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Dislocation damping and interstitial pinning in high purity single crystal tantalum
Seripta
METALLURGICA
V o l . 3, pp. 8 5 5 - 8 5 8 , 1969 P r i n t e d in t h e U n i t e d S t a t e s
Pergamon
Press,
Inc.
DISLOCATION DAMPING AND INTERSTITIAL PINNING IN HIGH PURITY SINGLE CRYSTAL TANTALUM*
R. C. Sanders + , S. R. Painter and T. J. Turner Department of Physics, Wake Forest University Winston-Salem, North Carolina 27109
(Received
September
22,
1969)
The purpose of this note is twofold, first to examine the application of the vibrating string model of dislocation damping to a bcc system, and second, to examine the use of the temperature dependence of the loop length to obtain dislocation-solute binding energies. Granato and L~cke (i) have reviewed the vibrating string model of dislocation damping and have pointed out a number of weaknesses of the theory.
Chief among these, for the present
investigation, are that it is an absolute zero theory and the interaction of the dislocations with the lattice is neglected.
That is, a very small Peierl's force is assumed.
The Granato-
LHcke theory, hereafter referred to as the G-L theory, has been applied with most success to fcc systems. difficulty.
Neglecting the Peierl's force in the bcc system seems to be a more serious In fact Chambers (2) has suggested that essentially all dislocations in bcc
tantalum must be thermally activated over potential barriers while in fcc systems only a small fraction must depend on thermal activation. Recently in this journal Dey (3) has described a method of interpreting dislocation damping data to obtain solute-atom dislocation binding energies.
He assumes, as did Granato
and L~cke (4) that the binding energy may be obtained from the equation a/L
c
=
C
=
C
o
exp(EB/kT)
(I)
where a is the lattice parameter, L c is the minor pinning length, C is the impurity concentration and EB is the binding energy. dence of the loop length L
c
Thus if this equation is valid the temperature depen-
will give EB.
The hysteretic decrement is given from the G-L theory as C AN
=
~
C exp(- ~
) c
(2)
o
Supported in part by the U. S. Atomic Energy Commission. Portions of this work were presented at the International Conference on Vacancies and Interstitials in Metals, J~lich, 1968. +Present address, The Western Electric Company, Winston-Salem, N. C.
855
856
DISLOCATION
DAMPING
AND
INTERSTITIAL
PINNING
IN Ta
Vol.
3, No.
ii
where e ° is the strain amplitude averaged over the length of the specimen (4) and the constants C 1 and C~ contain such factors as the dislocation density, the modulus of the material and the orientation of the specimen.
They are assumed constant for most studies.
There are
three ways that equation (2) can be used to obtain the temperature dependence of L .
The
C
first two "traditional" ways come from the slope and intercept of a G-L plot, that is a plot s/2 Thus the slope is given by Ce/L c and the intercept by Cl/L c .
of in AH ¢o ½ vrs. i/¢ o.
The third way to use this equation, suggested by Dey (3) is to plot Ins o vrs.i/T for a constant ~H"
If equation (i) is assumed to be valid and in s ° is negligible compared to ¢o then the
slope of such a plot is EB/k.
Dey suggested using ¢cr' the strain amplitude at which ampli-
tude dependence is observed, but this is not essential. The amplitude independent decrement $I is proportional to the fourth power of the loop length L
and a damping constant B, which is a measure of lattice resistance.
The tempera-
C
ture dependence of ~I has been used by Stewart and Fiore (5) to obtain E B by assuming equation (i) to be valid.
They also used the technique suggested by
Dey and found agreement in
a damping study on single-crystal Fe-18 Cr-20 Ni alloy. In this investigation these techniques are applied to data obtained on a three pass electron-beam zone refined single crystal of tantalum containing < 3.5 ppm oxygen supplied by the Materials Research Corporation. few degrees.
The axis of the specimen is along to within a
A Marx composite piezoelectric oscillator (6) was used with the specimen
resonating in its fundamental mode of 50 KHz (7).
T E M P --
376"
'!
K
O
394"
51 /
O
376 "K
4
o
338" K
3
o
32"t.'K
0
30e'K
2
V
')94"K
K
i
354"K 4 --
x
X ~
3
<~
5
6
. . . . . . . . . . . . . .
7
8
V
9
5
,6'
,6'
,6"
E s X I0
Es
FIG. I. Decrement of single crystal tantalum yrs. strain amplitude.
FIG. 2. The hysteretic decrement 5H yrs. strain at various temperatures.
- 1
Vol.
3,
No.
ii
DISLOCATION
DAMPING
The data reported here were obtained and brought
to a temperature
AND
INTERSTITIAL
PINNING
in the following manner.
