Phonon displacements theory

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

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 ... 

According to the theory of the Urbach absorption edge in crystals, the slope E is proportional to the thermal displacement of atoms r(7). The frozen phonon model assumes that an amorphous semiconductor has an additional temperature independent term, r , representing the displacements which originate from the static disorder, so that... 

The factor / indicates that F(r) is 90° out of phase with the local displacements. Such an electron potential, arising from phonons in crystals, is called an electron-phonon interaction. We saw that electrons may be freed in the crystal when impurities are present and may also be freed by thermal excitation even in the pure crystal. Any such free electrons contribute to the electrical conductivity, but that conductivity will in turn be limited by the scattering of the electrons by lattice vibrations or by defects. We will not go into the theories of such transport properties as electrical conductivity these arc discussed in most solid state physics texts- but will examine the origin of certain aspects of solids such as the electron-phonon interaction, which enter those theories. 

One of the most valuable features of Raman spectroscopy is the well-known effect of local strain on the optical phonons (at q k. 0). The most basic approach to the theory of lattice vibrations assumes that interatomic forces in the crystal are linear functions of the interatomic displacement so that they obey a form of Hooke s Law. Under this harmonic approximation, the frequency m for mode j is given by ... 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

The spectrum of polaritons can be found by means of Maxwell s macroscopic equations (see Ch. 4), provided that the dielectric tensor of the medium (44) is assumed to be known. Without going into details, we emphasize here that always a gap appears in the polariton spectrum (here we ignore spatial dispersion) in the region of the fundamental dipole-active vibration (C-phonon, exciton, etc.). At present, there is a sufficiently detailed theory for RSL by phonon-polaritons, taking many phonon bands into consideration. With this theory the RSL cross-section can be calculated for various scattering angles provided that the dielectric tensor of the crystal is known, as well as the dependence of the polarizability of the crystal on the displacement of the lattice sites and the electric field generated by this displacement (45). 

Let us assume that the light frequency uj is in the vicinity of a well-isolated dipole-allowed exciton resonance. In this case the operator of the exciton-phonon interaction, linear with respect to the operator of displacements of molecules from their equilibrium positions, has the form (3.155). It can be shown that in the first order of perturbation theory the real part of 7 uj, k) is given by the following formula (we assume here that the crystal temperature T = 0) ... 

To proceed further, three major approximations to the theory are made [44] First, that the transition operator can be written as a pairwise summation of elements where the index I denotes surface cells and k counts units of the basis within each cell second, that the element is independent of the vibrational displacement and, third, that the vibrations can all be treated within the harmonic approximation. These assumptions yield a form for w(kf, k ) which is equivalent to the use of the Bom approximation with a pairwise potential between the probe and the atoms of the surface, as above. However, implicit in these three approximations, and therefore also contained within the Bom approximation, is the physical constraint that the lattice vibrations do not distort the cell, which is probably tme only for long-wavelength and low-energy phonons. 

The atom-multiphonon component of the inelastic scattering can be obtained from the theory of Manson [44] by subtracting off the zero- and first-order terms from the power series expansion of the exponential displacement correlation function. With the assumption that the major contribution to the multiphonon scattering is due to low-energy, long-wavelength phonons, he is able to arrive at an expression with a very simple form for the transition rate. 

Figure 8. Expected shape of the electron-phonon part of p(T) according to Boltzmann theory. The curve of Laj g3Sro,X5 4 is displaced upward by an arbitrary amount See (M) for details.

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

In the dielectric screening method the electron density response due to the motion of the ions around their equilibrium positions is calculated in first order perturbation theory. The potential energy of the crystal for an arbitrary configuration of the ions is expanded to second order in the ionic displacements from equilibrium. The expansion coefficients of the second order term form a matrix. The Fourier transform of this matrix is the dynamical matrix whose eigenvalues yield the phonon frequencies. The dynamical matrix has an ionic and electronic part. The electronic part can be expressed in terms of the electron density response matrix and of the ionic potential. This method has the advantage over the total energy difference m ethod that the phonon frequencies for any arbitrary wave vector can be calculated without additional difficulties. Furthermore in this method the acoustic sum rule is automatically satisfied as a consequence of the way the dynamical matrix is derived. However the dielectric screening method is limited to harmonic phonons. 

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