[c6ca41e] | 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 is a trivial extension of the core_shell_sphere function to include |
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| 6 | *N* shells where the core is filled with solvent and the shells are interleaved |
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[e77872e] | 7 | with layers of solvent. For *N = 1*, this returns the same as the vesicle model, |
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| 8 | except for the normalisation, which here is to outermost volume. |
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| 9 | The shell thicknessess and SLD are constant for all shells as expected for |
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[c6ca41e] | 10 | a multilayer vesicle. |
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| 11 | |
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| 12 | .. figure:: img/multi_shell_geometry.jpg |
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| 13 | |
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| 14 | Geometry of the multilayer_vesicle model. |
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| 15 | |
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[9f60c06] | 16 | See the :ref:`core-shell-sphere` model for more documentation. |
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[c6ca41e] | 17 | |
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[041bc75] | 18 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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| 19 | |
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| 20 | .. math:: |
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| 21 | |
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| 22 | P(q) = \frac{\text{scale.volfraction}}{V_t} F^2(q) + \text{background} |
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| 23 | |
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| 24 | where |
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| 25 | |
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| 26 | .. math:: |
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| 27 | |
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| 28 | F(q) = (\rho_{shell}-\rho_{solv}) \sum_{i=1}^{n\_pairs} \left[ |
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| 29 | 3V(R_i)\frac{\sin(qR_i)-qR_i\cos(qR_i)}{(qR_i)^3} \\ |
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| 30 | - 3V(R_i+t_s)\frac{\sin(q(R_i+t_s))-q(R_i+t_s)\cos(q(R_i+t_s))}{(q(R_i+t_s))^3} |
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| 31 | \right] |
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| 32 | |
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| 33 | |
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| 34 | where $R_i = r_c + (i-1)(t_s + t_w)$ |
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| 35 | |
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| 36 | where $V_t$ is the volume of the whole particle, $V(R)$ is the volume of a sphere |
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| 37 | of radius $R$, $r_c$ is the radius of the core, $\rho_{shell}$ is the scattering length |
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| 38 | density of a shell, $\rho_{solv}$ is the scattering length density of the solvent. |
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| 39 | |
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| 40 | |
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[c6ca41e] | 41 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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| 42 | of the q vector which is defined as: |
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| 43 | |
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| 44 | .. math:: |
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| 45 | |
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| 46 | q = \sqrt{q_x^2 + q_y^2} |
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| 47 | |
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[3b681fa] | 48 | |
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| 49 | The outer most radius |
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[19e7ca7] | 50 | |
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| 51 | $radius + n\_pairs * thick\_shell + (n\_pairs- 1) * thick\_solvent$ |
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| 52 | |
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[3b681fa] | 53 | is used for both the volume fraction normalization and for the |
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| 54 | effective radius for *S(Q)* when $P(Q) * S(Q)$ is applied. |
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[c6ca41e] | 55 | |
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[40a87fa] | 56 | For information about polarised and magnetic scattering, see |
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[9a4811a] | 57 | the :ref:`magnetism` documentation. |
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[c6ca41e] | 58 | |
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| 59 | This code is based on the form factor calculations implemented in the NIST |
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| 60 | Center for Neutron Research provided c-library (Kline, 2006). |
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| 61 | |
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| 62 | References |
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| 63 | ---------- |
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| 64 | |
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| 65 | B Cabane, *Small Angle Scattering Methods*, |
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| 66 | in *Surfactant Solutions: New Methods of Investigation*, |
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| 67 | Ch.2, Surfactant Science Series Vol. 22, Ed. R Zana and M Dekker, |
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| 68 | New York, (1987). |
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| 69 | |
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| 70 | **Author:** NIST IGOR/DANSE **on:** pre 2010 |
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| 71 | |
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| 72 | **Last Modified by:** Piotr Rozyczko**on:** Feb 24, 2016 |
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| 73 | |
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| 74 | **Last Reviewed by:** Paul Butler **on:** March 20, 2016 |
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| 75 | |
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| 76 | """ |
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| 77 | |
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| 78 | from numpy import inf |
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| 79 | |
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| 80 | name = "multilayer_vesicle" |
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| 81 | title = "P(Q) for a Multi-lamellar vesicle" |
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| 82 | description = """ |
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| 83 | multilayer_vesicle model parameters; |
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| 84 | scale : scale factor for abs intensity if needed else 1.0 |
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| 85 | volfraction: volume fraction |
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| 86 | radius : Core radius of the multishell |
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| 87 | thick_shell: shell thickness |
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| 88 | thick_solvent: water thickness |
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| 89 | sld_solvent: solvent scattering length density |
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| 90 | sld: shell scattering length density |
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[e77872e] | 91 | n_pairs:number of "shell plus solvent" layer pairs |
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[c6ca41e] | 92 | background: incoherent background |
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| 93 | """ |
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| 94 | category = "shape:sphere" |
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| 95 | |
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| 96 | # pylint: disable=bad-whitespace, line-too-long |
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| 97 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 98 | parameters = [ |
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| 99 | ["volfraction", "", 0.05, [0.0, 1], "", "volume fraction of vesicles"], |
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[e77872e] | 100 | ["radius", "Ang", 60.0, [0.0, inf], "", "radius of solvent filled core"], |
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| 101 | ["thick_shell", "Ang", 10.0, [0.0, inf], "", "thickness of one shell"], |
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| 102 | ["thick_solvent", "Ang", 10.0, [0.0, inf], "", "solvent thickness between shells"], |
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| 103 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld", "solvent scattering length density"], |
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[42356c8] | 104 | ["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "sld", "Shell scattering length density"], |
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[e77872e] | 105 | ["n_pairs", "", 2.0, [1.0, inf], "", "Number of shell plus solvent layer pairs"], |
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[c6ca41e] | 106 | ] |
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| 107 | # pylint: enable=bad-whitespace, line-too-long |
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| 108 | |
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[925ad6e] | 109 | source = ["lib/sas_3j1x_x.c", "multilayer_vesicle.c"] |
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[c6ca41e] | 110 | |
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| 111 | polydispersity = ["radius", "n_pairs"] |
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| 112 | |
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| 113 | demo = dict(scale=1, background=0, |
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| 114 | volfraction=0.05, |
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| 115 | radius=60.0, |
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| 116 | thick_shell=10.0, |
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| 117 | thick_solvent=10.0, |
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| 118 | sld_solvent=6.4, |
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| 119 | sld=0.4, |
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| 120 | n_pairs=2.0) |
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| 121 | |
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| 122 | tests = [ |
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| 123 | # Accuracy tests based on content in test/utest_other_models.py |
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| 124 | [{'radius': 60.0, |
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| 125 | 'thick_shell': 10.0, |
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| 126 | 'thick_solvent': 10.0, |
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| 127 | 'sld_solvent': 6.4, |
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| 128 | 'sld': 0.4, |
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| 129 | 'n_pairs': 2.0, |
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| 130 | 'scale': 1.0, |
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| 131 | 'background': 0.001, |
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| 132 | }, 0.001, 122.1405], |
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| 133 | |
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| 134 | [{'volfraction': 1.0, |
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| 135 | 'radius': 60.0, |
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| 136 | 'thick_shell': 10.0, |
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| 137 | 'thick_solvent': 10.0, |
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| 138 | 'sld_solvent': 6.4, |
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| 139 | 'sld': 0.4, |
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| 140 | 'n_pairs': 2.0, |
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| 141 | 'scale': 1.0, |
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| 142 | 'background': 0.001, |
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| 143 | }, (0.001, 0.30903), 1.61873], |
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| 144 | ] |
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