This equation of motion describes phonons with u n ∝ ∝ exp ⁡ ⁡ [ i ( ω ω ph ( κ κ ) t − − κ κ n ) ] with the phonon dispersion relation ω ω ph 2 ( κ κ ) = ω ω min 2 + 2 g ( 1 − − cos ⁡ ⁡ κ κ ) with the dimensionless wavenumber | κ κ | ≤ ≤ π π . This shows that the frequency spectrum of the chain has a band gap ω ω min ≡ ≡ ω ω ph ( 0 ) = 1 with cut-off frequency ω ω max ≡ ≡ ω ω ph ( π π ) = ω ω min 2 + 4 g .