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

Подождите немного. Документ загружается.

442

Chapter

From the four possible triplets of resulting vectors (vl, v2,

to VZ, vj, v4),

the one that has shortest vectors is selected because the angles among these

vectors are closest to

90•‹. The Delaunay-Ito reduced primitive cells can be

classified into 24 types according to the relationships between unit cell

vectors and their scalar products. They are easily converted into one of the

14 Bravais lattices. For example, if

and

s13

S23

(al3

a23

90•‹),

the standard unit cell is orthorhombic C-centered, with lattice vectors, vioTfho,

calculated as follows:

vI)ltho

ortho

=-v1 +v2

The unit cell reduction using Delaunay-Ito method can be easily

automated as is done in the IT0 indexing computer code, which is discussed

in section 5.11. The Delaunay-Ito reduced unit cell, however, may not be the

one with the shortest possible vectors, although the latter is conventionally

defined as a standard reduced unit cell.

5.10.2

Niggli reduction

The Niggli approach defines the reduced unit cell in terms of the shortest

possible vectors.'

other words:

minimum

(5.26)

Unfortunately, this simple condition cannot be evaluated directly, and in

order to find a properly reduced unit cell it should be replaced by a total of

five simultaneous ineq~alities:~

P.M. de Wolff,

Gruber, Niggli lattice characters: Definition and graphical

representation, Acta Cryst.

A47,29

Zuo,

Muller,

M.-J.

Phillippe, and G. Esling, A refined algorithm for the reduced-cell

determination, Acta Cryst.

A51, 943 (1995).

Unit cell determination and rejnement

443

All angles between lattice vectors are then obtuse or are acute. The unit

cell defined by Eq.

5.27

is unique but the reduction algorithm is not as

straightforward as in the Delaunay-Ito method.' It is worth mentioning than

if all angles are obtuse, the Niggli unit cell coincides with the Delaunay-Ito

unit cell. When all angles are acute, the Delaunay-Ito unit cell contains only

two of the shortest vectors and the third vector is one of the diagonals in the

Niggli unit cell.

a similar way as Delaunay-Ito unit cells, the Niggli cells

form 44 classes (called characters) and each class is related to one of the 14

Bravais lattices.

Regardless of which indexing method was employed, the resulting unit

cell (especially when it is triclinic) shall be reduced using either

Delaunay-

Ito or Niggli method in order to enable the comparison of different solutions

and to facilitate database and literature searches. Furthermore, the

relationships between reduced unit cell parameters must be used to properly

determine the Bravais lattice. The Niggli-reduced cell is considered standard

and therefore, is preferable.

It is worth noting that not all automatic indexing computer codes perform

unit cell reduction and even when they do, the reduced unit cell may not be

standard. Therefore, it is advisable to always

checklreduce the obtained unit

cell using specialized software, for example the WLepage pr~gram,~ which

is available

on-line.3 This program not only reduces the unit cell but also

performs subcell and supercell search, which allows one to detect larger unit

cells with superlattice reflection extinctions, etc. This program is included

into the

ChekCell ~oftware,~ which also compares all possible unit cells

against the list of the observed Bragg peaks and evaluates the results in easy-

to-use graphical format.

5.11

Automatic

initio

indexing: computer codes

variety of first principles indexing methods has been implemented in

many different computer programs with various modifications. For example,

a total of eight different indexing routines are combined into a single suite

Kfiv);, B. Gruber, A unified algorithm for determining the reduced (Niggli) cell, Acta

Cryst. A32,297 (1976).

A.L. Spek, LEPAGE

an MS-DOS program for the determination of the metrical

symmetry of a translation lattice,

Appl. Cryst. 21, 578 (1988).

Available at

http://www.ccp14.ac.uWtutorial/lmgp/chekcell~lepage.html

Available at

http://www.ccpl4.ac.uk/tutorial/lmgp/index.html#chekcell.

LMGP suite of

programs for the interpretation of X-ray experiments, by Jean Laugier and Bernard Bochu,

ENSPILaboratoire des MatCriaux et du GCnie Physique, BP 46. 38042 Saint Martin

d'Hkes, France.

WWW:

http://www.inpg.fr/LMGP and http://www.ccpl4.ac.uk/

tutorial/lmgp/.

444 Chapter

called CrysFire' that can be downloaded from the International Union of

Crystallography or Collaborative Computational Project No.

Web sites

along with many other indexing programs.* According to its developer,

CrysFire is designed for a non-specialist but may be also useful for experts.

