[5d4777d] | 1 | r""" |
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[40a87fa] | 2 | For information about polarised and magnetic scattering, see |
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[9a4811a] | 3 | the :ref:`magnetism` documentation. |
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[19dcb933] | 4 | |
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| 5 | Definition |
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| 6 | ---------- |
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| 7 | |
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| 8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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| 9 | |
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| 10 | .. math:: |
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| 11 | |
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[eb69cce] | 12 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
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| 13 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
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[19dcb933] | 14 | \right]^2 + \text{background} |
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| 15 | |
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| 16 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
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[7e6bea81] | 17 | $r$ is the radius of the sphere and *background* is the background level. |
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[49da079] | 18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
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[7e6bea81] | 19 | scatterer and the solvent respectively, whose difference is $\Delta\rho$. |
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[19dcb933] | 20 | |
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| 21 | Note that if your data is in absolute scale, the *scale* should represent |
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| 22 | the volume fraction (which is unitless) if you have a good fit. If not, |
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| 23 | it should represent the volume fraction times a factor (by which your data |
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| 24 | might need to be rescaled). |
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| 25 | |
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| 26 | The 2D scattering intensity is the same as above, regardless of the |
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| 27 | orientation of $\vec q$. |
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| 28 | |
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| 29 | Validation |
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| 30 | ---------- |
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| 31 | |
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| 32 | Validation of our code was done by comparing the output of the 1D model |
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| 33 | to the output of the software provided by the NIST (Kline, 2006). |
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| 34 | |
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| 35 | |
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[eb69cce] | 36 | References |
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| 37 | ---------- |
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[19dcb933] | 38 | |
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| 39 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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| 40 | John Wiley and Sons, New York, (1955) |
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| 41 | |
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[ef07e95] | 42 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
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[5d4777d] | 43 | """ |
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| 44 | |
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[2d81cfe] | 45 | import numpy as np |
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[3c56da87] | 46 | from numpy import inf |
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[5d4777d] | 47 | |
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| 48 | name = "sphere" |
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[19dcb933] | 49 | title = "Spheres with uniform scattering length density" |
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[5d4777d] | 50 | description = """\ |
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[49da079] | 51 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
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[eb69cce] | 52 | /(qr)^3]^2 + background |
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| 53 | r: radius of sphere |
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[19dcb933] | 54 | V: The volume of the scatter |
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| 55 | sld: the SLD of the sphere |
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[49da079] | 56 | sld_solvent: the SLD of the solvent |
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[5d4777d] | 57 | """ |
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[a5d0d00] | 58 | category = "shape:sphere" |
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[5d4777d] | 59 | |
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[3e428ec] | 60 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 61 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[3e428ec] | 62 | "Layer scattering length density"], |
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[42356c8] | 63 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[3e428ec] | 64 | "Solvent scattering length density"], |
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| 65 | ["radius", "Ang", 50, [0, inf], "volume", |
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| 66 | "Sphere radius"], |
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| 67 | ] |
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[5d4777d] | 68 | |
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[925ad6e] | 69 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c"] |
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[5d4777d] | 70 | |
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| 71 | # No volume normalization despite having a volume parameter |
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| 72 | # This should perhaps be volume normalized? |
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| 73 | form_volume = """ |
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[ad90df9] | 74 | return sphere_volume(radius); |
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[5d4777d] | 75 | """ |
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| 76 | |
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| 77 | Iq = """ |
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[49da079] | 78 | return sphere_form(q, radius, sld, sld_solvent); |
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[5d4777d] | 79 | """ |
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| 80 | |
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| 81 | def ER(radius): |
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[c691551] | 82 | """ |
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[364d8f7] | 83 | Return equivalent radius (ER) |
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[c691551] | 84 | """ |
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[5d4777d] | 85 | return radius |
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| 86 | |
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[97d89af] | 87 | # VR defaults to 1.0 |
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| 88 | |
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[404ebbd] | 89 | def random(): |
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| 90 | radius = 10**np.random.uniform(1.3, 4) |
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| 91 | pars = dict( |
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| 92 | radius=radius, |
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| 93 | ) |
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| 94 | return pars |
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| 95 | |
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[7e6bea81] | 96 | tests = [ |
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| 97 | [{}, 0.2, 0.726362], |
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| 98 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
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| 99 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
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| 100 | 0.2, 0.228843], |
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| 101 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "ER", 120.], |
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| 102 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.], |
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| 103 | ] |
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