{"id":58,"date":"2018-11-30T10:24:04","date_gmt":"2018-11-30T10:24:04","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=58"},"modified":"2018-11-30T10:37:10","modified_gmt":"2018-11-30T10:37:10","slug":"elastic-interaction-of-x-rays-with-matter","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/elastic-interaction-of-x-rays-with-matter\/","title":{"rendered":"Elastic Interaction of X-rays with Matter"},"content":{"raw":"<div>\r\n\r\n<strong>\u00a0 \u00a0 Learning Outcomes<\/strong>\r\n\r\n&nbsp;\r\n\r\nAfter studying this module, you shall be able to\r\n<ul>\r\n \t<li>Define lattice planes and miller indices.<\/li>\r\n \t<li>Understand Bragg diffraction and x-ray diffraction technique<\/li>\r\n \t<li>Index x-ray diffraction peaks<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Elastic Scattering from a perfect lattice:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">X-rays are scattered elastically by electrons, which is named Thomson scattering. In this process the electron oscillates like a Hertz dipole at the frequency of the incoming beam and becomes a source of dipole radiation. The wavelength \u03bb of x-rays is conserved for Thomson scattering in contrast to the two inelastic scattering processes mentioned above. It is the Thomson component in the scattering of x-rays that is made use of in structural investigations by x-ray diffraction.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Materials are made of atoms. Knowledge of how atoms are arranged into crystal structures and microstructures is the foundation on which we build our understanding of the synthesis, structure and properties of materials. In a day to day work we talk about x-ray reflections from a series of parallel planes inside the crystal. The orientation and inter-planner spacing of these planes are defined by the three integers h, k, l called Miller indices. A given set of planes with indices h, k and l cut the a-axis of the unit cell in h sections, the b-axis in k sections and the c-axis in l sections. A zero indicates that the planes are parallel to the corresponding axis. For example the (220) planes cut the a-axis and the b-axis in half, but are parallel to c-axis. The procedure employed in determination of the <em>h<\/em>, <em>k<\/em>, and <em>l<\/em> index numbers is as follows:<\/p>\r\n\r\n<ol>\r\n \t<li style=\"text-align: justify\">If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell.<\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">At this point the crystallographic plane either intersects or parallels each of the three axes; the length of the planar intercept for each axis is determined in terms of the lattice parameters <\/span><em style=\"text-align: initial;font-size: 1em\">a<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">b<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">c<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have an infinite intercept, and, therefore, a zero index.<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If necessary, these three numbers are changed to the set of smallest integers by multiplication or division by a common factor.<\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Finally, the integer indices, not separated by commas, are enclosed within parentheses, thus: (<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">).<\/span><\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">\u00a0 \u00a0An intercept on the negative side of the origin is indicated by a bar or minus sign positioned over the appropriate index. Furthermore, reversing the directions of all indices specifies another plane parallel to, on the opposite side of and equidistant from, the origin. Mirror indices of different planes are shown in the figure below (Figure 4.1).<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-62\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22.png\" alt=\"\" width=\"693\" height=\"549\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Figure 4.1: Representation of Miller Indices<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Construct a ( 0 1 1) plane within a cubic unit cell. To solve this problem, carry out the procedure used in the preceding example in reverse order. To begin, the indices are removed from the parentheses, and reciprocals are taken, which yields \u00a5, -1, and 1.This means that the particular plane parallels the <em>x<\/em> axis while intersecting the <em>y<\/em> and <em>z<\/em> axes at \u2013<em>b<\/em> and <em>c<\/em>, respectively, as indicated in the accompanying sketch of figure 4.2 .This plane has been drawn in another sketch in figure 4.2. A plane is indicated by lines representing its intersections with the planes that constitute the faces of the unit cell or their extensions. For example, in this figure, line <em>ef<\/em> is the intersection between the ( 0 1 1) plane and the top face of the unit cell; also, line <em>gh<\/em> represents the intersection between this same ( 0 1 1) plane and the plane of the bottom unit cell face extended. Similarly, lines <em>eg<\/em> and <em>fh<\/em> are the intersections between ( 0 1 1) and back and front cell faces, respectively.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-63\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23.png\" alt=\"\" width=\"755\" height=\"306\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: center\">Figure 4.2: Construct a (011) plane within a cubic unit cell.<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are many different methods for measuring structure across this wide range of distances, but the more powerful experimental techniques involve diffraction. To date, most of our knowledge about the spatial arrangements of atoms in materials has been gained from diffraction experiments. In a diffraction experiment, an incident wave is directed into a material and a detector is typically moved about to record the directions and intensities of the outgoing diffracted waves.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Diffraction effects are observed when electromagnetic radiation impinges on periodic structures with geometrical variations on the length scale of the wavelength of the radiation. The inter-atomic distances in crystals and molecules amount to 0.15\u20130.4 nm which correspond in the electromagnetic spectrum with the wavelength of x-rays having photon energies between 3 and 8 keV. Accordingly, phenomena like constructive and destructive interference should become observable when crystalline and molecular structures are exposed to x-rays.