Another software suite is LMGP (see footnote No. 4 on page 443). It is

available with

tutorial and handles peak search, profile fitting, unit cell

refinement, and space group determination/evaluation in addition to

indexing. CrysFire, which lacks a graphical user interface, may be used

concurrently with LMGP and

ChekCell, assuming that there are some trial

unit cells fiom CrysFire in the output file.

Here we consider several of the most commonly used automatic indexing

programs: TREOR, DICVOL, and

IT0 in an attempt to make a novice

familiar with the

initio indexing capabilities and limitations in addition to

illustrating practical indexing examples. The following four sections are not

intended and should not be taken as substitutes for manuals to any of these

indexing utilities

each has a detailed manual describing format of input

data. Complete descriptions of indexing algorithms can be found in the

corresponding original references provided below.

All three computer codes (TREOR, DICVOL, and ITO) have been

extensively tested by hundreds of researchers. They have undergone multiple

revisions by both the original developers and experienced crystallographers,

and therefore, are quite reliable, provided adequate quality powder

diffraction data are employed. As already noted above, the following are the

keys to successful indexing from first principles: precision, precision, and

precision.

TREOR90 is a semi-exhaustive trial-and-error indexing program, which

is based on the permutation of indices in a selected basis set of lowest Bragg

angles

TREOR90 includes an analysis of the dominant axial zones

(i.e. hOO,

OkO

and 001).

the case of a monoclinic crystal system, the so-

R. Shirley, The CRYSFIRE System for automatic powder indexing: user's manual, The

Lattice Press,

Guildford Park Avenue, Guildford, Surrey GU2 7NL, England (2000).

Available

http://www.ccp14.ac.uk/tutorial/crys/index.html.

Available at

http:llwww.ccpl4.ac.uWsolution/indexing/.

P.-E. Werner,

Eriksson and

Westdahl, A semi-exhaustive trial-and-error powder

indexing program for all symmetries,

Appl. Cryst.

18,

367 (1985).

The algorithm is called semi-exhaustive because certain limitations on the possible

permutations

indices are incorporated into the program in order to increase its speed.

The numeral included after the name of the program (in this case 90) usually indicates

version number of year when the program was released as a public domain (in this case the

program was released in 1990).

Unit cell determination and refinement

445

called short-axis test and (020)-finding algorithm are employed to increase

chances of indexing when unit cell edges have unusually different lengths.

This program works with any crystal system but it is most effective in high

and medium symmetries: cubic, hexagonal, tetragonal and orthorhombic.

Triclinic indexing employs lower index limits and is exceedingly time

consuming. The basis set consists of up to six reflections, and the peaks

suspected to represent collinear reciprocal lattice vectors are automatically

excluded. The default combinations of peaks in the basis sets, maximum

values of indices and sums of indices are listed in

Table

5.20.

Table

5.20.

Order of peaks in basis sets and limitations imposed on the values of indices for

the basis-set peaks used in TREOR.

Crystal system

Basis set (in terms of peak number)

max

h,k,l

max

Cubic

Tetragonal

Hexagonal

Orthorhombic

Monoclinic

Triclinic

(12>3)> (1,2,4), (1 >2,5)>

(1,3,4), (2,3,4), (1,2,6)

(N,

A, (NJ ,3), (N,2,31b

(1,2,3,4), (1,2,3,5), (1,2,4,5)

(N,1,2,3)

1 A4), (N,

,3,4Ib

and additionally

(1,3,4,5), (1,2,3,6), (2,3,4,5), (1,2,3,7)'

(1,2,3,4,5,6), (1,2,3,4,5,7), (1,2,3,4,5,8),

(1,2,3,4,6,7), (1,2,3,4,6,8), (1,2,3,5,6,8),

(1,2,3,5,6,7), (1,2,3,5,7,8), (1,2,3,4,5,9)d

1,191 3

Absolute value of

h, k,

and

is assumed. Max

values are provided for each basis

set reflection, respectively.

is defined as a non-zero value in the keyword SELECT, e.g. in the orthorhombic crystal

system SELECT=5 results in the following basis sets

(1,2,5), (1,3,5),

and

(2,3,5).

Additional sets that are added to the basis set when MONOSET keyword value is greater

than

3,4,5

respectively.

The

IS'

observed peak is always

100

and the

2"d

observed peak always has positive indices.

The limiting values of indices may be freely modified by the user as

described in TREOR's manual, while the sequence of peaks in the basis set

may only be changed to a certain extent as described in the footnotes to

Table

5.20.

the majority of cases, program defaults work quite well and

any changes in them should be attempted only when no acceptable solution

has been found.

