[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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| 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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| 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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| 18 | $R$ is the radius of the sphere, *background* is the background level and |
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| 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 | Our model uses the form factor calculations as defined in the IGOR |
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| 31 | package provided by the NIST Center for Neutron Research (Kline, 2006). |
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| 32 | |
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| 33 | Validation |
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| 34 | ---------- |
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| 35 | |
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| 36 | Validation of our code was done by comparing the output of the 1D model |
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| 37 | to the output of the software provided by the NIST (Kline, 2006). |
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| 38 | Figure :num:`figure #sphere-comparison` shows a comparison of the output |
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| 39 | of our model and the output of the NIST software. |
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| 40 | |
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| 41 | .. _sphere-comparison: |
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| 42 | |
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| 43 | .. figure:: img/sphere_comparison.jpg |
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| 44 | |
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| 45 | Comparison of the DANSE scattering intensity for a sphere with the |
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| 46 | output of the NIST SANS analysis software. The parameters were set to: |
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| 47 | *scale* = 1.0, *radius* = 60 |Ang|, *contrast* = 1e-6 |Ang^-2|, and |
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| 48 | *background* = 0.01 |cm^-1|. |
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| 49 | |
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| 50 | |
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| 51 | Reference |
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| 52 | --------- |
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| 53 | |
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| 54 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, |
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| 55 | John Wiley and Sons, New York, (1955) |
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| 56 | |
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| 57 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
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[5d4777d] | 58 | """ |
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| 59 | |
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[3c56da87] | 60 | from numpy import inf |
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[5d4777d] | 61 | |
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| 62 | name = "sphere" |
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[19dcb933] | 63 | title = "Spheres with uniform scattering length density" |
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[5d4777d] | 64 | description = """\ |
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[19dcb933] | 65 | P(q)=(scale/V)*[3V(sld-solvent_sld)*(sin(qR)-qRcos(qR)) |
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| 66 | /(qR)^3]^2 + background |
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| 67 | R: radius of sphere |
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| 68 | V: The volume of the scatter |
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| 69 | sld: the SLD of the sphere |
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| 70 | solvent_sld: the SLD of the solvent |
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[5d4777d] | 71 | """ |
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[a5d0d00] | 72 | category = "shape:sphere" |
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[5d4777d] | 73 | |
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[3e428ec] | 74 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 75 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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| 76 | "Layer scattering length density"], |
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| 77 | ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "", |
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| 78 | "Solvent scattering length density"], |
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| 79 | ["radius", "Ang", 50, [0, inf], "volume", |
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| 80 | "Sphere radius"], |
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| 81 | ] |
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[5d4777d] | 82 | |
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| 83 | |
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| 84 | # No volume normalization despite having a volume parameter |
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| 85 | # This should perhaps be volume normalized? |
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| 86 | form_volume = """ |
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[994d77f] | 87 | return 1.333333333333333*M_PI*radius*radius*radius; |
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[5d4777d] | 88 | """ |
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| 89 | |
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| 90 | Iq = """ |
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[994d77f] | 91 | const double qr = q*radius; |
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[0fa687d] | 92 | const double qrsq = qr*qr; |
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[994d77f] | 93 | double sn, cn; |
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[5d4777d] | 94 | SINCOS(qr, sn, cn); |
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[0fa687d] | 95 | // Use taylor series for low Q to avoid cancellation error. Tested against |
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| 96 | // the following expression in quad precision: |
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| 97 | // 3.0*(sn-qr*cn)/(qr*qr*qr); |
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| 98 | // Note that the values differ from sasview ~ 5e-12 rather than 5e-14, but |
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| 99 | // in this case it is likely cancellation errors in the original expression |
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| 100 | // using double precision that are the source. Single precision only |
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| 101 | // requires the first 3 terms. Double precision requires the 4th term. |
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| 102 | // The fifth term is not needed, and is commented out below. |
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| 103 | const double bes = (qr < 1e-1) |
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| 104 | ? 1.0 + qrsq*(-3./30. + qrsq*(3./840. + qrsq*(-3./45360.)))// + qrsq*(3./3991680.)))) |
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| 105 | : 3.0*(sn/qr - cn)/qrsq; |
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[994d77f] | 106 | const double fq = bes * (sld - solvent_sld) * form_volume(radius); |
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| 107 | return 1.0e-4*fq*fq; |
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[5d4777d] | 108 | """ |
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| 109 | |
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| 110 | Iqxy = """ |
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| 111 | // never called since no orientation or magnetic parameters. |
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[994d77f] | 112 | //return -1.0; |
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| 113 | return Iq(sqrt(qx*qx + qy*qy), sld, solvent_sld, radius); |
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[5d4777d] | 114 | """ |
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| 115 | |
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| 116 | def ER(radius): |
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| 117 | return radius |
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| 118 | |
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[d547f16] | 119 | # VR defaults to 1.0 |
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| 120 | |
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[3e428ec] | 121 | demo = dict(scale=1, background=0, |
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| 122 | sld=6, solvent_sld=1, |
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| 123 | radius=120, |
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| 124 | radius_pd=.2, radius_pd_n=45) |
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[a503bfd] | 125 | oldname = "SphereModel" |
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| 126 | oldpars = dict(sld='sldSph', solvent_sld='sldSolv', radius='radius') |
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