^^ IGZTT Umted states Department of Commerce

NATL INST. OF STAND J.JKH.ij.lC,

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National Institute ot Standards and Technology

NISI

PUBLICATIONS

NIST Technical Note 1290

NIST * LATTICE

A Program to Analyze Lattice Relationships

Vicky Lynn Karen and Alan D. Mighell

QC

100

U5753

#1290

1991

C.2

Ni S K QC/Ot)

NIST Technical Note 1290 AV

NIST * LATTICE

A Program to Analyze Lattice Relationships

Spring 1991 Version

Vicky Lynn Karen

Alan D. Mighell

Reactor Radiation Division Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

September 1991

*** Patent pending on certain algorithms used in this program. ***

f WW %

o ,

*&i— V

S*TEi O* ^

U.S. Department of Commerce

Robert A. Mosbacher, Secretary

National Institute of Standards and Technology

John W. Lyons, Director
National Institute of Standards

and Technology

Technical Note 1290

Natl. Inst. Stand. Technol.

Tech. Note 1290

39 pages (Sept. 1991)

CODEN: NTNOEF

U.S. Government Printing Office

Washington: 1991

For sale by the Superintendent

of Documents

U.S. Government Printing Office

Washington, DC 20402
CONTENTS

I. Introduction 1

II. Relate Two Lattices via the Converse-Transform

A. General 2

B. Execution of program function 3

III. Reduction and Derivative Lattice

A. General 4

B. Execution of program function 5

IV. Symmetry Determination through Converse-Transformation Analysis

A. General 6

B. Execution of program function 21

V. Cell Transformation

A. General 22

B. Execution of program function 22

VI. Matrix Inversion

A. General 23

B. Execution of program function 23

VII. Description of Input Lines

A. FORMATS 24

B. PROGRAM Line 25

C. CELL Line 26

D. MATRIX Line 27

E. REL Line 28

F. RSS Line 29

G. SYM Line 30

VIII. Program Execution: Example

A. Input flowstream 31

B

.

Program output 33

IX. Appendix

Algorithm to generate converse-transformation matrices 34

iii

NIST*LATTICE

A Program to Analyze Lattice Relationships

I. Introduction

The NIST*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. The

present version of the program performs several functions including : 1)

the determination of symmetry, and the evaluation of experimental error,

through Converse-Transformation analysis; 2) the generation of

transformation matrices relating any two unit cells; 3) the calculation of

the reduced cell of the lattice, and the calculation and reduction of

specified derivative supercells and/or subcells; 4) unit cell

transformations; and 5) matrix inversions. It is planned to incorporate

additional features in forthcoming versions of this program.
II. RELATE TWO LATTICES via the Converse-Transform

A. General

This program function makes use cf the Converse-Transformation

operator, defined as

CT ] (Y.Z) == { (H,T) }

h,t i,j i=0,

j=0,m

where h,t represent the domains of H,T, respectively, and Y,Z,H,T represent

vector triples in R3 (Karen and Mighell, U.S. Patent Pending).

In this application of the CT operator, the REL program function allows

the user to relate ANY two cells using the specified matrix elements, h, and

input tolerances, t. Specifically, all transformation matrices. H, relating

input CELL 2 to input CELL 1 are generated. The output tolerance matrix,

associated with a matrix H, represents how closely the transformed CELL 2

agrees with CELL 1, and is of the form

tol a tol b tol c tol alpha tol beta tol gamma

Thus, if CELL 2 is defined by A2 , B2, C2, ALPHA2, BETA2, GAMMA2, then the

transformation of CELL 2 by a matrix H wi 1 1 give a transformed cell having

lattice parameters Al + tol a, ... GAMMA1 + tol gamma.
B. Execution of program function

num

1) REL probs

2)

I range

F range

M num of

el em

tol

a tol

b

tol

c

tol

alpha

tol

beta

tol

gamma

3) all al2 al3

(M option only)

a21 a22 a23 a31 a32 a33

(CELL 1) Al

(CELL 2) A2

(Lattice Group 1)

Bl

B2

CI

C2

ALPHA1

ALPHA2

BETA1

BETA2

GAMMA

GAMMA2

FORMATS

1) PROGRAM Line

2) REL Line

3) Possible matrix elements

(format = MATRIX Line)

(M-option only)

4) CELL Line

(A5,3X,I2)

(A1,2X,I2,5X,6F10.5)

(9F8.2)

(10X.6F10.5)

COMMENT

* Repeat (4) to equal number of problems.

