DISLOCATION BENDS I N HIGH STRESS
DEFORMED SILICON CRYSTALS
H. Gottschalk
To cite this version:
H. Gottschalk.
DISLOCATION BENDS I N HIGH STRESS DEFORMED SILICON CRYSTALS. Journal de Physique Colloques, 1983, 44 (C4), pp.C4-69-C4-74.
<10.1051/jphyscol:1983408>. <jpa-00222847>
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JOURNAL DE PHYSIQUE
Colloque C4, supplément au n°9, Tome 44, septembre 1983
page C4-69
DISLOCATION BENDS IN HIGH STRESS DEFORMED SILICON CRYSTALS
H. Gottschalk
Abt. f. Metallphysik,
II. Physikalisches
Inatitut,
Zulp-Ceher Sti>. 77, D-5000 Kdln 41, F.R.G.
Universitixt
KSln,
Résumé - Pour rendre compte des courbures de dislocations dans le silicium
différentes estimations de la tension de ligne sont présentées et
comparées aux résultats expérimentaux obtenus à faibles et
fortes
contraintes.
Abstract - Different approximations for the line tension of dislocation
bends in silicon are discussed and compared with experimental results
obtained for low stress and high stress deformation.
Introduction - In TEM-studies of deformed materials it is a widely used method to
estimate the local stress acting on a dislocation from the measured radius of its
curvature R. The static bow out of a dislocation between two obstacles is held in an
equilibrium position by the driving force
F =Tb
(1)
(T = res. shear stress, b = Burgers vector of the perfect dislocation) and the
backward directed force in the following text shortly called self force (S = line
tension)
F = S/R (2)
T = S/(Rb) (2a)
In silicon crystals deformed under high uniaxial stress the dislocations mainly
are lying straight along the <110>-directions with 60°-bends as transitions from one
<110>-Peierls trough to another one(Fig.l) /I/. This is also true for low stress
deformed specimens if the dislocation density is low (A. Tbnnessen, H. Gottschalk,
and H. Alexander, to be published; Fig.3). On cooling the crystals with the load
applied the dislocation motion is frozen in whereby the curvatures of the bends are
maintained.
It is reasonable now to try to apply the concept of line tension to these
dynamic bends as outlined for the static bow out. F. Louchet I'll used this method to
calculate the stress acting on dislocations in in-situ straining experiments in the
HVEM.
A problem, however, lies in the arbitrariness of the line energy of a
dislocation and therefore of the line tension which arises by the arbitrariness of
the cut-off radii. To achieve a reasonable choice for the cut-off parameters
different approaches to the line tension problem for this special case of 60 "-bends
are investigated and compared with experimental results in the low stress range. The
best matching parameters are used to apply the method to isolated partial
dislocations and to more complicated bend structures in the high stress range.
12
-3
Experimental - Silicon crystals (FZ-Si, 9-10
cm
B) were deformed in uniaxial
compression and cooled to room temperature with applied load.
Specimens Type A: Compression axis <213>,'T= 30 MPa, T = 650°C,£ = 0.2%, 0.8% and
2.5%.
Specimens Type B: Compression axis <213>,^= 250 MPa, T =420°C, after predeformation
t = 30 MPa, T = 750*C, e = 1.5%.
Specimens Type C: Compression axis <"2,1,11> , T = 420°C, after predeformation along
<213> , T = 750°C, £ = 1.5%.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983408
JOURNAL DE PHYSIQUE
F
s - Dislocation loop i n Si deformed
by high u n i a x i a l s t r e s s along (213)
Fig.2
-
Geometrical approximation of a
60 "-bend
D i f f e r e n t types of bends - After low s t r e s s deformation the s e p a r a t i o n of t h e
Therefore t h e
p a r t i a l s d i s small compared w i t h the r a d i u s of curvature R.
d i s l o c a t i o n s may be t r e a t e d Like p e r f e c t ones. For high s t r e s s e s t h i s condition i s
not f u l f i l l e d and t h e a c t u a l s i t u a t i o n of a l l p a r t i a l s i n t h e bend has t o be taken
i n t o account.
