[e7b3d7b] | 1 | #fcc paracrystal model |
---|
[3271e20] | 2 | #note model title and parameter table are automatically inserted |
---|
| 3 | #note - calculation requires double precision |
---|
[e7b3d7b] | 4 | r""" |
---|
[3c56da87] | 5 | Calculates the scattering from a **face-centered cubic lattice** with |
---|
| 6 | paracrystalline distortion. Thermal vibrations are considered to be |
---|
| 7 | negligible, and the size of the paracrystal is infinitely large. |
---|
| 8 | Paracrystalline distortion is assumed to be isotropic and characterized by |
---|
| 9 | a Gaussian distribution. |
---|
[3271e20] | 10 | |
---|
| 11 | Definition |
---|
| 12 | ---------- |
---|
| 13 | |
---|
[d138d43] | 14 | The scattering intensity $I(q)$ is calculated as |
---|
| 15 | |
---|
| 16 | .. math:: |
---|
[3271e20] | 17 | |
---|
[eb69cce] | 18 | I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q) |
---|
[3271e20] | 19 | |
---|
[d138d43] | 20 | where *scale* is the volume fraction of spheres, $V_p$ is the volume of |
---|
[eb69cce] | 21 | the primary particle, $V_\text{lattice}$ is a volume correction for the crystal |
---|
[d138d43] | 22 | structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$ |
---|
[3c56da87] | 23 | is the paracrystalline structure factor for a face-centered cubic structure. |
---|
[3271e20] | 24 | |
---|
[d138d43] | 25 | Equation (1) of the 1990 reference is used to calculate $Z(q)$, using |
---|
| 26 | equations (23)-(25) from the 1987 paper for $Z1$, $Z2$, and $Z3$. |
---|
[3271e20] | 27 | |
---|
[3c56da87] | 28 | The lattice correction (the occupied volume of the lattice) for a |
---|
[eb69cce] | 29 | face-centered cubic structure of particles of radius $R$ and nearest |
---|
[d138d43] | 30 | neighbor separation $D$ is |
---|
[3271e20] | 31 | |
---|
[d138d43] | 32 | .. math:: |
---|
| 33 | |
---|
| 34 | V_\text{lattice} = \frac{16\pi}{3}\frac{R^3}{\left(D\sqrt{2}\right)^3} |
---|
[3271e20] | 35 | |
---|
[3c56da87] | 36 | The distortion factor (one standard deviation) of the paracrystal is |
---|
[d138d43] | 37 | included in the calculation of $Z(q)$ |
---|
| 38 | |
---|
| 39 | .. math:: |
---|
[3271e20] | 40 | |
---|
[d138d43] | 41 | \Delta a = gD |
---|
[3271e20] | 42 | |
---|
[d138d43] | 43 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
---|
[3271e20] | 44 | |
---|
[2f0c07d] | 45 | .. figure:: img/fcc_geometry.jpg |
---|
[3271e20] | 46 | |
---|
[d138d43] | 47 | Face-centered cubic lattice. |
---|
[3271e20] | 48 | |
---|
| 49 | For a crystal, diffraction peaks appear at reduced q-values given by |
---|
| 50 | |
---|
[d138d43] | 51 | .. math:: |
---|
[3271e20] | 52 | |
---|
[d138d43] | 53 | \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2} |
---|
| 54 | |
---|
| 55 | where for a face-centered cubic lattice $h, k , l$ all odd or all |
---|
| 56 | even are allowed and reflections where $h, k, l$ are mixed odd/even |
---|
[3c56da87] | 57 | are forbidden. Thus the peak positions correspond to (just the first 5) |
---|
[3271e20] | 58 | |
---|
[d138d43] | 59 | .. math:: |
---|
| 60 | |
---|
| 61 | \begin{array}{cccccc} |
---|
| 62 | q/q_0 & 1 & \sqrt{4/3} & \sqrt{8/3} & \sqrt{11/3} & \sqrt{4} \\ |
---|
| 63 | \text{Indices} & (111) & (200) & (220) & (311) & (222) |
---|
| 64 | \end{array} |
---|
[3271e20] | 65 | |
---|
[d138d43] | 66 | **NB**: The calculation of $Z(q)$ is a double numerical integral that |
---|
| 67 | must be carried out with a high density of points to properly capture |
