1 | r""" |
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2 | This model provides the form factor, $P(q)$, for a spherical particle with a core-shell structure. |
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3 | The form factor is normalized by the particle volume. |
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4 | |
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5 | Definition |
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6 | ---------- |
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7 | |
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8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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9 | |
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10 | .. math:: |
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11 | |
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12 | P(q) = \frac{\text{scale}}{V} F^2(q) + \text{background} |
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13 | |
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14 | where |
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15 | |
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16 | .. math:: |
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17 | F^2(q)=\frac{3}{V_s}\left[V_c(\rho_c-\rho_s)\frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3}+ |
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18 | V_s(\rho_s-\rho_{solv})\frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}\right] |
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19 | |
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20 | |
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21 | where $V_s$ is the volume of the outer shell, $V_c$ is |
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22 | the volume of the core, $r_s$ is the radius of the shell, $r_c$ is the radius of the |
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23 | core, $\rho_c$ is the scattering length density of the core, $\rho_s$ is the scattering length |
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24 | density of the shell, $\rho_{solv}$ is the scattering length density of the solvent. |
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25 | |
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26 | The 2D scattering intensity is the same as $P(q)$ above, regardless of the |
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27 | orientation of the $q$ vector. |
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28 | |
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29 | NB: The outer most radius (ie, = radius + thickness) is used as the |
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30 | effective radius for $S(Q)$ when $P(Q) \cdot S(Q)$ is applied. |
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31 | |
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32 | Reference |
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33 | --------- |
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34 | |
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35 | A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) |
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36 | |
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37 | Validation |
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38 | ---------- |
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39 | |
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40 | Validation of our code was done by comparing the output of the 1D model to the output of |
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41 | the software provided by NIST (Kline, 2006). Figure 1 shows a comparison of the output of |
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42 | our model and the output of the NIST software. |
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43 | |
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44 | .. image:: img/core_shell_sphere_1d.jpg |
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45 | |
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46 | Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with |
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47 | the output of the NIST SANS analysis software. The parameters were set to: |
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48 | *scale* = 1.0, *radius* = 60 , *contrast* = 1e-6 |Ang^-2|, and |
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49 | *background* = 0.001 |cm^-1|. |
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50 | """ |
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51 | |
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52 | from numpy import pi, inf |
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53 | |
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54 | name = "core_shell_sphere" |
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55 | title = "Form factor for a monodisperse spherical particle with particle with a core-shell structure." |
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56 | description = """ |
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57 | F^2(q) = 3/V_s [V_c (core_sld-shell_sld) (sin(q*radius)-q*radius*cos(q*radius))/(q*radius)^3 |
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58 | + V_s (shell_sld-solvent_sld) (sin(q*r_s)-q*r_s*cos(q*r_s))/(q*r_s)^3] |
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59 | |
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60 | V_s: Volume of the sphere shell |
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61 | V_c: Volume of the sphere core |
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62 | r_s: Shell radius = radius + thickness |
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63 | """ |
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64 | category = "shape:sphere" |
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65 | |
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66 | # pylint: disable=bad-whitespace, line-too-long |
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67 | # ["name", "units", default, [lower, upper], "type","description"], |
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68 | parameters = [["radius", "Ang", 60.0, [0, inf], "volume", "Sphere core radius"], |
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69 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Sphere shell thickness"], |
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70 | ["core_sld", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Sphere core scattering length density"], |
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71 | ["shell_sld", "1e-6/Ang^2", 2.0, [-inf, inf], "", "Sphere shell scattering length density"], |
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72 | ["solvent_sld", "1e-6/Ang^2", 3.0, [-inf, inf], "", "Solvent scattering length density"]] |
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73 | # pylint: enable=bad-whitespace, line-too-long |
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74 | |
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75 | source = ["lib/sph_j1c.c", "core_shell_sphere.c"] |
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76 | |
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77 | demo = dict(scale=1, background=0, radius=60, thickness=10, |
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78 | core_sld=1.0, shell_sld=2.0, solvent_sld=0.0) |
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79 | |
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80 | oldname = 'CoreShellModel' |
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81 | oldpars = {} |
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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 | whole = 4.0 * pi / 3.0 * pow((radius + thickness), 3) |
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98 | core = 4.0 * pi / 3.0 * radius * radius * radius |
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99 | return whole, whole - core |
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100 | |
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101 | tests = [ |
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102 | [{'radius': 20.0, 'thickness': 10.0}, 'ER', 30.0 ], |
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103 | [{'radius': 20.0, 'thickness': 10.0}, 'VR', 0.703703704 ], |
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104 | |
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105 | # The SasView test result was 0.00169, with a background of 0.001 |
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106 | [{'radius': 60.0, |
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107 | 'thickness': 10.0, |
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108 | 'core_sld': 1.0, |
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109 | 'shell_sld':2.0, |
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110 | 'solvent_sld':3.0, |
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111 | 'background':0.0 |
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112 | }, 0.4, 0.000698838], |
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113 | ] |
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