[29da213] | 1 | r""" |
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| 2 | This model calculates the scattering from a gel structure, |
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| 3 | but typically a physical rather than chemical network. |
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[b8954d7] | 4 | It is modeled as a sum of a low-q exponential decay (which happens to |
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[48462b0] | 5 | give a functional form similar to Guinier scattering, so interpret with |
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[b8954d7] | 6 | care) plus a Lorentzian at higher-q values. See also the gel_fit model. |
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[29da213] | 7 | |
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| 8 | Definition |
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| 9 | ---------- |
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| 10 | |
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[40a87fa] | 11 | The scattering intensity $I(q)$ is calculated as (Eqn. 5 from the reference) |
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[29da213] | 12 | |
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[40a87fa] | 13 | .. math:: I(q) = I_G(0) \exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2) |
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[29da213] | 14 | |
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[40a87fa] | 15 | $\Xi$ is the length scale of the static correlations in the gel, which can |
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| 16 | be attributed to the "frozen-in" crosslinks. $\xi$ is the dynamic correlation |
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| 17 | length, which can be attributed to the fluctuating polymer chains between |
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| 18 | crosslinks. $I_G(0)$ and $I_L(0)$ are the scaling factors for each of these |
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| 19 | structures. Think carefully about how these map to your particular system! |
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[29da213] | 20 | |
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| 21 | .. note:: |
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| 22 | The peaked structure at higher $q$ values (Figure 2 from the reference) |
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| 23 | is not reproduced by the model. Peaks can be introduced into the model |
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[40a87fa] | 24 | by summing this model with the :ref:`gaussian-peak` model. |
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[29da213] | 25 | |
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| 26 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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| 27 | where the $q$ vector is defined as |
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| 28 | |
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[40a87fa] | 29 | .. math:: q = \sqrt{q_x^2 + q_y^2} |
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[29da213] | 30 | |
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| 31 | References |
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| 32 | ---------- |
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| 33 | |
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[0507e09] | 34 | .. [#] G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, 42 (2001) 2907-2913 |
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| 35 | |
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| 36 | Source |
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| 37 | ------ |
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| 38 | |
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| 39 | `gauss_lorentz_gel.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/gauss_lorentz_gel.py>`_ |
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| 40 | |
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| 41 | Authorship and Verification |
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| 42 | ---------------------------- |
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| 43 | |
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| 44 | * **Author:** |
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| 45 | * **Last Modified by:** |
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| 46 | * **Last Reviewed by:** |
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| 47 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[29da213] | 48 | """ |
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| 49 | |
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[2d81cfe] | 50 | import numpy as np |
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[2c74c11] | 51 | from numpy import inf, exp |
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[29da213] | 52 | |
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| 53 | name = "gauss_lorentz_gel" |
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| 54 | title = "Gauss Lorentz Gel model of scattering from a gel structure" |
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| 55 | description = """ |
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| 56 | Class that evaluates a GaussLorentzGel model. |
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| 57 | |
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| 58 | I(q) = scale_g*exp(- q^2*Z^2 / 2)+scale_l/(1+q^2*z^2) |
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| 59 | + background |
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| 60 | List of default parameters: |
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| 61 | scale_g = Gauss scale factor |
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| 62 | Z = Static correlation length |
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| 63 | scale_l = Lorentzian scale factor |
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| 64 | z = Dynamic correlation length |
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| 65 | background = Incoherent background |
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| 66 | """ |
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| 67 | category = "shape-independent" |
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[168052c] | 68 | # pylint: disable=bad-whitespace, line-too-long |
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[29da213] | 69 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[a807206] | 70 | parameters = [["gauss_scale", "", 100.0, [-inf, inf], "", "Gauss scale factor"], |
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| 71 | ["cor_length_static", "Ang", 100.0, [0, inf], "", "Static correlation length"], |
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| 72 | ["lorentz_scale", "", 50.0, [-inf, inf], "", "Lorentzian scale factor"], |
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| 73 | ["cor_length_dynamic", "Ang", 20.0, [0, inf], "", "Dynamic correlation length"], |
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[168052c] | 74 | ] |
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| 75 | # pylint: enable=bad-whitespace, line-too-long |
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[29da213] | 76 | |
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| 77 | def Iq(q, |
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[a807206] | 78 | gauss_scale=100.0, |
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| 79 | cor_length_static=100.0, |
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| 80 | lorentz_scale=50.0, |
