Theory of Dislocation Mobility in Pure Slip - Lothe1962

8/19/2019 Theory of Dislocation Mobility in Pure Slip - Lothe1962

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Theory of Dislocation Mobility in Pure Slip

Jens Lothe

Citation: Journal of Applied Physics 33, 2116 (1962); doi: 10.1063/1.1728907

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2/11

2116

R O T H M A N f O N E S G R A Y AND

H A R K N E S S

rial by Adda

et

al. 19 which is about what one would

expect if their values were slightly increased by dif

fusion

l o n g ~ g r i n

boundaries.20 Adda

et

at. did not

observe any anisotropy; however, visual observation

of

autoradiographs of polycrystalline samples is a less

reliable method of detecting anisotropy than the method

used by us. Our

D[lOOJ

and

D[ool]

are factors

of

ten and

twenty or sixty higher than those measured by Resnick

et

al.

l

on perfect single crystals. This difference

probably is not due to diffusion along mosaic boundaries

in our crystals for the reasons given above; the dis

crepancy is better attributed to the experimental

uncertainty of the data of Resnick

et

al.

19

Y.

Adda, A. Kirianenko, and C. Mairy, Compt. rend. 253,

445

(1961).

20

R. E. Hoffman and D. Turnbull,

J.

Appl. Phys.

22,

634 (1951).

21 R. Resnick and L.

L. Seigle,

J. Nuclear Materials 5,5 (1962).

J O U R N A L OF A P P L I E D

P H Y S I C S

CONCLUSIONS

We conclude that we have measured volume self

diffusion in alpha uranium, and that it is highly aniso

tropic, as expected from the structure. Diffusion in the

corrugated layers, where the jump distances are small

and the bonding is covalent and strong, is much faster

than diffusion out

of

such layers. However, there are

indications of fast diffusion between the corrugated

planes along dislocations.

ACKNOWLEDGMENTS

The assistance of M. Essling, E. S. Fisher, A. Hrobar,

S.

A.

Moore, M. H. Mueller, L.

J.

Nowicki, M. D. Odie,

and D. Rokop, and discussions with H. H. Chiswik,

E. S. Fisher, and

L

T. Lloyd are gratefully acknowl

edged. This project was begun by R. Wei .

V O L U M E 3 3 . N U M B E R

6

J U N E 1962

Theory of islocation Mobility in Pure Slip

JENS LOTHE

Metals Research Laboratory

Carnegie Institute of

Technology

Pittsburgh, Pennsylvania

(Received July 19, 1961; revised manuscript received November 9,

1961)

The mobility during glide of uniformly moving dislocations or dislocation segments supposed not to be

obstructed by any Peierls' barrier is estimated. For a straight freely moving dislocation, the strong an

harmonicities in the core region, the thermoelastic (edge dislocation) and the phonon viscosity effect give

rise to a drag stress

at

ordinary temperatures

T 'fJ,

f being the Debye temperature, of the order

,, ,,-,f.;.XV/c

in insulators.

In

metals the thermoelastic effect is negligible, while the core anharmonicity effect and the

phonon viscosity effect will be of the same order of magnitude as in insulators. In the above formula,

E=thermal energy density, V=dislocation velocity, and c=velocity of shear waves. The scattering of

phonons by the dislocation also causes a drag stress at ordinary temperatures of the order of magnitude

of

the above formula.

All of the above mentioned contributions to the drag stress go rapidly to zero with decreasing temperature.

However, if the dislocation s constrained by the Peierls' barrier except at freely moving kinks, the kink

mobility determines the dislocation mobility. It

is

shown that the scattering of phonons of a half-wavelength

longer than the kink width causes a drag stress which may outweigh all other contributions up to ordinary

temperatures, and which persists with decreasing temperature as

T

down to a temperature

'8b/D,

where

b=the lattice spacing constant and D

is

the kink width.

I. INTRODUCTION AND_OUTLINE

S

O far, a discussion and interrelation of the various

theories for dislocation mobility is lacking.

In

this

paper we briefly reconsider the various dissipative

mechanisms suggested and discuss, in order of magni

tude, their effect on dislocation mobility. Some estimates

of the dissipation that result because of the strong

anharmonicities in the core region are also presented.

Care is taken to represent the various contributions in

equations which are easily compared.

