Question

Choose the correct statement for the first Brillouin zone in two dimensions:
A. The region in k space that the electrons can occupy without being diffracted is called the First Brillouin zone.
B. For k<\(\pi\)/a electrons can not move freely in any direction inside the square without being diffracted.
C. For k = \(\pi\)/a electrons are prevented from moving in the x or y directions due to diffraction.
D. For k>\(\pi\)/a electrons can move perpendicularly inside the square.
Choose the correct answer from the options given below:

• A. A, B and D only
• B. A and B only
• C. A and C only
• D. B, C and D only

Hint:

Think of the Brillouin zone boundaries as "walls" in k-space. Inside the walls (the first BZ), electrons travel freely. When an electron's k-vector hits a wall, it gets diffracted, forming a standing wave that doesn't propagate. This is what creates the energy band gaps.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The first Brillouin zone (BZ) is a fundamental concept in solid-state physics, representing the primitive cell of the reciprocal lattice. Its boundaries are defined by the condition for Bragg diffraction. This question tests the physical meaning of the first BZ.
Step 2: Detailed Explanation:
Let's analyze each statement for a 2D square lattice of constant 'a'. The boundaries of the first BZ are at \(k_x = \pm \pi/a\) and \(k_y = \pm \pi/a\).


A. The region in k space that the electrons can occupy without being diffracted is called the First Brillouin zone. This is the essential physical definition. Electron states with wave vectors \(\mathbf{k}\) well inside the first BZ behave like free particles and do not undergo Bragg diffraction. This statement is correct.
B. For k<\(\pi\)/a electrons can not move freely... This is the opposite of statement A. For wave vectors with magnitude k smaller than the value at the boundary (\(\pi/a\)), electrons \textit{can} move freely without diffraction. This statement is incorrect.
C. For k = \(\pi\)/a electrons are prevented from moving in the x or y directions due to diffraction. When the wave vector component reaches the zone boundary (e.g., \(k_x = \pi/a\)), the condition for Bragg diffraction is met. This leads to the formation of standing waves instead of traveling waves. A standing wave has zero group velocity (\(v_g = d\omega/dk = (1/\hbar)dE/dk = 0\)), meaning there is no net propagation of the electron in that direction. Thus, they are prevented from moving. This statement is correct.
D. For k>\(\pi\)/a electrons can move perpendicularly inside the square. This statement is confusing and imprecise. While there are allowed energy states for \(k>\pi/a\) (in higher Brillouin zones), this statement doesn't add any clear, correct information. Compared to the precise and correct statements A and C, it is not a good description.

Step 3: Final Answer:
Statements A and C provide correct physical descriptions of the first Brillouin zone and the effect of its boundaries on electron propagation.