Define the Crystal Structure of Perovskites Superconductors Ferroelectrics (BaTiO 3 ) Colossal Magnetoresistance (LaSrMnO 3 ) Multiferroics (BiFeO 3 )

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)

A-site (Ca) Oxygen

B-site (Ti)

CaTiO3

eg

t2g

Perovskite formula – ABO3 A atoms at the corners B atoms (smaller) at the body-center 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 12-coordinate by O, Ti 6-coordinate by O, O distorted octahedral

PEROVSKITESA-site (Ca) Oxygen

B-site (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 real-space lattice.

This abstraction seems unnecessary. Why do we care?

hklK

1. The reciprocal lattice simplifies the interpretation of x-ray 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 real-space (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

length-1

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 real-space 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

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.

a

b

2π/a

2π/b(0,0)

Lattice waves

real space reciprocal space

(11) plane

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/crystal-rot.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 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.

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

1020

11

2212

02

01

21

00

(01)

(10)(11)

(21)

1a

2a

*b2

*b1

1a

(01)

(10)(11)

(21) Note perpendicularity of various vectors