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Xray Diffraction and Absorption Xray Di®raction and Absorption Experiment XDA University of Florida | Department of Physics PHY4803L | Advanced Physics Laboratory Objective cesses simultaneously produce xrays. First, the deceleration of beam electrons from colli- The di®raction of xrays from single crystals sions with the target produce a broad contin- and powdered crystalline samples will be in- uum of radiation called bremsstrahlung (brak- vestigated along with the emission and absorp- ing radiation) having a short wavelength limit tion of xrays by high Z metals. The di®rac- that arises because the energy of the photon tion patterns show sharp maxima (peaks) at hc=¸ can be no larger than the kinetic energy characteristic angles that depend on the wave- of the electron. Second, beam electrons can length of the xrays and the structure of the knock atomic electrons in the target out of in- crystal. A Geiger-MÄullertube is used to de- ner shells. When electrons from higher shells tect the intensity of the di®racted xrays versus fall into the vacant inner shells, a series of dis- scattering angle. Crystal properties are deter- crete xrays lines characteristic of the target mined from the angular position and intensity material are emitted. of the peaks. In our machine, which has a copper target, only two emission lines are of appreciable in- References tensity. Copper K® xrays (¸ = 0:1542 nm) Generally look in the QC481 and QD945 sec- are produced when an n = 2 electron makes tions. a transition to a vacancy in the n = 1 shell. A weaker K¯ xray with a shorter wavelength A. H. Compton and S. K. Allison, Xrays in (¸ = 0:1392 nm) occurs when the vacancy is Theory and Experiment ¯lled by an n = 3 electron. The reverse process of xray absorption by C. Kittel, Introduction to Solid State Physics an atom also occurs if the xray has either B. D. Cullity, Elements of Xray Di®raction an energy exactly equal to the energy di®er- ence between an energy level occupied by an Teltron manual, The Production, Properties, atomic electron and a vacant upper energy and Uses of Xrays level, or an energy su±cient to eject the atomic electron (ionization). For the xray energies Xray Emission and Absorption and metals considered in this experiment, the ionization of a K-shell electron is the domi- When an electron beam of energy around nant mechanism when the xray energy is high 20 kV strikes a metal target, two di®erent pro- enough, which leads to a complete absorption XDA 1 XDA 2 Advanced Physics Laboratory JODVVÃGRPH [UD\ÃWXEH 2θθθ θθθ θθθ d θθθ ÃPPÃVOLW &XÃDQRGH VDPSOH WDEOH FOXWFK d sin θθθ VFULEH SODWH OLQH FU\VWDO *0ÃWXEH θ GHWHFWRU Figure 1: The ray reflected from the second plane ÃPPÃVOLW DUP must travel an extra distance 2d sin θ. ÃPPÃVOLW of the initial xray photon and the ejection of 1 θ an electron|a process known as the photoelec- tric e®ect). If an xray does not have enough energy to cause a transition or to ionize an Figure 2: The xray di®raction apparatus. atom, the only available energy loss mecha- nism is Compton scattering. Constructive interference of the reflected waves occurs when this distance is an integral of the wavelength. The Bragg condition for The Xray Di®ractometer the angles of the di®raction peaks is thus: Thus, the spectrum of xrays from an xray n¸ = 2d sin θ (1) tube consists of the discrete lines superim- posed on the bremsstrahlung continuum. This where n is an integer called the order of di®rac- spectrum can be analyzed in much the same tion. Note also that the lattice planes, i.e., the way that a visible spectrum is analyzed using crystal, must be properly oriented for the re- a grating. Because xrays have much smaller flection to occur. This aspect of xray di®rac- wavelengths than visible light, the grating tion is sometimes used to orient single crystals spacing must be much smaller. A single crys- and determine crystal axes. tal with its regularly spaced, parallel planes Our apparatus is shown schematically in of atoms is often used as a grating for xray Fig. 2. The xrays from the tube are collimated spectroscopy. to a ¯ne beam (thick line) and reflect from the The incident xray wave is reflected spec- crystal placed on the sample table. The de- ularly (mirror-like) as it leaves the crystal tector, a Geiger-MÄuller(GM) tube, is placed planes, but most of the wave energy continues behind collimating slits on the detector arm through to subsequent planes where additional which can be placed at various scattering an- reflected waves are produced. Then, as shown gles 2θ. In order to obey the