Pecharsky V.K., Zavalij P.Y. Fundamentals of Powder Diffraction and Structural Characterization of Materials

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Preliminary data processing and phase analysis

The following functions are commonly used in background (bi)

interpolation,' where

varies from

and

is the total number of data

points measured in a whole powder diffraction pattern or in the region

included in the processing:

Polynomial function, which approximates the background as

Here and below, B,, are background parameters that can be refined and

is the order of the polynomial. The summation in

Eq.

4.1 is usually

carried out beginning from m

However, to account for the often

increased background at low angles, an additional (hyperbolic) factor

with m

-1 may also be included.

Chebyshev polynomial (type

or type 11, either shifted or not). It is

represented as a function of an argument xi, which is defined as

where

29n,i,

and

29,,,

are the minimum and maximum Bragg angles in

the powder diffraction pattern. The background is calculated as:

and the Chebyshev function, tm, is defined such that

where to

1 and

x. Values of the function

are calculated using

tabulated coefficients.

Fourier polynomial, in which the background is represented as the

following sum of cosines:

Many other functions can be used to approximate the background. All of them should be

continuous functions of Bragg angle in the processed range.

352

Chapter

Diffuse background function to account for a peculiar scattering from

amorphous phases (e.g. see

Figure

4.2,

specimen

or from a non-

crystalline sample holder:

where

=2ddi and the "d-spacing" is calculated for each point, 2Qi, of

the powder diffraction pattern.

Eqs.

4.1

to 4.6,

typically varies from 6 to 12 when the entire powder

diffraction pattern is of concern.

some instances, when profile fitting is

applied to short fragments of the powder diffraction pattern, the most

suitable background function is that given by Eq. 4.1 with

1 or 2, i.e. a

linear or parabolic background.

4.3.2

Smoothing

Smoothing is

numerical conditioning procedure employed to suppress

statistical noise, which is present in any powder diffraction pattern as a result

of random intensity measurement errors

(Eq.

3.8

in Chapter

3).

It improves

the appearance of the powder diffraction pattern. For example, smoothing

can make quickly collected data (say in a

min experiment) look similar to

a pattern collected in a longer (e.g. in an overnight) experiment, and may

help with certain automatic procedures, such as background subtraction,

Ka2

stripping and unbiased peak search.

Numerical conditioning, however, does not improve data quality.

Moreover, it causes broadening of Bragg peaks and loss of resolution

(e.g.

see

Figure

3.38

and

Figure

3.39

in Chapter

which illustrate broadening

caused by the varying receiving slit) and may result in the disappearance of

weak peaks when overdone. On the contrary, increasing experiment time

improves the pattern since it reduces statistical spreads.'

example of the

original and smoothed patterns is shown in

Figure

4.5.

The most typical smoothing approach is often called box car smoothing.

It involves averaging intensities of current and neighboring data points using

different weights. The weight is the largest for the point being smoothed and

it decreases rapidly for points located farther away. For example, when five

points are employed, the weights

(wi)

can be set at 1 for the point in the

middle (Yo),

0.5

for the nearest neighbors (Y+l), and 0.25 for the next nearest

neighbors

(Yk2).

As described in Chapter

section

3.7.1,

a two-fold reduction of relative statistical errors

requires a four-fold increase of data collection time.

Preliminary data processing and phase analysis

353

Bragg angle,

(deg.)

Figure

4.5.

The powder diffraction pattem from

Figure

4.3

with subtracted background prior

to (top) and after (bottom) smoothing by using an analogue of

Eq.

4.7

with

sequential data

points.

Thus, the smoothed intensity of each point is the weighted sum of the

intensities of 5 sequential data points divided by the sum of weights:'

smoothed

Y-2

~1'-1

WO~

2w2

2w1

where wo, w1 and w2 are

0.5 and 0.25, respectively.

this example, the

weights change linearly (weight of the next data point is reduced by two,

although different linear dependencies may be used as well). Furthermore,

nonlinear weighting schemes may be employed, e.g. a Gaussian distribution

of weights. The number of points and smoothing coefficients (weights)

varies for different smoothing attempts. Generally, the smoother pattern is

obtained when more points are employed or when the weights change less

drastically from one point to the next.

