Space reciprocal

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R  

It may be recalled that an alternative description for a crystal stmcture can be made in terms of sets of lattice planes, which intersect the unit cell axes at ua, VU2, and was-The reciprocals of the coefficients are transformed to the smallest three integers having the same ratios, h, k, and I, which are used to denote the plane (hkl). Of course, the lattice planes may or may not coincide with the layers of atoms. Any such set of planes is completely specihed by the interplanar spacing, dhU7 and the unit vector normal to the set, since the former is given by the projection of, for example, u ui onto n kh that is dhki = u ui- n ki- The reciprocal lattice vector is defined as  

The factor 2-77 is usually omitted from the definition of a reciprocal-lattice vector in crystallography. This is because Bragg s law defines the deviation of a diffracted ray from the direct ray in terms of the half-wavelength of the radiation and the quantity 1 jd, which, in crystallography, is taken as the reciprocal-lattice vector  

The factor 27r arises, however, when the relation A = iTrjk is used to express the periodicity of the incident radiation. Each vector G t/ of the reciprocal lattice corresponds 

TABLE 4.1. Primitive Translation Vectors of the Real-Space Cubic Lattices R = ua- + v32 + was 

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This expression holds true only in the large g region in reciprocal space (or small r in real space). Since... 

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a... 

In the Ewald summation method the initial set of charges are surrounded by a Gaussian distribution lated in real space) to which a cancelling change distribution must be added (calculated in reciprocal space). 

The original particle mesh (P3M) approach of Hockney and Eastwood [42] treats the reciprocal space problem from the standpoint of numerically solving the Poisson equation under periodic boundary conditions with the Gaussian co-ion densities as the source density p on the right-hand side of Eq. (10). Although a straightforward approach is to... 

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. 

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity at some point (hkl) in reciprocal space can be written as... 

Figure 2 View looking down on the real-space mesh (a) and the corresponding view of the reciprocal-space mesh (b) for a crystal plane with a nonrectangular lattice. The reciprocal-space mesh resembles the real-space mesh, but rotated 90°. Note that the magnitude of the reciprocal lattice vectors is inversely related to the spacing of atomic rows.

Figure 3 (a) Real-space lattice and reciprocal-space mesh for the GaAs (110) plane, (b)... 

Right Fig. 9. EEL spectra of an MWCNT obtained from the locations at 000, intermediate and 002 reflexions in the reciprocal space (modified from ref. 16). 

Let us give a brief summary of the LSGF method. We will consider a system of N atoms somehow distributed on the underlying primitive lattice. We start with the notion that if we choose an unperturbed reference system which has an ideal periodicity by placing eciuivalent effective scatterers on the same underlying lattice, its Hamiltonian may be calculated in the reciprocal space. Corresponding unperturbed Green s... 

The SSW form an ideal expansion set as their shape is determined by the crystal structure. Hence only a few are required. This expansion can be formulated in both real and reciprocal space, which should make the method applicable to non periodic systems. When formulated in real space all the matrix multiplications and inversions become 0(N). This makes the method comparably fast for cells large than the localisation length of the SSW. In addition once the expansion is made, Poisson s equation can be solved exactly, and the integrals over the intersitital region can be calculated exactly. 

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

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