Pseudo-Symmetry in Multiple Twinned CrystalsHaving M3M Point Group Symmetry

S A N T I A G O H A R R I A G U E A N D H A R R Y A. L E I B O V I C H

The pseudo-symmetry elements of the composite lattice of a twinned crystal having m3moint group symmetry are determined on the b a s i s of group theory. Twinning a c r o s s the11} planes is considered. The results for different composite lattices are g iven .

I N the last y e a r s , several authors have observedmultiple twinning in diffraction patterns of thin f i lmsof fcc m e t a l s .1-6 T h e s e m e t a l s twin a c r o s s the {111}planes. We thought that it would be useful a m a t h e -matical analysis, in o r d e r to determine the possiblepseudo-symmetries of twins.

The idea of pseudo-symmetry a r i s e s from the factthat , if the coincidence of points coming from dif-ferent orientations of the twin is neglected, the com-posi te lattice will mimic a point-group symmetry.However, the composite lattice will not conform tospace groups appropriate to t h e s e point groups.

The importance of pseudo-symmetry in diffractionpattern analysis has been shown.%B

We have considered the primary and secondarytwinning of a crystal having m3m point group sym-metry, and we have determined the pseudo-symmetryof composite structures formed by several twin o r i e n -tations.

M A T H E M A T I C A L P R O C E D U R E

P r i m a r y T w i n n i n g

W e consider twinning a c r o s s the {III] planes. W ehave considered the c o m p o s i t e structure f o r m e d bythe original orientation of the crystal and one, two,three, or the four twin orientations.

T h e p r o c e d u r e to d e t e r m i n e the p s e u d o - s y m m e t r yof the c o m p o s i t e lattice is similar to the one followedby Mokievskii et al.;9 the b a s i c relations a p p e a r in theclassic g r o u p t h e o r y bibliography, i.e. S c h a n k m a n ,I°and K u r o s h . n

T h e point g r o u p m 3 m of the crystal is i s o m o r p h i c tothe direct product:

{I} × {432} [1]

In a crystal with m3m point group symmetry, be-cause of its center of symmetry, the twin reflectionsare equivalent t o 180 deg rotations. Let these r o t a -tions be Mi, w h e r e i = 1,2, 3, 4 corresponds t o thetwin axes [111], [111], [111], and [111] respectively.

Let {Mi} be the group of o r d e r 2 f o r m e d by Mi andthe identity; and:

p i : {Mi} × {r~3rn} : {i} × Mi × {432} [2]

p i is not a group because M i does not commute withall the elements of {m3m}.

SANTIAGO HARRIAGUE and HARRY A. LEIBOVICH are withthe Grupo de Fisica del S61ido, Institutode Investigaciones Cientfficasy T6cnicas de las Fuerzas Armadas, Buenos Aires, Argentina.

Manuscript submitted April 3, 1971.

METALLURGICAL TRANSACTIONS

If we consider the lattice f o r m e d by the original andone twin orientation, the pseudo-symmetry operationsare defined as those in the corresponding set P* thatform the group G of highest o r d e r such that , for everyone of its elements gin, we have:

• i [ 3 ]gm P; = Pj'

for any j withP i -- { e / }

Relation [3] shows that , as i commutes with anyconsidered symmetry operation, it is only necessaryt o take into account the elements of {m3m} which a r ealso elements of {432}.

If we now consider the p r e s e n c e of more than onetwin orientation, the pseudo-symmetry elements ofthe composite lattice are a group of the union U of thesets p i corresponding t o the orientations present.

Let Ak be any element of {432}. It can be shown thatif Mi and Mj are conjugate u n d e r Ak, i.e.:

AkMiAk 1 = Mj [4]

then all the elements of p i and PJ fulfill [3].Therefore, t h e r e are two different kinds of elements

of the group U:a) Elements Ak belonging t o the group 432; pseudo-

symmetry elements of this kind must fulfill [4] for allthe orientations present.

b) Products MiAk: pseudo-symmetry elements ofthis kind must define a relation analogous t o [3] be-tween all the elements of U.

Secondary Twinning

If p r i m a r y twinned crystals twin aga in about t h e i r<111> axes , t h r e e new secondary twin orientations ap-pear for each of the p r i m a r y ones. The fourth onewill be coincident with the original orientation.

We can use the procedure described previously forprimary twinning if we now consider a p r i m a r y twinorientation as original. I ts four twin orientations willbe the t h r e e secondary ones, and the o r ig ina l orienta-tion.

R E S U L T S

P r i m a r y T w i n n i n g

T a b l e I s h o w s the equivalence defined b y the ele-m e n t s of {432} b e t w e e n the twin operations M i. T a b l eII s h o w s the p s e u d o - s y m m e t r y of the following c o m -posite structures:

V O L U M E 2, D E C E M B E R 1971-3485

Table I. Equivalence Between theTwin Operations M i as Defined bythe Elements of (432}

