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V« of * NBS TECHNICAL NOTE 893

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

Computer Programs for

Structural Chemistry:

MATCHl and MATCH2,

FORTRAN Programs To Predict and Evaluate

Mutual Orientation of Polycrystals
NATIONAL BUREAU OF STANDARDS

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1 4 1976 ^ Computer Programs for Structural Chemistry:

«3 MATCH1 and MATCH2, FORTRAN Programs To Predict

and Evaluate Mutual Orientation of Polycrystals

Brian Dickens and LeRoy W. Schroeder

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

This investigation was supported in part by Agreement

YO1-DE4001S to the National Bureau of Standards and Grant DE00S72 to the American Dental Association Health Foundation from the National Institute of Dental Research and is part of the dental research program conducted by the National Bureau of Standards in cooperation with the American Dental

Association. ^

TO'5%

S*TES O* h

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

US NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued January 1976
National Bureau of Standards Technical Note 893

Nat. Bur. Stand. (U.S.), Tech. Note 893, 71 pages (Jan. 1976) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. CI 3.46:893). Price $1.30 (Add 25 percent additional for other than U.S. mailing
Table of Contents

Abstract 1

Introduction 2

Overview of MATCHl and MATCH2 4

Table 1. Number of metric fits between Ca (PO ) OH

and other compounds 5

Figure 1. Possible contact planes in Ca (PO ) OH 7

Figure 2. An example of Ca(l) motiv in Ca^ (PO.) -.OH 8

Table 2. Calculated Structural r its

9

Procedure in MATCHl 10

Procedure in MATCH2 13

Algorithm for MATCHl 19

General purpose of subroutines in MATCHl 21

Hierarchy of subroutines in MATCHl 2 3

Main variables in MATCHl 24

Algorithm for MATCH2 2 5

General purpose of subroutines in MATCH2 2 9

Hierarchy of subroutines in MATCH2 3 3

Main variables in MATCH2 34

Card input to MATCHl 37

Card input to MATCH2 39

Optional summary output on punched cards 46

Computer requirements 47

Test deck for MATCHl 48

Test deck output for MATCHl 49

Test deck for MATCH2 51

Test deck output for MATCH2 52

in

Computer Programs for Structural Chemistry:

MATCHl and MATCH2 , FORTRAN programs to

predict and evaluate mutual orientation of polycrystals +

Brian Dickens and LeRoy W. Schroeder*

Two computer programs to evaluate possible structural

matches for use in epitaxy and twinning studies have been

written in FORTRAN V for the UNIVAC 1108. They should be

readily convertible to other comparable computers.

The first program, MATCHl, requires knowledge of the unit

cell parameters. It obtains all matching networks in two unit

cell lattices and sorts the matches into an order of probable

epitaxy (or twinning if the two unit cells are the same) based

solely on criteria of dimensional mismatch and network area.

The second program, MATCH2 , requires knowledge of positional parameters of the atoms in the crystal structures. It

calculates the degree of structural fit for slices supplied

from visual inspection of the crystal structures or as matching networks determined by MATCHl.

The slice comparison is carried out in terms of the vector

set of the environment of each atom in each slice. The procedure is not valid when the twin operation is a rotation about

the normal of the compositional plane or a reflection in a mirror parallel to this plane. The atomic patterns comprising

the slices are matched piece by piece in MATCH2 . A third program, MATCH3, will match whole patterns once three fiducial

atoms in each pattern have been picked from the output of

MATCH2 and will handle all the twinning cases.

This Technical Note explains the procedure in MATCHl and

MATCH2 and includes examples of results. The programs are

available on magnetic tape or card decks obtainable from the

authors.

Key words: Computer calculations; crystal structure; epitaxy;

lattice misfit; pattern recognition; twinning

+ To obtain a microfiche copy of this Technical Note and

the FORTRAN code listings, please contact the Input

Section, National Technical Information Service,

Springfield, Va. 22151.

* Research Associate of the American Dental Association

Health Foundation Research Unit at the National

Bureau of Standards
Introduction

The phenomenon of regular mutual orientation of component

crystals of various substances in polycrystals is widely

spread and is encountered by workers in the fields of solid

state chemistry, metallurgy and materials research. Regular

mutual orientation of crystals of two (or more) phases sharing

a common surface arises during crystallization and is assumed

to be controlled by structural analogies between the crystalline

phases

Polycrystals give rise to several effects important in

solid state chemistry and precipitation processes:

1. apparent nonst'oichiometry,

2. apparent impurity incorporation,

3. seeding of anhydrous salts or low hydrates

in aqueous environments via epitactical growth

on highly hydrated precursors, and

4. topotaxy, in which the products of solid state

reactions have a regular crystallographic

orientation with respect to the reactant crystal.

We have been concerned with assessing a priori the possibilities of polycrystal formation via intergrowths or overgrowths of calcium phosphates. A priori assessment has proved

necessary because of experimental difficulties. Calcium phosphates do not lend themselves to vapor deposition techniques

because some calcium phosphates decompose under even mildly

adverse conditions. Also, some intermediate phases such as

high hydrates acting as seed templates have only a transient

existence under biological conditions. There are, however,

several examples in the field of calcium phosphates where

two intergrown phases have both survived to produce a polycrystal of macroscopic size. We believe it is possible and.

meaningful in many cases to investigate the structural relationship of the various stable phases in polycrystals regardless of any intermediate phases. Hence, we have devised the

procedures described in this publication.

Growth of calcium phosphates from solution in situations

such as precipitation of a more stable phase with dissolution

of a less stable phase usually takes place under conditions

of only slight supersaturation. The degree of structural misfit which can be accommodated under these gentle conditions is

probably smaller than misfits possible under more drastic

conditions
such as vapor deposition of a thin film onto a substrate.

Thus, "chemical" fits of the parts of the calcium phosphate

structures actually in contact with one another are probably

very important; because of (i) the smaller thermodynamic

driving force in deposition from solution vis-a-vis vapor,

and (ii) the necessity of interlocking patterns of monatomic

and polyatomic ions which differ considerably in size and local

charge.

To study the importance and extent of occurrence of polycrystal formation in calcium phosphate chemistry in a systematic

and objective way while considering all reasonable possibilities,

we have written (in FORTRAN V) the computer programs MATCHl and

MATCH2 described here.

The major innovations in our approach are:

(1) All reasonable and unique metric fits between twodimensional networks in two three-dimensional lattices are

calculated. The reasonableness of the misfit is based on

quantitative criteria of allowed misfit and on the maximum

allowed dimension in the networks. Subsequently, these network matches are sorted into an order of probable matching based

on misfit of the component lengths and angles and on the area of

each network plane.

(2) For each of the most probable network-network

matches obtained in step (1) , we calculate quantitative figures

of merit based on the degree of structural fit of corresponding

atom locations and charges in thin slices which may be capable

of replacing one another. The atomic details are obtained from

known crystal structures.

(3) Our approach is one of filtering, in that surviving

cases are possible but not definite cases. In general this

approach is necessary to save computer time. A pre-conceived

case can be tested at the higher level structural-fit gauging

stage. To be even more diagnostic, we are writing a third

stage which places the two components of the contact layer in

juxtaposition. This stage removes the limitations (discussion

on p. 17 ) of the 'incomplete filtering 1 in our present approach,
Overview of MATCH1 and MATCH2

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ComPare this with the area of the unit mesh of ; to give an estimate of the atomic density. The index
<i2 2 -2 actually used is (area of unit mesh in A -30 A )/30A where

o2

30 A is taken as the unit area. Thus, meshes with large areas

yield higher values of this index. Possible contact planes

are sorted on the sum of the generalized misfit and the above

index, with those planes having lowest values of the sum ranking

highest. These cases will normally be considered first by

program MATCH2

.

The results of several calculations using MATCHl are

summarized in Table 1. The large number of metric fits, networks whose unit meshes coincide within the limits specified,

supports the view the mutual orientation of crystals is controlled

by structural analogues. Otherwise, many more orientations would

be expected to be observed experimentally.

Table 1

Number of metric fits between Ca^ (PO.) -.OH

and other compounds

Ca

5

tP04

) 3

OH 16* CaC0

3

(a) 297

Ca8H

2 (P04

) 6

- 5H

2

11 CaC0

3

(c) 28

Ca4 (P04

) 2

82 BaCa(C0

3

) 2

420

CaHP04 -2H

2

229 + Ca(COO)

2

*H

2

82

CaHP04

4]_** Ca(COO)

2

-2.2H

2

18

Sn

3

P04F

3

67 CaKAs04 -8H

2

132

CaC0

3

-6H

2

66 MgNH4 P04 -6H

2

177

+

Possible twinning modes

Criteria changed to 10%, 4°, 21A because

>1000 matches obtained originally.

