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 220
d_factor Paracrystal distortion factor None 0.06
radius Radius of sphere 40
sld Sphere scattering length density 10-6-2 3
sld_solvent Solvent scattering length density 10-6-2 6.3
theta Orientation of the a1 axis w/respect incoming beam degree 0
phi Orientation of the a2 in the plane of the detector degree 0
psi Orientation of the a3 in the plane of the detector degree 0

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

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 is used to calculate \(Z(q)\), using equations (13)-(15) from the 1987 paper 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


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)

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


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.


Fig. 51 1D and 2D plots corresponding to the default parameters of the model.

Reference 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)