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Dislocation configurations in AuCu3 and AuCu type superlattices
DISLOCATION
CONFIGURATIONS
IN A&u,
M. J. MARCINKOWSKIt,
AND
N. BROWNS
AuCu
and
TYPE
SUPERLATTICES*
R. M. FISHER?
Superlattice dislocations have been observed for the first time by transmission electron microscopy These consist of two pairs of partial dislocations held techniques carried out with thin films of AuCu,. The separation between the pairs is measured to be about together by an antiphase domain boundary. 130 A. A theoretical calculation has also been made of the expected spacing which involves the stacking fault energy, antiphase domain boundary energy and the elastic interaction energies between the The theoretical results are in good individual dislocations that constitute the superlattice dislocation. agreement with those observed experimentally. Further theoretical considerations of the AuCu type structure lead to two additional dislocation configurations characterized by the direction of their Burgers vectors with respect to the ordered lattice. This difference in the dislocation configurations between AuCu, and AuCu arises because the latter is a less symmetrical lattice. CONFIGURATION
DE
DISLOCATION DANS DES AuCu, et AuCu
SURSTRUCTURES
DU
TYP.E
Les auteurs observent pour la premiere fois par des techniques de transmission d’electrons sur des Ces super-dislocations sent constitue es de deux pellicules minces de AuCu, des super-dislocations. paires de dislocations partielles reunies par une frontier-e de domaine antiphase. La distance entre les paires mesure environ 13OA. Les auteurs calculent Bgalement theoriquement la distance. Dans ce calcul interviennent l’energie des fautes d’empilement, l’energie des frontieres des domaines antiphase et les energies d’interaction Blastique Les resultats theoriques de calcul entre les dislocations individuelles qui constituent la super-dislocation. sont en bon accord avec les valeurs trouvees experimentalement. Des considerations theoriques supplementaires sur la structure du type AuCu conduisent B deux configurations de dislocations additionnelles caracterisees par la direction de leurs vecteurs de Biirgers par rapport au reseau ordonne. Cette difference dans les configurations de dislocations entre AuCu, et AuCu provient de ce que ce dernier a un reseau moms symetrique. VERSETZUNGSANORDNUNGEN
IN
tfBERSTRUKTUREN
VOM
TYP
.hcu,
UND
AU&
Zum ersten Ma1 wurden mit elektronenmikroskopischen Durchstrahlungsmethoden in diinnen Filmen aus AuCu, ifberversetzungen beobachtet. Diese bestehen aus zwei Paaren van Halbversetzungen, die durch eine Antiphasen-Bereichsgrenze zusammengehalten wurden. Der Abstand zwischen den beiden Paaren wurde zu etwa 130 A gemessen. Eine theoretische Rechnung iiber den zu erwartenden Abstand wurde durchgefiihrt, in welche die Stapelfehlerenergie und die Antiphasen-Grenzflachenenergie eingehen sowie die elastischen Wechsel. Die welche die Uberversetzung bilden. wirkungsenergien zwischen den einzelnen Versetzungen, theoretischen Ergebnisse stimmen mit den experimentellen Beobachtungen gut uberein. Weitere theoretische Betrachtungen der Struktur vom Typ AuCu fiihren zu zwei zusatzlichen Versetzungsanordnungen, die durch die Richtung ihrer Burgersvektoren beziiglich dem geordneten Gitter charakterisiert sind. Dieser Unterschied in den Versetzungsanordnungen zwischen AuCu, und AuCu tritt auf, weil das letztere Gitter weniger symmetrisch ist.
INTRODUCTION
This paper is concerned
location
with the configuration
of
dislocations in the f.c.c. type of superlattice, e.g. Au&,. The original analysis of this type was treated theoretically superlattice, an ordered
by Brown
and Herman(l)
which consists
joined by an antiphase superlattice
dislocation
of two ordinary
dislocations
domain boundary. is similar
to
an
Thus, the extended
dislocation, which consists of two partial dislocations joined by a stacking fault. The superlattice disloca-
for the b.c.c.
e.g. CuZn. The perfect dislocation in lattice is the so-called superlattice dis-
tion in a f.c.c. lattice is complicated by the fact that it consists of a pair of extended dislocations connected
* Certain portions of this paper are based on a thesis by M. J. Marcinkowski presented to the Faculty of the School of Metallurgical Engineering, University of Pennsylvania. in partial fulfillment of the Degree of Doctor of Philosophy. Received April 4, 1960. t Edgar C. Bain Laboratory for Fundamental Research, U.S. Steel Corporation, Monroeville, Pennsylvania. $ Department of Metallurgical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania.
configuration depends on the elastic interaction among four partial dislocations, the energy of the stacking
by an antiphase
ACTA
METALLURGICA,
VOL.
9, FEBRUARY
1961
129
boundary,
and thus, its equilibrium
fault and the energy of the antiphase domain boundary. In treating the general case, the orientation of the dislocation line with respect to its Burgers vector is another important variable.
ACTA
130
We first make dislocation type
a theoretical
arrangements
alloys.
investigation
in the AuCu,
The results obtained
are compared
METALLURGICA,
and
from
with direct observakions
VOL.
of the
9,
(ill)
1961 SLIP PLANE
LIES
IN
PLANE
OF DRAWING
AuCu
the former
obtained
from
thin films of AuCu, using the method of transmission electron microscopy. basic not only for
The present investigation is the interpretation of plastic
behavior in ordered structures: the dislocation
contribution
but also in calculating
to residual resistivity.
The variation of the antiphase boundary (APB)
ewergy
with order (8) The calculation antiphase
begins
boundary
of long-range order, S. observations of the f.c.c. f.c.c.
lattice
cubic
consists
sublattices
with the variation
(APB)
energy
First, a few preliminary lattice are necessary. The
of four interpenetrating
corresponding
atoms per unit cell.
of the
with the degree
to each of the four
For the purpose
the rhombus
which connects
(Fig. 1). Thus I,,
four atomic
11A4,III,
and IV,
are lattice sites
which will be taken on the stationary II,,
III,
and IV,
sites from
and which lie in the same plane
are the lattice
plane, and I,, sites which
are
initially occpuied by atoms in the movable plane above A prior to slip. I,, II,, III, and IV, are not only the sites occupied by atoms in a plane twice removed from the A-plane in the f.c.c. lattice, but I,, II,, III, and IV, also represent the alternate atomic sites for atoms when the B-plane fault
by slipping
the immediate
has created
by an amount
problem
of disorder associated
a stacking
1/6a(ll2).
is to determine
Thus,
the amount
with a stacking fault.
Referring to Fig. 1, we will consider a [Oli] dislocation which consists of four partial dislocations in the following order: 1/6a![112], 1/6a[121], l/Sa[ll2] and 1/6a[121]. Only changes in energy produced by changes in nearest neighbors will be considered. When plane B is displaced by 1/6a[112],
the following
changes in nearest neighbors rhombus take place:
to the basic
~ _
Site
:f
I& IV,
~~~~~-~
TABLE 1
_
__-
relative
Change in nearest neighbors ___ (I, - 13) - (I, - 11,) (II, - II,) - (11, - IB) (III, - III,) - (III, - IV,) (IV, - IV,) - (IV, - III,)
+ 1 + + 1
+
The above table means, for example, that an atom at a stationary IA site gains a nearest neighbor from a I, site since I, + II, and loses a II, neighbor since II, + I,. Table 1 can be simplified by observing
JR
IC z_
UNIT RHOMBUS I&, THAT LIE
of accounting
for changes in nearest neighbors, it is useful to consider each of the sublattices
I I
simple
IS.~E,m0.Irc,.Ic,nC~m,,~, JUST ABOVE ----
CONNECTING SITES OF TYPE IA. ItA. IIIA AND IN THE PLANE OF THE DRAWING (PLANE A) :ATOM THE
BONDS CONNECTING SLIP PLANE
PLANE
OF THE
NEAREST
SITES
LYING
IN PLANE
E
DRAWING
NEIGHBOR
SITES
ACROSS
THE
FIG. 1. Variation in bond types r_wxoss slip plane, with respect to & unit rhombus due to slip displacements of the type 1/6a(112).
that physically
there are only two types of sites in the
AuCu, type of superlattice,
wrong sites and right sites.