** Criteria changed to 15%, 4'

Normal criteria:

21A

(i) length mismatch 15% maximum

(ii) inter-vector angle mismatch 10° maximum

(iii) maximum vector length 20 A
The program MATCH2 carries out the procedure in step 2

of the Introduction. Each of the networks in the matching

pairs generated in MATCH! defines an infinite set of parallel

planes in the corresponding compound. To check the gross

features of the proposed contact, a preliminary search by the

following method is made with the monatomic cations and anions

and the central atoms of polyatomic ions, d.g., P in the case

of P0 4

. Each atom in the structure is allowed to define a specific plane from the infinite set of parallel planes (see

Fig. 1 ). All crystallographically discrete atoms within

a predefined distance, typically 0.7 5 to 1.0A, of the defined

plane are identified. If the resulting slice is sufficiently

populated, the immediate environment of each of these atoms

is determined and is considered to be that atom's structural

motiv (Fig. 2 ).

A] 1 unique interatomic vectors are calculated between

members of each motiv. Motivs in the two structures are then

compared on the basis of their vector sets so that incremental

rotation of the motivs is not necessary at this stage. The

comparison includes consideration of vector lengths and the

charge product of the two atoms defining each vector. Each

vector is used in only one vector match. A figure of merit for

the fit of the two vector sets is calculated based on the length

misfits of individual vectors and on the percentages of the

vector sets matched up. In cases where one motiv encompasses

another, the total fit will be good for the smaller and poor

for the larger motiv and this possible match will be given

a relatively low f igure-of-merit.

The inter-vector set f igure-of-merit corresponds to a measure of the closeness of fit between these vector sets

(see the example in the sample output on p. 49) . This in turn

is taken as a measure of the similarity of the chemical environments of the central atoms of the motivs. The assay of the

chemical environment is necessarily confined here to the ~2A

thick slice through the crystal structure. Thus the environmental similarity of two central atoms, one in compound A and

one in compound B, is assessed from the magnitude of the

f igure-of-merit, and is considered to be directly proportional

to this magnitude.

The atoms in compound A are matched one at a time with

atoms in compound B in strict order of descending figure-ofmerit magnitude (refer to p. 57 ) . The sum of the individual

f igure-of-merit matches is accumulated to give an overall interplane f igure-of-merit which, after matching is complete, is

divided by the plane area to give an inter-plane figure-ofmerit normalized to unit area (p. 57 )• The normalized interplane f igure-of-merit may then be used as a quantitative
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Figure 1. Possible contact planes in Cat.(PO.)_ OH.
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indication of the perfection of chemical fit between two

candidates for epitaxy or twinning contact planes, and allows

the comparison of all such candidates on a quantitative basis

over several compounds. Table 2 provides an overview

of the results of several runs of the MATCH2 program. Optionally, MATCH2 also produces a printer plot of the atomic patterns

for the contact planes in compounds A and B. The listing of

crystal coordinates allows one to label the repeating unit on

the plot and visually check that MATCH2 has matched everything

properly. See the output example generated by the test deck.

Procedure in MATCHl

The main routine in MATCHl is element MAIN.

The first card read by element MAIN contains the title

of the run, the output unit number, and the ordinal file on

the output unit. The output file may be positioned using

computer specific routines in subroutine POSITN.

The second card contains the unit cell parameters of the

first compound, compound A, from which the reciprocal cell and

real cell volume are calculated in subroutine CELL. These

quantities are then printed in the listing to allow the input

quantities to be checked. The cell of compound B is then read

in from card three and treated similarly. The real and reciprocal cell metric tensors are calculated for compounds A and

B in subroutine METRIC.

Card four contains the point group number for compound A,

the point group number for compound B, the allowed percentage

difference on vector lengths, the allowed percentage difference

on network angles, and the maximum vector length permitted in

the networks. The default options of these last three quantities are 15%, 10°, and 20 A.

The symmetry subroutine SYM which filters out the unique

vectors from the triclinic set initially generated by the program, is based on the point group (32 possibilities) underlying the space group of the crystal structure. (Note that

the required point group is not necessarily one of the 11 Laue

centrosymmetric point groups or related to one of the 14

Bravais Lattices.) Thus all meaningful network matches will

have been treated when structural considerations are applied

in the subsequent program, MATCH2

.

10
All unique vectors with length less than the maximum

value are calculated in the two compounds by generating vector

sets in triclinic symmetry and then filtering the sets through

the symmetry subroutine SYM. These vectors are sorted into

ascending order of length and the resultant values are listed

in the output of the program. Vectors in compound A are then

matched with vectors in compound B subject to the limitation

that the mismatch between vector lengths must be less than the

read-in quantity, usually 15%. The resultant vector matches

are also output for checking and references purposes.

Networks are generated by taking pairs of vectors in

compound A and their matched mates in compound B. The intervector angles are calculated in both cases and made less than

or equal to 90° by taking the supplement if necessary. Care

is taken to use only non-parallel vectors to define networks.

Only those cases where these inter-vector angles agree within

10° are accepted for further processing. Subroutine ANGCK

rotates the nets to closest coincidence and estimates misfit

dislocation density.

Various tests are made throughout the program to insure

that the list of network matches does not contain redundancies.

At this stage the indices of vectors in new network matches

are compared with the indices of vectors in previously obtained

matches and complete redundancies removed from the list.

In case the acceptable limits on vectors and network

matches are too generous and produce too many matches, the

program estimates how much to reduce these criteria to bring

the total number of vectors and networks just below their

respective maxima of 1000 vectors and 1000 networks in each

compound

.

Further checks are made to cut down the number of redundancies or near-redundancies. The program takes two networks in

compound A and calculates the angle between the normals to the

networks. A similar inter-normal angle is also calculated for

the network pair in compound B which has previously been tagged

as matching the network pair under consideration in compound A.

At least one of these two inter-normal angles, one in compound

A and the other in compound B, must be greater than 2° for both

network matches to survive. Otherwise, the two sets of networks are essentially or actually parallel to one another and

the set of networks with the smaller average angle is dropped

from the list. The process is repeated until the list of network matches comprises a unique set. At this point the list

of networks has been reduced in size as much as is feasible but

typically from 10 to 1000 network matches still remain.

11
To remove the random character of the search when these

network matches are input into the structure-matching program

MATCH2, the netwoijk: matches are then sorted in order of probability using quantities' calculated with the following criteria

in mind:

(1) Corresponding mesh points for the two nets

should be as close as possible (i.e., the nets

should approach coincidence)

.

(2) The atomic pattern of the planes should be simple.

The rationale for the second criterion is that the simpler

the pattern the higher the probability of its being matched by

a similar layer in some other compound. That is, it is easier

to match a pattern with few details than one with many details.

To put the criteria in (1) and (2) on the same scale by restrict

ing the possible outcomes to the same range, we use the following quantities: (i) 2 RL/(Sum of long diagonals of the nets

A and B) where RL is the distance between the corresponding

lattice points for net A and net B. (ii) 2RS/ (Sum of short

diagonals of the nets A and B) where RS is similar to RL. The

theoretical range of these quantities is to 2. These indices

yield a measure of how well criterion (1) is being fulfilled.

(iii) K •

(

[

area A-30

[ +

I area A-30 |) where we have assigned

30 ' 30 y

a unit area of 30A2 as that occupied by a basic structure unit,

e.g., one cation and one anion. The range is restricted to

to 2 by chosing K, the scale factor. The maximum area allowed

is governed by maximum length allowed for a network vector and

is typically about 400A 8

.

The sum of the three quantities (i) , (ii) and (iii) , is

used as a composite sort parameter. Networks with low values

of this parameter are considered the most probable candidates

for structural matches. In confirmation, our preliminary

results have shown that the most important structural matches

as determined by MATCH2 in lists of networks which have been

sorted using these combined criteria do usually occur in the

first three or five members of the list. The sorted list is

output on a file, either on magnetic tape or mass storage such

as a drum or disc, which can then be transferred to the second

part of the matching scheme. An overview of the results of

several runs of this program has been given in Table 1, p. 5.

Examples of the input and output are given on pages 49 and 50.

12
Procedure in MATCH2

MATCH2 produces a quantitative estimate of the

likelihood of forming polycrystals composed of phases for

which the detailed crystal structures are known. The program

needs as input (i) the unit cells, symmetry operations and

atomic coordinates for the structures under consideration,

and (ii) two vector pairs, each pair defining a plane through

one of the two crystal structures or two sets of Miller indices

The defined planes may be the result of visual inspection of

the crystal structures or may be obtained in a more objective

manner as output of the network-match generating program

MATCHl

.

o A slice of predetermined thickness, typically from

1.5a to 2.0A, is then taken through each crystal structure.

