{'fraction_non_alphanumeric': 0.035590277777777776, 'fraction_numerical': 0.025173611111111112, 'mean_word_length': 4.4904761904761905, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Calculation of the Band Structure in 3D Phononic Crystal with Hexagonal Lattice

Abstract In this article, the propagation of acoustic waves in the phononic crystals (PCs) of three dimensions with the hexagonal (HEX) lattice is studied theoretically. The PCs are constituted of nickel (Ni) spheres embedded in epoxy. The calculations of the band structure and the density of states are performed using the plane wave expansion (PWE) method in the irreducible part of the Brillouin zone (BZ). In this study, we analyse the dependence of the band structures inside (the complete band gap width) on c/a and filling fraction in the irreducible part of the first BZ. Also, we have analysed the band structure of the ALHA and MLHKM planes. The results show that the maximum width of absolute elastic band gap (AEBG) (0.045) in the irreducible part of the BZ of HEX lattice is formed for c/a=6 and filling fraction equal to 0.01. In addition, the maximum of the first and second AEBG widths are 0.0884 and 0.0474, respectively, in the MLHKM plane, and the maximum of the first and second AEBG widths are 0.0851 and 0.0431, respectively, in the ALHA plane.