# Explanation: Determining the HKL (hkil) Indices for Direction B Let’s break down the process step by step: --- ## **Step 1: Understanding the Axes** In a **hexagonal crystal system**, we use: - **Three a-axes** (a₁, a₂, a₃) in the basal plane, 120° apart - **One c-axis** (vertical axis) ### **Four-Axis System Notation** For directions, we use **[uvtw]** (the Miller-Bravais indices), where: - **u, v, t** are for the a-axes - **w** is for the c-axis --- ## **Step 2: Find the HKL Indices for Direction B** Let's describe the **direction B** in the **three-axis system** (hkl): - **Direction B** starts from the origin and goes parallel to the **a₂ axis** (as seen in the figure). - In Cartesian coordinates, direction B would be along **a₂**. So, in terms of the three axes: - Along **a₁**: - Along **a₂**: 1 - Along **c**: So, the **indices in the three-axis (hkl) system** are: **[ 1 ]** --- ## **Step 3: Convert to Four-Axis (Miller-Bravais) System** ### **Conversion Formula** To convert from (h k l) to (h k i l): - **u = h** - **v = k** - **t = -(h + k)** - **w = l** So, for [ 1 ]: - **u = ** - **v = 1** - **t = -( + 1) = -1** - **w = ** So, in four-axis notation (**[uvtw]**): **[ 1 -1 ]** --- ## **Step 4: Final Answer** **The HKL (hkil) indices for direction B are:** \[ [\ 1\ \overline{1}\ ] \] --- ## **Summary Table** | System | Indices for B | |---------------------|-----------------| | Three-axis (hkl) | [ 1 ] | | Four-axis (hkil) | [ 1 -1 ] | --- ### **Key Points** - Always check the vector’s direction in terms of the axes. - Use the conversion formula for hexagonal systems. - The negative index is often written with a bar (e.g., \(\overline{1}\)). --- **If you need more examples or explanation for other directions, let me know!**