Analysis of the general structure of the E-I characteristic of high current superconductors with particular reference to a Nb-Ti SRM wire . Cryogenics 27 608

Analysis of the general structure of the E-I characteristic of high current superconductors with particular reference to a Nb-Ti SRM wire . Cryogenics 27 608
Analysis of the general structure of the E-I
characteristic of high current superconductors
~ i wire
with particular reference to a ~ b - SRM
D.P. Hampshire* and H. Jones
The Clarendon Laboratory, Parks Road, Oxford OX1 3PU. U K
Received 7 January 1987; revised 1 June 1987
In this Paper thecompatability between a three parameter fit, which assumes a normal density
distribution of critical currents, and the familiar empirical two parameter law ( V = atn) is
demonstrated by applying them both to the analysis of data obtained in the characterization
of a Nb-Ti standard reference material supplied by NBS. Data taken at temperatures of 4.24
and 2.21 K are used to illustrate that the three physical parameters obey general scaling laws
and to outline the physical interpretation which is derived for the empirical parameters. The
validity of flux flow in high current density materials being characterized by defect motion
with the flux line lattice is demonstrated.
Keywords: superconductors; Nb-Ti; critical currents; mathematical m o d e l s
This Paper presents a comparison between the three
parameter fit (3-P-F) the authors have considered before1
E = electric field along the superconductor;
I = transport current;
f = normal density function;
I,,+= mean critical current;
R,, = interaction length resistance;
p = synchronization constant (a measure of the spread);
and the empirical 2-P law which can be written in the
E =aln
(The parameters a and n are termed the alpha parameter
and the index.)
The first section addresses a brief physical interpretation of the 3-P fit. It demonstrates the compatibility
between this functional form and the empirical law. The
limiting forms of the E-l characteristic are derived in the
ranges of high, intermediate and low currents. The section
is completed with the derivation of simple relations
relating the empirical parameters of the 2-P law to the
physical parameters of the 3-P fit.
In the second section, data are presented on a Nb-Ti
wire which has been adopted by NBSS as their standard
reference material**. The analysis of data presented
previously, at 4.24 K is extended to include characterization using the empirical law and the physical interpretation of the empirical parameters explicitly outlined.
Additional data generated at 2.21 K are presented and are
again interpreted in terms of the 2-P and 3-P fits. The
scaling laws governing the three physical parameters
observed for Nb,Sn throughout its entire superconducting phase are shown to hold for this Nb-Ti.
Finally, the theoretical and experimental problems
associated with developing a more complete characterization of superconducting materials is discussed.
Three parameter fit
'Present address: Applied Superconductivity Centre, University of
Wisconsin-Madison, 917 Engineering Research Building, 1500
Johnson Drive, Madison, W1 53706. USA
t l n Reference l,theconstant b in Equation (1 ) was taken to be equal
to zero since clearly negative currents (i.e. T,< 0) are non-physical.
However. in this work we shall take b = - W for mathematical
tractability. assuming that P and I
,are such that the contribution to
< 0 is negligible. Physically this is equivalent to
the integral for I,
assuming that over the E-l range of interest, the effect due to the low
current tails of the distribution for li< 0 can be ignored. For the data
presented it will be seen that this is indeed appropriate
l ~ h E-lforms
for Equations (1 ) and (2) are used since these forms
explicitly demonstrate that for the purposes of comparison there is a
redundancy in the parameter: area of superconductor
001 1-2275/87/110608-09 $03.00
?) 1987 Butterworth 81 CO (Publlshers) Ltd
Lryogenics 1957 Vol 27 November
A detailed derivation of the 3-P-F has been given
elsewhere6. The basic physical concepts are as follows.
It has long been appreciated that in defect-free systems,
an ohmic-type law characterizes the state of flux flow of
the form
**It should be noted that one of the samples of 22.2 m, commercially available from NBS. was not used. Instead a longer length
( = 6m) from the same batch of material was used
E 1 characteristic of hi!7h current superconductors: D.P. Hampshire and H . Jones
where: RfL = flux flow length resistance; and I, =critical
For highly inhomogeneous materialstt, onc must consider the variation of I, along the length of the material. In
the simplest case, a distribution in critical currents is
considered and the resultant E-I characteristic is merely a
superposition of the ohmic-type relations given by Equation (3). Thus
E = R,,
( I ) For I + + MJ
This is the high voltage limit occurring in the high current
tails of the distribution function. The second term in
Equation (7) tends to zero. Substituting the normalization
we find
(1 - Ii)f(li)dIi
Lim E = RIJI - 7,)
(2) For I = 7,
where: li= local critical current in a given region; and RfL
has been replaced by R,, (=the interaction length resistance)since, as will be shown, R,, is not simply the flux
flow resistance.
