[be802cb] | 1 | r""" |
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| 2 | For information about polarised and magnetic scattering, click here_. |
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| 3 | |
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| 4 | .. _here: polar_mag_help.html |
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| 5 | |
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| 6 | Definition |
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| 7 | ---------- |
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| 8 | |
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| 9 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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| 10 | |
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| 11 | .. math:: |
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| 12 | |
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[eb69cce] | 13 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
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| 14 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
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[be802cb] | 15 | \right]^2 + \text{background} |
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| 16 | |
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| 17 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
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[eb69cce] | 18 | $r$ is the radius of the sphere, *background* is the background level and |
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[be802cb] | 19 | *sld* and *solvent_sld* are the scattering length densities (SLDs) of the |
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| 20 | scatterer and the solvent respectively. |
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| 21 | |
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| 22 | Note that if your data is in absolute scale, the *scale* should represent |
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| 23 | the volume fraction (which is unitless) if you have a good fit. If not, |
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| 24 | it should represent the volume fraction times a factor (by which your data |
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| 25 | might need to be rescaled). |
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| 26 | |
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| 27 | The 2D scattering intensity is the same as above, regardless of the |
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| 28 | orientation of $\vec q$. |
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| 29 | |
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| 30 | Validation |
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| 31 | ---------- |
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| 32 | |
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| 33 | Validation of our code was done by comparing the output of the 1D model |
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| 34 | to the output of the software provided by the NIST (Kline, 2006). |
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[d138d43] | 35 | Figure :num:`figure #spherepy-comparison` shows a comparison of the output |
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[be802cb] | 36 | of our model and the output of the NIST software. |
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| 37 | |
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[d138d43] | 38 | .. _spherepy-comparison: |
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[be802cb] | 39 | |
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| 40 | .. figure:: img/sphere_comparison.jpg |
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| 41 | |
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| 42 | Comparison of the DANSE scattering intensity for a sphere with the |
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| 43 | output of the NIST SANS analysis software. The parameters were set to: |
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| 44 | *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and |
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| 45 | *background* = 0.01 |cm^-1|. |
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| 46 | |
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| 47 | |
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[eb69cce] | 48 | References |
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| 49 | ---------- |
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[be802cb] | 50 | |
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| 51 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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| 52 | John Wiley and Sons, New York, (1955) |
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| 53 | |
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| 54 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
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| 55 | """ |
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| 56 | |
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[10576d1] | 57 | import numpy as np |
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[3c56da87] | 58 | from numpy import pi, inf, sin, cos, sqrt, log |
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[be802cb] | 59 | |
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[34375ea] | 60 | name = "sphere (python)" |
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[be802cb] | 61 | title = "Spheres with uniform scattering length density" |
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| 62 | description = """\ |
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[eb69cce] | 63 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qr)-qr cos(qr)) |
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| 64 | /(qr)^3]^2 + background |
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| 65 | r: radius of sphere |
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[be802cb] | 66 | V: The volume of the scatter |
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| 67 | sld: the SLD of the sphere |
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| 68 | solvent_sld: the SLD of the solvent |
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| 69 | """ |
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[a5d0d00] | 70 | category = "shape:sphere" |
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[be802cb] | 71 | |
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[3e428ec] | 72 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 73 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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| 74 | "Layer scattering length density"], |
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| 75 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
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| 76 | "Solvent scattering length density"], |
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| 77 | ["radius", "Ang", 50, [0, inf], "volume", |
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| 78 | "Sphere radius"], |
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| 79 | ] |
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[be802cb] | 80 | |
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| 81 | |
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| 82 | def form_volume(radius): |
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[3e428ec] | 83 | return 1.333333333333333 * pi * radius ** 3 |
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[be802cb] | 84 | |
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| 85 | def Iq(q, sld, solvent_sld, radius): |
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[b3f6bc3] | 86 | #print "q",q |
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| 87 | #print "sld,r",sld,solvent_sld,radius |
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[3e428ec] | 88 | qr = q * radius |
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[be802cb] | 89 | sn, cn = sin(qr), cos(qr) |
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[ade352a] | 90 | ## The natural expression for the bessel function is the following: |
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| 91 | ## bes = 3 * (sn-qr*cn)/qr**3 if qr>0 else 1 |
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| 92 | ## however, to support vector q values we need to handle the conditional |
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| 93 | ## as a vector, which we do by first evaluating the full expression |
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| 94 | ## everywhere, then fixing it up where it is broken. We should probably |
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| 95 | ## set numpy to ignore the 0/0 error before we do though... |
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[3e428ec] | 96 | bes = 3 * (sn - qr * cn) / qr ** 3 # may be 0/0 but we fix that next line |
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| 97 | bes[qr == 0] = 1 |
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[be802cb] | 98 | fq = bes * (sld - solvent_sld) * form_volume(radius) |
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[3e428ec] | 99 | return 1.0e-4 * fq ** 2 |
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[eb69cce] | 100 | Iq.vectorized = True # Iq accepts an array of q values |
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[be802cb] | 101 | |
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| 102 | def Iqxy(qx, qy, sld, solvent_sld, radius): |
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[3e428ec] | 103 | return Iq(sqrt(qx ** 2 + qy ** 2), sld, solvent_sld, radius) |
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[eb69cce] | 104 | Iqxy.vectorized = True # Iqxy accepts arrays of qx, qy values |
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[be802cb] | 105 | |
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[10576d1] | 106 | def sesans(z, sld, solvent_sld, radius): |
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| 107 | """ |
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| 108 | Calculate SESANS-correlation function for a solid sphere. |
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| 109 | |
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| 110 | Wim Bouwman after formulae Timofei Kruglov J.Appl.Cryst. 2003 article |
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| 111 | """ |
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[3e428ec] | 112 | d = z / radius |
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[10576d1] | 113 | g = np.zeros_like(z) |
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[3e428ec] | 114 | g[d == 0] = 1. |
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[10576d1] | 115 | low = ((d > 0) & (d < 2)) |
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| 116 | dlow = d[low] |
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[3e428ec] | 117 | dlow2 = dlow ** 2 |
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| 118 | 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] | 119 | return g |
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[ade352a] | 120 | sesans.vectorized = True # sesans accepts and array of z values |
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[10576d1] | 121 | |
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[be802cb] | 122 | def ER(radius): |
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| 123 | return radius |
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| 124 | |
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[b3f6bc3] | 125 | # VR defaults to 1.0 |
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[d547f16] | 126 | |
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[3e428ec] | 127 | demo = dict(scale=1, background=0, |
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| 128 | sld=6, solvent_sld=1, |
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| 129 | radius=120, |
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| 130 | radius_pd=.2, radius_pd_n=45) |
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[a503bfd] | 131 | oldname = "SphereModel" |
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| 132 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
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