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The magnetic field dependence of the flow stress in niobium
Scripta
METALLURGICA
V o l . II, pp. 4 5 S - 4 5 7 , Printed in t h e U n i t e d
1977 States
Pergamon
Press,
In¢
THE MAGNETIC FIELD DEPENDENCE OF THE FLOW STRESS IN NIOBIUM* J. H. Tregilgas ~ and J. M. Galligan Department of Metallurgy and The Institute of Materials Science University of Connecticut Storrs, Connecticut 06268 (Received February 18, 1 9 7 7 ) ( R e v i s e d A p r i l 4, ] 9 7 7 )
At low temperatures, the plastic deformation of metal crystals is influenced, to some extent, by drag processes ( i ) . For example, as shown in lead and lead alloys, the difference in stress necessary for plastic deformation between the normal state and the superconducting state, AoN_S, can be explained by treating the mobile dislocations as flexible strings (2,3). In treating the dislocation as a flexible string, it is necessary to consider inertial effects and in so doing it has been shown that the electron drag, on mobile dislocations is related to the attenuation of an acoustic wave in a crystal; the pertinent measurements which have shown the influence of inertial effects, have included measurements of the difference in the flow stress between the superconducting state and the so-called mixed state, AOM_ S (4,5). In the mixed state the magnetic flux penetrates the sample as a set of flux tubes, all parallel to the applied magnetic field, with the flux tubes forming a somewhat regular array. The flux, in the mixed state, penetrates the sample in a continuous manner, as opposed to a type I superconductor, in which the transition from the superconducting state to the normal state occurs rather discontinuously. At fields close to the upper critical field of a type II superconductor, HC2 it is found that the AOM_ S varies linearly with field (4,5). This observation is consistent with the field dependence of the attenuation of an acoustic wave in a mixed state crystal, but at variance with some previous measurements of AOM_ S in niobium (6). The previous measurements, as performed in niobium, were done without the benefit of magnetization measurements on the same crystal, and these measurements have led to the suggestion that AOM_ S varies as B/Hc 2 (6). This latter result is presumably consistent with treating the mixed state as an example of a composite material where it is suggested that a simple rule of mixture applied, the mixture being taken as flux tubes immersed in a superconducting matrix. Since the latter result is at variance with previous results on lead-indium alloys, where simultaneou~ measurements of the magnetization of the samples have been made, we have reexamined the question by making AOM_ S measurements of niobium crystals and magnetization measurements on the same crystal. As shown below, the measurements are much more consistent with the drag related to an acoustic attenuation model rather than a rule of mixtures model. The present experiments were carried out on zone-refined niobium crystals which were outgassed at 2000°C at a pressure of ~10-9Torr for fifty hours or so. Deformation was undertaken at 4.2°K in a laboratory scale tensile machine. The dewar used in this study included a superconducting magnet which was calibrated by many previous magnetization measurements of lead and lead alloys. The stress measurements were made by using a differential voltage technique in a bridge-type network which allowed high sensitivity, differential stress measurements to be made. The measurements themselves were made as follows: the crystal, after cooling to 4.2°K, was deformed in the superconducting state (field off) and then the field was switched to the mixed state, field on, at a predetermined field, H. The value of the field used in each measurement was determined from magnetization measurements made on the same specimen. The measured applied field and the magnetization measurement allow us to establish the magnetic
*Supported by the U.S.E.R.D.A. under Contract E(II-I) *Presently at Texas Instrument, Dallas, Texas.
455
2305.
456
FLOW
induction,
STRESS
IN N I O B I U M
B, on the basis of the standard
Vol.
ll,
No.
6
relation between B and H, B = H + 4~M
where M is the magnetization, Fig. i. The measured differences in stress AoM-S' obtained at various values of applied field, are then compared with the magnetic induction B/Hc2, Fig. 2, the comparison being made on the basis of a normalized plot. Quite clearly the present results show that AoMS , normalized to the difference in flow stress between the normal state and the superconducting state, does not vary as B/Hc 2 over the whole range of fields between the lower critical field Hc, and the upper critical field. This result which is typical of a number of samples, is at variance with previous measurements and a number of conjectures about the nature of the drag on mobile dislocation in superconductors (7). Indeed, the present results are similar to what one expects for a type II superconductor in the mixed state, in the case where the drag is related to the attenuation of an acoustic wave, Fig. 3; that is in the mixed state of a type II superconductor the attenuation of an elastic wave, over the range of fields between Hc, and Hc2, varies as B I/2 (2,8). In summary, then, we have shown that the difference in flow stress conducting state and the mixed state is more readily identifiable with an acoustic wave, than with a simple rule of mixtures. This result is qualitatively, with the dislocation moving as an underdamped oscillator the case of lead and lead alloys (9).
between the superthe attenuation of consistent, at least as has been shown for
References i.
