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

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

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

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

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

11. Systematic Extinctionsyb
Body Centered Cubic
x = ½ y = ½ z = ½
 = 2 (hx + ky + lz) = n with n odd
 = 2 (h/2 + k/2 + l/2) = n xa

12. Systematic Extinctionsyb xa
Body Centered Cubic
(1,1,1):  = 2 (1/2 + 1/2 + 1/2)
(1,1,1):  = 3 waves cancel
h + k + l = even for reflection to appear