Path Length and Phase Differences
d



(h,k)
a
A
a/h
a
xa
xa
_____________________________________________________________________________________________________________________
Unit cell side = a
Set of diffraction planes = (h,k)
New atom in unit cell at position = xa
Path length difference for rays reflecting from old and new planes = 
_____________________________________________________________________________________________________________________
a2
for large : d = h2 – A2
2
_____________________________________________________________________________________________________________________
for small : 2 = x2a2 – 2
_____________________________________________________________________________________________________________________
a
A = h sin 
 = xa sin 
_____________________________________________________________________________________________________________________
2 = x2a2 (1 – sin2)
a2
2
d = h2 (1 – sin2)
}
 xa
d = a = hx
h
_____________________________________________________________________________________________________________________
 = (hx) d
in general:  = (hx + ky + lz) d
_____________________________________________________________________________________________________________________
diffraction angle: 2 d sin  = 
sin  = /2d
_______________________________________________________________
extra path length: x = 2  sin  = /d

_______________________________________________________________
2

= 2 d

phase difference:  = 2 (hx + ky + lz)
phase difference: x
 sin


_______________________________________________________________
 sin
Colby College
plane wave amplitude:
2i (hx + ky + lz)
e