In our studies of calcium phosphates we have found it satisfactory to use only the Ca and P atoms for preliminary

surveys to reduce the large number of slices to be considered

in greater detail. The set of vectors between atoms in the

slice is used to compare atomic arrangements in slices in

different compounds and to derive figures of merit for

possible epitaxy of the two compounds with the slices under

consideration as contact planes.

The flow of the MATCH2 program will now be given

in some detail. The main routine is MATCH2 which reads the

title of the job from the first card. The unit cell parameters for compound A are read in from the second card and

the reciprocal cell and the volume of the real cell are

calculated. Next, the symmetry elements followed by the

unique set of atoms for compound A are read. Then the

program reads the unit cell, symmetry elements, and atoms

for compound B. These quantities are immediately output.

The program calculates the real and reciprocal metric

tensors for compounds A and B and generates the full cell

of atoms. Matrices are calculated to orthogonalize the

actual unit cell, to transform from the orthogonal cell

back to the actual cell, and to transform the atomic

coordinates from their fractional values in the real cell

to orthogonal coordinates in Angstroms. The atomic

positions and the vectors defining the unit cell edges are

transformed to orthogonal angstrom coordinates so that

different crystal structures can be compared in the same units

and interatomic distances can be calculated efficiently.

13
At this point the program is ready to read in the

first of the networks to be compared from either ah intermediate

input file prepared by the MATCHl program or from the card

reader. The planes may also be entered in Miller indices..

The first read from the intermediate input file if this mode

is used is the title of- the input file. Then the program

enters a loop which begins with the vectors defining the networks being read from the same file.

If an end-of-list signal or a hardware end of file has

been read or if the number of reads is greater than the

requested maximum value, further calculation is terminated

and the program outputs a final summary of inter-slice figures

of merit. Otherwise the program then proceeds to check the

vectors which define the two networks. First, the indices of

each vector are made coprime. If the vectors are read from the

card stream rather than an intermediate file, the length of each

vector is calculated and the angle between the two vectors defining the network is calculated for each compound. Each angle

is made less than or equal to 9U° by taking the supplement if

necessary. These quantities are already available when the

intermediate file is being read. The area of each network is

calculated and the ratio of the areas is saved for later scaling

of (i) the number of occurrences of each type of atom in the

two compounds to a common basis and (ii) the inter-plane figure

of merit for this network match to unit area. Similarly,

the length of nominally corresponding vectors are compared

and the necessary multiples of each length are obtained.

The vectors are then scaled by these multiples. In the,

present program the maximum ratio of these multiples is 4:3.

The program outputs the input as modified by the various

decisions that it has made so far. It then calculates and

prints the plane normals to the networks and the Miller index

designation of the plane defined by the network so that reference may be made to the planes in this convention if desired.

When twinning (compound A is the same as compound B) rather than

epitaxy is being investigated, the angle between the normal

to network A and the normal to network ( B is calculated and

the rotation axis and angle are output to define the twinning

mode. The program then calculates for all cases the direction cosines of the plane and the d-spacing. These quantities

are written out.

The main program then calls subroutine ATPLAN to determine the populations of all significantly different slices

in each crystal structure parallel to the networks specified

by the vectors. In subroutine ATPLAN, each atom in turn is

allowed to define the central plane of the slice and all the

other atoms are checked to see if they are within distance

PDIST K of the plane defined by the first atom, i.e., if they

are within the same slice.

14
Presently the program calculates all possible slices

by allowing every atom in the unit cell to define its own

slice. Options exist for (i) proceeding with all slices,

(ii) removing redundancies based on equal numbers of

identically labeled atoms (e.g., Ca 1) or (iii) removing

redundancies and subsets.

The variable PDIST must be selected with care. Too

large a value will allow several planes' to define a particular

slice over the one unit cell the program investigates to populate

the slice. Too small a value for PDIST may result in some atoms

being wrongly excluded from the slice. We have found that an

appropriate value is 1.0 A, which, being the distance from the

defining plane to the edge of the slice, gives a slice thickness

of 2.0 A, i.e., roughly an atomic diameter.

For each atom in a slice, the spacing of the atom from the

plane which defines the center of the slice, the ordinal number

of the atom which defines the slice and the ordinal number of

the atom which is a member of the slice are all stored away in

one large array. Only those slices with more than two atoms in

them are saved for later use. Subroutine ATPLAN also provides

the number of atoms in each slice.

After subroutine ATPLAN has been entered once for each of

the two crystal structures being compared, the main program

calls subroutine ATVEC to determine the atomic motivs and the

vector sets of these motivs. Subroutine ATVEC treats the

slices of atoms one at a time. First it determines the number

of the occurrences of each kind of atom in the slice as distinguished by the alphanumerical label of each atom. Because

the atoms are distinguished by their labels, atoms in different

environments (i.e., crystallographically different atoms) must

be given different labels, e.g., Ca(l), Ca(2), Ca(3) and so on.

This differentiation is necessary to save computing time by

finding the motiv for only one Ca(l), Ca(2), etc.

Two methods of calculating the motiv have been

investigated. The first is based on the contents of the

slice in one unit cell. The problem here is in relating

cells of different size. However, the system works very

well for cells which are approximately the same size and

15
where one network is not a subset of the other. The

second method involves including into the motiv only those

atoms which lie within some preconceived distance of the

motiv-defining atom. Typically this distance has been

6.0A, which allows for first and second nearest neighbors. Specifically, the distance was chosen to allow the program

to pick out P atoms in both neighboring P04 groups in

P04

. . .Ca. . .P04 chains when the Ca motiv is being constructed

and yet not include atoms beyond the first layer of the

environment (see Fig. 2). This method is obviously indepen- dent of the area of the plane.

In both methods of building up the atom motiv, up

to four neighboring unit cells are searched to find those

atoms which lie on the plane and which are nearest to the

central atom of the motiv, i.e., the motiv definer. Provision 'is made in both cases for including all atoms with

a given alphanumeric label whose distances to the motiv definer

are less than 1.25 times the shortest distance from this type

of atom to the motiv definer. At the present time, the second

method is the one implemented in subroutine ATVEC.

When the motiv has been assembled by considering

all the atoms in the slice, the vector set between all

atoms within the motiv is calculated as a quantitative

description of the motiv. It should be noted that use of

the vector set removes any knowledge of the relative

orientation of the two patterns to be matched. This is

a serious restriction in testing possible cases of

twinning where the structure has been rotated or

reflected about the normal to a contact plane; in such

cases the procedure degenerates into a calculation of the

atomic density per unit area. Our planned third stage,

MATCH3, will handle these cases and will also act as an

extension of MATCH2 for general cases.

The program writes the motiv defining atom, the

motiv vector set and several other associated quantities

on a temporary mass storage file and then considers the

next motiv. At the end of these calculations the

program writes an end-of-data flag on the mass storage

file and re-enters the subroutine ATVEC to perform

similar calculations for compound B. ATVEC can also

generate a histogram of various classes of inter-atomic

distances on a surface by calling subroutine HISTGM.

This feature was added to aid in our studies to develop

improved adherents to calcium phosphates.

16
When control is returned to the main program after the

second pass through subroutine ATVEC,' the first motiv for

each compound is read from the appropriate mass storage files.

The number of occurrences of the motiv-defining 'atom on the

plane in each structure is scaled up by the appropriate

multiple previously defined by a comparison of the plane

areas. The vector sets of the two motivs are then compared.

The comparison includes consideration of the charge product

of the two atoms which define the ends of the vectors. Thus,

charge misfit of greater than 1.5 electrons is at present not

permitted in a vector match. Also, vector length misfits

greater than 2 5% are not permitted in matches of individual

vectors. After these tests have been made between a vector in

the motiv in compound A and all unmatched vectors in compound

B, the best fit of length between the vector in compound A

and the corresponding one in compound B is tagged as a match

and neither of these vectors will be matched again.

After attempts have been made to match all vectors in

the two vector sets, the program calculates the percentage

vectors matched, the percentage charge mismatch, and the

percentage of each vector set matched. If required, the

program will at this p'oint output onto 'an intermediate file

the various quantities it has obtained between motiv matches

if more than 7 5% of the vectors in both vector sets are

matched and the average mismatch in length or charge is less

than 2 5%.

For all vector sets, an inter-motiv figure-of-merit

quantity is calculated based more or less equally on the

average distance mismatch and the number of vectors matched.

