STRUCTURAL UNIT/GRAIN BOUNDARY DISLOCATION MODEL FOR TWIST BOUNDARIES IN CUBIC CRYSTALS

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STRUCTURAL UNIT/GRAIN BOUNDARY

DISLOCATION MODEL FOR TWIST BOUNDARIES IN CUBIC CRYSTALS

P. Bristowe, R. Balluffi

To cite this version:

P. Bristowe, R. Balluffi. STRUCTURAL UNIT/GRAIN BOUNDARY DISLOCATION MODEL FOR

TWIST BOUNDARIES IN CUBIC CRYSTALS. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-

155-C4-170. �10.1051/jphyscol:1985419�. �jpa-00224668�

JOURNAL DE PHYSIQUE

Colloque C4, supplément au n04, Tome 46, avril 1985 page C4-155

STRUCTURAL UNIT/GRAI N BOUNDARY DISLOCATION MODEL FOR TWIST BOUNDARIES

I N CUBIC CRYSTALS

P.D. Bristowe and R.W. Balluffi

Department of Materials Science and Engineering, Massachusetts I n s t i t u t e of TechnoZogy, Cambridge, MA 022 39, U.S.A.

Abstract - The s y s t e m a t i c s of [O011 t w i s t boundary s t r u c t u r e i s presented formally i n terms of a s t r u c t u r a l u n i t l g r a i n boundary d i s l o c a t i o n h i e r a r c h i c a l model and t h e e a r l i e r model of Sutton i s generalized. By comparison with experimental observation and a t o m i s t i c c a l c u l a t i o n using p a i r - p o t e n t i a l models t h e physical s i g n i f i c a n c e of t h e individual members of t h e hierarchy i s d e t e r - mined. Comparison with experiment i n d i c a t e s a s t r o n g <110> t y p e primary

r e l a x a t i o n f o r û & 36.g0 and a s i g n i f i c a n t secondary r e l a x a t i o n near C 5 whicli must r e s u l t from "oblique" p e r t u r b a t i o n s i n t h e a r r a y of primary GBD's. On t h e o t h e r hand, comparison with a v a i l a b l e c a l c u l a t e d r e s u l t s i n d i c a t e s a strong

<110> type primary r e l a x a t i o n a t low angles but a progressively weaker relaxa- t i o n a t higher a n g l e s . Also, no evidence i s found f o r any s i g n i f i c a n t

secondary r e l a x a t i o n s when a t l e a s t one p a i r p o t e n t i a l i s employed. However, very r e c e n t s t u d i e s i n d i c a t e s t r o n g e r secondary r e l a x a t i o n s with o t h e r poten- t i a l s , and t h i s , i n f u t u r e work, should lead t o b e t t e r agreement between c a l c u l a t i o n s and experiment.

1. Introduction - In an attempt t o i n c r e a s e O u r understanding of t h e systematics of g r a i n boundary s t r u c t u r e and i t s r e l a t i o n s h i p t o g r a i n boundary d i s l o c a t i o n s a s t r u c t u r a l u n i t / g r a i n boundary d i s l o c a t i o n (SUIGBD) model has r e c e n t l y been devel- oped by Sutton and Vitek (1-4) which i s based on t h e r e s u l t s of t h e i r a t o m i s t i c cal cul a t i o n s and t h e e a r l i e r geometrical "ledge" model of Bishop and Chalmers ( 5 ) . The geometrical p r i n c i p l e underlying t h e Sutton and Vitek model i s t h a t any long period boundary may be constructed from units of s h o r t e r period boundaries. The s h o r t period boundaries can tlien be chosen a s "del i m i t i n g " boundaries (DB's) wliich f i x t h e l i m i t s on l o c a l m i s o r i e n t a t i o n ranges. The s t r u c t u r e s of t h e intervening long period boundaries i n each range of m i s o r i e n t a t i o n between two D B ' s w i l l then be made up of various mixtures of t h e DB u n i t s . The r u l e f o r mixing so as t o preserve c o n t i n u i t y of s t r u c t u r e i s t h a t t h e u n i t s i n minority be a s f a r a p a r t a s p o s s i b l e . Recently, B a l l u f f i and Bristowe ( 6 ) recognized t h a t a hierarchy of SU/GBD d e s c r i o - t i o n s e x i s t s corresponding t o t h e d i f f e r e n t p o s s i b l e choices of DB's, thereby pro- viding a g e n e r a l i z a t i o n of t h e model. Geometrically, t h e choice of DB's i s usually well defined; i . e . , t h e s h o r t e s t period boundaries (lowest C ) a r e chosen f i r s t and o t h e r s follow so t h a t t h e hierarchy of s t r u c t u r a l u n i t c o n f i g u r a t i o n s i s systerna- t i c a l l y formed. Also, t h e system i s n a t u r a l l y quantised on a f i n e r s c a l e a s t h e number of DB's i s increased. However, a t o m i s t i c c a l c u l a t i o n s have shown t h a t t h e choice of DB's, based on t h e simple geometry of r i g i d dichromatic p a t t e r n s , may not be t h e b e s t one i f s u b s t a n t i a l atomic rearrangement has occurred during r e l a x a t i o n . Therefore, t h e choice of physical l y s i g n i f i c a n t DB's can only be determined from examination o f equilibrium s t r u c t u r e s . I f a OB u n i t can be constructed from u n i t s of another DB i t i s c a l l e d a " m u l t i p l e u n i t r e f e r e n c e s t r u c t u r e " (MURS); otherwise i t i s c a l l e d "favored" ( F ) . Very few favored boundaries appear t o e x i s t , t h e most n o t a b l e being t h e s i n g l e c r y s t a l ( C I ) and simple twin boundary ( ~ 3 ) .

The s y s t e m a t i c d e s c r i p t i o n o f boundary s t r u c t u r e i n terms of mixtures of s t r u c - t u r a l u n i t s corresponds d i r e c t l y t o one i n terms of GBD's. For a long period t i l t boundary, each minority u n i t can be considered a s t h e c o r e of a p e r f e c t GBD having

