[da84551] | 1 | r""" |
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| 2 | This model calculates the scattering from fractal-like aggregates based |
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| 3 | on the Mildner reference. |
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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 scattering intensity $I(q)$ is calculated as |
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| 9 | |
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| 10 | .. math:: |
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[30b60d2] | 11 | :nowrap: |
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[da84551] | 12 | |
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[30b60d2] | 13 | \begin{align*} |
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[b716cc6] | 14 | I(q) &= \text{scale} \times P(q)S(q) + \text{background} \\ |
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| 15 | P(q) &= F(qR)^2 \\ |
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| 16 | F(x) &= \frac{3\left[\sin(x)-x\cos(x)\right]}{x^3} \\ |
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| 17 | S(q) &= \Gamma(5-D_S)\xi^{\,5-D_S}\left[1+(q\xi)^2 \right]^{-(5-D_S)/2} |
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| 18 | \sin\left[-(5-D_S) \tan^{-1}(q\xi) \right] q^{-1} \\ |
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[30b60d2] | 19 | \text{scale} &= \text{scale factor}\, N V^1(\rho_\text{particle} - \rho_\text{solvent})^2 \\ |
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[b716cc6] | 20 | V &= \frac{4}{3}\pi R^3 |
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[30b60d2] | 21 | \end{align*} |
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[da84551] | 22 | |
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[513efc5] | 23 | where $R$ is the radius of the building block, $D_S$ is the **surface** fractal |
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[b716cc6] | 24 | dimension, $\xi$ is the cut-off length, $\rho_\text{solvent}$ is the scattering |
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[5c94f41] | 25 | length density of the solvent and $\rho_\text{particle}$ is the scattering |
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| 26 | length density of particles. |
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[da84551] | 27 | |
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| 28 | .. note:: |
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[b716cc6] | 29 | |
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| 30 | The surface fractal dimension is only valid if $1<D_S<3$. The result is |
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| 31 | only valid over a limited $q$ range, $\tfrac{5}{3-D_S}\xi^{\,-1} < q < R^{-1}$. |
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| 32 | See the reference for details. |
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[da84551] | 33 | |
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| 34 | |
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[95441ff] | 35 | References |
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| 36 | ---------- |
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[da84551] | 37 | |
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| 38 | D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 |
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| 39 | |
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| 40 | """ |
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| 41 | |
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| 42 | from numpy import inf |
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| 43 | |
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| 44 | name = "surface_fractal" |
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| 45 | title = "Fractal-like aggregates based on the Mildner reference" |
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| 46 | description = """\ |
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| 47 | [The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
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| 48 | scale = scale_factor * V * delta^(2) |
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| 49 | p(x) = F(x*radius)^(2) |
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| 50 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
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| 51 | S(x) = [(gamma(5-Ds)*colength^(5-Ds)*[1+(x^2*colength^2)]^((Ds-5)/2) |
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| 52 | * sin[(Ds-5)*arctan(x*colength)])/x] |
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| 53 | where |
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| 54 | delta = sldParticle -sldSolv. |
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| 55 | radius = Particle radius |
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[a807206] | 56 | fractal_dim_surf = Surface fractal dimension (Ds) |
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[da84551] | 57 | co_length = Cut-off length |
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| 58 | background = background |
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| 59 | |
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| 60 | Ref. :Mildner, Hall,J Phys D Appl Phys(1986), 19, 1535-1545 |
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[a807206] | 61 | Note I : This model is valid for 1<fractal_dim_surf<3 with limited q range. |
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[da84551] | 62 | Note II: This model is not in absolute scale. |
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| 63 | """ |
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| 64 | category = "shape-independent" |
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| 65 | |
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[168052c] | 66 | # pylint: disable=bad-whitespace, line-too-long |
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[da84551] | 67 | # ["name", "units", default, [lower, upper], "type","description"], |
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[513efc5] | 68 | parameters = [["radius", "Ang", 10.0, [0, inf], "", |
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| 69 | "Particle radius"], |
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[a807206] | 70 | ["fractal_dim_surf", "", 2.0, [1, 3], "", |
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[513efc5] | 71 | "Surface fractal dimension"], |
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| 72 | ["cutoff_length", "Ang", 500., [0.0, inf], "", |
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| 73 | "Cut-off Length"], |
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[168052c] | 74 | ] |
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| 75 | # pylint: enable=bad-whitespace, line-too-long |
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[da84551] | 76 | |
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[925ad6e] | 77 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "surface_fractal.c"] |
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[da84551] | 78 | |
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[48462b0] | 79 | def random(): |
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| 80 | import numpy as np |
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| 81 | radius = 10**np.random.uniform(1, 4) |
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| 82 | fractal_dim_surf = np.random.uniform(1, 3-1e-6) |
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| 83 | cutoff_length = 1e6 # Sets the low q limit; keep it big for sim |
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| 84 | pars = dict( |
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| 85 | #background=0, |
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| 86 | scale=1, |
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| 87 | radius=radius, |
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| 88 | fractal_dim_surf=fractal_dim_surf, |
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| 89 | cutoff_length=cutoff_length, |
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| 90 | ) |
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| 91 | return pars |
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[da84551] | 92 | |
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[07a6700] | 93 | tests = [ |
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[168052c] | 94 | # Accuracy tests based on content in test/utest_other_models.py |
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| 95 | [{'radius': 10.0, |
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[a807206] | 96 | 'fractal_dim_surf': 2.0, |
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[168052c] | 97 | 'cutoff_length': 500.0, |
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[6dd90c1] | 98 | }, 0.05, 301428.66016], |
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[168052c] | 99 | |
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| 100 | # Additional tests with larger range of parameters |
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| 101 | [{'radius': 1.0, |
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[a807206] | 102 | 'fractal_dim_surf': 1.0, |
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[168052c] | 103 | 'cutoff_length': 10.0, |
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[6dd90c1] | 104 | }, 0.332070182643, 1125.00421004], |
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[168052c] | 105 | |
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| 106 | [{'radius': 3.5, |
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[a807206] | 107 | 'fractal_dim_surf': 0.1, |
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[168052c] | 108 | 'cutoff_length': 30.0, |
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| 109 | 'background': 0.01, |
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| 110 | }, 5.0, 0.00999998891322], |
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| 111 | |
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| 112 | [{'radius': 3.0, |
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[a807206] | 113 | 'fractal_dim_surf': 1.0, |
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[168052c] | 114 | 'cutoff_length': 33.0, |
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| 115 | 'scale': 0.1, |
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[6dd90c1] | 116 | }, 0.51, 2.50120147004], |
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[168052c] | 117 | ] |
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