Virtual phonons

J0(w + wq) — J°(w), reflecting processes which begin by the emission of bosons. This is quite different from the case of an isolated ring, where T = 0, and only the resonances contribution remains, which vanishes at T = 0. Here the effect comes from a single, virtual phonon, and will disappear as T / e —> 0. 

As the quantum mechanics language is more general and as a rule more appropriate in discussions of the JT effect, the interaction between JT centers should be considered as a virtual phonon exchange between the electrons in orbitally degenerate states. 

It is worthy to give a direct definition of CITE. CITE is virtual phonon exchange at electron orbital degeneracy, leading to the correlation of local distortions and selfconsistent correlation of electrons. The virtual phonon exchange is the result of electron-phonon (vibronic) interaction and of phonon dispersion. 

As it was mentioned in the Introduction, the central part of the CITE is the virtual phonon exchange interaction. There are different methods to get this effective interaction starting with the Hamiltonian (1). Sugihara [11] was first (1959) to show that this interaction could be obtained considering the vibronic interaction as a perturbation (week JT effect). 

As a result of this replacement the linear vibronic interaction in Hamiltonian (1) is cancelled, and the effective virtual phonon interaction operator appears. However the use of this method that formally could be applied to any Hamiltoninan is related to some practical difficulties. These difficulties take place in case of the presence of some non- commuting electron operators in the initial Hamiltonian. They are especially serious when the dynamics of the JT system is under discussion. 

The most general approach to the formation of the effective operator of the intersite virtual phonon exchange interaction is based on the canonical shift transformation of the Hamiltonian... 

The correlation of the electrons at different JT centers caused by the virtual phonon exchange leads at some temperatures to the ordering of the local distortions and of the self consistently coupled to them electronic states (orbitals). This happens when the loss in the elastic energy and entropy at the ordering is compensated by the gain in the energy of the crystal electronic subsystem. At this case the electronic order parameter (an average of a pseudospin operator) of the phase transition becomes different from zero and because of that the spontaneous lattice (sublattice) strain is also not zero. 

The coupling between the distortion and electric dipole moment structures is very tight and unusual. Both these orderings are the result of the virtual phonon exchange. The absence of the local center of inversion is the reason why the JT distortion is responsible for the formation of the electric dipole moment (see Fig. 8). At this situation the ordering of the distortions is accompanied by the ordering of the dipole moments. 

As it was discussed above in these systems the virtual phonon exchange is the leading interaction responsible for the structural phase transition and for the cor-... 

Weak-to-Intermediate Jahn-Teller Coupling. Virtual Phonon Exchange... 

Fig. 2 shows that if V < cs, the survival probability first decreases and then approaches the constant value of about 0.85, almost independent on V. This behavior reveals the physical reason for the short-time non exponential decay the initial conditions Eq. (2) imply that, initially, the impurity atom is surrounded by no virtual phonons, while in the steady state, the impurity atom must be surrounded by a cloud of virtual phonons (cf. the polaronic effect for electrons in a crystal, [Isihara 1971]). Thus the non-exponential stage of the decay is associated with the formation of such a phonon cloud. 

In the LOE, the current can be divided into the three terms, I°,8T1, and Ilnel, which are ballistic, elastic correction and inelastic terms, respectively. The last two terms are effects of inelastic transport. The elastic correction term relates to virtual phonon... 

The question arises naturally whether a bandwidth or transfer cut-off model is nearer to the physical reality. If the effective electron-electron interaction is the result of a virtual phonon exchange, two different situations may appear. Depending on whether we study the polarization bubble type diagrams or Cooper pair type processes (see Fig. la and lb, respectively), the elimination of the phonons from the problem leads to different types of effective interactions. In the first case the effective interaction has an energy transfer cut-off, given by the phonon energy, hi the second case the effective interaction can be written in the form... 

We have already illustrated Direct Exchange Coupling. The dd-dq exchange is very similar, except that the intensity is lower because of the oscillator strengths. Virtual phonon excheuige involves phonon wave coupling at the two sites where a part of the energy is transmitted... 

Finally, it should be noted that the phonon part in eq. (14) also leads to an effective quadrupolar interaction between 4f ions if the phonon coordinates are eliminated in second-order perturbation theory. This virtual phonon exchange mechanism is important in insulators, see Orbach and Tachiki (1967). In R intermetallics such as TmZn and TmCd, however, the electronic origin of effective interactions is now established, Levy et al. (1979). 

The more interesting case is the Green s function because the resonant w-behaviour of. S(/, w), which is due to virtual phonon emission, now becomes important. One has... 

It was not until 1957 that the phenomena of superconductivity was explained by a theory developed by Bardeen, Cooper, and Schreiffer (BCS theory) in which it was postulated that two electrons became loosely bound by exchanging a virtual phonon (another way of saying a cooperative interaction with the lattice ions). By becoming paired with opposite spins, the so-called Cooper pairs act as bosons and are no longer controlled by the Pauli principle therefore, many can exist in the same quantum state. This pairing creates an energy gap about the Fermi level in which no states are available for the Cooper pair to be scattered into hence, they can move through the lattice unimpeded, very much like the superfluid state of liquid He which exhibits zero viscosity. 

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