Nomenclature

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

• 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

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• 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. Measurement of strain and small grain size3. Determination of the orientation of single crystals4. Measurement of layer thickness5. Differentiation between crystalline and amorphous

materials6. Determination of electron distribution within the

atoms, and throughout the unit cell7. Determination of the texture of polygrained materials

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 Theory

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

The Bottom Line

If you do a whole bunch of math you can prove that the peaks only occur when (a1, a2, a3=lattice vectors):

n1, n2, n3 integers 11 2 nKa

22 2 nKa

33 2 nKa

Compare these relations to the properties of reciprocal lattice vectors:

laK

kaK

haK

hkl

hkl

hkl

2

2

2

3

2

1

2

)()( rrK Kr defI i0' kkK

The Laue Condition

Replacing n1 n2 n3 with the familiar h k l, we see that these three conditions are equivalently expressed as:

321 blbkbhK

The Laue condition (Max von Laue, 1911)

So, the condition for nonzero intensity is that the scattering vector K is a translation vector of the

reciprocal lattice.

n1, n2, n3 integers 11 2 nKa

22 2 nKa

33 2 nKa

From Laue to Bragg

Notice this angle is 2!

The magnitude of the scattering vector K depends on the angle between the incident wave vector and the scattered wave vector:

ok

'k

K

2

hkld

ok

'k

Elastic scattering requires: 2

' kkko

So from the wave vector triangle and the Laue condition we see:

sin

4sin2 kK

sin2 hkldLeaving Bragg’s law:

hkld

2

If the Bragg condition is not met, the incoming wave just moves through the lattice and emerges on the other side of the crystal (neglecting absorption)

0kkK Show vector

subtraction on the board

Bragg Equation

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

X-ray Diffraction

/hcE Typical interatomic distances in solid are of the order of 0.4 nm.

Upon substituting this value for the wavelength into the energy equation, we find that E is of the order of 3000 eV, which is a

typical x-ray energy. Thus x-ray diffraction of crystals is a standard diffraction probe.

dn 2

Above are 1st, 2nd, 3rd and 4th order “reflections” from the (111) face of NaCl. Orders of reflections are given as 111, 222, 333, 444, etc. (without parentheses!)

Bragg Equation: nd sin2The diffracted beams (reflections) from any set

of lattice planes can only occur at particular angles pradicted by the Bragg law.

A single crystal specimen in a Bragg-Brentano diffractometer (θin=θout) would produce only one family of peaks in the diffraction pattern.

At 20.6 °2 , Bragg’s law fulfilled for the

(100) planes, producing a

diffraction peak.

The (200) planes are parallel to the (100) planes. Therefore, they also diffract for this crystal. Since d200 is ½ d100, they appear at 42

°2 .

2q

The (110) planes would diffract at 29.3 °2 ; however, they are not properly aligned to produce a

diffraction peak (the perpendicular to those planes does not bisect the

incident and diffracted beams). Only background is observed.

20

THE EWALD SPHERE

Consider an arbitrary spherepassing through the reciprocal lattice,with the crystal arranged in the center of the sphere.

We specify two conditions:

(1)the sphere radius is 2 / - the inverse wavelength of X-ray radiation

(2)the origin of the reciprocal lattice lies on the surface of the sphere

X-rays are ON

O2/

2

diffracted ray

The diffraction spot will be observed when a reciprocal lattice point crosses the Ewald sphere

01

10

02

00 20

2

(41)

Ki

KD

DK

Reciprocal Space

The Ewald Sphere

The Ewald Sphere touches the reciprocal lattice (for point 41)

Bragg’s equation is satisfied for 41

A sphere of radius k Surface intersects a point in reciprocal space and its origin is at the tip of the incident wavevector.Sphere can be moved in reciprocal lattice space arbitrarily.Any points which intersect the surface of the sphere indicate where diffraction peaks will be observed if the structure factor is nonzero (later).

Only a few angles

1. Longitudinal or θ-2θ scanSample moves as θ, Detector follows as 2θ

k0 k’

0 10 20 30 40

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

K

0 10 20 30 40

Reciprocal lattice rotates by θ during scan

k0 k’

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

2

0 10 20 30 40

Kk0 k’

0 10 20 30 40

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

2K

k0 k’

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

2

0 10 20 30 40

Kk0 k’

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

0 10 20 30 40

2

0 10 20 30 40

Kk0 k’

1. Longitudinal or θ-2θ scanSample moves on θ, Detector follows on 2θ

0 10 20 30 400 10 20 30 40

•Provides information about relative arrangements, angles, and spacings between crystal planes.

2

0 10 20 30 40

Kk0

k’

Higher order diffraction peaks

http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/ewald.phphttp://www.physics.byu.edu/faculty/campbell/animations/x-ray_diffraction.html

3 COMMON X-RAY DIFFRACTION METHODS

X-Ray Diffraction Method

Laue

OrientationSingle Crystal

Polychromatic BeamFixed Angle

Rotating Crystal

Lattice constantSingle Crystal

Monochromatic BeamVariable Angle

Powder

Lattice ParametersPolycrystal/Powder

Monochromatic BeamVariable Angle

X-rays have wide wavelength range

(called white beam).

Back-reflection vs. TransmissionLaue Methods

The diffraction spots generally lay on: an ellipse

X-RayFilmSingle

Crystal

In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted backward are recorded.

