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Electron microscopy of a low stacking fault energy alloy
ELECTRON
MICROSCOPY H.
P.
OF A LOW
KARNTHALER,?
P.
M.
STACKING
FAULT
ENERGY
ALLOY*
and M. S. SPRING9
HAZZLEDINE:
The images of dissociated dislocations, simulated by a oomput>er controlled cathode ray display The comparison system are comprared with experiment.&ll~ observed images in Cir-10 at. oAAl crystals. allows the dislocations to be identified unambiguously and the details of the dislocation-disIoc~tion interactions to be deduced. It, is concluded thet in cry&& deformed into stage II of the work-hardening curve primary and secondary disiooations interact to form short lengths of Lamer-Cottrell dislocation as assumed in the forest theory of the flow stress. ETUDE
AU MICROSCOPE
ELECTROKIQUE D’UN ALLIAGE FAUTE D’EMPILFMENT
A FAIBLE
ENERGIE
DE
Les images des dislocations dissociiies, simulees par un systeme Q rayons c~thodiques control6 par un ordinrttenr, sont comparees sux images observees e~p~rimentalement drms des cristaux &-IO %a&. Al. LB eomprtraison permet d’identifier les dislocations sans ambiguit6 et den deduire les details des inter&&ions dislocations-dislocations. Los auteurs concluent que dans les cristaux deform& jusqu’au stade II de la courbe de consolidation, les dislocations prim&es et second&es agissent de faeon B former de courtes longueurs de dislocations de Lomer-Cottrell, comme le suppose la theorie des for& pour la. contrainte plastique. EL~KTRO~E~~~IKROSKOPISCH~
U~TERSUCHU~G ST~P~LFEHLERE~ERGIE
EIXER
LEGIERUXG
MIT KLEINER
Die auf einem computer-kontrollierten Kathodenstrahlsystem simulierten Bilder aufgespaltener Versetzungen wurden mit experimentell beobachteten Kontrastbildern van Versetzungen in Cu-10 Anhand dieses Vergleiches ist es miiglich, Versetzungen eindeutig zu At. ‘A Al-Kristallen verglichen. identifizieren und Einzelheiten der Wechselwirkung zwischen Versetzungen abzuleiten. Es zeigt sich, daB in den bis in den Bereich II der Verfestigungskurve verformten Kristallen durch Wechselwirkung zwischen Prim&r- und Sekundiirversetzungen kurze Lomer-Cottrell-Versetzungen entstehen, wie dies in der Theorie der ~~~~dvcrsetzun~n sngenommen wurde.
1. INTRODUCTION In transmission electron microscope studies of the deformation of low stacking fault energy alloy~‘~-~) the full inte~retation of the micrographs is often prevented by the fact that the contrast from the dissociated dislocations is not known. The dissociation of the dislocations creates difficulties in the interpretation only when the separation of the partial dislocations r is of the order of the extinction distance .&,_ When r < Is, the dislocations may be treated as undissociated and when r > 5, the partial dislocations may be considered separately. In both these cases the expected contrast is well known.@) In the intermediate range, r rti 0.5[,, the theoretical contrast should, in principle, be computed explicitly for each alloy. The object of the calculations and experiments in this paper is to determine to what extent it is possible, without recourse to elaborate computation (e.g. using many-beam theory or elastic anisotropy) to determine the Burgers vectors and the three dimensional arrangement of the dislocations in a deformed alloy. The alloy chosen is Cu-IO at.% Al in which r varies between O.l3E,,, and 0.67E,,, for the different * Received July 7, 1971. t II Physikalisches Institut, University of Vienna, Vienna, A-1090 Austria. $ Department of 3%etallurgy, University of Oxford, Oxford, England. Q Department of Architecture, University of Edinburgh, Edinburgh, Scotland. ACTA
METALLURGICA,
VOL.
20, MARCH
1972
dislocations considered. Particular attention is paid to glissile dislocations and to the composite dislocations which lie parallel to the intersection of two slip planes. The method is complementary to the weak-beam methodu’) in that the experiments may be performed more rapidly and hence larger areas of the specimen may be studied. 2.
CALCULATION AND DISPLAY
The following dislocations in Cu-10 at. y0 Al are considered : edge and screw glissile dislocations, 120’ (Lomer-Cottrell) locks and 90” (Hirth) locks.(le*lJi) The separation between partial dislocations (stacking fault ribbon width) was calculated using the stacking fault energy value of Howie and Swannos) and Brown,(13) taking the shear modulus C to be 4020 kg mrnm2 (from the measurements of Neighbours and Smith),(iQ and assuming that Poisson’s ratio = +. The stacking fault ribbon widths in screw and edge dislocations are 50 and 120 A, respectively. The width of each stacking fault in the 120” lock(i5) is 35 A and in the 90” lock it is 170 B (for the arrangement of the partials see Fig. 1). The contrast calculations have been made specifically for primary dislocatioIls (b = ~~~iOl~(lll)) and for composit,e dislocations formed from the primaries and dislocations on (lli) by the following 459
ACTA
460 ii1
IIT lir ili
ill
Iii 052
METALLtJRGl(‘A,
025
520
220
I02
205
_.1'2 y & ii0
ii2
002
VOL.
