X-Ray Diffraction
Lecture note-5
Materi Perkuliahan
Pertemuan Topik Subtopik Referensi
1 Scanning Electron Microscopy (SEM) PPT: NTU lecture & MIT lecture
Handbook: lihat SAP
2 Transmission Electron Microscopy (TEM) Brent Fultz · James Howe, Transmission
Electron Microscopy
and Diffractometry of Materials, Springer
2008
3 X-Ray Diffraction (XRD)-1 Sinar-X: sumber dan sifat- Y. Waseda et. al. X-Ray Diffraction
sifatnya Crystallography
4 X-Ray Diffraction (XRD)-2: Overview Struktur Sel satuan,Bravais lattice Y. Waseda et. al. X-Ray Diffraction
kristal Index Miller, d-spacing Crystallography
5 X-Ray Diffraction (XRD)-3: Interaksi Sinar-X dan Fenomena difraksi, Y. Waseda et. al. X-Ray Diffraction
Struktur kristal Hukum Bragg Crystallography
Fenomena scattering yang
berpengaruh thd intesitas
6 X-Ray Diffraction (XRD)-4: Aplikasi difraksi Identifikasi senyawa dgn Y. Waseda et. al. X-Ray Diffraction
Hanawalt index. Crystallography
Membuat diagram fasa
7 X-Ray Diffraction (XRD)-5: Aplikasi Difraksi Pengenalan texture, Y. Waseda et. al. X-Ray Diffraction
pengukuran tegangan sisa Crystallography
Bragg’s law is a simplistic model to understand what
conditions are required for diffraction.
l = 2d hkl sin q q q
dhkl dhkl
• For parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs
when Bragg’s law is satisfied.
• In our diffractometers, the X-ray wavelength l is fixed.
• Consequently, a family of planes produces a diffraction peak only at a specific angle q.
• Additionally, the plane normal must be parallel to the diffraction vector
• Plane normal: the direction perpendicular to a plane of atoms
• Diffraction vector: the vector that bisects the angle between the incident and diffracted beam
• The space between diffracting planes of atoms determines peak positions.
• The peak intensity is determined by what atoms are in the diffracting plane.
Jarak antar bidang pada sistem kristal
Penentuan Struktur:
Sistem Kubik
• Analysis of diffraction data is to say “indexing pattern analysis.”
• By combining the Bragg condition with the plane spacing for a cubic system, the diffraction peaks
with the sin2q values satisfy the following equation:
the sum of the square of plane indices, corresponding to the measured diffraction peaks is always an integer
and λ2/4a2 is found a constant for any X-ray diffraction pattern.
Another method using the following equation:
Penentuan Struktur: Sistem Kubik
DIFRACTOMETER SINAR X
KOMPONEN UTAMA MESIN XRD
1. sumber penghasil sinar X
2. optik primer ; mengatur sinar X yg akan menuju sample.
3. sample holder & stage ; tempat sample di simpan, holder bisa utk
serbuk, bulk, film tipis.
4. optik sekunder; menerima sinar X yg di difraksikan sample.
5. detektor; mendeteksi sinar X yg didifraksikan sampel
The X-ray beam produced by the X-ray tube is divergent. Incident-beam optics are used
to limit this divergence
l = 2d hkl sin q
• X Rays from an X-ray tube are:
• divergent
• contain multiple characteristic wavelengths as well as Bremmsstrahlung radiation
• neither of these conditions suit our ability to use X rays for analysis
• the divergence means that instead of a single incident angle q, the sample is actually illuminated
by photons with a range of incident angles.
• the spectral contamination means that the sample does not diffract a single wavelength of
radiation, but rather several wavelengths of radiation.
• Consequently, a single set of crystallographic planes will produce several diffraction peaks instead of one
diffraction peak.
• Optics are used to:
• limit divergence of the X-ray beam
• refocus X rays into parallel paths
• remove unwanted wavelengths
Divergence slits are used to limit the
divergence of the incident X-ray beam.
• The slits block X-rays that have too great a
divergence.
• The size of the divergence slit influences
peak intensity and peak shapes.
