(a) What are the direction indices for a vector that passes from point \(\frac{1}{3} \frac{1}{2} 0\) to point \(\frac{2}{3} \frac{3}{4} \frac{1}{2}\) in a tetragonal unit cell? (b) Repeat part (a) for a rhombohedral unit cell.

To determine the vector representing the direction between the two given points, we need to subtract the initial point coordinates from the final point coordinates. Let point A be \(\frac{1}{3} \frac{1}{2} 0\) and point B be \(\frac{2}{3} \frac{3}{4} \frac{1}{2}\). We can calculate the vector as follows: Vector AB = Point B - Point A

In a tetragonal unit cell, the a and b lattice parameters are equal, and the c lattice parameter is different. Let \(a_t\) and \(c_t\) be the lattice parameters for the tetragonal unit cell. Since we are not given specific values for \(a_t\) and \(c_t\), we can assume that they are equal to 1 for simplicity. Now we can find the direction indices (h, k, l) for the tetragonal unit cell using the vector AB: h = a_t * (Bx - Ax) = 1 * (\frac{2}{3} - \frac{1}{3}) = 1 * \frac{1}{3} k = a_t * (By - Ay) = 1 * (\frac{3}{4} - \frac{1}{2}) = 1 * \frac{1}{4} l = c_t * (Bz - Az) = 1 * (\frac{1}{2} - 0) = 1 * \frac{1}{2} So, the direction indices for the tetragonal unit cell are (h, k, l) = (1, 1, 2).

In a rhombohedral unit cell, all three lattice parameters are equal, and all three angles have the same value. Let \(a_r\) be the lattice parameter for the rhombohedral unit cell, and let \(\alpha_r\) be the angle between these lattice vectors. Since we are not given specific values for \(a_r\) and \(\alpha_r\), we can assume for simplicity that \(a_r = 1\) and \(\alpha_r = 90^\circ\) (which yields a cubic unit cell as a special case of the rhombohedral unit cell). Now we can find the direction indices (h', k', l') for the rhombohedral unit cell using the vector AB: h' = a_r * (Bx - Ax) = 1 * (\frac{2}{3} - \frac{1}{3}) = 1 * \frac{1}{3} k' = a_r * (By - Ay) = 1 * (\frac{3}{4} - \frac{1}{2}) = 1 * \frac{1}{4} l' = a_r * (Bz - Az) = 1 * (\frac{1}{2} - 0) = 1 * \frac{1}{2} So, the direction indices for the rhombohedral unit cell are (h', k', l') = (1, 1, 2).