---|
| 68 | the sharp peaks of the paracrystalline scattering. So be warned that the |
---|
[3c56da87] | 69 | calculation is SLOW. Go get some coffee. Fitting of any experimental data |
---|
| 70 | must be resolution smeared for any meaningful fit. This makes a triple |
---|
| 71 | integral. Very, very slow. Go get lunch! |
---|
[3271e20] | 72 | |
---|
[eb69cce] | 73 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
---|
[3c56da87] | 74 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
---|
| 75 | be accurate. Note that we are not responsible for any incorrectness of the |
---|
| 76 | 2D model computation. |
---|
[3271e20] | 77 | |
---|
[2f0c07d] | 78 | .. figure:: img/bcc_angle_definition.png |
---|
[d138d43] | 79 | |
---|
| 80 | Orientation of the crystal with respect to the scattering plane. |
---|
[3271e20] | 81 | |
---|
[eb69cce] | 82 | References |
---|
| 83 | ---------- |
---|
[3271e20] | 84 | |
---|
| 85 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
---|
| 86 | (Original Paper) |
---|
| 87 | |
---|
| 88 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
---|
| 89 | (Corrections to FCC and BCC lattice structure calculation) |
---|
| 90 | """ |
---|
| 91 | |
---|
[3c56da87] | 92 | from numpy import inf |
---|
[3271e20] | 93 | |
---|
[e7b3d7b] | 94 | name = "fcc_paracrystal" |
---|
| 95 | title = "Face-centred cubic lattic with paracrystalline distortion" |
---|
[3271e20] | 96 | description = """ |
---|
[e7b3d7b] | 97 | Calculates the scattering from a **face-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
---|
[3271e20] | 98 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
---|
| 99 | assumed to be isotropic and characterized by a Gaussian distribution. |
---|
| 100 | """ |
---|
[a5d0d00] | 101 | category = "shape:paracrystal" |
---|
[3271e20] | 102 | |
---|
[13ed84c] | 103 | single = False |
---|
| 104 | |
---|
[c0ccea8] | 105 | # pylint: disable=bad-whitespace, line-too-long |
---|
[3e428ec] | 106 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
| 107 | parameters = [["dnn", "Ang", 220, [-inf, inf], "", "Nearest neighbour distance"], |
---|
| 108 | ["d_factor", "", 0.06, [-inf, inf], "", "Paracrystal distortion factor"], |
---|
| 109 | ["radius", "Ang", 40, [0, inf], "volume", "Particle radius"], |
---|
| 110 | ["sld", "1e-6/Ang^2", 4, [-inf, inf], "", "Particle scattering length density"], |
---|
[02a0920] | 111 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "", "Solvent scattering length density"], |
---|
[3e428ec] | 112 | ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"], |
---|
| 113 | ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"], |
---|
| 114 | ["psi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"] |
---|
| 115 | ] |
---|
[c0ccea8] | 116 | # pylint: enable=bad-whitespace, line-too-long |
---|
[3e428ec] | 117 | |
---|
[02a0920] | 118 | source = ["lib/sph_j1c.c", "lib/gauss150.c", "lib/sphere_form.c", "fcc_paracrystal_kernel.c"] |
---|
[3271e20] | 119 | |
---|
| 120 | # parameters for demo |
---|
[3e428ec] | 121 | demo = dict(scale=1, background=0, |
---|
[02a0920] | 122 | dnn=220, d_factor=0.06, sld=4, sld_solvent=1, |
---|
[3e428ec] | 123 | radius=40, |
---|
| 124 | theta=60, phi=60, psi=60, |
---|
| 125 | radius_pd=.2, radius_pd_n=0.2, |
---|
| 126 | theta_pd=15, theta_pd_n=0, |
---|
| 127 | phi_pd=15, phi_pd_n=0, |
---|
| 128 | psi_pd=15, psi_pd_n=0, |
---|
| 129 | ) |
---|