^'"^ "' oo,

"i-^^. CO NBS TECHNICAL NOTE 1141

^fEM o^

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

NBS'^AIDSSO:

A FORTRAN Program for

Crystallographic Data Evaluation

)C

-00

15753

Jo.lM

-981

:.2
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The 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 per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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physical and chemical and materials measurement; coordinates the system with measurement

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physical and chemical measurement throughout the Nation's scientific community, industry,

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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 eftectiveness 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 A DP standards

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'Headquarters and Laboratories al Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
ITATIOMAL BOBBAO

OV STANBAXDai

LIBB4IT

NBS*AIDS80: ^^^^^^laai

A FORTRAN Program for

Crystallographic Data Evaluation

AJ6j

A. D. Mighell

C. R. Hubbard

J. K. Stalick

Center for Materials Science

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

^.< °^^o.

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1981
National Bureau of Standards Technical Note 1 141

Nat. Bur, Stand. (U.S.). Tech. Note 1141. 54 pages (Apr. I9SI) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Wasliiiigton, D.C. 20402

Price $3.75

(Add 25 percent for other than U.S. mailing)
TABLE OF CONTENTS

Abstract 1

1.0 Introduction 1

2.0 Research and Analysis Applications 2

2.1 Cell and Space Group Transformations 2

2.2 Determination of Metric Symmetry 3

2.3 Identification of Unknowns 5

2.4 Checking the Data for Consistency 6

2.5 Powder Pattern Analysis 6

3.0 Input Information 10

4.0 Output Information 20

5.0 Program Details 23

6.0 Acknowledgements 23

References 24

Appendix A. Space Groups and Aspects 26

Appendix B. Lattice Symmetry Determination 41

Appendix C. Examples of Input Decks and Output Listings 47

m

NBS*AIDS80: A FORTRAN PROGRAM FOR CRYSTALLOGRAPHIC DATA EVALUATION

A. D. Mighell, C. R. Hubbard, and J. K. Stalick

t

Center for Materials Science

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

Techniques for the computer-assisted evaluation of crystal! ographic

data have been developed to improve the data compilations of the

NBS Crystal Data Center and the JCPDS--International Centre for

Diffraction Data. The resulting computer program, NBS*AIDS80, can

be used for the analysis of unit-cell and powder diffraction data

by the general scientific community as well. Research and analysis

components include calculation of standard cells and space groups,

determination of metric symmetry, checking the data for consistency, generation of d-spacings and indices, and analysis of powder

patterns with calculation of figures of merit. Detailed input

and output specifications are given.

Key Words: Computer evaluation; crystal lographic data analysis;

JCPDS--International Centre for Diffraction Data; metric symmetry;

NBS Crystal Data Center; powder diffraction; unit cell.

1 .0 Introduction

The computer program NBS*AIDS80 is derived from several years of

experience at the National Bureau of Standards in the computer-assisted

evaluation of crystal lographic data. The program or its precursor has

been used routinely by the NBS Crystal Data Center for the analysis of

crystallographic data in the preparation of Crystal Data Determinative

Tables ,^ and it has recently been expanded in collaboration with the

JCPDS- -International Centre for Diffraction Data for analysis of powder

data for the Powder Diffraction File .^ The combined evaluation of

single-crystal and powder data in a common format enhances the quality

of the data in both data compilations.

NBS*AIDS80 was originally intended for use by the data base builders in the evaluation of literature data and the creation of master data

files. With such files, research studies can be carried out, indexes

can be generated, and publications can be prepared. In addition, the

files can be searched in an on-line or batch mode. However, it has

become apparent that the data evaluation routines in NBS*AIDS80 are of

value as a research and analysis tool for the general scientific community as well. In our evaluation of thousands of entries for the Crystal

k

^Submitted in part for inclusion in Volume 24 of "Advances in X-ray

Analysis," Plenum Press, New York, NY.

Raised figures indicate literature references on page 24.

1
Data and Powder Diffraction Files, we have noted an unexpectedly large

number of errors in the published crystal lographic data, especially with

respect to symmetry determination, unit cell transformation, density

calculation, and the indexing of powder data. It is our hope that by

using NBS*AIDS80 research scientists and journal editors will find and

correct such errors prior to publication, and that additional analysis

routines will be suggested for incorporation into the program.

2.0 Research and Analysis Applications

NBS*AIDS80 may be used as a research aid in conjunction with

routine diffractometry for materials identification and characterization,

in addition to the areas mentioned above. For example, in the determination of a crystal structure the program can be used once unit-cell

dimensions have been determined to establish the probable symmetry of

the lattice, the transformation matrix to the standard setting of the

cell and space group, the calculated density for a check on unit cell

contents, and the determinative ratios needed to find out if the same or

a similar compound previously has been solved. The program should be

used again at the end of the structure determination to give final

values for publication. NBS*AIDS80 is also valuable in powder diffraction analyses. The program can generate d-spacings for any symmetry and

input cell, and input powder data can be indexed based on a known unit

cell. Once the pattern has been indexed, the program compares the

observed and calculated d-spacings, flags systematic absences if present,

and calculates the figures of merit F(N) and M(20). In addition, a table of powder data for publication can be prepared. The research and

analysis applications of NBS*AIDS80 are summarized in the following

sections.

2.1 Cell and Space Group Transformations

In order to compare unit-cell and space-group data from separate

experimental determinations, it is essential that all data be reported

in a standard orientation. NBS*AIDS80 transforms the input cell and

space group to the Crystal Data setting, and the space-group number is

assigned. The Crystal Data cell provides a standard setting for the

publication and comparison of unit-cell parameters, and the determinative ratios can be used for direct entry into Crystal Data Determinative

Tables for identification of unknowns and location of isostructural

materials.

Errors are commonly made in the transformation of unit-cell axes

and space groups, especially when unusual centerings are encountered.

For example, SbSn and NiO crystallize in a rhombohedrally-distorted

sodium chloride structure. These cells are frequently reported using Fcentered rhombohedral axes; the corresponding hexagonal axes are calculated ignoring the face-centering, leading to incorrect axes and erroneous indexing of powder data. A more common use of NBS*AIDS80 would be

the calculation of hexagonal unit-cell axes from primitive rhombohedral

cell parameters.
Perhaps more importantly, it is essential that a given lattice be

characterized by the same cell before comparisons are made. De Camp^

has given an excellent example involving a lattice that has two different unit cells which appear to be based on the same three lattice translations. Crystals from two samples of condelphine hydroiodide gave apparently the same triclinic unit-cell dimensions when the cells v/ere re- duced using two separate reduction algorithms, with only the angle gamma

differing significantly (Figure 1). The other differences were assumed

to arise from experimental errors. The two cells are shown to define

the same lattice when they are transformed using the NBS*AIDS80 algorithm.

Cell 1 Cell 2 Cell 1 o o a 9.34A 9.32A 9.34A

b 17.39 17.45 17.39

c 9.10 9.09 9.10

a 94.85° 94.84 94.85°

3 119.15 118.83 119.15

Y 88.57 86.50 86.71

Figure 1. Unit cell dimensions for condelphine hydroiodide. Cell 1 is

transformed to cell 1' by the matrix (101,010,001).

2.2 Determination of Metric Symmetry

The reduced cell^ and the reduced form number are calculated by

NBS*AIDS80 to give the metric symmetry of the lattice. ^ The metric

symmetry is the highest symmetry possible for the lattice based solely

on geometric considerations. A systematic analysis of the Crystal Data

File has shown that the metric symmetry is almost always the same as the

true symmetry of the lattice.^ The metric symmetry is determined from

the Niggli reduced cell by the values of the following vector products^

known as the reduced form:

b-c

b-b C«C

-b

The reduced form can thus be used to indicate

of the lattice. If the reduced form indicates the

symmetry than the input lattice, the program will

the highest possible lattice symmetry. It is well

reduction provides a practical way to prove that a triclinic, as the crystal symmetry can never be hi

indicated by the reduced form. The use of the red

probable lattice symmetry is particularly importan

current trend towards determining the unit cell of

from diffractometer data. An example of the use o given in Figure 2.

3

the probable symmetry

possibility of higher

print the cell with

known that cell

lattice must be

gher than the symmetry

uced form to indicate

t in light of the

a compound solely

f this procedure is
Initial Cell Reduced Cell Metric Cell

a

b

ca

8.095A

8.096

30.62

88.67"

58.08

87.48

Figure 2.

8.096A

8.095

25.990

90.00°

90.00

92.52

Reduced Form

11.698A

25.990

11.194

90.00

90.01

90.00

65.5

0.0

Determination of the metric cell for an unknown antimony

tartrate. The metric cell was found by the program to be

orthorhombic B-centered, reduced form number 13.

Cell 1 Cell 2 Cell 3 Common Cell

a 9.407A 9.407A 9.416A 9.407A

b 13.773 13.800 13.768 9.407

c 12.829 12.801 12.822 9.416

a 90.06° 90.05° 90.00° 94.06°

B 96.15 95.99 96.14 94.30

y 89.84 90.01 89.84 94.07

Figure 3. Alternate A-centered cells for chabazite, all leading to a common cell

.

It should be emphasized that the reduced form as given by NBS*AIDS80

must be carefully analyzed in order to determine the true metric symmetry, as the assignment of symmetry is often dependent on the assessment

of experimental errors. A new routine for symmetry determination is

being developed and will be added to the program. This routine will

give all possible symmetries and pseudosymmetries. In addition, we plan

to add evaluation routines to handle specialized reduced forms that can result from, for example, the mis-indexing of a powder pattern, the

determination of the subcell of a lattice, or twinning.

One example of metric pseudosymmetry involves the mineral chabazite,

Ca2Al4Sie02i+*12H20. Chabazite has been reported as having rhombohedral

,

monoclinic, or tri clinic symmetry. Although the metric symmetry is

nearly rhombohedral, analysis in our laboratory reveals the symmetry to

be monoclinic or lower. Figure 3 gives three similar but distinct Acentered cells that can be related to a common pseudo-rhombohedral

cell.

7
Much of the effort involved in the solution of crystal structures

can be avoided if the metric symmetry is determined prior to data collection. In a systematic analysis of organic compounds, we noted a number of cases in which rhombohedral symmetry was incorrectly reported

as monoclinic or even triclinic synmetry. While the molecular geometries determined were correct, many more structural parameters than

necessary were determined and refined leading to a lower overall accura- cy of the structure parameters. Figure 4 gives one such example.

Rhombohedral Cell

Input Cell (Hexagonal Axes) Reduced Cell

a 19.900A 11.489A 11 .489A

b 11.489 11.489

c 21.258 60.590 21.258

a 74.32°

3 108.18° 74.32

Y 60.00

Reduced Form

132.0 451. 9\ /a-a a-a

66.0 66.0 / \ a-a a^a

2 2

Figure 4. Example of a rhombohedral cell reported as a C-centered

monoclinic cell. The reduced form (number 9) is clearly

rhombohedral

.

