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On the detection of expansion at large angle grain boundaries using electron diffraction
Scripta METALLURGICA
Vol. 18, pp. 617-620, 1984 Printed in the U.S.A.
Pergamon Press Ltd All rights reserve,
ON THE DETECTION OF EXPANSION AT LARGE ANGLE GRAIN BOUNDARIES USING ELECTRON DIFFRACTION R. W. B a l l u f f i and P. D. Bristowe Department of Materials Science and Engineering Massachusetts I n s t i t u t e of Technology Cambridge, Massachusetts 02139 (Received March 2, 1984) (Revised March 30, 1984) Introduction Lamarre and Sass (LS) [ I ] and Lamarre, et al. [2] have recently observed a new grain boundary electron d i f f r a c t i o n e f f e c t from a large angle [001] t w i s t boundary in gold and in nickel oxide which they suggest can be used in a simple way to probe changes in i n t e r p l a n a r spacing at the grain boundary in a d i r e c t i o n normal to the boundary plane. The observed e f f e c t consists of a length of r e l r o d in reciprocal space l y i n g p a r a l l e l to [002] j u s t inside the two superimposed (002) reciprocal l a t t i c e points of Lattices 1 and 2 as i l l u s t r a t e d in Fig. l ( a ) . LS argue that t h i s observation is evidence f o r the existence of a relrod of the form shown schem a t i c a l l y in Fig. l ( b ) even though t h i s cannot be confirmed because of the presence of the patch of heavy f i l m exposure near k = [00211,2 (due to the enormous i n t e n s i t y of the superimposed (002)1,2 r e f l e c t i o n s ) whichmasks everything in t h i s region of reciprocal space as i l l u s t r a t e d in Fig. l ( a ) . Here, k is the d i f f r a c t i o n vector. LS also assume that such an assumed d i f f r a c t i o n e f f e c t can b e - a t t r i b u t e d to a t h i n slab of grain boundary possessing a s l i g h t l y l a r g e r i n t e r p l a n a r spacing normal to the boundary ( i . e . , along [002]). They adopt a simple model in which such a grain boundary slab acts e f f e c t i v e l y as a separate t h i n crystal producing i t s own relrod which, due to i t s s l i g h t l y larger average i n t e r p l a n a r spacing, possesses an i n t e n s i t y maximum located nearer to the o r i g i n of reciprocal space than the r e f l e c tions from the unperturbed Lattices 1 and 2. From an estimate of the position of the assumed i n t e n s i t y maximum they estimate the average increase of the (002) i n t e r p l a n a r spacing in the grain boundary region. For a 8 = 22 ° [001] t w i s t boundary in gold the a p p l i c a t i o n of the model to the observed r e s u l t y i e l d s a value of about 5% f o r the average i n t e r p l a n a r d i l a t i o n . However, since no information on the thickness of the grain boundary slab could be deduced from the experiment the t o t a l expansion remains undetermined. As shown by Brokman and B a l l u f f i [3] a model of t h i s type should hold for grain boundary r e f l e c t i o n s from t w i s t boundaries occurring at positions in reciprocal space on elements of the "boundary d i f f r a c t i o n l a t t i c e " (BDL) which do not contain l a t t i c e r e f l e c t i o n s . In such cases the i n t e n s i t i e s scattered by the boundary region and the two unperturbed l a t t i c e s can be separated to a good approximation, and the s c a t t e r i n g from the boundary region can be considered independently. However, in the present case, where the i n t e n s i t y from the grain boundary region, considered separately, would appear close to l a t t i c e r e f l e c t i o n s on the same element of the BDL, i t is not clear that t h i s is the case. I t may, therefore, be necessary to perform a more complicated analysis taking account of the t o t a l s c a t t e r i n g from the grain boundary region and the two adjoining l a t t i c e s . The purpose of the present note is to present such an analysis. The r e s u l t s show that the more complicated analysis is indeed required and a more r e a l i s t i c explanation of the observed r e s u l t [Fig. l ( a ) ] is discussed. Model and Analysis We r e s t r i c t ourselves to a determination of the d i f f r a c t e d i n t e n s i t y along [002] in reciprocal space. In order to reveal the main features of the d i f f r a c t i o n effects to be expected, we employ the r e l a t i v e l y simple b i c r y s t a l model i l l u s t r a t e d in Fig. 2. The geometry corresponds closely to that of the LS experiment [ I ] where the region i l l u m i n a t e d by the electron
617 0036-9748/84 $3.00 + .00 Copyright (c) 1984 Pergamon Press
Ltd.