The functional forms of Equations (4)and (5)have been
presented before. The function f(1)has been interpreted as
defining the interaction between flux flow and stationary
flux -a unique single value for I, was assumed7.However,
in the present interpretation, this interaction is not
introduced. Following Jones et al.', the superconducting
material is considered to be divided up into regions of
unique and different critical current. The junction RI)
characterizes the distribution in critical currents.
It has been shown experimentally that f(1) approximates well to a normal density function1. This is in
agreement with the central limit theorem which predicts
this functional form, since superconductors consist of a
large number of different sources of inhomogeneity9.
Similarly, this normal density function can be justified in
the limit of a great number of defects actingjointly on each
vortex and contributing additively to the total pinning
drag forceI0. More recently, in a detailed deconvolution
of the E-I characteristic, the essentially Gaussian nature
of the distribution has been verified and some of the
deviations from the normal density function have been
The limiting functional forms of the E-l characteristic
are considered below. Equation (4) can be rewritten
Integrating by parts
This is the intermediate current limit where the greatest
proportion of regions are at criticality, changing from
regions of stationary flux to regions of flux flow.
If we consider the integral (S) in Equation (7) where
Lim S = Lim
Then, by letting X =
r/7. and using Stan-
dard expansions
Lim S =
l = 7c
J T ,
Lim E = RI,-----I,
r ~ l P(2nIf
(3) For
t t ~ h i sdescript~onof the material effectively defiues the dominant
mechanism for loss due to viscous flux flow [i.e. Equation (2!].
Clearly, in general, other mechanisms such as flux creep will need to
be incorporated. However, it is demonstrated that the results derived
for thisspecificcasearesufficient to explain thedata presented in this
I riiic)I2>>
/? -
This is the condition for the low voltage limit occurring
in the low current tails of the distribution function, which
can be considered as appropriate for I +0.
Using an expansion for the integral S12 we find
Cryogenics l987 Vol 27 November
E ! characteristic of high cur:.ent superconductors: D.P. Hampshire and H . Jones
Substituting into Equation (7)
Lirn E = R,,
1 . ~ 1>>l
Thus we have the expressionsfor the limiting functional
forms of the E - l characteristic assuming a normal density
function for the distribution in critical currents [Equations (9), (13) and (16)].
In Figures I and 2, general E-l characteristics have
been plotted for two values of 1, (150 and 300A) for
different values of j?. However, the value of RILhas been
left as a free parameter since it has been demonstrated for
Nb-Ti and Nb3Sn that R,, is typically five orders of
magnitude less than the normal resistivity of the respective materials6. The figures explicitly demonstrate the
functional form of the E-I characteristics. In Figures3 and
4, the functional form of these E-I characteristics have
Current, [ ( A )
F i g u r e 2 General E-/characteristic on a log-&
scale, assuming a
normal density function for the critical currents, where RIL is a free
parameter ( R m-'). I, = 300A and p (thesynchronization constant)
is a variable
been replotted on a log-log scale. It is clear that the 3-P
functional form predicts that the empirical law will hold
to a good approximation in the low current tail over a
number of decades of electric field13.
This section concludes using the equations derived. by
determining the relations between the empirical constants
a and n, and the physica! parameters R,,, I, and j?. In the
high current limit [i.e. limit ))l by equating Equation (9)
to Equation (2) it is conc ~idedthat Lirn a = RIL and
Lirn n = 1. In the low current limit [i.e. limit (l)], it is clear
from Figures 2 and 4 that the convex nature of the curves
implies Lim n -r co, and the insensitivity of the index n
l x l 2 U7
with current that is m e n t i o n e d G a implies Lim a-0.