In a hexagonal loop of a p e r f e c t d i s l o c a t i o n two types of bends occur determined
by t h e c h a r a c t e r of t h e tangent a t t h e apex (segment
Fig.2):
1. t h e 30"-type ( t r a n s i t i o n from a srew t o a 6 0 " - d i s l o c a t i o n )
2. t h e 90"-type ( t r a n s i t i o n from a (-60")- t o a(+6O0)-dislocation).
In a d i s s o c i a t e d d i s l o c a t i o n loop f o r both the leading and t h e t r a i l i n g p a r t i a l
o t h e r two types of bends occur:
1. t h e 0'-type ( t r a n s i t i o n from a (-30")- t o a ( + 3 O 0 ) - p a r t i a l )
2. t h e 60"-type ( t r a n s i t i o n from a 30"- t o a 9 0 ° - p a r t i a l ) .
m,
Three approaches t o t h e l i n e t e n s i o n
A rough approximation of t h e l i n e t e n s i o n which i s o f t e n given i n t h e l i t e r a t u r e
is
S = Gb
(G = shear modulus)
(3 )
Here the v a r i a t i o n of l i n e energy with t h e c h a r a c t e r of t h e d i s l o c a t i o n i s not taken
i n t o account-Since t h e r e i s a c l e a r evidence f o r an o r i e n t a t i o n dependence of t h e
l i n e t e n s i o n from t h e experimental r e s u l t s (Fig.3,
different curvatures f o r
d i f f e r e n t types of bends) t h i s approximation i s not u s e f u l .
1.
2. The o r i e n t a t i o n dependent l i n e t e n s i o n approximation f i r s t given by de W i t and
Koehler 131 and discussed by Hirth and h t h e / 4 / f o r a bow out of l e n g t h L i s given
i n t h e form ( V = Poisson's r a t i o )
with t h e energy of a d i s l o c a t i o n segment of l e n g t h L ,
2
2
2
W, = (Gb / 4 5 ) ( c o s @ + s i n p / ( l - v ) ) L l n ( L / e p )
(5)
Considering t h e d i s l o c a t i o n bend t h e parameter L i s not c l e a r l y d e f i n e d - b i s t h e
angle between d i s l o c a t i o n l i n e and Burgers vector and 9 i s a s u i t a b l e cut-off parameter which Hirth and Lothe assume t o be b / 8 / 5 / . Hirth and Lothe chose 3 so small
t o include t h e d i s l o c a t i o n core energy. In t h e l i t e r a t u r e some s i m i l a r r e l a t i o n s a r e
pres'ented v a r i i n g i n t h e logarithmic term. In most cases t h e cut-off
radius is
chosen l a r g e r . Weertman and Weertman / 6 / i n a simpler approximation not taking i n t o
account t h e o r i e n t a t i o n dependence of l i n e t e n s i o n use ln(R/5b). When t h e length L
i s i d e n t i f i e d with t h e segment % i n Fig.2 t h e comparison of both parameters y i e l d s :
L =
= R/(2tg1jo) = 2 ~ / ( 2 + 6 )
(6)
L / ( e p ) = Rl(1.866ep) = ~ J ( 5 . 0 7 ~=) ~ / ( 5 b ) ;
I t i s shown l a t e r t h a t t h e experimental r e s u l t s f o r
p cc b
=
b
are
(6a)
in
satisfying
agreement with t h i s c a l c u l a t i o n , i f t h e applied s t r e s s i s low and t h e r e f o r e the
r a d i i of curvature a r e l a r g e . That confirms t h a t t h e p r o p e r t i e s of t h e bend a r e
(Fig.2) i n t h e apex of the
determined by the c h a r a c t e r of t h e d i s l o c a t i o n segment
bend. Another simple choice f o r L i s L = R; i t s e f f e c t i s described l a t e r .
