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tha
ariicla.
EMPIRICAL SCALING FORMULAS
FOR
CRITICAL CURRENT
AND
CRITICAL FIELD
FOR
COMMERCIAL
NbTi*
M.
s.
Lubeii COUF-82H0C—35
Fus.-.on Energy Division
Oax Ridge National Laboratory
Oak Ridge, Tennessee 37830
DE84 004587
B
c2
(0)[l
-
(T/T
c
(0))
],
where
Abstract
Short sample
4.2 K
experimental facilities
are
plentiful,
but
equipment
for
measurements
of
current
as functions
of
temperature
and
field
is
scare.
An
analysis
has
been made
of
published data comprising
at
least
six
manufacturers
and
spanning
a
range
of
criti-
cal current density
at 4.2 K, 8 T of 50 to 108
kA/cm
2
,
and linear equations have been found
to fit the
data
over
a
wide range
of
field
B and
temperature
T. For
a constar.r temperature
of 4.2 K, the
following expres-
sion holds
for B in the
range
of 3 to 10 T:
j
(B, T = 4.2 K) = j [1 -
0.096B],
where
[B .(4.2 K)]"
1
c
o c2
=
0.096
with
a
standard deviation
of 3% for ten
samples.
The
constant
j can be
determined
for any
sample from
a
single point measurement
at a
convenient
field.
For a
constant field
of 8 T, the
following
expression holds
for T in the
range
of 2 to 5.5 K:
j
(B = 8 T, T) = j'[l -
0.177T],
where
[T (8
T)]"
1
CO C
=
0.177
with
a
standard deviation
of
less than
1%.
Linear equations have also been obtained
for
higher
fields
and
lower temperatures.
The
critical field
vs
temperature
is B (T)
B
_{0) = 14.5 T, T (0) = 9.2 K, and n = 1.7 (not 2,
c2
c
which
is
used
in
theoretical
derivations).
For
more
accurate critical Temperature calculations above
10 T,
this equation
can be
used with
the
modification
B
0
(0)
=
14.8 T. No one
simple power
law for the
upper
critical field holds over
the
whole temperature range.
Introduction
In many superconducting magnet design projects,
it
is
advantageous
to
perform
a
scoping study
for the
initial stage.
To
facilitiate this effort
it is
useful
to
have simple scaling rules
or
formulas
to
provide rapid results, which, although
not
exact,
are
accurate enough
in
their essential features
to
avoid
misleading conclusions.
This paper presents
the
results
of an
analysis
of
both published
and
unpublished critical current data
given
as a
function
of
both field
and
temperature.
Simple formulas have been obtained
for (1) the
critical
temperature
as a
function
of
field that
is
needed
to
obtain
an
estimate
of the
current sharing temperature
and hence temperature margin,
(2) the
critical current
density
for
constant temperature
as a
function
of
field,
and (3) the
critical current density
for con-
stant field
as a
function
of
temperature.
In this paper
the
following units
are
used:
T
(K), B (T), j
(kA/cm
2
),
C
(mJ/cm
3
-K).
All J
c
equations
are in 10
3
kA/cm
2
.
* Research sponsored
by the
Office
of
Fusion Energy,
U.S.
Department
of
Energy, under contract W-7405-eng-
26 with
the
Union Carbide Corporation.
Manuscript received December
3, 1982
Experimental source
Almost every superconducting magnet laboratory
is
equiped
to
perform short sample measurements (four-
probe voltage
vs
current)
at the
helium boiling
temperature
(4.2 K) in an
applied transverse magnetic
field.
Not
many laboratories
are
able
to
handle high
field measurements
of
very large conductors that
re-
quire high currents. Only
a
vory
few
laboratories
have temperature controlled cryostats needed
to per-
form short sample measurements over
a
wide range
of
field
and
temperature. Since
the
work
of
Hampshire,
Sutton,
and
Taylor
in
1969,
!
three other experimental
groups have published data
on
commercial NbTi
con-
ductor over
a
wide range
of
field
and
temperature
values.
2
~
1<
Yet,
surprisingly,
no one has
published
data covering both high temperature
(i.e.,
above
4.2 K) and low
temperature (i.e. ,
to
superfluid helium,
which also means high fields)
on the
same specimen.
It should
be
noted that
the
techniques
for
measur-
ing critical current density
are not
standardized
and
the criteria
on
voltage sensitivity used
by the
various
groups also
are
different.
