[9da407c] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[693570c] | 5 | The form factor for this bent disc is essentially that of a hyperbolic |
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| 6 | paraboloid and calculated as |
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[9da407c] | 7 | |
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| 8 | .. math:: |
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| 9 | |
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[693570c] | 10 | P(q) = (\Delta \rho )^2 V \int^{\pi/2}_0 d\psi \sin{\psi} sinc^2 |
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[9da407c] | 11 | \left( \frac{qd\cos{\psi}}{2} \right) |
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| 12 | \left[ \left( S^2_0+C^2_0\right) + 2\sum_{n=1}^{\infty} |
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[693570c] | 13 | \left( S^2_n+C^2_n\right) \right] |
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[9da407c] | 14 | |
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| 15 | where |
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| 16 | |
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| 17 | .. math:: |
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| 18 | |
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[58c3367] | 19 | C_n = \frac{1}{r^2}\int^{R}_{0} r dr\cos(qr^2\alpha \cos{\psi}) |
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[9da407c] | 20 | J_n\left( qr^2\beta \cos{\psi}\right) |
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| 21 | J_{2n}\left( qr \sin{\psi}\right) |
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| 22 | |
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| 23 | .. math:: |
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| 24 | |
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[58c3367] | 25 | S_n = \frac{1}{r^2}\int^{R}_{0} r dr\sin(qr^2\alpha \cos{\psi}) |
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[9da407c] | 26 | J_n\left( qr^2\beta \cos{\psi}\right) |
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| 27 | J_{2n}\left( qr \sin{\psi}\right) |
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| 28 | |
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[693570c] | 29 | and $\Delta \rho \text{ is } \rho_{pringle}-\rho_{solvent}$, $V$ is the volume of |
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| 30 | the disc, $\psi$ is the angle between the normal to the disc and the q vector, |
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| 31 | $d$ and $R$ are the "pringle" thickness and radius respectively, $\alpha$ and |
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| 32 | $\beta$ are the two curvature parameters, and $J_n$ is the n\ :sup:`th` order |
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[40a87fa] | 33 | Bessel function of the first kind. |
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[9da407c] | 34 | |
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[693570c] | 35 | .. figure:: img/pringles_fig1.png |
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[9da407c] | 36 | |
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[693570c] | 37 | Schematic of model shape (Graphic from Matt Henderson, matt@matthen.com) |
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[9da407c] | 38 | |
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| 39 | Reference |
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| 40 | --------- |
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| 41 | |
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[0507e09] | 42 | .. [#] Karen Edler, Universtiy of Bath, Private Communication. 2012. Derivation by Stefan Alexandru Rautu. |
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| 43 | .. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659 |
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| 44 | |
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| 45 | Authorship and Verification |
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| 46 | ---------------------------- |
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[693570c] | 47 | |
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[ef07e95] | 48 | * **Author:** Andrew Jackson **Date:** 2008 |
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| 49 | * **Last Modified by:** Wojciech Wpotrzebowski **Date:** March 20, 2016 |
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| 50 | * **Last Reviewed by:** Andrew Jackson **Date:** September 26, 2016 |
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[9da407c] | 51 | """ |
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| 52 | |
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[2d81cfe] | 53 | import numpy as np |
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[b297ba9] | 54 | from numpy import inf |
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[9da407c] | 55 | |
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[693570c] | 56 | name = "pringle" |
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| 57 | title = "The Pringle model provides the form factor, $P(q)$, for a 'pringle' \ |
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| 58 | or 'saddle-shaped' disc that is bent in two directions." |
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[9da407c] | 59 | description = """\ |
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| 60 | |
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| 61 | """ |
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| 62 | category = "shape:cylinder" |
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| 63 | |
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| 64 | # pylint: disable=bad-whitespace, line-too-long |
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| 65 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 66 | parameters = [ |
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| 67 | ["radius", "Ang", 60.0, [0, inf], "volume", "Pringle radius"], |
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| 68 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Thickness of pringle"], |
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[c047acf] | 69 | ["alpha", "", 0.001, [-inf, inf], "volume", "Curvature parameter alpha"], |
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| 70 | ["beta", "", 0.02, [-inf, inf], "volume", "Curvature paramter beta"], |
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[30fbe2e] | 71 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Pringle sld"], |
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[42356c8] | 72 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent sld"] |
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[9da407c] | 73 | ] |
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| 74 | # pylint: enable=bad-whitespace, line-too-long |
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| 75 | |
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[c047acf] | 76 | |
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[ee60aa7] | 77 | source = ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", |
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[b70cb3b] | 78 | "lib/sas_JN.c", "lib/gauss76.c", "pringle.c"] |
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[a34b811] | 79 | radius_effective_modes = [ |
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[b297ba9] | 80 | "equivalent cylinder excluded volume", |
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| 81 | "equivalent volume sphere", |
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| 82 | "radius"] |
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[9da407c] | 83 | |
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[8f04da4] | 84 | def random(): |
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[b297ba9] | 85 | """Return a random parameter set for the model.""" |
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[8f04da4] | 86 | alpha, beta = 10**np.random.uniform(-1, 1, size=2) |
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| 87 | radius = 10**np.random.uniform(1, 3) |
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| 88 | thickness = 10**np.random.uniform(0.7, 2) |
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| 89 | pars = dict( |
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| 90 | radius=radius, |
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| 91 | thickness=thickness, |
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| 92 | alpha=alpha, |
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| 93 | beta=beta, |
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| 94 | ) |
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| 95 | return pars |
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[9da407c] | 96 | |
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| 97 | tests = [ |
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| 98 | [{'scale' : 1.0, |
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| 99 | 'radius': 60.0, |
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| 100 | 'thickness': 10.0, |
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| 101 | 'alpha': 0.001, |
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| 102 | 'beta': 0.02, |
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[30fbe2e] | 103 | 'sld': 1.0, |
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[9da407c] | 104 | 'sld_solvent': 6.3, |
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[639042e] | 105 | 'background': 0.001, |
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[5fd684d] | 106 | }, 0.1, 9.87676], |
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[9da407c] | 107 | |
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| 108 | [{'scale' : 1.0, |
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| 109 | 'radius': 60.0, |
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| 110 | 'thickness': 10.0, |
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| 111 | 'alpha': 0.001, |
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| 112 | 'beta': 0.02, |
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[30fbe2e] | 113 | 'sld': 1.0, |
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[9da407c] | 114 | 'sld_solvent': 6.3, |
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[639042e] | 115 | 'background': 0.001, |
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[5fd684d] | 116 | }, 0.01, 290.56723], |
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[9da407c] | 117 | |
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| 118 | [{'scale' : 1.0, |
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| 119 | 'radius': 60.0, |
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| 120 | 'thickness': 10.0, |
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| 121 | 'alpha': 0.001, |
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| 122 | 'beta': 0.02, |
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[30fbe2e] | 123 | 'sld': 1.0, |
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[9da407c] | 124 | 'sld_solvent': 6.3, |
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[5fd684d] | 125 | 'background': 0.001, |
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| 126 | }, 0.001, 317.40847], |
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[91b8b80] | 127 | ] |
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