[8dca856] | 1 | r""" |
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[40a87fa] | 2 | For information about polarised and magnetic scattering, see |
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[9a4811a] | 3 | the :ref:`magnetism` documentation. |
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[8dca856] | 4 | |
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| 5 | Definition |
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| 6 | ---------- |
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| 7 | |
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[40a87fa] | 8 | The scattering intensity $I(q)$ is calculated as: |
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[8dca856] | 9 | |
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| 10 | .. math:: |
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| 11 | |
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[40a87fa] | 12 | I(q) = \frac{\text{scale}}{V}(\Delta \rho)^2 A^2(q) S(q) |
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| 13 | + \text{background} |
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[8dca856] | 14 | |
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[40a87fa] | 15 | |
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| 16 | where the amplitude $A(q)$ is given as the typical sphere scattering convoluted |
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[8dca856] | 17 | with a Gaussian to get a gradual drop-off in the scattering length density: |
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| 18 | |
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| 19 | .. math:: |
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| 20 | |
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| 21 | A(q) = \frac{3\left[\sin(qR) - qR \cos(qR)\right]}{(qR)^3} |
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[40a87fa] | 22 | \exp\left(\frac{-(\sigma_\text{fuzzy}q)^2}{2}\right) |
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[8dca856] | 23 | |
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[40a87fa] | 24 | Here $A(q)^2$ is the form factor, $P(q)$. The scale is equivalent to the |
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| 25 | volume fraction of spheres, each of volume, $V$. Contrast $(\Delta \rho)$ |
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| 26 | is the difference of scattering length densities of the sphere and the |
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| 27 | surrounding solvent. |
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[8dca856] | 28 | |
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[40a87fa] | 29 | Poly-dispersion in radius and in fuzziness is provided for, though the |
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| 30 | fuzziness must be kept much smaller than the sphere radius for meaningful |
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| 31 | results. |
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[8dca856] | 32 | |
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| 33 | From the reference: |
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| 34 | |
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| 35 | The "fuzziness" of the interface is defined by the parameter |
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[40a87fa] | 36 | $\sigma_\text{fuzzy}$. The particle radius $R$ represents the radius of the |
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[8dca856] | 37 | particle where the scattering length density profile decreased to 1/2 of the |
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[40a87fa] | 38 | core density. $\sigma_\text{fuzzy}$ is the width of the smeared particle |
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[8dca856] | 39 | surface; i.e., the standard deviation from the average height of the fuzzy |
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| 40 | interface. The inner regions of the microgel that display a higher density |
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| 41 | are described by the radial box profile extending to a radius of |
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[40a87fa] | 42 | approximately $R_\text{box} \sim R - 2 \sigma$. The profile approaches |
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| 43 | zero as $R_\text{sans} \sim R + 2\sigma$. |
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[8dca856] | 44 | |
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| 45 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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[40a87fa] | 46 | where the $q$ vector is defined as |
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[8dca856] | 47 | |
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[40a87fa] | 48 | .. math:: q = \sqrt{{q_x}^2 + {q_y}^2} |
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[8dca856] | 49 | |
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| 50 | References |
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| 51 | ---------- |
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| 52 | |
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| 53 | M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*, |
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| 54 | 20 (2004) 7283-7292 |
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| 55 | """ |
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| 56 | |
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| 57 | from numpy import inf |
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| 58 | |
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| 59 | name = "fuzzy_sphere" |
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| 60 | title = "Scattering from spherical particles with a fuzzy surface." |
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| 61 | description = """\ |
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| 62 | scale: scale factor times volume fraction, |
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| 63 | or just volume fraction for absolute scale data |
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| 64 | radius: radius of the solid sphere |
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[0cc31e1] | 65 | fuzziness = the standard deviation of the fuzzy interfacial |
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[8dca856] | 66 | thickness (ie., so-called interfacial roughness) |
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| 67 | sld: the SLD of the sphere |
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| 68 | solvend_sld: the SLD of the solvent |
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| 69 | background: incoherent background |
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| 70 | Note: By definition, this function works only when fuzziness << radius. |
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| 71 | """ |
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| 72 | category = "shape:sphere" |
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| 73 | |
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| 74 | # pylint: disable=bad-whitespace,line-too-long |
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| 75 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 76 | parameters = [["sld", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Particle scattering length density"], |
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| 77 | ["sld_solvent", "1e-6/Ang^2", 3, [-inf, inf], "sld", "Solvent scattering length density"], |
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[8dca856] | 78 | ["radius", "Ang", 60, [0, inf], "volume", "Sphere radius"], |
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[0cc31e1] | 79 | ["fuzziness", "Ang", 10, [0, inf], "", "std deviation of Gaussian convolution for interface (must be << radius)"], |
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[8dca856] | 80 | ] |
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| 81 | # pylint: enable=bad-whitespace,line-too-long |
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| 82 | |
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[925ad6e] | 83 | source = ["lib/sas_3j1x_x.c"] |
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[8dca856] | 84 | |
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| 85 | # No volume normalization despite having a volume parameter |
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| 86 | # This should perhaps be volume normalized? |
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| 87 | form_volume = """ |
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[3a48772] | 88 | return M_4PI_3*cube(radius); |
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[8dca856] | 89 | """ |
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| 90 | |
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| 91 | Iq = """ |
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| 92 | const double qr = q*radius; |
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[925ad6e] | 93 | const double bes = sas_3j1x_x(qr); |
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[8dca856] | 94 | const double qf = q*fuzziness; |
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[0cc31e1] | 95 | const double fq = bes * (sld - sld_solvent) * form_volume(radius) * exp(-0.5*qf*qf); |
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[8dca856] | 96 | return 1.0e-4*fq*fq; |
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| 97 | """ |
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| 98 | |
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| 99 | def ER(radius): |
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| 100 | """ |
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| 101 | Return radius |
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| 102 | """ |
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| 103 | return radius |
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| 104 | |
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| 105 | # VR defaults to 1.0 |
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| 106 | |
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| 107 | demo = dict(scale=1, background=0.001, |
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[0cc31e1] | 108 | sld=1, sld_solvent=3, |
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[8dca856] | 109 | radius=60, |
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| 110 | fuzziness=10, |
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| 111 | radius_pd=.2, radius_pd_n=45, |
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| 112 | fuzziness_pd=.2, fuzziness_pd_n=0) |
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| 113 | |
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| 114 | tests = [ |
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| 115 | # Accuracy tests based on content in test/utest_models_new1_3.py |
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| 116 | #[{'background': 0.001}, 1.0, 0.001], |
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| 117 | |
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| 118 | [{}, 0.00301005, 359.2315], |
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| 119 | |
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| 120 | ] |
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