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Zener relaxations in body-centred cubic iron-chromium solid solutions
Scripta
METALLURCICA
Vol. 8, pp. 5 5 3 - 5 5 8 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
ZENER IKELAXATIOHS lii m0DY-CEI~TRED CUBIC IRON-CIiROMIU~,! SOLID SOLVflOI~S
M.C. BORA}i and G.M. LEAK ~
(Received
February
25,
1974)
Introduction The Zener relaxation provides a powerful method for the study of solute diffusion in alloys at temperatures low compared with normal radio tracer methods.
A Zener relaxation in an
Fe-22.5% Cr alloy was first observed by Barrand (i); the present work was carried out on a series of Fe-Cr alloys to investigate the effect of chromium concentration on the relaxation and to relate this to diffusion in the alloys.
The diffusion rate first increases with
increasing chromium content with 8zl increase in activation energy. concentrated alloy (43.1% Cr alloy).
It then decreases in a very
Because of the complexity of the structure of concentrated
alloys it is difficult to interpret the difference between the activation energies for relaxation Qr and for diffusion.
The prediction (2) that for concentrated binary alloys Qr
should correspond to the higher of the two values QA and QB (where A and B refer to the activation energies for diffusion of the two species A and B) has not been supported by experiments (3,4).
On the other hand the problem with experiments on dilute alloys is that it
is difficult to detect Zener peaks since the relaxation strength varies as the square of the concentration.
The problem is even worse for polycr~stalline specimens which give high back-
grounds against which a small Zener peak ma~ not be ooserved.
It was necessary to use con-
centrated alloys in the present work since dilute alloys in the Fe-Cr system did not produce Zener peaks.
It was therefore not possible to determine Qr as a function of composition in
extreme dilution where the relaxation might be attributed to simple atom pairs. Experimental Procedu~_e Specimens (compositions shown in TABLE i) were kindly supplied by the British Iron and Steel Research Association.
The alloy of highest chromium content was too hard and brittle to
3wage and so was reduced to size by filing and grinding.
Wire specimens were then annealed at
1200°C for about 88 hours in pure hydrogen to produce large grain size. ~he contributions to damping from grain boundary relaxations.
This helped to eliminate
Two additional alloys containing
25.10 and 35.02 wt%. Cr were prepared by mixing and melting appropriate amounts of 19.2 and h3.1 wt%. Cr alloys in a high frequency induction furnace.
The resulting ingots were homogenized for
about 50 hours at lO00°C in vacuo before the final specimens were prepared for internal friction me as urement s.
553
554
ZENER RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID S O L U T I O N S
Vol.
8, No.
5
TABLE 1 Composition o f Fe-Cr Alloys in Wt%. Chromium
Carbon
Nitro6en
12.9
0.0028
O .0125
14.0
0.0026
0.0165
19.2
0.0026
0.004
43.1
0.008~
Measurement of logarithmic decrement was
carried out (5,6) in an evacuated, inverted,
torsion p e n d u l ~ by natural free decay at frequencies of 0.45 to 2.66 Hz. 9 cm. long and about 1 ram. in diameter were used.
Wire specimens about
Great care was necessary with the large grain
size specimens to ensure that they were not strained by mounting in the apparatus. Results No Zener peak was observed in alloys containing 12.9, 14.0 and 19.2 weig~ht per cent Cr. The main reason for this may lie in similar size and characteristics of the Fe and Cr atoms.
A
similar effect was observed by the authors in the Fe-Co system (5).
22 25.1%Cr
43.1% Cr
35.02%Cr
18
o
x
14
o~10 o
6 2 420
,.so
6' o oc
&o oc
;so
6;0 oc
FIG. 1 Typical Damping Curves
FIG. 1 shows damping curves for specimens containing 25.10, 35.02 8rid h3.10 wt.% Cr, (freqUency about 1 ~z).
An Arrhenius relationship between loglof and inverse peak temperature
for measurements at different frequencies, f, is shown in FIG. 2A for the 25.10 wt.% Cr alloy in agreement with the relationship: T =
TO exp (Qr/~Tp)
where 3o is the frequency factor and Qr is the activation energy for the relaxation process. The peaks were first norm~lised by assuming an exponential background for ascertaining the peak temperature Tp in the above expression for each frequency of vibration f.