Ixll<< I
In the intermediate current range [i.e. limit (2)] which is of
most interest in the analysis of the data presented, we have
from Equation (2)
Current. f ( A )
F i g u r e l General E-lcharacteristic o n a b g - h e a r scale. assuming
a normal density function for the critical currents. where RI, is a free
parameter ( R m - ' ) , I,= 150 A and p (the synchronization constant)
is a variable
This equation can be rewritten in terms of the standard
E-l characteristic of high current superconductors: D.P. Hampshire and H . Jones
deviation of the critical current distribution, a, where
From Equation (19) we have
aE 1
Lim -= - R,, = a n l n 1 7 a
It is pertinent to note that Equation (22) may be rewritten
Thus, from Equationq (20) and (23) we have the
empirical parameters in terms of the physical parameters.
This allows interpretation of the field and temperature dependence of the index n and the alpha parameter. This will
be explicitly outlined using data from the SRM Nb-Ti
wire in the next section.
Data on the SRM Nb-Ti wire
Experimental procedure
Loglo current. ( I ILoglo ( A l l
Figure 3 General E-/ characteristic on a log-log scale, assuming a
normal density function for the critical currents, where RI, is a free
parameter (Q m-'), I, = 150 A and (thesynchronization constant)
is a variable
L o p , current, ( I ) CLoqm (A) l
Figure 4 General E-/ characteristic on a log-log scale. assuming a
normal density function for the critical currents. where RIL is a free
parameter (Rm-'), l, = 300 Aand 1 (the synchronization constant)
is a variable
The equipment and experimental techniques for generating the E-I data have been detailed by the authors in the
preceding paper on this material14. The recommendations outlined by NBS Ibr measuring the electric fieldcurrent transitions of this material have been closely
followed to obtain these data. At each fixed temperature, a
direct current was slowly increased through the superconducting specimen and the voltage generated across the
potential taps measured. The ramp rate for the sample
current was chosen so that within experimental accuracy,
the E-I transition was reversible for I increasing and I
decreasing. A hard copy was produced of the voltage
versus current.
The self-field effects, which are only significant below
3 T were eliminated by taking two traces for opposite
direction of current flow. The mean value of current at
each voltage was used to define the zero self-field E-I
characteristic. No strain effects were observed.
During measurement the sample lay in intimate
thermal contact with the helium bath. The temperature
was stabilized by monitoring a manometer connected by a
static line to the pumped dewar. Standard vapour pressure curves were used to calibrate the temperature at 4.24
and 2.2 1 K..
The hard copy E-I characteristics were digitized between 10 and 50 pV m-' at 5 p V m-' intervals. Above
B,,(T) the resistance of the sample and sample holder was
measured and found to be field independent (the contribution of the superconductor in the normal state is
negligible). At each voltage measured, the appropriate
current being shunted through this parallel ohmic path
was subtracted and the corrected current through the
superconductor alone obtained as a function of voltage.
A full analysis of the errors in measurements generated
using these techniques has been presented elsewhere14.
This analysis suggests that the errors in the data can be
quoted in terms of uncertainty in temperature of k0.05 K.
Cryogenics 1987 Vol 27 November 611
E-l characteristic of high current superconductors: D.P. Hampshire and H. Jones
The analysis of the corrected E - I characteristics in terms
of both the empirical 2-P law and the 3-P fit is considered
below. The universality of the pinning force in agreement
with the Fietz-Webb15 scaling law is demonstrated as
well as the universality of the synchronization constant
and the interaction resistance as has been found for
For mathematical tractability, the normal distribution
function intrinsic to the 3-P fit can be rewritten to a good
approximation over the range of the distribution that is
sampled by
Thus the raw data was fitted to the following two
equations over the range 10-50 pV m- '
Reduced mopne~icfidd.B~r)(dimensionless units)
Figure 6 Reduced pinning force versus reduced magnetic field at:
0, 4.24 K; and B. 2.21 K for the Nb-Ti SRM
E = aI"
These parameters are considered in turn below.
Mean critical current. In Figure 5, the mean value of the
critical current is presented as a function of field at 4.24
and 2.21 K. The difference between the mean critical
current and the critical current at 30pVm-' for this
particular Nb-Ti material is less than 1 A (i.e. less than the
size of the symbol in Figure 5). It is of interest to note that
the functional form of the pinning force and the characteristic significant deviation from linearity in low field is
compatible with the model that predicts that the critical
current of Nb-Ti is determined by the shearing of defects
in the F-L-L past point pinning centres16.
In Figure 6, the critical current data has been replotted
as the reduced pinning force versus the reduced magnetic
field. The universality of these data in agreement with the
Fietz-Webb Scaling Law is explicitly demonstrated.
