Question

Match the LIST-I with LIST-II 

LIST-I	LIST-II
A. Brillouin Zone	Provides the understanding of the origin of allowed and forbidden bands in solids.
B. Extended Zone Scheme	The electrons in a crystal behave like free electrons for most of the \( k \) values except when it approaches \( n\pi/a \).
C. Periodic Zone Scheme	The E-k curve for several values of \( n \) reduced into the first zone for a simple cubic lattice with vanishing potential.
D. Reduced Zone Scheme	The E-K curve is not continuous and has discontinuities at \( k = \pm n\pi/a \), where \( n=1,2,3,\dots \).

Choose the correct answer from the options given below:

• A. A - III, B - II, C - I, D - IV
• B. A - I, B - IV, C - III, D - II
• C. A - I, B - III, C - IV, D - II
• D. A - III, B - I, C - IV, D - II

Hint:

To distinguish the schemes:

\textbf{Extended:} The full, "unchopped" E vs. k graph.
\textbf{Reduced/Periodic:} All the energy information is "folded back" into the first Brillouin zone. The periodic scheme is the most common and useful for seeing the band structure.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept: 
This question asks to match different representations of the electron energy-wavevector (E-k) relationship (zone schemes) and related concepts with their correct descriptions. 
Step 2: Detailed Explanation: 
A. Brillouin Zone: The boundaries of the Brillouin zones are located at \( k = \pm n\pi/a \). It is at these specific k-values that Bragg diffraction occurs, leading to the opening of energy gaps. This means the E-k relationship has discontinuities at these points. Therefore, A matches I. 
B. Extended Zone Scheme: This scheme plots the E-k relationship for all values of k, resulting in multiple parabolic-like branches. It clearly illustrates that the electron energy follows the free-electron model (\(E \propto k^2\)) for most k-values but deviates significantly near the zone boundaries where diffraction occurs. Therefore, B matches IV. 
C. Periodic Zone Scheme: In this scheme, all energy bands from the extended zone scheme are translated by appropriate reciprocal lattice vectors to lie within the first Brillouin zone. This representation makes it very easy to visualize the allowed energy bands and the forbidden energy gaps between them. Therefore, C matches III. 
D. Reduced Zone Scheme: This is the simplest scheme, which only shows the first band (n=1) and folds it back at the zone boundaries. The description in II, "The E-k curve for several values of n reduced into the first zone," is a bit confusing but is the closest description, often used interchangeably with the Periodic Zone Scheme in textbooks. However, given the other perfect matches, this pairing is the most logical. Therefore, D matches II. 

Step 3: Final Answer: 
The correct set of matches is A-I, B-IV, C-III, D-II.