1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model provides the form factor, $P(q)$, for a monodisperse hollow right |
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6 | angle circular cylinder (rigid tube) where the The inside and outside of the |
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7 | hollow cylinder are assumed to have the same SLD and the form factor is thus |
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8 | normalized by the volume of the tube (i.e. not by the total cylinder volume). |
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9 | |
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10 | .. math:: |
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11 | |
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12 | P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background} |
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13 | |
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14 | where the averaging $\left<\ldots\right>$ is applied only for the 1D |
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15 | calculation. If Intensity is given on an absolute scale, the scale factor here |
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16 | is the volume fraction of the shell. This differs from |
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17 | the :ref:`core-shell-cylinder` in that, in that case, scale is the volume |
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18 | fraction of the entire cylinder (core+shell). The application might be for a |
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19 | bilayer which wraps into a hollow tube and the volume fraction of material is |
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20 | all in the shell, whereas the :ref:`core-shell-cylinder` model might be used for |
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21 | a cylindrical micelle where the tails in the core have a different SLD than the |
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22 | headgroups (in the shell) and the volume fraction of material comes fromm the |
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23 | whole cyclinder. NOTE: the hollow_cylinder represents a tube whereas the |
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24 | core_shell_cylinder includes a shell layer covering the ends (end caps) as well. |
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25 | |
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26 | |
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27 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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28 | |
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29 | .. math:: |
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30 | |
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31 | P(q) &= (\text{scale})V_\text{shell}\Delta\rho^2 |
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32 | \int_0^{1}\Psi^2 |
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33 | \left[q_z, R_\text{outer}(1-x^2)^{1/2}, |
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34 | R_\text{core}(1-x^2)^{1/2}\right] |
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35 | \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\ |
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36 | \Psi[q,y,z] &= \frac{1}{1-\gamma^2} |
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37 | \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\ |
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38 | \Lambda(a) &= 2 J_1(a) / a \\ |
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39 | \gamma &= R_\text{core} / R_\text{outer} \\ |
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40 | V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\ |
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41 | J_1(x) &= (\sin(x)-x\cdot \cos(x)) / x^2 |
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42 | |
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43 | where *scale* is a scale factor, $H = L/2$ and $J_1$ is the 1st order |
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44 | Bessel function. |
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45 | |
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46 | **NB**: The 2nd virial coefficient of the cylinder is calculated |
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47 | based on the outer radius and full length, which give an the effective radius |
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48 | for structure factor $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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49 | |
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50 | In the parameters,the *radius* is $R_\text{core}$ while *thickness* |
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51 | is $R_\text{outer} - R_\text{core}$. |
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52 | |
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53 | To provide easy access to the orientation of the core-shell cylinder, we define |
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54 | the axis of the cylinder using two angles $\theta$ and $\phi$ |
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55 | (see :ref:`cylinder model <cylinder-angle-definition>`). |
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56 | |
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57 | References |
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58 | ---------- |
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59 | |
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60 | .. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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61 | Neutron Scattering*, Plenum Press, New York, (1987) |
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62 | |
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63 | Authorship and Verification |
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64 | ---------------------------- |
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65 | |
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66 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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67 | * **Last Modified by:** Paul Butler **Date:** September 06, 2018 |
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68 | (corrected VR calculation) |
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69 | * **Last Reviewed by:** Paul Butler **Date:** September 06, 2018 |
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70 | """ |
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71 | from __future__ import division |
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72 | |
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73 | import numpy as np |
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74 | from numpy import pi, inf, sin, cos |
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75 | |
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76 | name = "hollow_cylinder" |
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77 | title = "" |
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78 | description = """ |
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79 | P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude. |
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80 | radius = the radius of core |
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81 | thickness = the thickness of shell |
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82 | length = the total length of the cylinder |
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83 | sld = SLD of the shell |
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84 | sld_solvent = SLD of the solvent |
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85 | background = incoherent background |
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86 | """ |
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87 | category = "shape:cylinder" |
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88 | # pylint: disable=bad-whitespace, line-too-long |
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89 | # ["name", "units", default, [lower, upper], "type","description"], |
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90 | parameters = [ |
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91 | ["radius", "Ang", 20.0, [0, inf], "volume", "Cylinder core radius"], |
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92 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Cylinder wall thickness"], |
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93 | ["length", "Ang", 400.0, [0, inf], "volume", "Cylinder total length"], |
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94 | ["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Cylinder sld"], |
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95 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent sld"], |
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96 | ["theta", "degrees", 90, [-360, 360], "orientation", "Cylinder axis to beam angle"], |
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97 | ["phi", "degrees", 0, [-360, 360], "orientation", "Rotation about beam"], |
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98 | ] |
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99 | # pylint: enable=bad-whitespace, line-too-long |
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100 | |
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101 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "hollow_cylinder.c"] |
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102 | have_Fq = True |
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103 | effective_radius_type = [ |
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104 | "equivalent sphere", "outer radius", "half length", |
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105 | "half outer min dimension", "half outer max dimension", |
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106 | "half outer diagonal", |
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107 | ] |
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108 | |
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109 | def random(): |
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110 | length = 10**np.random.uniform(1, 4.7) |
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111 | outer_radius = 10**np.random.uniform(1, 4.7) |
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112 | # Use a distribution with a preference for thin shell or thin core |
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113 | # Avoid core,shell radii < 1 |
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114 | thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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115 | radius = outer_radius - thickness |
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116 | pars = dict( |
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117 | length=length, |
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118 | radius=radius, |
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119 | thickness=thickness, |
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120 | ) |
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121 | return pars |
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122 | |
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123 | # parameters for demo |
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124 | demo = dict(scale=1.0, background=0.0, length=400.0, radius=20.0, |
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125 | thickness=10, sld=6.3, sld_solvent=1, theta=90, phi=0, |
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126 | thickness_pd=0.2, thickness_pd_n=9, |
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127 | length_pd=.2, length_pd_n=10, |
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128 | radius_pd=.2, radius_pd_n=9, |
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129 | theta_pd=10, theta_pd_n=5, |
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130 | ) |
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131 | q = 0.1 |
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132 | # april 6 2017, rkh added a 2d unit test, assume correct! |
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133 | qx = q*cos(pi/6.0) |
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134 | qy = q*sin(pi/6.0) |
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135 | radius = parameters[0][2] |
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136 | thickness = parameters[1][2] |
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137 | length = parameters[2][2] |
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138 | # Parameters for unit tests |
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139 | tests = [ |
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140 | [{}, 0.00005, 1764.926], |
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141 | [{}, 0.1, None, None, |
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142 | (3./4*(radius+thickness)**2*length)**(1./3), # R_eff from volume |
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143 | pi*((radius+thickness)**2-radius**2)*length, # shell volume |
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144 | (radius+thickness)**2/((radius+thickness)**2 - radius**2), # form:shell ratio |
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145 | ], |
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146 | [{}, 0.001, 1756.76], |
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147 | [{}, (qx, qy), 2.36885476192], |
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148 | ] |
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149 | del qx, qy # not necessary to delete, but cleaner |
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