Calculate The First Six Diffraction-Peak Positions for Mgo Powder

Introduction

X-ray diffraction is a powerful technique used to analyze the crystalline structure of materials. When X-rays are incident on a crystalline material, they are diffracted at specific angles that depend on the material's lattice parameters. For MGO powder, we can calculate the positions of the first six diffraction peaks using Bragg's law.

The diffraction peaks correspond to specific planes in the crystal lattice, and their positions are determined by the spacing between these planes. The first six peaks typically correspond to the (111), (200), (220), (311), (222), and (400) planes in the face-centered cubic (FCC) structure of MGO.

X-ray Diffraction Theory

Bragg's law describes the condition for constructive interference of X-rays diffracted from a crystal lattice:

The interplanar spacing d for a given set of planes (hkl) in a cubic crystal is given by:

For MGO, the lattice parameter a is approximately 4.212 Å.

Calculation Method

The calculation involves the following steps:

1. Identify the Miller indices for the first six diffraction peaks in MGO (typically (111), (200), (220), (311), (222), and (400))
2. Calculate the interplanar spacing d for each set of planes using the lattice parameter
3. Use Bragg's law to calculate the diffraction angle θ for each d value
4. Convert the angles to 2θ values (2θ = 2θ) for the diffraction peaks

The calculator automates these steps using the known values for MGO.

Worked Example

Let's calculate the first diffraction peak for MGO using the (111) planes:

1. Miller indices: h = 1, k = 1, l = 1
2. Calculate d for (111):

1/d² = (1² + 1² + 1²)/4.212² = 3/17.744 ≈ 0.169

d ≈ 1/√0.169 ≈ 2.89 Å

3. Use Bragg's law with λ = 1.5406 Å (Cu Kα radiation):

1.5406 = 2 × 2.89 × sinθ

sinθ ≈ 0.268

θ ≈ arcsin(0.268) ≈ 15.5°

2θ ≈ 31.0°

The first diffraction peak for MGO is at approximately 31.0°.