3. When using t h e o r i e n t a t i o n dependent l i n e t e n s i o n eq.(4) t h e a c t u a l arrangement
of d i s l o c a t i o n segments o u t s i d e the bend i s n o t considered.
The f o r c e s of the
s t r a i g h t segments a d j a c e n t t o t h e bend, however, a c t i n g on t h e d i s l o c a t i o n a t t h e
apex a r e p a r t s of the s e l f f o r c e .
To improve t h e c a l c u l a t i o n of t h e l i n e t e n s i o n f o r an a p p l i c a t i o n t o d i s l o c a t i o n s a t
high s t r e s s a method described by Hirth and Lothe / 4 / i s applied t o t h e approximated
bend according t o Fig.2. The curved d i s l o c a t i o n segment ACE, 60"-sector of a c i r c l e
with r a d i u s R i s approximated by the s t r a i g h t segments
and
The energy of
an a r b i t r a r y segmented d i s l o c a t i o n arrangement considered by t h e authors i s composed
of t h e s e l f e n e r g i e s W
s i and t h e Wi n= t e r~a c t i own e n~e r g i~e s W+l. . J of& a l l~ segments
w
~
~
m, m,
z.
- -
For a d i s l o c a t i o n b e n d p a r t of a l a r g e loop (diameter D77R) t h e l i n e t e n s i o n r e s u l t s
to
S = l*(3W/al)/m
= const(7)
m = M
T = M4T (Fig.2); f o r d l , t h e d i f f e r e n t i a l change of t h e d i s l o c a t i o n l e n g t h 1,
one f i n d s (see Fig.2)
dl = d~(1-214)
For the f o u r bend types i n c o n s i d e r a t i o n t h e r e s u l t s a r e ( V ( S i ) = 0.217), s e t t i n g :
2
p=4/(2+&)
q=2(1-~)(&-2)
E=(Gb/4f)
-
-
,
F i g . 3 - Dislocation loop i n a s p e c i men with low s t r e s s a p p l i e d (30 MPa)
bend A: 30"-type; bend B: 90"-type
-
-
Calculated s t r e s s vs r a d i u s of
curvature R. Lines 2 t o 4:30e-type bends
Line 5: 90"-type bends (eq. (15) ,f =b)
Line 1: both types ( e q . ( 3 ) ) ;
2: e q . ( l l ) 9 = b / 8 ; 3: e q . ( l l ) q = b ;
4 : eq. (4) p=b, L from eq.(6).
&F
Comparison of t h e c a l c u l a t i o n s
The comparison of t h e pre-logarithmic f a c t o r s K i n
(15) with t h e f a c t o r s K 4 given by t h e o r i e n t a t i o n dependent l i n e t e n s i o n
eq. ( 8 )
c a l c u l a t i o n (eq.(4a))
shows t h a t they a r e i d e n t i c a l i n p r a c t i c e (Table 1 ) .
...
JOURNAL DE PHYSIQUE
C4-72
Table 1 - Pre-logarithmic f a c t o r s f o r t h e a c t u a l 60"-bend c a l c u l a t i o n K and f o r
o r i e n t a t i o n dependent l i n e t e n s i o n c a l c u l a t i o n K' ( e q ( 4 a ) ) .
Bend-type
K
K'
0"
1.517
1.554
30"
1.328
1.346
60"
0.949
0.931
90"
0.760
0.723
the
Both r e s u l t s f o r S only d i f f e r i n t h e logarithmic term i n a f a c t o r of t h e cut-off
p a r a m e t e r ? . S e t t i n g Q = b and c a l c u l a t i n g t h e s e l f s t r e s s f o r a bend of t h e
30"-type using eq.(2a) and L according t o eq.(6) y i e l d s a l a r g e r i n c r e a s e of t h e
s t r e s s f o r small r a d i i of curvature when using e q . ( l l )
i n s t e a d of eq. (4) (Fig.4,
l i n e s 3 and 4 ) . I f L = R i s used i n eq.(4) t h e r e s u l t s a r e n e a r l y equal t o those of
eq. 11 ( l i n e 3 i n Fig.4).
