1. Structure of Solids Objectives
By the end of this section you should be able to:
Construct a reciprocal lattice
Interpret points in reciprocal space
Determine and understand the Brillouin zone

2. Reciprocal Space
Also called Fourier space, k (wavevector)-space, or momentum space in contrast to real space or direct space.
The reciprocal lattice is composed of all points lying at positions from the origin. Thus, there is one point in the reciprocal lattice for each set of planes (hkl) in the real-space lattice.
This abstraction seems unnecessary. Why do we care?
The reciprocal lattice simplifies the interpretation of x-ray diffraction from crystals
The reciprocal lattice facilitates the calculation of wave propagation in crystals (lattice vibrations, electron waves, etc.)
We’ll come back to what that means.

3. Why Use The Reciprocal Space?
A diffraction pattern is not a direct representation of the crystal lattice
The diffraction pattern is a representation of the reciprocal lattice
There are many different ways to do diffraction. In one of them:
When we take an angle scan, that’s like taking a single line across the pattern
This is the reason why people often study polycrystals or powders. Explain what that is. Because, you don’t have to know the orientation, which you wouldn’t know if you just picked up a random material or are just starting to grow a material that isn’t well known. b2 b1

4. The Reciprocal Lattice
Crystal planes (hkl) in the real-space (or the direct lattice) are characterized by the normal vector and dhkl interplanar spacing z y
[hkl] x
Practice has shown the usefulness of defining a different lattice in reciprocal space whose points lie at positions given by the vectors
What plane is this? (010)
This was why we discussed the distance between planes before.
This vector has magnitude 2/dhkl, which is a reciprocal distance

5. Definition of the Reciprocal Lattice
Rn = n1 a1 + n2 a2 + n3 a3 (lattice vectors a1,a2,a3)
(h, k, l integers)
Suppose K can be decomposed into reciprocal lattice vectors:
Note: a has dimensions of length, b has dimensions of length-1
The basis vectors bi define a reciprocal lattice:
- for every real lattice there’s a reciprocal lattice
- reciprocal lattice vector b1 is perpendicular to plane defined by a2 and a3
The equivalent to breaking the real space lattice into a1,a2,a3 directions or x,y,z.
What does it mean that a1 dot b1 equals 2pi and a1 dot b2 or b3=0?
That means a1 is perpendicular to b2 and b3, but not necessarily parallel to a1 (but it could be).
Anyone know what a1 dot (a2 cross a3) is? Let’s look at it in the simple cubic case.
Continue to think about simple cubic structure: Lattice vectors are not unique, but the primitive unit cell always has the same volume.
+ cyclic permutations
is volume of unit cell
Definition of a’s are not unique, but the volume is.

6. Identify these planes 2D Reciprocal Lattice K in reciprocal space
A point in the reciprocal lattice corresponds to a set of planes planes (hkl) in the real-space lattice.
Identify these planes a2
Planes defined by perpendicular vector.
(01) Or (010), don’t have to put in third dimension
Compare distance in real space for (01) and (11): a to 1.41a/2 or about .7a
What planes are these? Easier to tell from k space. (12) a1
Real lattice planes (hk0)
K in reciprocal space
Khkl is perpendicular to (hkl) plane
Magnitude of K is inversely proportional to distance between (hkl) planes

7. Another Similar View: Lattice waves
real space
reciprocal space a b
2π/a
2π/b
Note that the reciprocal lattice has a longer dimension in the y direction, as opposed to the real lattice.
look at waves corresponding to the reciprocal lattice vectors.
if we change the place we look at by ANY real lattice vector, we have to get the same
Here fore K=0, infinite wave length.
(0,0)
There is always a (0,0) point in reciprocal space.
How do you expect the reciprocal lattice to look?

8. Red and blue represent different amplitudes of the waves.
Lattice waves
real space
reciprocal space a b
2π/a
2π/b
Here fore K=2pi/b, lambda=b
(0,0)
Red and blue represent different amplitudes of the waves.

9. Lattice waves real space reciprocal spaceb
2π/a
2π/b
Here fore K=4pi/b,lambda=b/2
(0,0)
Note that the vertical planes in real space correspond to points along the horizontal axis in reciprocal space.

