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

381

43-0318

and

NiMnO,(OH),

43-03

NIOKEL MANGANESE OXIDE HYDROXIDE

!j,

jj;

,I ,,I

-I

Bragg angle,

(deg.)

Figure

4.20.

Experimental powder diffraction pattern of NiMn02(OH) (top) compared with

the digitized PDF records 49-1 170 (middle, solid lines) and 43-03

(bottom, dashed lines).

Downward arrows indicate peaks present in the latter record but absent in the measured

pattern. Upward arrows shown on the experimental pattern indicate observed Bragg peaks that

are missing in the nickel manganese oxide hydroxide, ICDD record No. 43-03

18.

49-1170 NICKEL MANGANESE OXIDE

Careful analysis of

Figure

4.20

indicates that six peaks in this card (43-

18)

are clearly missing and three weak to medium intensity peaks have no

match in the experimental pattern. The second record 49-1 170, is almost a

perfect match, but a hydrogen atom is missing in its chemical formula, as

was determined later from neutron diffraction data.'

Achieving success in qualitative analysis by employing any search-match

utility becomes more and more challenging as the complexity of the powder

diffraction pattern increases, especially when a material is a mixture of

several phases. Positive phase identification can be performed by removing

peaks corresponding to all already known phases from the list and

continuing searches of the database. Nevertheless, matching all possible

records with the whole diffraction pattern may also work well. The first

approach increases chances to detect and identify minor phases, whereas the

second method avoids overlooking suitable records due to nearly complete

peak overlaps.

Chen, P.Y. Zavalij, M.S. Whittingham,

J.E.

Greedan,

N.P.

Raju, and M. Bieringer, The

hydrothermal synthesis of the new manganese and vanadium oxides, NiMn03H, MAV307

and MA0.75V4010~0.67H20

(MA

CH3NH3),

Mater. Chem.

93 (1999).

II.

.,..I

Chapter

43-0318 Quality: Doubtful quality

NiMn02(OH)

Nickel Manganese Oxide Hydroxide

RadPeKa Lamidx1.9373 Filter: d sp:Diffractometer

Cutoff: 1nt:Diffractometer IIIcor:

Ref:Yatnamoto, N., 33 48, (1986)

$4: SG.:

Ref;?

Dm: WFOM. Volume[CD]:O

ea:

SOP:

sign: 2v:

Prepared

hydrothermal treatment at 200-320 C

and

100 MPa of an equimolar mixture of \Ni

(

)2\ and $GG-MnOOH in a

IN \N H4

solution. Chemical analysis

(wt.%):

Ni 36.0, Mn 34.0,

0.58,

29.4.

assigned because unindexed.

21 reflections in oattern.

Int.

41.3

46.2

99.9

51.6

19.6

97.3

26.1

Int.

hkl

Figure

4.21.

Example of the unindexed PDF card (also see

Figure

4.20,

bottom). In this case,

all observed reflections are identified by their Bragg angles and relative intensities but without

the Miller indices. As a result, the "Doubtful quality" mark has been assigned to this record

by one of the ICDD editors. This usually points to the need for an independent verification

before the listed digitized pattern can be relied upon in a positive identification of a

polycrystalline material. Note that the original experimental data were collected using Fe Ka

radiation (see field No.

3).

Bragg angles, however, are listed for Cu

radiation, and these

were recalculated by the LookPDF utility using the Braggs equation.

Regardless of the chosen approach, specifying elemental composition of

the material is always helpful, as it imposes much needed constraints and

limits the number of feasible solutions for a visual analysis.

example of successful phase identification in a multiple phase sample

is shown in

Figure

4.23.

The search was conducted using the following

restrictions:

window of 0.06", two matching lines minimum, chemical

composition was restricted to all inorganic compounds containing silicon

and oxygen. No single match was adequate to interpret all strong observed

Bragg peaks. However, two records simultaneously, i.e. lithium silicate

Li2Si03 and quartz Si02, cover the majority of strong reflections. Most of the

remaining Bragg peaks correspond to a tridymite (Si02), which is present in

a lower concentration than both Li2Si03 and quartz.

few weak reflections

Preliminary data processing and phase analysis

383

in this pattern remain unidentified, likely due to the low quality of data (the

experimental pattern shown in

Figure

4.23

has been smoothed) or a small

amount of an impurity phase, which does not contain both silicon and

oxygen.

