1. No category

Reciprocal Lattice
&
Ewald Sphere Construction
Part of
MATERIALS SCIENCE
& A Learner’s Guide
ENGINEERING
AN INTRODUCTORY E-BOOK
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur- 208016
Email: [email protected], URL: home.iitk.ac.in/~anandh
http://home.iitk.ac.in/~anandh/E-book.htm
Reciprocal Lattice and Reciprocal Crystals
 A crystal resides in real space. The diffraction pattern of the crystal in Fraunhofer
diffraction geometry resides in Reciprocal Space. In a diffraction experiment (powder
diffraction using X-rays, selected area diffraction in a TEM), a part of this reciprocal
space is usually sampled.
 From the real lattice the reciprocal lattice can be geometrically constructed. The properties
of the reciprocal lattice are ‘inverse’ of the real lattice → planes ‘far away’ in the real
crystal are closer to the origin in the reciprocal lattice.
 As a real crystal can be thought of as decoration of a lattice with motif; a reciprocal crystal
can be visualized as a Reciprocal Lattice decorated with a motif* of Intensities.
 Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif*
 The reciprocal of the ‘reciprocal lattice’ is nothing but the real lattice!
 Planes in real lattice become points in reciprocal lattice and vice-versa.
* Clearly, this is not the crystal motif- but a motif consisting of “Intensities”.
Motivation for constructing reciprocal lattices
 In diffraction patterns (Fraunhofer geometry) (e.g. SAD), planes are mapped as spots (ideally points).
Hence, we would like to have a construction which maps planes in a real crystal as points.
 Apart from the use in ‘diffraction studies’ we will see that it makes sense to use reciprocal lattice
when we are dealing with planes.
 The crystal ‘resides’ in Real Space, while the diffraction pattern ‘lives’ in Reciprocal Space.
 As the index of the plane increases
→ the interplanar spacing decreases
→ and ‘planes start to crowd’ around the origin in
the real lattice (refer figure).
 Hence, it is a ‘nice idea’ to work in reciprocal
space (i.e. work with reciprocal lattice) when
dealing with planes.
As seen in the figure the diagonal is divided
into (h + k) parts.
We will construct reciprocal lattices in 1D and 2D
before taking up a formal definition in 3D
Let us start with a one dimensional lattice and construct the reciprocal lattice
Real Lattice
Reciprocal Lattice
How is this reciprocal lattice constructed?
One unit cell
 The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc.
 As the index of the plane increases it gets closer to the origin (there is a
crowding towards the origin)
 What do these planes wih fractional indices mean?
 We have already noted the answer in the topic on Miller indices and XRD.
Real Lattice
Note in 1D planes are points and
have Miller indices of single digit
(they have been extended into the
second dimension (as lines) for
better visibility
Reciprocal Lattice
Each one of these points
correspond to a set of
‘planes’ in real space
Note that in reciprocal space index
has NO brackets
Now let us construct some 2D reciprocal lattices
Example-1
Reciprocal Lattice
Each one of these points correspond to
a set of ‘planes’ in real space
02
(01)
12
1

a2
(11)
(21)

a1
a1
(10)
22
a2
11
01
*
11
*
b2
g
00
*
b1
21
*
g 21
10
1
g vectors connect
origin to reciprocal
lattice points
20
a1
The reciprocal lattice has an origin!
Overlay of real and
reciprocal lattices
Note that vectors in reciprocal
space are perpendicular to planes
in real space (as constructed!)
But do not measure distances from the figure!
Example-2
The real lattice

a2

a1
*
b2
*
b1
The reciprocal lattice
Note that vectors in reciprocal space
are perpendicular to planes in real
space (as constructed!)
But do not measure distances from the figure!
Reciprocal Lattice
Properties are reciprocal to the crystal lattice
The basis vectors of a reciprocal lattice are defined using the basis vectors of the crystal as below
BASIS
VECTORS
* 1  
b1  a2  a3 
V
* 1  
b2  a3  a1 
V
*
*
bi usuall written as ai
* 1  
b3  a1  a2 
V
*
b3
* 1  
b  b3  a1  a2 
V
Area (OAMB )
1