“0,
1972
For a dislocation at a particular depth in the foil the transmitted electron intensity was computed (using the displa~ement~sgiven above) at a number of points in the form of an image profile. Profiles were calculated for a number of dislocation depths and by interpolation between the profiles a tno dimensional array of intensity values was generated which, when displayed, simulat,es the image from an inclined dislocation.(lg*2*) Scar the ends of t,he dislocation the profile was assumed to remain the same from the foil surface to the closest comput’ed profile (a distance of 0.2[, in each case except, for 90” locks for which the distance was 0.5$,). For the purpose of display the dislocations were assumed to lie at an angle 0 of tan-l 4 to the foil surface but the pictnres may be demagnefied laterally (c.g. by viewing through a cylindrical lens) t’o represcut’ a dislocation at an angle 0 < tan-l 1.
ii2
y__________* : : -I'll2
FIG. 1. Computed micrographs of edge, screw, LomerCottrell and H&h dislocations for 12 different reflections. At the foot is the grey scale used for thr disptay.
reactions --i-
(fi2+2ll)+(iiZ+l2i)-ii2+iiO+ll~ (ii2+2ll)+(ll2+2il)+ii-2+002+112
(1207
(90”)
For each dislocation images have been calculated corresponding to those twelve (111) and (SZO} g vectors obtainable by tilting a foil specimen with [ill] normal through <20°, namely: ill, iii, lil, 220,202,OZZ and their negatives. The set of computed images may, however, be compared with micrographs of dislocations belonging to any glide system. 2.2 Image calculation The dislocation images were computed using the two-beam column approximation in the dynamical theory of electron diffraction.(16) The constants used were {in the notation of Wowie and ~~helal~) 1111= 250 11, &,, = 450 A, &l&’ = &/‘&,’ = 0.1, w = 0.3, t = 7%, and the following further approximations were made: isotropic elasticity theory was used, no allowance was made for surface relaxationcl’) and g mb x u wits approximat,ed by g * b, tan y. Thus the displacement R at a point r, + from each partial dislocation is R = & {b+ + b,[$ sin 24 + tan y x (4 log T -
g cos2&]~
A precision cathode-ray system in conjunction with a PDP7 computer was used to generate the displayed images from the computed intensities. The method is similar to that previously described for displaying bend contour cnlculations(21) and affords better area resolution and more intensit.y st,eps than earlier line-printer methods. In t,his ease, 21 approximately equally spaced steps of intensity were used and balanced by microdensiometry to give similar contrast on the recording film to that obtained in the microscope. The recording camera was defocussed by a standard amount to provide a smooth half-tone image. 2.4 Computer results The computed images are shown in Fig. 1. Of the 48 images in Fig. 1 only 24 were calculat,ed independently. For the edge dislocations the images in the ill, iii, 022, 022, 202 reflections are mirror images (mirror parallel to the long side of the picture} of the images in the 111, iii, 220, 220, 202 reflections, respectively. For the screw dislocation the same relationships hold but the images in the 022,022, 220 and 220 were not calculated; instead t#hey were assumed to be the same as the images in the Tii, 111, 111, lli reflections, respectively. This assumption was made because by comparing the computed profiles of the screw dislocation with those of undissociated screws(*) it is apparent that the slight dissociation of the screw dislocation has almost no effect on the image. The form of the image is controlled by the value of g - b for the total dislocation. For the Lomer-Cottrell and Hirth locks the dislocation images in the ill, iii, 022, 022 reflections are the same as those in lil, i11, 202, 202, respectively. In the
KARNTHALER
ELECTRON
et al.:
iWICROSCOPY (ii) sections
OF .A F.C.C. ALLOY
461
cut in such a way that specific
dis-
locations would both lie at an angle of 30” (Y tan-’ to the foil surface and be the shortest locations their
possible
4) dis-
in the foil (so that they could not change
character
by
shortening
their
length).
The
normals to the required sections are shown in Fig. 2 for edge (lll)[lOl] (lll)[iOl] locations to [liO]
dislocations
dislocations on primary
For
and conjugate
the vectors
dislocations
(E, E’) and for screw
(S, 8’).
composite
dis-
planes (parallel
are LC, LC’ and for composite
parallel to [Oli] the vectors are LC”, LC”‘.
Composite
dislocations
cannot
length but the “shortest
glide to shorten their
dislocation”
criterion is still
useful for identification. For each type of dislocation : edge, screw and com200 FIG. 2. Foil normals composite dislocations
for edge E, E’, screw S, S’ and LC, LC’ and LO’, LO. A circle
For edge dislocations
should be completely
foils which are 7.05, thick
(two physically
foils for g = (111) and g = (220)) the deviation
dislocations
from the Bragg position,
48 rectangles (8 x 6) of equal intensity. which is a fraction
different
w = 0.3.
0.02 to 0.49 in steps of 0.01 (48 steps). rectangles
includes several intensity steps.
to be obtained.
LC” was taken as the foil nor-
mal instead of E or E’ to increase the number flections
and in this case the dislocations
shows as an example the orientation of screw dislocations.
of re-
are at 25”
cates the possible
foil normals
in the grey scale
All intensities greater
and the available
re-
space
between the 70 and 110” small circles centred on S.
A
list of Iow order available reflectlions is given in Table 1. TABLE
from
simulation
used in the case
A circle of 20” around S indi-
flections therefore lie in the region of reciprocal
increases
Since only 21
of grey were used in the image
each of the 21 visible
into
The intensity,
of the incident intensity
from the top right to the top left boustrophedon shades
in
and in each case
The grey scale at the foot of Fig. 1 is divided
(20”), permit the
and their slip plane at 30” to the foil surface. Figure 2
invisible. All the images in Fig. 1 represent
which with
best selection of low order reflections
g * b and g . b, are zero for
each partial and the dislocation
and those
chosen (LC”, S, LC) are the foil normals the available tilt in the microscope
of 20” around S indicates the possible beam directions in the case of screw dislocations and the available reflections lie in the region of reciprocal space shown hatched. 220 and 220 reflections,
posite there are two possible foil normals
Dislocation
Line direction
Edge s xew Composite
1
Foil normal
[ii01
Reflections *t20, h220, 1022,
m
&202, 5022, ~202,
ilil, *iii, &iii,
*ill
*iii *iii
than 0.49 appear white and those below 0.02 appear black.
It should
3. EXPERIMENTS 3.1 Experimental
method
deviation
Single crystals of Cu-10 at. y. Al were grown by the Bridgman strained
in
technique, in an Instron
vacua
and
stage
shear
The
crystals
stress).
