[deb7ee0] | 1 | # rectangular_prism 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 | |
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[117090a] | 5 | This model provides the form factor, $P(q)$, for a hollow rectangular |
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| 6 | parallelepiped with a wall of thickness $\Delta$. |
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[deb7ee0] | 7 | It computes only the 1D scattering, not the 2D. |
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| 8 | |
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| 9 | Definition |
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| 10 | ---------- |
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| 11 | |
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| 12 | The 1D scattering intensity for this model is calculated by forming |
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| 13 | the difference of the amplitudes of two massive parallelepipeds |
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| 14 | differing in their outermost dimensions in each direction by the |
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[ab2aea8] | 15 | same length increment $2\Delta$ (Nayuk, 2012). |
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[deb7ee0] | 16 | |
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| 17 | As in the case of the massive parallelepiped model (:ref:`rectangular-prism`), |
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| 18 | the scattering amplitude is computed for a particular orientation of the |
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| 19 | parallelepiped with respect to the scattering vector and then averaged over all |
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| 20 | possible orientations, giving |
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| 21 | |
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| 22 | .. math:: |
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| 23 | P(q) = \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, |
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| 24 | \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi |
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| 25 | |
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[117090a] | 26 | where $\theta$ is the angle between the $z$ axis and the longest axis |
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| 27 | of the parallelepiped, $\phi$ is the angle between the scattering vector |
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| 28 | (lying in the $xy$ plane) and the $y$ axis, and |
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[deb7ee0] | 29 | |
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| 30 | .. math:: |
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[5111921] | 31 | :nowrap: |
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| 32 | |
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[30b60d2] | 33 | \begin{align*} |
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[deb7ee0] | 34 | A_{P\Delta}(q) & = A B C |
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| 35 | \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)} |
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| 36 | {\left( q \frac{C}{2} \cos\theta \right)} \right] |
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| 37 | \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)} |
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| 38 | {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right] |
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| 39 | \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)} |
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| 40 | {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\ |
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| 41 | & - 8 |
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| 42 | \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right) |
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| 43 | \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]} |
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| 44 | {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right] |
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| 45 | \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]} |
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| 46 | {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right] |
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| 47 | \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]} |
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| 48 | {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right] |
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[30b60d2] | 49 | \end{align*} |
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[deb7ee0] | 50 | |
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[117090a] | 51 | where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling |
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| 52 | $A \le B \le C$, and the volume $V$ of the parallelepiped is |
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[deb7ee0] | 53 | |
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| 54 | .. math:: |
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| 55 | V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta) |
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| 56 | |
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| 57 | The 1D scattering intensity is then calculated as |
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| 58 | |
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| 59 | .. math:: |
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[ab2aea8] | 60 | I(q) = \text{scale} \times V \times (\rho_\text{p} - |
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| 61 | \rho_\text{solvent})^2 \times P(q) + \text{background} |
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[deb7ee0] | 62 | |
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[ab2aea8] | 63 | where $\rho_\text{p}$ is the scattering length of the parallelepiped, |
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| 64 | $\rho_\text{solvent}$ is the scattering length of the solvent, |
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[deb7ee0] | 65 | and (if the data are in absolute units) *scale* represents the volume fraction |
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| 66 | (which is unitless). |
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| 67 | |
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| 68 | **The 2D scattering intensity is not computed by this model.** |
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| 69 | |
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| 70 | |
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| 71 | Validation |
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| 72 | ---------- |
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| 73 | |
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| 74 | Validation of the code was conducted by qualitatively comparing the output |
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| 75 | of the 1D model to the curves shown in (Nayuk, 2012). |
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| 76 | |
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[aa2edb2] | 77 | |
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| 78 | References |
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| 79 | ---------- |
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[deb7ee0] | 80 | |
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| 81 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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| 82 | |
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| 83 | """ |
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| 84 | |
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| 85 | from numpy import pi, inf, sqrt |
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| 86 | |
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| 87 | name = "hollow_rectangular_prism" |
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| 88 | title = "Hollow rectangular parallelepiped with uniform scattering length density." |