M. Kojima and T. Suzuki, Phys. Rev. Letters, 21, 896 (1968).
2.
M. Suenaga and J. M. Galligan, Physical Acoustics, R. N. Thurston) Academic Press, New York (1972).
3.
A.V.
4.
M. Suenaga and J. M. Galligan,
5.
C.S.
6.
G. Kostorz,
7.
R.W. Arsenault, Treatise on Materials Science and Technology, Academic Press, Inc., New York (1975).
8.
K. Maki, In Superconductivity
9.
J.H.
Granato,
Vol. 9 (Edited by W. P. Mason and
Phys. Rev., B4, 2196 (1971). Phys. Rev. Letters,
Pang, T. H. Lin and J. M. Galligan,
27, 721 (1971).
unpublished
results.
Scripta Met., ~, 95 (1970).
Tregilgas
Vol. 6 (Edited by H. Herman)
(R. Parks, ed.) PI035, Dekker, New York.
and J. M. Galligan,
Acta Met., 24, 1115 (1976).
T - 4.2 °K HCl
HCI = 1260 Oe
~E ¢1I
5
I0
15
20
25
Applied Field, H x 10- 2 0 e Fig. I
The mognetizofion curve for o Niobium crystol showing HCl ond HC2.
Vol.
iI, No.
6
FLOW STRESS
b
0 0
i' /
.8
/ /0
mt . 6
"
457
,m
1.0 (n I z
IN NIOBIUM
.4
/
/
/
/
/
0 0
0
8o
o
o oCb ~o
b
<1 U) !
= b
/
.2
/ /
0 0
0 I
,I
I
I
I
.2
.4
.6
~
1.0
B/Hc2 Fig. 2 A normalized plot showing variation
of ~
~O'N - e
with measured Magnetic
Induction; measured between Hcl and HCZ.
/e
•
o/o po
(/1 I
z
/
/
.e
go
/
1.0
.6
.4 /
.2
/ /•
0
1.0
.8
!
|
J
.6
.4
.2
J
( I-B/HC2 )l/Z Fig.3 A normalized plot of the variation of ~A O"M - 8 with Magnetic Induction,
B I/2 plotted between HCl and Hc2..
METALLURGICA
V o l . II, pp. 4 5 S - 4 5 7 , Printed in t h e U n i t e d
1977 States
Pergamon
Press,
In¢
THE MAGNETIC FIELD DEPENDENCE OF THE FLOW STRESS IN NIOBIUM* J. H. Tregilgas ~ and J. M. Galligan Department of Metallurgy and The Institute of Materials Science University of Connecticut Storrs, Connecticut 06268 (Received February 18, 1 9 7 7 ) ( R e v i s e d A p r i l 4, ] 9 7 7 )
At low temperatures, the plastic deformation of metal crystals is influenced, to some extent, by drag processes ( i ) . For example, as shown in lead and lead alloys, the difference in stress necessary for plastic deformation between the normal state and the superconducting state, AoN_S, can be explained by treating the mobile dislocations as flexible strings (2,3). In treating the dislocation as a flexible string, it is necessary to consider inertial effects and in so doing it has been shown that the electron drag, on mobile dislocations is related to the attenuation of an acoustic wave in a crystal; the pertinent measurements which have shown the influence of inertial effects, have included measurements of the difference in the flow stress between the superconducting state and the so-called mixed state, AOM_ S (4,5). In the mixed state the magnetic flux penetrates the sample as a set of flux tubes, all parallel to the applied magnetic field, with the flux tubes forming a somewhat regular array. The flux, in the mixed state, penetrates the sample in a continuous manner, as opposed to a type I superconductor, in which the transition from the superconducting state to the normal state occurs rather discontinuously. At fields close to the upper critical field of a type II superconductor, HC2 it is found that the AOM_ S varies linearly with field (4,5). This observation is consistent with the field dependence of the attenuation of an acoustic wave in a mixed state crystal, but at variance with some previous measurements of AOM_ S in niobium (6). The previous measurements, as performed in niobium, were done without the benefit of magnetization measurements on the same crystal, and these measurements have led to the suggestion that AOM_ S varies as B/Hc 2 (6). This latter result is presumably consistent with treating the mixed state as an example of a composite material where it is suggested that a simple rule of mixture applied, the mixture being taken as flux tubes immersed in a superconducting matrix. Since the latter result is at variance with previous results on lead-indium alloys, where simultaneou~ measurements of the magnetization of the samples have been made, we have reexamined the question by making AOM_ S measurements of niobium crystals and magnetization measurements on the same crystal. As shown below, the measurements are much more consistent with the drag related to an acoustic attenuation model rather than a rule of mixtures model. The present experiments were carried out on zone-refined niobium crystals which were outgassed at 2000°C at a pressure of ~10-9Torr for fifty hours or so. Deformation was undertaken at 4.2°K in a laboratory scale tensile machine. The dewar used in this study included a superconducting magnet which was calibrated by many previous magnetization measurements of lead and lead alloys. The stress measurements were made by using a differential voltage technique in a bridge-type network which allowed high sensitivity, differential stress measurements to be made. The measurements themselves were made as follows: the crystal, after cooling to 4.2°K, was deformed in the superconducting state (field off) and then the field was switched to the mixed state, field on, at a predetermined field, H. The value of the field used in each measurement was determined from magnetization measurements made on the same specimen. The measured applied field and the magnetization measurement allow us to establish the magnetic
*Supported by the U.S.E.R.D.A. under Contract E(II-I) *Presently at Texas Instrument, Dallas, Texas.
