| 1 | .. _fcc-paracrystal: |
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| 2 | |
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| 3 | Fcc paracrystal |
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| 4 | ======================================================= |
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| 5 | |
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| 6 | Face-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 **face-centered cubic lattice** with |
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| 27 | paracrystalline distortion. Thermal vibrations are considered to be |
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| 28 | negligible, and the size of the paracrystal is infinitely large. |
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| 29 | Paracrystalline distortion is assumed to be isotropic and characterized by |
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| 30 | a Gaussian distribution. |
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| 31 | |
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| 32 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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| 33 | |
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| 34 | Definition |
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| 35 | ---------- |
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| 36 | |
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| 37 | The scattering intensity *I(q)* is calculated as |
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| 38 | |
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| 39 | .. image:: img/image158.jpg |
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| 40 | |
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| 41 | where *scale* is the volume fraction of spheres, *Vp* is the volume of |
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| 42 | the primary particle, *V(lattice)* is a volume correction for the crystal |
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| 43 | structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* |
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| 44 | is the paracrystalline structure factor for a face-centered cubic structure. |
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| 45 | |
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| 46 | Equation (1) of the 1990 reference is used to calculate *Z(q)*, using |
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| 47 | equations (23)-(25) from the 1987 paper for *Z1*\ , *Z2*\ , and *Z3*\ . |
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| 48 | |
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| 49 | The lattice correction (the occupied volume of the lattice) for a |
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| 50 | face-centered cubic structure of particles of radius *R* and nearest |
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| 51 | neighbor separation *D* is |
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| 52 | |
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| 53 | .. image:: img/image159.jpg |
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| 54 | |
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| 55 | The distortion factor (one standard deviation) of the paracrystal is |
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| 56 | included in the calculation of *Z(q)* |
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| 57 | |
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| 58 | .. image:: img/image160.jpg |
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| 59 | |
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| 60 | where *g* is a fractional distortion based on the nearest neighbor distance. |
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| 61 | |
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| 62 | The face-centered cubic lattice is |
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| 63 | |
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| 64 | .. image:: img/image161.jpg |
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| 65 | |
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| 66 | For a crystal, diffraction peaks appear at reduced q-values given by |
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| 67 | |
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| 68 | .. image:: img/image162.jpg |
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| 69 | |
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| 70 | where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all |
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| 71 | even are allowed and reflections where *h*\ , *k*\ , *l* are mixed odd/even |
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| 72 | are forbidden. Thus the peak positions correspond to (just the first 5) |
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| 73 | |
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| 74 | .. image:: img/image163.jpg |
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| 75 | |
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| 76 | **NB: The calculation of** *Z(q)* **is a double numerical integral that |
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| 77 | must be carried out with a high density of** **points to properly capture |
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| 78 | the sharp peaks of the paracrystalline scattering.** So be warned that the |
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| 79 | calculation is SLOW. Go get some coffee. Fitting of any experimental data |
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| 80 | must be resolution smeared for any meaningful fit. This makes a triple |
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| 81 | integral. Very, very slow. Go get lunch! |
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| 82 | |
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| 83 | This example dataset is produced using 200 data points, *qmin* = 0.01 |Ang^-1|, |
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| 84 | *qmax* = 0.1 |Ang^-1| and the above default values. |
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| 85 | |
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| 86 | .. image:: img/image164.jpg |
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| 87 | |
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| 88 | *Figure. 1D plot in the linear scale using the default values (w/200 data point).* |
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| 89 | |
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| 90 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is |
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| 91 | approximated for 1d scattering. Thus the scattering pattern for 2D may not |
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| 92 | be accurate. Note that we are not responsible for any incorrectness of the |
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| 93 | 2D model computation. |
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| 94 | |
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| 95 | .. image:: img/image165.gif |
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| 96 | |
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| 97 | .. image:: img/image166.jpg |
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| 98 | |
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| 99 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
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| 100 | |
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| 101 | REFERENCE |
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| 102 | |
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| 103 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
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| 104 | (Original Paper) |
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| 105 | |
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| 106 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
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| 107 | (Corrections to FCC and BCC lattice structure calculation) |
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| 108 | |
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