X-ray Diffraction (XRD)

• 1.0 What is X-ray Diffraction

• 2.0 Basics of Crystallography
• 3.0 Production of X-rays
• 4.0 Applications of XRD
• 5.0 Instrumental Sources of Error
• 6.0 Conclusions
 Bragg’s Law

n =2dsin

English physicists Sir W.H. Bragg and his son Sir W.L. Bragg
developed a relationship in 1913 to explain why the cleavage
faces of crystals appear to reflect X-ray beams at certain angles of
incidence (theta, ). The variable d is the distance between atomic
layers in a crystal, and the variable lambda  is the wavelength of
the incident X-ray beam; n is an integer. This observation is an
example of X-ray wave interference
(Roentgenstrahlinterferenzen), commonly known as X-ray
diffraction (XRD), and was direct evidence for the periodic atomic
structure of crystals postulated for several centuries.
 Bragg’s Law

n =2dsin

The Braggs were awarded the Nobel Prize in

physics in 1915 for their work in determining
crystal structures beginning with NaCl, ZnS
and diamond.

Although Bragg's law was used to explain the interference pattern

of X-rays scattered by crystals, diffraction has been developed to
study the structure of all states of matter with any beam, e.g., ions,
electrons, neutrons, and protons, with a wavelength similar to the
distance between the atomic or molecular structures of interest.
 Deriving Bragg’s Law: n = 2dsin
X-ray 1
Constructive interference X-ray 2
occurs only when

n = AB + BC AB+BC = multiples of n

AB=BC

n = 2AB

Sin =AB/d

AB=dsin

n =2dsin

= 2dhklsin hkl
 Constructive and Destructive
Interference of Waves

Constructive Interference Destructive Interference

In Phase Out of Phase
 1.0 What is X-ray Diffraction ?
I

www.micro.magnet.fsu.edu/primer/java/interference/index.html
 Why XRD?

• Measure the average spacings between

layers or rows of atoms
• Determine the orientation of a single
crystal or grain
• Find the crystal structure of an unknown
material
• Measure the size, shape and internal
stress of small crystalline regions
 X-ray Diffraction (XRD)
The atomic planes of a crystal cause an incident beam of X-rays to
interfere with one another as they leave the crystal. The phenomenon is
called X-ray diffraction.

Effect of sample
incident beam
thickness on the
absorption of X-rays

crystal diffracted beam

film

http://www.matter.org.uk/diffraction/x-ray/default.htm
 Detection of Diffracted X-rays
by Photographic film
sample
film

X-ray
Point where Film
incident beam
enters
2 =0
2 = 180

Debye - Scherrer Camera

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 lines. The lines are seen as arcs
on the film.
 Bragg’s Law and Diffraction:
How waves reveal the atomic structure of crystals
n = 2dsin n-integer
Diffraction occurs only when Bragg’s Law is satisfied Condition for constructive
interference (X-rays 1 & 2) from planes with spacing d

X-ray1

X-ray2
l
=3Å
=30o
Atomic
plane
d=3 Å

2 -diffraction angle

http://www.eserc.stonybrook.edu/ProjectJava/Bragg/
 Planes in Crystals-2 dimension
= 2dhklsin hkl

Different planes
have different
spacings

To satisfy Bragg’s Law, must change as d changes

e.g., decreases as d increases.
 2.0 Basics of Crystallography
smallest building block
c
d3
b
a
Unit cell
(Å)
d1
Beryl crystals CsCl
(cm) Lattice d2
A crystal consists of a periodic arrangement of the unit cell into a
lattice. The unit cell can contain a single atom or atoms in a fixed
arrangement.
Crystals consist of planes of atoms that are spaced a distance d apart,
but can be resolved into many atomic planes, each with a different d-
spacing.
a,b and c (length) and ,  and  angles between a,b and c are lattice
constants or parameters which can be determined by XRD.
Seven Crystal Systems - Review
 Miller Indices: hkl - Review
Miller indices-the reciprocals of the
fractional intercepts which the plane
makes with crystallographic axes

