Reciprocal coordinate system

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). 

Both the components of E and the elements of the electric field gradient as given by Eqs. (8.30) and (8.32) are with respect to the reciprocal-lattice coordinate system. A transformation is required if the values in the direct-space coordinate systems are needed. To obtain the elements of the traceless V tensor, the quantity — (47t/3)pe(r) = — (47r/3K) F(H) exp ( — 27tiHT) must be subtracted from each of the diagonal elements VEU. 

An example of surfaces with tetragonal symmetry is the Cu(OOl) surface, as shown in Fig. 5.3. The top-layer nuclei form a two-dimensional square lattice on the x,y plane with lattice constant a. The origin of the coordinate system is chosen to be at one of the top-layer nuclei. The +z direction is defined as pointing into the vacuum. The reciprocal lattice is also shown in Fig. 5.3, with a lattice constant of... 

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

Figure 3 shows the plots of 1/(1 — U) vs. time. From the straight lines that start from the origin of the coordinate system used, values for ki were estimated. It can be seen easily that the values for fci are the reciprocal of the respective induction times U... 

When we rotate a contravariant nxl column vector (for position, velocity, momentum, electric field, etc.) we premultiply it by an n x n rotation tensor R. When, instead, we transform the coordinate system in which such vectors are defined, then the coordinate system and, for example, the V operator are covariant 1 x n row vectors, which are transformed by the tensor R 1 that is the reciprocal of R. A "dot product" or inner product a b must be the multiplication of a row vector a by a column vector b, to give a single number (scalar) as the result. This will be expanded further in the discussion of special relativity (Section 2.13) and of crystal symmetry (Section 7.10). 

Now we have two different coordinate systems. One, known as the laboratory axes , comprises three orthogonal axes of unit length, the directions of which are uniquely defined with respect to the axes of the diffractometer circles and the direction of the primary beam (although these conventions vary from one diffractometer type to another). A point in these coordinates is described by a vector x. The other system comprises three principal vectors of the reciprocal crystal lattice (see Section 2.1). In this system, a Bragg reflection is expressed by a vector h whose coordinates are the Miller indices hid (Section 2.2.1). The relation x = Ah between the two systems is defined by the orientation matrix (OM) A,... 

The direction of a HOLZ line is normal to the reciprocal lattice vector and its position is decided by the Bragg condition. In diffraction analysis, it is useful to express HOLZ lines using line equations in an orthogonal zone-axis coordinate system (x, y, z), with z parallel to the zone-axis direction. The x direction can be taken along the horizontal direction of the experimental pattern and y is normal to x. The Bragg diffraction equation (2) expressed in this coordinate is given by... 

Three additional concepts may be introduced by means of Fig. 2A. First, all measurements of the diffracted beams are made relative to the incident x-ray beam so that the diffracted beam leaves the crystal at an angle 20, as illustrated. Thus, while the diffraction condition is determined by the angle between the incident and diffracted beams relative to a lattice plane, the measured position of the diffracted beam is determined relative to the incident beam. Second, there is a reciprocal relation between the spacing between lattice planes and the positions of the diffraction spots small lattice spacings give diffraction spots with large values of 20. This leads to two types of space. The crystal coordinate system is in real space, whereas the diffrac-... 

Here the surface represents the probability to find the reciprocal lattice point (100) in the diffractometer coordinate system assuming Bragg-Brentano focusing geometry. The Z-axis is perpendicular to the sample, and X- and 7-axes are located in the plane of the sample. 

A standard set of reference axes and equations to describe spontaneous strains is now well established (Schlenker et al. 1978, Redfern and Salje 1987, Carpenter et al. 1998a). The orthogonal reference axes, X, Y and Z, are selected so that Y is parallel to the crystallographic y-axis, Z is parallel to the normal to the (001) plane (i.e. parallel to c ) and X is perpendicular to both. The +X direction is chosen to conform to a right-handed coordinate system. Strain is a second rank tensor three linear components, cn, 622 and 33 are tensile strain parallel to X, Y and Z respectively and co, 623, eu are shear strains in the XZ, YZ and XY planes, respectively. The general equations of Schlenker et al. (1978) define the strains in terms of the lattice parameters of a crystal (a, b, c, a, P, y, where P is the reciprocal lattice angle) with respect to the reference state for the crystal ( , bo, Co, cto, Po,Yo) ... 

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. 

Table 12-1. Points of Z)g lattice, reciprocal to Dg, split into four classes with respect to the Dg translations. The symbol e denotes an even integer and o an odd one. N.B. we use the fivefold coordinate system , see Figure 12-1 right). The authors in Refs. [2], [1], [17] use the threefold coordinate system , equivalent to the basis ej, 62, 3, 4, — 5, e - All these basis vectors, as in Figure 12-1 right) are projected in Eii, (e ) i...

Many of the complications of working in oblique coordinate systems can be simplified by the use of reciprocal basis vectors, as described in any crystallographic textbook. And, of course, they can be avoided by working, where possible, in Cartesian coordinates, based on unit vectors along mutually orthogonal directions. In this coordinate system the usual expressions for dot and cross product in terms of vector components apply ... 

Figure 3.25 The direction of the reciprocal lattice vector rj in the crystallite is specified by means of angles / and defined with reference to the coordinate system o-xyz fixed to the crystallite.

Nineteenth-century crystallography may be considered to be the mathematical branch of mineralogy. It is based on two empirical laws, the law of constancy of angle and the law of rational indices. These laws will be presented in the following pages after a discussion of some mathematical principles fundamental to crystallography, non-unitary coordinate systems and reciprocal coordinates. 

The reciprocal coordinate system was defined above (1.2). If a, b, c are the reciprocal base vectors of the vectors a,b,c, the vector r = Ha + Kh + Lc (H, K, L being coprime integers) represents the normal to the plane Hu + Kv-h Lw-= 1. Its norm is r = l/(d//KL) where is the distance of the plane from the origin, and hence the distance between consecutive planes in the same family. The reciprocal lattice is the set of vectors... 

The Pij are dimensionless numbers (as is the case for the atomic coordinates X ). The interpretation of the results of a structure determination is made easier if we choose another coordinate system based on the reciprocal lattice, ef = af/ af. The ef are unit vectors but, in general, they are not mutually perpendicular. 

It is instructive to note that the origin of these rules lies in the choice of coordinate system and that they may be derived without using structure factors. Any centered cell may be transformed into a primitive cell which, in general, does not convey the symmetry of the motif and which may thus be unsatisfactory from this point of view (Sections 1.4.1 and 2.6.1). Thus the transformation a = (a — b)/2, b = (a -h b)/2 transforms a C cell into a diamond-shaped primitive cell. The indices hkl transform in a covariant manner (Section 1.2.4), h = h — k)/2, k = h- - k)/2. As W and k must be integers according to the Laue equations, then h- -k must be an even number. The indices with h- -k odd do not correspond to a reciprocal lattice vector (Fig. 3.39). 

The axes of the unitary coordinate system are chosen in the following manner parallel to a, 02 parallel to b, 63 parallel to c. The reciprocal vectors are given in Exercise 5.4.2. The reciprocal vector a + 4c = + 4c C3 is coincident with the normal to the wave v. The... 

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