(
t’ensile tester
(1 .l kg mm-2)
II
be mentioned
that experimentally
very difficult to set the specimen
x
1O-5 torr),
into
stage
I
(2.3 kg mm-2
resolved
were sectioned
with
a
it is
at a predetermined
(w = 0.3) from the Bragg angle, especially
for higher order reflections. the diffraction
patterns
But an examination
showed
90 per cent of al1 the micrographs 3.2 Experimental
of
the result that for 0.2 < w < 0.4.
results and discussion
spark cutter and discs of diameter 2.3 mm were trepanned from the 1 mm thick sections, chemically
(a) Irzlined dislocations. The micrographs obtained from foils of type (ii) are directly comparable
thinned
with the computed images. In Figs. 3 and 4 the outlined computed dislocations have been given the
and
jet-electro-polished
niques and examined scope. The crystallographic
by standard
in a Siemens Elmiskop
techmicro-
length appropriate orientations
of the specimens
were of two types: (i) sections parallel to the primary slip plane (111) ;
to a foil thickness
of 75, and are
mounted on the micrographs in the correct orientation. In Fig. 3 the images of a set of dislocations close to edge orientation
((lll)[iOl])
may be compared
with
ACTA
462
FIG . 3. Computed
and observed
VOL.
20,
1972
(outlined) and el;perimental images for edge dislocations (lll)[lOl] showing in 8 different reflections. Foil normal of the micrographs is LC”.
the con tputed images. culated
METALLURGICA,
The general form of the calimages (e.g. “chain”
image for
strong double
image when g = ill
very slight doubling
t,he same area but in Fig. 3(g)
of the image may be seen.
The
g = 22c ), double image for g = 202, spotty image for g = lil and intense image for g = iii) is sufficiently
slight asymmetry from top to bottom of a dislocation whichismostnoticeableinthe weakimages (g = f lil)
close in each case that the Burgers vector of the dis-
and which does not appear in the computed images is due to the approximation used (see Section 2.2)
location s may
definitely
be
identified
as
@[iOl].
Althoug ;h the general forms match well and the width of the irnages agree, even with all the approximations
for g . b x u.@~) Figure 4 shows two sets of screw dislocations
used in the calculations, perfect.
with
match
is not
b = &&[iOl]
(the left and right sets in each picture
One feature which is never observed
is the
are of opposite
sign) in which the agreement between
the detailed
KARNTHALEK
FIQ. 4.
theory and experiment to
MICROSCOPY
their with
The screw dislocations
is close.
invisible
The dissociation partials
ELECTRON
when
very
g = jlil
slight
dissociation
of the screw dislocation some
edge
character
sufficient to give an asymmetry Figure
5(a)
shows
a typical
in stage II.
is,
(0.2E1,,). into
two
however.
Fig.
The
5(a)
reactions
in detail
and
the same Bmea
showi]
with the symmetry
Whelan.(16)
This
obviously
be true for glide dislocations,
true
Lomer-Cottrell
for
result
must
but it is also
dislocations
(which
are
on two planes) even though it is not true
that the Lomer-Cottrell
image is the same at the top
in Figs.
Figure 6 shows as an example b
are
at A
in
5(b-m).
foil thickness contrast.
the influence
of the
and the value of w on the dislocation
In Fig. 6(a) the deviation
from the Bragg position
w = 0.25 + 0.05
and the images of the dis-
sign
locations are in good agreement with the computed images where the foil is an even number of half extinc-
dislocations
of the same
tion distances thick (bright regions of the micrograph)
There are therefore two attractive
(C) and two
Two primary interact sign.
by dark field microscopy)
are analysed
of Howie
163
ALLOY
of a foil as at the bottom.
area of dislocation
Lomer-Cottrell
rules
dissociated
from top to bottom
The top t and the bottom
of the foil (determined indicated.
A F.C.C.
Figs. 5(h) and (k), in accordance
(g * b = 0)
of the foil through the g * b x u term. reactions
OF
images for screw dislocations (lll)[iOl] ComFlute d (outlined) and experimental in 8 different reflections. Foil normal of t,he micrographs is S.
are virtually owing
et al.:
(lll)[iOl]
dislocations
with two (Til)(OTi]
repulsive
(R)
type is visible.
junctions,
of opposite
and one junction
and the images are still readily identifiable where the foil has other
thicknesses.
in regions
In Fig.
6(b)
are indexed
in Fig.
w = 0.9 + 0.1 and the images show much less detail
5(l) and Fig. 5(m) is a view of the interactions
normal
and differ greatly from the computed images (Fig. 1, screw or edge, g = +202). For identification, there-
to (iii).
The dislocations
of each
In Figs. 5(j) and (k) the foil has been turned
upside down in the microscope so that dislocation ends which, in Figs. 5(h) and (i) are at the top of the foil are at the bottom
of the foil in Figs. 5(j) and (k).
It may be seen that the images at the ends of all the dislocations
are the same in Figs. 5(i) and (j) and in
fore, it is necessary
that the computed
and experi-
mental values of w should match closely but it is less important that the foil thicknesses should agree. (b) identify
Parallel
dislocations.
dislocations
in foils
It
is more
parallel
to
difficult
to
an active
5. (l), (m). ~~SIuc~t~on~eaet~o~~ at-d. (1) 8chemat~c of network in Figs. 5(b)-(i) with Burgers vectors of the various dislocations indicated. (m) View of t’he interact,ions normal to (ii 1) showing the attractive (C) and repulsive (R) junctions. FIG.
slip plane than in inclined foils because dislocations one character depths
in the foil.
However,
using the computed
images of Fig. 2. the conservation and the line directions network,
of
may appear quite different at different of b at a, junction,
of the dislocations,
a stage II
such as the one in Fig. 7, may be indexed
with very little ambign~ty Fig. *I(k). The dislocation structure consists of intersecting FIG. .5. (a) Region
of
inclined
dislocation
Top t and bottom b indiented.
reactions.
three slip planes (ill), diGcations
(lil)
slip lines parallel to
and (iii).
formed at attractive
The composite
junctions
are rarely
longer than 0.5y but are fre~~~e~~l~ placed end to end to give the impression of greater length. Cottrell dislocations never observed even
FIG. 5. (b)-(i)
Area. A of (ef in 8 reflections. Foil normal is LC. (j), (k) Area A but with foil inverted.
lower
Many Lomer-
are visible in Fig. 7, but we have
a Hirth lock ; their density
than
in
Cu. fz3j
The
must be
arra~~geme~~t of
FTC. 6, (a) Variation of image with foil thickness, 20 = 0.25 & 0.05. (b) Variation of image with foil thickness, W = 0.9 & 0.1.