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985419

C4-156 JOURNAL DE PHYSIQUE

a Burgers v e c t o r belonging t o t h e DSC l a t t i c e o f t h e d e l i m i t i n g boundary composed o f t h e m a j o r i t y u n i t s . Also, secondary GBD's (whose cores a r e a p p r o p r i a t e l y chosen m i n o r i t y u n i t s ) appear as p e r t u r b a t i o n s i n t h e spacing o f primary GBD's (whose cores a r e a l s o a p p r o p r i a t e l y chosen u n i t s ) ( 6 ) . Thus, a h i e r a r c h y o f GBD d e s c r i p t i o n s e x i s t s corresponding t o t h e h i e r a r c h y o f s t r u c t u r a l u n i t s d e s c r i p t i o n s , as deduced by B a l l u f f i and Bristowe (6). The h i e r a r c h i c a l nature o f t h e SU/GBD model i s one i l l u s t r a t i o n o f t h e non-uniqueness o f GBD d e s c r i p t i o n s (7,8) and choosing which member o f t h e h i e r a r c h y i s "best" o r has p a r t i c u l a r p h y s i c a l s i g n i f i c a n c e can o n l y be made by c a l c u l a t i o n o f a s p e c i f i c boundary p r o p e r t y such as t h e s t r e s s f i e l d o r d i f f r a c t i o n c o n t r a s t . I f t h e g r a i n boundary f o l l o w s t h e r i g i d l a t t i c e model, and no r e l a x a t i o n s are allowed i n t h e b i c r y s t a l near t h e core region, t h e s t r u c t u r a l u n i t s e x h i b i t considerable d i s t o r t i o n s , and a l 1 GBD's a r e completely d e l o c a l i z e d . However, i n a c t u a l cases a t l e a s t some r e l a x a t i o n s occur which improve t h e uniformity o f s t r u c t u r a l u n i t s and produce l o c a l i z e d GBD1s. The q u e s t i o n o f which GBD descrip- t i o n w i t h i n t h e h i e r a r c h y i s most a p p l i c a b l e was addressed by B a l l u f f i and Bristowe ( 6 ) by re-examining t h e data o f Sutton and V i t e k (1) i n terms o f t h e d i s t o r t i o n s o f t h e s t r u c t u r a l u n i t s . I t was concluded t h a t t h e d i s t o r t i o n s decreased as t h e number of DB's chosen increased, b u t t h a t nevertheless, d i f f e r e n t members o f t h e h i e r a r c h y of d e s c r i p t i o n s could be used advantageously f o r d i f f e r e n t purposes.

I n t h e present paper we use t h e SU/GBD h i e r a r c h i c a l model developed f o r t i l t boundaries (6) t o determine t h e systematics o f t w i s t boundary s t r u c t u r e i n FCC crys- t a l s . Primary focus w i l l be on [ O O l ] t w i s t boundaries s i n c e t h i s i s where most of t h e c o n t r o l l e d experimental i n f o r m a t i o n e x i s t s , b u t extension t o t h e [ O l l ] system i s a l s o i n d i c a t e d b r i e f l y . P r e l i m i n a r y work by Sutton (9) has shown t h a t t h e SU/GBD model f o r t w i s t boundaries i s i n h e r e n t l y more complicated than t i l t boundaries s i n c e a t h i r d core u n i t has t o be i n t r o d u c e d i n o r d e r t h a t t h e u n i t s f i l 1 a l 1 space, However, t h e general p r i n c i p l e s remain t h e same w i t h t h e m i n o r i t y u n i t s now being t h e i n t e r s e c t i o n o f t h e cores o f t h e screw GBD1s d e s c r i b i n g t h e boundary. More r e c e n t l y , Schwartz e t a l . (10) have examined a d d i t i o n a l [O011 t w i s t boundary s t r u c - t u r e s emphasizing t h e r e l a t i o n s h i p between v a r i o u s SU/GBD d e s c r i p t i o n s and c e r t a i n symmetry operations i n t h e boundary plane. We b e g i n here by p r e s e n t i n g t h e general geometrical h i e r a r c h y o f d e s c r i p t i o n s by reference t o r i g i d ( d i c h r o m a t i c ) [O01 1

p a t t e r n s where t h e d i s t o r t i o n o f t h e u n i t s i s o b v i o u s l y maximized. Special a t t e n t i o n i s p a i d t o the i d e a t h a t t h e d i s t o r t i o n s o f t h e s t r u c t u r a l u n i t s c h a r a c t e r i s t i c of t h e r i g i d model can be reduced, and GBD1s w i t h l o c a l i z e d s t r a i n f i e l d s can be formed, when r e l a x a t i o n i n t h e form o f l o c a l r o t a t i o n s occur i n t h e boundary. Next, t h e experimental i n f o r m a t i o n i s summarized and i n t e r p r e t e d , and f i n a l l y , some s e l e c t e d computer c a l c u l a t i o n s a r e shown and t h e i r appl i c a b i l i t y t o t h e model discussed.

2. Forma1 SU/GBD d e s c r i p t i o n o f [O011 t w i s t boundaries - I n order t o i n v e s t i g a t e t h e h i e r a r c h y o f SU/GBD d e s c r i ~ t i o n s t h a t e x i s t s f o r r O O l l t w i s t boundaries we have f i r s t e x a m i k d t h e r i g i d body dichromati c p a t t e r n s o f - s e v e r a l CSL c o n f i g u r a t i o n s i n t h e usual m i s o r i e n t a t î o n range 068645" (which generates a l 1 c r y s t a l l o g r a p h i c a l l y e q u i v a l e n t s t r u c t u r e s ) . For b r e v i t y we i l l u s t r a t e here o n l y t h r e e o f t h e l o n g p e r i o d s t r u c t u r e s which are viewed along t h e t w i s t a x i s and show, u s i n g d i f f e r e n t symbols, t h e two (002) planes immediately adjacent t o t h e boundary.

2.1 c l DB d e s c r i p t i o n - The most elementary SU/GBD d e s c r i p t i o n p o s s i b l e i s o b t a i n e d by s e l e c t i n g two 11 p e r f e c t c r y s t a l l a t t i c e "boundaries" as the DB's. These boun- d a r i e s , which must a l s o be favored, are chosen such t h a t t h e i r s t r u c t u r a l u n i t s possess t h e two s h o r t e s t p o s s i b l e p e r i o d v e c t o r s ( o r h a l f - p e r i o d v e c t o r ) 1/2<110>

and 1/2<100>. F i g . 1 ( a ) shows t h e c l DB u n i t s and t h e m i s o r i e n t a t i o n range over

which a l 1 o t h e r boundaries may be constructed by an appopriate m i x t u r e o f these

u n i t s . The l e f t h a l f o f t h e f i g u r e represents SU d e s c r i p t i o n s generated i n t h e

range 068645" b y s t a r t i n g w i t h a m a j o r i t y o f <110> type u n i t s and t h e r i g h t h a l f

represents SU d e s c r i p t i o n s generated i n t h e same range by s t a r t i n g w i t h a m a j o r i t y

o f <100> type u n i t s . Tlius, from O t o 45" two non-equivalent SU/GBD d e s c r i p t i o n s

e x i s t . However, o n l y one of t h e d e s c r i p t i o n s i s p h y s i c a l l y s i g n i f i c a n t and can be

chosen from examination of computed e q u i l i b r i u m s t r u c t u r e s . a n d / o r experimental ob-

s e r v a t i o n as described i n s e c t i o n s 3 and 4. N o t i c e a l s o t h a t a t 36.g0 ( ~ 5 ) on t h e

Fig. 1. Formal SU/GBD hierarchical description of [O011 t w i s t boundaries: ( a )

Cl DB description, ( b ) Ll/L5 DB description, and ( c ) Ll/C5/L13/L17 DB

description. The DB units and t h e i r period vectors a r e shown below the

misorientation range in each case. (For L13 and Cl7 the interna1 contents

of the units have been omitted f o r simplicity.) The shaded regions adja-

cent t o t h e DBs define the angular range over which t h e s t r u c t u r e can be

represented by the appropriate GBD of Burgers vector b. Structures which

have dual dislocation descriptions are indicated by 1 : l .

