Transmission Electron Microscopy 12. Reciprocal Spaceweb.eng.fiu.edu/wangc/EMA 6518 TEM 12.pdf ·...

Transmission Electron Microscopy

12. Reciprocal Space

EMA 6518Spring 2007

02/21/07

EMA 6518: Transmission Electron Microscopy C. Wang

Diffraction from Crystals


•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.


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.


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.


1c*cb*ba*a

0b*ca*ca*bc*bc*ab*a

=⋅=⋅=⋅

=⋅=⋅=⋅=⋅=⋅=⋅


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


• 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


• 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


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


• 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


The Laue Equations


•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


We assume that the crystal is infinitely large; we can always take the reciprocal lattice to be infinite.

The Laue Equations


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


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.


The Ewald Sphere of Reflection

Reciprocal Space



http://emaps.mrl.uiuc.edu/emaps.asp

Transmission Electron Microscopy 12. Reciprocal Spaceweb.eng.fiu.edu/wangc/EMA 6518 TEM 12.pdf ·...

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