FIG. ?(a).
FIG. 7 (c)I
FIG. 7(b).
FIG. 7(d).
ACTA
166
Fra. 7(e).
Fm. 7(f).
METALLURGICA,
VOL.
20,
1972
FIG. 7(g).
FIQ. 7(h).
KARNTHALER
et al.:
ELECTRON
MICROHCOPY
OF
A F.C.C.
467
ALLO\-
FIG. 7(i)
\
i
i FIG. 7(k).
FIG. 7. (a)-(j). Dislocation reactions. 10 different reflections showing the same area. [ill] foil normal. (k) Schematic of network with Burgers vectors of the dislocations.
The
only
dislocations
which
are the Lomer-Cottrell
completely
dislocations,
to their length (g = f%O),
disappear
when g is parallel
which is in accord with
the theory. 4. CONCLUSIONS
1. In an alloy such as Cu-10 at.% paratively Burgers
low anisotropy vectors
generally mental
approximation diffraction. dislocation reflections
FIG. 7(j)
compute
of the dissociated
be identified
images
Al with a com-
constant by
with those
(A = 3.66) dislocations
comparing calculated
of the dynamical
the
the may
experi-
in the lowest
theory
of electron
For a definite identification the same must be imaged using several different but for this purpose it is not necessary
to
the contrast for each micrograph.
2. The
images
of
dissociated
dislocations
differ
interacting primary and secondary dislocations in Fig. 7 is typical of that assumed in the forest model of
markedly from those of perfect dislocations separation of the partials >0.3E,.
work-hardening.(24-26)
3. In Cu-10 at. o/o Al many Lomer-Cottrell reactions are observed but Hirth locks have not been encoun-
Nearly all of the dislocations vectors sociation
which lie in (lil) many
are visible
in Fig. 7 have Burgers
but, because
of their dis-
when g = +lil
(e.g. at
B the [iOl] dislocation almost disappears in (g) and (h) where it is screw and is visible where it is edge).
when the
tered. 4. The arrangement slip planes
is precisely
of dislocations that
theory of strain hardening.
assumed
in intersecting in the forest
ACTA
465
METALLURCICA,
ACKNOWLEDGEMENTS
P. K. wishes to acknowledge financial support from the Royal Society European Exchange Programme and from the Oesterreichische Akademie der M. S. S. is grateful to the S.R.C. for a Wissenschaften. K.
grant.
would like to thank Drs. A. Howie, J. Goringe, MI.J. Whelan, D. Blaher and R. Perrin for many helpful discussions and Professor P. B. Hirsch F.R.S. for the provision of laboratory facilities. We
M.
REFERENCES I. M. J. WHELAN, Proc. R. Sot. A249, 114 (1958). 2. A. HOWIE, Direct O~eer~at~on of ~~perfect~o~~ in Crystals. Interscience (1961). 3. S. MADER and H-M.
Conference
for
THIERINGER, Fifth ~nt~r~at~onuZ Electron Microscopy. “Academic Press
(1962). 4. A. SEEOER,N.P.L.Symposium 15,~. 1. H.M.S.O. (1963). 5. J. W. STEEDS and P. M. HAZZLEDINE. Discuss. Paradav Sot. 38,103 (1964). 6. T. C. TISONE, J. 0. BRITTAIX and M. MESHII, Phil. Msg. 16,651 (1967). 7. P. B. HIRSCH and F. J. HUXPHREYS, Physics of Strength and PEastieity. M.I.T. Press (1969).
VOL.
20,
1972
8. P. B. HIRSCH, A. HOWIE, R. B. Krcaomow, D. W. PASHLEY and M. J. WHELAN, Electron &ficroseopy oj Thin Crystal. Butterworths (1965). 9. D. J. H. COCKAYNE, I. L. F. RAY and M. J. WHELAN. PhiE.Mug. 20, 1265 (1969). 10. J. FRIEDEL, PhiE. Mug. 46, 1169 (1955). 11. T. Jsssa~c, J. P. HIRTH md C. S. HARTLEY, J. Apple. Phys. 36, 2400 (1965). 12. A. HOWIE and P. R. S’CVAKN,Phil. Afag. 6, 1215 (1961). 13. L. M. BROWN, Phil. Hag. 10, 441 (1964). 14. J. R. NEIGHBOURS and C. S. SMITH, acfa Met. 2, 591 il9.541 i-“--I15. 4. S. STROH, Proc. Ph?/B.Sot. 6?B, 427 (1954). 16. A. HOWIE and M. J. WHELAN, Proc. R. Sot. A263, 215 (1961). 17. W. J. TUNSTALL, I’. 13.HIRSCH and J. W. STEEDS, Phil. Xag. 9, 99 (1964). 18. A. HOWIE and M. J. WHEL~~N, Proc R. Sot. A267, 206
(1962). 19. A. K. HEAD, AU&. J. Php. 20, 557 (1967). 20. P. !&!I\TBLE,AZ&. J. P&s. 21, 325 (1968). 21. M. S. SPRING and J. W. STEEDS, Phys. Status Solidi 37, 303 (1970). 22. J. SILCOCKand W. J. TIJNSTALL, Phil. Nag. 10,361 (1964). 23. H. STRUNK. Phil. Maa. 21. 857 (19701. 24. 2. S. BAS&KI, Phil. kigf6y:4, 39‘3 (19bS). 25. P. B. HIRSCH, Internal Stresses and Fatiglce in Metals. Elsevier (1959). 26. G. &ADA, A&z Met. 8, 541 (1960).