2.3 Identification of Unknowns

The standard Crystal Data cell and determinative ratios calculated

by NBS*AIDS80 can be used for the identification of unknown compounds.

For example, the unknown antimony tartrate mentioned above (see Figure

2) was identified using Volume 3 of Crystal Data Determinative Tables . The B-centered orthorhombic cell (a = il. 6941577 b = 25.961(9), c

= 11.192(5), a/b = 0.4504, c/b = 0.4311) was readily identified as potassium antimony tartrate tri hydrate using the portion of the Tables given

in Figure 5.

a/b c/b _ci _b _c Formula

0.4502 0.3664 8.76 19.43 7.13 CisHiqCIz

0.4506 0.3361 9.251 20.53 6.901 CisHasBr

> 0.4510 0.4316 11.696 25.932 11.192 K2Sb2C8H40i2-3H20

0.4511 0.4473 8.746 19.388 8.672 CaCeHi206.3H20

Figure 5. Identification of an unknown antimony tartrate using Crystal

Data Determinative Tables.
Even though the compound is supposedly known, it is always advisable to obtain the standard cell and check for identification prior to

data collection. The crystal used for data collection frequently does

not correspond to the analysis of the bulk material. For example, data

on orthorhombic sulfur has mistakenly been collected many times.

An alternative identification scheme will soon be available using

the reduced cells of all materials given in Crystal Data . This Identification File currently exists at NBS, and it will soon be available as part of the NIH-EPA Chemical Information System (CIS).^ It will also be

issued in a printed format, in which the reduced cell as given by

NBS*AIDS80 can be used for identification.

2.4 Checking the Data for Consistency

The program performs a check of the consistency of the unit cell,

Z, and formula through a density calculation. The molecular weight is

computed based on the input formula, and the cell volume is calculated.

Although these computations may seem trivial, our recent experience as editors reveals that the density calculation is incorrectly reported in

15% of the cases. The examples given in Figure 6 are taken from reports

of quantitative structure determinations, so there should be no questions

about the unit cell contents. The discrepancies often arise owing to

calculation of the density using preliminary cell dimensions or unit

cell contents. Other sources of error include incorrect computation of

molecular weight, cell volume, and Z or an approximate (or missing)

value for Avogadro's number.

Measured Calculated NBS*AIDS80

Formula Density Density Density

Cd2P207 4.90 5.04 4.886

Ba3P2 4.1 4.52 3.371

GeLi2Zn 4.035 4.15 4.353

Mn3(P04)2 4.1 3.92 3.798

Figure 6. Comparison of measured and calculated densities reported

in the literature with densities calculated by NBS*AIDS80.

2.5 Powder Pattern Analysis

The program can be used to generate 2e and d-spacings based on input-cell and space-group information for any crystal system. This

feature was derived from the program "Lattice Parameter Least Squares

Refinement"^ with minor modifications. If reflection data are input

(2e, d, sine/x, or Q) without hk£'s assigned to any or all of the

reflections, NBS*AIDS80 will assign the indices of the nearest reflection provided that |A(2e)| <0.5°. A small segment of the computer

printout from the hkj,/d-generation and indexing segment is shown in

Figure 7. The "C" following each observed d-spacing (D OBS) indicates

that the computer code has assigned the indices. If proper hk£'s were

given as input this column would be blank. However, if a given hki is

6
not allowed in the input space group the column would contain an "N".

Unindexable lines are marked with a "U". The maximum 2e value for

hk£/d-generation is an input parameter or, if left blank, determined by

the range of input lines. The extinction conditions are generated

automatically from the aspect symbol, space group symbol, or cell cen- tering symbol. They may optionally be entered as input data as well.

Following the hkj,, d and 2e table, a pattern summary is printed as shown in Figure 8. Included are the number of lines possible out to 2e

max, the number of lines resolvable based on an estimated resolution

factor, and the number of lines observed. This summary expresses the

coverage of the experimental measurements. Next is a tabulation of the

number of lines input, number indexed and number remaining unindexed.

For the indexed lines the average and maximum magnitude of A(2e) are

given. An average lA(2e)| of <0.02° should be expected for high quality

data. Data of this quality, or better, have been reported for nearly

two decades primarily due to the use of internal 2e standards. Another

assessment of the fit between the observed and calculated 2e's is the

number of lines with |A(2e)l greater than some arbitrary limit. The

JCPDS--International Centre for Diffraction Data currently uses a limit

of 0.05° for "•" quality patterns and a limit of 0.20° for "I" quality

patterns. (However, for a PDF card to be assigned either of these

quality marks, chemical and intensity data criteria must also be satisfied.)

Two other assessments of the fit of the input cell and observed

pattern are calculated by the program. M(20) is an indication of the

correctness of the unit cell. According to de Wolff^° a value of M(20)

>10 coupled with tv/o or fewer unindexed lines indicates that the cell is

substantially correct. A value of M(20) <5 and/or several unindexed

lines strongly suggests that the cell is not appropriate. The SmithSnyder figure of merit F(N)^^ assesses the overall fit of the experimental data and cell as well as the coverage or completeness of the

observations. For every one of the 105 patterns reported in Monograph

25, Section 16 and Section 17,^^ M(20) exceeded 15 and F(30) exceeded

13. However, in evaluating data in the literature, we often find indexed

lines with lA(2e)| >0.5°, and patterns with M(20) <5 and F(30) <5.

Often the reasons for the low quality of fit cannot be explained.

However sometimes large discrepancies are typographical in origin.

NBS*AIDS80 has a d/l/hkji/2e table generation subroutine to help eliminate these errors (Figure 9). The subroutine determines the appropriate

number of significant digits for each d-value based on the average

magnitude of error in 20 for the entire pattern. This keeps the roundoff errors due to formatting of d-values less than the average magnitude

of the error in 2e.
**MESSAGE**POWDER PATTERN GENERATION/ANALYSIS FOR SYSTEM TET

2-THETA(MAX) = 100.23 D(MIN) = 1.003868

2 CONDITIONS FOR NON-EXTINCTION CALLED FOR

CLASS CONDITION(S)

HKL H+K+L = 2N

HHL 2H+L = 4N

N D CALC D OBS INT

1 5.4508

** 5.4508 5.4502C 80

2 3.8489

** 3.8489 3.8472C 65

3 3.1485

4 3.1441 ** 3.1441 3.1434C 100

5 2.7254

** 2.7254 2.7250C 15

6 2.7216

** 2.7216 2.7194C 20

AUTHORS PROGRAM OBS CALC DIFF

HKL HKL

1 1 2-THETA 2-THETA

16.248

2-THETA

1 1 216.250 16.248

23.090

-.002

2

1 1 2 2 1 1

23.100 23.090

28.323

28.364

-.010

2 1 1 2 228.370 28.364

32.836

-.006

2 2

2 2 32.840 32.836

32.883

-.004

2 2 32.910 32.883 -.027

Figure 7. A segment of the hk£/d-generation output of NBS*AIDS80

for NHi,H2As04.

REFLECTION SUMMARY FOR ENTIRE PATTERN:

ESTIMATED RESOLUTION = .075 DEG. 2-THETA

THEORETICAL LINES TOTAL

THEORETICAL RESOLVABLE

UNIQUE OBSERVED LINES

TOTAL LINES INPUT = 39

NUMBER INDEXED = 39

NUMBER UNINDEXED =

75

47

39

*•*********•*• FOR INDEXED LINES•**•**•******

AVERAGE 2-THETA DIFFERENCE = .012

MAXIMUM 2-THETA DIFFERENCE = .030

# WITH DIFF>0.05 (STAR LIMIT) =

# WITH DIFF>0.20 (I LIMIT) = M(20) = 59.49 (DLIMIT = 1.3589, # POSSIBLE = 33)

F(30) = 47.69 (DELTA 2 THETA = .0123, # POSSIBLE = 51)

Figure 8. Pattern summary for NH4H2As04.
5.450 80 1 1 16.25

3.847 65 2 23.10

3.143 100 2 1 1 28.37

2.725 15 2 2 32.84

2.719 20 2 2 32.91

2.438 4 1 3 36.83

2.433 4 3 1 36.92

2.058 45 3 2 1 43.96

1.9291 1 4 47.07

1.9241 3 4 47.20

1.8166 9 3 3 50.18

Figure 9: d/I/hkJi/20 Table Generation for NH4H2As04.
3.0 Input Information

Any number of cells or powder patterns may be processed in a given

run. The input consists of a single CONTROL CARD followed by one or more sets of DATA CARDS. All data sets in a given run will be processed

as a homogeneous group as indicated by the control card. A typical

input deck thus consists of one control card followed by data set 1,

data set 2, etc. Examples of input decks and output listings are given

in Appendix C.

The following card types may be input for each data set:

CARD 1 Cell parameters and crystal system (always required)

CARD 2 Cell parameter standard deviations (optional)

CARD 3 Space group, Z, Dm, Dx (required)

CARD 6 Compound name (optional)

CARD 7 Chemical formula (optional)

CARD 8 Empirical formula (optional, but necessary for full

analysis)

CARD B Comments (optional)

CARD F Pattern information (1) (required if powder data is

input)

CARD G Pattern information (2) (optional if powder data is

input)

CARD H Extinction conditions (optional)

CARD I Powder pattern (required for powder data input)

CARD K Entry termination (always required)

Each set of data cards must begin with CARD 1 and must end with CARD K. In addition, CARD F and CARD I are required for powder data analysis.

The card types are indicated in column 80 and each data set must have

the cards ordered in the above sequence. The following table gives the

card types required for typical analyses:

10
Program analysis functions

Cell analysis (reduced cell. Crystal Data

cell, and metric symmetry determination)

Cell analysis, density, molecular weight

Cell analysis, powder pattern analysis

Cell analysis, density, molecular weight,

powder pattern analysis

Generation of d-spacings, cell analysis

Cards required

, 3, K

, 3, 7 or 8, K

, 3, F, I, K

, 3, 7 or 8, F, I, K

, 3, F, K

CONTROL CARD (processing codes)

Column

35

46-50

55

Format

II

n

15

II

Item

IJ0B - type of run

= data analysis (cell or cell and powder

pattern)

2 = d-spacing generation (no input powder data)

M0N25 - key for printing table of powder data in the

format of NBS Monograph 25

= no output

1 = print

ITHMX - 2e limit for d-spacing generation (if IJ0B = 2) given as an integer, right justified

IC0L - print control

Blank = 120 columns or greater

1 = 80 column printing

11
CARD 1 CELL PARAMETERS

Column Format

F9.5

Item

1-9

o a (original data) in A

10-18 F9.5 b

19-27 F9.5 c

28-35 F8.3 a (original data) in degrees

36-43 F8.3 3

44-51 F8.3 Y

73-78 6A1 Reference code, alphanumeric (ID for data set)

79 Al Crystal system code

A = anorthic (triclinic) H = hexagonal

M = monoclinic R = rhombohedral

= orthorhombic (H or R axes)

T = tetragonal C = cubic

80 Al 1 (card type 1)

Note: Important : To prevent program malfunction, enter only those

cell parameters required to define the cell:

anorthic (triclinic) a, b, c, a, 6, y

monoclinic (a-unique) a, b, c, a monoclinic (b-unique) a, b, c B monoclinic (c-unique) a, b, c y

orthorhombic a, b, c tetragonal a c hexagonal a c rhombohedral (H axes) a c rhombohedral (R axes) a a cubic a

For the rhombohedral crystal system, it is necessary to use the

crystal system code "R" (even when the input cell is based on hexagonal axes) in order to obtain the correct reduced cell and

metric symmetry analysis.

