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In this paper, we revisit the lattice vibration of one-dimensional monatomiclinear chain under open and periodic boundary conditions, and give the exactconditions for the emergence of the local vibration mode when one of the atomsis replaced by an impurity. Our motivation is twofold. Firstly, in deriving thedispersion relation of the atoms, periodic boundary condition is overwhelminglyutilized while open boundary condition is seldomly referred. Therefore wemanage to obtain the dispersion relation under both boundary conditionssimultaneously by Molinari formula. Secondly, in the presence of impurity,local vibration mode can emerge as long as the mass of the impurity $m'$ issmaller than the mass of the perfect atom $m$ to certain degree, which can bemeasured by the mass ratio $\delta=\frac{m-m'}{m}$. At periodic boundarycondition, the critical mass ratio is $0$ or $\frac{1}{N}$, depending onwhether the length $N$ of the chain is even or odd. At open boundary condition,the critical mass ratio is $\frac{N}{2N-1}$ if the impurity locates at the endof the chain, while it is $\frac{N}{(2N_l+1)(2N_r+1)}$ with $N_l$ and $N_r$ bethe number of atoms at the left and right hand sides of the impurity if theimpurity locates at the middle.

Local vibrational mode of impurity in a monatomic linear chain under open and periodic boundary conditions