996 ACTA METALLURGICA, VOL. 11, 1963

0.7 -

0.6 -

(&2) If2

I I I I , n , 1 0 5 IO IS 20 25 30

FIG. 2. Values of the particle size coefficient A,,8 and the root mean square strain {$)1/z obtained from the solid

lines of Fig. 1.

(a) (&p)1/2 = 0.0100; (b) (E~)~/~ = 0.0032.

strain distribution p(c) = (u/2) exp [--a 1 l 1 ] where

(e2) = 2/a2. To simplify the calculations, the strains

are taken to be uniform over distance so that 2, =

ne, and the crystals are assumed to have a uniform

size N, so that An8 = 1 - In//N,. For this model,

the Fourier coefficient becomes

A,(Z) = (1 - ]n]/NJ(l + 27r2n2(E2)Z2)-l (I)

Using a particle size N, = 100, and a rather large

root mean square strain (e2)l12 = 0.010, values of

A,(Z) are computed from equation (1). Fig. 1 shows

the usual plot of In A,(Z) as a function of Z2 for values

of n from O-25. The dashed curves show the correct

extrapolation to Z2 = 0, and the solid lines a linear

extrapolation using the first two orders only. Up to

about n = 6 the linear extrapolation leads to the

correct intercept (1 - ]n]/lOO), but for higher values

of n the linear extrapolation gives values which are

too small. The particle size coefficients A,’ and the

root mean square strains (e2)l12 given by the inter-

cepts and the slopes of the straight lines of Fig. 1 are

shown by Fig. 2(a). There is a downward curvature

in AnS and a falling off in (~~)l/~ qualitatively similar

to the effects observed by Wilkens. To show that

these effects are due solely to a linear extrapolation

when the strain distribution is non-Gaussian and the

mean square strain too high, the same calculations

have been made with a smaller strain (e)lj2 = 0.0032.

The results are shown by the curves of Fig. 2(b), in

this case AnS is free from curvature and (~~)l/~ is

constant out to much larger values of n.

For large (e2), curves of In A,(Z) vs. Z2 are straight

lines for a Gaussian strain distribution, they are

concave upward for a strain distribution such as

exp [-a 1~11 which falls off more slowly than a Gauss-

ian, and they are concave downward for a strain

distribution such as exp [ --a3 1 c3/] which falls off more

rapidly than a Gaussian. In some cases it is possible

to obtain experimentally enough points on the In A,(Z)

vs. Z2 curve to really determine the shape of the curve.@)

For these cases, the curves are found to be surpris-

ingly linear out to large values of Z2, justifying the

linear extrapolation against Z2, and showing that in

practice the strains in cold worked metals are ap-

preciably different from those postulated in the model

of Wilkens or in the model used here. The concave

downward curvature of AnS shown by Fig. 2(a) results from an unjustified linear extrapolation on

data from a hypothetical strain model, it has nothing

to do with the familiar “hook effect” which results

from the overlapping of the tails of the broad

reflections from cold worked metals.

B. E. WARREN

Massachusetts Institute of Technology

Cambridge, Massachusetts

References 1. M. WILKENS, 2. Naturf 17a, 277 (1962). 2. B. E. WARREN and B. L. AVERBACH, J. Appl. Phys. 21,

595 (1950). 3. B. E. WARREN, Progr. Met. Phys. 8, 155-157 (1959).

* Received February 4, 1963. Research sponsored by the U.S. Atomic Energy Com-

mission.

A dislocation model for twinning in f.c.c. metals*

In this note we propose a new dislocation model for

twinning in f.c.c. metals. At present there certainly

is no dearth of twinning mechanisms(1-3) for this

crystal system. Except in Ref. 1, these involve, in one

way or another, a pole mechanism. However Blewitt

and Redmanc4) recently have demonstrated that a twin

a few millimeters in thickness forms across a cross

section of a single crystal in less than a millisecond.

In order that a pole mechanism produces a thick twin

in this short time interval the operation of a large

number of poles is required. Each twinning dis-

location can climb up its pole only about 1000 atom

planes. (Assume the twinning dislocation moves with

the velocity of sound, ~3 x lo5 cm/set. In the case

LETTERS TO THE EDITOR 997

(III)

‘, = % [1121+ tC2iil

FIG. 1. Dissociation of dislocations in pile-ups on the (111) and (lli) planes.

of a crystal 1 cm dia., 10e5 set are required for the

twinning dislocation to sweep around the outer

diameter of the crystal. In a millisecond this

dislocation can climb 100 planes.) Since each pole

is severely limited in the number of planes through

which it can move a twinning dislocation, the

advantage of a pole mechanism over a mechanism

which regards twinning as brought about by the crea-

tion of large numbers of individual stacking faults(l)

is not large. We now consider how such large numbers

of faults may be created.

Consider the classic dislocation pile-up which is

brought about through a Cottrell-Lomer lock (shown

in Fig. 1). On the (111) plane we have (a/2)[10i]

dislocations which split into the two mobile Shockley

partials (a/6)[112] and (a/6)[211]. On the (lli) plane

the (a/2)[011] dislocations split into the two mobile

Shockley partials (a/6)[112] and (u/S)[iSl]. Now it is

possible for the (42)[011] dislocations to split into the

(u/S)@ l] mobile Shockley partials and the (43)[111]

Frank sessile partial dislocations. This split also is

shown in the figure. Normally this last dislocation

reaction should not occur because the dislocation

energy remains constant, whereas a split into two

Shockley partial dislocations lowers the total energy.

However near the head of a dislocation piled-up

dislocations are squeezed together. Thus it is possible

that the two Shockley partial dislocations are brought

-&-_________I “_----_____\ \

(III)

FIG. 2. Formation of a twin by dissociation of many dislocations in pile-ups on nearby slip planes in one slip

band.