* Relate cell with smaller volume to cell with larger volume;

that is, set CELL 1 = large volume and CELL 2 = small volume

(program calculates inverse matrix).

* Recommend routine use of I 8 or larger.

* CAUTION - Use F,M options with extreme caution. Start with

small tolerances and restrict number of matrix elements input

If necessary, gradually increase with subsequent executions

of the program.
III. Reduction and Derivative Lattice

A. General

The reduced cell is a unique, primitive cell that is based on the three

shortest noncoplanar vectors of the lattice and satisfies a specified set of

mathematical conditions. To be reduced, the cell must be in normal

representation and both the main and the special conditions for reduction

must be met.

Reduction is a mathematical procedure that leads to a unique cell,

provided there is no experimental error in the unit cell parameters.

Unfortunately, experimental error is always present for cells determined in

the laboratory. Furthermore, it appears that the reported experimental

error is too optimistic, even for unit cells refined by least-squares

analysis using modern diffractometry. 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. Thus, it is essential that the symmetry and

experimental error be evaluated through CT analysis BEFORE attempting to

obtain a standard or conventional cell.

There are many reasons for calculating derivative cells of the lattice.

For example, cells determined from single-crystal and powder data are sometimes derivative cells. In single-crystal work, a subcell in direct

space may be determined if reciprocal lattice nodes are missed on a diffraction photograph or a diff Tactometer . When using powder methods, a supercell in direct space may be determined if one does not find the

smallest cell consistent with a set of d-spacings. With NIST*LATTICE

,

supercells having volumes n (n = 2-9) times the volume of the reduced input

cell and/or subcells having 1/n (n = 2-9) times the volume of the reduced

input cell may be calculated.
B. Execution of program function

num

1) RSS probs

2) ty

pe

C ALPHA

( cell )

BETA GAMMA

FORMATS

1) PROGRAM Line

2) RSS Line

(A5,3X,I2)

( 1 1 , 2X , 21 1 , 3X , 2A1 , 6F 10 . 2

COMMENTS

* Repeat (2) to equal number of problems.

* CAUTION - RSS alone should NOT be used to determine symmetry or to find a transformation matrix to a conventional cell.
IV. SYMMETRY DETERMINATION through Converse-Transformation Analysis

A. General

In sharp contrast to other methods which focus on the consequences of

symmetry (such as dot products, d-spacings, etc.)- this approach deals with

symmetry in its most abstract form - represented as matrices. The basis of

the SYM program function is to generate a group of matrices reflecting the

holohedry of the lattice. This is accomplished through a specialized

application of the Converse-Transformation operator, defined as

CT ] (Y.Z) = { (H,T) }

h,t i,j i=0,n

j=0,m

where h,t represent the domains of H,T, respectively, and Y,Z,H,T represent

vector triples in R3 (Karen and Mighell, U.S. Patent Pending).

The SYM program function generates the matrices, H, that relate ANY

primitive cell of the lattice to itself. In theory, it is the nature of the

matrices themselves that defines the set to be analyzed (i.e., those

defining a symmetry group). In practice, however, the usual result is that

the tolerance matrices alone clearly define the groups and all that is

required to determine metric lattice symmetry and pseudosymmetry is to

count. The numbers of matrices for the seven lattice metric symmetries

are: triclinic, 1; monoclinic, 2; orthorhombic, 4; rhombohedral , 6;

tetragonal, 8; hexagonal, 12; and cubic, 24.

Generated with each symmetry matrix is a tolerance matrix of the form

tol a tol b tol c tol alpha tol beta tol gamma

Simply by averaging the group of tolerance matrices, an error matrix is

calculated that may be compared directly to the e.s.d.'s for the primitive

cell. More importantly, this error matrix (= averaged tolerance matrix) may

be used to calculate an idealized cell reflecting exact metric symmetry

(i.e., idealized cell = Al + avg.tol.a, ... GAMMA1 + avg. tol .gamma) . By

incorporating information from both sets of matrices, H and T, the idealized

cell provides a unique means of evaluating experimental error based on ALL

the symmetry operations of the lattice. This is crucial when determining

standard or conventional cells.

When evaluating symmetry using CT analysis, the experimentalist need

not rely solely on metric information. The group of H matrices may be

viewed as sets of equivalent (h,k,l)'s represented in matrix form. Write

the indices of a known reflection as a column matrix and premultiply by H to

generate sets of reflections that should have equivalent intensities if the

metric and crystal symmetry agree.
Once the symmetry and error have been evaluated, obtaining a transformation matrix to a standard or conventional cell may be accomplished

either through analysis of the H matrices themselves, or by inputing the

idealized cell reflecting exact metric symmetry into the RSS program

function with subsequent Table look-up (*Note* This is possible ONLY after

the symmetry and experimental error have been evaluated through CT

analysis) .