I f 9 = b / 8 a s assumed by Hirth and Lothe i s a p p l i e d t o e q . ( l l )
the stresses
become remarkably h i g h e r ( l i n e 2 i n Fig.4).
For t h e s e l f
Line 5 i s t h e s t r e s s c a l c u l a t e d f o r bends of 90"-type (eq.(15)).
s t r e s s e s of 90"-type bends c a l c u l a t e d by eq.(4) (L = R , ? = b ) s l i g h t l y higher values
a r e obtained (5% f o r l a r g e r a d i i of curvature (500nm) up t o 10% f o r small r a d i i
(25nm).
Additionally t h e r e s u l t s using t h e rough e s t i m a t e of eq.(3) ( l i n e 1) a r e shown
i n Fig.4.
Results of t h e measurements and d i s c u s s i o n
I. P e r f e c t d i s l o c a t i o n under a s t r e s s of 30 MPa (Fig.3)
I
1
Fig.5a Histogram of measured R of bends
Fig.5b Histogram of c a l c u l a t e d s t r e s s e s
of 30"-type i n samples of type A.
from Fig.5a using eq. (2a;4 ;6) ? = b
(From A. Tijnnessen, H. Gottschalk,and H. Alexander, t o be published)
Fig. 5b shows an accumulation a t 30 MPa,the deforming s t r e s s . Because of t h e
i n t e r a c t i o n of d i s l o c a t i o n s i n t h e deformed specimen it i s reasonable t h a t s t r e s s e s
below and above 30 MPa a r e found. In t h i s s t r e s s region t h e use of eq.(4) o r e q . ( l l )
does not change t h e r e s u l t s (Fig.4, l i n e s 3 and 4 ) i f 9 = b i s chosen. I f p = b / 8 i s
used i n e q . ( l l ) (Fig.4, l i n e 2 ) f o r a l l measured r a d i i (R < 800nm) t h e c a l c u l a t e d
s t r e s s e s l i e above t h e a p p l i e d s t r e s s 7 = 30 MPa which i s n o t reasonable. When using
the simple l i n e t e n s i o n approximation according t o eq.(3) t h e overestimation of t h e
s t r e s s e s becomes even higher.
Therefore t h e cut-off parameter Q = b which l e a d s t o a s a t i s f y i n g agreement
between t h e measurements and t h e c a l c u l a t i o n s f o r low s t r e s s e s s h a l l be used f o r t h e
high s t r e s s experiments,too.
For t h e bends A and B of d i f f e r e n t types i n Fig. 3 the r a d i i of curvature a r e : A
(30"-type bend), R = (540Z 30) nm;
B (90"-type bend) R = (250 Z 20)nm. Both
c u r v a t u r e s lead t o t h e same s t r e s s 2 = ( 3 2 Z 2 ) MPa which i s i n good agreement with
the a p p l i e d s t r e s s .
2. ~ i g hs t r e s s ( T =350 MPa) a c t i n g on i s o l a t e d p a r t i a l s (Fig.6)
In high s t r e s s deformed <2.1.11)-oriented
s ~ e c i m e n s d i s l o c a t i o n s w i t h very wide
, ,
s e p a r a t i o n s ( d > 1 pm) a r e found ( ~ i g . 6 ) . In such a case t h e p a r t i a l s can be i r e a t e d
a s i s o l a t e d and from t h e r a d i i of curvature of t h e i r bends t h e l o c a l t o t a l f o r c e on
the d i s l o c a t i o n can be c a l c u l a t e d which is t h e sum of t h e d r i v i n g f o r c e on t h e
d i s l o c a t i o n and t h e s t a c k i n g f a u l t energy ( r ( S i ) = 58 m ~ / m171).
