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Dislocation patterns in graphite
JOURNAL
A
OF NUCLEAR
det,ailed
patterns
analysis
observed
is
single crystal
dislocations
are ribbons
Shockloy
fault.
given
by means
in graphite the
type
of
the
1 (1962) 17-66, NORTH-HOLLAND
consisting
of two
by
a strip
can be interpreted
de fines
microscopy
flakes. It is found
lamelles
presentBe;
that the
partials
PUBLISHIXG
of
I1 est montre
que les dislocations
de rubans limit&
Tous de
modele
et en supposant
faults
consisting
stacked
lamella
of
i.e.
rhombohedrally
fautes
stacking
width
a
occur,
material.
The
fault
energy
of the dislocation
of extended vector. the
of
faults
nodes
ribbons
angle
between
with
this can be used to determine
of the ribbon
and
l’aide
de la forma
that
partielles
The
possible
pared
from
with
vectors
observed
were
1200” C annealing
found
to
3000” C and annealed
contain
vacancy
energy,
It
Burgers
vect)or
that
associated
and
with
the
of
the
theoretical
point
as to the nature
dislocations of view.
It
is discussed is shown
loops tend to form preferentially
that
formation
nature
mechanism
networks
sre
of the twin bounda-ies.
and the formation
end
discussed
the
mesures
Dislocation
of constrictions
&
et 8,
composes
de
connus. La depenque forme
son vecteur
du coefficient
de Burgers
de Poisson effe&f.
entre
rubans
sont
et comparees
Les vecteurs
disavec
de Burgers
$, I’aide
Zt 3000” C et reeuit
de boucles
h. l’aide
d’effets
de mesures
emmagasinee,
minstion
de leur
entre
boucles
les
of
qu’il
est possible
vecteur et
movement
La
are
electriques
&
de tirer
mecanisme
une eon-
de la d&er-
de Burgers.
L’interaction “glissile”
11 est montrb
de formation
est
que les boucles
B se former
de faute
de pre-
d’empilement,.
de reseaux
aussi bien que leur structure,
m6me que la nature de joints 17
et de
associe
dislocations
ont tendance
dans les rubans
cations,
discussed.
les
theoriquement.
dues 8. des lacunes f&ence
dues 8, C, trouve
est done
elusion sur la nature des boucles & partir
fault
ib 1200” C
de dislocations
de boucles.
Xl est montre a
as the
in the ribbons
avec
trempe
d’energie
la formation
vacancy
structure
as well
Btendus
de Burgers
possibles
la presence
discutee
dislocation
des
mesuree
des lacunes. Le pie de recuit B 1200°-1300’
vacancy
in the stacking
est
de dislocations
experimentalement
Du graphite
of the from
de nceuds
observees.
determines
rev&
ribbons. The
interactions
precedemment & determination
a conclusion
glissile
by
loops. from
0.it.d.
en empile-
de contraste.
and meas~~rements
loops can be reached. The int,era.ction between loops
sont
The
C, found previously
is therefore
of vacancy
is shown
loops.
Bnergie,
cutbes d’un point de vue theorique
at
ce
I1 est montrti que celle-ci peut etre utilisee
les configurations from
peak at 1200-1300’
formation
Les
making
de
seules des
de couches
d’empilement
pour la d~te~in&tion
Burgers
by
faible
de rubans
du ruban
est verifiee.
are
and com-
The
experimentally
means of eleet~rical measL~remonts stored
ribbons
of view
base
dance de la largeur d’un ruban et de l’engle
Poisson’s
effects.
quenched
is
between point
configurations.
determined
use of contrast Graphite
interactions a theoretical
de
aux vecteurs
la direction
discussed
etre inter-
sur la
que se presentent
faute
and it is shown
ratio.
of
de
de la largeur
an effective
peuvent
rhomboedrique.
L’energie l’aide
of the ribbon
observ&
de groupes
on
direction
is checked,
oomposees
ment
Burgers
une faute d’empile-
satisfa~~te
d’empilement
fautes
t,he
the shape
known
of the width
the
vector
from
and from
of partials
The dependence
the Burgers
is obtained
les r&eaux maniere
du micros-
par deux partielles
et contenant
ment.
stacking
est
cope Blectronique.
pret&
energy
dans
graphite
& l’aide
way on the basis of this model and on the assumption small
de
a 6th effect&e
du type de Shockley
in a consistent
monocristaux
celle-ci
sont con&it&es
of stacking
de
that
on&
CO., AMSTERDAM
Une anslyse detaillt3e de reseaux de dislocations
dislocation
of electron
separated
All patterns
5, No.
MATERIALS
de m&es.
de dislo-
est discute,
de
Res mouve-
fXe
Energie
t3reite
der
der
lijrtgwbr Xnaten Burgers
Voktor.
Bantlbreite richtung &I
und
ergibt md
DW dem
gezeigt, werden
dass kann,
sich
der
an Versetzungsteile~ Winkel dieaar um die
as16 der
Form
ver-
mit bekanntem
Zu~arnrnen~~~
zwischen
zwischen
und dem BU~~FS Vektor
wird
wend&
Stapelfehler
V~rs~t~~~n~sb~nder
wurde
tier
dor
Band-
unt~rs~leht*
Zusammenhang Paisson’sche
Zahl
VWXIL
b&immen.
Methods have been developed in recent years Co observe dislocations directly in the electron microscope lT2J). A number of papers, all discussing results obtained by me&n@ of electron microscopy and diffraction have been devoted to this subject recently 4*5$6).It is the purpose of this paper to give a more detailed account of our obse~~tions. Preliminary results have been published elsewhere 7,s). .Apart from their i~~~rt~a~~e for the mecfianical behaviour of graphite, the dislocations in this material present aspects of a more fundamental interest. In a sense graphite is a choice material to be used as a model for a crystal with ELsmall stacking fault energy and with a singte glide plane. It was therefore thought wo~hwhile to study in detail the geometrical features of such a model. A number of contrast effects, typicsl of parti& were noticed, but they will not be considered here. unless required for the .inb~ pretation.
Specimens were obtained by cleavage of natural single erSystals (originating from di~erent sources: Ceylon, Ticonderoga Limestone (N.Y.) and ~adag~,~~~r. ~ynt~~eti~ graphite prepared in the Philips Laboratories hy the graphiti~~tion of silicon carbide single crystals was also examined. For the qutinching experiments use was made of the Ticonderoga graphite which is e~e~pt~~~nal~~pure_ Graphite is t-l,choice materid for tr~~~sn~i~s~on electron microscopy hecause it transmits electrons very readily and because of the ease of preparation of thin foils. Its great ther~~~al stability and its reasonable heat, and electrical conducti~~it~ along the foil plane make it qnit*e stable under electron irradiation. The observations were made with the standard Philips microscope operated aL LOO ItV. In order to allow ~o~ltrolled tilting of the specimen, a de&e was eo~~str~~~~~~ which made it possible to turn very slowly the spelecimen holder. Since rotation about one axis only is possible in this way, i6 was necessary in a few cases to semount the specimen in another orientation with respect
DISLOCATION
PATTERNS
IN
19
GRAPHITE
Some graphites have however the rhombohedral
to the axis. Graphite flakes have a tendency to fall off the grids ; specimens were therefore
stacking
always
section through this structure is given in fig. 2b.
mounted
between
two
grids,
one
of
symbolized
by a b c a b c . . . a cross
which had a larger mesh size.
Most graphites
3.
as can be deduced from the occurrence of streaks in the X-ray diffraction patterns. This is
The Structure of Graphite The
consists
most
common
structure
graphite
of sheets of linked carbon atoms. A schematic
of a hexagonal
hexagonally
of
view of the structure
stacking
is presented
in fig. 1 in
especially
contain
many
stacking
true for ground and hence deformed
crystals. The amount of rhombohedrally material
faults
is known
It will be evident
to increase
stacked
with grinding.
from our observations
why
this is so. A consideration of the structure suggests immediately that the glide plane should be the c-plane since the binding between successive sheets is weak. The binding within the sheets
Fig. 1.
Schematic
view of the structure of graphite in space.
space, whilst fig. 2a and 2c give two projections of interest. It is clear that the stacking sequence can be symbolized by a b a b . . . . The two positions a and b are indicated in fig. 2c, it is further evident that a third position c is possible. The sequences a c a c . . . and b c b c . . . represent also hexagonal graphite structures.
Rhombohedral
Hexagonal.
cb 1
(a)
,t-‘-9
d’ b‘j
‘Ld
(c) Pig. 2.
Structure
of graphite.
graphite;
on the other hand is strong and breaking of C-C bonds on glide is highly improbable. Models for dislocations which do not involve breaking of C - C bonds will be presented below. Since the c-plane is also a cleavage plane, the foil plane will be parallel to the glide plane and we therefore expect dislocation arrangements in the plane of observation. With respect to glide such specimens should behave more like bulk material than foils, where the glide plane is steeply inclined with respect to the surface.
a...
.
.‘c’;?
..0 .....’ (d
1
a) Cross section of hexagonal graphite; b) Cross section of rhombohedral c) Projection on the c-plane; d) Direction of the Burgers vector,
20
P.
4.
DELAVIGNETTE
AND
types of stacking
AMELINCKX
i.e. one Iayer is in a wrong position.
Stacking Faults in Graphite Three different
S,
faults are
can only be eliminated
This fault by two glide motions,
a priori possible in graphite, if one excludes the
or by the glide of a dipole,
“a over a” stacking.
partial dislocations
The first type of stacking fault would involve only o?ze infraction
against the stacking
it can be represented
rule ;
L
The relative
(1)
a b a c b c b c b ... L-.-l
(2)
a b c a c a c ... I
In an .., a b a b . . . structure these two stacking faults result when either an a or a b layer is shifted into c position. This fault will be shown to occur between the partials of the dissociated dislocations responsible for basal slip ; it can therefore be generated by slip only. The third type of stacking fault finally is represent,ed by t,he sequence a b a e b a 1) a lt . . . or
YII!!+ a b a b c a I) a t) . , .
-II
lattice
planes
of the energies of
stacking
fault energy for faults of the second
type is very small ; we can therefore that all types have a small energy.
We will see later that this type of stacking fault occurs in vacancy loops. The second type contains two infractions against the hexagonal stacking rule, it can be symholized as :
Or
magnitudes
sign and same
these stacking faults are roughly as the numbers of infractions 1: 2: 3. We will show that the
ababaca,cac u
a b
Burgers vector in the adjacent indicated by arrows.
by the sequences
ahababcbcbc OS
i.e. two coupled
of opposite
-
Both faults involve three infractions against the stacking rule. They can be described as the result of the insertion of a c-layer either between “a and b” or “between b and a”. We will see that this type of stacking fault occurs in prismatic loops due to in~rstitials. In a few cases we will meet stacking faults of a fourth type: L+ ... a b a b c b a b (4) IL.-J
5. 5.1.
Different Kinds of Dislocations BASAL
infere
in Graphite
I?ISLOCAT1OSS
The dislocations are invariably parallel to the foil plane and they consist of ribbons of two (or three) partials. Wo will now see how this can be understood. The only observed glide plane being the c-plane, glissile ~sloeations have Burgers vectors in this plane. The shortest among the vectors connecting one atom to the next crystallographically equivalent one are AB, AC and AD, as well as their negatives (fig. 2d). The vectors can be decomposed into two partial vectors, according to the reaction AB + Acr+oB, as demonstrated in fig. 2c, d. In the language of dislocations this means that the dislocations can split into two partials of of the Shockley type. The stacking fault resulting from the displacement over a partial vector like Ao, Ba or Co consists of a lamella in rhombohedral stacking, with two violations of t,he hexagonal stacking rule, as visualized in fig. 3a. Since a rhombohedral variety of graphite exists this stacking fault has a low energy and an observable dissociation into partials actually takes place. as will be demonstrated by our observations. A second type of dissociat#ion is geometrically possible. AC + A@‘+- /E or act + Ao as shown also in fig. 2d. A partial dislocation now brings one layer on top of the second one; we will call this the “a over a” or “b over b” position. It is to be expected that the energy associated
DISLOCATION
PATTERNS
21
IN GRAPHITE
tb) Model of extended 60” basal dislocations in graphite. No carbon-carbon bonds are broken. a) The Fig. 3. dislocation has a Burgers vector AC. Notice the lamella of rhombohedrally stacked material; b) Projection on the c-plane;
the Burgers vectors of the partials are bl and bz.
with this stacking fault is considerably higher than in the first case. We will demonstrate that all observed patterns can be explained without the necessity of assuming the occurrence of this type of stacking. On the contrary in order to explain certain patterns we will have to infer the fact that an “a over a” stacking is avoided. This may be regarded as an indirect proof. We conclude that the second type of dissociation does not take place. A model of an extended dislocation having the required Burgers vectors is represented in fig. 3a, b; it is clear that no C-C bonds are broken. Only deformation of hexagones is required to take up the strain. The presence of two kinds of c-planes i.e. a and b planes within the unit cell gives rise to two classes of partials. For those dislocations located between “a and b” (a underneath, b on top) the dissociation reactions are of the type AB -+ Ao+ uB, i.e. 0 inside; for those located between “b and a” the possible dissociation reactions are of the type AB + (TB+ Aa i.e. u outside. The stacking fault associated with this first dissociation is described by the
sequence ababacbcb... and for the second
type of dissociation
by:
ababcaca... One of these stacking sequences is represented in fig. 3a. The movement of a partial Aa, Bo or Co between
“a and b”
(a underneath,
b on top)
changes the stacking according to the prescription a + b + c + a, whilst the partials aA, oB and aC have the opposite effect, i.e. they change the stacking according to the scheme a + c --f b --f a. On the other hand the movement of the partials like aA, aB and aC between “b and a” (b underneath, a on top) changes the stacking according to a + b + c + a ; the partials Ao, Bo, Ca do the reverse. 5.2.
NON
BASAL
DISLOCATIONS
One could also consider perfect dislocations with a Burgers vector [oooc]. In the edge orientation they are equivalent to the insertion of two supplementary c-planes. Glide on other
P.
DELAVICNETTE
AXD
planes than the c-plane would however involve
S.
AXELINCKS
for such glide. It is therefore doubtful that such
graphy of grown crystal faces 9) and from the study of cleavage faces of graphite 10). These dislocations may play a role in the growth of
dislocations will occur as glide dislocations,
polytypes
the breaking of C - C bonds ; there is no evidence
may
however
form
sessile
(fig. 4a). A dissociation
prismatic
they loops
into two partials with
Partial having
of graphite. dislocations
a component
a Burgers
vector
in the c-direction
with
result
vector c/2 would reduce the strain energy. This
from the insertion
would
c-plane. Consider first the insertion of a c-plane
imply
growth
of one of the vacancy
layers in fig. 4a. Such a process would however give rise to a stacking “a over a” or “b over b” depending on which layer is assumed to grow. There will therefore be a tendency for the two layers to grow simultaneously. Dislocations with Burgers vectors having a component in the c-direction and emerging in the c-face would give rise to spiral growth on this face. There is evidence for the occurrence of such dislocations from studies of the topo-
a-
a
-a
(al
or the removal
as would result on the precipitation of int,erst,itials. The layers will be formed in c-position between an a and a b layer (fig. 681)). A prismatic dislocation loop cont,aining a stacking fault of type three and having a Burgers vector (WO&) will result. The stacking fault of the t)ype threth could be converted into one of type one by the nucleation of a partial, but since t’he stacking fault energy is small the corresponding stress y/b is too small and this will not happen; t*he loop is stable. The removal of a part of the c-plane, as would occur in the precipitation of vacancies would give rise to a prismatic dislocation loop containing an “a over a” stacking. Since the energy of such a stacking fault is too high it will be removed by the nucleation of a glissile partial at the periphery of the loop. After the partial has swept the loop the Burgers vector will be inclined with respect to the c-plane. A cross section through such a loop is shown in fig. 68a. The stacking
--1-a -b-a-at-L-
(b)
Fig. 4.
NOR basal dislocations in graphite. a) Perfect (000~) dislocation; b) dissociation of the dislocation on the c-plane
loops;
of the dissociated
fault within the loop
is now of type one. It, is clear that both interstitial and vacancy loop are sessile II). We can now discuss the possibility of dis-
b-----b-b-____b-a-a--___a-
in (a) into two partial prismatic
of a single
(e projection loop.
sociation for the double loop of fig. 4c. If glissile partials are nucleated and accompany the edges of the growing layers, t,wo concentric loops, separated by an anular stacking fault region. could be formed, as shown in cross section in fig. 4b. The system of concentric loops has a smaller strain energy than the double loop. Concentric loops have been observed (see fig. 5 in B) but it was not possible to decide unambiguously whether they were of the type described here, or the superposition of two independent loops, which may lead to the same contrast effects.
DISLOCATION
PATTERNS
IN
23
GRAPHITE
perfect dislocations
in the face centered lattice.
In order to be stable, a network
should not
contain adjacent nodes of the same type (K or P), no adjacent nodes are further allowed to be both extended
or both contracted.
In addition
aII parallel segments must repel each other and they will usually have the same Burgers vector. In discussing the stability of nets we will also have to stacking
take into faults of
account the presence of different specific surface
energies. Faulted areas tend to shrink in surface as much as allowed by the increase in line energy. 7.
Determination
of Burgers Vectors of Basal
Dislocations
Fig. 5. Loops due to quenched in vacancies after annealing. The insets show some hexagonal loops.
6.
Nodes between Partial Dislocations Basal Plane. Stability of Nets
in the
When using the notation introduced in fig. 2d two kinds of threefold nodes among partial dislocations can be distinguished. Their lettering scheme is given in fig. 6. The corresponding node conditions are oA + (TBf OC= 0 and Aa + Ba + Ca = 0. A distinction is clearly only possible if one adopts a convention concerning the sense in which the symbols are read. We accept the same rule as formulated earlier by Frank 12) for the face centered cubic lattice. When looking out from the node point we read from left to right of the line. The two kinds of nodes given here correspond in fact to the K and P nodes introduced by Frank for
(a) Fig.
6.
The two possible types
(b)
of threefold
nodes.
a) Corresponding to the K node introduced by Frank; b) Corresponding to the P node introduced by Frank.
It has been shown by Hirsch et al. 1) and Whelan 13) that the intensity of the diffraction contrast at dislocations depends on the value of n = b .g, where g is the diffraction vector. For n=O the dislocation does not show any contrast. It is clear that this effect, can be used to determine the direction of the Burgers vectors. A knowledge of the structure then allows one to make a plausible guess as to the magnitude. In the particular case of graphite one knows that glissile dislocations have a
Burgers vector in the c-plane. It is therefore sufficient to determine one diffraction vector for which contrast disappears at a given dislocation. This can be done either in light or dark field. In the first method one tries to find an inclination
of the specimen
for which contrast
disappears at the dislocation of interest, the others being visible. Without changing the position of the specimen, a selected area diffraction pattern is made of the region of interest. Care should be taken to select a region at some distance from the extinction contour whose proximity is responsible for the contrast. This precaution is necessary in order to obtain meaningful results because the kinematical theory, on which the effect is based, may become inapplicable near the extinction contour. In the diffraction pattern the spot which causes the contrast will be the most intense one, and can therefore generally be recognized.
24
P.
Pig.
7.
Successive
families
pattc !rns for each case.
DELAVIQNETTE
of partials
AND
in a network
a) S,tacking fault contrast;
go
S.
out
AMELINCKX
of contrast,.
The
b), c) and d) One set’ of partial
On the ot’her hand it is possible to make a dark field image using a predetermined reflection. Those dislocations for which b is in the plane corresponding to the selected spot will then disappear. In graphite, partials disappear for reflections of the type (1120) showing that the Burgers vectors are Aa, Bo and Ca as predicted by the consideration of $4.1. The set of photographs of fig. 7 shows the disappearance of contrast successively at each family of partials in a network. The insets give in each case the diffraction pattern in the correct orientation with respect to the image. From these observations it can be concluded
insets
show
dislocations
the
cliEraot,ic)n
is ollt of cnontrast,.
that the partials have the screw orientation at the extended nodes, and, further, that the ribbons as a whole have screw character, since the nodes are symmetrical. This distribution of Burgers vectors is represented in fig. 7. In the majority of cases isolated nodes have this orientation. In one case an extended node of edge ribbons was identified; it is shown in fig. 8. Stacking fault contrast occurs when n is fractional, the most intense contrast of this kind is found for (lOi0) reflections i.e. if n= Q or 3. This is demonstrated in fig. $1. In this case two partials can go out of contrast simultaneously in certain circumstances, ap-
DISLOCATION
parently violating of contrast.
PATTXRNS
the g ’ L -I 0 criterion for lack
These contrast
effects,
which
are
IN
25
GRAPHITE
specific for partial dislocations,
will be discussed
in a separate paper. 8.
Width of the Ribbons According
dislocation
to
Read 14) the
width
d of
a
ribbon depends on its character i.e.
on the angle q~between its total Burgers vector and the direction
of the line. The theoretical
dependence, as given by theory is represented by:
isotropic
elasticity
where Y is Poisson’s ratio. This formula has yet been verified experimentally because of di~~ulty of measuring d. F~lrthermore variation of the width as a function of 9
Fig, 8. Extended node of edge ribbons. a) The t,hree sets of partial djsloe~tions are visible; b), c) and d) One set of partials is out of contrast. The insets show the diffracting plane for each case.
not the the can
yield a value for Y. Such measurements are however only possible in those crystal where the width of the ribbon is sufficiently large to allow a direct measurement; graphite is such a crystal. One should however take care to avoid a number of sources of errors. Because of spherical aberration, the electron microscope only gives an undeformed image, and hence a reliable width, in the center of the field. The segments of ribbons to be measured were therefore always photographed in the center of the screen. In view of the one-sided nature of dislocation contrast, and because of the displacement of the dark-line with respect to the actual position of the dislocations,
care must be taken when
estimating the widths of ribbons when different contrast conditions are used. In our measurements these difficulties were eliminated by using one long dislocation line which was slightly curved so that a wide range of orientations was available. The contrast was continuously adjusted, by tilting the specimen, in such a way that the contrast conditions were as constant as possible, at the position of the segment to be measured. Although the ~splacement of the dark line was taken into account, a constant displacement would not cause a serious error Fig. 9. Network showing stacking fault cont,rast,. since it would only produce a parallel displaceThe inset is the corresponding diffraction pattern. ment of the straight line in fig. 10. The method
P.
DELAVICNETTE
ANT)
S.
AMELINCKS
.
Pig.
10.
the total
Plot
of t,he width
Burgers do in eq.
vector
d of a dislocation
and the direction
(5) whilst
ribbon
as a function
of the ribbon.
the slope of t,he straight
used also avoids the eventual lack of reproducibility of the microscope magnification. The Burgers vectors of the ribbons were determined using the procedure outlined above. The diffraction pattern fixes at the same time the orientation of the foil and hence rp. Fig. I1 reproduces at small magnification one of the ribbons used. whilst the insets illustrate the
The
R,bbon
I
of COB 2 q where
intercept
with
line is 2v,/(2---I*),where
p is t,he angle between
the d axis gives v is Poisson’s
the quantity
v&o.
with specific surface energy I’, one obtains (14)
different segments used for the measurements. The insets are all at the same magnification. The error introduced by the curvature of the ribbon is negligable provided this is small enough with respect, to the accuracy of the measurements. The data points for this ribbon are given as black dots in fig. 10 the open dots referring to a second ribbon. A few isolated measure~~ents are also included in the plot. It is clear that the functional dependence is reasonably well obeyed. The value of v deduced from the slope of the st,raight line is 0.24 5 0.04 which is significantly smaller than +. The value for do is do N 0.1 x 10-4 cm it is used to determine the specific stacking fault energy. 9.
The Stacking
Fault
Energy
Expressing the equilibrium separation the two partials, bordering a stacking
do of fault
One of the ribbons used for the measurement Fig. 11. of the width. The insets show some of the segment’s used. Noticft
the visible change orientation.
in width
with
DISLOCATION
PATTERNS
27
IN GRAPHITE
Fig. 12. Contrast at triple ribbon: the three ribbons go out of contrast simultaneously. respectively a T-node and a Y-node are visible. a) Line contrast; b) Stacking fault contrast;
In T and Y c) One set of
part,&1 dislocat,ions is out of cont,rast,.
where do has been defined in 5 6 ; p is the shear modulus and b is the Burgers vector of the partial dislocation. The value of d 0, which is the width of a 45” ribbon, was found by taking the intercept with the d axis in fig. 10; do 1: 10-5 cm. With ,u= 2.3 x 1010 dynes/cm2 15) and b= 1.42 x lo-scmthisleads toy = 3.5 x lo-2erg/cm2. alternatively lowing
one can make use of the fol-
approximate
procedure to obtain stacking fault energies from extended nodes. The interaction between partials is taken into account to a certain extend and the total energy is ~nimized by means of a variational procedure. The specific stacking fault energy is then given by Y=
formula 16)
involving the radius of curvature R of an extended node such as fig. 12 and shown schematically in fig. 13. This relation expresses the equilibrium curvature of a dislocation line under an applied force y per unit length. This relation however does not take into account the interaction between the partials forming the node and it is therefore only a rough approximation. Since R= 1.2 x 10-4 cm we find y = 2 x 10-z erg/cmz. _ _ ^. Siems et al. 17) have indicated a better
KYdYd3
(8)
where K=
pb2 2-i-v --
8n
Fig. 13.