455
2305.
456
FLOW
induction,
STRESS
IN N I O B I U M
B, on the basis of the standard
Vol.
ll,
No.
6
relation between B and H, B = H + 4~M
where M is the magnetization, Fig. i. The measured differences in stress AoM-S' obtained at various values of applied field, are then compared with the magnetic induction B/Hc2, Fig. 2, the comparison being made on the basis of a normalized plot. Quite clearly the present results show that AoMS , normalized to the difference in flow stress between the normal state and the superconducting state, does not vary as B/Hc 2 over the whole range of fields between the lower critical field Hc, and the upper critical field. This result which is typical of a number of samples, is at variance with previous measurements and a number of conjectures about the nature of the drag on mobile dislocation in superconductors (7). Indeed, the present results are similar to what one expects for a type II superconductor in the mixed state, in the case where the drag is related to the attenuation of an acoustic wave, Fig. 3; that is in the mixed state of a type II superconductor the attenuation of an elastic wave, over the range of fields between Hc, and Hc2, varies as B I/2 (2,8). In summary, then, we have shown that the difference in flow stress conducting state and the mixed state is more readily identifiable with an acoustic wave, than with a simple rule of mixtures. This result is qualitatively, with the dislocation moving as an underdamped oscillator the case of lead and lead alloys (9).
between the superthe attenuation of consistent, at least as has been shown for
References i.
M. Kojima and T. Suzuki, Phys. Rev. Letters, 21, 896 (1968).
2.
M. Suenaga and J. M. Galligan, Physical Acoustics, R. N. Thurston) Academic Press, New York (1972).
3.
A.V.
4.
M. Suenaga and J. M. Galligan,
5.
C.S.
6.
G. Kostorz,
7.
R.W. Arsenault, Treatise on Materials Science and Technology, Academic Press, Inc., New York (1975).
8.
K. Maki, In Superconductivity
9.
J.H.
Granato,
Vol. 9 (Edited by W. P. Mason and
Phys. Rev., B4, 2196 (1971). Phys. Rev. Letters,
Pang, T. H. Lin and J. M. Galligan,
27, 721 (1971).
unpublished
results.
Scripta Met., ~, 95 (1970).
Tregilgas
Vol. 6 (Edited by H. Herman)
(R. Parks, ed.) PI035, Dekker, New York.
and J. M. Galligan,
Acta Met., 24, 1115 (1976).
T - 4.2 °K HCl
HCI = 1260 Oe
~E ¢1I
5
I0
15
20
25
Applied Field, H x 10- 2 0 e Fig. I
The mognetizofion curve for o Niobium crystol showing HCl ond HC2.
Vol.
iI, No.
6
FLOW STRESS
b
0 0
i' /
.8
/ /0
mt . 6
"
457
,m
1.0 (n I z
IN NIOBIUM
.4
/
/
/
/
/
0 0
0
8o
o
o oCb ~o
b
<1 U) !
= b
/
.2
/ /
0 0
0 I
,I
I
I
I
.2
.4
.6
~
1.0
B/Hc2 Fig. 2 A normalized plot showing variation
of ~
~O'N - e
with measured Magnetic
Induction; measured between Hcl and HCZ.
/e
•
o/o po
(/1 I
z
/
/
.e
go
/
1.0
.6
.4 /
.2
/ /•
0
1.0
.8
!
|
J
.6
.4
.2
J
( I-B/HC2 )l/Z Fig.3 A normalized plot of the variation of ~A O"M - 8 with Magnetic Induction,
B I/2 plotted between HCl and Hc2..