# Explanation of the (1̅ 2 1̅ ) Plane in a Hexagonal Lattice Let's break down **how to sketch and understand the (1̅ 2 1̅ ) plane** in the hexagonal close-packed (HCP) crystal structure shown in your image. ## 1. **Understanding Miller-Bravais Indices (h k i l)** - **h, k, i, l** are the Miller-Bravais indices for hexagonal crystals. - For the plane (1̅ 2 1̅ ): - **h = -1** - **k = 2** - **i = -1** - **l = ** The last index (**l**) refers to the direction along the *c-axis* (vertical axis in the drawing). Since **l = **, the plane is **parallel to the c-axis** (it doesn’t cut the height of the prism). ## 2. **Plotting the Plane** ### Step 1: **Identify Axes** - **h** and **k** are shown in the image. - The **i** direction is calculated by: \( i = -(h + k) \). - In hexagonal geometry, the three axes (a₁, a₂, a₃) are at 120° to each other in the base plane. ### Step 2: **Find Intercepts** - **Intercepts** are found by taking the reciprocal of the indices (ignoring signs for now): - Along **a₁ (h)**: intercept at \(-1\) - Along **a₂ (k)**: intercept at \(1/2\) - Along **a₃ (i)**: intercept at \(-1\) - Along **c (l)**: intercept at ∞ (since l=), so **no intercept on the vertical axis**. ### Step 3: **Draw the Plane** - On the **base hexagon** (bottom face): 1. **From the origin**: - Move **-1** unit along **h (a₁)** (this is one unit in the negative h direction). - Move **2** units along **k (a₂)** (this is two units in the positive k direction). - Move **-1** unit along **i (a₃)**. 2. **Connect these points** on the hexagonal base. - The plane will **cut through the corners** in a diagonal fashion on the bottom hexagon, and since l=, this plane will be **vertical** (parallel to the c-axis). ## 3. **Visual Summary** - **The (1̅ 2 1̅ ) plane** is a vertical plane that: - Passes through the corners as described above. - Stays parallel to the c-axis (vertical axis). - You would shade the region connecting those intercepts in the base hexagon and extend vertically upwards. --- ### **In Your Diagram** 1. **Start at one corner** of the base hexagon. 2. **Count -1 step along h (to the left, if h is rightward)**. 3. **Count +2 steps along k (up/left, depending on your orientation)**. 4. **Count -1 step along i (down/right, at 120° to h and k)**. 5. **Connect these points**—that’s the trace of the (1̅ 2 1̅ ) plane on the base. Draw a vertical plane passing through this trace up the height of the prism. --- ## **Summary Table** | Index | Direction | Intercept | |-------|-----------|-----------| | h | a₁ | -1 | | k | a₂ | 2 | | i | a₃ | -1 | | l | c | ∞ | --- ## **Tip** - In exams, **highlight or shade** the plane you traced for clarity. If you'd like, I can sketch this for you if you upload a blank version of the lattice! Let me know if you want a step-by-step drawing.