Schematic
1-Y
view of extended node in order
t,o illustrate the notations
used.
28
P.
for edge ribbons
DELAVIGNETTE
AND
and K=
for screw ribbons;
“fusion” and “intersection”. By the first term we mean that the two ribbons
/lb2 2-3v -8n 1-Y
yo is a dimensionless
meet, number
of screw
and we find ZJO=0.3 x 10d4 cm;
this
leads to y = 3.5 x 10-s erg/cm2 in good agreement with the value deduced from the width of the ribbon. Using anisotropic theory this value would presumably increase. + The small value of y explains why rhombohedral graphite is sometimes found, especially after deformation. Heavy shear along the c-plane results in large networks of the type shown in fig. 14, of which practically half of t~he surface area is rhombohedral. 10.
without
reaction
The meaning of ~0 is clear from
iig. 13. The node of fig. 12 consists ribbons
AMELXNCKX
tinguish between
which is y0==4.55 for an edge and 5.95 for a screw ribbon.
S.
Interaction between Ribbons: Fusion Reactions between Simple Ribbons
We will now discuss in detail how ribbons can interact with one another. We will dis-
only
crossing takes
over
place
one
another;
between
the
a last
partial of the first and the first partial of the second ribbon. We will distinguish cases whereby the ribbons are between the same or between adjacent
lattice planes. In the first case they
contain the same type of stacking fault, whereas in the second case they contain different stacking faults. 10.1.
TWO
RIBBONS
STACKING LATTICE 10. I. I.
WITH
FAULT
THE
SAME
AND
IN
TYPE
THE
OF
SAME
PLANE
The. two complete Burgers vectors fu~t~~ an obtuse angle of 120”
10.1.1.1.
The last partial of the first ribbon is the same as the first of the second, e.g. Ao -t-oB and Bo + aC such ribbons always attract
On meeting, the two partials crB + Bo annihilate each other mutually and a single ribbon Acri crc! results. If the two ribbons are however pinned somewhere, or attached to the rest of
(b)
Large network consisting in small stacking
Fig. 14.
fault triangles. In Tw, one twin produces the inversion of the stacking
fault t,riangles.
t According to calculations by Siems, Delavignette and Amelinekx 17) and by Spence 25) the value for y becomes
0,7 erg/cmz.
(Cl
(df
Big. 15. Fusion of two ribbons, containing the same staaking fault giving rise to an isolated extended node. a) and b) the node contains a stacking fauIt of type I; c) and d) the node cont,ains a stacking fault of type II.
DISLOCATION
the network
an isolated
extended
PATTERNS
comes the repulsion due to the central partials.
node forms
As a result
as shown in fig. 15a, b ; fig. 12 is an observed example. This process is not responsible for the formation of the regular networks since it can only
produce
isolated
extended
nodes.
29
IN GRAPHITE
The
narrower
the ribbons
as they approach
become
gradually
one another
and a
complete dislocations CB is formed by the reaction AB + CA --f CB this dislocation immedi-
interior of the curved triangle contains a stacking
ately separates again into two partials and an
fault of the same type as the ribbons giving rise to it. Fig. 15b represents the lettering
isolated
pattern
for two ribbons both of which contain a stacking
if the stacking
fault is of type I and
The but the e.g.
node is formed
as shown
fault of the second type, for example
fig. 15d if it is of type II. 10.1.1.2.
contracted
in fig. 16. The same type of reaction can occur
two ribbons as a whole attract the last partial of the first repels first partial of the second ribbon Aa+ aB and Ca+ aA.
The question arises first whether for parallel ribbons an equilibrium configuration is possible, whereby the four partials remain separated. For example the attraction of the outer partials might compensate for the repulsion of the inner partials. A detailed consideration of the forces acting on the different partials shows however that as the ribbons approach one another their width also diminishes in such a way that the attraction of the outer partials always over-
and UC+ Bo.
The complete Burgers acute angle of 60”
10.1.2.
aB + Ao
The result is given in fig. 16d. vectors form an
The dislocations now repel as a whole and no fusion reaction is possible; however, we will see further that they can intersect under the influence
of a shear stress. The complete Burgers vectors are at an angle of 180”
10.1.3.
The dislocations attract since they are of opposite sign; they will now annihilate each other mutually. 10.2.
THETWORIBBONS STACKING ADJACENT
CONTAINADIFFERENT
FAULT, THAT
IS THEY
ARE
IN
PLANES
10.2.1. The two ribbons as a whole repel
Two different 10.2.1.1.
situations
Formation
may occur.
of threefold
ribbons
In the first situation, the last partial of the first ribbon attracts the first partial of the (a)
(Cl
Fig. 16.
Cd)
Fusion of two ribbons containing the same
stacking fault producing an isolated contracted node the
second ribbon
(b)
two ribbons contain a type I stacking fault. a) to c) different stages; d) stable node.
;
as for example
in Au+ aB and
OC + Aa. Although the ribbons repel as a whole the inner partials of the quartet aB and UC attract. A detailed consideration of the equilibrium condition reveals that a metastable equilibrium position is possible without recombination of the inner partials (see appendix I). This can be understood intuitively as follows. The outer partials are kept at a distance by the repulsive forces between them ; the inner partials attract? but they are bound to the outer partials by the stacking faults and they are therefore kept separated. If the stacking fault energy is small the equilibrium is very unstable and re-
P.
DELAVIQNETTE
AND
S.
AMELINC’KX
a-
-a -b
b-
--P
a-
-a-------a
a-a-
I
l -b~ccb~b-
b-----ba------a
-b
-
b-----ba-a-
-b-b
--a-a al8
(b)
t.31
(C)
Fig.
l’i.
Nodels
of t)hreefold
asymmetrical
ribbonr.
ribbon;
a) cross section
c) Jn projection
of the symmetrical
combination will take place under a small shear stress: a threefold ribbon will therefore result. This ribbon is now stable and will only dissociate again under a strong shear stress in a suitable direction.
ribbon;
showing
on the c-plane,
b) cr’oss srlc:t,ion or tlift
the notation
nsc~l.
the phase shift at the fault plane is 1/3z whilst. it is 213~ for the other half of the ribbon. As will be discussed in a separate paper Is)% this
The geometrical structure of the threefold ribbon is represented schematically in fig. 1Ta. The three dislocations have the same resultant Burgers vector Aa. The central dislocation is in fact the close superposition of two dislocations with vectors aB and OC in adjacent lattice planes. The resultant
dislocation
behaves
like
one with vector Ao. This is shown by the fact that contrast is found to disappea,r at the three dislocations simultaneously as illustrated by fig. 12~. The width of a threefold ribbon is about five times that of a single ribbon. In appendix I the equilibrium separation in a threefold ribbon is discussed. If the two reacting ribbons are part of a network, nodes of the shape shown in fig. 18, may result. We will call them Y-nodes. Observed examples can be found on several of the micrographs e.g. in fig. 12 and 19 and fig. 76a. The different nature of the stacking faults in the two ribbons is strikingly illustrated by fig. 19b. where the two halfs of the t’riple ribbon exhibit a very different contrast. For one ribbon
(I)
a-L-b-+--+--a b!
b-b I-_a-a---a
c-.+--a-+--b
-b-b
b-b--b-b ~-a-a--1
Cdl
(0
Frlsion of two ribbolls containing Fig. IX. st,acking fault, resulting in the pro(lrl’:tion fold ribbon fold b)
ribbon Two
(Y-node). have
the
a tliffererrt of a tlrrvv-
The tlrrce par%ials in t,hcxthrclc:-
same Burgers \-vc+or. a) ant1 c) stable c:onfiguration: different stages; (11 cross section throueh the ribboll.
DISLOCATION
Fig. 19.
Threefold ribbons.
PATTERNS
may lead to significant differences in diffracted intensity. The interacting dislocations need not necessarily be in adjacent planes for t,he formation of the threefold ribbons. If only a few unit cells separate the glide planes, the interaction is probably strong enough to ensure the stability of the threefold ribbons, The level difference between t!he two halves could also give rise to a small shade difference in the stacking fault contrast. This is however not the main origin as mentioned before. The equilibrium distance of the outer partials decreases slightly as the distance between glide planes increases. The meeting
partials repel
If the ribbons repel as a whole and the meeting partials also repel as e.g. in Aa+ oB and oB -t-Crr, no fusion reaction is possible. The ribbons can however cross over and form a network by intersection as will be shown later. 10.2.2.
The two ribbons as a whale attract
The first possibility
31
GRAPHITE
a) line contrast; b) stacking fault cont,rast. Notice the difference in shade between
the two parts of the ribbon. In A is a symmetrical
10.2.1.2.
IN
is now that the meeting
ribbon; in R an asymmetrical
one.
partials attract as e.g. in Atr + aB and GC + Ba. The two inner partials recombine according to oB +Co + Ao; as shown in fig. 20b. This is however only an intermediate stage because Ao can further react with Ba to form a partial oC. Since AGF and aC now repel the final situation of fig. 20d is stable. We will call these T-nodes, because in a more symmetrical form they are T-shaped. A number of examples can be found; one is shown in fig. 12. In all other eases the ribbons do not react by fusion, but they will react by “cross over” or by intersection as will be discussed in the next chapter. The fusion reactions are “unrepeatable”. By this we mean that the reaction can only take place once, because it results in a transformation of one of the ribbons into another one which does not react in the same way as the first, with a set of similar dislocations. Let us take the specific example of fig. 15b. Once the extended node formed, a ribbou ACJ+ aC has now to react with the next ribbon Aa+ aB or Ba+ UC (fig. 15a). However no fusion reaction is possible and so the process
I’.
DELAVIGNETTE
AND
S.
AMELI1JCXX
fdt
A,U
6, B
-a--&b-.--&-~-b-cJb-4&b--&---+”
-b----t_c+b-V a-*1;
b&b-
-a-a.&&.--+ekalc
Fig.
20. Fusion
stacking
of the formation
of an extended
node
takes
place only once. The same applies to the other reactions so far discussed. It is therefore clear that regular networks will not be formed by these processes. We will show that they form by an intersection reaction or by “cross over”. In summary we can conclude that fusion reactions can give rise to four types of isolated nodes. If the two reacting ribbons are in the same lattice plane, and hence contain the same stacking fault isolated extended or contracted nodes can form ; if the two ribbons are in adjacent lattice planes. and hence contain different stacking fault, T and Y nodes can be formed. 11,
Intersection of Two Simple Ribbons
If a fusion
reaction
is possible
it will take
place
of two ribbons
fault. resulting
preferentially
containing
in the production
rather
than
a different of a T-node.
intersection.
It is thus clear that we will only have to consider the cases where no fusion reaction is possible. il.
1.
THE
TWO
BURGERS
11.1.1.
DISLOCATIONS
HAVE
THE
SAME
VECTOR
The two ribbons contain the.sam,e stacking fault
The two dislocations will annihilate as shown in fig. 2 1. The process is a little more complicated than in the case of complete dislocations. On intersecting the region of overlap is stacked “a over a” and will therefore immediately shrink in area. In practice it will not even form at, all. The region of “a over a” stacked material will finally be eliminated completely by the
DISLOCATION
recombination
PATTERNS
of the four partials OC and Ccr,
two independent
ribbons
Aa + oB result.
IN
33
GRAPHITE
annihilate completely
since they are in adjacent
planes. Instead of annihilation
“dipoles”
form.
One of the dipoles consists of Aa and aA; the The
11.1.2.
two
ribbons
contain
a
dif7erent
to a row of interst,itials, the other to a row of
~~a~k~ng Gaels The situation
is represented
ribbons with an extended formation
in fig. 22. Two
jog are formed.
The
process is in fact rather complicated
if analysed locations
in detail.
On intersecting
the dis-
with the same Burgers vector
cannot
Intersection
-a
within
the two dipoles
contains the stacking a b a b c b a b . . . i.e. two III stacking faults (fig. 22~); it will therefore tend to shrink, especially since there is no long range interactions
between the dipoles. In doing this
(C)
(b)
8,a
C/U
sic
U;A b
b-
.--+----a-
---.-a
c-
-?---&--bb-
-b-b x-a
The region
of two ribbons of t,he same Burgers vector and same stacking fault,
(Cl)
--b
vacancies.
(b)
(a)
Fig. 21.
other of aB and Bo. One of them is equivalent
i c
a-r b-
-1) U-C ----b
t 4
a-v b-
-b
b-
-----a
a-
IVJ
Fig. 22.
Interse&ion
of two ribbons of the same t*otal Burgers vector but containing & different st,acking fault.
34
P.
the layer c is transformed normal stacking.
UELAVIGNETTE
ALUD
int,o a, restoring the
AMELINCKX
the ribbons.
The line
energy
of the dipoles,
the two dipoles
En. increases of course with the distance between
combine into a single dipole and rest,ore perfect This can
the tZwo components. In the partio~lIar case shown here the dist,ance is not known ; i41~ is
most easily be seen by assuming that Ba and Aa
found to be about, 1.5-1.7 times the line energy
combine
of a ribbon.
material
into
between into
On meeting
S.
the two
OC whilst
dislocations
of
ribbons.
oB and OA combine
opposite
sign
Co.
liiach
ribbon contains some type of extended jog. It is assumed that in fig. 23 the angular dislocations
resulted from interseot8ions of this
type ; with
every
sharp
bent) in one
of
the
ribbons corresponds a bent, of opposite sense in another ribbon, st,rongly suggesting that “something” has to connect the corner p0int.s. It, is suggested that the ~onne~t,in~ entities are the dipoles of fig. 22. Such dipoles should give a weak or no contrast since the stress fields cancel approximately. From the angle enclosed by the ribbons and the dipoles t,he line energy of the latter can be estimated as a function of the line energy of
11.2.
THE
TWO
AND
ALSO
RIBBONS THE
TWO
REPEL
AS
MEETING
A
WHOLE,
PARTIALS
REPEL
11.2.1.
They contuin the same stacking fad mad they are in the Same latrine $ane for ~xa~~~l~ Ba--toA and (‘n-i OA
New segments will form as indicated by the dotted lines in fig. 24a. The two segments aA will further annihilate each other and give rise to an extended node. The central region contains a stacking “a on a” as will be seen by referring to fig. 24d. In fig. 240 is represented the stacking within the ribbon l&i-. aA along the line UV. The passage of a dislocation Ca on the a-plane causes a change in stacking according to the scheme a -+ 1) -+ o -+ a. The stacking along the cut XP becomes therefore as represented in fig. 24d i.e. the central area is stacked “a on a”. This region will shrink since ‘(a on a” is a high energy stacking fault, and the result is therefore a contracted node as shown in fig. 24b. The reasoning followed above is in fact somewhat idealized in a way. The real sequence of events is more like the following.
On meeting
the two inner partials repel. They cannot cross one over the other since this would create “a on a” stacked material. Since the two ribbons as a whole also repel a shear stress is required to make them react. If the shear stress is such that it causes the ribbons to approach, it may either cause the ribbon BA or the ribbon CA
Fig.
23.
L-shaped
divloeation
bents are presumably
ribbons.
The
connected.
sharp
to recombine. Suppose that the direction of the shear stress is such that BA recombines, and CA does not. The perfect dislocation BA can now cross the partial Ca without di~oulty, since the Rurgers vectors are perpendicular, there is no large elastic interaction and the stacking is not
I)ISLOCATION
PATTERNS
IN
(a)
U
816 -iI_= -b
(b)
CIA
b-b ,,&_-,~:~“,~~
-b_=
v
x
-C---+.--4i--J---C-
c-j-.---c,st_b--c
-a -b-b-b-
-b-b-b-
--B-------a--a-
--a--a-a-
-b-------b-b-
-b-b-b(Cl
Fig.
24.
Y
61A
616 -b--+&-+--b_b--&..-A&~b-
--a,b+---b=c+---a,b CT
35
GRAPHITE
Intersection
Cd)
of two ribbons giving rise to one contracted
affected by the perfect dislocation. Once BA has crossed over CG it can however dissociate into GA -/- Bcr, since this restores perfect material between the two p&Gals. The partials will now be re~stributed into ribbons in a different way than at the start. The result is the formation of an extended and 8 contracted node. The important feature
and one extended
node.
In this case the process is slightly different; it is now CA that crosses over Bo and then dissociates again into Co + GA. The final result is however 11.2.2.
identical.
The two rabble wnta~n a d~~~ren~ stacking fault, e.g. Ao + at2 and oB-+ Aa
The two meeting partials now always attract,
of this process, which is represented in fig. 25 is however that the sam? ribbons are again available for new interactions the process is therefore repeatable. An observed example of nodes probably formed by this process is visible in fig. 26. If the direction of the shear stress is ~fferent, the ribbon CA may be forced to recombine.
(a)
Fig.
(b)
25.
Real sequence of events for the interaction
of fzig. 24. b) BA
(Cl
a) the ribbon
Bo + crA is compressed;
crosses over C; c) BA
dissociates
again.
Fig. 26.
Probable
example
process described in 3 11.2.1.
of node formed
by the
a.nd pictured in fig. 25.
36
P.
At?
DELAVIG
ETTE
AND
S.
AMELINCKX
PIG
-b---+b+-r-
-b+b-b-1-1-a61A (Cl
Fig. 27.
Intersection
of two ribbons containing
but since they are not in the same lattice plane the possibility for crossing over exists and will be easier the farther the glide planes are spaced. After crossing the new segments indicated by dotted lines in fig. 27a will develop. The segments Aa will not annihilate each other directly since they are in different planes; instead, a dipole Aaf- aA is formed as shown in fig. 27b. The area where the ribbons overlap contains a double stacking fault, i.e. two layers in cubic stacking fig. 270. It will therefore be eliminated. This process proceeds by glide of the dipole towards the rest of the dislocations, with the formation of constrictions in the two ribbons as
a different hacking
fault.
illustrated in fig. 27d. The two constrictions are stabilized by the other dislocations present. There is no net repulsion between the dipole and the isolated partial aC, the stacking fault will therefore be eliminated completely. The final situation is described by the lettering pattern in fig. 27f; an isolated example is visible in fig. 76a. The main feature of this interaction is the formation of a threefold ribbon, the three partials having the same Burgers vectors. pu’umerous examples of such threefold ribbons can be found in the photographs, e.g. in fig. 12, 28. The real sequence of events is again
DISLOCATION
PATTERNS
37
IN GRAPHITE
stress, the two outer partials attract. The result will be that the complete
dislocation
AB and
AC, which repel will cross, and after crossing they will dissociate
into Ao + aC and crB + Aa.
We have now reached the situation which was used as a starting point of our previous reasoning, the constrictions
are however
already present
and all that is needed is a slight rearrangement to superpose 12.
12.1.
the dislocations.
Reactions between Threefold Single Ribbons FORMATION IWSION
OF MULTIPLE
Ribbons and
RIBBONS;
REACTION
Quartet ribbons can be formed by fusion of a triple ribbon and a single ribbon for example
Fig. 28. Network formed by the intersection of two sets of dislocations in successive planes,
somewhat different if one considers the actual process of meeting. Suppose that the left ribbon is aB + Ao and the right ribbon Aa+ SC. On approaching, under the influence of a shear
(AU+ Aat Au) and (Ba+ oC) gives rise to a ribbon (Aa + Au + aC -t- oC). In appendix II the equilibrium configuration of such a, ribbon is calculated (fig. 29a). The process can go on inde~nitely and more complicated ribbons can be formed, a possible fivefold ribbon can be formed by combining the fourfold ribbon Ao + Aa f UC+ Oc with OB + Au giving rise to Au + Ao + aC + Au + Ao (fig. 29b). If the reacting ribbons are part of a network,
Fig. 29. Fusion reaction of threefold ribbon with single ribbon of different Burgers vector. a) Formation of a fourfold ribbon; b) Formation of a fivefold ribbon; o) Possible node involving a fourfold ribbon.
P.
DELAVIGNETTE
AND
S.
AMELINCKX
nodes like the one shown formed.
in fig. 2%
can be
We will now discuss a few more typical nodes that may result from such processess.
Let the
threefold ribbon have Burgers vectors Aa t- Ao -t Ao and the single ribbon On meeting (a)
(fig.
(bl
(Cl
30b),
Bo+ GA, (fig. 3Oa).
Aa reacts with Ba and forms which
in turn reacts
with
OC
OA to
lb)
td)
of threefold ribbon with Fig. 30. Fusion reaction simple ribbon with one common Burgers vector (the
of threefold ribbon with Fig. 31. Fusion reaction simple ribbon with one common Burgers vector (the
adjacent
adjacent
stacking
fault ribbons
are of different
type).
stacking
fault ribbons
are of the same type).
”
”
% -a-a-
a-,-
_,-_,---.-a-
--b-b-
b--b-
-b-b-b-b-
(I!)
IO)
Fig.
32.
Interaction
of threefold
ribbon
with
simple
ribbon
without
common
Burgers
vector.
DISLOCATION
reform
Ba (fig. 30~). Finally
PATTERNS
Ba reacts
IN
GRAPHITE
with
Aa and forms again UC. The resulting configuration and the lettering pattern are shown in fig. 30d. The same threefold ribbon can react with the single ribbon Ao
and
aA
oA+Ba
(fig. 31a). On meeting
annihilate
one
another
whereas
Acr and Bo react to form UC. The resulting node is presented in fig. 31b. It is in fact equivalent to the configuration
shown in fig. 18 and which
resulted from the fusion of two single ribbons. 12.Z
INTERSECTION AND
SINGLE
0F
THREEFOLD
RIBBONS
RIBBONS
In view of the frequency of such interactions it seemed worthwhile to investigate their geometry. Different relative positions of threefold ribbons and single ribbons have to be considered. 12.2.1.
No common
Burgers vector
We will first analyse in detail the case in which the single ribbon is in the same lattice
Fig.
33.
type
Observed
described
examples
in fig.
32.
of intersection In
A
of the
superposition
of
ribbons.
I’
r’
/
’ /& A
/
Dl’
//
A/
Qf
/
‘I
/
‘_______-_Jf______J
(b)
A_-..-_
/ (d)
(Cl
Fig.
34.
Intersection
of
threefold
ribbon
with
simple
ribbon
having
one
Burgers
vector
in common.
40
.I?. DELAVIGNETTE
plane
as one
threefold
of the ribbons
that
build
ANI)
the
ribbon.
They
have
no
common
Burgers
vector e.g. Ao -!-AC -;- Aa and sB + Co. The interaction is shown schematically
From
contain
through
fig. 32a along XY
this drawing
(fig. 32~) it is
&CO layers
in cubic
stacking.
On intersecting, segments will develop as shown by dotted lines in fig. 32a. The situation is shown in space in fig. 32d a cross section along X’T’ is further represented in fig. 32e. From this it, can be concluded t*hat sweeping of the area of overlap by the dipole Bo + aB will eliminat’e the wrong stacking and restore perfect material. The other region containing the stacking “1) over b” shrinkage it into a the other
leads to more
of the processes
complicated
n~hich \vill now be described
patterns,
described some of
and anslysetl.
in fig. 32a, whilst,
evident that one of the overlapping regions contains a stacking “b on b” and the other regions
Combined patterns
of layers is given in fig. 32b. c:
as a cross section and UV.
13.
AMELIP;CKX
The combination
however
the arrangement
Y.
‘l’he most striking one is perhaps what could he called the “st’ar pat,tern”.
Two examples of
it are visible in fig. 36. The let,tering pattern is reproduced in fig. 37. The correctness of the l&tering pattern is proved by referring to the extinct~ion pat,terns of fig. 36d and e. In fig. S6d dislocations lett,eretl Ao show all lack of contrast,.
can only be eliminated by complete of the hexagonal mesh, t,ransforming eont.racted node. The perfect area on hand will extend as much as allowed
by the increase in length of the dislocations bordering it,. The final situation is represent,ed in fig. 32g and the lettering pattern in fig. 3211. Xxamples of observat’ion are visible in fig. 33.
12.2.2.
C’omrnon Burgers vector
If the single ribbon contains a dislocation which has the same Burgers vector as the dislocations
in the threefold
ribbon,
say Xo,
the result is quite different. If the simple ribbon approaches the threefold ribbon in such a way that the meeting partials repel (fig. 34), no fusion reaction takes place between the Aa dislocat,ions which are in the same plane ; t8hose which are in adjacent planes form dipoles (IX) Aa + aA. In one of the overlapping regions, I. a multiple stacking fault is formed; in II a stacking “a over a” would result. Both st,acking faults will be eliminated. The first by sweeping the dipole Ao-t_aA over the faulted area, the second by shrinking to zero surface. The final result is as shown in fig. 34d. An example of an observation is given in fig. 35.
