1. 3 COMMON X-RAY DIFFRACTION METHODS
Laue
Rotating Crystal
Powder
Orientation
Single Crystal
Polychromatic Beam
Fixed Angle
Lattice constant or Secondary Phases
Single Crystal
Monochromatic Beam
Variable Angle
Lattice Parameters
Polycrystal/Powder
Monochromatic Beam
Fixed Angle

2. Crystal structure determination by Laue method?
Although the Laue method can be used, several wavelengths can reflect in different orders from the same set of planes, making structure determination difficult (use when structure known for orientation or strain).
Rotating crystal method overcomes this problem. How?
(Was a question about what a synchrotron is)
You would interpret from the intensity of the spots, but if multiple spots are at the same location, that’s a problem.
Materials scientists typically already know the crystal structure for a material they’ve worked on a lot and then this method can be used to understand orientation or strain tensors.

3. ROTATING CRYSTAL METHOD
A single crystal is mounted with a rotation axis perpendicular to a monochromatic x-ray beam.
A cylindrical film is placed around it and the crystal is rotated. Or a detector is rotated.
Sets of lattice planes will at some point make the correct Bragg angle, and at that point a diffracted beam will be formed.  

4. Rotating Crystal Method
Reflected beams are located on imaginary cones. 
By recording the diffraction patterns (both angles and intensities), one can determine the shape and size of unit cell as well as arrangement of atoms inside the cell.
But around what axis should you rotate?
Requires various orientations measured and you may not know where those orientations are, so hard if that is the case.
You might get lucky if you guess or you could…
Film

5. THE POWDER METHOD Least crystal information needed ahead of time
If a powder is used, instead of a single crystal, then there is no need to rotate the sample, because there will always be some crystals at an orientation for which diffraction is permitted.
A monochromatic X-ray beam is incident on a powdered or polycrystalline sample.
Common method if you don’t know much about your material (or if you already have it in powder form).

6. The Powder Method
A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones.
A circle of film is used to record the diffraction pattern as shown.
Each cone intersects the film giving diffraction arcs.
If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.
If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones.
The cones may point both forwards and backwards.

7. Powder diffraction film
When the film is removed from the camera, flattened and processed, it shows the diffraction lines and the holes for the incident and transmitted beams. Likewise, detectors can show the same thing.

8. K

9. Useful for Phase Identification
The diffraction pattern for every phase is as unique as your fingerprint
Phases with the same element composition can have drastically different diffraction patterns.
Use the position and relative intensity of a series of peaks to match experimental data to the reference patterns in the database

10. Databases such as the Powder Diffraction File (PDF) contain dI lists for thousands of crystalline phases.
The PDF contains over 200,000 diffraction patterns.
Modern computer programs can help you determine what phases are present in your sample by quickly comparing your diffraction data to all of the patterns in the database.

11. Test September 12 in class
By the end of this section you should be able to:
Construct a reciprocal lattice
Interpret points in reciprocal space
Determine the Brillouin zone
There are several youtube videos on reciprocal space and other topics online. I recommend checking them out if you want to see another approach (which is often helpful). Remember that repetition in multiple forms is how our brains determine what is important to learn. Unfortunately, just thinking that something is important is not normally enough.

12. Reciprocal Space = https://www.youtube.com/watch?v=DFFU39A3fPY
Fourier transform of the real crystal lattice Also called Fourier space, k (wavevector)-space, or momentum space in contrast to real space or direct space.
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.)
Conversely, the fourier transform of the reciprocal lattice is the real lattice.

13. Why Use Reciprocal Space?
Sample
X-rays
Many different types of XRD
Purpose of this one?
A diffraction pattern (the interference of waves) is not a direct representation of the crystal lattice
The diffraction pattern is a representation of the reciprocal lattice
This pattern may not be what you think of when I say measuring the diffraction of a sample. You probably think about peaks. These are just scans along certain directions (where you rotate the sample and/or the incident x-rays). And the peaks will change based on which direction you go along.
What would one use this type of XRD pattern for? Determining distance between 001 planes. How would you do that? Average over the many shown. This reduces the noise/instrument calibration error from just looking at a few peaks.
Why do you think there are two different sets of 001 peaks? Could be two things. Picking up different materials or you have different phases in the material. Which do you think it is? Hard to say for sure. Typically one material phase would dominate, so other peaks would be weak. (Shown: XRD pattern of the YBCO film deposited on buffered silicon, so we have two different materials contributing peaks)
If you don’t know much about your sample yet, how would you even know which direction to scan along? This is the reason people often study powders/polycrystalline materials.
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 (note that the shown pattern also seems to have some other peaks which suggests its not perfectly single crystalline, meaning all in one orientation). 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
Is this what you think of when you hear diffraction? b1

14. 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.
K is also often called G
This vector has magnitude 2/dhkl, which is a reciprocal distance
A point in the reciprocal lattice corresponds to a set of planes (hkl) in the real-space lattice.