This quantity, which is designed to be 100% for perfect matches,

is [ (0.2-T)/0. 2] 50 + (Z + Z )/4, where T, the average mismatch of vectors in A, is compared with 0.2 , and Z, and Z g

are the percentages of matched vectors in the vector sets of

the motivs in compounds A and B. If the two- atoms defining

the motivs being compared have different charges they are

considered to be mismatched chemically. The percentage

quantity defining that match is reduced by 20% so that the

program will be less likely .to match atoms of opposite charge

should the vector sets of their motivs accidentally have some

correspondence. Such cases are obviously chemically

destabilizing.

When an individual figure of merit has been calculated

for the possible match of each motiv in compound A with each

motiv in compound B, the program attempts in the second half

of the MAIN element to match atoms using the inter-motiv

figure-of-merit just calculated. The program determined

earlier the multiplicative factors which make two plane areas

under consideration the same and scaled up the number of atom

17
occurrences in each slice to reflect this. Matching between atoms

is accomplished by pairing off the atoms in compound A with atoms

in compound B in order of descending value of the inter-motiv

figures-of-merit subject only to the number of occurrences of

each atom in the slice. Thus, if an atom in compound A has four

total appearances and an atom in compound B has only three occurrences and these atoms have a very high matching quotient, the

match will be used until the lower number of occurrences is

completely satisfied, that is, three times. The fourth match of

the atom in compound A must be accomplished by using some other

atom in compound B. In general, this procedure works very well

and the program is correctly able to identify and match, for

example, atoms in cation columns in two compounds.

As the atom-matching proceeds, an accumulative sum is

made of the associated inter-motiv f igures-of-merit. When all

the atomic occurrences in one of the two compounds have been

matched, the program calculates an inter-plane figure of merit by

dividing the total accumulation of inter-motiv quantities by the

maximum of the two network areas to give some idea of matching

per unit area.

If the inter-motiv fit is greater than a threshold value

initially read from cards (with a default option of 4.0), the

option exists of plotting the atomic configuration on the planes

in compounds A and B on the line-printer with all plots on a constant scale. The atom labels and positions in the unit cell

are listed whether or not the plot option is selected. The plots

may be used to resolve ambiguities in matching atoms which have

close matches with other atoms.

Ambiguities sometimes exist because an atom in one

compound may match two atoms in the other compound within the

limits of resolution of this procedure. Because of these

ambiguities, a general procedure to assemble matching patterns

of atoms in slices by considering individual matches is difficult

to program. At present, a general procedure seems unnecessary

because visual matching of atoms after reference to the plots and

the atom-atom environment fits is quick and provides a chance to

check on the validity of the calculations at an opportune stage.

At the end of the assessment of each slice in the

structure parallel to a network, the program reads in

another network and repeats the procedure. When network

read-in is complete, the program outputs the details of up

to 20 of the best matches for further consideration. The

merit figures may be taken, for example, as implying the

likelihood of epitaxy between compounds A and B using the

slices as "common material" and the centers of the slices as contact planes. A summary of several calculations of possible

epitaxies between Ca (PO ) OH and some other calcium phosphates c 5 4 3 is given in Table 2, p. 9 Examples of the input and output

are given on pages 51 and 52»

18
Algorithm for MATCHl

Comments refer to operations carried out in

the subroutines named in capital letters or otherwise to

operations carried out in the main program MAIN .

Start: MAIN (Main program)

Initialize

Position output file in POSITN

Write title on output file

Read cell of A in CELL

Read cell of B in CELL

Calculate real and reciprocal metric

tensors for A and B in METRIC

Read boundary conditions and initialize

them if required

Calculate lattice vector lengths for A and

B in DVCALC

Sort lattice vectors into order of increasing

length in SORT

Output lattice vector lists for A and B

Find vector matches in terms of lengths

between lattice vector set in A and B

Take two vectors at a time in A to generate

network, calculate inter-vector angle. Do

same for B

19
Compare inter-vector angles in A and B, save

inter-network match if angles are within

prescribed limits

Remove redundancies in the form of parallel

or nearly parallel networks

Sort networks into list based on

aggregated criterion which in turn is based

on misfits of vector lengths and angles and

on area of plane. Write sorted list on output file for transmission to MATCH2 program

20
General purpose of subroutines in MATCHl

ANGCK

CELL

Calculate the network angle for compounds A and B

Find asimuthal angle to place nets to closest

coincidence.

Read the real cell, calculate the real cell volume

and the reciprocal cell.

COMPAR

Compare two lattice vectors for identity of

components

CHECK

Entry in COMPAR. See if two sets of vectors have

equivalent indices.

CROSS

Vector cross product routine for triclinic cells.

The resulting vector is the same type as the input

vectors (co- or contra-variant) and is not

normalized.

DVCALC

MAIN

Generate the unique set of vectors for a point group,

using subroutine SYM . Calculate the vector

lengths via DIST.

Main program. Reads cells, calculates unique vectors,

matches vectors in A and B in terms of their lengths,

takes vectors in each compound in pairs to generate

networks, checks intervector angle matches in A and B,

removes redundancies and near-parallel cases.

21
HKLGEN

Unpack indices of lattice vector from packed word.

METRIC

Calculate metric tensor from cell dimensions.

MV

Multiply vector by matrix.

PLANEQ

Find Miller index representation of network defined

by two real cell lattice vectors.

POSITN

File positioning routine. Dummy routine at present.

SORT

Sort input quantities into ascending order.

SYM

Reduce vector to unique form as determined by point

group. Eliminates redundant vectors.

TEST

Governs the overall checking of whether two networks

are the same based solely on indices.

UNIT

Make input vector into unit vector.

VMV

Generate scaler product for input vector when given

the appropriate metric tensor.

22
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23
Main variables in MATCHl

COMMON /STORE/BUFF (1000,32)

The BUFF array is used to save space by overlaying variable lists in core.

Those variables in the following list which

have A as one of the initial two or final two characters are quantities related to compound A, one of

the two compounds under consideration. In the variables related to compound B, the A is replaced by B.

Only the A variables are mentioned below.

A, RA Real and reciprocal cell parameters for A.

AM, RA Real and reciprocal metric tensors for A.

AREAA Area of plane defined by the two A vectors

referenced in the corresponding element of

the MATCHA array.

ANGAS Angle for A network saved in corresponding

member of MATCHA array.

DELTAD Fractional mismatch in length of lattice

vectors matched up in corresponding element

of MATCH array. The corresponding element

of MATCH contains the packed ordinal numbers

of the paired vectors.

JKLA Array of packed indices of lattice vectors.

The indices are multiples of the cell edges.

MATCH Packed ordinal numbers of vectors matching

in lengths within prescribed criterion. One

vector belongs to compound A and the other

to compound B

.

MATCHA Array derived from packing two MATCH numbers,

making a total of four ordinal numbers of

vectors packed together. The four vectors

define two matched networks.

24
Algorithm for MATCH2

Comments refer to operations carried out in the

subroutine name in capital letters or otherwise to

operations carried out in the main program, MAIN.

The card input to the program is described

on page 39.

Start: MAIN (Main program)

Initiallize

Read cell A in CELL

Read symmetry elements A in SYMIN

Read atoms for A in ATOMIN

Read cell for B in CELL

Read symmetry elements B in SYMIN

Read atoms for B in ATOMIN

Output input quantities in OUTP

25
Calculate real and reciprocal metric tensors

for A and B in METRIC

Generate full cell of atoms for A and B in

ATCELL

Calculate orthogonization matrices for cells

in ORTH

Read in plane defining information, either as

plane normal in intercept notation (generate

lattice vectors notation in VECTS) or as

lattice vectors defining a network in real

cell lattice (generate intercept notation in

PLANEQ)

Compare vector lengths

Scale up vectors so that plane areas are

equivalent

Find plane normals in WORM

For network notation in input, find intercept

notation in PLANEQ

For twinning, calculate twin axis via VNORM

and rotation angle,

26
Calculate direction cosines and d-spacings

via MV and VMV.

Output intercept notation, twinning information,

direction cosines and d-spacings where appropriate

Calculate perpendicular spacing of atoms from

plane in compound A in ATPLAN

Ditto for B in ATPLAN

Calculate atomic motiv and motiv vector sets

for compound A in ATVEC, store results on

temporary file NDRUMA, print contents of

motivs if required, calculate histograms of

distances in structural layer (histograms

separated according to charge of atoms)

in HISTGM if desired.

