**Part b: Indices \([hkl]\) that define the rotation axes in a cube** A cube has **three types of rotation axes** (symmetry axes): 1. **Four-fold axes** (\(C_4\)) – through the centers of opposite faces 2. **Three-fold axes** (\(C_3\)) – through opposite corners (body diagonals) 3. **Two-fold axes** (\(C_2\)) – through the midpoints of opposite edges The Miller indices \([hkl]\) for each are: --- ### 1. **Four-fold axes (\(C_4\))** - **Direction**: Perpendicular to faces - **Indices**: Along the cube edges \[ [100],\ [010],\ [001] \] (and their negatives: \([-100],\ [0-10],\ [00-1]\)) --- ### 2. **Three-fold axes (\(C_3\))** - **Direction**: Along body diagonals (corner to opposite corner) - **Indices**: \[ [111],\ [1\bar{1}1],\ [11\bar{1}],\ [\bar{1}11] \] (and their negatives) --- ### 3. **Two-fold axes (\(C_2\))** - **Direction**: Through centers of opposite edges (edge diagonals) - **Indices**: \[ [110],\ [101],\ [011] \] (and their negatives: \([1\bar{1}0],\ [10\bar{1}],\ [0 1 \bar{1}]\)) --- ## **Final Answer (part b):** **The Miller indices \([hkl]\) that define the rotation axes in a cube are:** - **Four-fold axes:** \([100],\ [010],\ [001]\) - **Three-fold axes:** \([111],\ [1\bar{1}1],\ [11\bar{1}],\ [\bar{1}11]\) - **Two-fold axes:** \([110],\ [101],\ [011]\)