[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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| 5 | give a functional form similar to Guinier scattering, so interpret with |
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| 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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| 38 | |
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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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| 58 | parameters = [["gauss_scale_factor", "", 100.0, [-inf, inf], "", "Gauss scale factor"], |
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| 59 | ["static_cor_length", "Ang", 100.0, [0, inf], "", "Static correlation length"], |
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| 60 | ["lorentz_scale_factor", "", 50.0, [-inf, inf], "", "Lorentzian scale factor"], |
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| 61 | ["dynamic_cor_length", "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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[168052c] | 66 | gauss_scale_factor=100.0, |
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| 67 | static_cor_length=100.0, |
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| 68 | lorentz_scale_factor=50.0, |
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| 69 | dynamic_cor_length=20.0): |
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| 70 | """ |
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| 71 | |
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| 72 | :param q: Input q-value |
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| 73 | :param gauss_scale_factor: Gauss scale factor |
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| 74 | :param static_cor_length: Static correlation length |
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| 75 | :param lorentz_scale_factor: Lorentzian scale factor |
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| 76 | :param dynamic_cor_length: Dynamic correlation length |
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| 77 | :return: 1-D intensity |
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| 78 | """ |
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| 79 | |
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| 80 | term1 = gauss_scale_factor *\ |
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| 81 | exp(-1.0*q*q*static_cor_length*static_cor_length/2.0) |
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| 82 | term2 = lorentz_scale_factor /\ |
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| 83 | (1.0+(q*dynamic_cor_length)*(q*dynamic_cor_length)) |
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| 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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| 90 | demo = dict(scale=1, background=0.1, |
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| 91 | gauss_scale_factor=100.0, |
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| 92 | static_cor_length=100.0, |
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| 93 | lorentz_scale_factor=50.0, |
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| 94 | dynamic_cor_length=20.0) |
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| 95 | |
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[07a6700] | 96 | tests = [ |
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[168052c] | 97 | |
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| 98 | # Accuracy tests based on content in test/utest_extra_models.py |
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| 99 | [{'gauss_scale_factor': 100.0, |
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| 100 | 'static_cor_length': 100.0, |
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| 101 | 'lorentz_scale_factor': 50.0, |
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| 102 | 'dynamic_cor_length': 20.0, |
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[6dd90c1] | 103 | }, 0.001, 149.482], |
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[168052c] | 104 | |
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| 105 | [{'gauss_scale_factor': 100.0, |
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| 106 | 'static_cor_length': 100.0, |
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| 107 | 'lorentz_scale_factor': 50.0, |
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| 108 | 'dynamic_cor_length': 20.0, |
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[6dd90c1] | 109 | }, 0.105363, 9.1913], |
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[168052c] | 110 | |
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| 111 | [{'gauss_scale_factor': 100.0, |
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| 112 | 'static_cor_length': 100.0, |
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| 113 | 'lorentz_scale_factor': 50.0, |
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| 114 | 'dynamic_cor_length': 20.0, |
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[6dd90c1] | 115 | }, 0.441623, 0.633811], |
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[168052c] | 116 | |
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| 117 | # Additional tests with larger range of parameters |
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| 118 | [{'gauss_scale_factor': 10.0, |
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| 119 | 'static_cor_length': 100.0, |
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| 120 | 'lorentz_scale_factor': 3.0, |
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| 121 | 'dynamic_cor_length': 1.0, |
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[6dd90c1] | 122 | }, 0.1, 2.9712970297], |
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[168052c] | 123 | |
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| 124 | [{'gauss_scale_factor': 10.0, |
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| 125 | 'static_cor_length': 100.0, |
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| 126 | 'lorentz_scale_factor': 3.0, |
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| 127 | 'dynamic_cor_length': 1.0, |
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| 128 | 'background': 100.0 |
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[6dd90c1] | 129 | }, 5.0, 100.116384615], |
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[168052c] | 130 | |
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| 131 | [{'gauss_scale_factor': 10.0, |
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| 132 | 'static_cor_length': 100.0, |
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| 133 | 'lorentz_scale_factor': 3.0, |
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| 134 | 'dynamic_cor_length': 1.0, |
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[6dd90c1] | 135 | 'background': 0.0, |
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[168052c] | 136 | }, 200., 7.49981250469e-05], |
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| 137 | ] |
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