Only dislocations, or dislocation segments, that can

move freely without thermal activation, will be con-

On leave from Fysisk Institutt, Universitetet, Blindern,

Oslo,

Norway.

sidered. Internal friction experiments in copperl and

NaCl2 have shown that in these crystals there is a

modulus defect, quite constant with temperature, with

an accompanying internal friction that seems to tend

to zero as the temperature tends to zero, indicating

either freely moving dislocations or freely moving

kinks.3.4

At the end of the paper we shall make some separate

considerations on kinks.

t will

appear that the mobility

of a smooth dislocation will have a somewhat different

1 G. A. Alersand D.

O.

Thompson, J. Appl. Phys. 32,

283

(1961).

2

R.

B.

Gordon (private communication).

3

J.

Lothe and

J.

P. Hirth, Phys. Rev. 115,

543

(1959).

4 J. Lothe, Phys. Rev. 117, 704 (1960).

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3/11

D I SL O C A T I O N M O B I L I T Y IN PUR

SL I P

2117

temperature characteristic than the mobility of a

dislocation with a kinked core structure.

We think

that

the idea

of

some dislocation segments

being able to move freely, without any activation, need

not be an approximation. A possible "Peierls' barrier"

to the motion of a kink in a close-packed metal would

most likely be

so

small

that it

would be completely

obliterated

by

zero-point motion. Thus,

at

very low

temperatures,

we

would expect the kink to move

without friction.

II. RELAXATION EFFECTS

Relaxation effects in the matrix around the moving

dislocation leads to heat production. The rate

of

heat

production around the moving dislocation unequiv

ocally determines the dislocation mobility: The disloca

tion

will

move with constant velocity when the rate

of

heat production equals the rate

of

energy supply from

the external mechanism that provides a shear stress

acting on the dislocation.

A. ulk

Relaxations

Eshelb

y

5 has shown

that

the thermoelastic effect

will

give rise to heat production around a moving disloca

tion. The thermoelastic effect is appreciable only for

the moving edge-like dislocation, where irreversible

heat

flow will

take place between the compressional

and dilatational side. Recently, Mason

6

suggested

that

pure shear deformations should be accompanied by

thermal relaxation effects and energy dissipation. This

effect comes about when the vibrational frequencies in

a lattice do not change only with volume changes, as

assumed in the simple Gruneisen theory,

but

also

depend on shear strain. The phonon-viscosity effect,

as this effect

was

termed

by

Mason, would

be

equally

effective for both edge and screw dislocations.

In

the theories for the thermoelastic effect and the

phonon viscosity effect

it is

necessary to introduce a

cutoff, defining a cylinder around the dislocation core

within which the theories do not apply.

We

shall

reconsider what

is

the proper cutoff. In particular the

result for the phonon-viscosity effect depends sensitively

on the choice of cutoff.

1. The Thermoelastic Effect

Eshelby

5

calculated the thermoelastic heat production

per cycle for a vibrating edge dislocation. Because

of

a

term logarithmic in the frequency, it is not obvious

from Eshelby's result what the stress needed to keep

the dislocation in uniform motion

is.

Weiner

7

has

considered this problem, and for a rigorous analysis

Weiner's paper should be consulted. In order to get an

approximate,

but

simple and analytical, result

we

have

6 J. D. Eshelby, Proc. Roy. Soc. (London) A197 396 (1957).

6

W. P. Mason,

J.

Acoust. Soc. Am. 32, 458 (1960).

7

J. H. Weiner, J. App . Phys. 29, 1305 (1958).

calculated the problem by essentially the same pro

cedure as used

by

Eshelby. For simplicity the cutoff

radius

1was

introduced as a cutoff in

k

space,

k

ma x

=7r/l.

The resulting equation for the stress

u

needed to move

the edge dislocation

at

a speed

V is

with Poisson's

ration equal to

i

1

IJ b

Cp C .

u=- - - - ln 7 rK / lV)XV,

(1)

70 K C

p

when

7rK/lV»1.

Here the symbols are:

b

= magni tude of Burgers

vector; lJ.=shear modulus;

K=K cp=thermal

diffusiv

ity

(thermal conductivity divided by specific heat per

unit volume);

C

p

c.=specific heats per unit volume

at

constant pressure and volume, respectively; and

1= cutoff radius for the dislocation core.

An

asymptotic

correspondence between Eq. 1) and Eshelby's formula

8

as

1 >

0

can easily

be

demonstrated.