C4-158 JOURNAL DE PHYSIQUE

<110> branch t h e two descriptions become equivalent since a t t h i s angle there i s a 1 :1 mixture of each type of Cl u n i t . However, a t 36.9' (15) on the <100> branch there a r e four times a s many <100, units as <110> u n i t s . Figure 2 ( a - f ) shows the dichromatic patterns of the 1265 ( 8 = 21 . Z O ) , C281 (8 = 34.7") and C305 (8 = 42.7') configurations each in t h e i r two possible s t r u c t u r a l unit descriptions. I t i s seen t h a t the various structural units exhibit considerable d i s t o r t i o n s over the angular range in t h i s r i g i d representation as expected. The shaded regions represent the primary (Cl) GBD cores and the c i r c l e d symbols a r e t h e CSL points. Of course, the GBD arrays a r e purely forma1 (delocalized) in these r i g i d s t r u c t u r e s but become i d e a l l y localized l i n e defects when we visualize imposing " f u l l reversal" rotations

( i . e . , a rotation of each crystal by -8/2) about appropriate 'O' l a t t i c e elements normal t o the boundary t o recreate patches of perfect s i n g l e crystal centered on each ' 0 ' element. On the e110> branch the appropriate 'O' l a t t i c e i s the one parallel t o c110> w i t h l a t t i c e spacing given by d = 211/2<110>(/sin(8/2), and on t h e

<100> branch t h e ' 0 ' l a t t i c e i s parallel t o <100> with spacing d = 21<100>l/sin(8/2),.

As an example, some of the appropriate 'O' l a t t i c e elements a r e indicated by large + ' s in Fig. 2 ( a ) . W e note t h a t i f 'O' l a t t i c e relaxations occur which a r e l e s s than ideal ( i . e . , l e s s than f u l l reversa1 r o t a t i o n s ) a structural unit model r e s u l t s which consists of u n i t s possessing d i s t o r t i o n s which a r e intermediate between those of the highly d i s t o r t e d r i g i d model and the minimally distorted i d e a l l y relaxed model. In terms of GBDis, boundaries in the range 0-36.9' then consist of arrays of primary screw GBDis with e i t h e r Burgers vector 1/2<110> o r <100> depending on the choice of SU description. In the range 36.9"-45" the screw GBD's only have Burgers vector <100>. When the Burgers vector i s 1/2<110> t h e intersections of the screw dislocations a r e the minority <100> u n i t s and when the Burgers vector i s <100> the intersections of t h e screw dislocations a r e t h e minority <110> units. We note t h a t t h i s description i s more primitive than S u t t o n ' s (9) who a l s o took the C5 boundary t o be delimiting. W e consider t h a t description next.

2.2 Ll/Z5 DB description - The next most elementary description i s obtained by taking the 25 boundary as an additional DB, since i t s half-period vector i s the next l a r g e s t period o r half-period vector. Since, as described i n the previous section, two d i s t i n c t SU/GBD descriptions e x i s t i n the range 0<8$45", the DB's must be taken in p a i r s so t h a t f o r L5 the two crystallographically equivalent s t r u c t u r e s will each be represented by s t r u c t u r a l units with period vectors 1/2<210> and 1/2<310> as shown in Fig. l ( b ) . As f a r a s constructing boundaries by mixing units i s concerned, the range 0-45" on t h e <110> branch ( l e f t half of Fig. l ( b ) ) i s divided into two regions where from 0-36.9" <110> and <210> u n i t s are mixed and from 36.9"-45" <210>

and <310> units are mixed. On t h e other hand, f o r the <100> branch ( r i g h t half of Fig. 1 ( b ) ) <310> and <100> units a r e mixed in the range 0-36.g0, and <310> and <210>

uni t s a r e mixed i n the range 36 .go-45". A t 22.6", (Cl 3) on the <170> branch equal mixtures of units a r e obtained. Equal mixtures of appropriate units a r e also ob- tained a t 28.1" (117) and 43.6' ( ~ 2 9 ) on the <100> branch of t h i s Cl/C5 DB descrip- tion. The C5 DB's can now be considered a s MURS'S since they a r e a c t u a l l y composed of ( d i s t o r t e d ) Cl units. Fig. 3 ( a - f ) shows the C265, C281 and C305 dichromatic patterns each in t h e i r two possible Cl/C5 representations. The shaded regions again represent the GBD cores in t h e i r distorted inCipient form and a r e seen t o be quite d i f f e r e n t from Fig. 2 ( a - f ) . I f ideal ized rotational 'O' l a t t i c e relaxations are again visualized t o occur the s t r u c t u r e s may be interpreted in terrns of G B D ' s . For t h e L265 boundary t h e GBD1s are s t i l l primary w i t h Burgers vector 1/2<110> (Fig. 3a) or <100> (Fig. 3d), but the intersections of the GBDis a r e now the minority units

<210> and <310> respectively. For the C281 boundary the GBD arrays a r e no longer primary but secondary with Burgers vector 1/5<210> (Fig. 3b) o r 1/10<310> (Fig. 3e) with t h e intersection minority units'being of <110> o r <100> type respectively.

For the 1305 boundary the GBD arrays are again secondary with-Burgers vector1/54210>

(Fig. 3c) or 1/10<310> (Fig. 3f) but with the intersection minority u n i t s now of

<310> o r <210> type respectîvely. These r e s u l t s demonstrate the non-uniqueness of

the various GBD descriptions of boundaries within a given hierarchy in the manner

already demonstrated f o r t i 1 t boundaries ( 6 ) . Clearly, the GBD description most

appropriate f o r a p a r t i c u l a r boundary i s determined by how close i t i s t o an adja-

cent DB. As Fig. 1 (b) i l l u s t r a t e s , the Cl/C5 SU/GBD description quantised the

angular range i n t o additional dislocation regions where primary GBD's are found near

Fig. 2:

cl DB descriptions of [O011 t w i s t boundaries using r i g i d di chromatic patterns. The

<110> branch. Structures viewed along t w i s t axis show two (002) planes using d i f - f e r e n t symbol ^{S .} Ci r c l ed symbols define CSL points and large +s indicate some of the '0' l a t t i c e elements.

Shaded regions represent the GBD arrays. The minori t y u n i t s a r e located a t the intersections of t h e GBDs.

( a ) C265 (8 = 21.2') s t r u c - t u r e . Burgers vector of GBDs is 1/2<110>. Minority u n i t s a r e <100>. (b) C281

(9 = 34.7') s t r u c t u r e .

Burgers vector of GBDs i s

1/2<110>. Minori t y units

a r e <100>. ( c ) C305 ( 8 = 42.7')

s t r u c t u r e . Burgers vector

of GBDs 1s <100>. Minority

units a r e <110>.

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F i g . 2 (continued):

Ll DB descriptions of [O011 twist boundaries using r i g i d dichromatic patterns. The

<100> branch. Structures viewed along t w i s t axis show two (002) planes using d i f - f e r e n t symbol S. Ci rcl ed symbols define CSL points.