For additional details as well as discussions of other theoretical and

practical applications in crystallography, see Acta Cryst. (1987). A43,

375-384.
Table la. Metric Classification

Reduced

Form #

Monocl inic

10, 14 -1

-1

-1

17 1

1 -1 -1 -1

20, 25 -1

-1

-1

27 1

1 -1

1 -1

28 -1

-1

-1 1

29 -1

-1 1 -1

30 -1

-1

-1 1

Matri x
Table la. Metric Classification - continued

Reduced

Form #

Monocl inic

33 -1

1 -1

34 -1

-1

1

35 1 -1

-1

37 1 -1

-1 -1

39 1 -1 -1

-1

41 -1

1 -1 -1

43 -1

-1

1 1 1

Matri x
Table la. Metric Classification - continued

Reduced

Form = Matrix

Orthorhombic

8 -1 -1 -1

1

1

13 -1

-1

1

16 -1

-1

1 1 1

19 -1

-1 1 -1 1

23 -1

-1

-1

26 -1

-1 1 -1

32 -1

-1

1

1

1 -1 -1

1

-1

1 -1

-1

1 -1

•1 -1

-1

1

1 1

-1 -1

-1 1

•1

1 -1

1

1

1 -1 -1

1

1 -1

1 11 -1 -1

1

1 -1

1 -1

1 -1

-1

1

1 -1

1 -1

1 -1

-1

10
Table la. Metric Classification - continued

Reduced

Form w Matrix

Orthorhombic

36

38

40

42

-1

-1

-1

1

-i

1

1

1 -1

1

1 1 - i

1

1 -1 -1

1

1 -1 -1

1 -1

1 -1

1

1 -1

-1

1 -1

-1

1 -1

1 -1

11
Table la. Metric Classification - continued

Reduced Matrix

Form #

Rhomboheidral

2, 4 -1

-1

-1

1

1

1

9 -1 1 -1

-1 1

-1

1 -1

-1 1

24 -1

-1

1 1 1

1 -1 -1 -1

1

-1

1 -1

1

1

1

1 1

11 -1

1

1 -1

1 -1

• 1 -1

-1

1

1 -1 -1 -1

-1

-1

1

-1

1 -1

1

1 1 1 0-1

12
Table la. Metric Classification - continued

Reduced

Form #

Matrix

Tetragonal

6 -1 -1 -1

1

1

1 -1 -1 -1

1

7 -1 -1 -1

1

1

1 -1 -1 -1

1

11 -1

-1

1

1 -1

1

15 -1

-1

1 1 1

1 -1

1 1

1 -1

1 1 1

1 1

1 -1 -1

1 -1

-1

1

1•1 -1 -1

1

1 -1

1

1 -1

1

1 -1 -1

1

1 -1 -1 -1

-1

1 -1

1 1 1 -1

1

-1

1 1 1

1

1 1 1 -1

-1

-1

1 -1

1 -1

-1

1 -1 ! 1

-1

1 -1

1 -1

-1

1 1 1 -1

-1

11 1 1

-1

1

1

-1

1

1 1

13
Table la. Metric Classification continued

Reduced Matrix

Form #

Tetragonal

18 -1 -1 -1 -1

-1 1 -1 1 -1 -1

-1 1 -1 -1 1 -1

1 1 1

1 1 -1 1 -1 I

1 -1 1 -1 1

21 -1 -1 -1 -1

-1 -1 1 1

1 -1 1 -1

1 1 1 -1 -1 1 I -1 1 -1

14
Table la. Metric Classification - continued

Reduced

Form #

Matr ix

Hexagonal

12

22

1 -1

1 -1

-1

1 -1

1

1 -1

-1

•1 -1 -1

1

-1

1 -1 -1

1 -1

1 1

1 -1

1

1

-1

1 1

1

1 1

1

1

1 -1

1 1

1

1 1 -1

1

1 -1 -1

1 -1

1

1

1 -1

1

1 1 -1

-1

1 -1

-1

1 -1 -1

1

1

1 1 -1

1

1 1 -1

1

1 -1

1

1

1

1 -1

-1

15
Table la. Metric Classification - continued

Reduced

Form #

Matrix

Cubic

1 -1 -1 1 -1 1

1 1 -1 -1 1 -1 1

1 1 -1 -1 1

1 1 -1 1 -1 -1

1 1 -1 1 -1

1 1 1 -1 -1 -1 1

-1 1 -1 1 -1 -1

-1 1 -1 1 -1

1 -1 -1 1 -1 1 -1