Fig. 35. ~bservecl exa,mple of intersect,ion described in fig. 34, a) Line corltrast ; b) Partial in tJha threefold ribbons arc out of contrast.
DISLOCATION
PATTERNS
IN
41
GRAPHITE
a) Line contrast of star pattern B; b) Line contrast of star pattern A; Fig. 36. Examples of star patt,erns. c) Stacking fault contrast of the t’wo stars A and B; d) One set of partial dislocations shows lack of contrast in star A; central
e) Another
hexagons
set of partial
in t,he two
dislocations
is out of contrast
st,ars are of opposit,e sign, giving stacking
fault
within
in star A. Notice
an indicat)ion
that the curvature
as to the presence
of the
or not of a
the lrexagon.
-b-b’_b._
(Cl
Fig.
37.
Analysis
b) Cross section
(d)
of the star pattern.
along
XY,
showing
the partial
a) Two interacting extended nodes giving rise to the “star” pattern; the presence of one layer in wrong position; c) Arrangement in space of dislocations;
d) Lettering
scheme
of the star.
42
P.
DELAVIGNETTE
whereas in fig. 36e the dislocations of contrast.
exhibit
S.
AMELINCKX
CO are out
of the stacking fault can nevertheless be deduced
Pig. 36a and b show the full line
from the inward curvature of sides of the central
contrast and fig. 36~ the stacking fault contrast. It’ is clear
AND
that
stacking
the central fault
of the configuration
region
contrast.
does not
The stability
hexagon;
this proves moreover that the specific
stacking fault energy is somewhat larger for the region inside the hexagon
than outside.
is assured by the repulsion
If the glide planes of the two extended nodes
between all the parallel segments of the central
are spaced by more than one unit cell a stacking
hexagon
fault
on the one hand,
and the stacking
faults on the other hand. Alternating the central hexagon
which
contains
~~~
infractions
against
nodes of
the stacking rule is formed. The centra,l region
are of the K and P type
will be smaller in this case, and bhe curvature
and the regions of stacking faults are alternatively of type I and type II, as indicated by different cross-hatching in fig. 3’id. The simplest way in which this pattern could be generated is by the superposition of two extended nodes of a different type, and hence in successive lattice planes, as demonstrated by fig. 3’ia. The central hexagon then contains a stacking fault with two violations of the stacking rule, i.e. the sequence is a b a b a o a b a b. This stacking fault cannot easily be Ginated except by complete disappearance of the central hexagon. The stacking fault energy is however only of the same order of Inagnitude as y; however parallel segments of the central hexagon repel more strongly than the partials in a ribbon and its cross section will therefore be larger than the ribbon width. The stacking fault will not be visible since it only consists of one layer which is in a wrong position; there is therefore no phase difference between waves diffracted above and below the fault plane. The presence
of the sides of the hexagon more pronounced, but again no cont’rast will be observable. Star patterns, without a stacking fauIt in the central hole, were however also observed. The absence of a stacking fault can be deduced from the lack of contrast and from t)he outward curvature of the central hole, which gives it a rounded shape. An example is presented in fig. 36 in B. Although the lettering pattern remains identical, the generation mechanism is different. A configuration of ribbons giving rise to the observed pattern is shown in fig. 38. The reaction involves two fusion reactions of pairs of ribbons in the same plane as described in par. 10.1.1.1. and two intersection reactions of the kind described in $ 11.2.2. The central region is now free of st,acking faults. The intersection of two families of ribbons, for example (Aa + crB) and (oil -t Ccr), containing a different stacking fault, also gives rise to star patterns of this type. The slight variations in shape are due to the varying distance between the two planes. If these are separated by more than c/2 (the separation distance is *z+ +, where n is an integer), the interaction becomes less pronounced and “stars” with short or vanishingly small radial segments result, like the one shown in fig. 36 in B. We can prove this statement by referring to this photograph. prom the pronouuted inward curvature of the central hole in A we conclude that it must contain a fault with more than two violations of the stacking rule; the only alternative is four. The stacking i + sequence is then a II a b c a c b a b. It is easy
Pig. 38. Configuration of ribbons giving rise to a star pattern without a dacking fault in the center.
uu L--II
DISLOCATION
PATTERNS
IN
43
GRAPHITE
to see that this implies that the two planes in which the two parts of the pattern are situated are separated by at least SC (the location of these planes is indicated by arrows). Prom the inter~onne~tioI1 of the two patterns through
contracted
A and B,
nodes C, which are known
to be planar, can be deduced that in the star B too, the two levels are separated by the same distance, i.e. by more than c/2, proving our statement. A somewhat related pattern is shown in fig. 39 and the corresponding lettering scheme is given in fig. 40. The simplest way by which this configuration may result is by a fusion reaction between two contracted nodes of a different type as demonstrated by fig. 40a. The central “hole” is now free of a stacking fault, as can be deduced from the “outward” curvature and its extension at the expense of the stacking faults is therefore only limited by the increase in line tension. The actual configuration of ribbons giving rise to this pattern, is drawn in fig, 41. It is clear that four different kinds of ribbons are involved in the “reaction”, two pairs react by “fusion” giving rise to the threefold ribbons; the other reactions take place by intersection, Left and right of the central pattern the nodes are of different kind as a consequence of the presence left and right of different ribbons, in the vertical set.
Fig.
39.
Pattern
in threefold
ribbon.
As a final example we will discuss the remark-
sidered as resulting from the interaction between
(bl
resulting from the fusion of two contracted nodes in adjacent hole is free of stacking
“hole”
able pattern of fig. 42 of which fig. 43 represents the lettering scheme. The whole can be con-
(a)
Fig. 40.
Sgmmet,rical
faults.
An examplo
lattice planes,
was presented in fig. 39.
The central
dislocations,
with a different
Burgers
vector.
If the two sets are in the same lattice plane, the result> is a netgvork of extended tracted nodes as shown somewhat fig. 44. An observed
example
and con-
idealized in
is represented
in
fig. 45. As the mesh size decreases the curvature of
t,he sides
of
the
staeking
fault
t,riangles
becomes smaller and smaller and finally a net work consisting of triangular meshes is formed.
I R
fC>
three nodes of one type and one larger node of the second Qpe and hence in a different plane. The areas of overlap indicated in fig. 42 by A contain faults of larger specific energy than that, of the type t,wo stacking faults. This can be deduced from the curvature of the lines towards the area of overlap. Such a stacking fault is a natural consequence of t’he proposed model. If the two interacting configurations are in adjacent planes the stacking fault is of the type . . . abab+c’bab uu
. ..
and it contains two violations against the ab sequence, i.e. the same number as the other fault regions, it is however of a different nature and the energy may therefore be slightly larger, However separated
it is probable
that they are in planes
by gc, the fault is then of the type + . ..ababcacbab l__-l L.-..Jli
j.
.._
l..i
and it contains four violat,ions of the s&eking rule. The energy will now be s~~bstantially larger. In neither ease will it exhibit fault contrast, in accord with the observations. 14.
Regular
networks
The networks of ~slocatio~~s are in fact twist boundaries with a. c-rotation axis. They result from the i~terse~tion of two families of basal
st,ur paWorn. The ~&tern results Fig. 42. Nultiple from t’he intoract,ion of one extended nude of a given kind and t.tlree nodes of’ the other f‘auit
contrast;
e) Partiesis
b) Partiats
kind.
a) Sacking
Bo show lack of contrast;
A(r are otrt of contrast fig. 43).
(see notation
in
DISLOCATION
PATTERNS
IN
45
GRAPHITE
(11
%ig. 43.
Analysis in terms of Burgers vectors of the pattern in fig. 42. star;
b) Configuration
(b)
Fig. 44.
a) Lettering pattern of the multiple
of nodes giving rise to the star.
a) Curved extended and contracted nodes; b) Degenerated Idealized regular networks. if either the &aeking fault energy or the mesh size becomes small.
form of (a)
Alternative triangle are faulted. This is shown schematically in fig. 44b and an observed example is visible in fig. 14. If the two intersecting sets of dislocations are in adjacent lattice planes a regular array of nodes of the type shown in fig. 27 results. The corresponding network is shown with its lettering pattern in fig. 46. An observed example is presented in fig. 28. The same network also results from the intersection of a family of threefold ribbons, say with Burgers vector aA + OA + aA, and a set of single ribbons which have a Burgers vector in common with tShe threefold ribbon, say oB+Aa. A related type of regular network consist,s of an array of “stars”, it is drawn in an idealized fashion in fig. 46, a whilst fig. 47 gives an observed example. Its formation necessitates
Fig. 45.
Regular
network
contracted
of curved extended nodes.
and
46
P.
DELAVIGNETTE
AND
S.
AMELINCKX
(b)
Fig.
46.
Idealized
regular
also the intersection
n&works.
a) Network
of stars;
of at least two families of
ribbons containing different stacking faults. The perfectly regular networks are of course the exception; ingeneral “stranger”or “singular” dislocations meander through the nets and cause deviations from the regularity, we call We will present only one them “singularities”. network containing singularities, together with t,he analysis in terms of Burgers vect’ors, as
47.
Observed
example
of a network
of stars.
of
segments
of threcfoltf
ribbons.
summarized in the lettering pattern. In most cases the lettering patterns have been proved by making use of the contrast effects discussed in 97. The network is shown in fig. 48, the analysis is given in fig. 49. The most striking feature is perhaps the presence of a line of discontinuit,y XY across the field of view. Along this line the orientation of the extended nodes changes by
Fig.
Fig.
b) Network
48.
Dislocation
network
as seen
in
stacking
fault, contrast,; it contains a number of singularities which are analysed in t,erms of Hurgers x,ectors ill fig. 49.
DISLOCATION
Fig.
180”
(or 60’).
49.
Lettering
This suggests
pattern
PATTERNS
corresponding
that this line is
either an isolated partial in the plane of the net, or a twin boundary. The perfect straightness points to the second interpretation. It is clear that the partials
of the net interact
strongly
with this line, but nevertheless cross it. This observation is in agreement with the model of a twin boundary suggested by Kennedy 19) and which consists of a symmetrical wall of partials. A number of threefold ribbons is present in the pattern; the difference between the two stacking faults in these ribbons is evident from their contrast. The holes in the threefold ribbons (in A) can be shown to be caused by the intersection with a simple ribbon, which contains no partial with the same Burgers vector as the partials in the ribbon. A star pattern is visible in B.
IN
GRAPHITE
to the observed pattern
15.
shown in fig. 48.
The Formation Mechanism for the Hexagonal Networks
A mechanism for the formation of hexagonal networks containing extended and contracted nodes has been proposed by Whelan et al. z”) and applied to stainless steel. The mechanism involves cross slip and it is therefore a priori inapplicable to graphite, where there is only one prominent glide plane. Moreover there is direct evidence that networks in graphite form by glide on the basal plane only. In fig. 50 for example it is clearly visible that a few nodes have been formed in between the points A and B in the interval between the two exposures. We propose here a mechanism for the formation of such nodes based on glide along the c-plane only. Consider the situation pictured in fig. 51 whereby a ribbon BaA and a ribbon CoA meet
P.
Fig. 50.
Observed
DELAVIGNETTE
AND
stages in the process of node fornlation.
(b) two more nodes have been formed;
c) One partial
c*clntrast, although
S.
AMELINCKX
a) and b) Two succossivo stages;
is ollt, of contrast;
all partials
(b)
between
d) The entire, dislocation
(a) and
1’ is otlt, of
are in contrast,.
,c
)
a) The ljartials CIA and CO combine to mechanism of extended and conkacted nodes. Fig. 51. Formation form a perfect dislocation CA; b) Dislocation CA crosses over the partial Hn; r) ‘l’hc perfcot, dislocation CA dissociates again according to the scheme CA --f Co -+ nA.
DISLOCATION
PATTERNS
49
IN GRAPHITE
as would be the case when a first set of ribbons
have a direction in sector II the process will be
with the same Burgers vector is intersected
slightly Now
by
different. the partials
aA
are forced
which first meet have vectors uA and Co; they repel. But, since there is a driving force for
together to form a perfect dislocation
RA, which
movement
again is possible
a ribbon
of %Ldifferent
kind. The two partials
of the ribbon
CcrA, we can assume
that GA and Co will combine
to form a perfect
is now pushed
Bo
through
Ca. Because of the repulsion
CA as shown in fig. 51a. This whole
oA, BA will now move
~slocation
CA is repelled by the right partial
instead
CoA, there is however
only
a small interaction with the left partial Bo, since vector Bo is perpendicular to vector CA. The partial Ba is further bound to the partial aA by the stacking fault that separated them. It is therefore reasonable to accept that the complete dislocation CA will be pushed “over” or “through” the partial Bo without difficulty, creating the situation shown in fig. 51b. The situation is now such that the ribbon BaA has effectively been reformed at the other side of CA. The perfect dislocation CA may now eventually dissociate again giving rise to the situation of fig. 510 and the whole process can start over again. The process described here requires that the applied shear stress, all dislocations in the correct sense. This restricts the of favourable orientations for the shear to sector I of fig. 52. Should the shear
under move range stress stress
uA
same situation
further
eventually
to
between BA and over oA, but
dissociate
again,
Co-t aA, into oA+ Bo. The
as before
has now effectively
If y is relatively large with respect to the applied shear stress, sector II will extend somewhat at the expense of sector I. Successive stages of the process pictured in fig. 51 can be observed in the left bottom corner of fig. 50. Between the two exposures two more nodes have been formed as can be judged from the configuration of dust partials. The dislocation segment, marked P, and which according to process I should be perfect does in fact exhibit contrast effects which differ from those observed at partials. In particular in fig. 50 (a and d) the segment P is out of contrast, whereas all three sets of partials are in contrast.
Crossing of Dislocations
A
e
u
/I f-.
52. Illustrating the relative directions of partial
Burgers vectors and applied shear stress. The process represented in fig. 51 takes place if the applied shear stress has a direction in the range I. In the range different process takes place,
DISTANT CROSSINGS
We will speak about crossing when the interacting dislocations are situated in the plane
ci
e
//
c
a slightly
Co. This
been reached. It is clear that the kind of process that will occur depends on the orientation of the shear stress and may be slightly on the value of y.
16.1.
Fig.
will
within the ribbon
16.
A
A
e
it
-.1.;~ e*
the partial
since BA is perpendicular
dislocation
aA of the ribbon
and
II
separated by at least c. On crossing in not too distant planes the dislocations excert locally strong forces one on the other. Usually there is a, torque tending to twist them into the anti parallel orientation, where they attract. Many examples of the typical configurations that result from such interactions were observed for ribbons. The behaviour depends of course on the Burgers vector of the dislocations. The vertical set of ribbons of fig. 53 contains ribbons with two different Burgers vectors ; this is
50
Fig.
Y.
Crossing
53.
of two sets of dislocations;
t)he twist
c
DELAVIGNETTE
at the crossing
1,
/
1)
,
’
S.
AAWELTSCKX
notice
points.
*I(T T T
6’
AND
Fig.
66.
threefold more
I
deformed
crossing
/
Crossing ribbon.
of
one
Sotice
single
that) the
t,han t)hrx t,luxtifold
of single ribbons
thv largest
ribboll
with
single ribboll.
one
ribbon
is
For
t)lrtx
i trrcl~~t’is on 1~11~
/
Fig. Fig.
54.
Lettering
scheme of the crossing in fig.
illustrated
56.
a) Line
Crossing contrast
is ollt
53.
reflected in the different behaviour at the crossing points with the same dislocations. Fig. 54 represents the lettering pattern for this situation. If a threefold ribbon is crossed by a single ribbon the largest deformation is on the single ribbon as in fig. 55.
of ribbons
16.2.
c:LUSE
showing
; b) 0 no set of partial
constrictioll<. tlislorn!iorls
of contt’ast.
CROSSIiYGS
There is in fact) a continuous change in pattern from distant crossing to crossing in adjacent planes. For close crossing the configurations resemble those characteristic of intersection. In fig. 56 e.g. constrictions are formed at every crossing point. The reason for the formation of
DISLOCATION
PATTERNS
51
IN GRAPHITE
-a
b=a-
-b ---a -b ---+-&----b-b’-b X--a -b --a -b b
-b
(bl
(a)
Fig. 57.
LeMering scheme for the patterns of fig. 56.
the constrictions
is to avoid the formation
of
stacking faults containing four violations of the moreover stacking rule ; elastic interaction stabilizes the constrictions. This can be shown by referring to fig. 57 which is a cross section along XY. Suppose that Ba would leave the combinatio~l of three parGals Aa, aA and Ba. The region sweeped by Bo would contain a stacking fault with four violations. Since there is no longer a long range repulsion between BG and GA, because of the presence of Ao, Ba
a) configuration
in space;
b) cross section.
will remain together with the dipole Aa+ aA which is in stable equilibrium when the dislocations are one above the other (at 45”). As a further example we discuss fig. 58 (node A) which resembles a contracted and extended node pair. In reality however a segment bisects the extended node, this segment produces only a very weak contrast because it consists in fact of a “dipole”. The presence of the segment can however be inferred from the “arrow point” shape of the apparently extended node. The analysis is pictured in fig. 59. The line energy of the dipole can be deduced from the angles at the sharp bent; it is found to be roughly 16.3.
equal to the energy THREEFOLD
RIBBONS
of a partial. RESULTING
FROM:
CROSSING
An interesting
pattern
involving
crossing
is
shown in fig. 19. The lettering pattern is shown in fig. 60. The most important feature of this pattern and which requires an explanation, is
Fig. 58.
iVet,work showing arrow point nodes due to the presence of dipoles in A.
the inequality in spacing for one of the threefold ribbons. This can best be judged from the stacking fault contrast (fig. 19b) which gives the true width. From the lettering patterns it is evident that the ribbon Itr consists of the overlapping of two stacking faults of the same kind and hence separated by a distance c (or nc -n: integer). The layer sequence in such i + a stacking is a b a b c a b c b c where the planes I_ III
P.
DELAVIGNETTE
AND
S.
AMELINCKX
(a)
Fig.
Fig.
60.
Lett~ering pattern
59.
(b)
Wxhanism
of formation
illustx-ating the formation
of the ribbons have been indicated by arrows. Such a stacking fault contains four violations of the ab sequence, against two for the usual stacking fault and it has therefore a higher energy. The structure of this threefold ribbon is shown in fig. t7b. The line AG which consists of the superposition of crC and aB is now one of the outer partiafs (fig. l?‘b), instead of the central one. The two stacking fault energies yl and yz are moreover different. From the asymmetry the relative values of the stacking
of “arrow
point”
of the asymmdrical
nodes.
threefdd
ribbon
shown in fig. 1.9.
fault energies can be deduced using formula (I, 10); one finds ~~~~~~=~~5 (see appendix I). The presence of the high energy stacking fault in RI is also the reason why its surface is reduced to a minimum by a displacement of the segment S (see fig. 60). The formation of t*his type of ribbon is possible because crB on arriving at
DISLOCATION
PATTERNS
It is clear that the stacking fault in ribbon Ri gives
contrast
under
the same circumstances
as the fault within the ribbon
Rz, since they
both result from the same net displacement vector R =Aa. The lamella ca does not show up of course.
The equality
of contrast
in Rr
and Ra is evident from fig. 19b. The conclusion is that the same pattern shows the occurrence
of a second
type
IN
53
GRAPHITE
taining two stacking faults of different energy. This is clearly the reason why the two halves of the ribbons A, have a different width as opposed to the threefold ribbons in B of the same figure ; which are of the type that contains two stacking
faults of the same kind.
This pattern shows another feature of interest. In the points marked D (fig. 62) some protruding
of threefold
ribbons, which is however less common, since the combination can only take place under a
A16 --c-1-c--
relatively large shear stress. It is further noteworthy that the energy of the two superposed faults is smaller than the two times the energy of a single fault, proving that some interaction exists between faults in neighbouring planes. 16.4.
CROSSING
OF TWO
RIBBONS
OF THE
6jB
-b&f+b-b-C-b-
b-bc
-l-c-c---
g
-.-,-a--
Y
-b-b-b--1-,-,-
-b-b-bCb,
SAME
TYPE
Under a shear stress two ribbons with the same stacking fault, for example (Ac++B) and (Co -I-oB), in glide planes separated by a distance c (or a multiple of c) may erosa as shown in fig. 6la. Segments will tend to develop as indicated by dotted lines. The region of overlap now contains a fault of the type ababacbacac... III IL--J (fig. 61b), i.e. containing fonr violations against the rule. No mutual annihilation of the segment oB and Bo in the extended part of the node is possible; instead a dipole (Ba+aB) is formed. Movement of the dipole does not change the area of the stacking fault; moreover there is only a very weak interaction with the other dislocations. In an isolated node as in fig. 61~ there is no preference as to which way the dipole will go. In a network it will take such a shape as to minimize the energy, this is roughly the same as to reduoe its length as muoh as possible. The arrangement shown in fig. 6ld is probably the one that is observed in fig. 62 in A. The three dislocations Ao+Ca -+(TB and two other OB now form a short segment of threefold ribbon of the kind discussed in 8 16.3 i.e. con-
Fig. 61.
Crossing of two ribbons of the same type
but in different planes (distant
of nc).
54
P.
DELAVJGNETTE
AND
S.
AMELIIiCKX
this net and acquired
a bent shape.
On the
one hand it generated the segments of threefold ribbon, as discussed above. where
intersecting
(Co-i-(TB),
it created
On the other hand
dislocations the dipoles
of
t,he
marked
set, D.
Also in this case it is possible to estimate t’he line energy of the dipoles. It now t’urns out to be very
small since the angle of the cusp is
hardly visible;
a rough estimate leads to G_ :!
of tjhe energy of a partial.
The much smaller
value found here as compared to $ I 1. I._). is presumably a consequence of the much smaller distance between the component’s of the dipole. 16.5.
Fig.
62.
Network
showing
different
singularities
analyzed in 3 16.4 and in fig. 61. In A asymmetrical and in H symmetrical
threefold ribbons; in 1) dipoles.
parts have been developed at both sides of a mesh, which contains no stacking faults. It is clear that these protrusions should not be stable unless they are connected by some line that stabilizes them. This line cannot be a partial dislocation since all partials are in contrast; from the geometry follows that it cannot be a perfect dislocation either. It is suggested that this invisible line is due to a dipole. From its occurrence across a perfectly stacked mesh, can
,?UPERPOSITION
014’ HIBBOh-S
Some superpositions of ribbons give rise to features which are worthwhile discussing. An example of interest is for instance visible in fig. 33 in D and represented schematically in fig. 6Ya. Ribbon
has a Burgers vector Aa- (TB and ribbon 2 a vector Ca-I-GA. The partial with vector Ca of ribbon 2 is attracted towards the partial Ao of ribbon 1 and is repelled by the other partials. Combination Aa + Co + aB will therefore take place. The partial Co of ribbon 2 is equally attracted by both partials aB, it is 1
be concluded that it has to be a dipole of perfect dislocations. Such dipoles do not change the stacking on passage, as do dipoles of partials. They have therefore no strong tendency to go one way or the other; this explains why it could remain across a mesh. Such dipole results when two ribbons of the same total Burgers vector cross in planes at a distance c ; a drawing similar to the one presented in fig. 59 would show this. The whole pattern of fig. 62 can now be understood. The planar network in the right top corner results from the intersection of ribbons (Ao+ oC) and (Co+ oB) in the same plane. One singular dislocation with vector (Bo+ UC) roughly parallel to (Co + oB) but lying in a plane differing by C in level, crossed
(‘onfiguration of partial dislocations resulting Fig. 63. from ihr superposition of ribbons leading to thr pattern observed in fig. 33 in A. a) Configuration partials to be compared se&on
with fig. 33 in A;
of
b) Cross
of the inkial situat,ion; c) C*o and Au combine
and form aB ; d) flnal situation
: cross section along XY.
DISLOCATION
however
connected
to the left partial
PATTERNS
aB by
IN
55
GRAPHITE
It is clear that the symmetrical
process might
the stacking fault and it will therefore combine
also happen. The first step would then consist
with it; the resulting cross section is shown in
in the reaction
fig. 63d.
Ca+Aa + aB. Fig. 64 shows a somewhat the
pattern
aA + aB + Ca; the second step
is represented
related feature; schematically
in
fig. 65a. It is suggested that in this case we have a superposition
of a ribbon aA + Ca and Aa + aB.
