[be802cb] | 1 | r""" |
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[d7028dc] | 2 | For information about polarised and magnetic scattering, see |
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| 3 | the :doc:`magnetic help <../sasgui/perspectives/fitting/mag_help>` 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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| 35 | |
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[eb69cce] | 36 | References |
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| 37 | ---------- |
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[be802cb] | 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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| 42 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
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| 43 | """ |
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| 44 | |
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[10576d1] | 45 | import numpy as np |
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[3c56da87] | 46 | from numpy import pi, inf, sin, cos, sqrt, log |
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[be802cb] | 47 | |
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[49da079] | 48 | name = " _sphere (python)" |
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| 49 | title = "PAK testing ideas for Spheres with uniform scattering length density" |
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[be802cb] | 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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[be802cb] | 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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[be802cb] | 57 | """ |
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[a5d0d00] | 58 | category = "shape:sphere" |
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[be802cb] | 59 | |
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[3e428ec] | 60 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 61 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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| 62 | "Layer scattering length density"], |
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[49da079] | 63 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "", |
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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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[be802cb] | 68 | |
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| 69 | |
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| 70 | def form_volume(radius): |
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[3e428ec] | 71 | return 1.333333333333333 * pi * radius ** 3 |
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[be802cb] | 72 | |
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[49da079] | 73 | def Iq(q, sld, sld_solvent, radius): |
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[b3f6bc3] | 74 | #print "q",q |
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[49da079] | 75 | #print "sld,r",sld,sld_solvent,radius |
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[3e428ec] | 76 | qr = q * radius |
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[be802cb] | 77 | sn, cn = sin(qr), cos(qr) |
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[ade352a] | 78 | ## The natural expression for the bessel function is the following: |
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| 79 | ## bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1 |
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| 80 | ## however, to support vector q values we need to handle the conditional |
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| 81 | ## as a vector, which we do by first evaluating the full expression |
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| 82 | ## everywhere, then fixing it up where it is broken. We should probably |
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| 83 | ## set numpy to ignore the 0/0 error before we do though... |
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[3e428ec] | 84 | bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line |
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| 85 | bes[qr == 0] = 1 |
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[49da079] | 86 | fq = bes * (sld - sld_solvent) * form_volume(radius) |
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[3e428ec] | 87 | return 1.0e-4 * fq ** 2 |
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[eb69cce] | 88 | Iq.vectorized = True # Iq accepts an array of q values |
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[be802cb] | 89 | |
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[49da079] | 90 | def Iqxy(qx, qy, sld, sld_solvent, radius): |
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| 91 | return Iq(sqrt(qx ** 2 + qy ** 2), sld, sld_solvent, radius) |
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[eb69cce] | 92 | Iqxy.vectorized = True # Iqxy accepts arrays of qx, qy values |
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[be802cb] | 93 | |
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[49da079] | 94 | def sesans(z, sld, sld_solvent, radius): |
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[10576d1] | 95 | """ |
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| 96 | Calculate SESANS-correlation function for a solid sphere. |
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| 97 | |
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| 98 | Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article |
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| 99 | """ |
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[3e428ec] | 100 | d = z / radius |
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[10576d1] | 101 | g = np.zeros_like(z) |
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[3e428ec] | 102 | g[d == 0] = 1. |
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[10576d1] | 103 | low = ((d > 0) & (d < 2)) |
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| 104 | dlow = d[low] |
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[3e428ec] | 105 | dlow2 = dlow ** 2 |
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| 106 | g[low] = sqrt(1 - dlow2 / 4.) * (1 + dlow2 / 8.) + dlow2 / 2.*(1 - dlow2 / 16.) * log(dlow / (2. + sqrt(4. - dlow2))) |
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[10576d1] | 107 | return g |
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[d2950f4] | 108 | sesans.vectorized = True # sesans accepts an array of z values |
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[10576d1] | 109 | |
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[be802cb] | 110 | def ER(radius): |
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| 111 | return radius |
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| 112 | |
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[b3f6bc3] | 113 | # VR defaults to 1.0 |
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[d547f16] | 114 | |
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[3e428ec] | 115 | demo = dict(scale=1, background=0, |
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[49da079] | 116 | sld=6, sld_solvent=1, |
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[3e428ec] | 117 | radius=120, |
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| 118 | radius_pd=.2, radius_pd_n=45) |
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