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A computer program to transform planar indices
Metallography
279
SHORT C O M M U N I C A T I O N S
A Computer Program to Transform Planar Indices
MIN CHUNG JON AND R. J. DE ANGELIS~ University of Kentucky, Lexington, Kentucky
In determining solutions to electron diffraction patterns containing two phases which are crystallographically related, it is extremely helpful to know the transformation relating the planar indices of the two phases. It is then possible to obtain solutions to complex diffraction patterns once the orientation of a single component of the material has been determined. Transformation matrices have been determined for a number of solid-state diffusionless and diffusion-controlled transformations. Silcock et al. reported part of the transformation matrix for the beta-to-omega transformation in titanium? Manual computations of index transforms employing these matrices is laborious and time-consuming, especially when there are multiple habit planes as is the case when the parent phase is cubic. The purpose of this note is to present a simple computer program which transforms planar indices; the beta-to-omega transformation in titanium is used as an example in this presentation because of the current interest in this transformation. ~'3 The program, which is written in BASIC language, performs four transformations on the parent indices. I f a larger or smaller number of transforms are desired, this can be accomplished with very minor program changes. BASIC has the advantages that it requires very little knowledge of computer programming, it has "built in" matrix operators, and it is a common teleprocessing language The program also provides the user with the flexibility to select the number of indices he wishes to transform. a Min Chung Jon and R. J. De Angelis are Graduate Assistant and Associate Professor, respectively, in the Department of Metallurgical Engineering and Materials Science University of Kentucky.
Metallography, 4 (1971) 279-282 [Short Communication] Copyright © 1971 by American Elsevier Publishing Company, Inc.
280
M. C..7on and R. :7. De Angelis
The program below performs the computation
L LJ:r-
LA31 A3,,
AM
_.
where [HLK]e and [HKL]T are the Miller indices of the parent and transformed phases, A is the transformation matrix, and M is a constant. The elements Aii are easily determined from the crystallography of the transformation. 10 DIM A(3, 3), B(1, 3), C(1, 3), W(3, 3), X(3, 3), Y(3, 3), Z(3, 3) 20 GOSUB 270 30 MAT W = A 40 GOSUB 270 50 MAT X = A 60 GOSUB 270 70 MAT Y = A 80 GOSUB 270 90 MAT Z = A 100 FOR I = l TO 3 110 READ B(1, I) 120 NEXT I 130 PRINT ' H K L ' 140 PRINT USING 330, B(1, 1), B(1, 2), B(1, 3) 150 PRINT ' ', ' H K L ' 160 MAT C ~ B * W 170 GOSUB 350 180 MAT C=B*X 190 GOSUB 350 200 MAT C=B*Y 210 GOSUB 350 220 MAT C=B*Z 230 GOSUB 350 240 GO TO 100 250 STOP 260 END 270 MAT READ A 280 DATA 2, 0, 1, --2, 2, 1, 0, --2, 1 290 DATA 2, 0, 1, --2, 2, 1, 0, 2, --1 300 DATA 2, 0, 1, 2, --2, --1, 0, --2, 1 310 DATA 2, 0, --1, --2, 2, 1, 0, --2, 1 320 RETURN 330 IMAGE
281
P l a n a r Indices
340 I M A G E 350 P R I N T U S I N G 340, C(1, 1), C(1, 2), C(1, 3)
360 R E T U R N 370 D A T A 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 0, --2, 0, 0 380 D A T A 1, 1, 2, 1, 1, --2, 2, 2, 2, 2, --2, --2, 2 In this program the data given in statements 280, 290, 300, and 310 are the four sets of matrix elements describing the index relations in the body-centered cubic (beta) titanium to hexagonal (omega) transformation. This transformation has been shown (1, 4) to occur such that (111)~ is parallel to [0001]~, and (ll0)p is parallel to (1120),~. Thus there are four possible omega orientations, one TABLE I OUTPUT OF PROGRAM PRESENTED. PARENT BETA TITANIUM INDICES ARE INDICATED ON [,EFT, AND THE OMEGA TITANIUM 1NDICES ARE ON THE RIGHT.