(010)

a b c a b c
Axial length 4Å 8Å 3Å 4Å 8Å 3Å
Intercept lengths 1Å 4Å 3Å 8Å
Fractional intercepts ¼ ½ 1 0 1 0
Miller indices 4 2 1 0 1 0
h k l h k l
4/ =0
Several Atomic Planes and Their d-spacings in
a Simple Cubic - Review

a b c a b c
1 0 0 1 1 0
1 0 0 1 1 0

d100
Cubic
(100) a=b=c=a0
(110)

a b c
a b c
d012 0 1½
1 1 1
0 1 2
1 1 1

(111) (012)
Black numbers-fractional intercepts, Blue numbers-Miller indices
Planes and Spacings - Review
 Indexing of Planes and Directions -
Review

(111)
c c

b b
a a [110]

a direction: [uvw] a plane: (hkl)

<uvw>: a set of equivalent {hkl}: a set of equi-
directions valent planes
 3.0 Production of X-rays

Cross section of sealed-off filament X-ray tube

copper vacuum glass
X-rays
tungsten filament

electrons
cooling
water
to transformer
target
Vacuum

beryllium window X-rays metal focusing cap

X-rays are produced whenever high-speed electrons collide with a metal

target. A source of electrons – hot W filament, a high accelerating voltage
between the cathode (W) and the anode and a metal target, Cu, Al, Mo,
Mg. The anode is a water-cooled block of Cu containing desired target
metal.
 Characteristic X-ray Lines
K
K and K 2 will cause
extra peaks in XRD pattern,
and shape changes, but
can be eliminated by
K 1
adding filters.
<0.001Å
----- is the mass

Intensity
absorption coefficient of
K 2
Zr.
K

(Å)
Spectrum of Mo at 35kV
 Specimen Preparation

Powders: 0.1 m < particle size <40 m

Peak broadening less diffraction occurring
Double sided tape

Glass slide

Bulks: smooth surface after polishing, specimens should be

thermal annealed to eliminate any surface deformation
induced during polishing.
 JCPDS Card Quality of data

1.file number 2.three strongest lines 3.lowest-angle line 4.chemical

formula and name 5.data on diffraction method used 6.crystallographic
data 7.optical and other data 8.data on specimen 9.data on diffraction pattern.
Joint Committee on Powder Diffraction Standards, JCPDS (1969)
Replaced by International Centre for Diffraction Data, ICDF (1978)
A Modern Automated X-ray Diffractometer

Detector
X-ray Tube
2


Sample stage

Cost: $560K to 1.6M

Basic Features of Typical XRD Experiment

1) Production
X-ray tube
2) Diffraction

3) Detection

4) Interpretation
Detection of Diffracted X-rays
by a Diffractometer

C
Circle of Diffractometer

Recording

Amplifier
Focalization
Circle

Detector
Photon counter

Bragg - Brentano Focus Geometry, Cullity

 XRD Pattern of NaCl Powder

(Cu K )

Miller indices: The peak is due to X-

ray diffraction from the {220}
planes.

Diffraction angle 2 (degrees)

 4.0 Applications of XRD

• XRD is a nondestructive technique

• To identify crystalline phases and orientation
• To determine structural properties:
Lattice parameters (10-4Å), strain, grain size,
expitaxy, phase composition, preferred orientation
(Laue) order-disorder transformation, thermal
expansion
• To measure thickness of thin films and multi-layers*
• To determine atomic arrangement
• Detection limits: ~3% in a two phase mixture; can be
~0.1% with synchrotron radiation
Spatial resolution: normally none
 More Applications of XRD

a Diffraction patterns of three

(004) Superconducting thin films
annealed for different times.

a. Tl2CaBa2Cu2Ox (2122)
b b. Tl2CaBa2Cu2Ox (2122) +
Tl2Ca2Ba2Cu3Oy (2223)
b=a+c

Intensity
c. Tl2Ca2Ba2Cu3Oy (2223)
c
(004) CuO was detected by
comparison to standards
 High Temperature XRD Patterns of the
Decomposition of YBa2Cu3O7-

Intensity (cps)
T

2
 6.0 Conclusions

• Non-destructive, fast, easy sample prep

• High-accuracy for d-spacing calculations

• Can be done in-situ

• Single crystal, poly, and amorphous materials

• Standards are available for thousands of material

systems