The cutoff cannot be smaller than the lattice

distance b However, the conditions for the macroscopic

concept of thermoelastic relaxation to apply are not

satisfied that close to the core. A volume element to

which

we

apply macroscopic thermal concepts must

have a linear dimension

at

least

as

large

as

the phonon

mean free

path

in an insulator, or the electron mean

free path in a metal. Thus, it

is

natural to put

l=Ap

(insulator),

l=Ae (metal),

(2)

where

Ap

and

Ae

are the phonon mean free

path

and

electron mean free path, respectively. Applying thermo

elastic theory to closer distances

of

the core would,

for an insulator, lead to heat transmission faster than

sound, which is not possible.

Other factors equal, the thermoelastic effect

is

strongest in materials

of

low thermal conductivity.

Thus, for an estimate

of

the maximum contribution let

us consider a typical insulator. By making use

of

the

thermodynamic relation

(3)

where a

is

the volume expansion coefficient and

B

the

bulk modulus, combined with the approximate relations

(4)

and

(5)

where

Kp is

the lattice thermal conductivity and

'Y'"1.5 is

Gruneisen's constant,

we

can rewrite Eq.

(1)

to

U = ~ ~ C v T ) l n ~ ) x ~ ,

28Ap

V

C

(6)

8 In Eshelby's formula (4), reference 5 a factor 1/50 should be

substituted for the factor 1/10 occurring in

that

formula.

The

factor 1/50 can be derived from his formula (a24).

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4/11

2118

J NS LOTHE

when

V«c.

Poisson's ratio has been taken as in the

relation between J I and B c denotes the velocity of

sound.

9

At ordinary temperatures, in the region 0 to 0/2 it

is reasonable to put cvT

/2", E,

where

E

is the thermal

energy density and

Ap'" (S-30)b.

With

V

in say the

region

"'c/100,

the estimate of Eq. 6) for

T ,O

and

Ap ,Sb is

7)

In

metals, because of a higher thermal conductivity

by

a factor typically of about 30, the thermoelastic

contribution will be correspondingly smaller.

2 The Phonon Viscosity Effect

The

difference between the adiabatic and isothermal

bulk modulus is

(8)

For an isotropic body the shear modulus

J I

is related

to the bulk modulus B and Poisson's ratio v

by

the

equation

J.I.= 3(1-2v)B/[2(1+ )].

(9)

In

the elementary theory of Gruneisen's constant the

vibrational frequencies are taken, on the average, to

depend on the volume only, 'Y= d InO/d InV. In this

model

we

would have 11J 1 = J.l.ad - J.l.is = 0, corresponding to

v=

in Eq. (9) for the change. If, on the other hand,

we take the frequencies in a wave only to be modified

by

longitudinal strain parallel to the wave vector, we

should rather put

v=O

in Eq. (9) to calculate 11J 1 from

I1B;

(10)

Equation (10), thus, is a reasonable upper estimate

of the difference between the adiabatic and isothermal

shear modulus. It should be understood that, in this

context, adiabatic means that energy is not exchanged

between vibrational modes in the same volume element.

The additional shear stiffness during adiabatic deforma

tion comes about because phonons with wave vector

in the BC' direction become hotter during shear,

while those in the

AD'

direction become "cooler"

(Fig. 1).

In terms of three phonon processes it would take

Umklapp processes to establish equilibrium between

BC'

and AD' phonons. Thus, it should be approximately

A

B

FIG. 1. When the volume element

ABDC

is sheared

adiabatically

to

take

the shape

ABD'C',

phonons traveling

in

the direction

BC'

become hotter

and

those traveling

in the

direction

AD'

become

cooler.

9

Because

of the approximate nature of the

calculations, we

will

denote

the

shear wave

velocity

as

well

as

the

average sound

velocities

appropriate

to

the

various considerations in this

paper

by the same symbol c.

right, as done

by

Mason,6 to determine the relaxation

time from the equation

(11)

where

Ap is

given

by

Eq.

(4).

Then, employing Mason's analysis for the moving

screw dislocation,

but

only for the material outside a

cylinder of radius Ap around the core, we obtain

CT=7]b/ 47rAp2)X

V,

V«c,

where

7]

is the phonon viscosity

7]= TI1j t

(12)

(13)

By Eqs. (4), (9), and (11), and with 'Y' 1.5, Eq. (12)

can be written as

(14)

and, in the region T ,O, with the same approximation as

used for Eq. (7),

CT 'E/lOXV/c,

V«c.

