AlllDE Mfl3ST7

NAT'L INST OF STANDARDS & TECH R.I.C.

A1 11 02483597

HImes, Vicky L/NBS"Lattice : a program t PC100 .U5753 N0.1214 V1985 C.2 NBS-PUB-C

1<« NBS TECHNICAL NOTE 1214

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

QC

100

.U5753

No. 1214

1985

^ C. 2

NBS * LATTICE

A Program to Analyze

Lattice Relationships

Vicky L. Himes and Alan D. Mighell
m he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

j^ Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

basis for the nation's physical measurement system, (2) scientific and technological services for industry and

government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety.

The Bureau's technical work is performed by the National Measurement Laboratory, the National

Engineering Laboratory, the Institute for Computer Sciences and Technology, and the Institute for Materials

Science and Engineering

.

The National Measurement Laboratory

Provides the national system of physical and chemical measurement;

coordinates the system with measurement systems of other nations and

furnishes essential services leading to accurate and uniform physical and

chemical measurement throughout the Nation's scientific community, in- dustry, and commerce; provides advisory and research services to other Government agencies; conducts physical and chemical research; develops,

produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

• Basic Standards^

• Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement

capabilities; provides engineering measurement traceability services; develops

test methods and proposes engineering standards and code changes; develops

and proposes new engineering practices; and develops and improves

mechanisms to transfer results of its research to the ultimate user. The

Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical Engineering'^

Manufacturing Engineering

Building Technology

Fire Research

Chemical Engineering'

The Institute for Computer Sciences and Technology

Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and economy in Government

operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing

the Federal Information Processing Standards Program, developing Federal ADP standards guidelines, £md managing Federal participation in ADP

voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

• Programming Science and

Technology

• Computer Systems

Engineering

TJie Institute for Materials Science and Engineering

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information fundamental to the processing, structure, properties and performance of materials; addresses the scientific basis for new advanced materials technologies; plans

research around cross-country scientific themes such as nondestructive

evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and nondestructive evaluation; and broadly disseminates generic technical information resulting from

its programs. The Institute consists of the following Divisions:

• Ceramics

• Fracture and Deformation ^ • Polymers

• Metallurgy

• Reactor Radiation

'Headquarters and l,aboraiories at Gaithcrsburg, MD, unless otherwise noted; mailing address

Gaithersburg, MD 20899. 'Some divisions within the center are located at Boulder, CO 80303. 'Located at Boulder, CO, with some elements at Ciaithersburg, MD.
Lattice Relationships

Version of Summer, 1 985

Vicky L. Himes and Alan D. Mighell

NBS Crystal Data Center

Reactor Radiation Division

National Bureau of Standards

Gaithersburg, MD 20899

)ecember 1985

414 r *

v> Q

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

NBS

fiESEAfiCH INFOSMATIOIJ CENTEH /V8S fech/o ^ci\ I No re^ f^^^

^

\OX)

NBS* LATTICE >^S

A Program to Analyze '^^/

O '<7v
National Bureau of Standards Technical Note 1214

Natl. Bur. Stand. (U.S.), Tech. Note 1214, 79 pages (Dec. 1985)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

Page

I. Introduction 1

II. Identification by lattice matching 2 A. Introduction 2

B. Background and theory 2 C. Algorithm 5

1. Preparation of a search file of known compounds 5

2. Search strategy 6

3. Lattice-matching strategy 8 D. NBS Crystal Data File: selectivity and nature of the data 8 E. Concluding remarks 8 F. Operation of the program 10

1. General 10

2. Files and file assignments 10

3. Description of input lines 11

a. Program Control Line 11

b. LM Parameter Line 11

c. RSS Control Line 12

4. Examples of input flowstreams 14

5. Examples of computer output 15

III. Reduction and derivative supercell and subcell calculations 25

A. Introduction 25

B. Reduction 25

1. Theory and computer algorithm 25

2. Uses of reduced cells 25

3. Interpreting the results of a reduction program 26

4. Examples of interpreting the results of a reduction program 27

a. Further reduction is not necessary 27

b. Further reduction is necessary due to a normalization problem 28

c. Example requiring further reduction 29

C. Derivative supercells and subcells 37

1. Theory and computer algorithm 37

2. Uses of derivative supercells and subcells 37

D. Operation of the program 40

1. General 40

2. Description of input lines 40

a. Program Control Line 40

b. RSS Control Line 40

3. Example of input flowstream 42

4. Example of computer output 43

IV. Cell transformation 55 A. Introduction 55

B. Operation of the program 55

1

.

General 55

2. Description of input lines 55

a. Program Control Line 55

b. Matrix Parameter Line 56

c. Cell Parameter Line 56

3. Example of input flowstream 57

4. Example of computer output 57

m
Page

V. Matrix inversion 59

A. Introduction 59

B. Operation of the program 59

1. General 59

2. Description of input lines 59

a. Program Control Line 59

b. Matrix Parameter Line 60

3. Example of input flowstream 60

4. Example of computer output 61 VL Summary of program operations 62

A. General 62

B. Description of input lines. 62

1. Program Control Line 63

2. LM Parameter Line 63

3. RSS Control Line 64

4. Matrix Parameter Line 65

5. Cell Parameter Line 65

Vn. References 66

Appendix I. NBS Crystal Data File 67

Appendix II. 1-Line Search File 73

IV
NBS*LATTICE

A Program to Analyze Lattice Relationships

Version of Summer, 1985

Vicky L. Himes and Alan D. Migheli

NBS Crystal Data Center. Reactor Radiation Division, National Bureau ofStandards, Gaithersburg, MD 20899

A FORTRAN program to analyze lattice relationships has been written and is available for distribu- tion by the NBS Crystal Data Center. The present version of NBS*LATTICE performs several functions

including: 1) the characterization and identification of unknown materials using lattice-formula matching

techniques; 2) the calculation of the reduced cell of the lattice, and the calculation and reduction of

specified derivative supercells and/or subcells (i.e., this program function calculates the standard cells which are useful in the determination of metric lattice symmetry, in finding a matrix relating two unit

cells, etc.); 3) unit cell transformations; and 4) matrix inversions. It is planned to incorporate additional functions in forthcoming versions of this program. Among others, these functions will include a matrix method to determine metric lattice symmetry and a technique to find a transformation matrix relating any two unit cells.

Key words: computer program; lattice; identification; reduction; supercell; subcell; symmetry; trans- formation; data base.

I. Introduction

The NBS Crystal Data Center has been compiling crystallographic data for approximately 20 years. Over

the years, the Data Center has developed a number of evaluation procedures that can be used to analyze lattice relationships. These techniques would also be useful to the general scientific community. Consequently, a computer program based on these procedures has been written and is being distributed by the Data Center. This

work is being supported by the Office of Standard Reference Data at NBS and the JCPDS-Intemational Centre

for Diffraction Data. The NBS*LATTICE program is written in standard FORTRAN and is designed to be used in any

analytical laboratory. The software is multifunctional and can be used to analyze various types of lattice relationships. For example, when the program is used with the NBS Crystal Data File (1982), one can

characterize and identify unknown materials based on lattice-formula matching techniques. In addition, re- duced cells and reduced derivative supercells and/or subcells can be calculated by the program. Thus, this program function calculates the standard cells which are useful in determining metric lattice symmetry, in finding the relationship between two unit cells and in the identification of unknown materials. In addition to the more complex functions, the program performs simpler calculations which are useful in routine crys- tallography. With the program, for example, one can find the inverse of a three by three matrix and transform

a unit cell by a specified matrix. It is planned to incorporate additional program functions in forthcoming

versions of NBS*LATTICE. Among others, these include a matrix technique for metric symmetry deter- mination and a method to find a transformation matrix relating any two unit cells. The examples of lattice relationships discussed in this manual are actual problems. Many have been

published previously (Himes, 1983 and references cited therein).

1
II. Identification by lattice matching

A. Introduction

The lattice-matching program function is designed to be used in conjunction with the NBS Crystal Data

File (1982) for the characterization and identification of crystalline materials. The NBS Crystal Data Center

maintains a data base that contains evaluated crystallographic and chemical data on approximately 60,000

materials. The data fall into the following categories: organics, organometallics, metals, intermetallics, inorganics and minerals. There are two fundamental ways that large crystallographic data bases can be used. As

a source of critically evaluated data, the data base can be used as a basis for scientific research, or as an aid to

scientific research (e.g., to identify unknown compounds, to locate certain molecules, to obtain bibliographic

data, etc.). The type of data that can be obtained through search and retrieval programs include chemical name

and formula, cell parameters and cell volume, crystal system, space group symbol and number, density,

bibliographic data, plus additional data. Since the data base is formatted, many of these data items may be

searched readily using systems software available at a particular institution. However, general systems software

will not be adequate for certain types of information search and retrieval operations. One such example is the

identification of unknown compounds by matching lattices (as defined by unit cell parameters) and, if available, some chemical data. Although simple in principle, lattice-formula matching is a complex operation that requires

specialized scientific background in order to design a practical computer search algorithm. The following

section summarizes the lattice-matching technique developed at NBS. It is planned to incorporate the formulamatching operation in a subsequent version of this program.

Three relatively recent developments have given the lattice-formula method for compound identification great potential as a routine analytical tool. First, automated methods to determine a unit cell (and crystal

structure) are in widespread use. For the same reason, the data base of known crystalline compounds is large

and is rapidly growing. Second, new mathematical theories have practical applications in lattice-matching

procedures. These procedures permit fast and effective identification in spite of certain experimental errors made in unit cell determinations. Third, advancements in computer technology have greatly increased the

efficiency of search strategies through direct access of computer file records and increased mass storage

capability.

B. Background and theory

The identification of materials by powder diffraction is a well-established analytical technique. With the

powder method, identification is based on matching the many diffraction positions and intensities with those

of known materials. In contrast, the method described in this Section identifies materials based on their crystalline lattices as defined by the unit cell parameters. Matching unit cell parameters may be thought of as matching all possible positions of the diffraction lines (d-spacings) in a powder pattern. Practically, however,

there are many advantages to the lattice-matching approach, mainly due to the very compact nature of the unit

cell as compared to a powder pattern. For example, it is far easier to treat the experimental errors when

matching unit cell parameters than it is to evaluate the errors in all the observed (and unobserved) diffraction

positions in a powder pattern. In addition, the computer search times and mass storage requirements are

significantly decreased since there are fewer search parameters for the lattice-matching method. Finally, with

a given unit cell, mathematical procedures allow one to calculate derivative unit cells which could result from

certain errors made by the experimentalists. Since the lattice-matching method is efficient, one is able to routinely search the data base of known materials for the experimentally determined unit cell as well as for its calculated derivative cells. There are many unit cells that can be chosen to define a crystalline lattice. When identifying a material

using the powder method, unique sets of data are matched since any unit cell defining the lattice gives the same

calculated d-spacings. With the lattice-matching method, this uniqueness is guaranteed by always comparing

reduced cells. The reduced cell is a unique, primitive cell based on the three shortest lattice translations and

satisfying a specified set of mathematical conditions. (Section III describes the reduction and derivative lattice theory and applications.) For identification, each unknown cell and, if calculated, each of its associated

derivative cells is transformed to a reduced cell. These reduced cells are then checked against a reduced cell data base of known materials.
The NBS Crystal Data Center has prepared a data base that can be used for compound identification and

characterization. The NBS Crystal Data File is described more fully in Appendix I. In this File, all known cells have been transformed to a reduced cell and the File has been sorted first on increasing magnitudes of the unit

cell parameters. With this classification scheme, metrically similar lattices are located near each other in the

File, making it possible to design efficient lattice-search algorithms.

A summary of the lattice-formula matching procedure is presented in figure 1. From the unknown crystal, a unit cell is determined and reduced. The reduced cell is then checked against a file of known materials. If desired, one calculates derivative lattices which are also reduced and checked against the File of knowns.

Finally, the identification is verified using known chemical data. This procedure can be carried out either by

hand using a printed listing of knowns, or by computer program. Practical examples of this identification

procedure are illustrated in figures 2 and 3. In figure 2, only one single crystal of the sample was available. Thus, a chemical analysis was not possible.

The initial cell was determined on an automated diffractometer. This C-centered cell was transformed to a primitive cell and reduced. The reduced cell was then checked against the NBS Crystal Data File. The sample was found to be sodium sesquicarbonate dihydfate by a direct match of the reduced cell parameters. A full structure determination by neutron diffraction confirmed the identification (Choi & Mighell, 1982).

The example given in figure 3 shows how an identification can be made by using a derivative lattice procedure. In diffraction work, one may determine a supercell or subcell rather than the correct cell of the

lattice. For example, when using powder methods, a supercell in direct space may be determined if one does

not find the smallest cell consistent with the set of d-spacings. In single-crystal work, a subcell in direct space

is determined if reciprocal lattice nodes are missed on a diffraction photograph or a diffractometer. In either

case, it is possible to systematically calculate derivative cells and identify the material. Here, the unknown cell was correct and the known cell was a supercell. The initial cell and space group were determined on a

single-crystal diffractometer and a full set of diffraction data was collected. The empirical formula was known.

When checked against the File of known compounds, there were no matches. Next, the seven supercells of

twice the volume were calculated; only three were unique because the initial cell is metrically rhombohedral.

One of the three calculated supercells matched a cell in the NBS Crystal Data File. The known cell, which was

a supercell of the correct cell, had been determined from powder data only. Nevertheless, it was possible to verify the correct composition and to find appropriate literature references.

Figure 1. Identification by lattice-fiyrmula matching

Unknown Crystal

T

Unit Cell

T

Reduced Cell

Match

Unknown Lattices

against the NBS

Crystal Data File

Derivative

Supercells and

Subcells

I

Reduced

Derivative

Cells

Verify

Identification

based on

Known Chemistry
Figure 2. Identification: direct cell match

PROBLEM: Crystal of Unknown Composition

Full Set of Diffraction Data

Initial Cell

20 44 X

3.49

10.33

90.00 « 106.48

90.00

Reduced Cell

3.49 X

10.33

10.37

1 06.24 « 99.69

90.00

NBS CRYSTAL DATA FILE

3.49

3.49

3.49

10.13

10.31

10.43

12.08

10.35

12.17

90.0

106.1

90.0

90.0

99.7

90.0

90.0

90.0

90.0

CONCLUSION: Compound = Na3 H (C 03)2 • 2H2

* The cell parameters shown represent three adjacent file entries, not three separate matches.

Figure 3. Identification: match of a derivative cell

Initial Cell

6.619 6.619 6.619

62.25 62.25 62.25

1 Check File

1

^ Generate 7 Supercells of

.

Twice the Volume S. Only 3 are Unique

' \^

6.62 6.62 11.72

92.2 105.2 117.8

6.62 6.84 9.52

90.0 92.7 90.0

6.84 6.84 11.33

72.4 72.4 60.0

<

1 Check File

1 Match

' 1 Check File

No Matches

6.62 6.80 9.43

90.0 90.0 90.0

Check File

No Matches

I NH4SnF3
C. Algorithm

The theoretical basis for identification by lattice-formula matching was briefly discussed in the previous

sections. Here, emphasis is placed on the computer strategy used to match the unknown lattice(s) against the

NBS Crystal Data File.

1. Preparation of a search file of known compounds

NBS

Crystal Data File

1-Llne File

Search File

of Known

Compounds

Before using the NBS lattice-matching program function, a compact search file of known compounds must

be prepared. This is done to save computer mass storage, to increase the efficiency of a search, and to simplify

the program's input and output operations. Selected data from the NBS Crystal Data File are used to create a search flle in which each line of 132 characters corresponds to the data for one crystalline compound. For a given computer, the 1-line search file is prepared only once. Data Items in the Search File

of Known Lattices

Reduced Cells

b c a

2 *>2 ^2 ^2

Increasing

Ceil

Parameters

P2

y V SG # Ref. Formula

SG # Ref. Formula

a« b„ c„ a„ B^ y„ V SG # Ref. Formula

n n n n ' n ' n

In the 1-line search file, each known lattice, regardless of its original centering, is represented by its primitive, reduced cell. The entries are ordered by increasing magnitudes of the reduced cell parameters. The

space group, literature reference, chemical formula and other information are included in order to facilitate identification. A description of the data items and format specifications for the 1-line search file are given in Appendix II.
2. Search strategy

Sequential Access

1 KTM unknowns

1 Sorted Entry 1 . Unk1

on Cell Parameters Entry 2 / Unk2

Entry 3 / • Entry 4 / •

•

•

^ Unkn

• Entry m

When matching an unknown against a file of known compounds, two different strategies can be used to access the computer files: sequential and direct access. In the sequential access approach, one reads from the

beginning to the end of the file of knowns in a 'forward' manner only. After each read, a given file entry is matched against all the unknowns.

Direct Access

Unknowns File of

Knowns

Unkn

Entry

Sorted on Cell

Parameters

Unlike sequential access files, direct access enables one to go directly to a specified point in the file. One

can access the file in any order, 'forwards* or 'backwards'. The direct access of file records is efficient as it can

significantly decrease the number of file reads. The 1-line search file used for identification is a direct access computer file. In this direct access code, a knowledge of the data in the file of knowns is used to set up a system

of pointers which allows the program to start searching in about the right region of the data file. A sequential

search is used only for the regions of the file in which a match is possible. In the diagram, the search regions

are designated by boxes.
Grouped-Entry Direct Access

Unknowns File of

Knowns

Sorted on Ceil

Parameters

Unk6

Unk15 Group 1

-

Unk 2 I Group 2

Unk14

Unk 44 [Group 3

Unk 10

Unkn

Entry

Si

Entry

Entry

Entry

1

10,000

Sorted on Cell

Parameters

15,000

Ea

m

A 'grouped-entry' direct access search strategy is actually used in the NBS*LATTICE program. To carry out a grouped-entry direct access search, the unknowns are first sorted on increasing values of the reduced cell parameters. Duplicate lattices (i.e., those with exactly the same a,b,c,alpha,beta,gamma) are then deleted for the

sole purpose of eliminating repetitive output. Next, unknown cells with similar values of a (A) are searched as a group against the appropriate region of the data file. In the diagram, the regions that overlap are searched

only once. This approach is very efficient when one searches a number of unknowns against the 1-line search

file. Grouped-Entry Direct Access Search

Prepare

Search File

of Knowns

Prepare File

of Unknowns

• Reduced Cells

• Reduced

Derivative

Cells

Carry Out

Grouped-Entry

Direct Access

Search

The above diagram summarizes the lattice-matching procedure developed at NBS. First, the 1-line search

file of knowns is prepared from the NBS Crystal Data File . For a given computer, this direct access file of

knowns is prepared only once. Second, the file of unknowns is prepared. Each input cell may be a primitive or centered cell of the lattice. Derivative supercells and subcells may be calculated from the input cell. All cells in the file of unknowns are reduced and sorted on increasing values of the reduced cell parameters (see sec. Ill for details on the reduction and derivative lattice calculations). Third, the grouped-entry direct access search

is carried out. Each unknown cell as well as derivative supercells and subcells may be matched against the file of knowns in a single run.
3. Lattice-matching strategy

There are two basic strategies that can be used to determine whether two unit cells define the same

lattice: the cell and the matrix approaches. The present version of the NBS*LATTICE program uses the first strategy. The experimentally determined unit cells for both the known and unknown compounds are converted

to standard, reduced cells. The reduced cell parameters are then compared; if the reduced cells are the same,

then the original cells define the same lattice. Reduction is a mathematical procedure that leads to a unique cell in all cases, provided there is no experimental error in the unit cell parameters. However, as explained in section III, occasionally, the computer program may yield a geometrically reduced cell that requires further

reduction with respect to the cell angles because of the interactions of the experimental errors with the

normalization procedure and the process of satisfying the special conditions for reduction. Thus, in practice,

two experimentally determined cells defining the same lattice will always give the same a,b,c (within experimental error) for the reduced cell, but in certain cases, the reduced cell angles may differ. For this reason, the

program matches only the cell edges and cell volume of each unknown reduced cell against the known cells in the NBS Crystal Data File. It is possible that every match obtained in this way does not define the same

lattice. In the next version of the program, a mathematical procedure (Santoro, Mighell & Rodgers, 1980) will

be used to distinguish between lattices that have only a,b,c,V in common and lattices that are identical. For

those cases in which more than one match of a,b,c,V occurs, knowledge of the empirical formula or some other

chemical information is almost always sufficient to eliminate unwanted matches. This chemical screening will be automated in a later version of the computer program, completing the identification procedure by lattice- formula matching.

D. NBS Crystal Data File: selectivity and nature of the data

The NBS Crystal Data File contains evaluated crystallographic and chemical data on approximately 60,000

materials. For convenience, these data have been divided among two separate 1-line search files, one composed

of 'organic' data and the other containing 'inorganic' data. Each search file contains approximately 30,000

entries. Certain differences have been noted in the space group frequency distributions of the organic and

inorganic data (Mighell, Himes & Rodgers, 1983; Mighell & Rodgers, 1980). The compounds in the organic

search file usually crystallize in the lower symmetry crystal systems; while the space groups for materials in the

inorganic search file are skewed towards the higher symmetry crystal systems. This has practical implications

in the identification of compounds. For organic compounds, the cell alone (a,b,c,V) is usually sufficient to fully

characterize the material. However, because the inorganic search file generally contains more compounds from

the higher symmetry crystal systems and a greater percentage of isostructural data, some chemical information

is often needed to fully characterize these materials.

Experience has shown that the unit cell is highly characteristic of a compound and, like a powder pattern,

may be used for identification. On the average, only ~ 500 entries in either the organic or inorganic search file will separate two entries when the a-values of the reduced cells differ by —0.20 to —0.25 A. In the middle

region of each search file, —500 file entries will separate two reduced cells whose a-values differ by —0.11 A.

Thus, the lattice-matching procedure is relatively selective based on the a-cell parameter alone. When values

of b, c and V for the reduced cell are considered, the selectivity is dramatically increased. (The selectivity will be further enhanced in the next version of the program by the addition of a mathematical technique to

distinguish between lattices that have only a,b,c and V in common and lattices that are identical.)