MICROSCOPY H.
P.
OF A LOW
KARNTHALER,?
P.
M.
STACKING
FAULT
ENERGY
ALLOY*
and M. S. SPRING9
HAZZLEDINE:
The images of dissociated dislocations, simulated by a oomput>er controlled cathode ray display The comparison system are comprared with experiment.&ll~ observed images in Cir-10 at. oAAl crystals. allows the dislocations to be identified unambiguously and the details of the dislocation-disIoc~tion interactions to be deduced. It, is concluded thet in cry&& deformed into stage II of the work-hardening curve primary and secondary disiooations interact to form short lengths of Lamer-Cottrell dislocation as assumed in the forest theory of the flow stress. ETUDE
AU MICROSCOPE
ELECTROKIQUE D’UN ALLIAGE FAUTE D’EMPILFMENT
A FAIBLE
ENERGIE
DE
Les images des dislocations dissociiies, simulees par un systeme Q rayons c~thodiques control6 par un ordinrttenr, sont comparees sux images observees e~p~rimentalement drms des cristaux &-IO %a&. Al. LB eomprtraison permet d’identifier les dislocations sans ambiguit6 et den deduire les details des inter&&ions dislocations-dislocations. Los auteurs concluent que dans les cristaux deform& jusqu’au stade II de la courbe de consolidation, les dislocations prim&es et second&es agissent de faeon B former de courtes longueurs de dislocations de Lomer-Cottrell, comme le suppose la theorie des for& pour la. contrainte plastique. EL~KTRO~E~~~IKROSKOPISCH~
U~TERSUCHU~G ST~P~LFEHLERE~ERGIE
EIXER
LEGIERUXG
MIT KLEINER
Die auf einem computer-kontrollierten Kathodenstrahlsystem simulierten Bilder aufgespaltener Versetzungen wurden mit experimentell beobachteten Kontrastbildern van Versetzungen in Cu-10 Anhand dieses Vergleiches ist es miiglich, Versetzungen eindeutig zu At. ‘A Al-Kristallen verglichen. identifizieren und Einzelheiten der Wechselwirkung zwischen Versetzungen abzuleiten. Es zeigt sich, daB in den bis in den Bereich II der Verfestigungskurve verformten Kristallen durch Wechselwirkung zwischen Prim&r- und Sekundiirversetzungen kurze Lomer-Cottrell-Versetzungen entstehen, wie dies in der Theorie der ~~~~dvcrsetzun~n sngenommen wurde.
1. INTRODUCTION In transmission electron microscope studies of the deformation of low stacking fault energy alloy~‘~-~) the full inte~retation of the micrographs is often prevented by the fact that the contrast from the dissociated dislocations is not known. The dissociation of the dislocations creates difficulties in the interpretation only when the separation of the partial dislocations r is of the order of the extinction distance .&,_ When r < Is, the dislocations may be treated as undissociated and when r > 5, the partial dislocations may be considered separately. In both these cases the expected contrast is well known.@) In the intermediate range, r rti 0.5[,, the theoretical contrast should, in principle, be computed explicitly for each alloy. The object of the calculations and experiments in this paper is to determine to what extent it is possible, without recourse to elaborate computation (e.g. using many-beam theory or elastic anisotropy) to determine the Burgers vectors and the three dimensional arrangement of the dislocations in a deformed alloy. The alloy chosen is Cu-IO at.% Al in which r varies between O.l3E,,, and 0.67E,,, for the different * Received July 7, 1971. t II Physikalisches Institut, University of Vienna, Vienna, A-1090 Austria. $ Department of 3%etallurgy, University of Oxford, Oxford, England. Q Department of Architecture, University of Edinburgh, Edinburgh, Scotland. ACTA
METALLURGICA,
VOL.
20, MARCH
1972
dislocations considered. Particular attention is paid to glissile dislocations and to the composite dislocations which lie parallel to the intersection of two slip planes. The method is complementary to the weak-beam methodu’) in that the experiments may be performed more rapidly and hence larger areas of the specimen may be studied. 2.
CALCULATION AND DISPLAY
The following dislocations in Cu-10 at. y0 Al are considered : edge and screw glissile dislocations, 120’ (Lomer-Cottrell) locks and 90” (Hirth) locks.(le*lJi) The separation between partial dislocations (stacking fault ribbon width) was calculated using the stacking fault energy value of Howie and Swannos) and Brown,(13) taking the shear modulus C to be 4020 kg mrnm2 (from the measurements of Neighbours and Smith),(iQ and assuming that Poisson’s ratio = +. The stacking fault ribbon widths in screw and edge dislocations are 50 and 120 A, respectively. The width of each stacking fault in the 120” lock(i5) is 35 A and in the 90” lock it is 170 B (for the arrangement of the partials see Fig. 1). The contrast calculations have been made specifically for primary dislocatioIls (b = ~~~iOl~(lll)) and for composit,e dislocations formed from the primaries and dislocations on (lli) by the following 459
ACTA
460 ii1
IIT lir ili
ill
Iii 052
METALLtJRGl(‘A,
025
520
220
I02
205
_.1'2 y & ii0
ii2
002
VOL.
