Part 14' Diffraction Lineshapes

1
Part 14. Diffraction Lineshapes (From
Transmission Electron Microscopy and
Diffractometry of Materials, B. Fultz and J.
Howe, Springer-Verlag Berlin 2002. Chapter 8)
? Diffraction Line Broadening and Convolution
? Three sources of broadening of diffraction
peaks (1) small sizes of crystalline
(2) distributions of strains within individual
crystallites, or difference in
strains between crystallites (3) The
diffractometer (instrumental broadening)
? Crystallite Size Broadening ? Recall the
result of kinematical theory for the diffraction
lineshape of a small crystal. See part 9
Define deviation vector
2
Ignore the form factor

I
Half width half maximum (HWHM)
particular
usually small ?
Solve graphically
3
Define
The solution is x 1.392
1.392
L is the size of crystallite more accurately
the length of the column of N1
coherently-diffracting planes in the crystallite.
Convert the equation into a form more
appropriate for analysisof experimental data
acquired as a function of 2? angle.
Define
4
Use the FWHM
In X-ray, 2? is usually used, define
B in radians
Scherrer equation, K is Scherrer constant
If the ? is used instead of 2?, K should be
divided by 2.
5
? Strain broadening ? Uniform strain ?
lattice constant change ? Bragg peaks
shift. Assume the strain is ?, the interplanar
spacing is changed from d0 to d0(1 ?).
Laue condition
Along the direction normal to the
diffracting planes
Peak shift
Again, in terms of ?
Larger shift for the diffraction peaks of higher
order
6
In general, there is a distribution of strains
in a specimen. ? broadening of the diffraction
peaks, larger for higher order diffraction
peaks. In reality, the strain distribution is
continuous ? relate the mean square strain
to the width of the diffraction peak
?k
is the HWHM of the diffraction G along
? Measurements of Internal Stress From
elasticity, for an isotropic elastic solid, the
elastic constant E and v relate the stress
and strain tensors through
Kroeneckers delta
7
Written explicitly
Shear modulus
Stress normal to a free surface defined by
vector must be zero at the surface, i.e.,
The equation of equilibrium must be satisfied at
each point of the material
8
Transformation of the strain tensor from one
coordination system to another
where defines the cosine of the angle
between in the old coordinate system and
in the new coordinate system.
Consider the transformation of the sample
coordinate system to the laboratory
coordinate system .
Lets find out the transformation matrix for
the above configuration the transformation can
be achieved by first rotate along the axis by
an angle , followed by rotating an angle
along the (? means new axis after
first rotation)
9
The above two rotation matrix are
and
? transformation matrix for the coordinate system
10
Use the prime to denotes quantities defined in
the sample coordinate system and unprimed
quantities refer to the laboratory system
One can get
13 and 31
11
Look at the ?11 term, there are
Add and subtract one term
We get
Similar for ?22 term ?
For ?33 term ?
12
Lets group the sin2? into one term, and the
rest
? sign is used in Eq. 15-14 in textbook 2,
correct it
The quantity measured at angles ? and ?. ?
d-spacing in the stresses sample and
measured for the plane whose normal is
at angles ?, ? from the sample
coordinate system d-spacing for the
unstressed state is related
With these equations, one can start the stress
analysis via x-ray diffraction.
13
Three stress states of interests are uniaxial,
biaxial, and hydrostatic ststes. uniaxial
stress state
Eq.15-16 b, d in the textbook 2 are wrong
biaxial stress state
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Hydrostatic stress state
For hydrostatic stress state, a volumetric
strain is defined
The sum of the ?ii are invariant in any
coordinate system. The volumetric strain is
related to the hydrostatic stress, ?H, by
K bulk modulus
? Experimental observation of sin2?
dependence From the above discussion one
could measure the ?? using d- sin2? plot.
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Linear relation when the sample is in the biaxial
stress state.
Slope
When the sample in in the triaxial state ?
?-splitting
asymmetric
The shear stress can lead to compression of some
plane spacing and expansion of others
Presence of stress gradient, texture and/or
elastic and plastic anisotropic
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? Instrument broadening ? Main Sources
finite slit widths and variation in position
of the diffracting planes (misplacement of the
sample position on the goniometer, surface
roughness, partial transparency of the
specimen)
Combining all these broadening by the
convolution procedure ? asymmetric instrument
function
convolution
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? The Convolution Procedure Consider an
instrument function f(x) and the specimen
function g(x), the observed diffraction profile,
h(?). The convolution steps are Flip
f(x)? f(-x) Shift f(-x) with respect to
g(x) by ? f(-x) ? f(?-x) Multiply f
and g f(?-x)g(x) Integrate over x
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f(x)
3
2
1
0
1
2
-1
-2
0
4
g(x)
3
2
1
0
1
2
-1
-2
Assume f and g are the functions on the right,
the h(?) that we will get is
0
4
f(-x)
3
2
1
0
1
2
-1
-2
0
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? -1
? -2
4
4
A 0
3
3
A 4/5
(-2/5 6/5)
2
2
1
1
0
0
2
-2
2
-2
0
0
? 1
? 0
4
4
3
3
(-1/5 8/5)
A 1.75
A 11/5
2
2
1
1
0
0
2
-2
2
-2
0
0
? 2
h(?)
