Characterization of Materials Structure Using X-Ray Diffraction

Characterization of materials by
X-ray Diffraction
Dr. Sajjad Ullah
Institute of Chemical Sciences, University of Peshawar
December, 2018

X-ray Diffractometer
Dr. Sajjad Ullah ICS-UOP

X-rays Production

Sample Preparation

Crystalline materials are
characterized by the orderly periodic
arrangements of atoms.
• The unit cell is the basic repeating unit that defines a crystal.
• Parallel planes of atoms intersecting the unit cell are used to define
directions and distances in the crystal.
– These crystallographic planes are identified by Miller indices.
The (200) planes
of atoms in NaCl
The (220) planes
of atoms in NaCl

The atoms in a crystal are a periodic array of
coherent scatterers and thus can diffract light.
• Diffraction occurs when each object in a periodic array scatters
radiation coherently, producing concerted constructive interference at
specific angles.
• The electrons in an atom coherently scatter light.
– The electrons interact with the oscillating electric field of the light
wave.
• Atoms in a crystal form a periodic array of coherent scatterers.
– The wavelength of X rays are similar to the distance between
atoms.
– Diffraction from different planes of atoms produces a diffraction
pattern, which contains information about the atomic arrangement
within the crystal
• X Rays are also reflected, scattered incoherently, absorbed, refracted,
and transmitted when they interact with matter.

Bragg’s Law

Interference upon Scattering: An Example
= 1λ
If path difference= 0.5 λ???

XRD Analysis
1- Crystallinity and Phase Identification (Use of Search-match software)
2- Determination of Crystallite Size (Scherer Equation)
3- Determination of Unit cell size (inter-planer distances, unit cell dimension)
4- Indexing diffraction peaks (determination of Miller indices)

Crystallinity
and
Phase Identification
(Use of Search-match software)

X-Ray Powder Diffraction (XRPD) uses information about the
position, intensity, width, and shape of diffraction peaks in a
pattern from a polycrystalline sample.
The x-axis, 2theta, corresponds to the angular position of the
detector that rotates around the sample.

Zhu, T.; Gao, S.-P. The Stability, Journal of Physical Chemistry C, v. 118, n.
21, p. 11385–11396, 2014.
Crystalline strcture of ppolymorphs of TiO2: Rtuile, Anatse and Brookite

One of the most important uses of XRD!!!
• Obtain XRD pattern
• Measure d-spacings or 2θ values
• Obtain integrated intensities
• Compare data with known standards in the JCPDS file, which
are for random orientations (there are more than 50,000 JCPDS
cards of inorganic materials).
Another important source is American Mineralogist Crystal
Structure Database: http://rruff.geo.arizona.edu/AMS/amcsd.php
• Identify by matching (better to use search-Match software)
Phase Identification

2
bofore hydrothermal treatement
(112)
anatase
(b)
after hydrothermal treatement
amorphous part
(103) (105) (211)(200)(004)
(101)
16 20 24 28 32 36 40 44 48 52 56 60
intensity(a.u)
(c)
(a)
Figure 2: X-ray diffractograms of TiO2 before HTT (a) and after HTT at 105°C for 24h
(b) Diffractorgram of anatase reconsitituted from the Inorganic crystal structure
database (ICSD Collection code = 154609 )
(154609)
Identification of Phase and Crystallinity
Phase-pure TiO2

Using Search-Match software to identify phase

Powder Diffraction File (PDF)/crystallographic Card

Mixed Phase TiO2
Phase Identification…continued

Mixed Phase TiO2 (anatase, rutile)

Estimation of Crystallite Size by
XRD using Scherrer Formula
Thickness = 0.9 λ
B cosӨB

Effect of Crystali size on the Powder pattern—Crystallite size measurements
The Broadning of XRD peaks can arise because of two main factors:
1. Instrumental Broading due to imperfect optical geometry
and intrisic width of Kα line (polychromatic radiations)
2. Broading of X-ray beam due to small particle size: particle size (PS) effect
3. Other Factors: Stress or imperfections in the crystal (see next slide)
Since addiational broading occurs when PS < 200 nm, we can extract inforamtion on
PS by measuring this additional band broading (Factor-2) using Schrrer fromula:
Thickness = 0.9 λ/ B cosӨB
For more precise measurement of the PS, correction for instrumental peak-broading
should be made (Factor-1) accroding to Warren Formula:
B2= BM
2-BS
2 BM = measured width for sample
BS = measured width for standard
Thus a standar material with PS>>200nm and with a difraction peak near to relevant
peak of the sample is added.

