[5d4777d] | 1 | r""" |
---|
[40a87fa] | 2 | For information about polarised and magnetic scattering, see |
---|
[9a4811a] | 3 | the :ref:`magnetism` documentation. |
---|
[19dcb933] | 4 | |
---|
| 5 | Definition |
---|
| 6 | ---------- |
---|
| 7 | |
---|
| 8 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
| 9 | |
---|
| 10 | .. math:: |
---|
| 11 | |
---|
[eb69cce] | 12 | I(q) = \frac{\text{scale}}{V} \cdot \left[ |
---|
| 13 | 3V(\Delta\rho) \cdot \frac{\sin(qr) - qr\cos(qr))}{(qr)^3} |
---|
[19dcb933] | 14 | \right]^2 + \text{background} |
---|
| 15 | |
---|
| 16 | where *scale* is a volume fraction, $V$ is the volume of the scatterer, |
---|
[7e6bea81] | 17 | $r$ is the radius of the sphere and *background* is the background level. |
---|
[49da079] | 18 | *sld* and *sld_solvent* are the scattering length densities (SLDs) of the |
---|
[7e6bea81] | 19 | scatterer and the solvent respectively, whose difference is $\Delta\rho$. |
---|
[19dcb933] | 20 | |
---|
| 21 | Note that if your data is in absolute scale, the *scale* should represent |
---|
| 22 | the volume fraction (which is unitless) if you have a good fit. If not, |
---|
| 23 | it should represent the volume fraction times a factor (by which your data |
---|
| 24 | might need to be rescaled). |
---|
| 25 | |
---|
| 26 | The 2D scattering intensity is the same as above, regardless of the |
---|
| 27 | orientation of $\vec q$. |
---|
| 28 | |
---|
| 29 | Validation |
---|
| 30 | ---------- |
---|
| 31 | |
---|
| 32 | Validation of our code was done by comparing the output of the 1D model |
---|
| 33 | to the output of the software provided by the NIST (Kline, 2006). |
---|
| 34 | |
---|
| 35 | |
---|
[eb69cce] | 36 | References |
---|
| 37 | ---------- |
---|
[19dcb933] | 38 | |
---|
| 39 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
---|
| 40 | John Wiley and Sons, New York, (1955) |
---|
| 41 | |
---|
| 42 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
---|
[5d4777d] | 43 | """ |
---|
| 44 | |
---|
[3c56da87] | 45 | from numpy import inf |
---|
[5d4777d] | 46 | |
---|
| 47 | name = "sphere" |
---|
[19dcb933] | 48 | title = "Spheres with uniform scattering length density" |
---|
[5d4777d] | 49 | description = """\ |
---|
[49da079] | 50 | P(q)=(scale/V)*[3V(sld-sld_solvent)*(sin(qr)-qr cos(qr)) |
---|
[eb69cce] | 51 | /(qr)^3]^2 + background |
---|
| 52 | r: radius of sphere |
---|
[19dcb933] | 53 | V: The volume of the scatter |
---|
| 54 | sld: the SLD of the sphere |
---|
[49da079] | 55 | sld_solvent: the SLD of the solvent |
---|
[5d4777d] | 56 | """ |
---|
[a5d0d00] | 57 | category = "shape:sphere" |
---|
[5d4777d] | 58 | |
---|
[3e428ec] | 59 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
[42356c8] | 60 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
---|
[3e428ec] | 61 | "Layer scattering length density"], |
---|
[42356c8] | 62 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
---|
[3e428ec] | 63 | "Solvent scattering length density"], |
---|
| 64 | ["radius", "Ang", 50, [0, inf], "volume", |
---|
| 65 | "Sphere radius"], |
---|
| 66 | ] |
---|
[5d4777d] | 67 | |
---|
[925ad6e] | 68 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c"] |
---|
[5d4777d] | 69 | |
---|
| 70 | # No volume normalization despite having a volume parameter |
---|
| 71 | # This should perhaps be volume normalized? |
---|
| 72 | form_volume = """ |
---|
[ad90df9] | 73 | return sphere_volume(radius); |
---|
[5d4777d] | 74 | """ |
---|
| 75 | |
---|
| 76 | Iq = """ |
---|
[49da079] | 77 | return sphere_form(q, radius, sld, sld_solvent); |
---|
[5d4777d] | 78 | """ |
---|
| 79 | |
---|
| 80 | def ER(radius): |
---|
[c691551] | 81 | """ |
---|
[364d8f7] | 82 | Return equivalent radius (ER) |
---|
[c691551] | 83 | """ |
---|
[5d4777d] | 84 | return radius |
---|
| 85 | |
---|
[97d89af] | 86 | # VR defaults to 1.0 |
---|
| 87 | |
---|
[404ebbd] | 88 | def random(): |
---|
| 89 | import numpy as np |
---|
| 90 | radius = 10**np.random.uniform(1.3, 4) |
---|
| 91 | pars = dict( |
---|
| 92 | radius=radius, |
---|
| 93 | ) |
---|
| 94 | return pars |
---|
| 95 | |
---|
[7e6bea81] | 96 | tests = [ |
---|
| 97 | [{}, 0.2, 0.726362], |
---|
| 98 | [{"scale": 1., "background": 0., "sld": 6., "sld_solvent": 1., |
---|
| 99 | "radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, |
---|
| 100 | 0.2, 0.228843], |
---|
| 101 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "ER", 120.], |
---|
| 102 | [{"radius": 120., "radius_pd": 0.2, "radius_pd_n":45}, "VR", 1.], |
---|
| 103 | ] |
---|
| 104 | |
---|
| 105 | |
---|