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Vacancy formation and the thermal expansion of platinum
Scripta
METALLURGICA
Vol. 4, pp. 251-254, 1970 P r i n t e d in the U n i t e d States
VACANCY FORMATION
AND THE THERMAL
Pergamon
EXPANSION
OF PLATINUM
J. Van den Sype Department of M e t a l l u r g y a n d M a t e r i a l s U n i v e r s i t y of P e n n s y l v a n i a Philadelphia, Pa. 19104
(Received
Fig. 1 shows Platinum
some
in the temperature
surements
accurate
range
method.
1000 - 1 9 0 0 ° K . (1) o n t h e l i n e a r
They estimate
within 50 pprn. Kraftmakher
of m e a s u r i n g the results
the coefficient of K r a f t m a k h e r
Kraftmakher
data show increasing
other absolute
methods
present
is to show that these
paper
effect inherent
coefficient
of thermal
determining
vacancy
frequency
thermal
deviations
Curve
this technique.
T> 1400°K,
may be explained modulation
equilibrium
Technique
for Studying Thermal
data to be a method
In the temperature
For
the method(and
The purpose on the basis
technique
It is shown that this method
of t h e
of a v a c a n c y
for measuring
yields
the
a novel way of
conditions. Expansion
suitable conditions
the temperature
their
(Z) i n F i g . 1 s h o w s
t o Z6% a t 1900OK.
deviations
is heated under
of P t u s i n g t h e
from the optical comparison
s under
w. A s a r e s u l t ,
expansion
of
from the mea-
(Z) h a v e d e v e l o p e d
directly.
parameter
specimen
(1) i s d e r i v e d
are in good agreement.
in the temperature
expansion.
Modulation A wire
expansion
s e e r e f . 1), a m o u n t i n g
relaxation
Curve
expansion
the equation representing
(3) o n P t e m p l o y i n g
1000 - 1 4 0 0 ° K b o t h m e t h o d s
of t h e r m a l
and Cheremisina
of thermal
Science
2, 1970)
data o n t h e c o e f f i c i e n t
of K i r b y a n d R o t h r o c k
optical comparison
range
recent
February
Press,
of the specimen
of Metals
with an alternating exhibits
current
of
small oscillations
(at Zc~) about a m e a n value (TA) and this, in turn, produces periodic changes in its resistance (6R) and its length (54). The length fluctuations m a y be m e a s u r e d photometrically (Z). The resistance fluctuations give rise to a third harmonic component in the voltage drop across the specimen (4) i.e.,
zv~ =
where I and V s are the R M S
i
(i)
values of the current and 3rd harmonic voltage resp.
these considerations, it is clear that the coefficient of thermal expansion ~ m a y be
251
From
Inc
252
VACANCY
FORMATION
r e l a t e d to t h e t e m p e r a t u r e
AND THE THERMAL
c o e f f i c i e n t of r e s i s t a n c e
EXPANSION
OF PT
Vol.4,
(a) as f o l l o w s :
R'.I. 6~
8 :z~.v~
(z)
"
where R' and ~' are the resistance and length resp. of the specimen at 0°C. quantities on the right hand side of (P-)are readily measured, it.
No.
Since all
8 can be determined f r o m
In p a r t i c u l a r , a i s o b t a i n e d by d i f f e r e n t i a t i n g t h e r e s i s t a n c e v e r s u s t e m p e r a t u r e
c u r v e of t h e s a m p l e . E f f e c t s of V a c a n c y R e l a x a t i o n It i s c l e a r t h a t in o r d e r f o r ~ to r e p r e s e n t t h e e q u i l i b r i u m c o e f f i c i e n t of t h e r m a l e x p a n s i o n , t h e f r e q u e n c y of t h e t e m p e r a t u r e w h e r e ~ is the c h a r a c t e r i s t i c in t h e s a m p l e .
f l u c t u a t i o n s m u s t b e s u c h t h a t Z~<<~ -1
t i m e c o n n e c t e d with any r e l a x a t i o n p h e n o m e n o n o c c u r r i n g
F o r p u r e m e t a l s c l o s e to t h e i r m e l t i n g p o i n t , p a r t of t h e t h e r m a l
e x p a n s i o n i s due to t h e i n t r o d u c t i o n of v a c a n c i e s in t h e l a t t i c e and s i n c e t h i s p r o c e s s i s d i f f u s i o n l i m i t e d , t h e c o n d i t i o n Z~T<< 1 m a y n o t be s a t i s f i e d .
