1. Define the Crystal Structure of Perovskites
A-site (Ca)
Oxygen
B-site (Ti)
CaTiO3
Superconductors
Ferroelectrics (BaTiO3)
Colossal Magnetoresistance (LaSrMnO3)
Multiferroics (BiFeO3)
High εr Insulators (SrTiO3)
Low εr Insulators (LaAlO3)
Conductors (Sr2RuO4)
Thermoelectrics (doped SrTiO3)
Ferromagnets (SrRuO3) eg
t2g
They are drawn in a few ways, but this is one of the most common.
In addition to being among the most abundant minerals on earth, complex oxides give some of the most varied and interesting properties. These include their use as dielectric and superconducting materials. Yet, only recently has the research in the field of complex oxides flourished because they were long thought to be …well complex. This complexity comes from the strong coupling with charge, spin, and lattice dynamics, which often results in very full phase diagrams.
Though the coupling may lead to complex behaviors, the structure of these materials can be quite simple, such as the perovskite form shown here, where the A and B sites are typically different cations and X is an anion that bonds to both. This octahedral arrangement (imagine connecting the oxygens) gives rise to a crystal field potential, hinders the free rotation of the electrons and quenches the orbital angular momentum by introducing the crystal field splitting of the d orbitals. Even among only perovskite structures, we can see these varied behaviors.
Perovskite formula – ABO3
A atoms at the corners
B atoms (smaller) at the body-center
O atoms at the face centers

2. PEROVSKITES Lattice: Simple Cubic (idealized cubic structure)
A-site (Ca)
Oxygen
B-site (Ti)
CaTiO3
Lattice: Simple Cubic (idealized cubic structure)
1 CaTiO3 per unit cell
Cell Motif: Ti at (0, 0, 0); Ca at (1/2, 1/2, 1/2); 3 O at (1/2, 0, 0), (0, 1/2, 0), (0, 0, 1/2) could label differently
Nearest neighbors: Ca 12-coordinate by O, Ti 6-coordinate by O, O distorted octahedral
Motif is another word for basis.
Is it a bravais lattice? Yes, because it’s simple cubic. Don’t confuse atoms with lattice points.
Yes primitive—only one lattice point per unit cell. Don’t confuse lattice points with atoms, of which there are many.
Anion doesn’t have to be oxygen, even though that is common. For example, fluorides are also of interest. Current theory suggests that the properties in fluorides might be even better than oxides.
(B cell shows lines between nearest (and second nearest except for the center Ti atom, which would be along body diagonals) neighbors. Those probably aren’t shown because there typically aren’t strong interactions between the A and B cations in perovskites.

3. 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

4. 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.

5. Why Use The Reciprocal Space?
Many different types of XRD (later)
Purpose of this one?
A diffraction pattern 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

6. 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

7. Definition of the 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:
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
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 volume is.

8. 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

9. 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?

10. 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.

11. 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.

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

13. 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.

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

15. 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.

16. Examples of Image Fourier Transforms
Real Image
Fourier Transform
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

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

18. 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.
Look familiar?
Good websites:

19. 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.

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

21. 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.

22. 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.

23. 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.

24. Reciprocal Lattice
The reciprocal lattice has an origin!

25. Note perpendicularity of various vectors