Jahn-Teller phonons

The cooperative Jahn-Teller type interaction is obtained by the orbital-lattice coupled model [12,13]. Let us consider the Hamiltonian with the Jahn-Teller coupling g, the kinetic energy and the lattice potential for the Jahn-Teller phonon mode, the orbital-strain interaction, and the elastic-strain energy. The Jahn-Teller distortion mode Qi around a metal site i is represented by the Jahn-Teller phonon coordinates k- By introduce the canonical transformation for k, the orbital and lattice degrees of freedom are separated. The final form of the interaction between the inter-site orbitals given by... 

An interesting aspect of many structural phase transitions is the coupling of the primary order parameter to a secondary order parameter. In transitions of molecular crystals, the order parameter is coupled with reorientational or libration modes. In Jahn-Teller as well as ferroelastic transitions, an optical phonon or an electronic excitation is coupled with strain (acoustic phonon). In antiferrodistortive transitions, a zone-boundary phonon (primary order parameter) can induce spontaneous polarization (secondary order parameter). Magnetic resonance and vibrational spectroscopic methods provide valuable information on static as well as dynamic processes occurring during a transition (Owens et ai, 1979 Iqbal Owens, 1984 Rao, 1993). Complementary information is provided by diffraction methods. 

Figure 4.19 Model structures of (a) incommensurate and (b) commensurate phases of K2Pb[Cu(N02)6]. The displacement pattern of Jahn-Teller active phonons is shown by arrows. In (a) the phonon mode has wave-vector k — (0.425, 0.425, 0) and in (b), wave-vector of the phonon mode is k = (, j, ). (After Yamada, 1977.)...

Jahn-Teller effects,13 which are the chemist s analog of electron-phonon coupling mechanisms in molecules, are consequences of the application of perturbation theory. With respect to some distortion coordinate si the total electronic energy for a system may be expressed to second order as... 

I. B. Bersuker, in Electron-Phonon Dynamics and Jahn-Teller Effects (eds G. Bevilacqua, L. Martinelli and N. Terzi), World Scientific, Singapore, 1999, p. 63. 

Jahn-Teller Effect in the Excited State Anomalous Temperature Dependence of the Zero-Phonon Line... 

If there are several AP minima of close energy, then at low temperatures one should take into account two-phonon-assisted transitions between these minima. In Ref. [15] (see also Ref. [14]) it was found that the rate of these transitions depends on temperature as 7 3. However, as it was already mentioned above, in Ref. [9] it was found that the contribution of the two-phonon-assisted transitions between different Jahn-Teller minima of the AP to the ZPL width at low temperatures is described by the T5 law. Note that an increase of the Jahn-Teller interaction leads to a decrease of the rate of these transitions. Therefore, in the strong Jahn-Teller interaction limit this broadening mechanism becomes unimportant. 

In the present paper we assume that the molecule has the icosahedral symmetry. If one wants to consider a distortion of C 0+ or Cb0. the energy levels and their eigenvectors obtained here can be used as a starting point for the description of the Jahn-Teller effect in these systems. Indeed, the electron-phonon (or vibronic) coupling occurs if [.Tei]2 contains Fvib [19]. (Here Fd is the symmetry of an electronic molecular term, while J b is the symmetry of a vibrational normal mode.) Calculations using the terms in scheme of Ref. [4] have been performed in Ref. [20]. 

Already in the seminal paper of Bednorz and Muller [1], the guide to look for systems with a high superconductive transition temperature (Tc), has been the presence of strong electron-phonon interactions. Such interaction has been known to exist in a wide class of perovskite type oxides. The authors mention [1] the vibronic Jahn-Teller polaron effect [2] in this context. They also emphasize the fact that the Cu2+-ion is a well-known Jahn-Teller system and this circumstance preserves significance in the physics of cuprate superconductors [3-7]. As a microscopic cause for ferroelectric ordering the interband vibronic hybridisation has been supposed [8-11] enlargening the view on perovskites as Jahn-Teller systems. 

Fig. 3. Schematic view of a possible momentary orbital configuration in the dynamic Jahn-Teller state. The x2 — y2 and the 3z2 — r2 orbitals are randomly occupied and a phonon mode exists, indicated by the arrows at the left 3z2 r2 orbital, which causes an oscillation between the two orbital types.

Ground State of Quantum Jahn-Teller Model Selftrapping vs. Correlated Phonon-assisted... 

An effective Hamiltonian for a static cooperative Jahn-Teller effect acting in the space of intra-site active vibronic modes is derived on a microscopic basis, including the interaction with phonon and uniform strains. The developed approach allows for simple treatment of cooperative Jahn-Teller distortions and orbital ordering in crystals, especially with strong vibronic interaction on sites. It also allows to describe quantitatively the induced distortions of non-Jahn-Teller type. 

In this chapter an effective Hamiltonian for a static cooperative Jahn-Teller effect is proposed. This Hamiltonian acts in the space of local active distortions only and possesses extrema points of the potential energy equivalent to those of the full microscopic Hamiltonian, defined in the space of all phonon and uniform strain coordinates. First we present the derivation of this effective Hamiltonian for a general case and then apply the theory to the investigation of the structure of Jahn-Teller hexagonal perovskites. 

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