Using structure factor calculations, determine which planes the first six XRD peaks belong to in a face-centered cubic structured material.
To determine which planes the first six XRD (X-ray diffraction) peaks belong to in a face-centered cubic (FCC) structured material, we need to consider the formula for the structure factor (F_hkl) calculation for an FCC crystal. The structure factor relates to the intensity and position of the diffraction peaks.
The formula for the structure factor in an FCC crystal is:
F_hkl = f * (Z1 * exp(2πi(hx + ky + lz)) + Z2 * exp(2πi((h+l)x + (k+l)y + lz)) + Z3 * exp(2πi((h+l)x + ky + (k+l)z)))
Where:
- F_hkl is the structure factor for the specific set of planes with Miller indices (hkl).
- f is the atomic scattering factor, which depends on the nature of the atoms in the crystal.
- Z1, Z2, and Z3 are the number of atoms in the unit cell associated with each type of atom.
- x, y, z are the fractional coordinates of the atoms in the unit cell.
- h, k, l are the Miller indices of the diffraction peak.
To determine which planes correspond to the first six XRD peaks, we need to identify the set of planes with Miller indices (hkl) for which the structure factor F_hkl values are non-zero or significant. In an FCC crystal, the most intense diffraction peaks occur for specific values of (hkl) based on the values of h+k+l.
The FCC crystal has a stacking sequence of ABCABC..., where A, B, and C represent different atomic planes. The values of h, k, and l for strong reflections in the XRD pattern are related to the number of unit cells between equivalent planes in the stacking sequence.
For an FCC crystal, the following relationship holds: h + k + l must be an even number for non-zero structure factors.
Let's determine the planes corresponding to the first six XRD peaks by considering the values of (hkl) that satisfy the condition h + k + l = even.
1st peak (hkl): (111)
For (111), h + k + l equals 3. Since 3 is an odd number, this peak does not correspond to an FCC plane.
2nd peak (hkl): (200)
For (200), h + k + l equals 2. Since 2 is an even number, this peak corresponds to an FCC plane.
3rd peak (hkl): (220)
For (220), h + k + l equals 4. Since 4 is an even number, this peak corresponds to an FCC plane.
4th peak (hkl): (311)
For (311), h + k + l equals 5. Since 5 is an odd number, this peak does not correspond to an FCC plane.
5th peak (hkl): (222)
For (222), h + k + l equals 6. Since 6 is an even number, this peak corresponds to an FCC plane.
6th peak (hkl): (400)
For (400), h + k + l equals 4. Since 4 is an even number, this peak corresponds to an FCC plane.
Therefore, the planes corresponding to the first six XRD peaks in a face-centered cubic structured material are: (200), (220), (222), and (400).