FIG. 3. Twins seen on a (110) plane in a single crystal shock loaded to 435 kb in a (111) direction-polished

and etched. x 1700.

close together. A split into a Shockley and a Frank

partial then becomes possible because the stresses near

the head of the pile-up are such as to tear a Shockley

partial away from its Frank partial. (N.B., in Fig. 1

the partial (u/2)[211] is of opposite sign to the partial

(u/2)[2ii] and hence moves, as shown, in a direction

opposite to the dislocations in the pile-up on the (111)

plane.) Thus one stacking fault can be created. We

presume that a large number of dislocations in a

pile-up will split into Shockley and Frank partials

and thus produce a great number of stacking faults,

as indicated in Fig. 2. When we consider that there

are many Cottrell-Lomer locks on adjacent planes

in a slip band, we see that a vast number of stacking

faults will be produced, as in Fig. 2. In a cold worked

metal there are log-lOi dislocations/cm2. If it is

recalled that 3 x 10’ stacking faults in a cryst,al 1 cm

thick will completely transform it into a perfect

twinned crystal it can be recognized that only a small

fraction of the available dislocations need split up onto

(a)

FIG. 4. Slip of (211) partials in the matrix (a) producing the twin (b). Subsequent motion of similar partials as shown in (b) returns the twin to its original orientation.

998 ACTA METALLURGICA, VOL. 11, 1963

Shockley and Frank partials in order to completely a dislocation past an obstacle. Basinski concluded twin the crystal. that the temperature dependence of the thermal

This model has the following features: (1) It can activation energy is account for the production of a twin in microseconds. (2) It should become operative only after a certain aff a@ ap - amount of plastic strain. Experimentally it is known i 1 aT 4 = Tp a(l/kT)

(1)

that f.c.c. crystals form deformation twins only after they have been strainedta.5). (3) The twins produced

when dH is due entirely to the temperature dependence of the elastic constants. Here H is the thermal acti-

would be expected to be rather imperfect, i.e. not all the necessary faults in a stack of (111) plane need

vation energy, T the absolute temperature, o the

occur. Optical and transmission electron microscope applied stress, ,D the elastic shear modulus, Xc the

studies(st7) indicate that a twin often is made up of Boltzmann constant, and p = (a In g/a+ where E” is the strain rate.

segments of twinned and untwinned material. Examples are shown in Fig. 3 of shocked loaded

In their more sophisticated approach Conrad and Wiedersich considered the influence of a back stress

topper(7). At this moment, there does not seem to be any way

on the ~mperat~e dependence of the thermal

to decide between the various mechanisms. The activation energy. This back stress G* is considered

mechanism discussed here is most probable in systems to screen the active dislocations from the applied

with low fault energies, where pile-ups do occur. stress so that they see only a net stress 5 such that

As shown in Fig. 4, the partials can twin and 5=0--o P (2)

“untwin”. A considerably greater contribution to the shear strain is possible than is normally expected

Conrad and Wiedersich concluded that for their case

from a twin. This process can occur, through the a~ 54 ap mechanism described here, or with the other pole t i

- (3)

mechanisms. aT d = -Fyi a(l/kT)

J. B. COHEN It should be noted that this latter conclusion is

Materials Research Center J. WEERTMAN incompatible with that of Basinski for when the back

Department of Materials Science stress oP approaches zero, equation (3) does not reduce

The Technological Institute to equation (1). At present there is considerable

~o~h~~estern university interest in activation energies and it is important to

Eva.nston,~ ~~l~no~ reconcile this incompatibility of equations (1) and (3).

References Basinski’s result attributes the temperature depend-

1. P. B. HIRSCH, A. KELLY and J. W. MENTER, Proe. Phys. ence of the activation energy to the fact that an

See. Land. B68, 1132 (1955). increase in temperature causes a decrease in the elastic 2. H. SUZUKI and C. 5. BARRETT, Acta Met. 6, 156 (1958). 3. J. A. VENABLES, Phil. Mag. 6, 379 (1961).

constants which in turn reduces the effective size of

4. T. H. BLEWITT and J. K. REDMAN, Bull. Amer. Phys. the obstacle. The result of Conrad and Wiedersich SOS. Ser. II, Vol. 7, p. 199 (1962).

5. T. H. BLEWITT, R. R. COLTMAN and J. K. REDMAN, attributes the entire temperature dependence of the

J. Appt. Phys. 28, 651 (1957). activation energy to the fact that an increase in 6. J. A. VEXABLES, Proc. Eur. Reg. Conf on El. Micro. Vol. I,

p. 443 Delft (1960). temperature decreases the elastic constants and in

7. R. J. DEANGELIS and J. B. COHEN, to be publishd turn decreases the effectiveness of the ba,ck stress in

* The research upon which this paper was based was screening the active dislocations from the applied supported by the Advanced Research Projects Agency of the stress. Department of Defense, through the Northwestern University

Thus with an increase in temperature, at

Materials Research Center. constant applied stress, the net stress, 3, tending to Received February 11, 1963. assist activation is increased. Conrad and Wiedersich

neglect the influence of temperature on the effective size of the obstacle and this gives rise to t,hc above

Thermal activation energies for the low incompatibility. Such a treatment is justified only if

temperature deformation of metals* the back stresses are large with respect to the net stress for then equations (1) and (3) are essentially

Basinski and later Conrad and Wiedersichcs) equivalent. wrote notes on the thermal activation energy for the As the back stresses and the effective size of the

low temperature deformation of metals. Basinski’s obsticle are both dependent on temperature through

note referred specifically to the thermal act,ivation of the shear modulus, it would seem that when a back