.. _sc-paracrystal:

sc_paracrystal
=======================================================

Simple cubic lattice with paracrystalline distortion

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Source intensity                  None                     1
background  Source background                 |cm^-1|              0.001
dnn         Nearest neighbor distance         |Ang|                  220
d_factor    Paracrystal distortion factor     None                  0.06
radius      Radius of sphere                  |Ang|                   40
sld         Sphere scattering length density  |1e-6Ang^-2|             3
sld_solvent Solvent scattering length density |1e-6Ang^-2|           6.3
theta       c axis to beam angle              degree                   0
phi         rotation about beam               degree                   0
psi         rotation about c axis             degree                   0
=========== ================================= ============ =============

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


.. warning:: This model and this model description are under review following 
             concerns raised by SasView users. If you need to use this model, 
             please email help@sasview.org for the latest situation. *The 
             SasView Developers. September 2018.*
             
**Definition**

Calculates the scattering from a **simple 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.

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

.. math::

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

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 simple cubic structure.

Equation (16) of the 1987 reference\ [#CIT1987]_ is used to calculate $Z(q)$,
using equations (13)-(15) from the 1987 paper\ [#CIT1987]_ for $Z1$, $Z2$, and
$Z3$.

The lattice correction (the occupied volume of the lattice) for a simple cubic
structure of particles of radius *R* and nearest neighbor separation *D* is

.. math::

    V_\text{lattice}=\frac{4\pi}{3}\frac{R^3}{D^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.

The simple cubic lattice is

.. figure:: img/sc_crystal_geometry.jpg

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 simple cubic lattice any h, k, l are allowed and none are
forbidden. Thus the peak positions correspond to (just the first 5)

.. math::
    :nowrap:

    \begin{align*}
    q/q_0 \quad & \quad 1
                & \sqrt{2} \quad
                & \quad  \sqrt{3} \quad
                & \sqrt{4} \quad
                & \quad \sqrt{5}\quad \\
    Indices \quad & (100)
                  & \quad (110) \quad
                  & \quad (111)
                  & (200) \quad
                  & \quad (210)
    \end{align*}

.. note::

    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. Fitting of any experimental data
    must be resolution smeared for any meaningful fit. This makes a triple
    integral which may be very slow.

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 particularly at low $q$. For general details of the calculation
and angular dispersions for oriented particles see :ref:`orientation` .
Note that we are not responsible for any incorrectness of the
2D model computation.

.. figure:: img/parallelepiped_angle_definition.png

    Orientation of the crystal with respect to the scattering plane, when
    $\theta = \phi = 0$ the $c$ axis is along the beam direction (the $z$ axis).


.. figure:: img/sc_paracrystal_autogenfig.png

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

**Reference**

.. [#CIT1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765
   (Original Paper)
.. [#CIT1990] Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856
   (Corrections to FCC and BCC lattice structure calculation)

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** September 29, 2016
* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016

