The force between a dislocation dipole and a non-parallel dislocation

SCTA

60

METALLURGICA,

at 4°K. using standard d.c. potentiometric with the specimen in a magnetic Resistivity ducibly

field of 300 oersteds.

changes of lo-l2 ohm-cm

measured.

(The

techniques

could be repro-

experimental

details

are

given in Ref. 2.) The isochronal

annealing

are shown in Fig. 1.

data for 30 min anneals

Two distinct

annealing

stages

centered around 50°C and 130°C were observed. change of slope method of determining energies

which

characterized

stages was applied. not exhibit

these

two

1.0 f

annealing

datat2) the defect

stage was identified

mobile

as the monovacancy.

in this

The mag-

isochronal step is the divacancy. Attempts to measure the Ee$zctive m . the temperature range of 30-70°C in anomalous

resistivity

temperature

increases

was increased

when the

as shown

in

Fig. 2. These transients may have resulted from the breakup

of small vacancy

temperature

References 1. R.

W. BALLUFFI Quenched Metals,

and

0.

R.

693.

W.

SIEGEL,

Academic

Lattice Defects in Press, New York

(1965).

2. M. WUTTIG and H. K. BIRNBAIX, to bepuh1ishedin.J. Phys. Chem. Solids. 3. S. MADER, A. SEEGER and E. SIMSCH, 2. Metallk. 52, 785 (1961). 4. I. G. GREENBIELD and H. G. F. WILSDOR+, J. Phys. Sot. Japan 18, Suppl 3, 20 (1963). * Received Julv 6. 1965. This research was supported in part by Wright Patterson Air Force Base under contract AF 33(615)-1695.

a value of

netic data indicate that the defect mobile in the 50°C

annealing

1966

0.1 eV. On the basis of the magnetic

and the resistivity

resulted

14,

The data in the 130°C stage did

any transientsc2) and yielded

Eeffective = 1x

The

the activation

VOL.

clusters

was increased.

as the annealing

The increase in resistivity

The force between a dislocation dipole and a non-parallel dislocation* The influence of a dislocation dislocations

dipole D, composed

L, and L1’, on another dislocation

of

L, can

be described in detail by forces f on unit elements of L, or, very roughly,

by the total force F.

Some simple conclusions

will be drawn for the total

force F for the case when D and L, are non-parallel. Let us first mention

the results for the total force

would then require that pnv < npzv (where pnv is the

F2*l between two straight infinite non-parallel disloca-

resistivity

tions given

of a cluster of n divacancies).

Although of E~~‘tive defect

the difficulties in relating measured values the energy

to

will exist

vacancy

clusters

of motion

generally,

the instability

may be peculiar

et uZ.(~! have suggested

of a particular of small

to nickel.

that prismatic

Mader

loops may be

difficult to form in nickel below the Curie temperature because of magnetization however

form

during

effects. the

Some clusters may

quench

at temperatures

above the Curie temperature

and be retained to lower

temperatures.

then tend to dissociate

These would

into divacancies

on isothermal

culty of forming vacancy may be supported and Wilsdorf(4) and in irradiated mission electron temperatures interpreted mobility,

annealing.

The diffi-

clusters at low temperatures

by the observations

that vacancy

clustering

earlier

(Kroupa(l));

the forces

on unit

elements for this case were discussed in detail recently by Hartley Let

us

and Hirth,c2) and Bullough

assume

an

infinite

elastic

and Sharp.t3) medium

with

shear modulus 1~and Poisson ratio Y. The coordinates are chosen as in Fig. 1: the dislocation z axis, the dislocation

L, lies in the

L, is parallel to the xz plane,

intersects the y axis at the point y = (x (ial is thus the shortest distance between the non-parallel

lines L,, L, ;

we assume a + 0) and makes an angle /I with the z

Z

of Greenfield in quenched

nickel was not observed

by trans-

microscope

techniques

at annealing

below 200°C.

Greenfield

and Wilsdorf

their results on the basis of a low defect

which does not appear tenable in view of the

present results. M.

WUTTIG

H. K. BIRNBAUM Depart*ment of Mining,

Metallurgy

and Petroleum Engineering University of Illinois Urbana,, Illinois

FIG. 1. Dislocation dipole D, composed of dislocations and L,‘, and another straight dislocation L,.

L,

LETTERS

we assume 0 < /l < rrt.

direction;

of the dislocation

The Burgers vectors b(l)

and bc2) of L, and L, are general.

formula)

The total force F2*l

L, due to dislocation f 2~1(deduced

by integrating

The orientation

lines and the sense of the angle p is

shown in Pig. 1 by arrows. on dislocation

along L,.

