Reciprocal lattice wave

If nf - p/r is a fraction other than 2, the system is said to be weakly commensurate. There now appears in the LG free-energy functional a weak commensurability term [51] of the form id [ty(x))r + [iji ( )]r = d i >(x) r cos r0, coming from an r-vertex bubble in which r2kF is equal to a reciprocal lattice wave number (2ttpla). Here d 5rCDW( TrvF) 1( o) r+2 where... 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

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.

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Pt2,V and Pt y have been investigated at 1393 K and 1224 K respectively and we have explored the [100] and [110] planes of the reciprocal lattice. The measured Intensities have been Interpreted in a Sparks and Borie approach with first order displacements parameters and using a model Including 29 a(/ ) for PfsV and 21 for PtsV. In figure 1 is displayed the intensity distribution due to SRO a q) in the [100] plane. As for PdjV, the diffuse intensity of Pt V is spread along the (100) axes with maxima at the (100) positions, whereas the ground state is built on (1 j 0) concentration wave ( >022 phase). 

The wave vectors k can be expressed in terms of any basis vectors we choose. At the moment there is neither a direct nor a reciprocal lattice. Using (II.3a) in (II. 1) we see that the Fourier components of two indistinguishable densities can differ only by a phase factor ... 

The symmetry properties of the density show up experimentally as properties of its Fourier components p. If those components vanish except when the wave vector k equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave expansion of the density,... 

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. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

H(u) is the Fourier Transform of h(r) and is called the contrast transfer function (CTF). u is a reciprocal-lattice vector that can be expressed by image Fourier coefficients. The CTF is the product of an aperture function A(u), a wave attenuation function E(u) and a lens aberration function B(u) = exp(ix(u)). Typically, a mathematical description of the lens aberration function to lowest orders builds on the Weak Phase Approximation and yields the expression ... 

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

Here, gi is the length of the primitive reciprocal lattice vector. The constants a, p, and are determined either by considering leading Bloch waves or by fitting with first-principles calculations. The method of Harris and Liebsch is used extensively in the treatment of atom scattering data. With some modifications, the method of Harris and Liebsch is also applicable to calculate STM images. We will discuss it in detail in Chapter 5. 

Fig. 6.6 The wave-vector dependence of the energy-wavenumber characteristic, ( ) which has a node at q0 and a weak logarithmic singularity in its slope at q = 2kF. Also shown are a set of degenerate cubic reciprocal lattice vectors that are centred on q0. A tetragonal distortion would lift their degeneracy away from the node at q0 as shown, thereby lowering the band-structure energy. (After Heine and Weaire (1970).)...

If one examines the series in Equation (2.9) it will be seen that the terms will tend to reinforce whenever Q=2vl/d, where / denotes an integer. These values of Q define a one-dimensional reciprocal lattice and whenever Q takes on one of these values the diffracted waves will reinforce. These values of Q correspond to the familiar Bragg reflections given by 2d sin 0 =/A. It should be noted that for finite values of N the reciprocal lattice points have a finite width. 

Figure 2.1. Application of the reciprocal lattice to the analysis of electron diffraction data, (a) The vector corresponding to the incident wave is drawn through the origin of the reciprocal lattice, O, in the direction that the wave is travelling and has a length, XO, equal to 1/X. For diffraction to take place Q must correspond to 2ir times the vector joining O to another point in the reciprocal lattice, P, and distance XO must be equal to XP. Clearly this situation can only be satisfied by freak conditions. However, (b) illustrates what happens for the real case of a thin film in which the reciprocal lattice points become extended in the direction normal to the plane of the film.

The atoms of a crystal vibrate around their equilibrium position at finite temperatures. There are lattice waves propagating with certain wavelengths and frequencies through the crystal [7], The characteristic wave vector q can be reduced to the first Brillouin zone of the reciprocal lattice, 0 < q <7t/a, when a is the lattice constant. 

The (unfiltered) Fourier transform of this image is reproduced in Fig. 7. It shows a map of the k vectors that contribute to the standing wave pattern. Spots reflecting the reciprocal lattice of the Cu(lll) surface (originating from the atomic resolution) and circles corresponding to the 2D Fermi contour, i.e. the crossing of the Fermi level by the surface state, are... 

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