Three of the four sublattices
are physically
equivalent
so that we can designate a I, site as a I site, and IIB, III, and IV,, sites as a II site. Thus, substituting these designations in Table 1, the net change in nearest (I -
neighbors
I) + (II -
The following produced
for
a 1/6a[ll2]
II) -- 2[1 -
displacement
is
II].
table gives the change
in bonding
by each of the four partial dislocations: TABLE 2
Displacement
n[Oll
1. 1/6~~[112] -3. 1/6n[112] 2. 1/6n[l21] l/B@1211 4.
As expected, dislocation
Net change in bonding
1
(I -
I) + (II - II) -
2(1 -
II)
1 -81 - I) - (II ~ II) + 2(1 - II) the superlattice
which exactly
dislocation
reproduces
is a perfect
the lattice and
consequently produces no disorder. Fig. 2 shows a schematic of the superlattice dislocation, a[Oli]. The disorder associated with the APB is uniform between partial dislocations numbers 1 and 4 because partials 2 and 3 produce zero net change in bonding as shown in Table 2. It turns out that Table 2 is invariant with respect to the choice
of (011)
type
of dislocation
and the
MARCINKOWSKI, NUMBER
BROWN
AND
FISHER:
OF DISLOCATION
DISLOCATIONS
131
IN SUPERLATTICES
Similarly ANBB = S2 and ANll,
= --2S2.
(4)
The change in bond energy per unit rhombus is AE oR = S2( V&
-t V,,
-
2V,,)
(5)
where, for example,
V,, is the bond energy for a (Au-Au) bond. In order to get E,,, the energy per unit area of an APB, it is necessary to divide equation (5) by four times the area of the unit rhombus.
where a is the lattice Fro. Y.
sequence does
not
a[,Oli] superlattice dislocation in A&u,.
of its partial dislocations. hold
for
the
AuCu
This invariance
type
of
superlattice
Using the quasi-chemical
Yang(s)
which
the energy of the APB,
change in the number of Cu-Cu, Au-Cu and Au-Au The degree of long-range
order, R, in the AuCu, superlattice is defined as follows:
E OR- -
=
L-1 =
=
L-1=-
a l/2 a (Ill) same form(l)
when
5’) N
=
(1)
[-I--3(1 -
I
8) N
16
Average number of Cu atoms on 11 sites B =11[
l-
where the total number of atoms = N.
displacement
aN
for a l/Ba[llZ]
the quasi-chemical
Referring calculate
Substituting
to Fig.
approximation
S/l6 N2
is the width
of the 1/2a[Oli]
makes an arbitrary
16
When A! = 1,
= S2.
objective
is to
dislocation
and r the
angle, 8, with its Burgers vector. by the interaction
between pairs of dislocations,
3(3 + 8) N
stacking fault energy. The elastic interaction dislocations is given by
by E,, energy
energies
and by E,,
the
between
parallel
v is Poisson’s
ratio, R
E=2$~~[ln$-ilforedges
displacement E=g[ln:-l]forscrews
is a constant on the order of the crystal dimensions and r is the separation. The following table of
3116 N2
in terms of S, AN,,
of Lif3) is
spacings rr and r, where r,
where G is the shear modulus,
N2
by
on a -1 lo- plane has the
2, the present
the equilibrium
K12 + Kl' _,KlM* (2) AA =~___ l/l6
(8)
width of the superlattice dislocation. The caloulation is made for the general case where the dislocation
Thus, the total change in the number of Au-Au
bonds per unit rhombus is
92
--.--. a2
rl and r are determined
all the A atoms are on I sites, and when S = 0, all the atoms are randomly distributed between I and II sites.
1.41 kT,
(7)
The equilibrium cov$guration of the dislocation in AuCu,
16
Average number of Cu atoms on I sites B
= 2.44 kT,
used.
3(1 -
II
of
a2
Average number of Au atoms on II sites A
of
that
0.83 kT, S2 E OR=-----
1 + 3s --N. 16
I
with
The energy of an APB of a b.c.c. lattice produced
Average number of Au atoms on I sites A
approximation
agreement
2V,,)
( VA‘1 t- VBB -
the
bonds need to be calculated.
is in good
of the basic f.c.c.
Peierls
because it does not have the same degree of symmetry as the AuCu, type. In order to calculate
parameter
lattice.
(3)
interaction dislocations
energies is set up for the four partial with a separate column for the respective
ACTA
132
edge and screw interactions t,he interaction number
METALLURGICA,
and omitting
the part of
that does not depend on r or rl.
designating
each partial
The
is shown in Fig. 2 Screw interaction
in “*yt;Ifizz;
.
I co9 0 cm
l-2 1-3 1-4 2-3 2-4 3-4
(0 co92 e In (r co.! 0 cos (8 co* 0 cm (0
J
i
inuniteof[-_]
n/3) In rl VI) n/3) In r n/3) In (r - 29.1) cd (0 + */3 In (r - ?.I) COBB co9 (B + Tr/3)In 9.1
+ + +
where 0 is also defined. The total part of the energy which depends on rr and r is given by
e cos (e +
E total = 2 cos
[
+ +
sin
(
e
sin
1
n/3) __ 1-V
(e t
n/3)
VOL. I
so -
+
sin2
(e +
e
sin
(e +
47rEF rr -___-___. Gb2
Y
7~/3) In (rI
-
70
-
60
-
50
-
40
-
30
-
20
-
-10
-
-20
-
-30
-
-60
APPLY WHERE
I/
1
I
”
’ 200
’ 220
’ 240
TO 6 = 0
’ 40
’ 60
’ 60
’ 100
’ 120
’ 140
’ 160
(IN
Sri)]
(9) may be written in the form
E total = .w)
In rl + f2m
ln (r -- rA
has been assumed
=A(@
ln 5 +- rf2(e) + 2fs(e)] In (T -- ri)
-__ (2EF + E&277 Gb2
is the
only logical one that can be made at this time. from the equations
2rJl
(2EF + EofiP7 rl E,R 24r - 5) 662 __~ - ~ Gb2
Since no quantum mechanical between E, and E,,. calculations have been made for stacking fault energy assumption
’ 260
FIQ. 3. Graphical analysis of the equilibrium arrengement of dislocations in an A&u, type alloy over a wide range of Ep and EOR.
(9)
Gb2
’ 160
CM1
+ f3u3 In r + In (r -
r
or the ordering energy, the above
’ 20
Equation
1
It is to be noted that no interaction
r, and r are obtained
II
II
r X 106
ri)
57/3) [ln r + In (r -
27rE,,
I
ln rl
cos e c~s (e + n/3) I+-;
+ sin
I
RELATIONSHIPS DISLOCATION
60
0
e+
1961
1
[toss e + coss (e + n/3)] i_I’
+ sin2
9,
+
f,(e) ln
r-r1
Since -L
(r -
E,,
rl
hr(r -
rJ
Gb2
r12
rl)2 -
(r - rd2
(12)
~ 1 1
I.
< 1 (Fig. 3), thenln
and the last term in equation (12) may be neglected. Thus, r1 and r may be obtained from
The
above
equations
cannot
be solved
exactly.
However, a graphical solution was obtained for 0 = 0 as shown in Fig. 3 for a wide range of E, and E,,. Noting that the ratio of rJr is small compared to 1, an analytical solution of equations (10) and (11) may be obtained which is exceedingly accurate for all values of 0 and for a wide range of likely values of the physical parameters.
Gb2 ___-__2 rcos e r1 = 277(2EE, + EoIz) i X in,
-I- sin
cos (e + e
sin
‘IT/3)]
(8 +
n/3)]
1
(15)
MARCINKOWSKI.
(T - TX)= z-g-
numerical
(16) show excellent cal solution
corresponding action
sin (0 + r/3)12
calculations agreement
represented
of equation
. (16)
(15) and
with the exact graphi-
by Fig. 3.
Furthermore,
as
8 decreases, r increases more rapidly than r,, so that the degree of approximation increases. The accuracy of the approximate
solutions
the distance between 3(r - 2r,) is essentially
DISLOCATIONS
IN SUPERLATTICES
means physically
that
partial dislocations 2 and independent of the fact that
energy
between
presented,
in
edge
larger elastic intercompared
to
screw
dislocations, which in turn enables r to vary between 78 and 124 A when S = 1. The variation in rl with 0 is somewhat
smaller
although
appreciable.
Fig.
4
shows further tha>t the effect of order is to decrease ri by a fa’ctor of about 4 from its value in the disordered alloy.
