Equivalent wave vector

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

On the other hand, the evolution equations derived by Choh and Uhlenbeck (Eqs. 11 and 12) can be written in the wave vector space (see Eq. 92 with n = 3 and where is replaced by A ). Consequently, expression (114) establishes the equivalence between the result of Choh and Uhlenbeck (Eqs. 11 and 12) and the generalized Boltzmann equation in the Prigogine formalism (see Eq. (85)). 

Let us consider a wave vector k of an occupied state close to kp and its equivalent k — 2kp as indicated in Fig. 1.32. The Bloch functions of both states will be given by Eq. (1.31) ... 

Fig. 20. Effect of an uniaxial pressure on the relative magnetic Bragg peak intensities associated with the three equivalent k-vectors for UN, UAs and USb single crystals. The uniaxial pressure is applied along the (001) direction. Full circles correspond to magnetic peaks Ii, associated with a wave vector perpendicular to the stress and open circles to magnetic peaks I, associated to a wave vector parallel to the stress. (Rossat-Mignod et al. )...

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

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. 

Equation (3) shows that the space-group operator (R v) transforms a Bloch function with wave vector k BZ into one with wave vector R k, which either also lies in the BZ or is equivalent to ( ) a wave vector k in the first BZ. (The case Id = k is not excluded.) Therefore, as R runs over the whole R = P, the isogonal point group of G, it generates a basis ( 0kl for a representation of the space group G,... 

This is equivalent to the statement that any two-terminal transport phenomem can only have an even magnetic field dependence. In chiral systems, symmei allows all microscopic properties to have in principle an odd dependence on wave vector k of the moving particles. As the wave vector is also odd un< time-reversal, from Eq. (33) it follows that... 

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. 

Fig. 5. Schematic illustration of the zone folding effect (above figure). Even when the super cell increases to 4 unit cells in each direction, which is a total of 4x4x4 primary cells (equivalent to a 256-molecule super-cell, see the size C), there are still only 3 wave-vectors allowed in each direction. As shown at the bottom of the diagram, the wave-vector at q = 0 (i.e.. . = a) is not included.

The in-plane structure of the interface is probed by measuring the scattered intensity as a function of the angle 26 (fig. 3.59). or, equivalently, as a function of the horizontal component of the wave-vector, If the molecules in the interface are arranged in a two-dimensional periodic structure, diffraction occurs when coincides with a reciprocal lattice vector fulfilling the Bragg condition... 

Let us consider that the magnetic atom site j has equivalent atoms labeled as js (/I, j2,.. Jp) under the application of symmetry operators of the crystal space group G belonging to the wave vector group G. The group G is formed by the set of symmetry operators that leave invariant the propagation vector Gt = geG gk = k e L, where L is the crystallographic reciprocal lattice. 

The simplest example is that of a one-dimensional cluster of N equivalent molecules with nearest neighbor J interaction, periodic boundary conditions and intermolecular spacing r. The N states with a single local excitation are mixed to give a band of 1-exciton states with energies = 2Jcos(kr), and wave vector k = Q, TT/Nr, 2Tr/Nr,.Tr/r, as sketched in Fig. 1. 

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