Group of the wave vector

Therefore, k+bm and k label the same representation and are said to be equivalent (=). By definition, no two interior points can be equivalent but every point on the surface of the BZ has at least one equivalent point. The k = 0 point at the center of the zone is denoted by T. All other internal high-symmetry points are denoted by capital Greek letters. Surface symmetry points are denoted by capital Roman letters. The elements of the point group which transform a particular k point into itself or into an equivalent point constitute the point group of the wave vector (or little co-group of k) P(k) C P, for that k point. 

The little group (or group of the wave vector) G(k) is the space group... 

In Sections 16.4-16.6 the problem of finding the representations of a space group G at any particular k point was solved by reducing the size of the group of the wave vector... 

Equation (2) shows that the band energy has the symmetry of the point group of the wave vector. As R runs over the whole R P(k), it generates a set of degenerate eigenfunctions... 

Table 17.12. The point group of the wave vector P(k) at the symmetry pointsK = T, A, and Xfor the InSb and Si structures. 

The symmetry group of the wave vector for [001] propagation is the group of the square in the xy plane. This is the group A, of eight elements, with five irreducible representations. The five d states reduce to A, -i- A2 4- A 2 A, in this symmetry. There are three oxygens per primitive cell and, of the three linear... 

In earlier chapters, it was seen how a qualitative energy-level diagram for the smallest repeating chemical point group, or lattice point (known to crystallographers as the basis, or asymmetric unit), can be used to approximate the relative placement of the energy bands in a solid at the center of the BZ. This is so because the LCAO-MO theory is equivalent to the LCAO band scheme, minus consideration of the lattice periodicity. The present chapter will investigate how the orbital interactions vary for different values of the wave vector over the BZ. 

The resulting orderings are induced by the irreducible representation 4g(xi) of the wave vector group Gk as well as by the representation /ig(x3 + X5) of the point group Gk7=o- Indeed, in a general case the deviation can be written as... 

The sum over j concerns the atoms of the magnetic asymmetric unit for the wave vector k. So that j labels different sites. The sum over s concerns the symmetry operators of the wave vector group G. The phase factor. Ay. has two components ... 

The bare interaction energy is invariant with respect to scaling of the wave vector. This can be illustrated by selecting an arbitrary scale k and introducing the dimensionless group parameter... 

More detailed information about a possible mixing of different symmetry-group representations of pseudo-spin singlet or triplet order parameters can in principle be obtained by inspection of the wave-vector dependence in eq. (29). 

Figure 12.5 The group velocity as a function of the wave vector for the longitudinal branch in potassium.

So, a monochromatic traveling laser wave of frequency w interacts only with a group of particles that have a certain velocity projection v on the direction of the wave vector k within the limits of the homogeneous width ... 

A = amplitude, d = thickness of crystalline material, D = degeneracy factors, E, Ej, = energy levels. Eg = cohesive electric field intensity, E = laser field amplitude, /j = intensity of the fundamental, = vector sum of the wave vectors, = coherence length, L = interaction length, P = polarization induced by an incident electric field, T = carrier period, = group velocity, a = linear absorption coefficient, = dipolar susceptibility of order ,A = grating period, ft) =fre-quency, tOj = scattered wave frequency, h = Planck s constant/2jt. 

The formation of necklace like stmcmres due to poor solvents and counterion condensation was first simulated by the Kremer group and the structure factor was calculated as a function of the wave vector [131]. In a paper about end effects of... 

Figure 3 Snapshots of the shear wavefronts for different types of weld material to show the direction of the wave unit vector k (a phase velocity unit vector Cp ) and the group velocity unit vector...

The orbitals <]) j(k r) are Bloch functions labeled by a wave vector k in the first Brillouin zone (BZ), a band index p, and a subscript i indicating the spinor component. The combination of k and p. can be thought of as a label of an irreducible representation of the space group of the crystal. Thequantity n (k)is the occupation function which measures... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

Here it is our intention to show that for a system constituted by substrate phonons and laterally interacting low-frequency adsorbate vibrations which are harmonically coupled with the substrate, the states can be subclassified into independent groups by die wave vector K referring to the first Brillouin zone of the adsorbate lattice.138 As the phonon state density of a substrate many-fold exceeds the vibrational mode density of an adsorbate, for each adsorption mode there is a quasicontinuous phonon spectrum in every group of states determined by K (see Fig. 4.1). Consequently, we can regard the low-frequency collectivized mode of the adsorbate, t /(K), as a resonance vibration with the renormalized frequency and the reciprocal lifetime 7k-... 

If the origin of the group is situated at the midpoint between the centers of two chro-mophores, the vector r is described as the sum of the vector of the center a and the vector of the electron in the chromophore exhibited by the local coordinate r0 (Fig. 9). From the orthogonality of the wave-functions of each chromophore and unvariable condition of a, Eqs. (41) and (42) are reduced to ... 

---