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MAGNETIC MEASUREMENTS FOR

HIGH ENERGY PHYSICS

APPLICATIONS

Final Report

R. B. Goldfarb, Editor

United States Department of Commerce

National Institute of Standards and Technology

NISTIR 3975

MAGNETIC MEASUREMENTS FOR

HIGH ENERGY PHYSICS

APPLICATIONS

Electromagnetic Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303

August 1991

Prepared for:

U.S. Department of Energy

Division of High Energy Physics

Washington, DC 20545

Final Report

R. B. Goldfarb, Editor

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

Magnetic Measurements for

High Energy Physics Applications

Final Report

R. B. Goldfarb, Editor

Electromagnetic Technology Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

Boulder; Colorado 80303

Abstract

This report is a collection of papers describing magnetic measurements on

multifilamentary Nb-Ti superconductor wires and cables as a function of magnetic

field and time at liquid-helium temperatures. The papers deal with ac losses and

interfilament coupling by proximity effect and eddy currents. Flux creep was

investigated under different experimental conditions. A Hall-probe magnetometer,

which was used to measure magnetization and flux creep in the presence of a

transport current, is described. A method for increasing the critical current of superconducting cable by controlling twist pitch is demonstrated. A critical-state model for

the magnetization of superconductors was developed for samples with fielddependent critical current density and rectangular cross section.

Key words: ac losses; critical current; critical state; eddy currents; flux creep; Hall

sensors; hysteresis; magnetization; magnetometry; multipole fields; niobium-titanium;

proximity effect; Superconducting Super Collider; superconductors.

in

Magnetic Measurements for

High Energy Physics Applications

Final Report

Executive Summary

This report includes work done for the U. S. Department of Energy, Division

of High Energy Physics, during the period 15 June 1988 — 14 June 1991, under

interagency agreement No. DE-AI05-85ER40240. The topics investigated are

summarized below.

• “Magnetic Characteristics and Measurements of Filamentary Nb-Ti Wire for

the Superconducting Super Collider” - This paper discusses time-independent and

time-dependent ac losses that may be detected with magnetic instrumentation.

Different forms of magnetometry are reviewed. AC susceptibility measurements

demonstrate coupling in prototype SSC wires and cables containing Mn.

• “Hall Probe Magnetometer for SSC Magnet Cables: Effect of Transport

Current on Magnetization and Flux Creep” - This simple magnetometer can be used

to measure hysteresis loops and flux creep in superconductor wires and cables

carrying transport current. The presence of a transport current, on the order of 20%

of the critical current, did not greatly affect the magnetization or rate of flux creep.

• “Transport Current Effects on Flux Creep and Magnetization in Nb-Ti

Multifilamentary Cable Strands” — Flux creep measurements were made on an SSC

cable strand carrying a transport current equal to about 70% of the critical current.

There was a factor of four increase in the magnetization decay over a sample with

no transport current, and substantial decreases in the magnetization and fullpenetration field.

• “Enhanced Flux Creep in Nb-Ti Superconductors after an Increase in

Temperature” — An inadvertent increase in temperature in the operation of SSC

magnets could increase their rate of field decay. Measurements were made on short

cable samples. Magnetic relaxation measurements at 3.5 and 4.0 K, after field

cycling, showed a decay of 2.8% in 50 min. Relaxation measured after the critical

state was established at 3.5 K, followed by a temperature increase to 4.0 K, was 4.8%

in 50 min. A 0.1-K rise in temperature caused a magnetization decay of 3.2% in

50 min.

• “Investigation of Possible Causes for the Multipole Drift in Superconducting

Dipole Magnets” - Two experiments are described which address the problem of

multipole-field decay in dipole magnets: exposure of the cable to field gradients, and

the presence of solder joints. A sample cable’s magnetization decay was not affected

by either condition.

v
• “Effect of Mechanical Deformation on Nb-Ti Filament Proximity-Effect

Coupling at the Edges of SSC Cables” — Proximity-effect coupling in multifilament

wires may be induced by cabling. The rate of flux creep at low field is the same for

coupled and uncoupled filaments, which suggests that the proximity coupled matrix

is not a significant source of flux creep.

• “AC Loss Measurements of Two Multifilamentary NbTi Composite Strands” — This paper discusses magnetic hysteresis losses in two very different

multifilamentary Nb-Ti wires. One wire, a prototype SSC conductor, exhibited

interfilament coupling and high losses. The other, containing widely spaced filaments

in a resistive Cu-Ni matrix, had no coupling and lower losses. However, both wires

have similar figures of merit when the critical current density of the wires were

included in the analysis.

• “Effect of Cable and Strand Twist-Pitch Coincidence on the Critical Current

of Flat, Coreless Superconductor Cables” - The current-carrying capacity of a

Rutherford-type cable may be increased by about 10% simply by matching the cable

twist pitch to the strand twist pitch. Current transfer among filaments is minimized

and the voltage-current characteristic is improved.

• “Kim Model for Magnetization of Type-II Superconductors” - This paper

develops magnetization equations for hard superconductors using the Kim model for

critical current density as a function of internal magnetic field. The equations are for

an infinite sample with rectangular cross section in longitudinal field. In certain

limits, the equations are exact for cylinders and slabs. In other limits, the equations

reduce to those of the well known Bean model.

vi
CONTENTS

Abstract iii

Executive Summary v

I. Magnetic Characteristics and Measurements of Filamentary

Nb-Ti Wire for the Superconducting Super Collider

R. B. Goldfarb and R. L. Spomer 1

II. Hall Probe Magnetometer for SSC Magnet Cables:

Effect of Transport Current on Magnetization and Flux Creep

R. W. Cross and R. B. Goldfarb 9

III. Transport Current Effects on Flux Creep and Magnetization

in Nb-Ti Multifilamentary Cable Strands

R. W. Cross 12

IV. Enhanced Flux Creep in Nb-Ti Superconductors

after an Increase in Temperature

R. W. Cross and R. B. Goldfarb ........................ 18

V. Investigation of Possible Causes for the Multipole Drift

in Superconducting Dipole Magnets

R. M. Scanlan, R. W. Cross, and R. B. Goldfarb 20

VI. Effect of Mechanical Deformation on Nb-Ti Filament

Proximity-Effect Coupling at the Edges of SSC Cables

T. W. Petersen and R. B. Goldfarb 27

VII. AC Loss Measurements of Two Multifilamentary NbTi

Composite Strands

E. W. Collings, K. R. Marken, Jr., M. D. Sumption,

R. B. Goldfarb, and R. J. Loughran 29

VIII. Effect of Cable and Strand Twist-Pitch Coincidence on the

Critical Current of Flat, Coreless Superconductor Cables

J. W. Ekin 37

IX. Kim Model for Magnetization of Type-II Superconductors

D.-X. Chen and R. B. Goldfarb 48

Units for Magnetic Properties

R. B. Goldfarb and F. R. Fickett 61

Vll

MAGNETIC CHARACTERISTICS AND MEASUREMENTS OF

FILAMENTARY Nb-Ti WIRE FOR THE SUPERCONDUCTING SUPER COLLIDER

R. B. Goldfarb and R. L. Spomer

Electromagnetic Technology Division

National Institute of Standards and Technology

Boulder, Colorado 80302

ABSTRACT

In synchrotron accelerator applications, such as the Superconducting

Super Collider (SSC) , superconducting magnets are cycled in magnetic field.

Desirable properties of the magnets include field uniformity, field stability with time, small residual field, and fairly small energy losses upon

cycling. This paper discusses potential sources of problems in achieving

these goals, describes important magnetic characteristics to be considered,

and reviews measurement techniques for magnetic evaluation of candidate SSC

wires. Instrumentation that might be practical for use in a wire - fabrication

environment is described. We report on magnetic measurements of prototype

SSC wires and cables and speculate on causes for instability in multipole

fields of dipole magnets constructed with such cables.

INTRODUCTION

A typical field cycle for the proposed Superconducting Super Collider

(SSC) consists of an initial charge to a full field of 6.6 T, reduction of

field to 50 mT, increase to 0.33 T for proton injection, and a slow increase

of field to 6.6 T as the proton beam is accelerated. 1 The multifilamentary

Nb-Ti cables used in the construction of the superconducting dipole magnets

are themselves exposed to the field of adjacent windings, usually approximated as a transverse field. Electromagnetic characteristics of the wires and

cables are potential sources of difficulty in meeting magnet specifications.

Thus, requirements for the magnet imply design and performance specifications

for the wires and cables. In this paper we discuss magnetic parameters

useful for evaluating multifilar Nb-Ti superconductor wire and cable for the

SSC.

MAGNETIC CHARACTERISTICS

Superconductors under steady-state conditions are lossless except for

losses associated with thermally activated flux creep. With transient or ac currents and fields, there are several sources of energy dissipation. These

ac losses may be classified according to their mechanism and localization

within a wire composed of fine superconducting filaments in a normal -metal

matrix

.

Advances in Cryogenic Engineering (Materials), Vol. 36

Edited by R. P. Reed and F. R. Fickett Plenum Press, New York, 1990

1

215
In this section we describe ac loss effects that may be detected with

magnetic instrumentation. We discuss both time - independent and timedependent phenomena for the wire and cable, not the dipole magnets made with

these elements. Ideally, all of these ac losses should be minimized, subject

to the often conflicting requirements of high critical current and stability

against propagation of a normal zone (quench)

.

Time-Independent Effects

Hysteresis . Magnetic hysteresis upon field cycling is a major loss

mechanism. Hysteresis loss per field cycle is frequency independent. It

arises in type-II superconductors from irreversibility of the penetration of

flux vortices and shielding currents resulting from flux pinning in the

filament volume and at the filament surface. The energy dissipation per se might be viewed as a trivial problem in applications where field is only

occasionally cycled. However, large hysteresis leads to large remanent

magnetization from trapped flux in the filaments as the applied field is

reduced to zero. This remanent magnetization is the source of residual field

in a superconducting magnet. Even at low fields, such as the 0.33-T SSC

injection field, trapped flux acts as an offset to the field expected from

the magnet current. When it is predictable, the residual field may be

compensated. Field uniformity is usually achieved by strategic magnet

design. However, remanent magnetization due to trapped flux in the

superconductor wires makes it difficult to obtain uniform fields at low

currents

.

Because the ability of a superconductor to pin flux is an essential

requirement for high critical current, both small remanent magnetization and

small hysteresis may be achieved, not by reducing flux pinning, but by

reducing filament diameter, as predicted by the critical state model. 2 Hysteresis loss is generally higher for wire carrying transport current than

for an open-circuited sample, with the extra energy provided by the current

source, not the field. 3 Hysteresis loss is determined as the enclosed area

in a plot of magnetization vs. field. Measurements often are made with field

cycled from positive to negative values. The SSC cycle is such that fields

are always positive. In typical multifilamentary wires measured in

transverse field, positive-field hysteresis loops, with maximum applied

fields of 1 T, have about 45% of the loss associated with complete

hysteresis loops.

Self-field of transport current . When transport current changes,

moving self-field lines dissipate energy. 4 The use of small -diameter wires

reduces these virtually hysteretic losses. In f ine - filament wires, the selffield loss may be greater than the magnetic hysteresis loss. In accelerator

applications, cabling with fully transposed strands reduces self-field loss;

simple twisted strands would still have a large self-field. Three methods of

transposition are twisted rope, woven braid, and flattened twisted cable.

The last is planned for the SSC, but it results in mechanical damage to the

cable corners with a local reduction in critical current. 6

Coupling between filaments . There are two time - independent sources of

filament coupling. One is simply interfilament contact arising from metallurgical problems in processing. Coupled filaments act as a single filament

of large diameter, with its associated problems. The second is interfilamentary coupling by the proximity effect when filament spacing is on the

order of the coherence length. This type of coupling is important for wires

with closely spaced fine filaments. The addition of impurities, such as Ni

or Mn, to the matrix material is often effective in reducing the coupling. 6 In any event, proximity-effect coupling is disrupted when magnetic fields

approach the effective critical field of the coupling medium. The losses

associated with time - independent filament coupling are hysteretic.

216

2
Time-Dependent Effects

Coupling between filaments . An important coupling arises from eddy

currents driven by voltages induced by a changing applied field. Coupling

loss is caused by the transfer of current between filaments and dissipation

within the matrix. This relaxation phenomenon leads to time -varying

magnetization and field instability of the magnet. It may be reduced by

transposing the filaments, approximated by twisting the wire, during

manufacture . 7 This decreases the longitudinal distance over which the

transverse coupling currents can flow. Other ways to reduce coupling are by

increasing the resistivity of the Cu matrix and by increasing the distance

between filaments. The former strategy is consistent with other wire

requirements provided the stability of the conductor is not impaired.

Flux creep . Flux creep consists of thermally activated jumps of bundles

of flux vortices between pinning sites at constant field. Flux creep causes slow changes in magnetization and, in a superconducting magnet, changes in

magnet field. Flux creep is often ignored in strong-pinning superconducting

materials. In particular, the critical state model assumes that there is no flux creep . 2 However, flux creep has been found to be a problem in

accelerator dipole magnets . 8

' 10 The activation energy for flux creep is

reduced by the Lorentz force of the applied field on vortex currents . 11 Flux

flow results as a limiting case when flux vortices are no longer pinned at

high fields.

