Diffraction
2
ky = y
ky
 rows
y
ky
Real space
x, y
Reciprocal
space kx, ky
Reciprocal space versus real space
• Reciprocal space is seen in diffraction patterns .
• Reciprocal space is momentum space = k-space .
• The momentum p = ħ k is a key quantum number
of electrons, phonons, … in solids . |k| = 2/
• Everything is backwards in reciprocal space:
Large distances x in real space transform into
small k-vectors kx = 2 /x in reciprocal space.
• Waves are diffracted perpendicular to the
diffracting lines or planes. This generates a
900 rotation of diffraction patterns.
Rosalind Franklin’s x-ray diffraction pattern
of DNA, which led to the double-helix model
(Linus Pauling’s copy)
Test patterns for simulating diffraction from DNA
Single helix
Double helix
Explaining the diffraction pattern of a helix
Horizontal
line grating
Side view of a helix (screw):
Two tilted gratings
X
Horizontal
2/d
d
Vertical
dots
Diffraction pattern of
a green laser pointer
Vertical
X
X-ray diffraction pattern of DNA
Diffraction pattern (negative)
The DNA double helix

2
b
2
p
p = period of one turn
b = base pair spacing
 = slope of the helix
p b

X-ray diffraction image of the protein myoglobin
• This image contains about 3000 diffraction spots. All that information
is needed to determine the positions of thousands of atoms in myoglobin.
• Protein crystallography has become essential for biochemistry,
because the structure of a protein determines its function .
Reciprocal space
Low Energy Electron Diffraction (LEED)
at surfaces
k = 2/D
Real space
K=
2/d
D
d
1D chain structure
2D planar structure (7x7)
Diffraction conditions
k0
k
Connect the Bragg condition
with momentum conservation
k0
k
G
k = k0 + G
k0
Energy and momentum conservation in diffraction
G-vector
Origin
Energy conservation:
Ewald sphere
|k| = |k0|
Momentum conservation: Vector triangle k = k0 + G
G arrow starts at the origin (000)