As part of a practical test of the lattice-matching method, the entire 1-line organic search file (—30,000

compounds) was considered as a file of unknowns to be matched against itself. When tolerances of ±0.1 A for

a,b,c and ± 10 percent for V were used, most 'unknowns' gave only one or two matches. In the entire analysis, a maximum of 17 matches of a,b,c,V were found for a single 'unknown'. These contained data for the same or

isostructural compounds. As expected, a similar analysis of the inorganic search file yielded more matches on

the average. These two analyses demonstrated the efficiency of the computer program as well as the selectivity

of the lattice-matching method.

E. Concluding remarks

The identification of an unknown compound can be highly automated including the determination of the

unit cell. Both the unit cell determination and the identification process can be carried out in one instrument.

8
The lattice-matching method of identification is ideal for automated analyses since it is subject to precise

mathematical procedures. The method is efficient, highly selective, and various types of inter- and intra-lattice relationships can be established. In practice, the technique is reliable as the NBS Crystal Data File contains

evaluated data on well-characterized materials. Usually, the cells have been refined by least-squares techniques

and the chemistry is precisely known. For most crystals, the lattice is unique. This means that the cell or the

cell plus some chemical data is sufficient for identification. Experience with practical problems has shown that

identification by matching reduced cells is very straightforward and reliable when the correct cell of the lattice has been determined. However, a material can often be identified even if an error has been made in which a derivative cell of the unknown has been found. For example, if the cell centering is missed when indexing a powder pattern or, when using single-crystal methods, if rows of spots are missed on the diffraction photographs, this would result in the determination of a derivative cell of the correct lattice. To make an identification, one systematically calculates the derivative supercells and subcells and looks for matches in the NBS

Crystal Data File. When searching for matches, relatively large tolerances are used routinely. In addition,

checks for the 55 supercells of 2,3, and 4 times the volume of the unknown reduced cell and the 55 subcells of

1/2,1/3, and 1/4 times the volume are made. The ability to check routinely for matches of 1 10 extra derivative

cells for each unknown is only possible because of the efficiency of the computer program and the selectivity

of the method.

Lattice-Matching

Metliod for Identification

Unknown Sample

Powder

Diffractometer Films

Index

Single-Crystal Diffractometer

UnH Cell

Lattice-Matching

against the NBS Crystal Data File

As the above diagram illustrates, the lattice-formula method of identification is a general procedure that may be applied to any cell regardless of the technique used for its determination, including x-ray, neutron and

electron diffraction. The data may be recorded using a single-crystal or powder diffractometer, or by using film methods. In the past, lattices, as defined by unit cells, have not been used for the identification of materials.

Traditionally, the experimentalist would grind the sample or employ a Gandolfi camera to obtain a powder

pattern; the compound is then identified based on matching observed d-spacings and intensities. However,

there are many advantages to the lattice-matching method of identification as compared to the traditional approach. First, the cell gives more information about the metric lattice than a set of d-spacings. A unit cell may

be used to calculate a set of all possible d-spacings, but a set of d-spacings may be consistent with more than one cell. Practically, this situation is made worse since a limited set of d-spacings may be observed with the

powder method due to the nature of the compound and to other experimental conditions. Second, from the

unknown unit cell, mathematical procedures allow one to calculate derivative cells which can be checked

against the file of known lattices. Related materials can be found using this strategy, and identification can be

made in spite of certain errors made by the experimentalists. Third, it is far easier to treat experimental errors when matching unit cell parameters than it is to evaluate the errors in all the observed (and unobserved)

diffraction positions in a powder pattern. Finally, computer search times and mass storage requirements for
identification based on lattice matching are significantly decreased since there are fewer search parameters. In view of its many theoretical and practical advantages, we feel that the lattice-matching technique will change

the way in which powder experimental data will be used to identify materials. A unit cell will be determined

along with the powder diffraction pattern. The cell may be obtained from an indexing program using the

observed d-spacings, or it may be obtained directly using a single crystal taken from the sample. Once a cell has been determined, the identification may be carried out by lattice matching, followed by comparing the

powder intensity data and any available chemical information. The identification based on matching d-spacings

and intensities should be reserved for those cases in which a lattice match was not obtained, for mixtures, and

for those cases in which a unit cell could not be determined. Thus, the ability to identify crystalline materials

will be significantly improved due to the general nature of the lattice-matching approach, and the availability

of the NBS Crystal Data File, a large file containing known lattices from all areas of the solid state.

F. Operation of the Program

1. General

There are three types of input lines required to execute the lattice-matching (LM) program function. The

first input line is a Program Control Line that specifies the type of program function and the number of

independent problems to be considered. The second input line is an LM Parameter Line that specifies the

tolerances for a match of a,b,c, and V, the files to be searched (organic or inorganic), and whether full or limited

output is desired. The third type of input line is an RSS (reduction and derivative supercell and subcell; see sec. Ill) Control Line that defines the input lattice and specifies the derivative lattices to be calculated. For N

problems specified on the Program Control Line, one LM Parameter Line and N RSS Control Lines must

follow (1<N<20). Within a single computer run, the LM program function may be executed only once. Although at most 20 independent problems may be considered, up to 900 lattices, including the unknown

lattice(s) plus calculated derivative lattices, may be matched against data from the NBS Crystal Data File in a

single run. A description of the file assignments, formats for the input lines, and typical examples for the

execution of the LM program function follow.

2. Files and file assignments

Before the LM program function can be executed, the NBS Crystal Data File must be used to create the

direct access 1-line search files. For a given computer, the search files are prepared only once. The filename

NBSI must be associated with the inorganic data, while the filename NBSO must be associated with the organic

data. A summary of the assignments made when executing the LM program function is given below.

Unit Associated Description

Number Filename(s)

7 NBSI, NBSO Direct access 1-line search file

10 NBS10 Scratch file of reduced cell(s) and reduced derivative cells calculated by the program

10
3. Description of input lines

a. Program Control Line

The Program Control Line specifies the type of program function and the number of independent

problems to be considered. To execute the LM program function, one Program Control Line is required.

Program Control Line

Format(A5,3X,I2)

Column Format Item

1-5 A5 Type of program function

'LM ' = Lattice Matching

'RSS ' = Reduction and

Derivative

Supercell and Subcell

'TRANS' = Cell Transformation

INV ' = Matrix Inversion

6-8 3X Blank

9-10 12 Number of problems

Notes:

1 The LM program function is specified by 'LM ' in columns 1-5.

2 The number of independent problems specified for the execution of the LM

program function may range between 1 and 20. However, up to 900 lattices,

including the unknown lattice(s) plus calculated derivative lattices, can be

searched against the file(s) of known compounds in a single run. If more than

900 lattices are generated as a result of the data on the RSS Control Lines,

only the first 900 lattices will be used in the lattice-matching procedure.

b. LM Parameter Line

The LM Parameter Line specifies the tolerances for a match of a,b,c, and V, the file(s) to be searched, and

whether full or limited output is desired. Only one LM Parameter Line is allowed; this line must follow the

Program Control Line and precede the RSS Control Line(s).

LM Parameter Line

Format(2F10.2,5X,5Al,9X,Il)

Column

1-10

11-20

21-25

Format

F10.2

F10.2

5X

Item

Tolerance for a match of the cell edges (A)

Tolerance for a match of the cell volume (percentage of A^)

Blank

26-30 5A1 I/O/ Inorganic/Organic/

May specify one or more

files to be searched.

31-39 9X Blank

40 11 Blank/1/ Print/Do not print/

RSS output

11
Notes:

1 Although the minimum tolerance allowed by the program is 0.02 A, this value is NOT recommended for routine use. A tolerance of 0.10 A is a more

reasonable value to specify for most problems. The user may wish to increase

this tolerance in a second run depending on the results obtained. It must be

remembered that the known data in the search file(s) may not be as accurate and/or as precise as the experimental data input to the program.

2 Although the minimum tolerance allowed by the program is 5.0 percent, this value is NOT recommended for routine use. A tolerance of 10.0 percent is a more reasonable value to specify for most problems. The user may wish to increase this tolerance in a second run depending on the results obtained. It must be remembered that the known data in the search file(s) may not be as accurate and/or as precise as the experimental data input to the program.

3 One or more of the 1-line files may be searched in a single run. If no file is specified, the inorganic 1-line file will be searched.

4 In general, it is very useful to print the results of the reduction and derivative

lattice calculations. However, when operating the program in a 'demand'

environment or in cases where large numbers of derivative lattices are calculated, the user may wish to suppress this output.

c. RSS Control Line

The RSS Control Line defines the input lattice and the derivative lattices to be calculated. The lattice is defined by specifying the cell centering and all six unit cell parameters. The program will always carry out the

cell reduction calculations. To calculate derivative lattices, it is necessary to tell the program whether to calculate supercells, subcells, or both supercells and subcells. In addition, for derivative lattice calculations, it

is necessary to tell the program which multiples of the reduced cell volume (V) should be considered.

Supercells having 2 to 9 times the volume of the reduced input cell and/or subcells having 1/2 to 1/9 times the

volume of the reduced input cell may be calculated. There are 7 unique derivative lattices (supercells or

subcells, respectively) having 2 or 1/2 times the volume of the reduced input cell, 13 derivative lattices with

3V or 1/3V, 35 with 4V or 1/4V, 31 with 5V or 1/5V, 91 with 6V or 1/6V, 57 with 7V or 1/7V, 155 with 8V

or 1/8V, and 130 with 9V or 1/9V.

The RSS Control Line(s) must follow the Program Control Line and the LM Parameter Line. The number

of RSS Control Lines must be equal to the number of problems specified on the Program Control Line.

RSS Control Line

Format(Il,2X,2Il,3X,2Al,6F10.2)

Column Format Item

1 1 II Blank/1/2/3/

Blank = Reduction

1 = Reduction + supercells

2 = Reduction + subcells

3 = Reduction + supercells, subcells

2-3 2X Blank

2 4 II Blank/2/3/4/5/6/7/8/9/

Initial value (nl) to define the range

of volumes for calculated

derivative lattices.

12
II

6-8 3X

4 9 Al

5 10 Al

Blank/2/3/4/5/6/7/8/9/

Final value (n2) to define the range

of volumes for calculated derivative lattices.

Blank

P/A/B/C/F/I/R/ Cell centering

R/H/ Rhombohedral/Hexagonal/

metric axes. Used only for rhombohedral lattices.

6 11-20 F10.2 a (A)

7 21-30 F10.2 b

8 31-40 F10.2 c

9 41-50 F10.2 alpha n

10 51-60 F10.2 beta

11 61-70 F10.2 gamma

Notes:

2-3

6-8

9-11

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, then a 1, 2 or 3 is used for supercells, or subcells, or both supercells and subcells, respectively.

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, use an integer (nl) from 2 to 9.

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, use an integer (n2) from 2 to 9, but greater than

or equal to nl. These two integers (nl and n2) are used to define the range of volumes for the

derivative lattices to be calculated. The supercells to be calculated will have

volumes ranging from (nl)V to (n2)V while the volumes for the subcells to be calculated will range from (l/nl)V to (l/n2)V. Especially when some

chemical information is known, it is recommended to check routinely for the

55 supercells of 2,3, and 4 times the volume and the 55 subcells of 1/2,1/3, and

1/4 times the volume of the reduced input cell (i.e., nl = 2, n2 = 4, with a

'3' in column 1). To define the lattice, both the cell and cell centering are required. A symbol

specifying the cell centering must be placed in columns 9 and 10. A one

character symbol is used for P,A,B,C,F, and I centered cells. For a cell defining a rhombohedral lattice, a two character symbol is required (RR or RH); the first character of this symbol ('R') is placed in column 9. Column 10 is left blank unless the input cell defines a rhombohedral lattice. For a cell defining a rhombohedral lattice, an 'R' is placed in column 10 if the

lattice is defined by a primitive rhombohedral cell (rhombohedral axes), or an

'H' is used if the lattice is defined by a triply primitive rhombohedral cell (metrically hexagonal axes).

Cell edges (A).

Cell angles (°). Decimal numbers must be used for fractions of a degree. All

six cell parameters must always be specified regardless of the crystal symmetry.

13
4. Examples of input flowstreams

a. Example 1

1. . 1 . . .1. .

.

.1 . .1 . .1

1 LM 3

2 0.10 10.0 01

3 C 20 .44 3.49 10,,33 90.0 106.48 90.0

4 P 8.(D95 8.096 30,,667 88.69 57.95 87.48

5 1 22 P 13 .595 4.638 10,,321 90.0 98.28 90.0

6 LM 1

7 0.10 10.0

8 1 22 P 13 .595 4.638 10,,321 90.0 98.28 90.0

Notes:

1 The *LM ' in columns 1-5 indicates that the LM program function will be

executed; the '3' in column 10 specifies that three independent problems are to be

considered. This means that three RSS Control Lines must follow the Program

Control Line and the LM Parameter Line.

2 The tolerance for a match of the cell edges (A) is 0.10 (columns 1-10) while the

tolerance for a match of the cell volume (A') is 10.0 percent (columns 1 1-20). The 'OF

in columns 29-30 indicates that both the organic and inorganic data will be searched

for a match of a,b,c,V. Since column 40 is blank, the results of the reduction and

derivative lattice calculations will be printed.

3 Since column 1 is blank, the cell will be reduced and no derivative lattices will be

calculated. Columns 4-5 are also left blank for reduction-only calculations. The *C '

in columns 9-10 indicates that the input cell is C-centered. All six unit cell parameters

are specified in columns 11-70.

4 This RSS Control Line directs the program to reduce a primitive cell.

5 The '1' in column 1 specifies that supercells of the lattice will be calculated. The '22' in columns 4-5 signify that the 7 unique supercells of twice the volume of the reduced

input cell will be calculated. The 'P ' in columns 9-10 indicates that the input cell is primitive. All six unit cell parameters are specified in columns 11-70.

1-5 Although 3 independent problems are input to the LM program function, a total of 10

lattices (3 reduced input cells plus 7 reduced derivative supercells) are to be searched

against the organic and inorganic data from the NBS Crystal Data File.

6 The LM program function is to be executed a second time with one independent

problem to be considered. (This second operation will not be carried out by the

program.)

7 This LM Parameter Line specifies that tolerances of 0.10 A and 10.0 percent will be

used to define a match of a,b,c, and V, the organic data will be searched, and the RSS

output will be printed.

8 Same as 5.

14
b. Example 2

1 LM 4

2 0.10 10.0 I 1

3 P 3.0804 3.0806 15.122 89.96 89.99 119.99

4 C 6.282 15.203 5.674 90.0 114.04 90.0

5 P 5.674 6.282 8.225 67.55 81.05 65.96

6 P 7.751 7.393 7.395 63.93 72.22 103.27

Notes:

1 The 'LM ' in columns 1-5 indicates that the LM program function will be

executed; the '4' in column 10 specifies that four independent problems are to be

considered. This means that four RSS Control Lines must follow the Program Control Line and the LM Parameter Line.

2 Tolerances of 0.10 A (columns 1-10) and 10.0 percent (columjis 11-20) will be used

to define a match of a,b,c,V. In this case, only the inorganic data will be searched. The

T in column 40 indicates that the results of the reduction and derivative lattice calculations will not be printed.

3 Since column 1 is blank, the cell will be reduced and no derivative lattices will be

calculated. Columns 4-5 are also left blank for reduction-only calculations. The T

'

in columns 9-10 indicates that the input cell is primitive. All six unit cell parameters

are specified in columns 1 1-70.

4 This RSS Control Line directs the program to reduce a C-centered cell.

5-6 Same as 3.

5. Examples of computer output

a. Example 1

The computer output following this section results from the input flowstream discussed above for Example 1 (sec. n.F.4.).

1. Page I ofcomputer output

Since the results of the reduction and derivative lattice calculations are to be printed, a description

of the RSS output parameters is given.

2. Pages 2-5 ofcomputer output

There are three independent problems to be considered. Each of the input cells will be reduced

and, if indicated, derivative supercells and/or subcells will be calculated. These reduced cells will be

matched against data from the NBS Crystal Data File. See section III for a more detailed discussion

of the RSS results. The results of the reduction calculations for the first problem are given on the second page. This

problem was discussed previously in section II. B. and the identification process was illustrated in figure 2. The reduction output for the second independent problem is given on page 3. In this example, the

input cell was determined on an automated diffractometer from a twinned crystal. When the input cell

is reduced and the reduced cell matrix is analyzed, the orthorhombic nature of the lattice becomes

apparent. The reduced form corresponds to number 13 in table 3 (sec. III). The matrix

(1 10/-1 10/001) will transform the reduced cell to a C-centered orthorhombic cell.

15
The input cell for the third independent problem is reduced and the seven supercells having twice the

volume of the reduced cell are calculated. The output for these calculations is given on pages 4 and 5. The input

cell was determined on an automated diffractometer. The structure solution was not completely successful as there appeared to be an unusual amount of disorder. A cell was determined a second time and it was found to have twice the volume. When the structure was solved on the basis of the new cell, the disorder disappeared.

The third input cell is the subcell that was determined by missing nodes in reciprocal space. This example

illustrates how a compound can be identified in spite of certain experimental errors by systematically calculating derivative lattices and searching the files of known compounds.

3. Page 6 of computer output

This page summarizes the data input to the lattice-matching procedure. First, a list of unknowns is given.

The 'Original Sequence' column corresponds to the reduced cells and reduced derivative cells as they are calculated by the program. Thus, Original Sequence numbers 1, 2, and 3 correspond to the reduced cells for the first, second, and third independent problems, respectively. The Original Sequence numbers 4 through 10

correspond to the Supercells 1 through 7 given on pages 4 and 5 of the computer output. These lattices have

been sorted according to the increasing values of the reduced cell parameters. The 'Search Sequence' corre- sponds to the order of the lattices after this sort has been completed. There are 10 unknown lattices to be

searched against the file of known compounds. Tolerances of 0.10 A and 10.0 percent will be used to define a match of a,b,c, and V.

4. Page 7 ofcomputer output

The matches obtained when searching the inorganic data from the NBS Crystal Data File are presented

here. This data is sorted according to the 'Original Sequence' of the input cells. Thus, the first independent

problem (Original Sequence 1) was found to be sodium sesquicarbonate by a direct match of the reduced cell parameters. As explained in section II.B., this identification by lattice matching was confirmed by a full structure determination. The match of a,b,c,V for Original Sequence 3 is not valid since the cell was determined

from an organic compound.

5. Page 8 of computer output

The matches obtained when searching the organic data from the NBS Crystal Data File are given on this page. The second independent problem (Original Sequence 2) was found to be potassium antimony tartrate hydrate by a direct match of the reduced cell parameters. There are no direct matches of the reduced cell parameters for the third independent problem. However, a lattice match was found for one of the supercells

having twice the volume of the reduced input cell. The Original Sequence 5 corresponds to Supercell 2 in the

RSS output. For both the second and third independent problems, the identification made by lattice matching was verified on the basis of the known chemistry.

6. Page 9 of computer output

Within a single computer run, the LM program function may be executed only once. This restriction is intended to discourage the inappropriate use of the LM program function. For example, if data for five organic

compounds are to be analyzed, the user probably should not attempt five different searches. In general, since

the LM program function employs a grouped-entry strategy, the most efficient method would be to include the

data for all five organic compounds in a single computer run. In addition, the grouped-entry strategy enables

the user to check routinely for matches of the reduced input cells as well as for calculated derivative cells.

b. Example 2

The computer output following this section results from the input flowstream discussed above for Example 2 (sec. II.F.4.).

L Page 1 of computer output

Four independent problems are to be considered. The input cells will be reduced and matched against data

from the NBS Crystal Data File . For this example, the results of the RSS calculations have not been printed.

16
2. Page 2 of computer output

This page summarizes the data input to the lattice-matching procedure. First, a list of unknowns is given.

The 'Original Sequence' column corresponds to the reduced cells as they were calculated by the program.

Thus, Original Sequence numbers 1,2,3, and 4 correspond to the reduced cells for the first, second, third, and

fourth independent problems, respectively. These lattices have been sorted according to the increasing values

of the reduced cell parameters. The 'Search Sequence' corresponds to the order of the lattices after this sort has been completed and duplicate lattices have been deleted. Duplicate lattices are defmed as those having the same a,b,c within three decimal places and the same alpha, beta, gamma, within two decimal places. Duplicate

lattices are not considered in the lattice-matching procedure for the sole purpose of eliminating repetitive

output. There are several ways in which duplicate lattices can be input to the lattice-matching procedure. In

this example, independent problems 2 and 3 reduced to the same cell. Another common way is through the

generation of derivative supercells and/or subcells. For certain specialized triclinic lattices or for lattices having higher metric symmetry, some of the derivative lattices are metrically identical although they are oriented differently in space. In summary, there are three unknown lattices to be searched against the file of known compounds. One

duplicate lattice has been eliminated from the search procedure. Tolerances of 0.10 A and 10.0 percent will be

used to define a match of a,b,c, and V.

3. Page 3 ofcomputer output

The matches obtained when searching the inorganic data are presented here. This data is sorted according

to the 'Original Sequence' of the input cells. The first independent problem (Original Sequence 1) was found to be silicon carbide. For this example,

refined cell parameters were input to the program. When this cell is reduced, the resulting cell requires further

reduction when the experimental error is considered. This example is discussed in detail in section III.B.4.b. In

practice, two experimentally determined cells defining the same lattice will always give the same a,b,c (within

experimental error) for the reduced cell, but in certain cases, the reduced cell angles may differ due to the

interactions of the experimental errors with the normalization procedure and the process of satisfying the

special conditions for reduction (for details, see sec. III). For this reason, the program matches only the cell edges and volume of the unknown reduced cell against the reduced cell file of known compounds. In this example, even though the cell angles are different for the reduced input cell and the known cell, it is apparent

that the two cells define the same lattice and a transformation matrix relating them can be found by inspection. In more difficult cases, further reduction must be carried out (see sec. Ill) or a mathematical approach must

be used to determine whether the two cells define the same lattice. The second and third input cells were obtained using data from the same single crystal. As expected, these

input cells reduced to the same cell. Since the third reduced cell was deleted from the search procedure, no

computer output is expected. The second independent problem (Original Sequence 2) was identified as calcium

sulfate dihydrate on the basis of the lattice and known chemical information.

