From periodicity to quasiperiodicity Part II: Continuous periodic to quasiperiodic transition in superlattices

Scripta METALLURGICA et MATERIALIA

Vol. 25, pp. 533-538, 1991 Printed in the U.S.A.

PART II: CONTINUOUS

F R O M P E R I O D I C I T Y TO Q U A S I P E R I O D I C I T Y P E R I O D I C TO Q U A S I P E R I O D I C T R A N S I T I O N

Pergamon Press plc All rights reserved

IN S U P E R L A T T I C E S

U. D. Kulkarni, S. Banerjee and S. D. Kulkarni* Metallurgy Division Bhabha Atomic Research Center Trombay, Bombay 400 085, INDIA. *Department of Metallurgical Engineeering Indian Institute of Technology Powai, Bombay 400 076, INDIA. (Received August (Revised December

7, 1990) II, 1990)

Background We have seen in Part I of this paper that the structural models for quasicrystals are based on nonrepetitive arrangements of tiles of two different kinds. These models bring out two essential attributes of the quasicrystalline state, namely, the forbidden point symmetries and the quasiperiodic translational order. Structures, which are quasiperiodic and yet belong to crystallographic point groups, form two important classes - the incommensurate crystals (ICC) [I] and the quasiperiodic superlattices (QPSLs) [2]. Unlike the quasicrystals, these are derived from periodic lattices and can be generated by setting up either displacement modulations (ICC) or static concentration modulations (QPSL) whose periodicities are irrationally related to that of the basic lattice [3]. QPSLs, like quasicrystals, comprise quasiperiodic (QP) arrangements of tiles. The relationship between these two types of structures can be established using the strip projection method (cases (i) and (iii) respectively in Part I). Although the existence of QPSLs is not widely recognised, several structures, which can be r e g a r d e d as periodic manifestations of these (case (iv) in Part I), have been observed in c e r t a i n alloys, intermetallics and compounds. For instance, the vacancy ordered T-phases in Al-transition metal alloys constitute a series of one dimensional (l-D) Fibonacci superlattices of F C C [4]. In this part of the paper, we have extended the strip projection approach developed in Part I to 2-D periodic and quasiperiodic superlattices. The possibility of a continuous transition between two simple periodic superlattices via a series of Long Period Superlattices (LPSs), through a suitable control over the alloy stoichiometry, has been explored. Our model derives support from the electron microscopy observations on certain AI-Ti intermetallics and the relevant analyses of Loiseau et al. [5]. Quasiperiodic

Superlattices

and Their Periodic Approximants

It was mentioned in Part I that quasiperiodic tilings representing QPSLs can be recovered as projections from higher dimensional lattices, provided the projection space is made rational, while retaining the irrational orientation of the strip. Such QP tilings are built up of tiles of rational dimensions and are known, therefore, as rational modifications of the relevant quasicrystals. The two dimensional QPSLs, which form the subject of discussion in this paper, are r a t i o n a l modifications of the octagonal QP tiling. These belong to the 2-D crystallographic point symmetry group 4mm and can be obtained as projections of the points contained within a suitably defined strip in a 4-D simple cubic

533 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc

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lattice by setting o (which d e t e r m i n e s the o r i e n t a t i o n of the strip) equal to the irrational number 7 ( = 12 - 1) and 8 (which d e t e r m i n e s the orientation of the projection plane) equal to any r a t i o n a l number. For the choice of s u i t a b l e ® values, let us consider the " O c t a n a c c i " series in two d i f f e r e n t forms [6]. 0

1

2

5

12

29 .....

;

1

1

3

7

17

41 .....

The ratios of the successive n u m b e r s 0/i 1/2 2/5 5/12 ...... T ; I/I 1/3 3/7 7/17 tend to 7 in both the cases.

......

