Brillouin zone center

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Figure 4.5. Wave vectors around the center of the excitonic Brillouin zone for which coherent emission [solution of equations 4.10 and 4.25] is possible according to the disorder critical value Ac. We notice that r0 is the imaginary eigenvalue for K = 0 (emission normal to the lattice plane) and that K" and K1 indicate, respectively, components of K parallel and perpendicular to the transition dipole moment, assumed here to lie in the 2D lattice. The various curves for constant disorder parameter Ac determine areas around the Brillouin-zone center with (1) subradiant states (left of the curve) and (2) superradiant states (right of the curve). We indicate with hatching, for a large disorder (A,. r ), a region of grazing emission angles and superradiant states for a particular value of A.

This contrasting behavior of the two branches also can be seen in Fig. 5 where the displacements of the atoms G and H at points in the BZ are compared for examples, we have taken = 0 (also called the Brillouin zone center), q = irlAa, q = it/la, q = 2>TtlAa and q = itla (the zone boundary). From Eq. (5) the ratio of displacements for G and H in the /th cell with the same value of q is given by... 

Figure 31. Relative Brillouin-zone-center gap frequencies for translational in-plane modes for N2 on graphite as a function of temperature the gap frequencies A = A(T)/Ao are normalized with the corresponding ground-state gap frequency Aq, and the temperature T = T/T is normalized by the melting temperature T of the Vs solid. Circles (unfilled and filled circles refer to the a and y directions of the in-plane translations, respectively) constant temperature molecular dynamics simulations, Aq = 0.30 THz, and T = 73 K. Squares inelastic neutron scattering data [192], Aq = 0.40 THz, and T = 72 K. (Adapted from Fig. 15 of Ref. 140.)...

It is found [138] that the increase of the corrugation due to the inclusion of axially symmetric (experimentally determined bulk) quadrupole moments located at the carbon sites [361] which model the aspheiical charge distribution in the graphite substrate [see (3.9) and (3.10) in Section III.D.l] stabilizes the commensurate herringbone structure. This structure is head-tail-ordered as in Ref. 17 (see Fig. 53a or Fig. 54Z>, where the molecular axes have a systematic out-of-plane tilt) the unit cell is deformed because of the displacement of the molecular centers on the two sublattices. The Brillouin-zone-center frequency gap in the phonon spectrum is estimated [138] to amount to about 10 K in the ground state,... 

The space of electronic operators Op is naturally decomposed into subspaces, conforming to irreducible representations of a symmetry group of the Brillouin zone center and having the corresponding tensors i/(r,). To calculate the quantities (142), the parameters of an electron-deformation interaction (eq. 19) are necessary besides only coupling constants with odd optical vibrations are necessary, if acoustic vibrations are considered in the long-wave approximation. Parameters 5/(l 7g(u)) are linear combinations of 5 (f ) (eq. 112) with coefficients defining the expansion of the... 

Feibelman and Hamann (1979) found several unoccupied surface states near Ef in theoretical LDOS calculations for Sc(OOOl). Their surface state near the surface Brillouin zone center (F) was largely in character. Similar results were found in the LMTO... 

The Raman scattering of single-crystal graphite was found to contain a single narrow band (G-mode) at 1575 cm, that is assigned to the planar mode of the Brillouin zone center. However, jp -hybridized carbon, that is, single-crystal diamond, demonstrates a single first-order peak in the Raman spectrum, a narrow symmetric line at the frequency of 1332.5 cm (with a peak width of about 2.0 cm" ), which derives from the transverse TO phonon of symmetry. At... 

It is evident that acoustic modes of the Brillouin zone center cannot be detected by Raman scattering or infrared absorption, because they do not contribute to the vibrational energy. Indeed, they are the block translations of the crystal lattice along x, y, and z and their number is standard, equal to 3. 

T, Wilson Kondo scale of the PAM E-point Brillouin zone center... 

Because of the momentum conservation rule during the light scattering process, first-order Raman scattering is caused by phonons at the Brillouin zone center. The Raman scattering efficiency S with polarization detection is given by ... 

There are several other techniques that are sensitive to surface vibrations. Flowever, they are discussed here only very briefly, since they cannot measure the surface phonon dispersion. They are sensitive either to the Brillouin zone center only or... 

At the laser wavelengths used (generally 514.5 and 632.8 nm), the K and K wavevectors are very weak and the generated wavevector of the phonon is also very weak. This means that the Raman process involves a phonon of the Brillouin zone center. A crystalline defect (breaking of the periodicity or impurity) will modify the scattering spectrum compared to the spectrum of a perfect crystal in two ways ... 

The films used in electrochemistry are polycrystalline and the phonon confinement within one crystallite allows phonons with wavevectors slightly different but close to the Brillouin zone center to eventually participate in the observed Raman scattering process. Finally, the finite size of the crystallites can induce a partial breaking of the selection rules and allows the scattering of incident photons by phonons situated near the Brillouin zone center. Some species present in the films, such as monocrystalline graphite, are characterized by two peaks at 1350 and 1580 cm fi At a greater level of disorder, all of the selection rules can be broken and the total density of phonons can be observed. Characteristic phonons that can be observed are the following ... 

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