Wave vector units

Wave Vector (units of 2it/L) Wave Vector (units of 2tl/L) Wave Vector (units of 2tl/L)... 

In quasi-one-dimensional conductors, the one-dimensional electron gas instability is responsible for the formation of a charge density wave (CDW) with wave vector 2kF and/or 4kF via electron-phonon coupling. Thus the measurement, in reciprocal wave vector units, of the corresponding 4 and 2kF scattering wave vectors gives the value and twice the value of the charge transfer p, respectively (e.g., see Ref. 114). 

Salt Anion symmetry Transition temperature T (K) Wave vector" Unit cell References... 

In the above eqn, ID refers to the nanotubes whereas 2D refers to the graphene sheet, k is the ID wave vector, and t and Care unit vectors along the tubule axis and vector C, respectively, and p labels the tubule phonon branch. 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

Since the theory under examination works exclusively on scales essentially exceeding size a of a monomeric unit, the function D(Q) has a physical meaning only at Q wave vector can be calculated via the relationship... 

In passing from the first to the second problem, a feature of importance should be borne in mind. The periods of the orientational structure (3.1.9) can exceed those of the basic Bravais sublattice, Ai, A2. If this is the case, the unit cell A, A2 should be enlarged so that conditions (3.1.10) can be met and translations onto the new vectors R can reproduce the orientations of adsorbed molecules. Then the excitation Hamiltonian (3.1.3) can be represented in the Fourier form with respect to the wave-vector K as... 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

The mathematics necessary to understand the diffraction of X rays by a crystal will not be discussed in any detail here. Chapter 4 of reference 10 contains an excellent discussion. The arrangement of unit cells in a crystal in a periodic manner leads to the Laue diffraction conditions shown in equations 3.3 where vectors a, b, and c as well as lattice indices h, k, and l have been defined in Figure 3.5 and S is a vector quantity equal to the difference between the resultant vector s after diffraction and the incident X-ray beam wave vector So so that S = s - So-... 

Figure 2. Ewald construction for X-ray (soUd sphere) and electron (dotted sphere). ( kO, k wave-vectors, X - wave-length, a, b - parameters of reciprocal unit cell).

The oscillating dipole is a source of electromagnetic radiation of the same frequency, polarized in the direction of the oscillations. At large distances, the wave is spherical. According to the electromagnetic theory, the resulting electric vector at a point in the equatorial plane of the dipole is a>2/ r c2 times the moment of the dipole at time t — r /c. The amplitude of the spherically scattered wave at unit distance in the equatorial plane is therefore... 

The displacements of a particle j at r in unit cell /, subject to a phonon wave with wave vector q, obey the equation... 

Fig. 24. The dispersion curves for USb energy plotted against wave-vector transfer Q (in units of 2 3t/a). The dashed lines represent the phonon dispersion and are based on the measured open points as well as on knowledge of phonons in NaCl structures. The magnetic modes are represented by solid squares (the collective excitation) and the hatched area (excitonic level). (Lander and Stirling )...

The wave vector of a homogeneous wave may be written k = (k + zk")e, where k and k" are nonnegative and is a real unit vector in the direction of propagation. Equation (2.45) requires that... 

Consider an arbitrary particle illuminated by a plane wave with unit amplitude and linearly polarized along the direction q. The direction of propagation of this wave is specified by the unit vector e( and that of the scattered wave—that is, the direction of observation—by We denote by... 

Figure 5. Photorefractive two-beam coupling gain coefficient of undoped and doped BaTi03 crystals. Curves are theoretical fits to the data. Experimental conditions X = 515 nm, I = 3-5 W/cm2, beam ratio > 200, s-polarization, grating wave vector parallel to c-axis Concentrations refer to dopant atoms per BaTi03 formula unit in the melt.

When used as the dispersion formula for the phonons and polaritons in orthorhombic crystals, the symbols in Eq. (11.22) have the following meaning r)= 1,2,3 designates the three directions of the principal orthogonal axes. sv are the direction cosines of the normalized wave vector s = k/k with respect to the three principal axes of the crystal. If the unit vectors in the directions of these three principal axes are designated eue2,e3, one can write... 

We start from Maxwell equations. Using the MKS system of units, we have for wave vector k perpendicular to electric field E the equation [see Eq. (3)]... 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

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