The Effect Of Symmetry On Acoustic Band Gaps In Two-dimensional Phononic Crystals

Phononic crystal (PC) is a kind of periodic-structure material, in which there exist acoustic or elastic wave band-gaps (BG's). It belongs to a composite material. During the BG's sound and vibration are forbidden. One remarkable character of PC is that we can design these materials with BG's of different sizes and bounds by choosing different lattices, scatters, filling factors and materials with various parameters. The effects of symmetry on acoustic BG's in two-dimensional (2D) PC's are discussed in detail and systematically, which will help the artificial design of PC's.First, the concept of group is introduced, which is a mathematical tool for analyzing symmetry. And the application of group in crystal structure, and then the 17 kinds of 2D space groups are shown particularly.Next, the basic theory and method for calculating the band structure of PC are presented, especially the plane wave expansion method (PWE). The plane-wave-expansion formulas are deduced from the elastic wave equations. And then the application of PWE to the PC models considered in this paper is illustrated.Then, by considering the water-mercury and steel/air systems with five types of straight rods (with regular triangle (TRI), square (SQU), hexagonal (HEX), octagonal (OCT) and circular (CIR) cross-sections) arranged in two kinds of lattices (square and hexagonal), the effects of symmetry on acoustic BG's are studied at the below three different levels. First, for the given rods and lattice, we discussed the relationship between the width of BG and the system's symmetries which are changed by rotating the rods. Second, on the condition of a given lattice, we found the maximal BG value by adjusting the orientation of rods according to the above result, and then the dependence of BG on the rods' symmetry are analyzed. Third, the largest BG of any lattice can be obtained by selecting the rods with appropriate shape, size and orientation based on the above two conclusions. By comparing the width of these largest BG's we can see the effect of lattice's symmetry. Then the conditions underwhich the 2D PC's produce the largest BG can be clear through the above step-by-step analyse and comparison, which will be helpfiil to practice.