[9da407c] | 1 | r""" |
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| 2 | This model provides the form factor, $P(q)$, for a 'pringle' or 'saddle-shaped' |
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| 3 | object (a hyperbolic paraboloid). |
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| 4 | |
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| 5 | .. figure:: img/pringles_fig1.png |
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| 6 | |
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| 7 | (Graphic from Matt Henderson, matt@matthen.com) |
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| 8 | |
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[1805a9e] | 9 | The returned value is in units of cm^-1, on absolute scale. |
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[9da407c] | 10 | |
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| 11 | Definition |
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| 12 | ---------- |
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| 13 | |
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| 14 | The form factor calculated is |
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| 15 | |
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| 16 | .. math:: |
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| 17 | |
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| 18 | I(q) = (\Delta \rho )^2 V \int^{\pi/2}_0 d\psi \sin{\psi} sinc^2 |
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| 19 | \left( \frac{qd\cos{\psi}}{2} \right) |
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| 20 | \left[ \left( S^2_0+C^2_0\right) + 2\sum_{n=1}^{\infty} |
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| 21 | \left( S^2_0+C^2_0\right) \right] |
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| 22 | |
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| 23 | where |
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| 24 | |
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| 25 | .. math:: |
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| 26 | |
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| 27 | C_n = \int^{R}_{0} r dr\cos{qr^2\alpha \cos{\psi}} |
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| 28 | J_n\left( qr^2\beta \cos{\psi}\right) |
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| 29 | J_{2n}\left( qr \sin{\psi}\right) |
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| 30 | |
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| 31 | .. math:: |
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| 32 | |
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| 33 | S_n = \int^{R}_{0} r dr\sin{qr^2\alpha \cos{\psi}} |
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| 34 | J_n\left( qr^2\beta \cos{\psi}\right) |
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| 35 | J_{2n}\left( qr \sin{\psi}\right) |
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| 36 | |
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| 37 | |
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| 38 | .. figure:: img/pringle-vs-cylinder.png |
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| 39 | |
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| 40 | 1D plot using the default values (with 150 data points). |
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| 41 | |
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| 42 | Reference |
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| 43 | --------- |
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| 44 | |
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[af4da90] | 45 | Stefan Alexandru Rautu, Private Communication. 2012. |
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| 46 | As of 2016: stefanar@ncbs.res.in |
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[9da407c] | 47 | |
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| 48 | """ |
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| 49 | |
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| 50 | from numpy import inf, pi |
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| 51 | |
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| 52 | name = "pringles" |
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| 53 | title = "Pringles model for K Edler. Represents a disc that is bent in two directions." |
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| 54 | description = """\ |
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| 55 | |
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| 56 | """ |
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| 57 | category = "shape:cylinder" |
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| 58 | |
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| 59 | # pylint: disable=bad-whitespace, line-too-long |
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| 60 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 61 | parameters = [ |
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| 62 | ["radius", "Ang", 60.0, [0, inf], "volume", "Pringle radius"], |
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| 63 | ["thickness", "Ang", 10.0, [0, inf], "volume", "Thickness of pringle"], |
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[63e7f0d] | 64 | ["alpha", "", 0.001, [-inf, inf], "", "Curvature parameter alpha"], |
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| 65 | ["beta", "", 0.02, [-inf, inf], "", "Curvature paramter beta"], |
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| 66 | ["sld_pringle", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Pringle sld"], |
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| 67 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "", "Solvent sld"] |
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[9da407c] | 68 | ] |
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| 69 | # pylint: enable=bad-whitespace, line-too-long |
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| 70 | |
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[012b374] | 71 | source = ["lib/polevl.c", "lib/sas_J0.c", "lib/sas_J1.c", \ |
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[b70cb3b] | 72 | "lib/sas_JN.c", "lib/gauss76.c", "pringle.c"] |
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[9da407c] | 73 | |
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| 74 | def ER(radius, thickness): |
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| 75 | """ |
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| 76 | Return equivalent radius (ER) |
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| 77 | """ |
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| 78 | ddd = 0.75 * radius * (2 * radius * thickness + (thickness + radius) \ |
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| 79 | * (thickness + pi * radius)) |
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| 80 | return 0.5 * (ddd) ** (1. / 3.) |
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| 81 | |
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| 82 | demo = dict(background=0.0, |
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| 83 | scale=1.0, |
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| 84 | radius=60.0, |
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| 85 | thickness=10.0, |
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| 86 | alpha=0.001, |
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| 87 | beta=0.02, |
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| 88 | sld_pringle=1.0, |
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| 89 | sld_solvent=6.35) |
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| 90 | |
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| 91 | oldname = 'PringlesModel' |
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| 92 | oldpars = dict(background='background', |
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[012b374] | 93 | scale='scale', |
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| 94 | radius='radius', |
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| 95 | thickness='thickness', |
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| 96 | alpha='alpha', |
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| 97 | beta='beta', |
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| 98 | pringle_sld='sld_pringle', |
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| 99 | solvent_sld='sld_solvent') |
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[9da407c] | 100 | |
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| 101 | tests = [ |
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| 102 | [{'scale' : 1.0, |
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| 103 | 'radius': 60.0, |
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| 104 | 'thickness': 10.0, |
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| 105 | 'alpha': 0.001, |
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| 106 | 'beta': 0.02, |
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| 107 | 'sld_pringle': 1.0, |
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| 108 | 'sld_solvent': 6.3, |
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| 109 | 'background': 6.3, |
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| 110 | }, 0.1, 16.185532], |
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| 111 | |
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| 112 | [{'scale' : 1.0, |
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| 113 | 'radius': 60.0, |
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| 114 | 'thickness': 10.0, |
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| 115 | 'alpha': 0.001, |
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| 116 | 'beta': 0.02, |
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| 117 | 'sld_pringle': 1.0, |
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| 118 | 'sld_solvent': 6.3, |
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| 119 | 'background': 6.3, |
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| 120 | }, 0.01, 297.153496], |
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| 121 | |
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| 122 | [{'scale' : 1.0, |
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| 123 | 'radius': 60.0, |
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| 124 | 'thickness': 10.0, |
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| 125 | 'alpha': 0.001, |
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| 126 | 'beta': 0.02, |
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| 127 | 'sld_pringle': 1.0, |
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| 128 | 'sld_solvent': 6.3, |
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| 129 | 'background': 6.3, |
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| 130 | }, 0.001, 324.021256415], |
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| 131 | |
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| 132 | [{'scale' : 1.0, |
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| 133 | 'radius': 60.0, |
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| 134 | 'thickness': 10.0, |
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| 135 | 'alpha': 0.001, |
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| 136 | 'beta': 0.02, |
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| 137 | 'sld_pringle': 1.0, |
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| 138 | 'sld_solvent': 6.3, |
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| 139 | 'background': 6.3, |
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| 140 | }, (0.001, 90.0), 6.30000026876], |
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[91b8b80] | 141 | ] |
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