HKL
H KL
H
K
1, 1, 0
2, 0, 0
1,
1,--2
H
KL
O, 2, O, 2, 4,--2, --4, 2,
H
2 2 0 0
HKL
4, 4, 4, - 4, H
1, 0, 1
--2,
L
H
2 2 2 2 H
K
L
2,
2,
2
H K L
2 0 2 0
--4, --4, --4, 4, H
K L
O, 1, 1
1,
1, 2
HKL
0, 0, 0, 8, 8, --8, --8, 0, H
K
--2,--2,
0,--2, 0, 6, 4,--6, 4,--2,
L
H K L
0,--2 0,--2 0,--2 0, 2
H K L
2 0 0 2
K
O, 6, 0 0,--2, 4 4, 2 , - 2 --4, 6, --2
KL
HKL
--2, 0, --2, 4, 2,--4, --2, 0
O, O, O, O,
0, 0
HKL
2,--2, 2, 2, 2, --2, --2,--2,
K
L
6 2 2 2
L
2 H K L
4 0 2 2
0,--8,--2 0, 0,--6 --8, 0, 2 8,--8, 2
related to each of the four (111) directions in the beta phase. The complete list of indices to be transformed is placed in data statements starting at 370 using as many statements as required with numbers greater than 370. The output of the program as presented is given in Table I. Here the index to be transformed is on the left and the four possible resulting indices are on the right. The fourth hexagonal index (I : - - H - - K ) is not computed. Also, the value of
M. C. Jon and R. J. De Angelis
282
M was taken as unity, not one-half, therefore all indices are m u l t i p l i e d by a factor of two. T a b l e I I gives elements of the m a t r i x A w h i c h apply to the t i t a n i u m b e t a - t o o m e g a t r a n s f o r m a t i o n and b o d y - c e n t e r e d and f a c e - c e n t e r e d cubic twinning. T o apply this p r o g r a m to any solid-state t r a n s f o r m a t i o n only requires changing the data in statements 280 to 310 (that is, the elements of m a t r i x A). T A B L E II ELEMENTS OF THE TRANSFORMATION MATRIX
Transformation Beta Ti to Omega Ti*
Body-centered cubic twinning (These data can also be used for facecentered cubic twinning by considering the directions)
Crystallographic relation
Mt
Alt A21 Aat A~ A,~ A32 A~3 A~a Aaa
[111]M/[0001]co [lli]#//[0001]~ [lil]#/][0001]o~ [ill]#]/[0001]o~
½ ½ ½ ½
2 2 2 --2
0 1--2 2 1 0 1 --2 2 1 0 1 2 --2 --1 0 --1 --2 2 1
0--2 1 0 2 --1 0 --2 1 0 2 1
½
-- 1
2
2
2 -- 1
2
~
-- 1
2
2
2 -- 1 -- 2 -- 2 -- 2 -- 1
~
-- 1 -- 2
½
--1 --2 --2 --2 --1
[111 ] twin direction [1T1] twin direction [1il] twin direction [i11] twin direction
2
2 -- 2 -- 1 -- 2
2 -- 1
2 -- 2 -- 1
2 --2
2 --1
* Here the beta unit cell is taken to have a lattice parameter three times the normal body-centered cell (after reference 1). t M values are not included in the computation example presented. T h i s p r o g r a m has b e e n r u n on a I B M 360 c o m p u t e r f r o m a Conversational P r o g r a m m i n g T i m e Sharing S y s t e m originating f r o m a I B M 2741 T e l e p r o c e s s ing unit. T h e data in T a b l e I w e r e c o m p u t e d in 0.05 seconds of active c o m p u t e r time. Research was s p o n s o r e d by the Office of A e r o s p a c e Research, U n i t e d States Air Force, u n d e r C o n t r a c t F 33615-69-C-1027.
References 1. J. M. Silcock, M. H. Davies and H. K. Hardy, Monograph and Report Series No. 18, Institute of Metals, London, 1956, p. 93. 2. S. L. Sass, Acta Met., 17 (1969) 813. 3. D. De Fontaine, Acta Met., 18 (1970) 275. 4. M. J. Blackburn and J. C. Williams, Trans. AIME, 242 (1968) 2461.