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Coherent scattering preserves the precision of wave periodicity. Constructive or destructive interference then occurs <\/span>along<span style=\"text-align: initial;font-size: 1em\"> different directions as scattered waves are emitted by atoms of different types and positions. There is a profound geometrical relationship between the directions of waves that interfere constructively, which comprise the \u201cdiffraction pattern,\u201d and the crystal structure of the material. The diffraction pattern is a spectrum of real space periodicities in a material. Atomic periodicities with long repeat distances cause diffraction at small angles, while short repeat distances cause diffraction at high angles. It is not hard to appreciate that diffraction experiments are useful for determining the crystal structures of materials. Much more information about a material is contained in its diffraction pattern, however. Crystals with precise periodicities over long distances have sharp and clear diffraction peaks. Crystals with defects such as impurities, dislocations, planar faults, internal strains, or small precipitates are less precisely periodic in their atomic arrangements, but they still have\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">distinct diffraction peaks. Their diffraction peaks are broadened, distorted, and weakened, however, and \u201cdiffraction line shape analysis\u201d is an important method for studying crystal defects. Diffraction experiments are also used to study the structure of amorphous materials, even though their diffraction patterns lack sharp diffraction peaks.<\/span><\/p>\r\n\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-64\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-24.png\" alt=\"\" width=\"290\" height=\"266\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">Fig: 4.3 Bragg\u2019s X-ray diffraction<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">An often-used instrument for measuring the Bragg reflection of a thin film is the \u03b8\/2\u03b8 diffractometer. Let us introduce its operation principle by considering the results obtained with the question in mind as to how x-ray scattering experiments are preferably facilitated. What we are interested in is the measurement of Bragg reflections, i.e. their position, shape, intensity, etc., in order to derive microstructural information from them. The intensity variation that is associated with the reflection is included in the interference function, while the scattered intensity depends on the distance from the sample to the detection system <strong>R<\/strong>. We therefore should configure the instrument such that we can scan the space around the sample by keeping the sample\u2013detector distance <strong>R<\/strong> constant. This measure ensures that any intensity variation observed is due to the interference function and is not caused by a dependency on <strong>R<\/strong>. The detector should accordingly move on a sphere of constant radius <strong>R<\/strong> with the sample in the center of it. In addition, the sphere reduces to a hemisphere above the sample, since we are only interested in the surface layer and data collection will be performed in reflection mode. The geometry is shown in Fig. 4.3. Because the scattering of x-rays depends sensitively on the orientation of the crystal with respect to the scattering vector, we carefully have to define the various coordinate systems with which we are dealing.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-65\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-25.png\" alt=\"\" width=\"226\" height=\"157\" \/>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Figure 4.4: Schematic representation of a \u03b8\/2\u03b8 scan from the viewpoint of the sample reference frame.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">K0 is the incident wave vector and K is the reflected wave vector.<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The inter-planar spacing, d, sets the difference in path length for the ray scattered from the top plane and the ray scattered from the bottom plane. Figure 4.3 shows that this difference in path lengths is 2d sin \u03b8. Constructive wave interference occurs when the difference in path length for the top and bottom rays is equal to one wavelength, \u03bb:<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-66\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-26.png\" alt=\"\" width=\"197\" height=\"24\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The right hand side is multiplied by an integer, <em>n<\/em>, since this condition also provides constructive interference. For first order diffraction, <em>n<\/em> = 1. When there is a path length difference of <em>n<\/em>\u03bb between adjacent planes, we changed even though this new d may not correspond to a real interatomic distance. For example, when our diffracting planes are (100) cube faces, and<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-67\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-27.png\" alt=\"\" width=\"198\" height=\"29\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">then we speak of a (200) diffraction from planes separated by d200 = (d )\/2. A diffraction pattern from a material typically contains many distinct peaks, each corresponding to a different inter-planar spacing, <em>d<\/em>. For cubic crystals with lattice parameter<em> a<\/em><em>0<\/em>, the inter-planar spacing,<em> d<\/em>, of planes labeled by Miller indices (hkl) are<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-68\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-28.png\" alt=\"\" width=\"200\" height=\"38\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">which can be proved by the definition of Miller indices and the 3D Pythagorean theorem. From Bragg\u2019s law (eqn 2) we find that the (hkl) diffraction peak occurs at the measured angle 2\u03b8hkl:<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-69\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-29.png\" alt=\"\" width=\"257\" height=\"42\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are often many individual crystals of random orientation in the sample, so all possible Bragg diffractions can be observed in the \u201cpowder pattern.