Investigation in nanoscale

X-RAY DIFFRACTION
 X-ray generation
• Wavelength range:0.5-2.5 Å
 Bremsstruhlung radiation
Bremsstrahlung produced by rapid deceleration of a high-energy electron in
the electric field of an atomic nucleus.

1 2
KE  eV  mv
2
eV, in the order of 30,000 volts
m, electron mass
v, velocity of electron just befor impact

The decelerated electron emits energy ( x-ray).

http://en.wikipedia.org/wiki/Bremsstrahlung
White (polychromatic, continuous) radiation

Continuous X-ray spectrum of

molbdenum (Mo) as of a
function of applied voltage.
 Characteristics radiation
•When applied voltage excess a critical value Characteristic Radiation occurs .

•Characteristic emissions superimpose on continuous spectrum and they are

narrow and intense. They are used for x-Ray diffraction because of they are
approximate monochromatic.
 Characteristic radiation lines fall into
several sets, K, L, M., in the order of
increasing wavelength. Only K line is
used in x-ray diffraction.
2p 1s transition

3p 1s transition
Kα doublet
Unresolved Kα line is taken as
weighted average of wavelength of
Kα1 and Kα2

Electron Spin state 2P3/2

Kα1 is twice as strong as

Kα2

Electron Spin state 2P1/2

 X-Ray and Bragg’s law
• A crystal is viewed as a plane containing
several lattice point. At Bragg’s angle the
reflective beams are in phase and reinforced
beam is obtained.
 X-RAY spectroscopy
• A crystal with planes of known spacing or a grating of know spacing
can be set at particular angle Θ, thus, the incident wavelength of the
radiation can be determined.
• Intensity of diffraction at that particular angle Θ is recorded.
• Repeat this test for various diffraction angle Θ, a wavelength-x-ray
intensity curve can be plotted. Test sample
generates x-Ray
 X-Ray diffractometer
Crystal structure analysis:
• With known x-Ray irradiation, a crystal’s lattice spacing can be
determined by the diffraction pattern.
• Three methods :

method λ θ
Laue Variable Fixed
(polychromatic)

Rotating-crystal Fixed Variable in part

(monochromatic)

powder Fixed variable

(monochromatic)
 Laue method
• First x-ray diffraction ever used
• A white x-ray irradiation falls on a crystal
• Each set of planes in the crystal, selects and diffracts a
particular wavelength that satisfies Bragg’s law.

Spots lying on one curve are

transmitted diffractions
from planes belongs to one
zone.
 Rotating crystal method
• Uses the characteristic x-ray irradiations
• The crystal is mounted with one of its axes
or important crystallographic direction,
normal to the incident x-ray.
• The crystal axis or crystallographic
direction coincides with the axis of the
cylindrical film.
• The crystal rotates, a particular set of
lattice plane will satisfies Bragg’s law at a
particular angle and at that instant a
diffraction pattern will be generated.
• lattice planes that parallel/almost parallel
to the incident beam generates no
diffraction pattern.
 Powder method
• Sample is reduced to very fine powder. Each
microscopic grain is a crystal.
• Same principle as rotating crystal method. Only
that in powder form, each tiny crystal is orientated
at random angle with respect to incident beam.
The mass of powder in fact is equivalent to a single
crystal rotated at all possible axes. Therefore, all
lattice planes are capable of diffraction.
(a) The diffracted rays from a single crystal point to discrete directions each corresponding
to a family of diffraction planes. (Spot pattern)

(b) The diffraction pattern from a polycrystalline (powder) sample forms a series diffraction
cones if large number of crystals oriented randomly in the space are covered by the
incident x-ray beam. Each diffraction cone corresponds to the diffraction from the same
family of crystalline planes in all the participating grains. (Ring pattern)
 The composition, size and degree of
crystalline of the sample material can
be extracted from the ring positions
and radial widths of these rings.

John Hart, nanomanufacturing,

University of Michigan
S: X-Ray source
H: rotation table
A, B: X-Ray optics
F: receiving slit of the
detector

Diffraction patter, about one- half

of the entire range of 2.
Schematic drawing of
a X Ray Diffractometer
Only crystalline
material gives narrow
peaks.
From the locations of
these peaks (rings),
lattice spacing can be
calculated.
 Intensity of diffracted beam
Atom position in the unit cell affects the intensities
but not the directions of the diffracted beam.