There are nine matches of a,b,c,V for the fourth independent problem. Of these, two are direct cell matches. The identification of this unknown was made on the basis of the cell parameters and known chemical

data. The unknown was deduced to be Sr(C103)2 • H2O because it was isostructural with the barium analog in the NBS Crystal Data File. This example illustrates that every match of a,b,c,V does not necessarily define the same lattice. When a mathematical procedure (Santoro, Mighell & Rodgers, 1980) is used, it can be shown that only the two direct cell matches define the same lattice as the input cell, the remaining seven matches of a,b,c,V

do not. In the next version of the NBS*LATTICE program, this mathematical technique will be added to distinguish between lattices that have only a,b,c,V in common and lattices that are identical. At present,

knowledge of the empirical formula or some other chemical information is almost always sufficient to eliminate

unwanted matches.

17
Example 1

Section II.F.5.a.

Page 1

»»» NBS»LATTICE ••»

A PROGRAM TO ANALYZE LATTICE RELATIONSHIPS

Version of Spring, 1985

NBS Crystal Data Center

National Bureau of Standards Reactor Radiation Division

Gaithersburg, MD 20899

•• REDUCTION AND DERIVATIVE LATTICE ••

These calculations fall into two categories:

I. Reduction of an input cell.

CELL 1 « Input cell. This cell may be primitive or centered (A,B,C,I,F,RR,RH)

.

CELL 2 = Reduced primitive cell of the lattice.

T 1 -A matrix that transforms CELL 1 to a primitive cell of the lattice.

T 2 - A matrix that transforms CELL 1 to CELL 2.

II. Calculation and reduction of a series of derivative supercells and/or subcells. These derivative cells are calculated from the reduced cell of the lattice

(I.e. to oau-ry out the Type II calculation, the program first carries out

the Type I calculation).

CELL 1 = Reduced primitive cell (i.e. CELL 2 from Part I). CELL 2 = Reduced supercell or subcell.

T 1 = A matrix that transforms CELL 1 to a supercell or subcell of the lattice.

T 2 -A matrix that transforms CELL 1 to CELL 2.

For CELL 1 and CELL 2, the output parameters given are: a, b, c, alpha, beta, and volume. Cell edges are in angstroms and angles in degrees.

The reduced cell matrix is of the form: a.

a

b.c

b.b

Page 2

KEDUCTION AND DERIVATIVE LATTICE

Number of Independent problems to study - 3

Lattice Matching of reduced cell(s) with the NBS Crystal Data File

REDUCTION

•» Initial Cell Is C-Centered »»

T 1-

T 2-

T 2 INV-

.50 .00 1.00

-.50

1 .00

.00

.00/

.00/ 2.00/

.50

.00 1.00

.50 .00

.00

.00/ 1.00/

.00/

.00

.50 .00

.00 -.50 1.001.00

.00 .00 «» Cell Matrix »»

CELL CELL

20.MM00

3.t900

3. "4900 10.330010.3300

10.3679 90.000

106.238

1O6.M80 99.689

90 . 000 90.000

VIV2- 706.62

353.31 12.180

-29.9'49 106.709-6.090

107.193

.000

18
Page 3

2. REDUCTION

»» Initial Cell is Primitive «•

T 1 = 1.00 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

T 2= 1.00 .00 .00/ .00 -1 .00 .00/ 2.00 .00 -1 .00

T 2 INV= 1.00 .00 .00/ .00 -1 .00 .00/ 2.00 .00 -1 .00

CELL 1= 8.0950 8.0960 30.6670 88.690 57.950 87.180

CELL 2= 8.0950 8.0960 25.9930 90.021 90.185 92.520

»» Cell Matrix ••

VI- 1701.85 65.529 65.515 675.638

V2= 1701.85 -.087 -.678 -2.882

Page 4

SUPERLATTICES

»» Initial Cell is Primitive ««

T 1 = 1 .00 .00 .00/ .00 1.00 .00/ .00 .00 1 .00

T 2= .00 1 .00 .00/ .00 .00 -1.00/ -1.00 .00 .00 T 2 INV= .00 .00 -1.00/ l.OC .00 .00/ .00 -1.00 .00 »• Cell Matrix ««

CELL 1= 13.5950 1.6380 10.3210 90.000 98.280 90.000 V1 = 613.99 21.511 106.523 181.82 CELL 2= 1.6380 10.3210 13.5950 98.280 90.000 90.000 V2= 613.99 -20.207 .000 .00

Superlattioes for Delta = 2

«««fti' Supercell 1 'M«*«

T 1 = 1.00 .00 .00/ .00 1.00 .00/ .00 .00 2.00

T 2= 1 .00 .00 .00/ .00 1.00 .00/ .00 .00 2.00

T 2 INV= 1.00 .00 .00/ .00 1.00 .00/ .00 .00 .50 •• Cell Matrix ««

CELL 1= 1.6380 10.3210 13.5950 98.280 90 000 90.000 VI" 613.99 21.511 106.523 739.296

CELL 2- 1.6380 10.3210 27.1900 98.280 90 000 90.000 V2- 1287.99 -10.113 .000 .000

»»»«» Supercell 2 »»•»•

T 1 = 1.00 .00 .00/ .00 1.00 1.00/ .00 .00 2.00

T 2= -1.00 .00 .00/ .00 -1.00 -1.00/ .00 -1.00 1.00

T 2 INV= -1 .00 .00 .00/ .00 -.50 -.50/ .00 - . 50 .50 •• Cell Matrix t»

CELL 1= 1.6380 10.3210 13 5950 98.280 90.000 90.000 VI- 613.99 21.511 250.931 331 .761 CELL 2= 1.6380 15.8109 18 2113 105.716 90.000 90.000 V2- 1287.99 -78.301 .000 .000

19
«»»»« Supercell 3 *****

T 1- 1.00 .00 1.00/ .00 1.00 .00/ .00 .00 2.00

T 2= 2.00 .00 .00/ .00 1.00 .00/ -1.00 .00 1.00

T 2 INV= .50 .00 .00/ .00 1.00 .00/ .50 .00 1.00

CELL 1 - 1.6380 10.3210 13..5950 98.280 90.000 90.000

CELL 2= 9.2760 10.3210 11,.3611 97.831 108.837 90.000

V1»

V2- 613.991287.99

»» Cell Matrix »»

86.011 106.523 206.335

-20.207 -13.022 .000

«»»«» Supercell 1 »«»»»

T 1 = 1.00 .00 1.00/ .00 1.00 1.00/ .00 .00 2.00

T 2= -2.00 .00 .00/ -1.00 -1.00 .00/ -1.00 .00 1.00

T 2 INV- -.50 .00 .00/ .50 -1.00 .00/ -.50 .00 1.00 •• Cell Matrix ••

Pages

CELL

CELL

1.6380 10.3210 9.2760 11.3152 13.5950

11.3611 98.280

75.128

90.000 71.16390.000

65.802

VI- V2= 613.99

1287.99 86.01111.718

128.031 13.022 206.335

13.022

»«•»» Supercell

T 1 = 1.00 .00 .00/ .00 2.00 .00/ .00 .00 1 .00

T 2 = -1.00 .00 .00/ .00 .00 -1.00/ .00 -2.00 .00

T 2 INV= -1.00 .00 .00/ .00 .00 -.50/ .00 -1.00 .00 »• Cell Matrix ••

CELL 1= 1.6380 10..3210 13.5950 98.280 90.000 90.000 VI- 613.99 21.511 181.821 126.092 CELL 2= I.638O 13.,5950 20.6120 98.280 90.000 90.000 V2- 1287.99 -10.113 .000 .000

«*••<> Supercell 6 «»»•»

T 1 = 1.00 1 .00 .00/ .00 2.00 .00/ .00 .00 1 .00

T 2= 2.00 .00 .00/ -1.00 1.00 .00/ .00 .00 1 .00 T 2 INV= .50 .00 .00/ .50 1.00 .00/ .00 .00 1 .00

CEM. 1 = 1.6380 10.3210 13.595c 98.280 90.000 90.000 Vl = 613.99

CELL 2= 9.2760 11.3152 13.5950 97.518 90.000 111.198 V2= 1287.99

»• Cell Matrix »•

86.011 128.031 181.821 -20.207 .000 -13.022

««»«« Supercell Y •*•»»

T 1 = 2.00 .00 .00/ .00 1.00 .00/ .00 .00 1 .00 T 2= 2.00 .00 .00/ .00 1.00 .00/ .00 .00 1 .00

T 2 INV- .50 .00 .00/ .00 1.00 .00/ .00 .00 1 .00

CELL 1= 1.6380 10.3210 13..5950 98.280 90.000 90.000 CELL 2= 9.2760 10.3210 13.,5950 98.280 90.000 90.000VIV2= 613.99

1287.99

•» Cell Matrix »•

86.011 106.523 181.821 -20.207 .000 .000

20
Page 6

«« LATTICE MATCHING »»

Identification of unknown lattices using data from

the National Bureau of Standards Crystal Data File

•»» UNKNOWNS »•»

Search Original a b c Alpha Beta Gamma Vol Sequence Sequence

1 1 3.19000 10.33000 10.36790 106.23835 99.68915 90.00000 353.31

2 3 t. 63800 10.32100 13.59500 98.28000 90.00000 90.00000 613.99

3 4 1). 63800 10.32100 27.19000 98.28000 90.00000 90.00000 1287.99

i| 8 1.63800 13.59500 20.61200 98.28000 90.00000 90.00000 1287.99

5 5 1.63800 15.81088 18.21129 105.71631 90.00000 90.00000 1287.99

6 2 8.09500 8.09600 25.99303 90.02367 90.18155 92 . 52000 1701.85

7 10 9.27600 10.32100 13.59500 98.28000 90.00000 90.00000 1287.99

8 6 9.27600 10.32100 11.36137 97.83365 108.83732 90.00000 1287.99

9 9 9.27600 11.31521 13.59500 97.51803 90.00000 111.19796 1287.99

10 7 9.27600 11.31521 11.36137 75.12755 71.16268 65.80201 1287.99

Original number of unknowns •> 10 Number of duplicate unknowns = Number of unknowns for search = 10

Tolerance for cell edges » .10 Tolerance for cell volume - 10.00

Page 7

INORGANIC

»»• RESULTS »*•

Search Original a Sequence Sequence Alpha Beta Gamma Vol

1 1 3.190 10.330 10.368 106.21 99.69 90.00 353.3

3.19 10.31 10.35 106.1 99.7 90.0 352 15 MT 2180 ACCRA9 2 167 1919 Na3 H ( C 03 )2 12 H2

2 3 1.638 10.321 13.595 98.28 90.00 90.00 611.0

1.71 10.27 13.68 100.8 90.0 90.0 651 11 ML 9172 PRLAAZ 117 517 1928 Mg9 ( H , F )2 C SI 01 )1

21
Pages

ORGANIC

»•» RESULTS ••»

Search Original a b c Alpha Beta Gamma Vol Sequence Sequence

6 2 8.095 8.096 25.993 90.02 90.18 92.52 170U9

8.09 8.09 25.93 90.0 90.0 92.5 1697 20 OT 15513 ICHAA3 8 209 197') C8 HH 012 Sb2 -2 !2 ( K + ) !3 ( H2 ) 8.12 8.12 25.96 90.0 90.0 92.7 1708 20 OT 15653 KHTPAT 17 502 1966 C8 HH 012 Sb2 -2 !2 C K + ) !3 ( H2 )

5 5 1.638 15.811 18.211 105.75 90.00 90.00 1288.0

1.62 15.93 18.27 105.6 90.0 90.0 1296 11 MT 1 318 JACSAT 101 5858 1979 C12 H6 C12 I N 03

1.62 15.93 18.27 105.6 90.0 90.0 1296 11 MT 1 319 ACBCAR 36 2819 1980 C12 H6 C12 I N 03

Page 9

INPUT ERROR ... Only one Lattice Matching operation per run is allowed.

22
Example 2

Section II.F.5.b.

Page 1

••» NBS»LATTICE •»•

A PROGRAM TO ANALYZE LATTICE RELATIONSHIPS Version of Spring, 1985

NBS Crystal Data Center

National Bureau of Standards

Reactor Radiation Division

Gaithersburg, MD 20899

REDUCTION AND DERIVATIVE LATTICE

Number of independent problems to study - 4

Lattice Matching of reduced oell(s) with the NBS Crystal Data File

Page 2

»» LATTICE MATCHING •»

identification of unknown lattices using data from

the National Bureau of Standards Crystal Data File

UNKNOWNS

Search Original Sequence Sequence

Alpha Gamma Vol

1 1 3.08097 3.08060

2 2 5.671)00 6.28200 Delete 3 5. 671400 6.28200

3 K 7.39300 7.39500

15.12200 89.96000 89.95000 59.99178 121.29 8.22188 67.51920 81.05002 65.96000 217.15 8.22500 67.55000 81.05000 65.96000 217.15 7.71953 103.27236 107.76093 116.07000 328.06

Original number of unknowns « 1 Number of duplicate unknowns - 1 Number of unknowns for search = 3

Tolerance for cell edges Tolerance for cell volume

.10 10.00

23
Pages

INORGANIC

»»» RESULTS »»»

Search Original a b c Alpha Sequence Sequence

1 1 3.081 3.081 15.122 89.96 89.95 59.99 121.3

3.08 3.08 15.11 90.0 90.0 120.0 12H 186 HT 1000 BJAPAJ 1 171 1950 Si C

2 2 5.671 6.282 8.225 67.55 81.05 65.96 217.5

5.61 6.19 8.17 67.7 80.6 61.7 237 MN 11167 MIASA6 30 211 1953 ( Y , Er ) P 01 !2 H2 5.67 6.28 8.20 67.1 80.9 65.8 216 15 MT 11167 lECHAD 28 111 1936 Ca S 01 !2 H2

3 1 7.393 7.395 7.750 103.27 107.76 116.07 328.1 7.30 7.30 7.78 117.9 117.9 90.0 310 110 TL 20605 ZAACAB 339 52 1965 Ba3 ( Co 01 ) 7.31 7.31 7.69 90.0 90.0 120.0 355 HN 20610 BSCFAS 1805 1968 T10.33 ( WO. 67 NbO.33 ) 03

7.32 7.32 7.68 90.0 90.0 120.0 356 HN 20692 INOCAJ 5 1559 1966 Mo RbO.2703

7.32 7.32 7.80 117.9 117.9 90.0 312 110 TL 20693 ZAACAB 339 52 1965 Ba3 ( Cr 01 ) 7.32 7.32 7.83 117.8 117.8 90.0 315 110 TL 20691 ZAACAB 339 52 1965 Ba3 ( Fe 01 ) 7.31 7.31 7.77 118.1 118.1 90.0 311 110 TL 20793 ZAACAB 339 52 1965 Ba3 ( Ge 01 ) 7.35 7.35 7.69 103.2 107.9 116.0 322 15 MT 20833 ACCRA9 5 815 1952 Ba ( CI 03 )2 ! H2 7.36 7.36 7.72 61.6 61.6 60.0 302 166 RN 20863 ZAACAB 357 261 1968 K2 Pti Se6 7.10 7.10 7.75 103.1 107.9 116.1 328 15 MT 21025 JCPSA6 18 1883 1968 Ba ( CI 03 )2,! H2

24
III. Reduction and derivative supercell and subcell calculations

A. Introduction

The reduction and derivative supercell and subcell program function carries out two types of calculations:

1) the calculation of the reduced cell of the lattice; and 2) the calculation and reduction of specified derivative

supercells and/or subcells. Starting with any primitive or centered cell of the lattice, this program function

calculates the standard cells which are useful in the determination of metric symmetry, in finding the re- lationships between unit cells, and in the characterization and identification of unknown materials.

B. Reduction

1. Theory and computer algorithm

The reduced cell (Niggli, 1928) is defined as a unique, primitive cell that is based on the three shortest noncoplanar vectors of the lattice and satisfies a specified set of mathematical conditions. For a cell to be

reduced, the cell must be in normal representation and both the main and the special conditions for reduction

must be satisfied. These conditions are given in table 1. The main conditions assure that one has a cell based on the three shortest lattice translations, while the special conditions assure that the cell is unique. It has been

shown that in some lattices more than one cell is based on the three shortest lattice translations (Santoro &

Mighell, 1970); Gruber (1973) has shown that at most five different cells of this type may exist in the same

lattice. A summary of the reduction procedure is outlined in figure 1. The computer program calculates the

reduced cell starting with any primitive or centered cell of the lattice. To obtain the reduced cell, the program

carries out a series of cell transformations. The total transformation matrix from the initial to reduced cell, T„

is obtained by multiplying together the individual matrices in reverse order:

T, = (T„) (T„.,) (T„.2) ... (T,),

where Tn is the matrix used in the last cell transformation. If the initial cell is a primitive cell of the lattice, the

transformation matrix, T„ will have all integer elements and a determinant of one. Reduction is a mathematical procedure that leads to a unique cell in all cases, provided there is no

experimental error in the unit cell parameters. Unfortunately, experimental error is always present for cells determined in the laboratory. Furthermore, in our experimental work and in evaluating data for the NBS

Crystal Data Center, we have noted that the reported precision on the unit cell parameters is not necessarily a reliable indicator of the accuracy, and that it would be dangerous to estimate accuracy from the reported

precision in all cases. Therefore, when testing the conditions given in table 1, the present program does not use the reported experimental errors, but instead decides whether the conditions are satisfied to within standard

tolerances fixed in the program. These standard tolerances are set low with respect to routinely reported

estimated standard deviations (e.s.d.'s). The computer program reduces the cell based on its geometry and does

not, in general, evaluate the effects of experimental errors in the reduction procedure. Therefore, certain

precautions must be taken when interpreting the results of the reduction program; these are discussed in the

following sections. Additional details concerning the reduction procedure used in the program are given in Acta Crystallographica (Santoro & Mighell, 1970), and in the International Tables for X-ray Crystallography

(1969 a).

2. Uses of reduced cells

Since it is unique, the reduced cell has many practical applications. For example, the reduced cell can be

used in the characterization and identification of materials. By transforming two different cells of the lattice to the same, standard cell, reduction procedures provide a straightforward means of relating different experimental data for a compound. In a similar manner, experience in the NBS Crystal Data Center has shown that the reduced cell plus some chemical information, such as the empirical formula, can be used to identify

unknown materials. In fact, for most molecular compounds, the reduced cell alone is sufficient to identify the

material. This identification procedure is based on matching the unknown reduced cell against the entire NBS

Crystal Data File (1982). This File contains chemical and crystallographic data on approximately 60,000

25
materials; each lattice in the File is represented by its reduced cell. File surveys have shown that the lattice is highly characteristic of a material. This is not entirely surprising since matching reduced cell parameters may

be thought of as matching the positions of all possible d-spacings of a powder pattern. The powder method of

identification, based on matching observed d-spacings and intensities, is a widely used analytical technique.

However, there are many advantages to the lattice-matching approach. Additional information concerning the

lattice-formula identification procedure is given in section II. In addition to identification, the reduced cell may be used to determine the metric symmetry of the lattice. Table 3 defines the relationships between the 44 reduced forms, calculated from the reduced cell, and the

conventional Bravais lattice types. Analyses of the reduced cells in the NBS Crystal Data File have shown that the metric symmetry and the crystal symmetry are usually identical. When the two symmetries differ, it is not uncommon for the crystal symmetry to have been reported incorrectly in the literature. For details concerning

metric symmetry determination using the reduced ceil, see International Tablesfor X-ray Crystallography (1969

a), and Mighell & Rodgers (1980).

3. Interpreting the results of a reduction program

For each unit cell, the computer program always gives the unique, reduced cell defined by Niggli (1928).

However, in certain cases, after the experimental error has been considered, further reduction may be necessary. This is due to the interactions of the experimental errors with the normalization procedure and the process

of satisfying the special conditions for reduction. Therefore, after the reduced cell and reduced form have been

calculated, one should assume liberal experimental errors and inspect the conditions given in table 1. If there

is additional specialization in the reduced form (i.e., b • c = 0, a • b = l/2a • a, etc.), further reduction may be

possible. Table 2 gives a procedure for determining the reduced cell from an unreduced cell based on the three

shortest lattice translations. For cases in which further reduction may be possible, an alternate procedure may

be used. First, when inspecting table 1, look for extra specialization in the reduced form and set the dot products

to the ideal values; re-calculate the 'reduced' cell from the idealized reduced form. Second, re-run the reduction

program using the idealized 'reduced' cell. Third, check the new reduced form using tables I, 2 and 3 to see

that the reduction process is complete and that the new cell reveals the highest possible metric symmetry.