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| 89 | description = """ |
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[3d8283b] | 90 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background |
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[deb7ee0] | 91 | P(q,theta,phi) = (2/pi/V^2) * double integral from 0 to pi/2 of ... |
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| 92 | (AP1-AP2)^2(q)*sin(theta)*dtheta*dphi |
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| 93 | AP1 = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2) |
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| 94 | AP2 = S(q*C'*cos(theta)) * S(q*A'*sin(theta)*sin(phi)) * S(q*B'*sin(theta)*cos(phi)) |
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| 95 | C' = (C/2-thickness) |
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| 96 | B' = (B/2-thickness) |
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| 97 | A' = (A/2-thickness) |
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| 98 | S(x) = sin(x)/x |
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| 99 | """ |
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| 100 | category = "shape:parallelepiped" |
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| 101 | |
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| 102 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 103 | parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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[deb7ee0] | 104 | "Parallelepiped scattering length density"], |
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[42356c8] | 105 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[deb7ee0] | 106 | "Solvent scattering length density"], |
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[a807206] | 107 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[deb7ee0] | 108 | "Shorter side of the parallelepiped"], |
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| 109 | ["b2a_ratio", "Ang", 1, [0, inf], "volume", |
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| 110 | "Ratio sides b/a"], |
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| 111 | ["c2a_ratio", "Ang", 1, [0, inf], "volume", |
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| 112 | "Ratio sides c/a"], |
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| 113 | ["thickness", "Ang", 1, [0, inf], "volume", |
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| 114 | "Thickness of parallelepiped"], |
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| 115 | ] |
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| 116 | |
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[40a87fa] | 117 | source = ["lib/gauss76.c", "hollow_rectangular_prism.c"] |
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[deb7ee0] | 118 | |
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[a807206] | 119 | def ER(length_a, b2a_ratio, c2a_ratio, thickness): |
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[deb7ee0] | 120 | """ |
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[40a87fa] | 121 | Return equivalent radius (ER) |
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| 122 | thickness parameter not used |
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[deb7ee0] | 123 | """ |
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[a807206] | 124 | b_side = length_a * b2a_ratio |
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| 125 | c_side = length_a * c2a_ratio |
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[deb7ee0] | 126 | |
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| 127 | # surface average radius (rough approximation) |
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[a807206] | 128 | surf_rad = sqrt(length_a * b_side / pi) |
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[deb7ee0] | 129 | |
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| 130 | ddd = 0.75 * surf_rad * (2 * surf_rad * c_side + (c_side + surf_rad) * (c_side + pi * surf_rad)) |
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| 131 | return 0.5 * (ddd) ** (1. / 3.) |
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| 132 | |
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[a807206] | 133 | def VR(length_a, b2a_ratio, c2a_ratio, thickness): |
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[deb7ee0] | 134 | """ |
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[40a87fa] | 135 | Return shell volume and total volume |
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[deb7ee0] | 136 | """ |
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[a807206] | 137 | b_side = length_a * b2a_ratio |
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| 138 | c_side = length_a * c2a_ratio |
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| 139 | a_core = length_a - 2.0*thickness |
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[deb7ee0] | 140 | b_core = b_side - 2.0*thickness |
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| 141 | c_core = c_side - 2.0*thickness |
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| 142 | vol_core = a_core * b_core * c_core |
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[a807206] | 143 | vol_total = length_a * b_side * c_side |
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[deb7ee0] | 144 | vol_shell = vol_total - vol_core |
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| 145 | return vol_total, vol_shell |
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| 146 | |
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| 147 | |
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[31df0c9] | 148 | def random(): |
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| 149 | import numpy as np |
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| 150 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
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[8f04da4] | 151 | # Thickness is limited to 1/2 the smallest dimension |
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| 152 | # Use a distribution with a preference for thin shell or thin core |
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| 153 | # Avoid core,shell radii < 1 |
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| 154 | min_dim = 0.5*min(a, b, c) |
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| 155 | thickness = np.random.beta(0.5, 0.5)*(min_dim-2) + 1 |
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| 156 | #print(a, b, c, thickness, thickness/min_dim) |
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[31df0c9] | 157 | pars = dict( |
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| 158 | length_a=a, |
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| 159 | b2a_ratio=b/a, |
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| 160 | c2a_ratio=c/a, |
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| 161 | thickness=thickness, |
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| 162 | ) |
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| 163 | return pars |
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| 164 | |
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| 165 | |
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[deb7ee0] | 166 | # parameters for demo |
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| 167 | demo = dict(scale=1, background=0, |
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[ab2aea8] | 168 | sld=6.3, sld_solvent=1.0, |
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[a807206] | 169 | length_a=35, b2a_ratio=1, c2a_ratio=1, thickness=1, |
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| 170 | length_a_pd=0.1, length_a_pd_n=10, |
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[deb7ee0] | 171 | b2a_ratio_pd=0.1, b2a_ratio_pd_n=1, |
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| 172 | c2a_ratio_pd=0.1, c2a_ratio_pd_n=1) |
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| 173 | |
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[6dd90c1] | 174 | tests = [[{}, 0.2, 0.76687283098], |
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| 175 | [{}, [0.2], [0.76687283098]], |
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[deb7ee0] | 176 | ] |
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| 177 | |
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