| 1 | r""" |
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| 2 | .. warning:: This model and this model description are under review following |
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| 3 | concerns raised by SasView users. If you need to use this model, |
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| 4 | please email help@sasview.org for the latest situation. *The |
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| 5 | SasView Developers. September 2018.* |
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| 6 | |
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| 7 | Definition |
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| 8 | ---------- |
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| 9 | |
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| 10 | Calculates the scattering from a **simple cubic lattice** with |
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| 11 | paracrystalline distortion. Thermal vibrations are considered to be |
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| 12 | negligible, and the size of the paracrystal is infinitely large. |
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| 13 | Paracrystalline distortion is assumed to be isotropic and characterized |
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| 14 | by a Gaussian distribution. |
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| 15 | |
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| 16 | The scattering intensity $I(q)$ is calculated as |
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| 17 | |
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| 18 | .. math:: |
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| 19 | |
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| 20 | I(q) = \text{scale}\frac{V_\text{lattice}P(q)Z(q)}{V_p} + \text{background} |
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| 21 | |
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| 22 | where scale is the volume fraction of spheres, $V_p$ is the volume of |
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| 23 | the primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
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| 24 | structure, $P(q)$ is the form factor of the sphere (normalized), and |
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| 25 | $Z(q)$ is the paracrystalline structure factor for a simple cubic structure. |
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| 26 | |
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| 27 | Equation (16) of the 1987 reference\ [#CIT1987]_ is used to calculate $Z(q)$, |
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| 28 | using equations (13)-(15) from the 1987 paper\ [#CIT1990]_ for $Z1$, $Z2$, and |
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| 29 | $Z3$. |
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| 30 | |
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| 31 | The lattice correction (the occupied volume of the lattice) for a simple cubic |
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| 32 | structure of particles of radius *R* and nearest neighbor separation *D* is |
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| 33 | |
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| 34 | .. math:: |
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| 35 | |
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| 36 | V_\text{lattice}=\frac{4\pi}{3}\frac{R^3}{D^3} |
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| 37 | |
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| 38 | The distortion factor (one standard deviation) of the paracrystal is included |
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| 39 | in the calculation of $Z(q)$ |
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| 40 | |
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| 41 | .. math:: |
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| 42 | |
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| 43 | \Delta a = gD |
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| 44 | |
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| 45 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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| 46 | |
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| 47 | The simple cubic lattice is |
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| 48 | |
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| 49 | .. figure:: img/sc_crystal_geometry.jpg |
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| 50 | |
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| 51 | For a crystal, diffraction peaks appear at reduced q-values given by |
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| 52 | |
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| 53 | .. math:: |
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| 54 | |
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| 55 | \frac{qD}{2\pi} = \sqrt{h^2+k^2+l^2} |
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| 56 | |
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| 57 | where for a simple cubic lattice any h, k, l are allowed and none are |
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| 58 | forbidden. Thus the peak positions correspond to (just the first 5) |
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| 59 | |
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| 60 | .. math:: |
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| 61 | :nowrap: |
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| 62 | |
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| 63 | \begin{align*} |
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| 64 | q/q_0 \quad & \quad 1 |
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| 65 | & \sqrt{2} \quad |
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| 66 | & \quad \sqrt{3} \quad |
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| 67 | & \sqrt{4} \quad |
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| 68 | & \quad \sqrt{5}\quad \\ |
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| 69 | Indices \quad & (100) |
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| 70 | & \quad (110) \quad |
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| 71 | & \quad (111) |
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| 72 | & (200) \quad |
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| 73 | & \quad (210) |
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| 74 | \end{align*} |
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| 75 | |
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| 76 | .. note:: |
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| 77 | |
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| 78 | The calculation of *Z(q)* is a double numerical integral that must be |
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| 79 | carried out with a high density of points to properly capture the sharp |
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| 80 | peaks of the paracrystalline scattering. |
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| 81 | So be warned that the calculation is slow. Fitting of any experimental data |
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| 82 | must be resolution smeared for any meaningful fit. This makes a triple |
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| 83 | integral which may be very slow. |
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| 84 | |
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| 85 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is |
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| 86 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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| 87 | be accurate particularly at low $q$. For general details of the calculation |
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| 88 | and angular dispersions for oriented particles see :ref:`orientation` . |
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| 89 | Note that we are not responsible for any incorrectness of the |
