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Pressure and temperature dependence of defect formation and migration volumes
112
ACTA
~ETALL~RGICA.
~--Time (hr)
ctplutonium Slope
_--
( 1O-s ,uw cm/hr)
Normalised
0
52.5,
1.000
1000
33.0,
2000
17.3,
slope
TSlope -_.
4% Al 6 stabilized plutonium
(1O-s
Normalised slope _-._
,uw cm/hr)
11.46
1.000
.629
10.025
c.876)
.330
9.725
.849
3000
8.20
.156
8.46
.739
4000
3.71
.@706
6.78
.592
5000
1.67,
.0319
5.62,
.491
6000
0.715
.0136
4.39,
.384
7000
0.412
.00785
3.04,
.266
Slope
corrected for change in absoluta~mrtgnitude
a sample of S-plutonium nium at &W’K
stabilized
A.E.R.E.
with 3.4% alumi-
for 1200 hr without observing signs
Hurwell
Clarendon, Laboratory
E. KING J. A. LEE K. M[ENI)ELSSONN
D. A. WIGLEV References 1. J. A. LEE, K. MENDELSSOHN and D. A. WIOLEY, Phys. (1962). C. E. OLSEN 1129 (1963).
ELLIOTT,
Met. 11,
* floceived
and G. H. VINEYARD,
Acta
July 12, 1963.
Pressure and temperature dependence of defect formation and migration volumes* The volumes of defect formation and activation obtained from high pressure experiments can be used to estimate the contribution of point defects to the
.--
compressibility and thermal expansion coefficients of a crystal. We first consider the variation of vacancy formation volume with pressure. The formation volume at pressure P is V,(P)
=
V’(P)
-
V(P)
(1)
where V’ is the volume of a crystal containing a single vacancy and V is the volume of a corresponding perfect crystal. If p and p’ are the compressibilities of the perfect and imperfect crystals respectively, and if we assume that the compressibilities are independent of temperature, then equation (1) becomes VU(P) = V,(o)[l -
j?P] -
A,W(o)P
(2)
where A/3 = @’ - fi. The relative change in compressibility on introducing a vacancy can be taken as inversely proportional to the number of atoms in the crystal so that Aa -=_ P K, being a constant,
Oxford
Letters 1, 325
--
-due to a change in form factor of specimen.
of saturation. It may therefore be of interest to report that we have found clear evidence of saturation in a similar alloy (with 4% Al) after holding it at -5’K for over 7000 hr. This result is, however, not inconsistent with that of Elliott et al., since the accompanying Table 1 shows that in ~-plutonium saturation effects are hardly not*iceable after 1200 hr. It can be seen from the table that while after 7000 hr at -5’K, the rate of increase in resistance of u-plutonium has dropped to about 1 per cent of its initial value, that of S-plutonium has still over 25 per cent of its initial value. Both these specimens were made from the same stock and thus have the same self-heat except for the aluminium content. It may also be not.ed that these rates of resistance increase do not follow a true exponential law of the form ap/& (~~~~“~~~~*~-p). We are most grateful to Dr. Elliott for acquainting us with their results in advance of publication.
2. R. 0.
12, 1964
TABLE 1
~ -.-
t
VOL.
v,(P)
%
(3)
N
Equation (2) then becomes
= v,(o) -
/W,(o)
+
&VIP
(4)
wbeing the atomic volume. Precisely similar reasoning for the migration volume V’ leads to V:(P) = V:(o) -
/T[V:(o) + K,%]P
where Kpz is defined by
with A/3$ = /3: - ,!Y, p: being the compressibility of a crystal containing a saddle point configuration. Going through a similar procedure for the temperature dependence of V, and V: shows that the volumes
LETTERS
TO
THE
at T are related to those at T, by V,(T)
=
V,IT,)
+ dV,(T,)
V:(T) = V:( T,) + a[ V:( T,) + X,%# where K,
and fi,:
(7)
JMl(T - T,)
+
T -
27,) (8)
113
EDITOR
The ohserved activation volume depends on pressure according to vobs.
v
=
VIp”“~ therefore
VJo) - y [V,W + Kg]
(16)
are defined by &--CC oc
K, Z=Z----iV
(9)
(10) a, CL’and a$ being the thermal expansion coefficients (assumed to be independent of temperature) of a perfect crystal, a crystal containing a vacancy and a crystal containing a saddle point configuration respectively. Equations (4), (5), (7) and (8) show that the contribution of point defects, represented by the R’s, can be obtained if enough data are available. A relation between K, and K, can be obtained from Gruneisen’s equation
(11) y being Gruneisen’s constant and C, the heat capacity at constant volume. Writing equation (11) for the imperfect crystal and forming the ratio (CL’- tcffa shows that K,
= K,.
