.. _fcc-paracrystal:

fcc_paracrystal
=======================================================

Face-centred cubic lattic with paracrystalline distortion

=========== ================================== ============ =============
Parameter   Description                        Units        Default value
=========== ================================== ============ =============
scale       Source intensity                   None                     1
background  Source background                  |cm^-1|              0.001
dnn         Nearest neighbour distance         |Ang|                  220
d_factor    Paracrystal distortion factor      None                  0.06
radius      Particle radius                    |Ang|                   40
sld         Particle scattering length density |1e-6Ang^-2|             4
sld_solvent Solvent scattering length density  |1e-6Ang^-2|             1
theta       In plane angle                     degree                  60
phi         Out of plane angle                 degree                  60
psi         Out of plane angle                 degree                  60
=========== ================================== ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


Calculates the scattering from a **face-centered cubic lattice** with
paracrystalline distortion. Thermal vibrations are considered to be
negligible, and the size of the paracrystal is infinitely large.
Paracrystalline distortion is assumed to be isotropic and characterized by
a Gaussian distribution.

**Definition**

The scattering intensity $I(q)$ is calculated as

.. math::

    I(q) = \frac{\text{scale}}{V_p} V_\text{lattice} P(q) Z(q)

where *scale* is the volume fraction of spheres, $V_p$ is the volume of
the primary particle, $V_\text{lattice}$ is a volume correction for the crystal
structure, $P(q)$ is the form factor of the sphere (normalized), and $Z(q)$
is the paracrystalline structure factor for a face-centered cubic structure.

Equation (1) of the 1990 reference is used to calculate $Z(q)$, using
equations (23)-(25) from the 1987 paper for $Z1$, $Z2$, and $Z3$.

The lattice correction (the occupied volume of the lattice) for a
face-centered cubic structure of particles of radius $R$ and nearest
neighbor separation $D$ is

.. math::

   V_\text{lattice} = \frac{16\pi}{3}\frac{R^3}{\left(D\sqrt{2}\right)^3}

The distortion factor (one standard deviation) of the paracrystal is
included in the calculation of $Z(q)$

.. math::

    \Delta a = gD

where $g$ is a fractional distortion based on the nearest neighbor distance.

.. figure:: img/fcc_geometry.jpg

    Face-centered cubic lattice.

For a crystal, diffraction peaks appear at reduced q-values given by

.. math::

    \frac{qD}{2\pi} = \sqrt{h^2 + k^2 + l^2}

where for a face-centered cubic lattice $h, k , l$ all odd or all
even are allowed and reflections where $h, k, l$ are mixed odd/even
are forbidden. Thus the peak positions correspond to (just the first 5)

.. math::

    \begin{array}{cccccc}
    q/q_0 & 1 & \sqrt{4/3} & \sqrt{8/3} & \sqrt{11/3} & \sqrt{4} \\
    \text{Indices} & (111)  & (200) & (220) & (311) & (222)
    \end{array}

**NB**: The calculation of $Z(q)$ is a double numerical integral that
must be carried out with a high density of points to properly capture
the sharp peaks of the paracrystalline scattering. So be warned that the
calculation is SLOW. Go get some coffee. Fitting of any experimental data
must be resolution smeared for any meaningful fit. This makes a triple
integral. Very, very slow. Go get lunch!

The 2D (Anisotropic model) is based on the reference below where $I(q)$ is
approximated for 1d scattering. Thus the scattering pattern for 2D may not
be accurate. Note that we are not responsible for any incorrectness of the
2D model computation.

.. figure:: img/bcc_angle_definition.png

    Orientation of the crystal with respect to the scattering plane.


.. figure:: img/fcc_paracrystal_autogenfig.png

    1D and 2D plots corresponding to the default parameters of the model.

**References**

Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
(Original Paper)

Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
(Corrections to FCC and BCC lattice structure calculation)