Solid State

- The angular velocity vector is the vector sum of the components along the principal axes of the top.
- For precession, we use the relation for the precessional angular frequency: where , , and rad/s.
- Substituting the given values: Thus, the correct answer is (D) 7.

- The area of the first Brillouin zone for a two-dimensional hexagonal lattice (graphene) is given by the formula: This result comes from the reciprocal lattice vectors of the honeycomb structure.
Thus, the correct answer is (A).

- The energy dispersion relation for electrons in solids gives insight into the effective mass of the electrons. The curvature of the energy vs. wavevector plot ( ) is inversely related to the effective mass. The steeper the curvature, the smaller the effective mass.
- From the plots, we observe that the effective mass corresponds to the steepest curvature, indicating the smallest effective mass, and has the least curvature, indicating the largest effective mass. Thus, the correct order is .

- The transformation matrix for rotating a tensor by an angle is given by: - To apply the rotation to the original matrix, we use the transformation rule for tensor rotation: where is the rotation matrix and is the tensor matrix. - For the given tensor: - After applying the rotation matrix and calculating, we get:

- The Einstein frequency of a solid can be derived from the potential energy function. Using the standard procedure for harmonic oscillators and the form of the given potential energy, the Einstein frequency is found to be: - By analyzing the problem in terms of known constants and simplifying, the correct Einstein frequency formula is obtained as shown in option (B).

- In an FCC arrangement, there are 4 unit cells. Each unit cell has 8 octahedral voids and 12 tetrahedral voids.
- Number of Z ions = 4 (since Z occupies all FCC lattice sites).
- Number of X ions = of the tetrahedral voids = .
- Number of Y ions = of the octahedral voids = .
- The total number of Y ions is 3. Hence, the chemical formula is XYZ .