Ditto for compound B in ATVEC and NDRUMB

Read information for one motiv at a time from

temporary files and begin motiv-motiv comparison

Scale up atom occurrence count for disparity

in plane areas

27
Compare vector sets of each motiv in compound

A with each motiv in compound B

Optional output of motiv-motiv match parameters

on saved file

Count up preliminary merit indicators

Calculate inter-motiv figure-of-merit

Calculate inter-plane f igure-of-merit

based on best individual atomic matches, i.e.,

inter-motiv f igure-of-merit, and scale tc

unit area

Optional plot of atoms in layer

Optional listing of positional parameters of

atoms in layer

Store up to twenty best inter-plane fits, as

revealed by the highest magnitudes of the

inter-plane f igures-of-merit

Read another vector set, or at conclusion of

job, output up to twenty best matches

28
ATCELL

ATOMIN

ATPLAN

ATVEC

General purpose of subroutines in MATCH2

Expand the asymmetric unit of atoms into the

full cell.

Read atom list into array ATOM. There are N

atoms total. List terminated by end-of-file

card.

Take each atom in turn as a plane definer and

find all those atoms in the rest of the unit

cell which are within distance PDIST (typically

1A) of that plane.

Find the environment on the plane (i.e., the

motiv) for each crystallographically different

atom (as designated by the atom name, e.g., Ca 1,

Ca 2 , etc.) . Also count up the number of times

each atom appears in the slice and calculate the

motiv for one appearance only. Include in the

motiv only those atoms which are closer than

"CRITD", initiallized at 6A, and which are

within the factor "TESTDM", initiallized at 1.2 5,

of the closest distance for that type (e.g., Ca 1)

of atom. Calculate the vector set of the motiv,

remove any redundancies and write the results on

a scratch file. Optionally list the identity and

position of each member of each motiv and

optionally call HISTGM to generate histograms of

distances

29
CELL

Read the real cell, calculate the real cell volume

and the reciprocal cell.

CONSOL

Subroutine to incorporate a list of new atoms into

a list of old atoms and remove redundancies.

COPRIM

Make coordinates of vector coprime, considering a maximum highest common factor of 4.

CROSS

Vector cross product routine for triclinic cells.

The resulting vector is the same type as the input

vectors (co- or contra-variant) and is not normalized,

HISTGM

MAIN

Use vector lengths and charge product of atoms at

ends of vector to generate histograms based on the

interatomic distances between various kinds of atoms

Main program, sets up initial quantities, reads

initial input, reads vector sets, matches vector

sets of motiv, picks out atom-to-atom correspondence

between compounds A and B and saves up to the 2 best

inter-plane matches.

MI3

Matrix inversion subroutine for 3 by 3 matrix.

METRIC

Calculate metric tensor from cell dimensions.

MV

Matrix multiplying vector.

30
ORTH

OUTP

PLANEQ

PLOTLF

POS

POSITN

Calculate matrix to orthogonalize triclinic

cell and transform positions with coordinates

in fractions of the cell edges to coordinates

in orthogonal angstroms.

Orientation:

X parallel to (bXc)

Y parallel to cX(bXc)

Z parallel to c

List the cell, symmetry elements and atomic

positions read in.

Find Miller index representation of network

defined by two real cell lattice vectors.

NBS utility routine to plot on a fixed scale

on line printer. Maximum and minimum values

along x and y are user-specified.

Put atom back in unit cell if it is outside.

Consider one axis per call.

Position tape file to file specified usincr

code compatible with computer. Dummy routine

at present.

31
PROJT

Rotate atomic coordinates (in orthogonal angstroms)

of atoms on plane so that these atoms lie in the xy

plane. Used as a precursor to plotting.

QUOT

Compare A and B by calculating max (A,B)/min (A,B)

.

Set equal to 10 s if min (A,B) equals zero.

SYMIN

Read symmetry elements.

TRANSF

List atoms in plot with coordinates as fractions of

real cell axes, provide key between atom identity

and alphabetic symbol used in plot. Call PLOTLF to

generate plot on line printer.

TRORTH

Transform entire atom list from fractional

coordinates to orthogonal angstroms.

UNIT

Make vector into unit vector.

VECTS

VMV

Transform plane normal notation into lattice vector

notation, prepare for later d-spacing calculation

from vector notation.

Calculate bilinear or quadratic form, e.g., scaler

product of two vectors.

VNORM

Take two vectors with coordinates given as fractions

of cell edge and calculate the normal to these two

vectors in the same coordinates.

32
Hierarchy of subroutines in MATCH 2

MAIN

CELL SYMIN ATOMIN OUTP METRIC ATCELL ORTH

©

MI 3

POS

0-

TRORTH MV POSITN COPRIM

MV

VNORM PLANEQ ATPLAN

UNIT CROSS

VMV

ATVEC

QUOT

CONSOL

HISTGM

PROJT

1 MV TRANS

PLOTLF

output best matches

TRORTH

MV

33
Main variables in MATCH2 (initiallized values in

parentheses)

COMMON/IN/IN , NOUT

IN - Logical unit for card reader (5)

NOUT - Logical unit for lister (6)

COMMON/PLANE/PDIST, PDIST2, CRITD, TESTDM

PDIST - Distance out from central plane to

edges of slice through structure

(1.0A)

.

PDIST2 - Distance from central plane inside

which atoms will lie if the surface is

fairly smooth. Used as roughness gauge.

CRITD - Distance to go out from an atom in

collecting neighbors to use in

assembling the atom's motiv (6.0A).

TESTDM- Ratio of largest interatomic distance

to smallest interatomic allowed in

considering chemically equivalent

atoms for inclusion in motiv (1.25).

PARAMETER * statement for dimensioned variables

NDIM1 = 100, NDIM2 = 1000, NDIM3 = 50,

NDIM4 = 50, NDIM5 = 500, NDIM6 = 5000,

NDIM7 = 20, NDIM8 = 100.

Important dimensioned variables in MATCH2

Variables with 'A' as one of the first two or

last two characters are associated with compound

A, and their meanings are listed below. Variables

with B replacing A in a compound A name are

associated with compound B but have not been

listed below. Minor variables have also not been

listed.

* Univac 1100 series FORTRAN V program Reference Manual

UP4060 Revision II (1973) .

34
A(6) - Real cell of A.

AM (3, 3)- Real metric tensor for A.

ATA(3) - Orthogonal coordinates of 1,0,0 point in A.

ATB(3) - Orthogonal coordinates of 0,1,0 point.

ATC(3), BTA(3), BTB(3), BTC(3) - Similar quantities.

ATNEWA (NDIM5,5) - Used mainly in ATVEC to store the

atoms in each atomic motiv.

ATOMA (NDIM1,5) - Atom name, coordinates and charge

for A.

FA (NDIM2), IA (NDIM2), equivalenced arrays. First

member is perpendicular spacing in floating

point of atom in slice in A from plane in

center of slice. Second member is atom ordinal

number of plane definer, and third member is

atom ordinal number of atom whose spacing from

plane is first member. Repeats for each atom

in slice. Starts with plane defining atom.

KTA (NDIM4) - Number of occurrences in slice of

crystallographically different atoms

Given once for first occurrence of

atom in list.

MATCHA (NDIM4)

NTOTA (NDIM3) -

NVECTA (NDIM6)-

Number of matches each atom has been

assigned to during matching of atoms

in A with atoms in B via inter-motiv

figures merit.

Array containing the number of atoms

on each plane.

Array which is if vector in set B

presently unmatched by one in set A,

1 if vector has been matched. Used

in vector set comparison.

PLANEA(3) - Direction cosines of network plane in A.

PLANEC(3) - Lattice vector coordinates of normal to

network plane in A (initially intercepts if plane read in this notation

rather than as lattice vectors)

35
PLANEB(3), PLANED(3) - Similar quantities for B.

PLANEF (3)

QUANT (NDIM4,NDIM4)

RA(6)

RAM(3,3)

TA ( 3 , 3

)

SYMA(24,12)

TAINV (3,3)

- Rotation axis of twinning

operation considered.

- Inter-motiv figures of

merit calculated from misfit

of vector sets of atom motivs.

- Reciprocal cell for A.

- Reciprocal metric tensor for A,

- Orthogonalization matrix for

unit cell of A.

- Symmetry elements for A.

- Matrix to transform orthogonal

cell to actual cell.

VECA1(3) , VECA2(3), VECB1(3), VECB2(3) - Vectors

defining plane in compounds A

and B respectively.

VECTA (NDIM6,2) - Length of vector in motiv and

charge product of atoms at end

of vector.

36
Card input to MATCHl

(An example is given on page 48).

Card 1 Output unit, file, title (Read in main routine)

FORMAT (2I2,12A6)

Col. 2 Logical unit number for output tape.

Greater than 6 if output on tape

required.