EXPANSION AT LARGE ANGLE
618
GRAIN BOUNDARIES
Vol.
18, No.
6
beam consisted of a c y l i n d r i c a l region c o n t a i n i n g the transverse grain boundary. L a t t i c e s 1 and 2 are each h a l f c y l i n d e r s of radius R c o n t a i n i n g sets o f p a r a l l e l (002) planes o f spacing u n i t y . For c l a r i t y in t h i s f i g u r e we show a grain boundary region between the two l a t t i c e s which has a thickness o f f o u r (002) planes but in our c a l c u l a t i o n s s i x and e i g h t plane regions were also considered. For the f o u r plane model shown a d i l a t i o n parameter ~ is introduced such t h a t the spacing between planes 1 and -I is 1 + ci and the spacing between planes -2 and - I , and 1 and 2 is 1 + ~2. Thus, the t o t a l expansion E in t h i s case is ¢i + 2~2 but in general would be given by n E = cI + 2 ~ ¢i (in u n i t s o f i n t e r p l a n a r spacings) ( I ) i#l f o r a 2n plane boundary region, Also, i t is assumed, o f course, t h a t Ei decreases w i t h distance from the boundary. Since we s h a l l o n l y c a l c u l a t e the d i f f r a c t e d i n t e n s i t y along [002] we may regard the (002) planes as s t r u c t u r e l e s s in the x and y d i r e c t i o n s and w r i t e the scattered amplitude from the e n t i r e specimen (per u n i t area o f grain boundary) as Y =
~ exp [ i 2 ~ k z - Z ] a l l planes
.
(2)
For the four plane grain boundary model involving just two ¢i's this may be rewritten: N-2 Y = ~ {l - [l - ~L/(N-2)]2}I/2 • exp [i2~kz.C] + ~L=O exp[i2~kz(N-I + ~2)] + exp[i2~kz(N + ~2 + E l ) ] +
N-2 S {I - [ ~ / ( N - 2 ) ] 2 } I/2" • exp[i2~kz(~ + N + 1 + 2~2 + E l ) ] , ~=0
(3)
where N = one-half the total number of planes in the complete model. Similar expressions can be obtained for larger grain boundary regions. The scattered intensity is then given, as usual, by: I = YY* . (4) Values of I were calculated as a wide variety of N and ei. As is scale in the range between 0 < f and k=l. In order to smooth the were determined by averaging the
function of f = kz/k(o02 ) by means of these equations for a well known, the function of I ( f ) oscillates on a very fine < l and possesses at most 2N-2 subsidiary maxima between k=O results, average local values of I ( f ) , i . e . , values of T ( f ) , function over at least several subsidiary maxima. Results and Conclusions
All calculated curves showed the expected strong l a t t i c e reflection peaks near f = I. A series of typical results is plotted in Fig. 3. The major features of the calculated results may be summarized as follows: (1) for N ~ 200 the values of T(f) becameessentially independent of N for a l l values of f except for those very near f = l at the centers of the peaks; (2) the main effect of introducing expansion in the grain boundary region was to decrease the peak heights and increase the intensity in t h e i r v i c i n i t i e s as seen in Fig. 3. The increase in intensity was generally asymmetric and greater towards the o r i g i n ; (3) these effects were dependent on the thickness of the grain boundary region and magnitude of the interplanar d i l a tions; (4) the intensities f e l l o f f monotonically on both sides of the peaks; i . e . , no evidence was found for the existence of a distinguishable grain boundary relrod of the type assumed by LS; and (5) when negative values of ¢i were used, thereby introducing a boundary contraction, the asymmetry effect reversed and intensity was greater away from the origin (this appears to correspond to the d i f f r a c t i o n observation of Lamarre, et al. [2] for a boundary in germanium). Any physically r e a l i s t i c grain boundary expansion should f a l l within the range covered by the present calculations. Our results therefore indicate that one should not expect to find a distinguishable grain boundary relrod in the near v i c i n i t y of a strong l a t t i c e reflection
Vol.
18,
No.
