Wave Propagation in a One-dimensional Crystal Lattice

Wave Propagation in a One-dimensional Crystal Lattice.—Let us consider N atoms, each of mass m, equally spaced along a line, with distance d between neighbors. Let the x axis be along the line of atoms. We may conveniently take the positions of the atoms to be at x = d, 2d, 3d,. . . Nd, with y = 0, z = 0 for all atoms. These are the equilibrium positions of the atoms. To study vibrations, we must assume that each atom is displaced from its position of equilibrium. Consider the jth atom, which normally has coordinates x = jdy y = z 0, and assume that it is displaced to the position x = jd + /, y = Vi, z = f so that Vi, f / are the three components of the displacement of the atom. If the neighboring atoms, the (j — l)st and the j -f- l)st, are undisplaced, we assume that the force acting on the jth atom has the components 

Using the expressions (1.2) for the force, we can set up the equations of motion for the particles, using Newton s law that the force equals the mass times the acceleration. Thus we have 

Substituting Eqs. (1.5) and (1.6) in Eq. (1.3), we find that the factor A sin 2iwt sin is common to each term. Canceling this common factor, we have 

If the frequency v and the wave length X are related bv Ea. (1.7V the values of , in Eq. (1.5) will satisfy the equations (1.3). If the velocity were constant, we should have v = v/. the frequency being inversely proportional to the wave length. From Eq. (1.7) we can see that this is the case for long waves, or low frequencies, where we can approximate the sine by the angle. In that limit we have 

NextT wc must impose boundary conditions on our chainftf atoms as wc did with thp eontinnons solid. We are assuming N atoms, with undisplaced positions at x = d, 2d,. . . Nd. We shall assume that the chain is held at the ends, and to be precise we assume hypothetical atoms at x = 0, x = (N + l)d, which are held fast. As with the continuous solid, the precise nature of the boundary conditions is without effect on the higher harmonics. In Eq. (1.3), then, we assume that the equations can be extended to include terms f0 and fcv+i, but with the subsidiary conditions 

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