[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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[168052c] | 34 | G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, |
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| 35 | 42 (2001) 2907-2913 |
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[29da213] | 36 | """ |
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| 37 | |
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[2d81cfe] | 38 | import numpy as np |
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[2c74c11] | 39 | from numpy import inf, exp |
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[29da213] | 40 | |
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| 41 | name = "gauss_lorentz_gel" |
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| 42 | title = "Gauss Lorentz Gel model of scattering from a gel structure" |
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| 43 | description = """ |
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| 44 | Class that evaluates a GaussLorentzGel model. |
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| 45 | |
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| 46 | I(q) = scale_g*exp(- q^2*Z^2 / 2)+scale_l/(1+q^2*z^2) |
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| 47 | + background |
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| 48 | List of default parameters: |
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| 49 | scale_g = Gauss scale factor |
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| 50 | Z = Static correlation length |
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| 51 | scale_l = Lorentzian scale factor |
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| 52 | z = Dynamic correlation length |
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| 53 | background = Incoherent background |
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| 54 | """ |
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| 55 | category = "shape-independent" |
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[168052c] | 56 | # pylint: disable=bad-whitespace, line-too-long |
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[29da213] | 57 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[a807206] | 58 | parameters = [["gauss_scale", "", 100.0, [-inf, inf], "", "Gauss scale factor"], |
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| 59 | ["cor_length_static", "Ang", 100.0, [0, inf], "", "Static correlation length"], |
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| 60 | ["lorentz_scale", "", 50.0, [-inf, inf], "", "Lorentzian scale factor"], |
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| 61 | ["cor_length_dynamic", "Ang", 20.0, [0, inf], "", "Dynamic correlation length"], |
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[168052c] | 62 | ] |
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| 63 | # pylint: enable=bad-whitespace, line-too-long |
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[29da213] | 64 | |
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| 65 | def Iq(q, |
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[a807206] | 66 | gauss_scale=100.0, |
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| 67 | cor_length_static=100.0, |
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| 68 | lorentz_scale=50.0, |
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| 69 | cor_length_dynamic=20.0): |
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[168052c] | 70 | """ |
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| 71 | |
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| 72 | :param q: Input q-value |
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[a807206] | 73 | :param gauss_scale: Gauss scale factor |
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| 74 | :param cor_length_static: Static correlation length |
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| 75 | :param lorentz_scale: Lorentzian scale factor |
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| 76 | :param cor_length_dynamic: Dynamic correlation length |
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[168052c] | 77 | :return: 1-D intensity |
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| 78 | """ |
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| 79 | |
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[a807206] | 80 | term1 = gauss_scale *\ |
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| 81 | exp(-1.0*q*q*cor_length_static*cor_length_static/2.0) |
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| 82 | term2 = lorentz_scale /\ |
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| 83 | (1.0+(q*cor_length_dynamic)*(q*cor_length_dynamic)) |
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[168052c] | 84 | |
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| 85 | return term1 + term2 |
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[29da213] | 86 | |
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| 87 | Iq.vectorized = True # Iq accepts an array of q values |
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| 88 | |
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| 89 | |
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[48462b0] | 90 | def random(): |
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| 91 | gauss_scale = 10**np.random.uniform(1, 3) |
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| 92 | lorentz_scale = 10**np.random.uniform(1, 3) |
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| 93 | cor_length_static = 10**np.random.uniform(0, 3) |
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| 94 | cor_length_dynamic = 10**np.random.uniform(0, 3) |
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| 95 | pars = dict( |
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| 96 | #background=0, |
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| 97 | scale=1, |
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| 98 | gauss_scale=gauss_scale, |
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| 99 | lorentz_scale=lorentz_scale, |
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| 100 | cor_length_static=cor_length_static, |
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| 101 | cor_length_dynamic=cor_length_dynamic, |
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| 102 | ) |
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| 103 | return pars |
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| 104 | |
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| 105 | |
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[29da213] | 106 | demo = dict(scale=1, background=0.1, |
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[a807206] | 107 | gauss_scale=100.0, |
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| 108 | cor_length_static=100.0, |
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| 109 | lorentz_scale=50.0, |
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| 110 | cor_length_dynamic=20.0) |
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[29da213] | 111 | |
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[07a6700] | 112 | tests = [ |
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[168052c] | 113 | |
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| 114 | # Accuracy tests based on content in test/utest_extra_models.py |
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[a807206] | 115 | [{'gauss_scale': 100.0, |
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| 116 | 'cor_length_static': 100.0, |
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| 117 | 'lorentz_scale': 50.0, |
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| 118 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 119 | }, 0.001, 149.482], |
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[168052c] | 120 | |
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[a807206] | 121 | [{'gauss_scale': 100.0, |
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| 122 | 'cor_length_static': 100.0, |
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| 123 | 'lorentz_scale': 50.0, |
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| 124 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 125 | }, 0.105363, 9.1913], |
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[168052c] | 126 | |
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[a807206] | 127 | [{'gauss_scale': 100.0, |
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| 128 | 'cor_length_static': 100.0, |
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| 129 | 'lorentz_scale': 50.0, |
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| 130 | 'cor_length_dynamic': 20.0, |
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[6dd90c1] | 131 | }, 0.441623, 0.633811], |
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[168052c] | 132 | |
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| 133 | # Additional tests with larger range of parameters |
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[a807206] | 134 | [{'gauss_scale': 10.0, |
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| 135 | 'cor_length_static': 100.0, |
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| 136 | 'lorentz_scale': 3.0, |
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| 137 | 'cor_length_dynamic': 1.0, |
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[6dd90c1] | 138 | }, 0.1, 2.9712970297], |
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[168052c] | 139 | |
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[a807206] | 140 | [{'gauss_scale': 10.0, |
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| 141 | 'cor_length_static': 100.0, |
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| 142 | 'lorentz_scale': 3.0, |
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| 143 | 'cor_length_dynamic': 1.0, |
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[168052c] | 144 | 'background': 100.0 |
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[6dd90c1] | 145 | }, 5.0, 100.116384615], |
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[168052c] | 146 | |
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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 | 'background': 0.0, |
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[168052c] | 152 | }, 200., 7.49981250469e-05], |
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| 153 | ] |
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