Phonon states, -functions

The -functions of phonon states after the electronic transition induced by differently ses. (a) The 2-function at the minimal AA"+, that is, maximal squeezing (the chirp parameter see marker a in Fig. 1). (b) The AX+ is in its next maximum, (c) An intermediate state maximum and minimum. As we consider lower and lower chirp parameters, the maxima and ome less prominent (4), approaching the number state with equal distribution along a circle. 

We consider now the complete basis set of vibronic functions I f m, n) obtained as the direct product of the electronic functions l/+) and 1/1) by the phonons wave functions Im, n), where the integer numbers m and n (positive or zero) label the occupation numbers of the boson operators b b+ and b b, respectively. The Renner-Teller Hamiltonian (14) when applied to any vibronic function I f m,n) couples it with three states (at most) in fact it holds that ... 

Here, V(x,y,z) is the adiabatic potential and E is the energy of the ground phonon state. The vibrational co-ordinates x, y, z are treated as functions of s, the arc length of the path. Because the system has a zero-point energy E1 above the potential well, the boundary conditions are not unique. 

Here we apply the method described above (see also Ref. [13]) for the investigation of the temperature dependence of non-radiative transitions between electronic levels in a center. The method [13] is based on the calculation of the large t asymptotic of the phonon correlation function D,(f, r) = (l,0lx (f + T)xI(t)l 1,0) with t averaged over the vibrational period here 11,0) = 1110) is the product of the initial electronic state and the zero-point vibrational state. 

To apply described above non-perturbative method to the non-radiative transitions caused by the quadratic non-diagonal vibronic interaction qfli, one needs to find the phonon correlation functions Du(t, r) in the initial (non-stationary) state 11,0) for l > 0. To this end, we use the equation of motion xu + (ojxu + Veuqt j = 0 il, l > 0 / V l1) the integral form of it for t > 0 reads... 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

A suitable choice of the variational wave functions for various electron-phonon two-level systems is a long-standing problem in solid state physics as well as in quantum optics. For two-level reflection symmetric systems with intralevel electron-phonon interaction the approach with a variational two-center squeezed coherent phonon wave function was found to yield the lowest ground state energy. The two-center wave function was constructed as a linear combination of the phonon wave functions related to both levels introducing new VP. 

In these expressions, p — kT) E — fi k /2Mis the recoil energy of a free nucleus k the wave vector of the y-ray quantum and M the mass of the nucleus. The function g E) is the normalized density of phonon states ... 

In (34), the effective Hamiltonian H2) is still a matrix in the orbital space of electron wave functions. Similar to (17), it describes intersite orbital exchange coupling. In (34) and (35), the factor J(i - ) is the parameter of orbital exchange coupling, same as in (3.7). This time it is supported by the assumption of weak JT coupling. The matrix elements (0 Qy (i) n) are determined with zero-coupling oscillator wave functions. They take a nonzero value for = 1 only. In other words, in the effective Hamiltonian H2), the virtual excited states are one-phonon states. Therefore, the effective Hamiltonian (34) describes phonon-mediated orbital exchange. 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

The vibrational factors f0, gi and g j of (4.5) can be expressed in terms of creation operators at, of SSANMV operating on the ground state function of noninteracting phonons namely... 

The specific heat capacity of a substance comprises an electronic contribution Ce. and a contribution Cph of the phonons. The latter is dominant in carbon nanotubes, regardless of their structure. Cph is obtained from integration over the density function of phonon states and subsequent multiplication by a factor that considers the energy and the population of individual phonon levels. 

After considerable work, it was determined that the hosts which work best in this application are those in which vlbronic coupling is minimized. This was established by determining and comparing the phonon dispersion branches of the various compounds. It was found that those hosts which have low energy optical branches in the phonon spectrum function best for up-conversion phosphor applications. Note that this is akin to minimizing ground state perturbation at the activator site proper choice of host. The best hosts were found to be ... 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

Here, F and F are phonon spectral functions which describe phonons in the ground (g) and excited (e) electronic states of a chromofdiore. 

The function f(v) describes the weighted density of phonon states and we have already discussed the way this function can be found from the homogeneous PSB (Sect. 3.3). However the frmction f( v) can be found from PE experiments as well. 

Here again we find that the promoting mode contribute to the transition via (s V (l) ) and (( ) Vx (() )and that the contribution to the vibrational overlap is given by (h(n + l)/2Q) ) / (x, X g, Xsm-Xa ) with Wx = 0. The comparison with Equation H3 that applies to the case of resonant energy transfer and which and Ef are given by Equation H5 should be noted. The difference that arises from the additional vibronic interaction is manifested not only in the electronic part of the matrix element but also in a diminished initial energy gap h(Qs—cOx) in the density of states function (H4). If we make these modifications in Equation H1, we will immediately recover the appropriate expression for the phonon-assisted energy transfer ... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

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