I n d e e d , if t h e f r e q u e n c y
i s l a r g e r t h a n t h e v a c a n c y l i f e t i m e at t h e a v e r a g e t e m p e r a t u r e ,
the v a c a n c i e s will he
f r o z e n out and t h e a p p a r e n t t h e r m a l e x p a n s i o n c o e f f i c i e n t w i l l b e l e s s t h e e q u i l i b r i u m one. form.
If t h e v a c a n c i e s o b e y l i n e a r k i n e t i c s , it is e a s y to put t h e a b o v e in q u a n t i t a t i v e It m a y b e s h o w n t h a t E q . (g) i s a l w a y s v a l i d , p r o v i d e d b o t h a and 8 a r e t a k e n as
frequency dependent:
~-1 +
4 o~~
~"
~
~s)
and
1-13~ +4o~ ~- po~ (U)) = L 1 + 4w ~ ,'a
1 ~/~
(4)
=o and 8o refer to the equilibrium values, while ~== and 8~ are defined as ,~o = ~
l
~R
8®
(-~.-T.)
=
xD
(5) xD
x D b e i n g the a t o m f r a c t i o n of v a c a n c i e s , vacancy formation energy,
56 ~1 (~-#)
~ the v a c a n c y l i f e t i m e .
Denoting by Ef the
PD the a d d i t i o n a l s c a t t e r i n g p e r a t o m f r a c t i o n of v a c a n c i e s
and 6 t h e r a t i o of t h e v a c a n c y v o l u m e to the a t o m i c v o l u m e , one h a s
~o
_
=co
p,
=.£u.
k~T .~
XD(TA)
(6)
4
Vol.
4, No.
4
VACANCY
F O R M A T I O N A N D THE T H E R M A L
E X P A N S I O N OF PT
253
13o_130o = -~ 6 . kTAa Ef • xD(TA)
9' i s t h e r e s i s t i v i t y t i o n of t h e r e l a x a t i o n
(7)
of t h e s a m p l e a t 0 ° C .
Equations
e f f e c t s of v a c a n c i e s
on ~ a n d ~.
(3) to (7) g i v e a c o m p l e t e From
descrip-
E q u a t i o n (Z) it is e v i d e n t
that m e a s u r e m e n t s of 5~ and Va over a wide frequency range and as a function of TA will yield information on % xD(TA) , PD and ~ under equilibrium conditions.
Effects of the
relaxation on the specific heat have been discussed elsewhere (5). The Results of K r a f t m a k h e r on Pt (3) The data f r o m curve (Z) w e r e taken at a temperature modulation frequency of Z0 Hz.
If the vacancy lifetime in Pt is s o m e w h a t larger than I0 m s e c ,
the positive
deviations f r o m the equilibrium coefficient can be readily understood in the following way.
Since the vacancy fluctuations are frozen out, the quantity
reported by
K r a f t m a k h e r is (cf. Eq.(g) ):
~k =
13~o ~--e-° (100
(8)
This is due to the fact that ao data w e r e used to compute his "coefficient of thermal expansion" (cf. ref. 6).
Positive deviations will then occur if (Io/a¢o> ~ o / ~ .
More
quantitatively, the "false" thermal expansion data are related to the vacancy parameters (provided the frequency was large enough to freeze out the vacancy relaxation): ak
= L-?
