1. X-Ray Diffraction • Path Length • Phase Difference • For a new atom in a unit cell yb xa

2. Diffraction angle: • 2 d sin =  • sin = /2d  Bright spot  d (h,k,l)

3.  • Reflection from plane through new atom • Planes at same angle 

4.  • New plane at spacing •  = (hx + ky + lz) d  

5. Extra path length: • x =  sin • On each side    sin    sin

6. Extra path length: • x = 2  sin • But sin = /2dBragg spot • x = /d   sin  x  sin

7. phase difference: • 2 radians in one  • x 2/ = 2 /d   sin    sin

8. phase difference: •  = 2 (hx + ky + lz)   sin    sin

9. phase difference: •  = 2 (hx + ky + lz) 2ix/  e  sin  2ix/+ e  sin

10. phase difference: •  = 2 (hx + ky + lz) 2ix/  e  sin 2ix/ e e 2i(hx+ky+lz)   sin

11. Systematic Extinctions • Body Centered Cubic • x = ½ y = ½ z = ½ •  = 2 (hx + ky + lz) •  = 2 (h/2 + k/2 + l/2) yb xa

12. Systematic Extinctions • Body Centered Cubic • (1,1,1):  = 2 (1/2 + 1/2 + 1/2) • (1,1,1):  = 3 waves cancel • h+k+l = even yb xa