(15)

The result for an edge dislocation would not be much

different.

t must be borne in mind

that

Eq. (15), based on

Eq. (10), is an upper estimate.

According to Kittel,lO at ordinary temperatures T ,O,

the relaxation time for phonons is about the same in

insulators and metals.

In

metals the phonon-phonon

relaxation time

Tpp

is about the same as the phonon

electron relaxation time Tpe. Thus, the phonon mean

free

path

and the phonon viscosity in a metal should

typically be about the same as in an insulator, and the

phonon viscosity contribution to dislocation damping

should not be very different in insulators and metals.

B. Relaxations in the Core Region

t remains to estimate the relaxation contributions

inside the cutoff cylinder or radius

r=A.

The contribu

tions will be divided into two main classes:

(1) A volume contribution, for which the relaxation

strengths I1B and 11j.t Eqs. (8) and (10), are supposed

to apply on the average.

2) A misfit plane contribution, in which the an

harmonicities are

so

great and of such a nature

that

they must be considered separately.

This division corresponds to the Peierls-Nabarro

treatment of the dislocation, where two elastic solids

are joined

by

a misfit plane.

1.

The

Core Volume Contribution

In

the calculation on the thermoelastic effect for

an edge dislocation in a metal, only the matter outside

10

C. Kittel,

Introduction to Solid State Physics (John

Wiley

Sons, Inc., New York, 1956), 2nd ed., p. 149.

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5/11

DISLOCATION

MOBILITY

IN PURE SLIP

2119

a cylinder of radius

r=A.

was considered. Applying the

ordinary theory of thermal conduction within this

cylinder would lead to a phonon relaxation time

smaller

than

Tpe. As at ordinary temperatures T=Ap/c

"'Tpp'" Tpe, such a procedure must be wrong. Rather,

the various volume elements in the region we consider

have a single relaxation time

Tp •. In

each volume

element, phonon energy is transferred to the electron

gas and carried

out

of the region under consideration,

until equilibrium is established within a time T ' " Ap/ .

Thus, in analogy with Eq.

(13),

we define a bulk

viscosity

X,

X=T .B.

(16)

Then, for the region between r=A. and r=Ap for an

edge dislocation, with

,,= t

and

'Y' 1.5,

a simple

calculation following the treatment on phonon viscosity

gives, when A.»A

p

,

1 c.Tb V

(T '-

--X- V«c.

40

Ap

C

(17)

For relaxations in the region b


6/11

2120

J E NS

LOTHE

be zero. The second term

E av

F

vibr

2v

2

ax

(26)

gives rise to a change in vibrational energy.

I f

the

oscillator is moved with a velocity

V,

F

vibr

does work

at the rate

Fvibr V=-- V,

2v

2

ax

(27)

which goes to increase the energy of vibration. I f the

increase in vibrational energy radiates out, heat

is

supplied to the surrounding matrix. As the average

vibrational energy of the atoms in the misfit plane is

constant, the average value of Eq. (27) must determine

the rate at which heat is produced.

Each oscillator

is

coupled to its neighboring atoms,

with which vibrational energy

is

then exchanged.

Denote the relaxation time for energy exchange by T.

Because of the strong coupling to other atoms, the

relaxation time will only be a

few

periods, say roughly,

(28)

The differential equation determining the vibrational

energy of a moving oscillator is then

dE 1 E av

2

=

- - [E-Eeq T )J+ - - · v ,

(29)

dt T 2v

2

ax

where

Eeq(T)

is

the equilibrium value

Eeq(T) =hv/ (ehplkT -1 .

(30)

Now put

v= vo-ov cos(271 x/b).

(31)

By expanding to the first power in ov/vo

we

can rewrite

Eq. (29) as

dE 1

EO(

-+- E-Eo)= 1

dt T T

EoehVlkT)OV

-

cos(271 vt/b)

kT Vo

271 E

o

vov

- -

sin

(271 vt/b)

,

(32)

b

Vo

where Eo

is

Eq. (30) with v= Vo.

Only that part of the solution of Eq. (32) which is

in phase with sin (271 vt/b) contributes to the time

average of Eq. (27). Denoting this part

oE, we

obtain

when

vT/b


7/11

D I S L O C T I O N

MO B I L I T Y IN PURE SLI P

2121

sidered to move in an isotropic flux of phonons which

collide with the dislocation,

but

which do not collide

with each other, i.e., the phonon mean free

path is

assumed infinite. For such an approximation to be

adequate, the phonon mean free

path

must at least be

longer than the dislocation scattering width.