Flux jumps . Flux jumps are sudden unpinning of flux vortices in

response to instabilities, temperature increases, and breakdown in shielding

currents as the applied field is changed. In wires with insufficient Cu or Cu-alloy stabilizer, flux jumps could lead to a quench. Flux jumps result in

sudden drops in magnetization and could result in small changes in the field

of a magnet.

Eddy currents . Eddy currents arise in the normal matrix material in

response to a field change according to the classical mechanism dependent on

the skin depth. The time constant of the eddy currents, a function of

resistivity, is short. These eddy currents are differentiated from those

that couple filaments, discussed above.

MAGNETIC MEASUREMENTS

The magnetic parameters of interest in evaluating multifilar Nb-Ti wire

for the SSC may be obtained from measurements of magnetization as a function

of field, time, and transport current. Ideally, measurements on cable

samples should be also obtained. For a useful analysis, it is necessary to

know the critical current density of the wire at several fields. In

addition, the wire should be characterized by filament diameter, filament

spacing, filament twist pitch, number of filaments, sample volume, and

matrix- to- superconductor volume ratio. Magnetization values are usually

reported per unit volume of superconductor or of total composite.

Magnetization in filamentary superconductors is the signal from

superconductor shielding currents and other matrix currents discussed above.

The magnetization- field cycle, or hysteresis loop, should have a variable

cycle time, up to several hours, to extract time - dependent coupling

information. The field should be transverse to the wire axis and cycle from zero to positive value. Additional information may be obtained from

longitudinal-field measurements. Transport current could be controlled

independently of the applied field, though in actual SSC operation the

current would be approximately proportional to the field.

217

3
Total magnetic loss is the area enclosed by the loop in the

magnetization- field plane. In the limit where coupling currents have

decayed, the remaining area represents the magnetic hysteresis. The width of

the loop at relatively high fields may be used to compute an "effective"

filament diameter , 12 according to the Bean model , 13 if the critical current

density is known. If coupling currents have not decayed or if filaments are coupled by the proximity effect, a large loop area will result.

Magnetization vs. Field

Several methods of magnetometry might be used for the magnetization

measurements: integrated- flux , vibrating- sample (VSM) , vibrating-coil (VCM)

,

SQUID, and Hall-probe. We will discuss their advantages and disadvantages.

To our knowledge, VCM and Hall-probe magnetometers have not been used for

magnetic measurements of superconductors. They may be well suited for this

task in a wire - fabrication environment.

Integrated- flux . This method 14 ’ 16 is good for measurements on wires

carrying transport current. It detects flux jumps and frequency dependence.

It requires large samples to increase the signal- to-noise ratio. Integration

instrumentation limits the measurement to relatively fast field cycles.

VSM . Vibrating- sample magnetometer measurements are made with the

vibration axis longitudinal 16 or transverse 17 to the field. This method is

sensitive to small samples. It is a dc measurement when the field is stepped

and the signal is allowed to stabilize. However, if synchronous detection is

used or if the pick-up coils are well matched, data may be taken while

sweeping the field. It is difficult to vibrate samples with current leads

attached. We calibrate the pick-up coils with Ni wires, plates, cylinders,

or spheres in the same configuration as the superconductor samples.

VCM . A vibrating-coil magnetometer 18 ’ 19 would be good for measurements

on samples carrying transport current because the sample remains stationary.

As with a VSM, the VCM may be used in stepped or swept fields. A wire sample

could be formed into a coil, noninductively wound to avoid a magnetic signal

from the transport current in the sample. As with a VSM, the field is

supplied by an external magnet. The applied field should be uniform to avoid

field- induced signals. Calibration would be similar as for a VSM.

SQUID . This method20 ’ 21 is extremely sensitive and precise, suited to

small samples. Any current to the sample would disrupt the SQUID circuitry.

Field cycles are extremely slow.

Hall -probe . Two calibrated, cryogenic probes are used, one to measure

the applied field, the other to measure the flux density at the sample

surface . 22 " 24 The difference is the sample magnetization (after correcting

for demagnetizing field, if necessary) . The Hall probe could be positioned

so the Hall element is parallel to the azimuthal magnetic field from the

current, but perpendicular to the magnetic field and the magnetic moment from

the superconducting shielding currents. This method would be appropriate for

cable samples. Calibration may be achieved as for a VSM.

AC Susceptibility vs . Temperature

AC susceptibility is usually measured as a function of temperature in

constant ac field, with or without a dc bias field. Measurements are made

with a coaxial mutual - inductance system consisting of a primary excitation

field coil, a secondary pick-up coil, and a secondary compensation coil . 25 ’ 26

Susceptibility is an excellent tool for determining critical temperatures and

proximity-effect coupling in f ine - filament superconductors . 27 Low

frequencies are used to avoid eddy currents in the normal-metal matrix.

218

4
STUDY OF PROTOTYPE SSC WIRES AND CABLES

A disconcerting problem in accelerator magnets is field change over several hours at constant magnet current. This is often expressed as instability in multipole fields. 1 ’ 8 ’ 9 Possible mechanisms are flux creep and

eddy-current coupling between filaments. If the mechanism is flux creep,

there are two possibilities. One is flux creep intrinsic to the Nb-Ti

superconductor filaments or their surface. The other is flux creep in the

proximity-coupled matrix. (The proximity coupling itself is not time

dependent.) The presence of proximity-effect coupling in filamentary

superconducting wire may be deduced from measurements of magnetization vs. field or magnetic susceptibility vs. temperature.

Proximity-Effect Coupling

In hysteresis loops of magnetization vs. field, proximity coupling

causes a magnetization peak centered near zero field. 28 ’ 29 (The exact

position depends on a demagnetization correction of the field axis.) This

peak is different from the peak in the second and fourth quadrants which is seen experimentally and predicted by the Kim model for critical current

density. 30 The coupling peak arises from a large effective filament

diameter when filaments are coupled at low field. The proximity coupling is

destroyed at fields greater than about 0.2 T.

In ac susceptibility measurements, a large coupling peak in the

imaginary part may be seen as a function of temperature. This peak

represents hysteresis loss when the lower critical field of the proximitycoupled matrix becomes on the order of the measuring field as temperature

increases. Thus, the peak temperature is a strong function of measuring

field amplitude. We have used this technique to study intergranular coupling

in high- temperature superconductors. 31

5 6 7 8 9 Temperature (K)

Fig. 1. AC susceptibility (uncorrected for demagnetization factor) vs. temperature measured at 0.1 mT rms at 10 Hz for wire and cable

samples. The imaginary part shows intrinsic and coupling loss

peaks. The real part shows a broad transition which includes

intrinsic and coupling transitions.

219

5
Figure 1 shows the real and imaginary parts of external (not corrected

for demagnetization factor) ac susceptibility as functions of temperature

for samples consisting of small, sawed segments of prototype SSC wire and

cable with 0.5% Mn in the Cu matrix. The filament diameter is 5.3 nm and

the filament spacing is 0.53 nm. The susceptibility is plotted per unit

volume of Nb-Ti. The measuring field was 0.1 mT rms at 10 Hz applied

perpendicular to the wire and cable axes. The cable was the type used in the

fabrication of the inner layer of dipole magnet D-15A-6 in Ref. 1.

Intrinsic and coupling loss peaks (partially overlapping) appear in the

imaginary part of susceptibility. Correction for demagnetization factor

would change the apparent shapes of the peaks. For a measuring field of

0.01 mT, the coupling peaks move to higher temperature. For a 1-mT field,

the peaks are well separated as the coupling peaks move to lower temperature.

Much less pronounced susceptibility peaks were seen for fields applied

parallel to the wire and cable axes. But similar coupling peaks in the

imaginary part were seen for wire and cable used for the outer layer of

dipole magnet D-15A-6. The filaments are coupled at low temperature for low

fields despite the Mn doping of the matrix. As expected, magnetization vs. field at 4 K showed characteristic coupling peaks near zero field.

Multipole - field instability occurs at the relatively high fields that

would destroy proximity- effect coupling. However, there are portions of a dipole magnet which are exposed to very small fields where coupling is

presumably intact. In multifilamentary wires that are proximity coupled,

flux creep may occur in the interfilamentary matrix. We recently proposed

this mechanism to explain subtle intergranular frequency effects in the ac susceptibility of high- temperature superconductors. 32 To inhibit low-field

proximity coupling, more Mn could be added to the matrix, or a resistive

Cu-Ni matrix could be used.

Intrinsic Flux Creep

In addition to the possibility of flux creep in the proximity-coupled

matrix, there may be intrinsic flux creep in the superconducting filaments.

Flux creep 10 has been observed in Tevatron cable made with wire containing

presumably decoupled filaments separated by approximately 3 pm.

We measured positive-field VSM hysteresis loops for a small segment of

D-15A-6 outer- layer cable, filament diameter 4.3 nm and filament spacing

0.43 fxm, to see if there were time effects. The maximum field was 1 T, high

enough to uncouple the filaments, perpendicular to the cable axis and

parallel to the wide side. The field was stepped in units of 50 mT. For one cycle, magnetization was recorded after waiting 30 seconds after a field

change. The wait time was 3 minutes for the other cycle. There was no significant difference in the hysteresis loops. This suggests an absence of

intrinsic flux creep on the time scale of these measurements

.

Eddy-Current Filament Coupling

If multipole - field instability arises from eddy-current filament

coupling, more twist per unit length would help alleviate it. However,

because the coupling currents decay quickly in the resistive matrix, 33 they

should not be a factor for the drift of multipole fields over a period of

hours

.

To check for this time - dependent effect, we measured hysteresis loops,

using the two field cycles described’ above , for two coiled samples of D-15A-6

wire used in the inner- and outer- layer cables. The field was applied along

the axis of the coils, approximately transverse to the wire axis. These

particular samples had 0.5 and 1.5 twists per centimeter, respectively,

220

6
verified by etching in nitric acid the matrix of companion samples. Unlike

the short cable segment described above, the length of wire for each coil

sample was about 25 cm, long enough to contain many twists. There were no differences between long and short field cycles for the two coils. The

result suggests that 0.5 twist per centimeter is adequate to inhibit eddycurrent filament coupling for these wait times.

CONCLUSIONS

We have discussed several magnetic parameters to be considered in

testing multifilar Nb-Ti superconductor for the SSC, with the goal of

minimizing field nonuniformity, field instability, large residual fields,

and large energy losses. These parameters may be extracted from

measurements of magnetization vs. field. Several measurement methods, each

with certain advantages, were described.

Measurements of ac susceptibility of candidate SSC wires and cables

demonstrate proximity-effeet coupling. Flux creep in the proximity- coupled

matrix may be a source of time variations of multipole fields in prototype

SSC magnets. Intrinsic flux creep was not observed over a period of minutes.

Wires used for the inner and outer layers of the D-15A-6 prototype SSC dipole

magnet did not show serious eddy-current filament coupling for these sample

twist pitches.

ACKNOWLEDGMENTS

We had helpful discussions with C. E. Taylor, R. M. Scanlan, W. S.

Gilbert, E. W. Collings, and D. C. Larbalestier . R. J. Loughran assisted

with the VSM measurements. Sample wires and cables were generously provided

by R. M. Scanlan. This work was supported by the Department of Energy,

Division of High Energy Physics.

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measurements using small quantities of cerium magnesium nitrate,

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26. R. B. Goldfarb and J. V. Minervini, Calibration of ac susceptometer for

cylindrical specimens, Rev. Sci. Instrum. 55:761 (1984).

27. J. R. Cave, A. Fevrier, H. G. Ky, and Y. Laumond, Calculation of ac losses in ultra fine filamentary NbTi wires, IEEE Trans. Magn.

23:1732 (1987).

28. A. K. Ghosh, W. B. Sampson, E. Gregory, and T. S. Kreilick, Anomalous

low field magnetization in fine filament NbTi conductors, IEEE

Trans. Magn. 23:1724 (1987).

29. E. W. Collings, K. R. Marken Jr., M. D. Sumption, R. B. Goldfarb, and

R. J. Loughran, AC loss measurements of two multifilamentary NbTi

composite strands, paper AY-05, this conference.

30. D.-X. Chen and R. B. Goldfarb, Kim model for magnetization of type-II

superconductors, J. Appl. Phys. 66:2510 (1989).

31. R. B. Goldfarb, A. F. Clark, A. I. Braginski , and A. J. Panson,

Evidence for two superconducting components in oxygen- annealed

single-phase Y-Ba-Cu-0, Cryogenics 27:475 (1987).

32. M. Nikolo and R. B. Goldfarb, Flux creep and activation energies at the

grain boundaries of Y-Ba-Cu-0 superconductors, Phys. Rev. B 39:6615

(1989) . 33. M. N. Wilson, op. cit., pp . 176-181.

Contribution of the National Institute of Standards and Technology, not

subject to copyright.

