[8007311] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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[adc753d] | 4 | |
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[40a87fa] | 5 | This model provides the form factor for a circular cylinder with a |
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| 6 | core-shell scattering length density profile. Thus this is a variation |
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| 7 | of a core-shell cylinder or disc where the shell on the walls and ends |
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| 8 | may be of different thicknesses and scattering length densities. The form |
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| 9 | factor is normalized by the particle volume. |
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[8007311] | 10 | |
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| 11 | |
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| 12 | .. figure:: img/core_shell_bicelle_geometry.png |
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| 13 | |
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[a0fee3b] | 14 | Schematic cross-section of bicelle. Note however that the model here |
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| 15 | calculates for rectangular, not curved, rims as shown below. |
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| 16 | |
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| 17 | .. figure:: img/core_shell_bicelle_parameters.png |
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| 18 | |
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| 19 | Cross section of cylindrical symmetry model used here. Users will have |
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| 20 | to decide how to distribute "heads" and "tails" between the rim, face |
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| 21 | and core regions in order to estimate appropriate starting parameters. |
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[8007311] | 22 | |
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[adc753d] | 23 | Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, |
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| 24 | the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the |
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| 25 | scattering length density variation along the cylinder axis is: |
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| 26 | |
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| 27 | .. math:: |
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| 28 | |
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| 29 | \rho(r) = |
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| 30 | \begin{cases} |
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| 31 | &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] |
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| 32 | &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; |
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| 33 | L \lt z\lt (L+2t) \\[1.5ex] |
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| 34 | &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) |
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| 35 | \end{cases} |
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| 36 | |
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| 37 | The form factor for the bicelle is calculated in cylindrical coordinates, where |
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| 38 | $\alpha$ is the angle between the $Q$ vector and the cylinder axis, to give: |
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| 39 | |
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| 40 | .. math:: |
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| 41 | |
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| 42 | I(Q,\alpha) = \frac{\text{scale}}{V} \cdot |
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| 43 | F(Q,\alpha)^2 + \text{background} |
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| 44 | where |
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| 45 | |
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| 46 | .. math:: |
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| 47 | |
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| 48 | \begin{align} |
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| 49 | F(Q,\alpha) = &\frac{1}{V_t} \bigg[ |
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| 50 | (\rho_c - \rho_f) V_c \frac{J_1(QRsin \alpha)}{QRsin\alpha}\frac{2 \cdot QLcos\alpha}{QLcos\alpha} \\ |
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| 51 | &+(\rho_f - \rho_r) V_{c+f} \frac{J_1(QRsin\alpha)}{QRsin\alpha}\frac{2 \cdot Q(L+t_f)cos\alpha}{Q(L+t_f)cos\alpha} \\ |
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| 52 | &+(\rho_r - \rho_s) V_t \frac{J_1(Q(R+t_r)sin\alpha)}{Q(R+t_r)sin\alpha}\frac{2 \cdot Q(L+t_f)cos\alpha}{Q(L+t_f)cos\alpha} |
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| 53 | \bigg] |
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| 54 | \end{align} |
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| 55 | |
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| 56 | where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core, |
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| 57 | $V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius |
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| 58 | of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ |
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| 59 | the thickness of the rim and $J_1$ the usual first order bessel function. |
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| 60 | |
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[8007311] | 61 | The output of the 1D scattering intensity function for randomly oriented |
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[adc753d] | 62 | cylinders is then given by integrating over all possible $\theta$ and $\phi$. |
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[8007311] | 63 | |
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| 64 | The *theta* and *phi* parameters are not used for the 1D output. |
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| 65 | Our implementation of the scattering kernel and the 1D scattering intensity |
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| 66 | use the c-library from NIST. |
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| 67 | |
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[2f0c07d] | 68 | .. figure:: img/cylinder_angle_definition.jpg |
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[8007311] | 69 | |
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[2f0c07d] | 70 | Definition of the angles for the oriented core shell bicelle tmodel. |
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[8007311] | 71 | |
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[2f0c07d] | 72 | .. figure:: img/cylinder_angle_projection.jpg |
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[b0c4271] | 73 | :width: 600px |
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[8007311] | 74 | |
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| 75 | Examples of the angles for oriented pp against the detector plane. |
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| 76 | |
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| 77 | References |
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| 78 | ---------- |
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| 79 | |
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[adc753d] | 80 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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| 81 | lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available |
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| 82 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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| 83 | =26379>`_ |
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[b0c4271] | 84 | |
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| 85 | Authorship and Verification |
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| 86 | ---------------------------- |
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| 87 | |
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| 88 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[adc753d] | 89 | * **Last Modified by:** Paul Butler **Date:** September 30, 2016 |
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| 90 | * **Last Reviewed by:** Richard Heenan **Date:** October 5, 2016 |
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[8007311] | 91 | """ |
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| 92 | |
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| 93 | from numpy import inf, sin, cos |
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| 94 | |
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| 95 | name = "core_shell_bicelle" |
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| 96 | title = "Circular cylinder with a core-shell scattering length density profile.." |
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| 97 | description = """ |
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[a0fee3b] | 98 | P(q,alpha)= (scale/Vs)*f(q)^(2) + bkg, where: |
