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

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612

Chapter

low.' The corresponding plot of the observed and calculated powder

diffraction data is shown in

Figure

7.2,

from which it is obvious that the

intensities of nearly all observed Bragg peaks are lower than those of the

calculated reflections. As follows from

Eq.

7.4,

this is indeed the effect of an

overestimated scale factor. Thus, one of the critical parameters, which

should be determined at the beginning of every Rietveld refinement, is the

scale factor,

(also see

Table

7.1).

Several refinement cycles (all variables

except

are fixed) are usually sufficient to reach convergence (row

Table

7.3)

since in this case the least squares procedure, in fact, becomes linear.2

20 30 40 50 60 70 80 90 100 110 120

Bragg angle,

(deg.)

Figure 7.2.

The observed (open circles) and calculated (solid line) diffraction patterns of

LaNi4,85Sno.ls. Scattered intensity was calculated using instrumental and lattice parameters

determined during Le Bail's refinement and default scale factor

0.01). The difference,

~i~~

yicalc,

is displaced by -5,000 counts for clarity. Vertical bars in the lower part of the

figure indicate calculated positions of the Kal components of Bragg reflections in

LaNi, 8sSno,15. The data were collected from nearly spherical powder produced by high

pressure gas atomization (see

Figure

3.32) on a Rigaku TTRAX rotating anode powder

diffractometer using

Ka radiation.

This is usually not the case in the majority of Rietveld refinements because the default

value of the scale factor selected in LHPM-Rietica

0.01) is arbitrary. For example,

see

Table

7.5,

below, where initial residuals are much higher due to the inadequacy of the

default phase scale.

One cycle is usually enough to determine

additional cycles may be needed for a proper

determination of the intensities of overlapped Bragg peaks.

Crystal structure refinement

613

LaNi

,.,,

,,,,,

10.35%

R,,

13.21%

9.30%

6.36

20 30 40 50 60 70 80 90 100 110 120

Bragg angle,

(deg.)

Figure

7.3.

The observed and calculated diffraction patterns of LaNi4,85Sno.15. The scattered

intensity was calculated using scale factor, instrumental and lattice parameters determined

during Rietveld refinement, and guessed overall atomic displacement parameter

0.5

A'.

All notations are identical to

Figure

7.2.

Under normal circumstances,' the scale factor is one of the few least

squares parameters, which is always kept as a free variable in a Rietveld

refinement. Since in this particular example, all profile and lattice

parameters have been refined using Le Bail's approach, the next reasonable

step is to release all associated variables before proceeding with individual

atomic parameters (the scale factor remains a least squares free

~ariable).~

As expected, the resultant change in the figures of merit, listed in third row

Table

7.3,

is small because the background, peak shape and lattice

parameters are already quite accurate.

In rare cases, e.g. when all sites in the crystal structure are partially occupied, the scale

factor may become strongly correlated with the population parameters. This requires

special attention and detailed consideration of all possibilities exceeds the scope of this

text. One of the options is to use the known scale factor as a fixed parameter while refining

population parameters of all crystallographic sites. This option is, however, seldom

available because relative and not absolute intensities are customarily measured in a

powder diffraction experiment.

In this structure, only individual displacement and population parameters can be refined.

614

Chapter

7.3.2 Overall atomic displacement parameter

So far, the only "structural" parameter included in the refinement was the

scale factor. This particular crystal structure has no free coordinates of

individual atoms

all are fixed by the symmetry of the occupied special

positions. Thus, given the relatively low values of all residuals and recalling

that at the beginning we assumed an arbitrary value for the overall atomic

displacement parameter (B

0.5 A2), it is time to include it into the least

squares minimization. The inconsistency of the randomly chosen overall

atomic displacement parameter is also easily distinguishable fi-om the plot of

the observed and calculated intensities, which is shown in

Figure

7.3. The

intensities of the majority of Bragg peaks observed at low angles exceed

calculated intensities but at high angles (20

-65") the relationships become

just the opposite. Given the effect of the atomic displacement parameters on

the integrated intensity as a function of Bragg angle (see

Eq.

2.91), it is easy

to conclude that the actual overall B is considerably higher than 0.5

A2.

The refinement of the overall B results in a significant reduction of all

residuals (fourth line in

Table

7.3), which is also expected since now we

have a realistic global estimate of thermal motions of atoms in the crystal

lattice of LaNi4,s5Sno.15. The refined value of the overall atomic displacement

parameter is B

1.32(2) A2, where the number in parenthesis indicates a

standard deviation in the last significant digit.

7.3.3 Individual parameters, free and constrained variables

At this point, we may begin to refine atomic displacement parameters of

the individual atoms. This is done by substituting individual B's (which were

kept at

0) by the refined overall atomic displacement parameter.