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1972
For a dislocation at a particular depth in the foil the transmitted electron intensity was computed (using the displa~ement~sgiven above) at a number of points in the form of an image profile. Profiles were calculated for a number of dislocation depths and by interpolation between the profiles a tno dimensional array of intensity values was generated which, when displayed, simulat,es the image from an inclined dislocation.(lg*2*) Scar the ends of t,he dislocation the profile was assumed to remain the same from the foil surface to the closest comput’ed profile (a distance of 0.2[, in each case except, for 90” locks for which the distance was 0.5$,). For the purpose of display the dislocations were assumed to lie at an angle 0 of tan-l 4 to the foil surface but the pictnres may be demagnefied laterally (c.g. by viewing through a cylindrical lens) t’o represcut’ a dislocation at an angle 0 < tan-l 1.
ii2
y__________* : : -I'll2
FIG. 1. Computed micrographs of edge, screw, LomerCottrell and H&h dislocations for 12 different reflections. At the foot is the grey scale used for thr disptay.
reactions --i-
(fi2+2ll)+(iiZ+l2i)-ii2+iiO+ll~ (ii2+2ll)+(ll2+2il)+ii-2+002+112
(1207
(90”)
For each dislocation images have been calculated corresponding to those twelve (111) and (SZO} g vectors obtainable by tilting a foil specimen with [ill] normal through <20°, namely: ill, iii, lil, 220,202,OZZ and their negatives. The set of computed images may, however, be compared with micrographs of dislocations belonging to any glide system. 2.2 Image calculation The dislocation images were computed using the two-beam column approximation in the dynamical theory of electron diffraction.(16) The constants used were {in the notation of Wowie and ~~helal~) 1111= 250 11, &,, = 450 A, &l&’ = &/‘&,’ = 0.1, w = 0.3, t = 7%, and the following further approximations were made: isotropic elasticity theory was used, no allowance was made for surface relaxationcl’) and g mb x u wits approximat,ed by g * b, tan y. Thus the displacement R at a point r, + from each partial dislocation is R = & {b+ + b,[$ sin 24 + tan y x (4 log T -
g cos2&]~
A precision cathode-ray system in conjunction with a PDP7 computer was used to generate the displayed images from the computed intensities. The method is similar to that previously described for displaying bend contour cnlculations(21) and affords better area resolution and more intensit.y st,eps than earlier line-printer methods. In t,his ease, 21 approximately equally spaced steps of intensity were used and balanced by microdensiometry to give similar contrast on the recording film to that obtained in the microscope. The recording camera was defocussed by a standard amount to provide a smooth half-tone image. 2.4 Computer results The computed images are shown in Fig. 1. Of the 48 images in Fig. 1 only 24 were calculat,ed independently. For the edge dislocations the images in the ill, iii, 022, 022, 202 reflections are mirror images (mirror parallel to the long side of the picture} of the images in the 111, iii, 220, 220, 202 reflections, respectively. For the screw dislocation the same relationships hold but the images in the 022,022, 220 and 220 were not calculated; instead t#hey were assumed to be the same as the images in the Tii, 111, 111, lli reflections, respectively. This assumption was made because by comparing the computed profiles of the screw dislocation with those of undissociated screws(*) it is apparent that the slight dissociation of the screw dislocation has almost no effect on the image. The form of the image is controlled by the value of g - b for the total dislocation. For the Lomer-Cottrell and Hirth locks the dislocation images in the ill, iii, 022, 022 reflections are the same as those in lil, i11, 202, 202, respectively. In the
KARNTHALER
ELECTRON
et al.:
iWICROSCOPY (ii) sections
OF .A F.C.C. ALLOY
461
cut in such a way that specific
dis-
locations would both lie at an angle of 30” (Y tan-’ to the foil surface and be the shortest locations their
possible
4) dis-
in the foil (so that they could not change
character
by
shortening
their
length).
The
normals to the required sections are shown in Fig. 2 for edge (lll)[lOl] (lll)[iOl] locations to [liO]
dislocations
dislocations on primary
For
and conjugate
the vectors
dislocations
(E, E’) and for screw
(S, 8’).
composite
dis-
planes (parallel
are LC, LC’ and for composite
parallel to [Oli] the vectors are LC”, LC”‘.
Composite
dislocations
cannot
length but the “shortest
glide to shorten their
dislocation”
criterion is still
useful for identification. For each type of dislocation : edge, screw and com200 FIG. 2. Foil normals composite dislocations
for edge E, E’, screw S, S’ and LC, LC’ and LO’, LO. A circle
For edge dislocations
should be completely
foils which are 7.05, thick
(two physically
foils for g = (111) and g = (220)) the deviation
dislocations
from the Bragg position,
48 rectangles (8 x 6) of equal intensity. which is a fraction
different
w = 0.3.
0.02 to 0.49 in steps of 0.01 (48 steps). rectangles
includes several intensity steps.
to be obtained.
LC” was taken as the foil nor-
mal instead of E or E’ to increase the number flections
and in this case the dislocations
shows as an example the orientation of screw dislocations.
of re-
are at 25”
cates the possible
foil normals
in the grey scale
All intensities greater
and the available
re-
space
between the 70 and 110” small circles centred on S.
A
list of Iow order available reflectlions is given in Table 1. TABLE
from
simulation
used in the case
A circle of 20” around S indi-
flections therefore lie in the region of reciprocal
increases
Since only 21
of grey were used in the image
each of the 21 visible
into
The intensity,
of the incident intensity
from the top right to the top left boustrophedon shades
in
and in each case
The grey scale at the foot of Fig. 1 is divided
(20”), permit the
and their slip plane at 30” to the foil surface. Figure 2
invisible. All the images in Fig. 1 represent
which with
best selection of low order reflections
g * b and g . b, are zero for
each partial and the dislocation
and those
chosen (LC”, S, LC) are the foil normals the available tilt in the microscope
of 20” around S indicates the possible beam directions in the case of screw dislocations and the available reflections lie in the region of reciprocal space shown hatched. 220 and 220 reflections,
posite there are two possible foil normals
Dislocation
Line direction
Edge s xew Composite
1
Foil normal
[ii01
Reflections *t20, h220, 1022,
m
&202, 5022, ~202,
ilil, *iii, &iii,
*ill
*iii *iii
than 0.49 appear white and those below 0.02 appear black.
It should
3. EXPERIMENTS 3.1 Experimental
method
deviation
Single crystals of Cu-10 at. y. Al were grown by the Bridgman strained
in
technique, in an Instron
vacua
and
stage
shear
The
crystals
stress).
