[be802cb] | 1 | r""" |
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[2d81cfe] | 2 | For information about polarised and magnetic scattering, see |
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[ca4444f] | 3 | the :ref:`magnetism` documentation. |
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[be802cb] | 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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[be802cb] | 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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[eb69cce] | 17 | $r$ is the radius of the sphere, *background* is the background level and |
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[49da079] | 18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
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[be802cb] | 19 | scatterer and the solvent respectively. |
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| 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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[eb69cce] | 35 | References |
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| 36 | ---------- |
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[be802cb] | 37 | |
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| 38 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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| 39 | John Wiley and Sons, New York, (1955) |
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| 40 | |
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[0507e09] | 41 | Source |
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| 42 | ------ |
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| 43 | |
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| 44 | `_spherepy.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/_spherepy.py>`_ |
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| 45 | `sphere.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/sphere.c>`_ |
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| 46 | |
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| 47 | Authorship and Verification |
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| 48 | ---------------------------- |
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| 49 | |
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| 50 | * **Author: P Kienzle** |
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| 51 | * **Last Modified by:** |
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[ef07e95] | 52 | * **Last Reviewed by:** S King and P Parker **Date:** 2013/09/09 and 2014/01/06 |
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[0507e09] | 53 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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[be802cb] | 54 | """ |
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| 55 | |
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[10576d1] | 56 | import numpy as np |
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[3c56da87] | 57 | from numpy import pi, inf, sin, cos, sqrt, log |
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[be802cb] | 58 | |
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[49da079] | 59 | name = " _sphere (python)" |
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| 60 | title = "PAK testing ideas for Spheres with uniform scattering length density" |
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[be802cb] | 61 | description = """\ |
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[49da079] | 62 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
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[eb69cce] | 63 | /(qr)^3]^2 + background |
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| 64 | r: radius of sphere |
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[be802cb] | 65 | V: The volume of the scatter |
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| 66 | sld: the SLD of the sphere |
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[49da079] | 67 | sld_solvent: the SLD of the solvent |
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[be802cb] | 68 | """ |
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[a5d0d00] | 69 | category = "shape:sphere" |
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[be802cb] | 70 | |
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[3e428ec] | 71 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 72 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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| 73 | "Layer scattering length density"], |
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[49da079] | 74 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "", |
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[3e428ec] | 75 | "Solvent scattering length density"], |
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| 76 | ["radius", "Ang", 50, [0, inf], "volume", |
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| 77 | "Sphere radius"], |
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| 78 | ] |
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[be802cb] | 79 | |
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| 80 | |
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| 81 | def form_volume(radius): |
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[b297ba9] | 82 | """Calculate volume for sphere""" |
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[3e428ec] | 83 | return 1.333333333333333 * pi * radius ** 3 |
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[be802cb] | 84 | |
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[304c775] | 85 | def effective_radius(mode, radius): |
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[b297ba9] | 86 | """Calculate R_eff for sphere""" |
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[304c775] | 87 | return radius |
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| 88 | |
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[49da079] | 89 | def Iq(q, sld, sld_solvent, radius): |
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[b297ba9] | 90 | """Calculate I(q) for sphere""" |
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[b3f6bc3] | 91 | #print "q",q |
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[49da079] | 92 | #print "sld,r",sld,sld_solvent,radius |
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[3e428ec] | 93 | qr = q * radius |
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[be802cb] | 94 | sn, cn = sin(qr), cos(qr) |
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[ade352a] | 95 | ## The natural expression for the bessel function is the following: |
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| 96 | ## bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1 |
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| 97 | ## however, to support vector q values we need to handle the conditional |
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| 98 | ## as a vector, which we do by first evaluating the full expression |
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| 99 | ## everywhere, then fixing it up where it is broken. We should probably |
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| 100 | ## set numpy to ignore the 0/0 error before we do though... |
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[3e428ec] | 101 | bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line |
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| 102 | bes[qr == 0] = 1 |
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[49da079] | 103 | fq = bes * (sld - sld_solvent) * form_volume(radius) |
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[3e428ec] | 104 | return 1.0e-4 * fq ** 2 |
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[eb69cce] | 105 | Iq.vectorized = True # Iq accepts an array of q values |
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[be802cb] | 106 | |
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[49da079] | 107 | def sesans(z, sld, sld_solvent, radius): |
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[10576d1] | 108 | """ |
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| 109 | Calculate SESANS-correlation function for a solid sphere. |
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| 110 | |
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| 111 | Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article |
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| 112 | """ |
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[3e428ec] | 113 | d = z / radius |
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[10576d1] | 114 | g = np.zeros_like(z) |
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[3e428ec] | 115 | g[d == 0] = 1. |
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[10576d1] | 116 | low = ((d > 0) & (d < 2)) |
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| 117 | dlow = d[low] |
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[3e428ec] | 118 | dlow2 = dlow ** 2 |
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[2d81cfe] | 119 | g[low] = (sqrt(1 - dlow2/4.) * (1 + dlow2/8.) |
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| 120 | + dlow2/2.*(1 - dlow2/16.) * log(dlow / (2. + sqrt(4. - dlow2)))) |
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[10576d1] | 121 | return g |
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[d2950f4] | 122 | sesans.vectorized = True # sesans accepts an array of z values |
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[10576d1] | 123 | |
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[3e428ec] | 124 | demo = dict(scale=1, background=0, |
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[49da079] | 125 | sld=6, sld_solvent=1, |
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[3e428ec] | 126 | radius=120, |
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| 127 | radius_pd=.2, radius_pd_n=45) |
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