= 0/1 or 1/1 correspond to square l a t t i c e s w h i c h are r o t a t e d with respect to each other by 45 °, while ~ = T c o r r e s p o n d s to the o c t a g o n a l QP tiling. Whereas one extreme of the spectrum of rational modifications represents a periodic lattice, the other represents a q u a s i c r y s t a l [7]! The QP tilings c o r r e s p o n d i n g to e = 1/2 or 1/3 are shown in Figs. l(a) and (d). These are similar and are r o t a t e d w i t h r e s p e c t to each other by 45 ° . A c o m p a r i s o n of these with the octagonal QP tiling, shown in Fig. l(a) of Part I, reveals that the. three QP filings are b a s e d on an i d e n t i c a l tiling scheme. One of the two sets of p e r p e n d i c u l a r 45 ° r h o m b u s e s in the o c t a g o n a l QP tiling is r e p l a c e d by fat rhombuses and the other by lean rhombuses in the 1/2 and the 1/3 r a t i o n a l modifications. The fat and the l e a n r h o m b u s e s are tiles of rational dimensions characterised by minor to m a j o r d i a g o n a l ratios of 1/2 and 1/3 r e s p e c t i v e l y . Periodic Approximants (PAs) of the two r a t i o n a l m o d i f i c a t i o n s shown in Figs. l(a) and (d) can be obtained by a s s i g n i n g r a t i o n a l values to o (i.e., by m a k i n g the strip rational). 0/1 and 1/1 PAs of the 1/2 and the 1/3 rational m o d i f i c a t i o n s shown in Figs. l(b), (c) and l(e), (f) b r i n g out the c o m p l e m e n t a r y nature of these two rational m o d i f i c a t i o n s . The r a t i o n a l m o d i f i c a t i o n s and their PAs have an u n d e r l y i n g square lattice as d e p i c t e d in the figure. If the v e r t i c e s of the tiles and the r e m a i n i n g p o i n t s of the square lattice r e p r e s e n t the p o s i t i o n s of the two atomic species, say A1 and Ti respectively, these tilings can be thought of as superlattices of the s q u a r e lattice. The c o n s t i t u e n t tiles - the square, the fat rhombus and the lean r h o m b u s - can then be regarded as u n i m o l e c u l a r clusters of Ti4AI, TiaAl and T i 2 A I s t o i c h i o m e t r i e s r e s p e c t i v e l y . In the rest of this paper, the v a l u e of e will be assumed to be equal to 1/2 and t h e d i s c u s s i o n will be c e n t e r e d a r o u n d the three s t r u c t u r e s shown in Fig. l(a), l(b) and 1(c). The first of these is a q u a s i p e r i o d i c s u p e r l a t t i c e structure (referred to as the QPSL) c o n s i s t i n g of Ti4A1, TisA1 and TizA1 tiles. The other two structures r e p r e s e n t the s i m p l e s t and r a t h e r extreme PAs of this QPSL which, we believe, is a more f u n d a m e n t a l entity. The 0/1 PA (referred to as the 4-2 structure) in Fig. l(b), w h i c h has an o v e r a l l c o m p o s i t i o n TiaA1, consists of Ti4AI and TizAl tiles in equal numbers. The 1/1 PA (Fig. 1(c)) of s t o i c h i o m e t r y Ti~.sAI (referred to as the 4-3 structure) comprises Ti4AI and Ti3AI in an equal proportion. The QPSL, which contains Ti4AI, Ti3AI and Ti2AI tiles in the ratio 12 : 1 : I, has an i r r a t i o n a l c o m p o s i t i o n a p p r o x i m a t e l y equal to Ti~.I2AI. The underlying square lattice of the Q P S L s t r u c t u r e and its PAs can also be r e c o v e r e d as a p r o j e c t i o n from the 4-D lattice by s u i t a b l y e n l a r g i n g the window (the cross section of the strip) as per the scheme shown in Fig. 2. The vertices of the tiles in the p e r i o d i c and the q u a s i p e r i o d i c tilings (AI positions) can be obtained using an o c t a g o n s h a p e d w i n d o w w h i c h is the p r o j e c t i o n of a unit 4-D h y p e r c u b e onto the v i r t u a l plane. The square window, w h i c h yields the u n d e r l y i n g square lattice, c o r r e s p o n d s to the p r o j e c t i o n of a square in 4-D d e f i n e d by the p o s i t i o n vectors [m n ~ m] and [ n m m n], where m and n are n o n c o p r i m e integers, such that the two v e c t o r s and Xproj, Yproj are m u t u a l l y orthogonal. These two w i n d o w s have to be p o s i t i o n e d s u i t a b l y in the virtual space as d e t a i l e d in Part I of the paper in o r d e r to o b t a i n the d e s i r e d p e r i o d i c / q u a s i p e r i o d i c tilings. A simple decoration scheme in the h i g h e r d i m e n s i o n a l l a t t i c e thus exists for the QPSL and its PAs, w h e r e i n one d e c o r a t e s the o c t a g o n a l p o r t i o n of the w i n d o w with A1 atoms and the r e m a i n i n g p o r t i o n of the square w i n d o w w i t h Ti atoms.