Another commonly used smoothing approach is based on the fast Fourier

transformation (FFT) algorithm. The number of Fourier coefficients in the

Except for the first (the last) few points in the pattem, where smoothing is either truncated

to include only the points after (or before)

Yo,

or the point remains "unconditioned".

354

Chapter 4

original pattern is equal to the total number of the observed data points. In

this case, the reverse Fourier transformation results in the original pattern.

Each Fourier coefficient corresponds to a signal of specific frequency

observed in the original pattern: the higher order coefficients represent the

higher frequency signals. Thus, when high order coefficients are set to zero

or lowered, the reverse FFT produces a pattern similar to the original but

with removed or reduced high frequency noise, or in other words, a

smoother pattern. Setting more high order coefficients at zero produces

stronger smoothing. As a result, the removal of high frequency noise

"improves" the pattern but at the same time, more and more fine details

(weak or narrowly split peaks) are lost. The loss of weak features in a pattern

is a common problem in any smoothing algorithm.

Similar to background removal, smoothing should never be applied to a

powder diffraction pattern, which will be later used for profile fitting or

Rietveld refinement.' When performed, smoothing may improve certain

figures of merit

(e.g. R,,, R, and X2, see Eqs. 6.18, 6.19 and 6.22 in Chapter

6, below), but it will likely and considerably distort lattice parameters and

most certainly all intensity-sensitive structural parameters, including

coordinates, displacement and population parameters of the individual

atoms. The only reliable and justifiable way to improve the true quality of

the full profile fit is to perform a more accurate (i.e. careful sample

preparation and/or longer counting time) powder diffraction experiment.

4.3.3

Ka2

stripping

The presence of dual wavelengths in conventional x-ray sources, or in

other words the presence of the Ka2 component in both the incident and

diffracted beams, complicates powder diffraction patterns by adding a

second set of reflections from every reciprocal lattice point. They are located

at slightly different Bragg angles when compared with those of the

main

(Kal) component. This decreases resolution and increases overlapping of

Bragg peaks, both of which have adverse effect on an unbiased peak search.

Since every

KallKa2 double peak is caused by scattering from a single

reciprocal lattice point, the d-spacing remains constant and the scattered

intensity is proportional to the intensities of the two components in the

characteristic spectrum. Using Braggs' law the following equation reflects

the relationship between the positions of the

diffraction

peaks in the doublet:

Unlike background removal, where the subtraction of a constant background is permissible

(see the footnote on page

348),

this statement has no exceptions.

Preliminary data processing and phase analysis

355

Furthermore, the integrated intensities of the two peaks are related as:

I,,

:Ifi2

=2:1

Assuming that peaks in the doublet have identical shapes, which is

reasonable because the separation between the two is usually small, although

it increases with increasing Bragg angle, Eqs.

4.8

and

4.9

may be applied to

every point of the observed peak shape, but not to the background. The Ka2

stripping usually starts from the lowest Bragg angle point of the first

observed peak, which should not be in the range of any other diffraction

maximum, and moves towards the last point in the pattern.

example of

Ka2 stripping is shown in

Figure

4.6.

Bragg angle,

(deg.)

Figure

4.6.

An illustration of

Ka2

stripping (a). The expanded view of the range between

-20

and

40"

highlighting inaccuracies on the high Bragg angle sides of the strongest peaks (b).

The expanded view of the same region when the background subtraction and

Ka2

stripping

was preceded by a

point-smoothing (c). Although inaccuracies on high Bragg angle sides

are less visible after smoothing, the "improvement" is accompanied by the loss of resolution,

which is easily seen from comparison of the two peaks at

-36'

in (b) and (c).

356

Chapter

It is easy to see that this simple approach is far from ideal and the

removal of the

Ka2

contributions is far from perfect. The inaccuracies occur

because it is difficult to eliminate the background precisely.

higher quality

pattern usually results in a better

Ka2

stripping as illustrated in

Figure

4.6~

where the same pattern was smoothed before removing the background and

Ka2

components. Visible peak broadening due to the smoothing treatment is

noteworthy.

