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

powder diffraction data. The results are frequently employed to establish

andlor clarify relationships between crystal structures and properties of

materials; knowing these is truly critical in modem science and engineering.

The most typical sequences of data processing steps (or phases) are

therefore, visualized in a general schematic depicted in

Figure

4.1.

Correspondingly, these four major steps are the subjects of this and the

following three chapters in the book. As shown in

Figure

4.1,

the three

different quality levels of powder diffraction data are usually associated with

the expected outcomes of the experiment.'

Known

structure

Figure

4.1.

The flowchart illustrating common steps employed in a structural characterization

of materials by using the powder diffraction method. It always begins with the sample

preparation as a starting point, followed by a properly executed experiment; both are

considered in Chapter

Preliminary data processing and profile fitting are discussed in this

chapter in addition to common issues related to phase identification and analysis. Unit cell

determination, crystal structure solution and refinement are the subjects of Chapters

5,6,

and

respectively. The flowchart shows the most typical applications for the three types of

experiments, although any or all of the data processing steps may be applied to fast, overnight

and weekend experiments when justified by their quality and characterization goals.

The classification "fast", "overnight" and "weekend" experiments is usually applied to

laboratory powder diffractometers equipped with conventional sealed x-ray tube sources.

Obviously, when the brilliance of the available source increases dramatically, the time of

the actual experiment decreases. It is worth noting that since specialized beam time (e.g. a

synchrotron source) is limited, this normally implies that the majority of samples should

undergo a thorough preliminary examination using conventional x-ray sources.

342

Chapter

Thus, a fast experiment is routinely suitable for evaluation of the

specimen and phase identification, i.e. qualitative analysis. When needed, it

should be followed by a weekend experiment for a complete structural

determination.

overnight experiment is required for indexing and

accurate lattice parameters refinement, and a weekend-long experiment is

needed for crystal structure determination and refinement.

some instances,

e.g. when a specimen has exceptional quality and its crystal structure is

known or very simple, all relevant parameters can be determined using data

collected in an overnight experiment. Similarly, fast experiment(s) may be

suitable for unit cell determination in addition to phase identification.

any

case, one should use hislher own judgment and experience to assess both the

suitability of the experimental data and the reliability of the result.

When the sole goal of the experiment is to identify phases present in the

polycrystalline material, it may be achieved in a fast experiment, which may

be collected in as little as

min depending on the quality of the

specimen, brightness of the beam and geometry of the instrument. Another

important application of the rapidly collected data is visual examination of

Y(28) to evaluate both the crystallinity of the specimen and the complexity

of the pattern as illustrated by several distinct examples in

Figure

4.2.

Bragg angle,

(deg.)

Figure

4.2.

Fragments of powder diffraction patterns collected from three different materials

on a Rigaku TTRAX rotating anode powder diffractometer using Mo

radiation. Pattern

represents a material with excellent crystallinity: narrow and sharp Bragg reflections with

KallKa2

doublets becoming partially resolved at

14O,

low and nearly linear background.

Pattern

is also a crystalline material, however, its crystallinity is poor (andlor grain sizes are

extremely small, and/or the material has been strained), which is evident from broad but still

distinct Bragg reflections; certain amount of an amorphous component may be present in this

sample judging from a minor nonlinearity of the background between

-6

and

20"

20.

Pattern

is collected from a material, which is clearly non-crystalline in a conventional sense: long-

range order and periodicity are absent because no Bragg reflections have been observed.

Provided powder

is amorphous, broad halo(s) usually contain information about nearest

neighbors in the coordination spheres of atoms. Each pattern has been scaled individually.

Preliminary data processing and phase analysis

343

Considering the three examples illustrated in

Figure

4.2,

specimen

suitable for any kind of conventional structural characterization using

powder diffraction. Furthermore, judging from the relatively large separation

between Bragg reflections at low angles it is easy to conclude that the crystal

structure of this material is quite simple.' Specimen B in the current state,

may or may not be suitable for the determination of its crystal structure. If

possible, the material should be recrystallized or heat-treated (depending on

its nature) to increase grain sizes

and/or relieve strain. If successful, widths

of Bragg peaks will be reduced and pattern resolution will be improved.