\u201d There is a convention for labeling, or \u201cindexing,\u201d the different Bragg peaks in a powder diffraction pattern using the numbers (<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">). An example of an indexed diffraction pattern is shown in Fig. 4.5. The intensities of the different diffraction peaks vary <\/span>widely,<span style=\"text-align: initial;font-size: 1em\"> and are zero for some combinations of <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">. For this example of polycrystalline silicon, notice the absence of all combinations of <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\"> that are mixtures of even and odd integers, and the absence of all even integer combinations whose sum is not divisible by 4.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-70\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-30.png\" alt=\"\" width=\"235\" height=\"174\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">Fig: 4.5 Indexed powder x-ray diffraction pattern of polycrystalline silicon.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">One important use of x-ray powder diffractometry is for identifying unknown crystals in a sample. The idea is to match the positions and the intensities of the peaks in the observed diffraction pattern to a known pattern of peaks from a standard sample or from a calculation. There should be a one-to-one correspondence between the observed peaks and the indexed peaks in the candidate diffraction pattern. For a simple diffraction pattern as in figure 4.5, this tentative indexing still needs to be checked. To do so, the \u03b8 -angles of the diffraction peaks are obtained, and used with (eqn 3) to obtain the interplanar spacing for each diffraction peak. For cubic crystals it is then possible to use (eqn 4) to convert each interplanar spacing into a lattice parameter. The indexing is consistent if all peaks provide the same lattice parameters.<\/p>\r\n&nbsp;\r\n\r\nNon-cubic crystals usually require an iterative refinement of lattice parameters and angles. The relation between lattice constants and miller indices for non-cubic symmetry is given below in table 4.1\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Table 4.1:<\/span>\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-71 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31.png\" alt=\"\" width=\"450\" height=\"283\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Value Addition:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Do You Know?<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Bragg diffraction (also referred to as the <strong>Bragg formulation of X-ray diffraction<\/strong>) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">William Lawrence Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter <em>d<\/em>. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively. The interference is constructive when the phase shift is a multiple of 2 \u03c0; this condition can be expressed by Bragg's law and was first presented by Sir William Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society. Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond. They are the only father-son team to jointly win. William Lawrence Bragg was 25 years old, making him then, the youngest physics Nobel laureate.<\/p>\r\n&nbsp;\r\n\r\n<strong>1.\u00a0<\/strong><strong>Suggested Reading<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>For More Details (on this topic and othe r topics discussed in Text Module) See<\/strong>\r\n<ol>\r\n \t<li style=\"text-align: justify\">Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks\/Cole, Eastern Press Bangalore (India) 2005<\/li>\r\n \t<li>Charles Kittel, Introduction to Solid State Physics, John Wiley &amp; Sons, Singapore 1999<\/li>\r\n \t<li>X-ray Absorption Spectroscopy, J. E. Penner-hahn<span style=\"text-align: initial;font-size: 1em\">, MI, USA Page 159-186)<\/span><\/li>\r\n<\/ol>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Glossary:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Elastic Scattering:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">When a particle of radiation does not share energy with the matter in the scattering process it is called inelastic scattering. The outgoing particle may come out with <\/span>same<span style=\"text-align: initial;font-size: 1em\"> energy as its initial energy.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">XRD<\/strong><span style=\"text-align: initial;font-size: 1em\">: X-ray Diffraction<\/span><\/p>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div>\n<p><strong>\u00a0 \u00a0 Learning Outcomes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>After studying this module, you shall be able to<\/p>\n<ul>\n<li>Define lattice planes and miller indices.<\/li>\n<li>Understand Bragg diffraction and x-ray diffraction technique<\/li>\n<li>Index x-ray diffraction peaks<\/li>\n<\/ul>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Elastic Scattering from a perfect lattice:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">X-rays are scattered elastically by electrons, which is named Thomson scattering. In this process the electron oscillates like a Hertz dipole at the frequency of the incoming beam and becomes a source of dipole radiation. The wavelength \u03bb of x-rays is conserved for Thomson scattering in contrast to the two inelastic scattering processes mentioned above. It is the Thomson component in the scattering of x-rays that is made use of in structural investigations by x-ray diffraction.