6
EXPANSION AT LARGE ANGLE GRAIN BOUNDARIES
619
l y i n g on the same element of the BDL as in the present case. We conclude, therefore, that in such a case the c a l c u l a t i o n of the s c a t t e r i n g must include the c o n t r i b u t i o n s from the grain boundary region and the two adjoining l a t t i c e s , i n contrast to the model of LS. I t now remains to explain the observed d i f f r a c t i o n e f f e c t in Fig. l ( a ) . Our c a l c u l a t i o n s show that grain boundary expansion can produce considerable i n t e n s i t y in the region adjacent to the main peaks which is asjanmetrically d i s t r i b u t e d in a form which, f o r a certain range of i n t e r p l a n a r d i l a t i o n s and boundary thickness, appears to be in agreement with the observed results. I t is impossible to make a precise comparison because of the q u a l i t a t i v e nature of the observed i n t e n s i t y d i s t r i b u t i o n data. However, our c a l c u l a t i o n s indicate t h a t : ( I ) the boundary thickness has to be at least four (002) planes to generate any noticable asymmetric behavior in the peak i n t e n s i t y ; and (2) f o r boundary regions that are four to eight (002) planes t h i c k the t o t a l expansion E which best f i t s the data l i e s in the range of 30-45%, while the average i n t e r p l a n a r d i l a t i o n ( i . e . , E / ( 2 n - l ) ) l i e s in the range 5-15%. Numerous combinations of the ci can lead to these r e s u l t s . I t is seen that the 5% value f o r the average i n t e r p l a n a r d i l a t i o n deduced by LS on the basis of an o v e r s i m p l i f i e d model f a l l s at the lower end of our calculated r e s u l t s . I t is also i n t e r e s t i n g to compare our calculated t o t a l expansions with the 10-30% range obtained from various computer simulations for metals [ 4 , 5 , 6 ] and also the 41% value (concentrated at the boundary) f o r a hard-sphere model. We may conclude that the LS observations c e r t a i n l y provide useful q u a l i t a t i v e information regarding the existence of expansions (or contractions) at the boundary. I t is evident, however, that any serious e f f o r t to obtain a more detailed and exact i n t e r p r e t a t i o n of the observed d i f f r a c t i o n e f f e c t w i l l require q u a n t i t a t i v e measurements of the absolute i n t e n s i t i e s along [002] a n d , f u r t h e r , more accurate representations of the boundary structure and the scatt e r i n g from the e n t i r e b i c r y s t a l . In addition i t may be necessary to perform measurements in other regions of reciprocal space where the s c a t t e r i n g from the grain boundary region can be considered independently from the two adjoining l a t t i c e s , Acknowledgement This work was supported by the U.S. Department of Energy under Contract DE-ACO2-78ER05002. References I. 2. 3. 4. 5. 6.
P, Lamarre and S.L. Sass, Scripta Metall. 17, 1141 (1983), P. Lamarre, F. SchmUckle, K. Sickafus and S,L. Sass, Proceedings of the Conference on "Atomic-Scale Structure and Properties of I n t e r f a c e s , " Arizona, 1984, in press. A. Brokman and R.W. B a l l u f f i , Acta Metall. 3__~I, 1639 (1983). P.D. Bristowe and A.G. Crocker, P h i l . Mag. A 38, 487 (1978). P.D. Bristowe and S.L. Sass, Acta Metall. 28, 575 (1980). D. Wolf, Acta Metall. 32, 245 (1984).
(o)
I
FIGURE 1
I i I I
(002) 1,2
(b) I , I 018
~
09
J .
.
.
.
.
. . ~0
.
.
I I
f
(a) Schematic representation of electron d i f f r a c t i o n pattern observed in [ l ] . Hatched c i r c l e depicts region of heavy f i l m exposure due to intense l a t t i c e r e f l e c t i o n s (002)1, 2 • (b) Form of grain boundary relrod assumed by LS [ l ] to account f o r the observation.
620
E X P A N S I O N AT LARGE ANGLE GRAIN B O U N D A R I E S
18, No. 6
ELECTRONBEAM
Ill LATTICE R \2
Vol.
I00 -2-I
I 2
LATTICE
= 025
I
_
/
8O I~
60 4O
~-ic~LJ~L~
(002) PLl NE
It~ I*E I I÷E2
Bicrystal model used for present calculations.
iI / /
t;2:o.,o\
l//
;T~FT
/
/I
111 II ~
I p
/ ~ ;
i ~
2
! :r,,:o
~-vo ;,,=o~
2O
08
FIGURE 2
I / r~l=O. 15
09
I0
II
1,2
FIGURE 3 Locally averaged scattering intensity, T ( f ) , along [002] direction calculated for model in Fig. 2.