Judging from the number of Bragg peaks observed at low angles and from

the fact that the first Bragg reflection in specimen B occurs at a lower angle

when compared with specimen

the crystal structure of the former is more

complex that that of the latter.2 Finally, specimen

is unsuitable for the

analysis of its crystal structure and it should be appropriately modified, if

possible, to restore the crystallinity of the

material.3

Provided the specimen produces a satisfactory diffraction pattern and in

order to proceed with phase identification, both the positions and intensities

of Bragg peaks should be determined. This is usually achieved by an

automated peak search procedure using the pattern, which has been

processed to eliminate either or both the background and the

Ka2

components of the peaks. When the pattern is complex, automated peak

search procedures may result in numerous missed peaks, and then manual

peak hunting or even semi-manual profile fitting algorithms should be

employed in order to obtain the list of Bragg reflections suitable for phase

identification. The latter is accomplished by comparing the resulting list of

Bragg reflections with one or more databases

(e.g. the Powder Diffraction

~ile~~ maintained and distributed by the ICDD4) using different search-

match algorithms.

If phase identification was the only purpose of the experiment and the

obtained list of Bragg angles and intensities has no known match, the

Pattern

was collected from a LaNi4,s5Sno.15 powder prepared by high-pressure gas

atomization and then heat treated at 950•‹C for 5 min. The compound has a hexagonal

crystal structure with unit cell dimensions

5.04,

4.01

Pattern

was collected from an anhydrous FePO., powder obtained by decomposition

(holding at 80•‹C in vacuum for 12 h) of the hydrothermally prepared dihydrate

FeP04.2H20. The compound has a monoclinic crystal structure with unit cell dimensions

5.48,

7.48,

8.05

95.7'.

Pattern C was collected from a molecular compound (Ph3PCH2COPh-Br) ball-milled for

one hour, which resulted in the complete loss of crystallinity. The appropriate processing

in this case would be to slowly recrystallize the compound using a suitable solvent and

then gently grind the powder, if necessary, to prepare a specimen for a powder diffraction

experiment.

ICDDB

International Centre for Diffraction Data: http://www.icdd.com.

Chapter

collected data can be used as a fingerprint of a new crystalline substance.

The identity of the material should be confirmed using various experimental

techniques (e.g. differential thermal analysis, spectroscopic andlor

microscopic methods, etc.) to ensure that it is not a mixture of compounds. If

phase identification was only the first step in a powder diffraction

experiment, the next move is to proceed with the precise determination of

lattice parameters. This is easier done when unit cell dimensions are known

at least approximately, but they may also be established from first principles.

Since the highest possible accuracy of peak positions is essential, both

the indexing of the pattern and lattice parameters refinement impose high

demands on the quality of data, and overnight or weekend experiments

should be performed. Profile fitting of powder data is typically conducted

first and then it is followed by assignment of indices to individual Bragg

reflections. The latter can be accomplished using approximately

hown unit

cell dimensions or in a process of the so-called

ab initio

indexing.

Subsequent least squares refinement usually yields precise unit cell

dimensions.

some cases, when the suspected unit cell is quite accurate,

lattice parameters can be obtained in a process known as the full pattern

decomposition.

When the unit cell is established and refined, it may be the end of data

processing, but it may also be a part of a sequence leading to a solution of

the crystal structure from first principles. It is worthy of reminding that the

ab initio

structure determination may be performed in two quite different

ways. The first works in reciprocal space, where the intensities of Bragg

reflections are employed to recover their phases by applying either heavy

atom (Patterson) or direct methods. Both phase angles recovery methods

require highly accurate individual integrated intensities that may be obtained

from exceptional quality data (usually a weekend experiment)

via

full

pattern decomposition. The second group of techniques consists of

simulating the crystal structure in real space: from database searches and

simple geometrical modeling to energy minimization and random structure

generation. It may be necessary to employ several different approaches

andlor obtain a more accurate and better resolved powder diffraction pattern,

and/or examine different indexing solutions from the previous processing

step before the model of the crystal structure is judged acceptable. When the

crystal structure is solved, one final step, i.e. Rietveld refinement, is usually

needed in order to confirm and complete the structure.