The variation of r with 5’ is not shown. Howvaries as X2, r increases very ever, because E,, rapidly with decreasing S. The AuCu supcrlctttice The calculation
the &z[OlI] dislocations are extended. A few numerical results will be
133
to the pure edge and screw dislocations.
This results from the somewhat
1
kv + [sin 0 +
x
FISHER:
AND
[COBI9+ cos (0 + 77/3)12 OR
Detailed
JJBOWN
introduces
of E,,
for the AuCu supertattice
some new features
because
AuCu is less
particu1a.r for the case of AuCu,, to show the size of the various quantities which have been calculated.
symmetrical than AuCu,. In the AuCu lattice the I_* and III, sites are physically equivalent and may be
For this alloy E,,
referred to as I sites;
K&ter(3),
was assumed lo-l6
s
75 X2 while from the data of
G = 4.7 x lo11 dyn/cm2 and v = 0.33.
cm2.
to be 40 ergs/cm2
Consider
E,
while b2 = 2.34 x
first the variation
of r and r1
also physically
whereas the II,
equivalent
If a table similar to Table 2 is constructed,
and
minima
at
8 = -30”
and
60”,
respectively,
Displacement a(1101
Xet change in bonding
~..______
1. I/B@111
~
0
3. lp@ll] 4. 1/6rc[121] -.....
j /
0
2. 1/6a[121]
the
2((1 -
Table
5 100
-
following :
superdislocation
gives
a [Oli]
______-______
x
85 -
I. 2. 3. 4.
PURE EDGE DlSLOCPTlON T
1/6a./llT2j 1/6a[Bt] 1/6al.112] 1/6a[l21]
~ I -2[(I
’
’
1
-40 -30 -20 -10
’
0
’
’
’
’
10 20 JO 40 B (IN DEGREES)
’
SO
’
60
’
70
the same results
’
-.-
as the
.~. ~ _~ --___1
Ij + (II -- II) --. 2(1 - II)]
OF
I) + (11 -II] - 2(1 -II)] I) + (II - II) -- 2(1 - II)] DISLOCATION
4
FIG. 4. Variation of spacing between dislocations in A&u, with orientation of Burgers vector.
gives
as
4
f
80
II)]
symmetrical
s 0
2(1 -
- I) + (II - II) -. 2(1 - II)]
2[(1 -2[(I -
NUMBER
PURE SCREW DWCCATION
2(1 - II)]
Net change in bonding 21 (I -
I
II) -
displacement
-
Displacement a[Oli]
go-
75-
However,
is uot
TABLE
95-
80 -
3.
I) + (II -
--2[(I - I) + (II - II) -
shown in Fig. 5. A a[Oli] displacement -
L
j
-__
s 105 0
-2
it, is found
3
TABLE
Thus, ilO-
are
that
with 0. This is shown in Fig. 4 for several important cases. It will be noted that the separations are periodic with period rr, and that they possess maxima
and IV,
and are now called II sites.
90
Fro. 5. tc[ilOJ dislocation ik Au&.
ACTA
134 NUMBER
VOL.
9,
1961
OF DISLOCATION
1
T
i
METALLURGICA,
L-rr,4
i
I+--ri-4
FIG. 6. a[01 l] dish&ion in A&u. No coupling between 1!2a[OlT] dislocations.
The configuration
of the dislocation
The quasi-chemical
is shown in Fig. 6.
approximation
of Li(j) for an A3
f.c.0. lattice is 7 (k/U
i- V BB -
ZV,,)
FIG. 7. Dislocation arrang~~n~Atsindisordered AL&U,.
= 2.74 kT,.
gold.
The energy of the APB is
cooled
3.16 kT, 6 E OR = -_______--. a2 The calculation
for the equilibrium
l/Za[Olf]
dislocations
configuration
dislocations
of the type a[Oli]
of
dislocations extended
would
of the
the
width
in the disordered
state.
for the equilibrium
nonsymmetrical
more difficult
a[illO]
than for AuCu,.
by
The width of these
be less than
dislocations
The calculation
and the
consist of partials connected
both a stacking fault and APB.
of the
configuration
dislocation A graphical
will require a three surface analysis.
is much solution
An analytical
solution is not apparent because it requires the simultaneous solution of three equations, and one cannot
be assured that a simplifying
from
of the specimens
500°C
(T,
degree of disorder.
the a[Oli] displacement is trivial because there is no APB between dislocations 2 and 3. Thus, there are no superlattice
A portion
approximation
was then rapidly
= 390°C)
to insure a high
The remaining
specimens
were
then slowly cooled from 400 down to 180°C at a rate of 2.3’C/hr. Subsequent examination of t.he slowly cooled alloys by X-rays since the antiphase
showed S to be about 1, and
domain
size seas about
500 A,
nearly perfect long-range order exists. The specimens were then electropolished into thin sections with a chromic-acetic acid electrolyte using a procedure described by Fisher and Szirmae(a) which is a modificat.ion of that first proposed by Bollmann( These specimens were next examined by transmission electron microscopy’@ operating
using a Siemens Model 1 Elmiskop
with a double condenser at 100 kV.
Figure 7 shows an array of dislocations those observed
in the disordered
alloys.
typical
of
All of them
appear as single dislocations, however, they are probably dissociated into partials separated by a
will be discovered. The detailed calculation for the non-symmetrical case will be part of another paper,
distance rr. This distance, however, even in the most
however it appears qualitatively
38 A, or just below the resolution
that rr will be much
less than r2 and r. Direct observation
qf dislocatioxs
in both diswdered and
ordered AuCu, Copper of 99.9 per cent pnrity and an equal weight of 99.98 per cent pure gold were melted under vacuum in a zirconia crucible. The resulting ingot was rolled into &rips about 0.0015 in. thick and annealed for 2 hr at 900°C. The mean grain size resulting from this treatment was 0.1 mm and subsequent chemical analysis showed no detectable loss of either copper or gold, i.e. the final specimen contained 24.8 at.76
favorable
case (0 = 30”, S = 0, Fig. 4) is only about obtained
with the
present techniques. It will be further noted Fig. 7 represents a region within the specimen, has appa,rently undergone plastic deformation. of the dislocations
a considerable
The irregular such as those
that that
amount
of
shapes of many at point A are
probably explained by elastic interactions between dislocations on closely spaced slip planes. Jogs such as those at U are probably produced by combination with a dislocation in a different slip system as discussed by Whelan(g). Fig. 8 shows a slip band containing a large number of dislocations in a fully ordered specimen.
The norms1 to the surface is ]134]
MAHCINKOWKKI,
BROWX
AND
FISHER:
DISLOCATIONS
IN
SCPERLATTICES
135
TRACE OF SLIP PLANE
(UPPER SIDE OF FOIL)
PROJECTION OF DISLOCATION PAIR IN SLIP PLANE \\
t
FIG. 8.
Pile-up
of
superlatt,ice dislocations ordered AuCu,.
r, is too small to be resolved, and as Fig. 4 shows, r1 in the ordered alloy is expected to be nearly half that the
disordered
alloy.
between the dislocation
The
observecl
pairs is essentially
distance a measure
of r. A number of important features are to be not.ed in Fig. 8. First, the dislocations are quite irregular in shape,
while the This may
uniform. to their motion planes.
spacing be the
between them is not result of interference
by other dislocations
A few such dislocations
on different slip
which lie above
TRACE
DISLOCATION
bF
SLlP
PLANE
(UNDER
SIDE OF FOIL,
PAIR
FIG. 9. Geometrical relations needed to find true perpendicular distance y between coupled dislocation pairs in thin foils.
in fully
while the slip plane is (111). It can be seen that the dislocations are paired in agreement with the theory.
in
\
/
I/
COUPLED
+a+
the
r to compare with theory, the geometrical tion becomes quite important.
interpreta-
The three dimensional
analysis of the dislocation pairs giving rise to what is observed in Fig. 8 is shown in Fig. 9. From the angles and lengths
shown
in this figure,
given by the following
T is found
to be
relationship
r = a cos [tan-l (tan p cos y)]. Both a and /3 can be measured from the figure, while y can be found by a selected area diffraction
of this
same area. From Fig. 8, r is found to lie in most cases
main plane of slip are indicated
at point A in Fig. 8.
between
Most likely all the dislocations
were produced
since both y and 6 shown in Fig. 9 a.re known,
the ordering
transformation,
either
after
by handling
or
thickness
the values
of 90 and 135 8.
of the film can be measured,
In addition, the
and in its
by heating of the electron beam. Some of the irregularity in the spacing of the paired dislocations
present case is found to be 1000 8. Figure 10 shows a second array of coupled disloca-
may
tions in the ordered alloy lying in parallel slip planes.
be associated
interesting
feature
w-ith variations
in 8.
