1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model calculates the scattered intensity of a two-component system |
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6 | using the Teubner-Strey model. Unlike :ref:`dab` this function generates |
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7 | a peak. A two-phase material can be characterised by two length scales - |
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8 | a correlation length and a domain size (periodicity). |
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9 | |
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10 | The original paper by Teubner and Strey defined the function as: |
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11 | |
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12 | .. math:: |
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13 | |
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14 | I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background} |
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15 | |
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16 | where the parameters $a_2$, $c_1$ and $c_2$ are defined in terms of the |
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17 | periodicity, $d$, and correlation length $\xi$ as: |
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18 | |
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19 | .. math:: |
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20 | |
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21 | a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\ |
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22 | c_1 &= -2\xi^2\bigl(\frac{2\pi\xi}{d}\bigr)^2+2\xi^2\\ |
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23 | c_2 &= \xi^4 |
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24 | |
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25 | and thus, the periodicity, $d$ is given by |
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26 | |
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27 | .. math:: |
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28 | |
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29 | d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} |
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30 | - \frac14\frac{c_1}{c_2}\right]^{-1/2} |
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31 | |
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32 | and the correlation length, $\xi$, is given by |
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33 | |
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34 | .. math:: |
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35 | |
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36 | \xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} |
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37 | + \frac14\frac{c_1}{c_2}\right]^{-1/2} |
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38 | |
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39 | Here the model is parameterised in terms of $d$ and $\xi$ and with an explicit |
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40 | volume fraction for one phase, $\phi_a$, and contrast, |
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41 | $\delta\rho^2 = (\rho_a - \rho_b)^2$ : |
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42 | |
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43 | .. math:: |
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44 | |
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45 | I(q) = \frac{8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi} |
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46 | {a_2 + c_1q^2 + c_2q^4} |
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47 | |
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48 | where :math:`8\pi\phi_a(1-\phi_a)(\Delta\rho)^2c_2/\xi` is the constant of |
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49 | proportionality from the first equation above. |
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50 | |
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51 | In the case of a microemulsion, $a_2 > 0$, $c_1 < 0$, and $c_2 >0$. |
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52 | |
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53 | For 2D data, scattering intensity is calculated in the same way as 1D, |
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54 | where the $q$ vector is defined as |
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55 | |
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56 | .. math:: |
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57 | |
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58 | q = \sqrt{q_x^2 + q_y^2} |
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59 | |
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60 | References |
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61 | ---------- |
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62 | |
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63 | .. [#] M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195 |
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64 | .. [#] K V Schubert, R Strey, S R Kline and E W Kaler, *J. Chem. Phys.*, 101 (1994) 5343 |
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65 | .. [#] H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter, B Jakobs, T Sottmann, R Strey, and I Grillo, *J. Chem. Phys.*, 115 (2001), 580 |
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66 | |
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67 | Source |
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68 | ------ |
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69 | |
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70 | `teubner_strey.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/teubner_strey.py>`_ |
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71 | |
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72 | `teubner_strey.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/teubner_strey.c>`_ |
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73 | |
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74 | Authorship and Verification |
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75 | ---------------------------- |
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76 | |
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77 | * **Author:** |
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78 | * **Last Modified by:** |
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79 | * **Last Reviewed by:** |
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80 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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81 | """ |
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82 | from __future__ import division |
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83 | |
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84 | import numpy as np |
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85 | from numpy import inf, pi |
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86 | |
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87 | name = "teubner_strey" |
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88 | title = "Teubner-Strey model of microemulsions" |
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89 | description = """\ |
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90 | Calculates scattering according to the Teubner-Strey model |
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91 | """ |
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92 | category = "shape-independent" |
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93 | |
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94 | # ["name", "units", default, [lower, upper], "type","description"], |
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95 | parameters = [ |
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96 | ["volfraction_a", "", 0.5, [0, 1.0], "", "Volume fraction of phase a"], |
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97 | ["sld_a", "1e-6/Ang^2", 0.3, [-inf, inf], "", "SLD of phase a"], |
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98 | ["sld_b", "1e-6/Ang^2", 6.3, [-inf, inf], "", "SLD of phase b"], |
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99 | ["d", "Ang", 100.0, [0, inf], "", "Domain size (periodicity)"], |
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100 | ["xi", "Ang", 30.0, [0, inf], "", "Correlation length"], |
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101 | ] |
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102 | |
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103 | def Iq(q, volfraction_a, sld_a, sld_b, d, xi): |
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104 | """SAS form""" |
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105 | drho = sld_a - sld_b |
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106 | k = 2.0*pi*xi/d |
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107 | a2 = (1.0 + k**2)**2 |
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108 | c1 = 2.0*xi**2 * (1.0 - k**2) |
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109 | c2 = xi**4 |
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110 | prefactor = 8.0*pi * volfraction_a*(1.0 - volfraction_a) * drho**2 * c2/xi |
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111 | return 1.0e-4*prefactor / np.polyval([c2, c1, a2], q**2) |
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112 | |
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113 | Iq.vectorized = True # Iq accepts an array of q values |
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114 | |
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115 | def random(): |
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116 | """Return a random parameter set for the model.""" |
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117 | d = 10**np.random.uniform(1, 4) |
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118 | xi = 10**np.random.uniform(-0.3, 2)*d |
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119 | pars = dict( |
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120 | #background=0, |
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121 | scale=100, |
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122 | volfraction_a=10**np.random.uniform(-3, 0), |
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123 | sld_a=np.random.uniform(-0.5, 12), |
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124 | sld_b=np.random.uniform(-0.5, 12), |
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125 | d=d, |
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126 | xi=xi, |
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127 | ) |
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128 | return pars |
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129 | |
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130 | demo = dict(scale=1, background=0, volfraction_a=0.5, |
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131 | sld_a=0.3, sld_b=6.3, |
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132 | d=100.0, xi=30.0) |
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133 | tests = [[{}, 0.06, 41.5918888453]] |
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