The Rietveld method is employed both to finalize the model of the crystal

structure, when necessary, e.g. to locate a few missing atoms in the unit cell

by coupling it with Fourier series calculations, and to confirm the crystal

structure determination by refining positional and other relevant parameters

of individual atoms together with profile variables. The fully refined

Preliminary data processing and phase analysis

345

structural model must make both physical and chemical sense. If there are

doubts, additional investigations should be carried out; they may include a

better experiment and testing other feasible structural models in addition to

employing various experimental techniques, such as chemical,

thermogravimetric, spectroscopic, electron microscopic, neutron scattering,

and other. When structure determination is completed, it should be followed

by calculation of bond lengths and bond angles, structure drawing, and

preparation of the crystallographic data for publication in a suitable journal

and/or in a database.

Referring once again to

Figure

4.1, the flowchart presented there is quite

generic. It does not account for many other important applications of powder

diffraction such as microstructure analysis and determination of grain sizes

and microstrain, low angle scattering and structure of thin films, texture

analysis, and diffraction from polymers, fibers, liquids and amorphous

materials.

Furthermore, Rietveld refinement can be, and is indeed, broadly

used for quantitative phase and chemical analyses. The picture drawn in

Figure

4.1 should not be taken as a rigid set of processing steps. Many more

links between the boxes can be drawn, largely dependent on the type of the

material and the complexity of its crystal structure, quality of the

experiment, "non-crystallographic" knowledge about the material, and the

expectation or a major goal of a particular diffraction study.

4.3

Preliminary

data

processing

As we already know (e.g. see

Figure

3.43

in Chapter

3),

a typical powder

diffraction pattern is collected in a form of scattered intensity as a numerical

function of Bragg angle (also see

Figure

4.2

and

Figure

2.32

Figure

2.37).

What is required in many applications of the method (e.g. phase

analysis, unit cell dimensions, and structure solution), is a list of integrated

intensities or equivalent absolute values of structure factors associated with

the corresponding Miller indices of the reciprocal lattice points and the

observed Bragg angles

(e.g., see

Table

2.6

in Chapter

2).

Obviously, the

availability of such a list eliminates the effects of multiple instrumental and

specimen-related parameters from raw data but its creation requires certain

numerical processing of the pattern. Thus, raw powder diffraction data

converted into the list of observed Bragg angles and intensities are known as

the reduced or digitized powder diffraction patterns. It is worth noting that

reduced patterns may be employed in crystal structure refinement but only in

simple cases, when no substantial overlapping of Bragg reflections occurs.

order to obtain a reduced pattern, scattered intensity maxima should be

located and their relative intensities (either as peak height values or, most

commonly, as peak areas) should be established together with the

346

Chapter

corresponding values of Bragg angles. Depending on the required accuracy

and the availability of crystallographic information, e.g. unit cell dimensions

and symmetry, there are three common approaches to extracting intensities

and positions of Bragg peaks from raw powder diffraction data:

Peak search, which is unbiased by any kind of structural information.

This approach is based on automatic recognition of Bragg peaks. The

accuracy of peak positions, their intensities, and the completeness of the

search are the lowest, especially when weak Bragg reflections are of

concern, and it varies depending on both the employed algorithm and

quality of raw data. A typical automatic peak search produces reduced

patterns suitable for successful identification of phases when coupled

with a proper database search-match algorithm. This occurs because all

search-match utilities employ just a few of the strongest Bragg peaks; the

latter are conclusively detected by the majority of automatic peak search

algorithms. When the specimen is well crystallized, the results may

become suitable for a quantitative phase analysis, indexing and even for

lattice parameters refinement. However, the quality of a typical fast

experiment is usually insufficient and it may be necessary to conduct an

overnight experiment.

the majority of applications, the unbiased peak

search requires additional preliminary processing of the data, which

includes background subtraction,

Ka2

stripping, smoothing, and,

sometimes, other corrections, such as Lorenz-polarization and conversion

of variable slit data to a fixed slit experiment. All of these additional

processing steps may be required to improve the reliability of peak

detection.