Another
of Fig. 8 is that the dislocations
In this case, the dislocations
appear to be piled up against some barrier to the right.
in appearance,
The stress field from the dislocations
little or no interference
at the heat of
the pile-up is sufficient to bring these dislocations such
close
identity
proximity
with
as pairs is virtually
that the antiphase boundaries pairs do not show contrast
one
another
eliminated.
that
indicating
are much more uniform
that they have moved with from neighboring
dislocations
to
their
The reasons
between the dislocation is not yet completely
understood. However, it is not expected that conditions for disloca,tion resolution will in general be the same as those for antiphase boundary resolution. The next
objective
is to compare
the measured
values of r with the theoretical results. At this point, however, it must be remembered that Pig. 8 represents a projection of a three-dimensional dislocation network onto a plane surface, and what appear to be perpendicular distances between dislocation pairs are not the true perpendicular distances. Because of the desirability of obtaining an accurate measurement of
FIG. 10. Superlattice dislocations of nearly equilibrium spacing in fully ordered Ad&,.
ACTA
136
METALLURGIC-A,
VOL.
FIG.
therefore
should
analysis.
be better
for
comparing
the
The value of r, which is nearly
constant between all of the dislocations, is found to be 130 d and is in good agreement with the calculated results.
Since 8 is not known,
it is not possible
make a more critical comparison theoretical
to
with theory, although
the measured T values agree surprisingly
have on the superlattice
over which the separated above again
is apparently
SUMMARY
have
shown
AND
both
theoretically
highly irregular due to interaction all of the dislocations
are
with one another. remain paired with
spacing.
It is to be concluded
dislocations
which are held togethether
on the elastic
which
high
in fully
and
experi-
ordered
AuCu,
The superdislocations
by an antiphase boundary.
The spacing between the individual partial dislocations
dislocations,
the superlattice repulsive
forces
dislocation between
the stacking fault energy, and the anti-
phase boundary energy. By using isotropic elastic dislocation theory and the quasi-chemical approximation for the ordering energy, the separations
dislocations
the
travel
in pairs.
A
second
interesting feature of Fig. 11 is the band running from left to right, near the center of the figure. Possibly,
the alternate
from an antiphase
light and dark contours
boundary
contrast
which
arise should
depends
the partial
from these results that even after large deformations, in AuCu,
stress
CONCLUSION
are present as superlattice dislocations. dislocation consists of a pair of partial
that constitute
constant
local
dislocations,
that
less random
a surprisingly
due to some
concentration.
We
of dislocations
line at B
superlattice dislocation has been its equilibrium spacing r, which
mentally
one of the foils
in agree-
Fig. 12 is the portion of the long dislocation
treated in this manner. It will be noted that the large complex deformation has resulted in a more or
However,
undissociated
Another interesting feature of
what effect cold work might
was heavily deformed by repeated bending. Fig. 11 shows an array of dislocations observed in a specimen
distribution
are virtually
ment with the theory.
well with the
results shown in Fig. 4.
In order to determine
Interference to the motion of superlattice dislocations in fully ordered AuCu,.
12.
the dislocations
that was subsequently cold worked.
theoretical
1961
near the leading edge. As the pinning points are approached, however, this separation vanishes and
FIG. 11. Super-latticedislocations in fully ordered Au&,
and
9,
four
partial
dislocations
that
between
constitute
the
superlattice dislocation has been calculated for dislocation possessing an arbitrary Burgers vector. An alloy
of composition
bulk, has been electropolished
AuCu,,
fully
ordered
to films about
a in
1000 A
be similar to the contrast observed for stacking faults.(lO) The contrast appears most pronounced in
thick. Examining these by transmission microscopy, we have observed superlattice
the vicinity
tions for the first time. The separation between the two pairs of partial dislocations was measured to be about 130 A and is in good agreement with the
of the dark extinction
contours.
Finally, Fig. 12 shows some very interest’ing features of moving superlattice dislocations. A segment of the large dislocation line at A has been pinned at two points. Nevertheless, this segment is able to expand outward between the pinning points as a superdislocation with a well defined separation
electron disloca-
spacing calculated theoretically. The energy of the antiphase boundary discussed above has been calculated for one which has been produced
by slip and whose structure is not changed
MARCINKOWSKI,
by a subsequent
BROWN
diffusion of atoms.
FISHER:
AND
The structure
DISLOCATIONS
of
an antiphase boundary which exists during thermal equilibrium is different from a boundary produced only by slip in that the equilibrium ary has a thickness varies continuously.
antiphase bound-
over which the degree of order The calculation for the variation
in thickness of an antiphase
boundary
ture has been made by Brown’li)
with tempera-
for CuZn.
the free energy of the equilibrium
In general
antiphase boundary
will be less than that of the slip produced boundary
antiphase
and the difference will be greater the closer
the temperature Consideration
small resistance in
Ardley(12).
configura-
increases
in AuCu type alloys, the superlattice
disloca-
tion may consist of an asymmetric array of four bound partial dislocations or a pair of partial dislocations similar to that in the disordered alloy. The difference AuCu,
in the dislocation
arrangements
The configuration
of the dislocations
could
have
an important bearing on the plastic behavior of the crystal. For example, since there are two distinct types
of superlattice
dislocations
in AuCu and only
one type in AuCu,, it is expected that ordered AuCu would plastically deform in a more anisotropic fashion than ordered AuCu,. No experiments been made to test this conclusion. From the result,s described little doubt
that dislocations
here, there seems to be in fully ordered AU&I,
travel through the lattice as superlattice When
the
antiphase
500 A) these dislocations
have
boundaries
are
dislocations. large
(about
should move with negligibly
however,
that
rapidly.
of the
Thus, at low degrees of order, it independently
as moving
In this case, it would
follow
that
This would
type
of one another. these imperfect
must do work against the ordering forces,
since they will leave behind an antiphase mean
degrees higher
that the alloys of
yield
order
should
points,
than
boundary.
possessing be
inter-
stronger,
either
i.e.
the fully
ordered or the completely disordered alloys. These very interesting speculations are being presently investigated
in both AuCu, and Ni,Mn. ACKNOWLEDGMENT
We are indebted carrying
in the latter.
be recalled,
This is
results
to think of the $a(llO)
between
and AuCu is a result of the smaller degree of
symmetry
forces.
experimental
dislocations
possess
location
It will
the
may be more correct
ing on the Burgers
dis-
with
decreases this energy decreases very rapidly and the separation between the component dislocations also
mediate
of the superlattice
137
energy of the antiphase boundary bounded by the superlattice dislocation varies as S2 so that as S
tions in the AuCu type ordered alloys leads to two additional types of dislocation arrangements. Dependvector
SUPERLATTICES
due to the ordering
agreement
dislocations
is to To. of the possible dislocation
IN
to A. Szirmae for his assistance in
out the experimental
portion
of this work
and to D. S. Miller for many helpful discussions concerning the interpretation of the results. REFERENCES 1. 2. 3. 4. 5.
N.BRowN~~~M.HERMAN,TT~~~.A~~~. Imt.
Engrs. 206, 1353 (1956). C. N. YANG, J. Chevn. Phys. 13,66 (1945). Y. Y. LI, Phys. Rev. 76, 972 (1949). W. KBSTER, 2. Met&k. 32,145 (1940). Y. Y. LI, J. Chem. Phys. 17, 447 (1949).
Min. (Metoll.)
6. R. M. FISHER and A. SZIRMAE,Syllzposium on Electron Metallography, ASTM STP 262, 103 (1959). 7. W. BOLLMANN, Phys. Rev. 103,1588 (1956). 8. P. B. HIRSCH,MetaZZurg. Rev. 4, 101 (1959). 9. M. J. WHELAN, PTOC. Roy. Sot. 249, 114 (1959).
10. M. J. WHELAN, J. Inst. Met. 87, 392 (195!3). 11. N. BROW, Phil. Msg. 4, 693 (1959). 12. G. W. ARDLEY, Acta Met. 3, 525 (1955).