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Materials are made of atoms. Knowledge of how atoms are arranged into crystal structures and microstructures is the foundation on which we build our understanding of the synthesis, structure and properties of materials. In a day to day work we talk about x-ray reflections from a series of parallel planes inside the crystal. The orientation and inter-planner spacing of these planes are defined by the three integers h, k, l called Miller indices. A given set of planes with indices h, k and l cut the a-axis of the unit cell in h sections, the b-axis in k sections and the c-axis in l sections. A zero indicates that the planes are parallel to the corresponding axis. For example the (220) planes cut the a-axis and the b-axis in half, but are parallel to c-axis. The procedure employed in determination of the <em>h<\/em>, <em>k<\/em>, and <em>l<\/em> index numbers is as follows:<\/p>\n<ol>\n<li style=\"text-align: justify\">If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell.<\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">At this point the crystallographic plane either intersects or parallels each of the three axes; the length of the planar intercept for each axis is determined in terms of the lattice parameters <\/span><em style=\"text-align: initial;font-size: 1em\">a<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">b<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">c<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have an infinite intercept, and, therefore, a zero index.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If necessary, these three numbers are changed to the set of smallest integers by multiplication or division by a common factor.<\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Finally, the integer indices, not separated by commas, are enclosed within parentheses, thus: (<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">).<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify\">\u00a0 \u00a0An intercept on the negative side of the origin is indicated by a bar or minus sign positioned over the appropriate index. Furthermore, reversing the directions of all indices specifies another plane parallel to, on the opposite side of and equidistant from, the origin. Mirror indices of different planes are shown in the figure below (Figure 4.1).<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-62\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22.png\" alt=\"\" width=\"693\" height=\"549\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22.png 693w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22-300x238.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22-225x178.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-22-350x277.png 350w\" sizes=\"auto, (max-width: 693px) 100vw, 693px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">Figure 4.1: Representation of Miller Indices<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Construct a ( 0 1 1) plane within a cubic unit cell. To solve this problem, carry out the procedure used in the preceding example in reverse order. To begin, the indices are removed from the parentheses, and reciprocals are taken, which yields \u00a5, -1, and 1.This means that the particular plane parallels the <em>x<\/em> axis while intersecting the <em>y<\/em> and <em>z<\/em> axes at \u2013<em>b<\/em> and <em>c<\/em>, respectively, as indicated in the accompanying sketch of figure 4.2 .This plane has been drawn in another sketch in figure 4.2. A plane is indicated by lines representing its intersections with the planes that constitute the faces of the unit cell or their extensions. For example, in this figure, line <em>ef<\/em> is the intersection between the ( 0 1 1) plane and the top face of the unit cell; also, line <em>gh<\/em> represents the intersection between this same ( 0 1 1) plane and the plane of the bottom unit cell face extended. Similarly, lines <em>eg<\/em> and <em>fh<\/em> are the intersections between ( 0 1 1) and back and front cell faces, respectively.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-63\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23.png\" alt=\"\" width=\"755\" height=\"306\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23.png 755w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23-300x122.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23-65x26.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23-225x91.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-23-350x142.png 350w\" sizes=\"auto, (max-width: 755px) 100vw, 755px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: center\">Figure 4.2: Construct a (011) plane within a cubic unit cell.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are many different methods for measuring structure across this wide range of distances, but the more powerful experimental techniques involve diffraction. To date, most of our knowledge about the spatial arrangements of atoms in materials has been gained from diffraction experiments. In a diffraction experiment, an incident wave is directed into a material and a detector is typically moved about to record the directions and intensities of the outgoing diffracted waves.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Diffraction effects are observed when electromagnetic radiation impinges on periodic structures with geometrical variations on the length scale of the wavelength of the radiation. The inter-atomic distances in crystals and molecules amount to 0.15\u20130.4 nm which correspond in the electromagnetic spectrum with the wavelength of x-rays having photon energies between 3 and 8 keV. Accordingly, phenomena like constructive and destructive interference should become observable when crystalline and molecular structures are exposed to x-rays.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Coherent scattering preserves the precision of wave periodicity. Constructive or destructive interference then occurs <\/span>along<span style=\"text-align: initial;font-size: 1em\"> different directions as scattered waves are emitted by atoms of different types and positions. There is a profound geometrical relationship between the directions of waves that interfere constructively, which comprise the \u201cdiffraction pattern,\u201d and the crystal structure of the material. The diffraction pattern is a spectrum of real space periodicities in a material. Atomic periodicities with long repeat distances cause diffraction at small angles, while short repeat distances cause diffraction at high angles. It is not hard to appreciate that diffraction experiments are useful for determining the crystal structures of materials. Much more information about a material is contained in its diffraction pattern, however. Crystals with precise periodicities over long distances have sharp and clear diffraction peaks. Crystals with defects such as impurities, dislocations, planar faults, internal strains, or small precipitates are less precisely periodic in their atomic arrangements, but they still have\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">distinct diffraction peaks. Their diffraction peaks are broadened, distorted, and weakened, however, and \u201cdiffraction line shape analysis\u201d is an important method for studying crystal defects. Diffraction experiments are also used to study the structure of amorphous materials, even though their diffraction patterns lack sharp diffraction peaks.