theory of dislocation mobility in pure slip - lothe1962
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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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8/19/2019 Theory of Dislocation Mobility in Pure Slip - Lothe1962
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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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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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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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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
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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
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D I S L O C T I O N
MO B I L I T Y IN PURE SLI P
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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?-
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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
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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)
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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.
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MO I L I T Y IN P U R E SL I P
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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