The two partials aA and Aa form a dipole. One of the partials either aB or Co can now glide towards this dipole and eliminate the stacking fault. Say that aB moved towards the dipole, then Ca will be repelled and we will have the situation pictured in fig. 65b. The ribbon will have approximately the normal width. The partial Co may just as well be attracted towards the dipole and then the results would be as shown in fig. 65~. The presence of one ribbon stabilizes in fact the recombination of the other. In fig. 64 there is now apparently a change over from configuration 65b to configuration 65~ as represented in fig. 65a. Superposition may also lead to a narrow threefold ribbon. Suppose that a ribbon Aa + aB
Superposition
Fig. 64.
of ribbons giving rise to cross
over of part~ialn.
UIA AIU
UiB
and a ribbon Ba+ aC combine by fusion into a threefold ribbon Aa+ (Ba+ oB) +aC. The central line is now a dipole and hence does not repel very strongly the two outer partials. These two partials repel however and a ribbon having the width of a single ribbon
will result.
The
central line however should now give very little contrast. The narrow triple ribbon of fig. 19 may be of this nature.
C(U
lil ’ I
Even fourfold ribbons may result from superA model of a fourfold ribbon of this
position.
UIAAla clu UIB (iL)
Fig.
65.
Con6guration
superposition. in A;
b) Cross section
situation;
of
partials
resulting
from
a) Pattern to be compared with fig. 64 of one possible
equilibrium
c) Cross section of alternative
equilibrium
situations.
(8,
Fig.
66.
(1:)
Fourfold position
ribbon resulting from the superof two twofold ribbons.
P. DELAVICNETTE
56
Fig.
67. Different contrasts of the same fourfold ribbon R. The two outerrpartials have the same l3urgers vector.
type is shown in fig. 66. The strong repulsion between the two inner partials aB makes this central ribbon to be larger, than the outer ribbon. The contrast effects observed in fig. 67 suggest that this may be a ribbon of this type. 17. 17.1.
AND S. AMELINCKX
Quenched-in
prismatic loops
QUENCHING PROCEDURE
For these experiments single crystals of pure natural graphite originating from the Ticonderoga Limestone formation (N.Y.) were used. According to Hennig, who kindly supplied the material, these crystals have been heated at elevated temperature in chlorine gas in order to purify them by volatizing the chlorides of eventual impurities. The crystal flakes were mounted in a slit at the tip of an outgassed and purified thin graphite rod and heated in vacuum by means of electron bombardment. The peak temperature was about 3000” C as measured with an optical pyrometer. A considerable amount of sublimation took place, indicating that at the temperature reached, defect formation should be appreciable. The current was then switched off, and the crystal was allowed to cool under vacuum. It was estimated by pyrometry that during the first seconds the cooling rate was about 1000” C/see; which should be sufficient to trap a number of vacancies. When examined directly after the quench no
loops
were found. The crystals were then annealed at 1200” C. For this operation they were enclosed in a pure graphite holder and sealed off under vacuum in a quartz capsule. There was no contact between the graphite crystals and the quartz. After this heat treatment the crystals were re-examined and now circular, slightly hexagonal features situated in the c-plane, as shown in fig. 5, were found. 17.2.
THE OBSERVATIONOF LOOPS. BURGERS VECTOR DETERIWINATION
We will now demonstrate contrast
effects
b
are
in
b
that the observed
agreement
with
the
b
(4
a
(b) Fig. 68. Schematic view of prismatic loops. a) Loops due to the precipitation of vacancies; the Burgers vector is inclined with respect to the c-plane; b) Loops due to the precipitation of interstitials. The Burgers vector is perpendicular to the c-plane.
DISLOCATION
assumption
that
condensation dislocations exhibit
the
are
loops
of vacancies.
PATTERNS
due to the
That the loops are
is deduced from the fact that they
diffraction
contrast.
The best contrast
is obtained when the interior of the loops exhibits stacking fault contrast, fig. 5 is taken in these circumstances.
Some loops are lighter,
others are darker than the environment. other inclinations
For
of the specimens, darker loops
may become lighter and vice versa. On overlap two dark discs may produce a lighter sector, as e.g. in fig. 5 loops A. This behaviour is typical for the diffraction contrast due to a stacking fault parallel to the c-plane. As shown in fig. 68 vacancy loops are characterized by Burgers vectors inclined with respect to the c-plane, whereas interstitial loops should have a perpendicular Burgers vector. The Burgers vector of the loops was determined by using ( 1120) reflection to make dark field images as demonstrated in fig. 69. Those loops which have a Burgers vector not lying in the reflecting (1120) plane, will show up; those which have their Burgers vector in that plane will not show up; but they will for another (1130) reflection. It is clear from fig. 69 that most of the loops show up in a dark field line contrast, proving that they have an inclined Burgers vector and
57
IN GRAPHITE
17.3.
DISCUSSION
OF RESULTS
of the loops
Most
although
are single and circular,
some of them are slightly hexagonal.
In fig. 5 loop B two concentric are seen, the central contrast
as the environment.
coincidence,
the same
This may be a
and the feature would then simply
be the superposition also possible
circular loops
part exhibits
of two loops. It is however
that the central region is perfect
and that we have a loop of the kind discussed in fig. 4 and which dissociated into two partial prismatic loops. It is difficult to distinguish between both possibilities. From the fact that an inclined Burgers vector is observed it can be concluded that the “a over a” stacking has a large energy. If this were not the case the Burgers vector would be perpendicular both for interstitial and for vacancy loops. An implication of these observations is clearly that at about 1200” C vacancies seem to become mobile: the annealing stage observed at about 1200-1300” C by means of electrical measurements 22) and as a release of stored energy 23) is therefore 18.
probably
due to this process.
Interaction between Glissile Dislocations and Sessile Loops due to Point Defects
hence are vacancy loops. Similar observations confirming our point of view have been published
The interaction presents two aspects: on the one hand glissile dislocations are pinned by
recently
prismatic
by
Williamson
and
Baker 21). The
authors have more over been able to identify interstitial loops in irradiated material.
loops
in their glide
plane ; on the
other hand vacancy loops tend to form preferentially in the stacking fault ribbons of extended dislocations. We will discuss both points. 18.1.
PREFERENTIAL
LOOP
NUCLEATION
IN
RIBBONS
Fig.
69.
image
Quenched-in
but
inverted
dislocation
contrast;
loops.
b) Dark
a) normal field image
using the (1120) reflection. The dislocation lines show inverted
contrast except for one loop, which has its
Burgers vector in the (1120)
plane used.)
A cross section through a stacking fault ribbon is shown in fig. 70. As opposed to what happens in the perfect crystal, precipitation of vacancies in a plane marked by a rectangle in fig. 70b, within the ribbon, does not give rise to a stacking “a over a” but instead produces immediately the low energy stacking fault of type one without requiring the nucleation of a partial. One can therefore conclude that
P.
DELAVIGNETTE
AND
9.
AXELINCKX
--a-a-a-
W
(a)
I
i
-b-a+b_a+c
-.+-aa’bII r--___71 I ,L__;_r;IR
-b
s-a -b
-b
a
-i)
c-ab-
-b-a -c-v “-l-/b
c-
-
-ii
a
a-
-b
b
b-
-a-* -b
70.
a) Precipitation
of vacancies
of the high energy stacking in the plane
surrounded t,hrough
4-
inside a ribbon
by a rectangle
without XY;
diffuse
the
produces
producing
point
defects
towards
the
dislocations through long range elastic interaction. We will further present direct evidence that preferential nucleation in extended dislocations does happen. We will now show that such loops effectively pin the dislocations. Fig. 70 represents the geometry of a loop formed within a ribbon; on expanding the edges of the loop will soon meet the partials of the ribbon and form segments like AR and CD which have an inclined Burgers vector AT, BT or CT. The other segments of the loop AC and BD, have a perpendicular Burgers vector (OOOq’2). Cross sections through the loop are shown in fig. 70~ and d.
a vacancy
a stacking
d) Cross section
vacancies will find it energetically more favourable to precipitate inside the ribbon, rather than in perfect crystal, and hence there is an interaction energy, which is to be added to the Cottrell type interaction. The latter tends to make
b-
td)
fault,; b) Cross section RS through
the loop along
b-
b
(b) Fig.
b-
a high
loop within
fault ribbon,
energy
for removal
oan precipitate
fault;
c) Cross scct.ion
stmking
t~hrough the loop
necessity
vacancies
along
Uf’.
The partials are pinned to the loops not only because of elastic interaction but also if they should det.ach themselves from the loop by crossing it, they would have to transform the stacking within the loop into “a over a”. The latter process makes it difficult to cross the loop. 18.2.
PIXNIXG
OF GLISSILE
DISLOCATIOM
BY
LOOPS A partial, approaching a vacanoy loop from outside on a plane marked by a rectangle in fig. 70b, would also have to transform the stacking within the loop into “a over a” and will therefore be hampered in its movement by the loops. Recombination of the two partials may take place before cutting through the loop; this would avoid creating a bad stacking. Elastic interaction between glissile partials and prismatic loops may lead to further pinning. Such interactions have been described in detail for
DISLOCATION
PATTERNS
perfect dislocations
in zinc II), with some modi-
fication
be applied
they
can
to partial
59
IN GRAPHITE
dis-
locations in graphite. We will discuss now the different possibilities which are shown schematically
in fig. 71 and 72.
In noting the Burgers vectors
of the vacancy
loops,
the
we will only
consider
component
parallel to the c-plane. The glissile dislocations are supposed to be in one of the two planes that emerge in the loop. If they are situated in more will corredistant planes the interactions spondingly be weaker.
-
The vacancy loop is in the same plane
1X.2.1.
(fs inside) In
fig.
(b)
‘ila, the ribbon
as a whole is repelled
r
by the loop, whereas in fig. 710, the first arriving A
B+AU
0
-
UA
0
-B+u
c
t-
(Cl
Fig.
72.
Interaction
with the loop
c) Reactions
-
68
A
with
the
first
between
vacancy
loops
a) The ribbon as a whole is repelled; b) The horizontal component of the Burgers vectors of the loop is annihilated by the first arriving c) The fist second
arriving partial is repelled by the partial
narrowing
of the
and
dislocations.
the
component
Burgers vector of the loop is annihilated by the first arriving partial, transforming the loop into one with a vertical Burgers vector. The second arriving partial interacts weakly with the loop since the Burgers vectors are perpendicular.
(C)
loop,
but
strong shear stress, the
In fig. 71b, the horizontal
artial;
partial,
ribbon can pass the loop, the second partial will be held back, leading to a local widening of the ribbon. -
glissile
arriving
of the second one.
If, under a sufficiently
Interaction
and
partial is repelled, but the second is attracted, resulting in a local narrowing of the ribbon.
(b)
71.
loops
B
0
Fig.
vacancy
; b) Repulsion of the ribbon as a whole ;
repulsion
al B+Ba
between
glissile dislocations. a) Both partials react successively
(a.1
A6
-
a
is attracted
of the ribbon.
resulting
in
18.2.2.
The vacancy loop is in an adjacent plane (0 outside)
The situation represented in fig. 72b, leads to repulsion of the whole ribbon. In fig. 72a, on the contrary, both partials react successively
60
P. DELAVIGNETTE
with the loop, pinned.
and the ribbon
In fig. 72c, only
reacts with the loop, repelled.
will be firmly
one of the partials
the other partial
being
AND
S. AMELINCKX
18.3. OBSERVATIONS
Direct evidence ential nucleation
was obtained
for the prefer-
of loops in the stacking fault
In the latter case, a reversal
of the
ribbon of extended
shear stress would result in movement
of the
$ 18.1. Fig. 73 shows a linear arrangement of loops. Although the dislocation ribbon itself is
partial Aa, the partial aB being held back. From the foregoing discussions it will be clear that vacancy the movement
loops are effective of dislocation
in hampering
not
visible
the
dislocations
linear
as discussed in
arrangement
strongly
suggests its presence.
ribbons either by
repulsion, by reaction, or by holding them back after passage. In this discussion we neglected the additional pinning caused in certain cases by the necessity of creating a high energy stacking fault. as pointed out in a previous paragraph. Since the interstitial loops have a Burgers vector which is perpendicular to the Burgers vectors of the glissile ribbon, the elastic interact’ion will be smaller. Furthermore on arriving at the interstitial loop a partial can always choose a plane, either above or below the loop so as to avoid the formation of a high energy with stacking fault. The elastic interaction
19.
Twinning
in Graphite
It is well known that graphite forms mechanical twins very easily, the composition plane being {liol). Whereas the layer sequence in one crystal is a b a b . . ., it is a c a c . . . in the twin crystal. A model for the twin boundary has been proposed by Kennedy 19). It is made up of a pure symmetrical tilt boundary consisting of partial dislocations one every two layers. As shown by Kennedy this leads to the correct
ribbons in more distant planes is comparable to that caused by vacancy loops. Direct reactions. leading to effective pinning, is expected to be less pronounced for interstitial loops, since the Burgers vector of loops and partials are mutually perpendicular. The tendency for interstitials to precipitate in dislocation ribbons is weaker than that for vacancies because of the small specific stacking fault energy
in graphite.
With
an interstitial
loop formed in a position, between the planes c and b within the ribbon of fig. 70 would be + associated a stacking fault b a b a c a c . . . i.e. containing one violation of the stacking rule. On the other hand when formed in perfect material the interstitial loop would contain a stacking fault . . . a b ac b a b with three infractions against the stacking rule. On this basis a weaker interaction is to be expected than for vacancies.
Fig.
73.
presumably
Linear
arrangement
nucleated
of
preferentially
prismatic
loops
along a ribbon.
DISLOCATION
orientation
difference
between
PATTERNS
the two
com-
ponents of the twin. No direct experimental evidence for this model was given however. The direction of the twin boundary required
is also as
by the model, i.e. perpendicular
Burgers vector
to a more
direct evidence that the model is indeed correct. shows a network
to fig. 14, which
of extended
and contracted
nodes lying on both sides, of a strip of twin TW and also within the strip. It is therefore reasonable to assume that the whole network resulted from the interaction between the same two families of dislocations. It is clear that the stacking fault triangles within the extended nodes differ by 60” (or 180”) in orientations in both regions. Since the two twin crystals differ in orientation one might object that this is due to a trivial contrast effect, i.e. that one region exhibits inverted contrast which causes inversion of the triangle. This is however not the case ; a careful inspection of the triangles reveals a slight curvature
61
GRAPHITE
indicating unambiguously
which side the stack-
ing fault is on. We will now show that this behaviour
is just
what
of the
one would
twin boundary that
of a partial dislocation.
We will present here some additional, We refer for this purpose
IN
the
model
dislocations boundary, locations
expect
is exact.
if the model
First of all it is clear
allows
through interaction
the
the
propagation
boundary.
with the boundary
is of course to be expected,
does not prevent
At
of the dis-
but this
penetration.
The same ribbon which in crystal II (fig. 74) is for instance between the planes a and b will be between planes a and c in the second crystal. The consequence of this is that dissociation into partials will be different in both crystals if in both cases a low energy stacking fault is to be formed. A dislocation ribbon, say with Burgers vector AB will dissociate in crystal I according to the scheme Ao+ oB whereas in crystal II the same dislocation will dissociate according to the scheme Ao’ + o’B or referred to the same reference triangle aB + Ao. The notation will therefore have to change when crossing the twin line. \c
A/
C
6
I
B
\\
Fig.
74.
Analysis
orientation
of the influence of a twin on the presence of a triangular network as seen in fig. 14. The of the nodes formed
from the same ribbon in crystal I and II differs by 180”.
62
P.
Fig.
shows
74
the intersection
which leads to the formation contracted lettering
DELAVIGNETTE
of ribbons,
of extended
nodes in the two crystals. pattern
AND
and
In the
we have taken into account
the changes in notation
at t,he twin line. Using
the results of 5 9.2.1 these intersections
will give
rise to the nodes also shown in fig. 14. It is now
clear
orientation
that
these
exhibit
the
required
difference.
The model as presented
S. AMELINCKX
the loop is evident from the successive stages shown. Small particles apparently lying in the surface locally
of
the
foil,
dislocation
are sufficient
movement
to
inhibit
as demonstrated
by fig. 76. Depending on the direction of the shear stress the moving
ribbons
This is clearly
are widened
or narrowed.
due to the following
effect.
If
the shear stress tends to move the two partials here can in fact be
in opposite
site sense, away from each other.
considered as a kink band; the two twin boundaries always occur in pairs of opposite sign. It was further found that the angular
the partial on which the resolved shear stress is largest will impose the sense of motion of the ribbon, the other partial being pulled in the
differences are not constant’. The observation can also be considered as evidence that the high energy stacking fault i.e. “a over a” will not form. If this were the
same sense because of the stacking fault,. ‘This is a case where widening occurs. The direction of the shear stress may also be such as to push the partials together. The sense of movement of the ribbon will again be dictated by t’he partial which has the largest resolved shear stress. The ribbon will now be narrower than normal. In fig. 77 the ribbons are clearly
case the dissociation scheme would be the same on both sides of the twin and the orientation difference would not be observed. The change in dissociation scheme at the twin line can also be deduced from the singular node observed at this line. Evidence was found that the same dislocation line can propagate through a twin boundary in accord with the model. 20.
Movement Sources
of
The dislocations
Dislocations are extremely
-
Dislocation
mobile along
the c-plane, proving in the most direct way that this is the glide plane. After a few minutes of irradiation with the electron beam the dislocations sumably
start moving almost inevitably, preas pointed out by Hirsch 24), as a of the stresses caused by the consequence deposition of a carbon film. The edges of a crystal flake or cleavage steps are the usual nucleation sites for dislocations. No interval sources of the Frank-Read type were ever noticed, although the circumstances are optimum as far as we restrict ourselves to thin foils. 3Iovement of ribbons is relatively smooth; at the stress level caused by the electron irradiation the movement is slow enough to be followed visually. Fig. 75 represents a typical example of “edge sources”; the expansion of
widened. This figure is remarkable for another reason. In the points marked C the ribbons are locally much narrower. They have apparently constrictions. Such constrictions can e.g. be caused by jogs resulting from the intersection of t,he ribbons with a dislocation having a Burgers vector with a component in t,he c-direction. These constrictions apparently move along the ribbon as can be judged from their displacement between the two successive expos~lrc~.
Appendix
I
EQUILIBRI~,~ THREEFOLD
SEPARATIOS
OF
PAHTIALS
IN
.i
RIBBON
We will introduce
the notation
and
where p is the shear modulus, b the Burgers vector of the partials and v Poisson’s ratio. The
DISLOCATION
PATTERNS
1N
GRAPHITE
Example of an “edge source” a, b and c are photographs taken at intervals of time of approximately Fig. 75. 30 seconds. The movement is probably caused by the stress due to the carbon film which forms on t,he specimen as a consequence of the decomposit,ion
Fig.
76.
induced beam.
Example by
of the movement
changing
a) Medium
b) High intensity,
t’he intensity
int,ensity
of dislocations of the electron
of the electron
all the dislocations
the right side; c) Low intensity,
beam;
are moving to
all dislocations
are
moving to the left side; d) Again high intensity,
all
dislocations are moving again to the right side of the photograph.
of organic vapours.
Fig. 77. Example of an “edge source”, showing constrictions in C, probably caused by jogs. Notice that the constrictions
move
along the ribbons.
64
P.
DELAVICNETTE
AND
S. AMELINCKX
repel. A small decrease will cause the combination of the inner partials. (a)
After recombination threefold ribbons are formed (fig. 17), the three partials having the same Burgers vector. If yi =ya we have also x = y from symmetry considerations. The equilirium separation
(b)
Fig.
78.
threefold bination ribbon
Two
ribbons
ribbon, of
before
to illustrate
Burgers
vector
of edge character;
combination
notat’ions
a
used. a) Comto
a threefold
b) Combination
of Burgers
vectors leading to a threefold
leading
into
is then given by the equation
ribbon of screw character.
equilibrium condition can then be formulated by expressing that the forces on all dislocations are zero. We will first consider the combination
where A = a sin2 CJJ +p cosz q, CJY being the angle between the total Burgers vector and the direction of the ribbon. In the pure edge orientation v== 90” and x=i~~/y whilst in the pure screw orientation v=O” and x=3 ,/?/y. As compared to the corre-
of Burgers vectors shown in fig. 78a which would lead after combination of the inner partials to a threefold pure edge ribbon. The
sponding single ribbons the threefold ribbons are about five times larger. If the stacking fault energies yi and y2 are different in the two halves of the triple ribbons.
equilibrium
the equilibrium
Lx 1 -2v+w
conditions
-_;&p& I
1 a%+?
are
:y&+y=o
A [y-1+(~+y)-11-y2
(L2)
where v and w have the meaning illustrated in
is
v = 0.55 x 10-4 and w = 0.05 x lo-4
,U= 2.3 x 1010 dyne/cma,
of Burgers Another simple combination vectors is possible, which would lead to a pure screw threefold ribbon (fig. 78b). In this case the metastable equilibrium configuration is given by v=O.4xlO-4 cm, w=O.14~10-4 cm. These equilibrium configurations are metastable. A small increase in spacing will make them separate further since as a whole the ribbons
(1,5) (W)
x=3A / (2yl-yz+g)
(L7)
y=3A
(1.8)
/ (2yz-yl+g)
with (1,s)
g=Vy1”+y22-y1y2
assuming y = 3 x 10-Z erg/cma; b= 1.42 A and v=Q.
=o
(L4)
This set of three compatible equations with two unknowns can easily be solved:
fig. 78. This system of equations can easily be reduced to a system of two quadratic equations of which the only real solution
become
A [-x-l-(x+y)-l]+yl=O i A [x-1-y-11+y2-yl =o
(Ll)
,;;I-+;&w-Y=O
conditions
The ratio yl/ya can be expressed directly in terms of observed quantities x and y. From (1,4) and (1,5) this ratio follows directly when putting y/x = r Yl _- r(:!+r)
yz- 2r+
1
(1. IO)
Applying this to the ribbon of fig. 191) gives for r N 2; y1/y2 N X/5 which is about what one would expect.
DISLOCATION
PATTERNS
IN
Appendix II
Appendix III
The equ~brium separation of partials in a fourfold ribbon of the type shown in fig. 79 can be obtained by solving the set of equations
CURVATURE FAULTED
65
GRAPHITE
OF
A
PARTIAL
The radius of curvature of the partial is
pb2 %2
3011
1
-+T ( 2‘u
Vi-W p
p1 4 v+w
B1 4 2w
61 --=
4 21--w
()
TWO
REQIONS
&4=
3&l ___------4 ?J--w
SEPARATING
-
rd
(III, 1)
and hence if d
1 -l-+ u-w >
+;? (
---- l 2u
l +_L
V-EW
v-w >
A -y=()
I
El:”
which express that the net force on each dislocation vanishes. The solution of these equations using the same numerical values as in appendix I, is w=O.l7xlO-4 cm and v=O.53~10-4 cm.
8 : Fig. 80. Curvature of a partial separating two regions containing a different stacking fault,, ill~tr&ting the notations used.
Fig. 79.
Model of a fourfold ribbon, illustrating the notations used.
References l) P. B. Hirsch, R. W. Horne and M. 5. Whelan, Phil. Mag. 1 (1956) 677 P. B. Hirsch, J. Silcox, R. E. Smallmann and K. H. Westmacott, Phil. Msg. 3 (1958) 897 P. B. Hirsch, A. Howie and M. J. Whelan, Phil. Trans. Roy. Sot. London 252 (1960) 499 2) W. BoUrnarm, Phys. Rev. 103 (1956) 1588 3) A. Berghezan and A. Fourdeux, Compt. Rend. 248 (1969) 1333 A. Berghezan and A. Fourdeux, Vierter Int. Kongr. El. Miw., Berlin 1958 (Springer Verlag, 1960), p. 567 4) A. Grena.11,Nature 182 (1958) 448; J. Metals11 (1959) 60
V. A. Phillips and P. Cannon, G. E. Ree. Report nr 60-RL-( 1473M) (1960) 9 G. K. Williamson, Proc. Roy. Soo. A 257 (1960) 457 ?) P. Deitbvignette and S. Amelinekx, J. Appl. Phys. 31 (1960) 1691 9 S. Amelinckx and P. Delavignette, J. Appl. Phys. 31 (1960) 2126 9) F. Hubbard Horn, Nature 170 (1952) 581 lo) G. R. Hennig, J. de Chim. Phys. 58 (1961) 12 11) A. Berghezan, A. Fourdeux and S. Amelinokx, Acta Met. 9 (1961) 464 12 ) F. C. Frank, Rep. Conf. Defects in Crys~lI~e Solids (Phys. Sot., London, 1955) p. 159 13 ) M. J. Whelan, J. Inst. Metals 87 (1958-59) 392
7
66 14)
P.
\V. J. Read, Dislocations Book
15)
Cy.,
M. G. Bowman Chem.
16)
Inc.,
Solids
‘7)
R.
I*)
R.
(McGraw-Hill
1953),
Siems,
2’)
1’. 131
and J. A. Krllmhansl, Proc.