13

However,

no clear criterion for when the approximation of non

colliding phonons

is

valid has been established;

we

can

only assert

that

at lower temperatures this picture must

be right and offers an opportunity for complementary

and more reliable estimates than the previous core

considerations based on simple relaxation considera

tions. Fortunately, it will also turn

out

that the high

temperature values for

(J

derived in this picture are of

the same order of magnitude as the more important

contributions considered in the first

part

of this paper;

thus not too much ambiguity as to the correct order of

magnitude of (J

at

ordinary temperatures arises. At

low temperatures some of the effects to be studied give

a higher value for

(J

than the relaxation effects

and

should then be the dominant effects.

Consider a stationary dislocation placed in

an

isotropic gas of phonons. The phonons are scattered

by

the dislocation;

sayan

energy

W

is scattered

out

radially symmetric about the dislocation per unit time.

However, if the dislocation

is

moving with a velocity V

the scattered radiation

is

asymmetric and gives off a

net amount of quasi-momentum W/cXV/c per unit

time to the component parallel to the dislocation

motion, and thus a stress of the order

(38)

is needed

to

maintain uniform dislocation motion.4

The exact coefficient of proportionality in Eq. (38)

depends on the wavelength dependence of scattering

cross section. For our purposes, Eq.

(38) is

adequate,

and the problem is then to find W.

Two important cases

will

be studied: 1) the scatter

ing from a smooth infinite free dislocation, and (2) the

scattering from a free kink.

A.

The Infinite Smooth Dislocation Line

1 Scattering by Induced Vibrations

Consider a screw dislocation, and consider

it

to

interact with shear waves only. A shear wave whose

wave vector if makes an angle with the dislocation will

induce a wave motion of wave vector k cost?- on the

dislocation. Thus, if the velocity of a free wave motion

on the dislocation line

is

CD, shear waves for which

coSt?-


8/11

2122

l N S

LOTHE

I o e : = = = = = = = = ~ = =

m

0.1

2

3

4 5

6

7

8

O/T

FIG.

4. Curve

I: .IT,

normalized to 1 for

eiT.

Curve

II: ulT,

normalized to 1 for eIT=O, u is given by Eq. (43). Curve

III:

the constant

1.

where

E w,T)

iw

----

w=ck.

e w/kT-1

(44)

In the high temperature limit, with

E w,T)=kT

and E 'V3kT/b

3

,

Eq. (43) becomes

u -'E/10XV/c, 45)

which is precisely the formula originally given by

LeibfriedY

t

is important to note that

U

as given by Eq. (43)

decreases more slowly with temperature than the

thermal energy E, which is proportional· to

' fokmaxE ck,

T)k2dk.

The temperature dependences of

E

and

u

are compared

in Fig.

4.

f the phonon half-wavelength is small compared with

the width of the core misfit region, the phonon does not

tend to move the dislocation core as a whole. Neverthe

less, we shall expect Eq. (43) to be a fair estimate up

to higher temperatures because of another equally

effective scattering mechanism.

We

will

picture the core misfit strip in the glide plane

as a cut of some width 2a, over which shear stress

cannot be sustained. Consider shear waves of a wave

length A


9/11

D I S L O C T I O N MO B I L I T Y IN PURE SL I P

2123

which agrees well with Klemens'

20

estimate. With

k

m x

=7r/b and 'Y '1.5, the cross section for phonons of

highest energy becomes fJ ,b/2.

However, for dislocations for which the Peierls'

barrier is negligible, Eq. (46) would not apply to a

distance

r ,b

within the center (see Fig. 2). t is reason

able

to

suppose

the

core

to

be relaxed within a cylinder

of radius r '3b, and that the

perturbation

potential in

this region is approximately given by Eq. (46) with 3b

substituted

for r. Using

instead

of Eq. (46) a

perturba

tion potential

we

derive a scattering cross section

fJ='YWk/16(1 k2A2 i,

(49)

SO)

which for the high energy phonons, with A ,3b and

'Y' 1.5, gives a

constant

cross section

fJ ,b/40.

(51)

Thus, with

j

of the phonons being scattered, the drag

stress

at

ordinary temperatures will be

u 'e/60X V/e.

(52)

According

to

Klemens,20 Eq. (48) may be an under

estimate

by

as much as a factor of the order 10. The

uncertainty

is due to lack of precise knowledge about

the

anharmonic constants

and

the approximations

involved in the Born scattering formula.