12
CARD 2 CELL PARAMETER STANDARD D

Column Format

F9.5

Item

1-9 a(a)

10-18 F9.5 a(b)

19-27 F9.5 a(c)

28-35 F8.3 a(a)

36-43 F8.3 a(B)

44-51 F8.3 o{y)

80 Al 2 (card type 2)

13
CARD 3 SPACE (

Column Format

1-8 8A1

20-25 F6.0

30-35 F6.3

38-43 F6.3

80 Al

Note: Enter i

Item

Space group, aspect in Laue class, or cell centering

(left justified)

Z (may be entered as an integer, right justified, or

as a decimal number)

Dm (measured density)

Dx (calculated density)

3 (card type 3)

Enter the space group or aspect, left justified. The symbols

may be in upper and lower case, or all upper case.

P2i/c -> P21/C (or P21/c) R3m -^ R-3M (or R-3m)

P432i2 ^ P43212 I*/* ^ I*/*

If only the cell centering is known, enter the appropriate

symbol (P, C, F, etc.) in column 1 under space group. If no centering is indicated, a primitive cell is assumed. The

following symbols may be input:

anorthic (tri clinic) P, A, B, C, F, I monoclinic P, A, B, C, F, I orthorhombic P, A, B, C, F, I tetragonal P, C, F, I hexagonal P rhombohedral (H axes) R rhombohedral (R axes) R, F, I cubic P, F, I

CARD 6 COMPOUND NAME (optional)

Column Format Item

1-67 67A1 Compound name (left justified)

80 Al 6 (card type 6)

14
CARD 7 CHEMICAL FORMULA (optional)

Column Format Item

1-67 67A1 Chemical formula (left justified)

80 Al 7 (card type 7)

Note: If the chemical formula is input using the conventions specified below, the program will calculate the empirical formula

and Card 8 is not required. The formula must be consistent

with Z.

The formula is expressed as a sequence of discrete units. Each

unit ends with a space. The following symbols are allowed:

1. Chemical element and number. Each element is followed by the number of atoms of that element; the

element-number unit is terminated by a space. If

the number of atoms of a given element is 1, then

the 1 may be omitted. The element symbols may be in

upper and lower case, or all upper case. Decimal

multipliers are allowed.

Ba ZlZ (or BA CL2) Na2 S 04 (or NA2 S (34)

BiO.5 CaO.5 Sr (33 (or BIO. 5 CA0.5 SR 03)

2. Parentheses. A left parenthesis is always preceded

and followed by a blank. A right parenthesis is

always preceded by a blank and may be followed by a multiplier.

A5,2 ( S 04 )3 Co ( C )6 ( N H4 )2 S 04

3. Brackets. These are treated the same as parentheses.

Ni3 [ Co ( C N )6 ]2

Mg3 [ B ( H )4 ]2 F ( H ) S 04

4. Exclamation point. This symbol is used to indicate

a center dot. It may be followed by a multiplier.

Rb2 Zn ( Be F4 )2 !6 H2

2 ( N H4 )2 S 04 ! H2

La P 04 !0.5 H2

15
CARD 8 EMPIRICAL FORMULA (optional; not needed if Card 7 is input)

Column Format Item

1-67 67A1 Empirical formula (left justified)

80 Al 8 (card type 8)

Note: The empirical formula must be consistent with Z.

Follow each element symbol with a multiplier, and separate

each element-multiplier unit with a space. A multiplier of

1 may be omitted.

Ca3 H2 Mg (324 Si C22 H36 Cz

The element symbols may be in upper and lower case, or all

upper case; the first element symbol must begin in column 1.

CARD B COMMENTS (optional)

Column Format Item

Any alphanumeric comments

B (card type B)

1-67 67A1

80 Al

CARD F PATTERN

Column Format

7-15 F9.5

30 Al

PATTERN INFORMATION (1) (required only if powder data input or if d-generation is requested)

Item

o Wavelength in A (Blank = 1.5405)

Code for original data

Blank = no data to be read (calculate hkii's and

d's)

1 = e 2 = 2e

D = d-spacings

S = sin^e

Q = (d*)2 = l/d2

80 Al F (card type F)

16
CARD G PATTERN INFORMATION (2) (optional if powder data input)

Column Format Item

17-20 F4.2 2e tolerance used in marking poor fit between observed

and calculated 2e's (Blank = 0.05)

22 Al Rhombohedral -> hexagonal transformation code for

hk^'s

Blank = normal (no transformation)

R = input hkii's, based on rhombohedral

axes, are to be transformed to hk£'s

based on hexagonal axes

80 Al G (card type G)

17
CARD H

Column

21-47

80

EXTINCTION CONDITIONS (optional if powder data input or generated)

Format

2711

Al

Item

Extinction codes (see note)

H (card type H)

Note: Extinction conditions will automatically be read from the

extinction file for all space group and aspect settings given

in Appendix A. Conditions must be input on this card for unusual orientations (e.g., c-unique monoclinic) or if the

extinction file is not assigned.

If rhombohedral hkji's are to be transformed to hexagonal hk£'s

(column 22 on Card G = "R"), always input the special conditions for the rhombohedral cell based on hexagonal axes. The

general rhombohedral condition (-h+k+£=3n) need not be input.

See International Tables for X-ray Crystallography ,

Vol ume 1

,

for the necessary extinctions for eacn space group. For each

extinction, enter the integer value for "B" from the following

table in the appropriate column:

Indices Condition Col umn Indices Condition Col umn

hkii h+k=Bn 21 hOi h=Bn 34

hki h+£=Bn 22 hOji £=Bn 35

hks. k+2.=Bn 23 hO£ h+Ji=Bn 36

hk£ h+k=Bx,

h+£=By,

k+ii=Bz 24

hkO

hkO

h=Bn

k=Bn

37

38

hU h+k+Ji=Bn 25 hkO h+k=Bn 39

hki -h+k+Ji=Bn 26 hhO h=Bn 40

hhs. h=Bn 27 hOO h=Bn 41

hhiL £=Bn 28 OkO k=Bn 42

hhii h+£=Bn 29 00£ £=Bn 43

hh£ 2h+ii=Bn 30 hii h=Bn 44

Okji k=Bn 31 hii £=Bn 45

Ok£ £=Bn 32 hii h+2,=Bn 46

Oka k+£=Bn 33 hm h+2£=Bn 47

18
CARD I POWDER PATTERN (required only if powder data input)

Note: The data can be entered with either one, two, or three

lines per card, in increasing 2e (decreasing d). Up to

200 lines may be entered for a given powder pattern;

repeat Card I as necessary.

Column Format Item

1-7 F7.n e, 2e, d, sin^e, or Q for line 1 (Data must agree

with the data type indicated in column 30 of Card F. Give as many significant figures as possible.)

8-10 13 Relative intensity (any scale)

12-14 13 h

15-17 13 k

18-20 13 I

24-30 F7.n e, 26, d, sin^e, or Q for line 2

31-33 13 Relative intensity

35-37 13 h

38-40 13 k

41-43 13 I

47-53 F7.n e, 2e, d, sin^e, or Q for line 3

54-56 13 Relative intensity

58-60 13 h

61-63 13 k

64-66 13 I

80 Al I (card type I)

CARD K ENTRY TERMINATION

Column Format Item

80 Al K (card type K)

19
4.0 Output Information

Part 1. Warning and Error Messages

A warning message (**WARNING**) is printed if unusual data are entered. A warning is also given if the observed and calculated densities do not agree, if input hk£ indices are not consistent with space

group extinctions, or if the figure of merit indicates that the cell used

in indexing a powder pattern is suspect. If the analysis of the reduced

form indicates that a higher metric symmetry exists than was input, a warning message is printed and the cell parameters for the higher-symmetry cell are given along with the matrix for transformation of the

initial cell to the higher-symmetry cell.

An error message (**ERROR**) usually indicates that an illegal

parameter has been input or that the program has failed at some point.

The input cards should be checked for accuracy.

A message (**MESSAGE**) relays information only, and does not indicate a warning or an error.

Part 2. Cell Analysis

The input data and derived parameters are printed as a series of

card images, with the type of data output summarized. Some or all of the

following card images may be output:

CELL Initial cell, as input.

ESDS Standard deviations in the initial cell, if input; average

standard deviation in axial lengths in parts per 10^; and quality

index code. An "E" following the average standard deviation

indicates that the program has assumed a value of 5 in the least

significant digits.

Average standard

deviation QI

(parts per 10^) code

- 1 A

2 - 5 A6 - 10 B 11-50 B51 - 100 C

101 - 500 C501 - 1000 D 1001 - 5000 D5001 - 9999 F

(typical single-crystal diffractometer)

20
SG-1 Space group as input; space group or aspect number (see Appendix

A); Z, Dm, and Dx as input; initial cell volume.

SG-2 Space group for Crystal Data cell; space group number; Z for Crystal Data cell; Dx calculated by the program; molecular weight;

volume of Crystal Data cell. If no Z is input, then a value of 1 is assumed in the calculation of Dx; Dx is then followed by a "G".

NAME Compound name as input.

CHEM Chemical formula as input.

EMPR Empirical formula as input or as calculated from the chemical

formula.

COMM Comments as input.

MATX Transformation matrices:

Initial cell to reduced cell (T-2)

Determinant and initial cell to Crystal Data cell

Inverse of initial cell to Crystal Data cell, if not identity

matrix.

C-IN Initial cell and cell volume, as interpreted by the program.

C-RD Reduced cell, reduced cell volume, and reduced form number (see

Appendix B). An "X" following the reduced form number indicates

that the metric symmetry exceeds the input crystal symmetry.