The overall

after this substitution should be set to

The distribution of atoms in the

model of the crystal structure of LaNi4,~5Sn~.15 is such that two of the sites,

2(c) and 3(g), see

Table

7.2, are occupied by different types of atoms

simultaneously. Generally, atomic displacement parameters of different

atoms occupying the same crystallographic site should be constrained at

identical values.'

LHPM-Rietica and almost every other commonly available software

product, any dependent parameter, Pdependenr, may be constrained to any free

This is a reasonable simplification especially because it is nearly impossible to

differentiate properly between the vibrational motions of different atoms located in the

identical environment. Furthermore, individual displacement parameters of atoms

occupying the same crystallographic sites have a tendency to strong correlation, thus

causing severe instabilities of non-linear least squares. If a site has variable coordinate

parameters, the latter should be constrained equal for all atoms occupying the site due to

the same reasons.

Crystal structure

refinement

615

least squares parameter,

PPee.

The differentiation of the corresponding linear

relationship between the two variables results in the numerical constraint,

which is employed during the calculation of the normal equations matrix in

the least squares and is applied to the shift obtained for the free parameter

before adding it to the constrained variable:

Considering Eq. 7.9 and assuming than BNil

BSnl and BNi2

BSn2, the

proper constraints will be ABsnl

x&il and ABs~

xABNi2. This implies

that individual isotropic atomic displacement parameters of Ni atoms are

maintained as free least squares variables and the same for Sn atoms are

taken as constrained parameters.' The input file,

Ch'lExOlb.inp,

with all up-

to-the-point parameters refined and properly constrained is found on the

(consult both the LHPM-Rietica manual and the on-line tutorial2 for details

on how to introduce constraints into the input file).

The resulting refinement further reduces the residuals (see row five in

Table

7.3)

and yields the following distribution of the individual atomic

displacement parameters:

It is obvious that atoms in the 2(c) site have much larger atomic

displacement parameters than identical atoms in the

3(g) site. This situation

is quite unusual for a simple intermetallic compound and likely indicates that

our assumption about a statistical distribution of Ni and Sn in both

crystallographic sites was incorrect. The enhanced isotropic atomic

displacement parameter in the 2(c) site points to a lower scattering ability,

while the reduced atomic displacement parameter in the

3(g)

site points to

higher scattering factor when compared to the current distribution of atoms.

Indeed, we may speculate that only 3(g) sites contain Sn atoms, which have

greater scattering ability than Ni atoms. Another possibility is that the 2(c)

In the hexagonal crystal system,

and

unit cell parameters are constrained by symmetry.

b).

In LHPM-Rietica, lattice parameters are constrained automatically for all crystal

systems.

Brett Hunter, LHPM-Rietica Rietveld for Win95/NT. Tutorial is accessible

via

http:llwww.ccpl4.ac.uW, then "Tutorials" on the site map, then

"LHPM-

Rietica-Rietveld".

616

Chapter

sites are depleted in Sn, while some Sn atoms are still located there, and the

3(g)

sites are enriched in Sn.

When the precision of x-ray diffraction data is high, which appears to be

the case here, it is possible to refine the population of different

crystallographic sites to eliminate guesses and obtain a quantitative result.

The best way to do so is to return to the overall displacement parameter and

instead of refining individual atomic displacement parameters, include the

refinement of the individual population parameters in the corresponding

sites.'

Assuming

full occupancy of all sites, i.e.

gNil

gslll

and

g~i2

gsn2,

the corresponding constraints

(Eq.

7.9)

should be set at Ansnl

-1

xAnNil and

Ansn2.

-lxAnNi2. The file

Ch7ExOlc.inp,

in which all parameters are

properly constrained, is found on the

CD.

This Rietveld refinement step

results in the following occupancies of the two sites in question:

The negative occupancy by Sn of the

2(c) site has no physical sense,

especially given that the absolute value of the occupancy is comparable with

the standard deviation. Thus, this site appears to be pure Ni. On the other

hand, it is confirmed by the refinement that all Sn is segregated in the 3(g)

sites. It is worth noting that when the chemical composition of the forn~ula

unit is calculated from the refined occupancies, the result is

LaNi4.83c2,Sno.17(2), which matches the as-prepared chemical composition of

the material within one standard de~iation.~

We note that since one of the sites, l(a) seems to be fully occupied by La, the least squares

refinement of the population parameters of the two remaining sites, 2(c) and 3(g) may be

carried out together with the scale factor. Only when the population of all sites is in

question, special precautions should be taken to avoid severe correlation between the scale

factor and all population parameters. When all sites are occupied partially, diffraction data

alone normally do not provide an adequate answer because both

and are simple

multipliers, which affect the scattered intensity. Other experimental methods should be

employed to establish andlor prove that defects exist on all lattice sites. One of these are

precise gravimetric density data.

