XRD Line Broadening

1
XRD Line Broadening
With effects on Selected Area Diffraction (SAD)
Patterns in a TEM
2
Ideal versus real diffraction patterns

• Under an ideal diffraction scenario, the
  diffraction pattern will consist of ?-peaks in a
  dark background.
• In a practical situation, instrumental and
  sample related issues lead to? the presence
  of intensity between the Bragg peaks ? Bragg
  peaks with a certain profile (i.e. broadened).
• Let us assume that the instrumental origins of
  the non-ideality have been accounted for. Then,
  information about the sample can be obtained from
  the diffuse intensity between the Bragg peaks and
  the profile of the peaks.
• The diffuse intensity typically arises from
  defects like atomic disorder (point defects) and
  thermal vibrations of atoms.
• The broadening of Bragg peak can arise from
  defects in the sample (e.g. dislocations
  stacking faults) and due to small
  crystallite/grain size.

• In general truncation in real space and
  concomitant broadening in reciprocal space can
  arise from three sources as below. (i)
  Truncation of the wave-front (i.e. the wave-front
  has a finite extent? like a beam with a
  finite diameter).(ii) Truncation of the crystal
  (say due to small grain size). (iii) Truncation
  of the sample (finite sample sizes? may in
  addition lead to crystal truncation).
  Finite crystals can be features like
  precipitates, twins, etc.

3
Crystallite size and Strain

• Braggs equation assumes? Crystal is perfect
  and infinite? Incident beam is perfectly
  parallel and monochromatic.
• Actual experimental conditions are different from
  these leading various kinds of deviations from
  Braggs condition? Peaks are not ? curves ?
  Peaks are broadened (in addition to other
  possible deviations).
• There are also deviations from the assumptions
  involved in the generating powder patterns?
  Crystals may not be randomly oriented (textured
  sample) ? Peak intensities are altered w.r.t. to
  that expected.

• In a powder sample if the crystallite size lt 0.5
  ?m? there are insufficient number of planes to
  build up a sharp diffraction pattern? peaks
  are broadened

Funda Check
What is meant by the terms (i) particle size,
(ii) crystallite size, (iii) grain size.

• If a particle is amorphous or consists of many
  crystallites, the particle size cannot be
  directly measured by XRD.
• Crystallite is a small sized crystal and many
  such crystallites (i.e. now each particle is a
  single crystal of small size) can be used in
  powder diffraction to obtain crystallite size.
• In a solid polycrystalline sample (like a piece
  of Cu or Alumina), the grain size and crystallite
  size refer to the same thing.

4
When considering constructive and destructive
interference we considered the following points

• In the example considered ? was far away (at a
  larger angular separation) from ? (?Bragg) and it
  was easy to see the destructive interference
• In other words for incidence angle of ? the
  phase difference of ? is accrued just by
  traversing one d.
• If the angle is just away from the Bragg angle
  (?Bragg), then one will have to go deep into the
  crystal (many d) to find a plane (belonging to
  the same parallel set) which will scatter out of
  phase with this ray (phase difference of ?) and
  hence cause destructive interference
• If such a plane which scatters out of phase with
  a off Bragg angle ray is absent (due to
  finiteness of the crystal) then the ray will not
  be cancelled and diffraction would be observed
  just off Bragg angles too ? line
  broadening!(i.e. the diffraction peak is not
  sharp like a ??-peak in the intensity versus
  angle plot)
• This is one source of line broadening of line
  broadening. Other sources include residual
  strain, instrumental effects, stacking faults
  etc. (next slide).

5
Defects in crystals and their effect on the XRD
pattern

• In the context of the effect on the XRD pattern,
  defects have been traditionally classified as
  type-I and type-II defects. Recently concentrated
  disordered solid solutions have been categorized
  as a separate type of defect. A summary is as in
  the table below.

6

• In XRD (focussing on powder XRD for now), line
  broadening can come from many sources. They are
  as listed below. Instrumental broadening has to
  be subtracted to obtain broadening from other
  sources. This is done by using a standard
  sample with large grain size and low strain,
  wherein there is no crystallite size or strain
  broadening (sample is chosen such that the
  density of other defects is small).
• Macrostrain (e.g. arising from pulling a
  specimen) will result in peak shift, while
  microstrain will result in peak broadening.

