Lec. 6,7

1

• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

• Can’t resolve spacings

• Spacing is the distance between parallel planes of atoms.

2

X-Ray Diffractıon Methods

Von Laue Rotating Crystal Powder

OrientationSingle CrystalPolychromaticBeam, FixedAngle single

Lattice constantSingle Crystal

MonchromaticBeam, Variable

Angle Varied byrotation

Lattice ParametersPoly Crystal

MonchromaticBeam, VariableAngle Many s (orientations)

3

Laue Method• The Laue method is mainly used to determine the

orientation of large single crystals while radiation is

reflected from, or transmitted through a fixed crystal.

• The diffracted beams form arrays of

spots, that lie on curves on the film.

• The Bragg angle is fixed

for every set of planes in the

crystal. Each set of planes picks

out & diffracts the particular wavelength from the white radiation

that satisfies the Bragg law for

the values of d & θ involved. 4

• In the transmission Laue method, the film is

placed behind the crystal to record beams which

are transmitted through the crystal.

X-Rays

Film

Single

Crystal

Transmission Laue Method

• In the transmission Laue method,

the film is placed behind the

crystal to record beams which are

transmitted through the crystal.

• One side of the cone of Laue

reflections is defined by the

transmitted beam.

• The film intersects the cone,

with the diffraction spots

generally lying on an ellipse. 5

• The Laue method is mainly used to determine the

crystal orientation.

• Although the Laue method can also be used to

determine the crystal structure, several

wavelengths can reflect in different orders

from the same set of planes, with the different

order reflections superimposed on the same spot

in the film. This makes crystal structure

determination by spot intensity diffucult.

• The rotating crystal method overcomes this problem.

Crystal Structure Determination

by the Laue Method

6

• In the rotating crystal

method, a single crystal

is mounted with an axis

normal to a

monochromatic x-ray

beam. A cylindrical film

is placed around it & the

crystal is rotated about

the chosen axis.

• As the crystal rotates, Sets of lattice planes will at some

point make the correct Bragg angle

for the monochromatic incident beam, & at that point a diffracted

beam will be formed.

Rotatıng Crystal Method

7

The Lattice constant of the crystal can

be determined with this method. For a

given wavelength λ if the angle θ at

which a reflection occurs is known, d can

be determined by using Bragg’s Law.

2 2 2

ad

h k l

Rotatıng Crystal Method

2 sind n 8

The reflected beams are located on the surfaces

of imaginary cones. By recording the diffraction

patterns (both angles & intensities) for various

crystal orientations, one can determine the shape

& size of unit cell as well as the arrangement of

atoms inside the cell.

Film

Rotatıng Crystal Method

9

For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength

In crystals the typical interatomic spacing ~ 2-3 Å so thesuitable radiation is X-rays

Hence, X-rays can be used for the study of crystal structures

Beam of electrons TargetX-rays

An accelerating (/decelerating) charge radiates electromagnetic radiation10

Relationship of the Bragg angle (θ) and the experimentally measured diffraction angle (2θ).

X-ray intensity (from detector)

c

d n

2 sin c

11

Inte

nsi

ty

Wavelength ()

Mo Target impacted by electrons accelerated by a 35 kV potential

0.2 0.6 1.0 1.4

White radiation

Characteristic radiation → due to energy transitions

in the atom

K

K

12

Target Metal Of K radiation (Å)

Mo 0.71

Cu 1.54

Co 1.79

Fe 1.94

Cr 2.29

13

BRAGG’s EQUATION

d

The path difference between ray 1 and ray 2 = 2d Sin

For constructive interference: n = 2d Sin

Ray 1

Ray 2

Deviation = 2

14

15

16

Note that in the Bragg’s equation:

The interatomic spacing (a) along the plane does not appear

Only the interplanar spacing (d) appears

Change in position or spacing of atoms along the plane should not affect

Bragg’s condition !!

d

Note: shift (systematic) is

actually not a problem!

17

Bragg’s equation is a negative law

If Bragg’s eq. is NOT satisfied NO reflection can occur

If Bragg’s eq. is satisfied reflection MAY occur

Diffraction = Reinforced Coherent Scattering

Reflection versus Scattering

Reflection Diffraction

Occurs from surface Occurs throughout the bulk

Takes place at any angle Takes place only at Bragg angles

~100 % of the intensity may be reflected Small fraction of intensity is diffracted

X-rays can be reflected at very small angles of incidence18

n = 2d Sin, n= 1, 2, 3, …

n is an integer and is the order of the reflection

For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å

n Sin

1 0.34 20.7º First order reflection from (110)

2 0.69 43.92ºSecond order reflection from (110)

Also written as (220)

222 lkh

adhkl

8220

ad

2110

ad

2

1

110

220 d

d

19

sin2 hkldn

In XRD nth order reflection from (h k l) is considered as 1st order reflection

from (nh nk nl)

sin2n

dhkl

sin2 n n n lkhd

20

21

• If a powdered crystal is used instead of a single

crystal, then there is no need to rotate it, because

there will always be some small crystals at an

orientation for which diffraction is permitted.