222

8
1796 IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 2, MARCH 1991 Hall Probe Magnetometer for SSC Magnet Cables:

Effect of Transport Current on Magnetization and Flux Creep

R. W. Cross and R. B. Goldfarb

Electromagnetic Technology Division

National Institute of Standards and Technology

Boulder, Colorado 80303

Abstract

We constructed a Hall probe magnetometer to measure

the magnetization hysteresis loops of Superconducting Super

Collider magnet cables. The instrument uses two Hall-effect field sensors to measure the applied field H and the magnetic

induction B. Magnetization M is calculated from the difference

of the two quantities. The Hall probes are centered coaxially in the bore of a superconducting solenoid with the B probe against

the sample’s broad surface. An alternative probe arrangement,

in which M is measured directly, aligns the sample probe parallel

to the field. We measured M as a function of H and field cycle

rate both with and without a dc transport current. Flux creep as a function of current was measured from the dependence of ac

loss on the cycling rate and from the decay of magnetization with

time. Transport currents up to 20% of the critical current have

minimal effect on magnetization and flux creep.

Introduction

A consideration in the design of Superconducting Super

Collider (SSC) magnet cables is ac losses in the superconductor.

These ac losses may be either time-dependent or timeindependent [1]. Normally, ac loss measurements are made on

open-circuited samples. In actual use, however, the super- conductor carries a transport current which can influence the ac

loss of the sample [2-4], Another concern in SSC cable design

is magnetic relaxation in the filaments. Measurements of large

field decays in accelerator magnets, attributed to flux creep, have

been reported [5-8]. However, the observed field decay in the magnets is much larger than the relaxation measured in small open-circuited samples.

The objective of this study was to examine the effect of

transport current on the magnetization and the magnetic

relaxation in an SSC cable carrying transport current. We

describe a Hall probe magnetometer that can measure magnetization, ac losses, and flux creep, with and without transport current [9]. The magnetometer uses two Hall-effect field sensors. One measures the applied field H and the other measures the magnetic induction B or the magnetization M. Hall probe

magnetometers have been used to study ferromagnetic materials,

ferrites, anisotropy fields, superconducting tubes, and field profiles [10-13].

Experiment

A schematic diagram of the Hall probe magnetometer is shown in Fig. 1. The applied field is supplied by a super- conducting solenoid, 21 cm in length, 7.6 cm bore, with a maximum field of 7 T. Two cryogenic Hall-effect field sensors

are used; one measures the applied field H and the other

Contribution of the National Institute of Standards and

Technology, not subject to copyright.

Manuscript received September 24, 1990.

measures the magnetic induction B. Both probes are aligned

perpendicular to the applied field and centered coaxially in the bore of the solenoid. Their vertical separation is 4 cm. The

sample is mounted so that its broad face is perpendicular to the

field with the B probe mounted firmly against it. The self field from the transport current is parallel to the B probe and, ideally,

is not detected. Any self field actually sensed by the B probe

appears simply as a dc offset in the hysteresis loops.

The fields H and B are measured with two commercial

gaussmeters. Their analog outputs are low-pass filtered and input

to a differential amplifier, which subtracts the signals, and sent to

a computer, which calculates M. Magnetization can be calculated using the expression M = (B/hq-H)/(1-D), where D is the demagnetizing factor, approximately equal to 0.5 for the

transverse field orientation.

A U-shaped sample is used so that the current contacts can be soldered to the cable far enough from the Hall probe

sensing area to not affect the measurement. To obtain

repeatable results, the sample must be positioned accurately, with

the face of the sample firmly against probe B and properly

centered. This is achieved with three guide tubes and a Be-Cuspring-loaded mount for probe B. The insert is shown in Fig. 2. (Only two guide tubes are shown for clarity.) Samples are mounted to the sample plate using clamps at both ends of the sample. The sample plate is 6.1 cm in diameter, which allows for samples approximately 5.3 cm in length to be mounted. The

guide tubes are used for support and for sample positioning. The

tubes are symmetrically spaced so that the circular sample plate can slide firmly between them. The sample rod can easily be

removed and loaded while the probes and the solenoid remain

immersed in liquid helium.

The instrument may be calibrated with a Ni standard the same size as the sample or with a superconductor transfer standard which has been measured on a conventional magnetoTransport current

Figure 1. Schematic diagram of the Hall probe magnetometer.

The magnetic field Hq is supplied by a superconducting solenoid.

The sample is bent into a U-shape so that the solder contacts can

be made away from the Hall probe sensing area.

U.S. Government work not protected by U.S. Copyright.

9
Transport

Figure 2. Mechanical diagram of the Hall probe magnetometer

showing the sample mounting assembly. Probe B is in a springloaded holder which assures contact to the sample. The guide

tubes support the probes and direct the sample plate to probe B.

meter. The Hall-effect sensors must be calibrated before the measurements. This is done by removing the sample, setting the

field to a known value, and adjusting the gain of the meters until each gaussmeter displays the correct field value. A disadvantage

of Hall effect field sensors is their nonlinearity above 3 T,

typically 0.2%. For small values of M in large fields, the

nonlinearity can lead to distortion in M. The problem can be

mitigated if values of the background signal as a function of field are subtracted from the sample data during data processing.

An alternative Hall probe arrangement increases the sensitivity of the M measurement. In this arrangement, the sample

probe is aligned so that its sensing plane is parallel rather than

perpendicular to the applied field and M is measured directly.

We call this the “A/ configuration,” as distinct from the usual “5 configuration.” A disadvantage of this technique is that the alignment of the M probe must be precise to avoid sensing the applied field.

The sample used for the measurements was a Nb-Ti

multifilamentary SSC magnet cable with 23 strands. Each strand had approximately several thousand filaments, 4.2 gm in diameter. Only three of the 23 strands were used in the transport current measurements so that the applied current, limited by the power supply, would be closer to the critical current of the sample. All measurements were made at 4.0 K.

Results

Magnetization versus Field

A direct measure of magnetic hysteresis loss can be

obtained from magnetization-versus-field loops. Time dependent

losses, such as flux creep and eddy current coupling, can be

measured if the applied field is swept at different cycling rates

[1]. Figure 3 shows a typical plot of magnetization versus field for two field cycle rates. The sample was a short section of cable with no applied transport current. The response of the Hall

probes is fast enough to measure the higher frequencies with

excellent resolution. As seen in Fig. 3, the area of the hysteresis

loop is greater for the 0.1 Hz field cycle rate than for 1 mHz.

Figure 3. Plot of magnetization versus field at 0.1 Hz and

1 mHz. The difference in loss between the two curves is the

result of eddy current coupling of the filaments at the higher

frequency.

At 0.1 Hz, the eddy currents generated in the copper matrix have

not yet decayed and still couple the filaments, which leads to larger ac loss. At very low frequencies, other time-dependent

effects, such as flux creep, can be measured as a small decrease

in hysteresis as the field cycle rate decreases. For example, we

observed an average 2.5% reduction in magnetization on comparing curves at 5 and 0.5 mHz (not shown).

The effect of transport current on the magnetization

curves is presented in Fig. 4. The sample consisted of three

strands of wire. Curves are shown for 0 and 200 A of transport

current. The critical current for the three strands is estimated to be 1 kA at zero field. The effect of the applied current is seen

as a small decrease in magnetization at high fields as the critical current is reduced. This effect would be more pronounced at higher fields or currents [14].

Figure 4. Plot of magnetization versus field for 0 and 200 A of

transport current. The effect of the applied current is small,

showing a decrease in hysteresis only at the higher field values

due to the reduction in critical current.

10
1798 Flux Creep and Transport Current Effects

There should be an effect of transport current on flux creep, considering the increase of the Lorentz force acting on the pinned flux vortices. The Hall probe magnetometer is well suited

for flux creep measurements with and without transport current because it does not require sample motion for signal detection. Unexpected variations in field in other dc methods can give erroneous results in hysteretic materials. For example, VSM and

SQUID magnets might have enough field inhomogeneity to cause minor hysteresis loops to be traversed during sample motion.

The decay in magnetization as a function of time for 0 and

200 A of transport current are shown in Fig. 5. The measure- ments were made on three strands in the M configuration at a

field of 0.3 T, after a field cycle of 0 to 6 T, 6 to 0 T, and 0 to

0.3 T. The dc current was applied at the start of the field cycle.

After switching the magnet into persistent mode, magnetization was measured as a function of time. The field decay of the solenoid during the measurement was negligible as monitored at the H probe.

The data in Fig. 5 include the fast decay resulting from the

dissipation of eddy currents. The fast decay is large and nearly

complete in less than 100 sec. After that, there is a slow decay

of only a few percent of the initial magnetization that continues

beyond our measuring time. As seen from the graph, the effect of transport current is minimal. Field-decay effects observed in accelerator magnets occur at low current levels, usually less than

20% of the critical current. From these results, it seems that the

effect of transport current on flux creep is not large enough to explain the field decay in SSC magnets.

Figure 5. Plot of magnetization versus time at 0.3 T for 0 and

200 A of transport current Iv The left scale is for zero transport

current; the right scale is for 200 A. There is slight increase in decay rate with transport current.

Conclusion

A Hall probe magnetometer was constructed to measure

the effect of transport current on magnetization and flux creep

in SSC magnet cables. The Hall probe measurement is static; it does not require sample motion or field change to induce a

signal. The speed at which the field can be cycled is limited only

by the magnet inductance and the power supply compliance volt- age. However, the calibration of M is not direct. The instrument

sensitivity in the B configuration is less than that of other

magnetometers and very small samples cannot be measured.

Transport current causes a decrease in magnetization at high fields and current. The current has a small effect on flux creep, although not enough to explain the large decay observed

in SSC magnets. In the case of actual SSC dipole magnets,

however, the different current levels, field cycles, and field gradients to which the cable is exposed is more complicated than

in our experiments.

Acknowledgment

This work was supported by the U.S. Department of Energy, Division of High Energy Physics.

References

[1] R. B. Goldfarb and R. L. Spomer, Adv. Cryo. Eng.

(Materials) 36, 215 (1990).

[2] K. Kanbara, Cryogenics 27, 612 (1987).

[3] W. J. Carr, Jr., M. S. Walker, and J. H. Murphy, J. Appl.

Phys. 46, 4048 (1975).

[4] S. S. Shen and R. E. Schwall, IEEE Trans. Magn. 15, 232

(1979).

[5] D. A. Herrup, M. J. Syphers, D. E. Johnson, R. P. Johnson, A. V. Tollestrup, R. W. Hanft, B. C. Brown, M. J. Lamm.

M. Kuchnir, and A. D. Mclnturff, IEEE Trans. Magn. 25,

1643 (1989).

[6] R. W. Hanft, B. C. Brown, D. A. Herrup, M. J. Lamm, A. D. Mclnturff, and M. J. Syphers, IEEE Trans. Magn. 25,

1647 (1989).

[7] M. Kuchnir and A. V. Tollestrup, IEEE Trans. Magn. 25,

1839 (1989).

[8] W. S. Gilbert, R. F. Althaus, P. J. Barale, R. W.

Benjegerdes, M. A. Green, M. I. Green, and R. M. Scanlan,

Adv. Cryo. Eng. (Materials) 36, 223 (1990).

[9] R. W. Cross and R. B. Goldfarb, Department of Energy

Topical Workshop on Magnetic Effects of Persistent Currents

in Superconductors, 5-7 March 1990, Batavia, Illinois. [10] P. J. Flanders, IEEE Trans. Magti. 21. 1584 (1985).

[11] D. J. Craik, J. Phys. E: Sci. Instrum. 1, 1193 (1968).

[12] M. R. Cimberle, C. Ferdeghini, and A. S. Siri, Cryogenics

29, 69 (1989).

[13] D. D. Roshon, Jr., Rev. Sci. Instrum. 33, 201 (1962).

[14] M. N. Wilson, Superconducting Magnets, Clarendon Press, Oxford, U.K. (1983), pp. 171-174.

11
To be published in: Advances in Cryogenic Engineering (Materials), Vol. 38

Edited by F. R. Fickett and R. P. Reed

Plenum Press, New York, 1992

TRANSPORT CURRENT EFFECTS ON FLUX CREEP AND

MAGNETIZATION IN Nb-Ti MULTIFILAMENTARY CABLE STRANDS

R. W. Cross

Electromagnetic Technology Division

National Institute of Standards and Technology

Boulder, Colorado 80303

ABSTRACT

We used a Hall-probe magnetometer to measure the effect of transport current on magnetization

and flux creep in Nb-Ti multifilamentary cable strands. Large transport currents, up to 70% of the

critical current 7

C , were applied to the sample. The external field was applied transverse to the current and sample length. When the applied current approached the critical current of the strand,

the magnetization decreased and the Lorentz-force interaction between the field and the transport current dominated the creep. Both the short-time and long-time decay of magnetization increased.

The increase in the short-time decay was too large to be explained by eddy current decay. The

long-time decay was enhanced by a factor of 4 with a transport current of approximately 0.7/

c

.

