{"id":73,"date":"2018-11-30T10:37:37","date_gmt":"2018-11-30T10:37:37","guid":{"rendered":"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=73"},"modified":"2018-11-30T11:37:38","modified_gmt":"2018-11-30T11:37:38","slug":"reciprocal-latticeand-structure-factor","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/chapter\/reciprocal-latticeand-structure-factor\/","title":{"rendered":"Reciprocal Latticeand Structure Factor"},"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>Understand the need for reciprocal lattice to analyse diffraction phenomenon<\/li>\r\n \t<li>Learn the relationship and interatomic planes<\/li>\r\n \t<li>Construct the reciprocal lattice corresponding to various lattices for example, BCC, FCC, hexagonal lattice<\/li>\r\n \t<li>Define structure factor<\/li>\r\n \t<li>The relation of structure factor with the intensity of the diffraction peaks<\/li>\r\n \t<li>Understand why certain diffraction peaks are absent in some XRD patterns.<\/li>\r\n<\/ul>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Introduction:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To determine a real space lattice structure, the most popular experimental technique is some form of diffraction. Methods include x-ray, electron, atom, and neutron diffraction. The particles in these techniques have dual properties and behave as matter waves through the de Broglie relationship, \u03bb = h\/p, where \u03bb is wavelength, h is Planck\u2019s constant ( = 6.63 \u00d7 10-34J \u00b7 s), and p is the magnitude of momentum. The propagation of a wave is described by the advancement of a wave front. Assuming a plane wave propagation, the wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> is perpendicular to the plane wave front. The relationship between <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> is <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> = \u0127 <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">. From the de Broglie relationship (\u03bb =h\/p and <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> =[h\/(2\u03c0)] <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">), one obtains |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">|\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">=\u00a0 2\u03c0\/\u03bb. Note that the wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> in has a unit of inverse length. It is convenient to define a reciprocal space lattice in the momentum space that is related to the real space lattice. The symmetry of a real space lattice and the symmetry of its reciprocal space lattice are related. The unit vectors in the reciprocal space lattice have a reciprocal relationship with the unit vectors in the real space lattice. We have introduced real-space lattice points, basic unit vectors, the direction of a real-space plane, and interplanar spacing d. Reciprocal space also consists of reciprocal lattice points and reciprocal vectors. We can relate real space and reciprocal space using geometry in an actual diffraction experiment. The diffraction of a wave involves an incoming wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><strong style=\"text-align: initial;font-size: 1em\">in<\/strong><span style=\"text-align: initial;font-size: 1em\"> and a scattered wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out. The direction of <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out differs from the direction of <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">in (except in the forward scattering case). The difference in direction is the scattering angle 2\u03b8.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-77\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-32.png\" alt=\"\" width=\"275\" height=\"203\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: left\"><strong>Figure 5.1: <\/strong>Wave scattering from a sample<strong> a. <\/strong>The scattering angle is 2\u03b8, and the outgoing wavevector<strong> k<\/strong><strong>out<\/strong> has a momentum change<strong> K<\/strong>, relative to the incoming wavevector<strong> k<\/strong><strong>in<\/strong>.<strong> b <\/strong>Specular scattering where \u03b8out = \u03b8in = \u03b8<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">See Fig. 5.1. If the scattering is elastic then |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><strong style=\"text-align: initial;font-size: 1em\">in<\/strong><strong style=\"text-align: initial;font-size: 1em\">| = |k<\/strong><strong style=\"text-align: initial;font-size: 1em\">out<\/strong><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><span style=\"text-align: initial;font-size: 1em\"> = |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">| consider specular scattering, \u03b8in =\u03b8out = \u03b8. The change in wavevectors or the momentum transfer is defined as <\/span><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><span style=\"text-align: initial;font-size: 1em\">=<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out \u2013<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">in. Applying trigonometry as shown in Fig. 5.1, one obtains<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-78\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-33.png\" alt=\"\" width=\"145\" height=\"106\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In a scattering experiment, one knows the wavelength \u03bb and can measure the scattering angle 2\u03b8, and then the magnitude of <strong>K<\/strong> change can be obtained from above Eq.<\/p>\r\n&nbsp;\r\n\r\n<strong>Bragg Condition<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">We will see that this change of momentum <strong>K<\/strong> is related to interplanar spacing d in a real-space crystal as shown in figure 5.2. The incoming wavevector <strong>k<\/strong> , and it is scattered as ray 1. The outgoing wavevector <strong>k<\/strong> incident on the first plane (plane 1) with an angle \u03b8 out in is scattered specularly with \u03b8 out = \u03b8. The same wave scattering from the second plane is ray 2. The interplanar spacing d is the perpendicular distance between the first and second planes or a to b. The path length of ray 2 travels more than that of ray 1. The path length difference is in <em>hkl<\/em> in <em>hkl<\/em> is<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-79\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34.png\" alt=\"\" width=\"363\" height=\"302\" \/>\r\n<p style=\"text-align: justify\"><strong>Fig 5.2: <\/strong>Bragg scattering from two parallel planes. qin = \u03b8out = \u03b8 for specular diffraction. Wave 2 travels 2dhkl sin\u03b8 further than wave 1<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If this path-length difference is an integer number <\/span><em style=\"text-align: initial;font-size: 1em\">n<\/em><span style=\"text-align: initial;font-size: 1em\"> of wavelength \u03bb, then a constructed interference occurs to give the maximum intensity. This is Bragg\u2019s law.