1 | r""" |
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2 | .. _core_shell_sphere: |
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3 | |
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4 | This model provides the form factor, $P(q)$, for a spherical particle with |
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5 | a core-shell structure. The form factor is normalized by the particle volume. |
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6 | |
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7 | For information about polarised and magnetic scattering, see |
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8 | the :ref:`magnetism` documentation. |
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9 | |
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10 | Definition |
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11 | ---------- |
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12 | |
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13 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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14 | |
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15 | .. math:: |
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16 | |
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17 | P(q) = \frac{\text{scale}}{V} F^2(q) + \text{background} |
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18 | |
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19 | where |
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20 | |
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21 | .. math:: |
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22 | |
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23 | F^2(q) = \frac{3}{V_s}\left[ |
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24 | V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3} + |
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25 | V_s(\rho_s-\rho_\text{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3} |
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26 | \right] |
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27 | |
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28 | where $V_s$ is the volume of the whole particle, $V_c$ is the volume of the |
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29 | core, $r_s$ = $radius$ + $thickness$ is the radius of the particle, $r_c$ |
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30 | is the radius of the core, $\rho_c$ is the scattering length density of the |
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31 | core, $\rho_s$ is the scattering length density of the shell, |
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32 | $\rho_\text{solv}$, is the scattering length density of the solvent. |
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33 | |
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34 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
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35 | orientation of the $q$ vector. |
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36 | |
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37 | NB: The outer most radius (ie, = radius + thickness) is used as the |
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38 | effective radius for $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
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39 | |
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40 | References |
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41 | ---------- |
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42 | |
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43 | A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, |
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44 | John Wiley and Sons, New York, (1955) |
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45 | |
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46 | Validation |
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47 | ---------- |
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48 | |
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49 | Validation of our code was done by comparing the output of the 1D model to |
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50 | the output of the software provided by NIST (Kline, 2006). Figure 1 shows a |
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51 | comparison of the output of our model and the output of the NIST software. |
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52 | """ |
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53 | |
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54 | import numpy as np |
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55 | from numpy import pi, inf |
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56 | |
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57 | name = "core_shell_sphere" |
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58 | title = "Form factor for a monodisperse spherical particle with particle with a core-shell structure." |
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59 | description = """ |
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60 | F^2(q) = 3/V_s [V_c (sld_core-sld_shell) (sin(q*radius)-q*radius*cos(q*radius))/(q*radius)^3 |
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61 | + V_s (sld_shell-sld_solvent) (sin(q*r_s)-q*r_s*cos(q*r_s))/(q*r_s)^3] |
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62 | |
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63 | V_s: Volume of the sphere shell |
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64 | V_c: Volume of the sphere core |
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65 | r_s: Shell radius = radius + thickness |
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66 | """ |
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67 | category = "shape:sphere" |
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68 | |
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69 | # pylint: disable=bad-whitespace, line-too-long |
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70 | # ["name", "units", default, [lower, upper], "type","description"], |
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71 | parameters = [["radius", "Ang", 60.0, [0, inf], "volume", "Sphere core radius"], |
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72 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Sphere shell thickness"], |
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73 | ["sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "core scattering length density"], |
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74 | ["sld_shell", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", "shell scattering length density"], |
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75 | ["sld_solvent", "1e-6/Ang^2", 3.0, [-inf, inf], "sld", "Solvent scattering length density"]] |
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76 | # pylint: enable=bad-whitespace, line-too-long |
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77 | |
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78 | source = ["lib/sas_3j1x_x.c", "lib/core_shell.c", "core_shell_sphere.c"] |
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79 | |
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80 | demo = dict(scale=1, background=0, radius=60, thickness=10, |
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81 | sld_core=1.0, sld_shell=2.0, sld_solvent=0.0) |
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82 | |
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83 | def ER(radius, thickness): |
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84 | """ |
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85 | Equivalent radius |
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86 | @param radius: core radius |
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87 | @param thickness: shell thickness |
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88 | """ |
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89 | return radius + thickness |
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90 | |
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91 | def VR(radius, thickness): |
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92 | """ |
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93 | Volume ratio |
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94 | @param radius: core radius |
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95 | @param thickness: shell thickness |
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96 | """ |
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97 | return (1, 1) |
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98 | whole = 4.0/3.0 * pi * (radius + thickness)**3 |
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99 | core = 4.0/3.0 * pi * radius**3 |
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100 | return whole, whole - core |
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101 | |
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102 | def random(): |
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103 | outer_radius = 10**np.random.uniform(1.3, 4.3) |
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104 | # Use a distribution with a preference for thin shell or thin core |
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105 | # Avoid core,shell radii < 1 |
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106 | radius = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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107 | thickness = outer_radius - radius |
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108 | pars = dict( |
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109 | radius=radius, |
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110 | thickness=thickness, |
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111 | ) |
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112 | return pars |
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113 | |
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114 | tests = [ |
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115 | [{'radius': 20.0, 'thickness': 10.0}, 'ER', 30.0], |
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116 | # TODO: VR test suppressed until we sort out new product model |
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117 | # and determine what to do with volume ratio. |
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118 | #[{'radius': 20.0, 'thickness': 10.0}, 'VR', 0.703703704], |
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119 | |
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120 | # The SasView test result was 0.00169, with a background of 0.001 |
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121 | [{'radius': 60.0, 'thickness': 10.0, 'sld_core': 1.0, 'sld_shell': 2.0, |
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122 | 'sld_solvent': 3.0, 'background': 0.0}, 0.4, 0.000698838], |
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123 | ] |
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