Let's **find the Miller indices** for the directions A, B, C, and D in the cubic unit cell step by step. --- ## **Step 1: Understanding Miller Indices for Directions** For a direction \(\vec{r}\) in a cubic unit cell: 1. **Determine the vector's components** in terms of the unit cell dimensions (a, b, c). 2. **Subtract the tail coordinates from the head coordinates** to get the direction vector. 3. **Convert any fractions to whole numbers** by multiplying by the least common multiple (LCM). 4. **Enclose the indices in square brackets**: \([uvw]\). --- ## **Step 2: Analyze Each Direction** Let’s assume the origin is at the lower left front corner. ### **A. Direction A** - **Tail:** (, , ) - **Head:** (1/2, 1/2, ) **Direction vector:** \[ (1/2 - , \ 1/2 - , \ - ) = (1/2, 1/2, ) \] - Multiply by 2 to clear fractions: (1, 1, ) **Miller index:** \[ [110] \] --- ### **B. Direction B** - **Tail:** (1/2, , ) - **Head:** (1/2, 1/2, ) **Direction vector:** \[ (1/2 - 1/2, \ 1/2 - , \ - ) = (, 1/2, ) \] - Multiply by 2: (, 1, ) **Miller index:** \[ [010] \] --- ### **C. Direction C** - **Tail:** (1/2, , ) - **Head:** (1/2, 1/2, 1/2) **Direction vector:** \[ (1/2 - 1/2, \ 1/2 - , \ 1/2 - ) = (, 1/2, 1/2) \] - Multiply by 2: (, 1, 1) **Miller index:** \[ [011] \] --- ### **D. Direction D** - **Tail:** (1/2, 1/2, ) - **Head:** (, , 3/4) **Direction vector:** \[ ( - 1/2, \ - 1/2, \ 3/4 - ) = (-1/2, -1/2, 3/4) \] - Multiply by 4 (LCM of 2 and 4): \[ (-1/2 \times 4, -1/2 \times 4, 3/4 \times 4) = (-2, -2, 3) \] **Miller index:** \[ [\overline{2} \ \overline{2} \ 3] \] --- ## **Summary Table** | Direction | Vector Components | Miller Index | |-----------|-------------------|-------------| | A | (1/2, 1/2, ) | [110] | | B | (, 1/2, ) | [010] | | C | (, 1/2, 1/2) | [011] | | D | (-1/2, -1/2, 3/4) | [\(\overline{2}\) \(\overline{2}\) 3] | --- ## **Final Answers** - **A:** [110] - **B:** [010] - **C:** [011] - **D:** [\(\overline{2}\) \(\overline{2}\) 3] Let me know if you need further clarification or a breakdown for any specific direction!