The value of Qr
determined from the slope of the plot is 59.8 kcal. mole -I (2.6 ev) and the frequency factor T
is 10 -14"6 sec. O
Similar Arrhenius plots for 35.02 and 43.1 wt%. Cr alloys are shown in FIG. 2B and C
Vol,
8, No.
5
ZENER RELAXATIONS
IN B.C.C.
Fe-Cr S O L I D
SOLUTIONS
555
.4 .2 O @
@
--'2 !
I-6
!
11-8
!
12"0
I
12"2
12"4
104IT
FIG. 2 Variation of log (frequency) with inverse peak temperature
respectively.
A sunmm_v7 of the values of the main parameters investigated for the alloys, Qr'
To, Am and r2(6) are presented in TABLE 2. TABLE 2 Relaxation Parameters wt%. Cr
-lOgloTo
Am x 10 -3
Q Kcal
r2(8)
(ev) 25.10
1~.6
6.7
59.8 (2.6)
0.Th
35.02
15.9
9.6
6~.~ (2.8)
1.o3
~3.10
14.0
8.2
57.5 (2.5)
1.o0
The relaxation strength Am is calculated from the expression, (at ~ 6
= i),
~T m
where ~ is the angular frequency of vibration and 6 is the logarithmic decrement.
Anelastic
effects involve a redistribution of atoms to relieve an applied stress and if the atomic redistribution is a simple relaxation process involving a single time of relaxation, T, then A is m represented by the above equation. The value for the expression r2(~) shown in the last columm of TABLE 2 is calculated from the expression derived by Nowick and Berry (7), A
(T-1)
=
R 2.635 Qr r2(8)
r2(~) is the function of the distribution parameter B which is a measure of the departure of the observed relaxation effect from a unique process. calculated by Nowick and Berry (7).
Values of ~ can be obtained from tables
556
ZENER
RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID
SOLUTIONS
Vol.
8, No.
S
Discussion The trend in these results is that the activation energy for relaxation is higher t h ~ the diffusion activation energy of Wolfe and Pexton (8) and Bowen (9).
Bowen, in a series of
Fe-Cr alloys ranging from 2.02 to 19.2 wt.% Cr, found that progressive additions of chromium resulted in a small decrease in the activation energy, ranging from 58.4 kcal mole -1 (2.02 ~rt.% Cr) to 51.8 kcal mole -I (19.2 w%.% Cr). low concentration.
The present data are somewhat above those of Bowen for
A recent result of Barrend (i), 53 kcal mole -1 for an alloy with 22.5 wt.~
chromium, indicates an increase in activation energy with increase in chromium content.
The
highest value of activation energy for 35.02 wt.% Cr might be associated with the ferromagnetic curie temperature.
Mills (lO) and Stanley and Weft (ii) observed similar effects in Fe-V alloys
close to the Curie temperature.
Near this temperature
the measured activation energy for
diffusion is much higher then that for normal self diffusion which is associated with a smaller T
value.
This effect was attributed to the influence of ferromagnetic spin ordering on the
o
activation energy.
The 35.02 wt.% Cr alloy investigated in the present work has a Curie
temperature well within the region of the Zener peak and hence confirms the view of the above authors that high activation energies are due to the influence ol ferromagnetic spin ordering. Paxton and Kuniteke (12) studied diffusion in some concentrated alloys; their results
D v = 0.156 exp (- 48~00~ RT " for 25% Cr Dv =
~O exp (- 70000~RT " for 49% Cr
D v = 2~.6 These results seem to be inconsistent.
exp (- ~ 0 0 )
for 67% Cr
The presence of the o phase should not effect their
results since diffusion runs were carried out at temperatures above ll50°C at which there would be no o phase present.
Because of large differences it is not easy to compare our results with
those of Paxton and Kunitake.
Shinaeyev (13) pointed out that the large Q values may be due to
large binding forces resulting in increased hardness in concentrated alloys. that the increased hardness could be due to o phase ordering. also effect the result of Paxton and Kunitake at 25% Cr.
It seems likely
If this were true then it should
Due to the inconsistency in the
available diffusion results of Paxton and Kunitake in the concentrated region a re-investigation of diffusion measurements is needed before any conclusive explanations of these phenomena can be made.
The observation that the activation energy for Zener relaxation in the present work
is higher than the diffusion activation energy measured by Bowen in the relatively dilute alloys might be explained qualitatively since a solute which expands the solvent lattice will increase the work required to move an atom over the saddle point and hence Qr will increase as the amount of solute increases.
