[5d4777d] | 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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[40a87fa] | 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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[5d4777d] | 9 | |
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| 10 | .. math:: |
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| 11 | |
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[500128b] | 12 | I(q) = \text{scale}\frac{2\pi P(q)}{q^2\delta} + \text{background} |
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[5d4777d] | 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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[500128b] | 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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[5d4777d] | 21 | |
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[40a87fa] | 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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[5d4777d] | 24 | |
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[40a87fa] | 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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[5d4777d] | 30 | |
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[eb69cce] | 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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[5d4777d] | 33 | |
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| 34 | .. math:: |
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| 35 | |
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[eb69cce] | 36 | q = \sqrt{q_x^2 + q_y^2} |
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[5d4777d] | 37 | |
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| 38 | |
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[eb69cce] | 39 | References |
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| 40 | ---------- |
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[5d4777d] | 41 | |
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[0507e09] | 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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[5d4777d] | 44 | |
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[0507e09] | 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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[5d4777d] | 57 | """ |
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| 58 | |
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[2d81cfe] | 59 | import numpy as np |
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[3c56da87] | 60 | from numpy import inf |
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[5d4777d] | 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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[3e428ec] | 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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[5d4777d] | 73 | """ |
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[a5d0d00] | 74 | category = "shape:lamellae" |
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[5d4777d] | 75 | |
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[40a87fa] | 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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[5d4777d] | 84 | |
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| 85 | # No volume normalization despite having a volume parameter |
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[7c391dd] | 86 | # This should perhaps be volume normalized? - it is! |
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[5d4777d] | 87 | form_volume = """ |
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[994d77f] | 88 | return 1.0; |
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[5d4777d] | 89 | """ |
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| 90 | |
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| 91 | Iq = """ |
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[7c391dd] | 92 | const double sub = sld - sld_solvent; |
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[994d77f] | 93 | const double qsq = q*q; |
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[38d8774] | 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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[5d4777d] | 100 | """ |
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| 101 | |
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[404ebbd] | 102 | def random(): |
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[b297ba9] | 103 | """Return a random parameter set for the model.""" |
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[404ebbd] | 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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[3e428ec] | 110 | demo = dict(scale=1, background=0, |
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[7c391dd] | 111 | sld=6, sld_solvent=1, |
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[3e428ec] | 112 | thickness=40, |
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| 113 | thickness_pd=0.2, thickness_pd_n=40) |
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[40a87fa] | 114 | # [(qx1, qy1), (qx2, qy2), ...], [I(qx1,qy1), I(qx2,qy2), ...]], |
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[2d81cfe] | 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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