A free surface may be created in a crystal by starting with an infinite crystal and removing all interactions between particles on one side of the boundary and those on the other side. In general, the particles near the surface must move to new equilibrium positions and the force constants characterizing small oscillations of these particles will be different from those of the infinite crystal. Explicit calculations are presented of the new equilibrium positions in a semi-infinite, one-dimensional monatomic lattice with nearest and next-nearest neighbor interactions including an anharmonic interaction between nearest neighbors at the free end. The appearance of surface modes of vibration due to the changed force constants at the boundary is investigated in detail for a semi-infinite, one-dimensional monatomic lattice with nearest and next-nearest neighbor harmonic interactions. conditions of stability for such a lattice are established. In addition, the influence of a free surface on the localized mode due to a force-constant imperfection at some distance from the boundary is investigated. © 1964.