Similarly, it is easiIy shown that K,:=K
: P -+,
(13)
The & aris:es in equation (13) because the number of degrees OSfreedom is one less in the activated state than in a normal state. In equa,tions (12) and (13), it was assumed that C,, C,’ and Czl: were proportional to the number of degrees of freedom. The only data in existence to which any of these equations can be apphed are the pressure variation of the activation volume for difiusion in sodium.(i) In keeping with the small ion core and disordered vacancy structure in sodium, we will assume that Vt << V,. Differentiating equation (4) gives 8VQ &
= -B[V,Co,
(14)
+ IQ].
But the vacancy concentration depends on pressure according tot n,(P) = n~{o~ exp
i
-
PVvfof - - P2 av, -
___ kT
1
2Kr ( ap 1p=o
(15)
Plotting the observed activation volume for di~usion(l) against pressure, taking the initial slope and comparing with equation (16) gives K, = 4.8, i.e. a 1% vacancy concentration in sodium increases its compressibility by 4.8%. No other independent calculations are available for this quantity in sodium, but for f.c.c. metals Dienes@) estimates that a 1% vacancy concentration increases fi by 1%. It is to be expected that a vacancy would have a greater effect on the compressibility in sodium than in f.c.c. metals. The variation of activation volume wTithtemperature has not been measured, but the present results suggest that according to equation (12), a 1% vacancy concentration also increases the coefficient of thermal expansion by 4.8%. I,. A. GIRIFALCO School of Metallurgical Engineering and Laboratory for Research on the Structure of Matter University of Pennsylvania
Philadelphia, Pennsylvania References 1. N. H. NACRTRIEB, J. A. WEIL, E. CATALANO LAWSON, J. C’he’hem. P&p. 20, 1189 (1952). 2. G. J. DIENES, Phys. Rev. 86, 228 (1952).
and A. W.
* Received July 8, 1963. t The factor f in the second term in the exponential arises from the fact that equation (15) is obtained by expanding the Gibbs free energy of formation in a power series in P. The second derivative of the free energy is just the fcrst derivative of the volume.
Recovery of the electrical resistivity of copper, silver and gold after small plastic deformations at - 195°C * Nany experiments have been performed on the annealing of point defects after plastic deformation at low temperatures. In all these cases however, the degree of deformation is rather high, involving a large defect concentration, the annealing of which is easy to measure. See for instance Refs. l-6. In this work a first attempt was made to investigate the annealing after deformations amounting to a few per cent or less. The specimen material was 99.999% pure polycrystalline wire of 0.20-0.25 mm dia. Cu, Ag or Au. Prior to the deformation the wires were annealed for at least 1 hr at 450°C in air (Au) or in vac~o (Ag and Cu).
ACTA
~ETALL~RGICA.
~--Time (hr)
ctplutonium Slope
_--
( 1O-s ,uw cm/hr)
Normalised
0
52.5,
1.000
1000
33.0,
2000
17.3,
slope
TSlope -_.
4% Al 6 stabilized plutonium
(1O-s
Normalised slope _-._
,uw cm/hr)
11.46
1.000
.629
10.025
c.876)
.330
9.725
.849
3000
8.20
.156
8.46
.739
4000
3.71
.@706
6.78
.592
5000
1.67,
.0319
5.62,
.491
6000
0.715
.0136
4.39,
.384
7000
0.412
.00785
3.04,
.266
Slope
corrected for change in absoluta~mrtgnitude
a sample of S-plutonium nium at &W’K
stabilized
A.E.R.E.
with 3.4% alumi-
for 1200 hr without observing signs
Hurwell
Clarendon, Laboratory
E. KING J. A. LEE K. M[ENI)ELSSONN
D. A. WIGLEV References 1. J. A. LEE, K. MENDELSSOHN and D. A. WIOLEY, Phys. (1962). C. E. OLSEN 1129 (1963).
ELLIOTT,
Met. 11,
* floceived
and G. H. VINEYARD,
Acta
July 12, 1963.
Pressure and temperature dependence of defect formation and migration volumes* The volumes of defect formation and activation obtained from high pressure experiments can be used to estimate the contribution of point defects to the
.--
compressibility and thermal expansion coefficients of a crystal. We first consider the variation of vacancy formation volume with pressure. The formation volume at pressure P is V,(P)
=
V’(P)
-
V(P)
(1)
where V’ is the volume of a crystal containing a single vacancy and V is the volume of a corresponding perfect crystal. If p and p’ are the compressibilities of the perfect and imperfect crystals respectively, and if we assume that the compressibilities are independent of temperature, then equation (1) becomes VU(P) = V,(o)[l -
j?P] -
A,W(o)P
(2)
where A/3 = @’ - fi. The relative change in compressibility on introducing a vacancy can be taken as inversely proportional to the number of atoms in the crystal so that Aa -=_ P K, being a constant,
Oxford
Letters 1, 325
--
-due to a change in form factor of specimen.
of saturation. It may therefore be of interest to report that we have found clear evidence of saturation in a similar alloy (with 4% Al) after holding it at -5’K for over 7000 hr. This result is, however, not inconsistent with that of Elliott et al., since the accompanying Table 1 shows that in ~-plutonium saturation effects are hardly not*iceable after 1200 hr. It can be seen from the table that while after 7000 hr at -5’K, the rate of increase in resistance of u-plutonium has dropped to about 1 per cent of its initial value, that of S-plutonium has still over 25 per cent of its initial value. Both these specimens were made from the same stock and thus have the same self-heat except for the aluminium content. It may also be not.ed that these rates of resistance increase do not follow a true exponential law of the form ap/& (~~~~“~~~~*~-p). We are most grateful to Dr. Elliott for acquainting us with their results in advance of publication.