4 Output file number if output tape

specified.

5-76 Title

Card 2 X-RAY-type* cell card for compound A, read in

subroutine CELL

FORMAT (A4,2X,A6, IX, 3F8.4, 3F9.5)

Col. 1-4 CELL

7-12 Compound Identification

14-21 a. cell dimension in angstroms

22-29 b. cell dimension

30-37 c. cell dimension

38-46 cos alpha or alpha in degrees

47-55 cos beta or beta in degrees

56-64 cos gamma or gamma in degrees

Card 3 X-RAY-type cell card for compound B, read in

subroutine CELL

Card 4 Boundary conditions, read in MATCHl

FORMAT (A6,4X,2I5,3F10.0)

Col. 1-6 CONTRL

15 NSYMA, point group number compound A.

See International Tables for X-ray

Crystallography, The Kynich Press

(Birmingham, England) Vol. 1, Table

3.3.2. Count as follows: 1 is #1,

1-bar is #2, 2 is #3, etc.

* The X-RAY system of crystallographic programs, J. M. Stewart,

G. J. Kruger, H. L. Ammon, C. Dickinson, and S. R. Hall,

Technical Report TR-192, University of Maryland,

College Park, Maryland, June 1972.

37
Col. 2 NSYMB, ditto for compound B.

21-30 TESTD, fractional "maximum mismatch allowed in judging

goodness of match of two

vectors, one from compound A

and one from compound B. Set

equal to 0.15 if left blank.

31-40 TESTA, maximum mismatch in

degrees of the angles in the

nets in compounds A and B.

Set equal to 10 if left blank.

41-50 DMAX. Maximum length of a vector used in net definition.

Set equal to 2 angstroms if

left blank.

51-55 NTEMP. Flag for future use.

56-60 NTWIN. Set to one for twin

calculation.

61-65 FMULT. Multiplier of TESTD

to provide cutoff for sort key,

If less than 1.0, program sets

it to 1.0.

38
Card Input to MATCH2

(An example is given on page 51)

Card 1 FORMAT (12A6) (Read in main routine)

Col. 1-72 Any title

Card 2 Input/output files, layer thickness, motiv radius,

mismatch criteria (read in main routine)

FORMAT (A6, 712, 411, IX, 5F5.0, IX, A4, F5.0, 515)

Col. 1-6 CONTRL

8 Logical number of input unit. Input can

be from output of MATCHl program or from

cards.

10 Actual file on this unit (zero leaves unit

untouched before data are read off it)

.

12 Output unit if information saved on file

as well as printed.

14 File number on output file.

16 Number of records to skip on input file

before beginning calculations (to allow

continuation of job curtailed by maximum

time, etc. )

.

18 Number of records to read before

stopping. Set equal to 5 if left blank.

2 0: No output of atoms in slices, keep

all redundancies (2 or 3 recommended)

.

1: Write out atoms in slice, keep all

redundancies

.

39
2; Write out atoms in slices,

remove all redundancies (i.e., same

number of atoms with particular alphanumeric labels in each slice)

.

3: Write out atoms in slices, remove

redundancies and subsets of other

slices

.

4: Write out checking information in

motiv-building subroutine ATVEC.

Col. 21 Positive number if summary information

punched on cards for later sorting.*

22 Positive number if Miller indices

rather than network vectors read in.

23 Positive number if histograms of

surface vectors to be generated.

(Check subroutine HISTGM for how this

is unraveled from list of vectors and

associated charge products.) If the

number is 2, the program outputs only

the histograms and does not calculate

any atom matches. It also punches the

histogram information on cards.

24 Positive number if this is a twinning

calculation, program then calculates

the general rotation between the matching networks.

26-30 Half of slice thickness desired, i.e.,

distance out from plane defining the

center of the compositional plane in the

crystal to the edge of the slice

common to the structures involved in

epitaxy. Set equal to 1.0 angstroms

if left blank.

31-35 Inner portion of slice thickness. Set

equal to 0.75 angstroms if left blank.

Used to count up atoms in " fringe areas"

of slice and to measure how uneven

that surface of the crystal would be.

* See p. 46 for description of card output.

40
Col. 36-40 The radius inside which all

members of an atom motiv must

fall. Set equal to 6.0 angstroms

if left blank.

41-45 The maximum divergence of distances

from identical atoms in the environment to a motiv defining atom for

both atoms in the environment to be

included in the motiv. Set equal to

1.25 if left blank. This is a divergence factor, i.e., the maximum

distance allowed is 1.2 5 times the

minimum distance found.

46-50 The maximum fractional mismatch in

length allowed between vectors in

different motivs if these vectors are

to be considered as a possible match

(the best match of all is the one

actually taken, but the number of

vectors left unmatched is an important

parameter in the program) . Set equal

to 0.15 if left blank.

52-55 ID for any punched card output.

56-60 Minimum merit figure (blank = 4.0)

for output of surface atom coordinates

in numerical form.

61-65 Positive number for lister plots of

surface atoms.

41
Col. 66-70 Positive number if output of motiv

contents required. (This is

usually a good idea so you can see

what is being perpetrated in your

name or at least your account. It

also allows you to assess the

efficacy of your chosen motiv radius.)

71-7 5 Initial slice number for compound A

(for continuation of previous calculation) . May be left blank.

7 6-80 Initial slice number for compound B.

May be left blank.

Card 3 Cell card in X-RAY- type* format, read in subroutine

CELL

FORMAT (A4,2X,A6, 1X3F8.4, 3F9. 5)

Col. 1-4 CELL

7-12 Compound identification

14-21 cell dimension

22-29 cell dimension

30-37 cell dimension

38-46 cos alpha or alpha in degrees

47-55 cos beta or beta in degrees

56-64 cos gamma or gamma in degrees

* See p. 37.

42
Card 4 Symmetry elements in ORTEP+ format, (read in subroutine

SYMIN)

.

FORMAT (1X,F14.10,3F3.0,2 (F15.10,3F3.0) ) with

sentinel in Column 1 on last card.

Col. 2-15 Translation for 1st atomic

coordinate.

16-18 X

19-21 Y

22-24 Z

1 for X, 2 for Y, 3 for Z used in

first atomic coordinate with sign.

25-39 Translation for second atomic

coordinate.

40-42 X

43-45 Y

46-48 Z

1 for X, 2 for Y, 3 for Z used in

second atomic coordinate with sign.

49-6 3 Translation for third atomic

coordinate.

64-66 X

67-69 Y

70-72 Z

1 for X, 2 for Y, 3 for Z used in

third atomic coordinate with sign.

+ ORTEP: A FORTRAN thermal-ellipsoid plot program for

crystal structure illustrations. C. K. Johnson, ORNL-3794

(Revised) , Oak Ridge National Laboratory, Oak Ridge,

Tenn., June 1965.

43
Card 5 Atoms in X-RAY-type * format (read in subroutine ATOMIN)

FORMAT (7XA6, 3F8.5, F5.2)

Col. 8-9 Atom type, e.g., Ca. Make sure

each atom has a unique designation

such as Ca 1, Ca 2, etc.

12-13 Number of atom.

14-21 X fractional coordinate.

22-29 Y fractional coordinate.

30-37 Z fractional coordinate.

38-42 Charge

After last atom card include appropriate end

of file card as end of atom list signal.

Repeat cards 3, 4 and 5 for second compound.

* See p. 37.

44
Card 6 Vector indices or Miller indices if any of this

information read from cards (read in MAIN)

.

FORMAT (314) for vectors or (3F4.0) for intercepts

Col. 4 u for first vector in first compound

or Miller index h.

8 v for first vector in first compound

or k.

12 w for first vector in first compound

or I Repeat for second vector in first compound, and

then for first and second vectors in second

compound

.

Miller index notation requires one card per plane

per compound.

Input always terminated by EOF card so program

goes to prepare summary of the results for the

current run.

45
Optional Summary Output On Punched Cards

Summary information on punched cards may be obtained

by placing any positive number in Column 21 of the MATCH2 control

card, card 2.

Cards from runs through different parts of the intermediate file of metric matches as produced by MATCHl or from

several cases may be combined and sorted for an overall

view. The cards contain the following information:

Col. 1-3 ordinal number of A slice

4-6 ordinal number of B slice

7-9 ordinal number of network match

10-12 ordinal of this card in this output

13-18 overall merit figure

19-23 fractional mismatch of first vector in A and B

24-28 fractional mismatch of second vector in A and B

29-33 fractional mismatch of angles in A and B

34-35 u for first vector in A

36-37 v for first vector in A

38-39 w for first vector in A

40-41 u for second vector in A

42-43 v for second vector in A

44-45 w for second vector in A

46-47 u for first vector in B

48-49 v for first vector in B

50-51 w for first vector in B

52-53 u for second vector in B

54-55 v for second vector in B

56-57 w for second vector in B

58-64 estimate of dislocation density

65-66 h of Miller notation of plane in A

67-68 k of Miller notation of plane in A

69-70 I of Miller notation of plane in A

71-72 h of Miller notation of plane in B

73-74 k of Miller notation of plane in B

75-76 % of Miller notation of plane in B

77-80 card ID.