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| 90 | 2D model computation. |
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| 91 | |
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| 92 | .. figure:: img/parallelepiped_angle_definition.png |
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| 93 | |
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| 94 | Orientation of the crystal with respect to the scattering plane, when |
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| 95 | $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis). |
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| 96 | |
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| 97 | Reference |
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| 98 | --------- |
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| 99 | |
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| 100 | .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 (Original Paper) |
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| 101 | .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 (Corrections to FCC and BCC lattice structure calculation) |
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| 102 | |
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| 103 | Authorship and Verification |
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| 104 | --------------------------- |
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| 105 | |
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| 106 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 107 | * **Last Modified by:** Steve King **Date:** March 25, 2019 |
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| 108 | * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 |
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| 109 | """ |
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| 110 | |
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| 111 | import numpy as np |
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| 112 | from numpy import inf |
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| 113 | |
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| 114 | name = "sc_paracrystal" |
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| 115 | title = "Simple cubic lattice with paracrystalline distortion" |
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| 116 | description = """ |
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| 117 | P(q)=(scale/Vp)*V_lattice*P(q)*Z(q)+bkg where scale is the volume |
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| 118 | fraction of sphere, |
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| 119 | Vp = volume of the primary particle, |
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| 120 | V_lattice = volume correction for |
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| 121 | for the crystal structure, |
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| 122 | P(q)= form factor of the sphere (normalized), |
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| 123 | Z(q)= paracrystalline structure factor |
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| 124 | for a simple cubic structure. |
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| 125 | [Simple Cubic ParaCrystal Model] |
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| 126 | Parameters; |
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| 127 | scale: volume fraction of spheres |
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| 128 | bkg:background, R: radius of sphere |
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| 129 | dnn: Nearest neighbor distance |
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| 130 | d_factor: Paracrystal distortion factor |
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| 131 | radius: radius of the spheres |
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| 132 | sldSph: SLD of the sphere |
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| 133 | sldSolv: SLD of the solvent |
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| 134 | """ |
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| 135 | category = "shape:paracrystal" |
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| 136 | single = False |
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| 137 | # pylint: disable=bad-whitespace, line-too-long |
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| 138 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 139 | parameters = [["dnn", "Ang", 220.0, [0.0, inf], "", "Nearest neighbor distance"], |
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| 140 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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| 141 | ["radius", "Ang", 40.0, [0.0, inf], "volume", "Radius of sphere"], |
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| 142 | ["sld", "1e-6/Ang^2", 3.0, [0.0, inf], "sld", "Sphere scattering length density"], |
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| 143 | ["sld_solvent", "1e-6/Ang^2", 6.3, [0.0, inf], "sld", "Solvent scattering length density"], |
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| 144 | ["theta", "degrees", 0, [-360, 360], "orientation", "c axis to beam angle"], |
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| 145 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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| 146 | ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about c axis"] |
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| 147 | ] |
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| 148 | # pylint: enable=bad-whitespace, line-too-long |
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| 149 | |
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| 150 | source = ["lib/sas_3j1x_x.c", "lib/sphere_form.c", "lib/gauss150.c", "sc_paracrystal.c"] |
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| 151 | |
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| 152 | def random(): |
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| 153 | """Return a random parameter set for the model.""" |
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| 154 | # copied from bcc_paracrystal |
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| 155 | radius = 10**np.random.uniform(1.3, 4) |
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| 156 | d_factor = 10**np.random.uniform(-2, -0.7) # sigma_d in 0.01-0.7 |
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| 157 | dnn_fraction = np.random.beta(a=10, b=1) |
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| 158 | dnn = radius*4/np.sqrt(4)/dnn_fraction |
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| 159 | pars = dict( |
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| 160 | #sld=1, sld_solvent=0, scale=1, background=1e-32, |
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| 161 | dnn=dnn, |
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| 162 | d_factor=d_factor, |
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| 163 | radius=radius, |
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| 164 | ) |
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| 165 | return pars |
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| 166 | |
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| 167 | tests = [ |
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| 168 | # Accuracy tests based on content in test/utest_extra_models.py, 2d tests added April 10, 2017 |
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| 169 | [{}, 0.001, 10.3048], |
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| 170 | [{}, 0.215268, 0.00814889], |
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| 171 | [{}, 0.414467, 0.001313289], |
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| 172 | [{'theta': 10.0, 'phi': 20, 'psi': 30.0}, (0.045, -0.035), 18.0397138402], |
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| 173 | [{'theta': 10.0, 'phi': 20, 'psi': 30.0}, (0.023, 0.045), 0.0177333171285], |
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| 174 | ] |
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