• Narrow divergence slits:
• reduce the intensity of the X-ray beam
• reduce the length of the X-ray beam hitting
the sample
• produce sharper peaks
• the instrumental resolution is improved so
that closely spaced peaks can be resolved.
Other optics:
• limit divergence of the X-ray beam
• Divergence limiting slits Parallel Plate Collimator & Soller
• Parallel plate collimators Slits block divergent X-rays, but
• Soller slits do not restrict beam size like a
• refocus X rays into parallel paths divergent slit
• “parallel-beam optics”
• parabolic mirrors and capillary lenses
• focusing mirrors and lenses
• remove unwanted wavelengths
• monochromators
• Kb filters
Göbel Mirrors and capillary lenses
collect a large portion of the divergent
beam and refocus it into a nearly
Monochromators remove unwanted wavelengths of radiation from the incident or
diffracted X-ray beam.
• Diffraction from a crystal monochromator can be used to
select one wavelength of radiation and provide energy
discrimination.
• An incident-beam monochromator might be used to
select only Ka1 radiation for the tube source.
• A diffracted-beam monochromator, such as on the Rigaku
RU300, may be used to remove fluoresced photons, Kb, or
W-contimination photons from reaching the detector.
• Without the RSM slit, the monochromator removes ~75% of
unwanted wavelengths of radiation.
• When the RSM slit is used, over 99% of the unwanted
wavelengths of radiation can be removed from the beam.
Beam Divergence
Beam Divergence
Detectors
• point detectors
• observe one point of space at a time
• slow, but compatible with most/all optics
• scintillation and gas proportional detectors count all photons, within an energy window, that hit them
• Si(Li) detectors can electronically analyze or filter wavelengths
• position sensitive detectors
• linear PSDs observe all photons scattered along a line from 2 to 10° long
• 2D area detectors observe all photons scattered along a conic section
• gas proportional (gas on wire; microgap anodes)
• limited resolution, issues with deadtime and saturation
• CCD
• limited in size, expensive
• solid state real-time multiple semiconductor strips
• high speed with high resolution, robust
• selain tampilan kurva intensitas vs 2q (point detector)
, pola difraksi sinar x juga bisa ditampilkan dalam
bentuk titik (line detector).
PARAMETER PADA MESIN XRD
• scan range; sudut scan awal dan akhir (mis; dari 10 ke 80o)
• scan speed; kecepatan scan, semakin lambat semakin baik pola
difraksi yg dihasilkan.
• power, terdiri dari tegangan dan arus, biasanya 40 kv, 30 ma.
• step width
• slit width, mengatur intensitas dan resolusi sinar x
The diffraction pattern consists of a record of photon intensity
versus detector angle 2q.
• The position, intensity, width, and shape of the observed diffraction peaks tells us about the
crystal structure and, in some cases, microstructure of the sample.
A single crystal specimen in a Bragg-Brentano diffractometer would produce only
one family of peaks in the diffraction pattern.
2q
At 20.6 °2q, Bragg’s law The (110) planes would diffract at 29.3 The (200) planes are parallel to the (100) planes.
fulfilled for the (100) planes, °2q; however, they are not properly Therefore, they also diffract for this crystal. Since
producing a diffraction peak. aligned to produce a diffraction peak (the d200 is ½ d100, they appear at 42 °2q.
perpendicular to those planes does not
bisect the incident and diffracted beams).
Only background is observed.
A polycrystalline sample should contain thousands of crystallites.
All diffraction peaks should be observed.
2q 2q 2q
• For every set of planes, there will be a small percentage of crystallites that
are properly oriented to diffract (the plane perpendicular bisects the incident
and diffracted beams).
INTERPRETASI POLA DIFRAKSI
• umumnya pola difraksi ditampilkan dalam grafik intensitas vs 2q.
• untuk analisis, pola difraksi lebih baik di laporkan dalam grafik/tabel
intensitas relatif vs dhkl.
• dhkl bisa di hitung dari data 2q dengan menggunakan hukum bragg.
• intensitas relatif di hitung dgn cara membagi intensitas absolut oleh
intensitas maksimum.