Overall, phase identification in a multiple-phase material, which consist

of more than two phases is difficult and often has no reasonable solution in a

"blind" search, especially when none of the phases have been positively

identified prior to the search using a different experimental technique.

Furthermore, chances for success decrease proportionally to the increased

complexity of the measured powder diffraction pattern, unless the number of

possible components with different crystal structures in the mixture is

limited to just a few.

49-1170

Quality: Quality Data

Ni Mn 03

Nickel Manganese Oxide

Rad:CuKal Lambda:1.54056 Filter: d sp:Diffractometer

Cutoff: 1nt:Diffractometer lilcor:

Ref:Whittingham. S., SUNY at Binghamton, MaterialsResearch Center, NY, USA., (1997)

Sys:Orthorhombic S.G.:AZlam

a:2.8609*0.0001 b:14.6482*0.0005 c:5.2703+0.0002

"I:

2:4 mp

Ref2

Dx:4.861 Dm4.861 WFOM: F30=103.1(0.0081.36) VolumelCD1:220.86

SOP:

Sign:

Color:

Prepared by hydrothermal treatment

tetramethylammonium permanganate, nickel acetate and lithium carbonate for 2 days at

200 C. Pattern taken

23(1)

reflections in

ern.

Int.

Figure

4.22. Indexed PDF card (also see

Figure

4.20,

middle).' Every observed Bragg

reflection has been indexed and the corresponding

F33

figure of merit (see section

5.5.1

Chapter

is excellent. Based on these and other established criteria, the quality mark

assigned by the ICDD editor is "Quality Data", which usually is a good indicator that the

included digitized pattern may be trusted in positive phase identification.

Int.

Space group symmetry is listed as A2]arn, which has been transformed into Cmc2] in a

standard setting to produce the list of Miller indices.

0 10 1

1 9 1

2 2 1

2 4

0 0 4

0 2 4

2 4 1

0 8 3

2 0 2

19 2

2 2 2

2 6 0

1 7 3

1 I1 0

80.4338

80.7356

81.7659

83.0093

83.2543

88.1190

88.6939

89.1644

93.0153

95.6870

96.0050

Int.

Chapter

Smoothed data,

40 50

Bragg angle,

(deg.)

-1

Figure

4.23.

The results of a qualitative analysis of a multiple phase sample. Three crystalline

phases are clearly identifiable: lithium silicate

Li2Si03, silicon oxide

SiO, (quartz), and a

different polymorph of silicon oxide

tridymite.

low quality diffraction pattern collected

during a fast experiment was employed in this example. The data shown on top were

smoothed, the background was subtracted, and the

Ka2 components were stripped before the

digitized pattern (shown below the smoothed profile) was obtained using an automatic peak

search. Note, that many weak Bragg reflections were missed in the peak search.

4.4.3

Quantitative analysis

I,,

Quantitative phase analysis is used to determine the concentration of

various phases that are present in a mixture after the identity of every phase

has been established. Overall, the task may be quite complicated since

several critical requirements and conditions should be met in order to

achieve satisfactory accuracy of the analysis.

Proper alignment, and especially calibration of the diffractometer are

very important. Calibration should be performed by examining one or

several different mixtures arranged from carefully prepared and well

characterized materials. In general, any of the many available standard

Digitized pattern

I\b

29-0828

LITHIUM

SILICATE

47-1144

SILICON

OXIDE

Ill

18-1170

SILICON

OXIDE

ITRIDYMITE-M,

SYN

....

.I.

I..,

..,,

Preliminary data processing and phase analysis

reference materials (SRM) may be used,' including a specially developed

standard for phase identification and analysis. The latter is the SRM-674a,2 a

standard for powder

diffraction

intensity, which is a mixture of five stable

oxides: a-A1203 (corundum structure), CeO2 (fluorite structure), Crz03

(corundum structure), TiOz (rutile structure) and ZnO (wurtzite structure).

addition to instrumental factors, specimen preparation and properties

introduce several key features that may have a detrimental influence on the

accuracy of quantitative phase analysis. Sample-related factors cannot be

avoided completely, but their effects should be minimized as much as

possible

andlor accounted for in all calculations. The main problems in

quantitative analysis, borne by the nature and form of the employed sample

are as follows:

Preferred orientation, which may have a substantial effect on relative

intensities of various groups of Bragg reflections (see section 2.10.6,

Chapter 2). It should be minimized during sample preparation of both the

investigated sample and the standard, if the latter is employed;

Absorption (see section 2.10.5, Chapter 2), which is generally different

for phases with different chemical composition and gravimetric density.

