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In thewe discussed in detail the effects of lattice periodicity on the single-particle wavefunctions and the energy eigenvalues. We also touched on the notion that a crystal can have symmetries beyond the translational periodicity, such as rotations around axes, reflections on planes, and combinations of these operations with translations by vectors that are not lattice vectors, called “non-primitive” translations. All these symmetry operations are useful in calculating and analyzing the physical properties of a crystal. There are two basic advantages to using the symmetry operations of a crystal in describing its properties. First, the volume in reciprocal space for which solutions need to be calculated is further reduced, usually to a small fraction of the first Brillouin zone, called the “irreducible” part; for example, in the FCC crystals with one atom per unit cell, the irreducible part is of the full BZ.