INTRODUCTION

Previously, we reported experimental effects of transport current on magnetization in Superconducting Super Collider (SSC) cables at current and field values which simulated dipole

magnet operation at the injection field. 1 In actual use, the superconducting cables carry a transport current which can affect the ac losses of the sample.2-4 We found that applied currents up to 20%

of the critical current l

c

at the injection field only sightly affected the magnetization. The current reduced the magnetic hysteresis by less than 5%, which was seen only at high fields. In addition,

the magnetic relaxation of the shielding currents in the filaments was measured with and without a transport current at the injection field value of 0.3 T. We performed the experiment to determine

whether the presence of a transport current would increase the decay due to the Lorentz-force

interaction and to see if it could explain the large field decay observed in the model dipole magnets.

Others reported that flux creep in the filaments may be causing the enhanced field decay,5

" 8 but the

observed decay measured in open-circuited samples was too small to explain the field drift in the

magnets. We found that a current of approximately 0.2/

c at the injection field could not account

for the field drift in the magnets, although the current did slightly increase the decay.

In this study, we investigate the effect of transport current on flux creep and magnetization in Nb-Ti magnet cable strands at higher currents where the Lorentz-force interaction between the current and flux vortices starts to dominate the creep. In model SSC dipole magnets, the different

currents, field cycles, and field gradients to which the cable is exposed are very complicated. In operation the field is cycled from 0 to 6 T, where the applied current is close to the critical current of the cable, back to 0 T, and then finally to the injection field of 0.3 T. The objective of this study

is to measure magnetization and relaxation in the presence of a large transport current to determine

whether a large current causes a major increase in the decay.

12
EXPERIMENT

To measure the magnetic relaxation in SSC cables with a transport current, we developed a Hall

probe magnetometer, described in Ref. 1. For this experiment, we use a Hall-effect field sensor

to measure the applied field H and another to measure the magnetization M, as shown in Fig. 1. A U-shaped sample is used so that the current contacts can be soldered to the wire far from the Hall

probe’s sensing area. This way, any current redistribution at the contacts will not be detected. The

sample is a single strand of Nb-Ti multifilamentary SSC magnet cable with approximately 4000

filaments, each 4.2 /mi in diameter. We supply up to 200 A of transport current to the sample,

close to the critical current of the strand. The applied field is provided by a 7.6-cm bore,

superconducting solenoid with a maximum field of 7 T. All the measurements were made in liquid

He at 4.0 K.

Direct measurement of the magnetization M is much more sensitive than measuring the magnetic

induction B and subtracting the magnetic field H to obtain M. 9 This is especially true for

superconductors, where H and B can be very large and M may be small. However, to measure M

directly, it is necessary to accurately align probe Mparallel to the field, so it does not sense H. The

”M configuration" reduces the effect of nonlinear responses in the Hall-effect devices because the

M field stays small compared to B and H, and only one probe is used to measure M.

Transport

current

Figure 1. Schematic diagram of Hall probe magnetometer. The

magnetic field H is supplied by the superconducting solenoid. The M

probe is aligned parallel to the applied field near the U-shaped sample.

RESULTS AND DISCUSSION

Magnetization

To understand the effect of an applied current on magnetic hysteresis, we need to consider the

interaction of the current and field and its effect on the field profile in the sample. For simplicity,

we consider a superconducting slab carrying an applied current I

t and magnetic shielding currents

13
lm

. The transport current is maintained at a constant level so the only effect of the changing field

is to alter the magnetization currents.

10 The field penetration profile, based on the critical -state model, is shown in Fig. 2. In Fig. 2(a) the sample is exposed to a field and a current. The current

shifts the profile to one side, because on one side of the sample the applied current and the shielding currents add, and on the other side of the sample the currents subtract. As the field is reversed

[Fig. 2(b)-(d)], the magnetic shielding currents penetrate in the opposite direction to a full- penetration field H

p

[Fig. 2(c)], where the field change penetrates the entire slab, and then finally

to the maximum reverse field Hm [Fig. 2(d)]. Wilson10 showed that full penetration occurs at a lower field amplitude than the case with zero applied current. The penetration field H

p

is reduced

by a factor of (1 — /)» where i is the ratio of transport current to critical current I/Ic

. The

magnetization is reduced in the presence of a transport current by a factor of (1 — i 2

). In summary, the effect of a transport current is to shift the field profile, which causes a reduction of

the penetration field H

p

and a reduction of the magnetization M.

Figure 2. (a) Field profile for a superconducting slab carrying a dc transport current in an external field; (b) as the field is reduced; (c) when the field penetrates

the entire slab; (d) when the field reaches its opposite maximum value. After

Wilson. 10

In Fig. 3, we show a plot of magnetization versus applied field for the SSC cable strand at different applied currents. We show only the positive-field portion of the curve for clarity, though

the field was cycled between +0.5 T. A complete field cycle is traversed after the current is applied so that a critical-state profile can be established. There are two notable effects of the

transport current: (1) As the transport current is increased from 0 to 120 A (approximately 0.4/

c), the width of the magnetization curve AM decreases, especially at higher fields. This may be

understood from the (1 — i 2

) dependence of magnetization with current; as I

t increases, i approaches 1 and M goes to 0. The reduction of AM is proportionally larger at higher fields

because of the dependence of critical current on field; as H increases, I

c decreases and i approaches

1. (2) The field value for the minimum magnetization decreases with increasing applied current

(see inset). This results from the shift in the full penetration field H , which is reduced by

(1 - 0.

14
30 T T T T

- 30 1 1 1 1 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Applied field p.0H (T)

Figure 3. Plot of magnetization versus field for 0, 80, and 120 A of transport

current. The current causes the hysteresis width to decrease, especially at higher

fields. The field value for minimum magnetization also decreases with increasing

current (inset).

Flux creep

At temperature T — OK, vortices are pinned by material inhomogeneities and will not move

under action of a Lorentz force density J x B if the net current density J = V x BIhq is less than

the critical current density Jc

. At T > 0, some of the vortices are unpinned by thermal activation.

If the spatial dependence of the pinning potential is tilted by a flux-density gradient, the thermally

activated vortices may develop a drift velocity, causing dissipation. This process is called flux

creep. If a transport current is present, the increase in the Lorentz-force interaction between the current J (which includes both the transport and shielding currents) and the vortices can increase

the rate of relaxation. If the transport current approaches the critical current of the sample, the

interaction of the applied current and field can dominate the flux creep, leading to a large magnetic

relaxation.

The decay in magnetization as a function of time for 0 and 200 A of transport current is shown

in Fig. 4. The measurements were made on the single SSC cable strand at a field of 0.3 T, after a field cycle of 0 to 6 T, 6 to 0 T, and 0 to 0.3 T. The dc transport current was supplied at the

start of the field cycle. Magnetization was measured as a function of time after the magnet was

switched into persistent mode. The magnetization M{t) = AM(t)/2 is normalized to M(t) at time t = 0, where AM is the width of the hysteresis loop at a constant field (for a fully penetrated state).

Time t = 0 is defined as the time of the first measured value of M(t), which was taken immediately

after field stability. The field decay of the^olenoid magnet was negligible over the course of the

experiment.

The data in Fig. 4 show how a large transport current can influence flux creep. The decay in magnetization with time has two distinct features. The first is the large decay observed in the first 10 to 20 s. For the 0-A data, this decay can be ascribed to the relaxation of eddy currents in the

15
copper matrix, which is related to the resistivity of the copper. However, for the 200-A data, the

decay is much larger than can be attributed to eddy current decay, although the time constant is roughly the same.

1 The applied current may be causing a redistribution of the magnetic shielding

and eddy currents, which results in a fast decay. The second feature is the enhancement of the longtime decay. The relaxation for the 200-A data is much larger than for the 0-A data, decaying

approximately 13% in 500 s (not including the fast decay in the first 10 s) compared to the 3%

decay for /, = 0 A. Therefore, an application of a transport current of approximately 0.7/c causes

the long-time decay to increase by a factor of 4.

1 10 100 1000

Time (s)

Figure 4. Plot of magnetization vs. time at 0.3 T for 0 and 200 A of transport current 7

t

. The transport current increases the long-time decay by a factor of 4. The short-time decay in the first 5 s of the 200-A data is much larger than the eddycurrent decay seen in the 0-A data.

CONCLUSION

We used a Hall-probe magnetometer in the M configuration to measure the effect of a 200-A

transport current on magnetization and flux creep in SSC magnet-cable strands. This configuration

is more sensitive than measuring B and reduces the effect of nonlinear responses in the Hall-effect

devices. However, alignment of probe M in the field produced by the solenoid is difficult.

A transport current that is large with respect to the critical current of the wire reduced the

magnetization, especially at higher fields where the critical current is smaller. There is a qualitative

agreement with the (1 — i 2

) dependence on current. The field value for minimum magnetization

decreased with current and was in qualitative agreement with the (1 — i) dependence. For flux

creep measurements, the applied current caused an enhancement in both the short-time and long-time

decay. A current of approximately 0.7/c caused the decay to increase by a factor of 4. As the

applied current approaches the critical current, the Lorentz force acting on the vortices starts to

16
exceeded the pinning force, which leads to a large decay. In the limit /, -* I

c, the Lorentz force

completely exceeds the pinning force, flux begins to flow, and the magnetization decays to 0. The

field decay observed in the dipole magnets may be due to the complicated field and current cycle,

which actually traverses large fields and currents. As we noted in Ref. 1, the field-decay effects

in accelerator magnets occur at low transport currents.

ACKNOWLEDGMENTS

R. B. Goldfarb provided helpful discussions and suggestions. This work was supported by the

U.S. Department of Energy, Division of High Energy Physics.

REFERENCES

1. R. W. Cross and R. B. Goldfarb, Hall probe magnetometer for SSC magnet cables: Effects

of transport current on magnetization and flux creep, IEEE Trans. Magn. 27, 1796 (1991).

2. K. Kanbara, Hysteresis loss of a round superconductor carrying a dc transport current in an

alternating transverse field, Cryogenics 27, 621 (1987).

3. W. J. Carr, Jr., M. S. Walker, and J. H. Murphy, Alternating field loss in a multifilament

superconductor wire for weak ac fields superposed on a constant bias, J. Appl. Phys. 46,

4048 (1975).

4. S. S. Shen and R. E. Schwall, Interaction of transport current and transient external field in composite conductors, IEEE Trans. Magn. 15, 232 (1979).

5. W. S. Gilbert, R. F. Althaus, P. J. Barale, R. W. Benjegerdes, M. A. Green, M. I. Green,

and R. M. Scanlan, The effect of flux creep on the magnetization field in the SSC dipole

magnets, Adv. Cryo. Engr. (Materials) 36, 223 (1990).

6. D. A. Herrup, M. J. Syphers, D. E. Johnson, R. P. Johnson, A. V. Tollestrup, R. W. Hanft,

B. C. Brown, M. J. Lamm, M. Kuchnir, and A. D. Mclnturff, Time variations of fields

in superconducting magnets and their effects on accelerators, IEEE Trans. Magn. 25,

1643 (1989).

7. R. W. Hanft, B. C. Brown, D. A. Herrup, M. J. Lamm, A. D. Mclnturff, and M. J. Syphers, Studies of time dependence of fields in Tevatron superconducting dipole magnets,

IEEE Trans. Magn. 25, 1647 (1989).

8. M. Kuchnir and A. V. Tollestrup, Flux creep in a Tevatron cable, IEEE Trans. Magn. 25,

1839 (1989).

9. P. J. Flanders, A Hall sensing magnetometer for measuring magnetization, anisotropy,

rotational loss and time effects, IEEE Trans. Magn. 21, 1584 (1985).

10. M. N. Wilson, "Superconducting Magnets," Clarendon Press, Oxford, U.K. (1983),

pp. 171-174.

17
Enhanced flux creep in Nb-Ti superconductors after an increase

in temperature

R. W. Cross and R. B. Goldfarb

Electromagnetic Technology Division, National Institute ofStandards and Technology, Boulder,

Colorado 80303

(Received 25 July 1990; accepted for publication 29 October 1990)

The magnetic fields of Superconducting Super Collider (SSC) dipole magnets change with

time when the magnets are operated at constant current. The decay of the field is thought to be a consequence of flux creep in the Nb-Ti filaments in the superconducting

cables. However, measured magnetic relaxation of small samples of SSC cable as a function

of time is unlike the large decays that are observed in the fields of the actual magnets.

We have made relaxation measurements on sample SSC conductors at 3.5 and 4.0 K after

field cycling. The decay at both temperatures was 2.8% in 50 min. However, the

relaxation measured after a temperature increase from 3.5 to 4.0 K was 4.8% in 50 min. A

likely reason for the greater magnetization decay is that, after an increase in temperature,

the Nb-Ti is in a supercritical state, with shielding currents flowing at a density greater

than the new critical current density. This causes enhanced flux creep. We suggest that a small temperature rise during the operation of SSC magnets may contribute to the

unexpectedly large magnetic field decay.

Synchrotron accelerators, such as the Tevatron and the

proposed Superconducting Super Collider (SSC), use magnets made of cables of multifilamentary Nb-Ti superconductor wires. The magnetization of the Nb-Ti filaments

contributes to the magnetic field of the accelerator magnets. Flux creep, that is, thermally activated jumps of bundles of flux vortices, produces a decay in the magnetization

of superconductors with time, 1,2 and flux creep has been

presumed to cause a troublesome slow decay in the field, often measured as a multipole field, of accelerator dipole

magnets over a period of hours.

