# fcc_paracrystal

Face-centred cubic lattic with paracrystalline distortion

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

dnn

Nearest neighbour distance

220

d_factor

Paracrystal distortion factor

None

0.06

40

sld

Particle scattering length density

10-6-2

4

sld_solvent

Solvent scattering length density

10-6-2

1

theta

c axis to beam angle

degree

60

phi

degree

60

psi

degree

60

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 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

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

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[2] is used to calculate $$Z(q)$$, using equations (23)-(25) from the 1987 paper[1] 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

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

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

$\Delta a = gD$

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

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 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)

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

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 Oriented particles. Note that we are not responsible for any incorrectness of the 2D model computation.

Source

fcc_paracrystal.py $$\ \star\$$ fcc_paracrystal.c $$\ \star\$$ sphere_form.c $$\ \star\$$ gauss150.c $$\ \star\$$ sas_3j1x_x.c

References

Authorship and Verification

• Author: NIST IGOR/DANSE Date: pre 2010