Refinement of the crystal structure is, therefore, a powerful chemical analysis technique.

Unlike conventional chemical analysis, which only yields the bulk composition of the

sample, powder diffraction analysis facilitates accurate determination of the occupancies

of different crystallographic sites by various chemical elements, or in other words,

establishes precise chemical composition of the crystal at the atomic resolution. It should

be noted that the results may be considered reliable only when the difference in the

scattering ability of atoms in question is significant, in addition to a very high quality of

experimental data. This is indeed the case here because scattering factors of Sn and Ni are

related as

-1.8:

Crystal structure refinement 617

Proceeding with the refinement of the individual atomic displacement

parameters after removing Sn atoms from the 2(c) site and setting its

occupancy to

n~i~

0.08333 (the file

Ch7ExOld.inp

is located on the CD),

we find that the individual isotropic atomic displacement parameters of all

atoms become much closer to one another. The resultant residuals are listed

in row seven of Table 7.3.

7.3.4 Anisotropic atomic displacement parameters

The only remaining degree of freedom in this crystal structure is to refine

the displacement parameters of all atoms in the anisotropic approximation

(the presence of preferred orientation is quite unlikely since the used powder

was spherical and we leave it up to the reader to verify its absence by trying

to refine the texture using available experimental data). As noted in Chapter

special positions usually mandate certain relationships between the

anisotropic atomic displacement parameters of the corresponding atoms. In

the space group P6/mmm, the relevant constraints are as follows:

After the individual isotropic atomic displacement parameters were

replaced by the properly constrained individual anisotropic displacement

parameters (LHPM-Rietica uses

Pij,

see Eq. 2.93), the refinement converges

to the residuals listed in row

of Table 7.3. The parameters of the fully

refined structure are found in the file

Ch7ExOle.inp

on the CD.

7.3.5 Multiple phase refinement

The presence of a minor second phase impurity can be added either in the

form of the actual structural model of Ni or as a Le Bail's phase, where only

the unit cell and peak shape parameters are taken into account. The latter

option has been chosen since we are not interested in the crystal structure of

this minor impurity, and it may be a difficult task given its small

contribution to the total scattered intensity.

The refinement with both crystalline phases contributing to the

computation of the scattered intensity (row

Table 7.3) converges rapidly

and yields the residuals, which are only slightly higher than those, obtained

when no model of the crystal structure was present during a full pattern

618

Chapter

decomposition (see section

6.9

in Chapter

6).

The input file after the

completion of the refinement

(Ch7ExOlf.inp)

is also included on the

CD.

7.3.6

Refinement results

The final parameters in the crystal structure of the LaNi4.85Sno.15 as

determined fi-om the Rietveld refinement are listed in

Table

7.4.

The refined

model of the crystal structure of the LaNi4,85Sno.ls compound is shown in

Figure

7.4.

All interatomic distances are normal and the atomic

displacement parameters of all atoms show little anisotropy, which is typical

for many intermetallic compounds, structures of which are based on the

close packing of spheres. The resultant observed and calculated diffraction

patterns are plotted in

Figure

7.5

in a standard Rietveld format.

Table

7.4. Structural parameters of the major phase (LaNi4,ssSno.ls), fully refined by the

Rietveld technique employing powder diffraction data collected using Cu

radiation (see

Figure

7.5). The refined unit cell dimensions are:

5.04228(6),

4.01 170(5)

Atom Site

1 04pl

lo4p22 1 04~33b 1 04p12

La l(a)

0 0 1 156(2) 156(2) 258(3) 78(1)

Nil 2(c) 113 213 0 1 227(3) 227(3) 183(5) 1 13(1)

Ni2 3(g) 112

112 0.944(2) 178(3) 174(4) 187(5) 87(2)

Sn 3(g) 112 0 112 0.056(2) 178(3) 174(4) 187(5) 87(2)

The population parameters of the 3(g) site are listed as

i.e. they represent fractional

occupancies by Ni and Sn, refined assuming full overall occupancy of the site.

The following anisotropic displacement parameters are fixed

symmetry for all sites:

Figure

7.4. One unit cell of the crystal structure of LaNi4,85Sno.ls as determined from Rietveld

refinement. The model reflects different distribution of the Ni and Sn atoms between 2(c) and

3(g) sites (dark- and light-grey, respectively). The displacement ellipsoids are shown at 99%

probability. Compare this figure with

Figure

6.14.

Crystal structure refinement

619

LaNi

,,,,

,,,,,

5.85%

R,,

8.07%

3.10%

2.38

20 30 40 50 60 70 80 90 100 110 120

Bragg angle,

(deg.)

Figure

7.5.

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

completion of Rietveld refinement. Notations are identical to

Figure 7.2

and the second set of

vertical bars indicates the calculated positions

the

Ka,

components of Bragg peaks in the

impurity phase (solid solution of Sn in Ni).