Finally, apply the transformation matrix to the initial cell and evaluate the experimental errors. In the determination of metric symmetry, the special conditions for reduction are critical and care must

be taken to assure that the reduced cell revealing the highest metric symmetry has been obtained. Therefore,

after the reduced form has been calculated, liberal experimental errors should be used when inspecting tables

1, 2, and 3. Any extra specialization in the cell matrix (see table 1) may mean that further reduction is possible

and that the reduced form does not reveal the highest metric symmetry. Alternatively, we have found that a

matrix technique (Himes & Mighell, 1982) is extremely reliable in determining the metric symmetry of the

lattice to within any specified tolerances for the unit cell parameters. The matrix technique can be used

independently or in conjunction with reduction. In addition to symmetry determination, it provides a straightforward method to adjust the experimentally determined cell parameters so that one can calculate the 'ideal' reduced cell and reduced form (Himes, 1983). Clegg (1981) and Le Page (1982) provide alternative procedures

for obtaining a cell that reveals the highest metric symmetry of the lattice. When identifying an unknown using the reduced cell, it has been found that the three shortest vectors and

cell volume are usually sufficient to make the identification. In practice, a match of a,b,c,V of the unknown

reduced cell against an entry in the NBS Crystal Data File can be made with the angles of the two cells in disagreement. It is still likely that the two cells define the same lattice because, when the experimental errors are considered, one of the cells may not be normalized or may not be reduced with respect to the special

conditions. Sometimes it is obvious that the two cells define the same lattice and the transformation matrix

relating them can be found by inspection. In more difficult cases, further reduction must be carried out, or a mathematical approach (Santoro, Mighell & Rodgers, 1980) may be used in order to determine if the two cells define the same lattice. In summary, the computer program always calculates the reduced cell based on the geometry of the input

cell. When the reduced cell and reduced cell matrix are evaluated using liberal experimental errors, in certain

cases, further reduction may be indicated. In practice, the reduction program almost always obtains the

reduced cell, or a cell whose reduced form has one or more unsymmetrical scalars near zero and requires

re-normalization (see table 2) to obtain the reduced form revealing the highest possible symmetry of the lattice. Other instances in which further reduction is necessary are exceptional.

26
4. Examples of interpreting the results of a reduction program

a. Further reduction is not necessary

The compound dichlorobis(4-vinylpyridine)zinc(II) has been described in the triclinic space group PI with

the unit cell parameters

a = 7.501(4) b = 7.522(5) c = 14.482(6) A

a = 90.41(4) fi = 90.53(4) y = 105.29(5)°

(Steffen & Palenik, 1977; see also Marsh & Schomaker, 1979). In this case, the published unit cell is the reduced

cell. Inspection of the following reduced cell matrix reveals that a • a = b • b and b • c = a • c within a small

tolerance.

/ 56.265 56.580 209.728\

V -.780 -1.005 -14.879/

According to table 1, all of the main and special conditions for a type II cell are still satisfied if these equalities are true. Thus, further reduction is not indicated and it is not necessary to consult table 2. The specialized

reduced cell matrix corresponds to reduced form number 14 in table 3. The matrix (1 10/-1 10/001) may

be used to transform the reduced cell to the C-centered monoclinic cell with

a =9.115 b = 11.942 c = 14.482A

a = 89.93 /3 = 90.77 y = 89.83°

.

If more liberal tolerances are assumed, the reduced cell matrix is even more specialized with a • a = b • b and

b • c = a • c = 0. With these extra conditions, no further reduction is necessary (see table 1) and the reduced

form corresponds to number 13 in table 3. The reduced cell can be transformed to a C-centered orthorhombic

cell with the same cell parameters as the C-centered monoclinic cell. Thus, even though the interpretation of

the unit cell parameter errors does not result in further reduction, it does affect the analysis of the metric

symmetry for this compound. If a value of 0.04 degrees, obtained from the original triclinic cell, is used to estimate the errors for the angles of the transformed cells, the compound will have exact monoclinic metric

symmetry within approximately 5 e.s.d.'s and exact orthorhombic metric symmetry within approximately 20

e.s.d.'s. To summarize, we first assumed liberal experimental errors and looked for specialization in the reduced

cell matrix. Then, the conditions for reduction specified in table 1 were checked. In this example, the conditions were met. If all of the conditions were not satisfied due to the additional specialization, table 2 could be used

to obtain the appropriate transformation matrix to the reduced cell. Finally, table 3 was used for the deter- mination of metric symmetry.

An alternate approach that could be used is to look for specialization in the reduced cell matrix and set the

dot products to the ideal values. Idealized cell parameters are then calculated from the idealized cell matrix.

Table 1 must be checked to verify that the idealized cell is reduced; if not, table 2 may be used or the idealized

cell may be rerun through the reduction program. Finally, table 3 is used to determine the metric symmetry

of the idealized reduced cell. Data for the Zn(II) compound will be used to illustrate the second approach. For the Zn(II) compound

to have monoclinic metric symmetry, it was assumed that the reduced cell matrix was specialized with

a • a = b • b and b • c = a • c. One way to idealize the reduced cell matrix is to take the average of the

appropriate dot products. Thus, the idealized cell matrix becomes

/ 56.423 56.423 209.728>

892 -.892 -14.879>

and the idealized cell parameters are

a =7.5115 b =7.5115 c = 14.482 A

a = 90.47 fi = 90.47 y = 105.29 °.

27
For the compound to have orthorhombic metric symmetry, the reduced cell matrix must be specialized so that a • a = b • b and b • c = a • c = 0. The idealized cell matrix becomes

/ 56.423 56.423 209.728\

[ -14.879/

and the idealized cell parameters are

a =7.5115 b =7.5115 c = 14.482 A

a = 90.00 /3 = 90.00 y = 105.29 °.

When table 1 is checked or the idealized cell parameters are rerun through the reduction program, it can be

shown that, in this example, both the idealized monoclinic and the idealized orthorhombic cells are also the

idealized reduced cells. (This is not always true.) Since the original triclinic cell was the reduced cell, the

reported e.s.d.'s may be applied directly to the idealized reduced cells when estimating the deviations from

exact monoclinic or orthorhombic metric symmetry. The Zn(II) compound is metrically monoclinic since the

experimental errors estimated in this way correspond to a maximum of approximately 3 e.s.d.'s. In order to have

exact orthorhombic metric symmetry, the original triclinic cell parameters must be changed by a maximum of

approximately 13 e.s.d.'s. Although these two analyses of the reduced cell matrix were very similar, the estimate of the number of

e.s.d.'s required for exact monoclinic or orthorhombic metric symmetry varied. In general, there are several

ways to analyze the reduced cell matrices when considering the effects of the experimental errors. It should

be remembered that the method chosen may not be the one that minimizes the estimated errors. Therefore, all results should be considered even if they require relatively large experimental errors. In this example, experimental techniques should be used to determine whether the Zn(II) compound has triclinic, monoclinic or orthorhombic crystal symmetry.

b. Further reduction is required due to a normalization problem

The NBS*LATTICE program always calculates the reduced cell based on the geometry of the input cell. In the previous example, the cell obtained from the reduction program did not require further reduction when

rather large experimental errors were assumed. However, as this example illustrates, the experimental error need not be large to interact with the normalization requirements and the special conditions for reduction. It

is important to remember that the tolerances for testing the conditions given in table 1 are set low with respect

to routinely reported e.s.d.'s. The unit cell parameters,

a = 3.0804(6) b = 3.0806(8) c = 15.122(4) A

a = 89.96(2) fi = 89.99(2) y = 1 19.99(2) °,

were determined for alpha-silicon carbide using an automated diffractometer. The following reduced cell and

reduced cell matrix were calculated by the program:

reduced cell: a = 3.0810 b = 3.0806 c = 15.122 A

a = 89.96 fi = 89.95 7 = 59.99

°

/ 9.492 9.490 228.675\

, .033 .041 4.747/

reduced cell matrix =

Inspection of the reduced cell matrix reveals that it is specialized with a-a = b-b = 2a'b and

b • c = a • c = within a small tolerance. When table 1 is checked, all of the main and special conditions for a positive reduced form are satisfied. However, the cell is NOT reduced because it is neither a type I nor a type

II cell when the extra specialization is introduced. This is apparent when the idealized cell parameters are calculated from the idealized cell matrix.

Idealized cell matrix =

Ideahzed cell:

28

/ 9.492

I

9.492 228.675\

4.746,

a = 3.0809 a = 90.00b = 3.0809

a = 90.00

c = 15.122 A

7 = 60.00

°
The computer program calculated the geometrically reduced cell with all angles less than 90 degrees (type I cell). However, when small experimental errors are assumed, further reduction is indicated. Before using table 2 to find a matrix from the unreduced to the reduced cell, one must recognize that the cell type is 'non-standard'

because the angles are neither all less than 90 degrees (type I) nor all greater than or equal to 90 degrees (type

II). The special relations between the unsymmetrical scalars can be found in the fifth row of table 2, nonstandard condition (d), with b-c = 0, a-c = and a • b > 0. Using the table, the new cell matrix (idealized)

becomes

9.492 9.492 228.675

-4.746)•

At this point, reduction is complete. The resulting unit cell corresponds to a type II cell and all of the main and

special conditions for a negative reduced form have been satisfied (table 1). This reduced form, number 12 in table 3, reveals the hexagonal symmetry of the lattice. The matrix (1 10/010/001) from the initial to the

'reduced' cell (calculated by the program) may be multiplied by the matrix (-100/010/00-1) obtained

from table 2 to obtain a transformation matrix (-1 -1 0/0 1 0/00-1 ) from the initial cell to the reduced

cell after allowing for the experimental errors. The reduced cell calculated from the refined unit cell parameters had the a-cell parameter slightly greater

than the b-cell parameter. This occurred because they were equal within the small tolerance set in the computer

program. However, within the program, angles in the approximate range of 89.98 to 90.02 degrees are considered to be equal to 90 degrees. Since the value of alpha (89.96 degrees) was slightly outside of this range,

a reduced cell requiring further reduction resulted. If the initial value for alpha were not 89.96 but 89.981, the

reduced cell calculated by the program would require no further reduction. In this example, the need for further reduction arose because one or more of the unsymmetrical scalars was near zero. When these scalars are idealized to be zero, the cell matrix and, consequently, the unit cell are both non-standard. In practice, when

considering the effects of experimental errors, the most common reason for a cell reduced by the computer

program to require further reduction is this 'normalization' problem.

c. Example requiring further reduction

This example (Himes, 1983) shows that one must carefully inspect the relationships between the scalars in the reduced cell matrix output by a reduction program. Consider the reduced cell and reduced cell matrix

calculated from a C-centered monoclinic cell (a = 18.21(1), b = 10.509(1), c = 20.69(1)A, fi = 126.00(5)°

)

reported in the literature.

Reduced cell: a = 10.509

a= 107.12

b = 10.512

fi = 90.00

c = 17.798 A

y= 119.99°

/ 110.44

1-55.07

110.55 316.77 \

-55.22 i Reduced cell matrix =

With no experimental error, the cell is reduced and the reduced cell matrix corresponds to monoclinic reduced

form number 39. However, within a small tolerance, the cell matrix is speciaHzed so that a • a = b • b and

|b-c| = |a-b| =a- a/2. Inspection of table 1 reveals that the cell is not reduced when these conditions are

true. A transformation matrix to the reduced cell obtained after considering the effects of experimental error may be found by consulting table 2. First, the idealized cell matrix and cell are calculated.

C^: ,.,.,„,. , 110.48 110.48 316.77

Idealized cell matrix =

{^^^^^^ ^ _55_24

Idealized cell: a = 10.51 b = 10.51 c = 17.80 A

a = 107. 17 fi = 90.00 y = 120.00

°

29
Using this idealized cell matrix and table 2, the reduction can be completed in two steps. The idealized cell is a type II cell. The special relations between the scalars, a • a = b • b and | a • c | < | b • c | , are found in the

seventh row of the table. The transformation matrix (0-10/-100/00-l)istobe applied to the cell, and

the idealized cell matrix becomes:

/110.48 110.48 316.77 \ V -55.24 -55.24 )

'

According to table 1, special conditions (d) and (e) for a type II cell are not satisfied; thus, further reduction

is necessary. In this case, the special relations between the scalars can be found in row 13 for a type II cell with

a • c = -a • a/2 and a • b 7!^ 0. The transformation matrix to be applied to the cell is(-100/010/-10-l)

and the idealized cell matrix becomes:

/1 10.48 110.48 316.77 \ V 55.24 55.24 55.24 )

"

The conditions for a positive reduced form (calculated from a type I cell) now apply. Inspection of table 1 reveals that the reduction process has been completed. The idealized reduced cell, with

a = 10.51 b = 10.51 c = 17.80 A

a = 72.83 fi = 72.83 y = 60.00 °

, may be calculated from the idealized reduced cell matrix. The reduced form, number 9 in table 3, reveals the

rhombohedral nature of the lattice. A transformation matrix from the initial cell to the reduced cell may be obtained by multiplying together

the transformation matrix output by the computer program (010/ -.5 -.50/101) and the two matrices

obtained from table 2. Thus, the matrix (-.5 -.50/0-10/.5 -.5 1) may be used to transform the C-centered

monoclinic cell to the 'non-idealized' reduced cell with

a = 10.512 b = 10.509 c = 17.808 A

a = 72.84 fi = 72.78 y = 60.01 °. An alternate procedure would be to idealize the initial cell matrix output by the computer program, calculate

the idealized cell, and input this cell into the reduction program. In either case, a transformation matrix to the

'non-idealized' reduced cell (after the consideration of the experimental error) can be found.

30
Figure 1. Reduction sequence

1 Any Cell of the Lattice

i 2 Transform to a Primitive

Cell of the Lattice

i 3 Calculate a Cell Based on the Three Shortest

Lattice Translations

i 4 Order Cell Edges

so that a < b < c 1

5 Put the Cell in Normal Representation

i 6 Select a Unique Cell

i 7 Analyze and Interpret the Results

Notes:

1 The program reads in the cell and cell centering.

2 If other than primitive, the program transforms the initial cell to a primitive cell of the lattice. The required transformation matrices are stored in the program.

3-5 The program transforms the primitive cell calculated in step 2 to a primitive cell that passes the main conditions for reduction specified in table 1.

6 The program transforms the cell to one that passes the special conditions for reduction

specified in table 1. When this step has been completed, the computer program has reduced the

input cell based on its geometry.

7 The results of the reduction procedure must be analyzed and interpreted taking into account the effects of the experimental errors. First, one must be certain that the geometrically reduced cell output by the computer program has been correctly normalized and the special conditions have been satisfied when the experimental errors are considered. In certain cases, further reduction may be indicated (this procedure is given in table 2). Finally, the metric lattice symmetry and other properties of the lattice may be deduced from the reduced cell.

31
Table 1. Conditions for a reduced cell

Cell s is specified by three noncoplanar vectors a,b, and c and cell matrix S = (a-a b-b c-c/b-c a-c a-b)is defined by the dot products between these vectors. To be reduced, the cell must be in normal representation (type I or II) and all the main and special conditions for the given cell type must be satisfied. The main conditions are used to establish that a cell is based on the three shortest lattice translations. The special conditions are used to select a unique cell when two or more cells in the lattice have the same numerical values for the cell edges. If inspection of the table indicates that a cell which is based on the three shortest vectors is not reduced, one can complete the reduction with

the aid of the transformations given in table 2.

A. Positive reduced form, type I cell, all angles < 90 degrees

Main conditions: a-a<b-b<c«c

a • c < 1/2 a • a ;

b • c < 1/2 b . b ; a • b < 1/2 a • a

Special conditions: (a) if a • a = b • b

(b) if b . b = c . c

(c) if b • c = 1/2 b . b

(d) if a • c = 1/2 a • a

(e) if a • b = 1/2 a • a

then b •c < a-c

then a c < a-b

then a b < 2a-c

then a b<2b-c

then a c < 2b-c

B. Negative reduced form, type II cell, all angles > 90 degrees

Main conditions: (a)a-a<b-b<c-c; b-c<l/2b-b;

a • c < 1/2 a • a ; a • b < 1/2 a • a

(b) ( |b"c| + |a.c| + |a-b|)"< l/2(a.a + b.b)

Special conditions: (a) if a • a = b-b

(b) if b • b = C 'C

(c) if |b-c| = 1/2 b .

(d)if|a.c| = 1/2 a .

(e) if |a-b| = 1/2 a •

(f)if(lb.c 1 + |a-c| +

l/2(a

a + b . b)

then

then

then

then

then

|a.b|) =

|b-c| < |a.c|

|a-c| < |a-b|

a-b =

a-b =

a-c =

then a-a < 2 |a-c| + |a-b|

32
Table l,"*" Transformationsfor determining the reduced cellfrom an unreduced cell that is based on the shortest three translations ofthe lattice

This table includes the transformation matrices that are required to convert a non-standard cell to a type I or a type II cell and the matrices that are required to satisfy the special conditions for reduction.

The cells s and s' are defined by three noncoplanar vectors (a, b, c) and (A, B, C), respectively. The cell matrices S = (a-a b-b c-c/b-c a-c a-b)andS'=(A-A B-B C-C/BC AC A • B ) are defined by the dot products of the vectors defining the cells.

The relations between the scalars in column 4 are not intended to show the vector operations resulting from the application of a matrix

in column 3 to cell s. Instead, column 4 gives the numerical values of B • C, A • C, A • B in terms of the numerical values of b • c, a • c, and

a • b. The values for A • A, B • B, and C • C are not given in column 4 since numerically A-A = a-a, B-B = b-b, and C • C = c • c. These equalities hold only for those cases in which the relations between the scalars given in column 2 are exact.

To use this table, calculate the cell matrix S and check the scalar relations in column 2. If a given relationship is true, the cell is not reduced and it must be transformed by the matrix given in column 3 to give cell s' having the cell type specified in column 5. The resultant

cell matrix S' may be obtained from the data in column 4. The cell s' and cell matrix S' are relabeled as s and S, respectively. The process

is repeated in an iterative manner until the cell is reduced. In practice, the following strategy is convenient: 1) set the dot products in S

to ideal values (i.e., set the relations between the scalars for S to be exactly those given in the second column) considering the effects of

experimental error; 2) use the table in an iterative manner (described above) until the reduction process is complete; 3) obtain the unidealized reduced cell by applying the final transformation matrix to the original unreduced cell.

Type of Relations between Scalars Transformation Matrix Resultant Unsymmetrical Type of Cell Cells in Matrix S stos' Scalars in Matrix S' s'

non- b • c> 0, a-c<0, a-b<0 / 1 °\ BC = b-c I standard -1 ° AC = |a-c|

\ -1/ AB = |a-b|

non- b • c < 0, a-c>0, a.b<0 /-I o\ BC = |b-c| I standard 1 AC = a-c \ -1 / AB = |a.b|

non- b • c < 0, a-c<0, a-b>0 /-I o\ BC = |b-c| I standard -1 AC = |a-c|

\ 1/ AB = a-b

non- a) b • c < 0, a-c>0, a-b>0 or /

1 o\ BC = b-c II standard b) b • c < 0, a-c>0, a-b = or ° -1 AC = —a-c

c) b • c < 0, a-c = 0, a-b>0 or \ -1/ AB = -a-b

d) b c = 0, a-c>0, a-b =

non- a) b • c > 0, a-c<0, a-b>0 or /-^ 0\ BC = -b-c II standard b) b . c> 0, a-c<0, a.b = or 1 AC = a-c

c) b • c = 0, a-c<0, a-b>0 or \ -1/ AB = -a-b

d) b • c = 0, a-c = 0, a-b>0

non- a) b • c > 0, a-c>0, a-b<0 or /-I °\ BC = -b-c II standard b) b • c > 0, a-c = 0, a-b<0 or -1 A.C = — a - c

c) b c = 0, a-c>0, a-b<0 or \ 1/ AB = a-b

d)b-c> 0, a-c = 0, a-b =

I or II a-a = b- b and /

-1 0\ BC = a-c I or II, |a.c| < |b •c| [-' M AC = b-c respectively

\ -1/ AB = a-b

I or II b-b = c- c and /-I o\ BC = b-c I or II, |a-b| < |a •c| -1 A.C = ab respectively

\ -1 0/ AB = a - c

33
Table 2.+ Transformations for determining the reduced cell from an unreduced cell that is based on the shortest three translations of the

lattice—Continued

Type of

Cells

Relations between Scalars

in Matrix S

Transformation Matrix

stos'

Resultant Unsymmetrical

Scalars in Matrix S' Type of Cell

s'

II

II

II

II

be = b.b/2

2a-c < a-b

a • c = a • a/2 2b • c < a • b

a • b = a • a/2 2b • c < a • c

a • b = -a • a/2

ac#0

and

and

and

b-c = -b'b/2 and ab#0

a • c = —a • a/2 and a.b:?tO

and

a-a + b-b =

2(|b-c| + |a-c| + |a-b|) and

2|a-c| + |a-b| < a-a

(: :i •)

(;• ; ;)

\ -1 /

10-10

-1 -1 -10

1 -1 -1

( ; -i ;)

BC = b.b/2

AC = a-b—a- c AB = a-b

BC = a. b-b. c AC = a a/2

AB = ab

BC = a-c—b. c AC = a-c AB = a. a/2

BC = b.b/2

AC = |a.c| + la- b|

AB = |a.b|

BC = |b.c| + la b| AC = a.a/2

AB = |a.b|

BC = |b.c| + |a c| AC = |a.c|

AB = a •a/2 BC = |a.b|+ |b • cl-b.b

AC = |a.b| + |a c|—a.a AB = a-b

II

^ Table 5.1.2.2 of the International Tables for X-ray Crystallography (1969 a) modified to include published revisions and additional information.

34
Table 3. Metric classification of the 44 reduced forms

This Table is reprinted from Mighell & Rodgers, 1980; based on Table 5.1.3.1 prepared by A. D. Mighell, A. Santoro and J. D. H. Donnay for the International Tables for X-ray Crystallography (1969 a) and published revisions.

Reduced Form

Uo.

