XRD Simulation

The X-Ray Diffraction (XRD) module simulates the powder diffraction pattern for the loaded structure. It utilizes Kinematic Diffraction Theory, assuming ideal interactions without primary extinction or multiple scattering events.

Physical Model

1. Lattice Physics

The simulation first calculates the Reciprocal Lattice vectors ($b_1, b_2, b_3$) from the real space lattice ($a_1, a_2, a_3$): $$b_1 = 2\pi \frac{a_2 \times a_3}{a_1 \cdot (a_2 \times a_3)}$$ (and cyclic permutations for $b_2, b_3$).

2. Bragg Condition

The code iterates through integer Miller indices $(h, k, l)$ to find reciprocal lattice vectors $G_{hkl} = h b_1 + k b_2 + l b_3$. The d-spacing is calculated as: $$d_{hkl} = \frac{2\pi}{|G_{hkl}|}$$ Diffraction peaks are identified where the Bragg condition is met for the source wavelength ($\lambda = 1.5406 Å$, Cu K$\alpha$): $$\lambda = 2d \sin\theta$$

3. Structure Factor ($F_{hkl}$)

The intensity of each reflection is governed by the Structure Factor, calculated by summing over all $N$ atoms in the unit cell: $$F_{hkl} = \sum_{j=1}^{N} f_j(\theta) \exp\left[2\pi i (hx_j + ky_j + lz_j)\right]$$

• $x_j, y_j, z_j$: Fractional coordinates of atom $j$.
• $f_j(\theta)$: The atomic scattering factor.

4. Intensity Correction

The raw squared structure factor $|F_{hkl}|^2$ is corrected to obtain the observed intensity $I$. The primary correction applied in xrd.rs is the Lorentz-Polarization (LP) Factor for unpolarized radiation (standard laboratory diffractometer):

$$ \begin{align} LP(\theta) &= \frac{1 + \cos^2(2\theta)}{\sin^2\theta \cos\theta}\\ I_{calc} &= |F_{hkl}|^2 \times LP(\theta) \end{align}$$

5. Multiplicity and Merging

In powder diffraction, symmetry-equivalent planes (e.g., $(100)$ and $(010)$ in cubic systems) diffract at the same angle. The code algorithmically sorts peaks by $2\theta$ and merges peaks falling within a narrow tolerance ($0.05^\circ$), summing their intensities to account for multiplicity.

6. Match with Experiment

You can load the experimental ascii/xlsx file to compare the xrd using the "Load Experiment" button.

Settings

• Wavelength: Default fixed to Cu K$\alpha$ ($1.5406 Å$).
• Range: $10^\circ$ to $90^\circ$ $2\theta$.
• Filtering: Peaks with intensity $\leq 10^{-4}$ relative to the maximum are discarded to reduce noise.