Accepted November 27, 1970
279
SHORT C O M M U N I C A T I O N S
A Computer Program to Transform Planar Indices
MIN CHUNG JON AND R. J. DE ANGELIS~ University of Kentucky, Lexington, Kentucky
In determining solutions to electron diffraction patterns containing two phases which are crystallographically related, it is extremely helpful to know the transformation relating the planar indices of the two phases. It is then possible to obtain solutions to complex diffraction patterns once the orientation of a single component of the material has been determined. Transformation matrices have been determined for a number of solid-state diffusionless and diffusion-controlled transformations. Silcock et al. reported part of the transformation matrix for the beta-to-omega transformation in titanium? Manual computations of index transforms employing these matrices is laborious and time-consuming, especially when there are multiple habit planes as is the case when the parent phase is cubic. The purpose of this note is to present a simple computer program which transforms planar indices; the beta-to-omega transformation in titanium is used as an example in this presentation because of the current interest in this transformation. ~'3 The program, which is written in BASIC language, performs four transformations on the parent indices. I f a larger or smaller number of transforms are desired, this can be accomplished with very minor program changes. BASIC has the advantages that it requires very little knowledge of computer programming, it has "built in" matrix operators, and it is a common teleprocessing language The program also provides the user with the flexibility to select the number of indices he wishes to transform. a Min Chung Jon and R. J. De Angelis are Graduate Assistant and Associate Professor, respectively, in the Department of Metallurgical Engineering and Materials Science University of Kentucky.
Metallography, 4 (1971) 279-282 [Short Communication] Copyright © 1971 by American Elsevier Publishing Company, Inc.
280
M. C..7on and R. :7. De Angelis
The program below performs the computation
L LJ:r-
LA31 A3,,
AM
_.
where [HLK]e and [HKL]T are the Miller indices of the parent and transformed phases, A is the transformation matrix, and M is a constant. The elements Aii are easily determined from the crystallography of the transformation. 10 DIM A(3, 3), B(1, 3), C(1, 3), W(3, 3), X(3, 3), Y(3, 3), Z(3, 3) 20 GOSUB 270 30 MAT W = A 40 GOSUB 270 50 MAT X = A 60 GOSUB 270 70 MAT Y = A 80 GOSUB 270 90 MAT Z = A 100 FOR I = l TO 3 110 READ B(1, I) 120 NEXT I 130 PRINT ' H K L ' 140 PRINT USING 330, B(1, 1), B(1, 2), B(1, 3) 150 PRINT ' ', ' H K L ' 160 MAT C ~ B * W 170 GOSUB 350 180 MAT C=B*X 190 GOSUB 350 200 MAT C=B*Y 210 GOSUB 350 220 MAT C=B*Z 230 GOSUB 350 240 GO TO 100 250 STOP 260 END 270 MAT READ A 280 DATA 2, 0, 1, --2, 2, 1, 0, --2, 1 290 DATA 2, 0, 1, --2, 2, 1, 0, 2, --1 300 DATA 2, 0, 1, 2, --2, --1, 0, --2, 1 310 DATA 2, 0, --1, --2, 2, 1, 0, --2, 1 320 RETURN 330 IMAGE
281
P l a n a r Indices
340 I M A G E 350 P R I N T U S I N G 340, C(1, 1), C(1, 2), C(1, 3)
360 R E T U R N 370 D A T A 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 0, --2, 0, 0 380 D A T A 1, 1, 2, 1, 1, --2, 2, 2, 2, 2, --2, --2, 2 In this program the data given in statements 280, 290, 300, and 310 are the four sets of matrix elements describing the index relations in the body-centered cubic (beta) titanium to hexagonal (omega) transformation. This transformation has been shown (1, 4) to occur such that (111)~ is parallel to [0001]~, and (ll0)p is parallel to (1120),~. Thus there are four possible omega orientations, one TABLE I OUTPUT OF PROGRAM PRESENTED. PARENT BETA TITANIUM INDICES ARE INDICATED ON [,EFT, AND THE OMEGA TITANIUM 1NDICES ARE ON THE RIGHT.
