[7d4b2ae] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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[07300ea] | 4 | Calculates the scattering from a fractal structure with a primary building |
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| 5 | block of core-shell spheres, as opposed to just homogeneous spheres in |
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| 6 | the fractal model. It is an extension of the well known Teixeira\ [#teixeira]_ |
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| 7 | fractal model replacing the $P(q)$ of a solid sphere with that of a core-shell |
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| 8 | sphere. This model could find use for aggregates of coated particles, or |
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| 9 | aggregates of vesicles for example. |
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[7d4b2ae] | 10 | |
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| 11 | .. math:: |
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| 12 | |
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[07300ea] | 13 | I(q) = P(q)S(q) + \text{background} |
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[7d4b2ae] | 14 | |
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[07300ea] | 15 | Where $P(q)$ is the core-shell form factor and $S(q)$ is the |
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| 16 | Teixeira\ [#teixeira]_ fractal structure factor both of which are given again |
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| 17 | below: |
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[7d4b2ae] | 18 | |
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| 19 | .. math:: |
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| 20 | |
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[07300ea] | 21 | P(q) &= \frac{\phi}{V_s}\left[3V_c(\rho_c-\rho_s) |
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[7d4b2ae] | 22 | \frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3}+ |
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| 23 | 3V_s(\rho_s-\rho_{solv}) |
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[ca04add] | 24 | \frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}\right]^2 \\ |
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[efb7615] | 25 | S(q) &= 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q\xi)^2]^{(D_f-1)/2}} |
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[07300ea] | 26 | \frac{\sin[(D_f-1)\tan^{-1}(q\xi)]}{(qr_s)^{D_f}} |
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[7d4b2ae] | 27 | |
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[07300ea] | 28 | where $\phi$ is the volume fraction of particles, $V_s$ is the volume of the |
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| 29 | whole particle, $V_c$ is the volume of the core, $\rho_c$, $\rho_s$, and |
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| 30 | $\rho_{solv}$ are the scattering length densities of the core, shell, and |
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| 31 | solvent respectively, $r_c$ and $r_s$ are the radius of the core and the radius |
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[efb7615] | 32 | of the whole particle respectively, $D_f$ is the fractal dimension, and $\xi$ the |
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[07300ea] | 33 | correlation length. |
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[efb7615] | 34 | |
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[07300ea] | 35 | Polydispersity of radius and thickness are also provided for. |
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[7d4b2ae] | 36 | |
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[07300ea] | 37 | This model does not allow for anisotropy and thus the 2D scattering intensity |
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| 38 | is calculated in the same way as 1D, where the $q$ vector is defined as |
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[7d4b2ae] | 39 | |
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| 40 | .. math:: |
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| 41 | |
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| 42 | q = \sqrt{q_x^2 + q_y^2} |
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| 43 | |
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[07300ea] | 44 | Our model is derived from the form factor calculations implemented in IGOR |
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| 45 | macros by the NIST Center for Neutron Research\ [#Kline]_ |
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| 46 | |
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[aad336c] | 47 | References |
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| 48 | ---------- |
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[7d4b2ae] | 49 | |
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[07300ea] | 50 | .. [#teixeira] J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 |
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| 51 | .. [#Kline] S R Kline, *J Appl. Cryst.*, 39 (2006) 895 |
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| 52 | |
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| 53 | Authorship and Verification |
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| 54 | ---------------------------- |
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| 55 | |
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| 56 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 57 | * **Last Modified by:** Paul Butler and Paul Kienzle **on:** November 27, 2016 |
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| 58 | * **Last Reviewed by:** Paul Butler and Paul Kienzle **on:** November 27, 2016 |
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[7d4b2ae] | 59 | """ |
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| 60 | |
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[2d81cfe] | 61 | import numpy as np |
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[7d4b2ae] | 62 | from numpy import pi, inf |
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| 63 | |
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| 64 | name = "fractal_core_shell" |
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[07300ea] | 65 | title = "Scattering from a fractal structure formed from core shell spheres" |
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| 66 | description = """\ |
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| 67 | Model for fractal aggregates of core-shell primary particles. It is based on |
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| 68 | the Teixeira model for the S(q) of a fractal * P(q) for a core-shell sphere |
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| 69 | |
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| 70 | radius = the radius of the core |
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| 71 | thickness = thickness of the shell |
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| 72 | thick_layer = thickness of a layer |
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| 73 | sld_core = the SLD of the core |
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| 74 | sld_shell = the SLD of the shell |
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| 75 | sld_solvent = the SLD of the solvent |
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| 76 | volfraction = volume fraction of core-shell particles |
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| 77 | fractal_dim = fractal dimension |
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| 78 | cor_length = correlation length of the fractal like aggretates |
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| 79 | """ |
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[7d4b2ae] | 80 | category = "shape-independent" |