Such an explanation would account for the observed increase in
Qr since the atomic radius of iron is less than that of chromium.
Unfortunately, the internal
friction parameters such as Qr for Zener relaxations cannot be measured to study the effect of solute content in the Fe-Cr system due to the absence of Zener peaks in alloys containing less than 22.5 wt.% of Cr.
The drop in activation energy in ~3.1% Cr alloy is expected since this
alloy has a Curie temperature at about 400°C well below that of the Zener peak about 575°C and
Vol.
8,
No.
5
ZENER RELAXATIONS
IN B . C . C .
Fe-Cr
SOLID SOLUTIONS
557
therefore spin ordering should have been completed at lower temperatures. The pre-exponential constant, Xo' in the expression for relaxation time corresponds reasonably to that expected for processes involving unit atomic displacements.
However, the
relatively low Xo value in the 35.02% Cr alloy supports the view of Stanley and Wert (ll) and of Fischbach (14) that it arises because of the damping contribution due to ferromagnetic spin ordering since the peak temperature in this alloy is near the Curie temperature.
The relaxation
strength Am increases in proportion to c2(i-c) 2 as predicted by Leclaire and Lomer (15) FIG. 3.
A M x IO3 10 9 O 8 7 6 5
3
i
i
i
i
4
5
6
7
C 2 ( I - C 1 2 x 100 FIG. 3 Variation of Relaxation Strength with Concentration.
However, it drops rapidly in the alloy containing 43.1% Cr.
This alloy is almost inside the
uaiform o phase region at all temperatures of measurements.
No detailed information is
available regarding binding energies for this o phase.
However, the atomic binding in this
brittle compound is significantly stronger than that between the Fe and Cr atoms in the solid solutions.
The sharp drop in relaxation strength is almost certainly associated with the
formation of the o phase in this alloy.
Experiments with alloys containing more than 50 wt.%
Cr, which is beyond the uniform o phase region, will be needed to investigate this point. Temperature dependence of A m was not measured in these alloys.
Am values are calculated
from the height of the internal friction peak after subtracting the background aamping and so they are very sensitive to the assumed background.
Small scatter in the experimental points
in the internal friction spectrum or presence of some grain boundary damping will make considerable variation in the background damping and so the effect of temperature dependence on A will m be lost. The 8 values measured in these alloys are typical of those for a Zener relaxation except for the 25% Cr alloy.
This gives a negative 8 value which is unusual.
also observed by the authors in other b.c.c, alloy systems.
Similar results were
8 values are sensitive both to
558
ZENER
RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID
SOLUTIONS
Vol.
8, No.
I
Qr and A(T) values and a small error in measuring these quantities might reverse this observation. Acknowledgement One of the authors (M.C.B.) is indebted to the Science Research Council for support of this work. References 1.
P. Barrand, Metal Sci. J. i, 54 (1967).
2.
A.S. Nowick, Phys. Rev. 88, 925 (1952).
3.
D.P. Seraphim, Trans. Met. Soc., A/ME, 218, h85 (1960).
4.
T.J. Turner, G.F. Williams, Jnr., Acta Met. 10, 305 (1962).
5.
M.C. Borah, G.M. Leak, Phys.Stat.Sol. 15, 4hl (1973).
6.
M.C. Borah, G.M. Leak, Met. Sci. J.
7.
A.S. Nowick, B.S. Berry, I.B.M. Journal Res. Develop. 5, 297, 312 (1961).
8.
R.A. Wolfe, H.W. Paxton, Trans. Met. Soc. AIME, 230, 1426 (1964).
9.
A.W.L. Bowen, Ph.D. Thesis, Manchester University (1968).
i0.
B. Mills, J. De Physique, 32, C2-h3 (1971).
ii.
J. Stanley, C. Wert, J. Appl. Phys., 32, 267 (1961).
12.
H.W. Paxton, T. Kunitake, Trans. Met. Soc. AIME, 218, 1003 (1960).
13.
A. Shinaeyev, Phys. Met. and Metals, 20, 875 (1965).
14.
D.B. Fischbach, Acta Met., lO, 319 (1962).
15.
A.D. Leclaire, W.M. Lomer, Acta Met., 2, 731 (195h).
To be published.