<\/span><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-64\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-24.png\" alt=\"\" width=\"290\" height=\"266\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-24.png 290w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-24-65x60.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-24-225x206.png 225w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">Fig: 4.3 Bragg\u2019s X-ray diffraction<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">An often-used instrument for measuring the Bragg reflection of a thin film is the \u03b8\/2\u03b8 diffractometer. Let us introduce its operation principle by considering the results obtained with the question in mind as to how x-ray scattering experiments are preferably facilitated. What we are interested in is the measurement of Bragg reflections, i.e. their position, shape, intensity, etc., in order to derive microstructural information from them. The intensity variation that is associated with the reflection is included in the interference function, while the scattered intensity depends on the distance from the sample to the detection system <strong>R<\/strong>. We therefore should configure the instrument such that we can scan the space around the sample by keeping the sample\u2013detector distance <strong>R<\/strong> constant. This measure ensures that any intensity variation observed is due to the interference function and is not caused by a dependency on <strong>R<\/strong>. The detector should accordingly move on a sphere of constant radius <strong>R<\/strong> with the sample in the center of it. In addition, the sphere reduces to a hemisphere above the sample, since we are only interested in the surface layer and data collection will be performed in reflection mode. The geometry is shown in Fig. 4.3. Because the scattering of x-rays depends sensitively on the orientation of the crystal with respect to the scattering vector, we carefully have to define the various coordinate systems with which we are dealing.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-65\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-25.png\" alt=\"\" width=\"226\" height=\"157\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-25.png 226w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-25-65x45.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-25-225x156.png 225w\" sizes=\"auto, (max-width: 226px) 100vw, 226px\" \/><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Figure 4.4: Schematic representation of a \u03b8\/2\u03b8 scan from the viewpoint of the sample reference frame.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">K0 is the incident wave vector and K is the reflected wave vector.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The inter-planar spacing, d, sets the difference in path length for the ray scattered from the top plane and the ray scattered from the bottom plane. Figure 4.3 shows that this difference in path lengths is 2d sin \u03b8. Constructive wave interference occurs when the difference in path length for the top and bottom rays is equal to one wavelength, \u03bb:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-66\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-26.png\" alt=\"\" width=\"197\" height=\"24\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-26.png 197w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-26-65x8.png 65w\" sizes=\"auto, (max-width: 197px) 100vw, 197px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The right hand side is multiplied by an integer, <em>n<\/em>, since this condition also provides constructive interference. For first order diffraction, <em>n<\/em> = 1. When there is a path length difference of <em>n<\/em>\u03bb between adjacent planes, we changed even though this new d may not correspond to a real interatomic distance. For example, when our diffracting planes are (100) cube faces, and<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-67\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-27.png\" alt=\"\" width=\"198\" height=\"29\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-27.png 198w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-27-65x10.png 65w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">then we speak of a (200) diffraction from planes separated by d200 = (d )\/2. A diffraction pattern from a material typically contains many distinct peaks, each corresponding to a different inter-planar spacing, <em>d<\/em>. For cubic crystals with lattice parameter<em> a<\/em><em>0<\/em>, the inter-planar spacing,<em> d<\/em>, of planes labeled by Miller indices (hkl) are<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-68\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-28.png\" alt=\"\" width=\"200\" height=\"38\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-28.png 200w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-28-65x12.png 65w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">which can be proved by the definition of Miller indices and the 3D Pythagorean theorem. From Bragg\u2019s law (eqn 2) we find that the (hkl) diffraction peak occurs at the measured angle 2\u03b8hkl:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-69\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-29.png\" alt=\"\" width=\"257\" height=\"42\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-29.png 257w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-29-65x11.