~
The d r i v i n g f o r c e s
i n g e n e r a l a r e d i f f e r e n t f o r t h e leading and t h e t r a i l i n g p a r t i a l .
In t h e given
c r y s t a l l o g r a p h i c o r i e n t a t i o n of t h e s t r e s s a x i s one o b t a i n s f o r t h e d r i v i n g f o r c e
on t h e l e a d i n g p a r t i a l FDL and t h e d r i v i n g f o r c e on t h e t r a i l i n g p a r t i a l FDT :
FDL = (7112)bo
FDT = (5/12)bF
The t o t a l f o r c e on t h e l e a d i n g p a r t i a l F L and on t h e t r a i l i n g p a r t i a l FT i s :
r=
F = F
- f . = 2 . 0 . 1 0 - ~ N/m
+
1 1 . 4 . 1 0 - ~ N/m
('Z = 350 MPa)
F = F
L
DL
T
DT
These f o r c e s should be e q u a l t o t h e s e l f f o r c e s of t h e bends what i n f a c t i s found
from t h e experiment f o r t h e leading p a r t i a l s . For t h e average values of S/R of both
types of bends (0"-type and 60"-type) using eq. ( 2 ) , and (9) and (13) r e s p e c t i v e l y
( * = bp) one o b t a i n s : ( n i s t h e number of bends measured)
,.
0"-type: n = 20
SIR = (1.9 2 0 . 5 ) . 1 0 - ~ N/m
60"-type n = 1 5
S I R = ( 2 . 2 2 0 . 5 ) . 1 0 - ~ N/m
The bends of t r a i l i n g p a r t i a l s i n t h e i s o l a t e d form a r e l e s s
only a few measurements could have been performed u n t i l now.
frequent.
Therefore
SiR = (8/8/5.3/4.4/4.4).10-2
N/m
0"-type:
n=5
60"-type:
n=4
S/R = (6.3/6.1/5:2/4.6).10-2
N/m
For both types t h e measured r a d i i a r e found too l a r g e and t h e s e l f f o r c e s t o o small.
Though t h e number of measurements must be increased it may be concluded t h a t t h e r e
i s no e q u i l i b r i u m between d r i v i n g f o r c e and back f o r c e i n case of t h e t r a i l i n g
p a r t i a l s . As the d r i v i n g f o r c e i s higher than t h e back f o r c e one may assume
o b s t a c l e s beeing p r e s e n t a t t h e t r a i l i n g p a r t i a l 181.
F
s - I s o l a t e d p a r t i a l s (350 m a $ ,
s t r e s s axis (2,1,11>
3.
-
Fig.7
"Averaget1 d i s l o c a t i o n loop
W M P ~ ) , s t r e s s a x i s <213)
High s t r e s s ( T = 250 MPa) a c t i n g on complex bend c o n f i g u r a t i o n s (Fig.1)
For t h e p a r t i a l d i s l o c a t i o n s i n specimens of type B beeing deformed
along (213) t h e d r i v i n g f o r c e s a r e
by
compression
By taking t h e averages of t h e bend c u r v a t u r e s and of t h e s e p a r a t i o n s measured f o r a
l a r g e number of d i s l o c a t i o n loops an average d i s l o c a t i o n loop i s constructed which
i s shown i n Fig.7. According t o t h e d i f f e r e n t p o s i t i o n s of t h e p a r t i a l s t h r e e
d i f f e r e n t bend-types occur, c a l l e d A, B, C.
Taking the mean values of t h e o u t e r and inner bend f o r t h e t h r e e types though they
a r e found t o be d i f f e r e n t and considering t h e loop d i s l o c a t i o n a s a p e r f e c t one t h e
following s t r e s s e s a r e c a l c u l a t e d :
A
166 MF'a ( e q . ( 2 a ) , ( 4 ) )
190 MPa (eq. ( l a ) , ( 1 1 ) )
163 MPa (eq. ( 2 a ) , ( 1 5 ) )
B
145 MFa' ( e q . ( Z a ) , ( 4 ) )
220 MFa' ( e q . ( l a ) , ( l l ) )
C
189 MPa ( e q . ( 2 a ) , ( 4 ) )
With t h i s simple method a f a i r l y good agreement with the deformation s t r e s s i s only
calculation
achieved f o r t h e bend type C using the improved l i n e t e n s i o n
(eq. ( l l ) , (15)). It f a i l s f o r bend types A and B.