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">2<\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><sub> <em style=\"text-align: initial\">hkl<\/em><\/sub><span style=\"text-align: initial;font-size: 1em\"> sin <\/span><em style=\"text-align: initial;font-size: 1em\">q<\/em><sub><em style=\"text-align: initial\">B<\/em><\/sub><span style=\"text-align: initial;font-size: 1em\"><sub>\u00a0<\/sub> = <\/span><em style=\"text-align: initial;font-size: 1em\">n<\/em><em style=\"text-align: initial;font-size: 1em\">l<\/em><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Combining above equations, we obtain the reciprocal relationship between the change of wavevectors <strong>K<\/strong><strong>B<\/strong> and the interplanar spacing d<em>hkl<\/em> at the Bragg condition,<\/p>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter size-full wp-image-80\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-35.png\" alt=\"\" width=\"120\" height=\"59\" \/>\r\n\r\n<strong><em>Reciprocal Space Basic Vectors and their Relationship to Real Space Basic Vectors<\/em><\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In general, a diffraction experiment involves a 3D sample. The previous derivation is for one dimension. In the following, the real space and reciprocal space relationship in terms of vectors in three dimensions will be derived. We have defined the position vector of a lattice point in a 3D crystal in real space by<\/p>\r\n<img class=\"size-full wp-image-81 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-36.png\" alt=\"\" width=\"157\" height=\"30\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where n1, n2, and n3 are integers and <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong> are real space unit vectors. In Fig. 5.3, we sketched one unit cell in real space with basic vectors <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong>. Similarly, we can also define a reciprocal lattice vector <strong>G<\/strong>(<em>hkl<\/em>)<\/p>\r\n<img class=\"size-full wp-image-82 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-37.png\" alt=\"\" width=\"181\" height=\"28\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">where <em>h<\/em>, <em>k<\/em>, and <em>l<\/em> are the Miller indices of a crystal plane (<em>hkl<\/em>) and <strong>a<\/strong> are reciprocal unit vectors. Mathematically, one can show that <strong>G<\/strong>(<em>hkl<\/em>) \u2022 r is always an integer, n*, <strong>b<\/strong>*, and <strong>c<\/strong>*<\/p>\r\n<img class=\"size-full wp-image-83 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38.png\" alt=\"\" width=\"320\" height=\"101\" \/>\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<img class=\"aligncenter size-full wp-image-84\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39.png\" alt=\"\" width=\"260\" height=\"259\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Fig. 5.3: <\/strong><span style=\"text-align: initial;font-size: 1em\">Reciprocal basic vectors<\/span><strong style=\"text-align: initial;font-size: 1em\"> a<\/strong><span style=\"text-align: initial;font-size: 1em\">*,<\/span><strong style=\"text-align: initial;font-size: 1em\"> b<\/strong><span style=\"text-align: initial;font-size: 1em\">*, and<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong><span style=\"text-align: initial;font-size: 1em\">*andtheir relationship to <\/span>thereal<span style=\"text-align: initial;font-size: 1em\">-space basic vectors<\/span><strong style=\"text-align: initial;font-size: 1em\"> a<\/strong><span style=\"text-align: initial;font-size: 1em\">,<\/span><strong style=\"text-align: initial;font-size: 1em\"> b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nFrom vector algebra, when the previous equation is satisfied, one obtains the reciprocal lattice vectors\r\n\r\n<img class=\"size-full wp-image-85 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40.png\" alt=\"\" width=\"410\" height=\"48\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Where <strong>a<\/strong>\u00b7(<strong>b<\/strong> \u00d7 <strong>c<\/strong>) is the volume V of a unit cell in real space. See Fig. 3.3. The previous relationships can be rewritten as<\/p>\r\n<img class=\"size-full wp-image-86 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41.png\" alt=\"\" width=\"302\" height=\"56\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This means vector <strong>a<\/strong>* is the cross product of <strong>b<\/strong> and <strong>c<\/strong> or <strong>a<\/strong>* is perpendicular to the plane consisting of <strong>b<\/strong> and <strong>c<\/strong>. Using the right-hand rule, one obtains <strong>a<\/strong> Similarly, <strong>b<\/strong>* is obtained from the cross product of <strong>c<\/strong> and <strong>a<\/strong>, and<strong> c<\/strong>, shown in Fig. 5.3. c* is obtained from the cross product of<strong> a <\/strong>and<strong> b<\/strong>. Vectors<strong> a<\/strong>,<strong> b<\/strong>, and<strong> c <\/strong>are related to <strong>a<\/strong> *, <strong>b<\/strong> as*, and <strong>c<\/strong>*<\/p>\r\n<img class=\"size-full wp-image-87 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42.png\" alt=\"\" width=\"384\" height=\"72\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The magnitude of <strong>a<\/strong>* is inversely proportional to the magnitude of <strong>a<\/strong>. The same relationship is true for <strong>b<\/strong>* and<strong> c<\/strong>*. This means the size of a reciprocal lattice unit cell is inversely proportional to the size of the real space unit cell.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">One can obtain the reciprocal unit vectors <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">*, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">*, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">**, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">* from <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\"> in the previous relationships. A reciprocal lattice can be generated by <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">) =<\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">*+<\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">*+<\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">*, where <\/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\"> are integers. We illustrate the relationship between the reciprocal unit vectors and the real space unit vectors in a two-dimensional lattice shown in Fig. 3.4. A two-dimensional real space unit mesh consists of unit vectors <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> that are parallel to the page. The <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong>isperpendicular<span style=\"text-align: initial;font-size: 1em\"> to <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and the <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">* is perpendicular to <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">. Also, the length of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> Projected on <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> is 2\u03c0\/a and is the inverse of the length of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">. Also, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> * and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">) =<\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(001) =<\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\"> * since <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\"> =0, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\"> =0, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\"> =1. This means <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(001) is perpendicular to <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> (because of the cross product of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> \u00d7 <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">) and its magnitude is inversely proportional to the magnitude of<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong><span style=\"text-align: initial;font-size: 1em\">.