=o
_ 8 IE_/
"3]kT/~ xD (TA)
(9)
The observed deviation at 1900°K can be accounted for if w e take the following reason1 able values {at 1900°K): PD = 6 " 1 0 - 4 ~ c m / a t ° m fraction; 6 = ~; E f : I . 3 5 e V ; x D ( 1 9 0 0 ° K ) =0.3%.
This m a k e s our reinterpretation of the K r a f t m a k h e r data s e e m plausible.
It
is proposed that the modulation m e t h o d for m e a s u r i n g thermal expansion be extended to cover a wide enough frequency range so that both ~o/~o and ~=o/~=0m a y be determined oR the s a m e specimen.
It would then provide a powerful
tool
for
investigating
equilibrium properties of vacancies for materials w h e r e the quenching m e t h o d is difficult. A c k n o w l e d g m ent This is a contribution of the Laboratory for R e s e a r c h on the Structure of Matter, sponsored by the A d v a n c e d R e s e a r c h Project Agency, D e p a r t m e n t of Defense.
254
FORMATION AND THE THERMAL
VACANCY
EXPANSION
OF PT
Vol.
4, No. 4
}%eferences I.
R. K. Kirby and B. D. }%othrock, European Conference on Therrnophysical Properties of Solid Materials at High Temperatures (Nov. 1968).
2.
Ya. A. Kraftrnakher and I. M. Cheremisina, P M T F
3.
Ya. A. Kraftrnakher, Soy. Phys. Solid State 9, 1197 (1967).
4.
See eg., L. }%. Holland, ]. AppI. Phys. 3___4,Z350 (1963).
5.
J. Van den Sype, Phys. Star. Sol. (in press).
6.
Y. A. Kraftrnakher and E. B. Lanina, Soy. Phys. Solid State m7, 9Z (1965).
2__, 114 (1965).
24 22 20 T
O"
•
18
0 "-
16
-
2
o"
14 12 I0 8
I
I000
1200
I
I
I
1400 1600 1800 TEM PERATU RE, •K
2000
FIG. 1 Coefficient of thermal expansion of Platinum; line ( . . . . ): ref. Z.
solid line (
): r e f .
1; d a s h e d
METALLURGICA
Vol. 4, pp. 251-254, 1970 P r i n t e d in the U n i t e d States
VACANCY FORMATION
AND THE THERMAL
Pergamon
EXPANSION
OF PLATINUM
J. Van den Sype Department of M e t a l l u r g y a n d M a t e r i a l s U n i v e r s i t y of P e n n s y l v a n i a Philadelphia, Pa. 19104
(Received
Fig. 1 shows Platinum
some
in the temperature
surements
accurate
range
method.
1000 - 1 9 0 0 ° K . (1) o n t h e l i n e a r
They estimate
within 50 pprn. Kraftmakher
of m e a s u r i n g the results
the coefficient of K r a f t m a k h e r
Kraftmakher
data show increasing
other absolute
methods
present
is to show that these
paper
effect inherent
coefficient
of thermal
determining
vacancy
frequency
thermal
deviations
Curve
this technique.
T> 1400°K,
may be explained modulation
equilibrium
Technique
for Studying Thermal
data to be a method
In the temperature
For
the method(and
The purpose on the basis
technique
It is shown that this method
of t h e
of a v a c a n c y
for measuring
yields
the
a novel way of
conditions. Expansion
suitable conditions
the temperature
their
(Z) i n F i g . 1 s h o w s
t o Z6% a t 1900OK.
deviations
is heated under
of P t u s i n g t h e
from the optical comparison
s under
w. A s a r e s u l t ,
expansion
of
from the mea-
(Z) h a v e d e v e l o p e d
directly.
parameter
specimen
(1) i s d e r i v e d
are in good agreement.
in the temperature
expansion.