4ND
J. Phys.
22)
367 Roy.
Sots., London,
A 249 23)
114
Zeitschr. Phil.
in Crystals York
6 (1958)
M. ?J. Whelan, (1958)
New
DELAVIGNETTE
I’.
I>elavignette
fiir Phys.
(in the
Uevers,
S. Amelinckx
Mag.
(in the press)
I!‘)
A. *J. Kennedy,
2”)
M. J. Whelan, W. Bollmann,
and
Proc.
G. K.
Williamson
(1961)
313
G. H. Kinchin,
Second
Hennig
and
Conf.
1955,
W.
K.
Sot., R.
75 (1960)
607
TV. Horne
and
SOT. A 240 (1957)
524
25)
P/756
P. B. Hirsch, Wiley
G. B. Spcnce, State
P/751
Woods
Dislocations
I)elavignet,tr
and
G. R.
(John
Roy. Roy.
24)
P.
AMELINCKX
1955,
Amelinckx
lxess) and
P. B. Hirsch, Pmt.
S.
S.
Univ.
C. Baker, Geneva
Phil.
Mag.
Conf. P/442
J. E. HOVP, Second
6
(1956) Geneva
(1956)
et al., Second
Geneva
Conference
(1956) R. T%‘. Horne and Mechanical and Fifth 1961,
Sons,
and M. U. Whelan, Properties
New
York,
of Crystals 1957)
Bien. Conf. on Carbon, (to be published)
p. 92 Penn.
A
OF NUCLEAR
det,ailed
patterns
analysis
observed
is
single crystal
dislocations
are ribbons
Shockloy
fault.
given
by means
in graphite the
type
of
the
1 (1962) 17-66, NORTH-HOLLAND
consisting
of two
by
a strip
can be interpreted
de fines
microscopy
flakes. It is found
lamelles
presentBe;
that the
partials
PUBLISHIXG
of
I1 est montre
que les dislocations
de rubans limit&
Tous de
modele
et en supposant
faults
consisting
stacked
lamella
of
i.e.
rhombohedrally
fautes
stacking
width
a
occur,
material.
The
fault
energy
of the dislocation
of extended vector. the
of
faults
nodes
ribbons
angle
between
with
this can be used to determine
of the ribbon
and
l’aide
de la forma
that
partielles
The
possible
pared
from
with
vectors
observed
were
1200” C annealing
found
to
3000” C and annealed
contain
vacancy
energy,
It
Burgers
vect)or
that
associated
and
with
the
of
the
theoretical
point
as to the nature
dislocations of view.
It
is discussed is shown
loops tend to form preferentially
that
formation
nature
mechanism
networks
sre
of the twin bounda-ies.
and the formation
end
discussed
the
mesures
Dislocation
of constrictions
&
et 8,
composes
de
connus. La depenque forme
son vecteur
du coefficient
de Burgers
de Poisson effe&f.
entre
rubans
sont
et comparees
Les vecteurs
disavec
de Burgers
$, I’aide
Zt 3000” C et reeuit
de boucles
h. l’aide
d’effets
de mesures
emmagasinee,
minstion
de leur
entre
boucles
les
of
qu’il
est possible
vecteur et
movement
La
are
electriques
&
de tirer
mecanisme
une eon-
de la d&er-
de Burgers.
L’interaction “glissile”
11 est montrb
de formation
est
que les boucles
B se former
de faute
de pre-
d’empilement,.
de reseaux
aussi bien que leur structure,
m6me que la nature de joints 17
et de
associe
dislocations
ont tendance
dans les rubans
cations,
discussed.
les
theoriquement.
dues 8. des lacunes f&ence
dues 8, C, trouve
est done
elusion sur la nature des boucles & partir
fault
ib 1200” C
de dislocations
de boucles.
Xl est montre a
as the
in the ribbons
avec
trempe
d’energie
la formation
vacancy
structure
as well
Btendus
de Burgers
possibles
la presence
discutee
dislocation
des
mesuree
des lacunes. Le pie de recuit B 1200°-1300’
vacancy
in the stacking
est
de dislocations
experimentalement
Du graphite
of the from
de nceuds
observees.
determines
rev&
ribbons. The
interactions
precedemment & determination
a conclusion
glissile
by
loops. from
0.it.d.
en empile-
de contraste.
and meas~~rements
loops can be reached. The int,era.ction between loops
sont
The
C, found previously
is therefore
of vacancy
is shown
loops.
Bnergie,
cutbes d’un point de vue theorique
at
ce
I1 est montrti que celle-ci peut etre utilisee
les configurations from
peak at 1200-1300’
formation
Les
making
de
seules des
de couches
d’empilement
pour la d~te~in&tion
Burgers
by
faible
de rubans
du ruban
est verifiee.
are
and com-
The
experimentally
means of eleet~rical measL~remonts stored
ribbons
of view
base
dance de la largeur d’un ruban et de l’engle
Poisson’s
effects.
quenched
is
between point
configurations.
determined
use of contrast Graphite
interactions a theoretical
de
aux vecteurs
la direction
discussed
etre inter-
sur la
que se presentent
faute
and it is shown
ratio.
of
de
de la largeur
an effective
peuvent
rhomboedrique.
L’energie l’aide
of the ribbon
observ&
de groupes
on
direction
is checked,
oomposees
ment
Burgers
une faute d’empile-
satisfa~~te
d’empilement
fautes
t,he
the shape
known
of the width
the
vector
from
and from
of partials
The dependence
the Burgers
is obtained
les r&eaux maniere
du micros-
par deux partielles
et contenant
ment.
stacking
est
cope Blectronique.
pret&
energy
dans
graphite
& l’aide
way on the basis of this model and on the assumption small
de
a 6th effect&e
du type de Shockley
in a consistent
monocristaux
celle-ci
sont con&it&es
of stacking
de
that
on&
CO., AMSTERDAM
Une anslyse detaillt3e de reseaux de dislocations
dislocation
of electron
separated
All patterns
5, No.
MATERIALS
de m&es.
de dislo-
est discute,
de
Res mouve-
fXe
Energie
t3reite
der
der
lijrtgwbr Xnaten Burgers
Voktor.
Bantlbreite richtung &I
und
ergibt md
DW dem
gezeigt, werden
dass kann,
sich
der
an Versetzungsteile~ Winkel dieaar um die
as16 der
Form
ver-
mit bekanntem
Zu~arnrnen~~~
zwischen
zwischen
und dem BU~~FS Vektor
wird
wend&
Stapelfehler
V~rs~t~~~n~sb~nder
wurde
tier
dor
Band-
unt~rs~leht*
Zusammenhang Paisson’sche
Zahl
VWXIL
b&immen.
Methods have been developed in recent years Co observe dislocations directly in the electron microscope lT2J). A number of papers, all discussing results obtained by me&n@ of electron microscopy and diffraction have been devoted to this subject recently 4*5$6).It is the purpose of this paper to give a more detailed account of our obse~~tions. Preliminary results have been published elsewhere 7,s). .Apart from their i~~~rt~a~~e for the mecfianical behaviour of graphite, the dislocations in this material present aspects of a more fundamental interest. In a sense graphite is a choice material to be used as a model for a crystal with ELsmall stacking fault energy and with a singte glide plane. It was therefore thought wo~hwhile to study in detail the geometrical features of such a model. A number of contrast effects, typicsl of parti& were noticed, but they will not be considered here. unless required for the .inb~ pretation.
Specimens were obtained by cleavage of natural single erSystals (originating from di~erent sources: Ceylon, Ticonderoga Limestone (N.Y.) and ~adag~,~~~r. ~ynt~~eti~ graphite prepared in the Philips Laboratories hy the graphiti~~tion of silicon carbide single crystals was also examined. For the qutinching experiments use was made of the Ticonderoga graphite which is e~e~pt~~~nal~~pure_ Graphite is t-l,choice materid for tr~~~sn~i~s~on electron microscopy hecause it transmits electrons very readily and because of the ease of preparation of thin foils. Its great ther~~~al stability and its reasonable heat, and electrical conducti~~it~ along the foil plane make it qnit*e stable under electron irradiation. The observations were made with the standard Philips microscope operated aL LOO ItV. In order to allow ~o~ltrolled tilting of the specimen, a de&e was eo~~str~~~~~~ which made it possible to turn very slowly the spelecimen holder. Since rotation about one axis only is possible in this way, i6 was necessary in a few cases to semount the specimen in another orientation with respect
DISLOCATION
PATTERNS
IN
19
GRAPHITE
Some graphites have however the rhombohedral
to the axis. Graphite flakes have a tendency to fall off the grids ; specimens were therefore
stacking
always
section through this structure is given in fig. 2b.
mounted
between
two
grids,
one
of
symbolized
by a b c a b c . . . a cross
which had a larger mesh size.
Most graphites
3.
as can be deduced from the occurrence of streaks in the X-ray diffraction patterns. This is
The Structure of Graphite The
consists
most
common
structure
graphite
of sheets of linked carbon atoms. A schematic
of a hexagonal
hexagonally
of
view of the structure
stacking
is presented
in fig. 1 in
especially
contain
many
stacking
true for ground and hence deformed
crystals. The amount of rhombohedrally material
faults
is known
It will be evident
to increase
stacked
with grinding.
from our observations
why
this is so. A consideration of the structure suggests immediately that the glide plane should be the c-plane since the binding between successive sheets is weak. The binding within the sheets
Fig. 1.
Schematic
view of the structure of graphite in space.
space, whilst fig. 2a and 2c give two projections of interest. It is clear that the stacking sequence can be symbolized by a b a b . . . . The two positions a and b are indicated in fig. 2c, it is further evident that a third position c is possible. The sequences a c a c . . . and b c b c . . . represent also hexagonal graphite structures.
Rhombohedral
Hexagonal.
cb 1
(a)
,t-‘-9
d’ b‘j
‘Ld
(c) Pig. 2.
Structure
of graphite.
graphite;
on the other hand is strong and breaking of C-C bonds on glide is highly improbable. Models for dislocations which do not involve breaking of C - C bonds will be presented below. Since the c-plane is also a cleavage plane, the foil plane will be parallel to the glide plane and we therefore expect dislocation arrangements in the plane of observation. With respect to glide such specimens should behave more like bulk material than foils, where the glide plane is steeply inclined with respect to the surface.
a...
.
.‘c’;?
..0 .....’ (d
1
a) Cross section of hexagonal graphite; b) Cross section of rhombohedral c) Projection on the c-plane; d) Direction of the Burgers vector,
20
P.
4.
DELAVIGNETTE
AND
types of stacking
AMELINCKX
i.e. one Iayer is in a wrong position.
Stacking Faults in Graphite Three different
S,
faults are
can only be eliminated
This fault by two glide motions,
a priori possible in graphite, if one excludes the
or by the glide of a dipole,
“a over a” stacking.
partial dislocations
The first type of stacking fault would involve only o?ze infraction
against the stacking
it can be represented
rule ;
L
The relative
(1)
a b a c b c b c b ... L-.-l
(2)
a b c a c a c ... I
In an .., a b a b . . . structure these two stacking faults result when either an a or a b layer is shifted into c position. This fault will be shown to occur between the partials of the dissociated dislocations responsible for basal slip ; it can therefore be generated by slip only. The third type of stacking fault finally is represent,ed by t,he sequence a b a e b a 1) a lt . . . or
YII!!+ a b a b c a I) a t) . , .
-II
lattice
planes
of the energies of
stacking
fault energy for faults of the second
type is very small ; we can therefore that all types have a small energy.
We will see later that this type of stacking fault occurs in vacancy loops. The second type contains two infractions against the hexagonal stacking rule, it can be symholized as :
Or
magnitudes
sign and same
these stacking faults are roughly as the numbers of infractions 1: 2: 3. We will show that the
ababaca,cac u
a b
Burgers vector in the adjacent indicated by arrows.
by the sequences
ahababcbcbc OS
i.e. two coupled
of opposite
-
Both faults involve three infractions against the stacking rule. They can be described as the result of the insertion of a c-layer either between “a and b” or “between b and a”. We will see that this type of stacking fault occurs in prismatic loops due to in~rstitials. In a few cases we will meet stacking faults of a fourth type: L+ ... a b a b c b a b (4) IL.-J
5. 5.1.
Different Kinds of Dislocations BASAL
infere
in Graphite
I?ISLOCAT1OSS
The dislocations are invariably parallel to the foil plane and they consist of ribbons of two (or three) partials. Wo will now see how this can be understood. The only observed glide plane being the c-plane, glissile ~sloeations have Burgers vectors in this plane. The shortest among the vectors connecting one atom to the next crystallographically equivalent one are AB, AC and AD, as well as their negatives (fig. 2d). The vectors can be decomposed into two partial vectors, according to the reaction AB + Acr+oB, as demonstrated in fig. 2c, d. In the language of dislocations this means that the dislocations can split into two partials of of the Shockley type. The stacking fault resulting from the displacement over a partial vector like Ao, Ba or Co consists of a lamella in rhombohedral stacking, with two violations of t,he hexagonal stacking rule, as visualized in fig. 3a. Since a rhombohedral variety of graphite exists this stacking fault has a low energy and an observable dissociation into partials actually takes place. as will be demonstrated by our observations. A second type of dissociat#ion is geometrically possible. AC + A@‘+- /E or act + Ao as shown also in fig. 2d. A partial dislocation now brings one layer on top of the second one; we will call this the “a over a” or “b over b” position. It is to be expected that the energy associated
DISLOCATION
PATTERNS
21
IN GRAPHITE
tb) Model of extended 60” basal dislocations in graphite. No carbon-carbon bonds are broken. a) The Fig. 3. dislocation has a Burgers vector AC. Notice the lamella of rhombohedrally stacked material; b) Projection on the c-plane;
the Burgers vectors of the partials are bl and bz.
with this stacking fault is considerably higher than in the first case. We will demonstrate that all observed patterns can be explained without the necessity of assuming the occurrence of this type of stacking. On the contrary in order to explain certain patterns we will have to infer the fact that an “a over a” stacking is avoided. This may be regarded as an indirect proof. We conclude that the second type of dissociation does not take place. A model of an extended dislocation having the required Burgers vectors is represented in fig. 3a, b; it is clear that no C-C bonds are broken. Only deformation of hexagones is required to take up the strain. The presence of two kinds of c-planes i.e. a and b planes within the unit cell gives rise to two classes of partials. For those dislocations located between “a and b” (a underneath, b on top) the dissociation reactions are of the type AB -+ Ao+ uB, i.e. 0 inside; for those located between “b and a” the possible dissociation reactions are of the type AB + (TB+ Aa i.e. u outside. The stacking fault associated with this first dissociation is described by the
sequence ababacbcb... and for the second
type of dissociation
by:
ababcaca... One of these stacking sequences is represented in fig. 3a. The movement of a partial Aa, Bo or Co between
“a and b”
(a underneath,
b on top)
changes the stacking according to the prescription a + b + c + a, whilst the partials aA, oB and aC have the opposite effect, i.e. they change the stacking according to the scheme a + c --f b --f a. On the other hand the movement of the partials like aA, aB and aC between “b and a” (b underneath, a on top) changes the stacking according to a + b + c + a ; the partials Ao, Bo, Ca do the reverse. 5.2.
NON
BASAL
DISLOCATIONS
One could also consider perfect dislocations with a Burgers vector [oooc]. In the edge orientation they are equivalent to the insertion of two supplementary c-planes. Glide on other
P.
DELAVICNETTE
AXD
planes than the c-plane would however involve
S.
AXELINCKS
for such glide. It is therefore doubtful that such
graphy of grown crystal faces 9) and from the study of cleavage faces of graphite 10). These dislocations may play a role in the growth of
dislocations will occur as glide dislocations,
polytypes
the breaking of C - C bonds ; there is no evidence
may
however
form
sessile
(fig. 4a). A dissociation
prismatic
they loops
into two partials with
Partial having
of graphite. dislocations
a component
a Burgers
vector
in the c-direction
with
result
vector c/2 would reduce the strain energy. This
from the insertion
would
c-plane. Consider first the insertion of a c-plane
imply
growth
of one of the vacancy
layers in fig. 4a. Such a process would however give rise to a stacking “a over a” or “b over b” depending on which layer is assumed to grow. There will therefore be a tendency for the two layers to grow simultaneously. Dislocations with Burgers vectors having a component in the c-direction and emerging in the c-face would give rise to spiral growth on this face. There is evidence for the occurrence of such dislocations from studies of the topo-
a-
a
-a
(al
or the removal
as would result on the precipitation of int,erst,itials. The layers will be formed in c-position between an a and a b layer (fig. 681)). A prismatic dislocation loop cont,aining a stacking fault of type three and having a Burgers vector (WO&) will result. The stacking fault of the t)ype threth could be converted into one of type one by the nucleation of a partial, but since t’he stacking fault energy is small the corresponding stress y/b is too small and this will not happen; t*he loop is stable. The removal of a part of the c-plane, as would occur in the precipitation of vacancies would give rise to a prismatic dislocation loop containing an “a over a” stacking. Since the energy of such a stacking fault is too high it will be removed by the nucleation of a glissile partial at the periphery of the loop. After the partial has swept the loop the Burgers vector will be inclined with respect to the c-plane. A cross section through such a loop is shown in fig. 68a. The stacking
--1-a -b-a-at-L-
(b)
Fig. 4.
NOR basal dislocations in graphite. a) Perfect (000~) dislocation; b) dissociation of the dislocation on the c-plane
loops;
of the dissociated
fault within the loop
is now of type one. It, is clear that both interstitial and vacancy loop are sessile II). We can now discuss the possibility of dis-
b-----b-b-____b-a-a--___a-
in (a) into two partial prismatic
of a single
(e projection loop.
sociation for the double loop of fig. 4c. If glissile partials are nucleated and accompany the edges of the growing layers, t,wo concentric loops, separated by an anular stacking fault region. could be formed, as shown in cross section in fig. 4b. The system of concentric loops has a smaller strain energy than the double loop. Concentric loops have been observed (see fig. 5 in B) but it was not possible to decide unambiguously whether they were of the type described here, or the superposition of two independent loops, which may lead to the same contrast effects.
DISLOCATION
PATTERNS
IN
23
GRAPHITE
perfect dislocations
in the face centered lattice.
In order to be stable, a network
should not
contain adjacent nodes of the same type (K or P), no adjacent nodes are further allowed to be both extended
or both contracted.
In addition
aII parallel segments must repel each other and they will usually have the same Burgers vector. In discussing the stability of nets we will also have to stacking
take into faults of
account the presence of different specific surface
energies. Faulted areas tend to shrink in surface as much as allowed by the increase in line energy. 7.
Determination
of Burgers Vectors of Basal
Dislocations
Fig. 5. Loops due to quenched in vacancies after annealing. The insets show some hexagonal loops.
6.
Nodes between Partial Dislocations Basal Plane. Stability of Nets
in the
When using the notation introduced in fig. 2d two kinds of threefold nodes among partial dislocations can be distinguished. Their lettering scheme is given in fig. 6. The corresponding node conditions are oA + (TBf OC= 0 and Aa + Ba + Ca = 0. A distinction is clearly only possible if one adopts a convention concerning the sense in which the symbols are read. We accept the same rule as formulated earlier by Frank 12) for the face centered cubic lattice. When looking out from the node point we read from left to right of the line. The two kinds of nodes given here correspond in fact to the K and P nodes introduced by Frank for
(a) Fig.
6.
The two possible types
(b)
of threefold
nodes.
a) Corresponding to the K node introduced by Frank; b) Corresponding to the P node introduced by Frank.
It has been shown by Hirsch et al. 1) and Whelan 13) that the intensity of the diffraction contrast at dislocations depends on the value of n = b .g, where g is the diffraction vector. For n=O the dislocation does not show any contrast. It is clear that this effect, can be used to determine the direction of the Burgers vectors. A knowledge of the structure then allows one to make a plausible guess as to the magnitude. In the particular case of graphite one knows that glissile dislocations have a
Burgers vector in the c-plane. It is therefore sufficient to determine one diffraction vector for which contrast disappears at a given dislocation. This can be done either in light or dark field. In the first method one tries to find an inclination
of the specimen
for which contrast
disappears at the dislocation of interest, the others being visible. Without changing the position of the specimen, a selected area diffraction pattern is made of the region of interest. Care should be taken to select a region at some distance from the extinction contour whose proximity is responsible for the contrast. This precaution is necessary in order to obtain meaningful results because the kinematical theory, on which the effect is based, may become inapplicable near the extinction contour. In the diffraction pattern the spot which causes the contrast will be the most intense one, and can therefore generally be recognized.
24
P.
Pig.
7.
Successive
families
pattc !rns for each case.
DELAVIQNETTE
of partials
AND
in a network
a) S,tacking fault contrast;
go
S.
out
AMELINCKX
of contrast,.
The
b), c) and d) One set’ of partial
On the ot’her hand it is possible to make a dark field image using a predetermined reflection. Those dislocations for which b is in the plane corresponding to the selected spot will then disappear. In graphite, partials disappear for reflections of the type (1120) showing that the Burgers vectors are Aa, Bo and Ca as predicted by the consideration of $4.1. The set of photographs of fig. 7 shows the disappearance of contrast successively at each family of partials in a network. The insets give in each case the diffraction pattern in the correct orientation with respect to the image. From these observations it can be concluded
insets
show
dislocations
the
cliEraot,ic)n
is ollt of cnontrast,.
that the partials have the screw orientation at the extended nodes, and, further, that the ribbons as a whole have screw character, since the nodes are symmetrical. This distribution of Burgers vectors is represented in fig. 7. In the majority of cases isolated nodes have this orientation. In one case an extended node of edge ribbons was identified; it is shown in fig. 8. Stacking fault contrast occurs when n is fractional, the most intense contrast of this kind is found for (lOi0) reflections i.e. if n= Q or 3. This is demonstrated in fig. $1. In this case two partials can go out of contrast simultaneously in certain circumstances, ap-
DISLOCATION
parently violating of contrast.
PATTXRNS
the g ’ L -I 0 criterion for lack
These contrast
effects,
which
are
IN
25
GRAPHITE
specific for partial dislocations,
will be discussed
in a separate paper. 8.
Width of the Ribbons According
dislocation
to
Read 14) the
width
d of
a
ribbon depends on its character i.e.
on the angle q~between its total Burgers vector and the direction
of the line. The theoretical
dependence, as given by theory is represented by:
isotropic
elasticity
where Y is Poisson’s ratio. This formula has yet been verified experimentally because of di~~ulty of measuring d. F~lrthermore variation of the width as a function of 9
Fig, 8. Extended node of edge ribbons. a) The t,hree sets of partial djsloe~tions are visible; b), c) and d) One set of partials is out of contrast. The insets show the diffracting plane for each case.
not the the can
yield a value for Y. Such measurements are however only possible in those crystal where the width of the ribbon is sufficiently large to allow a direct measurement; graphite is such a crystal. One should however take care to avoid a number of sources of errors. Because of spherical aberration, the electron microscope only gives an undeformed image, and hence a reliable width, in the center of the field. The segments of ribbons to be measured were therefore always photographed in the center of the screen. In view of the one-sided nature of dislocation contrast, and because of the displacement of the dark-line with respect to the actual position of the dislocations,
care must be taken when
estimating the widths of ribbons when different contrast conditions are used. In our measurements these difficulties were eliminated by using one long dislocation line which was slightly curved so that a wide range of orientations was available. The contrast was continuously adjusted, by tilting the specimen, in such a way that the contrast conditions were as constant as possible, at the position of the segment to be measured. Although the ~splacement of the dark line was taken into account, a constant displacement would not cause a serious error Fig. 9. Network showing stacking fault cont,rast,. since it would only produce a parallel displaceThe inset is the corresponding diffraction pattern. ment of the straight line in fig. 10. The method
P.
DELAVICNETTE
ANT)
S.
AMELINCKS
.
Pig.
10.
the total
Plot
of t,he width
Burgers do in eq.
vector
d of a dislocation
and the direction
(5) whilst
ribbon
as a function
of the ribbon.
the slope of t,he straight
used also avoids the eventual lack of reproducibility of the microscope magnification. The Burgers vectors of the ribbons were determined using the procedure outlined above. The diffraction pattern fixes at the same time the orientation of the foil and hence rp. Fig. I1 reproduces at small magnification one of the ribbons used. whilst the insets illustrate the
The
R,bbon
I
of COB 2 q where
intercept
with
line is 2v,/(2---I*),where
p is t,he angle between
the d axis gives v is Poisson’s
the quantity
v&o.
with specific surface energy I’, one obtains (14)
different segments used for the measurements. The insets are all at the same magnification. The error introduced by the curvature of the ribbon is negligable provided this is small enough with respect, to the accuracy of the measurements. The data points for this ribbon are given as black dots in fig. 10 the open dots referring to a second ribbon. A few isolated measure~~ents are also included in the plot. It is clear that the functional dependence is reasonably well obeyed. The value of v deduced from the slope of the st,raight line is 0.24 5 0.04 which is significantly smaller than +. The value for do is do N 0.1 x 10-4 cm it is used to determine the specific stacking fault energy. 9.