**<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-88\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-43.png\" alt=\"\" width=\"266\" height=\"167\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\"><strong>Fig. 5.4 <\/strong>Relationship between real-space basic vectors<strong> a <\/strong>and<strong> b <\/strong>and reciprocal-space basic vectors<strong> a<\/strong>* and <strong>b<\/strong>*<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">space unit mesh consists of unit vectors <strong>a<\/strong> and <strong>b<\/strong> that are parallel to the page. The <strong>a<\/strong> is perpendicular to <strong>b<\/strong>, and the<strong> b <\/strong>* is perpendicular to<strong> a<\/strong>. Also, the length of<strong> a <\/strong>projected on<strong> a <\/strong>is 2p\/a and is the inverse of the length of <strong>a<\/strong>. Also, <strong>b<\/strong> * and <strong>b<\/strong> are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as <strong>G<\/strong>(<em>hkl<\/em>) =<strong>G<\/strong>(001) =<strong>c<\/strong> * since <em>h<\/em> =0, <em>k<\/em> =0, and <em>l<\/em> =1. This means <strong>G<\/strong>(001) is perpendicular to <strong>a<\/strong> and <strong>b<\/strong> (because of the cross product of <strong>a<\/strong> \u00d7 <strong>b<\/strong>) and its magnitude is inversely proportional to the magnitude of <strong>c<\/strong>.*<\/p>\r\n&nbsp;\r\n\r\n<strong>Reciprocal Lattice Vector and its Relationship<\/strong>\r\n\r\n<strong>to Interplanar Spacing<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">We can further examine the direction and magnitude of the reciprocal lattice vector <strong>G<\/strong>(<em>hkl<\/em>) for a general case. The (<em>hkl<\/em>) plane is defined as a plane intercepting the <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong> axes at a\/<em>h<\/em>,b\/<em>k<\/em>, and c\/<em>l<\/em>, respectively. The plane ABC shown in Fig. 3.5 represents the (<em>hkl<\/em>) plane. The vectors <strong>AB<\/strong> and <strong>AC<\/strong> equal <strong>b<\/strong>\/<em>k<\/em> -<strong>a<\/strong>\/<em>h<\/em> and <strong>c<\/strong>\/<em>l<\/em> -<strong>a<\/strong>\/<em>h<\/em>, respectively. The cross product <strong>AB<\/strong> \u00d7 <strong>AC<\/strong> is a vector <strong>G<\/strong>(<em>hkl<\/em>) perpendicular to the (<em>hkl<\/em>) plane or parallel to the normal of the (<em>hkl<\/em>) plane<\/p>\r\n<img class=\"size-full wp-image-89 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44.png\" alt=\"\" width=\"357\" height=\"47\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">The unit vector<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-90 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-45.png\" alt=\"\" width=\"111\" height=\"48\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">To obtain the shortest distance d<em>hkl<\/em> between a family of (<em>hkl<\/em>) planes, that is, the distance from point O to the (<em>hkl<\/em>) plane, one can take the dot product of <strong>OA<\/strong> or any vector in the (<em>hkl<\/em>) plane and <strong>n<\/strong>.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-91 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46.png\" alt=\"\" width=\"610\" height=\"523\" \/>\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\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<p style=\"text-align: justify\"><strong>Fig. 5.5 <\/strong>Reciprocal latticevector<strong> G<\/strong>(<em>hkl<\/em>) is perpendicular to the (<em>hkl<\/em>) plane consisting of vectors<strong> AB <\/strong>and<strong> AC <\/strong>with interception a\/<em>h<\/em>,b\/<em>k<\/em>, and c\/<em>l<\/em> on x-, y-, and z-axes<\/p>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Structure Factor<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In <\/span>condensed matter physics <span style=\"text-align: initial;font-size: 1em\">and <\/span>crystallography, <span style=\"text-align: initial;font-size: 1em\">the static structure factor (or structure factor) is a mathematical description of how a material scatters incident radiation. The structure factor is a particularly useful tool in the interpretation of <\/span>interference patterns <span style=\"text-align: initial;font-size: 1em\">obtained in <\/span>X-ray, electron and neutron diffraction <span style=\"text-align: initial;font-size: 1em\">experiments. The static structure factor is measured without resolving the energy of scattered photons\/electrons\/neutrons.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In <\/span>last chapter<span style=\"text-align: initial;font-size: 1em\"> we learned about x-ray diffraction in reciprocal space. Structure factor is the signature of the real structure expressed in reciprocal space. In other words, the structure factor depends on the arrangement of atoms in a unit cell. In x-ray diffraction we perform measurements in reciprocal space, therefore it is convenient to work in reciprocal space. Here we define a new parameter called structure factor which can predict the intensity of peaks and their extinction from the diffraction spectrum. The structure factor is defined as<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-92 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47.png\" alt=\"\" width=\"779\" height=\"216\" \/>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<em>\u00a0<\/em>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Structure Factor for BCC lattice<\/strong>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-93\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-48.png\" alt=\"\" width=\"300\" height=\"258\" \/>\r\n<p style=\"text-align: center\"><strong>Fig 5.6: <\/strong>A unit cell of BCC lattice<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The bcc basis referred to the cubic cell has identical atoms at <em>x<\/em><em>1<\/em> = <em>y<\/em><em>1<\/em> = <em>z<\/em><em>1<\/em> = 0 and <em>x<\/em><em>2<\/em> = <em>y<\/em><em>2<\/em> = <em>z<\/em><em>2<\/em> = \u00bd. Thus above equation becomes<\/p>\r\n<img class=\"size-full wp-image-94 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-49.png\" alt=\"\" width=\"292\" height=\"125\" \/>\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<p style=\"text-align: justify\">The idea is look for atoms per unit cell and carry the sum over all atoms in a unit cell. For example BCC structure has 2 atoms per unit cell and FCC structure has 4 atoms per unit cell. In BCC the sum will be carried over 2 atoms whereas in FCC it will be carried over 4 atoms.