Constructive destructive
interference from interference from
Base centered cubic Body centered cubic
Calculation of Diffraction Intensity
• Lattice is considered as gratings for the
determination of diffraction direction (Bragg’s law)
• In real X-Ray irradiation, lattice cannot be
considered a not a grating. For the calculation of
diffraction intensity, X-ray is considered as photons
of high energies and the intensity is the result of
photon-electron, photon-nucleus interaction. The
emission of diffracted beams is a result of
interference of scattered X-Rays, but not the
incident X-Ray,
Effects produced by the passage of X-ray
through matter

Forms Diffraction Contribute to

pattern background noise
 Factors of intensity calculation
• Structure factor F Determined by
the structure of
• Multiplicity factor p the crystal

• Absorption factor A  Account for sample

absorption of x-ray and
• Temperature factor e 2 M the resultant temperature
rise.

 2

 1  cos 2  
 A    e
2  2 M
I  F p 2
 sin  cos 
 
I, relative integrated intensity
Example of intensity calculation
Copper, face-centered cube

The intensity analysis gives

more details on the atom
arrangement of the unit cell.
Together with the lattice
spacing extracted from the
diffraction angle, the
structure of the crystal can
be predicted.
 Intensity in x-ray scattering
• Scattering factors is proportional to the atomic number. The scattering factor of
atoms of similar atomic numbers are close, therefore it is difficult to identify similar
atoms.

Problem materials: compound containing N and O(polymers)

aluminosilicates, Al, Si.

• the peak of maximum is taken as 100 and all the other peaks are scaled accordingly.
A set of peaks and their heights are adequate for phase identification. Sometimes
accurate measurement of peak positions is required.

• If preferred orientation exist for a material, it is likely that only those orientations
will be manifested.

• Sample preparation affects the scattering intensity. It is important that the sample
is finely grounded so that all crystal plane is observable.
Broadening of diffraction peak

B, full peak width at half-maximum

B  1 2
 Particle size and Peakwidth
0.9
t
B cos  B

• Peak width (B) is inversely proportional to

crystallite size (t)

P. Scherrer, “Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels
Röntgenstrahlen,” Nachr. Ges. Wiss. Göttingen 26 (1918) pp 98-100.
J.I. Langford and A.J.C. Wilson, “Scherrer after Sixty Years: A Survey and Some New
Results in the Determination of Crystallite Size,” J. Appl. Cryst. 11 (1978) pp 102-113.
http://prism.mit.edu/xray
 Particle size is not the only attribution to
peak boardening

Intensity (a.u.)

66 67 68 69 70 71 72 73 74

2q (deg.)
• These diffraction patterns were produced from the exact same sample
• Two different diffractometers, with different optical configurations, were used
• The apparent peak broadening is due solely to the instrumentation

http://prism.mit.edu/xray
 The Laue Equations describe the intensity of a diffracted peak
from a single parallelopipeden crystal
sin 2   /   s  sO   N 1 a1 sin 2   /   s  sO   N 2 a 2 sin 2   /   s  sO   N 3 a3
I  Ie F 2

sin 2   /   s  sO   a1 sin 2   /   s  sO   a 2 sin 2   /   s  sO   a3

• N1, N2, and N3 are the number of unit cells along the a1, a2, and a3 directions
• When N is small, the diffraction peaks become broader
• The peak area remains constant independent of N

5000 400
4500 N=99 350 N=20
4000 N=20
3500
300 N=10
N=10 250
3000 N=5
N=5 200
2500
N=2 N=2
2000 150
1500
100
1000
50
500
0 0
2.4 2.9 3.4 2.4 2.9 3.4

http://prism.mit.edu/xray
Stress measurement
Uniformed strain result in
shift in B, with no affect
on profile of the peak.
Used for measurement of
macrostress.

Non-uniformed
microstrain disturbs the
grain shape but no
distortion to the entire
volume of the sample.
The results is not shift in
B, but the profile of the
peak will be altered
 XRD Strain measurement
d  d o 1 2 
   sin     11   22 
do E E
2 2
biaxial stress     11 cos    22 sin 

In real test, will be changed several times and d

recorded.
Definition of 
 Stress measurement Example
Compressed thin gold film