / 3.0) |
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| 164 | |
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| 165 | |
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| 166 | demo = dict(scale=1, background=0, |
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[3556ad7] | 167 | sld=6, sld_solvent=1, |
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[a807206] | 168 | radius_polar=50, radius_equatorial=30, |
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[3e428ec] | 169 | theta=30, phi=15, |
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[a807206] | 170 | radius_polar_pd=.2, radius_polar_pd_n=15, |
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| 171 | radius_equatorial_pd=.2, radius_equatorial_pd_n=15, |
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[3e428ec] | 172 | theta_pd=15, theta_pd_n=45, |
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| 173 | phi_pd=15, phi_pd_n=1) |
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