Oxygen is assumed
Fig. i illustrates distinguished
temperatures.
plotted
in Fig. 4.
of the diffusion
to be pinning the dislocations
typical data and shows how ~H the amplitude
from ~ H the hysteretic
various
decrement.
The intercepts
If we assume equation
line give ¢
coefficient
(8).
independent
Fig. 2 shows the A H vrs.
of the dotted
tempera-
that the data were reproducible
This was necessary because of the magnitude
of oxygen in tantalum.
at decreasing
The specimen was allowed to equilibrate
for 6 to 12 hours and it was determined
at each temperature.
857
Damping characteristics
of the specimen as a function of strain amplitude were then determined
at each temperature
Ta
The system was evacuated
of 400 °K and held there for 24 hours.
tures from 400 ° to 200°K over a period of 72 hours.
IN
decrement
is
strain amplitude
yrs. temperature.
o (i) to be valid we obtain E B = 0.023 eV.
at
These are In Fig. 3
we have G-L plots whose slopes were used to obtain L also plotted
in Fig. 4.
The intercepts
Again assuming equation
as a function of temperature which is c (I) to be valid we have an E B of 0.028 eVo
of the G-L plots give a value for the product
of the nodal loop length L N
_S
cubed and the dislocation i
cm
density
of 2.8 x i0
(9) we have L N = 1.4 x I0
cm.
from 0.6 x i0
Assuming a dislocation
density
Values of B the damping constant calculated
_3
41 (4) range
cm.
of from
-3
to 2.1 x iC
dyne sec/cm ~ over the range 200 ° to 400°K. S
XIO
fs 6
'-
•
®
']\'%\\
394"K
o~
°
I
I
."r io
1.2
1.4
i.s
I.s
2.0
~
~
4
I0 / E s FIG, 3. Granato-L~cke temperatures.
plots for various
~
'~
o
,~
-4
LcXlO
(CM)
FIG. 4. a) Strain amplitude from intercept of FIG. 2 yrs. I/T. b) L c obtained from slopes of the G-L plots vrs. I/T.
858
DISLOCATION
DAMPING
AND
INTERSTITIAL
PINNING
IN T a
Vol.
3, No.
ii
These reasonable values for the parameters involved plus the linearity of the G-L plots lead us to suggest that a kink model, which considers the Peierl's stress,
is not required and
that the vibrating string model is adequate in this system in this temperature range.
The
agreement of the activation energies cited above is added support for the G-L theory.
How-
ever, in spite of this, we find a value of the dislocation interstitial binding energy which is unrealistic.
The value 0.025 ~ 0.004 eV is less than kT for some of the temperatures
employed, so the oxygen interstitial would not be bound.
The problem is not resolved with the
addition of a thermal force of the type given by Saul and Bauer (i0) since this involves a reinterpretation of the G-L slope with no significant change in EB.
Thus we conclude that the
methods described above do not necessarily yield EB and that equation (I) is not valid for this bcc system.
Further support for this contention comes from the measurements of Formby
and 0wen (ii) who found an EB of 0.45 eV by measuring the critical temperature at which discontinuous yield disappears in tantalum, and from an analysis of our
data following a method
of Saul and Bauer (i0) which yields an EB of approximately 0.3 eV (7). Apparently the energy of 0.025 eV obtained from the temperature dependence of Lc is an effective activation energy which results from two competing processes tending to pin and depin the dislocations.
Thus we do not measure EB since it is the energy associated with only
one of these processes. References i.
A. V. Granato and K. L~cke, Physical Acoustics, Vol. IV, ed. by W. P. Mason, Academic Press, N. Y., N. Y., 1966, p. 249.
2.
R. H. Chambers, App. Phys. Left., 2, 165 (1963).
3.
B. N. Dey, Scripta Met., 2, 279 (1968).
4.
A. V. Granato and K. LUcke, J. App. Phys., 27, 583, 789 (1956).
5.
D. C. Stewart and N. F. Fiore, Scripta Met., ~, 93 (1969).
6.
J. Marx, Rev. Sci. Instrum., 22, 503 (1951).
7.
Further details can be obtained from the thesis of R. C. Sanders. available from the authors.
8.
S. H. Carpenter and G. S. Baker, J. Appl. Phys., 36, 1733 (1965).
9.
W. A. Spitzig and T.E. Mitchell, Acta Met., 14, 1311 (1966).
i0. R. H. Saul and C. L. Bauer, J. Appl. Phys., 39, 1469 (1968). Ii. C. L. Formby and W. S. Owen, Phil. Mag., 13, 41 (1966).
