Define the Crystal Structure of Perovskites Superconductors Ferroelectrics (BaTiO 3 ) Colossal...

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Define the Crystal Structure of Perovskites• Superconductors
• Ferroelectrics (BaTiO3)
• Colossal Magnetoresistance (LaSrMnO3)
• Multiferroics (BiFeO3)
• High εr Insulators (SrTiO3)
• Low εr Insulators (LaAlO3)
• Conductors (Sr2RuO4)
• Thermoelectrics (doped SrTiO3)
• Ferromagnets (SrRuO3)
Asite (Ca) Oxygen
Bsite (Ti)
CaTiO3
eg
t2g
Perovskite formula – ABO3 A atoms at the corners B atoms (smaller) at the bodycenter O atoms at the face centers
• 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 12coordinate by O, Ti 6coordinate by O, O distorted octahedral
PEROVSKITESAsite (Ca) Oxygen
Bsite (Ti)
CaTiO3
Structure of SolidsObjectives
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
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 realspace lattice.
This abstraction seems unnecessary. Why do we care?
hklK
1. The reciprocal lattice simplifies the interpretation of xray diffraction from crystals
2. The reciprocal lattice facilitates the calculation of wave propagation in crystals (lattice vibrations, electron waves, etc.)
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
b2
b1
Many different types of XRD (later)
Purpose of this one?
Is this what you think of when you hear diffraction?
The Reciprocal Lattice
Crystal planes (hkl) in the realspace (or the direct lattice) are characterized by the normal vector and dhkl interplanar spacing
Practice has shown the usefulness of defining a different lattice in reciprocal space whose points lie at positions given by the vectors
hkln̂
x
y
z
hkld
hkln̂
hkl
hklhkl d
nK
ˆ2
This vector has magnitude 2/dhkl, which is a
reciprocal distance
[hkl]
Definition of the Reciprocal Lattice
Suppose K can be decomposed into reciprocal lattice vectors:321 blbkbhK
(h, k, l integers)
mRK n 2
ijji 2ab
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
Note: a has dimensions of length, b has dimensions of
length1
321
321 2
aaa
aa
b
+ cyclic permutations
321 aaa is volume of unit cell
Definition of a’s are not unique, but the volume is.
Rn = n1 a1 + n2 a2 + n3 a3 (real lattice vectors a1,a2,a3)
2D Reciprocal Lattice
Khkl is perpendicular to (hkl) plane
Real lattice planes (hk0) K in reciprocal space
Identify these planes
a1
a2
A point in the reciprocal lattice corresponds to a set of planes planes (hkl) in the realspace lattice.
Magnitude of K is inversely proportional to distance between (hkl) planes
a
b
2π/a
2π/b(0,0)
Another Similar View: Lattice waves
real space reciprocal space
There is always a (0,0) point in reciprocal space.How do you expect the reciprocal lattice to look?
a
b
2π/a
2π/b(0,0)
Lattice waves
real space reciprocal space
Red and blue represent different phases of the waves.
Lattice waves
real space reciprocal space
a
b
2π/a
2π/b(0,0)
Note that the vertical planes in real space correspond to points along the horizontal axis in reciprocal space.
a
b
2π/a
2π/b(0,0)
Lattice waves
real space reciprocal space
The real horizontal planes relate to points along R.S. vertical.In 2D, reciprocal vectors are perpendicular to opposite axis.
Group: What happens if the lattice is not rectangular?
• Determine the reciprocal lattice for:
a1
a2
Real space Fourier (reciprocal)
space
b1
b2
In 2D, reciprocal vectors are perpendicular to opposite axis.
Examples of Image Fourier Transforms
Brightest side points relating to the frequency of the stripes
Real Image
Fourier Transform
Examples of Image Fourier Transforms
http://www.users.csbsju.edu/~frioux/diffraction/crystalrot.pdf
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:Good websites:http://newton.umsl.edu/run//nano/reltutor2.htmlhttp://matter.org.uk/diffraction/geometry/plane_reciprocal_lattices.htm
321
321 2
aaa
aa
b
Look familiar?
Reciprocal Lattices to SC, FCC and BCCPrimitive Direct lattice Reciprocal latticeVolume of RL
SC
BCC
FCC
za
ya
xa
a
a
a
3
2
1
xza
zya
yxa
a
a
a
21
3
21
2
21
1
zyxa
zyxa
zyxa
a
a
a
21
3
21
2
21
1
zb
yb
xb
a
a
a
/2
/2
/2
3
2
1
yxb
zxb
zyb
a
a
a
23
22
21
zyxb
zyxb
zyxb
a
a
a
23
22
21
3/2 a
3/24 a
3/22 a
Direct Reciprocal
Simple cubic Simple cubic
bcc fcc
fcc bcc
We will come back to this if
time.
Extra SlidesAlternative Approaches
If you already understand reciprocal lattices, these slides might just confuse you. But, they can help if you are lost.
Construction of the Reciprocal Lattice
1. Identify the basic planes in the direct space lattice, i.e. (001), (010), and (001).
2. Draw normals to these planes from the origin.
3. Note that distances from the origin along these normals proportional to the inverse of the distance from the origin to the direct space planes
Above a monoclinic direct space lattice is transformed (the baxis 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.
Real space Fourier (reciprocal) space
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.
Reciprocal Lattice
(01)
(10)(11)
(21)
10 20
11
221202
01 21
00
The reciprocal lattice has an origin!
1a
2a
1a1
1a
*11g *
21g*b2
*b1