you can see the simulated XRD pattern suggests a high peak with hkl (121), and correspondingly I marked the plane (121) in the SC structure in Mecury. But I cannot see the evident, exact relationship between the two where there's no much atoms or something special, so why, the 121 plane leads to the high peak?
Liqin Ye The correspondence between the crystal structure and the hkl indices in an X-ray diffraction (XRD) pattern is governed by Bragg's law and the atomic arrangement within the crystal lattice. The intensity of a diffraction peak, such as the high peak at hkl (121) in your simulated XRD pattern, depends on several factors, not just the number of atoms in the plane. Here's an explanation:
1. Bragg's Law and hkl Indices: The hkl indices (121 in your case) represent the Miller indices of a specific set of lattice planes. These planes diffract X-rays when the condition nλ= 2dhkl sinӨ is met, where dhkl is the interplanar spacing, λ is the wavelength (1.54056 Å in your case, likely Cu Kα radiation), and Ө is the scattering angle. The (121) plane's spacing and orientation determine where the peak appears in the XRD pattern.
2. Structure Factor and Peak Intensity: The intensity of the diffraction peak is proportional to the square of the structure factor |F_{hkl}|^2 , which depends on:
- The number and type of atoms in the unit cell.
- The positions of atoms relative to the (121) plane.
- The scattering power of the atoms (related to atomic number).
A high peak at (121) suggests that the atoms are arranged in a way that constructively interferes strongly for this plane, even if the plane itself doesn't pass through many atoms.
3. Atomic Arrangement and Interference: The (121) plane may not appear to have many atoms when visualized (e.g., in Mercury), but the key is the distribution of atoms across multiple unit cells. The structure factor sums the contributions from all atoms in the unit cell, weighted by their phase factors. If the (121) plane aligns with a periodic arrangement that maximizes constructive interference (e.g., due to symmetry or a high density of scattering centers when considering the 3D lattice), it can produce a strong peak.
4. Symmetry and Crystal System: In a simple cubic (SC) structure, certain hkl planes (like 121) may exhibit enhanced intensity due to the symmetry of the lattice. For example, if the structure has a high density of atoms or specific atomic species along planes related to (121) by symmetry, this can amplify the peak. Additionally, the presence of heavy atoms or ordered defects can contribute.
5. Why (121) is Prominent: The high peak at (121) likely arises because this plane's spacing and the atomic positions in your SC structure lead to a large structure factor. Even if the plane doesn't cut through many atoms in a single 2D slice, the 3D periodicity and the phase relationships between atoms in the lattice can result in a strong diffraction signal. This is common in structures where certain planes align with the unit cell's symmetry elements or have favorable scattering conditions.
To confirm this, you could:
- Calculate the structure factor F_{121} using the atomic coordinates from your Mercury model.
- Check the symmetry operations (e.g., via space group) that might enhance the (121) reflection.
- Compare the simulated pattern with the theoretical intensities for an SC lattice.
Thank you for your kind words, Liqin Ye !. It’s always a pleasure to engage in meaningful discussions, and I’m glad we could share insights together. If you have any further thoughts or questions, I’d be happy to continue the conversation!
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