2. R. 0.
12, 1964
TABLE 1
~ -.-
t
VOL.
v,(P)
%
(3)
N
Equation (2) then becomes
= v,(o) -
/W,(o)
+
&VIP
(4)
wbeing the atomic volume. Precisely similar reasoning for the migration volume V’ leads to V:(P) = V:(o) -
/T[V:(o) + K,%]P
where Kpz is defined by
with A/3$ = /3: - ,!Y, p: being the compressibility of a crystal containing a saddle point configuration. Going through a similar procedure for the temperature dependence of V, and V: shows that the volumes
LETTERS
TO
THE
at T are related to those at T, by V,(T)
=
V,IT,)
+ dV,(T,)
V:(T) = V:( T,) + a[ V:( T,) + X,%# where K,
and fi,:
(7)
JMl(T - T,)
+
T -
27,) (8)
113
EDITOR
The ohserved activation volume depends on pressure according to vobs.
v
=
VIp”“~ therefore
VJo) - y [V,W + Kg]
(16)
are defined by &--CC oc
K, Z=Z----iV
(9)
(10) a, CL’and a$ being the thermal expansion coefficients (assumed to be independent of temperature) of a perfect crystal, a crystal containing a vacancy and a crystal containing a saddle point configuration respectively. Equations (4), (5), (7) and (8) show that the contribution of point defects, represented by the R’s, can be obtained if enough data are available. A relation between K, and K, can be obtained from Gruneisen’s equation
(11) y being Gruneisen’s constant and C, the heat capacity at constant volume. Writing equation (11) for the imperfect crystal and forming the ratio (CL’- tcffa shows that K,
= K,.
Similarly, it is easiIy shown that K,:=K
: P -+,
(13)
The & aris:es in equation (13) because the number of degrees OSfreedom is one less in the activated state than in a normal state. In equa,tions (12) and (13), it was assumed that C,, C,’ and Czl: were proportional to the number of degrees of freedom. The only data in existence to which any of these equations can be apphed are the pressure variation of the activation volume for difiusion in sodium.(i) In keeping with the small ion core and disordered vacancy structure in sodium, we will assume that Vt << V,. Differentiating equation (4) gives 8VQ &
= -B[V,Co,
(14)
+ IQ].
But the vacancy concentration depends on pressure according tot n,(P) = n~{o~ exp
i
-
PVvfof - - P2 av, -
___ kT
1
2Kr ( ap 1p=o
(15)
Plotting the observed activation volume for di~usion(l) against pressure, taking the initial slope and comparing with equation (16) gives K, = 4.8, i.e. a 1% vacancy concentration in sodium increases its compressibility by 4.8%. No other independent calculations are available for this quantity in sodium, but for f.c.c. metals Dienes@) estimates that a 1% vacancy concentration increases fi by 1%. It is to be expected that a vacancy would have a greater effect on the compressibility in sodium than in f.c.c. metals. The variation of activation volume wTithtemperature has not been measured, but the present results suggest that according to equation (12), a 1% vacancy concentration also increases the coefficient of thermal expansion by 4.8%. I,. A. GIRIFALCO School of Metallurgical Engineering and Laboratory for Research on the Structure of Matter University of Pennsylvania
Philadelphia, Pennsylvania References 1. N. H. NACRTRIEB, J. A. WEIL, E. CATALANO LAWSON, J. C’he’hem. P&p. 20, 1189 (1952). 2. G. J. DIENES, Phys. Rev. 86, 228 (1952).
and A. W.
* Received July 8, 1963. t The factor f in the second term in the exponential arises from the fact that equation (15) is obtained by expanding the Gibbs free energy of formation in a power series in P. The second derivative of the free energy is just the fcrst derivative of the volume.
Recovery of the electrical resistivity of copper, silver and gold after small plastic deformations at - 195°C * Nany experiments have been performed on the annealing of point defects after plastic deformation at low temperatures. In all these cases however, the degree of deformation is rather high, involving a large defect concentration, the annealing of which is easy to measure. See for instance Refs. l-6. In this work a first attempt was made to investigate the annealing after deformations amounting to a few per cent or less. The specimen material was 99.999% pure polycrystalline wire of 0.20-0.25 mm dia. Cu, Ag or Au. Prior to the deformation the wires were annealed for at least 1 hr at 450°C in air (Au) or in vac~o (Ag and Cu).