Profile fitting, which is usually biased by the user's decision about

whether or not a peak is present at a certain angle or by the results of an

automatic peak search. Found or manually marked Bragg peaks are fitted

to experimental data using one of the many available peak shape

functions (Chapter

section

2.9).

The fit can be performed using the

entire pattern at once or by splitting the pattern into several regions,

which are processed separately. Free parameters in these fits usually

include peak positions, their integrated intensity, and other relevant

variables required to describe peak shape functions. The latter may be

peak-specific of they may vary as certain analytical functions of Bragg

angle, common for the entire pattern or only a single range. This

approach works best when no information about the unit cell is available.

It produces highly accurate observed peak positions

(200bS)

and integrated

intensities

(Pb",

provided the quality of experimental data is adequate.

The obtained reduced patterns may be used in a quantitative phase

analysis, indexing from first principles and lattice parameters refinement,

all with higher than average probability of success. Profile fitting results

Preliminary data processing and phase analysis

347

may be suitable for a crystal structure determination, which however,

may be problematic when diffraction patterns are exceedingly complex

and contain numerous clusters of heavily overlapped Bragg reflections.

Full pattern decomposition, which is fundamentally biased by the chosen

unit cell dimensions. It relies on fitting the whole powder pattern at once.

In this method, positions of Bragg peaks are established from lattice

parameters and symmetry. Only unit cell dimensions are refined and the

resulting peak positions are not "observed" but rather they are calculated

from the refined lattice parameters. Peak shapes are dependent on a few

free variables in relevant analytical functions of Bragg angle, as was

described in Chapter 2, section

2.9.

The integrated intensities, however,

are determined individually for each Bragg reflection. This method

extracts quite reliable individual intensities and, hence, is a typical data

processing step that precedes the structure solution from first principles,

as will be discussed in Chapter

It is worth noting that the full pattern

decomposition approach is used increasingly often to obtain accurate

lattice parameters when a powder pattern has been indexed but the crystal

structure remains

unhown or is not of interest for a specific application.

Preliminary treatment of powder diffraction data and their conversion

into reduced powder patterns for phase identification and a database search

are nearly always included as parts of data processing software suites, which

are available with the purchase of a powder diffractometer. Perhaps this is

the main reason explaining the lack of comparable freeware. Thus, the

majority of examples found in this chapter have been obtained using the

DMSNT' software distributed by Scintag,

Inc.' We note, that software

developers use a range of data processing algorithms and therefore, we will

only be concerned with generic issues without getting into software-specific

details, which may be found in the corresponding manuals.

4.3.1

Background

Background (e.g. see

Figure

4.2 and

Figure

4.3) is unavoidable in

powder diffiaction, and each powder pattern has a different level of

background noise. The latter originates from inelastic scattering, scattering

from air, sample holder and particle surfaces, x-ray fluorescence, incomplete

monochromatization, detector noise, etc. As a result, the background must be

accounted for, which is usually done by either subtracting it during

preliminary processing of the data, or by adding its contribution

(e.g. see Eq.

2.48

in Chapter 2) to the calculated intensity, Y(B)~~'~, during profile fitting.

DMSNT: Data Management Software for Windows NTl2000 from Scintag Inc. Now

WinXRD: Data Collection and Analysis Package from Thermo ARL.

Formerly Scintag, Inc. of Cupertino, CA, now a subsidiary of Thermo ARL.

Chapter 4

Bragg angle,

(deg.)

Figure

4.3.

Powder diffraction pattern of the orthorhombic NiMn02(0H) collected on a

Scintag

XDS2000

powder diffractometer using Cu

radiation. In this example, the

background noise contributes a few percent to the highest measured scattered intensity.

Subtracting the background is considered mandatory during preliminary

data processing before eliminating (stripping)

Ka2

contributions for a

subsequent peak search. The background should be removed because

Ka2

stripping is based on the fixed

Kal

Ka2

intensity ratio, which is

(see

Chapter

section

2.2.4),

in addition to Bragg angle splitting due to the

difference in the wavelengths of these two components of the characteristic

x-ray spectrum. The

2:l

intensity ratio includes elastic scattering on

reciprocal lattice points, but not the background noise.