HKL
H KL
H
K
1, 1, 0
2, 0, 0
1,
1,--2
H
KL
O, 2, O, 2, 4,--2, --4, 2,
H
2 2 0 0
HKL
4, 4, 4, - 4, H
1, 0, 1
--2,
L
H
2 2 2 2 H
K
L
2,
2,
2
H K L
2 0 2 0
--4, --4, --4, 4, H
K L
O, 1, 1
1,
1, 2
HKL
0, 0, 0, 8, 8, --8, --8, 0, H
K
--2,--2,
0,--2, 0, 6, 4,--6, 4,--2,
L
H K L
0,--2 0,--2 0,--2 0, 2
H K L
2 0 0 2
K
O, 6, 0 0,--2, 4 4, 2 , - 2 --4, 6, --2
KL
HKL
--2, 0, --2, 4, 2,--4, --2, 0
O, O, O, O,
0, 0
HKL
2,--2, 2, 2, 2, --2, --2,--2,
K
L
6 2 2 2
L
2 H K L
4 0 2 2
0,--8,--2 0, 0,--6 --8, 0, 2 8,--8, 2
related to each of the four (111) directions in the beta phase. The complete list of indices to be transformed is placed in data statements starting at 370 using as many statements as required with numbers greater than 370. The output of the program as presented is given in Table I. Here the index to be transformed is on the left and the four possible resulting indices are on the right. The fourth hexagonal index (I : - - H - - K ) is not computed. Also, the value of
M. C. Jon and R. J. De Angelis
282
M was taken as unity, not one-half, therefore all indices are m u l t i p l i e d by a factor of two. T a b l e I I gives elements of the m a t r i x A w h i c h apply to the t i t a n i u m b e t a - t o o m e g a t r a n s f o r m a t i o n and b o d y - c e n t e r e d and f a c e - c e n t e r e d cubic twinning. T o apply this p r o g r a m to any solid-state t r a n s f o r m a t i o n only requires changing the data in statements 280 to 310 (that is, the elements of m a t r i x A). T A B L E II ELEMENTS OF THE TRANSFORMATION MATRIX
Transformation Beta Ti to Omega Ti*
Body-centered cubic twinning (These data can also be used for facecentered cubic twinning by considering the directions)
Crystallographic relation
Mt
Alt A21 Aat A~ A,~ A32 A~3 A~a Aaa
[111]M/[0001]co [lli]#//[0001]~ [lil]#/][0001]o~ [ill]#]/[0001]o~
½ ½ ½ ½
2 2 2 --2
0 1--2 2 1 0 1 --2 2 1 0 1 2 --2 --1 0 --1 --2 2 1
0--2 1 0 2 --1 0 --2 1 0 2 1
½
-- 1
2
2
2 -- 1
2
~
-- 1
2
2
2 -- 1 -- 2 -- 2 -- 2 -- 1
~
-- 1 -- 2
½
--1 --2 --2 --2 --1
[111 ] twin direction [1T1] twin direction [1il] twin direction [i11] twin direction
2
2 -- 2 -- 1 -- 2
2 -- 1
2 -- 2 -- 1
2 --2
2 --1
* Here the beta unit cell is taken to have a lattice parameter three times the normal body-centered cell (after reference 1). t M values are not included in the computation example presented. T h i s p r o g r a m has b e e n r u n on a I B M 360 c o m p u t e r f r o m a Conversational P r o g r a m m i n g T i m e Sharing S y s t e m originating f r o m a I B M 2741 T e l e p r o c e s s ing unit. T h e data in T a b l e I w e r e c o m p u t e d in 0.05 seconds of active c o m p u t e r time. Research was s p o n s o r e d by the Office of A e r o s p a c e Research, U n i t e d States Air Force, u n d e r C o n t r a c t F 33615-69-C-1027.
References 1. J. M. Silcock, M. H. Davies and H. K. Hardy, Monograph and Report Series No. 18, Institute of Metals, London, 1956, p. 93. 2. S. L. Sass, Acta Met., 17 (1969) 813. 3. D. De Fontaine, Acta Met., 18 (1970) 275. 4. M. J. Blackburn and J. C. Williams, Trans. AIME, 242 (1968) 2461.
Accepted November 27, 1970