• data dhkl dan intensitas relatif yg dihitung, kemudian diurutkan
berdasarkan nilai intensitas yg tertinggi.
Raw Data Reduced dI list
Position Intensity hkl dhkl (Å) Relative
[°2q] [cts] Intensity
25.2000 372.0000 (%)
25.2400 460.0000
25.2800 576.0000
{012} 3.4935 49.8
25.3200 752.0000 {104} 2.5583 85.8
25.3600 1088.0000
25.4000 1488.0000
{110} 2.3852 36.1
25.4400 1892.0000
{006} 2.1701 1.9
25.4800 2104.0000
25.5200 1720.0000 {113} 2.0903 100.0
25.5600 1216.0000
{202} 1.9680 1.4
25.6000 732.0000
25.6400 456.0000
25.6800 380.0000
25.7200 328.0000
What can we do with XRD?
• Identify phase composition
• Measure unit cell lattice parameters
• Estimate crystallite size, microstrain, and defect concentration
• Measure residual stress
• Measure texture and/or epitaxy
• Evaluate thin film quality
• Measure multilayer thin film thickness, roughness, and density
• Determine orientation of single crystals
• Solve or refine crystal structures
• Analyze ordered meso- and nanostructures
Mengapa menggunakan XRD?
• Powder XRD is a rapid method that can be used to routinely assess the nature of a
sample.
• Can be used for qualitative phase ID – What is in the sample?
And quantitative phase analysis. How much is in the sample?
• Each crystalline phase has a unique powder diffraction pattern
• You can distinguish between mixtures and compounds as the diffraction method is
sensitive to structure not just composition
• The powder pattern for the spinel MgAl2O4 looks different from the powder pattern of a
MgO, Al2O3 mixture
• You can distinguish between different polymorphic forms of the same compound
The diffraction pattern of every phase is a unique ‘fingerprint’
Cristobalite, SiO2
Intensity (a.u.)
Beta-Quartz, SiO2
Alpha-Quartz, SiO2
Amorphous
SiO2 Glass
10 20 30 40 50
2q (deg.)
Phase ID is often used for Quality Control and Analyses of Processes
• what did I make?
• did I make what I was trying to make?
• are my raw materials pure?
In the example below, the nanocrystalline Y2O3 did not match the reference pattern for yttria
00-041-1105> Y2O3 - Yttrium Oxide
Intensity(Counts)
15 20 25 30 35
2q (deg.)
Not only can the PDF provide phase matching, but it can also be a useful
data-mining source
• the sample contained a large fraction of monoclinic Y2O3
• the PDF card pointed to a reference describing this as a metastable phase
previously observed only in thin films of Y2O3
00-041-1105> cubic Y2O3
00-044-0399> monoclinic Y2O3
Intensity(Counts)
15 20 25 30 35
2q (deg.) B. Chivas and T.F. Morse, Boston University
We can use these phase analysis techniques to study reactions
ex situ • we revisit the sample of nanocrystalline Y O , now comparing the results of
2 3
annealing the original sample
• we see that annealing converted all of the monoclinic Y2O3 into cubic Y2O3
original, as-made
after annealing
Intensity(Counts)
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
2q (deg.)
XRD can discern between isostructural compounds as
well as between polymorphs
• we can map differences in electron density because X-rays scatter proportionally to Z2
• the cubic phases of CaTiO3 and SrTiO3 have identical crystal structures, with the A
cation replaced by Ca or Sr respectively
CaTiO3 cubic perovskite
Intensity(Counts) SrTiO3 cubic perovskite
30 35 40 45 50
2q (deg.)
–*the cubic phase of CaTiO3 is not normally stable at RT– we stabilized this phase by quenching the sample
Identifikasi Fasa
• Using powder X-ray diffraction for phase identification
• – this can be done by calculating the unit cell and then search the
NIST crystal data database for known compounds with the same or
similar unit cells
• Not very easy to do, but does not require that compound is in the ICDD powder
diffraction file
• – usually done by comparing the measured pattern against the
ICDD/JCPDS powder diffraction file data base.