Shaded regions represent t h e GBD arrays. The minority units a r e located a t the intersections of t h e GBDs.

( d ) L265 (8 = 21 .ZO) struc- t u r e . Burgers vector of GBDs i s <100>. Minority units a r e <110>. ( e ) C281 (8 = 34.7') s t r u c t u r e . Burgers vector of GBDs i s <100>. Minori ty units a r e <110>. ( f ) 1305 (O = 42.7') s t r u c t u r e . Burgers vector of GBDs i s

<100>. Minority units are

Fig. 3:

Cl/C5 DE descriptions of [OOl]

t w i s t boundaries using r i g i d dichromatic patterns. The

4 1 O> branch. Structures vlewed along t w i s t axis show two (002) planes using d i f - f e r e n t symbols. Circled symbols define CSL points.

Shaded regi ons represent the GBD arrays. The minority units a r e located a t the intersections of the GBDs.

( a ) C265 (e = 21.2') s t r u c - t u r e . Burgers vector of GBDs i s 1/2<110>. Minority units a r e <210>. ( b j c281 ( 0 = 34.7') s t r u c t u r e . Burgers vector of GBDs i s 1/5<210>. Minori t y uni t s are

< i i o > . ( C I ~ 3 0 5 ( e = 42.70)

s t r u c t u r e . Burgers vector of

GBDs i s 1/5<210>. Minority

units a r e <310>.

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Fig. 3 (continued) :

Cl/C5 DB descriptions of [OOl]

t w i s t boundaries using r i g i d dichromati c patterns. The

<100> branch . Structures viewed along t h e t w l s t axis show two (002) planes using di f f e r e n t symbols. Circled symbols define the CSL points.

Shaded regions represent t h e GBD arrays. The minoi-i t y units a r e located a t the intersections of t h e GBDs.

( d ) C265 (8 = 21.2') struc- t u r e . Burgers vector of GBDs i s <100>. Minority uni t s a r e <310>. (e) C281

(8 = 34.7') s t r u c t u r e .

Burgers vector of GBDs i s

1/10<310>. Minority uni t s

a r e <100>. ( f ) C305 (8 = 42.7')

s t r u c t u r e . Burgers vector of

GBDs i s 1/10<310>. Minori t y

units a r e <210>.

the Cl boundaries and secondary GBD's found near the C5 boundaries. A t the 1 :1 boundaries ( ~ 1 3 , C29 and C17) dual descriptions e x i s t .

2.3 Cl/C5/C13/C17 DB description - W e include here the t h i r d most elementary SU/GBD description t o i l l u s t r a t e f u r t h e r how t h e hierarchy i s formed. Two additional pairs of DB's (Cl3 and 117) with next s h o r t e s t period vectors are added t o t h e system as shown in Fig. l ( c ) . For brevity we do not show t h e dichromatic patberns f o r t h i s system. However, following the same rules of mixing appropriate units and i n t e r - preting the r e s u l t s in terms of GBD's we see t h a t the angular range i s quantised f u r t h e r so t h a t GBD's a r e formed around each DB with Burgers vector which i s e i t h e r a primitive o r non-primitive vector of the corresponding DSC l a t t i c e . W notice e t h a t the s t r u c t u r a l uni t s al t e r n a t e primi tivelnon-primi t i v e as the misorientation angle increases, a r e s u l t which follows naturally from the i n i t i a l choice of Cl units. Note f u r t h e r t h a t each DB except Cl i s a MURS and t h a t the 1 :1 boundaries, which have dual GBD descriptions and define the 'catchment areas" around each DB, increase i n number and a r e now C25, C89, 165, 129, 153, C85, and 137. I t i s easy t o see how the system, although systematic, becomes increasingly complex as f u r t h e r DB's are added and where, i n the extreme l i m i t , every boundary could be delimiting.

2.4 Secondary GBD's as perturbations in primary GBD arrays - I t was observed in the Introduction t h a t perturbations i n the primary GBD spacings correspond t o the secondary GBD content. That t h i s follows from the structural unit model can be seen by comparing the dichromatic patterns in Figure 2 and Figure 3. For example, on comparing Fiqure 2(b) and Figure 3(b) f o r the X281 boundary i t can be c l e a r l y seen t h a t the perturbation in the 1/2<110> primary GBD array corresponds exactly t o the secondary GBD array with Burgers vector 1/5<210>. Similarly, comparing Figs. 2 ( e ) and 3 ( e ) f o r C281 in the a l t e r n a t i v e s t r u c t u r a l unit model i t i s seen t h a t the per- turbation in the <100> primary GBD array corresponds t o the 1/10<310> secondary array. Clearly al1 secondary GBD's in the angular range 0-36.9' on the <110> branch can be considered perturbations in the 1/2<110> primary GBD's whereas al1 secondary GBD's in the range 0-45' on the <100> branch and 36.9-45' on the <110> branch can be considered perturbations in the <100> GBD's. Furthermore, perturbations can occur in the secondary GBD arrays themselves a s seen f o r example in Fig. 3 ( f ) where a non-uniform array of 1/10<310> secondary GBD's i s present corresponding t o pertur- bations in the <100> primary array in Fig. 2 ( f ) . In the hierarchy of descriptions such non-uniformities i n the secondaries would manifest themselves as higher order secondaries possessing DSC l a t t i c e vectors of an adjacent DB, C29 i n t h i s case. An i n t r i n s i c feature of t h e SU/GBD model presented so f a r i s t h a t t h e secondaries form a r a l l e l perturbations i n the primaries, and t h i s i s represented schematical l y in F i g 4 a where t i s the magnitude of the perturbations in the spacings of the as "e-77"

parallel GBD's, P . However, topologically there i s no reason why non-parallel per- turbations in the primary array cannot y i e l d secondary GBD's, and t h i s i s shown schematically i n Fig. 4(b) where t i s now a t 45' t o the primary array. Such an

"oblique" perturbation i s shown in Fig. 5 f o r C281 where the 1/10<310> secondary GBD's form as a non-parallel perturbation in the 1/2<110> primary array which may be

Fig. 4. Schematic representation of two d i f f e r e n t types of perturbations in the spacings of primary GBDs P , whicli y i e l d secondary GBDs S . ( a ) "para1 le1 " pertur- bations where t i s normal t o P and (b)

"oblique" perturbations where t i s a t 45'

t o P.

C4-164 JOURNAL DE PHYSIQUE

. Cl DB d e s c r i p t i o n of t h e C281 8 = 34.7") s t r u c t u r e i n which

-"t

t h e p e r t u r b a t i o n s of t h e primary GBDs of Burgers v e c t o r 1/2<110>

(shaded r e g i o n s ) a r e of t h e oblique ki nd . Resul ti ng secondary GBDs have Burgers vector 1/10<310>.

Compare with Figs. 2 ( e ) and 3 ( e ) .

compared with Fig. 2 ( e ) . Of course, an e q u i v a l e n t c o n s t r u c t i o n may be made f o r t h e

<210> r e p r e s e n t a t i o n of t h e C281 boundary where t h e 1/5<210> secondaries could be described a s oblique p e r t u r b a t i o n s of t h e <100> primaries but t h i s i s not shown here.