(
t’ensile tester
(1 .l kg mm-2)
II
be mentioned
that experimentally
very difficult to set the specimen
x
1O-5 torr),
into
stage
I
(2.3 kg mm-2
resolved
were sectioned
with
a
it is
at a predetermined
(w = 0.3) from the Bragg angle, especially
for higher order reflections. the diffraction
patterns
But an examination
showed
90 per cent of al1 the micrographs 3.2 Experimental
of
the result that for 0.2 < w < 0.4.
results and discussion
spark cutter and discs of diameter 2.3 mm were trepanned from the 1 mm thick sections, chemically
(a) Irzlined dislocations. The micrographs obtained from foils of type (ii) are directly comparable
thinned
with the computed images. In Figs. 3 and 4 the outlined computed dislocations have been given the
and
jet-electro-polished
niques and examined scope. The crystallographic
by standard
in a Siemens Elmiskop
techmicro-
length appropriate orientations
of the specimens
were of two types: (i) sections parallel to the primary slip plane (111) ;
to a foil thickness
of 75, and are
mounted on the micrographs in the correct orientation. In Fig. 3 the images of a set of dislocations close to edge orientation
((lll)[iOl])
may be compared
with
ACTA
462
FIG . 3. Computed
and observed
VOL.
20,
1972
(outlined) and el;perimental images for edge dislocations (lll)[lOl] showing in 8 different reflections. Foil normal of the micrographs is LC”.
the con tputed images. culated
METALLURGICA,
The general form of the calimages (e.g. “chain”
image for
strong double
image when g = ill
very slight doubling
t,he same area but in Fig. 3(g)
of the image may be seen.
The
g = 22c ), double image for g = 202, spotty image for g = lil and intense image for g = iii) is sufficiently
slight asymmetry from top to bottom of a dislocation whichismostnoticeableinthe weakimages (g = f lil)
close in each case that the Burgers vector of the dis-
and which does not appear in the computed images is due to the approximation used (see Section 2.2)
location s may
definitely
be
identified
as
@[iOl].
Althoug ;h the general forms match well and the width of the irnages agree, even with all the approximations
for g . b x u.@~) Figure 4 shows two sets of screw dislocations
used in the calculations, perfect.
with
match
is not
b = &&[iOl]
(the left and right sets in each picture
One feature which is never observed
is the
are of opposite
sign) in which the agreement between
the detailed
KARNTHALEK
FIQ. 4.
theory and experiment to
MICROSCOPY
their with
The screw dislocations
is close.
invisible
The dissociation partials
ELECTRON
when
very
g = jlil
slight
dissociation
of the screw dislocation some
edge
character
sufficient to give an asymmetry Figure
5(a)
shows
a typical
in stage II.
is,
(0.2E1,,). into
two
however.
Fig.
The
5(a)
reactions
in detail
and
the same Bmea
showi]
with the symmetry
Whelan.(16)
This
obviously
be true for glide dislocations,
true
Lomer-Cottrell
for
result
must
but it is also
dislocations
(which
are
on two planes) even though it is not true
that the Lomer-Cottrell
image is the same at the top
in Figs.
Figure 6 shows as an example b
are
at A
in
5(b-m).
foil thickness contrast.
the influence
of the
and the value of w on the dislocation
In Fig. 6(a) the deviation
from the Bragg position
w = 0.25 + 0.05
and the images of the dis-
sign
locations are in good agreement with the computed images where the foil is an even number of half extinc-
dislocations
of the same
tion distances thick (bright regions of the micrograph)
There are therefore two attractive
(C) and two
Two primary interact sign.
by dark field microscopy)
are analysed
of Howie
163
ALLOY
of a foil as at the bottom.
area of dislocation
Lomer-Cottrell
rules
dissociated
from top to bottom
The top t and the bottom
of the foil (determined indicated.
A F.C.C.
Figs. 5(h) and (k), in accordance
(g * b = 0)
of the foil through the g * b x u term. reactions
OF
images for screw dislocations (lll)[iOl] ComFlute d (outlined) and experimental in 8 different reflections. Foil normal of t,he micrographs is S.
are virtually owing
et al.:
(lll)[iOl]
dislocations
with two (Til)(OTi]
repulsive
(R)
type is visible.
junctions,
of opposite
and one junction
and the images are still readily identifiable where the foil has other
thicknesses.
in regions
In Fig.
6(b)
are indexed
in Fig.
w = 0.9 + 0.1 and the images show much less detail
5(l) and Fig. 5(m) is a view of the interactions
normal
and differ greatly from the computed images (Fig. 1, screw or edge, g = +202). For identification, there-
to (iii).
The dislocations
of each
In Figs. 5(j) and (k) the foil has been turned
upside down in the microscope so that dislocation ends which, in Figs. 5(h) and (i) are at the top of the foil are at the bottom
of the foil in Figs. 5(j) and (k).
It may be seen that the images at the ends of all the dislocations
are the same in Figs. 5(i) and (j) and in
fore, it is necessary
that the computed
and experi-
mental values of w should match closely but it is less important that the foil thicknesses should agree. (b) identify
Parallel
dislocations.
dislocations
in foils
It
is more
parallel
to
difficult
to
an active
5. (l), (m). ~~SIuc~t~on~eaet~o~~ at-d. (1) 8chemat~c of network in Figs. 5(b)-(i) with Burgers vectors of the various dislocations indicated. (m) View of t’he interact,ions normal to (ii 1) showing the attractive (C) and repulsive (R) junctions. FIG.
slip plane than in inclined foils because dislocations one character depths
in the foil.
However,
using the computed
images of Fig. 2. the conservation and the line directions network,
of
may appear quite different at different of b at a, junction,
of the dislocations,
a stage II
such as the one in Fig. 7, may be indexed
with very little ambign~ty Fig. *I(k). The dislocation structure consists of intersecting FIG. .5. (a) Region
of
inclined
dislocation
Top t and bottom b indiented.
reactions.
three slip planes (ill), diGcations
(lil)
slip lines parallel to
and (iii).
formed at attractive
The composite
junctions
are rarely
longer than 0.5y but are fre~~~e~~l~ placed end to end to give the impression of greater length. Cottrell dislocations never observed even
FIG. 5. (b)-(i)
Area. A of (ef in 8 reflections. Foil normal is LC. (j), (k) Area A but with foil inverted.
lower
Many Lomer-
are visible in Fig. 7, but we have
a Hirth lock ; their density
than
in
Cu. fz3j
The
must be
arra~~geme~~t of
FTC. 6, (a) Variation of image with foil thickness, 20 = 0.25 & 0.05. (b) Variation of image with foil thickness, W = 0.9 & 0.1.
