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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

$$ 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[#CIT1990]_ 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

$$ 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 $Z(q)$

$$ \Delta a = gD

$$

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

The simple cubic lattice is

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

$$ \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)

$$ \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 `orientation` . Note that we are not responsible for any incorrectness of the 2D model computation.

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).

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:** Steve King **Date:** March 25, 2019

**Last Reviewed by:** Richard Heenan **Date:** March 21, 2016

Created By |
sasview |

Uploaded |
Sept. 7, 2017, 3:56 p.m. |

Category |
Paracrystal |

Score |
0 |

Verified |
Verified by SasView Team on 07 Sep 2017 |

In Library |
This model is included in the SasView library by default |

Files |
sc_paracrystal.py sc_paracrystal.c |

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