It should always be accounted for.

Several different methods of the quantitative analysis have been

developed and extensively tested. They may be grouped into several broad

categories, and the most commonly used approaches are described below.

The absorption-diffraction method

employs a standard intensity (IOhkl)

from a pure phase and the intensity of the same Bragg peak (Ihkl)

observed in the mixture. The phase concentration in a mixture can be

calculated by using Klug's equation:

where:

Xa is the mass fraction of phase

in the mixture;

and

roa,hkl

are intensities of the selected Bragg reflection,

hkl,

for

phase

in the mixture and in the pure state, respectively;

E.g., see: Standard reference materials catalog 1992-93. NIST, Special publication 260,

NIST, Gaithersburg,

MD,

U.S.A. (1992), p.122. Up to date information about availability

and pricing of x-ray diffraction standards for powder diffraction may be found at

http:Nsrmcatalog.nist.gov/.

SRM-674a has been discontinued. Replacement should be available in the fall of 2002 as

SRM-674b.

386

Chapter

(CL/P)~,

are the mass absorption coefficients for phases

and

respectively.

Equation 4.14 makes use of the fact that the scattered intensity is

proportional to the amount of a particular phase,

e.g. see Eqs.

2.17

2.19 in Chapter 2, with a correction to account for different absorption of

x-rays by two components in the mixture. Since the ratio of intensities

from a pure phase and a mixture is employed, diffraction patterns from

both the pure material and from the analyzed mixture must be measured

at identical instrumental settings, in addition to identical sample

characteristics such as preparation, shape, amount, packing density,

surface roughness, etc.

Klug's equation becomes a simple intensity ratio

when two phases have identical absorption coefficients, i.e. when

p/p,

We note that the composition of the second phase (or a mixture of

all other phases) should be known in order to determine its mass

absorption coefficient. Otherwise, mass absorption should be determined

experimentally. When absorption effects are ignored, the accuracy of

quantitative analysis may be lowered drastically.

Method of standard additions

or spiking method consists of adding

known amounts of pure component

to a mixture containing X, and Xb

and b phases. It requires the preparation of several samples and

measurement of several diffraction patterns containing different yet

known additions

(Y,)

of phase

Other phases in the mixture are not

analyzed but at least one of them (b) should have a reference Bragg peak

(hkl)',

which does not overlap any reflection from phase

The intensity

ratio for this method is given as:

Assuming constant mass of phase b

(K=F/Xb),

Eq.

4.15 is converted

into:

where:

K is the slope of the plot of

I,,hkl

versus Ib,hkl' established during

measurements of mixtures with known additions of phase

is the intensity of the selected peak for phase

Ib,hkll

is the intensity of the selected peak for phase b.

Preliminary data processing and phase analysis

Figure

4.24.

Illustration of standard addition method of quantitative analysis. The plot of the

I,,

hkl/&,

intensity ratio as a function of the known amount

(Y,)

of added phase

The point

marked

Y?)

corresponds to the original two-phase mixture. The unknown amount of phase

in the original sample is

Thus, the unknown amount of phase a, X,, is determined from the

intercept of the calibration line with the

axis as shown in Figure

4.24.

The major advantage of this technique is that it enables quantitative

analysis in the presence of unknown phase(s) without the need to know

(or measure) absorption coefficients.

Internal standard method is likely the most commonly used approach in

a quantitative phase analysis. It is based on the following relationship:

where

is the slope of the plot of

I,,

hkl/Ib,

hklf

versus Xa/Yb.

Eq.

4.17, Xa

is the unknown amount of the analyzed phase a, and

is the known

amount of the added standard phase

that is different from that present

in the sample. In the same way as in the standard addition method,

several measurements are needed to determine the slope

individually

for each analyzed phase. The calibration line of Eq. 4.17 is then used to

determine the content of phase a by measuring the

I,,

hkl/Ib,

hkll

intensity

ratio for a mixture of the analyzed sample with the known amount of the

added internal standard Yb.

388

Chapter

The reference intensity ratio method

is based on the experimentally

established intensity ratio between the strongest Bragg peaks in the

examined phase and in a standard reference material. The most typical

reference material is corundum, and the corresponding peak is (1 13). The

reference intensity ratio

(k)

is quoted for a 5050

(wt.

mixture of the

material with corundum, and it is known as the "corundum number". The

latter is commonly accepted and listed for many compounds in the

ICDD's Powder Diffraction File. Even though this method is simple and

relatively quick, careful account and/or experimental minimization of

preferred orientation effects are necessary to obtain reliable quantitative

results.

