Anandh Subramaniam & Kantesh Balani Looking at Bragg’s equation a few tantalizing questions come to our mind:  There is no mention of interatomic spacing in the plane!  What about in-plane scattering wherein incident  scattered  but the waves are still in phase We take up these questions in this set of slides MATERIALS SCIENCE & ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

Note that in the Bragg’s equation: The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appears  Change in position or spacing of atoms along the plane should not affect Bragg’s condition !! d Note: shift (systematic) is actually not a problem!

Note: shift is actually not a problem Note: shift is actually not a problem!  Why is ‘systematic’ shift not a problem?

Consider the case for which 1  2 Constructive interference can still occur if the difference in the path length traversed by R1 and R2 before and after scattering are an integral multiple of the wavelength  (AY − XC) = h  (h is an integer)

This is looking at diffraction from atomic arrays and not planes Generalizing into 3D Laue’s equations S0  incoming X-ray beam S  Scattered X-ray beam This is looking at diffraction from atomic arrays and not planes

A physical picture of scattering leading to diffraction is embodied in Laue’s equations Bragg’s method of visualizing diffraction as “reflection” from a set of planes is a different way of understanding the phenomenon of diffraction from crystals The ‘plane picture’ (Bragg’s equations) are simpler and we usually stick to them Hence, we should think twice before asking the question: “if there are no atoms in the scattering planes (actually a subset of the planes being considered), how are they scattering waves?”