Vector of reciprocal lattice

When the crystal is irradiated by an X-ray beam, its lattice scatters the radiation selectively. A strong diffraction is observed when the wavevector of scattering for a particular angle (i.e. q) coincides with the vector of reciprocal lattice, as shown in the Ewald sphere. Fig. 5.10. The condition... 

The basic vectors of reciprocal lattice this way formed are orthogonal on the basic vector of direct lattice ... 

But which are vectorial expressions and the modules of crystallographic vectors of reciprocal lattice ... 

The statements (3.44) allow the scalar products calculation between the fundamental vectors of reciprocal lattice and those correspondents in direct lattice, with the results ... 

Therefore, any vector in reciprocal lattice can be written on the basic vectors of reciprocal lattice, generally as ... 

Yet, which is the relation between the vector of reciprocal lattice and... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

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. 

Construction of reciprocal lattice vectors a, b, c in a general triclinic cell of sides a, b, c. 

The commensurability of overlayers is characterized by the coincidence of reciprocal lattice vectors of adsorbate and substrate. Low or higher order coincidence leads to commensurate or higher order commensurate overlayer structures (Figs. 3.14a, b, c). K the reciprocal lattice vectors do not coincide, the overlayer is incommensurate (Fig. 3.14d). 

Now consider the orthorhombic crystal system. Simple analysis of Eq. 2.32 indicates that the following groups of reciprocal lattice points will have identical reciprocal lattice vector lengths and thus, are equivalent in terms of the corresponding Bragg angle ... 

Both approaches work in a similar way. In the case of platelet-like particles, the function is stretched along T (T > Tx), while in case of needles it is stretched along Tx (Ty < Tx). Therefore, in both cases x < 1 describes preferred orientation of the platelets and x > 1 describes preferred orientation of the needles. Obviously x = 1 corresponds to a completely random distribution of reciprocal lattice vectors and the corresponding radial distribution functions become a circle with unit radius (both Eq. 2.78 and 2.79 result in Tuu = 1 for any... 

Figure 2.50. The illustration of the complex distribution of reciprocal lattice vectors modeled using a spherical harmonic preferred orientation function for the (100) reflection.

Any reciprocal lattice vector, or reciprocal lattice point is uniquely specified by the set of three integers, hkl, which are the Miller indexes of the family of planes it represents in the crystal. Thus there is a one-to-one correspondence between reciprocal lattice points and families of planes in a crystal. It will be seen shortly that the reciprocal lattice is the Fourier transform of the real lattice, and vice versa. This was in fact demonstrated experimentally in Figure 1.7 of Chapter 1 by optical diffraction. As such, reciprocal space is intimately related to the distribution of diffracted rays and the positions at which they can be observed. Reciprocal space, in a sense, is the coordinate system of diffraction space. 

This is the relation given without proof in Sec. 2-6. By similar use of reciprocal-lattice vectors, other problems of crystal geometry, such as the derivation of the plane-spacing equations given in Appendix 3, may be greatly simplified. 

This shows that diffraction occurs when the magnitude of the scattering vector h is an integral number of reciprocal lattice spacings jd. We define a vector d ... 

It is common also to define a set of reciprocal lattice vectors , b, c, such as... 

In this case, the vectors G are members of the set of reciprocal lattice vectors and eqn (4.105) really amounts to a three-dimensional Fourier series. The task of solving the Kohn-Sham equations is then reduced to solving for the unknown coefficients ak+c- The only difficulty is that the potential that arises in the equation depends upon the solution. As a result, one resorts to iterative solution strategies. 

The second line contains data for four variables. The number of reciprocal-lattice vectors included in the lattice sums (6.58, 8.26) at each k point is determined by GMAX. It may be smaller than the corresponding GMAX... 

It is possible from measurements of a limited number of reciprocal-lattice vectors and observations of systematic absences of reflections to deduce not only the shape and dimensions of the unit cell, but also the complete symmetry of the crystal structure. The determination of the complete structure now requires the contents of the unit cell to be deduced from the intensities of the reflections . These are usually determined by using diffractometers rather than film to record the diffraction. A diffractometer is usually a device that allows the recording of the intensity of scattering in any particular direction in space. Modern types, using CCD arrays, can determine the intensity over a range of directions for one setting of the instrument. This greatly speeds up the collection of data but leads to some complication in terms of the need to calibrate the different... 

Figure 4.9 illustrates the construction of reciprocal lattice. The relationship between the reciprocal lattice vector and the interplanar distance in real-space is expressed as... 

The cell-periodic part of the wave function can be expanded using a discrete set of plane waves whose wave vectors are reciprocal lattice vectors of the crystal,... 

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