[cbb54e2] | 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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[caddb14] | 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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[cbb54e2] | 11 | |
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| 12 | .. math:: |
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| 13 | |
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[caddb14] | 14 | I(q) \propto \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background} |
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[cbb54e2] | 15 | |
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[caddb14] | 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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[cbb54e2] | 18 | |
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| 19 | .. math:: |
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| 20 | |
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[b3f2a24] | 21 | a_2 &= \biggl[1+\bigl(\frac{2\pi\xi}{d}\bigr)^2\biggr]^2\\ |
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[caddb14] | 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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[cbb54e2] | 24 | |
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[caddb14] | 25 | and thus, the periodicity, $d$ is given by |
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[cbb54e2] | 26 | |
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[caddb14] | 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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[cbb54e2] | 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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[caddb14] | 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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[cbb54e2] | 44 | |
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[caddb14] | 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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[cbb54e2] | 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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[eb69cce] | 60 | References |
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| 61 | ---------- |
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[cbb54e2] | 62 | |
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| 63 | M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195 |
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| 64 | |
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| 65 | K V Schubert, R Strey, S R Kline and E W Kaler, |
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| 66 | *J. Chem. Phys.*, 101 (1994) 5343 |
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| 67 | |
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[caddb14] | 68 | H Endo, M Mihailescu, M. Monkenbusch, J Allgaier, G Gompper, D Richter, |
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| 69 | B Jakobs, T Sottmann, R Strey, and I Grillo, *J. Chem. Phys.*, 115 (2001), 580 |
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[cbb54e2] | 70 | """ |
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[8393c74] | 71 | from __future__ import division |
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[cbb54e2] | 72 | |
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| 73 | import numpy as np |
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[4962519] | 74 | from numpy import inf, pi |
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[cbb54e2] | 75 | |
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| 76 | name = "teubner_strey" |
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| 77 | title = "Teubner-Strey model of microemulsions" |
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| 78 | description = """\ |
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[caddb14] | 79 | Calculates scattering according to the Teubner-Strey model |
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[cbb54e2] | 80 | """ |
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| 81 | category = "shape-independent" |
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| 82 | |
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[40a87fa] | 83 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 84 | parameters = [ |
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[caddb14] | 85 | ["volfraction_a", "", 0.5, [0, 1.0], "", "Volume fraction of phase a"], |
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| 86 | ["sld_a", "1e-6/Ang^2", 0.3, [-inf, inf], "", "SLD of phase a"], |
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| 87 | ["sld_b", "1e-6/Ang^2", 6.3, [-inf, inf], "", "SLD of phase b"], |
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| 88 | ["d", "Ang", 100.0, [0, inf], "", "Domain size (periodicity)"], |
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| 89 | ["xi", "Ang", 30.0, [0, inf], "", "Correlation length"], |
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[40a87fa] | 90 | ] |
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[cbb54e2] | 91 | |
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[8393c74] | 92 | def Iq(q, volfraction_a, sld_a, sld_b, d, xi): |
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[40a87fa] | 93 | """SAS form""" |
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[8393c74] | 94 | drho = sld_a - sld_b |
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[b3f2a24] | 95 | k = 2.0*pi*xi/d |
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[4962519] | 96 | a2 = (1.0 + k**2)**2 |
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| 97 | c1 = 2.0*xi**2 * (1.0 - k**2) |
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| 98 | c2 = xi**4 |
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[8393c74] | 99 | prefactor = 8.0*pi * volfraction_a*(1.0 - volfraction_a) * drho**2 * c2/xi |
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[caddb14] | 100 | return 1.0e-4*prefactor / np.polyval([c2, c1, a2], q**2) |
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| 101 | |
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[eb69cce] | 102 | Iq.vectorized = True # Iq accepts an array of q values |
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[cbb54e2] | 103 | |
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[48462b0] | 104 | def random(): |
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| 105 | import numpy as np |
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| 106 | d = 10**np.random.uniform(1, 4) |
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| 107 | xi = 10**np.random.uniform(-0.3, 2)*d |
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| 108 | pars = dict( |
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| 109 | #background=0, |
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| 110 | scale=100, |
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| 111 | volfraction_a=10**np.random.uniform(-3, 0), |
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| 112 | sld_a=np.random.uniform(-0.5, 12), |
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| 113 | sld_b=np.random.uniform(-0.5, 12), |
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| 114 | d=d, |
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| 115 | xi=xi, |
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| 116 | ) |
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| 117 | return pars |
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| 118 | |
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[caddb14] | 119 | demo = dict(scale=1, background=0, volfraction_a=0.5, |
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[48462b0] | 120 | sld_a=0.3, sld_b=6.3, |
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| 121 | d=100.0, xi=30.0) |
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[caddb14] | 122 | tests = [[{}, 0.06, 41.5918888453]] |
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