Materials engineering

1. Acknowledgment

2. X-ray Diffraction
• Key crystallographic concepts and basics of
diffraction
• Practical aspects of XRD
• Analysis of XRD diffraction data
• In situ studies and complementary techniques
• Examples

3. Crystallographic concepts
The (200) planes of atoms in NaCl The (220) planes of atoms in NaCl

4. Molecular structure of crystalline solids
LATTICE
BASIS
A Space LATTICE is an infinite, periodic array of mathematical points,
in which each point has identical surroundings to all others.
A CRYSTAL STRUCTURE is a periodic arrangement of atoms in the crystal
that can be described by a LATTICE + ATOM DECORATION (called a BASIS).

5. Rotational symmetry
Only 5 rotational symmetries (1, 2, 3, 4, 6) are allowed

6. Seven crystal systems
Decreasing
symmetry

7. Unit cells and unit cell vectors
a

b

c

Lattice parameters
axial lengths: a, b, c
interaxial angles: a, b, g
unit vectors:
In general: a ≠ b ≠ c
a ≠ b ≠ g
All period unit cells may be described
via these vectors and angles.
a

b

c


8. The 14 Bravais Lattices!

9. Crystallographic planes
Axis x y z
Intercept points 1 ∞ ∞
Reciprocals 1/1 1/∞ 1/∞
Smallest ratio 1 0 0
Miller indices (100)
Crystallographic planes are expressed by Miller indices

10. Crystallographic planes

12. Crystallographic concepts
How many periodic distances are there within a lattice?

13. • A lattice plane is a plane
which intersects atoms of a
unit cell across the whole
three‐dimensional lattice.
• There are many ways of
constructing lattice planes
through a lattice.
• The perpendicular
separation between each
plane is called the d-spacing.
How many periodic distances are there within a lattice?
Crystallographic concepts

14. • Each plane intersects the
lattice at a/h, b/k, and c/l.
• h, k, and l, are known as the
Miller indices (hkl) and are
used to identify each lattice
plane.
How do we describe lattice planes?
Crystallographic concepts

15. Relationship between d-spacing and lattice constants
Cubic:
Tetragonal:
Orthorhombic:
Hexagonal:
Rhombohedral:
Monoclinic:
Triclinic:
2
/
1
2
2
2
2
)
(
1









 l
k
h
a
dhkl
2
/
1
2
2
2
2
2 









c
l
a
k
h
dhkl
2
/
1
2
2
2
2
2
2 









c
l
b
k
a
h
dhkl

16. Crystallographic concepts--Summary
• Ordering in materials gives rise to structural periodicity which can
be described in terms of translational, rotational and other
symmetry relationships.
• Crystalline materials composed of “infinite” array of identical lattice
points.
• The unit cell is the smallest three dimensional box which can be
stacked to describe the 3D lattice of a solid. The edges (a, b, and c)
and inter‐edge angles (α, β, and g) of the unit cell are known as the
lattice parameters of a crystalline solid.
• Relationship between unit cell, crystal systems, and Bravais lattices
is described.
• Lattice planes are two dimensional planes which intersect the three
dimensional lattice in a periodic way, with a fixed perpendicular
separation known as the d‐spacing.
• Miller indices define the orientation of the plane within the unit
cell.

17. X-ray – a brief history
• In 1895 Roentgen discovered X-rays.
• Invisible, travels in straight line like visible light
• Can penetrate through human body, wood, metal
and other opaque objects
• In 1912, x-ray diffraction of crystals was
discovered
• Diffraction can reveal internal structure in the
order of 10-10 m (1 Å).
• These are electromagnetic radiation with shorter
wavelength (  ~0.5 -2.5 Å),  for visible light is
6000 Å.

18. X-ray generation
• Bombard beam of electrons at metal
target at high voltage
• Ionization of inner shell electrons results
in formation of an ‘electron hole’.
• Upon relaxation of electrons from upper
shells, a energy difference E (≈10‐10 m) =
h is released in the form of X‐rays of
specific wavelengths.
• Most of the kinetic energy of electrons is
converted into heat.
• Less than 1% transform into x-rays
• Commonly used metals are Cu Kα ( =
1.5418 Å) and Mo Kα ( = 0.71073 Å).
Two principal methods for X‐ray generation

19. Two principal methods for X‐ray generation:
2) Accelerate electrons in a particle accelerator
(synchrotron source). Electrons accelerated
at relativistic velocities in circular orbits. As
velocities approach the speed of light they
emit electromagnetic radiation in the X‐ray
region.
• The X‐rays produced have a range of
wavelengths (white radiation or
Bremsstrahlung).
• Results in high flux of X‐rays.
• Laue experiment
I

Minimum and maximum  determined
by initial kinetic energy
X-ray generation

20. X-ray diffraction pattern of a solid

21. Applications of XRD
• Phase determination: qualitative
– Identification of crystalline phases
• Quantitative phase analysis
– Relative composition of mixed phases
• Calculation of lattice parameters
– Structural variations under different conditions.
• Analysis of crystalline size and strain
• Structure solution
– Complete structure refinement of unknown phases
Crystal structure determines the properties of materials.

