john bargar 2nd annual ssrl school on hard x-ray scattering techniques in materials and...
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John Bargar2nd Annual SSRL School on Hard X-ray Scattering Techniques in
Materials and Environmental SciencesMay 15-17, 2007
What use is Reciprocal Space? An Introduction
a*
b*
x You are here
The reciprocal lattice is the Fourier transform of the electron density distribution in a crystal.
Scattering from a crystal occurs when the following expression evaluates to non-zero values:
F NOT!NOT!
OUTLINE
I. What is the reciprocal lattice?
1. Bragg’s law.
2. Ewald sphere.
3. Reciprocal Lattice.
II. How do you use it?
1. Types of scans:
Longitudinal or θ-2θ,
Rocking curve scan
Arbitrary reciprocal space scan
BUT…
• There are a gabillion planes in a crystal.
• How do we keep track of them?
• How do we know where they will diffract (single xtals)?
• What are their diffraction intensities?
BUT…
• There are a gabillion planes in a crystal.
• How do we keep track of them?
• How do we know where they will diffract (single xtals)?
• What are their diffraction intensities?
Starting from Braggs’ law…
Bragg’s Law: 2d sin = n
d
d
A
B
A’
B’
• Good phenomenologically• Good enough for a Nobel
prize (1915)
Better approach…
• Make a “map” of the diffraction conditions of the crystal. • For example, define a map spot for each diffraction condition.• Each spot represents kajillions of parallel atomic planes.
• Such a map could provide a convenient way to describe the relationships between planes in a crystal – a considerable simplification of a messy and redundant problem.
In the end, we’ll show that the reciprocal lattice provides such a map…
To show this, start again from diffracting planes…
Define unit vectors s0, s
d
d
A
B
A’
B’
• Notice that |s-s0| = 2Sinθ
• Substitute in Bragg’s law…
1/d = 2Sinθ/λ …
Diffraction occurs when
|s-s0|/λ = 1/d
(Note, for those familiar with q…
q = 2π|s-s0|
Bragg’s law: q = 2π/d = 4πSinθ/ λ
s0 ss – s0
s0
s – s0
To show this, start again from diffracting planes…
Define a map point at the end of the scattering vector at Bragg condition
d
d
A
B
A’
B’
Diffraction occurs when scattering vector connects to
map point.
Scattering vectors (s-s0/λ or q) have reciprocal lengths (1/λ).
Diffraction points define a reciprocal lattice.
Vector representation carries Bragg’s law into 3D.
Map point
s – s0
λ
Families of planes become points!
Single point now represents all planes in all unit cells of the crystal that are parallel to the crystal plane of interest and have same d value.
d
A
B
A’
B’
s0/λ s/λ
d
s – s0
λ
Ewald Sphere
A
Diffraction occurs only when map point intersects circle.
=1/d
A’
s – s0
λ s0/λ
s/λ
Circumscribe circle with radius 2/λ around scattering vectors…
Origin
s0
s
Thus, the RECIPROCAL LATTICE is obtained
1/d
Distances between origin and RL points give 1/d.
Reciprocal Lattice Axes:a* normal to a-b planeb* normal to a-c planec* normal to b-c plane
Index RL points based upon axes
Each point represents all parallel crystal planes. Eg., all planes parallel to the a-c plane
are captured by (010) spot.
Families of planes become points!
b*
a*(110)
(010)
(200)
s – s0
λ
Reciprocal Lattice of γ-LiAlO2
a*b*
a*
c*
Projection along c: hk0 layerNote 4-fold symmetry
Projection along b: h0l layer
a = b = 5.17 Å; c = 6.27 Å; P41212 (tetragonal)a* = b* = 0.19 Å-1; c* = 0.16 Å-1
general systematic absences (00ln; l≠4), ([2n-1]00)
c*a*
(200)
(400)
(600)
(020
) (0
40)
(060
) (110)
(200
) (4
00)
(600
)
(004)
(008)
In a powder, orientational averaging produces rings instead of spots
s0/λs/λ
OUTLINE
I. What is the reciprocal lattice?
1. Bragg’s law.
2. Ewald sphere.
3. Reciprocal Lattice.
II. How do you use it?
1. Types of scans:
Longitudinal or θ-2θ,
Rocking curve scan
Arbitrary reciprocal space scan
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
s0 s
0 10 20 30 40
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
s-s0/λ
0 10 20 30 40
Reciprocal lattice rotates by θ during
scan
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
s-s0/λ
0 10 20 30 40
0 10 20 30 40
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
s-s0/λ
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
0 10 20 30 40
s-s0/λ
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
0 10 20 30 40
s-s0/λ
0 10 20 30 40
1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ
0 10 20 30 40
s-s0/λ
0 10 20 30 40
• Note scan is linear in units of Sinθ/λ - not θ!• Provides information about relative arrangements, angles, and
spacings between crystal planes.
0 10 20 30 40
2. Rocking Curve scanSample moves on θ, Detector fixed
Provides information on sample mosaicity & quality of orientation
s-s0/λFirst crystallite
Second crystallite
Third crystallite
2. Rocking Curve scanSample moves on θ, Detector fixed
Provides information on sample mosaicity & quality of orientation
s-s0/λ
Reciprocal lattice rotates by θ during
scan
3. Arbitrary Reciprocal Lattice scansChoose path through RL to satisfy experimental need,
e.g., CTR measurements
s-s0/λ
A note about “q”
In practice q is used instead of s-s0
d
d
A
B
A’
B’
q
|q| = |k’-k0| = 2π * |s-s0|
|q| = 4πSinθ/λ
k0 k’
• Intensities of peaks (Vailionis)• Peak width & shape (Vailionis)• Scattering from non-crystalline
materials (Huffnagel)• Scattering from whole particles
or voids (Pople)• Scattering from interfaces
(Trainor)
What we haven’t talked about:
The End
The Beginning…
Graphical Representation of Bragg’s Law
• Bragg’s law is obeyed for any triangle inscribed within the circle: Sinθ = (1/d)/(2/λ)
A A’
s – s0 s0 s
s0
= 1/d
2/λ