3. Comparison with Experirnental Observations - We f i r s t summarize t h e TEM and X-ray d i f f r a c t i o n observations on low and high angle [OOl] t w i s t boundaries i n gold a s they r e l a t e t o t h e e x i s t e n c e of t h e p o s s i b l e primary and secondary GBD r e l a x a t i o n s which we have formally described above. For f u r t h e r d e t a i l s and r e f e r e n c e s s e e B a l l u f f i e t a l . ( 1 1 ) . The d i f f r a c t i o n c o n t r a s t observed i n TEM shows t h a t low a n g l e boundaries (<20°) possess a uniform a r r a y of primary l a t t i c e GBD's with Burgers v e c t o r 1/2<110>. A t high angles around C5 (36.9" f 2 " ) a r r a y s of secondary GBD's a r e observed wi t h p r i m i t i v e Burgers vector 1/10<31 O>. Other weaker secondary GBD a r r a y s a r e observed by TEM a t intermediate angles ( ~ 1 3 , C17) and a l s o have p r i m i t i v e DSC Burgers v e c t o r s . The X-ray d i f f r a c t i o n o b s e r v a t i o n s show s t r o n g <110> ' 0 ' l a t t i c e r e f l e c t i o n s f o r low angle boundaries (<IO0), intermediate angle boundaries ( ~ 1 3 , %22.6O) and high angle boundaries (15, ~ 3 6 . 9 ' ) . These r e s u l t s i n d i c a t e t h e e x i s t e n c e o f primary <110> r e l a x a t i o n s . Furthermore, i t has been deduced from t h e measured X-ray i n t e n s i t i e s t h a t t h e s e r e l a x a t i o n s a r e l a r g e , i . e . , they correspond e s s e n t i a l l y t o i d e a l f u l l - r e v e r s a 1 r o t a t i o n s , even a t high angles. Measurements of boundary energy versus t w i s t angle i n d i c a t e t h e e x i s t e n c e of only one pronounced secondary GBD energy cusp a t t h e C5 m i s o r i e n t a t i o n which must be due t o t h e 1/10

<310> GBD's (11). From t h i s information we conclude t h a t t h e primary r e l a x a t i o n i s

<110>, a t l e a s t up t o 36.g0, and t h a t around t h a t angle p e r t u r b a t i o n s i n t h e <110>

primaries g i v e r i s e t o t h e 1/10<310> secondaries. Comparing t h e s e experimental d a t a with t h e SU/GBD model we can make t h e following i n f e r e n c e s with regard t o t h e s t r u c - t u r e of twist boundaries i n gold. A t low angles t h e <110> d e s c r i p t i o n ( r a t h e r than

<100>) i s c l e a r l y t h e p r e f e r r e d primary GBD d e s c r i p t i o n and t h a t any p e r t u r b a t i o n s i n t h e d i s t r i b u t i o n of t h e primaries predicted by t h e model must be so weak as t o be undetectable by p r e s e n t techniques. A t high angles (around 36.g0) p e r t u r b a t i o n s i n t h e <110> primaries appear which a r e s t r o n g enough t o give secondary GBD's whose Burgers v e c t o r i s of t h e p r i m i t i v e 1/10<310> type. These p e r t u r b a t i o n s a r e evidently o f t h e oblique kind, a s f o r example shown i n Fig. 5 , and can be understood on ener- g e t i c grounds s i n c e t h e long range e l a s t i c energy o f t h e 1/10<310> GBD's i s lower than t h e corresponding energy of t h e 1/5<210> GBD's which could conceivably have been produced by a p a r a l l e l p e r t u r b a t i o n . In general , i f t h e Burgers vectors o f t h e observed secondary G B D ' s a r e always p r i m i t i v e vectors o f t h e DSC l a t t i c e then they w i l l have occurred by oblique p e r t u r b a t i o n s i n t h e 1/2<110> primary a r r a y i f t h e

<110> ' 0 ' l a t t i c e i s a t 45' t o t h e CSL and by p a r a l l e l p e r t u r b a t i o n s i f t h e <110>

'O' l a t t i c e i s p a r a l l e l t o t h e CSL,

4. Comparison w i t h A t o m i s t i c C a l c u l a t i o n s - We now summarize t h e r e s u l t s of a v a i l - a b l e a t o m i s t i c c a l c u l a t i o n s f o r [OOl] t w i s t boundaries o b t a i n e d b y means o f molecular s t a t i c s and a p a i r - p o t e n t i a l model. The purpose h e r e i s t o determine how w e l l t h e c a l c u l a t e d s t r b c t u r k s conform t o t h e e x p e r i m e n t a l d a t a c i t e d above and t o t h e s t r u c - t u r a l u n i t model . T t i s c o n v e n i e n t t o d i s c u s s f i r s t t h e c a l c u l a t e d p r i m a r y r e l a x a - t i o n and t h e n t h e c a l c u l a t e d secondary r e l a x a t i o n . Brokman and B a l l u f f i ( 1 2 ) have analyzed t h e e a r l i e r c a l c u l a t i o n s o f B r i s t o w e and Crocker , ( l 3 ) , who employed an e m p i r i c a l p o t e n t i a l r e p r e s e n t i n g copper, f o r s e v e r a l s h o r t p e r i o d [O01 ] t w i s t boun- d a r i e s i n terms o f t h e p a t t e r n o f p r i m a r y r e l a x a t i o n and found t h a t f o r a l 1 boun- d a r i e s up t o 36.9' t h e p r i m a r y r e l a x a t i o n i s r o u g h l y r o t a t i o n a l about <110> ' 0 ' l a t t i c e elements. T h i s c o n c l u s i o n may a l s o be reached f r o m more r e c e n t c a l c u l a t i o n s b y Wolf ( 1 4 ) employing a v a r i e t y o f p a i r p o t e n t i a l s . However, t h e <110> p r i m a r y r e l a x a t i o n s a r e s t r o n g e r a t l o w a n g l e s and become weaker a t h i g h angles i n apparent disagreement w i t h t h e X-ray o b s e r v a t i o n s . The weakness o f t h e p r i m a r y r e l a x a t i o n a t h i g h angles, which i s t h e r e s u l t f o r a l 1 p o t e n t i a l s used t o date, has n o t been under- stood. To i l l u s t r a t e t h e t y p i c a l p a t t e r n o f c a l c u l a t e d p r i m a r y r e l a x a t i o n s a t low and h i g h angles we p r e s e n t t h e e q u i l i b r i u m s t r u c t u r e o f a L I 4 5 ( 8 = 6.7") and a Cl09 ( 8 = 33.4') [O01 ] t w i s t boundary computed w i t h t h e same p a i r p o t e n t i a l used b y B r i s t o w e and Crocker ( 1 3 ) . Comparison o f F i g s . 6 ( a ) and 7 ( a ) , w h i c h show t h e com- p u t e d displacement f i e l d s , c o n f i r m s t h a t t h e r e l a x a t i o n i s much s m a l l e r a t h i g h angle and t h a t i n b o t h cases i t i s c e n t e r e d on <110> ' 0 ' l a t t i c e elements. (However, n o t e t h a t because t h e El09 displacements a r e so small i t i s d i f f i c u l t t o s e p a r a t e t h e p r i m a r y and secondary d i s p l acement f i e l d s o r determine unambiguously t h e c e n t e r s of r o t a t i o n . ) S t r u c t u r a l u n i t d e s c r i p t i o n s o f t h e s e boundaries a r e a l s o g i v e n i n F i g s . 6 and 7. To h e l p determine which member o f t h e SU/GBD h i e r a r c h y i s m o s t p h y s i - c a l l y s i g n i f i c a n t , p r e v i o u s w o r k e r s have computed h y d r o s t a t i c s t r e s s f i e l d d i s t r i -