One of the unfortunate results of

Ka2

elimination is the distortion of the

high angle slopes of all Bragg reflections. Therefore,

Ka2

stripping is a valid

step in the preparation of the powder pattern for the following automatic

peak search, but it

should never be

performed as a part of data conditioning

of the powder diffraction pattern for fitting andlor Rietveld refinement.

4.3.4

Peak

Fast and reliable peak search or peak localization is needed in order to

conduct either or both qualitative and quantitative phase analysis or database

search for matching

pattern(s). One of the most reliable (in terms of peak

recognition), but far from the fastest techniques, is locating peaks manually,

i.e. visually. This can be done in two ways: using the position of the peak

maximum or the mid-point of the peak's full width at half maximum. Both

cases require removing the

Ka2

contribution. If the latter is not done,

positions of both low angle peaks (where

Ka,

and

Ka2

contributions

essentially coincide) and high angle peaks (where these components are

nearly completely resolved) will be determined for the

Kal

components.

However, locations of peaks in the mid-angle range will be determined

somewhere between those of the

Kal

and

Ka2

components. Ideally, they

should correspond to the weighted average

wavelength (see

Table

2.1

Chapter

but in reality this is difficult to achieve, especially when peak

tops are used as their positions. When

Ka2

components are stripped before

locating peaks, this problem is avoided and the positions of all peaks

correspond to the

Kal

part of the characteristic spectrum.

automatic peak search is actually the simplest (one-dimensional) case

in the more general two- or three-dimensional image recognition problem.

Image recognition is easily done by a human eye and a brain but is hard to

formalize when random errors are present and, therefore, difficult to

automate. Many different approaches and methods have been developed;

two of them are most often used in peak recognition and will be discussed

here. These are: the second derivative method and the profile scaling

technique.

The

second derivative method

is actually a combination of background

subtraction,

Ka2

stripping and, if needed, smoothing, which are followed by

Preliminary data processing and phase analysis

357

the calculation of the derivatives. This method is extremely sensitive to

noise. As a result, when fast measured patterns with substantial random

errors are employed, smoothing becomes practically mandatory. The second

derivative method consists of calculating first and then second derivatives of

Y(28) with respect to 28, and utilizing them in the determination of peak

positions. The derivatives can be easily computed numerically as:

ay.

a2q

qt2

2y.t1

and

- -

820,

(320,)~

where Yi,

Yiil

and

Yit2

are the intensities of three consecutive data points and

is the data collection step. Instead of smoothing, it is possible to use a

polynomial fit in the vicinity of every data point with the point in question

located in the middle of a sequence. Once the coefficients of the polynomial

are determined, both the first and second order derivatives are easily

calculated analytically. For example, for a third order polynomial (a

minimum of five sequential data points should be employed)

the first and second derivatives, respectively, are

and

where

is the Bragg angle. When the argument of the polynomial is selected

such that a point

(for which the derivatives are calculated) is chosen in the

origin of coordinates along the 28 axis, i.e. when

2ei, then the

corresponding derivatives are simply

(Eq. 4.12) and 2b (Eq. 4.13).

Some peak search algorithms use the first derivative, which is reliable

only for simple, well resolved patterns. The second derivative method works

better with complex data.

example in

Figure

4.7,

top shows the profile

representing two partially resolved Bragg peaks together with the first

(middle) and second (bottom) derivatives. The first derivative is zero at the

peak maximum and it changes sign from positive to negative when the

Bragg angle increases. The second derivative reveals each peak in a much

more reliable fashion, i.e. as a sequence of negative values of the function,

which are hatched in

Figure

4.7

for clarity.

Chapter

Figure

4.7.

Intensity distribution in two partially resolved Bragg peaks (top) and the

corresponding first (middle) and second (bottom) derivatives. The second derivative forms

series of sequential negative regions (hatched), which represent both the maximum of each

Bragg peak (coinciding with the minimum of the corresponding negative region) and the

estimate of peak width near its half maximum (the width of the associated negative sequence).

To improve the detection, negative sequences in the second derivative are

usually fitted to a parabolic function, thus resulting in a better precision of

Bragg peak positions. The width of a Bragg peak can be estimated as the

range of the associated negative region, since the second derivative changes

its sign at each inflection point.