22. Diffractometer geometries
Transmission geometry
(Debye‐Scherrer geometry)
Best for samples with low absorption.
Capillaries can be used as sample holders
(measurement of air sensitive samples /
suspensions).
Reflection geometry
(Bragg‐Brentano geometry).
Best for strongly absorbing samples.
Requires flat sample surface.
Easily adapted for in situ investigations.

23. XRD data analysis
• The intensity of the diffraction signal is usually plotted against
the diffraction angle 2θ [°], but d [nm] or 1/d [nm‐1] may also be
used.
• A 2θ plot is pointless if the wavelength used is not stated
because the diffraction angle for a given d‐spacing is dependent
on the wavelength. The most common wavelength used in
PXRD is 1.54 Å (Cu Kα).
• The “signals” in a diffractogram are called (Bragg or diffraction)
peaks, lines, or reflections.
How do we present X‐ray diffraction data?

24. XRD data analysis

25. Bragg’s law
• In materials with a crystalline structure, X‐rays scattered
by ordered features will be scattered coherently
“in‐phase” in certain directions meeting the criteria for
constructive interference: signal amplification.
• The conditions required for constructive interference are
determined by Bragg’s law.
integer
plane
lattice
with
incidence
of
angle
planes
lattice
between
distance
ngth
ray wavele
-
X
where
sin
2





n
d
d
n




Bragg’s Law:

26. When will diffraction occur?
• The periodic lattice found in crystalline structures may act as a
diffraction grating for wave particles or electromagnetic radiation
with wavelengths of a similar order of magnitude (10‐10m / 1 Å).
• For solids there are three particles/waves with wavelengths
equivalent to interatomic distances and hence which will interact
with a specimen as they pass through it: X‐rays, electrons, and
neutrons.

27. What type of materials can we study?
• Gas: No structural order – see nothing.
• Liquid/Amorphous solids: Order over a few angstroms – broad
diffraction peaks
• Ordered solids: Extensive structural order – sharp diffraction peaks.
• Large crystal required.
• Crystal orientation known.
• Each lattice plane only present in one
orientation.
• No overlap of reflections.
• Reflection intensities may be accurately
measured.
•Most common in heterogeneous catalysis
•Assume all crystal orientations present.
• Each lattice plane present at all
orientations.
•Many overlapping peaks.
•Reflection intensities difficult to
determine.
Single
crystals
Polycrystalline
powders

28. What determines the intensity of the
diffraction peak?
• X‐rays scattered by electrons
• Greater the atomic number, Z, the higher the scattering factor
(structure factor) of a given element (directly proportional).
• Intensity proportional to sum of the scattering factors of atoms in a
given lattice plane.
• For Powder samples all planes with equivalent d‐spacing overlap.
• Intensity dependent on number of overlapping planes.
Electron density
Multiplicity
2
F
I 

29. How can we satisfy the Bragg equation?
• Vary θ
• Bragg condition met once at a time.
• With a fixed wavelength (laboratory source) data is collected as a function of
increasing diffraction angle up to a given value of d (resolution dependent).
Bragg’s Law: =2dsinθ
To observe diffraction from a given lattice plane, Bragg’s law may be satisfied
by varying either the wavelength, , or the Bragg angle, θ.
• Vary 
• Bragg condition met many times simultaneously.
• Faster but greater complexity in data analysis.
• Requires synchrotron which is expensive.
Laue diffraction
Monochromatic diffraction:

30. Diffraction pattern

31. Indexing a diffraction pattern of a cubic crystal
• For cubic material, interplanar spacing is given as
• Recall Bragg’s law:
• Combining eq (1) & (2)
• , a are constants, hence 2/4a2 is const.
• sin2θ is proportional to h2+k2+l2, i.e., planes with higher
Miller indices will diffract at higher θ.
)
1
(
1
2
2
2
2
2
a
l
k
h
d