10. Lattice waves
real space
reciprocal space a b
2π/a
2π/b
(0,0)

11. Lattice waves real space reciprocal spaceb
2π/a
2π/b
(0,0)
The real horizontal planes relate to points along R.S. vertical.
In 2D, reciprocal vectors are perpendicular to opposite axis.

12. Lattice waves real space reciprocal space (11) plane b a 2π/a (0,0)

13. Examples of Image Fourier Transforms
Real Image
Fourier Transform
Cannot see dots on projector-recreate the bottom left one.
A transformed image can be used for frequency filtering.
In images 1,3,5,7: Only three spots are shown to focus on change in distance between real and reciprocal, but more would appear just as in the first set of images.
Here is an image I took as a graduate student of the antiferromagnetic domains of a material. I could take a fourier transform to tell me about the average length scales of these domains and their varience.
Brightest side points relating to the frequency of the stripes

14. Examples of Image Fourier Transforms
Third set is most illustrative.
Note directions of spots in RL of third image. Not parallel to real space lattice.

15. Group: Reciprocal Lattice
Determine the reciprocal lattice for: a2 b2 a1 b1
Real space
Fourier (reciprocal) space

16. Group: Find the reciprocal lattice vectors of BCC
The primitive lattice vectors for BCC are:
The volume of the primitive cell is ½ a3(2 pts./unit cell)
So, the primitive translation vectors in reciprocal space are:
These lattice vectors should look familiar.
Good websites:

17. We will come back to this if time.
Reciprocal Lattices to SC, FCC and BCC
Primitive Direct lattice Reciprocal lattice Volume of RL SC
BCC
FCC
Direct
Reciprocal
Simple cubic
bcc
fcc
We will come back to this if time.
If we took b1 dot (b2 cross b3) we’d get the volume of the reciprocal cell, which would give these. Might come back and prove these values if time at the end of class.
Makes sense since real space volume was smaller for FCC in real space, so bigger in reciprocal space.

18. Volume of BZ In general the volume of the BZ is equal to (2 )3
Volume of real space primitive lattice

19. Discuss the reciprocal lattice in 1D
Real lattice x
What is the range of unique environments?
-/a
/a
Reciprocal lattice k
-6/a
-4/a
-2/a
2/a
4/a
Weigner Seitz Cell: Smallest space enclosed when intersecting the midpoint to the neighboring lattice points.
Why don’t we include second neighbors here (do in 2D/3D)?

20. The Brillouin Zone
-/a
/a
Reciprocal lattice k
4/a
-6/a
-4/a
-2/a
2/a
Is defined as the Wigner-Seitz primitive cell in the reciprocal lattice (smallest volume in RL)
Its construction exhibits all the wavevectors k which can be Bragg-reflected by the crystal
Got here in 50 minute class, XRD pattern took some time

21. Group: Draw the 1st Brillouin Zone of a sheet of graphene
Real Space
2-atom basis
The same perpendicular bisector logic applies in 3D
Wigner-Seitz Unit Cell of Reciprocal Lattice
= First Brillouin zone

22. Group: Polonium
Consider simple cubic polonium, Po, which is the closest thing we can get to a 1D chain in 3 dimensions. (a) Taking a Po atom as a lattice point, construct the Wigner-Seitz cell of polonium in real space. What is it’s volume? (b) Work out the lengths and directions of the lattice translation vectors for the lattice which is reciprocal to the real-space Po lattice. (c) The first Brillouin Zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice. Sketch the first Brillouin Zone of Po. (d) Show that the volume of the first Brillouin Zone is (2)3/V , where V is the volume of the real space primitive unit cell.
6N1p

23. Square Lattice (on board) Introduction of Higher Order BZs

24. Group: Determine the shape of the BZ of the FCC Lattice
How many sides will it have and along what directions? SC
BCC
FCC
# of nearest neighbors 6 8 12
Nearest-neighbor distance a
½ a 3
a/2
# of second neighbors
Second neighbor distance
a2
FCC Primitive and Conventional Unit Cells

26. WS zone and BZ Lattice Real Space Lattice K-space bcc WS cell
Bcc BZ (fcc lattice in K-space)
fcc WS cell
fcc BZ (bcc lattice in K-space)
The WS cell of bcc lattice in real space transforms to a Brillouin zone in a fcc lattice in reciprocal space while the WS cell of a fcc lattice transforms to a Brillouin zone of a bcc lattice in reciprocal space.