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| 81 | cor_length_dynamic=20.0): |
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[168052c] | 82 | """ |
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| 83 | |
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| 84 | :param q: Input q-value |
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[a807206] | 85 | :param gauss_scale: Gauss scale factor |
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| 86 | :param cor_length_static: Static correlation length |
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| 87 | :param lorentz_scale: Lorentzian scale factor |
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| 88 | :param cor_length_dynamic: Dynamic correlation length |
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[168052c] | 89 | :return: 1-D intensity |
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| 90 | """ |
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| 91 | |
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[a807206] | 92 | term1 = gauss_scale *\ |
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| 93 | exp(-1.0*q*q*cor_length_static*cor_length_static/2.0) |
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| 94 | term2 = lorentz_scale /\ |
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| 95 | (1.0+(q*cor_length_dynamic)*(q*cor_length_dynamic)) |
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[168052c] | 96 | |
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| 97 | return term1 + term2 |
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[29da213] | 98 | |
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| 99 | Iq.vectorized = True # Iq accepts an array of q values |
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| 100 | |
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| 101 | |
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[48462b0] | 102 | def random(): |
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[b297ba9] | 103 | """Return a random parameter set for the model.""" |
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[48462b0] | 104 | gauss_scale = 10**np.random.uniform(1, 3) |
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| 105 | lorentz_scale = 10**np.random.uniform(1, 3) |
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| 106 | cor_length_static = 10**np.random.uniform(0, 3) |
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| 107 | cor_length_dynamic = 10**np.random.uniform(0, 3) |
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| 108 | pars = dict( |
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| 109 | #background=0, |
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| 110 | scale=1, |
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| 111 | gauss_scale=gauss_scale, |
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| 112 | lorentz_scale=lorentz_scale, |
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| 113 | cor_length_static=cor_length_static, |
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| 114 | cor_length_dynamic=cor_length_dynamic, |
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| 115 | ) |
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| 116 | return pars |
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| 117 | |
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| 118 | |
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[29da213] | 119 | demo = dict(scale=1, background=0.1, |
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[a807206] | 120 | gauss_scale=100.0, |
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| 121 | cor_length_static=100.0, |
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| 122 | lorentz_scale=50.0, |
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| 123 | cor_length_dynamic=20.0) |
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[29da213] | 124 | |
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[07a6700] | 125 | tests = [ |
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[168052c] | 126 | |
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| 127 | # Accuracy tests based on content in test/utest_extra_models.py |
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[a807206] | 128 | [{'gauss_scale': 100.0, |
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| 129 | 'cor_length_static': 100.0, |
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| 130 | 'lorentz_scale': 50.0, |
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| 131 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 132 | }, 0.001, 149.482], |
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[168052c] | 133 | |
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[a807206] | 134 | [{'gauss_scale': 100.0, |
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| 135 | 'cor_length_static': 100.0, |
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| 136 | 'lorentz_scale': 50.0, |
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| 137 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 138 | }, 0.105363, 9.1913], |
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[168052c] | 139 | |
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[a807206] | 140 | [{'gauss_scale': 100.0, |
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| 141 | 'cor_length_static': 100.0, |
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| 142 | 'lorentz_scale': 50.0, |
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| 143 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 144 | }, 0.441623, 0.633811], |
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[168052c] | 145 | |
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| 146 | # Additional tests with larger range of parameters |
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[a807206] | 147 | [{'gauss_scale': 10.0, |
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| 148 | 'cor_length_static': 100.0, |
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| 149 | 'lorentz_scale': 3.0, |
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| 150 | 'cor_length_dynamic': 1.0, |
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[6dd90c1] | 151 | }, 0.1, 2.9712970297], |
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[168052c] | 152 | |
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[a807206] | 153 | [{'gauss_scale': 10.0, |
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| 154 | 'cor_length_static': 100.0, |
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| 155 | 'lorentz_scale': 3.0, |
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| 156 | 'cor_length_dynamic': 1.0, |
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[168052c] | 157 | 'background': 100.0 |
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[6dd90c1] | 158 | }, 5.0, 100.116384615], |
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[168052c] | 159 | |
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[a807206] | 160 | [{'gauss_scale': 10.0, |
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| 161 | 'cor_length_static': 100.0, |
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| 162 | 'lorentz_scale': 3.0, |
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| 163 | 'cor_length_dynamic': 1.0, |
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[6dd90c1] | 164 | 'background': 0.0, |
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[168052c] | 165 | }, 200., 7.49981250469e-05], |
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| 166 | ] |
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