Bragg Planes
How to do a Fourier transform on
paper with no calculations at all.
Bragg planes are always
perpendicular to S
s0
-s0
S

s

Since s0 and s are the same length and have the same
angle to the reflection plane, S = (s-s0)/ is normal to the
plane.
The length of S is 1/d
s0
-s0
S

s



The length of S is 2sin times the lengths of s and s0, which
is 1/. So |S| = 2sin/ = 1/d
Using Miller indeces: S = ha*+kb*+lc*
• Max von Laue says: The vectors S that have
•
amplitude > 0 are the ones where the Bragg planes
all line up with the unit cell origins.
This must be true for all unit cells in the crystal
(ta+ub+vc) to scatter with the same phase.
210
170
d from S using Miller indeces
1
1
1
d 

S (ha*)2  (kb*)2  (lc*)2 (h /a)2  (k /b)2  (l/c)2

Axes a,b,c are all orthogonal.
Where the first Bragg plane cuts the axes
• The n=1 Bragg plane (normal to S at
distance d) cuts the unit cell axes at
1/h
210
1/k
1/l
If indeces hkl are doubled, Bragg
distance d is halved.
• All unit cell origins have phase zero. But not all phase-zero
Bragg planes must go through a unit cell origin. For
example, the n=odd Bragg planes for the 0 2 0 reflection
does not touch a single unit cell origin.
010
020
210
420
3D Bragg planes
(2 3 3) Bragg planes
(4 6 6) Bragg planes
Phase-zero planes intersect the cell axes at fractional coordinates
(1/h,0,0), (0,1/k,0),(0,0,1/l)
Calculating the structure factors
• Draw a plane that intersects the unit cell
axes at 1/h, 1/k, and 1/l (careful to
consider the sign of h,k,l)
• Measure the phase of each atom as its
distance from the nearest Bragg plane,
divided by d and multiplied by 360°.
• Draw the scattering factor for that atom,
and sum the scattering factors to get the
structure factor.
In class exercise:
Calculate structure factors:
F( 1 1 0)
F(-1 1 0)
F(-2 1 0)
For a unit cell with two atoms:
carbon (amplitude 6) @ (0.5, 0.2, 0.0)
oxygen (amplitude 8) @ (0.3, 0.4, 0.0)
b
a
Calculating the density
• Given the structure factors F(hkl), find the
•
•
•
point(s) of maximum e-density. F(hkl) = |F(hkl)|ei
Draw Bragg planes with phase =  (Measure
phase in the direction (h,k,l))
Erase Bragg planes with phase = +180°
After drawing and erasing all F’s, the darkest
areas are the locations of the atoms.
In class exercise:
Find the maximum density point given
the following structure factors:
F( 1 1 0) = 1 ei(108°)
F(0 1 0) = 1 ei(180°)
F(1 1 0) = 1 ei(-60°)
F(-1 1 0) = 1 ei(60°)
F(-2 1 0) = 1 ei(-10°)
b
a
Terms we have learned
•
•
•
•
•
•
•
•
•
Reflection
Structure factor
Bragg planes
Scattering factor
Ewald sphere
Laue conditions
Reciprocal space
Miller indeces
Fourier transform