b u t i o n s (1, 10). Here we adopt a new and s i m p l e method f o r choosing t h e s t r u c t u r a l u n i t model i n which t h e k i n e m a t i c a l d i f f r a c t i o n c o n t r a s t i s c a l c u l a t e d f o r t h e boun- d a r y u s i n g t h e well-known column t e c h n i q u e ( 1 5 ) . T h i s c a l c u l a t i o n i s s e n s i t i v e t o t h e displacement f i e l d i n t h e l a t t i c e i n t h e v i c i n i t y o f t h e boundary, and i t essen- t i a l l y maps t h i s f i e l d as i t would be seen b y means o f d i f f r a c t i o n c o n t r a s t i n a TEM image. S i n c e t h e displacement f i e l d o f a square g r i d o f screw GBD's conforms t o t h e square g r i d o f screw GBD's t h e method images d i r e c t l y t h e geometry o f t h e GBD g r i d . The s c a t t e r e d a m p l i t u d e f r o m t h e model b i c r y s t a l was c a l c u l a t e d u s i n g t h e r e l a t i o n (15) :

Y ( S ) = C exp L i 2ng . R ( z ) ] . e x p [ i 2,rrsz)

c o l umn - -

where R(z) = displacement i n t h e l a t t i c e due t o t h e g r a i n boundary, z = d i s t a n c e a l o n g t h e column (which i s normal t o t h e boundary p l a n e ) , g = d i f f r a c t i o n v e c t o r , and s = d e v i a t i o n parameter. Since a l 1 displacements a r e E n w n f r o m t h e a t o m i s t i c c a l c u l a t i o n , t h e s c a t t e r e d i n t e n s i t y 1 = YY*, i s e a s i l y o b t a i n e d f r o m t h e above e q u a t i o n b y s i m p l e summation. The r e s u l t ( w i t h g = <220>) f o r t h e c l 4 5 boundary i s shown i n F i g . 6 ( c ) and p r o v i d e s c l e a r evidence f o r t h e e x i s t e n c e o f a s e t o f screw GBD's r u n n i n g e s s e n t i a l l y p a r a l l e l t o t h e d i f f r a c t i o n v e c t o r . A s i m i l a r b u t o r t h o - gonal r e s u l t was o b t a i n e d f o r a second g normal t o t h e f i r s t g. The d i r e c t i o n s and spacings o f t h e s e screw GBD's i s e n t i r e i y c o n s i s t e n t w i t h t h e " s t r u c t u r a 1 u n i t r e p r e - s e n t a t i o n o f t h e boundary shown i n F i g . 6 ( b ) which i s a member o f t h e Cl/C5 system ( t h e m i n o r i t y u n i t s a r e <210>C5 u n i t s and t h e m a j o r i t y u n i t s a r e <llO>Cl u n i t s ) . We n o t e t h a t i t would a l s o have been p o s s i b l e t o choose a C l d e s c r i p t i o n i n which t h e m i n o r i t y u n i t s a r e o f <100> t y p e b u t t h i s would l e a d t o s l i g h t l y l a r g e r d i s t o r t i o n s of t h e u n i t s . The Zl/C5 d e s c r i p t i o n i s t h e r e f o r e a reasonable c o m ~ r o m i s e choice.

The d i f f r a c t i o n c o n t r a s t c a l c u l a t i o n cou1 d a l s o g i v e i n f o r m a t i o n about t h e secondary GBD c o n t e n t o f t h e Cl09 boundary which i s expected t o have Burgers v e c t o r 1/10<310>. However, when t h e d i f f r a c t i o n c o n t r a s t c a l c u l a t i o n was performed w i t h

9 ⁼ <620> no c l e a r evidence f o r l i n e c o n t r a s t a l o n g <310> was found. (Also, w i t h

= <420> t h e r e was no evidence f o r l i n e c o n t r a s t a l o n g <210>.) Consequently i t was

i m p o s s i b l e t o choose a most p h y s i c a l l y s i g n i f i c a n t SU/GBD d e s c r i p t i o n f o r t h e c a l c u -

l a t e d boundary, and we t h e r e f o r e s i m p l y p r e s e n t t h e two most l i k e l y a l t e r n a t i v e s i n

C4-166 JOURNAL DE PHYSIQUE

Fig. 6 ( a ) : Computed displacement f i e l d of El45 t w i s t boundary.

Arrows show relaxations xl . Ci r c l e s a r e CSL points.

Fig. 6(c) : Scattered i n t e n s i t y 1 calculated f o r rel axed Cl45 struc- t u r e u s i n g Eqn. 1 wîth g parallel t o one s e t of screw dislocations a s shown by arrows.

Fig. 7 ( b ) : Zl/E5 DB description of computed Cl09 structure. Shaded regions represent the secondary GBDs of b ⁼ 1/5<210>. Minority u n i t s are<llO>.

FSg.: Cl/C5 DB description of computed Cl45 structure. Shaded regions represent primary GBDs of b = 1/2<110>. Minority units a r e - <210>.

Fig. !(a): Coniputed displacement f i e l d o f El09 twist boundary.

Arrows show r e l axati ans xl . Circl e s a r e CSL points.

@ ' ^xb

x % &y\ ^{, x ~:b&,} 2 ³

^O

x ' " ^x

x q ;

- /yxfi ^{kî>* *}

X A.. > X

x::

x ; :&,

,,*$y X;/=XyA;. \

AX A X -

-<,;y _,* z/xy/xxy ^*gx

A A

, y+;:..

Ly.;" $+fXA'S ^x .TA ^:

A

.,:<:;

x

x

; ^x

x ' ? Ax % A ' y ^,;;2 / = , ^;

@

,

, .*%Y: \>,;*- x

'@

Fig. 7 ( c ) : Cl/C5 OB description of

computed Cl 09 s t r u c t u r e . Shaded

regions represent secondary GBDs of

b = 1/10c310>. Minority u n i t s a r e

- <1 oo>.