Reduced Form Matrix Reduced Form

Type

Bravais

Lattice

Cell Transformation Reduced -> Conventional

First Row Second Row

a'a b-b cc b-c a-c a-b

a = b = c

1 a-a a-a a*a a-a2

a-a2

a-a2

+ Cubic F 111/111/111

2 a*a a*a a*a b-c b-c b-c + Rhombohedral hR iio/ioi/iii

3 a-a a*a a*a - Cubic P 100/010/001

4 a«a a-a a-a -Ib-cl -Ib-cl -Ib-cl - Rhombohedral hR 110/101 /ill

5 a-a a*a a'a

8-83

a-a3

a-a3

- Cubic 1 101/110/011

6 8*8 8*8 3*8 -a-a + la-bl 2

-a-a + |8-b| 2

-|8-b| - Tetragonal 1 011/101/110

7 8*8 a-a 8*8 -Ib-cl -a-a + lb-cl 2

-a-a + |b-cl 2

- Tetragonal 1 101/110/011

8 a-a a-a a-a -Ib-cl -la-cl -da-al-lb-ci-la-cl) - Orthorhombic 1 i 10/101 /oil

a = b 9 a-a 8'a c"c

a-a2

a-a2

8-82 + Rhombohedral hR 100/110/113

10 a-a a-a cc b-c b-c 8-b + Monodinic C** 110/iio/ooi

11 a-a 8-8 c*c - Tetragonal P 100/010/001

12 8*8 a*a cc

a-a2

- Hexagonal P 100/010/001

13 a-a a*a cc -la-bl - Orthorhombic C 110/110/001

14 8-8 8-8 cc -Ib-cl -Ib-cl -la-bl - Monodinic C** 110/110/001

15 a-a 8*8 cc

8-82

a-a2

- Tetragonal 1 100/010/112

16 8*8 8-8 cc -Ib-cl -Ib-cl -(a-8-2|b-c|) - Orthorhombic F iio/iio/112

17 a-a a-a cc -Ib-ci -la-cl -(a-a-|b-c|-|a-c|) - Monodinic I'*"*' ioi/ii 0/011

b=c

18 8-8 b-b b-b

8*84

a8

2

8*82 + Tetragonal 1 oii/iii/100

19 a-a b-b b-b b-c

a-a2

8-82 + Orthorhombic 1 ioo/oi 1/111

20 a-a b-b b-b b-c a-c a-c + Monodinic C+ 01 1 /oil /i 00

21 8-8 b-b b-b - Tetragonal P 010/001/100

22 a-a b-b b-b

b-b

2

- Hexagonal P 010/001/100

23 a-a b-b b-b -Ib-cl - Orthorhombic C 011/011/100

24 a-a bb b-b

2

a-a3

a-a

' 3

- Rhombohedral hR 121/011/100

25 a-a b-b b-b -Ib-cl -la-ci -la-cl - Monodinic C+ 011/011/100

35
Table 3. Metric classification of the 44 reduced forms —Continued

Reduced Form

No.

Reduced Form Matrix | Reduced Form

Type

Bravais

Lattice

Cell Transformation Reduced ^

Conventional

First Row+t* Second Row

a-a b*b c-c b-c a-c a-b

a«:b4:c 26 a-a b-b c*c

a-a4a-a2

a-a2

+ Orthorhombic F 100/120/102

27 a*a b-b c-c b-c

a-a 2a-a2 + Monoclinic 1*** 011/100/111

28 a*8 b-b c-c a-b2

a-a 2

a-b + Monodinic C iOO/1 02/010

29 a-a b'b cc

a-c2

a-c

a-a2

+ Monoclinic C 100/120/OOi

30 a-a b-b c-c b-b

2

a-b 2

a-b + Monoclinic C 010/012/iOO

31 a-a b-b c-c b-c a-c a-b + Triclinic P 100/010/001

32 a-a b-b c-c - Orthorhombic P 100/010/001

33 a-a b-b c-c -la-cl - Monoclinic P 100/010/001

34 a-a b-b c-c -la-bl - Monoclinic P ioo/ooi/oio

35 a-a b-b c-c -Ib-cl - Monoclinic P 010/ioo/ooi

36 a-a b-b c-c

a-a

' 2

- Orthorhombic C 100/102/010

37 a-a b-b c-c -Ib-cl

a-a

" 2

- Monoclinic C* 102/100/010

38 a-a b-b c-c

a-a2

- Orthorhombic C 100/120/001

39 a-a b-b c-c -Ib-cl

a-e2

- Monoclinic C" 120/100/001

40 a-a b-b c-c b-b

2

- Orthorhombic C OiO/012/100

41 a-a b-b c-c b-b

2

-la-cl - Monoclinic C'*' 012/010/100

42 a-a b-b c-c b-b

2

a-a2

- Orthorhombic 1 iOO/010/112

43 a-a b-b c-c b-b-|a-b|

2

a-a-la-bl

2

-Ia-b| - Monoclinic 1 ioo/i 12/010

44 a-a b-b c-c -Ib-cl -la-cl -la-bl - Triclinic P 100/010/001

+ lfa-a<4|a-cl \

* If bb<4|bc| > •* If c-c<4|b-cl J

Premultiply Table Matrix by 001/010/101 (I centered)

+t If 3a-a<c-c + 2|8-c| •••If 3b-b<c-c *^^J,l*'', 1 Premultiply Table Matrix by 101/010/100 (C centered) + 2lb-cl ) ttt No required relationships between symmetrical scalers for reduced forms 26-44.

36
C. Derivative supercells and subcells

1. Theory and computer algorithm

If a three by three upper-triangular matrix Q with integer elements and a determinant of n (n = a positive

integer greater than 1) is used to transform a primitive cell of the original lattice (PCOL), one obtains a derivative supercell of n times the volume of the PCOL. If the transpose of the inverse of Q is used to transform

a PCOL, one obtains a derivative subcell of 1/n times the volume of the PCOL (Santoro & Mighell, 1972). It has been shown that unique sets of upper-triangular matrices can conveniently be generated for each value of

the determinant (Santoro & Mighell, 1973). There are 7,13,35,31,91,57,155,130 unique upper-triangular matri- ces with determinants of 2,3,4,5,6,7,8,9, respectively. Each set of upper-triangular matrices can then be used to obtain all the unique supercells of n times the volume of the PCOL. By using the transpose of the inverse of

each of these matrices, one can calculate all the unique subcells of 1/n times the volume of the PCOL. In the

generation of the derivative cells, sets of these matrices can be used to transform any PCOL but they must be

applied to the same cell. (The NBS*LATTICE program applies these matrices to the reduced, primitive cell of the original lattice.) Table 4 gives the upper-triangular matrices for determinants with n of 2,3 and 4. For a general triclinic lattice with no specialization in the reduced form, the derivative supercells and subcells for each value of the determinant are metrically different. However, for certain specialized triclinic lattices or for

lattices of higher metric symmetry, some of the derivative cells are metrically identical although they are oriented differently in space. A summary of the procedure used by the program to calculate derivative supercells and subcells is outlined

in figure 2. The program first calculates the reduced cell of the original lattice, (RCOL) steps 1-6. Next, the

specified set(s) of upper-triangular matrices are generated (step 7). By using the upper-triangular matrices, set(s) of supercells and/or subcells are generated and reduced (step 8). With a given upper-triangular matrix, one can

determine a superlattice and a sublattice. In the program, each supercell is calculated in a two-step process: first, an intermediate supercell is calculated by transforming the RCOL by an upper-triangular matrix; second, the

intermediate cell is reduced to yield a reduced supercell of the lattice. Each subcell is calculated in a similar manner using a three-step process: first, the transpose of the inverse of each upper-triangular matrix is calculated; second, an intermediate subcell is calculated by transforming the RCOL by the transpose of the

inverse of the upper-triangular matrix; and third, the intermediate subcell is reduced. The present version of the

program will calculate all the upper-triangular matrices with determinants ranging from 2 to 9. Thus, supercells

with n (n = 2 to 9) times the volume of the RCOL and/or subcells with 1/n (n = 2 to 9) times the volume of

the RCOL can be determined.

2. Uses of derivative supercells and subcells

There are many reasons for calculating derivative cells of the lattice. For example, as outlined below, the

calculation of derivative cells can assist in obtaining the correct cell of the lattice and in the identification of

unknown compounds. Additional discussions and examples are given in section II.

a. Derivative cells can lead to the determination of the correct cell of the lattice.

Cells determined from single-crystal and powder data are sometimes derivative cells of the lattice. However, it is still possible to deduce the correct cell by systematically calculating derivative cells. For

example, a cell determined on a single-crystal diffractometer can be a subcell of the lattice; this occurs if one

has determined a supercell in reciprocal space (i.e. by missing rows of diffraction intensities, etc.). If a subcell having 1/2 the volume of the correct cell has been determined, the correct cell can be found by calculating the

7 possible supercells of the experimentally determined subcell. One of the 7 generated supercells is the correct

cell of the lattice. A cell determined by powder-indexing procedures can also be a derivative subcell, supercell, or composite cell of the lattice. By systematically calculating derivative cells from the initial powder cell, one

can often deduce the correct cell.

h. Derivative cells can assist in the identification of unknown compounds by using the NBS Crystal Data File.

The ability to calculate derivative cells is important in the identification of compounds. One must be

careful when identifying unknown materials by using the NBS Crystal Data File because the unknown cell or

the corresponding entry in the File may be in a derivative relationship (i.e., one may be the derivative lattice of the other). Therefore, one may wish to match routinely the reduced cell and selected derivative cells of the

unknown against the File.

37
Figure 2. Derivative supercells and subcells

1 Any Cell of the Lattice

2 Transform to a Primitive Cell of the Lattice

3 Calculate a Cell Based on the Three

Shortest Lattice Translations

4 Order Cell Edges so that a < b < c

5 Put the Cell in Normal Representation

6 Select a Unique Cell

7 Generate Upper-Triangular Matrices

8 Generate Derivative Lattices:

Transform the Reduced Cell of the Original

Lattice (Step 6) by the Appropriate Matrices;

Reduce the Derivative Lattices.

.a 7 Supercells of 2V

8b 13 Supercells of 3V

8c 35 Supercells of 4V

8d 31 Supercells of 5V

8e 91 Supercells of 6V

8f 57 Supercells of 7V

8g 155 Supercells of 8V

8h 130 Supercells of 9V

8a' 7 Subcells of 1/2V

8b' 13 Subcells of 1/3V

8c' 35 Subcells of 1/4V

8d' 31 Subcells of 1/5V

8e' 91 Subcells of 1/6V

8r 57 SubceUs of 1/7V

8g' 155 Subcells of 1/8V

8h' 130 Subcells of 1/9V

Analyze and Interpret the Results

8a 7 Supercells of 2V

8b 13 Supercells of 3V

8h 130 Supercells of 9V

8a' 7 Subcells of 1/2V

8b' 13 Subcells of 1/3V

8h' 130 Subcells of 1/9V

Notes:

1-6

7

Same steps as in the reduction procedure outlined in figure 1.

The required upper-triangular matrices generated by the computer program are stored in arrays. The program will not calculate upper-triangular matrices with determinants less than 2 or exceeding 9.

The specified derivative supercells and/or subcells are calculated by transforming the reduced

cell of the original lattice (RCOL) by the appropriate matrices. For each supercell, an uppertriangular matrix is used; for each subcell, the transpose of the inverse of the upper-triangular matrix is used. Each derivative cell is then transformed to the reduced cell of the derivative

lattice.

As indicated by the notations, the program can carry out calculations in either the supercell

(8a.. .8h), the subcell (8a'.. .8h'), or the supercell and subcell (8a...8h,8a'...8h') series. One can

calculate derivative lattices starting at any point and ending at any point within a given series. For example, one can carry out step 8a only, or all the steps 8a.. .8h, or two or more adjacent

steps, such as 8b, 8c and 8d.

38
Figure 2. Derivative supercells and subcells —Continued

8a 7 supercells of two times the volume of the RCOL (step 6) are calculated and reduced in a two-step process: first, the 7 unique upper-triangular matrices (step 7) with a determinant of 2 are used to transform the RCOL to 7 derivative supercells; second, the 7 derivative supercells are then reduced.

8a' 7 subcells of 1/2 the volume of the RCOL (step 6) are calculated and reduced in a three-step process: first, the transpose of the inverse of each of the 7 unique upper-triangular matrices (step 7) is calculated; second, the transposes of the inverses of these matrices are then used to transform

the RCOL to 7 derivative subcells with 1/2 the volume of the RCOL; third, the 7 derivative subcells are then reduced.

8h 130 supercells of 9 times the volume of the RCOL (step 6) are calculated and reduced in a two-step process as outlined in 8a except that 130 upper-triangular matrices are used.

8h' 130 subcells of 1/9 the volume of the RCOL (step 6) are calculated and reduced in a three-step process as outlined in 8a' except that 130 upper-triangular matrices are used.

The results must be analyzed and interpreted taking into account the effects of the experimental

errors.

Table 4. Unique Q matrices generating superlattices for |Q| = 2, 3, 4

For each value of Q , the matrices can be applied to any primitive cell of the original lattice, but they must be applied to the same cell.

|Q|=2 |Q|=4 |Q| = 4 (com.)

(100 / 010 / 002)

(100 / Oil / 002)

(101 / 010 / 002)

(101 / Oil / 002)

(100 / 020 / 001) (110 / 020 / 001)

(200 / 010 / 001)

IQI=3

(100 / 010 / 003) (100 / Oil / 003)

(100 / 012 / 003)

(101 / 010 / 003)

(101 / Oil / 003)

(101 / 012 / 003)

(102 / 010 / 003)

(102 / Oil / 003)

(102 / 012 / 003)

(100 / 030 / 001)

(110 / 030 / 001) (120 / 030 / 001) (300 / 010 / 001)

(100 /

(100 /

(100 /

(100 /

(101 /

(101 /

(101 /

(101 /

(102 /

(102 /

(102 /

(102 /

(103 /

(103 /

(103 /

(103 /

(100/

(100/

(101 /

(101 /

(110 /

(110 /

(111 /

010 / 004)

Oil / 004) 012 / 004) 013 / 004) 010 / 004) Oil / 004) 012 / 004) 013 / 004) 010 / 004)

Oil / 004) 012 / 004) 013 / 004) 010 / 004) Oil / 004) 012 / 004) 013 / 004) 020 / 002) 021 / 002) 020 / 002) 021 / 002) 020 / 002) 021 / 002) 020 / 002)

(111 /

(100 /

(110 /

(120 /

(130/

(200/

(200/

(201 /

(201 /

(200 /

(210 /

(400/

021 / 002) 040 / 001) 040 / 001) 040 / 001) 040 / 001) 010 / 002)

Oil / 002) 010 / 002)

Oil / 002) 020 / 001) 020 / 001) 010 / 001)

39
D. Operation of the Program

1. General

There are two types of input lines required to execute the reduction and derivative supercell and subcell (RSS) program function. The first input Hne is a Program Control Line that specifies the type of program

function and the number of problems to be solved. The second type of input line is an RSS Control Line that defines the input lattice and the calculations to be performed (i.e., various combinations of reduction, supercell, subcell calculations). For N problems specified on the Program Control Line, N RSS Control Lines must

follow (1 < N < 20). Within a single computer run, the RSS program function may be executed any number

of times. A description of the formats for the input lines and typical examples are given below.

2. Description of input lines

a. Program Control Line

The Program Control Line specifies the type of program function and the number of problems to be

solved. For each execution of the RSS program function within a single computer run, only one Program

Control Line is required.

Program Control Line

Format(A5,3X,I2)

Column Format Item

1-5 A5 Type of program function

'LM ' = Lattice Matching

'RSS ' = Reduction and

Derivative

Supercell and Subcell

'TRANS' = Cell Transformation

'INV ' = Matrix Inversion

6-8 3X Blank

9-10 12 Number of problems

Notes:

The RSS program function is specified by 'RSS ' in columns 1-5.

The number of problems specified in a single execution of the RSS program

function may range between 1 and 20.

b. RSS Control Line

The RSS Control Line defines the input lattice and the derivative lattices to be calculated. The lattice is defined by specifying the cell centering and all six unit cell parameters. The program will always carry out the

cell reduction calculations. To calculate derivative lattices, it is necessary to tell the program whether to calculate supercells, subcells, or both supercells and subcells. In addition, for derivative lattice calculations, it

is necessary to tell the program which multiples of the reduced cell volume (V) should be considered.

Supercells having 2 to 9 times the volume of the reduced input cell and/or subcells having 1/2 to 1/9 times the

volume of the reduced input cell may be calculated. There are 7 unique derivative lattices (supercells or

subcells, respectively) having 2 or 1/2 times the volume of the reduced input cell, 13 derivative lattices with

3V or 1/3V, 35 with 4V or 1/4V, 31 with 5V or 1/5V, 91 with 6V or 1/6V, 57 with 7V or 1/7V, 155 with 8V

or 1/8V, and 130 with 9V or 1/9V. For N problems specified on the Program Control Line, N RSS Control

Lines must follow (1 < N < 20).

40
Column

1 1

2-3

4

RSS Control Line

Format(Il,2X,2Il,3X,2Al,6F10.2)

Format

II

2X

II

II

6-8 3X

4 9 Al

5 10 Al

Item

Blank/1/2/3/

Blank = Reduction

1 = Reduction +supercells

2 = Reduction +subcells

3 = Reduction + supercells, subcells

Blank

Blank/2/3/4/5/6/7/8/9/

Initial value (nl) to define the

range of volumes for calculated

derivative lattices.

Blank/2/3/4/5/6/7/8/9/

Final value (n2) to define the range

of volumes for calculated derivative lattices.

Blank

P/A/B/C/F/I/R/ Cell centering

R/H/ Rhombohedral/Hexagonal/

metric axes. Used only for rhombohedral lattices.

6 11-20 F10.2 a (A)

7 21-30 F10.2 b

8 31-40 F10.2 c

9 41-50 F10.2 alpha i")

10 51-60 F10.2 beta

11 61-70 F10.2 gamma

Notes:

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, then a 1, 2 or 3 is used for supercells, or subcells, or both supercells and subcells, respectively.

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, use an integer (nl) from 2 to 9.

If only a reduction is to be carried out, this field is left blank. If derivative

lattices are to be calculated, use an integer (n2) from 2 to 9, but greater than or equal to nl.

41
Notes (cont.)

2-3 These two integers (n 1 and n2) are used to define the range of volumes for the

derivative lattices to be calculated. The supercells to be calculated will have

volumes ranging from (nl)V to (n2)V while the volumes for the subcells to be calculated will range from (l/nl)V to (l/n2)V.

4 To define the lattice, both the cell and cell centering are required. A symbol

specifying the cell centering must be placed in columns 9 and 10. A one

character symbol is used for P,A,B.C,F, and I centered cells. For a cell defining a rhombohedral lattice, a two character symbol is required (RR or RH); the first character of this symbol (*R') is placed in column 9.

5 Column 10 is left blank unless the input cell defines a rhombohedral lattice. For a cell defining a rhombohedral lattice, an 'R' is placed in column 10 if the

lattice is defined by a primitive rhombohedral cell (rhombohedral axes), or an

'H' is used if the lattice is defined by a triply primitive rhombohedral cell (metrically hexagonal axes).

6-8 Cell edges (A).

9-1 1 Cell angles (°). Decimal numbers must be used for fractions of a degree. All

six cell parameters must always be specified regardless of the crystal symmetry.

3. Example of input flowstream

, . .1 ] .1 . .1 ,1 .1

1 RSS 1

2 P 4.99 9.36 9.19 102.1 91.5 68.0

3 RSS 3

4 A 15.380 14.225 9.309 90.00 94.20 90.00

5 RR16.11 16.11 16.11 115.10 115.10 115.10

3 23 RH9.139 9.139 15.536 90.00 90.00 120.00

Notes:

1 The 'RSS ' in columns 1-5 indicates that the RSS program function will be exe- cuted; the T in column 10 specifies that one RSS Control Line will follow (one

problem is to be considered). At most 20 problems may be specified in a single

execution of the RSS program function.

2 Since column 1 is blank, the cell will be reduced and no derivative lattices will be

calculated. Columns 4-5 are also left blank for reduction-only calculations. The T

'

in columns 9-10 indicates that the input cell is primitive. All six unit cell parameters

are specified in columns 11-70.

3 The RSS program function will be executed a second time; in this case, three RSS

Control Lines will follow (three independent problems are to be considered).

42
Notes (cont.)

This RSS Control Line directs the program to reduce an A-centered cell. Note that

all six cell parameters are given even though this cell is metrically monoclinic. Even

in the cubic crystal system, all six cell parameters must be given. Unit cell parameters

are specified in columns 1 1-70.

The reduction of a rhombohedral cell will be carried out. A two character symbol

'RR' in columns 9-10 specifies that a primitive rhombohedral cell is used to define a rhombohedral lattice. All six unit cell parameters are given in columns 1 1-70.

The '3' in column 1 specifies that both supercells and subcells of the lattice will be

calculated. The '23' in columns 4-5 signify that the 7 upper-triangular matrices with

determinants of 2, and the 13 upper-triangular matrices with a determinant of 3 will

be calculated. These matrices will be used in the calculation of the 7 derivative

supercells of 2 times the volume of the reduced cell of the lattice and in the calculation

of the 13 derivative supercells of 3 times the volume of the reduced cell of the lattice. The transpose of the inverse of the appropriate upper-triangular matrix will be used

to calculate each of the 7 subcells of 1/2 times and the 13 subcells of 1/3 times the

volume of the reduced cell of the lattice. Since the input lattice is rhombohedral, some

of the derivative lattices will be metrically identical even though they are oriented

differently in space. A two character symbol 'RH' in columns 9-10 specifies that a

triply primitive rhombohedral cell (metrically hexagonal axes) is used to define a rhombohedral lattice. All six unit cell parameters are given in columns 11-70.

4. Example of computer output

The computer output following this section results from the input flowstream discussed above (sec.

III.D.3.).

a. Page 1 of computer output

A description of the output parameters is printed for the first execution of the RSS program function

within a computer run.

b. Page 2 of computer output

One problem is to be considered during the first execution of the RSS program function. This problem is the reduction of a primitive, triclinic unit cell (CELL 1). In the computer output, VI is the volume for the initial cell and V2 is the volume for the reduced cell (CELL 2). The matrix Tl is the identity matrix since a primitive

unit cell was input. The transformation matrix T2 = (100/-l 10/001) from the initial to the reduced

cell has all integers and a determinant of -)- 1. The reduced cell is of the type II with all angles greater than 90

degrees. The reduced cell matrix exhibits no specialization and corresponds to reduced form number 44 in table

3. The lattice is metrically triclinic; therefore, the crystal symmetry must be triclinic.

c. Page 3 of computer output

Three independent problems are to be considered in the second execution of the RSS program function.