27. Nomenclature
Usually, it is sufficient to know the energy En(k) curves - the dispersion relations - along the major directions.
Directions are chosen that lead aong special symmetry points. These points are labeled according to the following rules:
Direction along BZ
Points (and lines) inside the Brillouin zone are denoted with Greek letters.
Points on the surface of the Brillouin zone with Roman letters.
The center of the Wigner-Seitz cell is always denoted by a G
Energy or Frequency

28. Brillouin Zones in 3D fcc bcc hcp The BZ reflects lattice symmetry
Note: bcc lattice in reciprocal space is a fcc lattice
hcp
Note: fcc lattice in reciprocal space is a bcc lattice
The BZ reflects lattice symmetry
Construction leads to primitive unit cell in rec. space

29. Brillouin Zone of Silicon
Symbol
Description Γ
Center of the Brillouin zone
Simple Cubic M
Center of an edge R
Corner point X
Center of a face
FCC K
Middle of an edge joining two hexagonal faces L
Center of a hexagonal face C6 U
Middle of an edge joining a hexagonal and a square face W
Center of a square face C4
BCC H
Corner point joining 4 edges N P
Corner point joining 3 edges
Why do I say semiconductors? Metals are most freely so barely effected. Insulators don’t really conduct so don’t move around.
Points of symmetry on the BZ are important (e.g. determining bandstructure).
Electrons in semiconductors are perturbed by the potential of the crystal, which varies across unit cell.

30. Brillouin zone representations of graphene
Dirac cones
When you grow graphene on something, the unique cones go away and a gap is introduced.
She mentioned that the M point was in between two K points
Dr. Jessica McChesney at Argonne
Brillouin zone representations of graphene
Real space

31. Now that we are starting to understand reciprocal space, it’s time to take advantage of it. X-Ray Diffraction (XRD) Why do we use it?

32. Back to this Reciprocal Lattices to SC, FCC and BCC
Primitive Direct lattice Reciprocal lattice Volume of RL SC
BCC
FCC
Direct
Reciprocal
Simple cubic
bcc
fcc
Back to this

33. Brillouin Zone for fcc is bcc
Nomenclature
We use the following nomenclature: (red for fcc, blue for bcc):
The intersection point with the [100] direction is called X (H). The line G—X is called D. The intersection point with the [110] direction is called K (N). The line G—K is called S. The intersection point with the [111] direction is called L (P). The line G—L is called L.
Brillouin Zone for fcc is bcc
and vice versa.

34. Extra Slides Alternative Approaches If you already understand reciprocal lattices, these slides might just confuse you. But, they can help if you are lost.

35. Construction of the Reciprocal Lattice
Identify the basic planes in the direct space lattice, i.e. (001), (010), and (001).
Draw normals to these planes from the origin.
Note that distances from the origin along these normals proportional to the inverse of the distance from the origin to the direct space planes
If you already understand it, these slides might just confuse you. But, they can help if you are lost.

36. Fourier (reciprocal) space
Real space
Fourier (reciprocal) space
Above a monoclinic direct space lattice is transformed (the b-axis is perpendicular to the page). Note that the reciprocal lattice in the last panel is also monoclinic with * equal to 180°−.
The symmetry system of the reciprocal lattice is the same as the direct lattice.

37. Reciprocal lattice (Similar)
Consider the two dimensional direct lattice shown below. It is defined by the real vectors a and b, and the angle g. The spacings of the (100) and (010) planes (i.e. d100 and d010) are shown.
The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle g*. a* will be perpendicular to the (100) planes, and equal in magnitude to the inverse of d100. Similarly, b* will be perpendicular to the (010) planes and equal in magnitude to the inverse of d010. Hence g and g* will sum to 180º.

38. Reciprocal Lattice
The reciprocal lattice has an origin!

39. Note perpendicularity of various vectors