Wave vector definition

Fig. 4.16 Definition of the polar angles 9, c(). k is the wave vector of the emitted y-ray. The z-axis may be defined by the direction of a magnetic field...

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... 

Let us now specify the nature of the dynamical irreducibility condition in Eq. (56). The conservation rides of the wave vectors (Eq. 63) impose the condition that the k of certain particles is zero in certain intermediate states. For example, in Fig. 2a particles 2 and 3 have their k zero in the second state of propagation. It may be that the structure of the diagram is such that for one or many intermediate states the k of every particle is identically zero. The diagram is then reducible (see Fig. 2c) and is not contained in Eq. (56). This leads us to extend the definition of jijru. ) so as to include in it the reducible contributions. We shall define... 

We see from Fig. 2.5 that the Gaussian wave packet has its intensity, F 2, centred on x0 with a half width, W, whereas (k) 2 is centred on k0 with a half width, 1/W. Thus the wave packet, which is centred on x0 with a spread Ax — W, is a linear superposition of plane waves whose wave vectors are centred on k0 with a spread, A = jW. But from eqn (2.8), p = Hk. Therefore, this wave packet can be thought of as representing a particle that is located approximately within Ax = W of x0 with a momentum within Ap = h/W of po = hk0. If we try to localize the wave packet by decreasing W, we increase the spread in momentum about p0. Similarly, if we try to characterize the particle with a definite momentum by decreasing 1/W, we increase the uncertainty in position. 

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. 

Before presenting the applications, the theoretical background for the density matrix method should be constructed in molecular terms. For this purpose, a detailed definition of the density matrix of the system should be discussed. Let us consider a molecule at position Ry interacting with an applied laser field of bandwidth 8co and corresponding wave-vector width 8k — 8co/c. Now, it is assumed that the laser beam spot is similar to 8k and a system has only two states described by n> and m ) with respective energies hcontt and ha, within 8co. In this case, the dynamics of the system can be described as... 

Wilson ((>85) has pointed out that if a Brillouin zone is full, the electrons occupying the states of this zone can make no contribution to the electric current. This fact follows from the definition of the zone as a region enclosing all reduced wave vectors. Imagine all electrons of Figure 6 shifted Akx by an external field. The electrons in states within Akx of the zone boundary are reflected to the opposite zone boundary, so that the zone remains filled and there is no transfer of charge. This observation permits a sharp distinction between metallic conductors, semiconductors, and insulators. Because of the high density of states in a band, a crystal with partially filled bands is a metallic conductor. If all occupied zones (or bands) are filled, the crystal is a semiconductor if Ef kT, is an insulator if kT. 

There have been several measurements of the lattice dynamics of quartz by inelastic neutron scattering. Early results showed that the soft mode in the high-temperature phase is overdamped (Axe 1971). Other work on RUMs at wave vectors not directly associated with the phase transition showed that on cooling through the phase transition the RUMs rapidly increase in frequency since they are no longer RUMs in the low-temperature phase (Boysen et al. 1980). The most definitive study of the RUMs associated with the phase transition was that of Dolino et al. (1992). 

E ) must be found in the definition (10.4.20) of the J°(.. . ). Finally, there remains a factor (2jt) where Q is the number of loops in the diagram, i.e. the number of independent wave vectors. In other terms, a factor is associated with each independent vector appearing in the diagram. 

The definition, theoretical treatment, and experimental set-up of CARS, RIKES, and FWM are summarized in Tables 5.8, 5.9, and 5.10, respectively. The set-ups differ slightly for each technique, to satisfy the specific conditions of energy and wave vector matching. In... 

The effective interaction Kff is expected to depend not only on the wave vector but also on n, the number density of electrons in the medium. This, in turn, is related to the Fermi wave number fcp by the definition... 

A particular presentation of experimental data is used. In the theory of lattice vibration it is generally accepted to denote the phonon wave vector by q. The reduced wave vector 5 is plotted along abscissa. This dimensionless quantity equals to the ratio of the magnitude of the phonon wave vector to its maximal magnitude in the given crystallographic direction. By definition. 

Only bilinear variables with intermediate wave vector kc are to be included in the nonlinear Langevin equation, while the relations just derived contain sums over all wave vectors. Let us split up the sum in Eq. (67) and use the definition of the diffusion flux = ik /t, to obtain... 

A problem arises when trying to write Eq. (90) in vector form in the space of dynamical variables. If n linear variables are needed to form a complete set, then there exist possible bilinear variables. Thus, V must be an nxn matrix, and AA is a vector in an n -dimensional space as it turns out, these considerations would complicate the manipulations we are about to perform. Kawasaki has given a very simple alternate way to treat the vectorial structure of Eq. (90). This is to extend the definition of the wave vector of. a variable to include an index, which specifies the identity of the variable itself. Thus, instead of AL, we now just write Ak, where k is understood to contain both the Fourier transform variable and the index i. Sums over k are similarly reinterpreted. Now, combinations of the linear variables can always be chosen such that the matrix io)k-k is diagonal in the space of dynamical variables. Henceforth, we shall always assume that such a choice has been made. Then Eq. (90) holds as a vector equation with the new interpretation of wave vector. 

To find the dependence of the wave function and of the phase angle on the space vector r, the technique is the same as for the temporal oscillator. It consists of the definition of the wave vector operator in a similar manner as for the wave pulsation operator, that is, by taking a derivation operator, with a proportionality factor chosen for obtaining a real solution ... 

With these restrictions, the apparent space derivation defined in Equation 11.124 can be rewritten in terms of the damping vector in lieu of its operator and can be identified with the product of the imaginary number times the wave-vector, owing to definition 11.129 of the wave-vector operator ... 

Now consider the quantum dynamics of the same system. If the discs scatter a plane wave of wave vector k, the same matrix describes the scattering processes in the classical and semiclassical descriptions, but in the latter the parameters must be fixed as follows z = -ik,r = -l for Dirichlet boundary conditions,and P = 1/2 because the square of the wave function is the density of probabilityTherefore, we can analyze both the classical and the quantum dynamics of this model system with the same formalism, keeping in mind the different definitions of the parameters z, P and r. In the following we shall consider only the case lrl = l. 

Figure 3 In the upper part the particles density p vs position y — jcms citid time X is shown for three dijferent velocities (a),(h) v = 0.02, (c) v = 0.04, and (d) v — 0.4, respectively. The white and black colors indicate low and high density, respectively [ the gray scales are chosen independently for each suhfigure to maximise contrast]. The definition of the shearon wave vector q is examplified in (a) and (b), where the arrows indicate / 2q ) and j 2q2), respectively. In the lower part the resulting normal motion of the top plate Y is represented. The model parameters are ay = 1, = 0.5,...

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