Summaries are
METALLURGICA
V o l . 3, pp. 8 5 5 - 8 5 8 , 1969 P r i n t e d in t h e U n i t e d S t a t e s
Pergamon
Press,
Inc.
DISLOCATION DAMPING AND INTERSTITIAL PINNING IN HIGH PURITY SINGLE CRYSTAL TANTALUM*
R. C. Sanders + , S. R. Painter and T. J. Turner Department of Physics, Wake Forest University Winston-Salem, North Carolina 27109
(Received
September
22,
1969)
The purpose of this note is twofold, first to examine the application of the vibrating string model of dislocation damping to a bcc system, and second, to examine the use of the temperature dependence of the loop length to obtain dislocation-solute binding energies. Granato and L~cke (i) have reviewed the vibrating string model of dislocation damping and have pointed out a number of weaknesses of the theory.
Chief among these, for the present
investigation, are that it is an absolute zero theory and the interaction of the dislocations with the lattice is neglected.
That is, a very small Peierl's force is assumed.
The Granato-
LHcke theory, hereafter referred to as the G-L theory, has been applied with most success to fcc systems. difficulty.
Neglecting the Peierl's force in the bcc system seems to be a more serious In fact Chambers (2) has suggested that essentially all dislocations in bcc
tantalum must be thermally activated over potential barriers while in fcc systems only a small fraction must depend on thermal activation. Recently in this journal Dey (3) has described a method of interpreting dislocation damping data to obtain solute-atom dislocation binding energies.
He assumes, as did Granato
and L~cke (4) that the binding energy may be obtained from the equation a/L
c
=
C
=
C
o
exp(EB/kT)
(I)
where a is the lattice parameter, L c is the minor pinning length, C is the impurity concentration and EB is the binding energy. dence of the loop length L
c
Thus if this equation is valid the temperature depen-
will give EB.
The hysteretic decrement is given from the G-L theory as C AN
=
~
C exp(- ~
) c
(2)
o
Supported in part by the U. S. Atomic Energy Commission. Portions of this work were presented at the International Conference on Vacancies and Interstitials in Metals, J~lich, 1968. +Present address, The Western Electric Company, Winston-Salem, N. C.
855
856
DISLOCATION
DAMPING
AND
INTERSTITIAL
PINNING
IN Ta
Vol.
3, No.
ii
where e ° is the strain amplitude averaged over the length of the specimen (4) and the constants C 1 and C~ contain such factors as the dislocation density, the modulus of the material and the orientation of the specimen.
They are assumed constant for most studies.
There are
three ways that equation (2) can be used to obtain the temperature dependence of L .
The
C
first two "traditional" ways come from the slope and intercept of a G-L plot, that is a plot s/2 Thus the slope is given by Ce/L c and the intercept by Cl/L c .
of in AH ¢o ½ vrs. i/¢ o.
The third way to use this equation, suggested by Dey (3) is to plot Ins o vrs.i/T for a constant ~H"
If equation (i) is assumed to be valid and in s ° is negligible compared to ¢o then the
slope of such a plot is EB/k.
Dey suggested using ¢cr' the strain amplitude at which ampli-
tude dependence is observed, but this is not essential. The amplitude independent decrement $I is proportional to the fourth power of the loop length L
and a damping constant B, which is a measure of lattice resistance.
The tempera-
C
ture dependence of ~I has been used by Stewart and Fiore (5) to obtain E B by assuming equation (i) to be valid.
They also used the technique suggested by
Dey and found agreement in
a damping study on single-crystal Fe-18 Cr-20 Ni alloy. In this investigation these techniques are applied to data obtained on a three pass electron-beam zone refined single crystal of tantalum containing < 3.5 ppm oxygen supplied by the Materials Research Corporation. few degrees.
The axis of the specimen is along
A Marx composite piezoelectric oscillator (6) was used with the specimen
resonating in its fundamental mode of 50 KHz (7).
T E M P --
376"
'!
K
O
394"
51 /
O
376 "K
4
o
338" K
3
o
32"t.'K
0
30e'K
2
V
')94"K
K
i
354"K 4 --
x
X ~
3
<~
5
6
. . . . . . . . . . . . . .
7
8
V
9
5
,6'
,6'
,6"
E s X I0
Es
FIG. I. Decrement of single crystal tantalum yrs. strain amplitude.
FIG. 2. The hysteretic decrement 5H yrs. strain at various temperatures.
- 1
Vol.
3,
No.
ii
DISLOCATION
DAMPING
The data reported here were obtained and brought
to a temperature
AND
INTERSTITIAL
PINNING
in the following manner.