<\/p>\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">Structure Factor for FCC lattice<\/strong>\r\n\r\n<\/div>\r\n<img class=\"aligncenter size-full wp-image-95\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-50.png\" alt=\"\" width=\"294\" height=\"246\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\"><strong>Fig: 5.7: <\/strong>A unit cell of FCC lattice<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThe basis of the FCC structure referred to the cubic cell has identical atoms at (000); (0, 0.5, 0.5); (0.5, 0, 0.5); (0.5, 0.5, 0), Thus structure factor becomes\r\n\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-96 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51.png\" alt=\"\" width=\"503\" height=\"33\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<em>S<\/em><em>G<\/em> = 4<em>f <\/em>if<em> h, k, l <\/em>are either all odd or all even\r\n\r\n&nbsp;\r\n\r\n<em>S<\/em><em>G<\/em> = 0, otherwise\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When structure factor becomes zero that means the diffraction peak corresponding to that plane (<em>h k l<\/em>) will become extinct.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Value Addition:<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Do You Know?<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The concept of reciprocal lattice is very useful in x-ray diffraction measurements. For a measurement in a real <\/span>space<span style=\"text-align: initial;font-size: 1em\"> we can do it with meter scale, microscope and electron microscopy depending on the length scale which is in question. In real space <\/span>measurement<span style=\"text-align: initial;font-size: 1em\"> we scan the location in real space and measure the property. <\/span>Similarly<span style=\"text-align: initial;font-size: 1em\"> in x-ray diffraction <\/span>measurement<span style=\"text-align: initial;font-size: 1em\"> we scan the reciprocal space ( | <\/span><em style=\"text-align: initial;font-size: 1em\">K<\/em><span style=\"text-align: initial;font-size: 1em\"> |= 2 | <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><em style=\"text-align: initial;font-size: 1em\">in<\/em><span style=\"text-align: initial;font-size: 1em\"> | sin <\/span><em style=\"text-align: initial;font-size: 1em\">q<\/em><span style=\"text-align: initial;font-size: 1em\"> ) when we scan\u00a0<\/span>angle q and measure the property (Intensity). Therefore X-ray measurement is a measurement of reciprocal space. Thus we obtain the reciprocal lattice associated with the crystalline structure. The Fourier transform of the reciprocal lattice gives us the knowledge of real space lattice.<\/li>\r\n \t<li style=\"text-align: justify\">Structure factors are very useful in x-ray crystallography. Structure factors in solids are intimately related to the atomic arrangement of solids. Looking at the structure factor it is possible to determine if the material is crystalline or amorphous. Also<span style=\"text-align: initial;font-size: 1em\"> the intensity of the diffraction peak can be explained from <\/span>structure<span style=\"text-align: initial;font-size: 1em\"> factor. Using several diffraction patterns from various angles, the measured structure factors are used to determine the structure of very complex molecules such as proteins and DNA etc.<\/span><\/li>\r\n<\/ol>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Suggested Reading<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>For More Details (on this topic and other 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 style=\"text-align: justify\">Charles Kittel, Introduction to Solid State Physics, John Wiley &amp; Sons, Singapore 1999<\/li>\r\n \t<li style=\"text-align: justify\">Jens Als-Nielsen and Des McMorrow, John Wiley &amp; Sons, UK, 2011<\/li>\r\n \t<li style=\"text-align: justify\">Wikipedia<\/li>\r\n<\/ol>\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>Understand the need for reciprocal lattice to analyse diffraction phenomenon<\/li>\n<li>Learn the relationship and interatomic planes<\/li>\n<li>Construct the reciprocal lattice corresponding to various lattices for example, BCC, FCC, hexagonal lattice<\/li>\n<li>Define structure factor<\/li>\n<li>The relation of structure factor with the intensity of the diffraction peaks<\/li>\n<li>Understand why certain diffraction peaks are absent in some XRD patterns.<\/li>\n<\/ul>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 Introduction:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">To determine a real space lattice structure, the most popular experimental technique is some form of diffraction. Methods include x-ray, electron, atom, and neutron diffraction. The particles in these techniques have dual properties and behave as matter waves through the de Broglie relationship, \u03bb = h\/p, where \u03bb is wavelength, h is Planck\u2019s constant ( = 6.63 \u00d7 10-34J \u00b7 s), and p is the magnitude of momentum. The propagation of a wave is described by the advancement of a wave front. Assuming a plane wave propagation, the wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> is perpendicular to the plane wave front. The relationship between <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> is <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> = \u0127 <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">. From the de Broglie relationship (\u03bb =h\/p and <\/span><strong style=\"text-align: initial;font-size: 1em\">p<\/strong><span style=\"text-align: initial;font-size: 1em\"> =[h\/(2\u03c0)] <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">), one obtains |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">|\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">=\u00a0 2\u03c0\/\u03bb. Note that the wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\"> in has a unit of inverse length. It is convenient to define a reciprocal space lattice in the momentum space that is related to the real space lattice. The symmetry of a real space lattice and the symmetry of its reciprocal space lattice are related. The unit vectors in the reciprocal space lattice have a reciprocal relationship with the unit vectors in the real space lattice. We have introduced real-space lattice points, basic unit vectors, the direction of a real-space plane, and interplanar spacing d. Reciprocal space also consists of reciprocal lattice points and reciprocal vectors. We can relate real space and reciprocal space using geometry in an actual diffraction experiment. The diffraction of a wave involves an incoming wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><strong style=\"text-align: initial;font-size: 1em\">in<\/strong><span style=\"text-align: initial;font-size: 1em\"> and a scattered wavevector <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out. The direction of <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out differs from the direction of <\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">in (except in the forward scattering case). The difference in direction is the scattering angle 2\u03b8.<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-77\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-32.png\" alt=\"\" width=\"275\" height=\"203\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-32.png 275w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-32-65x48.