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| 81 | |
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| 82 | # pylint: disable=bad-whitespace, line-too-long |
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| 83 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 84 | parameters = [ |
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[6d96b66] | 85 | ["radius", "Ang", 60.0, [0.0, inf], "volume", "Sphere core radius"], |
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| 86 | ["thickness", "Ang", 10.0, [0.0, inf], "volume", "Sphere shell thickness"], |
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[42356c8] | 87 | ["sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Sphere core scattering length density"], |
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| 88 | ["sld_shell", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", "Sphere shell scattering length density"], |
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| 89 | ["sld_solvent", "1e-6/Ang^2", 3.0, [-inf, inf], "sld", "Solvent scattering length density"], |
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[6d96b66] | 90 | ["volfraction", "", 1.0, [0.0, inf], "", "Volume fraction of building block spheres"], |
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[217590b] | 91 | ["fractal_dim", "", 2.0, [0.0, 6.0], "", "Fractal dimension"], |
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[6d96b66] | 92 | ["cor_length", "Ang", 100.0, [0.0, inf], "", "Correlation length of fractal-like aggregates"], |
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| 93 | ] |
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[7d4b2ae] | 94 | # pylint: enable=bad-whitespace, line-too-long |
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| 95 | |
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[925ad6e] | 96 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/core_shell.c", |
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[217590b] | 97 | "lib/fractal_sq.c", "fractal_core_shell.c"] |
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[7d4b2ae] | 98 | |
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[404ebbd] | 99 | def random(): |
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[8f04da4] | 100 | outer_radius = 10**np.random.uniform(0.7, 4) |
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| 101 | # Use a distribution with a preference for thin shell or thin core |
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| 102 | # Avoid core,shell radii < 1 |
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| 103 | thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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| 104 | radius = outer_radius - thickness |
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| 105 | cor_length = 10**np.random.uniform(0.7, 2)*outer_radius |
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[404ebbd] | 106 | volfraction = 10**np.random.uniform(-3, -1) |
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| 107 | fractal_dim = 2*np.random.beta(3, 4) + 1 |
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| 108 | pars = dict( |
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| 109 | #background=0, sld_block=1, sld_solvent=0, |
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| 110 | volfraction=volfraction, |
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| 111 | radius=radius, |
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| 112 | cor_length=cor_length, |
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| 113 | fractal_dim=fractal_dim, |
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| 114 | ) |
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| 115 | return pars |
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| 116 | |
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[7d4b2ae] | 117 | demo = dict(scale=0.05, |
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| 118 | background=0, |
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| 119 | radius=20, |
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| 120 | thickness=5, |
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[aad336c] | 121 | sld_core=3.5, |
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| 122 | sld_shell=1.0, |
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| 123 | sld_solvent=6.35, |
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[7d4b2ae] | 124 | volfraction=0.05, |
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[a807206] | 125 | fractal_dim=2.0, |
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[7d4b2ae] | 126 | cor_length=100.0) |
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| 127 | |
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| 128 | def ER(radius, thickness): |
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| 129 | """ |
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| 130 | Equivalent radius |
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| 131 | @param radius: core radius |
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| 132 | @param thickness: shell thickness |
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| 133 | """ |
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| 134 | return radius + thickness |
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| 135 | |
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| 136 | def VR(radius, thickness): |
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| 137 | """ |
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| 138 | Volume ratio |
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| 139 | @param radius: core radius |
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| 140 | @param thickness: shell thickness |
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| 141 | """ |
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[4962519] | 142 | whole = 4.0/3.0 * pi * (radius + thickness)**3 |
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| 143 | core = 4.0/3.0 * pi * radius**3 |
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[7d4b2ae] | 144 | return whole, whole-core |
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| 145 | |
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| 146 | tests = [[{'radius': 20.0, 'thickness': 10.0}, 'ER', 30.0], |
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[d6f60c3] | 147 | [{'radius': 20.0, 'thickness': 10.0}, 'VR', 0.703703704]] |
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| 148 | |
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| 149 | # # The SasView test result was 0.00169, with a background of 0.001 |
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| 150 | # # They are however wrong as we now know. IGOR might be a more |
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| 151 | # # appropriate source. Otherwise will just have to assume this is now |
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| 152 | # # correct and self generate a correct answer for the future. Until we |
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| 153 | # # figure it out leave the tests commented out |
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| 154 | # [{'radius': 60.0, |
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| 155 | # 'thickness': 10.0, |
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| 156 | # 'sld_core': 1.0, |
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| 157 | # 'sld_shell': 2.0, |
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| 158 | # 'sld_solvent': 3.0, |
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| 159 | # 'background': 0.0 |
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| 160 | # }, 0.015211, 692.84]] |
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