L, was obtained(l)

from the Peach-Koehler

It has only B’f,’ component

zero which is a function

opposite

the dislocations

direction

sides or attractive

non-

of /3, bf$, bi’), br), ba”),u, v, and

sign a and does not depend on the distance intersecting

TO

Ial ; after

the force changes to the

(the force is repulsive

from both

61

EDITOR

distribution

of forces f on unit elements of L, changes

in a complicated

way but the total force F, which is an

integral of f along L,, is zero or, in the special case when L, intersects the inside of the dipole, generally a non-zero constant. A dislocation

in a crystal does not, of course, move

as a solid whole.

The motion

of a dislocation

trolled by forces f on the dislocation external stresses, different obstacles

L, let us now consider a

is con-

elements due to (e.g. the disloca-

tion dipoles), friction stress and also by the line tension of the dislocation

which acts against its curvature.

cases when the forces f are practically

from both sides).

Instead of one dislocation

THE

a short segment of a dislocation

concmtrated

In on

it is possible to take

dislocation

roughly the total force F as a point force or to calculate

location

the mean value off

dipole D, composed of L, and another disL1’: which is parallel to L, and has opposite

Burgers vector, b(l)’ = -b(l).

on this segment as f = F/l, where

1 is the length of the segment ; for the case of disloca-

The distance c between L, and L,’ and the angle p (Fig. 1) are arbitrary. From

tion L, in the close neighbourhood

the results mentioned

1 m c/sin b. This approach is justified by the action of

above it follows directly for the

total force F from the dipole dislocation

D on the non-parallel

L, that : (i) the total force F is zero when

the dislocation

L, does not intersect

tween L, and L,‘. sects the ribbon open interval

the ribbon

(ii) When the dislocation between

L, and L,’

be-

L, inter-

(a is from

(0, c sin CJJ) ; we assume p f

the

0, p # 7r)

the total force F has only the F, component

non-zero

or inside the dipole,

the line tension which decreases the sensitivity dislocation

to local variation

especially when the width c of the dipole is small and when the dislocation

L, moves under external

The total force from the dislocation L, on the dipole When the dislocation L2 is outside

D as a whole is -F.

(it acts in the direction of the shortest distance between

the dipole the total forces on individual L, and L,’ are opposite

or L,‘L,)

which does not

depend on the distances c and [a/ and is equal to 2Fis1. Using the value F$’

given in Ref. 1 it follows that

1FJ from equation

[( b(l) b(2) _ z

z

Z

2

1 (1)

The results given lead, therefore,

sign for a < 0 (37 < q < 27r).

Obviously, for CJJ = 0 or q = r the total force F = 0 also for a = 0. These simple but somewhat

surprising

with the use of the concept in an infinite medium. solution

to move the dipole with

it or, at least, to bend it locally.

where the + sign holds for a > 0 (0 < q~ < r) and the

locations

L, is inside the

L, moves under an external shear

stress it tries, in this position,

conclusion

not give a complete

When the dislocation

when the dislocation

x

connected

to l/2

(1)) ; they only try to make the

dipole the forces on Ll and L,’ have the same direction;

b(l) b(2)

-

dislocations

(and equal in magnitude

dipole locally wider or narrower and do not try to move it as a whole.

F,=f+

shea,r

stress.

the non-parallel

lines L,L,

of the

of f on short segments,

for the theories

the con-

tribution

to the hardening

of the elastic interaction

between

the

non-parallel

dipoles

and

dislocations,

insofar as they run outside the dipoles, can very well be neglected.

results are

of infinite dis-

For B +

to the following

of hardening:

F.KROUPA

0 they do

for the dislocation

L,

Institute of Physics

D; in this case the total force II and not only F Y but will also depend on b:) and b@)

Czechoslovak Academy of Sciences

total force : when a non-parallel dislocation L, moves as a rigid whole in the stress field of the dipole D, the

References

parallel to t’he dipole

Prague, Czechoslovakia

also F, can be infinite or zero. Let us stress once more the physical meaning of the

i The author is indebted to Dr. J. Hornstra for drawing his attention to the fact that the restriction 0 < p < n should be added in(l) to ensure the right sign of the total force.

1. F. KROUPA, Czech.J. Phys. Bll, 847 (1961). 2. C. S. HARTLEY, J. P. HIRTH, Acta Met.13,79 (1965). 3. R. BULLOU~H, J. V. SHARP, Phil. Mug. 11,605 (1965). * Received

May 20, 1965;

revised July 2, 1965.