1 | r""" |
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2 | This model provides the form factor, $P(q)$, for a monodisperse hollow right |
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3 | angle circular cylinder (tube) where the form factor is normalized by the |
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4 | volume of the tube |
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5 | |
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6 | .. math:: |
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7 | |
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8 | P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background} |
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9 | |
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10 | where the averaging $\left<\ldots\right>$ is applied only for the 1D calculation. |
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11 | |
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12 | The inside and outside of the hollow cylinder are assumed have the same SLD. |
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13 | |
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14 | Definition |
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15 | ---------- |
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16 | |
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17 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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18 | |
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19 | .. math:: |
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20 | |
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21 | P(q) &= (\text{scale})V_\text{shell}\Delta\rho^2 |
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22 | \int_0^{1}\Psi^2 |
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23 | \left[q_z, R_\text{shell}(1-x^2)^{1/2}, |
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24 | R_\text{core}(1-x^2)^{1/2}\right] |
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25 | \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\ |
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26 | \Psi[q,y,z] &= \frac{1}{1-\gamma^2} |
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27 | \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\ |
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28 | \Lambda(a) &= 2 J_1(a) / a \\ |
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29 | \gamma &= R_\text{core} / R_\text{shell} \\ |
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30 | V_\text{shell} &= \pi \left(R_\text{shell}^2 - R_\text{core}^2 \right)L \\ |
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31 | J_1(x) &= (\sin(x)-x\cdot \cos(x)) / x^2 |
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32 | |
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33 | where *scale* is a scale factor and $J_1$ is the 1st order |
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34 | Bessel function. |
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35 | |
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36 | To provide easy access to the orientation of the core-shell cylinder, we define |
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37 | the axis of the cylinder using two angles $\theta$ and $\phi$. As for the case |
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38 | of the cylinder, those angles are defined in Figure 2 of the CylinderModel. |
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39 | |
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40 | **NB**: The 2nd virial coefficient of the cylinder is calculated |
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41 | based on the radius and 2 length values, and used as the effective radius |
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42 | for $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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43 | |
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44 | In the parameters, the contrast represents SLD :sub:`shell` - SLD :sub:`solvent` |
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45 | and the *radius* is $R_\text{shell}$ while *core_radius* is $R_\text{core}$. |
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46 | |
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47 | .. figure:: img/hollow_cylinder_1d.jpg |
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48 | |
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49 | 1D plot using the default values (w/1000 data point). |
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50 | |
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51 | .. figure:: img/orientation.jpg |
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52 | |
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53 | Definition of the angles for the oriented hollow_cylinder model. |
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54 | |
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55 | .. figure:: img/orientation2.jpg |
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56 | |
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57 | Examples of the angles for oriented pp against the detector plane. |
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58 | |
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59 | References |
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60 | ---------- |
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61 | |
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62 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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63 | Neutron Scattering*, Plenum Press, New York, (1987) |
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64 | """ |
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65 | |
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66 | from numpy import pi, inf |
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67 | |
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68 | name = "hollow_cylinder" |
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69 | title = "" |
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70 | description = """ |
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71 | P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude. |
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72 | core_radius = the radius of core |
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73 | radius = the radius of shell |
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74 | length = the total length of the cylinder |
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75 | sld = SLD of the shell |
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76 | solvent_sld = SLD of the solvent |
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77 | background = incoherent background |
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78 | """ |
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79 | category = "shape:cylinder" |
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80 | # pylint: disable=bad-whitespace, line-too-long |
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81 | # ["name", "units", default, [lower, upper], "type","description"], |
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82 | parameters = [ |
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83 | ["radius", "Ang", 30.0, [0, inf], "volume", "Cylinder radius"], |
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84 | ["core_radius", "Ang", 20.0, [0, inf], "volume", "Hollow core radius"], |
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85 | ["length", "Ang", 400.0, [0, inf], "volume", "Cylinder length"], |
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86 | ["sld", "1/Ang^2", 6.3, [-inf, inf], "", "Cylinder sld"], |
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87 | ["solvent_sld", "1/Ang^2", 1, [-inf, inf], "", "Solvent sld"], |
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88 | ["theta", "degrees", 90, [-360, 360], "orientation", "Theta angle"], |
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89 | ["phi", "degrees", 0, [-360, 360], "orientation", "Phi angle"], |
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90 | ] |
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91 | # pylint: enable=bad-whitespace, line-too-long |
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92 | |
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93 | source = ["lib/J1.c", "lib/gauss76.c", "hollow_cylinder.c"] |
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94 | |
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95 | def ER(radius, core_radius, length): |
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96 | """ |
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97 | :param radius: Cylinder radius |
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98 | :param core_radius: Hollow core radius, UNUSED |
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99 | :param length: Cylinder length |
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100 | :return: Effective radius |
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101 | """ |
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102 | if radius == 0 or length == 0: |
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103 | return 0.0 |
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104 | len1 = radius |
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105 | len2 = length/2.0 |
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106 | term1 = len1*len1*2.0*len2/2.0 |
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107 | term2 = 1.0 + (len2/len1)*(1.0 + 1/len2/2.0)*(1.0 + pi*len1/len2/2.0) |
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108 | ddd = 3.0*term1*term2 |
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109 | diam = pow(ddd, (1.0/3.0)) |
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110 | return diam |
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111 | |
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112 | def VR(radius, core_radius, length): |
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113 | """ |
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114 | :param radius: Cylinder radius |
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115 | :param core_radius: Hollow core radius |
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116 | :param length: Cylinder length |
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117 | :return: Volf ratio for P(q)*S(q) |
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118 | """ |
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119 | vol_core = pi*core_radius*core_radius*length |
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120 | vol_total = pi*radius*radius*length |
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121 | vol_shell = vol_total - vol_core |
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122 | return vol_shell, vol_total |
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123 | |
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124 | # parameters for demo |
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125 | demo = dict(scale=1.0, background=0.0, length=400.0, radius=30.0, |
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126 | core_radius=20.0, sld=6.3, solvent_sld=1, theta=90, phi=0, |
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127 | radius_pd=.2, radius_pd_n=9, |
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128 | length_pd=.2, length_pd_n=10, |
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129 | core_radius_pd=.2, core_radius_pd_n=9, |
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130 | theta_pd=10, theta_pd_n=5, |
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131 | ) |
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132 | |
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133 | # For testing against the old sasview models, include the converted parameter |
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134 | # names and the target sasview model name. |
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135 | oldname = 'HollowCylinderModel' |
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136 | oldpars = dict(scale='scale', background='background', radius='radius', |
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137 | core_radius='core_radius', sld='sldCyl', length='length', |
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138 | solvent_sld='sldSolv', phi='axis_phi', theta='axis_theta') |
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139 | |
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140 | # Parameters for unit tests |
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141 | tests = [ |
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142 | [{"radius": 30.0}, 0.00005, 1764.926], |
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143 | [{}, 'VR', 1.8], |
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144 | [{}, 0.001, 1756.76] |
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145 | ] |
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