1 | r""" |
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2 | Calculates the scattering from fractal-like aggregates based on |
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3 | 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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11 | |
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12 | I(q) = scale \times P(q)S(q) + background |
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13 | |
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14 | .. math:: |
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15 | |
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16 | P(q) = F(qR)^2 |
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17 | |
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18 | .. math:: |
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19 | |
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20 | F(x) = \frac{3\left[sin(x)-xcos(x)\right]}{x^3} |
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21 | |
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22 | .. math:: |
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23 | |
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24 | S(q) = \frac{\Gamma(D_m-1)\zeta^{D_m-1}}{\left[1+(q\zeta)^2 |
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25 | \right]^{(D_m-1)/2}} |
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26 | \frac{sin\left[(D_m - 1) tan^{-1}(q\zeta) \right]}{q} |
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27 | |
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28 | .. math:: |
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29 | |
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30 | scale = scale\_factor \times NV^2(\rho_{particle} - \rho_{solvent})^2 |
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31 | |
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32 | .. math:: |
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33 | |
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34 | V = \frac{4}{3}\pi R^3 |
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35 | |
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36 | where $R$ is the radius of the building block, $D_m$ is the **mass** fractal |
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37 | dimension, | \zeta\| is the cut-off length, $\rho_{solvent}$ is the scattering |
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38 | length density of the solvent, |
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39 | and $\rho_{particle}$ is the scattering length density of particles. |
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40 | |
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41 | .. note:: |
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42 | |
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43 | The mass fractal dimension ( $D_m$ ) is only |
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44 | valid if $0 < mass\_dim < 6$. It is also only valid over a limited |
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45 | $q$ range (see the reference for details). |
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46 | |
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47 | |
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48 | References |
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49 | ---------- |
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50 | |
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51 | D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, |
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52 | 19 (1986) 1535-1545 Equation(9) |
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53 | |
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54 | |
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55 | """ |
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56 | |
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57 | from numpy import inf |
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58 | |
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59 | name = "mass_fractal" |
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60 | title = "Mass Fractal model" |
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61 | description = """ |
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62 | The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
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63 | scale = scale_factor * V * delta^(2) |
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64 | p(x)= F(x*radius)^(2) |
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65 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
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66 | S(x) = [(gamma(Dm-1)*colength^(Dm-1)*[1+(x^2*colength^2)]^((1-Dm)/2) |
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67 | * sin[(Dm-1)*arctan(x*colength)])/x] |
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68 | where delta = sldParticle -sldSolv. |
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69 | radius = Particle radius |
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70 | fractal_dim_mass = Mass fractal dimension |
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71 | cutoff_length = Cut-off length |
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72 | background = background |
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73 | Ref.:Mildner, Hall,J Phys D Appl Phys(1986), 9, 1535-1545 |
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74 | Note I: This model is valid for 1<fractal_dim_mass<6. |
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75 | Note II: This model is not in absolute scale. |
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76 | """ |
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77 | category = "shape-independent" |
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78 | |
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79 | # pylint: disable=bad-whitespace, line-too-long |
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80 | # ["name", "units", default, [lower, upper], "type","description"], |
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81 | parameters = [ |
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82 | ["radius", "Ang", 10.0, [0.0, inf], "", "Particle radius"], |
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83 | ["fractal_dim_mass", "", 1.9, [1.0, 6.0], "", "Mass fractal dimension"], |
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84 | ["cutoff_length", "Ang", 100.0, [0.0, inf], "", "Cut-off length"], |
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85 | ] |
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86 | # pylint: enable=bad-whitespace, line-too-long |
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87 | |
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88 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "mass_fractal.c"] |
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89 | |
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90 | demo = dict(scale=1, background=0, |
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91 | radius=10.0, |
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92 | fractal_dim_mass=1.9, |
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93 | cutoff_length=100.0) |
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94 | |
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95 | tests = [ |
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96 | |
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97 | # Accuracy tests based on content in test/utest_other_models.py |
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98 | [{'radius': 10.0, |
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99 | 'fractal_dim_mass': 1.9, |
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100 | 'cutoff_length': 100.0, |
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101 | }, 0.05, 279.59422], |
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102 | |
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103 | # Additional tests with larger range of parameters |
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104 | [{'radius': 2.0, |
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105 | 'fractal_dim_mass': 3.3, |
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106 | 'cutoff_length': 1.0, |
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107 | }, 0.5, 1.29116774904], |
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108 | |
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109 | [{'radius': 1.0, |
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110 | 'fractal_dim_mass': 1.3, |
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111 | 'cutoff_length': 1.0, |
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112 | 'background': 0.8, |
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113 | }, 0.001, 1.69747015932], |
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114 | |
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115 | [{'radius': 1.0, |
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116 | 'fractal_dim_mass': 2.3, |
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117 | 'cutoff_length': 1.0, |
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118 | 'scale': 10.0, |
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119 | }, 0.051, 11.6237966145], |
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120 | ] |
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