Crystallography reciprocal lattice

Figure 3.7 The Ewald sphere used to construct the direction of the scattered heam. The sphere has radius 1/ X. The origin of the reciprocal lattice is O. The incident X-ray heam is labeled So and the scattered heam is labeled s. (Adapted with permission from Figure 4.19 of Drenth, J. Principles of Protein X-Ray Crystallography, 2nd ed., Springer-Verlag, New York, 1999. Copyright 1999, Springer-Verlag, New York.)...

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. 

Electron crystallography of textured samples can benefit from the introduction of automatic or semi-automatic pattern indexing methods for the reconstruction of the three-dimensional reciprocal lattice from two-dimensional data and fitting procedures to model the observed diffraction pattern. Such automatic procedures had not been developed previously, but it is the purpose of this study to develop them now. All these features can contribute to extending the limits of traditional applications such as identification procedures, structure determination etc. 

Shmueli, U. and Wilson, A. J.C. (1996). Statistical properties of the weighted reciprocal lattice. In International Tables for Crystallography, Shmueli, U., ed., Vol. B, pp. 184-200. Kluwer Academic Pubhshers, Dordrecht. 

In X-ray crystallography, 2-A model" means that analysis included reflections out to a distance in the reciprocal lattice of 1/(2 A) from the center of the diffraction pattern. This means that the model takes into account diffraction from sets of equivalent, parallel planes spaced as closely as 2 A in the unit cell. (Presumably, data farther out than the stated resolution was unobtainable or was too weak to be reliable.) Although the final 2-A map, viewed as an empty contour surface, may indeed not allow us to discern adjacent atoms, structural constraints on the model greatly increase the precision of atom positions. The main constraint is that we know we can fit the map with groups of atoms — amino-acid residues — having known connectivities, bond lengths, bond angles, and stereochemistry. 

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 concept of the reciprocal lattice (see, e.g., Aschroft and Mermin 1976 Woolfson 1997) is central to understanding crystallography. Here it may suffice to remind the reader that each point in the reciprocal lattice refers to a given set of lattice planes of the crystal (in real space), with a well-defined orientation and spacing. A reciprocal lattice vector H is normal to the corresponding plane in real space and its length is the reciprocal value of the interplanar spacing. 

As can be seen from Eqs. 4.11 to 4.13, the knowledge of the Miller indices is crucial for the understanding of two-dimensional correlated strucmres. A simplistic method to determine the Miller indices of a given scattering pattern is to superimpose this pattern with a lattice. The lattice is applicable if reflexes only occur on points of intersection. If this condition is fulfilled, the position of the reflex in the lattice corresponds to its Miller indices. This superimposed lattice is called the reciprocal lattice and has a deeper meaning in crystallography. 

Reciprocal Lattice. A mathematical construct to express elegantly the conditions for an X-ray beam to be diffracted by a crystal lattice. If the directions of the crystal lattice are given by vectors a, b and c, then the reciprocal lattice vectors a, b and c are such that a is perpendicular to the plane of b and c, and a. a = 1, etc. Then a vector r from the origin to a point (h, k, 1) of the reciprocal lattice is perpendicular to the plane (h, k, 1) of the crystal lattice, and its length is the reciprocal of the spacing of the (h, k, 1) planes of the crystal lattice. Every point in the reciprocal lattice corresponds to a possible X-ray reflection from the crystal lattice. See also X-RAY CRYSTALLOGRAPHY. 

A construction due to Ewald illustrates the importance of the reciprocal lattice in X-ray crystallography. As Figure 4 shows, the Bragg equation is satisfied where the reflection sphere is cut by a lattice point of the reciprocal lattice constructed around the center of the cry.stal. Rotating the crystal together with the reciprocal lattice around a few different directions in the crystal fulfills the reflection condition for all points of the reciprocal space within the limiting sphere. The reciprocal lattice vector S is perpendicular to the set of net planes, and has the absolute magnitude 1/d. In vector notation ... 

Before we can get as far as discussing the adaptations we must make to the calculations, we must introduce two concepts commonly used in crystallography and materials science the reciprocal lattice and the Brillouin zone. If you are not already familiar with the basic concept of the crystallographic unit ceU, you should read Section 10.5 and its sub-sections before continuing with this section. 

Figure 2.43. Comparison between a real monoclinic crystal lattice (a = b c) and the corresponding reciprocal lattice. Dashed lines indicate the unit cell of each lattice. The magnitudes of the reciprocal lattice vectors are not in scale for example, la l = 1/dioo, lc l = 1/dooi, IGioil = dioi, etc. Note that fra-orthogonal unit cells (cubic, tetragonal, orthorhombic), the reciprocal lattice vectras will be aligned parallel to the real lattice vectors. 2009 From Biomolecular Crystallography Principles, Practice, and Application to Structural Biology by Bernard Rupp. Reproduced by permission of Garland Science/ Taylor Francis Group LLC.

To that we would add the reciprocal lattices, fc-space, and Brillouin zones of crystallography. Kittel (1966) says The crystal lattice is a lattice in real or ordinary space the reciprocal lattice is a lattice in Fourier space , and It is as important to be able to visualize the reciprocal lattice as it is to visualize the real crystal lattice. ... 

Here /i g(u) is the Fourier components of the image intensity function. u is the lattice vector in reciprocal space. F (u) is the Fourier spectmm of the projected potential y>,(x, y). In crystallography, F (u) is the structure factor of the crystal. 

Lattice. In crystallography, we encounter the term lattice in the context of both real space and reciprocal space. The real-space lattice (usually just referred to as the crystal lattice) is a conceptually infinite array of points that describes the positional and orientational... 

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