Reciprocal lattice Basis vectors

Reciprocal lattice basis vectors Given the basis vectors a, b, c in direct space, the reciprocal lattice basis vectors a b c are defined by the equation... 

To summarize, each spot on the diffraction pattern (Figure 8) can be uniquely referred to by integral indices b, k, and /, which are multipliers of the reciprocal lattice basis vectors a, b, and c. The position of each spot in this reciprocal space may then be expressed as a vector (ha + kW + lc) analogously to the positions of the atoms in the crystals, which can be defined by a vector (xa I jb I zc) in real space. Although the positions of the diffraction spots are defined by the FT of the crystal lattice, their intensities are defined by the FT of the contents of the unit cell. 

The construction of a reciprocal plane lattice is simple and is illustrated for the oblique plane (mp) lattice. Draw the plane lattice and mark the unit cell (Figure 2.6a). Draw lines perpendicular to the two sides of the unit cell. These lines give the axial directions of the reciprocal lattice basis vectors, (Figure 2.6b). Determine the perpendicular distances from the origin of the direct lattice to the end faces of the unit cell, d o and do (Figure 2.6c). The inverse of these distances, l/d o and I / do, are the reciprocal lattice axial lengths, a and b. ... 

Every direct lattice admits a geometric construction, the reciprocal lattice, by the prescription that the reciprocal lattice basis vectors (bi, hx, bs) obey the following important orthogonality rules relative to the direct lattice basis vectors (ai, ai, as) ... 

Let us return, for a moment, to Figure 10, the Bragg s law description of X-ray diffraction. X-rays are reflected by planes of lattice points, uniquely described by the three indices h, k, l. These three indices form the basis of another lattice, which we called the reciprocal lattice, where the distance from the origin to each point hkl was 1 /dhu, where dm was the distance between the Bragg planes. Each Bragg plane can be defined by its normal, which turns out to be a multiple of the reciprocal space basis vectors a, b, c. We can then refer to this plane, as well as to the Fourier term associated with it, by a reciprocal lattice vector d / = (ha + kb + lc ). Rewriting Equation (9) in terms of electron density, we get... 

The vector Ahki points in a direction perpendicular to a real space lattice plane. We would like to express this vector in terms of reciprocal space basis vectors a, b, c. ... 

Thirdly, a cutoff must be introduced on the reciprocal-lattice G-vectors to be included in the basis set. For a given k-point in the irreducible BZ the plane-wave kinetic energy j(4+G) should be less than some upper bound, typically in the range of 5-50 Ry. There exists an ambiguity when structures are calculated... 

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 ... 

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 ... 

Fig. 2.13. Two-dimensional Bravais lattice with the basis vectors a)s a2, and the reciprocal lattice vectors bi, b2. The solid and dashed arrows at angles A and 0A give the ferroelectric (k = 0) and antiferroelectric (k = bi/2) configurations of dipoles in the ground state.

The AO-basis Bloch functions are, as sum over reciprocal lattice vectors,... 

Figure 3.1 View of the real space and reciprocal space lattice vectors for the fee primitive cell. In the real space picture, circles represent atoms. In the reciprocal space picture, the basis vectors are shown inside a cube with side length At fa centered at the origin.

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

When calculating FE energy states along particular directions in the BZ it is often convenient to work in Cartesian coordinates, that is to use the (e basis rather than the (b basis. The matrix representation of a reciprocal lattice vector bm is... 

In the diffraction pattern from a crystalline solid, the positions of the diffraction maxima depend on the periodicity of the structure (i.e. the dimensions of the unit cell), whereas the relative intensities of the diffraction maxima depend on the distribution of scattering matter (i.e. the atoms, ions or molecules) within the repeating unit. Each diffraction maximum is characterized by a unique set of integers h, k and l (called the Miller indices) and is defined by a scattering vector h in three-dimensional space, given by h=ha +A b +Zc. The three-dimensional space in which the diffraction pattern is measured is called reciprocal space , whereas the three-dimensional space defining the crystal structure is called direct space . The basis vectors a, b and c are called the reciprocal lattice vectors, and they depend on the crystal structure. A given diffraction maximum h is completely defined by the structure factor F(h), which has amplitude F(h) and phase a(h). In the case of X-ray diffraction, F(h) is related to the electron density p(r) within the unit cell by the equation... 

A reciprocal-lattice vector can also be defined in terms of basis vectors ... 

Use of the reciprocal lattice unites and simplifies crystallographic calcnlations. The motivation for the reciprocal lattice is that the x-ray pattern can be interpreted as the reciprocal lattice with the x-ray diffraction intensities superimposed on it. See Section 14.2 for the definition of the reciprocal lattice vectors a b and c in terms of the direct basis vectors a, b, and c. Table 14.2 shows the parallel between the properties of the direct lattice and the reciprocal lattice, and Table 14.3 relates the direct and reciprocal lattices. 

Figure 2.29. The illustration of a single crystal showing the orientations of the basis vectors corresponding to both the direct (a, b and c) and reciprocal (a, b and c ) lattices and the Ewald s sphere. The reciprocal lattice is infinite in all directions but only one octant (where h>0,k>0 and / > 0) is shown for clarity.

Figure 2.40. The illustration of a reciprocal lattice vector, d /, as a vectorial sum of three basis unit vectors, a, b and c multiplied by h, k and /, respectively.

The exhaustive permutation technique may be improved by eliminating collinear reciprocal lattice vectors from the basis set, which can be done by analysis of the relationships between the observed low Bragg angle Q-values. As follows from Eq. 5.1, when two different ltd = are related to one another by a whole multiplier, there is a high probability that the two are collinear, and only the smallest is usually retained in the basis set. 

Figure 5.12. The illustration of the two-dimensional lattice with one long (b ) and one short (a ) reciprocal lattice vectors. If the three lowest Bragg angle peaks (filled circles) are selected as a basis set for indexing, all of them are collinear and only depend on a. The remaining two lattice parameters b and y ) cannot be determined from this basis set.

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