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

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Chapter

6.7

Figures of merit in full pattern decomposition

Since both Pawley and Le Bail full pattern decompositions are based on

finding a least squares solution of Eq. 6.6, the problem may be considered

solved and a pattern deconvoluted when the best possible fit between the

calculated, and observed intensities,

Y:~', is achieved. Indeed, the left

hand sides of each equation in Eq.

6.6

represent yrbS, and the right hand

sides represent

yicalc.

It is, therefore, of utmost importance to have certain numerical figures of

merit that quantify the quality of the least squares fit and therefore, may be

used to estimate the reliability of the extracted integrated 'intensities and

observed structure factors. The following figures of merit are customarily

used to characterize both the full pattern decomposition and Rietveld

refinement quality.

The profile residual (or reliability) factor, Rp:

The weighted profile residual,

R,:

The Bragg residual, RB (this figure of merit is quite important in Rietveld

refinement but has little to no use during full pattern decomposition because

only observed Bragg intensities are meaningful in both Pawley and Le Bail

methods):

Crystal structure solution

The expected profile residual,

%,,:

The goodness of fit,

(sometimes referred as chi-squared):

Eqs. 6.18 to 6.22, the following notations have been used:

is the total number of points measured in the experimental powder

diffraction pattern.

Y/'~Ys the observed intensity of the

ith

data point, i.e. the left hand side of

the

ith

equation in Eq. 6.6.

yiC"lC is the calculated intensity of the

ith

data point, i.e. the right hand side

of the

ith

equation in Eq. 6.6.

is the weight of the

ith

data point, which is usually taken as

110:

11~i"~"see Eq. 3.8 in section 3.7.1 for the definition of spread, oi).

is the number of independent Bragg reflections.

40bYs the "observed" integrated intensity of the

jth

Bragg peak, which has

been calculated after the observed intensity of every data point in the

powder

diffraction

profile has been partitioned according to the

calculated intensities of the contributing Bragg peaks (see Eq. 6.7).

J~.""'" is the calculated integrated intensity of the

jth

Bragg peak.

is the number of free least squares parameters.

All figures of merit except one (Eq. 6.20) include a contribution from the

background. This raises an interesting problem about the "reliability of

profile reliability factors" (pardon the tautology) when the correctness of the

structural model is of concern, which will be further discussed in Chapter 7.

A simple analysis of Eqs. 6.18 to 6.20 and 6.22 indicates that a better fit

results in lower values of all residuals. Unfortunately, there are no exact

thresholds for Rp, R, andlor RB, below which a fit is acceptable, good or

excellent. To a certain degree, the "absolute quality" of the result is

established by the relationship between R, and kXp, i.e. by the value of

x2.

The expected residual (Eq. 6.21) characterizes the quality of experimental

data because a larger denominator means a better counting statistics (see

Chapter

section 3.7.1). The last figure of merit

the goodness of fit

should

therefore, approach unity when

approaches

&,,.

practice, this is

rarely achieved even with high-quality powder diffraction data when

are

quite low, which is usually due to the inadequacy of peak shape and

background functions.

Another useful figure of merit is the Durbin-Watson d-statistic:

where

yiObS

yiCalc

and

is the corresponding statistical error (see Eq.

3.8).

According to Hill and Flack,' the weighted Durbin-Watson statistic

may be used to provide a sensitive measure of refinement progress and to

quantify serial correlation between adjacent least squares residuals in a full

pattern decomposition or Rietveld refinement based on step-scan

experimental data. It remains discriminating when other residuals fail,

e.g.

when comparing the results obtained at different step sizes. Ideally, the d-

parameter (Eq. 6.23) should be close to 2, thus indicating the absence of

serial correlation, although in practice it often deviates considerably

from

this ideal value.

general, if correlation is insignificant, d is closer to 2 than

another parameter,

which is defined as:

where

and

are the number of experimental observations and free least

squares parameters, respectively. When

2 serial correlation is

positive and sequential observations have the same sign, while in the case of

serial correlation is negative and sequential observations have

opposite signs.2

R.J. Hill and H.D. Flack, The use of the Durbin-Watson d-statistic in Rietveld analysis,

Appl. Cryst.

20,

356

(1987).

Ideally, experimental data (intensity) should be randomly distributed both above and

below the calculated intensity profile.

there are multiple sequences with all observed

points above or all below the calculated intensity values, it is said that serial correlation

occurs.

other words, the d-statistic reflects correlation between adjacent least-squares

residuals and it can be used as an indicator that refined parameters are unbiased.