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-32-225x166.png 225w\" sizes=\"auto, (max-width: 275px) 100vw, 275px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left\"><strong>Figure 5.1: <\/strong>Wave scattering from a sample<strong> a. <\/strong>The scattering angle is 2\u03b8, and the outgoing wavevector<strong> k<\/strong><strong>out<\/strong> has a momentum change<strong> K<\/strong>, relative to the incoming wavevector<strong> k<\/strong><strong>in<\/strong>.<strong> b <\/strong>Specular scattering where \u03b8out = \u03b8in = \u03b8<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">See Fig. 5.1. If the scattering is elastic then |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><strong style=\"text-align: initial;font-size: 1em\">in<\/strong><strong style=\"text-align: initial;font-size: 1em\">| = |k<\/strong><strong style=\"text-align: initial;font-size: 1em\">out<\/strong><strong style=\"text-align: initial;font-size: 1em\">|<\/strong><span style=\"text-align: initial;font-size: 1em\"> = |<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">| consider specular scattering, \u03b8in =\u03b8out = \u03b8. The change in wavevectors or the momentum transfer is defined as <\/span><strong style=\"text-align: initial;font-size: 1em\">K<\/strong><span style=\"text-align: initial;font-size: 1em\">=<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">out \u2013<\/span><strong style=\"text-align: initial;font-size: 1em\">k<\/strong><span style=\"text-align: initial;font-size: 1em\">in. Applying trigonometry as shown in Fig. 5.1, one obtains<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-78\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-33.png\" alt=\"\" width=\"145\" height=\"106\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-33.png 145w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-33-65x48.png 65w\" sizes=\"auto, (max-width: 145px) 100vw, 145px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In a scattering experiment, one knows the wavelength \u03bb and can measure the scattering angle 2\u03b8, and then the magnitude of <strong>K<\/strong> change can be obtained from above Eq.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Bragg Condition<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We will see that this change of momentum <strong>K<\/strong> is related to interplanar spacing d in a real-space crystal as shown in figure 5.2. The incoming wavevector <strong>k<\/strong> , and it is scattered as ray 1. The outgoing wavevector <strong>k<\/strong> incident on the first plane (plane 1) with an angle \u03b8 out in is scattered specularly with \u03b8 out = \u03b8. The same wave scattering from the second plane is ray 2. The interplanar spacing d is the perpendicular distance between the first and second planes or a to b. The path length of ray 2 travels more than that of ray 1. The path length difference is in <em>hkl<\/em> in <em>hkl<\/em> is<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-79\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34.png\" alt=\"\" width=\"363\" height=\"302\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34.png 363w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34-300x250.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34-225x187.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-34-350x291.png 350w\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" \/><\/p>\n<p style=\"text-align: justify\"><strong>Fig 5.2: <\/strong>Bragg scattering from two parallel planes. qin = \u03b8out = \u03b8 for specular diffraction. Wave 2 travels 2dhkl sin\u03b8 further than wave 1<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If this path-length difference is an integer number <\/span><em style=\"text-align: initial;font-size: 1em\">n<\/em><span style=\"text-align: initial;font-size: 1em\"> of wavelength \u03bb, then a constructed interference occurs to give the maximum intensity. This is Bragg\u2019s law.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">2<\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><sub> <em style=\"text-align: initial\">hkl<\/em><\/sub><span style=\"text-align: initial;font-size: 1em\"> sin <\/span><em style=\"text-align: initial;font-size: 1em\">q<\/em><sub><em style=\"text-align: initial\">B<\/em><\/sub><span style=\"text-align: initial;font-size: 1em\"><sub>\u00a0<\/sub> = <\/span><em style=\"text-align: initial;font-size: 1em\">n<\/em><em style=\"text-align: initial;font-size: 1em\">l<\/em><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Combining above equations, we obtain the reciprocal relationship between the change of wavevectors <strong>K<\/strong><strong>B<\/strong> and the interplanar spacing d<em>hkl<\/em> at the Bragg condition,<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-80\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-35.png\" alt=\"\" width=\"120\" height=\"59\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-35.png 120w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-35-65x32.png 65w\" sizes=\"auto, (max-width: 120px) 100vw, 120px\" \/><\/p>\n<p><strong><em>Reciprocal Space Basic Vectors and their Relationship to Real Space Basic Vectors<\/em><\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In general, a diffraction experiment involves a 3D sample. The previous derivation is for one dimension. In the following, the real space and reciprocal space relationship in terms of vectors in three dimensions will be derived. We have defined the position vector of a lattice point in a 3D crystal in real space by<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-81 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-36.png\" alt=\"\" width=\"157\" height=\"30\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-36.png 157w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-36-150x30.png 150w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-36-65x12.png 65w\" sizes=\"auto, (max-width: 157px) 100vw, 157px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where n1, n2, and n3 are integers and <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong> are real space unit vectors. In Fig. 5.3, we sketched one unit cell in real space with basic vectors <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong>. Similarly, we can also define a reciprocal lattice vector <strong>G<\/strong>(<em>hkl<\/em>)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-82 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-37.png\" alt=\"\" width=\"181\" height=\"28\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-37.png 181w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-37-65x10.png 65w\" sizes=\"auto, (max-width: 181px) 100vw, 181px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">where <em>h<\/em>, <em>k<\/em>, and <em>l<\/em> are the Miller indices of a crystal plane (<em>hkl<\/em>) and <strong>a<\/strong> are reciprocal unit vectors. Mathematically, one can show that <strong>G<\/strong>(<em>hkl<\/em>) \u2022 r is always an integer, n*, <strong>b<\/strong>*, and <strong>c<\/strong>*<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-83 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38.png\" alt=\"\" width=\"320\" height=\"101\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38.png 320w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38-300x95.