Area(OAMB )  Height of Cell OP
C
*
3
1
b 
d 001
*
3

a3

a2
P
B
B
*


b3 is  to a1 and a2
O
The reciprocal lattice is created by interplanar spacings
M
A

a1
Some properties of the reciprocal lattice and its relation to the real lattice
 A reciprocal lattice vector is  to the corresponding real lattice plane
*
*
*
*
g hkl  h b1  k b2  l b3
 The length of a reciprocal lattice vector is the reciprocal of the spacing of the
corresponding real lattice plane
g
*
hkl
*
1
 g hkl 
d hkl
 Planes in the crystal become lattice points in the reciprocal lattice
 ALTERNATE CONSTRUCTION OF THE REAL LATTICE
 Reciprocal lattice point represents the orientation and spacing of a set of planes
Going from the reciprocal lattice to diffraction spots in an experiment
 A Selected Area Diffraction (SAD) pattern in a TEM is similar to a section through the
reciprocal lattice (or more precisely the reciprocal crystal, wherein each reciprocal lattice
point has been decorated with a certain intensity).
 The reciprocal crystal has all the information about the atomic positions and the atomic
species.
 Reciprocal lattice* is the reciprocal of a primitive lattice and is purely geometrical
 does not deal with the intensities decorating the points
Physics comes in from the following:
 For non-primitive cells ( lattices with additional points) and for crystals having motifs (
crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the
corresponding scattering power (|Fhkl|2)
 Some of the Reciprocal lattice points go missing (or may be scaled up or down in
intensity)
 Making of Reciprocal Crystal: Reciprocal lattice decorated with a motif of scattering
power (as intensities)
 The Ewald sphere construction further can select those points which are actually observed
in a diffraction experiment
* as considered here
To summarize:
Real Lattice
Decoration of the lattice with motif
Real Crystal
Purely Geometrical Construction
Reciprocal Lattice
Decoration of the lattice with Intensities
Structure factor calculation
Reciprocal Crystal
Ewald Sphere construction
Selection of some spots/intensities from the
reciprocal crystal
Diffraction Pattern
Crystal = Lattice + Motif
 In crystals based on a particular lattice the intensities of particular
reflections are modified  they may even go missing
Position of the diffraction spots
Lattice
Intensity of the diffraction spots
Motif
Diffraction Pattern
Position of the diffraction spots
 RECIPROCAL LATTICE
Intensity of the diffraction spots
 MOTIF’ OF INTENSITIES
Making of a Reciprocal Crystal
 There are two ways of constructing the Reciprocal Crystal:
1) Construct the lattice and decorate each lattice point with appropriate intensity*
2) Use the concept as that for the real crystal**
 The above two approaches are equivalent for simple crystals (SC, BCC, FCC lattices
decorated with monoatomic motifs), but for ordered crystals the two approaches are
different (E.g. ordered CuZn, Ordered Ni3Al etc.) (as shown soon).
Real Lattice
Decorate with motif
Real Crystal
Use motif to compute
structure factor and hence
intensities to decorate
reciprocal lattice points
Take real lattice and construct
reciprocal lattice
Reciprocal Lattice
Reciprocal Crystal
* Point #1 has been considered to be consistent with literature– though this might be an inappropriate.
** Point #2 makes reciprocal crystals equivalent in definition to real crystals
SC
Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)
+
Selection rule: All (hkl) allowed
In ‘simple’ cubic crystals there are No missing
reflections
Single
sphere motif
Lattice = SC
001
101
011
111
000
010
=
100
SC crystal
110
Reciprocal Crystal = SC
SC lattice with Intensities as the motif at each
‘reciprocal’ lattice point
Figures NOT to Scale
BCC
Selection rule BCC: (h+k+l) even allowed
In BCC 100, 111, 210, etc. go missing
002
BCC crystal
x
022
x
x
202
x
222
x
Important note:
 The 100, 111, 210, etc. points in the
reciprocal lattice exist (as the corresponding
real lattice planes exist), however the
intensity decorating these points is zero.
101
x
x
x
000
x
100 missing reflection (F = 0)
x
011
200
020
x
110
x
220
Weighing factor for each point “motif”
F 4f
2
2
Reciprocal Crystal = FCC
FCC lattice with Intensities as the motif
Figures NOT to Scale
002
022
FCC
202
222
111
020
000
Lattice = FCC
200
100 missing reflection (F = 0)
220
110 missing reflection (F = 0)
Weighing factor for each point “motif”
Reciprocal Crystal = BCC
F  16 f
2
2
BCC lattice with Intensities as the motif
Figures NOT to Scale
Order-disorder transformation and its effect on diffraction pattern
 When a disordered structure becomes an ordered structure (at lower temperature), the
symmetry of the structure is lowered and certain superlattice spots appear in the Reciprocal
Lattice/crystal (and correspondingly in the appropriate diffraction patterns). Superlattice
spots are weaker in intensity than the spots in the disordered structure.
 An example of an order-disorder transformation is in the Cu-Zn system: the high
temperature structure can be referred to the BCC lattice and the low temperature structure
to the SC lattice (as shown next). Another examples are as below.
Disordered
Ordered
- NiAl, BCC
B2 (CsCl type)
- Ni3Al, FCC
L12 (AuCu3-I type)
Click here to know more about Superlattices & Sublattices
Click here to know more about Ordered Structures
Diagrams not to scale
Positional Order
In a strict sense this is not a crystal !!
High T disordered
BCC
Probabilistic occupation of each BCC lattice site:
50% by Cu, 50% by Zn
G = H  TS
470ºC
Sublattice-1 (SL-1)
Sublattice-2 (SL-2)
Low T ordered
SC
SL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (½, ½, ½)
Disordered
Ordered
- NiAl, BCC
B2 (CsCl type)
Click here to see structure factor
calculation for NiAl (to see why
some spots have weak intensity)
→ Slide 27
Click here to see XRD powder
pattern of NiAl → Slide 5
‘Diffraction pattern’ from
the ordered structure (3D)
Ordered
FCC
BCC
SC
For the ordered structure:
Reciprocal Crystal = ‘FCC’
FCC lattice with Intensities as the motif
Notes:
 For the disordered structure (BCC) the reciprocal crystal is FCC.
 For the ordered structure the reciprocal crystal is still FCC but with
a two intensity motif: ‘Strong’ reflection at (0,0,0) and superlattice
(weak) reflection at (½,0,0) .
 So we cannot ‘blindly’ say that if lattice is SC then reciprocal
lattice is also SC.
Reciprocal crystal
This is like the NaCl structure in
Reciprocal Space!
Disordered
Ordered
- Ni3Al, FCC
L12 (AuCu3-I type)
Click here to see structure factor
calculation for Ni3Al (to see
why some spots have weak
intensity) → Slide 29
Click here to see XRD powder
pattern of Ni3Al → Slide 6
Diffraction pattern from the
ordered structure (3D)
Ordered
BCC
FCC
SC
Reciprocal Crystal = BCC
BCC lattice with Intensities as the motif
Reciprocal crystal

