The Wavevector

The wavevector K can take on an arbitrary magnitude and an arbitrary direction within the 1st BZ. The 1st BZ of the naphthalene crystal (Fig. 5.5) follows from the usual rules and can be constructed in three steps as follows  

) The primitive vectors a, b, and c of the reciprocal lattice are derived from the primitive vectors a, b, and c of the crystal lattice e.g. 

From a, b, and c, the reciprocal lattice follows by means of periodic continuation. 

) The 1st Brillouin zone follows from the reciprocal lattice by construction of the planes which are perpendicular to the lines connecting neighbouring points in the reciprocal lattice at their midpoints. The smallest closed volume which is bounded by these planes is the 1st BZ. For the naphthalene crystal, we find from the lattice parameters at T = 300 K (Table 2.3) the following magnitudes for the reciprocal lattice vectors a = lit 0.145 A ) = 2jt 0.167 A c = 2jr 0.138 A and for the volume V of the primitive 

Each phonon in a molecular crystal is characterised by its phonon dispersion relation S2/(K) with j = 1,2. j. 6Z (Z = the number of molecules in the unit cell). The individual phonon dispersion relations are also termed phonon branches. At each point S2(K) along the dispersion relation curve, all the molecules oscillate with the same frequency and with the same amplitude, but with phase differences defined by K. At the F point, the molecules with the same orientation oscillate in phase in all the unit cells. An example is shown in Fig. 5.3. An example for K 0 and thus for a phase difference between neighbouring unit cells is shown schematically in Fig. 5.1. 

---

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

Electronic and optical excitations usually occur between the upper valence bands and lowest conduction band. In optical excitations, electrons are transferred from the valence band to the conduction band. This process leaves an empty state in the valence band. These empty states are called holes. Conservation of wavevectors must be obeyed in these transitions + k = k where is the wavevector of the photon, k is the... 

The quantity 1 + x is known as the dielectric constant, it is constant only in the sense of being independent of E, but is generally dependent on the frequency of E. Since x is generally complex so is the wavevector k. It is customary to write... 

This is not the case for stimulated anti-Stokes radiation. There are two sources of polarization for anti-Stokes radiation [17]. The first is analogous to that in figure B1.3.3(b) where the action of the blackbody (- 2) is replaced by the action of a previously produced anti-Stokes wave, with frequency 03. This radiation actually experiences an attenuation since the value of Im x o3 ) is positive (leading to a negative gam coefficient). This is known as the stimulated Raman loss (SRL) spectroscopy [76]. Flowever the second source of anti-Stokes polarization relies on the presence of Stokes radiation [F7]. This anti-Stokes radiation will emerge from the sample in a direction given by the wavevector algebra = 2k - kg. Since the Stokes radiation is... 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

Figure C2.16.5. Calculated plots of energy bands as a function of wavevector k, known as band diagrams, for Si and GaAs. Indirect (Si) and direct (GaAs) gaps are indicated. High-symmetry points of the Brillouin zone are indicated on the wavevector axis.

Equation (3.85) T is a translation vector that maps each position into an equivalent ition in a neighbouring cell, r is a general positional vector and k is the wavevector ich characterises the wavefunction. k has components k, and ky in two dimensions and quivalent to the parameter k in the one-dimensional system. For the two-dimensional lare lattice the Schrodinger equation can be expressed in terms of separate wavefunctions ng the X- and y-directions. This results in various combinations of the atomic Is orbitals, ne of which are shown in Figure 3.13. These combinations have different energies. The /est-energy solution corresponds to (k =0, ky = 0) and is a straightforward linear... 

Since it is the first derivative with respect to r that we are interested in, we only need the 1=1 term from this expansion. The angular part contributes only to the overall constant, but it is the spherical function j (kr) that sets the cutoff value of the wavevector, above which the phonons do not produce significant linear uniform stress on the domain. In Fig. 24, we plot the derivative dji x)/dx (or, rather, we plot the square of it, which enters into all the final expressions). 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

In order to calculate the phase properties of the melt, that expansion is cut off at some lower order. The order parameter T(r) in the ordered phase which is characterized by the set Q/ m of the wavevectors in the reciprocal space can be expanded as... 

Since the confined modes (whether they are waveguide modes or surface plasmons) are nonradiative (i.e., their wavevector parallel to the interface, is greater than the wavevector of the... 

An ordered monolayer of molecules having a large dynamical dipole moment must not be regarded as an ensemble of individual oscillators but a strongly coupled system, the vibrational excitations being collective modes (phonons) for which the wavevector q is a good quantum number. The dispersion of the mode for CO/Cu(100) in the c(2 x 2) structure has been measured by off-specular EELS, while the infrared radiation of course only excites the q = 0 mode. 

The Ti q) behave as wavevector-dependent relaxation times and the form of the wavevector dependence can provide a useful check on the consistency of models. Table 5 shows a comparison of the experimental coefficients for fresh apple tissue with those calculated with the numerical cell model. The agreement is quite reasonable and supports the general theoretical framework. It would be interesting to apply this approach to mealy apple and to other types of fruit and vegetable. 

Figure 1.3 The addition of waves scattered by an angle 2 from an atom at the origin and one at a vector r from the origin. The wavevectors k q and k jj are in the directions of the incident and diffracted beams, respectively, and k o = k h =l/...

We should first correct the wavevector inside the crystal for the mean refractive index, by multiplying the wavevectors by the mean refractive index (1 + IT). This expression is derived from classical dispersion theory. Equation (4. 18) shows us that is negative, so the wavevector inside the crystal is shorter than that in vacuum (by a few parts in 10 ), in contrast to the behaviom of electrons or optical light. The locus of wavevectors that have this corrected value of k lie on spheres centred on the origin of the reciprocal lattice and at the end of the vector h, as shown in Figure 4.11 (only the circular sections of the spheres are seen in two dimensions). The spheres are in effect the kinematic dispersion surface, and indeed are perfectly correct when the wavevectors are far from the Bragg condition, since if D 0 then the deviation parameter y, 0 from... 

---