The first problem is the reduction of an A-centered monoclinic unit cell. The matrix

Tl = (0.5-.5/0.5.5/100) transforms the initial A-centered cell (CELL 1) to a primitive cell of the lattice by removing the cell centering. The matrix T2 = (0.5-.5/0-.5-.5/-100) transforms the initial A-centered

cell to the reduced cell (CELL 2). In the computer output, VI is the volume of the A-centered cell and V2 is the volume of the reduced cell. The reduced cell matrix corresponds to reduced form number 14 in table 3. Thus, the matrix (1 10/-1 10/001) from table 3 will transform the reduced cell to a conventional

C-centered monoclinic cell.

43
d. Page 4 of computer output

The reduction of a rhombohedral cell is the second of three independent problems to be considered in this execution of the RSS program function. The transformation matrix T2 = (1 1 1/-100/0-10) from the

initial to the reduced cell has all integers and a determinant of + 1 since a primitive, rhombohedral cell was used

to define the rhombohedral lattice. The reduced cell matrix corresponds to reduced form number 24 in table

3. Note that, in this case, the reduced cell, based on the three shortest lattice translations, is not the primitive

rhombohedral cell, but one with the a-cell parameter different from b and c, and alpha different from beta and

gamma. The matrix (12 1/0-11/100) obtained from table 3 may be used to transform the reduced cell to a triply primitive rhombohedral cell based on metrically hexagonal axes.

e. Pages 5-12 of computer output

The last independent problem to be considered illustrates the computer output for a derivative lattice calculation. In this case, both superlattices and sublattices are to be calculated. First, the initial cell is reduced.

The matrix Tl transforms the input cell to a primitive cell of the lattice, while the matrix T2 transforms the

input cell to the reduced cell of the lattice. Since a triply primitive cell (metrically hexagonal axes) was used

to define the rhombohedral lattice (RH-centered), the reduced cell volume, V2, is one-third the volume for the

input cell, VI. The reduced cell matrix corresponds to reduced form number 2 in table 3. In this case, the

reduced cell is one with a = b = c and alpha = beta = gamma.

The output illustrates that, first, the initial cell (CELL 1) is transformed to the reduced cell of the original

lattice (CELL 2, or more conveniently, 'RCOL'). The RCOL then becomes CELL 1 for the calculation of the

derivative lattices. The derivative supercells are calculated by transforming the RCOL by the unique upper-triangular

matrices. First, the 7 supercells with volumes of 2 times the RCOL are determined; then the 13 supercells with

volumes of 3 times the RCOL are calculated. After each supercell is calculated, it is then reduced. The reduced

cell and reduced form corresponding to each superlattice are shown in the output. Two transformation

matrices, Tl and T2, are given in the output along with the supercells. The matrix Tl is a unique uppertriangular matrix that transforms the RCOL to a derivative supercell which is not reduced; the matrix T2

transforms the RCOL to a reduced derivative supercell. Note that Supercell 9 corresponds to the triply

primitive rhombohedral cell originally used to define the lattice (i.e., the input cell). The computer output for the derivative subcells follows that for the supercells. First, the 7 subcells with

volumes of 1/2 times the RCOL are determined; then the 13 subcells with volumes of 1/3 times the RCOL are

calculated. After each subcell is calculated, it is then reduced. The reduced cell and reduced form corresponding to each sublattice are shown in the output. The matrix Tl is the transpose of the inverse of a unique

upper-triangular matrix; Tl transforms the RCOL to a derivative subcell which is not reduced. The matrix T2

transforms the RCOL to a reduced derivative subcell. Many of the derivative lattices are metrically identical even though they are oriented differently in space. This occurs for certain specialized lattices and for lattices with higher metric symmetry.

44
Section III.D.4.

Page 1

*** NBS»LATTICE »»»

A PROGRAM TO ANALYZE LATTICE RELATIONSHIPS

Version of Spring, 1985

NBS Crystal Data Center

National Bureau of Standards

Reactor Radiation Division

Gaithersburg, MD 20899

** REDUCTION AND DERIVATIVE LATTICE

These calculations fall into two categories:

I. Reduction of an input cell.

CELL 1 = Input cell. This cell may be primitive or centered CA,B,C,I,F,RR,RH)

.

CELL 2 = Reduced primitive cell of the lattice.

T 1 = A matrix that transforms CELL 1 to d primitive cell of the lattice.

T 2 = A matrix that transforms CELL 1 to CELL 2.

II. Calculation and reduction of a series of derivative sup^iroeils ar.d/cr suboells. These derivative cells are calculated from the reduced cell of the lattice

(i.e. to carry out the Type II calculation, the prograffi first carries out

the Type I calculation).

CELL 1 = Reduced primitive cell (i.e. CELL 2 from Part I). CELL 2 = Reduced supercell or subcell.

T 1 = A matrix that transforms CELL 1 to a supercell or subcell of the lattice.

T 2 = A matrix that transforms CELL 1 to CELL 2.

For CELL 1 and CELL 2, the output parameters given are: a, b, c, alpha, beta, gamma

and volume. Cell edges are in angstroms and angles in degrees.

The reduced cell matrix is of the form: b.c

b.b a.c a.b

Page 2

REDUCTION AND DERIVATIVE LATTICE

Number of independent problems to study = 1

1 . REDUCTION

*• Initial Cell is I rimitive

T 1= 1.00 .00 .00/ .00 1.00 .00/

T 2-' 1.00 .00 .00/ -1.00 1.00 .00/

T 2 INV= 1.00 .00 .00/ 1.00 1.00 .00/

.00 .00 1.00

.00 .00 1.00

.00 .00 1 .00 Cell Matrix •»

CELL 1= CELL 2= 1.9900 9.3600 9.1900 1 OJ 00 91.500 68.000

1.9900 8.8011 9.1900 102. ,06 91.500 99.702

VI = 388.19 21.900 77.517 81.156

v?= 388.19 -16.831 -1.200 -7.101

45
''age 3

REDUCTION AND DERIVATIVE LATTICE

Number of independent problems to study = 3

»* Initial Cell is A-Centered **

T 1 = .00 .50 -.50/ .00 .50 .50/ 1.00 .00 .00

T 2 = .00 .50 -.50/ .00 -.50 -.50/ -1.00 .00 .00

T 2 INV= .00 .00 -1.00/ 1.00 -1.00 .00/ -1.00 --1.00 .00

CELL 1= 15.3800 1M.2250 9.3090 90.000 91.200 90 . 000 CELL 2= 8.5001 8.5001 15.3800 92.298 92.298 113.598

»» Cell Matrix »»

_ 2031 .16 72.252 72.252 236.511

= 1015.58 -5.213 -5.213 -28.92 3

Page 4

Initial Cell is Primitive **

T 1 = 1.00 .00 .00/ .00 1.00 .00/ .00 .00 1 .00

T 2= 1.00 1 .00 1.00/ -1.00 .00 .00/ .00 -1.00 .00

T 2 INV= .00 -1 .00 .00/ .00 .00 -1.00/ 1.00 1.00 1 .00

CELL 1= 16.1100 16.1100 16.1100 115.100 115.100 115.100 CELL 2' 10.8611 16.1100 16.1100 115.100 102.991 102.991

Cell Matrix »»

Vl = 2318.51 118.036 259.532 259.532

V2 = 2318.51 -110.093 -39.315 -39.315

46
Page 5

3. SUPERLATTICBS for a given delta followed by SUBLATTICES for 1/delta

»» Initial Cell Is RH-Centered »»

T 1= .33 -.33 -.33/

T 2- .33 -.33 -.33/

T 2 INV= 1.00 -1.00 .00/ -1.00 .00 1.00/ -1.00 -1.00 -1.00 »» Cell Matrix •»

CELL 1= 9.1390 9.1390 15.5360 90.000 90.000 120.000 Vl= 1123.71 5'i.(>59 51.659 51.659

CELL 2- 7.3932 7.3932 7.3932 76.351 76.351 76.351 V2= 371.58 12.898 12.898 12.898

.67 -.33 -.33/ .33 .67 -.33

.67 -.33 -.33/ .33 .67 -.33

.00 .00 1 .00/ -1.00 -1.00 -1.00

.00 .00/ .00 1.00 .00/ .00 .00 2.00

.00 .00/ .00 1.00 .00/ .00 .00 2.00

.00 .00/ .00 1.00 .00/ .00 .00 .50

Superlattlces for Delta

«»««» superoell 1 •»»»»

T 1" 1 .00 T 2- 1.00

T 2 INV= 1.00 .00 .00/ .00 1.00 .00/ .00 .00 .50 »* Cell Matrix »»

CELL 1. 7.3932 7.3932 7.3932 76.351 76.351 76.351 V1= 371.58 51.659 51.659 218.636

CELL 2- 7.3932 7.3932 11.7863 76.351 76.351 76.351 V2= 719.16 25.797 25.797 12.898

»»»»» Supercell 2 »»»»»

T 1= 1.00 .00 .00/ .00 1.00 1.00/ .00 .00 2.00

T 2= 1.00 .00 .00/ .00 -1.00 1.00/ .00 -1.00 "1.00

T 2 1NV= 1.00 .00 .00/ .00 -.50 -.50/ .00 .50 -750 •* Cell Matrix »»

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 Vl= 371.58 51.659 83.521 135.115

CELL 2= 7.3932 9.1390 11.6239 90.000 107.168 90.000 V2= 719.16 .000 -25.797 .000

»«»«» Supercell 3 »»»•»

T 1- 1.00 .00 1.00/ .00 .00 .00/ .00 .00 2.00

T 2- .00 1 . 00 . 00/ 1.00 .00 -1.00/ -1.00 .00 -1.00

T 2 INV- .00 .50 -.50/ 1.00 .00 .00/ .00 -.50 -.50

CELL 1 • 7.3932 7.3932 7.3932 76.351 76.351 76.351 VI = 371.58 CELL 2- 7.3932 9.1390 11.6239 90.000 107.168 90.000 V2 = 719.16

»» Cell Matrix »»

51.659 83.521 135.115

.000 -25.797 .000

»«»»« Supercell 1 «»»»»

T 1- 1.00 .00 1.00/ .00 1.00 1.00/ .00 .00 2.00

T 2- -1.00 1.00 .pO/ .00 1.00 -1.00/ -1.00 .00 -1.00 T 2 INV= -.50 .50 -.50'/ -50 .50 -.50/ .50 -.50 -.50 »* Cell Matrix ••

47
Page 6

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 9.1390 9.1390 11.6239 66.852 66.852 60.000

VI = 37H.58 83.521 83.521 135.115

V2 = 719.16 11.761 11.761 11.761

»»»»« supercell

T 1 = 1.00 .00 .00/ .00 2.00 .00/ .00 .00 1 .00 T 2= -1.00 .00 .00/ .00 .00 -1.00/ .00 -2.00 .00 T 2 INV= -1.00 .00 .00/ .00 .00 -.50/ .00 -1.00 .00

CELL 1= 7.3932 7.3932 7.3932 76 351 76.351 76.351 CELL 2= 7.3932 7.3932 11.7863 76 351 76.351 76.351

Vl = V2= 371.58 719.16

•» Cell Matrix ••

51.659 51.659 218.636

25.797 25.797 12.898

»»»»» Supercell

1.00 1.00 .00/ .00 2.00 .00/ .00 .00 1 .00

T 2 = .00 .00 -1.00/ -1.00 1.00 .00/ 1.00 1.00

T 2 INV= .00 .50 .50/ .00 .50 .50/ -1.00 .00

CELL 1- 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 7.3932 9.1390 11.6239 90.000 107.168 90.000Vl = V2= 371.58 719.16

»• Cell Matrix »»

51.659 83.521 135.115

.000 -25.797 .000

*»*»« Supercell 7 »«»»»

T 1 = 2.00 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

T 2 = .00 1 .00 .00/ .00 .00 1.00/ 2.00 .00 .00

T 2 XNV= .00 .00 .50/ 1.00 .00 .00/ .00 1.00 .00

CELL 1= 7.3932 7.3932 7.3932 76 351 76.351 76.351 VI = 371.58

CELL 2= 7.3932 7.3932 11.7863 76 351 76.351 76.351 V2 = 719.16

»» Cell Matrix »*

51.659 51.659 218.636

25.797 25.797 12.898

Superlattlces for Delta

»••» Supercell

.00 3.00

-.33 T 2 = 1 .00 .00 .00/ .00 - 1.00 .00/ .00 1 .00 - T 2 INV= 1.00 .00 .00/ .00 - 1.00 .00/ .00 -.33

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 7.3932 7.3932 21.6610 95.721 99.270 103.619

Vl= 371.58

V2= 1123.71

»» Cell Matrix »»

51.659 51.659 169.200

-15.961 -25.797 -12.898

•«»»« Supercell

T 1 = T 2=

1 .00

1 .00

.00

.00 .00/.00/ .00/

.00

.00

1 .00

1 .00

.67

1 .00/

1 .00/ -.33/

.00 .00.00 3.00

-1.00 2.00

.33 .33 ** Cell Matrix »»

CELL 1= CELL 2= 7.3932

7.3932

7.3932

11 .6239 7.3932

11.8896

76.351 67.025

76.351 83.27176.351 72.532

VI = V2= 371.58

1123.71 51.659 135.115 221.70267.557 12.898 25.797

48
Page?

»»»•» Supercell 3 »••»«

T 1= 1.00 .00 .00/ .00 1.00 2.00/ .00 .00 3.00

T 2= 1.00 .00 .00/ .00 1.00 -1.00/ "1.00 1.00 2.00

T 2 INV= 1.00 .00 .00/ .33 .67 .33/ .33 -.33 .33 »« Cell Matrix »»

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 7.3932 9.1390 17.3827 105.211 97.136 90.000Vl= 371.58

V2= 1123.71 51.659 83.521 302.157

-11.761 -15.961 .000

»»««» Supercell 1 »»»»»

T 1= 1.00 .00 1.00/ .00 1.00 .00/ .00 .00 3-00

T 2= .00 -1.00 .00/ -1.00 .00 -1.00/ 1.00 .00 -2.00

T 2 INV= .00 -.67 .33/ -1.00 .00 .00/ .00 -.33 -.33 »» Cell Matrix ••

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 7.3932 11.6239 11.8896 67.025 83.271 72.532

Vl= 371.58

V2= 1123.71 51.659 135.115 221.702

67.557 12.898 25.797

»»»»» Supercell 5 »»»»<

T 1- T 2=

T 2 INV

CELL 1 CELL 2

1.00 .00 1.00/ .00 1.00 1.00/ .00 .00 3.00

1.00 -1.00 .00/ -1.00 .00 -1.00/ 1.00 1.00 -1.00

.33 -.33 .33/ -.67 -.33 .33/ -.33 -.67 -.33

7.3932 7.3932 7.3932 76.351 76.351 76.351 9.1390 11.6239 11.7550 100.882 90.000 11 3. 118 vi= 371.58

V2= 1123.71

»• Cell Matrix •»

83.521 135.115 138.180

-25.797 .000 -11.761

»*»»» Supercell 6 »»»»»

T 1= 1.00 .00 1.00/ .00 1.00 2.00/ .00 .00 3.00

T 2= .00 1.00 -1.00/ 1.00 .00 1.00/ 1.00 -1.00 -1.00

T 2 INV= .33 .67 .33/ .67 .33 -.33/ -.33 .33 -.33 *» Cell Matrix »»

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 9.1390 11.6239 11.7550 100.882 90.000 II3.II8

V1= 371.58

V2= 1123.71

83.521 135.115 138.180

-25.797 .000 -11.761

»»»«» Supercell 7 »»«»»

T 1- T 2" T 2 INV

CELL 1 CELL 2

1.00 .00 2.00/ .00 1.00 .00/ .00 .00 3.00

.00 -1.00 .00/ -1.00 .00 1.00/ -1.00 1.00 -2.00

-.33 -.67 -.33/ -1.00 .00 .00/ -.33 .33 -.33

7.3932 7.3932 7.3932 76.351 76.351 76.351 7.3932 9.1390 17.3827 105.211 97.136 90.000

Vl= 371.58

V2= 1123.71

»» Cell Matrix »«

51.659 83.521 302.157

-11.761 -15.961 .000

»»«» Supercell 8 •»•»»

T 1 - 1 . 00

T 2= -1.00

T 2 INV= -.67

.00 2.00/ .00 1.00 1.00/ .00 .00 3.00

.00 1.00/ .00 -1.00 -1.00/ 1.00 -1.00 1.00

-.33 .33/ -.33 -.67 -.33/ .33 -.33 .33 »» Cell Matrix *»

CELL 1

=

7.3932 7.3932 7.3932 76.351 76.351 76.351 371.58 83.521 135.115 138. 180

49
Pages

CELL 2- 9.1390 11.6239 11.7550 100.882 90.000 113-118 V2- 1123. 7t -25.797 .000 -HI. 761

»»«»» Superoell 9

T 1= 1 .00 .00 2.00/

T 2= -1 .00 .00 1 .00/

T 2 INV= -.67 -.33 -.33/

.00 1 . 00

.00 1.00

.33 .67 -.33/

2.00/ .00 .00 3.00

1.00/ -1.00 -1.00 -1.00

.33 -.33 -.33 »» Cell Matrix •»

CELL 1= CELL 2= 7.3932 7.3932 7.3932 9.1390 -9.1390 15.5360. 76.351 90.000

76.351 76.351 90.000 120.000

VI- 371.58

V2= 1123.71

83.521 83.521 211.367

.000 .000 -HI. 761

»»»»» Superoell 10 «»»»»

T 1 = 1.00 .00 0.00/ .00

T 2= -1.00 .00 .00/ .00

T 2 INV- -1.00 .00 .00/ .00

CELL 1= 7.3932 7.3932 7.3932 CELL 2= 7.3932 7.3932 21.6610

3.00 .00/ .00 .00 1.00

.00 1.00/ .00 3.00 -1.00

.33 .33/ .00 1.00 .00

76.351

95.721 76.351 76.351 99.270 103.619

V1= 371.58

V2= 1123.71

»» Cell Matrix »»

51.659 51.659 169.200

-15.961 -25.797 -12.898

»»»»» Superoell 11 •»»»»

T 1- T 2-

T 2 INV=

1 . 00 1 . 00 .00 .00

.00/ .00 3.00

1.00/ 1.00

.67 -.33/ .00

1 .00

.33

.00/

.00/

.33/ 1.00

.00 .00 1.00 2.00

.00

1 .00 .00 .00 »» Cell Matrix •»

CELL 1= CELL 2= 7.3932 7.3932 7.3932 7.3932 11.6239 11.8896

76.351 76.351 67.025 83.27176.351 72.532

VI- 371.58

V2= 1123.71 51.659 135.115 221.702

67.557 12.898 25.797

»»»»» Superoell 12 »»»»»

T 1= 1 .00 2.00 .00/ .00 3.00

T 2= .00 .00 -1.00/ 1.00 -1.00

T 2 1NV= -.33 .33 -.33/ -.33 -.67

.00/ .00 .00 1.00

.00/ -2.00 -1.00 1.00

.33/ -1.00 .00 .00 •» Cell Matrix »»

CELL l=i 7.3932. 7.3932 7.3932 76.351 76.351 76.351 CELL 2- 7.3932 9.1390 17.3827 105. 2H1 97.136 90.000vi- 371.58

V2- 1123.71 51.659 83.521 302.157

-11.761 -15.961 .000

»»*«» Superoell 13 »»»»•

T 1= 3.00 .00 .00/

T 2= .00 1 .00 . .00/

T 2 INV= .00 -.33 -.33/

.00 1.00 .00/ .00 .00 1.00

.00 .00 -1.00/ -3.00 .00 1.00

1.00 .00 .00/ .00 -1.00 .00 »» Cell Matrix •»

CELL 1= 7.393? 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 7.3932 7.3932 21.6610 95.721 99.270 103.619

V1 = V2= 371.58

1123.-71 51.659 51.659 169.200

-15.961 -25.797 -12.898

Subl^ttlces for Delta "1/2

>»»»» Subcell 1 »»»»

50
Page 9

T 1= 1 .00 .00 .00/

T 2= .00 .00 .50/

T 2 INV= .00 1 .00 .00/

.00 1.00 .00/ .00 .00 .50 1.00 .00 .00/ .00 1.00 .00 .00 .00 1.00/ 2.00 .00 .00 »* Cell Matrix »»

CELL

CELL 7.3932

3.6966 7.3932

7.3932 7.39327.3932

76.351

76.351

76.351

76.351

76.351

76.351

VI > V2= 371.58

187.29

13.665 51.659 51.659

12.898 6.119 6.119

«»«»» Subcell 2 »*»»»

T 1 = 1 .00 .00 .00/ .00 1.00 .00/ .00 -.50 .50 T 2= .00 .50 .50/ .00 -.50 -.50/ 1.00 .00 .00 T 2 INV= .00 .00 1.00/ -1.00 -1.00 .00/ 1.00 -1.00 .00 »» Cell Matrix «•

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 VI = 371.58 20.880 33.779 51.659

CELL 2= 1.5695 5.8119 7.3932 107.168 90.000 90.000 V2= 187.29 -12.898 .000 .000

*»»»• Subcell 3 »»»»»

T 1= 1.00 .00 .00/ .00 1.00 .00/ -.50 .00 .50

T 2= .50 .00 -.50/ -.50 .00 -.50/ .00 1.00 .00 T2 1NV= 1.00 -1.00 .00/ .00 .00 1.00/ -1.00 -1.00 .00 »« Cell Matrix »»

CELL 1= CELL 2= 7.3932

1.5695 7.3932

5.8119 7.3932 7.3932

76.351 107.16876.351 90.00076.351 90.000

V1 = V2= 371.58

187.29 20.880 33.779 51.659

-12.898 .000 .000

»»«»• Subcell 1 «»»»»

T 1 = 1 . 00 .00 T 2= -.50 -.50

T 2 INV= -1.00 -1 .00

.00/ .00 1.00 .00/ -.50 -.50 .50 .50/ -.50 .50 -.50/ .50 -.50 -.50

.00/ -1.00 .00 -1.00/ .00 -1.00 -1.00 »» Cell Matrix »»