5
METALLURCICA
Vol. 8, pp. 5 5 3 - 5 5 8 , P r i n t e d in the U n i t e d
1974 States
Pergamon
Press,
Inc.
ZENER IKELAXATIOHS lii m0DY-CEI~TRED CUBIC IRON-CIiROMIU~,! SOLID SOLVflOI~S
M.C. BORA}i and G.M. LEAK ~
(Received
February
25,
1974)
Introduction The Zener relaxation provides a powerful method for the study of solute diffusion in alloys at temperatures low compared with normal radio tracer methods.
A Zener relaxation in an
Fe-22.5% Cr alloy was first observed by Barrand (i); the present work was carried out on a series of Fe-Cr alloys to investigate the effect of chromium concentration on the relaxation and to relate this to diffusion in the alloys.
The diffusion rate first increases with
increasing chromium content with 8zl increase in activation energy. concentrated alloy (43.1% Cr alloy).
It then decreases in a very
Because of the complexity of the structure of concentrated
alloys it is difficult to interpret the difference between the activation energies for relaxation Qr and for diffusion.
The prediction (2) that for concentrated binary alloys Qr
should correspond to the higher of the two values QA and QB (where A and B refer to the activation energies for diffusion of the two species A and B) has not been supported by experiments (3,4).
On the other hand the problem with experiments on dilute alloys is that it
is difficult to detect Zener peaks since the relaxation strength varies as the square of the concentration.
The problem is even worse for polycr~stalline specimens which give high back-
grounds against which a small Zener peak ma~ not be ooserved.
It was necessary to use con-
centrated alloys in the present work since dilute alloys in the Fe-Cr system did not produce Zener peaks.
It was therefore not possible to determine Qr as a function of composition in
extreme dilution where the relaxation might be attributed to simple atom pairs. Experimental Procedu~_e Specimens (compositions shown in TABLE i) were kindly supplied by the British Iron and Steel Research Association.
The alloy of highest chromium content was too hard and brittle to
3wage and so was reduced to size by filing and grinding.
Wire specimens were then annealed at
1200°C for about 88 hours in pure hydrogen to produce large grain size. ~he contributions to damping from grain boundary relaxations.
This helped to eliminate
Two additional alloys containing
25.10 and 35.02 wt%. Cr were prepared by mixing and melting appropriate amounts of 19.2 and h3.1 wt%. Cr alloys in a high frequency induction furnace.
The resulting ingots were homogenized for
about 50 hours at lO00°C in vacuo before the final specimens were prepared for internal friction me as urement s.
553
554
ZENER RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID S O L U T I O N S
Vol.
8, No.
5
TABLE 1 Composition o f Fe-Cr Alloys in Wt%. Chromium
Carbon
Nitro6en
12.9
0.0028
O .0125
14.0
0.0026
0.0165
19.2
0.0026
0.004
43.1
0.008~
Measurement of logarithmic decrement was
carried out (5,6) in an evacuated, inverted,
torsion p e n d u l ~ by natural free decay at frequencies of 0.45 to 2.66 Hz. 9 cm. long and about 1 ram. in diameter were used.
Wire specimens about
Great care was necessary with the large grain
size specimens to ensure that they were not strained by mounting in the apparatus. Results No Zener peak was observed in alloys containing 12.9, 14.0 and 19.2 weig~ht per cent Cr. The main reason for this may lie in similar size and characteristics of the Fe and Cr atoms.
A
similar effect was observed by the authors in the Fe-Co system (5).
22 25.1%Cr
43.1% Cr
35.02%Cr
18
o
x
14
o~10 o
6 2 420
,.so
6' o oc
&o oc
;so
6;0 oc
FIG. 1 Typical Damping Curves
FIG. 1 shows damping curves for specimens containing 25.10, 35.02 8rid h3.10 wt.% Cr, (freqUency about 1 ~z).
An Arrhenius relationship between loglof and inverse peak temperature
for measurements at different frequencies, f, is shown in FIG. 2A for the 25.10 wt.% Cr alloy in agreement with the relationship: T =
TO exp (Qr/~Tp)
where 3o is the frequency factor and Qr is the activation energy for the relaxation process. The peaks were first norm~lised by assuming an exponential background for ascertaining the peak temperature Tp in the above expression for each frequency of vibration f.
