[6272968] | 1 | r""" |
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[b0c4271] | 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[a5d0d00] | 5 | Calculates the scattering from a **body-centered cubic lattice** with |
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| 6 | paracrystalline distortion. Thermal vibrations are considered to be negligible, |
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| 7 | and the size of the paracrystal is infinitely large. Paracrystalline distortion |
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| 8 | is assumed to be isotropic and characterized by a Gaussian distribution. |
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[754c454] | 9 | |
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[a5d0d00] | 10 | The scattering intensity $I(q)$ is calculated as |
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[754c454] | 11 | |
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[eb69cce] | 12 | .. math:: |
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[754c454] | 13 | |
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[eb69cce] | 14 | I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q) |
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[754c454] | 15 | |
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| 16 | |
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[eb69cce] | 17 | where *scale* is the volume fraction of spheres, $V_p$ is the volume of the |
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| 18 | primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
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[a5d0d00] | 19 | structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$ |
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| 20 | is the paracrystalline structure factor for a body-centered cubic structure. |
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[754c454] | 21 | |
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[b0c4271] | 22 | Equation (1) of the 1990 reference\ [#CIT1990]_ is used to calculate $Z(q)$, |
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| 23 | using equations (29)-(31) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and |
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| 24 | $Z3$. |
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[754c454] | 25 | |
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[a5d0d00] | 26 | The lattice correction (the occupied volume of the lattice) for a |
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| 27 | body-centered cubic structure of particles of radius $R$ and nearest neighbor |
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| 28 | separation $D$ is |
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[754c454] | 29 | |
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[eb69cce] | 30 | .. math:: |
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[754c454] | 31 | |
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[a5d0d00] | 32 | V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3} |
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| 33 | |
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| 34 | |
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| 35 | The distortion factor (one standard deviation) of the paracrystal is included |
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| 36 | in the calculation of $Z(q)$ |
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| 37 | |
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[eb69cce] | 38 | .. math:: |
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[a5d0d00] | 39 | |
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| 40 | \Delta a = g D |
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| 41 | |
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| 42 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
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[754c454] | 43 | |
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| 44 | |
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[2f0c07d] | 45 | .. figure:: img/bcc_geometry.jpg |
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[d138d43] | 46 | |
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| 47 | Body-centered cubic lattice. |
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[754c454] | 48 | |
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| 49 | For a crystal, diffraction peaks appear at reduced q-values given by |
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| 50 | |
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[eb69cce] | 51 | .. math:: |
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[a5d0d00] | 52 | |
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| 53 | \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2} |
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| 54 | |
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| 55 | where for a body-centered cubic lattice, only reflections where |
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| 56 | $(h + k + l) = \text{even}$ are allowed and reflections where |
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| 57 | $(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions |
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| 58 | correspond to (just the first 5) |
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[754c454] | 59 | |
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[eb69cce] | 60 | .. math:: |
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[754c454] | 61 | |
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[eb69cce] | 62 | \begin{array}{lccccc} |
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| 63 | q/q_o & 1 & \sqrt{2} & \sqrt{3} & \sqrt{4} & \sqrt{5} \\ |
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| 64 | \text{Indices} & (110) & (200) & (211) & (220) & (310) \\ |
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| 65 | \end{array} |
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[754c454] | 66 | |
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[d138d43] | 67 | **NB**: The calculation of $Z(q)$ is a double numerical integral that must |
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[a5d0d00] | 68 | be carried out with a high density of points to properly capture the sharp |
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[d138d43] | 69 | peaks of the paracrystalline scattering. So be warned that the calculation |
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[a5d0d00] | 70 | is SLOW. Go get some coffee. Fitting of any experimental data must be |
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| 71 | resolution smeared for any meaningful fit. This makes a triple integral. |
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| 72 | Very, very slow. Go get lunch! |
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[754c454] | 73 | |
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[a5d0d00] | 74 | This example dataset is produced using 200 data points, |
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| 75 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values. |
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[754c454] | 76 | |
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[a5d0d00] | 77 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
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| 78 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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[b0c4271] | 79 | be accurate. |
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[754c454] | 80 | |
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[2f0c07d] | 81 | .. figure:: img/bcc_angle_definition.png |
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[d138d43] | 82 | |
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| 83 | Orientation of the crystal with respect to the scattering plane. |
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[754c454] | 84 | |
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[eb69cce] | 85 | References |
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| 86 | ---------- |
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[754c454] | 87 | |
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[b0c4271] | 88 | .. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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| 89 | (Original Paper) |
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| 90 | .. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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| 91 | (Corrections to FCC and BCC lattice structure calculation) |
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[754c454] | 92 | |
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[b0c4271] | 93 | Authorship and Verification |
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| 94 | ---------------------------- |
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| 95 | |
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| 96 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 97 | * **Last Modified by:** Paul Butler **Date:** September 29, 2016 |
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| 98 | * **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016 |
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[754c454] | 99 | """ |
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| 100 | |
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[3c56da87] | 101 | from numpy import inf |
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[754c454] | 102 | |
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[e166cb9] | 103 | name = "bcc_paracrystal" |
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[754c454] | 104 | title = "Body-centred cubic lattic with paracrystalline distortion" |
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| 105 | description = """ |
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[dcdf29d] | 106 | Calculates the scattering from a **body-centered cubic lattice** with |
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| 107 | paracrystalline distortion. Thermal vibrations are considered to be |
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| 108 | negligible, and the size of the paracrystal is infinitely large. |
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| 109 | Paracrystalline distortion is assumed to be isotropic and characterized |
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| 110 | by a Gaussian distribution. |
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[754c454] | 111 | """ |
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[485aee2] | 112 | category = "shape:paracrystal" |
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[13ed84c] | 113 | |
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[b0c4271] | 114 | #note - calculation requires double precision |
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[13ed84c] | 115 | single = False |
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| 116 | |
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[dcdf29d] | 117 | # pylint: disable=bad-whitespace, line-too-long |
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[485aee2] | 118 | # ["name", "units", default, [lower, upper], "type","description" ], |
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[dcdf29d] | 119 | parameters = [["dnn", "Ang", 220, [-inf, inf], "", "Nearest neighbour distance"], |
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| 120 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
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| 121 | ["radius", "Ang", 40, [0, inf], "volume", "Particle radius"], |
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[42356c8] | 122 | ["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Particle scattering length density"], |
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| 123 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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[dcdf29d] | 124 | ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"], |
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| 125 | ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"], |
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| 126 | ["psi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"] |
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[485aee2] | 127 | ] |
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[dcdf29d] | 128 | # pylint: enable=bad-whitespace, line-too-long |
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[485aee2] | 129 | |
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[0bef47b] | 130 | source = ["lib/sph_j1c.c", "lib/gauss150.c", "lib/sphere_form.c", "bcc_paracrystal.c"] |
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[754c454] | 131 | |
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| 132 | # parameters for demo |
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| 133 | demo = dict( |
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| 134 | scale=1, background=0, |
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[02a0920] | 135 | dnn=220, d_factor=0.06, sld=4, sld_solvent=1, |
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[c95dc908] | 136 | radius=40, |
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[754c454] | 137 | theta=60, phi=60, psi=60, |
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[cd3dba0] | 138 | radius_pd=.2, radius_pd_n=2, |
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[754c454] | 139 | theta_pd=15, theta_pd_n=0, |
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| 140 | phi_pd=15, phi_pd_n=0, |
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| 141 | psi_pd=15, psi_pd_n=0, |
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| 142 | ) |
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