Synchronization constant Bllndex n. In Figure 7 the
synchronization constant B is plotted as a function of field
at 4.24 and 2.2 1 K. The parameter B, is the value obtained
from the full 3-P fit using Equation (23).The parameter BE
is defined by
in accordance with Equation (20). In agreement with the
algebraic manipulations in the three parameter fit section,
we have
This equality has been observed experimentally by
Warnes and Larbalestier. Their data can be expressed by
BE = (1.2 0.2)bF*.
The difference between B, and BE that is observed in
Figure 7 occurs since the equality in Equation (29) only
holds rigorously at I,. In practice, these parameters have
been determined by fits taken over a range about I , .
It is shown in any elementary statistical text thatt2
Thus we now have a physical interpretation of the index n
through the synchronization constant, B.
In Figure 8, the universality of the synchronization
constant that has been found for 1.5 T S B S 15 T and
*The data in the paper referredto (Reference 12) has been presented
in te:ms of I(FWHM) where FWHM =full width half maximum.
From Equation (21)
Field. B ( T )
Figure 5 Mean value for the critical current distribution at 4.24 and
2.21 K as a function of field for the Nb-Ti SRM. 0 . 4 . 2 4 K; B, 2.21 K
Cryogenics 1987 'dot 27 November
E-l characteristic of high C'urrent
D.P. Hampshire and H. Jones
different ICi
have the same reduced field dependence but
havc different temperature dependences. More generally
it has been demonstrated that the universality of 11
demonstrates that the component regions of the distribution obey different scaling laws which are all of the
general Fietz-Webb type. The field and temperature
dependence of a material's universal synchronization
curve characterizes the different functional forms of f(T)
and g(h) that are found in these regions throughout the
Interaction length resistancelalpha parameter. In Figure
9 the interaction length resistance is plotted as a function
Field, B(T1
Figure 7 Synchronization constant as a function of field at 4.24
and 2.21 K for the N b T i SRM. W, PE(2.21 K); 0 , PE(4.24K);
flF(2.21 K); 0 , P ~ ( 4 . 2 4K)
of field at 4.24 and 2.21 K. The salient feature of these data
is that to within experimental accuracy the interaction
length resistance is a linear function of field through the
origin at low fields as has been found for the high current
material Nb,Sn by the authors and for many defect-free
The deviation
(low current)Type I1 superconductors'7~L8.
at the lowest field in the data (at 2 T) from the idealized
straight-line can be attributed to heating causing a
temperature increase of 0.05 K during the transition.
The universality of this parameter is demonstrated in
Figure 10 where the interaction coefficient, a*, is plotted as
a function of reduced field. The interaction coefficient, sl*,
is defined by
which normalizes R,,(B, T) using the idealized straightline through the origin.
Reduced mopnetic f 1 e l d . B dimensionless units)
Figure 8 Reduced synchronization constant versus reduced magnetic field at 4.24 and 2.21 K for the N b T i SRM. Symbols as for
Figure 7
2.5 K < T T, for Nb,Sn is demonstrated for this particular Nb-Ti wire. The reduced synchronization constant,
B*, is defined by
where B(0, 4.24 K) = 39.5 and @(O, 2.21 K) = 45.0. From
Equation (30)it can be seen that insofar as the synchronization constant, or equivalently the index n, is independent
of reduced field (for b < 0.4) but is a function ol temperature, the different critical currents that characterize this
Nb-Ti material are given (within this range) by
This equation implies that the component regions of
Field. B [ T )
Figure 9 lnteraction length resistance of the N b T i SRM versus
magnetic field at 4.24 and 2.21 K. W, RILL(2.21 K); a, RIEL(4.24 K);
D. RIL(2.21 K); 0. RIL(4.24 K)
Cryogenics l 981 I
c : 21 November
61 3
E-l characteristic of high current superconductors: D.P. Hampshire and H. Jones
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Reduced moqnet~cfield.