CELL 1= CELL 2= 7.3932

5.8775

7.3932

5.8775 7.39325.8775

76.351 102.05676.351 102.056

76.351 102.056

VI = V2= 371.58

187.29 31.515 31.515 31.515 -7.216 -7.216 -7.216

»»*»» Subcell 5 »»»»»

T 1 = T 2 = T 2 INV=

1.00

.00

.00 -1

.00

.50 ,00

.00/

.00/

.00/

.00 -1.00 -2,00

.50 .00/ .00

.00 .00/ .00

.00 .00/ .00

.00 1 .00 -1 .00 -1.00

.00 .00

CELL 1= CELL 2= 7.3932

3.6966

7

7 3932 3932 7.39327.3932

76.351 76.351

76.351 76.351 76.35176.351

Vl = V2= 371.58

187.29

Cell Matrix »»

13.665 51.659 51.659

12.898 6.119 6.119

««««« Subcell 6 <»»»«

T 1 = 1.00 .00 .00/ -.50 .50 .00/ .00 .00 1 .00 T 2= -.50 .50 .00/ .50 .50 .00/ .00 .00 -1 .00 T 2 1NV= -1.00 1 .00 .00/ 1.00 1 .00 .00/ .00 .00 -1 .00

CELL 1- 7.3932 7.3932 7.3932 76.351 76.351 76.351 CELL 2= 1.5695 5.8119 7.3932 107.168 90.000 90.000

VI = V2= 371.58

187.29

•• Cell Matrix •

20.880 33.779 51.659

-12.898 .000 .000

51
Page 10

**1Hi* Subcell »it«»«

T 1 = .50 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

T 2= .50 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

T 2 INV= 2.00 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

CELL 1= 7.3932 7.3932 7 3932 76.351 76 351 76.351 Vl = 371.58

CELL 2= 3.6966 7.3932 7 3932 76.351 76 351 76.351 V2= 187.29

*» Ceil Matrix **

13.665 51.659 5t.559

12.898 6.119 6.119

Sublattices for Delta =1/3

*»»« » Subcell *****

T 1 = 1.00 .00 .00/ .00 1.00 .00/ .00 .00 .33 T 2= .00 .00 .33/ -1.00 .00 .33/ .00 -1 .00 .33 T 2 INV= 1.00 -1 .00 .00/ 1.00 .00 -1.00/ 3.00 .00 .00

CELL 1 = 7.3932 7.3932 7.3932 76.351 76.351 76.351 VI = 371.58 CELL 2= 2.1611 7.2203 7.2203 78.521 81.279 81.279 V2 = 121.86

•» Cell Matrix »»

6.073 52.133 52.133

10.373 1.771 1.771

•» Subcell

T 1 = 1.00 .00 .00/ .00 1 .00 .00/ .00 -.33 .33

T 2 = .00 .33 .33/ .00 .33 -.67/ 1.00 .00 .00

T 2 INV= .00 .00 1 .00/ -2.00 -1 .Of: .00/ 1.00 -1.00 .00

CELL 1- 7.3932 7 3932 7.3932 76.351 76.351 76.351 CELL 2= 3.0163 6 0082 7.3932

*****

106.880 90.000

Subcell 3 *»*»»

101.686

T 1 = 1 .00 .00 .00/ .00 1 .00 .00/ .00 -.67 .33

T 2= .00 .33 -.33/ .00 .33 -.67/ 1.00 -.33 - • 33 T 2 INV= -1.00 .00 1.00/ -2.00 1 .00 .00/ -1.00 ~1 .00 .00

CELL 1= 7.3932 7 3932 7.3932 76.351 76.351 76.351 CELL 2= 3.8716 1 9632 7.2139 81.883 76,790 67.025

V1= 371.58

V2= 121.86

»» Cell Matrix »«

9.280 36.099 51.659

-12.898 .000 -1.610

VI = 371.58

V2= 121.86

*» Cell Matrix »»

15.013 21.631 52.171

3.207 6.111 7.506

*»**« Subcell 1 *****

T 1 = 1.00 .00 .00/ .00 1 .00 .00/ -.33 .00 .33

T 2 = .33 .00 -.33/ -.67 .00 -.33/ .00 1 .00 .00 T 2 INV- 1.00 -1 .00 .00/ .00 .00 1.00/ -2.00 -1.00 .00

CELL 1= 7.3932 7.3932 7.3932 76.351 76.351 76.351 Vl = 371.58 CELL 2= 3.0163 6.0082 7.3932 106.880 90.000 101.686 V2= 121.86

»» Cell Matrix **

9.280 36.099 51.659

-12.898 .000 -1.610

**»»« Subcell 5 »»«»»

1.00 .00 .00/ .00 1.00 .00/ -.33 .33 .33

52
Page 11

T 2- T 2 INV" -.33

-1.00 -.33

1.00

.33/

.00/

.67 -1.00

.33 .00 .33/ 1.00/ -.33

1.00

.67

1 .00 .33 1.00 •• Cell Matrix »»

CELL CELL

7.3932

3.9183

7.3932 5.79127.3932

5. 7912

76.351 101.11576.351 97.253

76.351 97.253

VI"

V2- 371.58121.86

15.353 33.573 33.573

-8.188 -2.866 -2.866

**««« suboell

T 1- T 2-

T 2 INV1.00

.33

.00

-.33 ,00/ -.33/

.00

.33 1.00

.67 1.00 1.00 1.00/ -1.00 1.00

.00/ -.33 -.33/ .33

.00/ -1.00

-.67

-.33 .00

.33

.67 1.00 »• Cell Matrix »»

CELL CELL

7.3932

3.9183 7.3932

5.7912 7.39325.7912

76.351 101.11576.351 97.253

76.351 97.253

VIV2- 371.58

121.66

15.353 33.573 33.573 -8.188 -2.866 -2.866

• *•«> Suboell 7 •••»»

T 1- T 2-

T 2 INV1.00

-.33 -1.00 -1

.00

.00

.00

.00/ -.33/

.00/

.00 1 = .67 -1.00

.00 .00/ -.67

.00 .33/ -.33

.00 1.00/ -2.00

.00 1.00 1.00

.33

.33 .00

CELL 1

.

CELL 2- 7.3932

3.8716

7.

1,.3932

.9632 7.3932 7.2139

76.351 76.351 81.883 76.790

Suboell 8 *****

76.351 67.025

VIV2=

371.

121,.58

.86

T 1-

T 2-

T 2 INV1.00

.33 - 1.00 -1

.00

.33 .00

.00/ .33/

.00/

.00 1 -.67 - -1.00 -1.00 .00/ -.67

.33 .33/ .33

.00 -1.00/ 1.00

-.33

-.33 - .00 -1

.33

.67

.00

CELL 1- CELL 2- 7.3932

3.9183

7.

5..3932

.7912 7.39325.7912

76.351 76.351 101.115 97.25376.351 97.253

VIV2-

371.

121,,58

,86

•» Cell Matrix •»

15.013 21.631 52.171

3.207 6.111 7.506

»» Cell Matrix ««

15.353 33.573 33.573 -8.188 -2.866 -2.866

•«»» » Suboell 9 »»•»»

T 1- T 2-

T 2 INV1.00

-.33 - -1.00 -1

.00 .00/

.33 -.33/

.00 .00/

.00 -.67 -1.001.00 .00/ -.67

.33 .33/ .33 1.00 1.00/ -1.00

-.67

.33 - .00 -1

.33

.67

.00

CELL 1 - CELL 2" 7.3932

5.1787

7.3932

5.2761

7.3932

5.2761

76.351 76.351 120.000 90.000

76.351

90 . 000 VIV2- 371.58

121.86

•• Cell Matrix »•

26.819 27.810 27.810

-13.920 .000 .000

*<I«IIK Suboell 10 »»»•»

T 1- 1.00 .00 .00/ .00 .33 .00/ .00 .00 1 .00

T 2- .00 .33 .00/ 1.00 .33 .00/ .00 -.33 1 .00 T 2 INV- -1.00 1 .00 .00/ -3.00 .00 .00/ -1.00 .00 1 .00

CELL 1 - 7.3932 7.3932 7.3932 76.351 76.351 76.351 Vl = 371.58 CELL 2- 2.1611 7.2203 7.2203 78.521 81.279 81.279 V2- 121.86

»• Cell Matrix •«

6.073 52.133 52.133

10.373 1.771 1.771

»»•»» Suboell 11 »•»•«

53
Page 12

T 1-

T 2-

T 2 INV1.00 -.331.00

.00

.33 1.00

.00/

.00/

.00/

-.33

.67 2.00

.33

.33 1.00

.00/

.00/

.00/

.00

.00

.00

.00 1.00

.00 -1.00

.00 -1.00 »» Cell Matrix •»

CELL

CELL

7.3932

3.0163

7.3932

6.0082 7.3932

7.3932

76.351 106.880

76.351 90.00076.351 104.686VI- V2- 371.58

121.86

9.280

-12.898 36.099

.000 51.659 -1.610

»»»•» subcell 12 »»»»»

T 1-

T 2-

T 2 INV1.00

.33 1.00 1

.00 .00/

.33 .00/ .00 .00/

-.67 .33 .00/ .00

.67 -.33 .00/ .33 2.00 -1.00 .00/ 1.00

.00 1 .33 -1 .00 -1.00

.00

.00 »» Cell Matrix »»

CELL 1- CELL 2- 7.3932

3.3716

7.3932

1.9632 7.3932 76.351 76.351 7.2139 81.883 76.790

76.351 67.025

Vl = V2" 371.58

121.86 15.013 21.631 52.171

3.207 6.111 7.506

• »•>• Subcell 13 ••••»

T 1- .33 .00 .00/ .00 1 .00 .00/ .00 .00 1 .00

T 2- .33 .00 .00/ .33 -1 .00 .00/ .33 .00 -1 .00

T 2 INV- 3.00 .00 .00/ 1.00 -1 .00 .00/ 1.00 .00 -1 .00 »» Cell Matrix »»

CELL 1 = 7.3932 7.3932 7.3932 76.351 76.351 76.351 VI = 371.58 6.073 52.133 52.133 CELL 2- 2.1611 7.2203 7.2203 78.521 81.279 81.279 V2- 121.86 10.373 1.771 1.771

54
IV. Cell transformation

A. Introduction

This program function enables the user to transform a unit cell by a specified three by three matrix.

B. Operation of the program

1. General

There are three types of input lines required to execute the cell transformation (TRANS) program

function. The first input line is a Program Control Line that specifies the type of program function and the

number of independent problems to be considered. The second type of input line is a Matrix Parameter Line

that defines the matrix to be used to transform the unit cell. The final type of input line is the Cell Parameter

Line that specifies the unit cell parameters a,b,c, alpha, beta, gamma. For N problems specified on the Program

Control Line, N sets of lines consisting of one Matrix Parameter Line followed by one Cell Parameter Line

must be input to the program. Within a single computer run, the TRANS program function may be executed

any number of times. A description of the formats for the input lines and typical examples are given below.

2. Description of input lines

a. Program Control Line

The Program Control Line specifies the type of program function and the number of problems to be

solved. For each execution of the TRANS program function within a single computer run, only one Program

Control Line is required.

Program Control Line

Format(A5,3X,I2)

Column Format Item

1-5 A5 Type of program function

'LM ' = Lattice Matching

'RSS ' = Reduction and

Derivative

Supercell and Subcell

'TRANS' = Cell Transformation

'INV ' = Matrix Inversion

6-8 3X Blank

9-10 12 Number of problems

Notes:

1 The TRANS program function is specified by 'TRANS' in columns 1-5.

2 The number of problems specified in a single execution of the TRANS

program function may range between 1 and 99.

55
b. Matrix Parameter Line

The Matrix Parameter Line deflnes the matrix to be used to transform the unit cell. The form of the three

by three transformation matrix required to execute the TRANS program function is :

(ail

321

a3i

a,2 ai3 a22 a23

a32 a33

The Matrix Parameter Line must precede the Cell Parameter Line.

Matrix Parameter Line

Format(9F8.2)

Column Format Item

1 1-8 F8.2 Matrix element an

2 9-16 F8.2 Matrix element ai2

3 17-24 F8.2 Matrix element ai3 4 25-32 F8.2 Matrix element a2i

5 33-40 F8.2 Matrix element a22

6 41-48 F8.2 Matrix element a23

7 49-56 F8.2 Matrix element 331

8 57-64 F8.2 Matrix element a^

9 65-72 F8.2 Matrix element 333

Notes:

1-9 Each matrix element must be repiesented as a decimal ni

c. Cell Parameter Line

This input line is used to define the unit cell to be transformed. All six unit cell parameters must always

be specified regardless of the crystal symmetry. The Cell Parameter Line must follow the Matrix

Parameter Line.

Cell Parameter Line

Format(10X,6F10.5)

Column Format Item

1-10 lOX Blank

1 11-20 F10.5 a(A)

2 21-30 F10.5 b

3 31-40 F10.5 c 4 41-50 F10.5 alpha

5 51-60 F10.5 beta

6 61-70 F10.5 gamma

56
Notes:

1-3 Cell edges (A).

4-6 Cell angles (degrees). Decimal numbers must be used for fractions of a degree.

3. Example of input flowstream

1 TRANS 2 2 .0 1.0 .0 .0 1.0

3 5.674 6.282 8.225

4 1.0 2.0 1.0 .0 -1.0

5 10.864 16.110 16.110

-2.0 -1.0 .0 .0

67.55 81.05 65.96

1.0 1.0 .0 .0

115.10 102.99 102.99

Notes:

1 The 'TRANS' in columns 1-5 indicates that the TRANS program function will be

executed. The '2' in column 10 specifies that two independent problems are to be

considered. This means that two sets of Hnes consisting of one Matrix Parameter Line

followed by one Cell Parameter Line must be input to the program.

2 This input line defines the matrix to be used to transform the first input cell.

3 All six unit cell parameters must be specified regardless of the crystal symmetry. Cell

edges are in angstroms and cell angles are in degrees. Decimal numbers have been

used for fractions of a degree. This unit cell will be transformed by the matrix defined on the second input line.

4 This input line defines the matrix to be used to transform the second input cell.

5 All six unit cell parameters are specified on this input line. Cell edges are in angstroms

and cell angles are in degrees. Decimal numbers have been used for fractions of a degree. This unit cell will be transformed by the matrix defined on the fourth input

line.

4. Example of computer output

The computer output following this section results from the input flowstream discussed above (sec. IV.B.3). A description of the output parameters is printed for the first execution of the TRANS program

function within a computer run. For the first independent problem (Cell Transformation 1), the input cell (CELL 1) was the reduced cell. The input matrix (T2), taken from table 3 in section III, transforms the reduced cell (reduced form number 30) to a C-centered monoclinic cell (CELL 2). The input cell for the second independent problem (Ceil Trans- formation 2) was also the reduced cell (reduced form number 24). The input matrix, T2, transforms the reduced

cell to the triply primitive rhombohedral cell (metrically hexagonal axes).

57
Example

Section IV.

••» NBS»UTTICE •»»

A PROGRAM TO ANALYZE LATTICE RELATIONSHIPS Version of Spring, 1985

NBS Crystal Data Center

National Bureau of Standards

Reactor Radiation Division

Gaithersburg, MD 20899

»• CELL TRANSFORMATION »»

CELL 1 » Input cell. CELL 2 " Transformed cell. T 2 - Input transformation matrix.

T 2 INV = Inverse matrix for T 2.

For CELL 1 and CELL 2, the output parameters given are:

a, b, c, alpha, beta, gamma and volume. Cell edges

are in angstroms and angles in degrees.

The cell matrix is of the form: a. a b.b c.c

b.c a.c a.b

CELL TRANSFORMATION

»» Cell Transformation 1 »•

T 2'

T 2 INV- .00 1.00 .00/ .00 1.00 -2.00/ -1.00 .00

.00 .00 -1.00/ 1.00 .00 .00/ .50 -.50

.00

.00 •» Cell Matrix «•

CELL 1= CELL 2= 5.6710 6.2820 8.2250 67.550 81.050 65.960 6.2820 15.2033 5.6710 90.000 111.010 90.000

VI = V2= 217.15

191.90

39.161 231.111 32.191

.000 -11.520 .001

»• Cell Transformation 2 »»

T 2- 1.00 2.00 1.00/

T 2 INV- .00 .00 1.00/

.00 -1.00 1.00/ 1.00 .00 .00

.33 -.33 -.33/ .33 .67 -.33 »» Cell Matrix ••

CELL 1- 10.8610 16.1100 16.1100 115.100 102.990 102.990

CELL 2- 27.1895 27.1892 10.8610 90.000 89.999 120.000

VIV2- 2318.15 6955.35739.268 739.251 118.026

.000 .001 -369.625

58
V. Matrix inversion

A. Introduction

This program function enables the user to calculate the inverse of a three by three matrix.

B. Operation of the program

1. General

There are two types of input lines required to execute the matrix inversion (INV) program function. The

first input line is a Program Control Line that specifies the type of program function and the number of

independent problems to be considered. The second type of input line is a Matrix Parameter Line that defines

the matrix whose inverse (if one exists) is to be calculated. For N problems specified on the Program Control

Line, N Matrix Parameter Lines must follow. Within a single computer run, the INV program function may

be executed any number of times. A description of the formats for the input lines and typical examples follow.

2. Description of input lines

a. Program Control Line

The Program Control Line specifies the type of program function and the number of problems to be

considered. For each execution of the INV program function within a single computer run, only one Program

Control Line is required.

Program Control Line

Format(A5,3X,I2)

Column Format Item

1-5 A5 Type of program function

'LM ' = Lattice Matching

'RSS ' = Reduction and

Derivative

Supercell and Subcell

'TRANS' = Cell Transformation

'INV ' = Matrix Inversion

6-8 3X Blank

9-10 12 Number of problems

Notes:

1 The INV program function is specified by 'INV ' in columns 1-5.

2 The number of problems specified in a single execution of the INV program

function may range between 1 and 99.

59
b. Matrix Parameter Line

The Matrix Parameter Line defines the matrix whose inverse is to be calculated. The form of the three by

three matrix is:

ai, ai2 an

azi a22 a23

.aai 332 a33

For N problems specified on the Program Control Line, N Matrix Parameter Lines must be input to the

program.

Matrix Parameter Line

Format(9F8.2)

Column Format Item

1 1-8 F8.2 Matrix element an

2 9-16 F8.2 Matrix element ai2

3 17-24 F8.2 Matrix element a,

4 25-32 F8.2 Matrix element azi

5 33-40 F8.2 Matrix element a22

6 41-48 F8.2 Matrix element a23

7 49-56 F8.2 Matrix element 331

8 57-64 F8.2 Matrix element a32

9 65-72 F8.2 Matrix element a33

Notes:

1-9 Each matrix element must be represented as a decimal number.

3. Example of input flowstream

. . .1 1. . 1. , .1 .

. .1 1.

1 INV 2

2 -1.0 .0 .0 .0 -1.0 1.0 -1.0 1.0 1.0

3 1.0 -1.0 .0 -1.0 .0 1.0 -1.0 -1.0 -1.0

Notes:

1 The 'INV ' in columns 1-5 indicates that the INV program function will be executed. The '2' in column 10 specifies that two independent problems are to be considered. Thus, two Matrix Parameter Lines follow the Program Control Line.

2-3 Each matrix element has been specified by a decimal number.

60
4. Example of computer output

The computer output following this section results from the input flowstream discussed above (sec, V.B.3.). A description of the output parameters is printed for the first execution of the INV program function

within a computer run. The two matrices used in this example were taken from table 3 in section III. The input

matrix (T2) for the first independent problem (Matrix Inversion 1) transforms a reduced cell having reduced

form number 19 to an I-centered orthorhombic cell. The input matrix for the second independent problem

(Matrix Inversion 2) transforms a reduced cell having reduced form number 2 to a triply primitive rhombohedral cell (metrically hexagonal axes).

Example

Section V.

NBS«LATTICE *»»

A PROGRAM TO ANALYZE LATTICE RELATIONSHIPS Version of Spring, 1985

NBS Crystal Data Center National Bureau of Standards Reactor Radiation Division

Gaithersburg, MD 20899

MATRIX INVERSION »»

T 2 = Input transformation matrix.

T 2 INV = Inverse matrix for T 2.

MATRIX INVERSION

*» Matrix Inversion 1 »«

T 2= -1.00 .00 .00/ .00 -1.00 1.00/ -1.00 1.00 1.00 T2 INV= -1.00 .00 .00/ -.50 -.50 .50/ -.50 .50 .50

»» Matrix Inversion 2 »»

T 2= 1.00 -1.00 .00/ -1.00 .00 1.00/ -1.00 -1.00 -1.00

T 2 INV= .33 -.33 -.33' -.67 -.33 ".33/ .33 .67 -.33

61
VI. Summary of program operations

A. General

The NBS*LATTICE program is designed for use in any analytical laboratory. The program uses metric

methods to analyze various types of lattice relationships. Whenever using metric methods, one must be

particularly careful that the unit cell parameters and errors have been properly determined and interpreted. In many cases, it appears that the reported experimental error is too optimistic, even for unit cell parameters

refined by least-squares analysis using modem diffractometry. Independent unit cell determinations of beta- clopenthixol illustrate this point. The x-ray crystal structure of beta-clopenthixol has been reported in the

literature (Jones, Sheldrick & Horn, 1981). Previously, a crystalline sample of beta-clopenthixol had been

obtained from one of the authors (A.H.) and the unit cell parameters were determined using both powder

(Morris, McMurdie, Evans, Paretzkin, Hubbard & Carmel, 1980) and single-crystal (Himes, 1983) diffraction

techniques.