Crystal structure solution

515

6.8

Structure solution from powder data

In the remainder of this chapter, we will consider multiple practical

structure solution examples. For the most part, individual intensities and

structure factors will be extracted by using Le Bail's algorithm of full pattern

decomposition. This technique is chosen instead of

Pawley's approach

because the former algorithm is usually more stable and it has been

incorporated into several freely available software programs, which are

coupled with Patterson and Fourier calculations. These are: LHPM-Rietical

and GSAS,' although other well-developed and tested computer codes are

a~ailable.~

some of the more complex examples, however, we will employ

manual profile fitting. The latter approach is less "user-friendly" in terms of

automation, but it avoids the unrestricted and sometimes unrealistic

determination of the intensities of

Ka,

and

Ka2

peaks.

By solving crystal structures of different classes of

material^,^

we will

illustrate only a few of the possible approaches to the

ab initio

structure

solution from powder diffraction data. Whenever possible the structure

factors obtained from full pattern decompositions should be used until the

coordinates of all atoms are established.

some cases it may be necessary

to re-determine individual structure factors based on the nearly completed

structural model, especially when locations of lightly scattering atoms are of

concern after all strongly scattering species have been correctly positioned in

the unit cell. This re-determination may be routinely performed during

Rietveld refinement and

will

be briefly discussed in Chapter

LHPM-Rietica (authors B.A. Hunter and C.J. Howard) may be downloaded from

ftp:Nftp.ansto.gov.au/pub/physics/neutron/rietveld/RieticaLHPM95/

via

a link at

http://www.ccp14.ac.uk.

GSAS (authors A.C. Larsen and R.B. Von Dreele) may be downloaded from

http://public.lanl.gov/gsas/

via

a link at http://www.ccpl4.ac.uk. A convenient graphic

user interface for GSAS (author Brian Toby) may be downloaded from

http://www.ncnr.nist.gov/programs/crystallography/

via

http://www.ccpl4.ac.uk.

One of these is EXPO

an integrated package for full pattern decomposition and for

solving crystal structure by direct methods (authors

Altomare, B. Carrozzini, G.

Cascarano, C. Giacovazzo, A. Guagliardi, A.G.G. Moliterni, R. Rizzi, M.C. Burla, G.

Polidori, and M. Camalli). For information on how to obtain the program, contact

sirware@area.ba.cnr.it or visit

http://www.irmec.ba.cnr.it/.

There is a variety of software that can be used to process "pseudo-single crystal"

experimental diffraction data represented in a form of individual structure factors or their

squares (consult http://www.iucr.org and/or

http://www.ccpl4.ac.uk).

The most commonly

used software products are SHELXS-90 and SHELXL-97 (author G.M. Sheldrick), which

are distributed free for academic use (consult SHELX home page at

http://shelx.uni-

ac.gwdg.de/SHELX/index.html

for details). Unless noted otherwise, processing of the

individual structure factor data, direct phase determination, Patterson-, Fourier- and E-map

calculations shown in this chapter were performed using WinCSD software, which is

available from STOE

Cie, Gmbh

(http://www.stoe.com/products/index.htm).

5 16 Chapter

6.9

Crystal structure

LaNi4.85Sn0.151

Consider a powder diffraction pattern of LaNi4.85Sno.ls, which is shown in

Figure

6.4.

Files Ch6ExOl-CuKa.dat and Ch6ExOl'-CuKa.xy, which

contain the same experimental data saved in two different formats, are found

on the CD accompanying this book. From Chapter 5 we already known that

the alloy is hexagonal and the lattice parameters are a

5.0421, c

4.0 1 18 A.

Analysis of Table

5.6

indicates that there are no forbidden

reflections, and from Table

2.14

and Table

2.15

the possible space group

symmetries for this material are as follows: P6/mmm, ~6m2, ~62m, P6mm,

P622, P6/m, ~6, P6, ~51m, P31m, P312, ~?;ml, P3m1, P321, ~5, and P3.

The fact that a total of 16

(!)

space groups are possible for this material

could make it a complicated choice for the ab

initio

structure solution.

However, its unit cell is quite small. Furthermore, LaNi4.s5Sno.ls is an

intermetallic compound and therefore, the highest symmetry space group

(P61mmm) is quite probable.

The measured gravimetric density of the alloy is 8.21 g/cm3. After

calculating the mass of its formula unit,

441.4 a.m.u., and the unit cell

volume,

d3a2c/2

88.327 A3,

Eq.

