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. |
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8 | |
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9 | .. math:: |
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10 | |
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11 | I(q) = \frac{1}{a_2 + c_1 q^2 + c_2 q^4} + \text{background} |
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12 | |
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13 | The parameters $a_2$, $c_1$ and $c_2$ can be used to determine the |
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14 | characteristic domain size $d$, |
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15 | |
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16 | .. math:: |
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17 | |
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18 | d = 2\pi\left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} |
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19 | + \frac14\frac{c_1}{c_2}\right]^{-1/2} |
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20 | |
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21 | |
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22 | and the correlation length $\xi$, |
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23 | |
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24 | .. math:: |
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25 | |
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26 | \xi = \left[\frac12\left(\frac{a_2}{c_2}\right)^{1/2} |
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27 | - \frac14\frac{c_1}{c_2}\right]^{-1/2} |
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28 | |
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29 | |
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30 | For 2D data, scattering intensity is calculated in the same way as 1D, |
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31 | where the $q$ vector is defined as |
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32 | |
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33 | .. math:: |
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34 | |
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35 | q = \sqrt{q_x^2 + q_y^2} |
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36 | |
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37 | |
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38 | .. figure:: img/teubner_strey_1d.jpg |
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39 | |
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40 | 1D plot using the default values (w/200 data point). |
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41 | |
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42 | References |
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43 | ---------- |
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44 | |
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45 | M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195 |
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46 | |
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47 | K V Schubert, R Strey, S R Kline and E W Kaler, |
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48 | *J. Chem. Phys.*, 101 (1994) 5343 |
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49 | |
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50 | """ |
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51 | |
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52 | import numpy as np |
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53 | from numpy import inf, sqrt |
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54 | |
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55 | name = "teubner_strey" |
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56 | title = "Teubner-Strey model of microemulsions" |
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57 | description = """\ |
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58 | Scattering model class for the Teubner-Strey model given by |
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59 | Provide F(x) = 1/( a2 + c1 q^2+ c2 q^4 ) + background |
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60 | a2>0, c1<0, c2>0, 4 a2 c2 - c1^2 > 0 |
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61 | """ |
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62 | category = "shape-independent" |
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63 | |
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64 | # ["name", "units", default, [lower, upper], "type","description"], |
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65 | parameters = [["a2", "", 0.1, [0, inf], "", |
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66 | "a2"], |
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67 | ["c1", "1e-6/Ang^2", -30., [-inf, 0], "", |
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68 | "c1"], |
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69 | ["c2", "Ang", 5000., [0, inf], "volume", |
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70 | "c2"], |
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71 | ] |
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72 | |
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73 | |
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74 | def form_volume(radius): |
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75 | return 1.0 |
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76 | |
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77 | def Iq(q, a2, c1, c2): |
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78 | return 1. / np.polyval([c2, c1, a2], q**2) |
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79 | Iq.vectorized = True # Iq accepts an array of q values |
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80 | |
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81 | def Iqxy(qx, qy, a2, c1, c2): |
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82 | return Iq(sqrt(qx**2 + qy**2), a2, c1, c2) |
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83 | Iqxy.vectorized = True # Iqxy accepts arrays of qx, qy values |
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84 | |
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85 | # ER defaults to 0.0 |
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86 | |
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87 | # VR defaults to 1.0 |
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88 | |
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89 | demo = dict(scale=1, background=0, a2=0.1, c1=-30.0, c2=5000.0) |
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90 | oldname = "TeubnerStreyModel" |
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91 | oldpars = dict(a2='scale') |
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92 | |
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93 | tests = [[{}, 0.2, 0.144927536232]] |
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