(2)
4
sin
or
sin
4
sin
2
2
2
2
2
2
2
d
d
d
n









)
(
4
sin 2
2
2
2
2
2
l
k
h
a






32. Indexing
• In cubic systems, the 1st peak in the XRD pattern will be
due to diffraction from planes with the lowest miller
indices
• For the close packed planes
– Simple cubic, (001), h2+k2+l2 =1
– BCC, (110), h2+k2+l2 = 2
– FCC, (111), h2+k2+l2 = 3
• Allowed reflection for cubic lattices
– Primitive, h2+k2+l2 = 1,2,3,4,5,6,8,9,10,11,12,13, 14,16…
– Body centered, h2+k2+l2 = 2,4,6,8,10,12,14,16…
– Face centered, h2+k2+l2 = 3,4,8,11,12,16,19,20,24,27,32…
– Diamond cubic, h2+k2+l2 = 3,8,11,16,19,24,27,32…
Allowed diffraction:
FCC: hkl all odd or all even
BCC: h+k+l = even

33. Calculation of lattice parameter

34. Steps
Step 1 : Identify the peaks and their proper 2θ values
Step 2: Determine sin2θ

35. Step 3: Calculate the ratio sin2θ/ sin2θmin and multiply by the appropriate integers.
Step 4: Select the result from (3) that yields h2 + k2 + l2 as a series of integers.

36. Step 5: Compare results with the sequences of h2 + k2 + l2 values
to identify the Bravais lattice
Step 6: Calculate lattice parameters

37. Phase determination (Qualitative)
• The matching process: Search - Match : Manual or
automated.
• Generally, all expected reflections should be seen in the
diffractogram, otherwise it is not a valid match.
• Low signal to noise ratio will give weak reflections, not
visible.
• If measured pattern contains more reflections than the
reference, more than one phase is present.
• So keep the reference pattern, continue searching for
references to explain the additional peaks.
• Proceed until all peaks are accounted for.
• Deviations in peak positions may be due to
– Thermal expansion
– Isostructural phase
– Presence of substitutional atom
– Impurities of similar Z but different atomic size

38. Phase determination (Qualitative)

40. Quantitative phase analysis
• Determining the relative proportions of
crystalline phases present in unknown
sample.
• Ratio of peak intensities varies linearly
as a function of weight fraction for any
two phases (A and B) in a mixture.
• Data on IA/IB value for all phases and
calibration with mixture of known
quantities is required.
• Determining amorphous content of
sample containing crystalline phase A.
• Use internal standard of known
crystallinity.
• Data on appropriate XRD standard B is
required.
• Known amount of std B is added to
specimen containing phase A.

41. Analysis of crystallite size and strain
Factors determining the peak profile (shape and width):
• Imperfections in Experimental setup :
– This lead to instrumental broadening, which is due to slit width and
goniometer radius, Instrumental broadening is fn of 2θ and is
determined by measurement of a suitable reference.
• Imperfections in Sample :
– This causes sample broadening.
– Periodicity in the crystal is not infinite (crystals have finite size), this
leads to size broadening.
– Lattice imperfections like dislocations, vacancies and substitutional
lead to strain broadening.
Ideal
Real

42. Analysis of crystallite size
• This related peak width to crystalline domain size.
• Crystals are assumed to have uniform size and shape.
• Scherrer const K depends on shape and is 0.94 for spherical crystals
with cubic symmetry.
Size broadening effect translates to crystal size:
• Volume averaged crystal thickness , L, depends on crystal shape
• For cubic crystallite : L = Lc = crystallite edge length (for reflections
of lattice planes parallel to cube) and
• For spherical crystallite, L < Lc = crystallite dia
• Lvol = 3Lc/4 for all reflections.
Scherrer equation
Relates peak width to
crystalline domain size

43. Analysis of microstrain broadening
• Lattice strain (microstrain) arises from displacement of the unit cells
about their normal position.
• Often caused by dislocations, surface restructuring, lattice vacancies,
interstitials, substitutionals etc.
• Very common in nanocrystalline materials.
• Strain is quantified as ε0 = Δd/d , where d is idealized d spacing and
Δd the most extreme deviation from d.
• The peak broadening due to strain:
• Hence, observed peak broadening Bobs is given as:
Bobs = Binstr + Bsample = Binstr + Bsize + Bstrain
• Instrumental broadening Binstr can be determined experimentally
with diffraction of standard or calculated with fundamental
parameters.
Bsize  1/ cosθ and Bstrain  tanθ
• Broadening may alter FWHM or integral breadth of the peak.

 tan
4
Bstrain o


44. Peak Position and Width