1.  In these set of slides we will try to visualize how constructive and destructive interference take place (using the Bragg’s view of diffraction as ‘reflection’ from a set of planes).  It is easy to ‘see’ as to how constructive interference takes place; however, it is not that easy to see how ‘rays’ of the Bragg angle ‘go missing’. Understanding constructive and destructive interference 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

2. Here we see waves scattered from two successive planes interfering constructively. (press page down button to see the successive graphics) Constructive Interference Note the phase difference of  introduced during the scattering by the atom.

4. Assuming that path difference of gives constructive interference: Similar to the path difference of, path difference of 2, 3 … n also constructively interfere. All Constructively interfere Also to be noted is the fact that if the path difference between Ray-1 and Ray-2 is then the path difference between Ray-1 and Ray-3 is 2 and Ray-1 and Ray-4 is 3 etc. Going across planes

5. Destructive Interference Exact destructive interference (between two planes, with path difference of /2) is easy to visualize. The angle is not Bragg’s angle (let us call it  d ).

6. At a different angle  ’ the waves scattered from two successive planes interfere (nearly) destructively Warning: this is a schematic Destructive Interference

7.  In the previous example considered  ’ was ‘far away’ (at a larger angular separation) from  (  Bragg ) and it was easy to see the (partial) destructive interference.  In other words for incidence angle of  d (couple of examples before) the phase difference of  is accrued just by traversing one ‘d’.  If the angle is just away from the Bragg angle (  Bragg ), then one will have to go deep into the crystal (many ‘d’) to find a plane (belonging to the same parallel set) which will scatter out of phase with this ray (phase difference of  ) and hence cause destructive interference.  In the example below we consider a path difference of /10 between the first and the second plane (hence, we will have to travel 5 planes into the crystal to get a path difference of /2).

8.  If such a plane (as mentioned in the page before) which scatters out of phase with a off Bragg angle ray is absent (due to finiteness of the crystal) then the ray will not be cancelled and diffraction would be observed just off Bragg angles too  line broadening! (i.e. the diffraction peak is not sharp like a  -peak in the intensity versus angle plot)  Line broadening can be used to calculate crystallite size (grain size).  This is one source of line broadening of line broadening. Other sources include: residual strain, instrumental effects, stacking faults etc. Click hereClick here to know more about peak broadening. Click hereClick here to know more about peak broadening.