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Extra info for Lattice Dynamical Foundations of Continuum Theories
Com by UNIVERSITY OF BIRMINGHAM on 06/09/16. For personal use only. 11) has not been used yet. e. 18) in an even better form. Since V_p=V p according to the third invariance relation in Eq. 20) This latter expression ;s symmetric with respect to the particles before and after the central one at R,. 20) has an easy interpretation as a system of coupled oscillators where the forces between a particle and its p-th neighbors are defined through the spring constants vp and detennined by the relative displacements uR,+p - uR, and uR,_p - uR, between the R,-th particle and its two p-th order neighbors.
39 particles move in opposite directions with their center of gravity remaining fixed. On the first branch therefore, in the long wave limit, the identity of the particles is not noticed and the dynamics is governed by a wave equation just like in the monatomic case and the frequencies are in the audible range. For this reason, the first branch is called "Acoustic Branch" and the mode of wave propagation as the "Acoustic Mode". 12 illustrates the two types of modes. The name of the upper branch as, "Optical Branch" ;s due to the frequencies involved being typically in the infrared, close to the optical frequencies.
Com by UNIVERSITY OF BIRMINGHAM on 06/09/16. For personal use only. 22) Noting that 2a 3 is the volume of the unit cell, PI and P2 are the partial mass densities and P is the average (or total) mass density. 21) can be rearranged to yield a fonm that has a clearer physical meaning. 25) For the diatomic lattice at the long wave limit (see Eq. 58)), it was found that u2 -+- u1 for the acous tic mode and m1u1 + m2u2 -+- 0 on the optical mode. These two limits correspond respectively to v-+-O and u-+-O.