1 1 1 -1 1 -1

1 1 1 -1 1 1 -1

-1 1 1 1 -1

1 1 1 -1 1 -1

1 1 -1 -1 1 1 -1

1 -1 1 -1 1

1 -1 1 -1 1 -1 1 1 -1 I

1 -1 1 -1 1 -1

16
Table la. Metric Classification - continued

Reduced

Form #

Matrix

Cubi c

1 -1 -1 -1

-1 -1 1 1

1 -1 1 -1

-1 -1 -1 -1

1 -1 1 1 -1 1 -1 1

-1 -1 -1 -1

1 -1 1 1

1 -1 1 -1

1 1 1 1 1 -1 1 1 -1 1 -1 1

1 1 1 1

• 1 -1 1 1

1 -1 1 -1

1 1 1 -1 -1 1 I -1 1 -1

17
Table la. Metric Classification - continued

Reduced

Form #

Matr i x

Cubic

1 -1 -1 -1 -1 -1 -1 -1 -1 -1

1 1 1 -1

1 1 1 1 1 1

1 -1 -1 -1

-1 1 1 1 -1 -1

-1 -1 -1 1 1 1

-1 -1 -1 -1

1 1 1 -1 -1 1 1 1 •1 1 1 1 -1 -1

1 1 1 1

• 1 -1 -1 1 1 -1 -1 -1

1 -1 -1 -1 1 1

1 1 1 1

1 1 -1 -1 -1 1

1 -1 -1 -1 1 -1 -1 -1

1 1 1 1 1 1 1 1 1 -1 -1 -1 I -1 -1 -1

18
Table lb. Metric Classification

* CAUTION * The Reduced Form Matrix and Type must be calculated

from the idealized cell reflecting exact metric symmetry.

Reduced Form Matrix Reduced Cell Reduced Form

No.

First Row Second Row Form

Type

Bravait

Lattice Transformation Reduced -* Conventional

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

1 a-a2

a-a2

a-a2 + Cubic F 111/111/111

2 b-c b-c b-c + Rhombohedral hR 110/101/111

3 - Cubic P 100/010/001

4 -lb-el -Ib-cl -Ib-cl - Rhombohedral hR 110/1 oi /iii

5

a-a3

a-a

3

a-a3

- Cubic I 101/110/011

6

-a-a + la-bl 2

-a-a + la-bl 2

-la-bl - Tetragonal I 011/101/110

7 -lb-el -a-a + lb-el 2

-a-a + lb-cl 2

- Tetragonal I 101/110/011

8 -lb-c| -la-cl -da-al-lb-cl-la-cl) - Orthorhombie I iio/ioi/oii

a-b

9 a-a a-a c*c

a-a2

a-a2

a-a2

+ Rhombohedral hR 100/110/113

10 a-a a-a c*c b-c b-c a-b + Monodinic C** 110/110/001

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

12 a*a a*a c*c

a-a2

- Hexagonal P 100/010/001

13 -la-bl - Orthorhombie C 110/110/001

14 a-a a-a c-c -ib-cl -Ib-cl -la-bl - Monodinic C** 110/110/001

15 a-a a-a c*c

a-a

" 2

a-a2

- Tetragonal I 100/010/112

16 a-a a-a c*c -Ib-cl -lb-el -<a-a-2lb-c!) - Orthorhombie F iio/iio/ii2

17 a-a a-a c-c -Ib-cl -la-cl -(a-a-|b-cl-|a-cl) - Monodinic l T* ioi/iio/011

b-e

18 a-a b-b b-b

a-a4a-a2

a-a2

+ Tetragonal I 011/H1/100

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

a-a2a-a2

+ Orthorhombie 1 ioo/oi 1/iii

20 a-a b-b b-b b-c a-e a-e + Monodinic C+ 011/oii/ioo

21 a-a 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 - Orthorhombie C 011/011/100

24 a-a b-b b-b *- T

2

a-a3

a-a

" 3

- Rhombohedral hR 121/011/100

25 a-a b-b b-b -Ib-cl -la-el -la-cl - Monodinic C

+ 011/011/100

19
Table lb. Metric Classification (continued)

Reduced Form Matrix Reduced Call Reduced Form

No.

