# X-ray scattering simulation¶

The module for simulation of X-ray scattering properties from the atomic level. The approach works only for finite systems, so that periodic boundary conditions and cell shape are ignored.

## Theory¶

The scattering can be calculated using Debye formula [Debye1915] :

$I(q) = \sum_{a, b} f_a(q) \cdot f_b(q) \cdot \frac{\sin(q \cdot r_{ab})}{q \cdot r_{ab}}$

where:

• $$a$$ and $$b$$ – atom indexes;

• $$f_a(q)$$$$a$$-th atomic scattering factor;

• $$r_{ab}$$ – distance between atoms $$a$$ and $$b$$;

• $$q$$ is a scattering vector length defined using scattering angle ($$\theta$$) and wavelength ($$\lambda$$) as $$q = 4\pi \cdot \sin(\theta)/\lambda$$.

The thermal vibration of atoms can be accounted by introduction of damping exponent factor (Debye-Waller factor) written as $$\exp(-B \cdot q^2 / 2)$$. The angular dependency of geometrical and polarization factors are expressed as [Iwasa2007] $$\cos(\theta)/(1 + \alpha \cos^2(2\theta))$$, where $$\alpha \approx 1$$ if incident beam is not polarized.

### Units¶

The following measurement units are used:

• scattering vector $$q$$ – inverse Angstrom (1/Å),

• thermal damping parameter $$B$$ – squared Angstrom (Å2).

## Example¶

The considered system is a nanoparticle of silver which is built using FaceCenteredCubic function (see ase.cluster) with parameters selected to produce approximately 2 nm sized particle:

from ase.cluster.cubic import FaceCenteredCubic
import numpy as np

surfaces = [(1, 0, 0), (1, 1, 0), (1, 1, 1)]
atoms = FaceCenteredCubic('Ag', [(1, 0, 0), (1, 1, 0), (1, 1, 1)],
[6, 8, 8], 4.09)


Next, we need to specify the wavelength of the X-ray source:

xrd = XrDebye(atoms=atoms, wavelength=0.50523)


The X-ray diffraction pattern on the $$2\theta$$ angles ranged from 15 to 30 degrees can be simulated as follows:

xrd.calc_pattern(x=np.arange(15, 30, 0.1), mode='XRD')
xrd.plot_pattern('xrd.png')


The resulted X-ray diffraction pattern shows (220) and (311) peaks at 20 and ~24 degrees respectively.

The small-angle scattering curve can be simulated too. Assuming that scattering vector is ranged from $$10^{-2}=0.01$$ to $$10^{-0.3}\approx 0.5$$ 1/Å the following code should be run:

xrd.calc_pattern(x=np.logspace(-2, -0.3, 50), mode='SAXS')
xrd.plot_pattern('saxs.png')


The resulted SAXS pattern:

## Further details¶

The module contains wavelengths dictionary with X-ray wavelengths for copper and wolfram anodes:

from ase.utils.xrdebye import wavelengths
print('Cu Kalpha1 wavelength: %f Angstr.' % wavelengths['CuKa1'])


The dependence of atomic form-factors from scattering vector is calculated based on coefficients given in waasmaier dictionary according [Waasmaier1995] if method of calculations is set to ‘Iwasa’. In other case, the atomic factor is equal to atomic number and angular damping factor is omitted.

### XrDebye class members¶

class ase.utils.xrdebye.XrDebye(atoms, wavelength, damping=0.04, method='Iwasa', alpha=1.01, warn=True)[source]

Class for calculation of XRD or SAXS patterns.

Initilize the calculation of X-ray diffraction patterns

Parameters:

atoms: ase.Atoms

atoms object for which calculation will be performed.

wavelength: float, Angstrom

X-ray wavelength in Angstrom. Used for XRD and to setup dumpings.

dampingfloat, Angstrom**2

thermal damping factor parameter (B-factor).

method: {‘Iwasa’}

method of calculation (damping and atomic factors affected).

If set to ‘Iwasa’ than angular damping and q-dependence of atomic factors are used.

For any other string there will be only thermal damping and constant atomic factors ($$f_a(q) = Z_a$$).

alpha: float

parameter for angular damping of scattering intensity. Close to 1.0 for unplorized beam.

warn: boolean

flag to show warning if atomic factor can’t be calculated

calc_pattern(x=None, mode='XRD', verbose=False)[source]

Calculate X-ray diffraction pattern or small angle X-ray scattering pattern.

Parameters:

x: float array

points where intensity will be calculated. XRD - 2theta values, in degrees; SAXS - q values in 1/A ($$q = 2 \pi \cdot s = 4 \pi \sin( \theta) / \lambda$$). If x is None then default values will be used.

mode: {‘XRD’, ‘SAXS’}

the mode of calculation: X-ray diffraction (XRD) or small-angle scattering (SAXS).

Returns

list of intensities calculated for values given in x.

get(s)[source]

Get the powder x-ray (XRD) scattering intensity using the Debye-Formula at single point.

Parameters:

s: float, in inverse Angstrom

scattering vector value ($$s = q / 2\pi$$).

Returns

Intensity at given scattering vector $$s$$.

get_waasmaier(symbol, s)[source]

Parameters:

symbol: string

atom element symbol.

s: float, in inverse Angstrom

scattering vector value ($$s = q / 2\pi$$).

Returns

Intensity at given scattering vector $$s$$.

Note

for hydrogen will be returned zero value.

plot_pattern(filename=None, show=False, ax=None)[source]

Plot XRD or SAXS depending on filled data

Uses Matplotlib to plot pattern. Use show=True to show the figure and filename=’abc.png’ or filename=’abc.eps’ to save the figure to a file.

Returns

matplotlib.axes.Axes object.

set_damping(damping)[source]

set B-factor for thermal damping

write_pattern(filename)[source]

Save calculated data to file specified by filename string.

## References¶

Debye1915
1. Debye Ann. Phys. 351, 809–823 (1915)

Iwasa2007
1. Iwasa, K. Nobusada J. Phys. Chem. C, 111, 45-49 (2007) doi:10.1021/jp063532w

Waasmaier1995
1. Waasmaier, A. Kirfel Acta Cryst. A51, 416-431 (1995)