The background

should never be

subtracted prior to full pattern

decomposition and full profile-based Rietveld refinement.'

these cases it

is approximated using various analytical functions with coefficients, which

are refined along with other parameters. Thus computed background is then

added to the intensity calculated as a function of Bragg angle.

The only exception is the subtraction of a constant component of the background, i.e. the

lowest observed intensity, to improve the visibility and enhance contributions from Bragg

peaks, if required. Constant background subtraction can be described by the following

analytical expression:

y,'

min(Y,)

where

is the measured intensity,

is the

intensity with constant background subtracted, unity is added to avoid zero intensity and a

potential for a division by

and

varies from

where

is the total number of

collected data points.

Preliminary data processing and phase analysis

349

In the majority of preliminary data processing algorithms, the

background is handled in two different ways:

Automatic, when the background points are chosen based on certain

criteria and then these points are employed to approximate the

background by means of certain smooth analytical function.

Manual, when the user selects points that belong to the background (i.e.

they do not belong to any of the visible Bragg peaks), which are then

used in the same way as in automatic background removal.

Both approaches have their advantages and disadvantages. Manual

subtraction is slow but generally yields a more accurate or reasonable

background approximation, while an automatic algorithm is faster and easier

but sometimes lacks the required accuracy. A typical background is

enhanced at low Bragg angles. It has a broad minimum in the mid-angles

range and then gradually increases towards high Bragg angles. Actual

behavior, however, depends on many factors, such as the material of the

sample holder, incident beam and monochromator geometry, sample

chemistry and microstructure, and other parameters.

the powder

diffraction pattern shown in

Figure

4.3, the background looks nearly linear

but in reality, this is not the case as can be seen in

Figure

4.4, where the

ordinate has been rescaled to reveal low-intensity details.

As seen in

Figure

4.4a, the background determined automatically using

the default box value of

1.5

is far from the best choice in this case because

greater than needed curvature results in accepting broad bases of the

strongest peaks as the background. This distorts intensities of Bragg peaks

and therefore,

Ka2

stripping or other following treatments cannot be

performed satisfactorily.

the next example

(Figure

4.4b), the background

was also treated automatically, but this time with box value increased to

4.0.

The result is a much better approximation but a careful examination of the

bases of the strong peaks indicates that the background remains

overestimated in these critical regions. Manual selection of points for a

background approximation is shown in

Figure

4.4c, which appears to be the

best choice for this particular powder diffraction pattern. Both examples b

and

are suitable for a subsequent

Ka2

stripping. It is worth noting that

neither the automatic nor the manual background subtraction works well

when there is a substantial overlapping and clustering of Bragg reflections.

Once again, the background should be eliminated only when the next step

involves the removal of the

Ka2

components or other unwanted peaks from

the pattern to perform a peak search using algorithms that do not account for

its presence (e.g. as in the DMSNT). Other methods that also yield both peak

positions and their intensities with a better precision (e.g. profile fitting), do

not require

Ka2

stripping and, therefore, background subtraction in these

cases should be avoided. The background is interpolated using various

350

Chapter

analytical functions and their coefficients are refined simultaneously with

other profile parameters, including structural parameters during Rietveld

refinement. Hence, the background is accounted in a more flexible way,

experimental data are not subjected to an irreversible modification.'

and

10 20

40 50

Bragg angle,

(deg.)

Figure

4.4.

The powder diffraction pattern from

Figure

4.3

shown with a different intensity

scale. The background (thin lines at the bottom of each plot) has been approximated by means

of a) an automatic algorithm with the default box width2 of

So;

b) an automatic algorithm

with a

4.0"

box width; c) background points selected manually as indicated by small circles.

Background approximations have been performed using the DMSNT software.

Indeed, it is possible and always recommended to save a backup copy of the unmodified

experimental data file. If for any reason the background should be reevaluated, it is easy to

do so, provided a copy of the original data file exists.

Width (in degrees 20) of the box (window) used in the Box Car Curve Fit method for

background removal. The window determines how close the background curve follows the

data. An underestimated box width results in humps under the peaks, as seen in (a). An

overestimated box width translates in the background that does not follow the data. A

proper box width (usually established experimentally) results in the background that

follows the data closely without following the peaks, as seen in (b).