4  Symmetry and Reciprocal Space

NoteLearning Objectives

• Define symmetry operations, point groups, and the crystallographic restriction in a way that connects directly to crystal physics.
• Explain why point symmetry constrains macroscopic tensors and leads to selection rules for allowed processes.
• Construct the reciprocal lattice from the direct lattice and interpret reciprocal vectors geometrically.
• Relate Miller planes, reciprocal-lattice vectors, and interplanar spacings.
• Define the first Brillouin zone and explain why physically distinct dispersions are plotted in \(\mathbf k\)-space.
• Prepare for later lectures on phonon branches and the density of states by understanding why reciprocal space is the natural language of lattice-periodic excitations.