The Stacking
Fault
Energy
Expressing the equilibrium separation the two partials, bordering a stacking
do of fault
One of the ribbons used for the measurement Fig. 11. of the width. The insets show some of the segment’s used. Noticft
the visible change orientation.
in width
with
DISLOCATION
PATTERNS
27
IN GRAPHITE
Fig. 12. Contrast at triple ribbon: the three ribbons go out of contrast simultaneously. respectively a T-node and a Y-node are visible. a) Line contrast; b) Stacking fault contrast;
In T and Y c) One set of
part,&1 dislocat,ions is out of cont,rast,.
where do has been defined in 5 6 ; p is the shear modulus and b is the Burgers vector of the partial dislocation. The value of d 0, which is the width of a 45” ribbon, was found by taking the intercept with the d axis in fig. 10; do 1: 10-5 cm. With ,u= 2.3 x 1010 dynes/cm2 15) and b= 1.42 x lo-scmthisleads toy = 3.5 x lo-2erg/cm2. alternatively lowing
one can make use of the fol-
approximate
procedure to obtain stacking fault energies from extended nodes. The interaction between partials is taken into account to a certain extend and the total energy is ~nimized by means of a variational procedure. The specific stacking fault energy is then given by Y=
formula 16)
involving the radius of curvature R of an extended node such as fig. 12 and shown schematically in fig. 13. This relation expresses the equilibrium curvature of a dislocation line under an applied force y per unit length. This relation however does not take into account the interaction between the partials forming the node and it is therefore only a rough approximation. Since R= 1.2 x 10-4 cm we find y = 2 x 10-z erg/cmz. _ _ ^. Siems et al. 17) have indicated a better
KYdYd3
(8)
where K=
pb2 2-i-v --
8n
Fig. 13.
Schematic
1-Y
view of extended node in order
t,o illustrate the notations
used.
28
P.
for edge ribbons
DELAVIGNETTE
AND
and K=
for screw ribbons;
“fusion” and “intersection”. By the first term we mean that the two ribbons
/lb2 2-3v -8n 1-Y
yo is a dimensionless
meet, number
of screw
and we find ZJO=0.3 x 10d4 cm;
this
leads to y = 3.5 x 10-s erg/cm2 in good agreement with the value deduced from the width of the ribbon. Using anisotropic theory this value would presumably increase. + The small value of y explains why rhombohedral graphite is sometimes found, especially after deformation. Heavy shear along the c-plane results in large networks of the type shown in fig. 14, of which practically half of t~he surface area is rhombohedral. 10.
without
reaction
The meaning of ~0 is clear from
iig. 13. The node of fig. 12 consists ribbons
AMELXNCKX
tinguish between
which is y0==4.55 for an edge and 5.95 for a screw ribbon.
S.
Interaction between Ribbons: Fusion Reactions between Simple Ribbons
We will now discuss in detail how ribbons can interact with one another. We will dis-
only
crossing takes
over
place
one
another;
between
the
a last
partial of the first and the first partial of the second ribbon. We will distinguish cases whereby the ribbons are between the same or between adjacent
lattice planes. In the first case they
contain the same type of stacking fault, whereas in the second case they contain different stacking faults. 10.1.
TWO
RIBBONS
STACKING LATTICE 10. I. I.
WITH
FAULT
THE
SAME
AND
IN
TYPE
THE
OF
SAME
PLANE
The. two complete Burgers vectors fu~t~~ an obtuse angle of 120”
10.1.1.1.
The last partial of the first ribbon is the same as the first of the second, e.g. Ao -t-oB and Bo + aC such ribbons always attract
On meeting, the two partials crB + Bo annihilate each other mutually and a single ribbon Acri crc! results. If the two ribbons are however pinned somewhere, or attached to the rest of
(b)
Large network consisting in small stacking
Fig. 14.
fault triangles. In Tw, one twin produces the inversion of the stacking
fault t,riangles.
t According to calculations by Siems, Delavignette and Amelinekx 17) and by Spence 25) the value for y becomes
0,7 erg/cmz.
(Cl
(df
Big. 15. Fusion of two ribbons, containing the same staaking fault giving rise to an isolated extended node. a) and b) the node contains a stacking fauIt of type I; c) and d) the node cont,ains a stacking fault of type II.
DISLOCATION
the network
an isolated
extended
PATTERNS
comes the repulsion due to the central partials.
node forms
As a result
as shown in fig. 15a, b ; fig. 12 is an observed example. This process is not responsible for the formation of the regular networks since it can only
produce
isolated
extended
nodes.
29
IN GRAPHITE
The
narrower
the ribbons
as they approach
become
gradually
one another
and a
complete dislocations CB is formed by the reaction AB + CA --f CB this dislocation immedi-
interior of the curved triangle contains a stacking
ately separates again into two partials and an
fault of the same type as the ribbons giving rise to it. Fig. 15b represents the lettering
isolated
pattern
for two ribbons both of which contain a stacking
if the stacking
fault is of type I and
The but the e.g.
node is formed
as shown
fault of the second type, for example
fig. 15d if it is of type II. 10.1.1.2.
contracted
in fig. 16. The same type of reaction can occur
two ribbons as a whole attract the last partial of the first repels first partial of the second ribbon Aa+ aB and Ca+ aA.
The question arises first whether for parallel ribbons an equilibrium configuration is possible, whereby the four partials remain separated. For example the attraction of the outer partials might compensate for the repulsion of the inner partials. A detailed consideration of the forces acting on the different partials shows however that as the ribbons approach one another their width also diminishes in such a way that the attraction of the outer partials always over-
and UC+ Bo.
The complete Burgers acute angle of 60”
10.1.2.
aB + Ao
The result is given in fig. 16d. vectors form an
The dislocations now repel as a whole and no fusion reaction is possible; however, we will see further that they can intersect under the influence
of a shear stress. The complete Burgers vectors are at an angle of 180”
10.1.3.
The dislocations attract since they are of opposite sign; they will now annihilate each other mutually. 10.2.
THETWORIBBONS STACKING ADJACENT
CONTAINADIFFERENT
FAULT, THAT
IS THEY
ARE
IN
PLANES
10.2.1. The two ribbons as a whole repel
Two different 10.2.1.1.
situations
Formation
may occur.
of threefold
ribbons
In the first situation, the last partial of the first ribbon attracts the first partial of the (a)
(Cl
Fig. 16.
Cd)
Fusion of two ribbons containing the same
stacking fault producing an isolated contracted node the
second ribbon
(b)
two ribbons contain a type I stacking fault. a) to c) different stages; d) stable node.
;
as for example
in Au+ aB and
OC + Aa. Although the ribbons repel as a whole the inner partials of the quartet aB and UC attract. A detailed consideration of the equilibrium condition reveals that a metastable equilibrium position is possible without recombination of the inner partials (see appendix I). This can be understood intuitively as follows. The outer partials are kept at a distance by the repulsive forces between them ; the inner partials attract? but they are bound to the outer partials by the stacking faults and they are therefore kept separated. If the stacking fault energy is small the equilibrium is very unstable and re-
P.
DELAVIQNETTE
AND
S.
AMELINC’KX
a-
-a -b
b-
--P
a-
-a-------a
a-a-
I
l -b~ccb~b-
b-----ba------a
-b
-
b-----ba-a-
-b-b
--a-a al8
(b)
t.31
(C)
Fig.
l’i.
Nodels
of t)hreefold
asymmetrical
ribbonr.
ribbon;
a) cross section
c) Jn projection
of the symmetrical
combination will take place under a small shear stress: a threefold ribbon will therefore result. This ribbon is now stable and will only dissociate again under a strong shear stress in a suitable direction.
ribbon;
showing
on the c-plane,
b) cr’oss srlc:t,ion or tlift
the notation
nsc~l.
the phase shift at the fault plane is 1/3z whilst. it is 213~ for the other half of the ribbon. As will be discussed in a separate paper Is)% this
The geometrical structure of the threefold ribbon is represented schematically in fig. 1Ta. The three dislocations have the same resultant Burgers vector Aa. The central dislocation is in fact the close superposition of two dislocations with vectors aB and OC in adjacent lattice planes. The resultant
dislocation
behaves
like
one with vector Ao. This is shown by the fact that contrast is found to disappea,r at the three dislocations simultaneously as illustrated by fig. 12~. The width of a threefold ribbon is about five times that of a single ribbon. In appendix I the equilibrium separation in a threefold ribbon is discussed. If the two reacting ribbons are part of a network, nodes of the shape shown in fig. 18, may result. We will call them Y-nodes. Observed examples can be found on several of the micrographs e.g. in fig. 12 and 19 and fig. 76a. The different nature of the stacking faults in the two ribbons is strikingly illustrated by fig. 19b. where the two halfs of the t’riple ribbon exhibit a very different contrast. For one ribbon
(I)
a-L-b-+--+--a b!
b-b I-_a-a---a
c-.+--a-+--b
-b-b
b-b--b-b ~-a-a--1
Cdl
(0
Frlsion of two ribbolls containing Fig. IX. st,acking fault, resulting in the pro(lrl’:tion fold ribbon fold b)
ribbon Two
(Y-node). have
the
a tliffererrt of a tlrrvv-
The tlrrce par%ials in t,hcxthrclc:-
same Burgers \-vc+or. a) ant1 c) stable c:onfiguration: different stages; (11 cross section throueh the ribboll.
DISLOCATION
Fig. 19.
Threefold ribbons.
PATTERNS
may lead to significant differences in diffracted intensity. The interacting dislocations need not necessarily be in adjacent planes for t,he formation of the threefold ribbons. If only a few unit cells separate the glide planes, the interaction is probably strong enough to ensure the stability of the threefold ribbons, The level difference between t!he two halves could also give rise to a small shade difference in the stacking fault contrast. This is however not the main origin as mentioned before. The equilibrium distance of the outer partials decreases slightly as the distance between glide planes increases. The meeting
partials repel
If the ribbons repel as a whole and the meeting partials also repel as e.g. in Aa+ oB and oB -t-Crr, no fusion reaction is possible. The ribbons can however cross over and form a network by intersection as will be shown later. 10.2.2.
The two ribbons as a whale attract
The first possibility
31
GRAPHITE
a) line contrast; b) stacking fault cont,rast. Notice the difference in shade between
the two parts of the ribbon. In A is a symmetrical
10.2.1.2.
IN
is now that the meeting
ribbon; in R an asymmetrical
one.
partials attract as e.g. in Atr + aB and GC + Ba. The two inner partials recombine according to oB +Co + Ao; as shown in fig. 20b. This is however only an intermediate stage because Ao can further react with Ba to form a partial oC. Since AGF and aC now repel the final situation of fig. 20d is stable. We will call these T-nodes, because in a more symmetrical form they are T-shaped. A number of examples can be found; one is shown in fig. 12. In all other eases the ribbons do not react by fusion, but they will react by “cross over” or by intersection as will be discussed in the next chapter. The fusion reactions are “unrepeatable”. By this we mean that the reaction can only take place once, because it results in a transformation of one of the ribbons into another one which does not react in the same way as the first, with a set of similar dislocations. Let us take the specific example of fig. 15b. Once the extended node formed, a ribbou ACJ+ aC has now to react with the next ribbon Aa+ aB or Ba+ UC (fig. 15a). However no fusion reaction is possible and so the process
I’.
DELAVIGNETTE
AND
S.
AMELI1JCXX
fdt
A,U
6, B
-a--&b-.--&-~-b-cJb-4&b--&---+”
-b----t_c+b-V a-*1;
b&b-
-a-a.&&.--+ekalc
Fig.
20. Fusion
stacking
of the formation
of an extended
node
takes
place only once. The same applies to the other reactions so far discussed. It is therefore clear that regular networks will not be formed by these processes. We will show that they form by an intersection reaction or by “cross over”. In summary we can conclude that fusion reactions can give rise to four types of isolated nodes. If the two reacting ribbons are in the same lattice plane, and hence contain the same stacking fault isolated extended or contracted nodes can form ; if the two ribbons are in adjacent lattice planes. and hence contain different stacking fault, T and Y nodes can be formed. 11,
Intersection of Two Simple Ribbons
If a fusion
reaction
is possible
it will take
place
of two ribbons
fault. resulting
preferentially
containing
in the production
rather
than
a different of a T-node.
intersection.
It is thus clear that we will only have to consider the cases where no fusion reaction is possible. il.
1.
THE
TWO
BURGERS
11.1.1.
DISLOCATIONS
HAVE
THE
SAME
VECTOR
The two ribbons contain the.sam,e stacking fault
The two dislocations will annihilate as shown in fig. 2 1. The process is a little more complicated than in the case of complete dislocations. On intersecting the region of overlap is stacked “a over a” and will therefore immediately shrink in area. In practice it will not even form at, all. The region of “a over a” stacked material will finally be eliminated completely by the
DISLOCATION
recombination
PATTERNS
of the four partials OC and Ccr,
two independent
ribbons
Aa + oB result.
IN
33
GRAPHITE
annihilate completely
since they are in adjacent
planes. Instead of annihilation
“dipoles”
form.
One of the dipoles consists of Aa and aA; the The
11.1.2.
two
ribbons
contain
a
dif7erent
to a row of interst,itials, the other to a row of
~~a~k~ng Gaels The situation
is represented
ribbons with an extended formation
in fig. 22. Two
jog are formed.
The
process is in fact rather complicated
if analysed locations
in detail.
On intersecting
the dis-
with the same Burgers vector
cannot
Intersection
-a
within
the two dipoles
contains the stacking a b a b c b a b . . . i.e. two III stacking faults (fig. 22~); it will therefore tend to shrink, especially since there is no long range interactions
between the dipoles. In doing this
(C)
(b)
8,a
C/U
sic
U;A b
b-
.--+----a-
---.-a
c-
-?---&--bb-
-b-b x-a
The region
of two ribbons of t,he same Burgers vector and same stacking fault,
(Cl)
--b
vacancies.
(b)
(a)
Fig. 21.
other of aB and Bo. One of them is equivalent
i c
a-r b-
-1) U-C ----b
t 4
a-v b-
-b
b-
-----a
a-
IVJ
Fig. 22.
Interse&ion
of two ribbons of the same t*otal Burgers vector but containing & different st,acking fault.
34
P.
the layer c is transformed normal stacking.
UELAVIGNETTE
ALUD
int,o a, restoring the
AMELINCKX
the ribbons.
The line
energy
of the dipoles,
the two dipoles
En. increases of course with the distance between
combine into a single dipole and rest,ore perfect This can
the tZwo components. In the partio~lIar case shown here the dist,ance is not known ; i41~ is
most easily be seen by assuming that Ba and Aa
found to be about, 1.5-1.7 times the line energy
combine
of a ribbon.
material
into
between into
On meeting
S.
the two
OC whilst
dislocations
of
ribbons.
oB and OA combine
opposite
sign
Co.
liiach
ribbon contains some type of extended jog. It is assumed that in fig. 23 the angular dislocations
resulted from interseot8ions of this
type ; with
every
sharp
bent) in one
of
the
ribbons corresponds a bent, of opposite sense in another ribbon, st,rongly suggesting that “something” has to connect the corner p0int.s. It, is suggested that the ~onne~t,in~ entities are the dipoles of fig. 22. Such dipoles should give a weak or no contrast since the stress fields cancel approximately. From the angle enclosed by the ribbons and the dipoles t,he line energy of the latter can be estimated as a function of the line energy of
11.2.
THE
TWO
AND
ALSO
RIBBONS THE
TWO
REPEL
AS
MEETING
A
WHOLE,
PARTIALS
REPEL
11.2.1.
They contuin the same stacking fad mad they are in the Same latrine $ane for ~xa~~~l~ Ba--toA and (‘n-i OA
New segments will form as indicated by the dotted lines in fig. 24a. The two segments aA will further annihilate each other and give rise to an extended node. The central region contains a stacking “a on a” as will be seen by referring to fig. 24d. In fig. 240 is represented the stacking within the ribbon l&i-. aA along the line UV. The passage of a dislocation Ca on the a-plane causes a change in stacking according to the scheme a -+ 1) -+ o -+ a. The stacking along the cut XP becomes therefore as represented in fig. 24d i.e. the central area is stacked “a on a”. This region will shrink since ‘(a on a” is a high energy stacking fault, and the result is therefore a contracted node as shown in fig. 24b. The reasoning followed above is in fact somewhat idealized in a way. The real sequence of events is more like the following.
On meeting
the two inner partials repel. They cannot cross one over the other since this would create “a on a” stacked material. Since the two ribbons as a whole also repel a shear stress is required to make them react. If the shear stress is such that it causes the ribbons to approach, it may either cause the ribbon BA or the ribbon CA
Fig.
23.
L-shaped
divloeation
bents are presumably
ribbons.
The
connected.
sharp
to recombine. Suppose that the direction of the shear stress is such that BA recombines, and CA does not. The perfect dislocation BA can now cross the partial Ca without di~oulty, since the Rurgers vectors are perpendicular, there is no large elastic interaction and the stacking is not
I)ISLOCATION
PATTERNS
IN
(a)
U
816 -iI_= -b
(b)
CIA
b-b ,,&_-,~:~“,~~
-b_=
v
x
-C---+.--4i--J---C-
c-j-.---c,st_b--c
-a -b-b-b-
-b-b-b-
--B-------a--a-
--a--a-a-
-b-------b-b-
-b-b-b(Cl
Fig.
24.
Y
61A
616 -b--+&-+--b_b--&..-A&~b-
--a,b+---b=c+---a,b CT
35
GRAPHITE
Intersection
Cd)
of two ribbons giving rise to one contracted
affected by the perfect dislocation. Once BA has crossed over CG it can however dissociate into GA -/- Bcr, since this restores perfect material between the two p&Gals. The partials will now be re~stributed into ribbons in a different way than at the start. The result is the formation of an extended and 8 contracted node. The important feature
and one extended
node.
In this case the process is slightly different; it is now CA that crosses over Bo and then dissociates again into Co + GA. The final result is however 11.2.2.
identical.
The two rabble wnta~n a d~~~ren~ stacking fault, e.g. Ao + at2 and oB-+ Aa
The two meeting partials now always attract,
of this process, which is represented in fig. 25 is however that the sam? ribbons are again available for new interactions the process is therefore repeatable. An observed example of nodes probably formed by this process is visible in fig. 26. If the direction of the shear stress is ~fferent, the ribbon CA may be forced to recombine.
(a)
Fig.
(b)
25.
Real sequence of events for the interaction
of fzig. 24. b) BA
(Cl
a) the ribbon
Bo + crA is compressed;
crosses over C; c) BA
dissociates
again.
Fig. 26.
Probable
example
process described in 3 11.2.1.
of node formed
by the
a.nd pictured in fig. 25.
36
P.
At?
DELAVIG
ETTE
AND
S.
AMELINCKX
PIG
-b---+b+-r-
-b+b-b-1-1-a61A (Cl
Fig. 27.
Intersection
of two ribbons containing
but since they are not in the same lattice plane the possibility for crossing over exists and will be easier the farther the glide planes are spaced. After crossing the new segments indicated by dotted lines in fig. 27a will develop. The segments Aa will not annihilate each other directly since they are in different planes; instead, a dipole Aaf- aA is formed as shown in fig. 27b. The area where the ribbons overlap contains a double stacking fault, i.e. two layers in cubic stacking fig. 270. It will therefore be eliminated. This process proceeds by glide of the dipole towards the rest of the dislocations, with the formation of constrictions in the two ribbons as
a different hacking
fault.
illustrated in fig. 27d. The two constrictions are stabilized by the other dislocations present. There is no net repulsion between the dipole and the isolated partial aC, the stacking fault will therefore be eliminated completely. The final situation is described by the lettering pattern in fig. 27f; an isolated example is visible in fig. 76a. The main feature of this interaction is the formation of a threefold ribbon, the three partials having the same Burgers vectors. pu’umerous examples of such threefold ribbons can be found in the photographs, e.g. in fig. 12, 28. The real sequence of events is again
DISLOCATION
PATTERNS
37
IN GRAPHITE
stress, the two outer partials attract. The result will be that the complete
dislocation
AB and
AC, which repel will cross, and after crossing they will dissociate
into Ao + aC and crB + Aa.
We have now reached the situation which was used as a starting point of our previous reasoning, the constrictions
are however
already present
and all that is needed is a slight rearrangement to superpose 12.
12.1.
the dislocations.
Reactions between Threefold Single Ribbons FORMATION IWSION
OF MULTIPLE
Ribbons and
RIBBONS;
REACTION
Quartet ribbons can be formed by fusion of a triple ribbon and a single ribbon for example
Fig. 28. Network formed by the intersection of two sets of dislocations in successive planes,
somewhat different if one considers the actual process of meeting. Suppose that the left ribbon is aB + Ao and the right ribbon Aa+ SC. On approaching, under the influence of a shear
(AU+ Aat Au) and (Ba+ oC) gives rise to a ribbon (Aa + Au + aC -t- oC). In appendix II the equilibrium configuration of such a, ribbon is calculated (fig. 29a). The process can go on inde~nitely and more complicated ribbons can be formed, a possible fivefold ribbon can be formed by combining the fourfold ribbon Ao + Aa f UC+ Oc with OB + Au giving rise to Au + Ao + aC + Au + Ao (fig. 29b). If the reacting ribbons are part of a network,
Fig. 29. Fusion reaction of threefold ribbon with single ribbon of different Burgers vector. a) Formation of a fourfold ribbon; b) Formation of a fivefold ribbon; o) Possible node involving a fourfold ribbon.
P.
DELAVIGNETTE
AND
S.
AMELINCKX
nodes like the one shown formed.
in fig. 2%
can be
We will now discuss a few more typical nodes that may result from such processess.
Let the
threefold ribbon have Burgers vectors Aa t- Ao -t Ao and the single ribbon On meeting (a)
(fig.
(bl
(Cl
30b),
Bo+ GA, (fig. 3Oa).
Aa reacts with Ba and forms which
in turn reacts
with
OC
OA to
lb)
td)
of threefold ribbon with Fig. 30. Fusion reaction simple ribbon with one common Burgers vector (the
of threefold ribbon with Fig. 31. Fusion reaction simple ribbon with one common Burgers vector (the
adjacent
adjacent
stacking
fault ribbons
are of different
type).
stacking
fault ribbons
are of the same type).
”
”
% -a-a-
a-,-
_,-_,---.-a-
--b-b-
b--b-
-b-b-b-b-
(I!)
IO)
Fig.
32.
Interaction
of threefold
ribbon
with
simple
ribbon
without
common
Burgers
vector.
DISLOCATION
reform
Ba (fig. 30~). Finally
PATTERNS
Ba reacts
IN
GRAPHITE
with
Aa and forms again UC. The resulting configuration and the lettering pattern are shown in fig. 30d. The same threefold ribbon can react with the single ribbon Ao
and
aA
oA+Ba
(fig. 31a). On meeting
annihilate
one
another
whereas
Acr and Bo react to form UC. The resulting node is presented in fig. 31b. It is in fact equivalent to the configuration
shown in fig. 18 and which
resulted from the fusion of two single ribbons. 12.Z
INTERSECTION AND
SINGLE
0F
THREEFOLD
RIBBONS
RIBBONS
In view of the frequency of such interactions it seemed worthwhile to investigate their geometry. Different relative positions of threefold ribbons and single ribbons have to be considered. 12.2.1.
No common
Burgers vector
We will first analyse in detail the case in which the single ribbon is in the same lattice
Fig.
33.
type
Observed
described
examples
in fig.
32.
of intersection In
A
of the
superposition
of
ribbons.
I’
r’
/
’ /& A
/
Dl’
//
A/
Qf
/
‘I
/
‘_______-_Jf______J
(b)
A_-..-_
/ (d)
(Cl
Fig.
34.
Intersection
of
threefold
ribbon
with
simple
ribbon
having
one
Burgers
vector
in common.
40
.I?. DELAVIGNETTE
plane
as one
threefold
of the ribbons
that
build
ANI)
the
ribbon.
They
have
no
common
Burgers
vector e.g. Ao -!-AC -;- Aa and sB + Co. The interaction is shown schematically
From
contain
through
fig. 32a along XY
this drawing
(fig. 32~) it is
&CO layers
in cubic
stacking.
On intersecting, segments will develop as shown by dotted lines in fig. 32a. The situation is shown in space in fig. 32d a cross section along X’T’ is further represented in fig. 32e. From this it, can be concluded t*hat sweeping of the area of overlap by the dipole Bo + aB will eliminat’e the wrong stacking and restore perfect material. The other region containing the stacking “1) over b” shrinkage it into a the other
leads to more
of the processes
complicated
n~hich \vill now be described
patterns,
described some of
and anslysetl.
in fig. 32a, whilst,
evident that one of the overlapping regions contains a stacking “b on b” and the other regions
Combined patterns
of layers is given in fig. 32b. c:
as a cross section and UV.
