PHYSICAL REVIEW B VOLUME 42, NUMBER 4 1 AUGUST 1990 Vibrational relaxation and dephasing of two-phonon bound states in molecular crystals Franco Bogani Dipartimento di Fisica, UniUersita degli Studi di Firenze, Largo E. Fermi 4, 50125 Firenze, Italy Gianni Cardini and Vincenzo Schettino' Dipartimento di Chimica, Universita degli Studi di Firenze, Via G. Capponi 9, 50121 Firenze, Italy Pier Lorenzo Tasselli Dipartimento di Fisica, UniUersita degli Studi di Firenze, Largo E. Fermi 4, 50125 Firenze, Italy (Received 5 May 1988; revised manuscript received 26 February 1990) The vibrational relaxation and dephasing of two-phonon states in the region of combinations or overtones of intramolecular vibrations in molecular crystals are discussed. The model Hamiltonian used includes (1) a harmonic part comprehensive of two-body intermolecular interactions that are responsible for the phonon dispersion; (2) a single-site intramolecular anharmonic term that can give rise to the formation of resonances or bound states; (3) cubic and quartic terms coupling inter- nal modes to the lattice phonons. These latter terms give rise to depopulation and dephasing pro- cesses, respectively, of the bound states. Explicit expressions for the shift and linewidth of the bound states are obtained. It is shown that the relaxation processes are mixed processes involving both the intra- and the intermolecular anharmonicity. The relaxation follows multistage routes. Besides contributions due to scattering diagrams of single-phonon lines there are relaxation process- es that are peculiar to the bound states. Calculations of the shift and linewidths of bound states have been performed on a model system and the general trend of the shift and linewidth as a func- tion of temperature and of the strength of anharmonic interactions has been studied. It is shown that the contribution of depopulation and dephasing processes can be comparable. The model cal- culations are discussed in connection with available experimental results. I. INTRODUCTION Two-phonon bound states have been observed in a number of molecular crystals in the region of combina- tions and overtones of internal vibrations. The spectros- copy of these states is well accounted for by the presently available theory' and detailed calculations have been performed in crystals such as CO~, NzO, ' HCl, HBr, and CS&. In recent years the dynamics of bound states in some of these crystals has been probed by coherent time-resolved Raman experiments " or by high-resolution Raman spectroscopy. ' ' The data re- ported so far (relaxation times or linewidthsi show con- siderable peculiarities. For instance, the low-temperature relaxation times of different bound states in the Fermi resonance region in the CO& crystal may differ by as much as 4 orders of magnitude. " ' This and other features are not fully understood. The interpretation of the available experiments is of considerable interest to clarify the dynamics of two-phonon bound states and their role in matters such as the phonon instabilities. ' A theory of the dynamics of two-phonon bound states poses some novel problems. Indeed, the well-known ex- pressions for the depopulation' and dephasing process- es' of single-phonon states are not valid for bound states. These latter, in fact, are not easily described in terms of crystal normal coordinates. They are rather collective excitations of the crystal and most likely share the char- aeter of soliton waves. At least, this kind of interpreta- tion has been suggested in the case of two-magnon bound states. ' It is, therefore, necessary to formulate a theory of the relaxation of two-phonon bound states ex novo. This can be accomplished using two different approaches. According to the first, one can look for a definition of a collective normal coordinate appropriate for the bound state and then apply the usual expressions for the phonon linewidths with a proper redefinition of the anharmonic coefficients. This kind of approach, as outlined in Ref. 12, leads to simple and compact expressions for the linewidths but leaves some conceptual difficulties in- herent to the definition of the collective coordinate. In the second approach, which is the object of the present paper, the choice is to work with the harmonic normal coordinates of the crystal. The bound states are then considered as strongly perturbed two-phonon states that are further, but weakly, coupled by anharmonic in- teractions to the phonon bath. According to this view the interaction of the bound states with the phonon bath is mediated by the phonons composing the collective ex- citation. The strong perturbation producing the bound states arises from the intramolecular anharmonicity' and the coupling to the phonons and the vibrational re- laxation have an intermolecular origin. The two types of interaction must be considered simultaneously. Owing to the different strength of the two types of terms, the inter- molecular anharmonicity can be discussed within a per- 42 2307 1990 The American Physical Society