Firit Rowt+t Second Row Form

Type

Bravait

Lattice Transformation Reduced -» Conventional

••• b-b c-c b-c a-e a-b i«b«c

26 a-a b-b c-c

a-a4

a-a2

a-a2

+ Orthorhombic F 100/120/102

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

a-a 2

a-a2

+ Monodinic I*** 011/100/111

28 a- a b'b c-c a-b

2

a-a2

a-b + Monodinie C 1 00/1 02/010

29 a-a b-b cc

a-c

2b-b

a-c

a-b

a-a2

+ Monodinic C 100/120/001

30 a-a b-b c-c

2 2 a-b + Monodinic C 010/012/100

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

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

33 a-a b-b c-c -la-el - Monodinic P 100/010/001

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

35 a-a b-b c-c -lb-el - Monodinic P oio/ioo/ooi

36 a-a b-b c-c

a-a

2

- Orthorhombic C 100/102/010

37 a-a b-b c-c -lb-el

a-a2

- Monodinic C* 102/100/010

38 a-a b-b c-c

a-a2

a-a

- Orthorhombic C 100/1 20/001

39 a-a b-b c-c -lb-el

b-b "T Monodinic C** 120/100/001

40 a-a b-b c-c "T"

- Orthorhombic C OlO/012/iOO

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

2

-la-el - Monodinic Cf 012/010/100

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

2

a-a

" 2

- Orthorhombic 1 100/010/112

43 a-a b-b c-c b-b-la-bl

2

a-a-la-bl

2

-la-bl - Monodinic 1 100/112/010

44 a-a b-b c-c -lb-c| -la-el -la-bl - Tridinic P 100/010/001

If a-a<4|a-cl

)

• If b-b<4lb-cl > •• lfc-e<4|b-cl )

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

... !! ^"VS

e *l'.VCl

, I Premultiply Table Matrix by 101/010/100 (C centered) •"If 3b-b<c-c + 2lb-cl ) t+t No required relationship! between symmetrical scalers for reduced forms 26-44.

20
3. Execution of program function

num

1) SYM probs

I range

2) RRl

tol

a

tol

b tol

c tol

alpha

tol

beta

tol

gamma

3) RSS Line

information

(only for R,R1

opt ions)

Al Bl CI ALPHA1 BETA1 GAMMA

1

FORMATS

1) PROGRAM Line

2) SYM Line

3) CELL Line

-- or -- RSS Line

(A5,3X,I2)

(A1,I1,1X,I2,5X,6F10.5)

(10X,6F10.5)

(I1,2X,2I1,3X,2A1.6F10.2)

COMMENTS

* Repeat (3) to equal number of problems.

* Recommend routine use of large tolerances, typically

1.0 Angstroms and 4.0 degrees, in order to obtain a menu

of all possible symmetries and pseudosymmetr ies.

* Recommend R option (large tolerances) for routine use, and

I 24 option for skewed cells.

* For all executions except R or Rl, a PRIMITIVE cell is assumed.

* For R, Rl matrix element codes, tolerances are applied to

REDUCED cells

21
V. Cell Transformation

A. General

This program function enables the user to transform a unit cell by a specified three by three matrix. This transformation is defined by the

equation :

(matrix) CELL A = CELL B

with a matrix of the form :

( all al2 al3 )

( a21 a22 a23 )

( a31 a32 a33 )

B. Execution of program function

num

1) TRANS probs

2) all al2 al3 a21 a22 a23 a31 a32 a33

3) Al Bl CI ALPHA1 BETA1 GAMMA1

FORMATS

1) PROGRAM Line (A5.3X.I2)

2) MATRIX Line (9F8.2)

3) CELL Line (10X.6F10.5)

COMMENTS

* Repeat (2,3) to equal number of problems.

22
VI. Matrix Inversion

A. General

This program function enables the user to calculate the inverse of a three by three matrix of the form :

( all al2 al3 )

( a21 a22 a23 )

( a31 a32 a33 )

B. Execution of program function

num

1) INV probs

2) all al2 al3 a21 a22 a23 a31 a32 a33

FORMATS

1) PROGRAM Line (A5,3X.I2)

2) MATRIX Line (9F8.2)

COMMENTS

* Repeat (2) to equal number of problems.

23
VII. Description of Input Lines

Each individual Section II through VI contains additional information

concerning the theory and operations for each program function.