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TRANSITION

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535

®=~

(a)

I F (d) o = T

(c)

(b)

°o Q

(e) C = 0/I

(f) O = I/I

Fig. 1 Quasiperiodic superlattices and their Periodic Approximants (PAs): a) I/2 rational modification of octagonal quasiperiodic tiling; b) 0/i PA of (a); c) I/I PA of (a); d) 1/3 rational modification of octagonal quasiperiodic tiling; e) 0/I PA of (d); f) I/I PA of (d). All tilings have an underlying square lattice as depicted. The three types of tiles - the square, the fat and the lean rhombuses represent unimolecular clusters of Ti4AI, Ti3AI and TizAl stoichiometries respectively.

Fig. 2 Arrangement of "windows" for the quasiperiodic superlattice (Fig. l(a)) and its periodic approximants. The octagonal and the square windows correspond to the vertices of the tiles (AI atoms) and the points of the underlying square lattice respectively.

A l t h o u g h the Q P S L s t r u c t u r e has not b e e n e n c o u n t e r e d in any a l l o y s y s t e m to date, an u n m i s t a k a b l e t e n d e n c y for its f o r m a t i o n , as we w i l l see, is e v i d e n c e d in the AI-Ti system w h i c h a l s o s h o w s a c o n t i n u o u s t r a n s i t i o n f r o m the 4-2 s t r u c t u r e to the 4-3 s t r u c t u r e [5]. The 4-2 s t r u c t u r e has a l s o b e e n p r o p o s e d as a candidate for the short range ordered state in the w e l l - k n o w n 1 M 0 a l l o y s [8,9]. The s t r u c t u r e of the n o v e l H - p h a s e in A I - H f - S i a l l o y s c a n a l s o be e x p l a i n e d in terms of a t r i a x i a l s t a c k i n g of l a y e r s of the 4-2 s t r u c t u r e [10]. A s t r u c t u r e b a s e d on a t i l i n g s i m i l a r to the 4-3 s t r u c t u r e is o b s e r v e d in c e r t a i n p e r o v s k i t e s [11].

536

TRANSITION

IN SUPERLATTICES

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25, No. 3

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3 Computed Diffraction Patterns (CDPs): Fundamental Reflections; o - * type superlattice reflections; • - * type superlattice reflections. a) u = M; CDP of the 4-2 structure; b ) a = 7 ; CDP of the QPSL structure; c) u = W3; CDP of the 4-3 structure. The positions of the fundamental and the superlattice reflections together form nonspacefilling periodic approximants(PAs) of certain quasiperiodic structures (cf. Part I). Figs. 3(a),(b) and (c) are 0/i PAs of the ½ rational modification, the octagonal quasiperiodic structure and the Ii3 rational modification respectively. •

-

Periodic t o Q u a s i p e r i o d i c

Transition

in S u p e r l a t t i c e s

The computed diffraction patterns corresponding to the three superlattice s t r u c t u r e s show certain i n t e r e s t i n g features. The d i f f r a c t i o n pattern of the QPSL is d e n s e l y filled with an i n f i n i t e n u m b e r of s u p e r l a t t i c e reflections. However, if one considers only the two s t r o n g e s t types of superlattice reflections at ~* and M* and e q u i v a l e n t p o s i t i o n s of the square lattice, the d i f f r a c t i o n p a t t e r n (Fig. 3(b)) of the Q P S L appears to be intermediate between those of the 4-2 and the 4-3 structures (Fig. 3(a) and 3(c)). These two s t r u c t u r e s are r e m a r k a b l y similar in that they b e l o n g to the same 2-D space g r o u p P4,® and are b a s e d on an i d e n t i c a l a r r a n g e m e n t of tiles. The r e l a t i o n s h i p b e t w e e n these and the QPSL becomes apparent, if one allows a c o n t i n u o u s change in a from 0/1 to 1/1. Such a change in a is tantamount to a continuous change in s t o i c h i o m e t r y from TisAl to T i ~ . ~ A I and r e s u l t s in superlattice structures of varying periodicities. In general, if a is expressed as the ratio of two n o n c o p r i m e integers, p/q, the c o m p o s i t i o n (X,I) changes as (qi+pq)/(p+2q)Z, while the d i m e n s i o n of the square unit cell of the p r o j e c t e d s t r u c t u r e equals (2p+4q} times the l a t t i c e p a r a m e t e r of the u n d e r l y i n g square lattice. The c o r r e s p o n d i n g d i f f r a c t i o n p a t t e r n s show a square grid of s u p e r l a t t i c e r e f l e c t i o n s w h i c h divides any unit r e c i p r o c a l square (say 00 - I0 - 11 01) of the underlying square lattice into (2p+4q) equal parts. The relatively stronger superlattice r e f l e c t i o n s are i n v a r i a b l y l o c a t e d at M* and M* (where, u = 2q/(2p+4q)) positions of the square lattice. V a r i a t i o n in u from 0/1 to i/I, causes a c o n t i n u o u s shift in the p o s i t i o n s of the s u p e r l a t t i c e r e f l e c t i o n s from u = ~ ( c o r r e s p o n d i n g to the 4-2 structure) to u = '13 (the 4-3 structure). As a limiting case, when a equals T, the unit cell b e c o m e s i n f i n i t e l y large and the r e s u l t i n g s t r u c t u r e is the QPSL w h o s e d i f f r a c t i o n p a t t e r n (u = T) is i n t e r m e d i a t e b e t w e e n those of the 4-2 and the 4-3 s t r u c t u r e s . V a l u e s of c i n t e r m e d i a t e b e t w e e n 0/1 and 1/1 represent superlattice structures whic h contain both TiIAl and Ti3AI r h o m b u s e s in a d d i t i o n to the Ti4AI squares. The ratio of the n u m b e r of T i 3 A I r h o m b u s e s to the total number of rhombuses increases continuously from 0 to 1, as a is v a r i e d from 0/1 to 1/1. The QPSL s t r u c t u r e appears to be p r e c i s e l y m i d w a y b e t w e e n the 4-2 and the 4-3 structures in that it c o n t a i n s r h o m b u s e s of the two types in equal proportion. The e a r l y stages of the p e r i o d i c to q u a s i p e r i o d i c s u p e r l a t t i c e t r a n s i t i o n b e t w e e n the 4-2 (or the 4-3) s t r u c t u r e and the QPSL merit a special attention. These c o n s t i t u t e