With

the same

uncertainty in Eq. (52), the drag stress due

to

strain

field scattering might be as high as

u '-'e/5X V/e. 53)

B. Scattering

at

a Kink. Kink Mobilit

y

21

When the Peierls' barrier is significant, the moving

element will be the kink, which is a short dislocation

segment taking the dislocation from one Peierls' valley

to a neighboring valley (Fig. 6). The kink is supposed

to

be able to

translate

freely along

the

dislocation, and

the

action of thermal waves of wavelength >./2>D,

where

D

is the kink width, will be

to

vibrate the entire

kink and make it radiate energy.

Consider a

kink in

a screw dislocation.

The

action of

the kink on

the

elastic waves, for small kink displace

ments, can be deduced from a

Hamiltonian

22

20 P. G. Klemens in Solid State Physics, edited

by

F. Seitz and

D.

Turnbull

(Academic Press Inc., New York, 1958), Vol. 7,

p.22.

21 The author

acknowledges gratefully

that Dr.

Eshelby gave

him the

opportunity

to compare with unpUblished results obtained

by Dr. Eshelby by methods different from those employed in

this paper.

22 t can be shown, by use of the definition of a force on a

dislocation

and the

reciprocal theorem of linear elasticity,

that

this procedure is right.

Peierls

volle

Peierls volley

FIG. 6. A kink of width D brings the dislocation from one Peierls'

valley into a neighboring one.

so that the equations of motion for the waves are

(55)

Here,

x

is the displacement of the kink,

u

is the stress

amplitude of the elastic wave, U is the stress amplitude

resolved

onto the

kink, and

V

is the

total

volume.

For

forced vibrations

mCY w

2

y)=Fo singt,

(56)

the rate at which energy is given to the oscillator when

y=O

and

y=O at t=O is, taking the

dominant

term,

F02

sin(g-w)t

7rF02

F · y = -

- -8 g-w) .

(57)

4m g-w 4m

By treating Eq. (55) this way and summing over all

shear lattice waves, it is found that for an oscillation

x= A singt,

the

rate of energy radiation is

(58)

The

kink vibrations are caused

by

incident thermal

waves

(59)

where mk is the effective kink mass. From Eqs. (58) and

(59) we deduce that all

the

incident waves up to some

k

m x

cause

the kink

to radiate at a rate

(60)

Considering only waves for which Aj2> D to vibrate the

kink as a whole,

we

must

put

(61)

According to Lothe and Hirth,3 the width of the

kink is

7r(

S

1

D=; 21rCTp ,

(62)

where S is the line tension and

Up

the Peierls' barrier,

and the kink energy is

(63)


] IP: 130.113.76.6 On: Wed, 03 Dec 2014 20:48:43


10/11

2124

l N S

L O T H E

when

T>6b/D,

Eq. 60) becomes

and the corresponding stress is

b

T>f:J

v'

b

T>(}-.

64)

(65)

In a close packed metal, say copper, the Bordoni peak

experiments

23

indicate as typical values

Wk ' 10-

2

j t

3

and D , 7b,

yielding

(J''''110(3kT/b

3

)XV/c,

T>f:J/7.

(66)

An estimate of the effect of thermal waves of half

wavelength appreciably shorter than

D

is needed.

To

this end it should be fair to consider the kink as a

segment of length

D

with a velocity normal to itself.

The

thermoelastic effect and the phonon viscosity effect

are negligible for a kink because of the short range of

the stress field.

The

most important damping mecha

nisms are then those of

Sec. III A,

and a likely value of

the stress due to those sources at a higher temperature is

( J ' , ~ ( 3 k T ) ~ X ~

5 b

3

D c'

67)

which is seen to be smaller

than

Eq. 66)

by

a factor

2/7.

As

W

k

,l/D, it is general that Eq. (65) will be the

more important contribution. This rough estimate of

the importance of the short wavelength phonons ignores

that

the kink is of finite length and

that

important

scattering effects might arise at the transition from the

kink

part

of the dislocation to that

part

lying in the

Peierls' valley. However, the kink width

D

will be of the

same order of magnitude as the wavelength of waves

of the frequency with which the dislocation vibrates in

the Peierls' barrier. Thus, elastic waves of shorter

wavelength will vibrate the dislocation lying in the

Peierls' valley beyond resonance, i.e., the vibrations

will be controlled

by

the mass and the line tension

rather

than

the Peierls' barrier. t follows

that

the

short wavelength phonons will make the entire disloca

tion radiate quite uniformly, with little distinction

between the kink segment and

that part of

the disloca

tion lying in the Peierls' valley, and thus no significant

scattering effects for short wavelength phonons

at

the

transition between the kink and the Peierls' barrier

locked dislocation would be expected.