Here a monochromatic X-ray beam is incident on

a powdered or polycrystalline sample.

• Useful for samples that are difficult to obtain in

single crystal form.

• The powder method is used to determine the

lattice parameters accurately. Lattice parameters

are the magnitudes of the primitive vectors a, b

and c which define the unit cell for the crystal.

The Powder Method

22

• For every set of crystal planes, by chance,

one or more crystals will be in the correct

orientation to give the correct Bragg angle

to satisfy Bragg's equation. Every crystal

plane is thus capable of diffraction.

• Each diffraction line is made up of a large

number of small spots, each from a separate

crystal. Each spot is so small as to give the

appearance of a continuous line.

The Powder Method

23

• If a monochromatic X-ray beam is directed

at a single crystal, then only one or two

diffracted beams may result. See figure• For a sample of several randomly orientated

single crystals, the diffracted beams will lie

on the surface of several cones. The cones

may emerge in all directions, forwards and

backwards. See figure• For a sample of hundreds of crystals

(powdered sample), 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 lines. The lines are seen as arcs

on the film. See figure

The Powder Method

24

THE POWDER METHOD

Cone of diffracted rays

25

Diffraction cones and the Debye-Scherrer geometry

Film may be replaced with detector

POWDER METHOD

Different cones for different reflections

26

• A small amount of powdered material is sealed into a fine

capillary tube made from glass that does not diffract X-Rays.

• The sample is placed in the Debye Scherrer camera and

is accurately aligned to be in the center of the camera. X-

Rays enter the camera through a collimator.

• The powder diffracts the X-Rays

in accordance with Braggs Law to

produce cones of diffracted

beams. These cones intersect a

strip of photographic film located

in the cylindrical camera to

produce a characteristic set of

arcs on the film.

Debye Scherrer Camera

27

• When the film is removed from the

camera, flattened & processed, it shows

the diffraction lines & the holes for the

incident & transmitted beams.

Powder Diffraction Film

28

Some Typical Measurement Results• Laue - “white” X-rays

– Yields stereoscopic projection of reciprocal lattice

• Rotating-Crystal method: monochromatic X-rays

– Fix source & rotate crystal to reveal reciprocal lattice

• Powder diffraction - monochromatic X-rays– Powder sample to reveal “all” directions of RL

0

0.5

1

20 30 40 50 60 70 80 90

Ce0.8

Y0.2

CoIn5

I/ImaxCeCoIn5 Theory

Norm

aliz

ed C

ounts

229

Photograph of a

XRD Diffractometer

(Courtesy H&M Services.)

30

(b) Diffraction Pattern

from a sample of

gold powder.

(a) Diagram of a

diffractometer

showing a powdered

sample, incident &

diffracted beams.

31

The results of a diffraction experiment using X-Rays

with λ = 0.7107 Å (radiation obtained from a

molybdenum, Mo, target) show that diffracted peaks

occur at the following 2θ angles:

Example (From the Internet)

Find: The crystal structure, the indices of the plane

producing each peak, & the lattice parameter of the

material. 32

First calculate the sin2 θ value for each peak, then

divide through by the lowest denominator, 0.0308.

Example (Solution)

33

Example (Solution Continued)Then use the 2θ values for any of the peaks to

calculate the interplanar spacing & thus the lattice

parameter.

Picking Peak 8: 2θ = 59.42° or θ = 29.71°

Ǻ

So, for example:

868.2)4)(71699.0(

71699.0)71.29sin(2

7107.0

sin2

2224000

400

lkhda

d

This is the lattice parameter for body-centered cubic iron.

Ǻ

34

Note: XRD is a nondestructive technique!

Some uses of XRD include:1. Distinguishing between crystalline & amorphous

materials.

2. Determination of the structure of crystalline

materials.

3. Determination of electron distribution within the

atoms, & throughout the unit cell.

4. Determination of the orientation of single crystals.

5. Determination of the texture of polygrained

materials.

6. Measurement of strain and small grain size…..etc.

Applications of XRD

35

Advantages• X-Rays are the least expensive, the most

convenient & the most widely used method to

determine crystal structures.

• X-Rays are not absorbed very much by air, so the

sample need not be in an evacuated chamber.

Disadvantages• X-Rays do not interact very strongly with lighter

elements.

Advantages & Disadvantages of XRD

Compared to Other Methods

36

X-Rays

λ ~ 1 Ǻ

E ~ 104 eV

interact with

electrons,

penetrating

Neutrons

λ ~ 1 Ǻ

E ~ 0.08 eV

interact with

nuclei, highly

penetrating

Electrons

λ ~ 1 Ǻ

E ~ 150 eV

interact with

electrons, less

penetrating

Diffraction Methods

37