There are two ways of constructing the Reciprocal Crystal:
1) Construct the lattice and decorate each lattice point with appropriate intensity
2) Use the concept as that for the real crystal (lattice + Motif)
1) SC + two kinds of Intensities decorating the lattice
2) (FCC) + (Motif = 1FR + 1SLR)
Motif
 FR  Fundamental Reflection
 SLR  Superlattice Reflection
1) SC + two kinds of Intensities decorating the lattice
2) (BCC) + (Motif = 1FR + 3SLR)
Motif
Example of superlattice spots in a TEM diffraction pattern
The spots are
~periodically arranged
[112]
[111]
Superlattice spots
[011]
SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4Zn94Y2 alloy showing important zones
d [nm]
2 [deg.]
I
hkl
1
0.2886
30.960
163.1
100
2
0.2041
44.360
1000.0 1 1 0
12
3
0.1666
55.080
33.2
111
8
4
0.1443
64.520
147.1
200
6
5
0.1291
73.280
31.7
210
24
6
0.1178
81.660
276.5
211
24
7
0.1020
98.040
91.0
220
12
8
0.0962
106.400
2.5
300
6
9
0.0962
106.400
10.1
221
24
10
0.0913
115.140
161.7
310
24
11
0.0870
124.560
9.4
311
24
12
0.0833
135.220
64.8
222
8
13
0.0800
148.460
13.4
320
24
Example of superlattice peaks in XRD pattern
NiAl pattern from 0-160 (2)
Superlattice reflections (weak)
Mul.
6
The Ewald Sphere
* Paul Peter Ewald (German physicist and crystallographer; 1888-1985)
 Reciprocal lattice/crystal is a map of the crystal in reciprocal space → but it does not tell
us which spots/reflections would be observed in an actual experiment.
 The Ewald sphere construction selects those points which are actually observed in a
diffraction experiment
Circular of a Colloquium held at Max-Planck-Institut für Metallforschung (in honour of Prof.Ewald)
7. Paul-Peter-Ewald-Kolloquium
Freitag, 17. Juli 2008
organisiert von:
Max-Planck-Institut für Metallforschung
Institut für Theoretische und Angewandte Physik,
Institut für Metallkunde,
Institut für Nichtmetallische Anorganische Materialien
der Universität Stuttgart
Programm
13:30
Joachim Spatz (Max-Planck-Institut für Metallforschung)
Begrüßung
13:45
Heribert Knorr (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg
Begrüßung
14:00
Stefan Hell (Max-Planck-Institut für Biophysikalische Chemie)
Nano-Auflösung mit fokussiertem Licht
14:30
Antoni Tomsia (Lawrence Berkeley National Laboratory)
Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone
15:00
Pause
Kaffee und Getränke
15:30
Frank Gießelmann(Universität Stuttgart)
Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanoröhren: Aktuelle Themen der
Flüssigkristallforschung
16:00
Verleihung des Günter-Petzow-Preises 2008
16:15
Udo Welzel (Max-Planck-Institut für Metallforschung)
Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind
ab 17:00
Sommerfest des Max-Planck-Instituts für Metallforschung
The Ewald Sphere
 The reciprocal lattice points are the values of momentum transfer for which the
Bragg’s equation is satisfied.
 For diffraction to occur the scattering vector must be equal to a reciprocal lattice
vector.
 Geometrically  if the origin of reciprocal space is placed at the tip of ki then
diffraction will occur only for those reciprocal lattice points that lie on the surface
of the Ewald sphere.
 Here, for illustration, we consider a 2D section thought the Ewald Sphere (the
‘Ewald Circle’)
See Cullity’s book: A15-4
Bragg’s equation revisited
n   2 d hkl Sin hkl
Rewrite
Sin hkl 
 2
d hkl