3-6 Magnetometer measure- ments of flux creep on samples of multifilamentary superconductor wire and cable used in the construction of these

magnets show some flux creep.

7,8 However, after an initial rapid decay of magnetization, the flux creep is about one

order of magnitude smaller than the field decay in the

actual superconductor magnets.

Recently, Sun el al

9 showed that flux creep in superconductors could be reduced or even eliminated by operating in a subcritical state achieved by lowering the conductor’s temperature after the critical state is achieved.

Clem has suggested that such a scheme, applied to accel- erator magnets, might lessen the field-decay problem. 10 In

this letter, we consider the inverse theorem that the en- hanced flux creep seen in accelerator magnets may be a consequence of an increase in operating temperature,

which forces the conductor into a supercritical state. We

find that an increase in sample temperature of 0.5 K, after a typical SSC field cycle, almost doubles the rate of magnetization decay.

The sextupole fields of model SSC dipole magnets

change with time when the magnets are operated at constant current under conditions similar to SSC accelerator

use. Large field decays have been observed, 6,11 and such

decays can result in beam loss during the SSC injection

period of several hours. The logarithmic time dependence

of field decay and the temperature dependence of relaxation in different dipole magnets suggest a flux creep mechanism. For example, the reported rate at 1.8 K was less than at 4.2 K.6 However, the measured relaxation of samples of cable is generally less than the field relaxation re- ported for the magnets.

To examine the magnetic decay process, we made a

series of three relaxation measurements at 3.5 and 4.0 K.

The measurements were made with a superconducting

quantum interference device magnetometer using a scan

length of 2 cm, which corresponds to a field variation of

<0.01%. The field from a superconducting solenoid was

applied perpendicular to the flat side of a 0.7 cm sample of

multifilamentary Nb-Ti superconductor cable. The sample

had 23 strands, each with approximately 10 000 4.2-/zmdiam filaments. To simulate the original SSC field cycle, for comparison with existing field-decay data, all measure- ments were made after the following field cycle: 0-5 T, 2 min pause; 5-0 T, 2 min pause; 0-0.3 T, 2 min pause. The

final 2 min pause was included to avoid the fast decay

resulting from eddy-current coupling and to establish a reproducible initial starting time, / = 0. The pauses en- sured field stability before the measurements were taken

and were in addition to the time required to ramp to each

field and to switch into persistent mode.

Curves (a) and (b) in Fig. 1 for 3.5 and 4.0 K show

magnetization M(t) as a function of time scaled by the

magnetization M(0) at t = 0. The decay rates due to ther- mal activation are nearly the same, about 2.8% after 3000

s, but the value of A/( 0) is approximately 10% less at 4.0

K. The rate of decay, R = [kM(t)/M(0)]/& In /, for the

decade of time 300-3000 s, is 0.008, in good agreement

with measurements by Ghosh on SSC wires.

8 It is about

one-third to one-tenth the sextupole-field decay rate ob8 served in SSC model magnets after a similar field cycle.

Magnetization decay was then measured after a temperature step from 3.5 to 4.0 K. The field cycle was the

same, but after stabilizing at 0.3 T at 3.5 K, the tempera415 Appl. Phys. Lett. 58 (4), 28 January 1991 0003-6951 /91/040415-02S02.00 © 1991 American Institute of Physics 415

18
FIG. 1. Decay of magnetization as a function of time measured with a SQUID magnetometer at (a) 3.5 K, (b) 4.0 K, (c) 4.0 K after an

increase in temperature from 3.5 K, and (d) 3.5 K after a decrease in temperature from 4.0 K.

ture was increased to 4.0 K. During the temperature sta- bilization, which took 2 min, the magnetization decayed by

about 10%. After temperature stabilization, magnetization

versus time was measured using the same routine as before,

beginning at t = 0. The results are shown as curve (c) in Fig. 1. The decay is 4.8% in 3000 s, R = 0.014, significantly greater than for curves (a) and (b). The decay rate was also measured after a decrease in temperature from 4.0

to 3.5 K using the same protocol. As expected, the decay

rate was less, 1.8% in 3000 s, R = 0.006, shown as curve

(d). Similar measurements were made for smaller temperature steps. After a 0.1 K increase from 3.9 K, the magnetization decayed by 3.2% in 3000 s, R = 0.010. Measurements on samples from other SSC cables showed similar

trends.

The enhancement in magnetic relaxation after an in- crease in temperature may be related to the temperature

dependence of the critical current density Jr Sun et al?

showed that flux creep in high-temperature superconductors could be reduced by a decrease in temperature. A sample initially in the critical state has magnetic shielding currents equal to the critical current density. Upon decreasing

the temperature, Jc increases and the sample is in what we

416 Appl. Phys. Lett., Vol. 58, No. 4, 28 January 1991

call a subcritical state, with shielding currents less than the

critical current density. This reduces flux creep. We expect that an increase in temperature would in- crease flux creep. After the magnetic field is cycled, the

sample is initially in the critical state. When the sample is warmed, Jc decreases and the sample is in what we call a

supercritical state, with shielding currents greater than the

critical current density. To restore equilibrium, the shielding currents and associated pinned flux must redistribute,

leading to enhanced flux creep. Considering the operating

conditions for SSC magnets, we suggest that a temperature

increase may contribute to the measured sextupole-field

decay with time.

We thank R. M. Scanlan, Lawrence Berkeley Laboratory, for helpful discussions. This work was supported by

the U. S. Department of Energy, Division of High Energy

Physics.

1 P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39 (1964).

2 M. R Beasley, R. Labusch, and W. W. Webb, Phys. Rev. 181, 682

(1969).

3 D. A. Herrup, M J. Syphers, D E. Johnson, R P Johnson, A. V.

Tollestrup, R. W. Hanft, B. C. Brown, M. J Lamm, M. Kuchnir, and

A. D. Mclnturff, IEEE Trans. Magn. 25, 1643 (1989).

4 R. W. Hanft, B C. Brown, D. A. Herrup, M. J. Lamm, A D. Mclnturff, and M. J. Syphers, IEEE Trans. Magn. 25, 1647 (1989).

5 M. Kuchnir and A. V. Tollestrup, IEEE Trans. Magn. 25, 1839 ( 1989).

6 W. S. Gilbert, R. F. Althaus, P. J. Barale, R. W. Benjegerdes, M. A

Green, M. I. Green, and R. M. Scanlan, Adv. Cryo. Engr. (Materials)

36, 223 (1990).

7 E. W. Collings, A. J. Markworth, K. R Marken, M. D. Sumption, and

R. M. Scanlan, Department of Energy Topical Workshop on Magnetic

Effects of Persistent Currents in Superconductors, 5-7 March 1990, Batavia, Illinois.

8 A. K. Ghosh, Department of Energy Topical Workshop on Magnetic

Effects of Persistent Currents in Superconductors, 5-7 March 1990, Batavia, Illinois.

9

J. Z. Sun, B. Lairson, C B Eom, J Bravman, and T H Geballe, Science 247, 307 ( 1990); M R. Beasley, R. Labusch, and W. W. Webb,

Phys. Rev. 181, 682 (1969).

I0 J. R. Clem, Department of Energy Topical Workshop on Magnetic

Effects of Persistent Currents in Superconductors, 5-7 March 1990, Batavia, Illinois.

11 W. S. Gilbert, R. F. Althaus, P. J Barale, R W. Benjegerdes, M A

Green, M. I. Green, and R. M. Scanlan, IEEE Trans. Magn. 25, 1459

(1989).

R. W. Cross and R. B. Goldfarb 416

19
Investigation of Possible Causes for the

Multipole Drift in Superconducting Dipole Magnets

R. M. Scanlan

Lawrence Berkeley Laboratory

Berkeley, California 94720

R. W. Cross and R. B. Goldfarb

National Institute of Standards and Technology

Boulder, Colorado 80303

Introduction

A time-dependent drift or relaxation of the higher order multipoles has been

observed at low fields in many superconducting dipole magnets. These include

dipoles used in the Tevatron at Fermi National Accelerator Laboratory (FNAL), 1

the Hadron Electron Ring Accelerator (HERA) at Deutsches Elektronen-Synchrotron

(DESY),2 and prototype magnets built for the Superconducting Super Collider

(SSC) project.3 These results were reviewed recently at a workshop,4 and a

number of possible explanations were discussed. Among these is the decay of

magnetization due to flux creep.

Although measurements which show flux creep occurring in superconducting wires

and cables were presented, the magnitude of the creep was a factor of five or 10

lower than the multipole decay in the dipole magnets. Several mechanisms which

may cause an enhanced decay in the environment of the dipole magnets were

proposed. W. Fietz (Department of Energy) and C. Taylor (Lawrence Berkeley

Laboratory) suggested that the time dependence could arise from the current and flux

redistribution which occurs as the cable strands traverse from the edge of the cable

in the high field region to the edge in the low field region. Taylor also suggested that

current redistribution due to the presence of solder joints could give rise to a time

dependent change in the multipoles. More recently, experiments at DESY showed

that the magnitude of the multipoles varied along the axis of the HERA dipoles, and

the period of this variation is identical to the strand pitch length in the cable.5 This

result suggests that currents of different magnitude are flowing in different strands

of the cable. Several mechanisms have been proposed to explain this phenomenon,

including nonuniform resistance due to cold welds in the strands. This note describes

the results of several experiments which attempt to simulate some of these conditions

and to observe the resulting magnetization change as a function of time.

20
Experiment

A number of techniques have been used to observe the time dependence of the

magnetization in superconductors. They include vibrating-sample magnetometry

(VSM),6 superconducting-quantum-interference-device (SQUID) magnetometry,7

and Hall-probe magnetometry.8 A Hall-probe magnetometer was chosen for these

experiments because we wanted to measure a full cross section SSC cable, 10 mm

wide and several centimeters long, and this could not be done with conventional

SQUID or VSM equipment. Furthermore, we wanted to make measurements in a

field gradient.

The experimental arrangement for the Hall-probe magnetometer has been

described in Ref. 8 and is shown in Fig. 1, where the sample is placed in a uniform

transverse field. To simulate the condition of the cable in a dipole magnet, the

background field was cycled from 0 to 6 T and back to 0, followed by a ramp to a set

field, after which the background field was held constant by switching the magnet to

persistent mode. A number of studies of magnetization versus time were performed,

and the results are in Fig. 2-7. Magnetization as a function of time t was normalized

to the value at t = 0, defined as the time the magnet was placed into persistent

mode.

For the experiments described here, several different configurations were studied.

First, the cable was measured in the “standard” configuration illustrated in Fig. 1.

The data appear in Fig. 2. Next, the sample ends were extended into the high field

gradient region near the top of the solenoid (Figs. 3—6). To examine the influence

of solder joints, a sample was prepared with an overlap solder joint. The joint was

in a fairly uniform field, but the sample tails extended out of the solenoid and were

thus in a field gradient (Fig. 7).

All runs showed basically the same type of behavior, characterized by two

different regimes. The first is a fast decay which is associated with eddy currents,

6

and a second, slower decay which is associated with flux creep the superconducting

filaments. The decay rates were not affected by the two variables introduced in these

experiments: a field gradient along the cable and a solder joint in the cable. No

significant change was observed which could help explain the large multipole drift

seen in the dipole magnets. The Hall-probe technique appears to be a useful

technique for studying large-volume samples such as the Rutherford-type cables used

in accelerator dipole magnets.

21
Transport

current

Figure 1. Schematic diagram of the Hall-probe magnetometer. The

magnetic field Hq is supplied by a superconducting solenoid. The

sample is bent into a U-shape so that solder contacts can be made

away from the Hall sensing area. After Ref. 8.

Figure 2. Normalized magnetization as a function of time. Measuring

field 0.3 T. Control experiment: sample tails not in a field gradient;

no solder joint.

22
Figure 3. Normalized magnetization as a function of time. Measuring

field 0.3 T, same as in Fig. 2. Sample tails extend to region of nearzero field.

Figure 4. Normalized magnetization as a function of time. Measuring

field 0.01 T, much smaller than in Fig. 3. Sample tails extend to region

of near-zero field.

23
Figure 5. Normalized magnetization as a function of time. Measuring

field 0.1 T, somewhat smaller than in Fig. 3. Sample tails extend to

region of near-zero field.

1.000

:0.975 -

G

o 0.950

• pH

03N• pH m-> OJ£ 0.925

b£

03

Field = 0.5 T

Field gradient

0.900 _i i i i

i i i i i i i 10 100

Time (s)

1000

Figure 6. Normalized magnetization as a function of time. Measuring

field 0.5 T, somewhat larger than in Fig. 3. Sample tails extend to

region of near-zero field.

24
1.000 t

i—

r

T

0.875 1 1 1—L - j - 1 1 1 1 1 1 1—L - - 1 -L-L - i -J—

10 100 1000

Time (s)

Figure 7. Normalized magnetization as a function of time. Measuring

field 0.3 T. Sample tails extend to region of near-zero field, similar to

Fig. 3, but sample has Sn-Pb solder joint in a region of fairly uniform

field. Magnetization measured on a segment of cable far from the

solder joint.

