Phonon expansion

The normalized probability of nuclear inelastic absorption W(E) can be decomposed in terms of a multi-phonon expansion (222,223) ... 

Since the corrugation of adsorbed layer may be neglected, the scattering probar bility is given by (2.1.1) with the dynamic structural factor (2.1.2). It is convenient to use the phonon expansion of the dynamic structural factor, because one can identify individuail processes of phonons creation and annihilation in experiments (Gibson and Sibener 1988 Moses et al. 1992). 

A further distinction arises because for many solids under a wide variety of state conditions the constituent particles execute small amplitude vibrations about well defined equilibrium positions (noteworthy exceptions are plastic crystals). If this situation pertains it is possible to express rj,(t) = Rj,+uj,(t) the instantaneous position of particle i in terms of its mean position and its time dependent displacement uj (t). If uj is in some sense small with respect to Rj one can develop F(Q,t) as a power series in g u. This is the so-called phonon expansion (10)... 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

Since it is the first derivative with respect to r that we are interested in, we only need the 1=1 term from this expansion. The angular part contributes only to the overall constant, but it is the spherical function j (kr) that sets the cutoff value of the wavevector, above which the phonons do not produce significant linear uniform stress on the domain. In Fig. 24, we plot the derivative dji x)/dx (or, rather, we plot the square of it, which enters into all the final expressions). 

To estimate the above expression, consider a particular case of the symmetric two-well potential in which the two-phonon interaction //- can approximately be derived from expansion (A2.32) ... 

This is equivalent to the initial Hamiltonian in equation (1) when the extrema are considered, but depends on active one-centre nuclear displacements only. It does not mean, however, that only these local distortions are different from zero in extrema points. Some non-active intra-site distortions, will be non-zero as well. Indeed, using for them the same Van Vleck expansion, with the coefficients a Jkj), and the equilibrim values of phonon coordinates (6), we obtain... 

It is generally accepted in the theory of the cooperative Jahn-Teller effect to include the interaction with uniform strains in the way proposed by Kanamori [14], i.e., as additional terms of vibronic interaction at each Jahn-Teller ion. On the other hand, within one-centre-coordinate approach used here the vibronic interaction is fully described by means of one-centre active nuclear displacements qn. Therefore the interaction with uniform strains can be included implicitfy as additional terms in the Van Vleck expansion (3). Since phonons and uniform strains are independent degrees of freedom this new expansion is written as follows ... 

Here, M is the reduced mass for the optic oscillation in the cell, or the mass of crystal cell for the acoustic phonons mk the mode frequency and N the number of cells in the macro-crystal. It is easy to see that the sum in Eq. (11) converges for all types of the displacements 8Rt due to the rapid decrease in 8Rt with the increasing distance from the center Rt. Therefore, the lattice particles located near the center only give the real contribution to the sum. The number N is very large, so the displacement 8qg for each mode is very small. Then, one may take into account the first few terms only in the expansion of the final phonon wave function on displacement 8q ... 

The possibility of negative thermal-expansion coefficient (TEC) values along a direction of strong coupling in layered or chain structures (the so-called membrane effect") was suggested for the first time by Lifshitz [4] for strongly anisotropic compounds. In the phonon spectra of such compounds... 

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