C-CD Crystal Data cell and determinative ratios. For the conventions

used to define the Crystal Data cell see the Introduction to

Volume 3 or 4 of Crystal Data Determinative Tables .^ The first

determinative ratio is a/b^ for anorthic, monoclinic, and orthorhombic; £/a^ for tetragonal, hexagonal, and rhombohedral (hexagonal

axes); and a^ for cubic. The second determinative ratio is Qjb_

for anorthic, monoclinic, and orthorhombic.

E-CD Standard deviations of the Crystal Data cell parameters, if initial

standard deviations were input.

DOTM Dot product matrix (^'^j j^*^, ^'C, b«c, ^'jc, ^'b) of the reduced

cell. The values given for the aot'^products should be compared

with the data in Appendix B to determine metric symmetry.

PDFl Pattern information (1) as input (card F).

PDF2 Pattern information (2) as input (card 6).

EXTC Extinction conditions as input (card H).

21
HIST Date processed; number of warnings generated; number of errors generated; year-version of NBS*AIDS80.

Part 3. Powder Pattern Analysis

The data are sorted on increasing 2e, and intensities are rescaled

to I (max) = 100. Observed and calculated d-spacings, 2e's and hki's are

given, as well as a(20) for indexed lines. Any line that is unindexed

on input will be assigned the indices of the closest calculated line

provided that the observed and calculated 2e's agree within 0.5°. The

following editorial marks may be assigned to a given line by the program:

C = hkji assigned by the program

U = unindexed

N = not permitted by the space group

The reflections exceeding the |A(2e)| tolerance are listed. A

reflection summary is given for the entire pattern, including the following: the theoretical number of lines possible, the number of lines

resolvable, the number of unique observed lines, the total number of

lines input, and the number of lines indexed and unindexed. For the

indexed lines, the average and maximum |A(2e)| are given, along with the

number of lines exceeding the tolerance of the PDF "•" limit (lA(2e)| < 0.05°) and "I" limit (|A(2e)| ^0.20°).

The figures of merit M(N) and F(N) proposed by de Wolff ^° and by

Smith and Snyder, ^^ respectively, are calculated for indexed powder patterns. Unindexed lines are not included in the calculation of either

figure of merit. M(N) is defined by the equation

*^

M(N) = Q(N)

2
 N

where N is 20 or the number of indexed lines input if less than 20, Q(N)

is the Q-value ( = l/d^ ) of the N*"" line, and <IaQ|> is the average

magnitude of the difference between Q(obs) and Q(calc). According to de

Wolff, a value of M(20) > 10 will indicate the essential correctness of

the indexing provided that there are not more than two unindexed lines

with 2e < 2e(N). The Smith-Snyder figure of merit is defined by

F(N) = ^ ^ 
 N(poss)

where N is 30 or the number of indexed lines input if less than 30, and

N(poss) is the number of theoretical lines possible (excluding spacegroup extinct lines) up to and including the N^" observed and indexed

line. Co-positional lines (e.g. cubic indices 221 and 300) are counted

as one possible line. The recommended format for reporting this figure

of merit is F(N) [ 
, N(poss) ]. The factor N/N(poss) is a 22
measure of the completeness of the measurements while <lA2e|> indicates

the consistency of the unit cell with the experimental 2e values. For

both M(N) and F{N) the average error values are based on the assigned

hkji's. If a line is input with multiple indices, then all possible

differences (obs - calc) for that line are used in calculating the

average error value.

Part 4. Table of Powder Data for Publication

A table of d-spacings, intensities, hk£ indices, and 2e's is prepared if M0N25 = 1 on the control card. The number of significant

figures given for the spacings is determined by the overall |A(2e)|

for the pattern.

5.0 Program Details

Examples of input card decks and output listings for typical data

analysis runs are given in Appendix C. NBS*AIDS80 is written in FORTRAN

and is currently running on a UNIVAC 1100 series computer. As overlayed,

it occupies about 21 K words of memory and requires from one second for

cell analysis to five seconds for cell and powder data analysis. To improve portability of the program, we have grouped all ENCODE and DECODE

statements together in a single subroutine. Laboratories with compilers

that do not handle ENCODE and DECODE statements may have to write assembly language subroutines. The program makes use of the full 128 character ASCII character set. However, input may be restricted to only

upper case for formulas and space group symbols. The program will be

provided free of charge provided that a new, unlabeled 9 track magnetic

tape, certified to 1600 bpi , accompanies the request. The supplied tape

will be written unblocked in ASCII at 1600 bpi.

6.0 Acknowledgements

Work on the program was sponsored by the Office of Standard Reference Data (NBS), by the Ceramics, Glass, and Solid State Science Division

(NBS), and by the JCPDS--International Centre for Diffraction Data. The

following scientists made invaluable contributions to the design and

development of NBS*AIDS80: R. J. Boreni of the Crystal Data Center,

L. Calvert of the Metals Data Center (Ottawa, Canada), E. Evans of the

JCPDS Associateship at NBS, M. Holmany of the JCPDS, F. McClune of the

JCPDS, H. McMurdie JCPDS--Ceramics Editor, J. R. Rodgers of the Cambridge

Crystallographic Data Centre, A. Santoro of the Reactor Division (NBS),

R. Snyder of Alfred University, and D. Watson of the Cambridge Crystallographic Data Centre.

23
REFERENCES

1. "Crystal Data Determinative Tables." 3rd ed. U.S. Department of

Commerce, National Bureau of Standards and the JCPDS--International

Centre for Diffraction Data, Swarthmore, PA, USA (1972, 1973,

1978); Vols. 1-4.

2. "Powder Diffraction File", JCPDS— International Centre for Diffraction Data, 1601 Park Lane, Swarthmore, PA 19081.

3. W. H. De Camp, The Existence of Metrically Similar Unit Cells Based

on the Same Lattice: A Precautionary Note, Acta Cryst . , B32 : 2257

(1976).

4. A. Santoro and A. D. Mighell, Determination of Reduced Cells, Acta

Cryst ., A26 : 124 (1970).

5. "International Tables for X-ray Crystallography." Kynoch Press,

Birmingham, Vol. 1, 3rd ed: 530 (1969).

6. A. D. Mighell and J. R. Rodgers, Lattice Symmetry Determination,

Acta Cryst ., A36 : 321 (1980).

7. V. Himes, Data in Figures 2 and 3 were taken from a dissertation to

be submitted to the Graduate School, the Catholic University of

America, Washington, D. C. 20064, in partial fulfillment of the

requirements for the Ph.D. Degree in Chemistry.

8. CIS Project, Information Sciences Corp., 2135 Wisconsin Ave. N.W.,

Washington, D. C. 20007.

9. H. T. Evans, Jr., D. E. Appleman, and D. S. Handwerker, Report #PB

216188, U.S. Department of Commerce, National Technical Information

Center, 5285 Port Royal Rd., Springfield, VA 22151.

10. P. M. de Wolff, A Simplified Criterion for the Reliability of a Powder Pattern Indexing, J. Appl . Cryst ., 1_: 108 (1968).

11. G. S. Smith and R. L. Snyder, F(N): A Criterion for Rating Powder

Diffraction Patterns and Evaluating the Reliability of PowderPattern Indexing, J. Appl . Cryst. , 12^: 60 (1979).

12. M. C. Morris, H. F. McMurdie, E. H. Evans, B. Paretzkin, J. H.

de Groot, C. R. Hubbard, and S. J. Carmel , Standard X-ray Diffraction

Powder Patterns, NBS Monograph Section 16 and Section 17, National

Bureau of Standards, Washington, D. C. 20234.

13. J. D. H. Donnay and 0. Kennard, Diffraction Symbols, Acta Cryst .

,

Vj 1337 (1964).

24
14. B. Gruber, The Relationship between Reduced Cells in a General

Bravais Lattice, Acta Cryst ., A29 : 433 (1973).

15. A. D. Mighell, A. Santoro, and J. D. H. Donnay, Addenda and Errata

to International Tables for X-ray Crystallography, Acta Cryst . , B27

1837 (1971); Acta Cryst ., B31 : 2942 (1975).

16. E. Parthe and J. Hornstra, Corrections to the Tables in Chapter

5.1, Reduced Cells, given in the 1969 edition of Volume 1 of International Tables, Acta Cryst ., A29 : 309 (1973).

17. A. Santoro and A. D. Mighell, Coincidence-Site Lattices, Acta

Cryst ., A29: 169 (1973). See Appendix for the algorithm used to

generate the Q matrices.

25
APPENDIX A

SPACE GROUPS AND ASPECTS

The following tables give the space groups and diffraction aspects

allowed by NBS*AIDS80 for the assignment of space group and aspect numbers and for the generation of systematic absences. Although any valid

space group or diffraction aspect may be entered on Card 3, only those

listed in these tables (in addition to cell centering symbols) will

result in the automatic generation of systematic absences. For the

monoclinic system the b^-axis is assumed to be unique.

Part I (Space Groups)

The space group numbers assigned by the program correspond to those

given in the International Tables for X-ray Crystallography .^ The as- signed number is followed by an orientation code when more than one orientation is permitted for a given space group. In the monoclinic

system these codes are A, B, or C, and in the orthorhombic system they

are A, B, C, D, E, or F.

Part II (Aspects)

The diffraction aspect, if known, should be entered on Card 3 when

the space group is not uniquely defined. The diffraction aspects for

each Laue class have been given by Donnay and Kennard^^ ^^^1 are reprinted

in Supplement II of the 3rd Edition of Crystal Data Determinative

Tables ,^ Volumes 1 and 2. The four aspects that begin with the Laue

group symmetry should be rewritten so that the lattice type appears

first, followed by the Laue group:

TP* ^ P*,-l 3P* -> P*,-3

3mP*cl -> P*cl,-3m 3mP*lc -> P*lc,-3m

An aspect number assigned by the program is preceded by an asterisk(*),

and this number corresponds to the number of the highest symmetry space

group that is consistent with the diffraction aspect. The aspects for

the enantiomorphic space groups (e.g. P4i2i2 and P432i2) are assigned the

number corresponding to the lower-numbered space group of the pair. The

orientation codes for the monoclinic and orthorhombic systems are the

same as those assigned to the space groups.

26
PART I

SPACE GROUPS

Space

Group

No.

1

Aspect

No.

2

2

PI

PT

Anorthic (TricHm'c) System

Space Group Symbols

Monoclinic System

Space

Group

No.

Aspect

No.

10

A*

Space

B GroupC

3 P2

4 11 P2i

5 12 C2 A2 12

6 10 Pm

7 13 Pc Pa Pn

8 12 Cm Am Im

9 15 Cc Aa la

10 10 P2/m

11 11 P2i/m

12 12 C2/m A2/m I2/m

13 13 P2/C P2/a P2/n

14 P2i/c P2i/a P2i/n

15 15 C2/C A2/a I2/a

Orientation code

27
Orthorhombic System

Space

Group

No.

