X-Ray Diffraction

• Path Length

• Phase Difference

• For a new atom in a unit cell

xa

yb

• Diffraction angle:• 2 d sin = • sin = /2d

Bright spot d

(h,k,l)

• Reflection from plane through new atom

• Planes at same angle

• New plane at spacing = (hx + ky + lz) d

• Extra path length: x = sin• On each side

sin

sin

• Extra path length: x = 2 sin • But sin = /2d Bragg spot

x = /d

sin

sin

x

• phase difference:• 2 radians in one • x 2/ = 2 /d

sin

sin

• phase difference: = 2 (hx + ky + lz)

sin

sin

• phase difference: = 2 (hx + ky + lz)

sin

sin

2ix/e

e2ix/+

• phase difference: = 2 (hx + ky + lz)

sin

sin

2ix/e

e2ix/e2i(hx+ky+lz)

Systematic Extinctions

• Body Centered Cubic

• x = ½ y = ½ z = ½ = 2 (hx + ky + lz) = 2 (h/2 + k/2 + l/2)

xa

yb

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

xa

yb