CONFIGURATIONS
IN A&u,
M. J. MARCINKOWSKIt,
AND
N. BROWNS
AuCu
and
TYPE
SUPERLATTICES*
R. M. FISHER?
Superlattice dislocations have been observed for the first time by transmission electron microscopy These consist of two pairs of partial dislocations held techniques carried out with thin films of AuCu,. The separation between the pairs is measured to be about together by an antiphase domain boundary. 130 A. A theoretical calculation has also been made of the expected spacing which involves the stacking fault energy, antiphase domain boundary energy and the elastic interaction energies between the The theoretical results are in good individual dislocations that constitute the superlattice dislocation. agreement with those observed experimentally. Further theoretical considerations of the AuCu type structure lead to two additional dislocation configurations characterized by the direction of their Burgers vectors with respect to the ordered lattice. This difference in the dislocation configurations between AuCu, and AuCu arises because the latter is a less symmetrical lattice. CONFIGURATION
DE
DISLOCATION DANS DES AuCu, et AuCu
SURSTRUCTURES
DU
TYP.E
Les auteurs observent pour la premiere fois par des techniques de transmission d’electrons sur des Ces super-dislocations sent constitue es de deux pellicules minces de AuCu, des super-dislocations. paires de dislocations partielles reunies par une frontier-e de domaine antiphase. La distance entre les paires mesure environ 13OA. Les auteurs calculent Bgalement theoriquement la distance. Dans ce calcul interviennent l’energie des fautes d’empilement, l’energie des frontieres des domaines antiphase et les energies d’interaction Blastique Les resultats theoriques de calcul entre les dislocations individuelles qui constituent la super-dislocation. sont en bon accord avec les valeurs trouvees experimentalement. Des considerations theoriques supplementaires sur la structure du type AuCu conduisent B deux configurations de dislocations additionnelles caracterisees par la direction de leurs vecteurs de Biirgers par rapport au reseau ordonne. Cette difference dans les configurations de dislocations entre AuCu, et AuCu provient de ce que ce dernier a un reseau moms symetrique. VERSETZUNGSANORDNUNGEN
IN
tfBERSTRUKTUREN
VOM
TYP
.hcu,
UND
AU&
Zum ersten Ma1 wurden mit elektronenmikroskopischen Durchstrahlungsmethoden in diinnen Filmen aus AuCu, ifberversetzungen beobachtet. Diese bestehen aus zwei Paaren van Halbversetzungen, die durch eine Antiphasen-Bereichsgrenze zusammengehalten wurden. Der Abstand zwischen den beiden Paaren wurde zu etwa 130 A gemessen. Eine theoretische Rechnung iiber den zu erwartenden Abstand wurde durchgefiihrt, in welche die Stapelfehlerenergie und die Antiphasen-Grenzflachenenergie eingehen sowie die elastischen Wechsel. Die welche die Uberversetzung bilden. wirkungsenergien zwischen den einzelnen Versetzungen, theoretischen Ergebnisse stimmen mit den experimentellen Beobachtungen gut uberein. Weitere theoretische Betrachtungen der Struktur vom Typ AuCu fiihren zu zwei zusatzlichen Versetzungsanordnungen, die durch die Richtung ihrer Burgersvektoren beziiglich dem geordneten Gitter charakterisiert sind. Dieser Unterschied in den Versetzungsanordnungen zwischen AuCu, und AuCu tritt auf, weil das letztere Gitter weniger symmetrisch ist.
INTRODUCTION
This paper is concerned
location
with the configuration
of
dislocations in the f.c.c. type of superlattice, e.g. Au&,. The original analysis of this type was treated theoretically superlattice, an ordered
by Brown
and Herman(l)
which consists
joined by an antiphase superlattice
dislocation
of two ordinary
dislocations
domain boundary. is similar
to
an
Thus, the extended
dislocation, which consists of two partial dislocations joined by a stacking fault. The superlattice disloca-
for the b.c.c.
e.g. CuZn. The perfect dislocation in lattice is the so-called superlattice dis-
tion in a f.c.c. lattice is complicated by the fact that it consists of a pair of extended dislocations connected
* Certain portions of this paper are based on a thesis by M. J. Marcinkowski presented to the Faculty of the School of Metallurgical Engineering, University of Pennsylvania. in partial fulfillment of the Degree of Doctor of Philosophy. Received April 4, 1960. t Edgar C. Bain Laboratory for Fundamental Research, U.S. Steel Corporation, Monroeville, Pennsylvania. $ Department of Metallurgical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania.
configuration depends on the elastic interaction among four partial dislocations, the energy of the stacking
by an antiphase
ACTA
METALLURGICA,
VOL.
9, FEBRUARY
1961
129
boundary,
and thus, its equilibrium
fault and the energy of the antiphase domain boundary. In treating the general case, the orientation of the dislocation line with respect to its Burgers vector is another important variable.
ACTA
130
We first make dislocation type
a theoretical
arrangements
alloys.
investigation
in the AuCu,
The results obtained
are compared
METALLURGICA,
and
from
with direct observakions
VOL.
of the
9,
(ill)
1961 SLIP PLANE
LIES
IN
PLANE
OF DRAWING
AuCu
the former
obtained
from
thin films of AuCu, using the method of transmission electron microscopy. basic not only for
The present investigation is the interpretation of plastic
behavior in ordered structures: the dislocation
contribution
but also in calculating
to residual resistivity.
The variation of the antiphase boundary (APB)
ewergy
with order (8) The calculation antiphase
begins
boundary
of long-range order, S. observations of the f.c.c. f.c.c.
lattice
cubic
consists
sublattices
with the variation
(APB)
energy
First, a few preliminary lattice are necessary. The
of four interpenetrating
corresponding
atoms per unit cell.
of the
with the degree
to each of the four
For the purpose
the rhombus
which connects
(Fig. 1). Thus I,,
four atomic
11A4,III,
and IV,
are lattice sites
which will be taken on the stationary II,,
III,
and IV,
sites from
and which lie in the same plane
are the lattice
plane, and I,, sites which
are
initially occpuied by atoms in the movable plane above A prior to slip. I,, II,, III, and IV, are not only the sites occupied by atoms in a plane twice removed from the A-plane in the f.c.c. lattice, but I,, II,, III, and IV, also represent the alternate atomic sites for atoms when the B-plane fault
by slipping
the immediate
has created
by an amount
problem
of disorder associated
a stacking
1/6a(ll2).
is to determine
Thus,
the amount
with a stacking fault.
Referring to Fig. 1, we will consider a [Oli] dislocation which consists of four partial dislocations in the following order: 1/6a![112], 1/6a[121], l/Sa[ll2] and 1/6a[121]. Only changes in energy produced by changes in nearest neighbors will be considered. When plane B is displaced by 1/6a[112],
the following
changes in nearest neighbors rhombus take place:
to the basic
~ _
Site
:f
I& IV,
~~~~~-~
TABLE 1
_
__-
relative
Change in nearest neighbors ___ (I, - 13) - (I, - 11,) (II, - II,) - (11, - IB) (III, - III,) - (III, - IV,) (IV, - IV,) - (IV, - III,)
+ 1 + + 1
+
The above table means, for example, that an atom at a stationary IA site gains a nearest neighbor from a I, site since I, + II, and loses a II, neighbor since II, + I,. Table 1 can be simplified by observing
JR
IC z_
UNIT RHOMBUS I&, THAT LIE
of accounting
for changes in nearest neighbors, it is useful to consider each of the sublattices
I I
simple
IS.~E,m0.Irc,.Ic,nC~m,,~, JUST ABOVE ----
CONNECTING SITES OF TYPE IA. ItA. IIIA AND IN THE PLANE OF THE DRAWING (PLANE A) :ATOM THE
BONDS CONNECTING SLIP PLANE
PLANE
OF THE
NEAREST
SITES
LYING
IN PLANE
E
DRAWING
NEIGHBOR
SITES
ACROSS
THE
FIG. 1. Variation in bond types r_wxoss slip plane, with respect to & unit rhombus due to slip displacements of the type 1/6a(112).
that physically
there are only two types of sites in the
AuCu, type of superlattice,
wrong sites and right sites.
