Transmission Electron Microscopy
12. Reciprocal Space
EMA 6518Spring 2007
02/21/07
EMA 6518: Transmission Electron Microscopy C. Wang
Diffraction from Crystals
EMA 6518: Transmission Electron Microscopy C. Wang
•A crystal is a three dimensional diffraction grating•The lattice periodicity of the crystal determines the sampling
regions of the diffraction pattern
•Where the peaks appear•The unit cell contents give you the envelope function
•The intensity of the peaks
Double slits experiment
Another Lattice-Reciprocal lattice
• Reciprocal lattice vectors
• Reciprocal lattice: a lattice in reciprocal space
• Reciprocal space: Think of any crystal as having two lattices, one real and the other reciprocal.
“real” space vs. “reciprocal” space
if something is large in real space, then it’s small in reciprocal space
• The reciprocal lattice gives us a method for picturing the geometry of diffraction; it gives us a “pictorial representation” of diffraction.
EMA 6518: Transmission Electron Microscopy C. Wang
Another Lattice-Reciprocal lattice
• k and g --- reciprocal lattice vectors
• In the reciprocal lattice, sets of parallel (hkl) atomic
planes are represented by a single point located a distance 1/dhkl from the lattice origin.
EMA 6518: Transmission Electron Microscopy C. Wang
Kd
nB
==λ
θsin2
The vector K is reciprocally related to d, and vice versa.
Reciprocal Lattice• In real space, we can define any lattice vector, rn, by
rn=n1a+n2b+n3c
where the vectors a, b, and c are the unit-cell translations in real space while n1, n2, and n3 are all
integers.
• Any reciprocal lattice vector, r*, can be defined in a similar manner
r*=m1a*+m2b*+m3c*
where a*, b*, and c* are the unit-cell translations in
reciprocal space while m1, m2, and m3 are all integers.
EMA 6518: Transmission Electron Microscopy C. Wang
1c*cb*ba*a
0b*ca*ca*bc*bc*ab*a
=⋅=⋅=⋅
=⋅=⋅=⋅=⋅=⋅=⋅
EMA 6518: Transmission Electron Microscopy C. Wang
Reciprocal Lattice
•Be careful, this result does not mean that a* is parallel to a.•The direction of a* is completely defined by
• a* is perpendicular to both b and c and must therefore be the normal to the plane containing b and c.
• defines the length of the
vector a* in terms of the length of the vector a. It gives
the scale or dimension of the reciprocal lattice. The product of the projection of a* on the vector a multiplied by
the length of a is unity.
0b*ca*ca*bc*bc*ab*a =⋅=⋅=⋅=⋅=⋅=⋅
1c*cb*ba*a =⋅=⋅=⋅
Reciprocal Lattice
EMA 6518: Transmission Electron Microscopy C. Wang
• The vector g:
ghkl
=ha*+kb*+lc*where h, k and l are all integers and together define the plane (hkl)
• The definition of the plane (hkl) is that it cuts the a, b, and c axes at 1/h, 1/k, and 1/l, respectively.
Reciprocal Lattice
EMA 6518: Transmission Electron Microscopy C. Wang
• AB=b/k-a/h. This vector and all vectors (AB, BC and CA) in the (hkl) plane are normal to the vector ghkl
0*)c*b*a()ab
(
0gAB
=++⋅−
=⋅
lkhhk
hkl
• The unit vector, n, parallel to g is simply
The shortest distance from the origin O to the plane is the
dot product of n with vector OB
gg/
g
1a
g
c*)*b*a(a
g
gan =⋅
++=⋅=⋅
h
lkh
hh
Reciprocal Lattice
EMA 6518: Transmission Electron Microscopy C. Wang
g
1a
g
c*)*b*a(a
g
gan =⋅
++=⋅=⋅
h
lkh
hh
g
1=
hkld
• The definition of the hkl indices is OA=a/h; OB=b/k; OC=c/l
•The plane ABC can then be represented as (hkl)
Reciprocal Lattice
EMA 6518: Transmission Electron Microscopy C. Wang
• Reciprocal lattice is so called because all lengths are in reciprocal units.
• Reciprocal-space notation:
(hkl) is shorthand notation fro a particular vector in reciprocal space, {hkl}is then the general form for these reciprocal lattice vectors. [UVW] is a particular plane in reciprocal space.
• Warning: The real-lattice vectors and the reciprocal-lattice vectors with the same indices (e.g., [123] and the normal to the plane (123)) are parallel only in the case of cubic materials. In other material, some special vectors may be parallel to one another, but most pairs will not be parallel.
Reciprocal Lattice
EMA 6518: Transmission Electron Microscopy C. Wang
The Laue Equations
EMA 6518: Transmission Electron Microscopy C. Wang
•The Bragg equation does not explicitly tell us aboutthe directions in which diffraction occurs
•We have to remember that the line bisecting the incoming and outgoing beams is always perpendicular
to the planes responsible for diffraction
•Laue equations make the directionality of the processmore obvious as we have a set of three equations, one
for each crystallographic axis that must be
simultaneously satisfied
The Laue Equations
EMA 6518: Transmission Electron Microscopy C. Wang
We assume that the crystal is infinitely large; we can always take the reciprocal lattice to be infinite.
The Laue Equations
EMA 6518: Transmission Electron Microscopy C. Wang
K=g•This equation represents the Laue conditions for constructive interference; so we will refer to this as the
condition fro Laue, or Bragg, diffraction.
•Laue conditions: N=⋅ nrKWe must satisfy certain conditions on K in order to have Bragg (or Laue) diffraction.
l
k
h
=⋅
=⋅
=⋅
cK
bK
aK
The Ewald Sphere of Reflection
EMA 6518: Transmission Electron Microscopy C. Wang
The reciprocal lattice is a 3D array of points, each of
which we will now
associate with a reciprocal-lattice rod, or
“relrod” for short, which is centered on the point.
• When the sphere cuts through the reciprocal lattice point the Bragg condition is satisfied. When it cuts through a rod you still see a diffraction spot, even though the Bragg condition is not satisfied.
• The value for this intensity is such that if the Ewald sphere cuts through that point in reciprocal space, then the diffracted beam, g, will have that intensity.
• If the Ewald sphere moves, the intensity will change.
• The vector CO is kI and has length 1/λ; this defines where C is located, i.e., we start with O and measure back to C.
EMA 6518: Transmission Electron Microscopy C. Wang
The Ewald Sphere of Reflection
Reciprocal Space
EMA 6518: Transmission Electron Microscopy C. Wang
Reciprocal Space
EMA 6518: Transmission Electron Microscopy C. Wang
Reciprocal Space
EMA 6518: Transmission Electron Microscopy C. Wang
Reciprocal Space
EMA 6518: Transmission Electron Microscopy C. Wang
EMA 6518: Transmission Electron Microscopy C. Wang
http://emaps.mrl.uiuc.edu/emaps.asp
EMA 6518: Transmission Electron Microscopy C. Wang
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