Which is this?

a hyperbola

X-Ray Film

SingleCrystal

32

LAUE METHOD

The diffracted beams form arrays of spots, that lie on curves on the film.

Each set of planes in the crystal picks out and diffracts a particular wavelength from the white radiation that satisfies the Bragg law for the values of d and θ involved.

Laue Pattern

The symmetry of the spot pattern reflects the symmetry of the crystal when viewed along the direction of the incident

beam.

Great for symmetry and orientation determination

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?

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. Sets of lattice planes will at some point make the correct Bragg angle, and at that point a diffracted beam will be formed.

Rotating Crystal Method

Film

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.

Reflected beams are located on imaginary cones.

THE POWDER METHODLeast 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.

38

The Powder Method

• 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.

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.

39

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.

Useful for Phase IdentificationThe 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

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.

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

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

X(phase a)/X(phase b)I(p

hase

a)/I(p

hase

b) ..

Reference Intensity Ratio Method

Applications of Powder Diffractometry-phase analysis (comparison to known patterns)-unit cell determination (dhkl′s depend on lattice parameters)-particle size estimation (line width)-crystal structure determination (line intensities and profiles)

XRD: “Rocking” Curve Scan

• Vary ORIENTATION of K relative to sample normal while maintaining its magnitude.How? “Rock” sample over a very small angular range.

• Resulting data of Intensity vs. Omega ( , w sample angle) shows detailed structure of diffraction peak being investigated. Can inform about quality of sample.

ikfk

“Rock” Sample

Sample normalK K

XRD: Rocking Curve Example

• Rocking curve of single crystal GaN around (002) diffraction peak showing its detailed structure.

16.995 17.195 17.395 17.595 17.7950

8000

16000

GaN Thin Film(002) Reflection

Inte

nsity

(C

ount

s/s)

Omega (deg)

Compare to literature to see how good (some

materials naturally easier than others)

Generally limited by quality of substrate

X-Ray Reflectivity (XRR)

• A glancing, but varying, incident angle, combined with a matching detector angle collects the X rays reflected from the samples surface

• Interference fringes in the reflected signal can be used to determine:– thickness of thin film layers– density and composition of thin

film layers– roughness of films and interfaces

X-ray reflectivity measurement

Si

Mo

Mo

Mo

r t [Å] s[Å]0.68 19.6 5.8

0.93 236.5 34.0

1.09 14.1 2.71.00 5.0 2.7

1.00 2.8

Calculation of the electron density, thickness and interface roughness for each particular layer

W

The surface must be smooth (mirror-like)

0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,010

0

101

102

103

104

105

106

Inte

nsity

(a.

u.)

Diffraction angle (o2)

Edge of TER

Kiessig oscillations (fringes)

Lots of extra slides

• There is a lot of useful information on diffraction. Following are some related slides that I have used or considered using in the past.

• A whole course could be tough focusing on diffraction so I can’t cover everything here.

XRD: Reciprocal-Space Map

• Vary Orientation and Magnitude of k.• Diffraction-Space map of GaN film on AlN buffer

shows peaks of each film.

/2

GaN(002) AlN

Preferred Orientation (texture)

• Preferred orientation of crystallites can create a systematic variation in diffraction peak intensities– can qualitatively analyze using a 1D diffraction pattern– a pole figure maps the intensity of a single peak as a

function of tilt and rotation of the sample• this can be used to quantify the texture

(111)

(311)(200)

(220)

(222)(400)

40 50 60 70 80 90 100Two-Theta (deg)

x103

2.0

4.0

6.0

8.0

10.0

Inte

nsity

(Cou

nts)

00-004-0784> Gold - Au

Diffracting crystallites

The X-ray Shutter is the most important safety device on a diffractometer

• X-rays exit the tube through X-ray transparent Be windows.

• X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays.

• Being aware of the status of the shutters is the most important factor in working safely with X rays.

Cu

H2O In H2O Out

e-

Be

XRAYS

windowBe

XRAYS

FILAMENT

ANODE

(cathode)

AC CURRENT

window

metal

glass

(vacuum) (vacuum)

Primary

Shutter

Secondary

Shutter

Solenoid

SAFETY SHUTTERS

Neutron

λ = 1A°

E ~ 0.08 eV

interact with nucleiHighly Penetrating

Electron

λ = 2A°

E ~ 150 eV

interact with electronLess Penetrating

Diffraction Methods

• 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

54

Electron Diffraction(Covered in Chapter

18)If low electron energies are used, the penetration depth will be very small (only about 50 A°), and the beam will be reflected from the surface. Consequently, electron diffraction is a useful technique for surface structure studies.

Electrons are scattered strongly in air, so diffraction experiment must be carried out in a high vacuum. This brings complication and it is expensive as well.

55

Electron Diffraction

Electron diffraction has also been used in the analysis of crystal structure. The electron, like the neutron, possesses wave properties;

Electrons are charged particles and interact strongly with all atoms. So electrons with an energy of a few eV would be completely absorbed by the specimen. In order that an electron beam can penetrate into a specimen , it necessitas a beam of very high energy (50 keV to 1MeV) as well as the specimen must be thin (100-1000 nm)

02AeVm

h

m

kE

ee

4022 2

222