(dimensionless units)
Figure 10 Interactioncoefficient versus reduced magnetic field at
4.24 and 2.21 K for the Nb-Ti SRM. m, ~ ~ ( 2 . K);
2 1 a.~ ~ ( 4 . K);
0, aF(2.21 K); 0 ,aF(4.24 K)
R N L = 6 . 9 7 R m-' f 0.1 R m - '
Thus Figure 10 shows
RIL(B. T ) = RIL(0) b .K(b)
distribution of inhomogeneitiesY.In this case the supercurrent increases through the superconductor until criticality just before flux flow is about to occur. Beyond
criticality. because of the superconductor's high differential resistivity. any further increase in currents is shunted
through the low resistivity copper matrix. In this case R,,
becomes simply the matrix resistivity and the field
dependence is due merely to the magnetoresistance of the
high resistivity ratio copper necessary for stability in this
wire. However, this model is not appropriate here since
although it could be argued that the field dependence in
Figure 9 is the magnetoresistance of the copper matrix,
one would expect R,, to be independent of temperature,
which is clearly not the case. The data on Nb,Sn
presented by the authors shows R,, changing by an order
of magnitude at 2 T between 2 and 10 K. The very low
value for the interaction resistivity found above demonstrates that in the flux flow state when most of the
component regions in the normal density distribution of
critical currents are above criticality,only a small fraction
of the total fluxons present are in motion. This has been
interpreted by the authors as direct experimental evidence
that flux flow is characterized by the motion of defects
within the flux line lattice. This is analogous to the role of
defects in real crystals reducing the critical stress by many
orders of magnitude from that of the ideal crystal. In
conclusion: the function f describes the distribution in
critical currents associated with the different defects
within the flux line lattice; RI, is a field and temperature
dependent length resistance, characterizing the superconductor when all the defects within the flux line lattice
are in motion.
In Figure l 1 the length resistance of Nb-Ti (i.e. with the
matrix etched off) is plotted as a function of temperature.
Thestandard Kohler law predicts there is no intrinsic field
dependence of the normal state resistivity for this Nb-Ti.
This figure demonstrates that in the normal state above T,
the length resistance RN, is given by
- 4TIT,)
or equivalently
Theoretical and experimental analyses of defect-free Type
I1 superconductors assert that above criticality, in the flux
b = BIBcz(T);
Lim K(b) = l ;
Lim YT/T,) = l ; and
R,,(O) = a constant characteristic length resistance of the
material in the flux flow state.
It is interesting to note that for this Nb-Ti Y T )
'l3. By extrapolating to T = 0
= 1/(1 - T/T,)'I3*
R,,(O) = Bcz(0). Lim 8 R d B . 0) = 33.g f l m - l
Or equivalently for this wire
l . . . .
1 0 20
where p, is the interaction resistivity.
A possible model of the interaction length resistance
can be made in terms of it being characteristic of random
Cryogenics 1987 Vol 27 November,
30 4 0
50 6 0 7 0
L _
Temperoture. T ( K )
Figure 11 Length resistance of the Nb-Ti SRM as a function of
E I characteristic of high current superconductors: D.P. Hampshire and H . Jones
Row state
Lim RfL(B,T) =
- . R,,
where RfLis defined in Equation (3).
It is important to note that Equation (35) is derived for a
homogeneous system with randomly distributed point
scattering centres. Certainly the data here demonstrates
that the interaction resistance is some five orders of
magnitude less than the flow resistance, although the
linearity in fields common to both seems highly
It is certainly clear that more theoretical input is
required to generalize Equation (35) from its appropriate
idealized system to more complex systems such as Nb-Ti.
Equally, more measurements of the flux flow resistivity
and interaction resistivity are required to characterize the
functional form of, and relationship between, these two
In Figures 9 and 10, the empirical derivation of the
interaction length resistance, R,,,, is also presented. The
3-P fits show that for this particular Nb-Ti material, I, is
found at a voItage criterion of z 30pV m-'. If we use this
electric field criterion to arbitrarily define I,, then either
using the alpha parameter directly through Equation (23)
or equivalently through Equation (24) we have
2n30 pV m-'
Lim . I
The good agreement between RI,, and RI, can be seen
in these two figures. It is interesting to note that for two
high current samples, Nb,Sn and Nb-Ti, I, has been
found at a fixed electric field criterion throughout the field
temperature range ( z 5 0 and zz 30pV m-', respectively).
