Phonon Creation and Annihilation

Nuclear absorption of incident X-rays (from the synchrotron beam) occurs elastically, provided their energy, y, coincides precisely with the energy of the nuclear transition, Eq, of the Mossbauer isotope (elastic or zero-phonon peak at = E m Fig. 9.34). Nuclear absorption may also proceed inelasticaUy, by creation or annihilation of a phonon. This process causes inelastic sidebands in the energy spectrum around the central elastic peak (Fig. 9.34) and is termed nuclear inelastic scattering (NIS). 

Complementary to other methods that constimte a basis for the investigation of molecular dynamics (Raman scattering, infrared absorption, and neutron scattering), NIS is a site- and isotope-selective technique. It yields the partial density of vibrational states (PDOS). The word partial refers to the selection of molecular vibrations in which the Mossbauer isotope takes part. The first NIS measurements were performed in 1995 to constitute the method and to investigate the PDOS of 

9 Nuclear Resonance Scattering Using Synchrotron Radiation 

So far, NIS applications to the study of molecular dynamics have been performed mainly with Fe-containing systems. NIS with Sn has been used to investigate the dynamics of tin ions chelated by DNA [84]. The technique has also 

---

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form... 

The next phase for the theorists in connection with this work lies in predictions of helium atom scattering intensities associated with surface phonon creation and annihilation for each variety of vibrational motion. In trying to understand why certain vibrational modes in these similar materials appear so much more prominently in some salts than others, one is always led back to the guiding principle that the vibrational motion has to perturb the surface electronic structure so that the static atom-surface potential is modulated by the vibration. Although the polarizabilities of the ions may contribute far less to the overall binding energies of alkali halide crystals than the Coulombic forces do, they seem to play a critical role in the vibrational dynamics of these materials. 

Figure 45. TOF spectrum transformed to energy transfer distribution for CO/Rh(lll). The top panel shows the single-phonon creation and annihilation peaks for the fmstrated translational motion of CO at 5.75 meV along with a difiuse elastic peak at zero energy transfer. In the lower panel, the shift in energy to 5.44 meV due to the heavier mass of the C 0 isotope is clearly discernible (dashed vertical line). (Reproduced fiom Fig. 3 of Ref. 130, with permission.)...

The first two terms are diagonalized by introducing the phonon creation and annihilation operators (see (Cohen-Tannoudji et al. 1977)) ... 

We shall always be interested in longitudinal vibrations of quasi-one-dimensional systems, so the vector notation can mostly be omitted in subsequent discussions if one refers to the component of a vector parallel to the chain axis. The phonon amplitudes m, can be expressed in terms of phonon creation and annihilation operators as ... 

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). 

For paramagnetic spin systems, there are two major processes of relaxation (55). One relaxation mode involves spin-flipping accompanied by lattice phonon creation and/or annihilation (spin-lattice relaxation), and the other mode is due to the mutual flipping of neighboring spins such that equilibrium between the spins is maintained (spin-spin relaxation). For the former mode of relaxation, th decreases with increasing temperature, and the latter relaxation mode, while in certain cases temperature dependent, becomes more important (th decreases) as the concentration of spins increases. 

The terms creation and annihilation arise in applications where the system of interest is a gr oup of harmonic oscillators with a given distribution of frequencies. Photons in the radiation field and phonons in an elastic field (see Chapters 3 and 4 respectively) correspond to excitations of such oscillators. uj, is then said to create a phonon (or a photon) of frequency > and cia, destroys such a particle. ... 

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations ... 

To take the interaction between phonons and photons into consideration, it is necessary to add to the Hamiltonian (6.32), the Hamiltonian Ho(a) of the free field of transverse photons and the Hamiltonian Hint for the interaction of the field of transverse photons with phonons. The linear transformation from the operators a and C to the polariton creation and annihilation operators, i.e. to the operators t(k) and p(k), diagonalizes the quadratic part of the total Hamiltonian. The two-particle states of the crystal, corresponding to the excitation of two B phonons, usually have a small oscillator strength and the retardation for such states can be neglected. In view of the afore-said, the quadratic part of the total Hamiltonian with respect to the Bose operators can be written in the form of the sum H0(B) + where... 

Here, ancj are the creation and annihilation operators, respectively, for phonons in mode q = (q, r), where q denotes the vector of the phonon and r is the branch label. The energy of these phonon modes is given by u>q. Furthermore, the single-molecule Hamiltonian as well as the intermolecular transfer interaction are still considered to be operators in phonon space. 

Here the superscript 0 represents the trace with respect to the non-interacting density matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from the lead-molecule coupling (V. ) vanish because they are odd in creation and annihilation operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution... 

The creation and annihilation operators provide alternative forms for many quantum mechanical expressions, and they are used widely for phonons (vibrational quanta) as well as photons. Eor example, the Hamiltonian operator for an harmonic oscillator can be written... 

Assuming nearly-free electrons with plane-wave states i//(k) = G cxp(ikr) and energies E(k) = h k l2m, and introducing electron creation and annihilation operators and 4> which satisfy anticommutation relations (8.6), the Hamiltonian of the electron-phonon interac-... 

The single phonon approximation is obtained by expanding the exponential expression for the influence functional F into series and keeping two terms only. The fist term describes the elastic scattering while the second one corresponds to single phonon scattering and can be splitted into two first parts, corresponding to the creation and annihilation of a phonon ... 

If sample temperature T 0, the phototransition is accompanied by processes of creation and annihilation of phonons. In this realistic case, the cumu-lant function (p (t) is [8,12] ... 

The quadratic interaction causes processes of creation or annihilation of two phonons. These processes do not contribute to ZPL broadening. Besides of these two-phonon processes, the quadratic interaction causes the two-phonon Raman-like processes which arc characterized by the simultaneous creation and annihilation of one phonon. The probability of such two-quantum processes is proportional to n(n + 1) = [2sh(hv/2kT)] where n is the average phonon number. 

Here, e is the energy of TLS in the ground ( and excited (e) electron states of a chromophore. The second term in Eq. (122) describes an operator yklding tunneling transitions in TLS. Th e transitions result from the jrfionon modulation of a barrier. An exdtation of TLS with an energy 8 we shaU call a twmelon. It is obvious that tunnelons in contrast to phonons are the excitations of Fermi-type, i.e. the operators and C of the creation and annihilation of a tunnelon obey the Fermi-relation CC -i- C C = 1. These operators relate to the Pauli matrices as follows ... 

We have already seen in Sect. 3.4 that the homogeneous halfwidth of ZPL is due to two-phonon Raman-like processes one phonon is created and the other phonon is annihilated. It is obvious that the quadratic interaction AC C yields two-tunnelon processes of such a type in the electron-tunnelon system and the probability of a simultaneous creation and annihilation of a tiumdon is proportional to (1 — f)f = (2ch(E/2kT)). Therefore, the effect of the quadratic electron-tunnelon interaction on the homogeneous halfwidth of ZPL is given by the following formula [89, 90]... 

---