25
References

1. D. A. Herrup, M. J. Syphers, D. E. Johnson, R. P. Johnson, A. V. Tollestrup, R.

W. Hanft, B. C. Brown, M. J, Lamm, M. Kuchnir, and A. D. Mclnturff, Time

variations of fields in superconducting magnets and their effects on accelerators,

IEEE Trans. Magn. 25, 1643-1646 (1989).

2. H. Briick, Zhengkuan Jiao, D. Gall, G. Knies, J. Krzywinski, R. Meinke, H.

Preissner, and P. Schmuser, Time dependence of persistent current effects in the

superconducting HERA magnets, 11th Int. Conf. Magnet Tech. (MT-11), T.

Sekiguchi and S. Shimamoto, eds., Elsevier Science, Essex, U.K. (1990), pp. 141-

146.

3. W. S. Gilbert, R. F. Althaus, P. J. Barale, R. W. Benjegerdes, M. A. Green, M.

I. Green, and R. M. Scanlan, The effect of flux creep on the magnetization field

in the SSC dipole magnets, Adv. Cryo. Eng. (Materials) 36, 223-230 (1990).

4. Department of Energy Topical Workshop on Magnetic Effects of Persistent

Currents in Superconductors, March 5-7, 1990, Fermi National Accelerator

Laboratory, Batavia, Illinois.

5. H. Briick, D. Gall, J. Krzywinski, R. Meinke, H. Preissner, A. Freter, M.

Halemeyer, P. Schmuser, R. Stiening, D. ter Avest, and L. J. M. van de Klundert,

Observation of a periodic pattern in the persistent-current fields of the

superconducting HERA dipole magnets, DESY-HERA Report 91-01 (1991).

6. K. R. Marken, A. J. Markworth, M. D. Sumption, and E. W. Collings, Eddycurrent effects in twisted and wound SSC strands, IEEE Trans. Magn. 27, 1791-

1795 (1991).

7. T. W. Petersen and R. B. Goldfarb, Effect of mechanical deformation on Nb-Ti

filament proximity-effect coupling at the edges of SSC cables, IEEE Trans. Magn.

27, 1809-1810 (1991).

8. R. W. Cross and R. B. Goldfarb, Hall probe magnetometer for SSC magnet

cables: Effect of transport current on magnetization and flux creep, IEEE Trans.

Magn. 27, 1796-1798 (1991).

26
IEEE TRANSACTIONS ON MAGNETICS. VOL. 27, NO. 2, MARCH 1991 1809 Effect of Mechanical Deformation on Nb-Ti Filament

Proximity-Effect Coupling at the Edges of SSC Cables

T. W. Petersen and R. B. Goldfarb

Electromagnetic Technology Division

National Institute of Standards and Technology

Boulder, Colorado 80303

Abstract

Magnetization as a function of transverse magnetic field and time was measured for short strands extracted from the

centers and edges of five Nb—Ti Rutherford cables designed for use in Superconducting Super Collider dipole magnets. The

multifilamentary strands all had 6-jim diameter filaments. Edge

samples, which had severe mechanical deformation, showed small magnetic coupling losses at low fields, compared to no coupling

losses for undeformed center strands. Sharp strand bends at cable edges decreases the interfilament spacing to the order of

the coherence length in the normal matrix material which

increases the effective filament diameter and hysteresis loss at low fields. Microscopic studies of the cables’ cross sections

confirmed smaller interfilament separations in these samples.

Flux creep measurements, represented by the time dependence

of magnetization, showed little difference between edge and

center samples. This indicates that the proximity-coupled matrix

in edge samples is not a significant source of flux creep.

Introduction

Superconducting Super Collider (SSC) dipole magnets are required to have highly uniform fields. The magnetization of the magnets’ superconducting filaments results in field nonuniformity.

This effect is largest at low fields where the magnetization is highest. To reduce the magnetization of superconductor wires,

multifilamentary conductors are used. According to the critical state model, for a large transverse field [1], the superconductor

magnetization M depends on the filament diameter d and the

critical current density Jc: M = 2 Jc d / (3-7r).

Efforts to decrease the size of the filaments have

succeeded in producing wires with filament diameters smaller

than 1 jim. However, to maintain the critical current of the wire, the distance between adjacent filaments is also decreased with

decreasing filament size. Once the separation between the filaments is on the order of the coherence length in the normal

matrix material, the filaments become coupled through supercurrents in the matrix [1,2]. This increases the effective filament

diameter and, as a result, the magnetization. Uncabled SSC wire

shows very little coupling due to the proximity effect. However,

recent work has shown a degradation of the cable’s currentcarrying capacity at the edges, presumably due to mechanical

deformations resulting from the cabling process [3,4].

This paper examines the effect of the cabling process on

the magnetic properties of SSC cables. Measurements of magnetization as a function of both time and applied field were

performed on samples taken from the edges and centers of SSC

cables. Filament separation was also studied by microscopic

evaluation of the samples.

Contribution of the National Institute of Standards and

Technology, not subject to copyright.

Manuscript received September 24, 1990.

Experimental Details

Samples for magnetization as a function of time and

applied field were extracted from five different 23-strand

Rutherford cables designed for use in SSC dipole magnets. Most

samples had a Cu/superconductor volume ratio of 1.36 and

keystone angles varying from 1.6 to 3.0 degrees. A cable was

fixed in epoxy to prevent movement of the strands while a 4-mm

section was cut with a diamond saw. The epoxy was removed

from the sample and strand segments were extracted from the centers and the edges. Strand samples were weighed and the

total volume of Nb-Ti was calculated from the mass of the

strand, the densities of Cu and the Nb-Ti, and the Cu/superconductor ratio. Some specimens were cast in epoxy and

polished prior to microscopic studies.

Measurements of magnetization per unit volume of Nb-Ti

were made at 5 K using a SQUID magnetometer with the applied field perpendicular to the axis of the strand. Magnetization as a function of applied field was carried out for maximum fields of 2 T (1.6 MA/m). A magnetometer scan

length of 4 cm was used with a field variation of 0.19%. Flux

creep measurements, represented by the time dependence of the magnetization, were made after a field cycle of 0 to 2 T to 25 mT. This field cycle was chosen to maximize the proximity

coupling between filaments. An external field of 25 mT

corresponds to zero internal field for a demagnetizing factor of

0.5. A scan length of 2 cm was used with a field variation of less than 0.01%. Due to the finite time required to perform a magnetization measurement, it was impossible to monitor the decay of the magnetization that occurred in the first 2 min

following the field change.

Results and Discussion

The magnetization of samples taken from the edges of the cables show small coupling peaks centered around zero field, whereas data for samples taken from the center of the cables show no such losses. These results are represented in Fig. 1. The results shown are representative of the five cables studied.

In each case, the edge sample showed hysteresis losses not observed in the center sample. These losses are characteristic of

proximity-effect coupling [2]. This suggests that the separation

between some of the filaments in the edge samples is on the order of the coherence length in the Cu matrix. We attribute the change in the filament separation to mechanical deformation

resulting from the cabling process.

Following the derivation of Collings [5], we estimate the coherence length in the matrix material to range between 0.21 and 0.36 jim for the samples studied. Therefore, to avoid

coupling of the filaments at zero field, the spacing should be

about 0.7 jtm or greater. Microscopic studies of the cables’ cross sections showed spacings between the filaments of approximately

1 jim for undeformed center strands. In edge samples, however, we observed smaller interfilamentary spacing, approximately 0.2 jim, for filaments in the deformed regions of the strand.

U.S. Government work not protected by U.S. Copyright.

27
1810

Internal Field (kA/m)

Figure 1. Magnetization as a function of internal field (corrected for demagnetizing factor) for center and edge samples. The left graph shows the low field magnetization for a center strand of SSC cable. The right graph shows an edge strand from the same cable. Note the small coupling peaks (arrows) for the edge sample near zero internal field.

Measurements of magnetization as a function of time were

carried out for the two types of samples. The decay of the magnetization in the edge samples showed no appreciable

difference from the center samples as can be seen in Fig. 2. The

results shown are representative of the five samples measured.

Additional measurements made at higher fields also did not show

differences between samples. Despite the fact that some sections of magnet cable are located in regions of small field, flux creep

Figure 2. Normalized magnetization as a function of time for a

field cycle of 0 to 2 T to 25 mT. After this field cycle the

internal field is near zero, which allows proximity coupling

between filaments.

in the proximity-coupled matrix is probably not a source of the

field decay seen in SSC dipole magnets for the following reasons.

(1)

The fraction of proximity-coupled filaments in a cable, and

the associated coupling peak in the hysteresis loop, is small. (2) The rate of decay of magnetization is the same for coupled and

uncoupled samples. (3) The time constant of the magnetization

decay observed in these samples is short (15-18 min) compared

to the decay observed in accelerator magnets (1-2 h) [6-9].

Acknowledgment

This work was supported by the U.S. Department of

Energy, Division of High Energy Physics.

References

[1] W. J. Carr, Jr., J. H. Murphy, and G. R. Wagner, Adv. Cryo.

Engr. 24, 415 (1978).

[2] V. R. Karasik, N. G. Vasil’ev, and V. G. Ershov, Z/z. Eksp.

Teor. Fiz. 59, 790 (1970) [Sov. Phys. JETP 32, 433 (1971)].

[3] L. F. Goodrich, S. E. Pittman, J. W. Ekin, and R. M.

Scanlan, IEEE Trans. Magn. 23, 1642 (1987).

[4] L. F. Goodrich and S. L. Bray, IEEE Trans. Magn. 25, 1949

(1989).

[5] E. W. Collings, Adv. Cryo. Engr. (Materials) 34, 867 (1987).

[6] D. A. Herrup, M. J. Syphers, D. E. Johnson, R. P. Johnson,

A. V. Tollestrup, R. W. Hanft, B. C. Brown, M. J. Lamm,

M. Kuchnir, and A. D. Mclnturff, IEEE Trans. Magn. 25,

1643 (1989).

[7] R. W. Hanft, B. C. Brown, D. A. Herrup, M. J. Lamm, A. D. Mclnturff, and M. J. Syphers, IEEE Trans. Magn. 25,

1647 (1989).

[8] M. Kuchnir and A. V. Tollestrup, IEEE Trans. Magn. 25,

1839 (1989).

[9] W. S. Gilbert, R. F. Althaus, P. J. Barale, R. W.

Benjegerdes, M. A. Green, M. I. Green, and R. M. Scanlan,

Adv. Cryo. Engr. (Materials) 36, 223 (1990).

28
AC LOSS MEASUREMENTS OF TWO MULTTFILAMENTARY NbTi

COMPOSITE STRANDS

E. W. Collings*, K. R. Marken, Jr.*, M. D. Sumption*!,

R. B. Goldfarb**, and R. J. Loughran**

* Battelle Memorial Institute, Columbus, OH 43201

t Permanent address: Ohio University, Athens, OH 45701

** National Institute of Standards and Technology, Boulder, CO 80303

ABSTRACT

As part of an interlaboratory comparative testing program conducted in support of

the Versailles Agreement on Advanced Materials and Standards (VAMAS), transversefield DC hysteresis loss measurements were made at liquid-helium temperatures at fields

of up to 3 T (30 kG) on two samples of multifilamentary NbTi composite. The strands

differed widely in filament number, were comparable in filament diameter, and one of

them was provided with a Cu-Ni barrier between the filaments. The results have been

analyzed, and magnetically deduced critical current density values obtained (for

comparison with directly measured transport data) using various standard techniques.

Based on these studies, a figure -of-merit for AC loss is recommended. The Cu-matrix

strand, with its interfilamentary spacing of less than 1 pm, exhibited pronounced

proximity-effect-induced coupling losses; this was not observed in the mixed-matrix strand

which possessed not only a Cu-Ni barrier but also an interfilamentary spacing of typically

4 pm.

INTRODUCTION

In support of the Versailles Agreement on Advanced Materials and Standards

(VAMAS), DC hysteresis loss measurements were made at liquid-helium temperatures in

transverse magnetic fields of up to 3 T (30 kG) on samples prepared from two types of

multifilamentary NbTi composite superconductors. The strands, designated herein as Sample E (henceforth SL-E) and Sample D (henceforth SL-D), had been manufactured in

Japan and in the U.SA., respectively. SL-E is a “mixed-matrix” strand in which the

filaments are surrounded first by Cu and then by a thin (15 pm) eddy-current barrier of

Cu-Ni alloy. SL-D consists of closely spaced NbTi filaments separated only by Cu.

Photomicrographs of the strands are presented in Figs. 1 and 2. Noticeable in the

magnified cross sections are the irregular shapes of the filaments, particularly in SUE.