Although no strict limits have been set, usually no more than

lowest

Bragg angle peaks total should be used in the indexing attempt. TREOR90

446

Chapter

may successfully index powder diffraction patterns containing a few

impurity peaks by automatically skipping up to three Bragg reflections,

which do not fit into a found unit cell. If some peaks were skipped, one

should always remember the very important statement (capitalized in the

manual): do not accept unindexed peaks unless you are able to explain them.

The key to a successful indexing is not a complete absence of impurity

peaks (a few may be present) but it is the accuracy with which peak

positions have been determined and the absence of significant systematic

errors. Yet another important piece of advice, given in the manual, should

always be followed: do not waste computer time on bad data. Since the cost

of computer time continuously lowers, but the cost of labor continuously

rises, this statement could be rephrased: do not waste

your

time on bad data.

The latter is indeed applicable to any type of data analysis.

Several other items should be considered while using TREOR. Indexing

starts from the cubic crystal system and may proceed through all crystal

systems down to triclinic. By default, however, the indexing stops at the

orthorhombic symmetry. Therefore, in order to check monoclinic solutions,

the value of a keyword MONO should be set to the maximum desired

monoclinic angle,

e.g. MONO

130 will result in searching for solutions

with angles

varying from 90 to 130'. Indexing in a triclinic unit cell can be

attempted by including the instruction TRIC

1. Another way of testing all

crystal systems is to specify a negative value of the maximum unit cell

volume using keyword VOL, e.g. VOL

-2000. TREOR reduces the

triclinic unit cell only for the best solution and therefore, numerous non-

reduced variations of the same unit cell may be produced.

When an acceptable solution is found, the program terminates and

therefore, indexing in lower symmetry unit cells will not be performed.

solution is considered acceptable if the minimum figure of merit, MN, where

is the smallest of 20 and the number or present peaks, is reached or

exceeded. It is advisable to check lower symmetry indexing in order to avoid

situations when accidental relationships between the real unit cell parameters

result in successful high symmetry indexing. In the majority of cases, the

incorrect high symmetry unit cell has a large volume that cannot be

explained by the increased symmetry of the lattice. This can be done by

disabling any of the high symmetry crystal systems: for example, cubic

indexing can be suppressed by using the keyword KS

0, and tetragonal and

hexagonal indexing are not attempted when THS

0 (consult the manual).

alternative way of continuing the indexing in low symmetry lattices is

to increase the minimum figure of merit, MN, by adding or changing the

keyword MERIT. If this keyword is missing (default), the minimum MN is

set to

which is quite low given the accuracy of both the modern

diffractometers and data processing algorithms (e.g. see the corresponding

Unit cell determination and rejkement

447

figures of merit described in section 5.5.2). Any solution with

greater

than

is considered to be a true unit cell and the program execution

terminates.

order to continue the search for other possible solutions, the

value of MERIT should be increased, e.g. by setting MERIT

50 or higher.

By default, TREOR adjusts the data for possible systematic errors in the

first seven peaks by using higher order peaks. Sometimes this procedure may

be not quite right, and when no solution has been found, it is recommended

to suppress this correction by adding the instruction IDIV

The successful indexing result (if any)

the one with

greater than a

set minimum

is stored in a condensed output file. All intermediate

solutions with

less than the specified minimum but greater than

and no

more than 3 unindexed lines (unless different limits are specified) are stored

in a general output file. Therefore, using unrealistically high MERIT

1000

and VOL

-2000 would result in checking all crystal systems and unit cells

with volumes smaller than 2000

A3.

5.11.2

DICVOL'

DICVOL91 is an exhaustive trial-and-error indexing program with

variation of parameters by successive dichotomy and partitioning of the unit

cell volume. This program works with all crystal systems, however, low

symmetry monoclinic and especially triclinic indexing may take some time.

The indexing strategy is based on searching for a solution from high to low

symmetry using partitioning of the unit cell volume in

400

increments,

except for a triclinic case, when the maximum volume is estimated from the

density of diffraction peaks in the pattern.

Similar to TREOR, any crystal system can be included or excluded from

the indexing process, and increasing a minimum figure of merit,

(the

default is

5),

can be used to generate low symmetry results. Input data may

also be used to set maximum lengths of the unit cell edges, monoclinic angle

and unit cell volume. Errors in peak positions can be supplied with each

Bragg reflection separately or assumed identical for all peaks. The default is

0.03" of

29, which is somewhat excessive for good quality diffi-actometer

data, and the value of 0.01 or 0.02", depending on the resolution, is

recommended. For more details, consult the manual available along with the

program. As a rule, indexing should be performed in three steps by selecting

the following crystal systems: cubic through orthorhombic; monoclinic; and

finally triclinic, if needed.

