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The concept of periodic boundary conditions (PBCs) is immensely significantin treating an ideal lattice of infinite extent as a finite lattice. Anexplicit usage of PBCs is often found missing in undergraduate texts onanalytical treatment of lattice dynamics. The aim of the present work is tocover this gap by illustrating the application of Born-von Karman PBCs inlattice dynamical calculations using a computational approach. The equations ofmotion are set up for a linear diatomic lattice with a basis, using the nearestneighbour approximation. The solution is obtained by implementing fourth orderRunge-Kutta algorithm in python. Fast Fourier Transform (FFT) technique is thenused to obtain the phonon frequency spectrum corresponding to the computedsolutions. Similar computations are extended to obtain the phonon spectrum formonatomic square and honeycomb lattices under the second nearest neighbourapproximation. The computed results are validated against the analytical onesin each case. The target group of the present work are the students andeducators at the undergraduate level.

An exploration of the phonon frequency spectrum and Born-von Karman  periodic boundary conditions in 1D and 2D Lattice systems using a  computational approach