X-Ray Diffraction
yb
• Path Length
• Phase Difference
• For a new atom in a unit cell
xa

d
• Diffraction angle:
• 2 d sin = 
• sin = /2d
Bright
spot 
(h,k,l)

• Reflection from plane
through new atom
• Planes at same angle



• New plane at spacing
•  = (hx + ky + lz) d


 sin
• Extra path length:
• x =  sin
• On each side


 sin


 sin

 sin
•
•
•
•
Extra path length:
x = 2  sin
But sin = /2d Bragg spot
x = /d
x

 sin

 sin
• phase difference:
• 2 radians in one 
• x 2/ = 2 /d

• phase difference:
•  = 2 (hx + ky + lz)

 sin

 sin

• phase difference:
•  = 2 (hx + ky + lz)

 sin

 sin
2ix/
e
2ix/+
e
• phase difference:
•  = 2 (hx + ky + lz)
2ix/

 sin

 sin
e
e
2ix/ 2i(hx+ky+lz)
e
•
•
•
•
yb
Systematic Extinctions
Body Centered Cubic
xa
x=½ y=½ z=½
 = 2 (hx + ky + lz) = n with n odd
 = 2 (h/2 + k/2 + l/2) = n
yb
Systematic Extinctions
•
•
•
•
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