Three of the four sublattices
are physically
equivalent
so that we can designate a I, site as a I site, and IIB, III, and IV,, sites as a II site. Thus, substituting these designations in Table 1, the net change in nearest (I -
neighbors
I) + (II -
The following produced
for
a 1/6a[ll2]
II) -- 2[1 -
displacement
is
II].
table gives the change
in bonding
by each of the four partial dislocations: TABLE 2
Displacement
n[Oll
1. 1/6~~[112] -3. 1/6n[112] 2. 1/6n[l21] l/B@1211 4.
As expected, dislocation
Net change in bonding
1
(I -
I) + (II - II) -
2(1 -
II)
1 -81 - I) - (II ~ II) + 2(1 - II) the superlattice
which exactly
dislocation
reproduces
is a perfect
the lattice and
consequently produces no disorder. Fig. 2 shows a schematic of the superlattice dislocation, a[Oli]. The disorder associated with the APB is uniform between partial dislocations numbers 1 and 4 because partials 2 and 3 produce zero net change in bonding as shown in Table 2. It turns out that Table 2 is invariant with respect to the choice
of (011)
type
of dislocation
and the
MARCINKOWSKI, NUMBER
BROWN
AND
FISHER:
OF DISLOCATION
DISLOCATIONS
131
IN SUPERLATTICES
Similarly ANBB = S2 and ANll,
= --2S2.
(4)
The change in bond energy per unit rhombus is AE oR = S2( V&
-t V,,
-
2V,,)
(5)
where, for example,
V,, is the bond energy for a (Au-Au) bond. In order to get E,,, the energy per unit area of an APB, it is necessary to divide equation (5) by four times the area of the unit rhombus.
where a is the lattice Fro. Y.
sequence does
not
a[,Oli] superlattice dislocation in A&u,.
of its partial dislocations. hold
for
the
AuCu
This invariance
type
of
superlattice
Using the quasi-chemical
Yang(s)
which
the energy of the APB,
change in the number of Cu-Cu, Au-Cu and Au-Au The degree of long-range
order, R, in the AuCu, superlattice is defined as follows:
E OR- -
=
L-1 =
=
L-1=-
a l/2 a (Ill) same form(l)
when
5’) N
=
(1)
[-I--3(1 -
I
8) N
16
Average number of Cu atoms on 11 sites B =11[
l-
where the total number of atoms = N.
displacement
aN
for a l/Ba[llZ]
the quasi-chemical
Referring calculate
Substituting
to Fig.
approximation
S/l6 N2
is the width
of the 1/2a[Oli]
makes an arbitrary
16
When A! = 1,
= S2.
objective
is to
dislocation
and r the
angle, 8, with its Burgers vector. by the interaction
between pairs of dislocations,
3(3 + 8) N
stacking fault energy. The elastic interaction dislocations is given by
by E,, energy
energies
and by E,,
the
between
parallel
v is Poisson’s
ratio, R
E=2$~~[ln$-ilforedges
displacement E=g[ln:-l]forscrews
is a constant on the order of the crystal dimensions and r is the separation. The following table of
3116 N2
in terms of S, AN,,
of Lif3) is
spacings rr and r, where r,
where G is the shear modulus,
N2
by
on a -1 lo- plane has the
2, the present
the equilibrium
K12 + Kl' _,KlM* (2) AA =~___ l/l6
(8)
width of the superlattice dislocation. The caloulation is made for the general case where the dislocation
Thus, the total change in the number of Au-Au
bonds per unit rhombus is
92
--.--. a2
rl and r are determined
all the A atoms are on I sites, and when S = 0, all the atoms are randomly distributed between I and II sites.
1.41 kT,
(7)
The equilibrium cov$guration of the dislocation in AuCu,
16
Average number of Cu atoms on I sites B
= 2.44 kT,
used.
3(1 -
II
of
a2
Average number of Au atoms on II sites A
of
that
0.83 kT, S2 E OR=-----
1 + 3s --N. 16
I
with
The energy of an APB of a b.c.c. lattice produced
Average number of Au atoms on I sites A
approximation
agreement
2V,,)
( VA‘1 t- VBB -
the
bonds need to be calculated.
is in good
of the basic f.c.c.
Peierls
because it does not have the same degree of symmetry as the AuCu, type. In order to calculate
parameter
lattice.
(3)
interaction dislocations
energies is set up for the four partial with a separate column for the respective
ACTA
132
edge and screw interactions t,he interaction number
METALLURGICA,
and omitting
the part of
that does not depend on r or rl.
designating
each partial
The
is shown in Fig. 2 Screw interaction
in “*yt;Ifizz;
.
I co9 0 cm
l-2 1-3 1-4 2-3 2-4 3-4
(0 co92 e In (r co.! 0 cos (8 co* 0 cm (0
J
i
inuniteof[-_]
n/3) In rl VI) n/3) In r n/3) In (r - 29.1) cd (0 + */3 In (r - ?.I) COBB co9 (B + Tr/3)In 9.1
+ + +
where 0 is also defined. The total part of the energy which depends on rr and r is given by
e cos (e +
E total = 2 cos
[
+ +
sin
(
e
sin
1
n/3) __ 1-V
(e t
n/3)
VOL. I
so -
+
sin2
(e +
e
sin
(e +
47rEF rr -___-___. Gb2
Y
7~/3) In (rI
-
70
-
60
-
50
-
40
-
30
-
20
-
-10
-
-20
-
-30
-
-60
APPLY WHERE
I/
1
I
”
’ 200
’ 220
’ 240
TO 6 = 0
’ 40
’ 60
’ 60
’ 100
’ 120
’ 140
’ 160
(IN
Sri)]
(9) may be written in the form
E total = .w)
In rl + f2m
ln (r -- rA
has been assumed
=A(@
ln 5 +- rf2(e) + 2fs(e)] In (T -- ri)
-__ (2EF + E&277 Gb2
is the
only logical one that can be made at this time. from the equations
2rJl
(2EF + EofiP7 rl E,R 24r - 5) 662 __~ - ~ Gb2
Since no quantum mechanical between E, and E,,. calculations have been made for stacking fault energy assumption
’ 260
FIQ. 3. Graphical analysis of the equilibrium arrengement of dislocations in an A&u, type alloy over a wide range of Ep and EOR.
(9)
Gb2
’ 160
CM1
+ f3u3 In r + In (r -
r
or the ordering energy, the above
’ 20
Equation
1
It is to be noted that no interaction
r, and r are obtained
II
II
r X 106
ri)
57/3) [ln r + In (r -
27rE,,
I
ln rl
cos e c~s (e + n/3) I+-;
+ sin
I
RELATIONSHIPS DISLOCATION
60
0
e+
1961
1
[toss e + coss (e + n/3)] i_I’
+ sin2
9,
+
f,(e) ln
r-r1
Since -L
(r -
E,,
rl
hr(r -
rJ
Gb2
r12
rl)2 -
(r - rd2
(12)
~ 1 1
I.
< 1 (Fig. 3), thenln
and the last term in equation (12) may be neglected. Thus, r1 and r may be obtained from
The
above
equations
cannot
be solved
exactly.
However, a graphical solution was obtained for 0 = 0 as shown in Fig. 3 for a wide range of E, and E,,. Noting that the ratio of rJr is small compared to 1, an analytical solution of equations (10) and (11) may be obtained which is exceedingly accurate for all values of 0 and for a wide range of likely values of the physical parameters.
Gb2 ___-__2 rcos e r1 = 277(2EE, + EoIz) i X in,
-I- sin
cos (e + e
sin
‘IT/3)]
(8 +
n/3)]
1
(15)
MARCINKOWSKI.
(T - TX)= z-g-
numerical
(16) show excellent cal solution
corresponding action
sin (0 + r/3)12
calculations agreement
represented
of equation
. (16)
(15) and
with the exact graphi-
by Fig. 3.
Furthermore,
as
8 decreases, r increases more rapidly than r,, so that the degree of approximation increases. The accuracy of the approximate
solutions
the distance between 3(r - 2r,) is essentially
DISLOCATIONS
IN SUPERLATTICES
means physically
that
partial dislocations 2 and independent of the fact that
energy
between
presented,
in
edge
larger elastic intercompared
to
screw
dislocations, which in turn enables r to vary between 78 and 124 A when S = 1. The variation in rl with 0 is somewhat
smaller
although
appreciable.
Fig.
4
shows further tha>t the effect of order is to decrease ri by a fa’ctor of about 4 from its value in the disordered alloy.