The example from

Figure

4.7

is simulated and therefore, unaffected by

noise. Thus, the ranges where the second derivative becomes negative can be

detected with confidence. When processing real data, false peaks will be

found, especially in the background regions. To avoid finding an excessive

number of false peaks, an automatic peak search is usually coupled with

several limiting parameters, which should be established empirically for a

given quality and complexity of the data. For example, the minimum

observed intensity above which the search will be conducted excludes

incompletely removed background; the minimum intensity above which a

detected maximum would be considered as a peak excludes weak peaks; the

minimum number of sequential negative values of the second derivative,

which would be considered as the manifestation of a peak, excludes noise.

Preliminary data processing and phase analysis

359

The results of an automatic peak search can be further improved by adding

real andlor removing false peaks manually.

example of such a peak

search is shown in

Figure

4.8.

The

profile scaling

peak search algorithm employs a realistic analytical

peak shape, e.g. Pearson

VII

or pseudo-Voigt functions. This approach does

not require

Ka2

stripping or smoothing. Background subtraction may still be

necessary if the fitting algorithm does not account for its presence. The

simplest method uses the chosen peak shape function with preset parameters,

which may be adjusted manually.

improved approach is to use a well

resolved and strong Bragg peak to determine the peak shape parameters,

which are better suited to the actual data. The resulting normalized analytical

shape is then moved along the diffraction pattern and its intensity (which is

simply a multiplier, i.e. scale factor) is calculated by means of a linear least

squares technique (see Chapter

section

5.13.1)

to produce the best fit.

Regions, which meet certain criteria, are stored as observed peaks.

Figure

4.8.

Automatic peak search conducted using a second derivative method (top) and

manually corrected reduced pattern (bottom). The upward arrow placed on the digitized

pattern shows a false peak (which was eliminated manually) and the downward arrows show

the missed peaks (which were added manually).

3 60

Chapter 4

If necessary, an automatic peak search may be repeated using the

difference between the observed data and the sum of profiles of all detected

peaks. As a result, weak

andlor poorly resolved Bragg reflections, missed in

the previous search, may be found. This simple profile scaling method yields

relatively accurate peak positions and integrated intensities and, if

anticipated by the algorithm and realized in a computer code, their

FWHM's.

Its use is growing proportionally to the increasing computer speed. Since the

search remains automatic it may still require the adjustment of several

empirical parameters to exclude the excessive appearance of false peaks or

to improve the detection of weak Bragg reflections.

4.3.5

Profile

fitting

Profile fitting is the most accurate, although the slowest and the most

painstaking procedure resulting in observed peak positions, full widths at

half maximum, and integrated intensities of individual Bragg reflections. It

is based on minimization of the difference between observed and calculated

profiles using a non-linear least squares technique (see Chapter 6, section

6.6). The calculated profile is represented as a sum of scaled profiles of all

individual Bragg reflections detected in the whole pattern or in any part of

the pattern plus an appropriate background function (see section 4.3.1).

Individual peak profiles are described by one of the common peak shape

functions, typically pseudo-Voigt or Pearson-VII (see Chapter 2, section

2.9).

For conventional neutron diffraction data, a pure Gaussian function

may be employed.' Generally, three types of parameters can be adjusted

during the least squares fit:

Peak positions (28), are normally refined for the Kal components. If

present, the locations of the Ka2 constituents are established by Eq. 4.8.

Peak shape parameters, which include full width at half maximum

(H),

asymmetry (a), and exponent (P) for Pearson-VII or mixing parameter

(q) for pseudo-Voigt functions. All peak shape parameters are typically

refined for Kal reflections. The corresponding Kaz components are

assumed to have

(or q) identical to Kal. In some applications,

peak shape parameters may be fixed at certain commonly observed

values, or they may only be adjusted manually.

Integrated intensity

(I),

which is simultaneously a scaling factor for each

individual peak shape, see Eq. 2.48 in Chapter 2. Typically the integrated

intensity of the Kal reflection is a free parameter and the intensity of the

Ka2 part (if present) is restricted as given in Eq. 4.9.

It is worth noting, that when software on hand does not employ a Gauss peak shape

function, it can be easily modeled by the pseudo-Voigt function using the fixed mixing

parameter,