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| 99 | f(q)= Vt(sld_rim - sld_solvent)* sin[qLt.cos(alpha)/2] |
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| 100 | /[qLt.cos(alpha)/2]*J1(qRout.sin(alpha)) |
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| 101 | /[qRout.sin(alpha)]+ |
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| 102 | (sld_core-sld_face)*Vc*sin[qLcos(alpha)/2][[qL |
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| 103 | *cos(alpha)/2]*J1(qRc.sin(alpha)) |
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| 104 | /qRc.sin(alpha)]+ |
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| 105 | (sld_face-sld_rim)*(Vc+Vf)*sin[q(L+2.thick_face). |
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| 106 | cos(alpha)/2][[q(L+2.thick_face)*cos(alpha)/2]* |
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| 107 | J1(qRc.sin(alpha))/qRc.sin(alpha)] |
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[8007311] | 108 | |
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| 109 | alpha:is the angle between the axis of |
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| 110 | the cylinder and the q-vector |
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[a0fee3b] | 111 | Vt = pi.(Rc + thick_rim)^2.Lt : total volume |
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| 112 | Vc = pi.Rc^2.L :the volume of the core |
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| 113 | Vf = 2.pi.Rc^2.thick_face |
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| 114 | Rc = radius: is the core radius |
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[8007311] | 115 | L: the length of the core |
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[a0fee3b] | 116 | Lt = L + 2.thick_face: total length |
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| 117 | Rout = radius + thick_rim |
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| 118 | sld_core, sld_rim, sld_face:scattering length |
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| 119 | densities within the particle |
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[aad336c] | 120 | sld_solvent: the scattering length density |
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[8007311] | 121 | of the solvent |
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| 122 | bkg: the background |
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| 123 | J1: the first order Bessel function |
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| 124 | theta: axis_theta of the cylinder |
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| 125 | phi: the axis_phi of the cylinder... |
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| 126 | """ |
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| 127 | category = "shape:cylinder" |
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| 128 | |
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| 129 | # pylint: disable=bad-whitespace, line-too-long |
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| 130 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 131 | parameters = [ |
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| 132 | ["radius", "Ang", 20, [0, inf], "volume", "Cylinder core radius"], |
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[2222134] | 133 | ["thick_rim", "Ang", 10, [0, inf], "volume", "Rim shell thickness"], |
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| 134 | ["thick_face", "Ang", 10, [0, inf], "volume", "Cylinder face thickness"], |
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[8007311] | 135 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], |
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[42356c8] | 136 | ["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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| 137 | ["sld_face", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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| 138 | ["sld_rim", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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| 139 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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[8007311] | 140 | ["theta", "degrees", 90, [-inf, inf], "orientation", "In plane angle"], |
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| 141 | ["phi", "degrees", 0, [-inf, inf], "orientation", "Out of plane angle"], |
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| 142 | ] |
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| 143 | |
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| 144 | # pylint: enable=bad-whitespace, line-too-long |
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| 145 | |
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[40a87fa] | 146 | source = ["lib/Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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| 147 | "core_shell_bicelle.c"] |
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[8007311] | 148 | |
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| 149 | demo = dict(scale=1, background=0, |
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| 150 | radius=20.0, |
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[2222134] | 151 | thick_rim=10.0, |
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| 152 | thick_face=10.0, |
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[8007311] | 153 | length=400.0, |
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[aad336c] | 154 | sld_core=1.0, |
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| 155 | sld_face=4.0, |
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| 156 | sld_rim=4.0, |
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| 157 | sld_solvent=1.0, |
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[8007311] | 158 | theta=90, |
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| 159 | phi=0) |
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| 160 | |
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| 161 | qx, qy = 0.4 * cos(90), 0.5 * sin(0) |
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| 162 | tests = [ |
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| 163 | # Accuracy tests based on content in test/utest_other_models.py |
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| 164 | [{'radius': 20.0, |
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[2222134] | 165 | 'thick_rim': 10.0, |
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| 166 | 'thick_face': 10.0, |
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[8007311] | 167 | 'length': 400.0, |
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[aad336c] | 168 | 'sld_core': 1.0, |
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| 169 | 'sld_face': 4.0, |
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| 170 | 'sld_rim': 4.0, |
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| 171 | 'sld_solvent': 1.0, |
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[8007311] | 172 | 'background': 0.0, |
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| 173 | }, 0.001, 353.550], |
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| 174 | |
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| 175 | [{'radius': 20.0, |
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[2222134] | 176 | 'thick_rim': 10.0, |
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| 177 | 'thick_face': 10.0, |
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[8007311] | 178 | 'length': 400.0, |
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[aad336c] | 179 | 'sld_core': 1.0, |
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| 180 | 'sld_face': 4.0, |
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| 181 | 'sld_rim': 4.0, |
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| 182 | 'sld_solvent': 1.0, |
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[8007311] | 183 | 'theta': 90.0, |
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| 184 | 'phi': 0.0, |
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| 185 | 'background': 0.00, |
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| 186 | }, (qx, qy), 24.9167], |
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| 187 | |
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| 188 | # Additional tests with larger range of parameters |
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| 189 | [{'radius': 3.0, |
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[2222134] | 190 | 'thick_rim': 100.0, |
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| 191 | 'thick_face': 100.0, |
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[8007311] | 192 | 'length': 1200.0, |
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[aad336c] | 193 | 'sld_core': 5.0, |
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| 194 | 'sld_face': 41.0, |
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| 195 | 'sld_rim': 42.0, |
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| 196 | 'sld_solvent': 21.0, |
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[8007311] | 197 | }, 0.05, 1670.1828], |
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| 198 | ] |
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