For this reason, accurate filament cross-sectional areas (for the purpose of Jc calculation

from the results of I

c measurement) could only be obtained by an etching-and-weighing

procedure. Some specifications of the strands are listed in Table 1. HYSTERESIS-LOSS MEASUREMENT

Measurements were made at Battelle, Columbus Division (BCD) and the National

Institute of Standards and Technology (NIST) using vibrating-sample magnetometry. At

both places the instruments were calibrated against pure Ni standards. At BCD,

Advances in Cryogenic Engineering (Materials), Vol. 36 169

Edited by R. P. Reed and F. R. Fickett Plenum Press, New York, 1990

29
Fig. 1. Scanning electron micrograph of Fig. 2. Scanning electron micrograph of

Sample E (SL-E). Sample D (SL-D).

magnetization was measured in the slowly swept field (amplitudes 0.1 to 1.6 T; sweep rates

1.7 to 28 mT/s) of an iron-core electromagnet; data were recorded automatically at a field

resolution of about 1/200^ of the field-sweep amplitude. At NIST, point-by-point

measurements were taken in stepped fields (up to 3 T) provided by a superconducting

solenoid. The typical BCD sample consisted of an epoxy-potted 3-mm-diameter bundle of

6-mm-long pieces of strand. The typical NIST sample was formed by winding a strand or group of strands along the thread of a 5-mm-diameter nylon screw.

Table 1. Specifications of Strands under Investigation

Sample Code Sample E (SL-E) Sample D (SL-D)

Type Mixed matrix Copper matrix

Configuration NbTi/Cu/CuNi NbTi/Cu

Volume Ratio 21.5/44.1/34.4 42.0/58.0

Twist pitch, mm 6 13

Strand diameter, D, mm 0.35 0.742

Fil. diameter*, w, /im 5.79 4.62

Number of filaments 760 10,980

I

c at 3 tesla, A 50 675 **

Jc *** at 3 tesla, IC^A/cm2 2.50 3.67

*

*****

Measured by the etching-and-weighing technique on 304-cm (SL-E) and 149-cm

(SL-D) lengths of strand. Measured density of bulk Nb-46.5Ti = 6.097 g/cm3

. Straight-sample measurement at NIST (manufacturer’s supplied value, 653 A).

Based on I

c and the above-measured NbTi cross-sectional area.

170

30
Magnetization,

10' 3

A

m

(1(T 2

emu

cm'

1)

Magnetization,

HF 4A

m

(HT 3

emu

cm' 1)

Fig. 3. Magnetization per unit length

of strand at 4.2 K for SL-E

-- typical BCD data.

Applied Field Strength, KT1 T (kOe)

-5.0 -2J5 0 2.5 5j0

Fig. 5. Magnetization per unit volume

of strand at 4 K for SL-E - typical NIST data.

Fig. 4. Magnetization per unit length

of strand at 4.2 K for SL-D — typical BCD data.

Applied Field Strength, lO-1 T (kOe)

-5J0 -15 0 15 5D

Fig. 6. Magnetization per unit volume

of strand at 4 K for SL-D

-- typical NIST data.

171

31
RESULTS AND DISCUSSION

Some typical hysteresis loops are presented in Figs. 3 to 6. Two features are noteworthy: (1) Fig. 3 (in which a large field-sweep amplitude has been deliberately

selected) exhibits a pronounced paramagnetic slope due to the presence of the CuNi

barrier material of which 34.4 vol.% is present; (2) Fig. 4 exhibits a shoulder and a peak

near the origin, structure which is not resolved in Fig. 6. The shoulder is due to the NbTi

filament magnetization while the sharp peak, which also dominates Fig. 6, results from

coupling currents flowing in the Cu matrix (cf. Ref. 1). Coupling is not present in SL-E

with its larger filament separation and resistive barrier layer.

The results of AC-loss measurements can be represented in several ways. A direct

approach, and one which is useful from an applications standpoint, is to tabulate or plot

the energy loss per cycle per unit length of wire, Q^, as function of the field-sweep

amplitude, This is done in Fig. 7. According to the figure, there is good general

agreement between the BCD and NIST data over the entire field-amplitude range, in spite

of the fact that the two sets of measurements were made on samples differing widely in

configuration. Furthermore, when measurements were made in both laboratories on the

BCD samples, the results differed by less than 7% *.

AC-LOSS REPRESENTATIONS AND FIGURES OF MERIT

Representations of AC Loss

Depending on the purpose in mind, various “levels” of refinement can be adopted

in reporting and comparing of AC-loss information. In general, the hysteretic loss of a multifilamentary strand per unit volume of superconductor per unit field-sweep amplitude

may be regarded as a function, Q(JC

, w, P, A), of: (i) the critical current density, Jc, of the

superconductor, (ii) the diameter, w, of the filaments, (iii) P, the influence of proximityeffect coupling between the filaments, and (iv) an “addenda”, A, which includes the effect

of imponderables such as filament-cross-section irregularity.

Level-0. The simplest representational level, referred to here as Level-0, is (Fig.

7) which derives directly from hysteresis-loss measurements on a known length, £, of

strand. At this level the various possible contributions to loss remain unspecified.

According to Fig. 7, under the Level-0 criterion SL-D is the more lossy -- but it also

possesses more filaments than SL-E and is able to carry, for this and other reasons, a

larger critical current.

Level1

. A more significant indicator of hysteretic loss, especially from an

engineering standpoint with a particular application in mind, would be to normalize to the I

c of the strand (taken, perhaps, at some operating field of interest). In this next level of refinement hysteretic loss, gauged by Q£/Ic

, responds only to differences in w and A.

Level-2. Finally, under Level-2 (to be discussed below), differences in w would also

be absorbed, and the resulting criterion would be positioned to emphasise loss due to proximity-effect coupling and the addenda.

* At 1^ = 0.97 T, the per-cycle hysteresis loss in SL-E as measured at NIST was 6.7%

lower than the BCD-measured value; also at = 0.97 T, the NIST-reported loss in SLD was 1.4% higher than the BDC value.

172

32
Fig. 7. The Level-0 criterion (loss per cycle per unit length of strand)

applied to Sample D (SL-D) and Sample E (SL-E).

Figures of Merit for Hysteretic Loss

The QJJr (Level-1) Approach. Following the well-known critical-state model2 as applied by Carr and co-authors (whose work is summarized in Ref. 3) to a multifilamentary composite strand in a transverse applied field, the hysteresis loss per unit volume of

the superconducting component of the strand, is given in SI units by:

Qv = (8/3*) w Jc pQ Hn,

(or Qv = (0.8/3 *) w Jc F^, in practical units) (1)

i.e., Rq = Qy/Jc = (8/3*) w /i0 ^

(or Rq = = (0.8/3*) w in practical units) (2)

It follows that a plot of Rq versus should, under the Bean approximation, be linear

with slope proportional to w, the filament diameter. Departure of Rq from its expected

value, possibly as a result of proximity-effect coupling between the filaments, could then

be expressed in terms of some effective filament diameter4

, we£f

. Within this framework,

the following figures-of-merit (FOM) might be selected: (i) at a given F^, the ratio

weff/w; (ii) at a given F^ and H, the quotient Rq = Qyj^H). The use of JC(H)

recognises that in practice Jc is not independent of field, and that for the purpose of a

criterion may have to be measured at some field different from F^. Finally, we note that

in obtaining an experimental value of Rq, it is convenient to replace Qy/Jc by its identical

equivalent, Qg/Ic

, the latter being a ratio of directly measured quantities, Fig. 8.

For a pair of “ideal” multifilamentary strands, say A and B, the quotient

Rq^/Rq b

at a given F^ should be simply wA

/wB, the filament-diameter ratio. In the case of the present strands, direct measurements have shown that wE

/wD = 1.25 (see Table 1 and its footnote *). At Hjjj = 3 tesla, this may be compared to the magnetically obtained

value of Rq E/Rq D = 1.35. The interfilamentary coupling present in SL-D (see Fig. 4)

should cause Rq E/Rq D

to be less than the filament-diameter ratio. That it is not,

suggests that factors not yet taken into consideration are masking the ability of the Rq

criterion to properly represent the presence of coupling. One such factor could well be a difference between the field-dependences of Jc for the two strands.

173

33
The QJL. (Level-2) Approach. According to the critical-state model for a cylinder of diameter w in a transverse applied field, the total height, AM^H), of the NbTivolume-normalized hysteresis loop (measured between the shielding and trapping

branches) at some field H is related to JC (H), in SI units, by5:

AMy(H) = (4/3t) Jc w

(or AMyCH) = (0.4/3t) Jc w, in practical units) (3)

Bean’s model was of course based on the premise that Jc was independent of field; nevertheless, even Bean2 and many others to follow, employed Eqn. (3) to determine the

field-dependence of Jc

, as for example in Fig. 9. A more rigorous treatment of the critical state would be to introduce a field-dependent Jc at the outset -- as in the work of Ohmer

and Heinrich6

, who constructed a critical state model (for cylinders in the field-parallel

orientation) based on a modified Kim7 equation.

Fig. 8. The Level-1 criterion (loss per cycle per unit length per unit

critical current at reference field) applied to SL-D and SL-E.

Fig. 9. Magnetization-determined Jc versus applied field strength

for SL-E and SL-D.

174

34
Fig. 10. The Level-2 criterion as function of M(H)-loop amplitude, H^.

The proportionality between AMV(H) and JC (H) provides an opportunity to convert

Rq into a quotient that can be derived solely from the hysteresis loop, without recourse to a separate Jc measurement. From Eqns. (1) and (3), and referring to either unit-volume

or unit-length quantities, respectively:

Rm = CyAM^H) = Qt/AMfc(H) = 2 /i0 (or Rm = CyAM^H) = Q^/AM^(H) = 2Ftm, in practical units) (4)

Under a strict Bean criterion (J

c = const.), AM is independent of H and RM

is linear with

slope 2.0. Fig. 10 indicates the extent to which this is obeyed in practice. In formulating

Rm as a new FOM it will be necessary to normalize to a loop height measured at some arbitrarily selected reference field, Href . In Fig. 10, RM is plotted versus for

three values of : (1/3)1^, (l/2)Hm, and (9/10)1^, respectively. An important

advantage of RM as an FOM lies in its independence of both Jc (as in the Level-0

criterion) and w (Level-1). Thus, RM is expected to respond directly to the influence of

proximity-effect coupling. Indeed, in Fig. 10, the curves for SL-D (which is coupled) all lie above their SL-E counterparts.

SUMMARY AND CONCLUSION

The hysteretic loss of two multifilamentary strands has been measured by vibratingsample magnetometry. The results emphasize the importance of acquiring high-resolution

data at low fields where Jc

, and hence, the hysteresis-loop-height and the loss per unit

field increment is greatest. Furthermore, if interfilamentary proximity-effect is present,

high-resolution data are useful for resolving the coupling peak and its nearby NbTi

shoulder.

Several representations of AC-loss criteria have been suggested. The simplest is Qv or (Level-0), which leaves unspecified all possible contributions to loss and which

derives directly from hysteresis measurements on known amounts of superconductor. The

next level of refinement takes current-carrying capacity into account, resulting in a

criterion that responds to filament diameter and what might be termed “high-order”

hysteretic losses such as those due to proximity-effect coupling, and perhaps filamentshape distortions and other “addenda”. The final level of refinement yields a criterion

that includes both I

c and w, and which emphasizes proximity-effect coupling and the

addenda.

175

35
The criterion Rq = Q£/Ic = QJJC (Level-1), normalizes hysteretic loss to currentcarrying capacity and is thus useful from a design standpoint. Since filament diameter, w,

is not absorbed into the criterion, the strand with the largest w (for given I

c) will have the

largest Rq. Next, the presence of an interfilamentary-coupling contribution to AC loss

could be identified directly by introducing the concept of an effective filament diameter

4

, wcff = Rq/^/Ox)!^ (in SI units, see Eqn. (2)); the corresponding FOM (a quantity

intended to increase beyond 1 in proportion to increased coupling loss) would be weff/w.

Finally, a Level-2 form of FOM that includes both Jc and w has been suggested; it is Qg/AMj^ (see Eqn. (4)). Under a strict Bean criterion, in which Jc

is assumed independent

of H, AM£ is constant and equal to AM^H^,); a dimensionless Qg/fAM^HjJ.HjJ is then

equal to 2.0 (see Fig. 10). But in practice Jc and hence AM^, decreases with H.

Consequently Qg/fAM^Hj.H,,,] becomes equal to 2 only when AM^ is measured at some

intermediate value (H < F^) of the applied field. An advantage of the Q/AM quotient is that it can be obtained entirely by magnetometry, AM^ substituting for an auxiliary Jc determination. Another advantage of this approach lies in the fact (indicated in the first paragraph of this discussion) that the largest contribution to incremental hysteretic loss occurs at low fields, a region in which Jc may be out of the range of the current-transport

measuring equipment.

In conclusion it is important to recognize, as a comparison of Figs. 8 and 10

indicates, the more inclusive the AC-loss criterion or FOM, the smaller is its variation

from sample to sample. In general, when defining an FOM, normalization should be

carried only far enough to achieve some specific objective, be it engineering or scientific in

nature.

ACKNOWLEDGEMENTS

At BCD, the epoxy-potted magnetization samples were prepared by R. D. Smith.

At NIST, L. F. Goodrich measured transport critical currents on SL-D, and R. L. Spomer

assisted with acquiring the magnetization data. The research was sponsored by the U.S.

Department of Energy, Division of High-Energy Physics.

REFERENCES

1. E. W. Collings, K. R. Marken Jr., M. D. Sumption, E. Gregory, and T. S. Kreilick,

“Magnetic studies of proximity-effect coupling in a very closely spaced fine-filament

NbTi/CuMn composite superconductor”, paper in this conference.