Modulation A wire
expansion
s e e r e f . 1), a m o u n t i n g
relaxation
Curve
expansion
the equation representing
(3) o n P t e m p l o y i n g
1000 - 1 4 0 0 ° K b o t h m e t h o d s
of t h e r m a l
and Cheremisina
of thermal
Science
2, 1970)
data o n t h e c o e f f i c i e n t
of K i r b y a n d R o t h r o c k
optical comparison
range
recent
February
Press,
of the specimen
of Metals
with an alternating exhibits
current
of
small oscillations
(at Zc~) about a m e a n value (TA) and this, in turn, produces periodic changes in its resistance (6R) and its length (54). The length fluctuations m a y be m e a s u r e d photometrically (Z). The resistance fluctuations give rise to a third harmonic component in the voltage drop across the specimen (4) i.e.,
zv~ =
where I and V s are the R M S
i
(i)
values of the current and 3rd harmonic voltage resp.
these considerations, it is clear that the coefficient of thermal expansion ~ m a y be
251
From
Inc
252
VACANCY
FORMATION
r e l a t e d to t h e t e m p e r a t u r e
AND THE THERMAL
c o e f f i c i e n t of r e s i s t a n c e
EXPANSION
OF PT
Vol.4,
(a) as f o l l o w s :
R'.I. 6~
8 :z~.v~
(z)
"
where R' and ~' are the resistance and length resp. of the specimen at 0°C. quantities on the right hand side of (P-)are readily measured, it.
No.
Since all
8 can be determined f r o m
In p a r t i c u l a r , a i s o b t a i n e d by d i f f e r e n t i a t i n g t h e r e s i s t a n c e v e r s u s t e m p e r a t u r e
c u r v e of t h e s a m p l e . E f f e c t s of V a c a n c y R e l a x a t i o n It i s c l e a r t h a t in o r d e r f o r ~ to r e p r e s e n t t h e e q u i l i b r i u m c o e f f i c i e n t of t h e r m a l e x p a n s i o n , t h e f r e q u e n c y of t h e t e m p e r a t u r e w h e r e ~ is the c h a r a c t e r i s t i c in t h e s a m p l e .
f l u c t u a t i o n s m u s t b e s u c h t h a t Z~<<~ -1
t i m e c o n n e c t e d with any r e l a x a t i o n p h e n o m e n o n o c c u r r i n g
F o r p u r e m e t a l s c l o s e to t h e i r m e l t i n g p o i n t , p a r t of t h e t h e r m a l
e x p a n s i o n i s due to t h e i n t r o d u c t i o n of v a c a n c i e s in t h e l a t t i c e and s i n c e t h i s p r o c e s s i s d i f f u s i o n l i m i t e d , t h e c o n d i t i o n Z~T<< 1 m a y n o t be s a t i s f i e d .
I n d e e d , if t h e f r e q u e n c y
i s l a r g e r t h a n t h e v a c a n c y l i f e t i m e at t h e a v e r a g e t e m p e r a t u r e ,
the v a c a n c i e s will he
f r o z e n out and t h e a p p a r e n t t h e r m a l e x p a n s i o n c o e f f i c i e n t w i l l b e l e s s t h e e q u i l i b r i u m one. form.
If t h e v a c a n c i e s o b e y l i n e a r k i n e t i c s , it is e a s y to put t h e a b o v e in q u a n t i t a t i v e It m a y b e s h o w n t h a t E q . (g) i s a l w a y s v a l i d , p r o v i d e d b o t h a and 8 a r e t a k e n as
frequency dependent:
~-1 +
4 o~~
~"
~
~s)
and
1-13~ +4o~ ~- po~ (U)) = L 1 + 4w ~ ,'a
1 ~/~
(4)
=o and 8o refer to the equilibrium values, while ~== and 8~ are defined as ,~o = ~
l
~R
8®
(-~.-T.)
=
xD
(5) xD
x D b e i n g the a t o m f r a c t i o n of v a c a n c i e s , vacancy formation energy,
56 ~1 (~-#)
~ the v a c a n c y l i f e t i m e .
Denoting by Ef the
PD the a d d i t i o n a l s c a t t e r i n g p e r a t o m f r a c t i o n of v a c a n c i e s
and 6 t h e r a t i o of t h e v a c a n c y v o l u m e to the a t o m i c v o l u m e , one h a s
~o
_
=co
p,
=.£u.
k~T .~
XD(TA)
(6)
4
Vol.
