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XRD Line Broadening

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Title: 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
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