1. Anandh Subramaniam & Kantesh Balani
Reciprocal Lattice
&
Ewald Sphere Construction
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide
The concepts of reciprocal lattice (and sometimes Ewald sphere construction) often ‘strike terror’ in the hearts of students.
However, these concepts are not too difficult if the fundamentals are understood in 1D and then extended to 3D.

2. Reciprocal Lattice and Reciprocal Crystals
Why study reciprocal lattices?
Often the concepts related to reciprocal lattice strikes terror in the minds of students. As we shall see this is not too difficult if concepts are first understood in 1D.
A crystal resides in real space. The diffraction pattern 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.
The diffraction pattern from a crystal (in Fraunhofer diffraction geometry), consists of a periodic array of spots (sharp peaks of intensity). (Click here to know the conditions under which this is satisfied).
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.c
 I.e. the information needed is the geometry of the lattice.
* Clearly, this is not the crystal motif- but a motif consisting of “Intensities”.

3. As index of the plane increases, the interplanar spacing decreases
One motivation for constructing reciprocal lattices
In diffraction patterns (Fraunhofer geometry & under conditions listed here: (Click here to know the conditions under which this is satisfied).) (e.g. SAD), planes are mapped as spots (ideally points). This as you will remember is the Bragg’s viewpoint of diffraction. 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’ in the real lattice (refer figure).  Hence, it is a ‘nice idea’ to work in reciprocal space (i.e. work with the reciprocal lattice), especially when dealing with planes.
As index of the plane increases, the interplanar spacing decreases

4. 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 O
Reciprocal Lattice
The periodic array of points with lattice parameter ‘a’ is transformed to a reciprocal lattice with periodicity of ‘1/a’.
The reciprocal lattice point at a distance of 1/a from the origin (O), represents the whole set of points (at a, 2a, 3a, 4a,….) in real space.
The reciprocal lattice point at ‘2/a’ comes from a set of points with fractional lattice spacing a/2 (i.e. with periodicity of a/2). The lattice with periodicity of ‘a’ is a subset of this lattice with periodicity of a/2. (Refer next slide).

5. How is this reciprocal lattice constructed?
To construct the reciprocal lattice we need not ‘go outside’ the unit cell in real space! (We already know that all the information we need about a crystal is present within the unit cell– in conjunction with translational symmetry).
Just to get a ‘feel’ for the planes we will be dealing with in the construction of 3D reciprocal lattices, we ‘extend’ these points perpendicular to the 1D line and treat them as ‘planes’.
Note there is only one “Miller” index in 1D
The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc.
As the index of the plane increases the interplanar spacing decreases and the first in the setgets closer to the origin (there is overall crowding).
What do these planes with fractional indices mean?  We have already noted the answer in the topic on Miller indices and XRD.
One unit cell
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 and for the reason stated above).
Each one of these points correspond to a set of ‘planes’ in real space
Note that the indices in reciprocal space have no brackets
Reciprocal Lattice
Note that in reciprocal space index has NO brackets

6. Funda Check
What do the various points (with indices 1, 2, 3, 4… etc.) represent in real space?
‘1’ represents these set of planes in reciprocal space (interplanar spacing ‘a’)
Reciprocal Lattice
Real Lattice a
Real Lattice
Real Lattice
‘4’ represents these set of planes in reciprocal space (interplanar spacing a/4)
‘2’ represents these set of planes in reciprocal space (interplanar spacing a/2)
Reciprocal Lattice
‘1’ represents these set of planes in reciprocal space (interplanar spacing ‘a’)
Real Lattice
‘3’ represents these set of planes in reciprocal space (interplanar spacing a/3)
Real Lattice
Note again: 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 and for the reason stated before).

7. Now let us construct some 2D reciprocal lattices
Each one of these points correspond to a set of ‘planes’ in real space
Example-1
Real Lattice
Reciprocal Lattice
g vectors connect origin to reciprocal lattice points
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!

8. The reciprocal lattice
The real lattice
Example-2
Reciprocal Lattice
Real Lattice
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!

9. Reciprocal Lattice B 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 B
The reciprocal lattice is created by interplanar spacings

10. Some properties of the reciprocal lattice and its relation to the real lattice
A reciprocal lattice vector is  to the corresponding real lattice plane
The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane
Planes in the crystal become lattice points in the reciprocal lattice  Note that this is an alternate geometrical construction of the real lattice.
Reciprocal lattice point represents the orientation and spacing of a set of planes.

