In an ideal infinite crystal, the equilibrium position of each individual atom is determined by the forces exerted by all the other atoms in the crystal, resulting in a periodic structure.
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In general, Bloch oscillations are a consequence of the periodic structure of the lattice potential and the existence of a maximum value of the Bloch wave vector k _ { max }.
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More generally, for periodic structures having discrete translational symmetry, the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations.
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Recently, a graphene-based Bragg grating ( one-dimensional photonic crystal ) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a prism coupling technique.
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A periodic structure can support both forward and backward space harmonics, which are not modes of the field, and cannot exist independently, even if a beam can be coupled to only one of them.
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The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the Laue conditions the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector.
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Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic.
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A graphene-based Bragg grating ( one-dimensional photonic crystal ) has been fabricated and demonstrated its capability for excitation of surface electromagnetic waves in the periodic structure by using He Ne laser as the light source.
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A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity, but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms.
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Another strand of research has tried to construct three-dimensional photonic structures from self-assembly essentially letting a mixture of dielectric nano-spheres settle from solution into three-dimensionally periodic structures that have photonic band-gaps.