• This contains powder patterns for a very large number of compounds.
Langkah2 Identifikasi Fasa
• Collect the XRD pattern over a range that is suitable for the material we are studying
• typically 20 to 70 °2θ for inorganic specimens
• typically 5 to 40 °2θ for organic specimens
• data collection time ranges from 5 min to 1 hour for our instruments
• in the “real world”, these times are typically 30 min to 2 hours
• Compare the experimental data to a reference database of powder diffraction patterns
• Subscribe to the Powder Diffraction File
• 172,360 inorganic diffraction patterns
• 107,507 inorganic crystal structures
• 30,728 organic diffraction patterns
• We also have the Cambridge Structural Database
• ~400,000 organic crystal structures
Index untuk PDF
• There are several different types of indices available for the powder
diffraction file in addition to software for doing computer searches
• – Alphabetical index gives card number of a given composition, or
mineral name
• – Hanawalt and other search indices can be used to find a card that
matches a diffraction pattern that you have measured
Indeks Hanawalt
• The Hanawalt index can be used to identify components in your sample based on the d-spacings of
the three most intense lines arising from a phase.
• Index is organized by d-spacing of the strongest lines
» Index contains 8 strongest lines for each phase, but search itself is initially based on only the
three strongest
» As this index makes use of intensity information, the search procedure can be impaired if the
intensities that you measure are different from those in the PDF due to absorption, changes in
wavelength, preferred orientation etc.
Prosedur menentukan fasa
• Locate d-spacing group for most intense line
• In second column, find best match to the d-spacing of the second
strongest line (agreement should be ±0.01 Å
• Look in column three to find match for third line
• Compare relative intensities of the strongest lines for you best match and
for you data
• If things look good, compare all lines on PDF card with what you observed.
All lines that are on the PDF card should appear in you pattern and the
relative intensities should be similar
• – Note agreement of d-spacings for high angle lines should be better than
those at low angle as they are in general measured more accurately
When is a match a match?
• If you sample is a pure phase, every line that you recorded should
match well with one on the PDF card
– However, you may have a second phase in your sample
– You may have some kb lines in you pattern
– The PDF card may not have any high angle lines on it
– A line may be very weak
– Intensity differences may arise due to absorption, wavelength
differences, preferred orientation etc.
Identification of components in a mix
• Identification of components in a mixture can be difficult as the three
strongest lines in your powder pattern may not all come from the
same phase.
• – Need to examine different combinations of strong lines when using
the Hanawalt index until a phase is tentatively identified. One one
component is identified, ignore the lines from that phase a look for a
match to the remaining lines
Andi menemukan sebuah botol berlabel “standar
metal” yang mengandung serbuk logam. Kemudian
andi melakukan analisa dengan difraksi sinar X,
sampai dengan sudut (2θ) 97° menggunakan sinar x
dengan panjang gelombang 1.542Å. Hasil dari analisa
XRD yang dilakukan andi adalah sbb: 2θ (derajat) dan
Intensitas (intensitas dalam count/second)
Pertanyaan :
Identifikasi sampel yang dimaksud dan berikan
komentar atas hasil yang diperoleh
Computerized search match
• Identification of components in a mix is often done using
computerized matching algorithms. They are not fool proof. The
results depend on the data quality, the data base quality and the
criteria used in the search
• – Matches are usually ranked using a figure of merit
• – Large figure of merit would indicate a good match using the
definition below
Problems
• Phase in your sample may not be in PDF
• – Not common, but you have a big problem!
• Intensities in your pattern may be poor due to grainy sample, preferred
orientation etc
• – Try multiple packings of sample. If you see intensity variations consider
using a different sample preparation method or grinding the sample.
• Positions in you pattern may be wrong
• – Collect your data with an internal standard present so that you can
correct any angular errors
• Your sample is a solid solution with a composition that does not
correspond to a pattern in the database. Some software will contract or
expand a lattice during the search match procedure to try and find matches
in cases like this
• Data base pattern may be of poor quality
• untuk sistem dengan banyak fasa / multi senyawa,
maka cara manual akan sulit dilakukan.