However, more data will be ,required to decide whether
Equation (22) can be generally used as a technical tool to
obtain an order of magnitude estimate of the interaction
length resistance using the empirical 2-P law without
having to embark on the significant input involved in the
full 3-P fit.
uniquely associate the function derived in deconvolution
with the distribution of critical currents intrinsic to the
sample [using Equation (5)], it isclear that the ohmic-type
relation [Equation ( 3 1 must hold in all regions where Ii is
less than I. It has been shown that this linear relation will
break down at large currents and electric fields since: liis
not independent of transport current at high currents and
in p a r t i ~ u l a r ' ~Lim li= 0; and for very high current
density materials at very high electric fields there is
significant power dissipation which will increase Rli.
These inevitable non-linear effects will cause a non-zero
value of d2E/d12 beyond the linear region which is not
attributable to the intrinsic distribution in critical currents. Physically this argument can be seen as follows: the
E-I characteristic can be seen to be a superposition of
'ohmic-type' characteristics defined by Equation (3). It is
not possible when deconvoluting the E-I characteristic to
distinguish non-linearity in this equation from intrinsic
variations in the distribution function unless the nonlinear terms can either be corrected for or shown to be
If we now consider a more extreme case, where nonlinearity has started to occur before all regions are above
their local critical current, it is clear that there will be no
linear region in the E-l characteristic. One of the
fundamental difficulties in the analysis of these systems
will be due to the impossibility of proving a priori, from
the E-I characteristic alone, when non-linearity becomes
Certainly to address these questions comprehensively
accurate measurement of the double-differential of the
E-J characteristic will be required. Recently Warnes and
Larbalestier have been addressing this. It would be of
interest to complement their approach by measuring this
quantity (and hence the distribution function) directly
using, for example, a pair of coupled circuits, most
importantly to test for the non-reversibility of the distribution function (which is to be expected to at least some
degree since critical current density is not a fundamental
thermodynamic function), to reduce the time of analysis,
and in order to investigate the non-linear effectsdiscussed
More complete crlaracterization of
superconducting materials
In this final section, the theoretical and experimental
problems associated with developing a more complete
characterization of superconducting materials and in
particular the field and temperature dependence of their
distribution function is discussed.
Consider an idealized case consisting of a highly
stoichiometric material which has a normal density
function distribution ofcritical currents. It is clear that the
first three regions of the E-I characteristic can be
described as follows: 1, an initial low voltage region
where there is very little flux flow and the conductivity is
essentially infinite; 2, an intermediate region of high
curvature in the characteristic occurring over a range I,/P
(flux flow is just beginning in many regions); and 3, a linear
high voltage region where aE/dl= R,, and where this
linearity extrapolates through I,. (At higher transport
currents there are other regions characterized, for
example, by the motion of all unpinned fluxons or the
bulk motion of all fluxons.) Necessarily, in order to
Data has been generated characterizing a Nb-Ti SRM
at 4.24 and 2.21 K.
The compatibility between the 2-P empirical law and
the 3-P fit has been demonstrated.
The universality of RI,, the interaction length resistance, and /3, the synchronization constant, has been
demonstrated for this particular Nb-Ti.
The interaction length resistance is a field and temperature dependent parameter characteristic of the superconductor. The E-I characteristic is characteristic of
the bulk superconductor and not due to damaged
regions along its length.
The universality of the P constant demands that the
component regions of the superconductor obey different scaling laws which are all of the Fietz-Webb
functional form.
A physical interpretation of the empirical index, n, has
Cryogenics 1987 \Jol 27 November
61 5
E-l characteristic of high current superconductors: D.P. Hampshire and H. Jones
been given through the equation
or equivalently
7 An estimate of the interaction length resistivity can be
made through the equation
The magnitude of this parameter demonstrates that
the mechanism for flux flow is characterized by defect
motion within the flux line lattice.
8 An outline of the more exacting measurements required for a more comprehensive understanding of
high current superconductors and some of the associated analytical problems have been discussed.
The authors are grateful to Mr J. Cosier for his help
during the production of the data for Figure 11, Dr W.
Warnes and Professor D.C. Larbalestier for copies of
their work prior to publication, Dr A.F. Clark for
providing the Nb-Ti material, and Mr Alan Day and
Miss Penny Jackson for their help in the preparation of
this publication.
1 Hampshire, D.P. and Jones, H. Critical current ofa NbTi reference
material as a function of field and temperature Proc Ninrh Inr
Magner Technology Conf (Eds Marinucci. C. and Weymuth, P.)
Swiss Institute of Nuclear Research, Villigen, Switzerland (1986)
53 1
S1 6
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