| 1 | .. _bcc-paracrystal: |
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| 2 | |
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| 3 | Bcc paracrystal |
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| 4 | ======================================================= |
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| 5 | |
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| 6 | Body-centred cubic lattic with paracrystalline distortion |
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| 7 | |
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| 8 | =========== ================================== ============ ============= |
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| 9 | Parameter Description Units Default value |
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| 10 | =========== ================================== ============ ============= |
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| 11 | scale Source intensity None 1 |
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| 12 | background Source background |cm^-1| 0 |
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| 13 | dnn Nearest neighbour distance |Ang| 220 |
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| 14 | d_factor Paracrystal distortion factor None 0.06 |
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| 15 | radius Particle radius |Ang| 40 |
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| 16 | sld Particle scattering length density |1e-6Ang^-2| 4 |
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| 17 | solvent_sld Solvent scattering length density |1e-6Ang^-2| 1 |
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| 18 | theta In plane angle degree 60 |
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| 19 | phi Out of plane angle degree 60 |
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| 20 | psi Out of plane angle degree 60 |
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| 21 | =========== ================================== ============ ============= |
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| 22 | |
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| 23 | The returned value is scaled to units of |cm^-1|. |
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| 24 | |
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| 25 | |
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| 26 | Calculates the scattering from a **body-centered cubic lattice** with |
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| 27 | paracrystalline distortion. Thermal vibrations are considered to be negligible, |
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| 28 | and the size of the paracrystal is infinitely large. Paracrystalline distortion |
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| 29 | is assumed to be isotropic and characterized by a Gaussian distribution. |
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| 30 | |
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| 31 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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| 32 | |
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| 33 | Definition |
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| 34 | ---------- |
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| 35 | |
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| 36 | The scattering intensity $I(q)$ is calculated as |
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| 37 | |
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| 38 | .. math: |
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| 39 | |
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| 40 | I(q) = \frac{\text{scale}}{V_P} V_\text{lattice} P(q) Z(q) |
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| 41 | |
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| 42 | |
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| 43 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the |
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| 44 | primary particle, *V(lattice)* is a volume correction for the crystal |
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| 45 | structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$ |
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| 46 | is the paracrystalline structure factor for a body-centered cubic structure. |
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| 47 | |
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| 48 | Equation (1) of the 1990 reference is used to calculate $Z(q)$, using |
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| 49 | equations (29)-(31) from the 1987 paper for *Z1*\ , *Z2*\ , and *Z3*\ . |
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| 50 | |
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| 51 | The lattice correction (the occupied volume of the lattice) for a |
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| 52 | body-centered cubic structure of particles of radius $R$ and nearest neighbor |
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| 53 | separation $D$ is |
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| 54 | |
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| 55 | .. math: |
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| 56 | |
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| 57 | V_\text{lattice} = \frac{16\pi}{3} \frac{R^3}{\left(D\sqrt{2}\right)^3} |
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| 58 | |
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| 59 | |
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| 60 | The distortion factor (one standard deviation) of the paracrystal is included |
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| 61 | in the calculation of $Z(q)$ |
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| 62 | |
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| 63 | .. math: |
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| 64 | |
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| 65 | \Delta a = g D |
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| 66 | |
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| 67 | where $g$ is a fractional distortion based on the nearest neighbor distance. |
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| 68 | |
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| 69 | The body-centered cubic lattice is |
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| 70 | |
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| 71 | .. image:: img/bcc_lattice.jpg |
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| 72 | |
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| 73 | For a crystal, diffraction peaks appear at reduced q-values given by |
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| 74 | |
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| 75 | .. math: |
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| 76 | |
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| 77 | \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2} |
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| 78 | |
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| 79 | where for a body-centered cubic lattice, only reflections where |
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| 80 | $(h + k + l) = \text{even}$ are allowed and reflections where |
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| 81 | $(h + k + l) = \text{odd}$ are forbidden. Thus the peak positions |
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| 82 | correspond to (just the first 5) |
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| 83 | |
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| 84 | .. math: |
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| 85 | |
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| 86 | \begin{eqnarray} |
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| 87 | &q/q_o&&\quad 1&& \ \sqrt{2} && \ \sqrt{3} && \ \sqrt{4} && \ \sqrt{5} \\ |
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| 88 | &\text{Indices}&& (110) && (200) && (211) && (220) && (310) |
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| 89 | \end{eqnarray} |
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| 90 | |
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| 91 | **NB: The calculation of $Z(q)$ is a double numerical integral that must |
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| 92 | be carried out with a high density of points to properly capture the sharp |
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| 93 | peaks of the paracrystalline scattering.** So be warned that the calculation |
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| 94 | is SLOW. Go get some coffee. Fitting of any experimental data must be |
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| 95 | resolution smeared for any meaningful fit. This makes a triple integral. |
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| 96 | Very, very slow. Go get lunch! |
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| 97 | |
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| 98 | This example dataset is produced using 200 data points, |
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| 99 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above default values. |
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| 100 | |
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| 101 | .. image:: img/bcc_1d.jpg |
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| 102 | |
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| 103 | *Figure. 1D plot in the linear scale using the default values |
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| 104 | (w/200 data point).* |
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| 105 | |
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| 106 | The 2D (Anisotropic model) is based on the reference below where $I(q)$ is |
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| 107 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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| 108 | be accurate. Note that we are not responsible for any incorrectness of the 2D |
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| 109 | model computation. |
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| 110 | |
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| 111 | .. image:: img/bcc_orientation.gif |
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| 112 | |
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| 113 | .. image:: img/bcc_2d.jpg |
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| 114 | |
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| 115 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
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| 116 | |
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| 117 | REFERENCE |
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| 118 | --------- |
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| 119 | |
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| 120 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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| 121 | (Original Paper) |
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| 122 | |
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| 123 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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| 124 | (Corrections to FCC and BCC lattice structure calculation) |
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| 125 | |
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