Phonon representations ( lattice / spectrum )

The gap between the nonlocalized latticephonon description and the localized Einstein oscillator treatment is filled by transforming the phonon Hamiltonian back to the particle variables. The particle-coordinate, normalized, wave function for the phonon vacuum state is exhibited. The lattice phonon is a spatially nonlocalized concept. There are circumstances, however, that require its combination with other, localized, phenomena. Then, one might wish to return to a description that uses the more localized interactions among the lattice constituents. Such interactions may not be very well known, however, and, in contrast with the phonon spectrum, are not subject to direct measurement. It would be of some utility, therefore, to transform back from the phonon description in order to realize the localized treatment of the lattice particles. In the interests of minimizing notational problems, the distinction between longitudinal and transverse polarizations will be set aside, so that each phonon mode, labeled 4, and characterized by angular frequency wct, and propagation vector kt, will be three-dimensionally degenerate. This is conveyed by the following Hamiltonian, referring to the linear displacement regime, which is expressed in terms of the phonon annihilation operator yO, and the adjoint creation operator yt,, as X hwcOyty4,. [1] Given N equilibrium positions, r,, one introduces the dynamical position vectors ra = r~a + I p,(el* ro"y4 + e "y along with the momenta for particles of mass M, Pa = I MWOp,4(e1Y.rOay4, 'Iei [2] [3] 1 [rak, Pbl] = 6kl6ab' fth [6] as a consequence of the phonon property 11 en (rOrob) = (ik-(rO.-rOb) = ,ab [7] The notation introduced here for a spectral average, (), will occur again. The inverses of the relations 2 and 3 are given by [8] Ye =4 AX[(r-ro)a +M-pe-w4I a _Mop along with its adjoint, ye = a Z[(r-ro)a pale, a MWO where o-,o = -12_p(M\= [9]