1 | r""" |
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2 | Polydispersity in the bilayer thickness can be applied from the GUI. |
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3 | |
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4 | Definition |
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5 | ---------- |
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6 | |
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7 | The scattering intensity $I(q)$ for dilute, randomly oriented, |
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8 | "infinitely large" sheets or lamellae is |
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9 | |
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10 | .. math:: |
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11 | |
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12 | I(q) = \text{scale}\frac{2\pi P(q)}{q^2\delta} + \text{background} |
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13 | |
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14 | |
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15 | The form factor is |
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16 | |
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17 | .. math:: |
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18 | |
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19 | P(q) = \frac{2\Delta\rho^2}{q^2}(1-\cos(q\delta)) |
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20 | = \frac{4\Delta\rho^2}{q^2}\sin^2\left(\frac{q\delta}{2}\right) |
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21 | |
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22 | where $\delta$ is the total layer thickness and $\Delta\rho$ is the |
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23 | scattering length density difference. |
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24 | |
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25 | This is the limiting form for a spherical shell of infinitely large radius. |
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26 | Note that the division by $\delta$ means that $scale$ in sasview is the |
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27 | volume fraction of sheet, $\phi = S\delta$ where $S$ is the area of sheet |
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28 | per unit volume. $S$ is half the Porod surface area per unit volume of a |
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29 | thicker layer (as that would include both faces of the sheet). |
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30 | |
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31 | The 2D scattering intensity is calculated in the same way as 1D, where |
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32 | the $q$ vector is defined as |
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33 | |
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34 | .. math:: |
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35 | |
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36 | q = \sqrt{q_x^2 + q_y^2} |
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37 | |
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38 | |
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39 | References |
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40 | ---------- |
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41 | |
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42 | .. [#] F Nallet, R Laversanne, and D Roux, *J. Phys. II France*, 3, (1993) 487-502 |
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43 | .. [#] J Berghausen, J Zipfel, P Lindner, W Richtering, *J. Phys. Chem. B*, 105, (2001) 11081-11088 |
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44 | |
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45 | Source |
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46 | ------ |
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47 | |
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48 | `lamellar.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/lamellar.py>`_ |
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49 | |
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50 | Authorship and Verification |
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51 | ---------------------------- |
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52 | |
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53 | * **Author:** |
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54 | * **Last Modified by:** |
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55 | * **Last Reviewed by:** |
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56 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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57 | """ |
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58 | |
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59 | import numpy as np |
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60 | from numpy import inf |
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61 | |
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62 | name = "lamellar" |
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63 | title = "Lyotropic lamellar phase with uniform SLD and random distribution" |
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64 | description = """\ |
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65 | [Dilute Lamellar Form Factor](from a lyotropic lamellar phase) |
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66 | I(q)= 2*pi*P(q)/(delta *q^(2)), where |
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67 | P(q)=2*(contrast/q)^(2)*(1-cos(q*delta))^(2)) |
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68 | thickness = layer thickness |
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69 | sld = layer scattering length density |
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70 | sld_solvent = solvent scattering length density |
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71 | background = incoherent background |
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72 | scale = scale factor |
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73 | """ |
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74 | category = "shape:lamellae" |
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75 | |
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76 | # pylint: disable=bad-whitespace, line-too-long |
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77 | # ["name", "units", default, [lower, upper], "type","description"], |
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78 | parameters = [ |
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79 | ["thickness", "Ang", 50, [0, inf], "volume", "total layer thickness" ], |
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80 | ["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Layer scattering length density" ], |
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81 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density" ], |
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82 | ] |
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83 | # pylint: enable=bad-whitespace, line-too-long |
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84 | |
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85 | # No volume normalization despite having a volume parameter |
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86 | # This should perhaps be volume normalized? - it is! |
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87 | form_volume = """ |
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88 | return 1.0; |
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89 | """ |
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90 | |
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91 | Iq = """ |
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92 | const double sub = sld - sld_solvent; |
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93 | const double qsq = q*q; |
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94 | // Original expression |
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95 | //return 4.0e-4*M_PI*sub*sub/qsq * (1.0-cos(q*thickness)) / (thickness*qsq); |
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96 | // const double alpha = fmod(q*thickness+0.1, 2.0*M_PI)-0.1; |
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97 | // Use small angle fix 1-cos(theta) = 2 sin^2(theta/2) |
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98 | const double sinq2 = sin(0.5*q*thickness); |
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99 | return 4.0e-4*M_PI*sub*sub/qsq * 2.0*sinq2*sinq2 / (thickness*qsq); |
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100 | """ |
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101 | |
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102 | def random(): |
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103 | """Return a random parameter set for the model.""" |
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104 | thickness = 10**np.random.uniform(1, 4) |
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105 | pars = dict( |
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106 | thickness=thickness, |
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107 | ) |
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108 | return pars |
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109 | |
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110 | demo = dict(scale=1, background=0, |
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111 | sld=6, sld_solvent=1, |
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112 | thickness=40, |
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113 | thickness_pd=0.2, thickness_pd_n=40) |
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114 | # [(qx1, qy1), (qx2, qy2), ...], [I(qx1,qy1), I(qx2,qy2), ...]], |
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115 | tests = [ |
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116 | [{'scale': 1.0, 'background': 0.0, 'thickness': 50.0, |
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117 | 'sld': 1.0, 'sld_solvent': 6.3, 'thickness_pd': 0.0}, |
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118 | [0.001], [882289.54309]] |
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119 | ] |
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120 | # ADDED by: converted by PAK? (or RKH?) |
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121 | # ON: 16Mar2016 - RKH adding unit tests from sasview to early 2015 conversion |
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