Diffraction scattering vector

For an assembly of several free point electrons, interference occurs between radiation scattered by different centers. If the incident and diffracted beams are defined by two unit vectors, s0 and s, respectively (Fig. 1.1), the phase difference of the radiation scattered by two points, separated by the vector r, equals 27tS-r, where S is the scattering vector, equal to (s — s0)/A. Vector S bisects s and s0, and has the length 2 sin 0/X. In the physics literature, an alternative notion is commonly used. The incident and diffracted beams are defined by the vectors k... 

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). 

In the diffraction pattern from a crystalline solid, the positions of the diffraction maxima depend on the periodicity of the stmcmre (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 or molecules) within the unit cell. In the case of XRD, the scattering matter is the electron density within the unit cell. Each diffraction maximum is characterized by a unique set of integers h, k and I (Miller indices) and is defined by a scattering vector H in three-dimensional... 

To get constructive interference, we have to fulfill the Bragg condition. Inserting Eq. (A.l) into Eq. (A.6) leads to q = 2it/d. In other words we observe a diffraction peak, if the scattering vector is perpendicular to any of the lattice planes and its norm is equal to... 

Let us come back to our task to find all possible diffraction peaks for a given crystal lattice. What are the possible scattering vectors that lead to constructive interference This question can be answered in an elegant way by defining the so-called reciprocal lattice If a, a2, and <23 are primitive vectors of the crystal lattice, we choose a new set of vectors according to... 

Figure A.4 The Ewald construction. Given an incident wave vector ki, a sphere of radius k, is drawn around the end point of kz. Diffraction peaks are observed only if the scattering vector q ends on this sphere.

Figure A.5 Ewald construction for surface diffraction, a) a side view of the reciprocal lattice at the surface. Constructive interference occurs for all intersection points of the vertical rods with the Ewald sphere. This is equivalent to the condition when the component qj of the scattering vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice, b) the top view of the reciprocal surface lattice. The circle is the projection of the Ewald sphere. If we disregarding the radiation scattered into the crystal, the number of lattice points within the circle (corresponding to the intersections of the rods with the Ewald sphere) is identical to the maximum number of observed diffraction peaks.

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... 

Figure 6.3 Schematic of the resolution function in the reciprocal space. Two vectors ki and kd represent the incident and the diffracted wave vectors and H represents the scattering vector. The divergence of the incident X-ray and the acceptance window of the diffracted beam side optic are represented as Suj and 6(29).

A polycrystalline thin film does not have any preferred orientation (Figure 6.4 (c)). In such a case, the diffraction from the crystal is not a spot but a so-called Debye-Scherrer ring. Therefore, the sample does not have to be inclined to obtain the diffraction pattern. Conventional 2 0-6 scans move the scattering vector H in the radial direction along the film surface normal. Thus, these scans give sufficient information when the film is polycrystalline. The obtained diffracted intensity must be corrected in terms of the absorption and the Lorentz polarization. These two terms and the obtained diffracted intensity have the following relation ... 

A typical diffraction curve is shown in Fig. 2. It gives the measured intensity from a solution as a function of the scattering vector... 

Liquids, Gases and Disordered Solids. Liquids, disordered solids, gases, and single crystals can diffract X rays. For liquids and disordered solids, where there is no long-range order, and the short-range order extends from 0 to maybe 1 or 2 nm, the diffraction consists of very broad maxima in the intensity function 7(s), where s is the scattering vector defined in Eq. (11.23.2). The one-dimensional Fourier transform of I(s) is the radial distribution function R(r) ... 

It can be shown that what is actually measured in a neutron diffraction experiment is the component of S (or J), which is normal to the scattering vector (which is in turn normal to the scattering plane, hkl), and the total diffiacted intensity is then proportional to... 

Figure 3.59. Top and side-view of the experimental set-up for grazing incidence X-ray diffraction measurements on Langmuir monolayers. The footprint of the Incident beam is indicated by the darker area. Only the crossed-beam area ABCD contributes to the detected scattering. The Seller eollimator selects a horizontal scattering angle of 20 the position-sensitive detector (PSD) has its axis vertical and measures the Bragg rod profiles. In this geometry one has the ability to determine the lateral (q ) as well as the vertieal (q ) components of the scattering vector. (Redrawn from J. Als-Nielsen and K. KJaer, loc. clt.

FIGURE 5.3 The object in this optical diffraction experiment is the five point array in (a). Its diffraction pattern is seen in (b). Each point in (a) is characterized by a position x, y, z. and each point in diffraction space (b) is characterized by a diffraction vector s and a resultant wave Fs, which produces an intensity. The diffraction pattern in (b) is simply the collection of Fs for all scattering vectors s. 

From Equation (2), we deduce that diffraction is observed only when the indices h, k, l in d take integral values. These reciprocal space vectors form a lattice, the reciprocal lattice, and the mathematical relationship between the real and reciprocal lattices (and between other aspects of the diffraction pattern) is a FT, as we will explain below. The interpretation of the Ewald construction is that diffraction is observed when the scattering vector s-s0 is equal to a reciprocal space vector A bki with integral indices h, k, l. This occurs whenever such a... 

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