6.4 results in

0.99

1. Thus, one

unit cell of the compound contains 1 La, 4.85 Ni and 0.15 Sn atoms. Because

of the fractional amounts in the unit cell, the Ni and Sn atoms are likely

distributed statistically in their respective crystallographic sites (they have

similar atomic radii:

r~i

1.35, rs,

1.45

A),

but much larger La atoms (rL,

1.95

should occupy a separate crystallographic site. Thus, before

proceeding with the full pattern decomposition it is advisable to check

whether some of the space groups that are possible for this compound can be

eliminated.2 Analysis of the multiplicities of site positions of the 16 possible

space groups indicates that none of the groups can be excluded outright

because every space group symmetry has I-, 2- and 3-fold sites (some of the

trigonal space groups have only 1- and 3-fold sites), which are sufficient to

accommodate one La and a total of five Ni+Sn atoms.

Ting,

I.E.

Anderson, and

V.K.

Pecharsky, unpublished.

Generally, it is unnecessary to have the exact space group symmetry information during

the full pattern decomposition. In fact, it is more practical to select the most symmetrical

space group in the established diffraction group and avoid multiple and complete Bragg

peak overlaps. For example when both trigonal and hexagonal symmetry (primitive lattice)

groups are possible, full pattern decomposition is best performed in the hexagonal crystal

system assuming Laue class

Glmrnm. This eliminates Bragg doublets (e.g.

hkl

and

khl,

which are present in the Laue class Glm) and quadruplets (e.g.

hkl, hk

khl

and

which

should be taken into account in the Laue class

3),

which are indistinguishable using both

Le Bail and Pawley techniques. For structure solution, it may be necessary to add the

missing combinations of indices to the array of structure amplitudes obtained in the

highest symmetry space group.

Crystal structure solution

517

80 100 120

Bragg angle,

(deg.)

Figure

6.4.

The powder diffraction pattern collected from a sample of LaNi,,8SSno,Is using

Cu Ka radiation on a Rigaku

TTRAX

rotating anode diffractometer. The divergence slit was

0.75' and the receiving slit was 0.03O. The experiment was carried out in a continuous

scanning mode with a rate 0.5

deglmin and with a sampling step 0.02'. The powder used in

this experiment was prepared by gas atomization from the melt and therefore, particles were

nearly spherical (see inset in

Figure

3.32).

this example,' we will use LHPM-Rietica. For readers that have no

experience with the program, we suggest using both the manual, which is

available for download with the software, and a Web-based tutorial2 in

combination with the experimental data found on the

CD.

Ideally, the results

presented below should be reproduced nearly exactly, although small

deviations may occur due to the nature of the non-linear least squares.

Full pattern decomposition usually begins with the refinement of the

background while keeping peak shape and instrumental parameters fixed at

their default values, and unit cell dimensions fixed at their best-known

values. We begin with only two parameters representing the background as a

Due to its simplicity, this example is purely academic, however, it is an excellent sample

to gain initial experience in deconvoluting powder diffraction patterns using Le Bail's

technique. Therefore, it will be considered in detail.

Brett Hunter, LHPM-Rietica Rietveld for Win951NT. Tutorial is accessible

via

http://www.ccpl4.ac.uk/,

then "Tutorials" on the site map, then "LHPM-

Rietica-Rietveld".

Chapter 6

straight line, which is sufficient as a first approximation (see Figure

6.4).

Peak shapes are represented using a pseudo-Voigt function.' The progression

of the full pattern decomposition is illustrated by the corresponding figures

of merit, which are assembled in Table 6.2.

Initial refinement of just two background parameters serves an important

function: it enables preliminary determination of the individual integrated

intensities, which are initially set to unity (see section

6.5),

and the

calculated diffraction pattern begins to resemble the observed diffraction

data (Figure 6.5). When this initial refinement step is completed and the

refinement appears stable, it is possible to begin releasing other relevant

parameters. The refinement has been performed in the order shown in Table

6.2; the results are illustrated in Figure 6.6 to Figure 6.11 and in Table 6.3.

The initial refinement of a linear background (Figure 6.5) is far from an

ideal fit. Although the unit cell dimensions are quite precise, we know that

the experimental data are affected by a small systematic error in Bragg

angles (see Table 5.6, which indicated a zero shift of

0.032' or an equivalent

sample displacement error). Furthermore, peak shape parameters have been

set at their defaults, which may be unsuitable to describe the observed profile

quite well. A relatively poor fit is reflected in high residuals as seen in row

two of Table 6.2. When major peak shape

(U,

and

qo)

and lattice

parameters were refined along with zero shift, the fit improves considerably

as seen in rows 3 and

in Table 6.2 and visualized in Figure 6.6 and Figure

6.7, respectively. For such a simple pattern, the order in which these

parameters have been refined is practically irrelevant, but for the cases that

are more complex it may be necessary to examine the differences,

Zalc,

and to decide which set of parameters is more likely to have the largest

effect on the improved fit: they should be refined first.