Rietveld refinement

(Chapter 7, section 7.3.5 and the example in section

7.3.8) of multiple phase samples may be used for relatively accurate

quantitative analysis. It requires knowledge of the atomic structure for

each phase present in the mixture. Structural data are needed to calculate

corresponding intensities. Scale factors for every phase present in the

mixture, which are determined quite accurately during Rietveld

refinement, are proportional to the fraction of the unit cells present in the

irradiated volume of the sample. The latter directly follows from Eqs.

2.18 and 2.19 after recalling that the proportionality coefficient,

constant in any given powder diffraction experiment. Thus, the scale

factors can be easily converted into weight, molar or volume fractions of

the respective phase. This method appears to be one of the fastest and,

perhaps, the most reliable tool in quantitative phase analysis, especially

because it offers a possibility to introduce a preferred orientation

correction in addition to quantitative analysis with respect to all present

phases.

Full pattern decomposition

using Le Bail's or Pawley's techniques (see

Chapter

does not require the atomic structure to be known and it

produces intensities of individual Bragg peaks. Thus, multiple reflections

from each phase can be used to compute intensity ratios required in

methods described in items

through

above, which increases the

accuracy of the analysis. The use of multiple Bragg peaks in evaluating

an average intensity ratio, to some extent diminishes the detrimental

influence of preferred orientation as long as it remains small to moderate.

This method, however, requires lattice parameters and therefore, is

applicable to indexed patterns only. The phase composition is actually

determined using any of the first four methods listed above by using

intensities of several strong or all Bragg peaks instead of a single

reflection.

Both the accuracy and limits of detection in a quantitative analysis are

dependent on the method used, the quality of the experimental data, and

Preliminary data processing and phase analysis

other factors.

lower limit of detection is usually accepted as the

concentration equivalent of two standard deviations of the observed

background level. For example, if the average background is at 100 counts

and the maximum observed Bragg peak has a peak intensity of

1100 counts,

then two standard deviations of the background are: 2xd100

20. Thus, the

detection limit for this phase is estimated at 2041 100-100)x100%

2%. The

accuracy of the quantitative phase analysis is difficult to estimate rigorously.

It varies considerably and is often claimed to be between 1 and

5%.

The

Rietveld profile fitting method is probably the best for a realistic estimate of

the accuracy because standard uncertainties in calculated concentrations can

be easily estimated from standard uncertainties in the corresponding scale

factors.

3 90

Chapter

4.5

Additional reading

L.S.

Zevin and

Kimmel, Quantitative x-ray diffractometry, Springer,

New York (1995).

2. R. Jenkins and R.L. Snyder, Introduction to x-ray powder diffractometry,

John Wiley and Sons, New York (1996).

R.J. Hill, Data collection strategies: fitting the experiment to the need, in:

R.A. Young, Ed., The Rietveld method. IUCr monographs on

crystallography, 5, Oxford University Press, Oxford, New York (1 993).

H.P. Klug, and L.E. Alexander, X-ray diffraction procedures for

polycrystalline and amorphous materials, 2nd ed. John Wiley, New York

(1 974).

5. B.D. Cullity, Elements of x-ray diffraction, 2"d ed. Addison-Wesley,

Reading, MA (1 978).

6. J. Faber and T. Fawcett, The Powder Diffraction File: present and future,

Acta

Cryst. B58, 325 (2002).

7. S.N. Kabekkodu, J. Faber, and T. Fawcett, New Powder Diffi-action File

(PDF-4) in relational database format: advantages and data-mining

capabilities, Acta

Cryst. B58,

333

(2002).

8. M.A. Van Hove, K. Hermann and P.R. Watson, The NIST Surface

Structure Database

SSD version 4, Acta Cryst. B58,338 (2002).

9. P.S. White, J.R. Rodgers, and Y. Le Page, CRYSTMET: a database of

the structures and powder patterns of metals and intermetallics, Acta

Cryst. B58,

343

(2002).

10. A. Belsky, M. Hellenbrandt, V.L. Karen, and P. Luksch, New

developments in the Inorganic Crystal Structure Database (ICSD):

accessibility in support of materials research and design, Acta

Cryst.

B58,364 (2002).

11. F.H. Allen, The Cambridge Structural Database: a quarter of a million

crystal structures and rising, Acta Cryst. B58, 380 (2002).