[5d4777d] | 1 | r""" |
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[b0c4271] | 2 | Definitions |
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| 3 | ----------- |
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| 4 | |
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[5d4777d] | 5 | Calculates the scattering from a cylinder with spherical section end-caps. |
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[eb69cce] | 6 | Like :ref:`barbell`, this is a sphereocylinder with end caps that have a |
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| 7 | radius larger than that of the cylinder, but with the center of the end cap |
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[e65a3e7] | 8 | radius lying within the cylinder. This model simply becomes a convex |
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[eb69cce] | 9 | lens when the length of the cylinder $L=0$. See the diagram for the details |
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| 10 | of the geometry and restrictions on parameter values. |
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[5d4777d] | 11 | |
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[eb69cce] | 12 | .. figure:: img/capped_cylinder_geometry.jpg |
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[5d4777d] | 13 | |
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[eb69cce] | 14 | Capped cylinder geometry, where $r$ is *radius*, $R$ is *bell_radius* and |
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| 15 | $L$ is *length*. Since the end cap radius $R \geq r$ and by definition |
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| 16 | for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
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| 17 | $h = - \sqrt{R^2 - r^2}$ |
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[5d4777d] | 18 | |
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[eb69cce] | 19 | The scattered intensity $I(q)$ is calculated as |
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[5d4777d] | 20 | |
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| 21 | .. math:: |
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| 22 | |
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[fcb33e4] | 23 | I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right> |
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[5d4777d] | 24 | |
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[fcb33e4] | 25 | where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as |
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[5d4777d] | 26 | |
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| 27 | .. math:: |
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| 28 | |
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[eb69cce] | 29 | A(q) =&\ \pi r^2L |
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[fcb33e4] | 30 | \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} |
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| 31 | {\tfrac12 qL\cos\alpha} |
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| 32 | \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ |
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[19dcb933] | 33 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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[fcb33e4] | 34 | \cos\left[ q\cos\alpha |
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[19dcb933] | 35 | \left(Rt + h + {\tfrac12} L\right)\right] |
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| 36 | \times (1-t^2) |
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[fcb33e4] | 37 | \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} |
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| 38 | {qR\sin\alpha \left(1-t^2\right)^{1/2}} |
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[5d4777d] | 39 | |
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[eb69cce] | 40 | The $\left<\ldots\right>$ brackets denote an average of the structure over |
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| 41 | all orientations. $\left< A^2(q)\right>$ is then the form factor, $P(q)$. |
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[5d4777d] | 42 | The scale factor is equivalent to the volume fraction of cylinders, each of |
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[eb69cce] | 43 | volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length |
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| 44 | densities of the cylinder and the surrounding solvent. |
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[5d4777d] | 45 | |
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| 46 | The volume of the capped cylinder is (with $h$ as a positive value here) |
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| 47 | |
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| 48 | .. math:: |
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| 49 | |
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[19dcb933] | 50 | V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h) |
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[5d4777d] | 51 | |
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| 52 | |
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[eb69cce] | 53 | and its radius of gyration is |
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[5d4777d] | 54 | |
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[19dcb933] | 55 | .. math:: |
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| 56 | |
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| 57 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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| 58 | + R^4\left(6h+\tfrac32 L\right) |
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| 59 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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| 60 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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| 61 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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| 62 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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| 63 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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| 64 | |
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[5d4777d] | 65 | |
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[19dcb933] | 66 | .. note:: |
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[5d4777d] | 67 | |
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[eb69cce] | 68 | The requirement that $R \geq r$ is not enforced in the model! |
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[19dcb933] | 69 | It is up to you to restrict this during analysis. |
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[5d4777d] | 70 | |
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| 71 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
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| 72 | |
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[9802ab3] | 73 | .. figure:: img/cylinder_angle_definition.png |
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[5d4777d] | 74 | |
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| 75 | Definition of the angles for oriented 2D cylinders. |
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| 76 | |
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| 77 | |
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[eb69cce] | 78 | References |
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| 79 | ---------- |
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[5d4777d] | 80 | |
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[b0c4271] | 81 | .. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 |
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[31df0c9] | 82 | .. [#] H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda |
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[b0c4271] | 83 | and errata) |
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| 84 | |
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| 85 | Authorship and Verification |
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| 86 | ---------------------------- |
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[5d4777d] | 87 | |
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[b0c4271] | 88 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 89 | * **Last Modified by:** Paul Butler **Date:** September 30, 2016 |
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[fcb33e4] | 90 | * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017 |
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[5d4777d] | 91 | """ |
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[2d81cfe] | 92 | |