A. FORMATS

* PROGRAM Line

* CELL Line

* MATRIX L ine

* REL L ine

* RSS Line

* SYM Line

(A5.3X.I2)

(10X.6F10.5)

(9F8.2)

(A1,2X,I2,5X,6F10.5)

(I1,2X,2I1,3X,2A1,6F10.2)

(A1,I1,1X,I2,5X,6F10.5)

24
B. PROGRAM Line

The PROGRAM Line specifies the type of program function and the number

of indepenaent problems to be considered.

PROGRAM Line

Format(A5,3X.I2)

Column Format Item

1-5 A5 Type of program function

Relate Two Lattices

Reduction and Derivative

Supercell and Subcell

Symmetry Determination

Cell Transformation

Matrix Inversion

'REL

'RSS

'SYM

'TRANS

'INV

6-8 3X Blank

9-10 12 Number o

25
C. CELL 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 Line

Format(10X,6F10.5)

Column Format Item

1-10 10X Blank

11-20 F10.5 a (Angstroms)

21-30 F10.5 b

31-40 F10.5 c

41-50 F10.5 alpha (degrees)

51-60 F10.5 beta

61-70 F10.5 gamma

26
D. MATRIX Line

The MATRIX Line defines the three by three matrix to be used when

executing the TRANS or INV program functions.

MATRIX Line

Format(9F8.2)

Column Format Item

1-8 F8.2 Matrix element all

9-16 F8.2 Matrix element al2

17-24 F8.2 Matrix element al3

25-32 F8.2 Matrix element a21

33-40 F8.2 Matrix element a22

41-48 F8.2 Matrix element a23

49-56 F8.2 Matrix element a31

57-64 F8.2 Matrix element a32

65-72 F8.2 Matrix element a33

27
E. REL Line

The REL Line defines the limits for the matrix elements and unit cell

tolerances .

REL Line

Format(Al,2X,I2,5X.6F10.5)

Column Format Item

Al I/F/M/ Matrix elements to be considered

by the program

I = Integers generated

F = Integers generated plus fractions

M = Manual input

2-3 2X Blank

4-5 12 1-40/1-10/2-99/ for I,F,M options

I,F options = Range of integers generated

M option = Number of matrix elements input

6-10 5X Blank

11-20 F10.5 tol a

21-30 F10.5 tol b

31-40 F10.5 tol c

41-50 F10.5 tol alpha

51-60 F10.5 tol beta

61-70 F10.5 tol gamma

28
F. RSS Line

The RSS 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.

RSS Line

Format (I 1,2X,2I1,3X,2A1,6F10.2)

Column Format Item

1 II Blank/1/2/3/

Blank = Reduction

1 = Reduction + supercells

2 = Reduction + subcells

3 = Reduction + supercells,

subcel Is

2-3 2X Blank

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

Initial value (nl) to define the

range of volumes for calculated

derivative lattices.

5 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.

6-8 3X

9 Al

10 Al

11-20 F10.2 a (Angstroms)

21-30 F10.2 b

31-40 F10.2 c

41-50 F10.2 alpha (degrees)

51-60 F10.2 beta

61-70 F10.2 gamma

29
G. SYM Line

The SYM Line defines the limits for the matrix elements and unit ceil

tolerances, as well as the type of lattice transformations to be performed

(i.e., the calculation and analysis of reduced and/or derivative lattices)

SYM Line

Format(Al,Il,lX,I2,5X,6F10.5)

Column Format Item

Al I/R/

I = Integers generated

R = RSS, then integers generated

II Blank/1/ Normal/abbreviated

RSS output

3 IX Blank

4-5 12 1-40/ Range of integers generated

6-10 5X Blank

11-20 F10.5 tol a

21-30 F10.5 tol b

31-40 F10.5 tol c

41-50 F10.5 tol alpha

51-60 F10.5 tol beta

61-70 F10.5 tol gamma

30
VIII. Program Execution: Example

A. Input flowstream

1) SYM 1 2) R 1.0 1.0 1.0 4.0 4.0 4.0

3) 00 C 14.259 22.539 8.741 90.00 114.10 90.00

4) RSS 2 5) 1 22 P 13.595 4.638 10.321 90.00 98.28 90.000

6) 2 22 P 15.928 4.623 18.271 90.00 105.58 90.000

7) REL 1 8) F 10 0.2 0.2 0.2 1.0 1.0 1.0

9) 6.003 6.007 23.389 97.04 92.61 110.16

10) 6.030 11.471 11.711 94.72 97.72 99.49

11) TRANS 1 12) -1.0 .00 .00 .50 .50 .00 -.5 -.5 -2.0

13) 6.030 11.471 11.711 94.72 97.72 99.49

14) INV 2 15) -1.0 .00 .00 .50 .50 .00 -.50 -.50 -2.0

16) -1.0 .00 .00 1.0 2.0 .00 .00 -.50 -.50

Notes

The 'SYM ' in columns 1-5 indicates that the SYM program

function will be executed. The ' 1' in columns 9-10

specifies that one independent problem is to be considered.