4
4
3
3
A 0
2
2
1
1
0
0
?
2
-2
2
-2
0
0
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Convolution of Gaussians
A convenient measure of the breadth, B ( 1/a),
of a Gaussian function is its half width at the
height e-1 of its maximum. In typically defined
FWHM ? B. Assume two functions f(?) and g(?)
having breadths Bf and Bg respectively ? h(?)
f(?)g(?) having a breadths Bh . The relation is
Convolution of Lorentzuans
HWHM, B ( 1/a), Assume two Lorentzian functions
f(?) and g(?) having breadths Bf and Bg
respectively ? h(?) f(?)g(?) having a breadths
Bh . The relation is
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? 1/a 10 HWHM
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? Fourier Transform and Deconvolutions ?
Mathematical Features The following shows
how to remove the blurring, caused by the
instrument function, with the inverse
procedure of deconvolution (Stokes correction).
Instrument broadening function f(k) (k is
function of ?) True specimen diffraction
profile g(k) Measured by the
diffractometer h(K) Fourier transform the
above three functions
l 1/length, the range in k of the Fourier
series is the interval l/2 to l/2.
These Fourier transform repeat themselves with a
period of l. Confine to one period and Require f
and g vanish at its ends.
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The function f and g vanished outside of the k
range ? Integration from -? to ? is replaced by
l/2 to l/2
Orthogonality condition
vanishes by symmetry
24
Convolution in k-space is equivalent to a
multiplication in real space (with variable n/l).
The converse is also true. Important result of
the convolution theorem!
Deconvolution
When we have the full sets of Fourier
coefficients F(n) and H(n), we perform a
division in n-space for each Fourier coefficient.
F(n) is obtained from
Orthogonality relationship
25
H(n) is obtained similarly
Data from perfect specimen
Rachinger Correction (optional)
f(k)
Corrected data free of instrument broadening
Stokes Correction G(n) H(n)/F(n)
F.T.-1
F.T.
Data from actual specimen
Rachinger Correction (optional)
h(k)
g(k)
Perfect specimen chemical composition,
shape, density similar to the actual specimen (?
specimen roughness and transparency broadening
are similar) E.g. For polycrystalline alloy,
th specimen is usually obtained by annealing
26
Unless f(k), g(k), and h(k) are symmetric and
located at the center of the interval, their F.T.
are complex.
g(k) is real and can be reconstructed as
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real part
Effects of Noise on Deconvolution
Noise in data causes serious problems for
numerical deconvolution. Typically,
noise-to-signal ratio decrease as the
square root of the number of
counts. 100 counts noise band ?10
The random noise function r(k) is added to the
signal. Properties of r(k) in digital data 1.
r(k) ? r(mk0), 0 ? m ? N, k0channel interval 2.
average value of r(k) 0
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3. Statistical independence between r(mk0) and
r(m?k0)
The inverse Fourier transform of r(mk0) is the
Fourier transform of the noise R(n)
l interval length Nk0
The Noise Problem Return to stokes correction,
with noise
Fourier transform of a Gaussian function
29
Fourier transform of a Lorentzian function
For large n, G(n) swings randomly between large
positive and negative values ? disaster for
the Deconvolution process ? noise filter is
required!
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? Simultaneous Strain and Size Broadening ?
Usually, X-ray diffraction peaks are broadened by
strain distribution, small grain sizes, and
instrument effects. The instrument effects
can be corrected using the process
discussed in the previous section. In
principle, if one knows either the strain
distribution or the crystalline size
distribution (e.g. may be from TEM dark
field), deconvolute to get the other!
Usually, we dont know either one and is it
possible to get both of them?
Crystalline size broadening is independent of G.
Strain broadening depends linearly on G.
? an extrapolation procedure to obtain
the lineshape of a hypothetical
diffraction peak at G0 which would
be broadened only by size effects.
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? Extrapolations of Peak Width vs. G
(Williamson-Hall ) Easiest way! Requires an
assumption about the shape of the
diffraction peaks
Gaussian function characteristic of the strain
broadening
convolution
Kinematical crystal shape factor intensity
Hardly a rigorous choice, possible
corelation between size and strain are
ignored (most of real sample, larger size,
smaller strain) Assume a Gaussian strain
distribution (quick falloff for strain
larger than the yield strain) ?(?)