Aspect

No.

47

A

Space Group Symbols BCD E F

16 P222

17 P222i P2i22 P22i2

18 P2i2i2 P22i2i P2i22i

19 P2i2i2i

20 C222i A2i22 B22i2

21 65 C222 B222 A222

22 69 F222

23 71 1222

24 71 I2i2i2i

25 47 Pmm2 Pm2m P2mm

26 51 Pmc2i P2ima P2iam Pb2im Pcm2i Pm2ib

27 49 Pcc2 Pb2b P2aa

28 51 Pma2 P2cm P2mb Pbm2 Pc2m Pm2a

29 57 Pca2i P2ica Pb2ia P2iab Pc2ib Pbc2i

30 53 Pnc2 Pcn2 Pn2b Pb2n P2na P2an

31 59 Pmn2i Pnm2i P2inm Pn2im Pm2in P2imn

32 55 Pba2 P2cb Pc2a

33 62 Pna2i Pbn2i P2icn P2inb Pn2ia Pc2in

34 58 Pnn2 Pn2n P2nn

35 65 Cmm2 Bm2m A2mm

36 63 Cmc2i A2iam A2ima Bm2ib Bb2im Ccm2i

37 66 Ccc2 A2aa Bb2b

38 65 Amm2 Am2m B2mm

28

C2mm Bmm2 Cm2m
Orthorhombic System (cont.)

Space

Group

No.

Aspect

No.

67

A

Space

B

Group

C

Symbol

s

D E F

39 Abm2 Bma2 C2ma Ab2m Cm2a B2am

40 63 Ama2 C2cm Bbm2 B2mb Cc2m Am2a

41 64 Aba2 Ab2a C2ca B2ab Cc2a Bba2

42 69 Fmm2 Fm2m F2mm

43 Fdd2 Fd2d F2dd

44 71 Imm2 Im2m 1 2mm

45 72 Iba2 Ib2a I2aa

46 74 Ima2 Im2a Ibm2 I2am Ib2m I2ma

47 47 Pmmm

48 Pnnn

49 49 Pccm Pmaa Pbmb

50 Pban Pncb Pcna

51 51 Pmma Pmcm Pcmm Pmam Pbmm Pmmb

52 Pnna Pnnb Pnan Pncn Pbnn Pcnn

53 53 Pmna Pncm Pbmn Pman Pcnm Pnmb

54 Pcca Pbaa Pbcb Pccb Pbab Pcaa

55 55 Pbam Pcma Pmcb

56 Pccn Pbnb Pnaa

57 57 Pbcm Pcmb Pbma Pcam Pmab Pmca

58 58 Pnnm Pnmn Pmnn

59 59 Pmmn Pnmm Pmnm

60 Pbcn Pnab Pnca Pbna Pcan Pcnb

61 Pbca Pcab

29
Orthorhombic System (cont.)

Space

Group

No.

Aspect

No.

62

A

Space

B

Group Symbols

C D E F

62 Pnma Pmnb Pbnm Pmcn Pcmn Pnam

63 63 Cmcm Bbmm Amam Amma Ccmm Bmmb

64 64 Cmca Abam Ccma Abma Bmab Bbam

65 65 Cmmm Bmmm Ammm

66 66 Cccm Amaa Bbmb

67 67 Cmma Abmm Bmam

68 Ccca Bbab Abaa

69 69 Fmmm

70 Fddd

71 71 Immm

72 72 I bam Imaa Ibma

73 Ibca

74 74 Imma Ibmm Imam

Tetragonal System

Space

Group

No.

Aspect

No.

Space

Group

Symbols

Space

Group

No.

Aspect

No.

Space

Group

Symbol

s

75 83 P4 76 76 P4i

77 84 P4, 78 76 P43

79 87 14 80 I4i

81 83 P4 82 87 14

83 83 P4/m 84 84 P42/m

85 P4/n 86 P42/n

87 87 I4/m 88 I4i/a

30
Tetragonal System (cont.)

Space

Group

No.

Aspect

No.

Space

Group

Symbols

89 123 P422

91 91 P4i22

93 P4222

95 91 P4322

97 139 1422

99 123 P4mm

101 132 P42cm

103 124 P4cc

105 131 P42mc

107 139 1 4mm

109 122 I4imd

111 123 P42m

113 113 P42im

115 123 P4m2

117 127 P4b2

119 139 I4m2

121 139 I42m

123 123 P4/mmm

125 P4/nbm

127 127 P4/mbm

129 P4/nmm

131 131 P42/mmc

133 P42/nbc

Space

Group

No.

Aspect

No.

Space

Group

Symbols

90 113 P42i2

92 92 P4i2i2

94 P422i2

96 92 P432i2

98 I4i22

100 127 P4bm

102 136 P42nm

104 128 P4nc

106 135 P42bc

108 140 1 4cm

110 I4icd

112 131 P42c

114 P42iC

116 132 P4c2

118 136 P4n2

120 140 I4c2

122 122 I42d

124 124 P4/mcc

126 P4/nnc

128 128 P4/mnc

130 P4/ncc

132 132 P42/mcm

134 P42/nnm

31
Tetragonal System (cont.)

Space

Group

No.

Aspect

No.

Space

Group

Symbols

135 135 P42/mbc

137 P42/nmc

139 139 I4/mmm

141 I4i/amd

Space

Group

No.

Aspect

No.

Space

Group

Symbol

s

136 136 P42/mnm

138 P42/ncm

140 140 I4/mcm

142 I4i/acd

Hexagonal and Rhombohedral System

Space

Group

No.

Aspect

No.

Space

Group

Symbols

143 147 P3

145 144 P32

147 147 P3

149 162 P312

151 151 P3il2

153 151 P3212

155 166 R32

157 162 P31m

159 163 P31c

161 167 R3c

163 163 P31c

165 165 P3cl

167 167 R3c

169 169 P6i

171 171 P62

Space

Group

No.

Aspect

No.

Space

Group

Symbols

144 144 P3i

146 148 R3

148 148 R3

150 164 P321

152 152 P3i21

154 152 P3221

156 164 P3ml

158 165 P3cl

160 166 R3m

162 162 P31m

164 164 P3ml

166 166 R3m

168 175 P6

170 169 P65

172 171 P61,

32
Hexagonal and Rhombohedral System (cont.)

Space

Group

No.

Aspect

No.

Space

Group

Symbols

Space

Group

No.

Aspect

No.

Space

Group

Symbol

s

173 176 P63 174 175 P6

175 175 P6/m 176 176 P63/m

177 191 P622 178 178 P6i22

179 178 P6522 180 180 P6222

181 180 P6^22 182 P6322

183 191 P6mm 184 192 P6cc

185 193 P63cm 186 194 P6 3mc

187 191 P6m2 188 193 P6c2

189 191 P62m 190 194 P62c

191 191 P6/mmm 192 192 P6/mcc

193 193 P63/mcm 194 194 P63/mmc

Cubic System

Space

Group

No.

Aspect

No.

Space

Group

Symbols

Space

Group

No.

Aspect

No.

Space

Group

Symbol

s

195 200 P23 196 202 F23

197 204 123 198 P2i3

199 204 I2i3 200 200 Pm3

201 Pn3 202 202 Fm3

203 Fd3 204 204 Im3

205 Pa3 206 Ia3

207 221 P432 208 P4232

209 225 F432 210 F4i32

211 229 1432 212 212 P4332

33
Cubic System (cont.)

Space

Group

No.

Aspect

No.

Space

Group

Symbols

213 212 P4i32

215 221 P43m

217 229 I43m

219 226 F43c

221 221 Pm3m

223 223 Pm3n

225 225 Fm3m

111 Fd3m

11^ 229 Im3m

Space

Group

No.

Aspect

No.

Space

Group

Symbol

s

214 I4i32

216 225 F43m

218 223 P43n

220 I43d

222 Pn3n

224 Pn3m

226 226 Fm3c

228 Fd3c

230 Ia3d

34
PART II

ASPECTS

Anorthic (Triclinic) System

Aspect Number Space Groups (Space Group Numbers)

P*,l

Aspect

P*/*

P2i/*

C*/*

A*/*

I*/*

P*/c

P*/a

P*/n

C*/c

A*/

a

I*/a

Aspect

Pec*

P*aa

Pb*b

P**a

P*c*

2 PI (2), PI (1)

Monoclinic System

Number Space Groups (Space Group Numbers)

10

n

12A

12B

12C

13A

13B

13C

15A

15B

15C

Number

47

49A

49B

49C

51A

51B

P2/m (10), P2 (3), Pm (6)

P2i (4)

(5A), Cm (8A)

(5B), Am (8B)

(5C), Im (8C)

(7A)

(7B)

(70

(9A)

(9B)

(9C)

OrthorhombJc System

Space Groups (Space Group Numbers)

1 £-X/ Ml \ 1 1 / ,

C2/m (12A),

• '-I

C2 (

A2/m (12B), A2 (

I2/m (12C), 12 (

P2/C (13A), Pc (

P2/a (13B), Pa (

P2/n (13C), Pn (

C2/C (15A), Cc (

A2/a (15B), Aa (

I2/a (15C), la (

Pmmm (47), P222 (16), Pmm2 (25A), Pm2m (25B), P2mm (25C)

Pccm (49A), Pcc2 (27A)

Pmaa (49B), P2aa (27C)

Pbmb (49C), Pb2b (27B)

Pmma (51A), P2ima (26B), Pm2a (28F)

Pmcm (51 B), Pmc2i (26A), P2cm (28B)

35
Orthorhombic System (cont.)

Aspect Number Space Groups (Space Group Numbers)

PC** 51C

P*a* 51

D

Pb** 51

E

P**b 51

F

P*na 53A

Pnc* 53B

Pb*n 53C

P*an 53D

Pen* 53E

Pn*b 53F

Pba* 55A

Pc*a 55B

P*cb 55C

Pbc* 57A

Pc*b 57B

Pb*a 57C

Pea* 57D

P*ab 57E

P*ca 57F

Pnn* 58A

Pn*n 58B

P*nn 58C

P**n 59A

Pn** 59B

Pcmm ( 51C),

Pmam ( 51D),

Pbmm ( 51E),

Pmmb ( 51 F),

Pmna ( 53A),

Pncm ( 53B),

Pbmn ( 53C),

Pman 1 53D),

Pcnm 1 53E),

Pnmb ( 53F),

Pbam ( 55A),

Pcma (.55B),

Pmcb ( 55C),

Pbcm :57A),

Pcmb [57B),

Pbma [570,

Pcam [57D),

Pmab (57E),

Pmca (57F),

Pnnm (58A),

Pnmn (58B),

Pmnn (58C),

Pmmn (59A),

Pnmm (59B),

PcmZi (26E), Pc2m (28E)