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-29-225x37.png 225w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">There are often many individual crystals of random orientation in the sample, so all possible Bragg diffractions can be observed in the \u201cpowder pattern.\u201d There is a convention for labeling, or \u201cindexing,\u201d the different Bragg peaks in a powder diffraction pattern using the numbers (<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">). An example of an indexed diffraction pattern is shown in Fig. 4.5. The intensities of the different diffraction peaks vary <\/span>widely,<span style=\"text-align: initial;font-size: 1em\"> and are zero for some combinations of <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">. For this example of polycrystalline silicon, notice the absence of all combinations of <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\">, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\"> that are mixtures of even and odd integers, and the absence of all even integer combinations whose sum is not divisible by 4.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-70\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-30.png\" alt=\"\" width=\"235\" height=\"174\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-30.png 235w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-30-65x48.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-30-225x167.png 225w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">Fig: 4.5 Indexed powder x-ray diffraction pattern of polycrystalline silicon.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">One important use of x-ray powder diffractometry is for identifying unknown crystals in a sample. The idea is to match the positions and the intensities of the peaks in the observed diffraction pattern to a known pattern of peaks from a standard sample or from a calculation. There should be a one-to-one correspondence between the observed peaks and the indexed peaks in the candidate diffraction pattern. For a simple diffraction pattern as in figure 4.5, this tentative indexing still needs to be checked. To do so, the \u03b8 -angles of the diffraction peaks are obtained, and used with (eqn 3) to obtain the interplanar spacing for each diffraction peak. For cubic crystals it is then possible to use (eqn 4) to convert each interplanar spacing into a lattice parameter. The indexing is consistent if all peaks provide the same lattice parameters.<\/p>\n<p>&nbsp;<\/p>\n<p>Non-cubic crystals usually require an iterative refinement of lattice parameters and angles. The relation between lattice constants and miller indices for non-cubic symmetry is given below in table 4.1<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Table 4.1:<\/span><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-71 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31.png\" alt=\"\" width=\"450\" height=\"283\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31.png 450w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31-300x189.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31-65x41.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31-225x142.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-31-350x220.png 350w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div>\n<p><strong>\u00a0 \u00a0 Value Addition:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Do You Know?<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Bragg diffraction (also referred to as the <strong>Bragg formulation of X-ray diffraction<\/strong>) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">William Lawrence Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter <em>d<\/em>. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively. The interference is constructive when the phase shift is a multiple of 2 \u03c0; this condition can be expressed by Bragg&#8217;s law and was first presented by Sir William Lawrence Bragg on 11 November 1912 to the Cambridge Philosophical Society. Although simple, Bragg&#8217;s law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond. They are the only father-son team to jointly win. William Lawrence Bragg was 25 years old, making him then, the youngest physics Nobel laureate.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>1.\u00a0<\/strong><strong>Suggested Reading<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>For More Details (on this topic and othe r topics discussed in Text Module) See<\/strong><\/p>\n<ol>\n<li style=\"text-align: justify\">Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks\/Cole, Eastern Press Bangalore (India) 2005<\/li>\n<li>Charles Kittel, Introduction to Solid State Physics, John Wiley &amp; Sons, Singapore 1999<\/li>\n<li>X-ray Absorption Spectroscopy, J. E. Penner-hahn<span style=\"text-align: initial;font-size: 1em\">, MI, USA Page 159-186)<\/span><\/li>\n<\/ol>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Glossary:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Elastic Scattering:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">When a particle of radiation does not share energy with the matter in the scattering process it is called inelastic scattering. The outgoing particle may come out with <\/span>same<span style=\"text-align: initial;font-size: 1em\"> energy as its initial energy.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">XRD<\/strong><span style=\"text-align: initial;font-size: 1em\">: X-ray Diffraction<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":5,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["dr-amarjeet-singh"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-58","chapter","type-chapter","status-publish","hentry","contributor-dr-amarjeet-singh"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/58","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":4,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions"}],"predecessor-version":[{"id":72,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/58\/revisions\/72"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/58\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=58"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=58"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=58"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=58"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}