Vol. 18, pp. 617-620, 1984 Printed in the U.S.A.
Pergamon Press Ltd All rights reserve,
ON THE DETECTION OF EXPANSION AT LARGE ANGLE GRAIN BOUNDARIES USING ELECTRON DIFFRACTION R. W. B a l l u f f i and P. D. Bristowe Department of Materials Science and Engineering Massachusetts I n s t i t u t e of Technology Cambridge, Massachusetts 02139 (Received March 2, 1984) (Revised March 30, 1984) Introduction Lamarre and Sass (LS) [ I ] and Lamarre, et al. [2] have recently observed a new grain boundary electron d i f f r a c t i o n e f f e c t from a large angle [001] t w i s t boundary in gold and in nickel oxide which they suggest can be used in a simple way to probe changes in i n t e r p l a n a r spacing at the grain boundary in a d i r e c t i o n normal to the boundary plane. The observed e f f e c t consists of a length of r e l r o d in reciprocal space l y i n g p a r a l l e l to [002] j u s t inside the two superimposed (002) reciprocal l a t t i c e points of Lattices 1 and 2 as i l l u s t r a t e d in Fig. l ( a ) . LS argue that t h i s observation is evidence f o r the existence of a relrod of the form shown schem a t i c a l l y in Fig. l ( b ) even though t h i s cannot be confirmed because of the presence of the patch of heavy f i l m exposure near k = [00211,2 (due to the enormous i n t e n s i t y of the superimposed (002)1,2 r e f l e c t i o n s ) whichmasks everything in t h i s region of reciprocal space as i l l u s t r a t e d in Fig. l ( a ) . Here, k is the d i f f r a c t i o n vector. LS also assume that such an assumed d i f f r a c t i o n e f f e c t can b e - a t t r i b u t e d to a t h i n slab of grain boundary possessing a s l i g h t l y l a r g e r i n t e r p l a n a r spacing normal to the boundary ( i . e . , along [002]). They adopt a simple model in which such a grain boundary slab acts e f f e c t i v e l y as a separate t h i n crystal producing i t s own relrod which, due to i t s s l i g h t l y larger average i n t e r p l a n a r spacing, possesses an i n t e n s i t y maximum located nearer to the o r i g i n of reciprocal space than the r e f l e c tions from the unperturbed Lattices 1 and 2. From an estimate of the position of the assumed i n t e n s i t y maximum they estimate the average increase of the (002) i n t e r p l a n a r spacing in the grain boundary region. For a 8 = 22 ° [001] t w i s t boundary in gold the a p p l i c a t i o n of the model to the observed r e s u l t y i e l d s a value of about 5% f o r the average i n t e r p l a n a r d i l a t i o n . However, since no information on the thickness of the grain boundary slab could be deduced from the experiment the t o t a l expansion remains undetermined. As shown by Brokman and B a l l u f f i [3] a model of t h i s type should hold for grain boundary r e f l e c t i o n s from t w i s t boundaries occurring at positions in reciprocal space on elements of the "boundary d i f f r a c t i o n l a t t i c e " (BDL) which do not contain l a t t i c e r e f l e c t i o n s . In such cases the i n t e n s i t i e s scattered by the boundary region and the two unperturbed l a t t i c e s can be separated to a good approximation, and the s c a t t e r i n g from the boundary region can be considered independently. However, in the present case, where the i n t e n s i t y from the grain boundary region, considered separately, would appear close to l a t t i c e r e f l e c t i o n s on the same element of the BDL, i t is not clear that t h i s is the case. I t may, therefore, be necessary to perform a more complicated analysis taking account of the t o t a l s c a t t e r i n g from the grain boundary region and the two adjoining l a t t i c e s . The purpose of the present note is to present such an analysis. The r e s u l t s show that the more complicated analysis is indeed required and a more r e a l i s t i c explanation of the observed r e s u l t [Fig. l ( a ) ] is discussed. Model and Analysis We r e s t r i c t ourselves to a determination of the d i f f r a c t e d i n t e n s i t y along [002] in reciprocal space. In order to reveal the main features of the d i f f r a c t i o n effects to be expected, we employ the r e l a t i v e l y simple b i c r y s t a l model i l l u s t r a t e d in Fig. 2. The geometry corresponds closely to that of the LS experiment [ I ] where the region i l l u m i n a t e d by the electron
617 0036-9748/84 $3.00 + .00 Copyright (c) 1984 Pergamon Press
Ltd.