Bragg condition, in the ray diagram of Fig. 1 where the plane the crystal must rotate to an angle θ when the spacing is denoted d, the path length di®er- detector is at an angle 2θ. This θ : 2θ relation- ence for waves reflected from successive planes ship is maintained by gears under the sample is 2d sin θ. Note that the scattering angle (the table. angle between the original and outgoing rays) Assuming d is known (from tables of crystal is 2θ. spacings), the wavelength ¸ of xrays detected February 12, 2007 Xray Di®raction and Absorption XDA 3 at a scattering angle 2θ can be obtained from Eq. 1. Powder Di®raction An ideal crystal is an in¯nite, 3-dimensional, periodic array of identical structural units. The periodic array is called the lattice. Each y point in this array is called a lattice point. ( y The structural unit|a grouping of atoms or ( molecules|attached to each lattice point is y ( called the basis and is identical in composi- tion, arrangement, and orientation. A crystal thus consists of a basis of atoms at each lattice Figure 3: The simple cubic lattice points (dots) point. This can be expressed: and connecting lines showing the cubic structure. lattice + basis = crystal (2) points at the same positions as those of the The general theoretical treatment for the sc lattice but it has additional lattice points determination of the di®racted xrays|their at the center of each unit square (cube face) angles and intensities|was ¯rst derived by de¯ned by the grid. Laue. Starting with Huygens' principal, it in- volves constructions such as the Fourier trans- A primitive unit cell is a volume contain- form of the crystal's electron distribution and ing a single lattice point that when suitably develops the concept of the reciprocal lattice. arrayed at each lattice point completely ¯lls The interested student is encouraged to ex- all space. The number of atoms in a primi- plore the Laue treatment. (See Introduction to tive unit cell is thus equal to the number of Solid State Physics, by Charles Kittel.) How- atoms in the basis. (There are many ways of ever, we will only explore crystals that can be choosing a primitive unit cell.) The primitive described with a cubic lattice for which a sim- unit cell for the sc lattice is most conveniently pler treatment is su±cient. taken as a cube of side a0. a0 is called the There are three types of cubic lattices: lattice spacing or lattice constant. the simple cubic (sc), the body-centered cu- The bcc and fcc primitive unit cells (rhom- bic (bcc) and the face-centered cubic (fcc). bohedra) will not be used. Instead, bcc and The simple cubic has lattice points equally fcc crystals will be treated using the sc lat- spaced on a three dimensional Cartesian grid tice and sc unit cell (which would then not be as shown by the dots in Fig. 3. As a viewing primitive). aid the lattice points are connected by lines A crystal is completely described by speci- showing the cubic nature of the lattice. fying the lattice (sc, bcc, or fcc) and the po- The body-centered cubic lattice has lattice sition of each atom in the basis relative to a points at the same positions as those of the single lattice point. For the sc lattice, the basis sc lattice and additional lattice points at the atom positions are speci¯ed by their relative center of each unit cube de¯ned by the grid. Cartesian coordinates (u; v; w) within the sc The face-centered cubic lattice also has lattice unit cell. u, v, and w are given in units of the February 12, 2007 XDA 4 Advanced Physics Laboratory lattice spacing a0, so that, for example, the turns out that the spacings for all possible lat- body-centered position would be described by tice planes in the sc lattice can be represented the relative coordinates (u; v; w) = ( 1 ; 1 ; 1 ). by 2 2 2 a When a sc lattice is used to describe a bcc d(hkl) = np 0 (3) or fcc crystal, one still speci¯es the position h2 + k2 + l2 of atoms using relative coordinates (u; v; w). where hkl are integers{called the Miller in- But all atoms within the sc unit cell must be dices and n is an integer. Together with Eq. 1 given|including the extra basis atoms that this leads to the Bragg scattering relation giv- would occur at the body-centered or face- ing the angular position of the Bragg peaks centered positions, respectively. 2a ¸ = p 0 sin θ: (4) If the unit cell is taken so that its corners h2 + k2 + l2 are at lattice points, each corner lattice point The Miller indices are used to classify the is shared among the eight cells that meet at possible reflections as shown in Fig.
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