Table 6.2.

Figures of merit obtained at different stages during the full pattern decomposition

of the powder diffraction pattern of LaNi4,85Sno.ls using Le Bail approach incorporated in

LHPM-Rietica. Wavelengths used:

hKa,

S4O59

hKaz

1 S4441

Re,,

5.24%.

Refined parameters Illustration

Rw;

Experimental data

Figure

6.4

Background (linear)

Figure 6.5

22.75 28.25 29.02

Figure 6.6

19.40 24.13 21.20

zero shift

Figure

6.7

7.75 12.23 5.44

Background (3'd order), asymmetry

Figure 6.8

5.61 9.16 3.06

All, plus broader base width

Fiwre 6.10

4.82 8.31 2.52

second ~hase. all ~arameters

~Lre 6.11

4.38 6.47 1.53

Here and in all other similar tables, the residuals are listed in percent and

dimensionless (see Eqs. 6.18 to 6.20 and 6.22).

Asymmetry was treated in the approximation given

by:

C.J. Howard, The approximation

of asymmetric neutron powder diffraction peaks by sums of Gaussians,

Appl. Cryst.

15,

615 (1982). See also Eq. 2.63.

Crystal structure solution

519

LaNi

,,,,

,,,,,

20 40 60 80 100 120

Bragg angle,

(deg.)

Figure

6.5.

The observed (circles) and calculated (lines) powder patterns of LaNi4.s5Sno,15

after a linear background has been refined. Peak shape parameters were kept at their default

values and lattice parameters were fixed at

5.0421,

4.01 18

The vertical bars located

just below the background level indicate calculated positions of Bragg peaks for

Ka,.

The

curve in the bottom part of the plot represents the difference

qobS

Y,Ca''.

me scales for the

observed, calculated and difference plots are identical. The inset clarifies details in the

vicinity of the strongest Bragg peak. Same notations are maintained in all similar figures

below and throughout the book.

this example, lattice parameters and the zero shift correction have a

substantial impact on the quality of the fit and the weighted profile residual,

R,,

decreases nearly two-fold (from

-24

-12

%),

while the refinement of

peak shape parameters decreases

by only

-4

Therefore, in this case

lattice parameters should have been refined first. However, it is not always

obvious which parameters are more important and should be released at a

particular stage of the least squares refinement. Because of this, in complex

cases a trial-and-error approach is often employed.'

specific of the Le Bail's approach nearly always requires that early refinement includes

the simplest suitable background function with all other relevant parameters kept fixed at

their default or approximately known values. This ensures the proper determination of the

individual integrated intensities, which are initially set to identical values, and the overall

success of the full pattern decomposition.

Chapter

LaNi

,,,,

,,,,,

20 40 60 80 100 120

Bragg angle,

(deg.)

Figure

6.6.

The observed and calculated powder diffraction patterns of LaNi4,85Sno.ls after

peak shape parameters,

Wand qo, were refined together with linear background.

Bragg angle,

(deg.)

Figure

6.7.

The observed and calculated powder diffraction patterns of LaNi4,85Sno

after

zero shift and lattice parameters were included in the refinement.

Crystal structure solution

The result illustrated in Figure 6.7 shows a satisfactory fit and the next

groups of parameters included in the refinement were: a more complex

background (third order polynomial instead of a linear function) and peak

asymmetry. The fit further improves and when the calculated and observed

profiles match quite well

(i.e.

Y?"

approach

0),

as shown in Figure

6.8, it is useful to examine the bases of the strongest Bragg peaks to establish

whether or not the selected peak range is adequate. The peak width at the

base was

FWHM's so far (default) and, as seen in Figure 6.9, it is too

narrow. Therefore, from this point on it was increased to

FWHM's, which

further improves the fit (Figure

6.10

and row six in Table

6.2).

The proper

selection of peak widths at the base is best established visually, i.e. it should

be gradually increased until "steps", which are indicated by arrows in Figure

6.9 become indistinguishable. While it is possible to select an even larger

peak width at the base, it is always better to choose a realistic minimum

value because as the complexity of the pattern increases, the excessive

widths of peaks may have a deleterious effect on the background function,

especially at high Bragg angles.

LaNi,,,,Sn,,,,,

R,,

9.16

20 40 60 80 100 120

Bragg angle,

(deg.)

Figure

6.8.

The observed and calculated powder diffraction patterns of LaNi4,85Sno.15 after the

polynomial background and asymmetry were refined together with lattice parameters, zero

shift and peak shape parameters

(U, V,

Wand qo).