EXPANSION AT LARGE ANGLE
618
GRAIN BOUNDARIES
Vol.
18, No.
6
beam consisted of a c y l i n d r i c a l region c o n t a i n i n g the transverse grain boundary. L a t t i c e s 1 and 2 are each h a l f c y l i n d e r s of radius R c o n t a i n i n g sets o f p a r a l l e l (002) planes o f spacing u n i t y . For c l a r i t y in t h i s f i g u r e we show a grain boundary region between the two l a t t i c e s which has a thickness o f f o u r (002) planes but in our c a l c u l a t i o n s s i x and e i g h t plane regions were also considered. For the f o u r plane model shown a d i l a t i o n parameter ~ is introduced such t h a t the spacing between planes 1 and -I is 1 + ci and the spacing between planes -2 and - I , and 1 and 2 is 1 + ~2. Thus, the t o t a l expansion E in t h i s case is ¢i + 2~2 but in general would be given by n E = cI + 2 ~ ¢i (in u n i t s o f i n t e r p l a n a r spacings) ( I ) i#l f o r a 2n plane boundary region, Also, i t is assumed, o f course, t h a t Ei decreases w i t h distance from the boundary. Since we s h a l l o n l y c a l c u l a t e the d i f f r a c t e d i n t e n s i t y along [002] we may regard the (002) planes as s t r u c t u r e l e s s in the x and y d i r e c t i o n s and w r i t e the scattered amplitude from the e n t i r e specimen (per u n i t area o f grain boundary) as Y =
~ exp [ i 2 ~ k z - Z ] a l l planes
.
(2)
For the four plane grain boundary model involving just two ¢i's this may be rewritten: N-2 Y = ~ {l - [l - ~L/(N-2)]2}I/2 • exp [i2~kz.C] + ~L=O exp[i2~kz(N-I + ~2)] + exp[i2~kz(N + ~2 + E l ) ] +
N-2 S {I - [ ~ / ( N - 2 ) ] 2 } I/2" • exp[i2~kz(~ + N + 1 + 2~2 + E l ) ] , ~=0
(3)
where N = one-half the total number of planes in the complete model. Similar expressions can be obtained for larger grain boundary regions. The scattered intensity is then given, as usual, by: I = YY* . (4) Values of I were calculated as a wide variety of N and ei. As is scale in the range between 0 < f and k=l. In order to smooth the were determined by averaging the
function of f = kz/k(o02 ) by means of these equations for a well known, the function of I ( f ) oscillates on a very fine < l and possesses at most 2N-2 subsidiary maxima between k=O results, average local values of I ( f ) , i . e . , values of T ( f ) , function over at least several subsidiary maxima. Results and Conclusions
All calculated curves showed the expected strong l a t t i c e reflection peaks near f = I. A series of typical results is plotted in Fig. 3. The major features of the calculated results may be summarized as follows: (1) for N ~ 200 the values of T(f) becameessentially independent of N for a l l values of f except for those very near f = l at the centers of the peaks; (2) the main effect of introducing expansion in the grain boundary region was to decrease the peak heights and increase the intensity in t h e i r v i c i n i t i e s as seen in Fig. 3. The increase in intensity was generally asymmetric and greater towards the o r i g i n ; (3) these effects were dependent on the thickness of the grain boundary region and magnitude of the interplanar d i l a tions; (4) the intensities f e l l o f f monotonically on both sides of the peaks; i . e . , no evidence was found for the existence of a distinguishable grain boundary relrod of the type assumed by LS; and (5) when negative values of ¢i were used, thereby introducing a boundary contraction, the asymmetry effect reversed and intensity was greater away from the origin (this appears to correspond to the d i f f r a c t i o n observation of Lamarre, et al. [2] for a boundary in germanium). Any physically r e a l i s t i c grain boundary expansion should f a l l within the range covered by the present calculations. Our results therefore indicate that one should not expect to find a distinguishable grain boundary relrod in the near v i c i n i t y of a strong l a t t i c e reflection
Vol.
18,
No.