• biasanya analisa dibantu dengan software ; misal
jade, topaz, xpowder, match, pcpdfwin, dst..
• software tsb harus dilengkapi dengan database xrd yg
lengkap, misal pdf2 atau pdf4. harga database tsb
sangat mahal, dan tiap tahun biasanya diupdate.
APLIKASI XRD
The diffraction pattern of every phase is a unique ‘fingerprint’
Cristobalite, SiO2
Intensity (a.u.)
Beta-Quartz, SiO2
Alpha-Quartz, SiO2
Amorphous
SiO2 Glass
10 20 30 40 50
2q (deg.)
SELAIN IDENTIFIKASI FASA, BISA JUGA MENENTUKAN BERAPA BANYAK FASA YG ADA
(KURANG AKURAT)
Red Paint Pigment Mixture
28 wt% Hematite, 21 wt%
Fe2O3 Anatase, TiO2
Intensity(Counts)
51 wt% Rutile, TiO2
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
2q (deg.)
The Scherrer Equation was published in 1918
Kl
B(2q ) =
L cosq
• Peak width (B) is inversely proportional to crystallite size (L)
• 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.
Many factors may contribute to the observed peak
profile
• Instrumental Peak Profile
• Crystallite Size
• Microstrain
• Non-uniform Lattice Distortions
• Faulting
• Dislocations
• Antiphase Domain Boundaries
• Grain Surface Relaxation
• Solid Solution Inhomogeneity
• Temperature Factors
• The peak profile is a convolution of the profiles from all of these contributions
Crystallite Size Broadening
0.94l
B(2q ) =
L cosq
• Peak Width B(2q) varies inversely with crystallite size
• The constant of proportionality, K (the Scherrer constant) depends
on the how the width is determined, the shape of the crystal, and
the size distribution
• the most common values for K are 0.94 (for FWHM of spherical crystals with
cubic symmetry), 0.89 (for integral breadth of spherical crystals with cubic
symmetry, and 1 (because 0.94 and 0.89 both round up to 1).
• K actually varies from 0.62 to 2.08
• For an excellent discussion of K, refer to JI Langford and AJC Wilson, “Scherrer
after sixty years: A survey and some new results in the determination of
crystallite size,” J. Appl. Cryst. 11 (1978) p102-113.
The Scherrer Constant, K
Kl 0.94l
B(2q ) = B(2q ) =
L cosq L cosq
• The constant of proportionality, K (the Scherrer constant) depends on the how the width is determined,
the shape of the crystal, and the size distribution
• the most common values for K are:
• 0.94 for FWHM of spherical crystals with cubic symmetry
• 0.89 for integral breadth of spherical crystals w/ cubic symmetry
• 1, because 0.94 and 0.89 both round up to 1
• K actually varies from 0.62 to 2.08
• For an excellent discussion of K, refer to JI Langford and AJC Wilson, “Scherrer after sixty years: A survey
and some new results in the determination of crystallite size,” J. Appl. Cryst. 11 (1978) p102-113.
Methods used to Define Peak Width
• Full Width at Half Maximum (FWHM)
• the width of the diffraction peak, in
radians, at a height half-way FWHM
between background and the peak
Intensity (a.u.)
maximum
• Integral Breadth
• the total area under the peak divided
by the peak height 46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9
2q (deg.)
• the width of a rectangle having the
same area and the same height as
the peak
• requires very careful evaluation of
Intensity (a.u.)
the tails of the peak and the
background
46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9
2q (deg.)
Remember, Crystallite Size is Different than
Particle Size
• A particle may be made up of several different crystallites
• Crystallite size often matches grain size, but there are exceptions
Examples: Particle Size Calculation
Estimation of Residual Stress
• Definition: Stresses that remain in material or body without
application of an external load (applied force, displacement of
thermal gradient).
• Origin: Usually originates during manufacturing and processing of
materials due to heterogeneous plastic deformations, thermal
contractions and phase transformations.
Origin of Residual Stress
• Mechanical
• Thermal
• Chemical
Mechanically Generated
• Occurs due to manufacturing process that produce non-uniform
plastic deformation.