DB (MURS) r

219 227 2 9 P l i

Fig. 8. Forma1 SU/GBD hierarchical description of [Oll] twist boundaries: ( a ) 11/Z3 DB description, (b) 11/13/19/Cll DB description, and ( c ) Cl/C3/19/Cll/117/

119/127 DB description. The DB units and t h e i r period vectors are shown below the misorientation range i n each case. (For 19, C11, 117, Cl9 and C27 the interna1 contents of t h e u n i t s have been omitted f o r simplicity.) The shaded regions adjacent t o the DBs define the angular range over which the s t r u c t u r e can be represented by the appropriate GBD of Burgers vector b.

Structures which have dual dislocation descriptions a r e indicated by 1 :1.

C4-168 JOURNAL DE PHYSIQUE

t h e Cl/C5 system. F i g u r e 7(b) shows t h e 1/5<210> d e s c r i p t i o n i n which t h e m i n o r i t y u n i t s a r e <110> and Fig. 7 ( c ) shows t h e 1/10<310> d e s c r i p t i o n i n which t h e m i n o r i t y u n i t s a r e <100>. I n b o t h cases a l 1 t h e u n i t s show considerable d i s t o r t i m and a r e f a r from " i d e a l . " For t h i s boundary, then, any secondary r e l a x a t i o n s appear t o be considerably weaker than the primary r e l a x a t i o n s . These c a l c u l a t i o n s m y be explained i n terms o f d e l o c a l i z e d secondary GBD cores o r p o s s i b l y t h e d i s s o c i a t i o n o f p e r f e c t secondary GBD's i n t o p a r t i a l GBD1s. Such d i s s o c i a t i o n c o u l d be p r a c t i c a l l y i n d i s t i n - guishable from more u n i f o r m core spreading f o r a boundary o f t h i s p e r i o d i c i t y . I t

i s known (13) t h a t t h e t h r e e h i g h e s t symmetry t r a n s l a t i o n s t a t e s o f 15 have very s i m i l a r energies f o r t h e copper p o t e n t i a l employed and t h e r e f o r e l a r g e domains o f them may c o e x i s t i n t h e boundary separated by p a r t i a l GBD1s. I n f a c t , i n c i p i e n t forms o f these domains n a t u r a l l y occur i n t h e dichromatic p a t t e r n o f any high-C boundary near C5. (We n o t e t h a t a s i m i l a r s t r u c t u r e c o n t a i n i n g these t h r e e domains, suggested by V i t e k e t a l . (16) f o r t h e Cl09 boundary, i s unacceptable s i n c e they a r e n o t separated by p a r t i f i GBD1s.) As an a d d i t i o n a l e x p l a n a t i o n i t i s a l s o p o s s i b l e t h a t t h e Cl09 boundary i s n o t c l o s e enough t o t h e DB (C5) t o d e t e c t i n d i v i d u a l w i d e l y spaced secondary GBD's.

I n f u r t h e r work, very r e c e n t c a l c u l a t i o n s on GBD core s ~ r e a d i n g , described i n ( I l ) , using the " s p l i n e 1" p o t e n t i a l f o r Cu and a pseudopotential f o r Al both used by Wo I f (14), have produced r e s u l t s i n d i c a t i n g s t r o n u e r r e l a x a t i o n s (and more l o c a l -

i z e d secondary GBD cores) i n near-C5 boundaries than those described above. Since these p o t e n t i a l s appear t o e x h i b i t small cusps i n the energy curve a t 15 (14) whereas t h e e m p i r i c a l p o t e n t i a l used by Bristowe and Crocker does n o t , t h e r e t h e r e f o r e seems t o be a d i r e c t c o r r e l a t i o n between t h e existence o f cusps and the l o c a l i z a t i o n o f GBD1s as m i g h t be expected. C a l c u l a t i o n s of t h e s t r u c t u r e o f t h e Cl09 boundary and o t h e r high-C boundaries using these p o t e n t i a l s (and p o s s i b l y o t h e r more r e a l i s t i c p o t e n t i a l s ) are therefore c l e a r l y necessary b u t have n o t y e t been performed. How- ever, on t h e b a s i s of these r e c e n t r e s u l t s , i t seems l i k e l y t h a t such c a l c u l a t i o n s would a l l o w a more d i s t i n c t SU/GBD d e s c r i p t i o n which would be o f t h e 1/10 <310>type.

5. Discussion and Concl usions

Several caveats should be mentioned a t t h i s p o i n t . The forma1 SU/GBD models presented i n 53 were based on dichromatic p a t t e r n s w i t h f u l l c r y s t a l d e n s i t y and w i t h the two c r y s t a l s i n the coincidence p o s i t i o n (131, i .e., no vacant s i t e s and a l s o no d i f f e r i n g t r a n s l a t i o n s t a t e s were a l lowed. Several a t o m i s t i c c a l c u l a t i o n s (4,13) have r e p o r t e d the importance o f i n t r o d u c i n g these a d d i t i o n a l v a r i a b l e s t o c r e a t e m u l t i p l i c i t i e s o f s t r u c t u r e s and p a r t i a l GBD's. While i t i s b e l i e v e d t h a t these are i m p o r t a n t p o s s i b i 1 it i e s the general framework o f t h e model w i l l remain t h e same.

We a l s o remark t h a t the [ O O l ] t w i s t system, on which Our a t t e n t i o n has been focused, i s a somewhat s p e c i a l system because o f i t s h i g h symmetry ( c u b i c ) and n o t e t h a t some of the f e a t u r e s described above may n o t occur i n more general t w i s t boun- daries, e.g., the p o s s i b i l i t y o f having b o t h p r i m i t i v e and n o n - p r i m i t i v e DSC l a t t i c e GBD's. As an example o f another system w i t h l e s s symmetry c u r r e n t l y b e i n g i n v e s t i - gated we i l l u s t r a t e i n F i g . 8 t h e forma1 h i e r a r c h y o f GBD d e s c r i p t i o n s f o r [ O l l ] t w i s t boundaries. I n t h i s system, i n c o n t r a s t t o t h e case o f [ O O l ] t w i s t boundaries, secondary GBD1s have been observed e x p e r i m e n t a l l y near 13 and c9, and a l s o evidence f o r r e l a t i v e l y deep c a l c u l a t e d secondary GBD cusps on t h e energy curve has been obtained (1 1 ) .

F i n a l l y Our conclusions are:

( i ) The systematics o f [ O O l ] t w i s t boundary s t r u c t u r e can be understood f o r m a l l y i n terms o f a s t r u c t u r a l u n i t / g r a i n boundary d i s l o c a t i o n (SU/GBD) h i e r a r - c h i c a l model .

( i i ) The physical s i g n i f i c a n c e o f i n d i v i d u a l members of t h e h i e r a r c h y can be

determined by a t o m i s t i c c a l c u l a t i o n and a l s o comparison w i t h experimental obser-

v a t i o n .

( i ii ) Comparison o f t h e SU/GBD model w i t h a v a i l a b l e experimental r e s u l t s f o r g o l d i n d i c a t e t h a t t h e p r i m a r y r e l a x a t i o n i s o f t h e <110> t y p e and i s s t r o n g (cor- responding t o almost f u l l - r e v e r s a 1 r o t a t i o n s ) f o r 0 < 36.9'. I n a d d i t i o n , observed secondary GBD's near Z5 must r e s u l t from " o b l i q u e " p e r t u r b a t i o n s i n the a r r a y o f primary GBD1s.