13.
AMELIP;CKX
The combination
however
the arrangement
Y.
‘l’he most striking one is perhaps what could he called the “st’ar pat,tern”.
Two examples of
it are visible in fig. 36. The let,tering pattern is reproduced in fig. 37. The correctness of the l&tering pattern is proved by referring to the extinct~ion pat,terns of fig. 36d and e. In fig. S6d dislocations lett,eretl Ao show all lack of contrast,.
can only be eliminated by complete of the hexagonal mesh, t,ransforming eont.racted node. The perfect area on hand will extend as much as allowed
by the increase in length of the dislocations bordering it,. The final situation is represent,ed in fig. 32g and the lettering pattern in fig. 3211. Xxamples of observat’ion are visible in fig. 33.
12.2.2.
C’omrnon Burgers vector
If the single ribbon contains a dislocation which has the same Burgers vector as the dislocations
in the threefold
ribbon,
say Xo,
the result is quite different. If the simple ribbon approaches the threefold ribbon in such a way that the meeting partials repel (fig. 34), no fusion reaction takes place between the Aa dislocat,ions which are in the same plane ; t8hose which are in adjacent planes form dipoles (IX) Aa + aA. In one of the overlapping regions, I. a multiple stacking fault is formed; in II a stacking “a over a” would result. Both st,acking faults will be eliminated. The first by sweeping the dipole Ao-t_aA over the faulted area, the second by shrinking to zero surface. The final result is as shown in fig. 34d. An example of an observation is given in fig. 35.
Fig. 35. ~bservecl exa,mple of intersect,ion described in fig. 34, a) Line corltrast ; b) Partial in tJha threefold ribbons arc out of contrast.
DISLOCATION
PATTERNS
IN
41
GRAPHITE
a) Line contrast of star pattern B; b) Line contrast of star pattern A; Fig. 36. Examples of star patt,erns. c) Stacking fault contrast of the t’wo stars A and B; d) One set of partial dislocations shows lack of contrast in star A; central
e) Another
hexagons
set of partial
in t,he two
dislocations
is out of contrast
st,ars are of opposit,e sign, giving stacking
fault
within
in star A. Notice
an indicat)ion
that the curvature
as to the presence
of the
or not of a
the lrexagon.
-b-b’_b._
(Cl
Fig.
37.
Analysis
b) Cross section
(d)
of the star pattern.
along
XY,
showing
the partial
a) Two interacting extended nodes giving rise to the “star” pattern; the presence of one layer in wrong position; c) Arrangement in space of dislocations;
d) Lettering
scheme
of the star.
42
P.
DELAVIGNETTE
whereas in fig. 36e the dislocations of contrast.
exhibit
S.
AMELINCKX
CO are out
of the stacking fault can nevertheless be deduced
Pig. 36a and b show the full line
from the inward curvature of sides of the central
contrast and fig. 36~ the stacking fault contrast. It’ is clear
AND
that
stacking
the central fault
of the configuration
region
contrast.
does not
The stability
hexagon;
this proves moreover that the specific
stacking fault energy is somewhat larger for the region inside the hexagon
than outside.
is assured by the repulsion
If the glide planes of the two extended nodes
between all the parallel segments of the central
are spaced by more than one unit cell a stacking
hexagon
fault
on the one hand,
and the stacking
faults on the other hand. Alternating the central hexagon
which
contains
~~~
infractions
against
nodes of
the stacking rule is formed. The centra,l region
are of the K and P type
will be smaller in this case, and bhe curvature
and the regions of stacking faults are alternatively of type I and type II, as indicated by different cross-hatching in fig. 3’id. The simplest way in which this pattern could be generated is by the superposition of two extended nodes of a different type, and hence in successive lattice planes, as demonstrated by fig. 3’ia. The central hexagon then contains a stacking fault with two violations of the stacking rule, i.e. the sequence is a b a b a o a b a b. This stacking fault cannot easily be Ginated except by complete disappearance of the central hexagon. The stacking fault energy is however only of the same order of Inagnitude as y; however parallel segments of the central hexagon repel more strongly than the partials in a ribbon and its cross section will therefore be larger than the ribbon width. The stacking fault will not be visible since it only consists of one layer which is in a wrong position; there is therefore no phase difference between waves diffracted above and below the fault plane. The presence
of the sides of the hexagon more pronounced, but again no cont’rast will be observable. Star patterns, without a stacking fauIt in the central hole, were however also observed. The absence of a stacking fault can be deduced from the lack of contrast and from t)he outward curvature of the central hole, which gives it a rounded shape. An example is presented in fig. 36 in B. Although the lettering pattern remains identical, the generation mechanism is different. A configuration of ribbons giving rise to the observed pattern is shown in fig. 38. The reaction involves two fusion reactions of pairs of ribbons in the same plane as described in par. 10.1.1.1. and two intersection reactions of the kind described in $ 11.2.2. The central region is now free of st,acking faults. The intersection of two families of ribbons, for example (Aa + crB) and (oil -t Ccr), containing a different stacking fault, also gives rise to star patterns of this type. The slight variations in shape are due to the varying distance between the two planes. If these are separated by more than c/2 (the separation distance is *z+ +, where n is an integer), the interaction becomes less pronounced and “stars” with short or vanishingly small radial segments result, like the one shown in fig. 36 in B. We can prove this statement by referring to this photograph. prom the pronouuted inward curvature of the central hole in A we conclude that it must contain a fault with more than two violations of the stacking rule; the only alternative is four. The stacking i + sequence is then a II a b c a c b a b. It is easy
Pig. 38. Configuration of ribbons giving rise to a star pattern without a dacking fault in the center.
uu L--II
DISLOCATION
PATTERNS
IN
43
GRAPHITE
to see that this implies that the two planes in which the two parts of the pattern are situated are separated by at least SC (the location of these planes is indicated by arrows). Prom the inter~onne~tioI1 of the two patterns through
contracted
A and B,
nodes C, which are known
to be planar, can be deduced that in the star B too, the two levels are separated by the same distance, i.e. by more than c/2, proving our statement. A somewhat related pattern is shown in fig. 39 and the corresponding lettering scheme is given in fig. 40. The simplest way by which this configuration may result is by a fusion reaction between two contracted nodes of a different type as demonstrated by fig. 40a. The central “hole” is now free of a stacking fault, as can be deduced from the “outward” curvature and its extension at the expense of the stacking faults is therefore only limited by the increase in line tension. The actual configuration of ribbons giving rise to this pattern, is drawn in fig, 41. It is clear that four different kinds of ribbons are involved in the “reaction”, two pairs react by “fusion” giving rise to the threefold ribbons; the other reactions take place by intersection, Left and right of the central pattern the nodes are of different kind as a consequence of the presence left and right of different ribbons, in the vertical set.
Fig.
39.
Pattern
in threefold
ribbon.
As a final example we will discuss the remark-
sidered as resulting from the interaction between
(bl
resulting from the fusion of two contracted nodes in adjacent hole is free of stacking
“hole”
able pattern of fig. 42 of which fig. 43 represents the lettering scheme. The whole can be con-
(a)
Fig. 40.
Sgmmet,rical
faults.
An examplo
lattice planes,
was presented in fig. 39.
The central
dislocations,
with a different
Burgers
vector.
If the two sets are in the same lattice plane, the result> is a netgvork of extended tracted nodes as shown somewhat fig. 44. An observed
example
and con-
idealized in
is represented
in
fig. 45. As the mesh size decreases the curvature of
t,he sides
of
the
staeking
fault
t,riangles
becomes smaller and smaller and finally a net work consisting of triangular meshes is formed.
I R
fC>
three nodes of one type and one larger node of the second Qpe and hence in a different plane. The areas of overlap indicated in fig. 42 by A contain faults of larger specific energy than that, of the type t,wo stacking faults. This can be deduced from the curvature of the lines towards the area of overlap. Such a stacking fault is a natural consequence of t’he proposed model. If the two interacting configurations are in adjacent planes the stacking fault is of the type . . . abab+c’bab uu
. ..
and it contains two violations against the ab sequence, i.e. the same number as the other fault regions, it is however of a different nature and the energy may therefore be slightly larger, However separated
it is probable
that they are in planes
by gc, the fault is then of the type + . ..ababcacbab l__-l L.-..Jli
j.
.._
l..i
and it contains four violat,ions of the s&eking rule. The energy will now be s~~bstantially larger. In neither ease will it exhibit fault contrast, in accord with the observations. 14.
Regular
networks
The networks of ~slocatio~~s are in fact twist boundaries with a. c-rotation axis. They result from the i~terse~tion of two families of basal
st,ur paWorn. The ~&tern results Fig. 42. Nultiple from t’he intoract,ion of one extended nude of a given kind and t.tlree nodes of’ the other f‘auit
contrast;
e) Partiesis
b) Partiats
kind.
a) Sacking
Bo show lack of contrast;
A(r are otrt of contrast fig. 43).
(see notation
in
DISLOCATION
PATTERNS
IN
45
GRAPHITE
(11
%ig. 43.
Analysis in terms of Burgers vectors of the pattern in fig. 42. star;
b) Configuration
(b)
Fig. 44.
a) Lettering pattern of the multiple
of nodes giving rise to the star.
a) Curved extended and contracted nodes; b) Degenerated Idealized regular networks. if either the &aeking fault energy or the mesh size becomes small.
form of (a)
Alternative triangle are faulted. This is shown schematically in fig. 44b and an observed example is visible in fig. 14. If the two intersecting sets of dislocations are in adjacent lattice planes a regular array of nodes of the type shown in fig. 27 results. The corresponding network is shown with its lettering pattern in fig. 46. An observed example is presented in fig. 28. The same network also results from the intersection of a family of threefold ribbons, say with Burgers vector aA + OA + aA, and a set of single ribbons which have a Burgers vector in common with tShe threefold ribbon, say oB+Aa. A related type of regular network consist,s of an array of “stars”, it is drawn in an idealized fashion in fig. 46, a whilst fig. 47 gives an observed example. Its formation necessitates
Fig. 45.
Regular
network
contracted
of curved extended nodes.
and
46
P.
DELAVIGNETTE
AND
S.
AMELINCKX
(b)
Fig.
46.
Idealized
regular
also the intersection
n&works.
a) Network
of stars;
of at least two families of
ribbons containing different stacking faults. The perfectly regular networks are of course the exception; ingeneral “stranger”or “singular” dislocations meander through the nets and cause deviations from the regularity, we call We will present only one them “singularities”. network containing singularities, together with t,he analysis in terms of Burgers vect’ors, as
47.
Observed
example
of a network
of stars.
of
segments
of threcfoltf
ribbons.
summarized in the lettering pattern. In most cases the lettering patterns have been proved by making use of the contrast effects discussed in 97. The network is shown in fig. 48, the analysis is given in fig. 49. The most striking feature is perhaps the presence of a line of discontinuit,y XY across the field of view. Along this line the orientation of the extended nodes changes by
Fig.
Fig.
b) Network
48.
Dislocation
network
as seen
in
stacking
fault, contrast,; it contains a number of singularities which are analysed in t,erms of Hurgers x,ectors ill fig. 49.
DISLOCATION
Fig.
180”
(or 60’).
49.
Lettering
This suggests
pattern
PATTERNS
corresponding
that this line is
either an isolated partial in the plane of the net, or a twin boundary. The perfect straightness points to the second interpretation. It is clear that the partials
of the net interact
strongly
with this line, but nevertheless cross it. This observation is in agreement with the model of a twin boundary suggested by Kennedy 19) and which consists of a symmetrical wall of partials. A number of threefold ribbons is present in the pattern; the difference between the two stacking faults in these ribbons is evident from their contrast. The holes in the threefold ribbons (in A) can be shown to be caused by the intersection with a simple ribbon, which contains no partial with the same Burgers vector as the partials in the ribbon. A star pattern is visible in B.
IN
GRAPHITE
to the observed pattern
15.
shown in fig. 48.
The Formation Mechanism for the Hexagonal Networks
A mechanism for the formation of hexagonal networks containing extended and contracted nodes has been proposed by Whelan et al. z”) and applied to stainless steel. The mechanism involves cross slip and it is therefore a priori inapplicable to graphite, where there is only one prominent glide plane. Moreover there is direct evidence that networks in graphite form by glide on the basal plane only. In fig. 50 for example it is clearly visible that a few nodes have been formed in between the points A and B in the interval between the two exposures. We propose here a mechanism for the formation of such nodes based on glide along the c-plane only. Consider the situation pictured in fig. 51 whereby a ribbon BaA and a ribbon CoA meet
P.
Fig. 50.
Observed
DELAVIGNETTE
AND
stages in the process of node fornlation.
(b) two more nodes have been formed;
c) One partial
c*clntrast, although
S.
AMELINCKX
a) and b) Two succossivo stages;
is ollt, of contrast;
all partials
(b)
between
d) The entire, dislocation
(a) and
1’ is otlt, of
are in contrast,.
,c
)
a) The ljartials CIA and CO combine to mechanism of extended and conkacted nodes. Fig. 51. Formation form a perfect dislocation CA; b) Dislocation CA crosses over the partial Hn; r) ‘l’hc perfcot, dislocation CA dissociates again according to the scheme CA --f Co -+ nA.
DISLOCATION
PATTERNS
49
IN GRAPHITE
as would be the case when a first set of ribbons
have a direction in sector II the process will be
with the same Burgers vector is intersected
slightly Now
by
different. the partials
aA
are forced
which first meet have vectors uA and Co; they repel. But, since there is a driving force for
together to form a perfect dislocation
RA, which
movement
again is possible
a ribbon
of %Ldifferent
kind. The two partials
of the ribbon
CcrA, we can assume
that GA and Co will combine
to form a perfect
is now pushed
Bo
through
Ca. Because of the repulsion
CA as shown in fig. 51a. This whole
oA, BA will now move
~slocation
CA is repelled by the right partial
instead
CoA, there is however
only
a small interaction with the left partial Bo, since vector Bo is perpendicular to vector CA. The partial Ba is further bound to the partial aA by the stacking fault that separated them. It is therefore reasonable to accept that the complete dislocation CA will be pushed “over” or “through” the partial Bo without difficulty, creating the situation shown in fig. 51b. The situation is now such that the ribbon BaA has effectively been reformed at the other side of CA. The perfect dislocation CA may now eventually dissociate again giving rise to the situation of fig. 510 and the whole process can start over again. The process described here requires that the applied shear stress, all dislocations in the correct sense. This restricts the of favourable orientations for the shear to sector I of fig. 52. Should the shear
under move range stress stress
uA
same situation
further
eventually
to
between BA and over oA, but
dissociate
again,
Co-t aA, into oA+ Bo. The
as before
has now effectively
If y is relatively large with respect to the applied shear stress, sector II will extend somewhat at the expense of sector I. Successive stages of the process pictured in fig. 51 can be observed in the left bottom corner of fig. 50. Between the two exposures two more nodes have been formed as can be judged from the configuration of dust partials. The dislocation segment, marked P, and which according to process I should be perfect does in fact exhibit contrast effects which differ from those observed at partials. In particular in fig. 50 (a and d) the segment P is out of contrast, whereas all three sets of partials are in contrast.
Crossing of Dislocations
A
e
u
/I f-.
52. Illustrating the relative directions of partial
Burgers vectors and applied shear stress. The process represented in fig. 51 takes place if the applied shear stress has a direction in the range I. In the range different process takes place,
DISTANT CROSSINGS
We will speak about crossing when the interacting dislocations are situated in the plane
ci
e
//
c
a slightly
Co. This
been reached. It is clear that the kind of process that will occur depends on the orientation of the shear stress and may be slightly on the value of y.
16.1.
Fig.
will
within the ribbon
16.
A
A
e
it
-.1.;~ e*
the partial
since BA is perpendicular
dislocation
aA of the ribbon
and
II
separated by at least c. On crossing in not too distant planes the dislocations excert locally strong forces one on the other. Usually there is a, torque tending to twist them into the anti parallel orientation, where they attract. Many examples of the typical configurations that result from such interactions were observed for ribbons. The behaviour depends of course on the Burgers vector of the dislocations. The vertical set of ribbons of fig. 53 contains ribbons with two different Burgers vectors ; this is
50
Fig.
Y.
Crossing
53.
of two sets of dislocations;
t)he twist
c
DELAVIGNETTE
at the crossing
1,
/
1)
,
’
S.
AAWELTSCKX
notice
points.
*I(T T T
6’
AND
Fig.
66.
threefold more
I
deformed
crossing
/
Crossing ribbon.
of
one
Sotice
single
that) the
t,han t)hrx t,luxtifold
of single ribbons
thv largest
ribboll
with
single ribboll.
one
ribbon
is
For
t)lrtx
i trrcl~~t’is on 1~11~
/
Fig. Fig.
54.
Lettering
scheme of the crossing in fig.
illustrated
56.
a) Line
Crossing contrast
is ollt
53.
reflected in the different behaviour at the crossing points with the same dislocations. Fig. 54 represents the lettering pattern for this situation. If a threefold ribbon is crossed by a single ribbon the largest deformation is on the single ribbon as in fig. 55.
of ribbons
16.2.
c:LUSE
showing
; b) 0 no set of partial
constrictioll<. tlislorn!iorls
of contt’ast.
CROSSIiYGS
There is in fact) a continuous change in pattern from distant crossing to crossing in adjacent planes. For close crossing the configurations resemble those characteristic of intersection. In fig. 56 e.g. constrictions are formed at every crossing point. The reason for the formation of
DISLOCATION
PATTERNS
51
IN GRAPHITE
-a
b=a-
-b ---a -b ---+-&----b-b’-b X--a -b --a -b b
-b
(bl
(a)
Fig. 57.
LeMering scheme for the patterns of fig. 56.
the constrictions
is to avoid the formation
of
stacking faults containing four violations of the moreover stacking rule ; elastic interaction stabilizes the constrictions. This can be shown by referring to fig. 57 which is a cross section along XY. Suppose that Ba would leave the combinatio~l of three parGals Aa, aA and Ba. The region sweeped by Bo would contain a stacking fault with four violations. Since there is no longer a long range repulsion between BG and GA, because of the presence of Ao, Ba
a) configuration
in space;
b) cross section.
will remain together with the dipole Aa+ aA which is in stable equilibrium when the dislocations are one above the other (at 45”). As a further example we discuss fig. 58 (node A) which resembles a contracted and extended node pair. In reality however a segment bisects the extended node, this segment produces only a very weak contrast because it consists in fact of a “dipole”. The presence of the segment can however be inferred from the “arrow point” shape of the apparently extended node. The analysis is pictured in fig. 59. The line energy of the dipole can be deduced from the angles at the sharp bent; it is found to be roughly 16.3.
equal to the energy THREEFOLD
RIBBONS
of a partial. RESULTING
FROM:
CROSSING
An interesting
pattern
involving
crossing
is
shown in fig. 19. The lettering pattern is shown in fig. 60. The most important feature of this pattern and which requires an explanation, is
Fig. 58.
iVet,work showing arrow point nodes due to the presence of dipoles in A.
the inequality in spacing for one of the threefold ribbons. This can best be judged from the stacking fault contrast (fig. 19b) which gives the true width. From the lettering patterns it is evident that the ribbon Itr consists of the overlapping of two stacking faults of the same kind and hence separated by a distance c (or nc -n: integer). The layer sequence in such i + a stacking is a b a b c a b c b c where the planes I_ III
P.
DELAVIGNETTE
AND
S.
AMELINCKX
(a)
Fig.
Fig.
60.
Lett~ering pattern
59.
(b)
Wxhanism
of formation
illustx-ating the formation
of the ribbons have been indicated by arrows. Such a stacking fault contains four violations of the ab sequence, against two for the usual stacking fault and it has therefore a higher energy. The structure of this threefold ribbon is shown in fig. t7b. The line AG which consists of the superposition of crC and aB is now one of the outer partiafs (fig. l?‘b), instead of the central one. The two stacking fault energies yl and yz are moreover different. From the asymmetry the relative values of the stacking
of “arrow
point”
of the asymmdrical
nodes.
threefdd
ribbon
shown in fig. 1.9.
fault energies can be deduced using formula (I, 10); one finds ~~~~~~=~~5 (see appendix I). The presence of the high energy stacking fault in RI is also the reason why its surface is reduced to a minimum by a displacement of the segment S (see fig. 60). The formation of t*his type of ribbon is possible because crB on arriving at
DISLOCATION
PATTERNS
It is clear that the stacking fault in ribbon Ri gives
contrast
under
the same circumstances
as the fault within the ribbon
Rz, since they
both result from the same net displacement vector R =Aa. The lamella ca does not show up of course.
The equality
of contrast
in Rr
and Ra is evident from fig. 19b. The conclusion is that the same pattern shows the occurrence
of a second
type
IN
53
GRAPHITE
taining two stacking faults of different energy. This is clearly the reason why the two halves of the ribbons A, have a different width as opposed to the threefold ribbons in B of the same figure ; which are of the type that contains two stacking
faults of the same kind.
This pattern shows another feature of interest. In the points marked D (fig. 62) some protruding
of threefold
ribbons, which is however less common, since the combination can only take place under a
A16 --c-1-c--
relatively large shear stress. It is further noteworthy that the energy of the two superposed faults is smaller than the two times the energy of a single fault, proving that some interaction exists between faults in neighbouring planes. 16.4.
CROSSING
OF TWO
RIBBONS
OF THE
6jB
-b&f+b-b-C-b-
b-bc
-l-c-c---
g
-.-,-a--
Y
-b-b-b--1-,-,-
-b-b-bCb,
SAME
TYPE
Under a shear stress two ribbons with the same stacking fault, for example (Ac++B) and (Co -I-oB), in glide planes separated by a distance c (or a multiple of c) may erosa as shown in fig. 6la. Segments will tend to develop as indicated by dotted lines. The region of overlap now contains a fault of the type ababacbacac... III IL--J (fig. 61b), i.e. containing fonr violations against the rule. No mutual annihilation of the segment oB and Bo in the extended part of the node is possible; instead a dipole (Ba+aB) is formed. Movement of the dipole does not change the area of the stacking fault; moreover there is only a very weak interaction with the other dislocations. In an isolated node as in fig. 61~ there is no preference as to which way the dipole will go. In a network it will take such a shape as to minimize the energy, this is roughly the same as to reduoe its length as muoh as possible. The arrangement shown in fig. 6ld is probably the one that is observed in fig. 62 in A. The three dislocations Ao+Ca -+(TB and two other OB now form a short segment of threefold ribbon of the kind discussed in 8 16.3 i.e. con-
Fig. 61.
Crossing of two ribbons of the same type
but in different planes (distant
of nc).
54
P.
DELAVJGNETTE
AND
S.
AMELIIiCKX
this net and acquired
a bent shape.
On the
one hand it generated the segments of threefold ribbon, as discussed above. where
intersecting
(Co-i-(TB),
it created
On the other hand
dislocations the dipoles
of
t,he
marked
set, D.
Also in this case it is possible to estimate t’he line energy of the dipoles. It now t’urns out to be very
small since the angle of the cusp is
hardly visible;
a rough estimate leads to G_ :!
of tjhe energy of a partial.
The much smaller
value found here as compared to $ I 1. I._). is presumably a consequence of the much smaller distance between the component’s of the dipole. 16.5.
Fig.
62.
Network
showing
different
singularities
analyzed in 3 16.4 and in fig. 61. In A asymmetrical and in H symmetrical
threefold ribbons; in 1) dipoles.
parts have been developed at both sides of a mesh, which contains no stacking faults. It is clear that these protrusions should not be stable unless they are connected by some line that stabilizes them. This line cannot be a partial dislocation since all partials are in contrast; from the geometry follows that it cannot be a perfect dislocation either. It is suggested that this invisible line is due to a dipole. From its occurrence across a perfectly stacked mesh, can
,?UPERPOSITION
014’ HIBBOh-S
Some superpositions of ribbons give rise to features which are worthwhile discussing. An example of interest is for instance visible in fig. 33 in D and represented schematically in fig. 6Ya. Ribbon
has a Burgers vector Aa- (TB and ribbon 2 a vector Ca-I-GA. The partial with vector Ca of ribbon 2 is attracted towards the partial Ao of ribbon 1 and is repelled by the other partials. Combination Aa + Co + aB will therefore take place. The partial Co of ribbon 2 is equally attracted by both partials aB, it is 1
be concluded that it has to be a dipole of perfect dislocations. Such dipoles do not change the stacking on passage, as do dipoles of partials. They have therefore no strong tendency to go one way or the other; this explains why it could remain across a mesh. Such dipole results when two ribbons of the same total Burgers vector cross in planes at a distance c ; a drawing similar to the one presented in fig. 59 would show this. The whole pattern of fig. 62 can now be understood. The planar network in the right top corner results from the intersection of ribbons (Ao+ oC) and (Co+ oB) in the same plane. One singular dislocation with vector (Bo+ UC) roughly parallel to (Co + oB) but lying in a plane differing by C in level, crossed
(‘onfiguration of partial dislocations resulting Fig. 63. from ihr superposition of ribbons leading to thr pattern observed in fig. 33 in A. a) Configuration partials to be compared se&on
with fig. 33 in A;
of
b) Cross
of the inkial situat,ion; c) C*o and Au combine
and form aB ; d) flnal situation
: cross section along XY.
