Reciprocal Lattice and X-Ray Diffraction

Figure 7.16. Illustration of Ewald sphere construction, and diffraction from reciprocal lattice points. This holds for both electron and X-ray diffraction methods. The vectors AO, AB, and OB are designated as an incident beam, a diffracted beam, and a diffraction vector, respectively.

To determine if the C o molecules form a regular lattice, we performed electron and X-ray diffraction studies on the individual crystals and on the powder. A typical X-ray diffraction pattern of the C o powder is shown in Fig. 3. To aid in comparing the electron diffraction results with the X-ray results we have inset the electron diffraction pattern in Fig. 3. From the hexagonal array of diffraction spots indexed as shown in the figure, a d spacing of 8.7 A was deduced corresponding to the (100) reciprocal lattice vector of a hexagonal lattice. The... 

The study of X-ray diffraction in thin films is essentially a onedimensional problem and so we have been able to avoid any awkward geometry in that case. This is not true for the case of electron diffraction and it is thus worthwhile introducing the reciprocal lattice explicitly. We return once more to Equation (2.3)... 

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

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... 

The lattice wave numbers defined in Eq. (16-16) arc the reciprocal lattice vectors familiar in diffraction theory. Only if the change in wave number resulting from the diffraction is equal to a lattice wave number can a wave, whether it be an X-ray or an electron, be diffracted otherwise the wavelets scattered by the different ions interfere with each other and reduce the diffracted intensity to zero. Only for diffraction by q equal to a lattice wave number do the scattered waves add in phase. Thus a wave having wave number k can only be diffracted to final states of wave number k that can be written as k = k -I- q, where q is a lattice wave number. Furthermore, the diffracted wave will have the same frequency as the incident wave if it is an X-ray, or the same energy if it is an electron, from which it follows that k = k. Combining the two conditions, k -I- qp = gives the Bragg condition for diffraction. 

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. 

A three-dimensional lattice, reciprocal to the crystal lattice, is very useful in analyses of X-ray diffraction patterns it is called the reciprocal lattice. Earlier in this chapter the diffraction pattern of a series of regularly spaced slits was considered to be composed of an envelope profile, the diffraction pattern of a single slit, and sampling regions, ... 

FIGURE 7.1. The relative orientations of the reciprocal lattice of a crystal (expressed as a and b ), and its indexed X-ray diffraction pattern (expressed as h and k). In the diffraction pattern the intensities of the diffracted beams (/) (the blackness of spots on X-ray film, for example) and the directions of travel (sin 6) (positions of spots on the X-ray film) are measured. Note the relationship of a to h, and b to k. From the positions of spots on the photographic film it is possible to deduce the dimensions of the reciprocal lattice, hence of the crystal lattice, hence the indices hkl of each Bragg reflection. 

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