FIG. ?(a).
FIG. 7 (c)I
FIG. 7(b).
FIG. 7(d).
ACTA
166
Fra. 7(e).
Fm. 7(f).
METALLURGICA,
VOL.
20,
1972
FIG. 7(g).
FIQ. 7(h).
KARNTHALER
et al.:
ELECTRON
MICROHCOPY
OF
A F.C.C.
467
ALLO\-
FIG. 7(i)
\
i
i FIG. 7(k).
FIG. 7. (a)-(j). Dislocation reactions. 10 different reflections showing the same area. [ill] foil normal. (k) Schematic of network with Burgers vectors of the dislocations.
The
only
dislocations
which
are the Lomer-Cottrell
completely
dislocations,
to their length (g = f%O),
disappear
when g is parallel
which is in accord with
the theory. 4. CONCLUSIONS
1. In an alloy such as Cu-10 at.% paratively Burgers
low anisotropy vectors
generally mental
approximation diffraction. dislocation reflections
FIG. 7(j)
compute
of the dissociated
be identified
images
Al with a com-
constant by
with those
(A = 3.66) dislocations
comparing calculated
of the dynamical
the
the may
experi-
in the lowest
theory
of electron
For a definite identification the same must be imaged using several different but for this purpose it is not necessary
to
the contrast for each micrograph.
2. The
images
of
dissociated
dislocations
differ
interacting primary and secondary dislocations in Fig. 7 is typical of that assumed in the forest model of
markedly from those of perfect dislocations separation of the partials >0.3E,.
work-hardening.(24-26)
3. In Cu-10 at. o/o Al many Lomer-Cottrell reactions are observed but Hirth locks have not been encoun-
Nearly all of the dislocations vectors sociation
which lie in (lil) many
are visible
in Fig. 7 have Burgers
but, because
of their dis-
when g = +lil
(e.g. at
B the [iOl] dislocation almost disappears in (g) and (h) where it is screw and is visible where it is edge).
when the
tered. 4. The arrangement slip planes
is precisely
of dislocations that
theory of strain hardening.
assumed
in intersecting in the forest
ACTA
465
METALLURCICA,
ACKNOWLEDGEMENTS
P. K. wishes to acknowledge financial support from the Royal Society European Exchange Programme and from the Oesterreichische Akademie der M. S. S. is grateful to the S.R.C. for a Wissenschaften. K.
grant.
would like to thank Drs. A. Howie, J. Goringe, MI.J. Whelan, D. Blaher and R. Perrin for many helpful discussions and Professor P. B. Hirsch F.R.S. for the provision of laboratory facilities. We
M.
REFERENCES I. M. J. WHELAN, Proc. R. Sot. A249, 114 (1958). 2. A. HOWIE, Direct O~eer~at~on of ~~perfect~o~~ in Crystals. Interscience (1961). 3. S. MADER and H-M.
Conference
for
THIERINGER, Fifth ~nt~r~at~onuZ Electron Microscopy. “Academic Press
(1962). 4. A. SEEOER,N.P.L.Symposium 15,~. 1. H.M.S.O. (1963). 5. J. W. STEEDS and P. M. HAZZLEDINE. Discuss. Paradav Sot. 38,103 (1964). 6. T. C. TISONE, J. 0. BRITTAIX and M. MESHII, Phil. Msg. 16,651 (1967). 7. P. B. HIRSCH and F. J. HUXPHREYS, Physics of Strength and PEastieity. M.I.T. Press (1969).
VOL.
20,
1972
8. P. B. HIRSCH, A. HOWIE, R. B. Krcaomow, D. W. PASHLEY and M. J. WHELAN, Electron &ficroseopy oj Thin Crystal. Butterworths (1965). 9. D. J. H. COCKAYNE, I. L. F. RAY and M. J. WHELAN. PhiE.Mug. 20, 1265 (1969). 10. J. FRIEDEL, PhiE. Mug. 46, 1169 (1955). 11. T. Jsssa~c, J. P. HIRTH md C. S. HARTLEY, J. Apple. Phys. 36, 2400 (1965). 12. A. HOWIE and P. R. S’CVAKN,Phil. Afag. 6, 1215 (1961). 13. L. M. BROWN, Phil. Hag. 10, 441 (1964). 14. J. R. NEIGHBOURS and C. S. SMITH, acfa Met. 2, 591 il9.541 i-“--I15. 4. S. STROH, Proc. Ph?/B.Sot. 6?B, 427 (1954). 16. A. HOWIE and M. J. WHELAN, Proc. R. Sot. A263, 215 (1961). 17. W. J. TUNSTALL, I’. 13.HIRSCH and J. W. STEEDS, Phil. Xag. 9, 99 (1964). 18. A. HOWIE and M. J. WHEL~~N, Proc R. Sot. A267, 206
(1962). 19. A. K. HEAD, AU&. J. Php. 20, 557 (1967). 20. P. !&!I\TBLE,AZ&. J. P&s. 21, 325 (1968). 21. M. S. SPRING and J. W. STEEDS, Phys. Status Solidi 37, 303 (1970). 22. J. SILCOCKand W. J. TIJNSTALL, Phil. Nag. 10,361 (1964). 23. H. STRUNK. Phil. Maa. 21. 857 (19701. 24. 2. S. BAS&KI, Phil. kigf6y:4, 39‘3 (19bS). 25. P. B. HIRSCH, Internal Stresses and Fatiglce in Metals. Elsevier (1959). 26. G. &ADA, A&z Met. 8, 541 (1960).