For a d e t a i l e d c a l c u l a t i o n of t h e l i n e t e n s i o n t h e complicated a c t u a l s i t u a t i o n
of t h e p a r t i a l s i n t h e bend has t o be taken i n t o account. The geometrical
approximation according t o Fig.:! i s shown i n Fig.8. A f u r t h e r approximation i s made
by taking t h e s e l f f o r c e (eq. ( 2 ) , ( 9 ) o r ( 1 3 ) ) of t h e i s o l a t e d p a r t i a l c o n t r i b u t i n g t o
t h e f o r c e on t h e segment L and L1 r e s p e c t i v e l y . Then t h e f o r c e s of t h e t h r e e p a r t i a l
segments l y i n g opposite t o t h e segment L o r L o i n c o n s i d e r a t i o n must be added. For
t h e c a l c u l a t i o n of t h e i n t e r a c t i o n f o r c e s of t h e s e segmented p a r t i a l d i s l o c a t i o n s a
C4-74
JOURNAL DE PHYSIQUE
=\
method
3 d3
given
by
Hirth
and
Lothe
191
is
applied.
The
rl eo sc ua l tl iyn gv a fr oi irnc ge s b on
u t ot hnel y segments
t h e value aL t and
t h e apex
L'
of
a r e e a cs hl i g bend
htly
i s c o n s i d e r e d h e r e . Table 2 c o n t a i n s t h e r e s u l t s of t h e
computations f o r t h e t h r e e l e a d i n g and t h e t h r e e t r a i l i n g
bends. The i n t e r a c t i o n f o r c e Fij means t h e f o r c e e x e r t e d by
The d i s l o c a t i o n s
are
d i s l o c a t i o n i on d i s l o c a t i o n j .
numbered a c c o r d i n g t o Fig.8.
The f o r c e s p o i n t i n g i n t h e
d i r e c t i o n of t h e d r i v i n g f o r c e s a r e counted p o s i t i v e .
Fig.8 - Dissociated
d i s l o c a t i o n bend
.
Table 2 - Forces on t h e d i s l o c a t i o n s i n a d i s s o c i a t e d d i s l o c a t i o n bend (lo-' Nlm)
Bend type
A(1ead.)
A ( t r a i 1 . ) B(1ead.)
B ( t r a i 1 . ) C(1ead.)
C(trai1-j
SIR
-2 -10
-2.92
-3 -07
-3 - 5 1
-3 -58
-2.28
FZ5 o r F 52
~2.61
-2.31
+4.63
-5.60
+6.65
-6.55
F15 o r F42
+0.01
-0 -45
+O .53
-0.66
+0.18
-0.30
'35 Or '62
-0 -17
-0.09
+0.71
+0.10
-0.62
-0 -28
f
-5 -80
+5 -80
-5.80
+5.80
-5.80
+5.80
+5.60
+5.60
+4.00
+4.00
+5.60
+4.00
D F
+5 -98
+1.00
+4.53
Sum
-1-45
+O. 74
+1.99
~
- -
-
~- -
The sum i n t h e l a s t l i n e of Table 2 should y i e l d zero. While t h e r e s u l t s f o r t h e
l e a d i n g p a r t i a l s i n bend B and C a r e f a i r l y s a t i s f y i n g , i f t h e e r r o r i s c o n s i d e r e d
which a r i s e s when adding up s o many c o n t r i b u t i o n s , t h e r e s u l t s f o r t h e t r a i l i n g
p a r t i a l s i n p a r t i c u l a r do n o t f i t t o t h e assumption of an e q u i l i b r i u m between t h e
d r i v i n g f o r c e and t h e back f o r c e i n t h e bend.