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(a) O = I/B

I I

-

-

~

(b) c = 7/8

537

\ .....

Fig. 4 Unit Cells of Long Period Antiphase Boundary modulated (LPAPB) structures. LPAPB derivatives of the 4-2 structure (a); of the 4-3 structure (b). The nonconservative APBs and the octagonal regions of uncertainty are outlined with thick lines. The corners of the unit cells are marked with solid squares. the w e l l l k n o w n long p e r i o d a n t i p h a s e b o u n d a r y m o d u l a t e d (LPAPB) structures. The s t r u c t u r e in Fig.4(a) (o = t~), which contains marginally higher Ti than the stoichiometric and u n m o d u l a t e d 4-2 structure, comprises four "out of phase" d o m a i n s (translational variants) of the 4-2 s t r u c t u r e s e p a r a t e d by TisAl tiles which form antiphase b o u n d a r i e s (APBs) in <1 0 0> o r i e n t a t i o n s . The c o u n t e r p a r t of t h i s for the 4-3 structure, which contains marginally lower Ti than the unmodulated 4-3 structure, (Fig. 4(b)) shows <1 1 0> APBs c o n s i s t i n g of Ti2AI tiles. The APBs in Fig. 4 should be r e g a r d e d as APBs of the n o n c o n s e r v a t i v e type [12], since their i n t r o d u c t i o n in the r e l e v a n t u n m o d u l a t e d s u p e r l a t t i c e s leads to a d e p a r t u r e from the e x a c t s t o i c h i o m e t r i e s . LPAPB s t r u c t u r e s of this kind, thus represent possible ways of accommodating offstoichiometry in superlattice s t r u c t u r e s without p r o d u c i n g disorder. An i n t e r e s t i n g feature of these 2-D LPAPB s t r u c t u r e s is that the APBs t h e m s e l v e s become "out of phase" (discontinuous) at the r e g i o n s of uncertainty. These regions, it may be recalled, can be tiled in several d i f f e r e n t ways as d i s c u s s e d in Part I. The period of the LPAPB s t r u c t u r e s and, therefore, the s p a c i n g b e t w e e n the APBs change as o is varied. This in turn results in a shift in the p o s i t i o n s of s u p e r l a t t i c e r e f l e c t i o n s as d i a c u s s e d earlier. This o b s e r v a t i o n is in c o n f o r m i t y w i t h the c l a s s i c a l work of Fujiwara [13] on the diffraction effects from superlattices containing periodic a r r a n g e m e n t s of APBs. A p e r i o d i c to q u a s i p e r i o d i c s u p e r l a t t i c e t r a n s i t i o n similar to the one described above has been o b s e r v e d in c e r t a i n AI-Ti i n t e r m e t a l l i c s [5]. A [200} plane of FCC comprises a square l a t t i c e w h e r e i n the sides of the square are d e f i n e d by ~<011> vectors. In the e q u i a t o m i c AITi phase the a l t e r n a t e (200) layers are occupied e x c l u s i v e l y by A1 or Ti atoms. A r e p l a c e m e n t of the Ti layers in AITi by the 4-2 s t r u c t u r e yields the Al~Tis phase. P r o g r e s s i v e additions of Ti to this phase lead to the formation of superlattices which are apparently incommensurate (quasiperiodic). This is r e v e a l e d by a continuous shift of superlattice r e f l e c t i o n s from the simple r a t i o n a l p o s i t i o n s (a = M) towards the irrational positions (a = T) of the QPSL structure. A l t h o u g h the extent of the o b s e r v e d s h i f t w a s rather limited and the truly quasiperiodic QPSL structure was not realised, high r e s o l u t i o n e l e c t r o n m i c r o g r a p h s showed r e g i o n s c o n t a i n i n g various c o n f i g u r a t i o n s of Ti4AI, TisAl and Ti2AI tiles. It is s i g n i f i c a n t that Loiseau et al. [5] have i n t e r p r e t e d the o c c u r r e n c e of this apparent " i n c o m m e n s u r a b i l i t y " in terms of a real space s t r u c t u r e c o n t a i n i n g "islands" of various sizes of the 4-2 and the 4-3 structures, a l t h o u g h the p o s s i b i l i t y that such s t r u c t u r e s could be interpreted as periodic approximants of the QPSL structure has not been c o n s i d e r e d by them. The s t r u c t u r e s g e n e r a t e d by projection from 4-D with a relatively larger deviation in a (0 < o < T) from 0/I show a close resemblance with the real space m o d e l s p r e s e n t e d by L o i s e a u et al.