7
XRD Line Broadening

• Unresolved ?1 , ?2 peaks. ? Non-monochromaticity
  of the source (finite width of ? peak).
• Imperfect focusing, etc.

Instrumental
Bi
Crystallite size
BC

• In the vicinity of ?B the -ve of Braggs
  equation not being satisfied

Strain

• Residual Strain arising from dislocations,
  coherent precipitates etc. leading to broadening

BS
Stacking fault
BSF

• In principle every extended defect contributes to
  some broadening.
• Localized defects (e.g. isolated point defects)
  cause diffuse scattering.

Other defects
It has been recently demonstrated that
concentrated solid solutions lead to Bragg peak
broadening.1
The net broadening is the sum of all sources of
broadening
We will see soon as how we add or subtract
broadening from various sources (this depends of
the peak profile used).
8

• Full Width at Half-Maximum (FWHM) is typically
  used as a measure of the broadening of the peak.
  Other measures have also been used.

The diffraction peak we see is a result of
various broadening mechanisms at work
Full Width at Half-Maximum (FWHM) is typically
used as a measure of the peak width
9
Fitting of Peak profiles

• An important point related to peak broadening is
  the fitting of a profile to the peak. Standard
  curves used are Gaussian (G), Lorentzian (L), a
  combination of Lorentzian and Gaussian (called
  pseudo-Voigt (PS)), Pearson-VII (Lorentzian
  function to power m). The most popular
  currently are the pseudo-Voigt function (wherein
  the mix of G L can be varied).

For a peak with a Lorentzian profile

• Bi ? Instrumental broadening
• Bc ? Crystallite size broadening
• Bs ? Strain broadening

Longer tail
For a peak with a Gaussian profile
A geometric mean can also used
This formula is used when pseudo-Voigt function
is used for the peak profile fitting.
10
Subtracting Instrumental Broadening

• Instrumental broadening has to be subtracted to
  get the broadening effects due to the sample.
  This is typically done using the steps below.

A) (1) Mix specimen with known coarse-grained (
10?m), well annealed (strain free) ? does not
give any broadening due to strain or crystallite
size (the broadening is due to instrument only
(Instrumental Broadening)). A brittle material
which can be ground into powder form without
leading to much stored strain is good for this
purpose. (2) If the pattern of the test sample
(standard) is recorded separately then the
experimental conditions should be identical (it
is preferable that one or more peaks of the
standard lies close to the specimens peaks).
B) Use the same material as the standard as the
specimen to be X-rayed but with large grain size
and well annealed
11
Scherrers formula crystallite size broadening

• The Scherrers formula is used for the
  determination of grain size from broadened peaks.
• This works best for Gaussian line profiles and
  cubic crystals.
• The formula is not expected to be valid for very
  small grain sizes (lt10 nm). At very large grain
  sizes also the accuracy of the method suffers (as
  the broadening is small).
• Instrumental broadening has to be subtracted
  first. This formula can be used only if strain
  and other sources of broadening are small. If
  considerable strain broadening is expected then
  the Williamson Hall method can be used
  (considered soon).
• The accuracy of the method is of the order of
  only 10.

• ? ? Wavelength
• L ? Average crystallite size (? to
  surface of specimen)
• k ? 0.94 k ? (0.89, 1.39) 1 (the accuracy
  of the method is only 10?)

This formula can perhaps take the credit of
being the least carefully used formula in
research in materials science!!!
12
Strain broadening

• The micro-strain (?) in the material (due to
  dislocations and other strain fields) can lead to
  peak broadening. This broadening (Bs) is a
  function of the Bragg angle of the peak? varies
  as Tan(?B) (Fig.1).
• If we plot the FWM arising from these two sources
  (Fig.2) Bc BS we see that Bc is dominant at
  low angles and can be used to separate
  crystallite and strain broadening. This can be
  done using the Williamson Hall plot as
  considered next.