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-38-225x71.png 225w\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-84\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39.png\" alt=\"\" width=\"260\" height=\"259\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39.png 260w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39-150x150.png 150w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39-65x65.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-39-225x224.png 225w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Fig. 5.3: <\/strong><span style=\"text-align: initial;font-size: 1em\">Reciprocal basic vectors<\/span><strong style=\"text-align: initial;font-size: 1em\"> a<\/strong><span style=\"text-align: initial;font-size: 1em\">*,<\/span><strong style=\"text-align: initial;font-size: 1em\"> b<\/strong><span style=\"text-align: initial;font-size: 1em\">*, and<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong><span style=\"text-align: initial;font-size: 1em\">*andtheir relationship to <\/span>thereal<span style=\"text-align: initial;font-size: 1em\">-space basic vectors<\/span><strong style=\"text-align: initial;font-size: 1em\"> a<\/strong><span style=\"text-align: initial;font-size: 1em\">,<\/span><strong style=\"text-align: initial;font-size: 1em\"> b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>From vector algebra, when the previous equation is satisfied, one obtains the reciprocal lattice vectors<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-85 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40.png\" alt=\"\" width=\"410\" height=\"48\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40.png 410w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40-300x35.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40-65x8.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40-225x26.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-40-350x41.png 350w\" sizes=\"auto, (max-width: 410px) 100vw, 410px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Where <strong>a<\/strong>\u00b7(<strong>b<\/strong> \u00d7 <strong>c<\/strong>) is the volume V of a unit cell in real space. See Fig. 3.3. The previous relationships can be rewritten as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-86 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41.png\" alt=\"\" width=\"302\" height=\"56\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41.png 302w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41-300x56.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-41-225x42.png 225w\" sizes=\"auto, (max-width: 302px) 100vw, 302px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This means vector <strong>a<\/strong>* is the cross product of <strong>b<\/strong> and <strong>c<\/strong> or <strong>a<\/strong>* is perpendicular to the plane consisting of <strong>b<\/strong> and <strong>c<\/strong>. Using the right-hand rule, one obtains <strong>a<\/strong> Similarly, <strong>b<\/strong>* is obtained from the cross product of <strong>c<\/strong> and <strong>a<\/strong>, and<strong> c<\/strong>, shown in Fig. 5.3. c* is obtained from the cross product of<strong> a <\/strong>and<strong> b<\/strong>. Vectors<strong> a<\/strong>,<strong> b<\/strong>, and<strong> c <\/strong>are related to <strong>a<\/strong> *, <strong>b<\/strong> as*, and <strong>c<\/strong>*<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-87 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42.png\" alt=\"\" width=\"384\" height=\"72\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42.png 384w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42-300x56.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42-225x42.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-42-350x66.png 350w\" sizes=\"auto, (max-width: 384px) 100vw, 384px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The magnitude of <strong>a<\/strong>* is inversely proportional to the magnitude of <strong>a<\/strong>. The same relationship is true for <strong>b<\/strong>* and<strong> c<\/strong>*. This means the size of a reciprocal lattice unit cell is inversely proportional to the size of the real space unit cell.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">One can obtain the reciprocal unit vectors <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">*, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">*, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">**, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">* from <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and <\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\"> in the previous relationships. A reciprocal lattice can be generated by <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">) =<\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">*+<\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">*+<\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\">*, where <\/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\"> are integers. We illustrate the relationship between the reciprocal unit vectors and the real space unit vectors in a two-dimensional lattice shown in Fig. 3.4. A two-dimensional real space unit mesh consists of unit vectors <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> that are parallel to the page. The <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong>isperpendicular<span style=\"text-align: initial;font-size: 1em\"> to <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">, and the <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">* is perpendicular to <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">. Also, the length of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> Projected on <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> is 2\u03c0\/a and is the inverse of the length of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\">. Also, <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> * and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(<\/span><em style=\"text-align: initial;font-size: 1em\">hkl<\/em><span style=\"text-align: initial;font-size: 1em\">) =<\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(001) =<\/span><strong style=\"text-align: initial;font-size: 1em\">c<\/strong><span style=\"text-align: initial;font-size: 1em\"> * since <\/span><em style=\"text-align: initial;font-size: 1em\">h<\/em><span style=\"text-align: initial;font-size: 1em\"> =0, <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\"> =0, and <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\"> =1. This means <\/span><strong style=\"text-align: initial;font-size: 1em\">G<\/strong><span style=\"text-align: initial;font-size: 1em\">(001) is perpendicular to <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> and <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\"> (because of the cross product of <\/span><strong style=\"text-align: initial;font-size: 1em\">a<\/strong><span style=\"text-align: initial;font-size: 1em\"> \u00d7 <\/span><strong style=\"text-align: initial;font-size: 1em\">b<\/strong><span style=\"text-align: initial;font-size: 1em\">) and its magnitude is inversely proportional to the magnitude of<\/span><strong style=\"text-align: initial;font-size: 1em\"> c<\/strong><span style=\"text-align: initial;font-size: 1em\">.