1 d hkl
2
 Draw a circle with diameter 2/
 Construct a triangle with the diameter as the hypotenuse and 1/dhkl as a side
(any triangle inscribed in a circle with the diameter as the hypotenuse is a right angle
triangle: APO = 90): AOP
 The angle opposite the 1/d side is hkl (from the rewritten Bragg’s equation)
 Now if we overlay ‘real space’ information on the Ewald Sphere. (i.e. we are going to ‘mixup’ real and reciprocal space information).
 Assume the incident ray along AC and the diffracted ray along CP. Then automatically the
crystal will have to be considered to be located at C with an orientation such that the dhkl
planes bisect the angle OCP (OCP = 2).
 OP becomes the reciprocal space vector ghkl (often reciprocal space vectors are written without the ‘*’).
g
*
hkl
*
1
 g hkl 
d hkl
The Ewald Sphere construction
Which leads to spheres for various hkl reflections
Crystal related information is present in the reciprocal crystal
Sin hkl 
 2
d hkl
1 d hkl

The Ewald sphere construction generates the diffraction pattern
2
Chooses part of the reciprocal crystal which is observed in an experiment
Radiation related information is present in the Ewald Sphere
Ewald Sphere
 When the Ewald Sphere (shown as circle in 2D below) touches the reciprocal lattice point
 that reflection is observed in an experiment (41 reflection in the figure below).
Reciprocal Space
2
Ki
KD
The Ewald Sphere touches the
reciprocal lattice (for point 41)
 Bragg’s equation is satisfied for 41
02
01
00
DK
10
20
(41)
DK = K =g = Diffraction Vector
Ewald sphere  X-rays
Diffraction from Al using Cu K radiation
Row of
Rows
of reciprocal
reciprocal lattice
lattice points
points
The 111 reflection is observed at a
smaller angle 111 as compared to the
222 reflection
(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1
Ewald sphere  X-rays
Now consider Ewald sphere construction for two different crystals
of the same phase in a polycrystal/powder (considered next).
Click to compare them
(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1
POWDER METHOD
Diffraction cones and the Diffractometer geometry
 In the powder method  is fixed but  is variable (the sample consists of crystallites in
various orientations).
 A cone of ‘diffraction beams’ are produced from each set of planes (e.g. (111), (120) etc.)
(As to how these cones arise is shown in an upcoming slide).
 The diffractometer moving in an arc can intersect these cones and give rise to peaks in a
‘powder diffraction pattern’.
Click here for more details regarding the powder method
‘3D’ view of the ‘diffraction cones’
Different cones for different reflections
Diffractometer moves in a semi-circle to capture the intensity of the diffracted beams
THE POWDER METHOD
Understanding the formation of the cones
Cone of diffracted rays
In a power sample the point P can
lie on a sphere centered around O
due all possible orientations of the
crystals
The distance PO = 1/dhkl
Circular Section through the spheres
made by the hkl reflections
The 440 reflection is not observed
(as the Ewald sphere does not intersect
the reciprocal lattice point sphere)
Ewald sphere construction for Al
Allowed reflections are those for
h, k and l unmixed
The 331 reflection is not observed
Ewald sphere construction for Cu
Allowed reflections are those for
h, k and l unmixed