Space coordinates, reciprocal

The magnitude of k corresponds to a wave number 2n/X and therefore is measured with a unit of reciprocal length. For this reason k is said to be a vector in a reciprocal space or k space . This is a space in a mathematical sense, i.e. it is concerned with vectors in a coordinate system, the axes of which serve to plot kx, ky and kz. The directions of the axes run perpendicular to the delimiting faces of the unit cell of the crystal. 

Because the orientation of the reciprocal space coordinate system is rigidly coupled to the orientation of the real-space coordinate system of the sample, the reciprocal space can be explored8 by tilting and rotating the sample in the X-ray beam (cf. Chap. 9). 

The indexing proeedure of any diffraction pattern requires the knowledge of positions of some reflections on the pattern. The relations between the position of each peak on a two-dimensional diffraction pattern and the corresponding point in reciprocal space can be established only after a successful indexing. The combined use of the reciprocal coordinates and the determined peak-shape are essential for extracting integrated intensities from diffraction data. 

Firstly, coordinates of relatively heavy atoms in the examined crystal are read out from the deconvoluted image, hence the partial structure factor Fp(u) containing only the contribution from those heavy atoms in one unit cell are calculated. Secondly, divide the reciprocal space into a number of circular zones and the intensity of each beam within the /-th zone is corrected as... 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

A convenient data collection strategy is to perform a loop scan in which the specimen is set at a particular angle and then a -2 scan is performed. The specimen position is then incremented and another -2 or longitudinal scan is performed. In this way, one collects data in a square grid in reciprocal space. Appropriate software is then nsed to convert the data to reciprocal space coordinates nsing the eqnations given in the last section. 

Fig. 13a and b. Intensity contour maps around the 5.9-nm and 5.1-nm actin layer lines (indicated by arrows) a resting state b contracting state. Z is the reciprocal-space axial coordinate from the equator. M5 to M9 are myosin meridional reflections indexed to the fifth to ninth orders of a 42.9-nm repeat, (c) intensity profiles (in arbitrary units) of the 5.9- and 5.1-nm actin reflections. Dashed curves, resting state solid curves, contracting state. Intensity distributions were measured by scanning the intensity data perpendicular to the layer lines at intervals of 0.4 mm. The area of the peak above the background was adopted as an integrated intensity and plotted as a function of the reciprocal coordinate (R) from the meridian... 

Coordinate system Description Components of direct space vector Components of reciprocal space vector... 

Each reflection can be assigned three coordinates or indices in the imaginary three-dimensional space of the diffraction pattern. This space, the strange land where the reflections live, is called reciprocal space. Crystallographers usually use h, k, and l to designate the position of an individual reflection in the reciprocal space of the diffraction pattern. The central reflection (the round solid spot at the center of the film in Fig. 2.11) is taken as the origin in reciprocal... 

That T is a series of Bessel rather than trigonometric functions is merely a consequence of using cylindrical polar coordinates (r, j, cz ) for atoms in real space and (R, iji, i/a for points in reciprocal space. Not only is this a convenient framework for describing a helical molecule, but it can lead to economies in computing T. For helices, only Bessel terms with... 

Figure 1. Reciprocal space coordinates (R, i/<, 1) are the cylindrical polar coordinates...

Figure 1. Relationship between reciprocal space position vector D and the reciprocal space coordinates (1)...

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

Figure 10. Ewald sphere construction showing the relationship between the reciprocal space position vector D, the reciprocal space coordinates (R,Z), and the angles n and which define the direction of a scattered ray...

Structure solution in powder diffraction is approached by two different methodologies. One is using the conventional reciprocal space methods. The second is by real space methods where all the known details about the sample (say, molecular details such as bond distances, angles, etc., for an organic molecule, and coordination spheres such as octahedral, tetrahedral etc., in case of inorganic compounds) in question are exploited to solve the structure. 

Here Mj is the Madelung constant based on I as unit distance, n is the number of molecules in the unit cell, zy is the charge number of atom j, V is the volume of the unit cell and h is the magnitude of the vector (hi, h2, ha) in reciprocal space or the reciprocal of the spacing of the planes (hihjha). The coordinates of atom j are a i/, X2, Xay. The sums over j are taken over all the atoms in the imit cell. F(h) is the Fourier transform of the Patterson function and (h) is the Fourier transform of the charge distribution /(r). F h) is given by... 

For every family of planes having integral Miller indexes hkl, a vector can be drawn from a common origin having the direction of the plane normal and a length 1 /d, where d is the perpendicular distance between the planes. The coordinate space in which these vectors are gathered, as in Figure 3.19, is called reciprocal space, and the end points of the vectors for all of the families of planes form a lattice that is termed the reciprocal lattice. 

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. 

Remember further that each reciprocal lattice point represents a vector, which is normal to the particular family of planes hkl (and of length 1 /d u) drawn from the origin of reciprocal space. If we can identify the position in diffraction space of a reciprocal lattice point with respect to our laboratory coordinate system, then we have a defined relationship to its family of planes, and the reciprocal lattice point tells us the orientation of that family. In practice, we usually ignore families of planes during data collection and use the reciprocal lattice to orient, impart motion to, and record the three-dimensional diffraction pattern from a crystal. Note also that if we identify the positions of only three reciprocal lattice points, that is, we can assign hkl indexes to three reflections in diffraction space, then we have defined exactly the orientation of both the reciprocal lattice, and the real space crystal lattice. 

We can see the diffraction pattern with our own eyes when we collect X-ray data because we obtain the image, the pattern of diffraction spots, on the face of our detector or film. We can t directly see the families of planes in the actual crystal, but we know, through the Ewald construction, how the diffraction pattern is related to the crystal orientation, and hence to the dispositions of the planes that pass through it. We also know from Ewald how to move the crystal about its center, once we know its orientation with respect to our laboratory coordinate system, in order to illuminate various parts of reciprocal space. In data collection we watch the diffraction pattern, not the crystal, and let the pattern of intensities guide us. 

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