Phonon spectrum and the maintenance condition of crystalline beams

It has been shown that the maintenance condition for a crystalline beam requires that there be no resonance between the crystal’s phonon frequencies, and the frequency associated with a beam moving through a lattice of N_{sp} periods. This resonance can be avoided provided that the phonon frequencies all are below half the lattice frequency. Here, we study in detail the phonon modes of several crystalline beams. The analytic results obtained in the smooth approximation are compared with numerical evaluations employing Fourier transform of the molecular dynamics (MD) modes. The stability of various crystalline structures is examined through systematic MD simulations based on several different lattice designs. The maintenance condition, when combined with either the simple analytic theory or the numerical evaluation of phonon modes, exhibits excellent agreement with the MD calculations of crystal stability. A confirmed maintenance condition, derived from linear-resonance criteria, is that the lattice frequency must not equal the sum of any two phonon frequencies.