( i v ) Comparison o f t h e SU/GBD model w i t h corresponding a t o m i s t i c pair-poten- t i a l model c a l c u l a t i o n s which are a v a i l a b l e i n d i c a t e s a s t r o n g primary <110> r e l a x a - t i o n a t low angles ( 8 < 22.6O) b u t a p r o g r e s s i v e l y weaker r e l a x a t i o n a t h i g h e r angles. The weakness o f t h e primary r e l a x a t i o n a t h i g h angles, which i s a t variance w i t h t h e experimental r e s u l t s , has n o t been understood. Separately. no evidence i s found f o r t h e existence o f s i g n i f i c a n t secondary r e l a x a t i o n s when a t l e a s t one e m p i r i c a l p a i r p o t e n t i a l i s employed. However, v e r y r e c e n t GBD c o r e c a l c u l a t i o n s i n d i c a t e t h a t s t r o n g e r secondary r e l a x a t i o n s can be obtained w i t h o t h e r p o t e n t i a l s and t h e r e f o r e i t i s l i k e l y t h a t b e t t e r agreement w i t h observations can be achieved i n f u t u r e work.

Acknoviledgment - This research was supported by the U.S. Department o f Energy under Contract NO. DE-FG02-84-ER-45116.

References

( 1 ) SUTTON, A.P., and VITEK, V., P h i l . Trans. Roy. Soc. Lond. (1983) 1.

(2) SUTTON, A.P., and VITEK, V., P h i l . Trans. Roy. Soc. Lond. A309 (1983) 37.

( 3 ) SUTTON, A.P., and VITEK, V., P h i l . Trans. Roy. Soc. Lond. A309 (1983) 55.

( 4 ) WANG, G.J., SUTTON, A.P., and VITEK, V., Acta Met. 32 ( 1 9 8 v 0 9 3 . ( 5 ) BISHOP, G.H., and CHALMERS, B., S c r i p t a Met. 2 ( 1 9 6 v 133.

( 6 ) BALLUFFI, R.W., and BRISTOWE, P.D., S u r f . S c i . 144 (1984) 28.

( 7 ) SUTTON, A.P., Met. Reviews i n press.

( 8 ) BALLUFFI, R.W., and OLSON, G.B., Trans. AIME i n press.

( 9 ) SUTTON, A.P., P h i l . Mag. A46 (1982) 171.

(10) SCHWARTZ, D., SUTTON, A . p T a n d VITEK, V., P h i l . Mag. i n press.

(1 1) BALLUFFI , R.W., BRISTOWE, P.D., BABCOCK, S.E., CHAN, S.W., KVAM, E.P., and LIU, J.S., 3. de Physique, t h i s volume.

(12) BROKMAN, A., and BALLUFFI, R.W., Acta Met. 29 (1981) 1703.

(13) BRISTOWE, P.D., and CROCKER, A.G., P h i l . M a c A 3 8 (1978) 487.

(14) WOLF, D., Acta Met. (1984) 245; (1984) 7 3 5 .

(15) HIRSH, P.B., HOWIE, A., NICHOLSON, R.B., PASHLEY, D.W., and WHELAN, M.J., i n

" E l e c t r o n Microscopy o f Thin C r y s t a l s u (Butterworths, London, 1965) Ch. 7.

(161 VITEK, V., SUTTON, A.P., WANG, G.J., and SCHWARTZ, D., S c r i p t . Met. (1983) 183.

DISCUSSION

D. Wolf: I n my own c a l c u l a t i o n s on (100) CSL t w i s t boundaries i n A l , Cu, Ag, and Au (Acta Meta11 , ^1984), some p o t e n t i a l s y i e l d e d minor cusps near t h e Z =5 o r i e n t a t i o n s whereas o t h e r gave completely smooth energy vs. m i s f i t - a n g l e curves. Have you checked whether t h e p o t e n t i a l you used g i v e s a s m a l l energy cusp f o r Z = 5 ? Would you expect t h a t i f a p o t e n t i a l does n o t show such a cusp you would a l s o n o t expect secondary g r a i n boundary d i s l o c a t i o n s nearby?

P.D. Bristowe: F o r t h e copper e m p i r i c a l p o t e n t i a l used i n t h i s paper no

c a l c u l a t i o n s o f h i g h - 2 boundaries w i t h i n do o f 2=5 have been made b u t a l 1

i n d i c a t i o n s a r e t h a t t h e energy curve i s smooth and t h a t t h e r e f o r e t h e secondary

C4-170 JOURNAL DE PHYSIQUE

GBDs would be d e l o c a l i z e d and t h i s is supported by Our d i f f r a c t i o n c o n t r a s t c a l c u l a t i o n s . However, a s mentioned i n t h e t e x t , r e c e n t GBD c o r e c a l c u l a t i o n s employing o t h e r p o t e n t i a l s used by y o u r s e l f which do g i v e small cusps have e s t a b l i s h e d a c o r r e l a t i o n between GBD l o c a l i z a t i o n and t h e e x i s t e n c e o f cusps.

H. Sawhill: 1 t h i n k t h a t t h e discrepancy between observed secondary d i s l o c a t i o n s near .2=5 and t h e l a c k o f c o n t r a s t p r e d i c t e d using kinematic d i f f r a c t i o n theory is n o t such a major discrepancy. I f you look a t Bloch Wave s c a t t e r i n g (dynamic d i f f r a c t i o n ) you w i l l s e e t h a t t h e r e s u l t s i n f a c t do p r e d i c t t h a t t h e s e small p e r t u r b a t i o n s w i l l be seen i n t h e b r i g h t f i e l d images.

P.D. Bristowe: F i r s t l y , it should be emphasized t h a t we have used t h e kinematic c a l c u l a t i o n o f t h e d i f f r a c t i o n c o n t r a s t a s a simple t y p e o f "phase i n t e g r a l probe1>

f o r i n v e s t i g a t i n g t h e p o s s i b l e e x i s t e n c e o f a s y s t e m a t i c displacement f i e l d

a d j a c e n t t o t h e boundary which would be a s s o c i a t e d w i t h t h e presence o f p h y s i c a l l y

s i g n i f i c a n t GBDs. The method c l e a r l y revealed t h e presence o f primary GBDs i n t h e

1145 boundary (Fig. 6 c ) . The f a c t t h a t no d e t e c t a b l e d i f f r a c t i o n c o n t r a s t was

p r e s e n t i n t h e s i m i l a r c a l c u l a t i o n f o r t h e E l 0 9 boundary must i n d i c a t e t h a t only

displacement f i e l d due t o secondary GBDs i n t h i s boundary is exceedingly weak by

c m p a r i s o n a s concluded i n t h e t e x t . We f e e l it u n l i k e l y t h a t t h i s r e s u l t w i l l be

overturned by t h e r e s u l t s o f dynamical c a l c u l a t i o n s . Nevertheless, we plan t o

confirm t h i s i n f u t u r e work.