When powder diffraction data, included in the indexing, contain impurity

Bragg peaks, the likely outcome of this program will be the incorrect unit

Boutlif and

Louer, Indexing of powder diffraction patterns for low symmetry lattices

by the successive dichotomy method,

Appl. Cryst.

24,987

(1991).

448

Chapter

cell as no attempt to skip one or more non-conforming peaks is foreseen.

addition, DICVOL does not perfonn any systematic error correction and if

zero shift, sample displacement or other errors are significant, they should be

minimized beforehand. No formal limits on the amount of input Bragg peaks

has been established in this program, but it is recommended that 20 or more

lowest Bragg angle peaks are used for a reliable indexing. It is worth noting

that if the true unit cell belongs to one of the high symmetry crystal systems

(e.g. tetragonal), an attempt to index the diffraction pattern in lower

symmetry crystal systems (e.g. orthorhombic, monoclinic and triclinic) will

usually result in the solution with

(orthorhombic),

and

90"

(monoclinic) and/or

and

90" (triclinic). Independent unit

cell reduction (see section 5.10) should be employed to test any resulting low

symmetry unit cell in order to compare it with the literature of database

records.

This algorithm realizes a zone search indexing method combined with the

Delaunay-Ito technique (see section 5.10.1) for the reduction of the most

probable unit cell. The most commonly used versions of computer codes are

IT013 and

IT015. The program arrives at a solution by using the following

algorithm:

Finds and reduces potential zones always using 20 lowest Bragg angle

peaks, tests them, and refines by using a least-square method.

Builds lattices by finding zones with a common row, and calculating

angles between them.

Reduces the resultant unit cell by using the Delaunay-Ito method and

then transforms it into one of the 14 Bravais lattices if the lattice is not

primitive.

Finally, attempts to index all available peaks and if the indexing is

successful, calculates the figure(s) of merit.

The program works quite effectively with low symmetry crystal systems:

triclinic, monoclinic and orthorhombic. The resulting centered lattice in the

monoclinic and orthorhombic symmetry (if centering has been found) is an

important argument in accepting such a solution, since it is less likely to be a

random unit cell when compared with any primitive lattice. Lattice centering

analysis is not performed in higher symmetry cases; nevertheless, they are

suggested at the end of the output list and should be carefully analyzed, even

though the list often contains large high symmetry unit cells.

J.W. Visser,

fully automatic program for finding the unit cell from powder data,

Appl.

Cryst.

(1969).

Unit cell determination and rejnement

449

The program requires at least

lowest Bragg angle peaks for finding a

reasonable solution and it will not work with fewer Bragg peaks. A total of

consecutive Bragg reflections are recommended. The maximum

number of peaks is

40.

Peaks in excess of

are indexed using only the best

found solution. Similar to TREOR, IT0 may successfully index powder

diffraction patterns, which contain several impurity peaks. The maximum

number of skipped unindexed peaks is one of the user-specified parameters.

IT0 can check for potential zero-shift errors (by default no checking is

performed), and allows a zero-shift correction that will be applied to all

peaks before indexing. In addition, in the final least squares refinement of

the unit cell dimensions, the zero-shift error in Bragg angles can be

determined (by default).

The program works in a straightforward fashion, and only a few

parameters can be varied in the case of failure. These are: several tolerances,

the number of peaks for zone search, and the zero-shift correction. As a last

resort, low intensity (potential impurity peaks)

and/or high angle peaks (if

they were included in the unsuccessful run) may be removed from the list

one-by-one. The default values for tolerances for two- and three-dimensional

zone search are set to

3.0

and 4.5, respectively, in the units of

which are

defined in this program as

104/d2.

However, it is recommended to set them as

low as

2.0

and

2.5

when the accuracy in peak positions is at or better than

0.01"

28.

The final advice, given by the author

(J.W.

Visser) should

always be remembered: "Finding the unit cell depends for 95% on the

quality of the input data. A random error of

0.03" 28

can usually be

tolerated, but a systematic zero point error of

0.02"

is probably disastrous".

5.11.4

Selecting

solution

TREOR and IT0 programs have an option to input intensity for each

observed diffraction peak but they are included only for informative

purposes and are never analyzed by the code. Nevertheless, it is often helpful

to see the intensity next to indices when analyzing the results, especially for

those Bragg peaks, which do not fit into the selected lattice.