DISLOCATION
however
connected
to the left partial
PATTERNS
aB by
IN
55
GRAPHITE
It is clear that the symmetrical
process might
the stacking fault and it will therefore combine
also happen. The first step would then consist
with it; the resulting cross section is shown in
in the reaction
fig. 63d.
Ca+Aa + aB. Fig. 64 shows a somewhat the
pattern
aA + aB + Ca; the second step
is represented
related feature; schematically
in
fig. 65a. It is suggested that in this case we have a superposition
of a ribbon aA + Ca and Aa + aB.
The two partials aA and Aa form a dipole. One of the partials either aB or Co can now glide towards this dipole and eliminate the stacking fault. Say that aB moved towards the dipole, then Ca will be repelled and we will have the situation pictured in fig. 65b. The ribbon will have approximately the normal width. The partial Co may just as well be attracted towards the dipole and then the results would be as shown in fig. 65~. The presence of one ribbon stabilizes in fact the recombination of the other. In fig. 64 there is now apparently a change over from configuration 65b to configuration 65~ as represented in fig. 65a. Superposition may also lead to a narrow threefold ribbon. Suppose that a ribbon Aa + aB
Superposition
Fig. 64.
of ribbons giving rise to cross
over of part~ialn.
UIA AIU
UiB
and a ribbon Ba+ aC combine by fusion into a threefold ribbon Aa+ (Ba+ oB) +aC. The central line is now a dipole and hence does not repel very strongly the two outer partials. These two partials repel however and a ribbon having the width of a single ribbon
will result.
The
central line however should now give very little contrast. The narrow triple ribbon of fig. 19 may be of this nature.
C(U
lil ’ I
Even fourfold ribbons may result from superA model of a fourfold ribbon of this
position.
UIAAla clu UIB (iL)
Fig.
65.
Con6guration
superposition. in A;
b) Cross section
situation;
of
partials
resulting
from
a) Pattern to be compared with fig. 64 of one possible
equilibrium
c) Cross section of alternative
equilibrium
situations.
(8,
Fig.
66.
(1:)
Fourfold position
ribbon resulting from the superof two twofold ribbons.
P. DELAVICNETTE
56
Fig.
67. Different contrasts of the same fourfold ribbon R. The two outerrpartials have the same l3urgers vector.
type is shown in fig. 66. The strong repulsion between the two inner partials aB makes this central ribbon to be larger, than the outer ribbon. The contrast effects observed in fig. 67 suggest that this may be a ribbon of this type. 17. 17.1.
AND S. AMELINCKX
Quenched-in
prismatic loops
QUENCHING PROCEDURE
For these experiments single crystals of pure natural graphite originating from the Ticonderoga Limestone formation (N.Y.) were used. According to Hennig, who kindly supplied the material, these crystals have been heated at elevated temperature in chlorine gas in order to purify them by volatizing the chlorides of eventual impurities. The crystal flakes were mounted in a slit at the tip of an outgassed and purified thin graphite rod and heated in vacuum by means of electron bombardment. The peak temperature was about 3000” C as measured with an optical pyrometer. A considerable amount of sublimation took place, indicating that at the temperature reached, defect formation should be appreciable. The current was then switched off, and the crystal was allowed to cool under vacuum. It was estimated by pyrometry that during the first seconds the cooling rate was about 1000” C/see; which should be sufficient to trap a number of vacancies. When examined directly after the quench no
loops
were found. The crystals were then annealed at 1200” C. For this operation they were enclosed in a pure graphite holder and sealed off under vacuum in a quartz capsule. There was no contact between the graphite crystals and the quartz. After this heat treatment the crystals were re-examined and now circular, slightly hexagonal features situated in the c-plane, as shown in fig. 5, were found. 17.2.
THE OBSERVATIONOF LOOPS. BURGERS VECTOR DETERIWINATION
We will now demonstrate contrast
effects
b
are
in
b
that the observed
agreement
with
the
b
(4
a
(b) Fig. 68. Schematic view of prismatic loops. a) Loops due to the precipitation of vacancies; the Burgers vector is inclined with respect to the c-plane; b) Loops due to the precipitation of interstitials. The Burgers vector is perpendicular to the c-plane.
DISLOCATION
assumption
that
condensation dislocations exhibit
the
are
loops
of vacancies.
PATTERNS
due to the
That the loops are
is deduced from the fact that they
diffraction
contrast.
The best contrast
is obtained when the interior of the loops exhibits stacking fault contrast, fig. 5 is taken in these circumstances.
Some loops are lighter,
others are darker than the environment. other inclinations
For
of the specimens, darker loops
may become lighter and vice versa. On overlap two dark discs may produce a lighter sector, as e.g. in fig. 5 loops A. This behaviour is typical for the diffraction contrast due to a stacking fault parallel to the c-plane. As shown in fig. 68 vacancy loops are characterized by Burgers vectors inclined with respect to the c-plane, whereas interstitial loops should have a perpendicular Burgers vector. The Burgers vector of the loops was determined by using ( 1120) reflection to make dark field images as demonstrated in fig. 69. Those loops which have a Burgers vector not lying in the reflecting (1120) plane, will show up; those which have their Burgers vector in that plane will not show up; but they will for another (1130) reflection. It is clear from fig. 69 that most of the loops show up in a dark field line contrast, proving that they have an inclined Burgers vector and
57
IN GRAPHITE
17.3.
DISCUSSION
OF RESULTS
of the loops
Most
although
are single and circular,
some of them are slightly hexagonal.
In fig. 5 loop B two concentric are seen, the central contrast
as the environment.
coincidence,
the same
This may be a
and the feature would then simply
be the superposition also possible
circular loops
part exhibits
of two loops. It is however
that the central region is perfect
and that we have a loop of the kind discussed in fig. 4 and which dissociated into two partial prismatic loops. It is difficult to distinguish between both possibilities. From the fact that an inclined Burgers vector is observed it can be concluded that the “a over a” stacking has a large energy. If this were not the case the Burgers vector would be perpendicular both for interstitial and for vacancy loops. An implication of these observations is clearly that at about 1200” C vacancies seem to become mobile: the annealing stage observed at about 1200-1300” C by means of electrical measurements 22) and as a release of stored energy 23) is therefore 18.
probably
due to this process.
Interaction between Glissile Dislocations and Sessile Loops due to Point Defects
hence are vacancy loops. Similar observations confirming our point of view have been published
The interaction presents two aspects: on the one hand glissile dislocations are pinned by
recently
prismatic
by
Williamson
and
Baker 21). The
authors have more over been able to identify interstitial loops in irradiated material.
loops
in their glide
plane ; on the
other hand vacancy loops tend to form preferentially in the stacking fault ribbons of extended dislocations. We will discuss both points. 18.1.
PREFERENTIAL
LOOP
NUCLEATION
IN
RIBBONS
Fig.
69.
image
Quenched-in
but
inverted
dislocation
contrast;
loops.
b) Dark
a) normal field image
using the (1120) reflection. The dislocation lines show inverted
contrast except for one loop, which has its
Burgers vector in the (1120)
plane used.)
A cross section through a stacking fault ribbon is shown in fig. 70. As opposed to what happens in the perfect crystal, precipitation of vacancies in a plane marked by a rectangle in fig. 70b, within the ribbon, does not give rise to a stacking “a over a” but instead produces immediately the low energy stacking fault of type one without requiring the nucleation of a partial. One can therefore conclude that
P.
DELAVIGNETTE
AND
9.
AXELINCKX
--a-a-a-
W
(a)
I
i
-b-a+b_a+c
-.+-aa’bII r--___71 I ,L__;_r;IR
-b
s-a -b
-b
a
-i)
c-ab-
-b-a -c-v “-l-/b
c-
-
-ii
a
a-
-b
b
b-
-a-* -b
70.
a) Precipitation
of vacancies
of the high energy stacking in the plane
surrounded t,hrough
4-
inside a ribbon
by a rectangle
without XY;
diffuse
the
produces
producing
point
defects
towards
the
dislocations through long range elastic interaction. We will further present direct evidence that preferential nucleation in extended dislocations does happen. We will now show that such loops effectively pin the dislocations. Fig. 70 represents the geometry of a loop formed within a ribbon; on expanding the edges of the loop will soon meet the partials of the ribbon and form segments like AR and CD which have an inclined Burgers vector AT, BT or CT. The other segments of the loop AC and BD, have a perpendicular Burgers vector (OOOq’2). Cross sections through the loop are shown in fig. 70~ and d.
a vacancy
a stacking
d) Cross section
vacancies will find it energetically more favourable to precipitate inside the ribbon, rather than in perfect crystal, and hence there is an interaction energy, which is to be added to the Cottrell type interaction. The latter tends to make
b-
td)
fault,; b) Cross section RS through
the loop along
b-
b
(b) Fig.
b-
a high
loop within
fault ribbon,
energy
for removal
oan precipitate
fault;
c) Cross scct.ion
stmking
t~hrough the loop
necessity
vacancies
along
Uf’.
The partials are pinned to the loops not only because of elastic interaction but also if they should det.ach themselves from the loop by crossing it, they would have to transform the stacking within the loop into “a over a”. The latter process makes it difficult to cross the loop. 18.2.
PIXNIXG
OF GLISSILE
DISLOCATIOM
BY
LOOPS A partial, approaching a vacanoy loop from outside on a plane marked by a rectangle in fig. 70b, would also have to transform the stacking within the loop into “a over a” and will therefore be hampered in its movement by the loops. Recombination of the two partials may take place before cutting through the loop; this would avoid creating a bad stacking. Elastic interaction between glissile partials and prismatic loops may lead to further pinning. Such interactions have been described in detail for
DISLOCATION
PATTERNS
perfect dislocations
in zinc II), with some modi-
fication
be applied
they
can
to partial
59
IN GRAPHITE
dis-
locations in graphite. We will discuss now the different possibilities which are shown schematically
in fig. 71 and 72.
In noting the Burgers vectors
of the vacancy
loops,
the
we will only
consider
component
parallel to the c-plane. The glissile dislocations are supposed to be in one of the two planes that emerge in the loop. If they are situated in more will corredistant planes the interactions spondingly be weaker.
-
The vacancy loop is in the same plane
1X.2.1.
(fs inside) In
fig.
(b)
‘ila, the ribbon
as a whole is repelled
r
by the loop, whereas in fig. 710, the first arriving A
B+AU
0
-
UA
0
-B+u
c
t-
(Cl
Fig.
72.
Interaction
with the loop
c) Reactions
-
68
A
with
the
first
between
vacancy
loops
a) The ribbon as a whole is repelled; b) The horizontal component of the Burgers vectors of the loop is annihilated by the first arriving c) The fist second
arriving partial is repelled by the partial
narrowing
of the
and
dislocations.
the
component
Burgers vector of the loop is annihilated by the first arriving partial, transforming the loop into one with a vertical Burgers vector. The second arriving partial interacts weakly with the loop since the Burgers vectors are perpendicular.
(C)
loop,
but
strong shear stress, the
In fig. 71b, the horizontal
artial;
partial,
ribbon can pass the loop, the second partial will be held back, leading to a local widening of the ribbon. -
glissile
arriving
of the second one.
If, under a sufficiently
Interaction
and
partial is repelled, but the second is attracted, resulting in a local narrowing of the ribbon.
(b)
71.
loops
B
0
Fig.
vacancy
; b) Repulsion of the ribbon as a whole ;
repulsion
al B+Ba
between
glissile dislocations. a) Both partials react successively
(a.1
A6
-
a
is attracted
of the ribbon.
resulting
in
18.2.2.
The vacancy loop is in an adjacent plane (0 outside)
The situation represented in fig. 72b, leads to repulsion of the whole ribbon. In fig. 72a, on the contrary, both partials react successively
60
P. DELAVIGNETTE
with the loop, pinned.
and the ribbon
In fig. 72c, only
reacts with the loop, repelled.
will be firmly
one of the partials
the other partial
being
AND
S. AMELINCKX
18.3. OBSERVATIONS
Direct evidence ential nucleation
was obtained
for the prefer-
of loops in the stacking fault
In the latter case, a reversal
of the
ribbon of extended
shear stress would result in movement
of the
$ 18.1. Fig. 73 shows a linear arrangement of loops. Although the dislocation ribbon itself is
partial Aa, the partial aB being held back. From the foregoing discussions it will be clear that vacancy the movement
loops are effective of dislocation
in hampering
not
visible
the
dislocations
linear
as discussed in
arrangement
strongly
suggests its presence.
ribbons either by
repulsion, by reaction, or by holding them back after passage. In this discussion we neglected the additional pinning caused in certain cases by the necessity of creating a high energy stacking fault. as pointed out in a previous paragraph. Since the interstitial loops have a Burgers vector which is perpendicular to the Burgers vectors of the glissile ribbon, the elastic interact’ion will be smaller. Furthermore on arriving at the interstitial loop a partial can always choose a plane, either above or below the loop so as to avoid the formation of a high energy with stacking fault. The elastic interaction
19.
Twinning
in Graphite
It is well known that graphite forms mechanical twins very easily, the composition plane being {liol). Whereas the layer sequence in one crystal is a b a b . . ., it is a c a c . . . in the twin crystal. A model for the twin boundary has been proposed by Kennedy 19). It is made up of a pure symmetrical tilt boundary consisting of partial dislocations one every two layers. As shown by Kennedy this leads to the correct
ribbons in more distant planes is comparable to that caused by vacancy loops. Direct reactions. leading to effective pinning, is expected to be less pronounced for interstitial loops, since the Burgers vector of loops and partials are mutually perpendicular. The tendency for interstitials to precipitate in dislocation ribbons is weaker than that for vacancies because of the small specific stacking fault energy
in graphite.
With
an interstitial
loop formed in a position, between the planes c and b within the ribbon of fig. 70 would be + associated a stacking fault b a b a c a c . . . i.e. containing one violation of the stacking rule. On the other hand when formed in perfect material the interstitial loop would contain a stacking fault . . . a b ac b a b with three infractions against the stacking rule. On this basis a weaker interaction is to be expected than for vacancies.
Fig.
73.
presumably
Linear
arrangement
nucleated
of
preferentially
prismatic
loops
along a ribbon.
DISLOCATION
orientation
difference
between
PATTERNS
the two
com-
ponents of the twin. No direct experimental evidence for this model was given however. The direction of the twin boundary required
is also as
by the model, i.e. perpendicular
Burgers vector
to a more
direct evidence that the model is indeed correct. shows a network
to fig. 14, which
of extended
and contracted
nodes lying on both sides, of a strip of twin TW and also within the strip. It is therefore reasonable to assume that the whole network resulted from the interaction between the same two families of dislocations. It is clear that the stacking fault triangles within the extended nodes differ by 60” (or 180”) in orientations in both regions. Since the two twin crystals differ in orientation one might object that this is due to a trivial contrast effect, i.e. that one region exhibits inverted contrast which causes inversion of the triangle. This is however not the case ; a careful inspection of the triangles reveals a slight curvature
61
GRAPHITE
indicating unambiguously
which side the stack-
ing fault is on. We will now show that this behaviour
is just
what
of the
one would
twin boundary that
of a partial dislocation.
We will present here some additional, We refer for this purpose
IN
the
model
dislocations boundary, locations
expect
is exact.
if the model
First of all it is clear
allows
through interaction
the
the
propagation
boundary.
with the boundary
is of course to be expected,
does not prevent
At
of the dis-
but this
penetration.
The same ribbon which in crystal II (fig. 74) is for instance between the planes a and b will be between planes a and c in the second crystal. The consequence of this is that dissociation into partials will be different in both crystals if in both cases a low energy stacking fault is to be formed. A dislocation ribbon, say with Burgers vector AB will dissociate in crystal I according to the scheme Ao+ oB whereas in crystal II the same dislocation will dissociate according to the scheme Ao’ + o’B or referred to the same reference triangle aB + Ao. The notation will therefore have to change when crossing the twin line. \c
A/
C
6
I
B
\\
Fig.
74.
Analysis
orientation
of the influence of a twin on the presence of a triangular network as seen in fig. 14. The of the nodes formed
from the same ribbon in crystal I and II differs by 180”.
62
P.
Fig.
shows
74
the intersection
which leads to the formation contracted lettering
DELAVIGNETTE
of ribbons,
of extended
nodes in the two crystals. pattern
AND
and
In the
we have taken into account
the changes in notation
at t,he twin line. Using
the results of 5 9.2.1 these intersections
will give
rise to the nodes also shown in fig. 14. It is now
clear
orientation
that
these
exhibit
the
required
difference.
The model as presented
S. AMELINCKX
the loop is evident from the successive stages shown. Small particles apparently lying in the surface locally
of
the
foil,
dislocation
are sufficient
movement
to
inhibit
as demonstrated
by fig. 76. Depending on the direction of the shear stress the moving
ribbons
This is clearly
are widened
or narrowed.
due to the following
effect.
If
the shear stress tends to move the two partials here can in fact be
in opposite
site sense, away from each other.
considered as a kink band; the two twin boundaries always occur in pairs of opposite sign. It was further found that the angular
the partial on which the resolved shear stress is largest will impose the sense of motion of the ribbon, the other partial being pulled in the
differences are not constant’. The observation can also be considered as evidence that the high energy stacking fault i.e. “a over a” will not form. If this were the
same sense because of the stacking fault,. ‘This is a case where widening occurs. The direction of the shear stress may also be such as to push the partials together. The sense of movement of the ribbon will again be dictated by t’he partial which has the largest resolved shear stress. The ribbon will now be narrower than normal. In fig. 77 the ribbons are clearly
case the dissociation scheme would be the same on both sides of the twin and the orientation difference would not be observed. The change in dissociation scheme at the twin line can also be deduced from the singular node observed at this line. Evidence was found that the same dislocation line can propagate through a twin boundary in accord with the model. 20.
Movement Sources
of
The dislocations
Dislocations are extremely
-
Dislocation
mobile along
the c-plane, proving in the most direct way that this is the glide plane. After a few minutes of irradiation with the electron beam the dislocations sumably
start moving almost inevitably, preas pointed out by Hirsch 24), as a of the stresses caused by the consequence deposition of a carbon film. The edges of a crystal flake or cleavage steps are the usual nucleation sites for dislocations. No interval sources of the Frank-Read type were ever noticed, although the circumstances are optimum as far as we restrict ourselves to thin foils. 3Iovement of ribbons is relatively smooth; at the stress level caused by the electron irradiation the movement is slow enough to be followed visually. Fig. 75 represents a typical example of “edge sources”; the expansion of
widened. This figure is remarkable for another reason. In the points marked C the ribbons are locally much narrower. They have apparently constrictions. Such constrictions can e.g. be caused by jogs resulting from the intersection of t,he ribbons with a dislocation having a Burgers vector with a component in t,he c-direction. These constrictions apparently move along the ribbon as can be judged from their displacement between the two successive expos~lrc~.
Appendix
I
EQUILIBRI~,~ THREEFOLD
SEPARATIOS
OF
PAHTIALS
IN
.i
RIBBON
We will introduce
the notation
and
where p is the shear modulus, b the Burgers vector of the partials and v Poisson’s ratio. The
DISLOCATION
PATTERNS
1N
GRAPHITE
Example of an “edge source” a, b and c are photographs taken at intervals of time of approximately Fig. 75. 30 seconds. The movement is probably caused by the stress due to the carbon film which forms on t,he specimen as a consequence of the decomposit,ion
Fig.
76.
induced beam.
Example by
of the movement
changing
a) Medium
b) High intensity,
t’he intensity
int,ensity
of dislocations of the electron
of the electron
all the dislocations
the right side; c) Low intensity,
beam;
are moving to
all dislocations
are
moving to the left side; d) Again high intensity,
all
dislocations are moving again to the right side of the photograph.
of organic vapours.
Fig. 77. Example of an “edge source”, showing constrictions in C, probably caused by jogs. Notice that the constrictions
move
along the ribbons.
64
P.
DELAVICNETTE
AND
S. AMELINCKX
repel. A small decrease will cause the combination of the inner partials. (a)
After recombination threefold ribbons are formed (fig. 17), the three partials having the same Burgers vector. If yi =ya we have also x = y from symmetry considerations. The equilirium separation
(b)
Fig.
78.
threefold bination ribbon
Two
ribbons
ribbon, of
before
to illustrate
Burgers
vector
of edge character;
combination
notat’ions
a
used. a) Comto
a threefold
b) Combination
of Burgers
vectors leading to a threefold
leading
into
is then given by the equation
ribbon of screw character.
equilibrium condition can then be formulated by expressing that the forces on all dislocations are zero. We will first consider the combination
where A = a sin2 CJJ +p cosz q, CJY being the angle between the total Burgers vector and the direction of the ribbon. In the pure edge orientation v== 90” and x=i~~/y whilst in the pure screw orientation v=O” and x=3 ,/?/y. As compared to the corre-
of Burgers vectors shown in fig. 78a which would lead after combination of the inner partials to a threefold pure edge ribbon. The
sponding single ribbons the threefold ribbons are about five times larger. If the stacking fault energies yi and y2 are different in the two halves of the triple ribbons.
equilibrium
the equilibrium
Lx 1 -2v+w
conditions
-_;&p& I
1 a%+?
are
:y&+y=o
A [y-1+(~+y)-11-y2
(L2)
where v and w have the meaning illustrated in
is
v = 0.55 x 10-4 and w = 0.05 x lo-4
,U= 2.3 x 1010 dyne/cma,
of Burgers Another simple combination vectors is possible, which would lead to a pure screw threefold ribbon (fig. 78b). In this case the metastable equilibrium configuration is given by v=O.4xlO-4 cm, w=O.14~10-4 cm. These equilibrium configurations are metastable. A small increase in spacing will make them separate further since as a whole the ribbons
(1,5) (W)
x=3A / (2yl-yz+g)
(L7)
y=3A
(1.8)
/ (2yz-yl+g)
with (1,s)
g=Vy1”+y22-y1y2
assuming y = 3 x 10-Z erg/cma; b= 1.42 A and v=Q.
=o
(L4)
This set of three compatible equations with two unknowns can easily be solved:
fig. 78. This system of equations can easily be reduced to a system of two quadratic equations of which the only real solution
become
A [-x-l-(x+y)-l]+yl=O i A [x-1-y-11+y2-yl =o
(Ll)
,;;I-+;&w-Y=O
conditions
The ratio yl/ya can be expressed directly in terms of observed quantities x and y. From (1,4) and (1,5) this ratio follows directly when putting y/x = r Yl _- r(:!+r)
yz- 2r+
1
(1. IO)
Applying this to the ribbon of fig. 191) gives for r N 2; y1/y2 N X/5 which is about what one would expect.
DISLOCATION
PATTERNS
IN
Appendix II
Appendix III
The equ~brium separation of partials in a fourfold ribbon of the type shown in fig. 79 can be obtained by solving the set of equations
CURVATURE FAULTED
65
GRAPHITE
OF
A
PARTIAL
The radius of curvature of the partial is
pb2 %2
3011
1
-+T ( 2‘u
Vi-W p
p1 4 v+w
B1 4 2w
61 --=
4 21--w
()
TWO
REQIONS
&4=
3&l ___------4 ?J--w
SEPARATING
-
rd
(III, 1)
and hence if d
1 -l-+ u-w >
+;? (
---- l 2u
l +_L
V-EW
v-w >
A -y=()
I
El:”
which express that the net force on each dislocation vanishes. The solution of these equations using the same numerical values as in appendix I, is w=O.l7xlO-4 cm and v=O.53~10-4 cm.
8 : Fig. 80. Curvature of a partial separating two regions containing a different stacking fault,, ill~tr&ting the notations used.
Fig. 79.
Model of a fourfold ribbon, illustrating the notations used.
References l) P. B. Hirsch, R. W. Horne and M. 5. Whelan, Phil. Mag. 1 (1956) 677 P. B. Hirsch, J. Silcox, R. E. Smallmann and K. H. Westmacott, Phil. Msg. 3 (1958) 897 P. B. Hirsch, A. Howie and M. J. Whelan, Phil. Trans. Roy. Sot. London 252 (1960) 499 2) W. BoUrnarm, Phys. Rev. 103 (1956) 1588 3) A. Berghezan and A. Fourdeux, Compt. Rend. 248 (1969) 1333 A. Berghezan and A. Fourdeux, Vierter Int. Kongr. El. Miw., Berlin 1958 (Springer Verlag, 1960), p. 567 4) A. Grena.11,Nature 182 (1958) 448; J. Metals11 (1959) 60
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