Reciprocal Lattices to SC, FCC and BCCPrimitive Direct lattice Reciprocal lattice Volume 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

How would you find the

Volume of the Brillouin Zone (BZ)

In general the volume of the BZ is equal to

(2 )3

Volume of real space primitive lattice

Thus the volume of the reciprocal lattice is equal to the volume of the BZ.

Discuss the reciprocal lattice in 1D

a

Wigner 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)?

Real lattice

Reciprocal lattice

k

0 2/a 4/a-2/a-4/a-6/a

x

-/a /a What is the range of unique environments?

Look familiar?

The Brillouin Zone

• Is defined as the Wigner-Seitz primitive cell in the reciprocal lattice (smallest volume/area/distance in RL)

• Its construction exhibits all the wavevectors k which can be Bragg-reflected by the crystal

Reciprocal lattice

k

0 2/a 4/a-2/a-4/a-6/a

-/a /a

Group: Draw the 1st Brillouin Zone of a sheet of graphene

Real Space

2-atom basis

a2

a1

a2*

a1*

Wigner-Seitz Unit Cell of Reciprocal Lattice= First Brillouin zone

In what directions do b1 and b2 point?

Only overlaying the grid as a visual aid (not part of the

lattice).

Group: PoloniumConsider simple cubic polonium, Po, which is similar to a 1D chain in 3 dimensions.

(a) Determine the lengths and directions of the lattice translation vectors for the lattice which is reciprocal to the real-space Po lattice.

(b) The first Brillouin Zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice. Sketch the first Brillouin Zone of Po. (Identify the location of the faces of the shape.)

(c) 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.

Square Lattice(on board)

Introduction of Higher Order BZs(HOBZs will seem more important in Ch.9)

Group: Determine the shape of the BZ of the FCC Lattice

FCC Primitive and Conventional Unit Cells

How many sides will it have and along what directions?How might you approach this?

SC BCC FCC

# of nearest neighbors 6 8 12

Nearest-neighbor distance a ½ a 3 a/2

# of second neighbors 12 6 6

Second neighbor distance a2 a a

Reciprocal Space to the FCC Lattice

WS zone and BZLattice 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 BZ of fcc is the WS cell of bcc.The BZ of bcc is the WS cell of fcc.

Labelling the BZ

Directions are chosen that lead along special symmetry points. These points are labeled according to the following rules:

• Points on the surface (red) of the Brillouin zone are Roman letters.

• Points (and lines) inside the Brillouin zone are denoted with Greek letters.

• The center of the Wigner-Seitz cell is always denoted by a G

Usually, it is sufficient to know the energy En(k) curves - the dispersion relations - along the major directions.

Ener

gy o

r Fre

quen

cy

Direction along BZ

Brillouin Zones in 3Dfcc

hcp

•The BZ reflects lattice symmetry•Construction leads to primitive unit cell in rec. space

bcc

Note: fcc lattice in reciprocal space is a bcc lattice

Note: bcc lattice in reciprocal space is a fcc lattice

What kind of crystal structure is

Si?

Brillouin Zone of Silicon

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.

Symbol DescriptionΓ Center of the Brillouin zone

Simple CubicM Center of an edgeR Corner pointX 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 Corner pointX Center of a square face C4

BCC

H Corner point joining 4 edges

N Center of a faceP Corner point joining 3 edges

Learning Objectives for Diffraction

After our diffraction class you should be able to:• Explain why diffraction occurs• Utilize Bragg’s law to determine angles of diffraction• Briefly discuss some different diffraction techniques• (Next time) Determine the lattice type and lattice

parameters of a material given an XRD pattern and the x-ray energy

• Alternative reference: Ch. 2 Kittel

Continuum limit:Where the wavelength is bigger than the spacing between

atoms. Otherwise diffraction effects dominate.

Application of XRD

1. Determination of the structure of crystalline materials2. Determination of the orientation of single crystals3. Differentiation between crystalline and amorphous

materials 4. Determination of the texture of polygrained materials5. Measurement of layer thickness6. Measurement of (epitaxial) strain 7. Determination of electron distribution within the

atoms, and throughout the unit cell

XRD is a nondestructive and cheap technique. Some of the uses of x-ray diffraction are:

DIFFRACTION• Diffraction is a wave phenomenon in which the apparent

bending and spreading of waves when they meet an obstruction is measured.

• Diffraction occurs with electromagnetic waves, such as light and radio waves, and also in sound waves and water waves.

• X-ray diffraction is optimally sensitive to the periodic nature of the solid’s atomic structure.

When X-rays interact with atoms, you get scattering

Scattering is the emission of X-rays of

the same frequency/energy as the incident X-rays in all

directions (but with much lower intensity)

Similar to the double slit experiment, this scattering will

sometimes be constructive

Incident beam

Zeroth Order

Second order

Will

look

at t

his

agai

n sh

ortly

Physical Model for X-ray Scattering

Consider a plane wave scattering on an atom.

ok

'kAtom

)( tRkioincident

oeA

)'( tRkioscattered eA

R

'R

Diffraction In a Crystal

ko

Detector

Pi

ri

To calculate amplitude of scattered waves at detector position, sum over contributions of all scattering centers Pi with scattering amplitude (form factor) f:

R’)()()( ii

iiInDet ef rRkrr R’-ri

Generic incoming radiation amplitude is:)(

00 ii

In eA rRk

R

)'()'(0 )( kkrRkRk 00 r ii

i

iDet efeA

The intensity that is measured (can’t measure amplitude) is

2

)()( rrK Kr defI i0' kkK

source

R, R’ >> ri

The book calls K, but G is another common notation.Scattering vector

Diffraction Theory

ko

Detector

Pi

ri

R’ R’-ri

)(0

0 iiIn eA rRk

R

The intensity that is measured (can’t measure amplitude) is

2

)()( rrK Kr defI i0' kkK

source

The book calls K, but G is another common notation.Scattering vector

ko

k’K=k’-ko

How does this limit ?

where, d is the spacing of the planes and n is the order of diffraction.

• Bragg reflection can only occur for wavelength

• This is why we cannot use visible light. No diffraction occurs when the above condition is not satisfied.

ndhkl sin2

dn 2

Neutron

λ = 1A°

E ~ 0.08 eV

interact with nucleiHighly Penetrating

Electron

λ = 2A°

E ~ 150 eV

interact with electronLess Penetrating

Non-xray Diffraction Methods(more in later chapters)

• Any particle will scatter and create diffraction pattern

• Beams are selected by experimentalists depending on sensitivity– X-rays not sensitive to low Z elements, but neutrons are– Electrons sensitive to surface structure if energy is low– Atoms (e.g., helium) sensitive to surface only

• For inelastic scattering, momentum conservation is important

X-Ray

λ = 1A°

E ~ 104 eV

interact with electronPenetrating