P2iam (26C), Pma2 (28A)

Pb2im (260), Pbm2 (28D)

Pm2ib (26F), P2mb (280)

P2na (30E)

Pnc2 (30A)

Pb2n (30D)

P2an (30F)

Pcn2 (30B)

Pn2b (30C)

Pba2 (32A)

Pc2a (32C)

P2cb (32B)

Pbc2i (29F)

Pc2ib (29E)

Pb2ia (29C)

Pca2i (29A)

P2iab (29D)

P2ica (29B)

Pnn2 (34A)

Pn2n (34B)

P2nn (34C)

Pm2in (31E), P2imn {31F)

Pnm2i (31B), Pn2im (31D)

36
Orthorhombic System (cont.)

Aspect Number Space Groups (Space Group Numbers)

P*n* 59C

Pn*a 62A

P*nb 62B

Pbn* 62C

P*cn 62D

Pc*n 62E

Pna* 62F

C*c* 63A

Bb** 63B

A*a* 63C

A**a 63D

Cc** 63E

B**b 63F

C*ca 64A

Aba* 64B

Cc*a 64C

Ab*a 64D

B*ab 64E

Bba* 64F

c*** 65A

B*** 65B

^*** 65C

Ccc* 66A

A*aa 66B

Pmnm ( 59C),

Pnma ( 62A),

Pmnb ( 62B),

Pbnm ( 62C),

Pmcn 1 62D),

Pcmn ( 62E),

Pnam i 62F),

Cmcm ( 63A),

Bbmm i 63B),

Amam i 63C),

Amma ( 63D),

Ccmm 63E),

Bmmb .63F),

Cmca :64A),

Abam .64B),

Ccma [640,

Abma [640),

Bmab [64E),

Bbam [64F),

Cmmm (65A),

Bmmm (65B),

Ammm (65C),

Cccm (66A),

Amaa (66B),

Pmn2i (3ia; , P2inm (31C)

Pn2ia (33E;

P2inb (33D:

Pbn2i (33B:

P2icn (33C;

Pc2in (33F^

Pna2i (33A^

Cmc2i (36A ), C2cm (40B)

Bb2im (36E^), Bbm2 (40C)

A2iam (36B ), Ama2 (40A)

A2ima (36C ), Am2a (40F)

Ccm2i (36F ), Cc2m (40E)

Bm2ib (36D ), B2mb (40D)

C2ca ( 41C)

Aba2 ( 41A)

Cc2a ( 41E)

Ab2a ( 41B)

B2ab ( 41D)

Bba2 <.41F)

C222 ;21A) , Cmm2 (35A),

B222 [21B) , Bm2m (35B),

A222 [210 , A2mm (35C),

Ccc2 (37A)

A2aa (37B)

37
Aspect

Bb*b

C**a

Ab**

B*a*

I ***

Iba*

I*aa

Ib*a

I**a

lb**

I*a*

Aspect

Orthorhombic System (cont.)

Number Space Groups (Space Group Numbers)

66C

67A

67B

67C

69

71

Bbmb

Cmma

Abmm

Bmam

Fmmm

Immm

ImZm

66C), Bb2b (37C)

67A), C2ma (39C), Cm2a (39E)

67B), Abm2 (39A), Ab2m (39D)

67C), Bma2 (39B), B2am (39F)

69), F222 (22), Fmm2 (42A), Fm2m (42B), F2mm (42C)

71), 1222 (23), I2i2i2i (24), Imm2 (44A),

44B), I2mm (44C)

P4l,3 76

P4/* 83

P42/* 84

14/* 87

P4i,322 91

P4i,32i2 92

P42i* 113

I4*d 122

P4/*** 123

72A Ibam (72A), Iba2 (45A)

72B Imaa (72B), I2aa (45C)

72C Ibma (72C), Ib2a (45B)

74A Imma (74A), Im2a (46B), I2ma (46F)

74B Ibmm (74B), Ibm2 (46C), Ib2m (46E)

74C Imam (74C), Ima2 (46A), I2am (46D)

Tetragonal System

Number Space Groups (Space Group Numbers)

P4i (76), P43 (78)

P4/m (83), P4 (75), P4 (81)

P42/m (84), P42 (77)

I4/m (87), 14 (79), 14 (82)

P4i22 (91), P4322 (95)

P4i2i2 (92), P432i2 (96)

P42im (113), P42i2 (90)

I42d (122), I4imd (109)

P4/mmm (123), P422 (89), P4mm (99), P42m (111),

P4m2 (115)

38
P4/*cc 124

P4/*b* 127

P4/*nc 128

P4/**c 131

P4/*c* 132

P4/*bc 135

P4/*n* 136

j4 /*** 139

Tetragonal System (cont.)

Aspect Number Space Groups (Space Group Numbers)

P4/mcc (124), P4cc (103)

P4/mbm (127), P4bm (100), P4b2 (117)

P4/mnc (128), P4nc (104)

P42/mmc (131), P42mc (105), P42c (112)

P42/mcm (132), P42cm (101), P4c2 (116)

P42/mbc (135), P42bc (106)

P42/mnm (136), P42nm (102), P4n2 (118)

I4/mmm (139), 1422 (97), I4mm (107), I4m2 (119),

I42m (121)

I4/*c* 140 I4/mcm (140), I4cm (108), I4c2 (120)

Hexagonal and Rhombohedral System

Aspect Number Space Groups (Space Group Numbers)

P3i,2 144 P3i (144), P32 (145)

P*,3 147 P3 (147), P3 (143)

R* 148 R3 (148), R3 (146)

P3i^2l2 151 P3il2 (151), P3212 (153)

P3i^221 152 P3i21 (152), P3221 (154)

P31* 162 P31m (162), P312 (149), P31m (157)

P*lc,3m 163 P31c (163), P31c (159)

P3*l 164 P3ml (164), P321 (150), P3ml (156)

P*cl,3m 165 P3cl (165), P3cl (158)

R** 166 R3m (166), R32 (155), R3m (160)

R*c 167 R3c (167), R3c (161)

P6i^5 169 P6i (169), P65 (170)

P62,i, 171 P62 (171), P64 (172)

39
Hexagonal and Rhombohedral System (cont.)

Aspect Number Space Groups (Space Group Numbers)

P6/* 175 P6/m (175), P6 (168), P6 (174)

P63/* 176 P63/m (176), P63 (173)

P6i,522 178 P6i22 (178), P6522 (179)

P62,422 180 P6222 (180), P6i,22 (181)

P6/*** 191 P6/mmm (191), P622 (177), P6mm (183), P6m2 (187),

P62m (189)

P6/*cc 192 P6/mcc (192), P6cc (184)

P6/*c* 193 P63/mcm (193), P63cm (185), P6c2 (188)

P6/**c 194 P63/mmc (194), P63mc (186), P62c (190)

Cubic System

Aspect Number Space Groups (Space Group Numbers)

Pm3 (200), P23 (195)

Fm3 (202), F23 (196)

Im3 (204), 123 (197), 12^3 (199)

P4332 (212), P4i32 (213)

Pm3m (221), P432 (207), P43m (215)

223 Pm3n (223), P43n (218)

Fm3m (225), F432 (209), F43m (216)

Fm3c (226), F43c (219)

Im3m (229), 1432 (211), I43m (217)

P*3 200

F*3 202

1*3 204

P4i,332 212

P*3* 221

P*3n 223

F*3* 225

F*3c 226

1*3* 229

40
APPENDIX B

LATTICE SYMMETRY DETERMINATION

There exists a close correlation between the metric symmetry (i.e.

the symmetry determined from the cell parameters alone) and the crystal

symmetry of the lattice. It has been shown that for 97 percent of the

entries in the NBS Crystal Data File, the metric symmetry is the same as the crystal symmetry reported by the authors. ^ Thus it was concluded

that more emphasis should be placed on cell determination and refinement

algorithms used in routine diffractometry because of the basic importance

of the metric lattice in crystallography. A convenient procedure to

determine crystal symmetry is as follows:

Determine any cell defining

the lattice

Transform to reduced cell

Determine the reduced form

type (1-44)

Transform reduced cell to

conventional cell

Verify by checking equivalence of intensities

The algorithm used by NBS*AIDS80 to calculate the reduced cell

guarantees that both the main and special conditions for reduction are

satisfied.'* If only the main conditions for reduction are satisfied, one obtains a cell based on the shortest three non-coplanar translations.

However, the special conditions must also be satisfied since in certain

lattices two or more cells may be based on the shortest three translations.'* In fact, Gruberi'* has shown that as many as five different cells

of this type may exist in the same lattice. Thus the special conditions

are necessary to define one and only one of these cells as the reduced

cell. Table 1 gives the main and special conditions used by NBS*AIDS80

to determine the reduced cell.

Based on the reduced cell, NBS*AIDS80 calculates' the vector products

41
defining the reduced form:

/ ci'd b'b C'C \

\ b'C a»c a'b / From the reduced form, the reduced form type (1-44) is assigned. Table

2 gives the conditions for the 44 reduced forms, as well as the matrices

relating each reduced form to a corresponding conventional cell of the

Bravais lattice. This table^ is derived from Table 5.1.3.1 of the

International Tables for X-ray Crystallography ,^ including the published

addenda and errata. is »i6 Thus to find the metric symmetry of an unknown,

one can introduce any cell of the lattice on Card 1; if the cell is not

primitive, the centering must be specified on Card 3. The program

calculates the reduced cell, reduced form, and reduced form type. If

the metric symmetry exceeds the input crystal symmetry, both the higher

symmetry cell and the transformation matrix (input cell to Crystal Data

conventional celP) will be output. In practice we have found it con- venient to determine a primitive triclinic cell on the diffractometer

and use NBS*AIDS80 to suggest the lattice symmetry.

It is essential that the input cell be determined as accurately as possible. Usually the program will find the correct metric symmetry

with the input of a refined cell determined from a diffractometer.

However, caution should be used in the interpretation of metric symmetry.

In its present form NBS*AIDS80 has the practical and theoretical limitation of determining only one metric cell when, in certain cases, more

than one answer may be possible owing to experimental errors in the

input cell. Thus the reduced cell and reduced form should be checked

using Tables 1 and 2 to determine if an alternative symmetry might be

considered. For example, the program may indicate a symmetry lower than

the lattice symmetry because of inexact determination of cell angles;

idealization of the experimental angles (e.g. 90") may be necessary to

obtain the correct analysis. Also, if the reduced form exhibits more specialization than is required to define the lattice type, it may

signify that a subcell of the lattice has been input or that the material is twinned.