5
In
addition,
the
NbTi
alloy compositions
of the
various vendors differ,
with
the
nominal range
of 44 to 50.5 wt % Ti
being
covered
by the
present data
(the
Fermilab composition
is Nb-46.5
wt % Ti).
Nevertheless,
the
analysis presented here shows
a remarkable consistency
in the
functional dependence
of
the
NbTi data, which span
a
period
of 13
years
and
include conductoi from
at
least
six
manufacturers.
Perhaps,
on
reflection,
the
consistency should
not
be
surprising.
It is
well known that
the
magnitude
of
the
critical current density depends
on the
metal-
lurgical properties (e.g., degree
of
cold working,
amount
of
dislocations,
etc.),
whereas
the
values
of
the
upper critical field
and
critical temperature
are properties
of the
alloy composition
and are
inde-
pendent
of the
metallurgical state.
For
NbTi alloys
in
the
range
of 44 to 50.5 wt % Ti, the B „ and T
c2
c
values
do not
vary
to any
significant degree.
5
"
7
Therefore,
the
extrapolation procedure used
to
find
the linear
j
c
equations merely reflects these facts,
and
one
does
not
find much fluctuation
in B . and T
c2
c
over time
or
from different manufacturers.
Critical Temperature
and
Upper Critical Field
Hawksworth
and
Larbalestier
8
have pointed
out
that
the
most accurate method
for
determining
the
bulk
upper critical field
is to
plot
the
pinning force
(j
x B) vs B at
fixed temperature
and to
extrapolate
the high field linear falloff with increasing field
to
zero pinning force. While
we
don't quarrel with this
assessment,
in the
present work
the
linear portion
of
the critical current density
is
extrapolated
to
zero
value
for the
upper critical field
at
fixed temperature.
The slight curvature
of j in j vs B
grap'ns that occurs
at
the
lowest values
of
critical current density
is
neglected
as
I
ot
being
of
technological importance.
Likewise,
the
sharp increase
in
critical current
DISCLAIMER
This report was prepared as an account of work sponsored by an agency of the United States
Government. Neither the United States Government nor any agency
thereof,
nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsi-
bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or
process disclosed, or represents that its use would not infringe privately owned rights. Refer-
ence herein to any specific commercial product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-
mendation, or favoring by the United States Government or any agency
thereof.
The views
and opinions of authors expressed herein do not necessarily state or reflect those of the
United States Government or any agency
thereof.
density at the very low fields (<3 T) is also neglected,
because functional dependences and scaling rules are
rarely needed in this field range.
ORNL-OWG 82-20(6 FED
Fig.
1. Upper critical field vs temperature for NbTi
commercial conductor of nominal composition 44 wt % Ti
to 48 wt % Ti.
The upper critical field data are shown in Fig. 1.
The best fit to the data up to B = 10 T is
I
(1)
where B
2
<0) = 14.5 T, T (0)
9.2 K, and n = 1.7.
Note that in theoretical calculations one usually sees
Eq. (1) used with n = 2. However, for commercial Nb-
46.5 wt % Ti, the 1.7 power law fits the data better
than a quadratic dependence on temperature. For more
accurate critical temperature calculations above 10 T,
a small adjustment to Eq. (1) is needed, namely, a
slight increase in the upper critical field to B (0)
= 14.8 T. No one simple power law holds over the
whole temperature range. The primary usefulness of
such a figure is to find out what the critical tempera-
ture is for a known maximum field. Rearranging Eq. (1)
to solve for T yields a useful formula for critical
temperature as a function of field,
T
c
(B) = 9.2[1 -
(B/14.5)]
0
'
59
for B < 10 T, (2a)
T
c
(B) = 9.2[1 -
(B/14.8)]
0
'
59
for B > 10 T. (2b)
Current Sharing Tempera
i:
re_
The critical temperature for NbTi varies from
9.2 K in zero fiel to the value given by Eq. (2) in
field B. The curi .nt sharing temperature varies from
this value for zei.o current to the bath temperature
for transport current I = I . If we make the
op c
plausible assumption that the current sharing tempera-
ture is a linear function of I /I , then the following
expression can provide the current sharing temperature
as a function of field with the reduced current I /I
op c
and the bath temperature T as parameters:
[T
C
CB> -
(3)
Enthalpy Calculation
One useful application of Eqs. 2 and 3 is the cal-
culation for the amount of energy suddenly deposited
in a specimen that is needed to raise the temperature
from the bath temperature to T
For specific heat.
we will combine the accepted value for copper with the
recent field dependent measurements of Elrod et al.