• May develop naturally during processing or treatment or may be
introduced deliberately to develop a particular stress profile in a
component.
• Operations that produce “undesirable” surface tensile stresses or
residual stress gradients are rod or wire drawing, welding, machining
(turning, milling) and grinding.
• Compressive residual stresses can be introduced by shot peening,
autofrettage of pressure vessels, toughening of glass or cold
expansion of holes
Thermally Generated
• Macroscopically: Occurs as a result of non-uniform heating or cooling
operations
• Microscopic level: Can also develop in a material during manufacture
and processing as a consequence of Coefficient of thermal expansion
mismatch between different phases or constituents.
Chemically Generated
• Develops due to volume changes associated with chemical reactions,
precipitation or phase transformations.
• Chemical surface treatments and coatings can lead to the generation
of substantial residual stress gradients in the surface layers of
components.
• For example, nitriding produces compressive stress in the diffusion
region due to expansion of the lattice and precipitation of nitrides.
Residual stress
• Residual stress causes small changes in d and shifts the diffraction
angle.
• Residual stresses are determined from the diffraction data by
calculating the residual strain from the diffraction peak positions.
Measuring peak positions at different tilts allows us to
quantify residual strains
• Measuring the same diffraction
peak at different tilts allows us to
Intensity(Counts)
measure the d-spacing between
planes (hkl) at different
orientations within the physical
sample 103.0 103.5 104.0 104.5 105.0 105.5
2q (deg.)
106.0 106.5 107.0
• With no strain, (hkl) planes have
a d-spacing d0
Intensity(Counts)
• With a uniaxial in-plane stress,
planes (hkl) that are parallel to
the sample surface (tilt=0°) have 103.0 103.5 104.0 104.5 105.0 105.5
2q (deg.)
106.0 106.5 107.0
a dhkl distorted by Poisson’s ratio
• With the same uniaxial in-plane
stress, planes (hkl) that are tilted
Intensity(Counts)
with respect to the sample
surface are strained by a greater
amount determined by Young’s
modulus and Poisson’s ratio 103.0 103.5 104.0 104.5 105.0 105.5
2q (deg.)
106.0 106.5 107.0
Estimation by XRD
• One of the most widely used non-destructive techniques for residual
stress measurement.
• Residual stress in the material causes the interplanar spacing of the
material to change.
• Changes in the interplanar spacing “d” can be used with the Bragg’s
equation to detect elastic strain “ε” through a change in the Bragg
scattering angle Δθ
Accurate determination
of stress free spacing “d0”
is required
• Stress is evaluated from strain values using Young’s modulus, Poisson
ratio and taking into consideration Elastic Anisotropy of the material.
• Typically single peak, available at highest value of 2θ, is used for
analysis.
• Diffraction is selective and hence biased towards a particular sets of
grains.
• The peak shift sample both Type I and average Type II stresses, while
Type III stresses give peak broadening.
Instrumentation
• Parallel X-ray Beam
• Goniometer capable of rotation and tilt
• ψ- tilting is performed either in iso-inclination mode or sideinclination
mode
• Side-inclination is preferred because :
• The effect of misalignments of the sample height on the stress result is less
pronounced
• The range of tilt angles is not restricted by small Bragg Angles
• The measurements on samples with concave surface are less hindered by shading
effects
• Stress free sample of the material under investigation is needed
X=tilt angles
Resolving Multi-axial Stresses
When the sample is level, only one set of planes diffract. But when
the sample is tilted, a different sets of crystallites diffract. Thus each
set of crystallites resolve a different directional component of the
stress to which the sample is subjected. If the sample is compressed
from the sides, the lattice spacing of crystallites oriented parallel to
the surface will increase more than those oriented at an angle to the
surface. The peak position will shift during tilting, since crystallites
are subjected to different magnitude of stress
Residual Stress by X-ray Diffraction
Techniques
Unstressed
The compressive stress spreads
apart the planes of atoms parallel
to the surface (and perpendicular
to the stress), which results in
diffraction peaks at lower angles of
incidence
Strain-Sin2 ψ Curves