The variation of r with 5’ is not shown. Howvaries as X2, r increases very ever, because E,, rapidly with decreasing S. The AuCu supcrlctttice The calculation
the &z[OlI] dislocations are extended. A few numerical results will be
133
to the pure edge and screw dislocations.
This results from the somewhat
1
kv + [sin 0 +
x
FISHER:
AND
[COBI9+ cos (0 + 77/3)12 OR
Detailed
JJBOWN
introduces
of E,,
for the AuCu supertattice
some new features
because
AuCu is less
particu1a.r for the case of AuCu,, to show the size of the various quantities which have been calculated.
symmetrical than AuCu,. In the AuCu lattice the I_* and III, sites are physically equivalent and may be
For this alloy E,,
referred to as I sites;
K&ter(3),
was assumed lo-l6
s
75 X2 while from the data of
G = 4.7 x lo11 dyn/cm2 and v = 0.33.
cm2.
to be 40 ergs/cm2
Consider
E,
while b2 = 2.34 x
first the variation
of r and r1
also physically
whereas the II,
equivalent
If a table similar to Table 2 is constructed,
and
minima
at
8 = -30”
and
60”,
respectively,
Displacement a(1101
Xet change in bonding
~..______
1. I/B@111
~
0
3. lp@ll] 4. 1/6rc[121] -.....
j /
0
2. 1/6a[121]
the
2((1 -
Table
5 100
-
following :
superdislocation
gives
a [Oli]
______-______
x
85 -
I. 2. 3. 4.
PURE EDGE DlSLOCPTlON T
1/6a./llT2j 1/6a[Bt] 1/6al.112] 1/6a[l21]
~ I -2[(I
’
’
1
-40 -30 -20 -10
’
0
’
’
’
’
10 20 JO 40 B (IN DEGREES)
’
SO
’
60
’
70
the same results
’
-.-
as the
.~. ~ _~ --___1
Ij + (II -- II) --. 2(1 - II)]
OF
I) + (11 -II] - 2(1 -II)] I) + (II - II) -- 2(1 - II)] DISLOCATION
4
FIG. 4. Variation of spacing between dislocations in A&u, with orientation of Burgers vector.
gives
as
4
f
80
II)]
symmetrical
s 0
2(1 -
- I) + (II - II) -. 2(1 - II)]
2[(1 -2[(I -
NUMBER
PURE SCREW DWCCATION
2(1 - II)]
Net change in bonding 21 (I -
I
II) -
displacement
-
Displacement a[Oli]
go-
75-
However,
is uot
TABLE
95-
80 -
3.
I) + (II -
--2[(I - I) + (II - II) -
shown in Fig. 5. A a[Oli] displacement -
L
j
-__
s 105 0
-2
it, is found
3
TABLE
Thus, ilO-
are
that
with 0. This is shown in Fig. 4 for several important cases. It will be noted that the separations are periodic with period rr, and that they possess maxima
and IV,
and are now called II sites.
90
Fro. 5. tc[ilOJ dislocation ik Au&.
ACTA
134 NUMBER
VOL.
9,
1961
OF DISLOCATION
1
T
i
METALLURGICA,
L-rr,4
i
I+--ri-4
FIG. 6. a[01 l] dish&ion in A&u. No coupling between 1!2a[OlT] dislocations.
The configuration
of the dislocation
The quasi-chemical
is shown in Fig. 6.
approximation
of Li(j) for an A3
f.c.0. lattice is 7 (k/U
i- V BB -
ZV,,)
FIG. 7. Dislocation arrang~~n~Atsindisordered AL&U,.
= 2.74 kT,.
gold.
The energy of the APB is
cooled
3.16 kT, 6 E OR = -_______--. a2 The calculation
for the equilibrium
l/Za[Olf]
dislocations
configuration
dislocations
of the type a[Oli]
of
dislocations extended
would
of the
the
width
in the disordered
state.
for the equilibrium
nonsymmetrical
more difficult
a[illO]
than for AuCu,.
by
The width of these
be less than
dislocations
The calculation
and the
consist of partials connected
both a stacking fault and APB.
of the
configuration
dislocation A graphical
will require a three surface analysis.
is much solution
An analytical
solution is not apparent because it requires the simultaneous solution of three equations, and one cannot
be assured that a simplifying
from
of the specimens
500°C
(T,
degree of disorder.
the a[Oli] displacement is trivial because there is no APB between dislocations 2 and 3. Thus, there are no superlattice
A portion
approximation
was then rapidly
= 390°C)
to insure a high
The remaining
specimens
were
then slowly cooled from 400 down to 180°C at a rate of 2.3’C/hr. Subsequent examination of t.he slowly cooled alloys by X-rays since the antiphase
showed S to be about 1, and
domain
size seas about
500 A,
nearly perfect long-range order exists. The specimens were then electropolished into thin sections with a chromic-acetic acid electrolyte using a procedure described by Fisher and Szirmae(a) which is a modificat.ion of that first proposed by Bollmann( These specimens were next examined by transmission electron microscopy’@ operating
using a Siemens Model 1 Elmiskop
with a double condenser at 100 kV.
Figure 7 shows an array of dislocations those observed
in the disordered
alloys.
typical
of
All of them
appear as single dislocations, however, they are probably dissociated into partials separated by a
will be discovered. The detailed calculation for the non-symmetrical case will be part of another paper,
distance rr. This distance, however, even in the most
however it appears qualitatively
38 A, or just below the resolution
that rr will be much
less than r2 and r. Direct observation
qf dislocatioxs
in both diswdered and
ordered AuCu, Copper of 99.9 per cent pnrity and an equal weight of 99.98 per cent pure gold were melted under vacuum in a zirconia crucible. The resulting ingot was rolled into &rips about 0.0015 in. thick and annealed for 2 hr at 900°C. The mean grain size resulting from this treatment was 0.1 mm and subsequent chemical analysis showed no detectable loss of either copper or gold, i.e. the final specimen contained 24.8 at.76
favorable
case (0 = 30”, S = 0, Fig. 4) is only about obtained
with the
present techniques. It will be further noted Fig. 7 represents a region within the specimen, has appa,rently undergone plastic deformation. of the dislocations
a considerable
The irregular such as those
that that
amount
of
shapes of many at point A are
probably explained by elastic interactions between dislocations on closely spaced slip planes. Jogs such as those at U are probably produced by combination with a dislocation in a different slip system as discussed by Whelan(g). Fig. 8 shows a slip band containing a large number of dislocations in a fully ordered specimen.
The norms1 to the surface is ]134]
MAHCINKOWKKI,
BROWX
AND
FISHER:
DISLOCATIONS
IN
SCPERLATTICES
135
TRACE OF SLIP PLANE
(UPPER SIDE OF FOIL)
PROJECTION OF DISLOCATION PAIR IN SLIP PLANE \\
t
FIG. 8.
Pile-up
of
superlatt,ice dislocations ordered AuCu,.
r, is too small to be resolved, and as Fig. 4 shows, r1 in the ordered alloy is expected to be nearly half that the
disordered
alloy.
between the dislocation
The
observecl
pairs is essentially
distance a measure
of r. A number of important features are to be not.ed in Fig. 8. First, the dislocations are quite irregular in shape,
while the This may
uniform. to their motion planes.
spacing be the
between them is not result of interference
by other dislocations
A few such dislocations
on different slip
which lie above
TRACE
DISLOCATION
bF
SLlP
PLANE
(UNDER
SIDE OF FOIL,
PAIR
FIG. 9. Geometrical relations needed to find true perpendicular distance y between coupled dislocation pairs in thin foils.
in fully
while the slip plane is (111). It can be seen that the dislocations are paired in agreement with the theory.
in
\
/
I/
COUPLED
+a+
the
r to compare with theory, the geometrical tion becomes quite important.
interpreta-
The three dimensional
analysis of the dislocation pairs giving rise to what is observed in Fig. 8 is shown in Fig. 9. From the angles and lengths
shown
in this figure,
given by the following
T is found
to be
relationship
r = a cos [tan-l (tan p cos y)]. Both a and /3 can be measured from the figure, while y can be found by a selected area diffraction
of this
same area. From Fig. 8, r is found to lie in most cases
main plane of slip are indicated
at point A in Fig. 8.
between
Most likely all the dislocations
were produced
since both y and 6 shown in Fig. 9 a.re known,
the ordering
transformation,
either
after
by handling
or
thickness
the values
of 90 and 135 8.
of the film can be measured,
In addition, the
and in its
by heating of the electron beam. Some of the irregularity in the spacing of the paired dislocations
present case is found to be 1000 8. Figure 10 shows a second array of coupled disloca-
may
tions in the ordered alloy lying in parallel slip planes.
be associated
interesting
feature
w-ith variations
in 8.