**<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-88\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-43.png\" alt=\"\" width=\"266\" height=\"167\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-43.png 266w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-43-65x41.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-43-225x141.png 225w\" sizes=\"auto, (max-width: 266px) 100vw, 266px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><strong>Fig. 5.4 <\/strong>Relationship between real-space basic vectors<strong> a <\/strong>and<strong> b <\/strong>and reciprocal-space basic vectors<strong> a<\/strong>* and <strong>b<\/strong>*<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">space unit mesh consists of unit vectors <strong>a<\/strong> and <strong>b<\/strong> that are parallel to the page. The <strong>a<\/strong> is perpendicular to <strong>b<\/strong>, and the<strong> b <\/strong>* is perpendicular to<strong> a<\/strong>. Also, the length of<strong> a <\/strong>projected on<strong> a <\/strong>is 2p\/a and is the inverse of the length of <strong>a<\/strong>. Also, <strong>b<\/strong> * and <strong>b<\/strong> are related in a similar way. For example, if one has the (001) plane in real space, the reciprocal lattice direction in the reciprocal space can be determined as <strong>G<\/strong>(<em>hkl<\/em>) =<strong>G<\/strong>(001) =<strong>c<\/strong> * since <em>h<\/em> =0, <em>k<\/em> =0, and <em>l<\/em> =1. This means <strong>G<\/strong>(001) is perpendicular to <strong>a<\/strong> and <strong>b<\/strong> (because of the cross product of <strong>a<\/strong> \u00d7 <strong>b<\/strong>) and its magnitude is inversely proportional to the magnitude of <strong>c<\/strong>.*<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Reciprocal Lattice Vector and its Relationship<\/strong><\/p>\n<p><strong>to Interplanar Spacing<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We can further examine the direction and magnitude of the reciprocal lattice vector <strong>G<\/strong>(<em>hkl<\/em>) for a general case. The (<em>hkl<\/em>) plane is defined as a plane intercepting the <strong>a<\/strong>, <strong>b<\/strong>, and <strong>c<\/strong> axes at a\/<em>h<\/em>,b\/<em>k<\/em>, and c\/<em>l<\/em>, respectively. The plane ABC shown in Fig. 3.5 represents the (<em>hkl<\/em>) plane. The vectors <strong>AB<\/strong> and <strong>AC<\/strong> equal <strong>b<\/strong>\/<em>k<\/em> &#8211;<strong>a<\/strong>\/<em>h<\/em> and <strong>c<\/strong>\/<em>l<\/em> &#8211;<strong>a<\/strong>\/<em>h<\/em>, respectively. The cross product <strong>AB<\/strong> \u00d7 <strong>AC<\/strong> is a vector <strong>G<\/strong>(<em>hkl<\/em>) perpendicular to the (<em>hkl<\/em>) plane or parallel to the normal of the (<em>hkl<\/em>) plane<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-89 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44.png\" alt=\"\" width=\"357\" height=\"47\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44.png 357w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44-300x39.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44-225x30.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-44-350x46.png 350w\" sizes=\"auto, (max-width: 357px) 100vw, 357px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">The unit vector<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-90 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-45.png\" alt=\"\" width=\"111\" height=\"48\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-45.png 111w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-45-65x28.png 65w\" sizes=\"auto, (max-width: 111px) 100vw, 111px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">To obtain the shortest distance d<em>hkl<\/em> between a family of (<em>hkl<\/em>) planes, that is, the distance from point O to the (<em>hkl<\/em>) plane, one can take the dot product of <strong>OA<\/strong> or any vector in the (<em>hkl<\/em>) plane and <strong>n<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-91 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46.png\" alt=\"\" width=\"610\" height=\"523\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46.png 610w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46-300x257.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46-65x56.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46-225x193.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-46-350x300.png 350w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><\/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<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 style=\"text-align: justify\"><strong>Fig. 5.5 <\/strong>Reciprocal latticevector<strong> G<\/strong>(<em>hkl<\/em>) is perpendicular to the (<em>hkl<\/em>) plane consisting of vectors<strong> AB <\/strong>and<strong> AC <\/strong>with interception a\/<em>h<\/em>,b\/<em>k<\/em>, and c\/<em>l<\/em> on x-, y-, and z-axes<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Structure Factor<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In <\/span>condensed matter physics <span style=\"text-align: initial;font-size: 1em\">and <\/span>crystallography, <span style=\"text-align: initial;font-size: 1em\">the static structure factor (or structure factor) is a mathematical description of how a material scatters incident radiation. The structure factor is a particularly useful tool in the interpretation of <\/span>interference patterns <span style=\"text-align: initial;font-size: 1em\">obtained in <\/span>X-ray, electron and neutron diffraction <span style=\"text-align: initial;font-size: 1em\">experiments. The static structure factor is measured without resolving the energy of scattered photons\/electrons\/neutrons.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In <\/span>last chapter<span style=\"text-align: initial;font-size: 1em\"> we learned about x-ray diffraction in reciprocal space. Structure factor is the signature of the real structure expressed in reciprocal space. In other words, the structure factor depends on the arrangement of atoms in a unit cell. In x-ray diffraction we perform measurements in reciprocal space, therefore it is convenient to work in reciprocal space. Here we define a new parameter called structure factor which can predict the intensity of peaks and their extinction from the diffraction spectrum. The structure factor is defined as<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-92 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47.png\" alt=\"\" width=\"779\" height=\"216\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47.png 779w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47-300x83.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47-768x213.png 768w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47-65x18.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47-225x62.