Force constants renormalized

In this equation, the interaction between active local distortions up to three next nearest neighbour octahedra along the chain was taken into account, while the force constants Km include the effects of interchain interactions. If the interchain contribution is confined by nearest neighbours then only the elastic constant corresponding to the interaction between nearest neighbour octahedra within the chain is renormalized ... 

Due to the dependence of / on active nuclear distortions, the Heisenberg exchange interaction, equation (38), summed up over all nearest neighbour centres in a chain, should be added from the start to the energy functional in equation (32). Equation (41) shows, however, that this addition merely leads to the renormalization of the effective force constant Kt in equation (32) by... 

We can conclude that orbital-dependent exchange interaction in hexagonal perovskites does not contribute essentially to the ordering of orbitals and Jahn-Teller distortions on sites but only results in a slight renormalization of the force constant Kt and, therefore, of A2 and, 42 parameters of the basic model in equation (37). 

Because of the term 5K, when we evaluate dco(q,a)/dV, as demanded by eqn (5.72) in our deduction of the thermal expansion coefficient, it will no longer vanish as it would if we were only to keep K. The phonon frequencies have acquired a volume dependence by virtue of the quasiharmonic approximation which amounts to a volume-dependent renormalization of the force constant matrix. 

Figure 4. Phonon dispersion for fee FesNi (renormalized with the eleetron-phonon coupling). Force constants have been obtained from ab initio calculations. Diamonds mark experimental results .

Here the anharmonic nature appears only in the equilibrium distances a changed by thermal expansion, see (5.52), and the altered vibrational frequencies w corresponding to the renormalized force constant f, see (5.55,57). We therefore approximate the atomic motion by a system of uncoupled normal vibrations but with different equilibrium positions and vibrational frequencies. This is, of course, an approximation since the anharmonic terms of the potential energy really lead to interactions of the normal vibrations. From (5.68,69), we obtain... 

The quasiharmonic approximation studied in Sects.5.2,3 gives reasonable results for the thermodynamic properties of crystals in which the anhar-monicity is weak and the force constants are renormalized by thermal expansion only. In crystals with very strong enharmonic interactions, however, this approximation breaks down. Examples are the vibrations in rare-gas solids, in particular solid helium, soft modes in ferro-electric phase transitions and melting processes. For these cases a method has been developed, the self-consistent harmonic approximation (SCHA), which allows a qualitative description of the effects of strong anharmonicity. 

The SCHA is formally again harmonic, that is, the true lattice system is to be approximated by some other effective harmonic lattice whose force constants and lattice parameter are to be optimally adjusted. The renormalized force constants are obtained from a self-consistency condition. Self-consistency is achieved by replacing the normal harmonic force constants by effective force constants which are thermal averages with respect to the effective harmonic Hamiltonian. 

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