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| 93 | import numpy as np |
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[0b56f38] | 94 | from numpy import inf, sin, cos, pi |
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[5d4777d] | 95 | |
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| 96 | name = "capped_cylinder" |
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| 97 | title = "Right circular cylinder with spherical end caps and uniform SLD" |
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| 98 | description = """That is, a sphereocylinder |
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[485aee2] | 99 | with end caps that have a radius larger than |
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| 100 | that of the cylinder and the center of the |
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| 101 | end cap radius lies within the cylinder. |
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| 102 | Note: As the length of cylinder -->0, |
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[e65a3e7] | 103 | it becomes a Convex Lens. |
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[2222134] | 104 | It must be that radius <(=) radius_cap. |
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[485aee2] | 105 | [Parameters]; |
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| 106 | scale: volume fraction of spheres, |
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| 107 | background:incoherent background, |
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| 108 | radius: radius of the cylinder, |
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| 109 | length: length of the cylinder, |
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[2222134] | 110 | radius_cap: radius of the semi-spherical cap, |
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[485aee2] | 111 | sld: SLD of the capped cylinder, |
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[e65a3e7] | 112 | sld_solvent: SLD of the solvent. |
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[5d4777d] | 113 | """ |
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[a5d0d00] | 114 | category = "shape:cylinder" |
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[dcdf29d] | 115 | # pylint: disable=bad-whitespace, line-too-long |
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[485aee2] | 116 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[42356c8] | 117 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder scattering length density"], |
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| 118 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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[dcdf29d] | 119 | ["radius", "Ang", 20, [0, inf], "volume", "Cylinder radius"], |
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| 120 | |
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[485aee2] | 121 | # TODO: use an expression for cap radius with fixed bounds. |
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| 122 | # The current form requires cap radius R bigger than cylinder radius r. |
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| 123 | # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between |
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| 124 | # cylinder and cap in [0,90]. The problem is similar for the barbell |
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| 125 | # model. Propose r/R in [0,1] in both cases, with the model specifying |
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| 126 | # cylinder radius in the capped cylinder model and sphere radius in the |
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| 127 | # barbell model. This leads to the natural value of zero for no cap |
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| 128 | # in the capped cylinder, and zero for no bar in the barbell model. In |
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| 129 | # both models, one would be a pill. |
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[2222134] | 130 | ["radius_cap", "Ang", 20, [0, inf], "volume", "Cap radius"], |
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[dcdf29d] | 131 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], |
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[9b79f29] | 132 | ["theta", "degrees", 60, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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| 133 | ["phi", "degrees", 60, [-360, 360], "orientation", "rotation about beam"], |
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[485aee2] | 134 | ] |
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[dcdf29d] | 135 | # pylint: enable=bad-whitespace, line-too-long |
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[485aee2] | 136 | |
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[26141cb] | 137 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "capped_cylinder.c"] |
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[485aee2] | 138 | |
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[31df0c9] | 139 | def random(): |
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| 140 | # TODO: increase volume range once problem with bell radius is fixed |
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| 141 | # The issue is that bell radii of more than about 200 fail at high q |
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[2d81cfe] | 142 | volume = 10**np.random.uniform(7, 9) |
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| 143 | bar_volume = 10**np.random.uniform(-4, -1)*volume |
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| 144 | bell_volume = volume - bar_volume |
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[31df0c9] | 145 | bell_radius = (bell_volume/6)**0.3333 # approximate |
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| 146 | min_bar = bar_volume/np.pi/bell_radius**2 |
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| 147 | bar_length = 10**np.random.uniform(0, 3)*min_bar |
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| 148 | bar_radius = np.sqrt(bar_volume/bar_length/np.pi) |
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| 149 | if bar_radius > bell_radius: |
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| 150 | bell_radius, bar_radius = bar_radius, bell_radius |
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| 151 | pars = dict( |
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| 152 | #background=0, |
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| 153 | radius_cap=bell_radius, |
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| 154 | radius=bar_radius, |
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| 155 | length=bar_length, |
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| 156 | ) |
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| 157 | return pars |
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| 158 | |
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| 159 | |
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[485aee2] | 160 | demo = dict(scale=1, background=0, |
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[e65a3e7] | 161 | sld=6, sld_solvent=1, |
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[2222134] | 162 | radius=260, radius_cap=290, length=290, |
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[485aee2] | 163 | theta=30, phi=15, |
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| 164 | radius_pd=.2, radius_pd_n=1, |
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[2222134] | 165 | radius_cap_pd=.2, radius_cap_pd_n=1, |
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[485aee2] | 166 | length_pd=.2, length_pd_n=10, |
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| 167 | theta_pd=15, theta_pd_n=45, |
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| 168 | phi_pd=15, phi_pd_n=1) |
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[0b56f38] | 169 | q = 0.1 |
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| 170 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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| 171 | qx = q*cos(pi/6.0) |
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| 172 | qy = q*sin(pi/6.0) |
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[2d81cfe] | 173 | tests = [ |
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| 174 | [{}, 0.075, 26.0698570695], |
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| 175 | [{'theta':80., 'phi':10.}, (qx, qy), 0.561811990502], |
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| 176 | ] |
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| 177 | del qx, qy # not necessary to delete, but cleaner |
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