Question

The vector form of Bragg's law is used in the construction of:

• A. Brillouin zone
• B. Extended Zone
• C. Reduced Zone
• D. Periodic zone

Hint:

Associate "reciprocal space" with concepts like Bragg's law in vector form, reciprocal lattice vectors, and Brillouin zones. The Brillouin zone is essentially the "unit cell" of the reciprocal lattice, and its boundaries represent points where waves are strongly diffracted.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question connects Bragg's law of diffraction with the concept of Brillouin zones in reciprocal space. Brillouin zones are fundamental to understanding the behavior of waves (like electron wavefunctions or phonons) in a periodic crystal lattice.
Step 2: Detailed Explanation:
Bragg's law in its scalar form is \( 2d \sin\theta = n\lambda \). In reciprocal space, the condition for Bragg diffraction can be expressed in vector form. If \(\vec{k}\) is the wave vector of the incident wave and \(\vec{k'}\) is the wave vector of the diffracted wave, the diffraction condition is \( \vec{k'} - \vec{k} = \vec{G} \), where \(\vec{G}\) is a reciprocal lattice vector.
Since diffraction is an elastic scattering process, \( |\vec{k'}| = |\vec{k}| \). This leads to the vector form of Bragg's law:
\[ 2\vec{k} \cdot \vec{G} = G^2 \]
Geometrically, this equation describes a plane that is the perpendicular bisector of the reciprocal lattice vector \(\vec{G}\).
The first Brillouin zone is the Wigner-Seitz primitive cell of the reciprocal lattice. Its boundaries are formed by the set of these perpendicular bisector planes for the shortest reciprocal lattice vectors. Therefore, the vector form of Bragg's law directly defines the boundaries of the Brillouin zones.
Step 3: Final Answer:
The vector form of Bragg's law defines the planes that form the boundaries of the Brillouin zones.