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-47-350x97.png 350w\" sizes=\"auto, (max-width: 779px) 100vw, 779px\" \/><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><em>\u00a0<\/em><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Structure Factor for BCC lattice<\/strong><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-93\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-48.png\" alt=\"\" width=\"300\" height=\"258\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-48.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-48-65x56.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-48-225x194.png 225w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p style=\"text-align: center\"><strong>Fig 5.6: <\/strong>A unit cell of BCC lattice<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The bcc basis referred to the cubic cell has identical atoms at <em>x<\/em><em>1<\/em> = <em>y<\/em><em>1<\/em> = <em>z<\/em><em>1<\/em> = 0 and <em>x<\/em><em>2<\/em> = <em>y<\/em><em>2<\/em> = <em>z<\/em><em>2<\/em> = \u00bd. Thus above equation becomes<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-94 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-49.png\" alt=\"\" width=\"292\" height=\"125\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-49.png 292w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-49-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-49-225x96.png 225w\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" \/><\/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 style=\"text-align: justify\">The idea is look for atoms per unit cell and carry the sum over all atoms in a unit cell. For example BCC structure has 2 atoms per unit cell and FCC structure has 4 atoms per unit cell. In BCC the sum will be carried over 2 atoms whereas in FCC it will be carried over 4 atoms.<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">Structure Factor for FCC lattice<\/strong><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-95\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-50.png\" alt=\"\" width=\"294\" height=\"246\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-50.png 294w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-50-65x54.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-50-225x188.png 225w\" sizes=\"auto, (max-width: 294px) 100vw, 294px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\"><strong>Fig: 5.7: <\/strong>A unit cell of FCC lattice<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>The basis of the FCC structure referred to the cubic cell has identical atoms at (000); (0, 0.5, 0.5); (0.5, 0, 0.5); (0.5, 0.5, 0), Thus structure factor becomes<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-96 alignleft\" src=\"http:\/\/msp07.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51.png\" alt=\"\" width=\"503\" height=\"33\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51.png 503w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51-300x20.png 300w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51-65x4.png 65w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51-225x15.png 225w, https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-content\/uploads\/sites\/107\/2018\/11\/Untitled-51-350x23.png 350w\" sizes=\"auto, (max-width: 503px) 100vw, 503px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><em>S<\/em><em>G<\/em> = 4<em>f <\/em>if<em> h, k, l <\/em>are either all odd or all even<\/p>\n<p>&nbsp;<\/p>\n<p><em>S<\/em><em>G<\/em> = 0, otherwise<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When structure factor becomes zero that means the diffraction peak corresponding to that plane (<em>h k l<\/em>) will become extinct.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Value Addition:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Do You Know?<\/strong><\/p>\n<ol>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The concept of reciprocal lattice is very useful in x-ray diffraction measurements. For a measurement in a real <\/span>space<span style=\"text-align: initial;font-size: 1em\"> we can do it with meter scale, microscope and electron microscopy depending on the length scale which is in question. In real space <\/span>measurement<span style=\"text-align: initial;font-size: 1em\"> we scan the location in real space and measure the property. <\/span>Similarly<span style=\"text-align: initial;font-size: 1em\"> in x-ray diffraction <\/span>measurement<span style=\"text-align: initial;font-size: 1em\"> we scan the reciprocal space ( | <\/span><em style=\"text-align: initial;font-size: 1em\">K<\/em><span style=\"text-align: initial;font-size: 1em\"> |= 2 | <\/span><em style=\"text-align: initial;font-size: 1em\">k<\/em><em style=\"text-align: initial;font-size: 1em\">in<\/em><span style=\"text-align: initial;font-size: 1em\"> | sin <\/span><em style=\"text-align: initial;font-size: 1em\">q<\/em><span style=\"text-align: initial;font-size: 1em\"> ) when we scan\u00a0<\/span>angle q and measure the property (Intensity). Therefore X-ray measurement is a measurement of reciprocal space. Thus we obtain the reciprocal lattice associated with the crystalline structure. The Fourier transform of the reciprocal lattice gives us the knowledge of real space lattice.<\/li>\n<li style=\"text-align: justify\">Structure factors are very useful in x-ray crystallography. Structure factors in solids are intimately related to the atomic arrangement of solids. Looking at the structure factor it is possible to determine if the material is crystalline or amorphous. Also<span style=\"text-align: initial;font-size: 1em\"> the intensity of the diffraction peak can be explained from <\/span>structure<span style=\"text-align: initial;font-size: 1em\"> factor. Using several diffraction patterns from various angles, the measured structure factors are used to determine the structure of very complex molecules such as proteins and DNA etc.<\/span><\/li>\n<\/ol>\n<div>\n<p><strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Suggested Reading<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>For More Details (on this topic and other 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 style=\"text-align: justify\">Charles Kittel, Introduction to Solid State Physics, John Wiley &amp; Sons, Singapore 1999<\/li>\n<li style=\"text-align: justify\">Jens Als-Nielsen and Des McMorrow, John Wiley &amp; Sons, UK, 2011<\/li>\n<li style=\"text-align: justify\">Wikipedia<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":6,"template":"","meta":{"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-73","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\/73","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":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/73\/revisions"}],"predecessor-version":[{"id":98,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapters\/73\/revisions\/98"}],"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\/73\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/media?parent=73"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/pressbooks\/v2\/chapter-type?post=73"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/contributor?post=73"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/msp07\/wp-json\/wp\/v2\/license?post=73"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}