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No simple analytical form for
Approximate the size broadening part with a
Gaussian function having a characteristic width
The approximation is good only when strain
broadening is much larger than the size
broadening
The convolution of two Gaussian is another
Gaussian
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Plot ?k2 vs G2
Slope
?k2
G2
Alternatively, assume the size and strain
broadening are Lorentzian functions. The
convolution of two lorentzians ?
Plot ?k vs G
Slope
?k
G
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from P. Lamparter
Ball-milled Mo
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Nanocrystalline CeO2 Powder
from P. Lamparter
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Nb film, WH plot
from P. Lamparter
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from P. Lamparter
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Due to anisotropy of shape or elastic
constants, strains and sizes are not the same
in all crystallographic direction ? ?k2 vs G2
or ?k vs G seldom fall on a straight line.
Use a series of diffraction along the same
direction, e.g. (200), (400), (600 overlap
with 442, can not be used) ? provide a
characteristic size and characteristic
mean-square strain for each crystallographic
direction! Elastic constants in some materials
vary significantly with crystallographic
orientation ? different crystal planes produce
different strain ? a plot of E?k to see what
happen!
39
E?k fit better than ?k in this case ? elastic
anisotropic is the main reason for the
deviation of ?k to G.
Ball-milled bcc Fe-20Cu
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? Fourier Methods with Multiple Orders ?
Another much powerful and elegant method to exam
a series of diffraction peaks, such as
(002), (004), (006), or (111), (222),
(333), , is available. ? Method of Warren
and Averbach. The method illustrates the
fundamentals of diffraction phenomena.
Lets start with the diffracted wave, ?(?k)
Sum over all N unit cells, each with Structure
factor F, and located at Rm.
The intensity of the diffracted bean is
Lets work with orthorhombic, tetragonal, or
cubic lattices and use these reciprocal space
variables
reciprocal lattice vectors
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real lattice basis vectors m1,
m2 and m3 integer
displacement vector at site m from strain in the
material
Lets consider only an (0 0 l) diffraction and
its higher order diffractions (0 0 2l), (0 0
3l). ? h 0, k 0 ? ? along a3.
? Correction for Peak intensity in practice we
measure the diffraction intensity along
in terms of 2?.
42
Lorentz-polarization factor
Intensity factor due to absorption,
large absorption coefficients do not permit
deep X-ray penetration
Other factors
wavelength dependent orientation effect
(similar to tile sensitivity factor)
density factor
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In order that all the diffracted power belonging
to the reflection (0 0 l) be included, it is
necessary to integrate the above equation over
h1 and h2 from -1/2 to 1/2
if
1
1
0
0
if
Summation of m3 and m3? ? the contribution of
all pairs of cells in a given column m1m2.
Summation over m1 and m2 ? the contribution of
all columns
m3
na3
m1
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and
Let
Let Nn represent the number of cells in the
whole sample with an nth neighbor in the same
column.
an average over all pairs
of nth neighbors in the same column through out
the entire sample.
Na average number of cells per column, N
number of cells in the sample The fourfold
summation has been replaced by a single
summation over all values of n.
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The imaginary term will be cancelled out. For
n term For n term
The Real term will not vanish.
Define
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The above equation is like do the Fourier
transform of the diffracted intensity.
Fourier coefficient for the cosine term
Size factor
Strain factor
When lZn is small
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n 0
n 1
lnAn(l)
n 2
n 3
l2
1
4
9
Strain information is obtained from the slope of
the curves
? Column Lengths distribution of small
crystallite sizes in the material ?
Lorentzian peaks (a consequence of a
particular type of length distribution (an
exponential function). Assume a
random termination of column, the probability
of not terminating a column before distance n
is defined as P(n)
the probability that termination occurs in dn
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For random termination probability ?
The probability of finding a column with length
between n and n dn ?(n)dn equals to the
probability that the column terminates in this
interval)
Normalization condition require
? characteristic inverse number
Na
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n
Na
In term of length L nd
Diffraction plane spacing
Each cell in a column has itself as a 0th
neighbor
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Convert the sum into integral
Convert the sum into integral
0
1
With size broadening only, Zn 0
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Fourier cosine transform
Fourier cosine transform of exponential ?
Lorentian lineshape.
Nanostructured crystallites with heterogeneous
size distributions often have Lorentzian-like
diffraction peaks!
Fourier coefficient for the sine term sine term
is usually small, if sine term is not zero? the
diffraction peak is asymmetric. For example
strain distribution is non- uniform.
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from P. Lamparter
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from P. Lamparter
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from P. Lamparter
n
i
?(i)
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from P. Lamparter