The 'R ' in columns 1-2 instructs the program first

to reduce the cell, with normal RSS output, and

then to generate the symmetry matrices. The tolerances

for the cell edges and angles are 1.0 Angstroms and

4.0 degrees, respectively.

This RSS Line instructs the program to calculate the

reduced cell of a C-centered input cell. Regardless

of the symmetry, all six unit cell parameters are specified in columns 11-70. Decimal numbers have been

used for fractions of a degree.

The 'RSS ' in columns 1-5 indicates that the RSS program

function will be executed; the ' 2' in columns 9-10 specifies

that two RSS Lines will follow (two problems are to be

considered) .

The '1' in column 1 specifies that supercells will

be calculated. The '22' in columns 4-5 are integers

(nl and n2) that define the range of volumes for

derivative cells to be calculated. In this case,

the supercells will have volumes ranging from (nl)V

to (n2)V. All six unit cell parametes are given

given in columns 11-70.

31
6 The '2' in column 1 specifies that subcells

will be calculated. The '22' in columns 4-5 are integers

(nl and n2) that are used to define the range of

volumes for the derivative cells to be calculated.

The subcells will have volumes ranging from ( 1/n 1 )

V

to (l/n2)V.

7 The ' REL ' in columns 1-5 indicates that the REL program

function will be executed. The ' 1' in columns 9-10 specifies

one problem.

8 The 'F' in column 1 and the '10' in columns 4-5 specify

that only integers in the range -10 to +10 plus simple

rational numbers will be considered as possible matrix elements

The tolerances for the cell edges and angles are 1.0

Angstroms and 4.0 degrees, respectively.

9-10 The program will calculate the H matrices relating

the two eel Is .

11 The 'TRANS' in columns 1-5 indicates that the TRANS program

function will be executed. The ' 1' in columns 9-10 specifies

that one independent problem will be considered.

12 This input line defines the matrix to be used to transform

the input cell. Each matrix element has been specified

by a decimal number.

13 All six unit cell parameters must be given regardless

of the crystal symmetry.

14-16 The ' INV ' in columns 1-5 indicates that the INV program

function will be executed. The ' 2' in columns 9-10 specifies

that the inverses for two matrices will be calculated.

32
B. Program output

1-3 The SYM program function can be used in the routine determination and

evaluation of symmetry. For this C-centered cell, the LATTICE program

generated a group of 6 symmetry matrices showing that the metric

symmetry is rombohedral. The crystal structure was originally

reported in a C-centered monoclinic space group. Later, the structure

was reanalyzed and republished in a rhombohedral space group.

4-6 The RSS program function may be used to establish the relationship

between two cells determined on a diff Tactometer

.

The first input cell ( Cell A ) was determined on an automated

diffractometer , diffraction data were collected, and the structure was solved. The structure solution was not completely successful as there

appeared to be an unusual amount of disorder. A cell was determined a second time (second input cell; Cell B) and it was found to have twice

the volume of the initial cell. When the data were recollected and

the structure was solved on the basis of Cell B, the disorder

di sappeared.

Cell B was found to be a supercell of Cell A as a matrix relating the

two cells has integer elements and a determinant of 2. In this case, a subcell of the correct cell was determined by missing nodes in

reciprocal space. This example illustrates how the calculation of

derivative lattices may prove useful in routine crystal lographic

analyses .

7-10 The REL program function can be used to determine derivative

subcel 1 /supercel 1 as well as composite relationships between cells.

In this example, CELL 1 was determined from a single crystal using an automated diffractometer. while CELL 2 was deduced by indexing a powder pattern.

When determining a cell from powder data using an indexing program,

one may obtain a cell that is related to the correct cell, especially

if too few lines have been used or if the d-spacings have large

experimental errors.

The output for this problem shows that CELL 2 is related to CELL 1 by

a matrix whose elements are integers and simple rational numbers. In

this case, therefore, the relationship is not that of a subcell or supercell, but is a composite one. The ability to establish such a relationship is very useful when characterizing materials.