The value of Qr
determined from the slope of the plot is 59.8 kcal. mole -I (2.6 ev) and the frequency factor T
is 10 -14"6 sec. O
Similar Arrhenius plots for 35.02 and 43.1 wt%. Cr alloys are shown in FIG. 2B and C
Vol,
8, No.
5
ZENER RELAXATIONS
IN B.C.C.
Fe-Cr S O L I D
SOLUTIONS
555
.4 .2 O @
@
--'2 !
I-6
!
11-8
!
12"0
I
12"2
12"4
104IT
FIG. 2 Variation of log (frequency) with inverse peak temperature
respectively.
A sunmm_v7 of the values of the main parameters investigated for the alloys, Qr'
To, Am and r2(6) are presented in TABLE 2. TABLE 2 Relaxation Parameters wt%. Cr
-lOgloTo
Am x 10 -3
Q Kcal
r2(8)
(ev) 25.10
1~.6
6.7
59.8 (2.6)
0.Th
35.02
15.9
9.6
6~.~ (2.8)
1.o3
~3.10
14.0
8.2
57.5 (2.5)
1.o0
The relaxation strength Am is calculated from the expression, (at ~ 6
= i),
~T m
where ~ is the angular frequency of vibration and 6 is the logarithmic decrement.
Anelastic
effects involve a redistribution of atoms to relieve an applied stress and if the atomic redistribution is a simple relaxation process involving a single time of relaxation, T, then A is m represented by the above equation. The value for the expression r2(~) shown in the last columm of TABLE 2 is calculated from the expression derived by Nowick and Berry (7), A
(T-1)
=
R 2.635 Qr r2(8)
r2(~) is the function of the distribution parameter B which is a measure of the departure of the observed relaxation effect from a unique process. calculated by Nowick and Berry (7).
Values of ~ can be obtained from tables
556
ZENER
RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID
SOLUTIONS
Vol.
8, No.
S
Discussion The trend in these results is that the activation energy for relaxation is higher t h ~ the diffusion activation energy of Wolfe and Pexton (8) and Bowen (9).
Bowen, in a series of
Fe-Cr alloys ranging from 2.02 to 19.2 wt.% Cr, found that progressive additions of chromium resulted in a small decrease in the activation energy, ranging from 58.4 kcal mole -1 (2.02 ~rt.% Cr) to 51.8 kcal mole -I (19.2 w%.% Cr). low concentration.
The present data are somewhat above those of Bowen for
A recent result of Barrend (i), 53 kcal mole -1 for an alloy with 22.5 wt.~
chromium, indicates an increase in activation energy with increase in chromium content.
The
highest value of activation energy for 35.02 wt.% Cr might be associated with the ferromagnetic curie temperature.
Mills (lO) and Stanley and Weft (ii) observed similar effects in Fe-V alloys
close to the Curie temperature.
Near this temperature
the measured activation energy for
diffusion is much higher then that for normal self diffusion which is associated with a smaller T
value.
This effect was attributed to the influence of ferromagnetic spin ordering on the
o
activation energy.
The 35.02 wt.% Cr alloy investigated in the present work has a Curie
temperature well within the region of the Zener peak and hence confirms the view of the above authors that high activation energies are due to the influence ol ferromagnetic spin ordering. Paxton and Kuniteke (12) studied diffusion in some concentrated alloys; their results
D v = 0.156 exp (- 48~00~ RT " for 25% Cr Dv =
~O exp (- 70000~RT " for 49% Cr
D v = 2~.6 These results seem to be inconsistent.
exp (- ~ 0 0 )
for 67% Cr
The presence of the o phase should not effect their
results since diffusion runs were carried out at temperatures above ll50°C at which there would be no o phase present.
Because of large differences it is not easy to compare our results with
those of Paxton and Kunitake.
Shinaeyev (13) pointed out that the large Q values may be due to
large binding forces resulting in increased hardness in concentrated alloys. that the increased hardness could be due to o phase ordering. also effect the result of Paxton and Kunitake at 25% Cr.
It seems likely
If this were true then it should
Due to the inconsistency in the
available diffusion results of Paxton and Kunitake in the concentrated region a re-investigation of diffusion measurements is needed before any conclusive explanations of these phenomena can be made.
The observation that the activation energy for Zener relaxation in the present work
is higher than the diffusion activation energy measured by Bowen in the relatively dilute alloys might be explained qualitatively since a solute which expands the solvent lattice will increase the work required to move an atom over the saddle point and hence Qr will increase as the amount of solute increases.