Conclusion - It i s shown t h a t t h e c a l c u l a t i o n of t h e l i n e t e n s i o n and t h e s t r e s s e s
i n a bend i s v e r y s e n s i t i v e t o t h e choice of t h e c u t - o f f
parameter q . S e t t i n g i t
e q u a l t o t h e Burgers v e c t o r e x c e l l e n t agreement of t h e s t r e s s d e r i v e d from t h e bend
c u r v a t u r e s w i t h t h e a p p l i e d s t r e s s i s achieved. Using an improved c a l c u l a t i o n of t h e
l i n e t e n s i o n t a k i n g t h e a c t u a l d i s l o c a t i o n arrangement i n t o a c c o u n t t h e a p p l i e d
s t r e s s can be d e r i v e d from t h e bend c u r v a t u r e s f o r i s o l a t e d l e a d i n g p a r t i a l s , t o o .
That f a i l s , however, completely f o r t h e t r a i l i n g p a r t i a l s .
These r e s u l t s mean t h a t f o r t h e d i s l o c a t i o n under low s t r e s s c o n d i t i o n s and f o r
t h e i s o l a t e d l e a d i n g p a r t i a l s under h i g h s t r e s s c o n d i t i o n s a s t e a d y s t a t e motion of
t h e bends can be assumed f o r which a p e r t u r b a t i o n of t h e e q u i l i b r i u m s t a t e can r e l a x
i n a s h o r t r e l a x a t i o n time. This seems n o t t o be v a l i d f o r t h e t r a i l i n g p a r t i a l s f o r
which i n t e r a c t i o n w i t h u n i d e n t i f i e d o b s t a c l e s may occur.
These assumptions a r e
confirmed by t h e r e s u l t s f o r d i s s o c i a t e d d i s l o c a t i o n bends.
As i n t h e s e c a s e s t h e
measured r a d i i of c u r v a t u r e f o r t h e l e a d i n g and t h e t r a i l i n g p a r t i a l a r e f a i r l y
s i m i l a r a breakdown of t h e computations because of q u i t e d i f f e r e n t r a d i i of
c u r v a t u r e i s impossible.
The s p e c i a l r o l e t h e t r a i l i n g p a r t i a l s a p p a r e n t l y p l a y should be f u r t h e r
i n v e s t i g a t e d by a d d i t i o n a l measurements of i s o l a t e d d i s l o c a t i o n bends and by a n
improvement of t h e c a l c u l a t i o n of t h e complicated c o n f i g u r a t i o n of d i s s o c i a t e d
bends. The a p p l i c a t i o n of t h e l i n e a r a n i s o t r o p i c e l a s t i c i t y t h e o r y t o curved
d i s l o c a t i o n s , however, a s o u t l i n e d e.g. by Scattergood / l o / , should o n l y r e s u l t i n
s l i g h t m o d i f i c a t i o n s s i n c e t h e a n i s o t r o p y f a c t o r of S i i s o n l y A = 1.56.
-
Acknoledgements
The a u t h o r wishes t o thank P r o f . D r . H. Alexander f o r s t i m u l a t i n g
d i s c u s s i o n s and Dipl. Phys. W. WeiB f o r h i s e x p e r i m e n t a l c o n t r i b u t i o n s .
References
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/ 2 / Louchet, F., P h i l o s . Mag. 43, (1981) 1289
131 de W i t , G. and Koehler, J.S., Phys. Rev.
(1959) 1113
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C h a p t . 6 ; ~ . 213
/ 6 / Weertman,J. and Weertman, J.R., Elementary D i s l o c a t i o n Theory (Macmillan), 1964
171 G o t t s c h a l k , H., J. Physique Colloq. 40, (1979) C6-127
181 Gottschalk,H., t h i s conference
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116,