Macro-stress: Uniform
(a) Tensile= d increase OR
(b) Compressive: d decrease
Peak location changes
Micro-stress:
Tensile + Compressive
Peak-width increases

Size and stress broadening generally symmetric broadening
Instrumental factors cause asymmetric broadening
X-ray Characterization of Materials, by Eric Lifshin

The Scherrer Formula
λ = the X-ray wavelength used (Cu/K α = 0.154 nm)
ӨB = The Bragg angle. It is obtained by dividing by 2 the 2Ө value of the
corresponding peak.
B = The line broadning (in terms of angular spread) measured from the
extra peak width at half the peak height and is obtained from
Warrem formula:
B2= BM
2-BS
2
Note: The Value of B is to be converted to radian before putting in Schreer formula
k = is a shape factor which is 0.9 for spherical particles
𝑡 =
𝒌λ
𝑩 𝒄𝒐𝒔θB
A.R. West, Solid State Chemistry and its Applications, Wiley, Chichester [West Sussex] New York, 1984.

Worked Example
2ӨB = 25.3⁰
ӨB = 12.56⁰
CosӨB = 0.975
BM = 0.86⁰ = 0.015 rad
BS = 0.10⁰ = 0.0017 rad
B2= BM
2 - BS
2
B2= 2.2211x10-4
B= 0.0149
𝑡 =
𝒌λ
λ (Cu-Kα) = 0.154 nm
𝑡 =
𝟎. 𝟗 x 0.154
𝟎. 𝟎𝟏𝟗𝟒 𝒙 𝟎. 𝟗𝟕𝟓
= 9.5 nm
360° = 2 x 3.142 radians
0.86° = .86 x 2 x 3.142/(360) = .015 radians
k= shape factor (0.9 for spherical particles)
𝑡 =
𝒌λ

A.R. West, Solid State Chemistry and its Applications, Wiley, Chichester [West Sussex] New York, 1984.

Given: B = 0.5°, λ = 0.154 nm, 2θ = 27°
θ = 13.5 °; cos (13.5) = 0.972
360° = 2 x 3.142 radians
0.5° = .5 x 2 x 3.142/(360) = .00873 radians
t = 16.3 nm
Worked Example
A sample shows diffraction peak at 2θ = 27° which has FWHM of 0.5°.
Calculate the crystallite size if Cu/Kα X-ray radiation (0.154 nm) were
used.
𝑡 =
𝒌λ
𝑩 𝒄𝒐𝒔 θ
𝑡 =
0.9x 0.154
𝟎.𝟎𝟎𝟖𝟕𝟑 𝒙 𝟎.𝟗𝟕𝟐
= 16.3 nm

Given: λ = 0.154 nm, 2θ = 27°
θ = 13.5 °; cos (13.5) = 0.972
The Scherrer Equation
Worked Example
If t = 16.3 nm, what is B for the same reflection?All other conditions are similar
to previous example
𝐵 =
𝒌λ
𝒕 𝒄𝒐𝒔 θ
B =
𝟎.𝟗 𝒙 𝟎.𝟏𝟓𝟒
𝟏𝟔.𝟑 𝒙 𝟎.𝟗𝟕𝟐
= 0.00874 radians = 0.5°
6.284 (2π) radians = 360°
0.00874 radian = 0.00874 x 360/(6.284) = 0.5°
If t = 300 nm, B = 0.027°
If t = 500 nm, B = 0.016°

Unit Cell Size (inter-planer
distances, unit cell dimension) from
Diffraction Data
The Bragg equation is:
nλ = 2dsin(θ)

EXAMPLE: Unit Cell Size (inter-planer distances) from Diffraction Data
The diffraction pattern of copper metal was measured with x-ray radiation
of wavelength of 1.315Å. The first order Braggs diffraction peak was found
at an angle 2θ of 50.5°. Calculate the spacing between the diffracting planes
in the copper metal.
d =
nλ
𝟐 sinθ
θ = 25.25 degrees
n =1
λ = 1.315Å
Rearranging this equation for the unknown
spacing d:
d =
1 x 1.315
0.4266 x 2 = 1.541 Å
The Bragg equation is:
nλ = 2dsin(θ)

d =
nλ
𝟐 sinθ
2θ = 25.3
Θ= 12.65
Sin (12.65) = 0.219
d =
1 x 1.54056
2x 0.219
d= 3.517

EXAMPLE: Indexing of planes in X-ray Diffractogram
X-ray diffraction of Cu (Cu has fcc structure) is done using Cu/Kα X-ray
radiation ( 0.154 nm ). One prominent peak appears at 2θ= 43.2°. What are the
Miller indices for this peak?
d =
0.154
𝟐 sin (43.2/2)
= 0.209 nm
From Braggs equation is:
d =
λ
𝟐 sinθ
For FCC we know that dhkl =
𝒂
ℎ2+𝑘2+𝑙2 (from geometry)
𝒂= 2 r 𝟐 = 2 x 0.128 𝟐 = 0.362 nm
ℎ2 + 𝑘2 + 𝑙2 =
𝑎
𝑑ℎ𝑘𝑙
=
0.362
0.209
= 1.732
ℎ2
+ 𝑘2
+ 𝑙2 = 3
For FCC, the principal diffraction planes are those whose indices are all odd or all even
(e.g., (111), (222), (200), (002))
Thus ℎ= 𝑘= 𝑙 = 1 and the diffraction peak is due to (111) plane of FCC of Cu
𝒂= 2 r 𝟐
𝑟= atomic radius (Cu) = 0.128 nm
𝒂= unit cell dimension
(from geometry)