Published Single- Powder

single-crystal cell crystal cell cell

a = 6.493(2) A 6.4978(7) A 6.518(4) A

b = 7.758(3) 7.7701(8) 7.773(3) c = 21.881(8) 21.871(2) 21.939(11)

a= 90.11(2)° 90.011(8)° 90.06(4)

° /3= 91.48(2) 91.501(9) 91.60(5)

y= 92.81(2) 93.129(9) 93.06(4)

A comparison of the unit cell parameters reveals that they are in reasonable agreement with each other. However, depending on which values are used, the gamma angles for the two cells refined on diffractometers

differ by approximately 16 to 35 estimated standard deviations. Thus, in general, one should assume rather large

experimental errors when analyzing lattice relationships. This is especially true when working with cells prior

to the final cell, or for cases in which unit cell parameters were determined from irregularly shaped crystals

having an absorption problem.

The present version of NBS*LATTICE performs several functions including: 1) the characterization and

identification of unknown materials using lattice-formula matching techniques; 2) the calculation of the reduced

cell of the lattice, and the calculation and reduction of specified derivative supercells and/or subcells; 3) unit

cell transformations; and 4) matrix inversions. To execute the lattice-matching (LM) program function, three

types of input lines are required. For N problems specified on the Program Control Line, one LM Parameter

Line and N RSS Control Lines must follow (1 < N < 20). Within a single computer run, the LM program

function may be executed only once. Although at most 20 independent problems may be considered, up to 900

lattices may be matched against data from the NBS Crystal Data File in a single run. To execute the reduction

and derivative supercell and subcell (RSS) program function, two types of input lines are required. For N

problems specified on the Program Control Line, N RSS Control Lines must follow (1 < N < 20). There are three types of input lines required to execute the cell transformation (TRANS) program function. For N

problems specified on the Program Control Line, N sets of lines consisting of one Matrix Parameter Line

followed by one Cell Parameter Line must be input to the program (1 < N < 99). The execution of the matrix

inversion (INV) program function requires two types of input lines. For N problems specified on the Program

Control Line, N Matrix Parameter Lines must follow (1 < N < 99). The RSS, TRANS, and INV program

functions may be executed any number of times within a single computer run. A description of the formats for the input lines is given below.

B. Description of input lines

Each individual section II through V contains additional information concerning the theory and operations for each program function.

62
1. Program Control Line

The Program Control Line specifies the type of program function and the number of independent

problems to be considered. One Program Control Line is required for each program function to be executed.

Column

Program Control Line

Format(A5,3X,I2)

Format Item

1-5 A5 Type of program function

•LM

'RSS

TRANS'

'INV '

= Lattice Matching = Reduction and

Derivative

Supercell and Subcell = Cell Transformation = Matrix Inversion

6-8 3X Blank

9-10 12 Number of problems

2. LM Parameter Line

The LM Parameter Line specifies the tolerances for a match of a,b,c, and V, the file(s) to be searched, and

whether full or limited output is desired. Tolerances of 0. 10 A and 10.0 percent are reasonable values to specify

for most problems. The user may wish to increase these tolerances in a second run depending on the results obtained. Only one LM Parameter Line is allowed when executing the LM program function.

LM Parameter Line

Format(2F10.2,5X,5Al,9X,Il)

Column Format Item

1-10

11-20

F10.2

F10.2

Tolerance for a match of the cell edges (A)

. Tolerance for a match of the cell volume (percentage of A')

21-25 5X Blank

26-30 5A1 I/O/ Inorganic/Organic/

May specify one or more

files to be searched.

31-39 9X Blank

40 11 Blank/1/' Print/Do not print/

RSS output

63
3. RSS Control Line

The RSS Control Line defines the input lattice and the derivative lattices to be calculated. To define the

lattice, the cell centering and all six unit cell parameters (regardless of the crystal symmetry) must be specified.

When executing the LM program ftinction, especially when some chemical information is known, it is recom- mended to check routinely for the 55 supercells of 2, 3, and 4 times the volume and the 55 subcells of 1/2, 1/3,

and 1/4 times the volume of the reduced input cell.

RSS Control Line

Format(Il,2X,2Il,3X,2Al,6F10.2)

Column Format Item

1 11 Blank/1/2/3/

Blank = Reduction

1 = Reduction + supercells 2 = Reduction + subcells

3 = Reduction + supercells, subcells

2-3 2X Blank

4 11 Blank/2/3/4/5/6/7/8/9/

6-8 3X

9 Al

10 Al

Initial value (nl) to define the

range of volumes for calculated

derivative lattices.

II Blank/2/3/4/5/6/7/8/9/

Final value (n2) to define the range

of volumes for calculated derivative lattices.

Blank

P/A/B/C/F/I/R/ Cell centering

R/H/ Rhombohedral/Hexagonal/

metric axes. Used only for rhombohedral lattices.

11-20 F10.2 a (A)

21-30 F10.2 b

31-40 F10.2 c

41-50 F10.2 alpha (°)

51-60 F10.2 beta

61-70 F10.2 gamma

64
4. Matrix Parameter Line

The Matrix Parameter Line defines the three by three matrix to be used when executing the TRANS or INV program functions.

Matrix Parameter Line

Format(9F8.2)

Colunm Format Item

1-8 F8.2 Matrix element an

9-16 F8.2 Matrix element ai2

17-24 F8.2 Matrix element a^

25-32 F8.2 Matrix element aai

33-40 F8.2 Matrix element aja

41-48 F8.2 Matrix element An

49-56 F8.2 Matrix element as,

57-64 F8.2 Matrix element a32

65-72 F8.2 Matrix element 333

5. Cell Parameter Line

This input line is used to define a unit cell. All six unit cell parameters must always be specified regardless

of the crystal symmetry.

Cell Parameter Line

Format(10X,6F10.5)

Column Format Item

1-10 lOX Blank

11-20 F10.5 a (A)

21-30 F10.5 b

31-40 F10.5 c

41-50 F10.5 alpha

51-60 F10.5 beta

61-70 F10.5 gamma

65
VII. References

Choi, C. S. & Mighell, A. D. (1982). Acta Cryst. B38, 2874-2876.

Clegg, W. (1981). Acta Cryst. A37, 913-915.

Crystal Data: Determinative Tables (1972, 1973, 1978, 1984). Vols. 1-6, U. S. Department of Commerce,

National Bureau of Standards, Gaithersburg, MD, and the JCPDS-International Centre for Diffraction Data, Swarthmore, PA.

Gruber, B. (1973). Acta Cryst. A29, 433-440.

Himes, V. L. (1983). Ph.D. Thesis. Department of Chemistry, The Catholic University of America,

Washington, D.C. 20064.

Himes, V. L. & Mighell, A. D. (1982). Acta Cryst. A38, 748-749.

International Tables for X-ray Crystallography (1969 a). Vol. I, 3rd ed., pp. 530-535. Reduced-cells section by

A. D. Mighell, A. Santoro & J. D. H. Donnay. Birmingham: Kynoch Press. International Tablesfor X-ray Crystallography (1969 b). Vol. I, 3rd ed., pp. 74-346. Birmingham: Kynoch Press. Jones, P. G., Sheldrick, G. M. & Horn, A. S. (1981). Acta Cryst. B37, 906-910.

Le Page, Y. (1982). J. Appl. Cryst. 15, 255-259.

Marsh, R. E. & Schomaker, V. (1979). Inorg. Chem. 18, 2331-2336.

Mighell, A. D., Himes, V. L. & Rodgers, J. R. (1983). Acta Cryst. A39, 737-740.

Mighell, A. D., Hubbard, C. R. & Stalick, J. K. (1981). National Bureau of Standards (U.S.) Technical Note

1141, U.S. Department of Commerce, National Bureau of Standards, Washington, D.C. 20234.

Mighell, A. D. & Rodgers, J. R. (1980). Acta Cryst. A36, 321-326.

Molecular Structures and Dimensions (1970-1984). Published for the Crystallographic Data Centre, Cambridge

and the International Union of Crystallography.

Morris, M. C, McMurdie, H. F., Evans, E. H., Paretzkin, B., Hubbard, C. R. & Carmel, S. J. (1980). National

Bureau of Standards (U.S.) Monograph 25, Section 17. NBS Crystal Data File (1982). A magnetic tape of crystallographic and chemical data compiled and evaluated

by the NBS Crystal Data Center, National Bureau of Standards, Gaithersburg, MD 20899.

Niggli, P. (1928). Handbuch der Experimentalphysik , Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft.

Santoro, A. & Mighell, A. D. (1970). Acta Cryst. A26, 124-127.

Santoro, A. & Mighell, A. D. (1972). Acta Cryst. A28, 284-287.

Santoro, A. & Mighell, A. D. (1973). Acta Cryst. A29, 169-175.

Santoro, A., Mighell, A. D. & Rodgers, J. R. (1980). Acta Cryst. A36, 796-800.

Steffen, W. L. & Palenik, G. J. (1977). Inorg. Chera. 16, 1119.

66
Appendix I. NBS Crystal Data File

A large data base with chemical and crystallographic data is available for lease in magnetic tape form. The

data base contains evaluated data on approximately 60,000 materials. The data are collected from the published

literature. To avoid duplication of effort, the NBS Crystal Data Center collaborates with other data centers including the Cambridge Crystallographic Data Centre (England); the JCPDS-International Centre for Diffraction Data (Swarthmore, PA); the Metals Data Center (Ottawa, Canada); and the Inorganic Structural Data

Center (Bonn, Germany). The NBS Crystal Data Center is especially interested in the nature of the lattice and

the chemical composition. The data for each compound are processed, evaluated, and put into standard form.

Derivative parameters are calculated such as the reduced cell, the empirical formula and the calculated density.

Each entry consists of the reduced cell and volume, crystal system, space group, space group number, density,

chemical name, chemical formula, reference to Crystal Data , journal reference, plus additional data. The work

was supported by the Office of Standard Reference Data at the National Bureau of Standards.

Use of the Data Base

The data base will prove to be a practical analytical tool for compound identification because the reduced

cell in the lower symmetry crystal systems or the reduced cell plus some chemical data in the higher symmetry

crystal systems is unique for most compounds. Unknowns can conveniently be identified by the following

sequence:

1. Determine a primitive or centered cell for the unknown crystal;

2. Calculate the reduced cell;

3. Check the NBS Crystal Data File;

4. Identify the unknown compound on the basis of the lattice (as defined by the unit cell parameters)

and/or chemical information.

In addition to routine analytical work, the data base will be useful in research studies and in preventing

redeterminations of published structures.

Coverages and Updates

The data base contains data for most commonly occurring materials. All types of substances with known

unit cells are included. The data fall into the following categories: organics, organometallics, metals, inter- metallics, inorganics, and minerals. New entries are being added on a regular basis and a revised version of the

data base will be made available to users on a periodic basis.

Information

For questions concerning the scientific content of the File, write the NBS Crystal Data Center. Distribution of the NBS Crystal Data File is being handled by:

JCPDS-International Centre for Diffraction Data

1601 Park Lane

Swarthmore, PA 19081

Telephone: 215-328-9400.

Structure of an Entry

Each entry consists of the data for one crystalline compound. Each entry consists of the reduced cell, cell volume, crystal system, space group, space group number, calculated density, chemical name, chemical

formula, and journal reference. There are four record types (i.e., four 80 character records) per entry:

Record 1 Reduced Cell and References

Record 2 Space Group, Calculated Density, Chemical Class, Reduced Cell Volume

Record 3 Chemical Formula

Record 4 Chemical Name.

Record 3 and record 4 can be continued. All records in a given entry have the same identification number.

67
Format Specifications for Each Record in an Entry

RECORD 1

Column

REDUCED CELL and REFERENCES

Format Item

1-6 F6.2 a (reduced cell) 7-12 F6.2 b

13-18 F6.2 c 19-24 F6.1 alpha

25-30 F6.1 beta

31-36 F6.1 gamma

37 IX Blank

38 11 Crystal Data volume number

39 Al Crystal system (A, M, O, R, T, H, C)

40-48 F9.4 Crystal Data ratio 49 IX Blank

50-55 A6 CODEN for journal reference

56-59 A4 Volume

60 IX Blank

61-64 A4 Page

65 IX Blank

66-69 14 Year

70 IX Blank

71 Al Structure code (N, L, T)

72 Al * (If quant, structure occurs in a different

- reference; inorganic file only)

73-78 6A1 ID number

79 IX Blank

80 Al 1

RECORD 2 SPACE GROUP, D;c, CLASS, REDUCED CELL VOLUME

Column Format Item

1-3 13 Space group number

4 IX Blank

5-12 A8 Crystal Data sipace group

13 IX Blank

14-18 F5.2 Dx

19 IX Blank

20-22 A3 Chemical class 23 IX Blank

24-32 19 Reduced cell volume

33-34 2X Blank

35-42 A8 Accession date

43-44 2X Blank

45-52 A8 Modification date

53-72 20X Blank

73-78 6A1 ID number

79 IX Blank

80 11 2

68
RECORD 3 CHEMICAL FORMULA

Column Format Item

1-67 67A

Chemical formula

68-69 2X Blank

70 Al C (formula continuation code)

71 11 Sequence number

72 IX Blank

73-78 6A1 ID number

79 IX Blank

80 11 3

RECORD 4 CHEMICAL NAME

Column Format Item

1-67 67A

Chemical name

68-69 2X Blank

70 Al C (name continuation code)

71 11 Sequence number

72 IX Blank

73-78 6A1 ID number

79 IX Blank

80 11 4

Description of Data Parameters for Record 1

1. Reduced Cell Cols. 1-36

The reduced cell is a unique, primitive cell that is based on the three shortest vectors of the lattice. The cell edges are set so that a < b < c. The mathematical conditions for full reduction and procedures to obtain

the reduced cell have been published (Santoro & Mighell, 1970; International Tables for X-ray Crystallography, 1969 a).

2. Crystal Data Volume Number Col. 38

This is the volume number (1-6) in the Crystal Data: Determinative Tables (1972, 1973, 1978, 1984) series in which additional data on the compound may be found. This number is given only for inorganic entries.

3. Crystal System Col. 39

A = Anorthic (Triclinic)

M = Monoclinic

O = Orthorhombic

R = Rhombohedral

T = Tetragonal

H = Hexagonal

C = Cubic.

69
4. Crystal Data Ratio Cols. 40-48

The first determinative ratio for the cell reported on the top line in Crystal Data: Determinative Tables is given. The first determinative ratio is: a/b = Anorthic, monoclinic, and orthorhombic

c/a = Tetragonal and hexagonal

c/a = Rhombohedral (metrically hexagonal axes) a = Cubic.

5. Journal Reference Cols. 50-69

The journal reference to the original literature from which the data were extracted is given as: CODEN

(i.e. a code representing the journal name), volume, page, year. Most CODENS correspond to those used

by Chemical Abstracts, a few have been made up by the data center. A file of CODENS and journal names

is distributed with the NBS Crystal Data File.

6. Structure Code (N, L, T) Col. 71

'T' signifies that a full structure determination has been reported in the original literature. Usually all parameters except hydrogen have been refined.

*N' signifies structural data have not been given.

'L' signifies that limited structural data have been reported.

7. Full Structure Determination Reported

in a Different Reference Col. 72

If a

'*' occurs in col. 72, it signifies that another reference reports a full structure determination on the same

material. This symbol is used only for inorganic data.

8. ID Number Cols. 73-78

Each entry has a unique identification number. All records in a given entry have the same identification

number.

9. Record Sequence Number Col. 80

A sequence number of 1,2,3,4 is used for record types 1-4.

Description of Data Parameters for Record 2

1. Space Group Number Cols. 1-3

A number (1-230) as given in Volume 1 of the International Tables for X-ray Crystallography (1969 b).

2. Crystal Data Space Group Cols. 5-12

The conventional space group symbol which corresponds to the cell setting as given on the top line of each

entry in Crystal Data: Determinative Tables. A list of all possible symbols is given in NBS Technical Note

1141 (Mighell, Hubbard & Stalick, 1981).

3. Dx (Calculated Density) Cols. 14-18

70
4. Chemical Class Cols. 20-22

For the inorganic compounds, an 'M' is given if the compound is a mineral. For the organic compounds,

the chemical class is specified by a number 1-86. Each number corresponds to a chemical class as listed in Molecular Structures and Dimensions (1970-1984).

Examples: 18 = Benzoquinones; 51 = Steroids

5. Reduced Cell Volume Cols. 24-32

Cell volume (A^) of the primitive reduced cell given on record 1.

6. Accession and Modification Dates Cols. 35-52

These two dates are used by the Data Center for internal bookkeeping.

7. ID Number Cols. 73-78

Same unique identification number as on record 1.

8. Record Sequence Number Col. 80

Always a

'2'.

Description of Data Parameters for Record 3

1. Chemical Formula Cols. 1-67

The formula is expressed by a sequence of discrete units. For a complete description of the conventions

used in the data base, see NBS Technical Note 1141.

2. Continuation Code and Sequence No. Cols. 71-72

If more than one record is required for the formula, the symbols in columns 70 and 71 are nonblank.

Example:

Col. 70 Col. 71 Formula C 1 Formula C 2

Formula 3

3. ID Number Cols. 73-78

Same unique identification number as on records 1 and 2.

4. Record Sequence Number Col. 80

Always a

'3'.

71
Description of Data Parameters for Record 4

1. Chemical Name Cols. 1-67

For inorganic compounds, the numbers following the name indicate the number of atoms (or radicals)

corresponding to each word in the name. Example:

Iron Nitride (3 1) = Fej N

Zirconium Bromide Nitride (1 1 1) = Zr Br N

For organic compounds, the names as given in the literature have not been modified.

2. Continuation Code and Sequence No. Cols. 71-72

Same as for record 3 (item 4),

3. ID Number Cols. 73-78

Same unique identification number as on records 1,2, and 3.

4. Record Sequence Number Col. 80

Always a *4'.

72
Appendix II. 1-Line Search File

The 1-Line Search File contains crystallographic and chemical data on approximately 60,000 materials.

This data file contains selected lattice and chemical data which were derived from the NBS Crystal Data File

(see app. I).

Structure of an Entry

Each entry corresponds to the data for one crystalline compound. Each entry consists of the reduced cell, cell volume, space group number, crystal system, structure code, sequence number, journal reference, and

chemical formula. There is one record (132 characters) per entry.

Format Speciflcations

Column Format Item

1-6 F6.2 a (reduced cell)

7-12 F6.2 b

13-18 F6.2 c 19-24 F6.1 alpha

25-30 F6.1 beta

31-36 F6.1 gamma

37-42 16 Reduced cell volume

43 IX Blank

44-46 13 Space group number

47 IX Blank

48 Al Crystal system (A, M, O, R, T, H, C)

49 Al Structure code (N, L, T)

50 IX Blank

51-55 15 Sequence number

56 IX Blank

57-62 A6 CODEN for journal reference

63-66 A4 Volume

67 IX Blank

68-71 A4 Page

72 IX Blank

73-76 A4 Year

77 IX Blank

78-131 A54 Chemical formula

132 Al Truncation symbol for chemical formula

Description of Data Parameters

1. Reduced Cell Cols. 1-36

The reduced cell is a unique, primitive ceil that is based on the three shortest vectors of the lattice. The cell edges are set so that a < b < c. The mathematical conditions for full reduction and procedures to obtain

the reduced cell have been published.

2. Reduced Cell Volume Cols. 37-42

Cell volume (A^) of the primitive reduced cell.

3. Space Group Number Cols. 44-46

A number (1-230) as given in V ilume 1 of the International Tables for X-ray Crystallography (1969 b).

73
4. Crystal System Col. 48

A = Anorthic (Triclinic)

M = Monoclinic

O = Orthorhombic

R = Rhombohedral

T = Tetragonal

H = Hexagonal

C = Cubic.

5. Structure Code (N, L, T) Col. 49

T' signifies that a full structure determination has been reported in the original literature. Usually all parameters except hydrogen have been refined.

'N' signifies structural data have not been given.

'L' signifies that limited structural data have been reported.

6. Sequence Number Cols. 51-55

The position of the entry in the 1-Line Search File.

7. Journal Reference Cols. 57-76

The journal reference to the original literature from which the data were extracted is given as: CODEN

(i.e., a code representing the journal name), volume, page, year. Most CODENS correspond to those used

by Chemical Abstracts, a few have been made up by the data center. A file of CODENS and journal names

is distributed with the NBS Crystal Data File.

8. Chemical Formula Cols. 78-131

The formula is expressed by a sequence of discrete units. For a complete description of the conventions

used in the data base, see NBS Technical Note 1 141.

9. Truncation Symbol Col. 132

A '*' signifies that the chemical formula has been truncated to 54 characters.

itU.S. GOVERNMENT PRINTING OFFICE:, 1986't9107020322

74
FOKM NBS-1 14A (REV.) 1-84)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS/TN-1214

2. Performing Organ. Report No, 3. Publication Date December 1985

4. TITLE AND SUBTITLE

NBS*LATTICE A Program to Analyze Lattice Relationships

5. AUTHOR(S)

Vicky L. Himes and Alan D. Mighell

6. PERFORMING ORGANIZATION Of joint or other thon N6S, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as item 6

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

A FORTRAN program to analyze lattice relationships has been written and is available for distribu- tion by the NBS Crystal Data Center. The present version of NBS*LA TTICE performs several functions

including: 1) the characterization and identification of unknown materials using lattice-formula matching

techniques; 2) the calculation of the reduced cell of the lattice, and the calculation and reduction of

specified derivative supercells and/or subcells (i.e., this program function calculates the standard cells which are useful in the determination of metric lattice symmetry, in fmding a matrix relating two unit

cells, etc.); 3) unit cell transformations; and 4) matrix inversions. It is planned to incorporate additional functions in forthcoming versions of this program. Among others, these functions will include a matrix method to determine metric lattice symmetry and a technique to find a transformation matrix relating any two unit cells.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

computer program; lattice; identification; reduction; supercell; subcell;

S3niraietry; transformation; data base

13. AVAILABILITY

[53Unlimited

For Official Distribution. Do Not Release to NTIS

SB Order From Superintendent of Documents, U.S. Government Printing Office, Washington, DC

20402.

[^3] Order From National Technical Information Service (NTIS), Springfield, VA 22161

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PRINTED PAGES

79

15. Price

USCO/MM-DC 8J-M06

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