Another
of Fig. 8 is that the dislocations
In this case, the dislocations
appear to be piled up against some barrier to the right.
in appearance,
The stress field from the dislocations
little or no interference
at the heat of
the pile-up is sufficient to bring these dislocations such
close
identity
proximity
with
as pairs is virtually
that the antiphase boundaries pairs do not show contrast
one
another
eliminated.
that
indicating
are much more uniform
that they have moved with from neighboring
dislocations
to
their
The reasons
between the dislocation is not yet completely
understood. However, it is not expected that conditions for disloca,tion resolution will in general be the same as those for antiphase boundary resolution. The next
objective
is to compare
the measured
values of r with the theoretical results. At this point, however, it must be remembered that Pig. 8 represents a projection of a three-dimensional dislocation network onto a plane surface, and what appear to be perpendicular distances between dislocation pairs are not the true perpendicular distances. Because of the desirability of obtaining an accurate measurement of
FIG. 10. Superlattice dislocations of nearly equilibrium spacing in fully ordered Ad&,.
ACTA
136
METALLURGIC-A,
VOL.
FIG.
therefore
should
analysis.
be better
for
comparing
the
The value of r, which is nearly
constant between all of the dislocations, is found to be 130 d and is in good agreement with the calculated results.
Since 8 is not known,
it is not possible
make a more critical comparison theoretical
to
with theory, although
the measured T values agree surprisingly
have on the superlattice
over which the separated above again
is apparently
SUMMARY
have
shown
AND
both
theoretically
highly irregular due to interaction all of the dislocations
are
with one another. remain paired with
spacing.
It is to be concluded
dislocations
which are held togethether
on the elastic
which
high
in fully
and
experi-
ordered
AuCu,
The superdislocations
by an antiphase boundary.
The spacing between the individual partial dislocations
dislocations,
the superlattice repulsive
forces
dislocation between
the stacking fault energy, and the anti-
phase boundary energy. By using isotropic elastic dislocation theory and the quasi-chemical approximation for the ordering energy, the separations
dislocations
the
travel
in pairs.
A
second
interesting feature of Fig. 11 is the band running from left to right, near the center of the figure. Possibly,
the alternate
from an antiphase
light and dark contours
boundary
contrast
which
arise should
depends
the partial
from these results that even after large deformations, in AuCu,
stress
CONCLUSION
are present as superlattice dislocations. dislocation consists of a pair of partial
that constitute
constant
local
dislocations,
that
less random
a surprisingly
due to some
concentration.
We
of dislocations
line at B
superlattice dislocation has been its equilibrium spacing r, which
mentally
one of the foils
in agree-
Fig. 12 is the portion of the long dislocation
treated in this manner. It will be noted that the large complex deformation has resulted in a more or
However,
undissociated
Another interesting feature of
what effect cold work might
was heavily deformed by repeated bending. Fig. 11 shows an array of dislocations observed in a specimen
distribution
are virtually
ment with the theory.
well with the
results shown in Fig. 4.
In order to determine
Interference to the motion of superlattice dislocations in fully ordered AuCu,.
12.
the dislocations
that was subsequently cold worked.
theoretical
1961
near the leading edge. As the pinning points are approached, however, this separation vanishes and
FIG. 11. Super-latticedislocations in fully ordered Au&,
and
9,
four
partial
dislocations
that
between
constitute
the
superlattice dislocation has been calculated for dislocation possessing an arbitrary Burgers vector. An alloy
of composition
bulk, has been electropolished
AuCu,,
fully
ordered
to films about
a in
1000 A
be similar to the contrast observed for stacking faults.(lO) The contrast appears most pronounced in
thick. Examining these by transmission microscopy, we have observed superlattice
the vicinity
tions for the first time. The separation between the two pairs of partial dislocations was measured to be about 130 A and is in good agreement with the
of the dark extinction
contours.
Finally, Fig. 12 shows some very interest’ing features of moving superlattice dislocations. A segment of the large dislocation line at A has been pinned at two points. Nevertheless, this segment is able to expand outward between the pinning points as a superdislocation with a well defined separation
electron disloca-
spacing calculated theoretically. The energy of the antiphase boundary discussed above has been calculated for one which has been produced
by slip and whose structure is not changed
MARCINKOWSKI,
by a subsequent
BROWN
diffusion of atoms.
FISHER:
AND
The structure
DISLOCATIONS
of
an antiphase boundary which exists during thermal equilibrium is different from a boundary produced only by slip in that the equilibrium ary has a thickness varies continuously.
antiphase bound-
over which the degree of order The calculation for the variation
in thickness of an antiphase
boundary
ture has been made by Brown’li)
with tempera-
for CuZn.
the free energy of the equilibrium
In general
antiphase boundary
will be less than that of the slip produced boundary
antiphase
and the difference will be greater the closer
the temperature Consideration
small resistance in
Ardley(12).
configura-
increases
in AuCu type alloys, the superlattice
disloca-
tion may consist of an asymmetric array of four bound partial dislocations or a pair of partial dislocations similar to that in the disordered alloy. The difference AuCu,
in the dislocation
arrangements
The configuration
of the dislocations
could
have
an important bearing on the plastic behavior of the crystal. For example, since there are two distinct types
of superlattice
dislocations
in AuCu and only
one type in AuCu,, it is expected that ordered AuCu would plastically deform in a more anisotropic fashion than ordered AuCu,. No experiments been made to test this conclusion. From the result,s described little doubt
that dislocations
here, there seems to be in fully ordered AU&I,
travel through the lattice as superlattice When
the
antiphase
500 A) these dislocations
have
boundaries
are
dislocations. large
(about
should move with negligibly
however,
that
rapidly.
of the
Thus, at low degrees of order, it independently
as moving
In this case, it would
follow
that
This would
type
of one another. these imperfect
must do work against the ordering forces,
since they will leave behind an antiphase mean
degrees higher
that the alloys of
yield
order
should
points,
than
boundary.
possessing be
inter-
stronger,
either
i.e.
the fully
ordered or the completely disordered alloys. These very interesting speculations are being presently investigated
in both AuCu, and Ni,Mn. ACKNOWLEDGMENT
We are indebted carrying
in the latter.
be recalled,
This is
results
to think of the $a(llO)
between
and AuCu is a result of the smaller degree of
symmetry
forces.
experimental
dislocations
possess
location
It will
the
may be more correct
ing on the Burgers
dis-
with
decreases this energy decreases very rapidly and the separation between the component dislocations also
mediate
of the superlattice
137
energy of the antiphase boundary bounded by the superlattice dislocation varies as S2 so that as S
tions in the AuCu type ordered alloys leads to two additional types of dislocation arrangements. Dependvector
SUPERLATTICES
due to the ordering
agreement
dislocations
is to To. of the possible dislocation
IN
to A. Szirmae for his assistance in
out the experimental
portion
of this work
and to D. S. Miller for many helpful discussions concerning the interpretation of the results. REFERENCES 1. 2. 3. 4. 5.
N.BRowN~~~M.HERMAN,TT~~~.A~~~. Imt.
Engrs. 206, 1353 (1956). C. N. YANG, J. Chevn. Phys. 13,66 (1945). Y. Y. LI, Phys. Rev. 76, 972 (1949). W. KBSTER, 2. Met&k. 32,145 (1940). Y. Y. LI, J. Chem. Phys. 17, 447 (1949).
Min. (Metoll.)
6. R. M. FISHER and A. SZIRMAE,Syllzposium on Electron Metallography, ASTM STP 262, 103 (1959). 7. W. BOLLMANN, Phys. Rev. 103,1588 (1956). 8. P. B. HIRSCH,MetaZZurg. Rev. 4, 101 (1959). 9. M. J. WHELAN, PTOC. Roy. Sot. 249, 114 (1959).
10. M. J. WHELAN, J. Inst. Met. 87, 392 (195!3). 11. N. BROW, Phil. Msg. 4, 693 (1959). 12. G. W. ARDLEY, Acta Met. 3, 525 (1955).