4, No.
4
VACANCY
F O R M A T I O N A N D THE T H E R M A L
E X P A N S I O N OF PT
253
13o_130o = -~ 6 . kTAa Ef • xD(TA)
9' i s t h e r e s i s t i v i t y t i o n of t h e r e l a x a t i o n
(7)
of t h e s a m p l e a t 0 ° C .
Equations
e f f e c t s of v a c a n c i e s
on ~ a n d ~.
(3) to (7) g i v e a c o m p l e t e From
descrip-
E q u a t i o n (Z) it is e v i d e n t
that m e a s u r e m e n t s of 5~ and Va over a wide frequency range and as a function of TA will yield information on % xD(TA) , PD and ~ under equilibrium conditions.
Effects of the
relaxation on the specific heat have been discussed elsewhere (5). The Results of K r a f t m a k h e r on Pt (3) The data f r o m curve (Z) w e r e taken at a temperature modulation frequency of Z0 Hz.
If the vacancy lifetime in Pt is s o m e w h a t larger than I0 m s e c ,
the positive
deviations f r o m the equilibrium coefficient can be readily understood in the following way.
Since the vacancy fluctuations are frozen out, the quantity
reported by
K r a f t m a k h e r is (cf. Eq.(g) ):
~k =
13~o ~--e-° (100
(8)
This is due to the fact that ao data w e r e used to compute his "coefficient of thermal expansion" (cf. ref. 6).
Positive deviations will then occur if (Io/a¢o> ~ o / ~ .
More
quantitatively, the "false" thermal expansion data are related to the vacancy parameters (provided the frequency was large enough to freeze out the vacancy relaxation): ak
= L-?
=o
_ 8 IE_/
"3]kT/~ xD (TA)
(9)
The observed deviation at 1900°K can be accounted for if w e take the following reason1 able values {at 1900°K): PD = 6 " 1 0 - 4 ~ c m / a t ° m fraction; 6 = ~; E f : I . 3 5 e V ; x D ( 1 9 0 0 ° K ) =0.3%.
This m a k e s our reinterpretation of the K r a f t m a k h e r data s e e m plausible.
It
is proposed that the modulation m e t h o d for m e a s u r i n g thermal expansion be extended to cover a wide enough frequency range so that both ~o/~o and ~=o/~=0m a y be determined oR the s a m e specimen.
It would then provide a powerful
tool
for
investigating
equilibrium properties of vacancies for materials w h e r e the quenching m e t h o d is difficult. A c k n o w l e d g m ent This is a contribution of the Laboratory for R e s e a r c h on the Structure of Matter, sponsored by the A d v a n c e d R e s e a r c h Project Agency, D e p a r t m e n t of Defense.
254
FORMATION AND THE THERMAL
VACANCY
EXPANSION
OF PT
Vol.
4, No. 4
}%eferences I.
R. K. Kirby and B. D. }%othrock, European Conference on Therrnophysical Properties of Solid Materials at High Temperatures (Nov. 1968).
2.
Ya. A. Kraftrnakher and I. M. Cheremisina, P M T F
3.
Ya. A. Kraftrnakher, Soy. Phys. Solid State 9, 1197 (1967).
4.
See eg., L. }%. Holland, ]. AppI. Phys. 3___4,Z350 (1963).
5.
J. Van den Sype, Phys. Star. Sol. (in press).
6.
Y. A. Kraftrnakher and E. B. Lanina, Soy. Phys. Solid State m7, 9Z (1965).
2__, 114 (1965).
24 22 20 T
O"
•
18
0 "-
16
-
2
o"
14 12 I0 8
I
I000
1200
I
I
I
1400 1600 1800 TEM PERATU RE, •K
2000
FIG. 1 Coefficient of thermal expansion of Platinum; line ( . . . . ): ref. Z.
solid line (
): r e f .
1; d a s h e d