15. Definition of the Real/Reciprocal Lattice
Rn = n1 a1 + n2 a2 + n3 a3 (real lattice vectors a1,a2,a3)
(h, k, l integers)
Suppose K can be decomposed into reciprocal lattice vectors:
b’s are defined such that:
Note: a has dimensions of length, b has dimensions of length-1
Implication: b1 is always perpendicular to a2 and a3
The basis vectors bi define a reciprocal lattice:
- for every real lattice there’s a reciprocal lattice
Removed K dot R = 2 pi times integer. Seemed more direct to just say b’s defined this way. (perhaps leaving it in will make the diffraction click better later, but I don’t think it makes a big difference)
We discussed before that any vector in real space pointing from one lattice point to another can be written in terms of integers times the real space lattice vectors. We will find we can do the same thing in reciprocal space.
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 primitive volume is.

16. Identify these planes 2D Reciprocal Lattice K in reciprocal space
A point in the reciprocal lattice corresponds to a set of 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

17. Another Similar View: Lattice waves
real space
reciprocal space a b
2π/a
2π/b
(0,0)
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.
There is always a (0,0) point in reciprocal space.
How do you expect the reciprocal lattice to look?

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

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

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

21. 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, a reciprocal lattice vector is  to opposite R.S. axis.

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

23. Group: What happens if the lattice is not rectangular?
Determine the reciprocal lattice for: a2 b2 a1 b1
Real space
Fourier (reciprocal) space
In 2D, reciprocal vectors are perpendicular to opposite axis.

24. Group: Find the reciprocal lattice vectors of BCC
B F – E C
D C – A F
A E – B C
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.
Look familiar?
Good websites:

25. 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.
Got here (slide 19) in talkative class. Spend a lot of time talking about Perovskites to review basic crystal structure concepts.
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.

26. Volume of the Brillouin Zone (BZ)
In general the volume of the BZ is equal to
(2 )3
Volume of real space primitive lattice
Real lattice
Real Volume
BZ Volume
Simple cubic a3
8 3/a3
bcc
a3/2
16 3/a3
fcc
a3/4
32 3/a3

27. The reciprocal lattice in one dimension
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)?

28. 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 unique set of distance/area/volume in reciprocal space)
Its construction exhibits all the wavevectors k which can be Bragg-reflected by the crystal
Also critical for understanding energy diagrams
Got here in 50 minute class, XRD pattern took some time

29. 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

31. 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)
Ran out of time on this slide (slide 26 on current version)
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.

32. 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
Remember that directions in reciprocal space refer to planes of atoms in real space.
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

33. Brillouin Zones in 3D fcc bcc hcp
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 (future class)
Construction leads to primitive unit cell in rec. space

34. 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
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.

35. 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.
6N1p

36. Square Lattice (on board) Introduction of Higher Order BZs (these will make more sense when we get to energy band diagrams)

37. Extra Slides Alternative Approaches and Other Stuff for which we ran out of time If you already understand reciprocal lattices, these slides might just confuse you. But, they can help if you are lost.

38. Group: Draw the 1st Brillouin Zone of a sheet of graphene
(atoms at corners)
a1* a1 a2
a2*
This is a lot like the one we did a few classes back. Only difference is that the other one had a1 and a2 lengths of different magnitude. Note I show this time a different way of defining the lattice vectors. Either one is fine, and you can always move vectors around, as long as you don’t change the magnitude nor direction.
The dots are lattice points. Remember that atoms do not necessarily have to occur at lattice points, though it is very common. I don’t do that here because I think it makes it more obvious how the real and reciprocal lattice relate to the BZ.
Real Space
2-atom basis
(many ways to define vectors)
Wigner-Seitz Unit Cell of Reciprocal Lattice
= First Brillouin zone
The same perpendicular bisector logic applies in 3D

39. Brillouin zone representations of graphene
Dirac cones
When you grow graphene on something, the unique cones go away and a gap is introduced.
Brillouin zone representations of graphene
Real space

40. Quantitative Phase Analysis
With high quality data, you can determine how much of each phase is present
The ratio of peak intensities varies linearly as a function of weight fractions for any two phases in a mixture
RIR method is fast and gives semi-quantitative results
Whole pattern fitting/Rietveld refinement is a more accurate but more complicated analysis
Reference Intensity Ratio Method
RIR method: need to know the constant of proportionality

41. Examples of Image Fourier Transforms
Real Image
Fourier Transform
Moved from just before BCC example, due to timing constraints
Cannot see dots on projector-recreate the bottom left one. (tried to change contrast to help)
What if you take fourier transform of something that isn’t lines? What is the fading due to? This is edging effects from the fact that our image size is finite.
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

42. Examples of Image Fourier Transforms
Note directions of spots in RL of third image. Not parallel to real space lattice.

43. 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.

44. 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.

45. 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.

46. Reciprocal Lattice
The reciprocal lattice has an origin!

47. Note perpendicularity of various vectors

48. Brillouin Zone for fcc is bcc
More on 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.