Fcc Paracrystal

#fcc paracrystal model #note model title and parameter table are automatically inserted #note - calculation requires double precision Definition

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.

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

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

.. note:: At this point the GUI does not return $V_\text{lattice}$ separately so that the user will need to calculate it from the equation given and the appropriate returned parameters.

.. warning:: As per the equations below, this model will return I(q)=0 for all q if the distortion factor is equal to 0. The model is not meant to support perfect crystals.

Face-centered cubic (FCC) lattice taken from reference [#Matsuoka1987]_.

Following the derivation from reference [#Matsuoka1987]_, as corrected in reference [#Matsuoka1990]_, and based on the above figure, the primitive unit cell vectors $\vec{a_1},\vec{a_2}$, and $\vec{a_3}$, which enclose the smallest possible unit cell for the fcc lattice, are defined below:

$$ \vec{a_1} = \frac{1}{2} (\vec{b_2} + \vec{b_3}) \\ \vec{a_2} = \frac{1}{2} (\vec{b_1} + \vec{b_3}) \\ \vec{a_3} = \frac{1}{2} (\vec{b_1} + \vec{b_2}).
$$

Where the $\vec{b_1},\vec{b_2}$, and $\vec{b_3}$ are the conventional unit cell vectors. As defined by reference [#Matsuoka1987]_, the constant $a$ is the lattice parameter of the conventional unit cell with $|\vec{b_1}|=|\vec{b_2}|=|\vec{b_3}|=a$. Using this definition, the nearest-neighbor distance ($D$) is given by $D=|\vec{a_1}|=|\vec{a_2}|=|\vec{a_3}|=\sqrt{(a/2)^2+(a/2)^2}=\frac{a}{\sqrt2}$.

The volume of the primitive unit cell $V_u$ is given by:

$$ V_u = |(\vec{a_1}\times \vec{a_2})\cdot\vec{a_3}| \\ = (\frac{a^2}{4},\frac{a^2}{4},-\frac{a^2}{4})\cdot(\frac{a}{2},\frac{a}{2},0)\\ =a^3/4
$$
In this case, the volume fraction ($V_{lattice}$) of spherical particles with radius $R$ sitting on the fcc lattice is given by:

$$ V_{lattice} = \frac{4/3 \pi R^3}{a^3/4}\\ = \frac{16\pi R^3}{3a^3}\\ = \frac{8 \pi R^3}{3\sqrt{2}D^3}.
$$
Now, continuing to follow [#Matsuoka1987]_, the structure (lattice) factor $Z(\vec{q})$ for a 3D paracrystal can be written as:

$$ Z(\vec{q}) = \prod_{k=1}^{3}Z_k(\vec{q})
$$
with

$$ Z_k(\vec{q}) = \frac{1-|F_k|^2}{1-2|F_k|\cos(\vec{a_k}\cdot\vec{q})+|F_k|^2}
$$
and where $F_k(\vec{q})$ is the structure factor of the primitive unit cell defined as:

$$ F_k(\vec{q}) = e^{-\frac{1}{2} \Delta a^2_k q^2} \times e^{-i\vec{q}\cdot\vec{a_k}}.
$$
Here, $\vec{a_k}$ are the primitive unit cell vectors $\vec{a_1}$, $\vec{a_2}$, and $\vec{a_3}$. Furthermore, $\Delta a_k$ is the isotropic distortion of the lattice point from its ideal position and can be defined by a constant factor $g=\Delta a / |\vec{a_1}| = \Delta a / |\vec{a_2}| = \Delta a / |\vec{a_3}|=\Delta a/D$.

Finally, assuming the definitions presented in this document, the authors of reference [#Matsuoka1987]_ have derived the lattice factors which are given by:

$$ Z_1(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qa}{2}(\sin\theta \cos\phi + \sin\theta \sin\phi + \cos\theta)] + e^{-q^2\Delta a^2}\}\\ Z_2(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qa}{2}(-\sin\theta \cos\phi - \sin\theta \sin\phi + \cos\theta)] + e^{-q^2\Delta a^2}\}\\ Z_3(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qa}{2}(-\sin\theta \cos\phi + \sin\theta \sin\phi - \cos\theta)] + e^{-q^2\Delta a^2}\}\\
$$
Note that Sasview is using the nearest-neighbor parameter ($D$) as an input instead of the conventional unit cell parameter $a$. In this case, Using $a=\sqrt{2}D$, we rewrite $Z_1(q)$, $Z_2(q)$, and $Z_3(q)$ in terms of $D$ instead of $a$, which leads to:

$$ Z_1(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qD}{\sqrt{2}}(\sin\theta \cos\phi + \sin\theta \sin\phi + \cos\theta)] + e^{-q^2\Delta a^2}\}\\ Z_2(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qD}{\sqrt{2}}(-\sin\theta \cos\phi - \sin\theta \sin\phi + \cos\theta)] + e^{-q^2\Delta a^2}\}\\ Z_3(q,\theta,\phi)=[1-e^{-q^2\Delta a^2}]/\{1-2e^{-\frac{1}{2}q^2\Delta a^2}\cos[\frac{qD}{\sqrt{2}}(-\sin\theta \cos\phi + \sin\theta \sin\phi - \cos\theta)] + e^{-q^2\Delta a^2}\}\\
$$
Finally, the position of the Bragg peaks for the fcc lattice are indexed by (reduced q-values):

$$ \frac{qa}{2\pi}=\frac{qD}{\sqrt{2}\pi}=\sqrt{h^2+k^2+l^2}.
$$
where the scattering condition imposes that h,k, and l are all odd or all even. Thus the peak positions correspond to (just the first 5)

$$ \begin{array}{cccccc} q/q_0 & 1 & \sqrt{4/3} & \sqrt{8/3} & \sqrt{11/3} & \sqrt{4} \\ \text{Indices} & (111) & (200) & (220) & (311) & (222) \end{array}
$$
.. 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. If a double-precision GPU with OpenCL support is available this may improve the speed of the calculation.

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).

References

.. [#Matsuoka1987] Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 (Original Paper) .. [#Matsuoka1990] 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:** Jonathan Gaudet **Date:** September 26, 2022
**Last Reviewed by:** Paul Butler **Date:** November 2, 2022