[a84a0ca] | 1 | r""" |
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| 2 | Calculates the scattering from a **simple cubic lattice** with |
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| 3 | paracrystalline distortion. Thermal vibrations are considered to be |
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| 4 | negligible, and the size of the paracrystal is infinitely large. |
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| 5 | Paracrystalline distortion is assumed to be isotropic and characterized |
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| 6 | by a Gaussian distribution. |
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| 7 | |
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| 8 | Definition |
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| 9 | ---------- |
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| 10 | |
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| 11 | The scattering intensity $I(q)$ is calculated as |
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| 12 | |
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| 13 | .. math:: |
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| 14 | |
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| 15 | I(q) = \frac{scale}{V_p}V_{lattice}P(q)Z(q) |
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| 16 | |
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| 17 | where scale is the volume fraction of spheres, $V_p$ is the volume of |
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| 18 | the primary particle, $V_{lattice}$ is a volume correction for the crystal |
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| 19 | structure, $P(q)$ is the form factor of the sphere (normalized), and |
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| 20 | $Z(q)$ is the paracrystalline structure factor for a simple cubic structure. |
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| 21 | |
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| 22 | Equation (16) of the 1987 reference is used to calculate $Z(q)$, using |
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| 23 | equations (13)-(15) from the 1987 paper for Z1, Z2, and Z3. |
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| 24 | |
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| 25 | The lattice correction (the occupied volume of the lattice) for a simple |
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| 26 | cubic structure of particles of radius *R* and nearest neighbor separation *D* is |
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| 27 | |
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| 28 | .. math:: |
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| 29 | |
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| 30 | V_{lattice}=\frac{4\pi}{3}\frac{R^3}{D^3} |
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| 31 | |
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| 32 | The distortion factor (one standard deviation) of the paracrystal is included |
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| 33 | in the calculation of $Z(q)$ |
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| 34 | |
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| 35 | .. math:: |
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| 36 | |
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| 37 | \Delta a = gD |
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| 38 | |
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| 39 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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| 40 | |
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| 41 | The simple cubic lattice is |
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| 42 | |
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| 43 | .. figure:: img/sc_crystal_geometry.jpg |
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| 44 | |
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| 45 | For a crystal, diffraction peaks appear at reduced q-values given by |
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| 46 | |
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| 47 | .. math:: |
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| 48 | |
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| 49 | \frac{qD}{2\pi} = \sqrt{h^2+k^2+l^2} |
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| 50 | |
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| 51 | where for a simple cubic lattice any h, k, l are allowed and none are |
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| 52 | forbidden. Thus the peak positions correspond to (just the first 5) |
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| 53 | |
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| 54 | .. math:: |
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| 55 | :nowrap: |
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| 56 | |
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| 57 | \begin{align*} |
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| 58 | q/q_0 \quad & \quad 1 |
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| 59 | & \sqrt{2} \quad |
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| 60 | & \quad \sqrt{3} \quad |
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| 61 | & \sqrt{4} \quad |
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| 62 | & \quad \sqrt{5}\quad \\ |
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| 63 | Indices \quad & (100) |
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| 64 | & \quad (110) \quad |
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| 65 | & \quad (111) |
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| 66 | & (200) \quad |
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| 67 | & \quad (210) |
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| 68 | \end{align*} |
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| 69 | |
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| 70 | .. note:: |
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| 71 | |
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| 72 | The calculation of *Z(q)* is a double numerical integral that must be |
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| 73 | carried out with a high density of points to properly capture the sharp |
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| 74 | peaks of the paracrystalline scattering. |
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| 75 | So be warned that the calculation is SLOW. Go get some coffee. |
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| 76 | Fitting of any experimental data must be resolution smeared for any |
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| 77 | meaningful fit. This makes a triple integral. Very, very slow. |
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| 78 | Go get lunch! |
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| 79 | |
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| 80 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is |
