Brillouin zone lattice dynamics

Rush (1967) measured the inelastic incoherent neutron scattering from solid benzene and determined the phonon frequency distribution. Nakamura and Miyazawa (1969) carried out a complete Brillouin zone lattice dynamics calculation using the potential constants of Harada and Shimanouchi (1967), assuming the benzene molecule to be a nonvibrating rigid body. These authors first obtained agreement to better than 6% with Harada and Shimanouchi for the q = 0 modes. They then calculated the frequency distribution and were able to fit the experimental specific heat to better than 2 % by taking into account the internal modes as well and... 

Recently, we hav measured the surface phonon dispersion of Cu(l 10) along the rx, rF, and F5 azimuth of the surface Brillouin zone (Fig. 13) and analyzed the data with a lattice dynamical slab calculation. As an example we will discuss here the results along the TX-direction, i.e. the direction along the close-packed Cu atom rows. 

In contrast, neutron spectroscopy is a more powerful probe, its results are directly proportional to the phonon density of states (DOS) (see Fig. 2) which can be vigorously calculated by lattice dynamics (LD) and molecular dynamics (MD). Applying these simulation techniques provide an excellent opportunity for constructing and testing potential functions. Because optical selection rules are not involved, INS measures all modes (IR/Raman measure the modes at the Brillouin Zone (BZ) q = 0, see Fig. 2) and is particularly suitable for studying disordered systems (or liquids). It hence provides direct information on the hydrogen bond interactions in water and ice. 

Ii3/2 multiplets. Frequencies and polarization vectors of phonons in the LiYp4 crystal were obtained at 8000 points in the irreducible part of the Brillouin zone using the rigid ion model of lattice dynamics derived on the basis of neutron scattering data. Matrix elements of electronic operators Vds) were calculated with the wave functions obtained from the crystal-field calculation. The inverse lifetimes of the crystal-field sublevels determine the widths of corresponding absorption lines. 

Another interesting result from this simple picture of lattice dynamics for an alkali-halide-type crystal is that when the atoms have the same mass. Mg = Mh = M, and there are no differentiating forces between the atoms (such as different next-nearest-neighbor interactions), then G and H are identical within the context of this dynamical model. The crystal is then indistinguishable from a monatomic crystal that has a unit cell length a = all and corresponding Brillouin zone boundary at = -kIu = I-kIu. (See Fig. 6.) The consequence for the lattice dynamics is that the optical and acoustic mode frequencies become the same at = +7r/a, to = If lM), and there is no aeoustic-optical band gap just as in Fig. 4 (lower panel). From a... 

Lattice Frequencies [19] (in cm ) at the F Point of the Two-Dimensional Brillouin Zone for Orientationally Ordered N2 Monolayer on Graphite flora Time-Dependent Hartree Lattice Dynamics Including Up to Quartic Displacement Terms, the Spherical Expanded Ab Initio N2-N2 Potential [23], and an Empirical N2-Graphite Potential [326] ... 

It is the measurement, in a representative number of directions in the Brillouin zone, of the coy(q) and its variation with temperature and pressure which provides the most direct information on the interatomic forces. The forces are inferred by adjusting the frequencies calculated from a crystal model with assumed interatomic interactions to achieve agreement with measured vibrational frequencies obviously, the more frequencies measured, the better the test of the assumed force model. These aspects of lattice-dynamic studies are discussed in some detail in the Appendix. It should be emphasized, however, that almost without exception crystal force models have been obtained in a... 

Further comments should be made before discussing lattice dynamical models. Because the medium in which the phonon waves travel is discrete, there is a minimum wavelength and thus a maximum value of q in any given direction. The planes which bound that region in momentum space define the limits of q and the Brillouin zone. If the eigenfrequencies are summed over the Brillouin zone according to... 

In the case of a periodic solid the vibrational modes become phonons and the dynamical matrix becomes a function of a reciprocal lattice vector k chosen from the Brillouin zone. This means that in constructing D(k) all interactions are multiplied by the phase factor exp(ikrjj), where rp is the interatomic vector. A more detailed discussion of the theory of phonons can be found elsewhere (Dove 1993 Chapter 13 by Kubicki). 

In all lattice dynamics treatments for librational degrees of freedom discussed in Section IIC, interactions between these coordinates and translations must be considered. As already pointed out, interaction matrix elements vanish only at the zone center and some very special points on the zone boundary and this only for centro-symmetric solids. The first calculation of the dispersion curves for a molecular solid throughout the Brillouin zone was carried out by Cochran and Pawley (1964) for hexamethylenetetramine (hexamine). Once the librational displacement coordinates have been defined and a potential function chosen, the interaction force constants O, can be calculated. The subscripts refer to the displacement coordinate components, where we use i to designate a translational displacement component and a to designate a librational displacement component m,. The corresponding dynamical matrix elements are Mi, analogous to the M,-, defined in (2,9). In general, the matrix elements are complex, and the matrix is hermitian. 

Schnepp and Ron (1969) carried out a complete lattice dynamical calculation for a-Na throughout the Brillouin zone, using the potential model of Kuan, Warshel, and Schnepp (1969). This model contains three parameters which were calibrated for the optical modes [librational assignments of Brith, Ron, and Schnepp (1969)] and the equilibrium properties of the solid. The lattice dynamics was formulated in terms of Eulerian angle librational displacements. Dispersion curves for the symmetry directions of the Brillouin zone and density of states functions were reported with and without translation-libration interactions. The results clearly demonstrated the importance of these interactions. Schnepp... 

Suzuki and Schnepp (1971) have carried out a complete lattice dynamical treatment throughout the Brillouin zone. Dispersion curves. 

APPENDIX. GROUP THEORY AND SYMMETRY IN THE BRILLOUIN ZONE AS APPLIED TO LATTICE DYNAMICS OF MOLECULAR SOLIDS... 

Several good sources treat space groups and group theory of the Brillouin zone (Koster, 1957 Slater, 1965). In particular, applications of lattice dynamics have been treated by Maradudin and Vosko (1968) and by Montgomery (1969). Venkataraman and Sahni (1970) have included a discussion of symmetry in their review. In recent years, several books have appeared which contain comprehensive listings of the irreducible representations of space groups (Zak, 1969 Slater, 1965 Kovalev, 1965 Miller and Love, 1967). 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

Brillouin-zone, of the thiee-dim isioiial crystal are lequiied. A dynamical theory of crystal lattices was established by Bom and Huang (1954) crystal vibrations were treated with Cartesian synunetry-coordinates whidi were constructed with respect to the translational-symmetry of crystal lattices. 

In order to probe the entire Brillouin zone of k-space, inelastic neutron scattering experiments must be carried out, where the excitation of lattice phonons draws whatever momentum is needed from the high momentum of the neutron quanta. In addition, the k-vector of the incoming radiation can be properly matched with the phonon k-vector in modulus and direction, so that k-dependent intensities can be obtained, the various branches can be probed separately, and the entire dispersion curves can be experimentally determined. Such experiments have been carried out only for some particularly simple crystals. For a review see e.g. Gonze, X. Rignanese, G.-M. Caracas, R. First-principle studies of the lattice dynamics of crystals and related properties, Z. Krist. 2(X)5,220,458 72. 

We start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

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