7.3.7

Different radiation

The x-ray powder diffraction data were also collected from the same

specimen of LaNi4,85Sno.ls using Mo

radiation. The experimental data are

shown in

Figure

7.6.

One of the most obvious benefits of using higher

energy photons, is the greater volume of the reciprocal lattice, which can be

examined experimentally.

important advantage of including the data

measured at higher sinelh into the Rietveld refinement, is in obtaining more

accurate values of the individual atomic displacement parameters together

with the more precise site populations in this crystal structure. The better

accuracy, when compared to Cu

radiation, becomes obvious from a

simple analysis of Eqs. 2.89 and 2.91, which indicate that the effect of

varying

is independent of the Bragg angle, while the varying

has the

largest influence on the intensity, scattered at high sinO/L. Thus, by including

intensity scattered at high sin91h we reduce the correlation between the

atomic displacement and population parameters.

Chapter

10 20 30

50 60

Bragg angle,

(deg.)

Figure

7.6.

The x-ray powder diffraction pattern of LaNi4,85Sno collected on a Rigaku

TTRAX

rotating anode powder diffractometer using Mo Ka radiation with a step

A20

O.OlO.

The nearly spherical LaNi,8SSno

powder was produced by high pressure gas

atomization. The two short thick vertical arrows show the range of sinelk examined using Cu

Ka radiation (e.g. see

Figure

7.2),

and the long thin vertical arrow indicates the fundamental

upper limit of the reciprocal space

(20

180')

accessible when using Cu

radiation.

Since we did not perform Le Bail's decomposition of this powder

diffraction

pattern, we will illustrate the Rietveld refinement sequence in this

case beginning from all profile parameters selected at their default values

and the same starting model of the crystal structure as was used in the case

of Cu

radiation data. The input file, Ch7ExOlg.inp, with initial variables

and the experimental data file, Ch7ExOl-MoKa.dat, are found on the

CD.

The progression of the refinement is shown in

Table

7.5.

When peak shape and background parameters are unknown, the Rietveld

refinement strategy usually changes. At the beginning (first nine rows in

Table

7.5), major attention has been paid to a sensible refinement of the peak

shape, background and lattice parameters along with the zero shift to achieve

the best possible agreement between the observed and calculated scattered

intensity. Only then, the overall displacement and properly constrained

individual population parameters were included into the refinement. At this

point, the examination of low Bragg angle peak shapes indicated that

Crystal structure reJinement

Howard's asymmetry approximation is not suitable and asymmetry was

changed to a more realistic Finger, Cox and Jephcoat model, as indicated in

the footnotes to

Table

7.5.

Table

7.5.

The progress of Rietveld refinement of the crystal structure of LaNi4.ssSno.,s using

powder diffraction data shown in

Figure

7.6.

Wavelengths used:

hKal

0.70932

hKu2

0.71361

Refined parameters

RD R,D RB

Initial (all default, model from

Table

7.2)"

617.8 808.3 608.7 3x104

Scale factor

Scale factor plus linear background

Scale, linear background plus

U, V,

Scale, linear background,

Wplus qo

Scale, linear background,

W, qo,

and

Scale, linear background,

qo,

plus

asymmetry

Scale, linear background,

qo,

asymmetry,

plus zero shift

Scale,

W, qo,

asymmetry, zero shift, plus

background (third order)

Scale,

W, qo,

asymmetry, zero shift, background

(third order) plus overall

Scale,

qo,

asymmetry, zero shift, background

(third order), overall

plus individual population

parameters

Scale,

qo,

zero shift, background (third

order), overall

individual population parameters.

Asymmetry in

FCJ

approximation

Scale,

qo,

asymmetry, zero shift, background

(third order), individual population parameters plus

individual anisotropic displacement parameters

b,C

Scale, all peak shape, lattice, zero shift, background (third

order), individual population and anisotropic parameters

b,C

All as above plus a second phase (Le Bail's

decom~osition)

b'C

Pseudo-Voigt peak shape function with Howard's asymmetry, see

Eq.

2.63.

Pseudo-Voigt peak shape function. Asymmetry was changed to Finger, Cox and Jephcoat

approximation, which better represents peak shapes measured on this powder

diffractometer using Mo

radiation, see section 6.10.

Population of the 2(c) site was set to pure Ni, while the population of the 3(g) site

remained free parameter.

Since the quality of this powder diffkaction pattern is excellent, we can

immediately switch to the refinement of the individual anisotropic

parameters. Furthermore, the presence of diffraction data at high

sinelh

enables us to refine the individual displacement and population

in the

3(g)

site simultaneously. Finally, the presence of an impurity was also

accounted as the "Le Bail's'' phase. The fully refined input data file,