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| 81 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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| 82 | be accurate. Note that we are not responsible for any incorrectness of the 2D |
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| 83 | model computation. |
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| 84 | |
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| 85 | .. figure:: img/sc_crystal_angle_definition.jpg |
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| 86 | |
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| 87 | Reference |
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| 88 | --------- |
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| 89 | Hideki Matsuoka et. al. *Physical Review B,* 36 (1987) 1754-1765 |
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| 90 | (Original Paper) |
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| 91 | |
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| 92 | Hideki Matsuoka et. al. *Physical Review B,* 41 (1990) 3854 -3856 |
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| 93 | (Corrections to FCC and BCC lattice structure calculation) |
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| 94 | |
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| 95 | """ |
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| 96 | |
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| 97 | from numpy import inf |
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| 98 | |
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[02a0920] | 99 | name = "sc_paracrystal" |
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[a84a0ca] | 100 | title = "Simple cubic lattice with paracrystalline distortion" |
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| 101 | description = """ |
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| 102 | P(q)=(scale/Vp)*V_lattice*P(q)*Z(q)+bkg where scale is the volume |
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| 103 | fraction of sphere, |
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| 104 | Vp = volume of the primary particle, |
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| 105 | V_lattice = volume correction for |
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| 106 | for the crystal structure, |
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| 107 | P(q)= form factor of the sphere (normalized), |
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| 108 | Z(q)= paracrystalline structure factor |
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| 109 | for a simple cubic structure. |
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| 110 | [Simple Cubic ParaCrystal Model] |
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| 111 | Parameters; |
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| 112 | scale: volume fraction of spheres |
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| 113 | bkg:background, R: radius of sphere |
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| 114 | dnn: Nearest neighbor distance |
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| 115 | d_factor: Paracrystal distortion factor |
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| 116 | radius: radius of the spheres |
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| 117 | sldSph: SLD of the sphere |
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| 118 | sldSolv: SLD of the solvent |
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| 119 | """ |
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| 120 | category = "shape:paracrystal" |
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| 121 | single = False |
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| 122 | # pylint: disable=bad-whitespace, line-too-long |
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| 123 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 124 | parameters = [["dnn", "Ang", 220.0, [0.0, inf], "", "Nearest neighbor distance"], |
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| 125 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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| 126 | ["radius", "Ang", 40.0, [0.0, inf], "volume", "Radius of sphere"], |
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[02a0920] | 127 | ["sld", "1e-6/Ang^2", 3.0, [0.0, inf], "", "Sphere scattering length density"], |
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| 128 | ["sld_solvent", "1e-6/Ang^2", 6.3, [0.0, inf], "", "Solvent scattering length density"], |
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[a84a0ca] | 129 | ["theta", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a1 axis w/respect incoming beam"], |
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| 130 | ["phi", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a2 in the plane of the detector"], |
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| 131 | ["psi", "degrees", 0.0, [-inf, inf], "orientation", "Orientation of the a3 in the plane of the detector"], |
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| 132 | ] |
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| 133 | # pylint: enable=bad-whitespace, line-too-long |
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| 134 | |
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[02a0920] | 135 | source = ["lib/sph_j1c.c", "lib/sphere_form.c", "lib/gauss150.c", "sc_paracrystal_kernel.c"] |
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[a84a0ca] | 136 | |
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| 137 | demo = dict(scale=1, background=0, |
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| 138 | dnn=220.0, |
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| 139 | d_factor=0.06, |
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| 140 | radius=40.0, |
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[02a0920] | 141 | sld=3.0, |
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| 142 | sld_solvent=6.3, |
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[a84a0ca] | 143 | theta=0.0, |
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| 144 | phi=0.0, |
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| 145 | psi=0.0) |
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| 146 | |
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| 147 | tests = [ |
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| 148 | # Accuracy tests based on content in test/utest_extra_models.py |
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| 149 | [{}, 0.001, 10.3048], |
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| 150 | [{}, 0.215268, 0.00814889], |
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| 151 | [{}, (0.414467), 0.001313289] |
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| 152 | ] |
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| 153 | |
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| 154 | |
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