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Discussion It has been demonstrated in P a r t s I and II of this paper that the four types of structures that can be r e c o v e r e d as p r o j e c t i o n s from h i g h e r d i m e n s i o n a l l a t t i c e s [2,14] are not mere theoretical p o s s i b i l i t i e s , since they can occur in real systems. It has also been s h o w n that one simple p e r i o d i c s u p e r l a t t i c e s t r u c t u r e can transform continuously into another through a series of long period superlattices. These periodic structures may be regarded as periodic m a n i f e s t a t i o n s of a more f u n d a m e n t a l e n t i t y - the QPSL. The early stages of this continuous transformation represent long period antiphase boundary modulated (LPAPB) structures. It s h o u l d be m e n t i o n e d here that s i m i l a r p o s s i b i l i t i e s have also been considered in the case of c e r t a i n i n c o m m e n s u r a t e c r y s t a l s [15]. An alternative way of looking at the p e r i o d i c to q u a s i p e r i o d i c s u p e r l a t t i c e transitions is the c o n c e n t r a t i o n wave a p p r o a c h [16]. In the framework of this approach, the 4-2 structure can be regarded as a superimposition of two ~ e r p e n d i c u l a r M* c o n c e n t r a t i o n w a v e s [8]. The c o n c e n t r a t i o n k-vectors for the 4-3 and the QPSL structures are ~* and ~* r e s p e c t i v e l y . The q u a s i p e r i o d i c nature of the l a t t e r is a c o n s e q u e n c e of the i r r a t i o n a l i t y of these c o n c e n t r a t i o n waves. The c o n t i n u o u s c h a n g e in p e r i o d i c i t y can be viewed as a c o n t i n u o u s change in these c o n c e n t r a t i o n k-vectors. This has been b r o u g h t out by T a k e d a et al. [17] in their t r e a t m e n t of long period superlattices in Cu-Pd alloys. These long p e r i o d superlattices are d e r i v e d from an LI2 (Cu3Au-type) s t r u c t u r e in much the same way as the long p e r i o d s u p e r l a t t i c e s d i s c u s s e d in this paper are d e r i v e d from the 4-2 and the 4-3 structures [18]. Although the thermodynamic model of T a k e d a et al., w h i c h is based on the concept of o r d e r i n g b e y o n d the Lifshitz point [19], does not deal w i t h long p e r i o d s u p e r l a t t i c e s per s_ee but with a related form of short r a n g e order g e n e r a t e d by spinodal ordering, certain features like the concentration dependent shift of superlattice reflections from the r a t i o n a l p o s i t i o n s of the simple u n m o d u l a t e d s u p e r l a t t i c e s t r u c t u r e appear to be common to b o t h - the long p e r i o d superlattices in Cu-Pd and the 2-D long period s u p e r l a t t i c e s d e s c r i b e d in this paper. This aspect of the periodic to q u a s i p e r i o d i c t r a n s i t i o n will be dealt with in a greater detail in a s u b s e q u e n t publication. References

1) 2) 3) 4) 5) 6)

7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

18) 19)

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