Only small temperature excursions
(i.e.,
up to 10 K)
are considered. The density at absolute zero is used
to obtain the heat capacity per unit volume.
-,-3
C =
10"
f + 1
[(6.75f +
50.55)^
+ (97.43f + 69.81BJT]
(in mJ/cm
3
-K) ,
(4)
where f is the copper/superconducting ratio. Using Eqs.
(2a),
(3), and (4), a calculation was made to determine
the enthalpy needed to raise a NbTi conductor from
T, = 4.2 K to T for I /I =0.5 and I /I = 0.8
b cs op c op c
as a function of copper/superconducting ratio. The
shape of the curve in Fig. 2 and the magnitude are not
surprising, but perhaps one forgets that it takes only
3 mj/cm
3
or less of energy density to start producing
Joule heating in a NbTi conductor in pool boiling
helium. One interesting feature is that stability
tests performed on a conductor operating at high
reduced current in a 5-T field perhaps can be corre-
lated with 8-T uerformance at more modest values of
reduced operating current.
ORNL-DWG8f-<7428 FED
6
5 —
E
u
-5
THAL
LJ
1 1 1
1
\
\
~\\
\
\
1 ! 1
1
J 1 1
V
-——
• 111
1
' !
Nb
\5
—- —
l l 1 1
1 i 1 1 |
-46.5 % Ti
T (i = 0.5 !
-—•
)
8T(i
—
8T(i
=
1 1 1 1
i ! >
—
= O.5)
0.6)
1 1
1
—
|
4 —
3 —
o
5 10 15
COPPER/SUPERCONDUCTING RATIO
20
Fig.
2. Enthalpy of NbTi conductor vs copper/supercon-
during ratio for 5-T and 8-T fields and with operating
currents of 0.5 and 0.8 of the critical current.
Critical Current Density
Variation With Field
Although there is no theoretical expression for
the dependence of the critical current density on
field at constant temperature for type II superconduc-
tors,
a linear equation is an excellent approximation
over the field range of most interest, 3 T < B < 10 T.
For each of the samples reported in the litera-
ture,
1-'*,8,1<J
th
e
appropriate linear equation has been
determined. Examples for two of the investigations
(Hudson et al.
3
and Spencer et si.
2
) are given in
units of 10
3
kA/cm
2
:
j = 550 - 50 B (5a)
j = 371 - 36.9 B (5b)
These and others not shown are not as distinct as
might first appear. A general form for j (B) at
T = 4.2 K (a corresponding equation can be determined
for other bath temperatures) is
j (B, T = 4.2 K) = j (1 -
0.096
B). (6)
The constant j can be found for any sample from a
single measurement. The coefficient of field
0.096
=
[B
^(T)]"
1
represents an effective upper critical
field for T = 4.2 K, in this case B (T = 4.2 K) =
10.4 T. This value is the mean of the ten measure-
ments analyzed and has a standard deviation of 3%.
Note that the samples measured span a wide spectrum
of critical current density; j (B = 8 T, T = 4.2 K)
varies from 50 zo 108 kA/cm
2
.
Variation With Temperature
A linear variation with temperature for the
critical current density at constant field was estab-
lished a long time ago and is generally accepted as a
reliable assumption. The appropriate linear equations
for some of the samples studied have been determined
for B = 8 T over the widest temperature range avail-
able (the case of super"luid helium is considered
separately).
j (B = 8 T, T) = j'(l - 0.177 T),
c o
(7)
over the temperature range 2K<T<5.5K, where
again the constant j' can be determined from a single
o
measurement at 8 T (corresponding equations can be
determined for other field
values).
The coefficient,
0.177, has a standard deviation of less than 1% over
the data analyzed and represents the effective criti-
cal temperature, 0.177 = [T
(B)]"
1
or T
c
(B = 8 T) =
5.65 K.
The field value of interest for high-energy
physics applications is 5 T. The temperature varia-
tion at this field value is
j(B = 5 T, T) = j'(l - 0.14 T)
(8)
which holds over the range 2 K < T < 7 K. The effec-
tive critical temperature is T (B = 5 T) =
(0.14)""
1
=
7.1 K. The standard deviation is 2%.
Low Temperature, High Field Variation
There is much less information available in the
range of superfluid helium (T < 2.17 K) and in fields
above 10 T. As might be anticipated, the data have
somewhat more scatter in this range as
well.
While a
great deal of interest has focused on the alloys of
NbTi (particularly NbTi with tantalum additions) for
applications at 1.8 K, Hirabayashi et al.