6
EXPANSION AT LARGE ANGLE GRAIN BOUNDARIES
619
l y i n g on the same element of the BDL as in the present case. We conclude, therefore, that in such a case the c a l c u l a t i o n of the s c a t t e r i n g must include the c o n t r i b u t i o n s from the grain boundary region and the two adjoining l a t t i c e s , i n contrast to the model of LS. I t now remains to explain the observed d i f f r a c t i o n e f f e c t in Fig. l ( a ) . Our c a l c u l a t i o n s show that grain boundary expansion can produce considerable i n t e n s i t y in the region adjacent to the main peaks which is asjanmetrically d i s t r i b u t e d in a form which, f o r a certain range of i n t e r p l a n a r d i l a t i o n s and boundary thickness, appears to be in agreement with the observed results. I t is impossible to make a precise comparison because of the q u a l i t a t i v e nature of the observed i n t e n s i t y d i s t r i b u t i o n data. However, our c a l c u l a t i o n s indicate t h a t : ( I ) the boundary thickness has to be at least four (002) planes to generate any noticable asymmetric behavior in the peak i n t e n s i t y ; and (2) f o r boundary regions that are four to eight (002) planes t h i c k the t o t a l expansion E which best f i t s the data l i e s in the range of 30-45%, while the average i n t e r p l a n a r d i l a t i o n ( i . e . , E / ( 2 n - l ) ) l i e s in the range 5-15%. Numerous combinations of the ci can lead to these r e s u l t s . I t is seen that the 5% value f o r the average i n t e r p l a n a r d i l a t i o n deduced by LS on the basis of an o v e r s i m p l i f i e d model f a l l s at the lower end of our calculated r e s u l t s . I t is also i n t e r e s t i n g to compare our calculated t o t a l expansions with the 10-30% range obtained from various computer simulations for metals [ 4 , 5 , 6 ] and also the 41% value (concentrated at the boundary) f o r a hard-sphere model. We may conclude that the LS observations c e r t a i n l y provide useful q u a l i t a t i v e information regarding the existence of expansions (or contractions) at the boundary. I t is evident, however, that any serious e f f o r t to obtain a more detailed and exact i n t e r p r e t a t i o n of the observed d i f f r a c t i o n e f f e c t w i l l require q u a n t i t a t i v e measurements of the absolute i n t e n s i t i e s along [002] a n d , f u r t h e r , more accurate representations of the boundary structure and the scatt e r i n g from the e n t i r e b i c r y s t a l . In addition i t may be necessary to perform measurements in other regions of reciprocal space where the s c a t t e r i n g from the grain boundary region can be considered independently from the two adjoining l a t t i c e s , Acknowledgement This work was supported by the U.S. Department of Energy under Contract DE-ACO2-78ER05002. References I. 2. 3. 4. 5. 6.
P, Lamarre and S.L. Sass, Scripta Metall. 17, 1141 (1983), P. Lamarre, F. SchmUckle, K. Sickafus and S,L. Sass, Proceedings of the Conference on "Atomic-Scale Structure and Properties of I n t e r f a c e s , " Arizona, 1984, in press. A. Brokman and R.W. B a l l u f f i , Acta Metall. 3__~I, 1639 (1983). P.D. Bristowe and A.G. Crocker, P h i l . Mag. A 38, 487 (1978). P.D. Bristowe and S.L. Sass, Acta Metall. 28, 575 (1980). D. Wolf, Acta Metall. 32, 245 (1984).
(o)
I
FIGURE 1
I i I I
(002) 1,2
(b) I , I 018
~
09
J .
.
.
.
.
. . ~0
.
.
I I
f
(a) Schematic representation of electron d i f f r a c t i o n pattern observed in [ l ] . Hatched c i r c l e depicts region of heavy f i l m exposure due to intense l a t t i c e r e f l e c t i o n s (002)1, 2 • (b) Form of grain boundary relrod assumed by LS [ l ] to account f o r the observation.
620
E X P A N S I O N AT LARGE ANGLE GRAIN B O U N D A R I E S
18, No. 6
ELECTRONBEAM
Ill LATTICE R \2
Vol.
I00 -2-I
I 2
LATTICE
= 025
I
_
/
8O I~
60 4O
~-ic~LJ~L~
(002) PLl NE
It~ I*E I I÷E2
Bicrystal model used for present calculations.
iI / /
t;2:o.,o\
l//
;T~FT
/
/I
111 II ~
I p
/ ~ ;
i ~
2
! :r,,:o
~-vo ;,,=o~
2O
08
FIGURE 2
I / r~l=O. 15
09
I0
II
1,2
FIGURE 3 Locally averaged scattering intensity, T ( f ) , along [002] direction calculated for model in Fig. 2.