11. 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 (i.e. I have to look into both the positions of the points and the intensities decorating them).
Revision+
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) (Where F is the structure factor).  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

12. Real Lattice Real Crystal Reciprocal Lattice Reciprocal Crystal
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

13. 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
Is determined by
Intensity of the diffraction spots
Motif
Diffraction Pattern
Position of the diffraction spots
 RECIPROCAL LATTICE
Intensity of the diffraction spots
 MOTIF’ OF INTENSITIES

14. Take real lattice and construct reciprocal lattice
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
Take real lattice and construct reciprocal lattice
Use motif to compute structure factor and hence intensities to decorate reciprocal lattice points
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

15. SC
Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)
Reciprocal Space
Real Space
Selection rule: All (hkl) allowed In ‘simple’ cubic crystals there are No missing reflections +
Single sphere motif
001
011
Lattice = SC
101
111
000
010 =
100
110
Reciprocal Crystal = SC
SC crystal
SC lattice with Intensities as the motif at each ‘reciprocal’ lattice point
Figures NOT to Scale

16. BCC BCC crystal 002 x 022 x x 202 x 222 x 011 x 101 x x 000 x 020 x
Selection rule BCC: (h+k+l) even allowed In BCC 100, 111, 210, etc. go missing
002 x
022
BCC crystal x x
202 x
222 x
011 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
000 x
020 x
110 x
100 missing reflection (F = 0)
200 x
220
Weighing factor for each point “motif”
Reciprocal Crystal = FCC
FCC lattice with Intensities as the motif
Figures NOT to Scale

17. FCC Lattice = FCC Reciprocal Crystal = BCC 002 022 202 222 111 020 000
200
220
100 missing reflection (F = 0)
110 missing reflection (F = 0)
Weighing factor for each point “motif”
Reciprocal Crystal = BCC
BCC lattice with Intensities as the motif
Figures NOT to Scale

18. 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 Ordered Structures
Click here to know more about Superlattices & Sublattices

19. In a strict sense this is not a crystal !!
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) SC
Low T ordered
SL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (½, ½, ½)

20. Reciprocal Crystal = ‘FCC’
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
Reciprocal crystal
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.
This is like the NaCl structure in Reciprocal Space!

21. Diffraction pattern from the ordered structure (3D)
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 SC
Click here to see XRD powder pattern of Ni3Al → Slide 6
Diffraction pattern from the ordered structure (3D)
Ordered
BCC
FCC
Reciprocal Crystal = BCC
BCC lattice with Intensities as the motif
Reciprocal crystal

22. 1) SC + two kinds of Intensities decorating the lattice
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

23. The spots are ~periodically arranged
Example of superlattice spots in a TEM diffraction pattern
The spots are ~periodically arranged
[112]
[111]
[011]
Superlattice spots
SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4Zn94Y2 alloy showing important zones

24. NiAl pattern from 0-160 (2)
Example of superlattice peaks in XRD pattern
NiAl pattern from 0-160 (2)
Superlattice reflections (weak)

25. The Ewald Sphere
* Paul Peter Ewald (German physicist and crystallographer; )
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

26. 7. Paul-Peter-Ewald-Kolloquium
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

27. 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

28. This is Bragg’s equation in reciprocal space
Bragg’s equation revisited
This is Bragg’s equation in reciprocal space
Rewrite
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)

29. Now if we overlay ‘real space’ information on the Ewald Sphere. (i. e
Now if we overlay ‘real space’ information on the Ewald Sphere. (i.e. we are going to ‘mix-up’ 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 ‘*’).

30. The Ewald Sphere construction
Which leads to spheres for various hkl reflections
Crystal related information is present in the reciprocal crystal
The Ewald sphere construction generates the diffraction pattern
Chooses part of the reciprocal crystal which is observed in an experiment
Radiation related information is present in the Ewald Sphere

31. K = K =g = Diffraction Vector
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).
The Ewald Sphere touches the reciprocal lattice (for point 41)
 Bragg’s equation is satisfied for 41 41
K = K =g = Diffraction Vector

32. Ewald sphere  X-rays Diffraction from Al using Cu K radiation
Rows of reciprocal lattice points
Row of reciprocal lattice 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

33. 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

34. 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

35. ‘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

36. Understanding the formation of the cones
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

37. Ewald sphere construction for Al
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

38. The 331 reflection is not observed
Ewald sphere construction for Cu
Allowed reflections are those for h, k and l unmixed