11-13 In lattice analyses, it is often necessary to transform one unit cell

to another. In this example, the TRANS program function is used to

transform the unit cell obtained by powder indexing to the correct

cell of the lattice.

14-16 The INV program function calculates the inverses of these matrices.

33
Appendix

Synopsis of Algorithm to Generate All Converse-Transformation

Matrices (H) Relating Two Lattices. The Transformation is

Defined by the Equation {Y = (H Matrix) Z}.

Part 1: Find Matrix Triples which Statisfy Transformation of Cell

Edges (Only). — Generate one at a time all potential matrix triples.

Input matrix elements are used to form a row of a transformation matrix. Test each triple as outlined below. — Calculate symmetrical dot products for transformed Cell Z. Dot product may be a. a, b.b, or c.c depending on order of

matrix row in the final transformation matrix H. —- Calculate transformed cell edge for cell Z. — Test whether matrix triple transforms a-cell edge for

Z to Ya within the specified tolerance.

-- If yes, matrix triple is a possible 1st row

in a matrix H, save matrix row, actual tolerance,

transformed cell edge for Z. — Test whether matrix triple transforms b-cell edge for

Z to Yb within the specified tolerance. — If yes, matrix triple is a possible 2nd row

in a matrix H, save matrix row, actual tolerance,

transformed cell edge for Z. — Test whether matrix triple transforms c-cell edge for

Z to Yc within the specified tolerance.

-- If yes, matrix triple is a possible 3rd row

in a matrix H, save matrix row, actual tolerance,

transformed cell edge for Z. — Repeat cycle until all potential matrix triples have been

generated and tested. Then go to Part 2 (next page).

34
Part 2: Starting with Matrix Triples Satisfying Transformation of Cell

Edges (For Z to Y), Determine which Combinations also Transform

Cell Angles. — Generate all combinations of matrix triples for first 2 rows

of H Matrix. Test one at a time as outlined below. — Calculate gamma for transformed Cell Z and check

whether it is within specified tolerances. — If yes, then test one at a time all possible

3rd rows with a given row 1 and 2 combination.

-- Determinant of matrix H will be positive — if

not check next combination of matrix triples. — Calculate alpha for transformed cell Z and

check whether it is within specified tolerances.

-- Calculate beta for transformed cell Z and

check whether it is within specified tolerances. — If alpha and beta are acceptable, then

this combination of three matrix triples

transforms cell Z to cell Y within the

specified tolerances -- assign the elements

to matrix H. Output H. — Repeat cycle until all possible 3rd rows have been

combined with a given row 1 & 2 combination. — Repeat cycle until all potential H matrices (combinations of

matrix triples from Part 1) have been generated and tested.

In NIST*LATTICE, this algorithm has been encoded in standard, structured

FORTRAN. A detailed discussion of the strategy and applications to

phase characterization and to the automation of single-crystal

diffractometers has been set forth in a Patent Disclosure (Karen, V. L. and Mighell, A.D. (1989) U.S. Patent Application 07/434,383 ).

U.S. G.p.o. :1 9 91 _28l-557:40303

"
NIST-114A

(REV. 3-90) U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBUCATION OR REPORT NUMBER

NIST/TN 1290

2. PERFORMING ORGANIZATION REPORT NUMBER

3. PUBUCATION DATE September 1991

4. TITLE AND SUBTITLE

NIST * LATTICE A Program to Analyze Lattice Relationships

5. AUTHOR(S)

Vicky Lynn Karen and Alan D. Mighell

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

GAITHERSBURG, MD 20899

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

Same as item #6.

10. SUPPLEMENTARY NOTES

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. UTERATURE SURVEY, MENTION IT HERE.)IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

The NIST*LATTICE

to be used in any ana and can be used to an present version of th

the determination of

through Converse-Tran

transformation matric

the reduced eel 1 of t specified derivative

transformations; and

additional features i

program is written in standard FORTRAN and is designed

lytical laboratory. The software is multifunctional

alyze various types of lattice relationships. The

e program performs several functions including : 1)

symmetry, and the evaluation of experimental error,

sformation analysis; 2) the generation of

es relating any two unit cells; 3) the calculation of

he lattice, and the calculation and reduction of

super-cells and/or subcells; 4) unit cell

5) matrix inversions. It is planned to incorporate

n forthcoming versions of this program.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

computer program; derivative lattice calculation, latttice relationships, reduced cells;

symmetry determination

13. AVAILABILITY UNLIMITED FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

X

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

39

15. PRICE

ELECTRONIC FORM

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