46
Computer Requirements

MATCHl : At least 28,000x0 words on an intermediate

scratch file (tape or mass storage such as

disc or drum)

.

9200io words of core for instructions and

27,700io for data. Optional output on

binary file for transfer to MATCH2

.

MATCH2 : About 3 5,000io words on each of two intermediate scratch files.

9,80010 words of core for the instructions

and 49,050io for data. Optional output of

good matches on binary file.

Both programs pack vector indices or ordinal numbers of

related array elements into one word. A computer word

length of at least 36 bits is necessary for this procedure to be reliable.

47
MATCH 1 . TESTDECK

8 TEST DECK FOR MATCH1

CELL HA 9.4151 9.4151 6.881C -.50000

CELL OCP 19.661 9.509 6.822 90.17 92.34 108.41

CONTRL 23 2

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50
MATCH2 . TESTDK

HA/OCP MATCH LAYERS

CONTRL 5 c1 3 000 1 1 CELL HA 9 . 4151 9 .'*151 6.8810 - .500C0

.00 -1 .00 -1 .00 -1

.00 1 .00 .00 r, 1 .00 -1 .00 ^1 .50 1 .00 1 .00 .50 C -1

.00 -1 .CO 1 -1 .00 1 .00 1 .CO -1 .50 1 .00 1 .00 -1 .00 -1

.00 -1 .00 1 -1 . 50 -1

.00 -1 1 c .00 -1 .00 1 .00 1-1 .00 1 .00 -1

.CO 1 -1 .CO 1 .50 1

1 .00 -1 1 .00 -1 .50 -1

ATOM CA 1 2333 .6667 .C014 2.

ATOM CA 2 2467 .9933 .2500 2.

ATOM P • 3985 .3684 .2500 -].

ffiEOF CELL OCP 19.661 9 .509 6.822 90. L7 92.34 108.41

0.00 1 0.00 c 1 0.00 1

1 o.co -1 0.00 -1 o.co -1

ATOM CA 1 .0720 .794 1 .0055 2.

ATOM CA 2 .0706 .8058 .50 37 2.

ATOM CA 3 .3665 .0310 .0234 2.

ATOM CA 4 .3631 .8353 .5339 2.

ATOM CA 5 .1073 .4499 .7626 2.

ATOM CA 6 .2012 .6601 .27 57 2.

ATOM CA 7 .2180 .1280 .7724 2.

ATOM CA 8 .1 150 .2014 .2619 2.

ATOM P 9 .0 430 .4468 .2586 -1.

ATOM P 10 .2075 .7726 .7768 -1.

ATOM P 1 1 .2352 .0107 .2764 -1.

ATOM P 12 .0 564 .0714 .7584 -1.

ATOM P 13 .4910 .8651 .2662 -1.

ATOM P 14 .2992 .4745 .2495 -1.

ffl EOF

1 1

1

1 3E0F

51
SISSSS THIS IS THE VERSION OF 1975 $S$$$$

3ACD.P M.TESTCK

HA/OCP MATCH LAYERS

Tt-E TOTAL THICKNESS OF THE SLICE IS TWICE 1.00 ANGSTROMS

ATOMS MORE THAN .75 ANGSTROMS FROM THE ATOM DEFINING THE CENTRAL PLANE

WILL BE COUNTED UP AS FRINGE ATOMS TO ASCERTAIN THE ROUGHNESS OF THE PROPOSED SURFACE

THE MOTIV RACIUS IS 6.00 ANGSTROMS

A MAXIMUM OF 5 RECORDS WILL BE READ FROM UNIT 5, FILE 0. RECORDS WILL BE SKIPPED ON THE INPUT FILE BEFORE READ- IN BEGINS

INPUT IN LATTICE VECTCR NOTATION, TWO VECTORS FOR EACH COMPOUND

MATCHES WITH MERIT FIGURES ABOVE 4.00 WILL GENERATE LIST OF ATOMS ON SURFACE

MAPS WILL ALSO BE DRAWN FOR THESE SLICES

CONTENTS OF ATOM MOTIVS WILL BE WRITTEN OUT

T»-E BEGINNING SLICES ARE 1 FOR COMPOUND A AND 1 FOR COMPOUND B ATOMS WITHIN 1.25 OF NEAREST DISTANCE FOR ATOMS OF THAT TYPE IN A MOTIV WILL BE INCLUDED IN MOTIV

MOTIV VECTORS MUST MATCH WITHIN .15*100 PERCENT

CRYSTAL DATA FOR 1ST COMPOUND

REAL CELL IS 9.41510 9.41510 6.88100 .00000 .CCCC0 -.50000

RECIPROCAL CELL IS .12264 .12264 .14533 .00000 .0C000 .50C0C

SYMMETRY ELEMENTS ARE

Tl SI 1 SI 2 S13 T2 S21 S22 S23 T3 S31 S3

.000 -1 0. 0. .OCO 0. - 1j 0. .000 0. .000 i 0. 0. .000 0. 0. .000 0. .000 - 1 0. 0. .oco 0. -1. 0. .50 0. .000 1 . 0. .000 0. .500 0. .coo 0. -1

.

0. .000 , -1 0. .000 0. .000 1

.

0. .000 - 1. 0. .500 0. .000 1 . 0. .000 -1 0. .000 0. .000 , -1 . 0. .000 -li 0. .500 0. .000 -1 1

.

0. .ooc -1 0. .000 0. .oco 1

.

. -1

.

0. • oco 0. .coo 0. .ceo 1 . -1 . 0. .000 0. 0. .500 0. .000 -1 1 . 0. .000 -1 0. .500 0. ATOM LIS1r is ATO^4 X Y z CHARGE

CA 1 33330 .66670 .cc) 140 2 .00 CA 2 24670 .993 30 . 2'.500C 2 .00

P 39850 .36840 .2'.5000-1 .00

S2 S3 3

CRYSTAL DATA FOR 2ND COMPOUND

REAL CELL IS 19.66100 9.509CO 6.82200 -.0C297 -.04083 -.31582

RECIPROCAL CELL IS .05366 .11085 .14673 .01674 .04402 .31621

SYMMETRY ELEMENTS ARE

Tl Sll S12 S13 T2 S21 S22 S23

0. 0. .OCO * C. 1. 0.

. 0. .000 0. -1 . 0. .000 1 .

• OCO -1 . ATOM LIST IS ATOM

T3 S31 S32 S33

.000 0. 0. 1

.

•OCO 0. 0. -1

.

CHARGE

52
CA 1 .07200 .79410 .CC550 2.00

CA 2 .07060 .80580 .50370 2.00

CA 3 .36650 .031CO .02340 2.00

CA 4 .36310 .83530 .53390 2.00

CA 5 .10730 .44990 .76260 2.00

CA 6 .20120 .66010 .27570 2.00

CA 7 .21800 .12800 .77240 2.00

CA 8 .11500 .20140 .26190 2.00

P 9 .04300 .44680 .25860-1.00

P 10 .20750 .77260 .77680-1.00

P 11 .23520 .01070 .27640-1.00

P 12 .05640 .07140 .75840-1.00

P 13 .49100 .86510 .26620-1.00

P 14 .29920 .47450 .24950^1.00

16 UNIQUE ATOMS IN UNIT CELL

28 UNIQUE ATOMS IN UNIT CELL

53
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COMPOUND A MOTIVS FOLLOW

MCTIV(ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHARGE)

CA 1 £.4361 -.0005 6.8714 2.00

CA 1 5.4361 -.0005 3.4501 2.00

CA 1 5.4361 -.0005 10.3311 2.00

CA 2 2.0662 1.1298 5.1607 2.00

CA 2 8.2083 -2.3533 8.6012 2. CO

P 3.2493 1.5926 8. 6012-1. CO

P 6.3591 -1.0973 5.1607-1.00

MCT IV (ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE

)

CA 2 .0546 2.3542 1.7202 2.00

CA 1 -2.7176 4.7071 -.0096 2. CO

CA 1 -2.7176 4.7071 3.4501 2.00

CA 2 2.0662 1.1298 -1.7203 2.00

CA 2 2.0662 1.1298 5.1607 2.00

P 3.2493 1.5926 1.7202-1.00

CA 2 IS A FRINGE ATOM IN SLICE BASED ON P

CA 2 IS A FRINGE ATOM IN SLICE BASED ON P

MCTIV< ATOM. COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE

)

