[5d4777d] | 1 | r""" |
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[19dcb933] | 2 | For information about polarised and magnetic scattering, click here_. |
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[5d4777d] | 3 | |
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[19dcb933] | 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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[19dcb933] | 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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[19dcb933] | 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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| 35 | Figure :num:`figure #sphere-comparison` shows a comparison of the output |
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| 36 | of our model and the output of the NIST software. |
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| 37 | |
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| 38 | .. _sphere-comparison: |
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| 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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[19dcb933] | 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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[5d4777d] | 55 | """ |
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| 56 | |
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[3c56da87] | 57 | from numpy import inf |
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[5d4777d] | 58 | |
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| 59 | name = "sphere" |
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[19dcb933] | 60 | title = "Spheres with uniform scattering length density" |
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[5d4777d] | 61 | description = """\ |
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[eb69cce] | 62 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qr)-qr cos(qr)) |
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| 63 | /(qr)^3]^2 + background |
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| 64 | r: radius of sphere |
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[19dcb933] | 65 | V: The volume of the scatter |
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| 66 | sld: the SLD of the sphere |
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| 67 | solvent_sld: the SLD of the solvent |
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[5d4777d] | 68 | """ |
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[a5d0d00] | 69 | category = "shape:sphere" |
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[5d4777d] | 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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| 74 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
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| 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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[5d4777d] | 79 | |
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[9c461c7] | 80 | source = ["lib/sph_j1c.c"] |
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[5d4777d] | 81 | |
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| 82 | # No volume normalization despite having a volume parameter |
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| 83 | # This should perhaps be volume normalized? |
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| 84 | form_volume = """ |
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[994d77f] | 85 | return 1.333333333333333*M_PI*radius*radius*radius; |
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[5d4777d] | 86 | """ |
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| 87 | |
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| 88 | Iq = """ |
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[994d77f] | 89 | const double qr = q*radius; |
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[9c461c7] | 90 | const double bes = sph_j1c(qr); |
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[994d77f] | 91 | const double fq = bes * (sld - solvent_sld) * form_volume(radius); |
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| 92 | return 1.0e-4*fq*fq; |
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[5d4777d] | 93 | """ |
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| 94 | |
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| 95 | Iqxy = """ |
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| 96 | // never called since no orientation or magnetic parameters. |
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[994d77f] | 97 | //return -1.0; |
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| 98 | return Iq(sqrt(qx*qx + qy*qy), sld, solvent_sld, radius); |
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[5d4777d] | 99 | """ |
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| 100 | |
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| 101 | def ER(radius): |
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| 102 | return radius |
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| 103 | |
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[d547f16] | 104 | # VR defaults to 1.0 |
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| 105 | |
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[3e428ec] | 106 | demo = dict(scale=1, background=0, |
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| 107 | sld=6, solvent_sld=1, |
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| 108 | radius=120, |
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| 109 | radius_pd=.2, radius_pd_n=45) |
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[a503bfd] | 110 | oldname = "SphereModel" |
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| 111 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
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