Theory of one-phonon-assisted energy transfer between rare-earth ions in crystals

A theoretical framework and the method of application are presented to describe nonresonant energy transfer processes between rare-earth ions of the f(N) electronic configuration at centrosymmetric sites of crystals, in which the energy mismatch is made up by the emission or absorption of one nondiagonal phonon. The established theory of Holstein, Lyo, and Orbach (HLO) is applicable to, for example, an (EQ<->EQ,V) energy transfer process which is composed of an allowed pure electric quadrupole-electric quadrupole (EQ-EQ) nonradiative transition and a vibrational transition in which one diagonal phonon emission or absorption occurs from a definite electronic state of the donor or acceptor. By contrast, the theory applies to an (EQ<->EDV) process where an EQ transition occurs at one site and one nondiagonal phonon in an electric dipole vibronic (EDV) transition is involved at the other site. We find that for the (EQ<->EDV) process the coherent cancellations occurring in the conventional diagonal HLO theory of one-phonon-assisted processes, which lead to the dominance of two-phonon energy transfer processes, do not occur in the nondiagonal one-phonon-assisted case. First, the Debye phonon model used by HLO theory has been employed, in which the crystal is assumed to consist of an isotropic, continuous medium. This model is only applicable to acoustic phonons with small wave vector q. The energy transfer rate obtained for the nondiagonal one-phonon-assisted process increases quadratically with increasing intersite energy mismatch, when it is small compared with the average thermal energy k(B)T at temperature T. Second, to take into account the crystal structure on the atomic scale which usually has anisotropic properties and to consider optical phonons, etc., the phonon involved in the diagonal and nondiagonal energy transfer process has been described by a running lattice wave, as an irreducible representation basis component of the space group and of the solution of lattice dynamical equations. The transition element and transition rate thus obtained show that the significant difference between the coherence effects of the diagonal and nondiagonal cases still occurs. Furthermore, some new points arise, especially the contributions from flat parts of the dispersion curves of optical phonon branches, to the studied processes. Therefore, contrary to the conclusion of HLO theory, optical phonons with q=0 can make important contributions to one-phonon-assisted energy transfer processes for the nondiagonal case. In addition, the approximations inherent in the widely used spectral overlap model are pinpointed, and the selection rules and coherence effect of lattice waves are briefly discussed. Noticeably, although we focus upon centrosymmetric systems, however, the nondiagonal processes and the related results obtained in this paper are also applicable to noncentrosymmetric systems.