Such an explanation would account for the observed increase in
Qr since the atomic radius of iron is less than that of chromium.
Unfortunately, the internal
friction parameters such as Qr for Zener relaxations cannot be measured to study the effect of solute content in the Fe-Cr system due to the absence of Zener peaks in alloys containing less than 22.5 wt.% of Cr.
The drop in activation energy in ~3.1% Cr alloy is expected since this
alloy has a Curie temperature at about 400°C well below that of the Zener peak about 575°C and
Vol.
8,
No.
5
ZENER RELAXATIONS
IN B . C . C .
Fe-Cr
SOLID SOLUTIONS
557
therefore spin ordering should have been completed at lower temperatures. The pre-exponential constant, Xo' in the expression for relaxation time corresponds reasonably to that expected for processes involving unit atomic displacements.
However, the
relatively low Xo value in the 35.02% Cr alloy supports the view of Stanley and Wert (ll) and of Fischbach (14) that it arises because of the damping contribution due to ferromagnetic spin ordering since the peak temperature in this alloy is near the Curie temperature.
The relaxation
strength Am increases in proportion to c2(i-c) 2 as predicted by Leclaire and Lomer (15) FIG. 3.
A M x IO3 10 9 O 8 7 6 5
3
i
i
i
i
4
5
6
7
C 2 ( I - C 1 2 x 100 FIG. 3 Variation of Relaxation Strength with Concentration.
However, it drops rapidly in the alloy containing 43.1% Cr.
This alloy is almost inside the
uaiform o phase region at all temperatures of measurements.
No detailed information is
available regarding binding energies for this o phase.
However, the atomic binding in this
brittle compound is significantly stronger than that between the Fe and Cr atoms in the solid solutions.
The sharp drop in relaxation strength is almost certainly associated with the
formation of the o phase in this alloy.
Experiments with alloys containing more than 50 wt.%
Cr, which is beyond the uniform o phase region, will be needed to investigate this point. Temperature dependence of A m was not measured in these alloys.
Am values are calculated
from the height of the internal friction peak after subtracting the background aamping and so they are very sensitive to the assumed background.
Small scatter in the experimental points
in the internal friction spectrum or presence of some grain boundary damping will make considerable variation in the background damping and so the effect of temperature dependence on A will m be lost. The 8 values measured in these alloys are typical of those for a Zener relaxation except for the 25% Cr alloy.
This gives a negative 8 value which is unusual.
also observed by the authors in other b.c.c, alloy systems.
Similar results were
8 values are sensitive both to
558
ZENER
RELAXATIONS
IN B.C.C.
Fe-Cr
SOLID
SOLUTIONS
Vol.
8, No.
I
Qr and A(T) values and a small error in measuring these quantities might reverse this observation. Acknowledgement One of the authors (M.C.B.) is indebted to the Science Research Council for support of this work. References 1.
P. Barrand, Metal Sci. J. i, 54 (1967).
2.
A.S. Nowick, Phys. Rev. 88, 925 (1952).
3.
D.P. Seraphim, Trans. Met. Soc., A/ME, 218, h85 (1960).
4.
T.J. Turner, G.F. Williams, Jnr., Acta Met. 10, 305 (1962).
5.
M.C. Borah, G.M. Leak, Phys.Stat.Sol. 15, 4hl (1973).
6.
M.C. Borah, G.M. Leak, Met. Sci. J.
7.
A.S. Nowick, B.S. Berry, I.B.M. Journal Res. Develop. 5, 297, 312 (1961).
8.
R.A. Wolfe, H.W. Paxton, Trans. Met. Soc. AIME, 230, 1426 (1964).
9.
A.W.L. Bowen, Ph.D. Thesis, Manchester University (1968).
i0.
B. Mills, J. De Physique, 32, C2-h3 (1971).
ii.
J. Stanley, C. Wert, J. Appl. Phys., 32, 267 (1961).
12.
H.W. Paxton, T. Kunitake, Trans. Met. Soc. AIME, 218, 1003 (1960).
13.
A. Shinaeyev, Phys. Met. and Metals, 20, 875 (1965).
14.
D.B. Fischbach, Acta Met., lO, 319 (1962).
15.
A.D. Leclaire, W.M. Lomer, Acta Met., 2, 731 (195h).
To be published.
5