All three programs TREOR, DICVOL and

IT0 allow optional input of

the information about the measured gravimetric density and formula weight

in order to estimate the number of formula units expected in the found unit

cell (see section 6.3 in Chapter 6). The latter should be an integer number

compatible with the unit cell symmetry, e.g. in a primitive monoclinic lattice

it normally should be a multiple of

or 4. The agreement between the

number of formula units in the unit cell and lattice symmetry may be used as

an additional stipulation when selecting the most probable solution.

Chapter

Automatic indexing programs usually produce more than one solution

and the following criteria should be considered when deciding which one is

the best, or in other words, which solution we believe represents the true unit

cell:

1. The correct solution must result in a high figure of merit (although not

necessarily the highest) and most, if not all diffraction peaks should be

indexed. Unindexed Bragg reflections must be explained, for example, by

low intensity, by noticeably different peak widths, or they should be

identified as corresponding to an impurity phase.

As is stated in the DICVOL manual:

a solution from the first 20 (or

lines

must index the complete powder diffraction pattern

. . .".

The

best way to evaluate the quality of the indexing is to perform lattice

parameters refinement and to calculate a figure of merit using all

measured and indexed Bragg peaks.

The presence of distinct systematic absences and an unambiguous

determination of the diffraction group makes a particular indexing

solution especially probable. Exceptionally encouraging are centered

lattices and, for example, such space groups as P2&, Pbca, 14,/arnd,

R%, and many others, which contain multiple glide planes and/or screw

axes, distinguishable from the list of unobserved (extinct or forbidden)

Bragg reflections.

4. When the unit cell is found, it can be used to search multiple databases in

addition to the ICDD's Powder Diffraction File. A positive search result

usually confirms indexing, while a negative outcome does not necessarily

mean that the indexing is wrong

a particular crystal structure may be

novel andlor not yet included in a database.

5. Finally, in the case of a new material, the correctness of the

initio

indexing is only ensured by solving and refining the crystal structure,

which makes both chemical and physical sense, i.e. has correct bond

distances and angles, reasonable coordination polyhedra, oxidation states,

etc.

All three of the briefly described computer programs may only work

consistently when the quality of input data is very high. This implies that the

accuracy of peak positions is outstanding, normally at least 0.02' or better,

which is usually achieved by maintaining a well aligned goniometer and by

using a proper profile fitting procedure. It is important to ensure that all

possible systematic errors are eliminated beforehand, since they can lead to

unpredictable results andlor failure to find a solution. When experimental

data are affected by a small systematic error, reducing the number of Bragg

peaks in the initial indexing attempt may be helpful.

Each program is distributed with ready to use files containing

testldemo

example(s). We encourage the reader to familiarize hidherself with them

Unit

cell

determination

and

refinement

after reading the manuals before trying to perform an

initio

indexing of a

new diffraction pattern.

order to minimize hand input when creating source data files for use

by any of the three programs described above, a simple conversion utility

(pks2ind.exe) is included on the CD accompanying this book. The

pks2ind

utility converts any ASCII table containing at least one column with Bragg

angles of observed peaks

(28)

into formats acceptable by TREOR90,

DICVOL91 and IT0 programs. It also enables correction of experimental

data for a sample displacement error, if the latter is known. A short manual

(pks2ind.pdf) describing this conversion utility is included on the CD.

Slightly modified versions of

TREOR90, DICVOL91 and IT01

may be

downloaded from the website of one of the authors.' The modifications

enable acceptance of the input data by specifying source file name as a

parameter of an MS DOS command line or by dragging-and-dropping files

over an appropriate program icon placed on the desktop, thus simplifying the

dialog.

5.12

initio

indexing

examples

Below we describe three practical indexing examples. The first one will

be considered in detail, with all input data files included on the CD

accompanying this book. The remaining two examples are illustrated

without providing all input data files since we expect that the reader should

be able to use existing data and reproduce indexing results independently.

5.12.1 Hexagonal indexing:

LaNi4.85Sno.15

Experimental data from the LaNi4,ssSno.ls sample are especially useful for

this illustration because as established earlier, this diffraction pattern has

been successfully indexed manually. We also know that the data are affected

by a small systematic error, which can be eliminated by introducing a zero

shift correction of 0.032", and that there are two low intensity Bragg peaks,

which belong to an impurity phase (see sections

5.4

and

5.8).

5.12.1.1 TREOR90

The input data file for the TREOR90 program (Ch5Ex04-Treor90-

All.dat) is found on the CD and it contains the following information, where

the comments after the exclamation symbols are for information purpose

only and are not included in the data file:

Peter Zavalij's web site at

http://materials.binghamton.edu/zavalij;

follow the "Software"

link.