We have noted from our experience with obtaining cells from an automated diffractometer as well as from analysis of the data in the NBS

Crystal Data File that the determination of a subcell of the lattice is

a common source of error. A number of structure determinations have

been carried out using only half of the available unique data; this error often results in apparent disorder in the refined structure. If

it is suspected that a subcell of the lattice has been determined, one

can obtain the cells of 2, 3, and 4 times the volume by applying the

upper triangular g matrices given in Table 3 to any primitive cell of

the lattice. 17 The seven cells of twice the original volume, thirteen

cells of three times the volume, or 35 cells of four times the volume

can be used to calculate the positions of the nodes in reciprocal space

to be checked for diffraction intensity. Alternatively, one can apply

the transpose of the inverse of the Q matrices to the reciprocal cell to

determine the nodes.

'^

42
Table 1. Conditions for a Reduced Cell

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

Main conditions: a«a < b«b < c-c ; b«c < \ b«b ; a«c < h a«a ;

a«b < ^_ a«a

then

•Xi '\, < v^

then v% < ^'k

then a«b < 2^-^

then l'\ < 2 b-c

then l"i < 2H

Special conditions: (a) if a«a = b-b

(b) if b-b = c-c

(c) if b-c = h b-b

(d) if a«c = h a^a

(e) if a«b = h a«a

Negative reduced form, type II cell, all angles >^ 90°

Main conditions: (a) a-a < b-b < c«c ; ib«c| < h b«b;

|a«cl < h a^a ; la^bl < h a^a

(b)(|b'c| + la-cl + |a«b|) < ii ( a-a + b-b )

Special conditions: (a) if a-a = b-b then |b-c| < |a-c|

(b) if b-b = c-c then |a-c| < |a-b|

(c) if |b-c| = h b-b then a-b =

(d) if la-cl = h a-a then a-b =

^ ' ''\j'V-' '\j 'Xj \, 'Xj

(e) if la-b] = h ^'^ then a-c =

(f) if (Ib.^l + l^.^l + la.bl) -H (^.^+fe.b )

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

• For further details see the discussion in Reference 5.

43
Table 2. Metric Classification of Reduced Forms''

Reduced Form

No.

Reduced Form Matrix Reduced Form

Type

Bravais

Lattice

Cell Transformation Reduced -> Conventional

First Row Second Row

a-a b*b c*c 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 110/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 110/1 01 /ill

5 a-a a-a a-a

a-a3

a-a3

a-a3

- Cubic 1 101/110/011

6 a-a a-a a-a -a-a + la-bl 2

-a-a + |a-b| 2,

-Ia-b| _ Tetragonal 1 011/101/110

7 a-a a-a a-a -Ib-cl -a-a + |b-c| 2

-a-a + |b-cl 2

- Tetragonal 1 101/110/011

8 a-a a-a a-a -Ib-cl -la-cl -(|a-a|-|b-c|-|a-c|) - Orthorhombic 1 iio/ioi/oii

a = b

9 a-a a-a c*c

a-a2a-a2a-a2

+ Rhombohedral hR 100/110/113

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

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

12 a-a a-a c-c

a-a2

- Hexagonal P 100/010/001

13 a-a a-a c-c -la-b| - Orthorhombic C 110/110/001

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

15 a-a a-a c-c

a-a '~2'a-a '~2 - Tetragonal 1 100/010/112

16 a-a a-a c-c -Ib-cl -Ib-cl -(a-a-2|b-c|) - Orthorhombic F 110/110/112

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

b=c

18 a-a b-b b-b

a-a4a-a2a-a2 + Tetragonal 1 011/iii/ioo

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

a-a2

a-a2

+ Orthorhombic 1 100/011/111

20 a-a b-b b-b b-c a-c a-c + Monoclinic 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 - Orthorhombic C 011/011/100

24 a-a b-b b-b

2

a-a3a-a "T - Rhombohedral hR 121/011/100

25 a-a b-b b-b -Ib-cl -la-cl -la-cl - Monoclinic C' 011/011/100

44
Table 2. (continued)

Reduced Form

No.

Reduced Form Matrix Reduced Form

Type

Bravais

Lattice

Cell Transformation Reduced -* Conventional

First Rowttt Second Row

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

26 3-a b-b c-c

a-a4

a-a2a-a2

+ Orthorhombic F 100/120/102

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

a-a 2a-a2

+ IVIonoclinic 1*** oTi/ioo/111

28 a-a b-b c-c ab

2

a-a2 a-b + Monoclinic C 100/102/010

29 a-a b-b c-c

a-c2

a-c

a-a2

+ Monoclinic C 100/120/001

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

2

a-b

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

31 a-a b-b c-c b-c a-c a-b + Tnclinic 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-b - Monoclinic P i 00/001 /oio

35 a-a b-b c-c -Ib-cl - Monoclinic P 010/100/001

36 a-a b-b c-c aa

2

- Orthorhombic C 100/102/010

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

a-a2

- Monoclinic C* 102/100/010

38 a-a b-b c-c

a-a "T - Orthorhombic C 100/120/001

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

a-a

- 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 aa "T - Orthorhombic 1 100/010/112

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

2

a-a - la-bl 2 -la-bl - Monoclinic 1 100/112/010

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

f a-a -' 4|a-cl \

f b-b^ 4lb-cl > Premult

f c-c''4lb-cl )

f 3a-a '' C-C + 2la-cl

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

Premultiply Table Matrix by 101/010/100 (C centered! *•• If 3b-b'-c-c+2lb-cl

+ ^t No required relationships between symmetrical scalars for reduced forms 26-44.

Reprinted from Reference 6. Based on Table 5.1.3.1 prepared by

A. Mighell, A. Santoro, and J. D. H. Donnay for the International

Tables for X-ray Crystallography ^ and published revisions. i^»^^

45
Table 3. Unique Q Matrices Generating Superlattices for |Q| = 2, 3, 4

191 = 2 191 -'

IQ = 4

100/010/002 100/010/004 111/020/002

100/011/002 100/011/004 111/021/002

101/010/002 100/012/004 100/040/001

101/011/002 100/013/004 110/040/001

100/020/001 101/010/004 120/040/001

110/020/001 101/011/004 130/040/001 I

200/010/001 101/012/004 200/010/002 ;

101/013/004 200/011/002 :

1

91 - 3 102/010/004

102/011/004

201/010/002 '

100/010/003 201/011/002

100/011/003 102/012/004 200/020/001

100/012/003 102/013/004 210/020/001

101/010/003 103/010/004 400/010/001

101/011/003 103/011/004

101/012/003 103/012/004

J

102/010/003 103/013/004 H

102/011/003 100/020/002

102/012/003 100/021/002

100/030/001 101/020/002

110/030/001 101/021/002

120/030/001 110/020/002

300/010/001 110/021/002 \

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.

46
APPENDIX C

EXAMPLES OF INPUT DECKS AND OUTPUT LISTINGS

Example 1. Cell analysis for two data sets

Input 10deck:

1 2 12.83 9.026 13.44 123.0 TESTOIMI

3 .02 .009 .05 .5 2

4 CC 4 1.46 3

5 Sodtuin carbonate decahj/dra + e 6

6 NA2 C 03 J 10 H2 O 7

7 Example cf cell analysis and density calculation B 8 K 9 8.095 8.096 30.62 88.67 58.08 87.48 TESTOSAl

10 P 3 11 Example of symmetrv/ determination B 12 K

Output listing:11121314151617 *** MCNOCLINIC ***** TESTOl **

12.83 9.026 13.44 123.0 CELL

.02 .009 .05 .5 209 C- ESDS

Cc 9A 4 1.460 1305.31 SG-1

Cc 9A 4 1.456 286.14 1305.31 SG-2

Sodium carbonate decahydrate NAME

Na2 C 03 1 10 H2 O ' CHEM

C H20 Na2 C13 EMPR Example of cell analysis and density calculation COMM

T-2 .50 .50 .00 / -.50 .50 .00 / .50 -.50 1.00 MATX

i.oo: i.oc .00 .00 / .00 -i.oo .00 /-i.oo .00 -1.00 matx

INV 1.00 .00 .00 / .00 -1.00 .00 /-I.oo .00 -1.00 MATX

12.830 9.026 13.440 90.00 123.00 90.00 1305.31 C-IN

7.843 7.843 12.175 98.77 105.91 109.75 652.65 17 C-RD

12.830 9.026 12.546 90.00 116.05 90.00 1.4214 1.3900 C-CD

.020 .009 .104 .00 .35 .00 E-CD

61.519 ei.519 148.238 -14.562 -26.172 -20.785 DOTM

01/12/81 45 16/12/80 HIST

** ANORTHIC ***** TEST02 *** **WARNING**AIt BI AND/OR CI > 30A

**WARNING**METRIC SYMMETRY EXCEEDS CRYSTAL SYMMETRY

CRYSTAL DATA BRAVAIS LATTICE = AP BRAVAIS LATTICE FROM REDUCED CELL=OB

OB CELL parameters: 11.698 25.990 11.194 90.00 90.01 90.00

MATRIX (INITIAL TO OB ) : ,- 1

.

1112 .0-1 .0 .0/-2.0 .0 1 .0/- 1.0 1.0 .0 13 14 15 1 6 1 7 8.095 e.096 30.62 88.67 58.08 87.48 CELL

95E C ESDS

P 1701 .65 SG-1

P 1701 .65 SG-2

Example of symmetry determination COMM

T-2 .00 1.00 .00 /-I.OO .00 .00 /-2.00 .00 1.00 MATX

I.oo: -I.OC .00 .00 /-2.00 .00 1.00 / .00 1 .00 .00 MATX

INV -1.00 .00 .00 / .00 .00 1.00 /-2.00 1.00 .00 MATX 8.095 e.096 30.620 88.67 £8.08 87.48 170 1 .65 C-IN

8.096 8.095 25.990 90.00 90.00 92.52 1701 .65 13X C-RO

8.095 25.990 8.096 90.00 92.52 90.00 .3115 .3115 C-CD

65.545 65.529 675.473 .001 -.0 09 -2.38 2 DOTM

01/12/61 2 45 16/12/80 HIST

47
Example 2. Generation of d-spacings and hkl indices

Input 12deck:

40 1 2 12.214 TEST03C1

3 PA3 4 1.670 3 A SODIUM ALUMINUM SULFATE HYORATt (SODA ALUM) 6

5 NA AL ( S 04 ) 2 I 12 H2 O 7 6 Example of hk l/d- ge nep a+ I on B

7 1.540598 F

8 K

Output listing:

T-2 1.0 .00 .00 / .00 1 .00 .00 / .00 .00 1 .00

1. oo: 1.00 .00 .00 / .00 1 .00 .00 / .00 .00 1.00

12.214 12.214 12.214 90.00 90.00 90.00 1822.11

12.214 12.214 12.214 90.00 iO.OO 90.00 1822.11

12.214 12.214 12.214 90.00 90.00 90.00 12.2140

149.182 149.182 149. 182 .000 .000 .000

l.J 405^8

01/12/81 45 16/12/80

11121314151617 *** CUBIC *** TtST03 ***

12.214 CELL

40E B- ESDS

Pa3 205 4" 1.670 1822.11 SG1

Pa3 205 4 1.671 458.27 1822.11 SG-2

SODIUM ALLMNUM SULFATE HYJRATE (SCDA ALUM) NAME

Ne Al ( S C4 ) 2 I 12 H2 O CHEM

Al H24 Na C20 S2 EMPR Example of h k L /d-gen era+ I on COMM

MATX

MATX C-IN

3 C-RD

C-CD

DOTM

PDFl

HIST

*»MESSAGE**PCWDER PATTERN GENEF AT ION/ ANAL YS I S FOR SYSTEM CUB

2-THETA(MAX)= 40.00 D(MIN)= 2.252212

3 CONDITIONS FCR NON-E XT I NCT I C N CALLED FOR

CLASS CONDITICN(S)

CKL K =2N

HOL L =2N

HKO H =2N

QBS CALC DIFF

CALC D OBS INT HKL HKL 2-THETA 2-THETA 2-THETA

12.542

14.492

16.214

17.773

20.551

2 1.812

24.147

25.238

26.287

27.298

29.223

30.144

31 .039

31 .912

32.764

33.597

34.412

35.994

36 .762

37.517

38.259

39.708

1 7.05 18

2 6. 1C70

3 5.4€23

4 4.9ee3

5 4.3163

6 4.0713

7 3.6627

8 3.5 2 59

9 3.3676

1 3.2643

1 1 3.0=35

1 2 2.9t 23 13 2.8769

14 2. 8021

1 5 2.7311

16 2.66=3

1 7 2.6040

18 2.4932

19 2.4428

20 2.39S4

21 2.3506

22 2.2661

<AUTHORS> <PROGRAM> HKL H K L

1 1 1 2

2 1

1 2 1 2 2 2 2 1

1 3 1 2 2 2

2 3

2 3 1 4 2 3 2

1 4 1 3 3 1 2 4

2 4 1 3 3 2 2 4 2 4 3 3 4 1

1 5 1 2 5 48
Example 3. Powder pattern analysis with calculation of

figures of merit

Input deck:

1 I 1 2 8.8659 5.0433 TEST04H1

3 .0006 .0005 2 4 P321 3 2 .729 3

5 So di urn St li con FLucrt d e 6

6 NA2 SI F6 7 7 Ex amp le of powd e r p a+t e rn an o lvs« s Inc ludi ng ca(.c of f t g ur e s of me r It B 8 1 .540598 2 F

9 17.58 13 1 20.03 100 1 1 21 .07 95 1 1 1 I 10 26.74 90 1 1 1 29.20 50 2 1 30.79 9 2 1 2 I 11 35.05 5 3 35.58 7 2 35.66 15 2 1 1 3 1 12 37.54 I 1 2 39.47 90 3 1 40.68 6 2 2 4 I 13 41.16 4 1 1 2 42.42 3 3 1 42. 88 4 2 2 5 T 14 44.61 9 2 2 1 46.22 6 3 1 1 47.72 10 2 1 2 6 I 15 50.79 55 3 2 51.86 4 3 2 54.56 2 3 7 1 16 55.19 20 3 2 1 55.96 4 1 3 56.52 9 3 1 2 8 I 17 57.94 20 4 1 1 58.69 4 1 1 3 60.01 3 2 3 9 I 18 60.60 3 4 2 62.84 11 3 3 63.23 5 5 1 10 I 19 63.94 8 2 1 3 64. 11 8 4 2 66.50 3 3 3 11 I 20 67.03 8 4 1 2 6 7.03 8 4 2 1 21 K

Output listing:

1 8. 8659

.0006

P3 21 P321

2 1 5.0433

.0005

3

3

HEXAGONAL

3 1 4 TEST04 ***5 1

8 2.729

2.729 1 88.06

343.31

34 3.3 1 150

150 Sodium Silicon Fluoride

Na2 SI F6

F6 Nb2 si Example of powder pattern analysis including calc of figures of merit

22

T-2 .00 .00 1 .00 /I.OO .00 .00 / .00 1 .00 .00

i.oo: 1 .OC .00 .00 / .00 1 .00 .00 / .00 .00 1.00

8.866 8.866 5.043 90.00 90.00 120.00 343.31

5.043 8.8 66 8.866 120-00 90.00 90.00 343.31

8.866 8.866 5.043 90.00 90.00 120.00 .5688

.001 .001 .000 .00 .00 .00

25.435 78.604 78.604 -39.302 .000 300 1.54C598 2 01/12/ei +5 16/12/80

CELL

ESDS

SG-1

SG-2

NAME

CHEM

EMPR

COMM MATX

MATX

C-IN

C-RO

C-CO

E-CD

OOTM

PDFl

HIST

**MESS AGE**PCWDER PATTERN GENERATION/ANALYSIS FOR SYSTEM HEX £-THETA(MAX)= 68.03 D(M1N)= 1.376990

CONDITICNS FCR NON-EXT I NCT ICN CALLED FOR

CLASS CaNDITICN(S)

<A0THORS> <PROGRAM>

N D CALC OBS INT H K L H K L

1 7.6761 1 2 5.0433 1 ** 5.0433 5.0408 13 1 3 4.4229 I 1 *** 4.4329 4.4294 100 1 1 4 4.2153 1 1 *** 4.2153 4.2131 95 1 1 5 3.8390 2 6 3.3296 1 1 1 ** 3.3296 3.3312 90 1 1 1 7 3.0547 2 I *** 3. 0=47 3.0559 50 2 1 8 2.9020 2 1 *** 2.9020 2.9016 9 2 1

OBS CALC DIFF 2-THETA 2-THETA 2-THETA

1 1.516

17.571

17.580 17.571 -.009

20.014

20.030 20.014 -.0 16

21 .059

21.070 21.059 -.011

23.150

26.753

26.740 26.753 .013

29.212

29.200 29.212 .012

30.785

30.790 30.785 -.005

49
9 2.5594

*** 2.55S4 2.5581

1 2.5£ 16 *** 2.5216 2.5212

1

1

2.5 1=3 *** 2.5153 2.51 57

1 2 2.3«58

*f* 2.3^58 2.39 39

13 2.2623

*** 2.2623 2.28 12 14 2.2165

*** 2.2165 2.2161

15 2. 1S18 *** 2. 1916 2.1914

15

90

3 35 .032

3

2 35, 050 3535

.032

.573

-.013

2

2 1 1 35. 580 35

35

.573

.666

-.007

2 1 1 1 2 35. 660 35

37

.666

.511

.006

1 2 3 1 37. 54 3739.511

.451

-.029

5 1 2 2 39, 470 39

40

.451 .673-.019

2 2

1 1 2 40. 68 4041 .673

.151 -.007

1 1 2 41. 160 41 .151 -.009

32 1.5719 1 1 3 58.688

*** 1 .57 19 1.57 18 4 1 ^ I 3 58.69 53.688 -.002

33 1.53S9 2 3 60.029

*** 1.5359 1 .5404 T 2 3 60.01 60.029 .019

34 1.5356 5 60.215

35 1.5274 4 2 60.574

*** 1.5274 1.5268 3 4 2 60.600 60.574 -.026

36 1.4777 3 3 62.839

*** 1.4777 1.4776 1 1 a 3 62.840 62.839 -.001

37 1. 4650 5 1 63.250

*** 1.4690 1.469 5 5 5 1 63.230 63.250 .020

38 1.4547 2 1 3 63.949

** 1.4547 1 .4548 a 2 1 3 63.940 63.949 .009

39 1.45 10 4 2 64.128

*** 1.45 10 1.4514 e 4 2 64.1 1 64.128 .018

40 1.4440 3 2 2 64.475

41 1.4160 3 3 1 65.806

42 1 .4051 3 3 66.490 ** 1.4051 1.4049 3 3 3 66.500 66.490 -.010

43 1.3955 4 1 2 67.005

*** 1.3955 1.3951 8 4 1 2 67.030 67.005 -.025 ** 1.3945 1 .3951 8 4 2 1 67.030 67.064 .034

44 1.3945 4 2 1 67.064

45 1.3790 5 1 67.916

REFLECTICN SUMMARY FOR ENTIRE PATTERN:

ESTIMATED PESCLUTION= .075 DEG. 2-THETA

ThECRTICAL LINES TOTAL= 45

THEORETICAL RESCLVABLE= 42

UNIQUE OBSERVED LI NES= 34

TOTAL LINES INPUT= 35

NUMEER INDEXED= 35

NUMBEP UNINDEXEO=

4c4c:^:tc«:4"t»KFOF INDEXED LINES********

AVERAGE ABS<DEL 2-THETA)= .013

LARGEST 2-THETA DIFFERENCE= .034

« WITI- DIFF>0.05 (STAR L1MIT)=

* WITh DIFF>0.20 (I LIMIT)=

AVERAGE 2-THETA DIFFERENCE= -.001

« WITH DIFF > +0.02(2-THETA)= 3

« WITh DIFF < -0.02<2-THETA)= 3 M(20) = 67.17 (DLIMIT = 1.7584, « POSSIBLE = 24)

F<30) = 70.27 (DELTA 2 THETA = .0115t * POSSIBLE = 37)

50
NBS-n4A iREv. 2-eci U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS Tech. Note 1141

2. Performing Organ. Report No. 3. Publication Date

April 1981

4. TITLE AND SUBTITLE

NBS*AIDS80: A FORTRAN Program for

Crystal lographic Data Evaluation

5. AUTHOR(S)

A.D. Mighell, C.R. Hubbard, and J.K. Stalick

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, O.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as No. 6

10. SUPPLEMENTARY NOTES

I^J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bibliography or literature survey, mention it here)

Techniques for the computer-assis

have been developed to improve the d

Center and the JCPDS--International

computer program, NBS*AIDS80, can be

powder diffraction data by the gener

and analysis components include calc

determination of metric symmetry, ch

tion of d-spacings and indices, and

tion of figures of merit. Detailed

ted evaluation of crystal lographic data

ata compilations of the NBS Crystal Data

Centre for Diffraction Data. The resulting

used for the analysis of unit-cell and

al scientific community as well. Research

ulation of standard cells and space groups,

ecking the data for consistency, generaanalysis of powder patterns with calculainput and output specifications are given.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Computer evaluation; crystallographic data analysis; JCPDS--International

Centre for Diffraction Data; metric symmetry; NBS Crystal Data Center;

powder diffraction; unit cell.

13. AVAILABILITY

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