Wave vector space

Examination of the EXAFS formulation in wave vector form reveals that it consists of a sum of sinusoids with phase and amplitude. Sayers et al32 were the first to recognize the fact that a Fourier transform of the EXAFS from wave vector space (k or direct space) to frequency space (r) yields a function that is qualitatively similar to a radial distribution function and is given by ... 

Let us start by writing Cohen s results (24) in wave vector space. We obtain... 

Let us begin by writing <0 fl18B(i2S) 0> explicitly in wave vector space. Starting from Eq. (25) and taking into account the conservation rules for the wave vectors (Eq. 63), we see that the intermediate state between the two-particle streaming operators is k = 0, which gives ... 

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)). 

Fig. 420 S AXS pattern for bulk samples of a P(F)S49-i>-PLA 192 and b PS50-i>-PLA214 annealed at 150 °C (a) and 173 °C (b). For the voided st5rrenic scaffold the scattered intensity for small wave vectors q increased significantly due to enhanced contrast (c). Curve a corresponds to a disordered worm-like morphology as confirmed by SEM. Peaks with scattering wave vector spacing-ratios consistent with a laid gyroid phase are discernible in case of sample b and c. The peaks of curve c located in the q range from 13 to 18 A can be attributed to lemtiining PLA etchant, sodium hydroxide...

The effect of adding periodicity to the nematic has been described by Brazovskii [7], [8]. The nematic-isotropic transition is an example of a transition between two spatially uniform phases, which means that the transition takes place at the origin of wave-vector space. Fluctuations of the system... 

In discussing the effect of thermal fluctuations, let us consider wave-vector space... 

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via detennination of the wave vector transferred between the photon/neutron and the sample this is a Fourier transfonn relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give infonnation on the time dependence of the motions involved. 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

Ciccotti et aV° have evaluated shear viscosity by applying a shearing force that is periodic in space and has by necessity a finite wave 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... 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

Mermin [9, 18] has given a recipe for the construction of a set of Fourier components for a density characterised by a certain space group. The space group is then specified by a point group G, a lattice of wave vectors in the sense discussed above, and a set of phase... 

This section introduces the basic mathematics of linear vector spaces as an alternative conceptual scheme for quantum-mechanical wave functions. The concept of vector spaces was developed before quantum mechanics, but Dirac applied it to wave functions and introduced a particularly useful and widely accepted notation. Much of the literature on quantum mechanics uses Dirac s ideas and notation. 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

For the calculation of the hydrodynamic thickness we divide the profile artificially into elementary layers, the result being independent of the division chosen provided it is sufficiently fine. The s.a.n.s. data is obtained as a function of Q, the wave vector (4it/A sin(0/2), where X is the neutron wavelength and 0 the scattering angle. The Q resolution corresponds in real space to a fraction of a bond length which is small enough for defining an elementary layer. 

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

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