semiconductor physics and Devices

Week 4: Reciprocal Lattice (Part 2)

Lecture Topics:

1. Ewald Construction

   • Introduction to the Ewald construction, a geometric method used in crystallography to understand diffraction in reciprocal space.

   • Visualization of reciprocal lattice points and diffraction conditions using the Ewald sphere.

   • How the Ewald construction helps in determining the possible diffraction angles based on the wavelength of incident radiation and crystal structure.

   • Application of Ewald construction to interpret X-ray, neutron, and electron diffraction experiments.

2. Brillouin Zones

   • Definition of Brillouin zones as the Wigner-Seitz cell in reciprocal space.

   • The first Brillouin zone: a key concept in describing electronic band structure and wave propagation in crystals.

   • Geometric construction of Brillouin zones for common crystal lattices (SC, BCC, FCC).

   • Understanding the importance of Brillouin zones in the study of electronic properties and phonon behavior.

3. Bragg's Law in Reciprocal Space

   • Recap of Bragg's Law: nλ=2dsin⁡θn\lambda = 2d\sin\thetanλ=2dsinθ.

   • Expressing Bragg’s Law in terms of the reciprocal lattice: diffraction occurs when a reciprocal lattice point lies on the Ewald sphere.

   • Relation between real space and reciprocal space: how lattice planes in real space correspond to points in reciprocal space.

   • How the reciprocal lattice provides a more convenient framework for understanding diffraction and wave interactions with solids.

4. Application of Reciprocal Lattice in X-ray Diffraction

   • Using the reciprocal lattice to interpret diffraction patterns.

   • Determining lattice parameters from X-ray diffraction data.

   • Comparison of real-space and reciprocal-space techniques for analyzing crystal structure.

   • Case study: Determining the structure of simple cubic and FCC lattices using X-ray diffraction.

Examples:

• Constructing the first Brillouin zone for a 2D square lattice.

• Applying the Ewald construction to visualize diffraction conditions in an FCC crystal.

• Using reciprocal space to explain why diffraction occurs at certain angles and not others.

Homework/Exercises:

1. Construct the first Brillouin zone for a simple cubic lattice.

2. Show how Bragg's Law can be derived using the concept of reciprocal lattice vectors.

3. Explain the relationship between real-space lattice planes and reciprocal lattice points for a body-centered cubic (BCC) structure.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 2: Reciprocal Lattice (continued).

Key Takeaways:

• The Ewald construction is a powerful visual tool for understanding diffraction in reciprocal space.

• Brillouin zones are essential for analyzing electronic properties and phonon dynamics in solid-state systems.

• Reciprocal lattice theory simplifies the understanding of diffraction and provides a clear connection between real and reciprocal space.

This week builds on the concept of the reciprocal lattice introduced in Week 3, with a focus on advanced topics like Ewald construction and Brillouin zones, which are crucial for understanding diffraction and electronic properties in solid-state physics.