Oxygen is assumed
Fig. i illustrates distinguished
temperatures.
plotted
in Fig. 4.
of the diffusion
to be pinning the dislocations
typical data and shows how ~H the amplitude
from ~ H the hysteretic
various
decrement.
The intercepts
If we assume equation
line give ¢
coefficient
(8).
independent
Fig. 2 shows the A H vrs.
of the dotted
tempera-
that the data were reproducible
This was necessary because of the magnitude
of oxygen in tantalum.
at decreasing
The specimen was allowed to equilibrate
for 6 to 12 hours and it was determined
at each temperature.
857
Damping characteristics
of the specimen as a function of strain amplitude were then determined
at each temperature
Ta
The system was evacuated
of 400 °K and held there for 24 hours.
tures from 400 ° to 200°K over a period of 72 hours.
IN
decrement
is
strain amplitude
yrs. temperature.
o (i) to be valid we obtain E B = 0.023 eV.
at
These are In Fig. 3
we have G-L plots whose slopes were used to obtain L also plotted
in Fig. 4.
The intercepts
Again assuming equation
as a function of temperature which is c (I) to be valid we have an E B of 0.028 eVo
of the G-L plots give a value for the product
of the nodal loop length L N
_S
cubed and the dislocation i
cm
density
of 2.8 x i0
(9) we have L N = 1.4 x I0
cm.
from 0.6 x i0
Assuming a dislocation
density
Values of B the damping constant calculated
_3
41 (4) range
cm.
of from
-3
to 2.1 x iC
dyne sec/cm ~ over the range 200 ° to 400°K. S
XIO
fs 6
'-
•
®
']\'%\\
394"K
o~
°
I
I
."r io
1.2
1.4
i.s
I.s
2.0
~
~
4
I0 / E s FIG, 3. Granato-L~cke temperatures.
plots for various
~
'~
o
,~
-4
LcXlO
(CM)
FIG. 4. a) Strain amplitude from intercept of FIG. 2 yrs. I/T. b) L c obtained from slopes of the G-L plots vrs. I/T.
858
DISLOCATION
DAMPING
AND
INTERSTITIAL
PINNING
IN T a
Vol.
3, No.
ii
These reasonable values for the parameters involved plus the linearity of the G-L plots lead us to suggest that a kink model, which considers the Peierl's stress,
is not required and
that the vibrating string model is adequate in this system in this temperature range.
The
agreement of the activation energies cited above is added support for the G-L theory.
How-
ever, in spite of this, we find a value of the dislocation interstitial binding energy which is unrealistic.
The value 0.025 ~ 0.004 eV is less than kT for some of the temperatures
employed, so the oxygen interstitial would not be bound.
The problem is not resolved with the
addition of a thermal force of the type given by Saul and Bauer (i0) since this involves a reinterpretation of the G-L slope with no significant change in EB.
Thus we conclude that the
methods described above do not necessarily yield EB and that equation (I) is not valid for this bcc system.
Further support for this contention comes from the measurements of Formby
and 0wen (ii) who found an EB of 0.45 eV by measuring the critical temperature at which discontinuous yield disappears in tantalum, and from an analysis of our
data following a method
of Saul and Bauer (i0) which yields an EB of approximately 0.3 eV (7). Apparently the energy of 0.025 eV obtained from the temperature dependence of Lc is an effective activation energy which results from two competing processes tending to pin and depin the dislocations.
Thus we do not measure EB since it is the energy associated with only
one of these processes. References i.
A. V. Granato and K. L~cke, Physical Acoustics, Vol. IV, ed. by W. P. Mason, Academic Press, N. Y., N. Y., 1966, p. 249.
2.
R. H. Chambers, App. Phys. Left., 2, 165 (1963).
3.
B. N. Dey, Scripta Met., 2, 279 (1968).
4.
A. V. Granato and K. LUcke, J. App. Phys., 27, 583, 789 (1956).
5.
D. C. Stewart and N. F. Fiore, Scripta Met., ~, 93 (1969).
6.
J. Marx, Rev. Sci. Instrum., 22, 503 (1951).
7.
Further details can be obtained from the thesis of R. C. Sanders. available from the authors.
8.
S. H. Carpenter and G. S. Baker, J. Appl. Phys., 36, 1733 (1965).
9.
W. A. Spitzig and T.E. Mitchell, Acta Met., 14, 1311 (1966).
i0. R. H. Saul and C. L. Bauer, J. Appl. Phys., 39, 1469 (1968). Ii. C. L. Formby and W. S. Owen, Phil. Mag., 13, 41 (1966).
Summaries are