1
' have
shown recently that the binary alloys are every bit
as good as the ternaries at fields above 10 T at
1.8 K. In fact, the highest critical current density
they report on at B = 12 T, T = 1.8 K is a binary
NbTi with a value of 100 kA/cm
:
. This is comparable
with the best NbTi data measured at B = 8 T, T = 4.2 K.
The field variation at T = 1.8 K (another diffi-
culty in low temperature data is that some samples are
reported at 2 K while others are reported at 1.8 K) is
j (B, T = 1.8 K) = j (1 -
0.0728
B),
(9)
which is good over the range 5 T < B < 13 T. The
coefficient corresponds to an effective upper critical
field of
B
c2
(T
= 1.8 K) =
(0.0728)-'
= 13.7 T. The
standard deviation for these data is 3%. The tempera-
ture variation at B = 11 T is
j (B = 11 T, T) = j'(l -
0.245
T), (10)
which holds over the temperature range 1.8 K < T <
3.8 K. The coefficient corresponds to an effective
critical temperature of T (11 T) =
(0.245)"
1
= 4.1 K.
The standard deviation is 6%.
Conclusion
Except for the highest fields and lowest temperature
or for some special purpose, there is sufficient good
critical current density data on NbTi in the literature,
in the author's opinion, to make further measurements
unnecessary. A single short sample measurement at a
convenient field and bath temperature is all that is
needed to supply the magnitude of the critical current
density and therefore the constant j or j' in the
o o
equations for the functional dependences given in the
paper.
Acknowledgments
The author is grateful to Dr. Y. Iwasa for
useful comments and for transmitting data prior to
publication and to Dr. P. A. Hudson for enlarged
figures of published data.
References
1. R. Hampshire, J. Sutton, and M. T. Taylor,
"Effect of Temperature on the Critical Cu- rent
Density of Nb-44 wt % Ti alloy," Low Temperature
and Electric Power, London: International
Institute of Refrigeration Commission I, Annexe
M69-I,
pp. 251-257, 1969.
2.
C. R. Spencer, P. A. Sanger, and M.Young, "The
Temperature and Magnetic Field Dependence of
Superconducting Critical Current Densities of
Multifilamentary Nb-jSn and NbTi Composite Wires,"
IEEE Trans. Magn., Vol. MAG-15, pp.
76-79,
January 1979.
3. P. A. Hudson, F. c. Yin, and H. Jones, "Evaluation
of the Temperature and Magnetic Field Dependence
of Critical Current Densities of Multifilamentary
Superconducting Composites," IEEE Trans. Magn.,
Vol.
MAG-17,
pp. 1649-1652, September 1981.
4.
Y. Iwasa and M. J. Leupold, "Critical Current
Data of NbTi Conductors at Sub-4.2 K Temperatures
and High Fields," Cryogenics, Vol. 22, pp. 477-
479,
September 1982.
5. L. F. Goodrich and F. R. Fickett, "Critical
Current Measurements: A Compendium of Experi-
mental Results," Cryogenics, Vol. 22, pp. 225-
241,
May 1982.
6. M. S. Lubell, "State of the Art of Superconducting
Magnets/'
Cryogenics, Vol. 12, pp. 340-355,
October 1972.
7.
D. C. Larbalestier, "Nb-Ti Alloy Superconductors
— Present Status and Potential for Improvement,"
Advances in Cryogenic Engineering/Materials,
(ed. by A. F. Clark and R. P.
Reed),
New York:
Plenum Publishing Corporation, 1980, Vol. 26,
pp.
10-36.
8. D. G. Hawksworth and D. C. Larbalestier, "The
High Field j and Scaling Behavior in Nb-Ti and
Alloyed Nb-Ti Superconductors," Proceedings of
the 8th Symposium on Engineering Problems
of Fusion Reseach, New York: IEEE, 1979,
pp.
249-254.
9. S. A. Elrod, J. R. Miller, and L. Dresner,
Advances in Cryogenic Engineering/Materials,
(ed. by
R7~P.
Reed and A. F.
Clark),
New York:
Plenum Publishing Corporation, 1982, Vol. 28,
pp.
601-610.
10.
R. Aymar et al., "Tore-Supra Programme of Develop-
ment Qualifying Tests," document EUR-CEA-FE-1143,
February 1982.
11.
H. Hirabayashi, M. Kobayashi, T. Shintomi,
M. Takasaki, et al., "Alloy Conductors for the
Use of High Field Magnet Construction," paper
presented at the Ninth International Cryogenic
Engineering Conference, Kobe, Japan, May
11-14,
1982.