A satisfactory theoretical treatment

of

the effect

of short wavelength phonons would require a model

which not only involves the translation of the kink with

preservation of shape, but which also includes all the

other degrees of freedom of the entire dislocation.

23

D.

O.

Thompson and D. K. Holmes

J

Appl Phys 30 525

(1959). • .

.

Before such a rather complicated analysis is attempted,

we cannot do much better

than

to introduce a cutoff of

the order of magnitude of Eq. 61) and estimate the

effect of phonons of short wavelength in the manner

explained above.

Finally, we want to present some considerations on

the self-consistency of the treatment of kink mobility.

t

must be required that the damping stress Eq. 67)

does not damp the kink to the extent

that

for

k k

max

=rr/D the kink motion is not mass controlled. Thus,

it must be required that

1(3kT)

b

3

W

max

mkWmai>- X .

5 b

3

D

c

68)

This inequality reduces to

1>3kTD/j.tb4,

which is well

fulfilled up to ordinary temperatures for reasonable

kink widths, say

D ,7b

and

3kT ,1O-2

j t

3

• Similarly,

it can be asserted that the direct radiation Eq. 64) is

much more important than the energy radiation taking

place because the oscillations are damped

by

the

stress Eq. (67).

The

radiation Eq. 58) gives rise to a frequency

dependent back force on the kink,

69)

For this force to be less than the inertial reaction, the

inequality

70)

must be fulfilled. This inequality reduces to

2c/D>n.

For

Qmax=ck

max

[Eq

(61)J,

this inequality fails to be

fulfilled by a factor 2/7r. Thus, for Qmax, the oscillations

are controlled about equally much

by

inertia and radia

tion resistance. This circumstance corresponds to the

well-known fact that, in the interaction of an electron

with radiation of a wavelength shorter

than

the

radius

of

the electron, the electron motion is radiation

resistance controlled.

Taking the effect of radiation resistance on the

kink oscillations into account would not change the

order of magnitude of our estimate.

The

fact remains

that the kink oscillations induced

by

the waves Qmax

are largely independent of the effect of shorter wave

lengths on the kink and give rise to a scattering over

shadowing the scattering due to the short wavelength

phonons. The mobility will be proportional to T down

to

T ,f:Jb/D.

IV. SUMMARY, DISCUSSION, AND CONCLUSION

There is evidence

that

under some conditions disloca

tions can move freely without thermal activation.

1

,2

This behavior is to be expected when the Peierls'

barrier for the straight dislocation or for the kink

segment is broken down by zero-point motion. The

various factors determining the mobility of dislocations

experiencing no Peierls' barrier have been discussed in

some detail.


] IP: 130.113.76.6 On: Wed, 03 Dec 2014 20:48:43


11/11

D I S LO C A TI O N

MO I L I T Y IN P U R E SL I P

2125

Two bulk relaxation processes are important: the

thermoelastic effect and the shear viscosity effect.

In

insulators at temperatures of the order the Debye

temperature

J

the thermoelastic effect and the shear

viscosity effect each give rise to a drag stress of the

order

(71)

but it should be kept in mind that the above estimate

for the shear-viscosity contribution may be an over

estimate. In metals the thermoelastic effect

is

unimport

ant, while the shear viscosity effect should be of about

the same magnitude as in insulators. With decreasing

temperature these effects go rapidly to zero, as cp

Ap.

Estimates of relaxation contributions in or near the

core, where the theories of thermoelasticity and

phonon viscosity do not readily apply, have been

attempted. Only the relaxations associated with the

strong anharmonicities

of

the Peierls-Nabarro structure

of the slip plane were found to give an appreciable

contribution, again typically of the order

(72)

at ordinary temperatures. This contribution also goes

rapidly to zero with decreasing temperature.

The phonon scattering processes can be divided into

two main types: scattering by the dislocation strain

field and scattering by dislocation vibrations. The first

process would cause a drag stress in the region

E/60X V c

Documents

Theory of Dislocation Mobility in Pure Slip - Lothe1962