0.3615 nm
0.3615 nm
The Pythagorean Theorem: a2 +b2 = c2
c =4r
a= 0.3615 nm
b=0.3615 nm
(4r)2= (0.3615)2 + (0.3615)2
r= 0.1278 nm
𝒂= 2 r 𝟐
𝒂 = 2 x 0.128 𝟐 = 0.362 nm
r1
r4
r2
4r
r3
Calculate the Radius (r ) of Cu atoms in the FCC if the unit cell dimension is 0.3615 nm

EXAMPLE: Unit Cell dimension from X-ray Diffractogram
X-ray diffraction of a Metal sample (FCC) is done using Cu/Kα X-ray
radiation (1.54Å). One prominent peak appears at 2θ = 32.72°. If this peak
correspond to (110) planes of the samples, what is the unit cell dimension?
d =
1.54Å
𝟐 sin (43.2/2)
= 2.73 Å
From Braggs equation is:
d =
λ
𝟐 sinθ
For FCC we know that dhkl =
a
ℎ2+𝑘2+𝑙2 (from geometry)
a = 2.73 x 2 = 3.87Å

Some Example of the Application of XRD

Study of Bismuth vanadate
(BiVO4)
S. Ullah et. al., Applied Catalysis B: Environmental 243 (2019) 121–135, doi:10.1016/j.apcatb.2018.09.091

Figure 3: XRD patterns of BV samples prepared using (a) different microwave treatment times,
(b) different pH values while keeping all other conditions constant and (c,d) Rietveld refinement
of powder XRD data for BV(pH6) and BV(pH9.5), respectively. For comparison, the standard
diffraction patterns of tetragonal BV (PDF no. 14-133, red dotted lines) and monoclinic BV (PDF
no. 75–2480, black solid lines) have also been inserted at the bottom of a and b.
XRD study of
Bismuth vanadate
(BiVO4)

BiVO4 Photocatalyst
Visible (LEDs) UV-Visible (Xe lamp)
S. Ullah et. al., Applied Catalysis B: Environmental 243 (2019) 121–135, doi:10.1016/j.apcatb.2018.09.091

Studies of NaYF4 Materials
S. Ullah, et al., Microwave-assisted synthesis of NaYF4 :Yb 3+ /Tm 3+ upconversion particles with tailored
morphology and phase for the design of UV/NIR-active NaYF4 :Yb3+ /Tm3+ @TiO2 core@sh, CrystEngComm. 19
(2017) 3465–3475. doi:10.1039/C7CE00809K. Dr. Sajjad Ullah ICS-UOP

SEM image, and (c) XRD of NaYbF4: Tm3+ upconversion particles (UCPs). The inset in a is a
magnified view of UCPs.
S. Ullah et. al., Applied Catalysis B: Environmental 243 (2019) 121–135, doi:10.1016/j.apcatb.2018.09.091Dr. Sajjad Ullah ICS-UOP

Studies of polyhedral oligomeric silsesquioxanes (POSS)
Figure: Sketch for polyhedral oligomeric silsesquioxanes (POSS) structure formed during sol-
gel process
E.P. Ferreira-Neto et al. Materials Chemistry and Physics 153 (2015) 410-421

(a) Sketch for polyhedral oligomeric silsesquioxanes (POSS) structure formed during sol-gel
process
(b) correlation length observed by powder X-ray diffraction
(c) XRD patterns of POSS materials
(c)
E.P. Ferreira-Neto et al. Materials Chemistry and Physics 153 (2015) 410-421

The XRD patterns for powder laponite (LA) and laponite-Erbium (LA-Er) samples are shown in
Fig. 2. Comparing these XRD patterns, it was found that the basal refection corresponding to the
(001) plane at 2θ value of 6.2˚ is shifted to a 5.2˚ in LA-Er sample. This shift could be more
clearly seen in the magnified region (3-14 ˚) of the diffractograms shown as inset in Figure 2.
5 10 15 20 25 30 35 40 45 50
2degree
6.2
5.2
LA
LA-Er
4 6 8 10 12 14
Intensity(a.u)
2
6.2
5.2
Intensity(a.u)
The inter-planar distance
between the (001) plan
calculated using Braggs equation
comes out to be 14.4 Å and 17.4
Å for Lap, Lap@Eu. An increase
of 3.0 Å was observed for
Lap@Eu and Lap@[Eu(tta)n]
compared to the pristine
Laponite. This increase in inter-
planar distances (3 Å) confirms
intercalation of Er (between the
layers of LA.
XRD studies of Laponite (LA) Clay before and after
intercalation of Er+3 ions.
Reference: DOI: https://doi.org/10.1515/aot-2018-0030 Dr. Sajjad Ullah ICS-UOP

Characterization of Materials Structure Using X-Ray Diffraction

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