Smaller angle peaksshould be used to separate
Bs and Bc

• ? ? Strain in the material

Fig.1
Fig.2
13
Separating crystallite size broadening and strain
broadening
Williamson-Hall method G.K.Williamson W.H. Hall

• The total broadening due to strain and
  crystallite size can be added to get Br.
• As in the equations below we plot BrCos? as a
  function of Sin?. The slope will be ? (strain)
  and from the intercept (k?/L) we can compute the
  crystallite size (L). Overall this method gives
  an estimate of strain and crystallite size, but
  is not very accurate.
• An example of this plot is considered next.

Crystallite size broadening
Strain broadening
Plot of Br Cos? vs Sin?
14
Example of the use of Williamson-Hall (W-H) method

• To compute strain broadening we take a reference
  sample (Annealed Al sample with low dislocation
  density) and a cold worked sample (high
  micro-strain) and obtain powder patterns.
• For the three peaks in the plot (111, 200, 220)
  we generate the W-H plot.
• From the slope and the intercept we determine the
  strain and crystallite size.

Sample Annealed AlRadiation Cu k? (? 1.54 Å)
Sample Cold-worked AlRadiation Cu k? (? 1.54
Å)
15
Annealed Al
Cold-worked Al
Peak 2? (?) hkl Bi FWHM (?) Bi FWHM (rad)
1 38.52 111 0.103 1.8 ? 10-3
2 44.76 200 0.066 1.2 ? 10-3
3 65.13 220 0.089 1.6 ? 10-3
2? (?) Sin(?) hkl B (?) B (rad) Br Cos? (rad)
38.51 0.3298 111 0.187 3.3 ? 10-3 2.8 ? 10-3 2.6 ? 10-3
44.77 0.3808 200 0.206 3.6 ? 10-3 3.4 ? 10-3 3.1 ? 10-3
65.15 0.5384 220 0.271 4.7 ? 10-3 4.4 ? 10-3 3.7 ? 10-3
16
Spot/ring Broadening in SAD patterns in the TEM

• In a TEM Selected Area Diffraction (SAD) pattern,
  with decreasing crystallite size the effects as
  listed below are observed on the pattern
  obtained.
• SAD patterns from single crystalline regions give
  rise to spots, which are approximately a section
  of the reciprocal crystal. (Diagrams on next
  page).

• Size gt 10 ?m ? Spotty ring (no. of grains
  in the irradiated portion insufficient to produce
  a ring).
• Size ? (10, 0.5)? ? Smooth continuous ring
  pattern.
• Size ? (0.5, 0.1)? ? Rings are broadened.
• Size lt 0.1 ? ? No ring pattern. (irradiated
  volume too small to produce a diffraction ring
  pattern diffraction occurs only at low angles).

Increasing Size
Spotty ring
Rings
0.5 ?
10 ?m
Tending to single crystal/grain chosen by SAD
aperture
0.1 ?
0.5 ?
Zoom in of small sizes
Broadened Rings
Diffuse
17
Effect of crystallite size on SAD patterns

• If the grain size is large then the SAD aperture
  can chose a single grain to give rise to a
  single crystal pattern (Fig.1). Fig.2 shows the
  path of rotation of spots along one axis.
• Fig.3 4 show increasing number of
  crystallites/grains being chosen by the SAD
  aperture, giving rise to a spotty pattern.

Rotation has been shown only along one axis for
easy visualization ? Rotation in along all axes
should be considered to simulate random
orientation
Fig.2
Fig.1
Fig.3
Fig.4
Schematics
Single crystal
Few crystals in the selected region
Spotty pattern
Spotty rings from Pd nanocrystals
18

• If a huge number of crystallites are chosen the
  pattern becomes a ring pattern (Fig.5).
• If the crystallite size is further reduced, then
  the rings get broadened due to relaxation in
  Braggs condition (crystallite size broadening,
  Bc) (Fig.6).
• In amorphous materials a broad halo is obtained
  (Fig.7).

Fig.5
Fig.6
Ring pattern
Broadened Rings
Diffuse halo from glass