Elements Conjugate Conjugate Conjugate Conjugateof {432} to Ma toM2 to M3 toM~

1[100] ~ Ms M4 MI M:[01O]~ M4 M3 M2 i l[001] ~ M~ ML 344 Ms

1[011 ]~ Ms M2 MI M4[011 ] ~ M~ M4 Ms M2[101]~ M4 M2 M3 MI[ 10] ]~ MI M3 M2 /144

Ii!0j ! M,[110] ~ M1 M2 3'/4 M3[ll l]~ MI M3 3/4 M2[]]1]~ M4 M2 M~ Ms[1i l l ~ M2 M4 M3 M,[] 1]] ~ Ms MI M2 M4[!111~ M, M4 M2 M3[ll l]~ M4 3/2 M1 Ms[1]~]~ M4 Mt Ma M2[] 11]a5 M2 M3 Mt M 4

t100] ~ /144 M1 M2 M3[010] ~ M2 M4 Mt Ma[001 ] ~ M3 M 4 M2 M I

[100]~ M2 M3 M4 M1[010] ~ M3 M~ M4 M2[001] ~ M4 M3 M1 M2

1) O r i g i n a l and one twin o r i e n t a t i o n : The p s e u d o -s y m m e t r y m i m i c s p o i n t g r o u p 6 / r n m m ; the 6 - f o l daxis c o i n c i d e s wi th the twin a x i s . T h i s r e s u l t a g r e e swi th one f o u n d p r e v i o u s l y .%~

2) O r i g i n a l a n d two twin o r i e n t a t i o n s : T h e p s e u d o -s y m m e t r y m i m i c s p o i n t g r o u p m m m ; two p e r p e n d i c u -l a r b i n a r y a x e s b e l o n g to the p l a n e d e t e r m i n e d b y thetwo twin a x e s , a n d t h e t h i r d b i n a r y axis i s n o r m a l tot h e m .

3) O r i g i n a l a n d t h r e e twin o r i e n t a t i o n s : T h e p s e u d o -s y m m e t r y m i m i c s p o i n t g r o u p 3 ( 2 / m ) ; the 3 - f o l d a x i sc o i n c i d e s wi th the a b s e n t twin a x i s .

4) O r i g i n a l a n d the f o u r twin o r i e n t a t i o n s : a s T a b l e Is h o w s , the o r i g i n a l c u b i c s y m m e t r y i s r e s t o r e d .

S e c o n d a r y T w i n n i n g

T a b l e II s h o w s the p s e u d o - s y m m e t r y of the follow-ing structures:

5) Original, one p r i m a r y , and one of its s e c o n d a r ytwin o r i e n t a t i o n s .

Table II. Pseudo-Symmetry Elements and Pseudo-Symmetry Point Groups ofSeveral Composite Lattices. Cases are Numbered as They are in Section

RESULTS. (Indexes Refer to Cubic Axes in the Original Orientation).

Case

Pseudo-Symmetry

6-Fold 4-Fold PointAxis Axis 3-FoldAxis 2-Fold Axis Group

1 <111> - - <:110> and <112> 6 / m m m2 - - <100> and < I 10> mmm3 - - <111> <110> 3 2/m4 - <100> <111> <110> m3m5 - - - <122> and <114> mmm6 - - <511> <114>and<110> "3 2/m7 <122> <511> and <111> <114> and <1 t0> m3m8 . . . . <100> and<110> mmm9 - - <111> <110> 5 2/m

10 - <100> <111> <110> m3m

6) Original, one p r i m a r y and two of its s e c o n d a r ytwin orientations.

7) Original, one p r i m a r y , and its t h r e e s e c o n d a r ytwin orientations.

8) Original, two p r i m a r y , and their six s e c o n d a r ytwin orientations: it can be s h o w n that the p s e u d o -s y m m e t r y point g r o u p is m m m , with two b i n a r y axesin the p l a n e d e t e r m i n e d by the two twin axes.

9) Original, three p r i m a r y , and their nine s e c o n d a r ytwin orientations: the p s e u d o - s y m m e t r y point g r o u pis 3 ( 2 / m ) ; the 3-fold axis coincides with the absenttwin axis.

10) Original, four p r i m a r y , and their t w e l v e s e c o n d a r ytwin orientations: cubic s y m m e t r y appears.

A C K N O W L E D G E M E N T

T h e a u t h o r s wish to t h a n k Dr. O s c a r Wittke, of theUniversity of Chile, for helpful discussions.

R E F E R E N C E S

1. R. D. Burbank and R. D. Heidem'eich:Phil. Mag., 1960, vol.5, p. 373.2. D. W.Pashley and M. J. Stowell: Phil. Mag., 1963,vol. 8, p. 1605.3. S. lno, D. Watanabe, and S. Ogawa: £ Phys. Soc. Japan, 1964, vol. 19, no. 6,

p. 881.4. S. lno: Z Phys. Soc. Japan, 1966,vol. 21, no. 2, p. 347.5. S. Ino and S. Ogawa: J. Phys. Soc. Japan, 1967,vol. 22, no. 6, p. 1365.6. J. G. Allpress and J. V. Sanders: Surface Sci., 1967,voI. 7, p. 1.7. N. F. Henry, H. Lipson and W. A. Wooster: The Interpretation o f X-rayDif-

fraction Photographs, p. 117,The Macmillan Co., New York, 1951.8. J. D. Dunitz: Acts Cryst., Internat., 1964,vol. 17,pt. 10,p. 1299.9. V. A. Mokievskii, I. I. Shafranovskii,P. V. Vovk, and I. I. Afanas'ev: Krystal-

log., USSR, 1966,vol. 11, no. 4, p. 539.10. B. Schankman: Group Theory, Van Nostrand, Princeton, 1965.11. A. G. Kurosh: The Theory o fGroups, Chelsea PuN. Co., New York, 1955.

3486-VOLUME 2, DECEMBER 1971 METALLURGICAL TRANSACTIONS