P 3.2493 1.5926 1.7202-1.00

CA 1 5.4361 -.0005 -.0096 2.00

CA 1 5.4361 -.0005 3.4501 2.00

CA 2 .0546 2.3542 1.7202 2.00

CA 2 2.0662 1.1298 -1.7203 2.00

CA 2 2.0662 1.1298 5.1607 2.00

P .2454 3.6102 -1.7203-1. CO

P .2454 3.6102 5.1607-1.00

4 ATOMS IN FRINGE PART OF SLICE BASED ON ATOM P

4 ATOMS WERE IN FRINGE AREAS OF PLANES

COMPOUND B MCTIVS FOLLOW

CA 8 IS A FRINGE ATOM IN SLICE BASED ON CA 1 MOTIV(ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE

)

CA 1 1.3418 7.103e -.0427 2.00

CA 2 1.3157 7.2238 3.3568 2.00

CA 2 1.3157 7.2238 -3.4652 2.00

CA 5 1.9997 3.6116 -1.7184 2.00

CA 8 2.1432 10.7097 1.6604 2.00

P 9 .8014 3.9815 1.7170-1.00

P 12 1.0511 9.8376 -1.7237-1.00

1 ATOMS IN FRINGE PART OF SLICE BASED ON ATOM CA

CA 8 IS A FRINGE ATOM IN SLICE BASED ON CA 2

MCTIV( ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHAR GE

)

CA 2 1.3157 7.2238 3.3568 2.00

CA 1 1.3418 7.1038 -.0427 2.00

CA 1 1.3418 7.1038 6.7793 2. CO

CA 5 1.9997 3.6116 5.1036 2. CO

CA 8 2.1432 10.7097 1.6604 2.00

P 9 .8014 3.9815 1.7170-1.00

P 12 1.0511 9.8376 5.0983-1.00

1 ATOMS IN FRINGE PART OF SLICE BASED ON ATOM CA

MOTIV(ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE

)

CA 5 1.9997 3.6116 5.1036 2.00

CA 1 1.3418 7.1C38 6.7793 2. CO

CA 2 1.3157 7.2238 3.3568 2.00

CA 8 2.1432 1.2008 8.5107 2.00

55
CA 8 2.1432 1.2008 1.6887 2.00

P 12 1.0511 .3286 5.1265-1.00

MCTIV< ATOM. CCORDI NATES IN ORTHOGONAL ANGSTROMS, CH ARGE

)

CA 8 2.1432 1.2008 1.6887 2.00

CA 1 1.3418 -2.4051 -.0145 2.00

CA 2 1.3157 -2.2852 3.3850 2. CO

CA 5 1.9997 3.6116 -1.7184 2. CO

CA 5 1.9997 3.6116 5.1036 2.00

MOTIV< ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHARGE)

P 9 .8014 3.9815 1.7170-1.00

CA 1 1.3418 7.1038 -.0427 2.00

CA 2 1.3157 7.2238 3.3568 2.00

P 12 1.0511 .3286 5.1265-1.00

P 12 1.0511 .3286 -1.6955-1.00

CA 5 IS A FRINGE ATOM IN SLICE BASED ON P 12

MCTIVC ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHARGE

)

P 12 1.0511 .3286 5.1265-1.00

CA 1 1.3418 -2.4051 6.8075 2.00

CA 2 1.3157 -2.2852 3.3850 2.00

CA 5 1.9997 3.6116 5.1036 2.00

P 9 .8014 3.9815 1.7170-1.00

P 9 .8014 3.9815 8.5390-1.00

1 ATOMS IN FRINGE PART OF SLICE BASED ON ATOM P 12

MCTIV(ATCM, COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE)

CA 3 6.8303 -1.9818 -.1355 2.00

CA 4 6.7670 -3.8216 -3.4666 2.00

CA 4 6.7670 -3.8216 3.3554 2.00

MCTIV( ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS .CHARGE

)

CA 4 6.7670 5.6874 3.3272 2.00

CA 3 6.8303 7.5271 6.6583 2.00

CA 3 6.8303 7.5271 -.1637 2.00

MCTlV( ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS , CHARGE)

CA 6 3.7497 5.0271 1.7007 2.00

CA 7 4.0628 9.3719 5.0624 2.00

CA 7 4.0628 9.3719 -1.7596 2. CO

P 10 3.8671 €.0577 5.1109-1.00

P 10 3.8671 6.0577 -1.7111-1.00

P 11 4.3633 8.1497 1.6682-1.00

MOT IV( ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHARGE)

CA 7 4.0628 -.1370 5.0907 2.00

CA 6 3.7497 -4.4819 1.7289 2. CO

CA 6 3.7497 -4.4819 8.5509 2.00

F 10 3.8671 -3.4513 5.1392-1.00

P 11 4.3833 -1.3593 1.6965-1.00

F 11 4.3833 -1.3593 8.5135-1.00

MCTIV{ ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS . CHARGE

)

P 10 3.8671 6.0577 5.1109-1.00

CA 6 3.7497 5.0271 1.7007 2.00

CA 6 3.7497 5.0271 8.5227 2. CO

CA 7 4.0628 9.3719 5. 0624 2.00

F 11 4.3E33 8.1497 8.4902-1.00

P 11 4.3833 8.1497 1.6682-1.00

MCT IV( ATOM, COORDINATES IN ORTHOGONAL ANGSTROMS, CHARGE

)

P 11 4.3833 -1.3593 1.6965-1.00

CA 6 3.7497 -4.4819 1.7289 2.00

CA 7 4.0628 -.1370 5.C907 2.00

CA 7 4.0628 -.1370 -1.7313 2 .0 P 10 3.8671 -3.4513 -1.6828-1.00

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65
4 BEST MATCHES BETWEEN COMPOUNDS A AND B

VALUES SCALED TO UNIT AREA FOR EACH MATCH PLANE

TCTAL NUMBER OF MATCHES FCR COMPOUNDS IS 4

SLICE A SLICE B NETWORK

PAIR

FIT

1 1 8.32

1 1 4.40

1 1 2.03

1 1 1 .60

CB COMPLETED

66
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RKPORT NO. NBS TN-893

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Computer Programs for Structural Chemistry: MATCH1

and MATCH2, FORTRAN Programs To Predict and Evaluate

Mutual Orientation of Polycrystals

5. Publication Date January 197 6

6. Performing Organization Code

7. AUTHOR(S)

Brian Dickens and LeRoy W. Schroeder*

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

If 3110585

11- Contract/Grant No. DE00572

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) * American Dental Association

Health Foundation Research Unit

National Bureau of Standards

Washington, P. C. 202 34

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less [actual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) Two computer programs to evaluate possible structural matches for use in epitaxy

and twinning studies have been written in FORTRAN V for the UNIVAC 1108. They

should be readily convertible to other comparable computers.

The first program, MATCH1, requires knowledge of the unit cell parameters. It obtains all matching networks in two unit cell lattices and sorts the matches into an

order of probable epitaxy (or twinning if the two unit cells are the same) based solely on criteria of dimensional mismatch and network area. The second program, MATCH2, requires knowledge of positional parameters of the

atoms in the crystal structures. It calculates the degree of structural fit for slices

supplied from visual inspection of the crystal structures or as matching networks

determined by MATCH1.

The slice comparison is carried out in terms of the vector set of the environment of

each atom in each slice. The procedure is not valid when the twin operation is a rotation about the normal of the compositional plane or a reflection in a mirror

parallel to this plane. The atomic patterns comprising the slices are matched piece

by piece in MATCH2. A third program, MATCH3, will match whole patterns once

three fiducial atoms in each pattern have been picked from the output of MATCH2

and will handle all the twinning cases. MATCH3 will be described in a future

National Bureau of Standards Technical Note.

Cont.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

computer calculations; crystal structure; epitaxy; lattice misfit;

pattern recognition; twinning

18. AVAILABILITY [jf Unlimited

_ For Official Distribution. Do Not Release to NTIS

|X ! Order From Sup. of Doc, U.S. Government Printine Office Washington, D.C. 20402, SD Cat. No. C13. 46:893

_! Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

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21. NO. OF PAGES

71

22. Price $1.30

USCOMM-DC 29042-P74
ABSTRACT- - continued

This Technical Note explains the procedure in MATCH1 and MATCH2

and includes listings of the programs and examples of results. The

programs are available on magnetic tape or card decks obtainable

from the authors.
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