2. C. P. Bean, “Magnetization of high-field superconductors”, Rev. Mod. Phys. 36, 31-39

(1964).

3. E. W. Collings, Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys, Vol. 1, Plenum Press, New York, 1988, p. 353.

4. S. S. Shen, “Magnetic properties of multifilamentary Nb3Sn composites”, in

Filamentary A15 Superconductors, ed. by M. Suenaga and A. F. Clark, Plenum

Press, New York, 1980, pp. 309-320.

5. W. J. Carr, Jr., and G. R. Wagner, “Hysteresis in a fine filament NbTi composite”,

Adv. Cryo. Eng. 30, 923-930 (1984).

6. M. C. Ohmer and J. P. Heinrich, “Magnetization of hysteretic superconductors for

complete field penetration and critical state model with JC(H) = a/H”, J. Appl.

Phys. 44, 1804-1809 (1973).

7. Y. B. Kim, C. F. Hempstead, and A. R. Stmad, “Magnetization and critical supercurrents”, Phys. Rev. 129, 528-535 (1963).

176

36
Effect of Cable and Strand Twist-Pitch Coincidence

on the Critical Current of Flat, Coreless Superconductor Cables

J. W. Ekin

Electromagnetic Technology Division

National Institute of Standards and Technology

Boulder; Colorado 80303

Abstract

Data are presented which indicate that a very simple technique for enhancing the

critical current in flat, coreless superconductor cables is to match the cable twist pitch

with the strand twist pitch. In this manner the same group of filaments within each

strand is degraded at each successive bend at the cable edges. This coincident twist

method minimizes current transfer among filaments, enhances the slope of the

voltage-current characteristic, consistently improves the critical current by about 10%

in these tests, and is easy to apply.

Introduction

This paper presents a very simple method to enhance the transport critical

current density Jc in flat, coreless superconductor cables, such as the “Rutherford”

cables utilized in construction of Superconducting Super Collider (SSC) accelerator

magnets. The technique recognizes and compensates for the fact that the

superconductor filaments within each strand are periodically degraded where the

strands are bent at the cable edges, 1,2 (see Fig. 1). The degradation in Jcis not

uniform across the strand where it is bent, but affects the filaments on the outside of

the bend (where the tensile strain is greatest) significantly more than the filaments

on the inside of the bend.

When the degraded filaments within a strand at one bend do not match up with

the same degraded filaments at the next bend, current transfer across the strand

matrix material is required to redistribute the current among the filaments and

optimize the total current that can be carried by the strand. This produces voltages

within the strand as the current redistributes across the normal matrix material,

degrading the overall critical current of the cable (especially when measured at

sensitive electric-field detection levels).

3

37
125 X

Filaments on

Outside of Bend

1250 X

Filaments on

Inside of Bend

1250 X

Figure 1. Cross-sectional view of a bent NbTi strand, showing greater

filament area reduction for the filaments on the outside of the bend

(left side of the figure), than on the inside of the bend (right side of

the figure).

38
This situation presents the opportunity for a conceptually very simple method to

minimize the bend degradation in flat, coreless cables. The technique is to match the

cable twist pitch with the strand twist pitch such that the same group offilaments within

each strand are degraded at each successive bend (see Fig. 2). In this way current

transfer among the filaments within the strand is minimized, current-transfer voltages

are minimized and the Jc

is not degraded as much by the edge bending. The

technique only requires that the cabling twist pitch coincides with an integer multiple

of the strand twist pitch.

Experiment

To test this technique, we studied a series of NbTi strands that are candidate

superconductors for the construction of the SSC dipole magnets (0.65 mm diameter,

filament twist pitch of 1.27 cm). The conductors were wound on flattened mandrels

in such a manner that the spacing between bends at the mandrel edges occurred at

either integer or half integer multiples of the strand twist pitch (see Fig. 2). The

thickness of the mandrel and the finite length of strand wrapped around the mandrel

edge were taken into account in calculating the width of the mandrel needed to

match cable and twist pitches. That is, as shown in Fig. 2, we desired that the total

edge-to-edge length of the strand’s neutral axis, L, to be an integer or half integer of

strand twist pitches, S. As seen in Fig. 2, this is just:

L = w + (7r/2)(r + D)=RS, (1)

where w is the mandrel width, t is the mandrel thickness, D is the strand diameter,

and R is the cable/strand pitch ratio. R was set to either an integer or half integer

value by varying the mandrel width, w.

The sample/mandrel assembly was then placed in the bore of a high-field

solenoidal magnet such that the winding axis of the sample was concentric with the

central axis of the bore of the test magnet. The voltage taps were counterwound

along the superconductor in order to minimize the loop area for inductive pickup in

the high field test magnet. The samples were held in place with varnish, which in

most cases was sufficient to support the Lorentz forces experienced by the strand

during testing. Jc data were obtained with transport current applied along the strand

in both directions and the results averaged to make a first order correction for the

self-field effect. Precision of the Jr measurements is about ±2%.

39
Twist Pitch Coincidence Effect

R = 1/2

filaments

i within strand

i (diametrically

opposed)

R = 1

R = 2

w

Superconductor

Mandrel

Figure 2. Test geometry, showing bend-to-bend separations that are an

integer or half-integer multiple (R) of the strand twist pitch.

40
Results

Fig. 3 presents a logarithmic plot of a set of voltage-current (V—I) characteristics

obtained on sample 1 when the strand bends are spaced by 1.5 and 2 strand twist

pitches, S. Focusing on the data for a bend spacing of 1.5 S, we see that the slope

of the V—I curve is relatively low at high electric field. In contrast, for the other set

of data where the bend spacing is 2 5, the slope of the V—I characteristic is

significantly higher at high electric fields and the critical current remains considerably

higher, especially at electric fields below 10'7 V/cm. Furthermore, when the test is

repeated at a smaller bend spacing of 1 S, the J

c and slope recover to higher values.

Thus, the degradation in these quantities occurs only when the bend spacing is not

an integer multiple of S.

This is seen more clearly in Fig. 4 where Jc has been determined at an electric

field criterion of 10"7 V/cm and plotted as a function of the bend spacing (expressed

as the cable/strand pitch ratio, R) imposed on the strand for magnetic fields ranging

from 3 T to 8 T. Also shown in Fig. 4 are the results for several other NbTi strands

having different local copper-to-superconductor area ratios (that is, the ratio in the

immediate vicinity of each filament, not the overall ratio) and diffusion barrier

materials. As seen in Fig. 4, the Jc for all samples periodically increases and

decreases as the bend spacing alternately matches up with an integer or half-integer

number of strand twist pitches. That is, an integer cable/strand twist pitch ratio

consistently produced a —10% Jc improvement in this test (corresponding to about

a factor of 10 difference in voltage, as seen from Fig. 3) under widely varying

conditions of magnetic field, local area ratio, and diffusion barrier material.

Similar results were obtained for the logarithmic slope of the V—I curves. The

slope of the V—I curve can be represented by n,

3' 5 defined as

n = d\nV / d\nl . (2)

Plots of n versus the bend spacing are presented in Fig. 5. This figures shows that

n periodically increases and decreases with the bend spacing, in direct

correspondence to the variation in Jc

. The highest values of the slope, n, are

obtained when the bend spacing coincides with an integer number of strand twist

pitches. The difference in n can be more than 60%.

Discussion

The logarithmic slope of the V—I curve, n, is an index of the amount of

inhomogeneity in Jc along the filaments within a strand.

6 As discussed in Ref. 6,

41
oo

in

IDI o

in

CVJ

£C QC

toonEto©nE

>. CO CO C -O

CD =

Q.

O o w J—L . oooo

COo

C\J

(wo/a) 3

42

Figure

3.

Logarithmic

plot

of

the

voltage-current

characteristics

ofa

NbTi

strand

showing

significant

degradation

inJc

andn

when

bends

are

spaced

apart

by

1.5

twist

pitches,

in

contrast

to 2

twist

pitches.
J

C

(R)/J

C

(R

=

0.5)

Figure 4. Critical current of three NbTi strands having different copperto-superconductor local area ratios as a function of the cable-tostrand twist-pitch ratio, R. The data show a significant improvement

in critical current for integer values of R.

43
50

Local Cu:SC=0.40, Nb barrier

Q L _ . ' , ,_j

0.5 1 1.5 2

Cable/Strand Pitch Ratio (R)

Figure 5. Logarithmic slope n of the voltage-current characteristics of

the same three NbTi strands as in Fig. 4, showing a similar

improvement in n for integer values of R.

44
however, n is affected only by inhomogeneities along a given filament, and not by

differences between different filaments. The data presented here illustrate this point.

For the half integer case, many filaments within the strand experience alternate high

and low bend degradation at the points of bending. This forces considerable current

transfer to occur among the filaments in order to redistribute the current among the

filaments between bends in order to optimize the total current carried by the strand.

However, when the bend spacing is matched to the strand twist pitch, it is always

the same group of filaments that are most severely degraded at each bend, and a

large redistribution of current among the filaments is not needed to optimize the total

current carried by the strand. Slight mismatches in the cable-to-strand twist pitch are

not important, as long as the affected group of filaments changes slowly over many

twist lengths so that a long length of strand is available for current to transfer from

the degraded group of filaments to the less degraded group of filaments.

The strand length, x, needed to accommodate the transfer of current among the

-7 filaments is given by

x = (0.1/nf (pm/p*)

v> D , (3)

where D is the strand diameter, pmis the resistivity of the matrix material, and p* is

the resistivity criterion. Substituting values for these NbTi samples, we find that the

current transfer length x becomes greater than the strand twist pitch at a strand

resistivity p* of about 1.4 x 10' 13

fl cm (an electric field of 10"8 V/cm at 6 T), which

explains the degraded Jc seen at low electric field levels in Fig. 3 for half integer

cable/strand pitch ratios. However, the transfer length* is less than a tenth the strand

twist pitch at a high electric field of 10"6 V/cm, for example, corresponding to p* =

1.4 x 10' 11

fl cm at 6 T. In this regime, the current can redistribute between

successive bends and there is little effect, as seen in Fig. 3 at high electric fields. From

a practical standpoint, however, strand resistivities for magnet applications should be

less than 10" 14

fl cm, and thus this coincidence twist effect is expected to make a

significant difference in such applications.

Application

These experiments were performed on strands wound around a mandrel with a

very small cable twist pitch to have a significant number of bends within the small test

volume of a research magnet. Equation 1, however, can be simply adapted to the

practical Rutherford cable configuration where the strand progresses along the cable

with a reasonable cable pitch. The strand length between bends L is then given

approximately by:

45
( 4 L

) = [(P/2)2 + W2)'A = RS

where P is the cable twist pitch and W is the overall cable width. (If we make the

reasonable assumption that the extra small amount of length at the strand’s neutral

axis needed to round the corner at the cable edge is approximately equal to twice the

strand width, then this equation holds exactly.) As before, we desire that L be an

integer number of strand twist pitches S (that is, the ratio L/S = R is an integer).

In the case of SSC cables, the strand twist pitch S is typically 12.7 mm and the

cable pitch P is about 76 mm. The overall cable width is 12.3 mm for the inner cable

and 11.7 mm for the outer cable. Substituting these values into Eq. 4 results in the

coincident cable pitches given in Table 1. These data thus suggest that, instead of

a cable twist pitch of 76 mm, setting the cable twist pitch to, for example, 72.1 mm

for the inner cable, or 72.5 mm for the outer cable may be a simple way to improve

the overall Jc in the these cables. Cables with different strand twist pitches or cable

widths are readily determined using Eq. 4. Thus, this coincident twist method

presents the possibility of enhancing the overall critical current simply with little

additional cost, only an adjustment in twist pitch during cabling.

Table 1. Optimum Cable Twist Pitches for SSC Superconductor Cables

Cable/Strand Pitch Ratio Inner Cable Outer Cable

2 44.4 mm (1.75 in) 45.1 mm (1.78 in)

3 72.1 mm (2.84 in) 72.5 mm (2.86 in)

4 98.6 mm (3.88 in) 98.9 mm (3.89 in)

5 124.6 mm (4.90 in) 124.8 mm (4.91 in)

Acknowledgments

The author is grateful to G. Reinacker, N. Bergren, and C. Clickner for assistance

with the measurements; N. Bergren, S. Bray, and R. Gerrans for help in preparing

the cross-sectional photomicrograph; R. Goldfarb and R. Scanlan for helpful

discussions. This work was supported by the U. S. Dept, of Energy, Div. of High

Energy Physics, and the NIST superconductor program.

46
References

1. L. F. Goodrich, E. S. Pittman, J. W. Ekin, and R. M. Scanlan, Studies of NbTi

strands extracted from coreless Rutherford cables, IEEE Trans. Magn. 23, 1642

(1987).

2. J. W. Ekin, S. L. Bray, and N. F. Bergren, The effect of local area ratio and

filament diffusion-barrier material on critical-current bend degradation of

multifilamentary NbTi wires, NISTIR 89-3912, National Institute of Standards

and Technology, Boulder, Colorado (1989).

3. J. W. Ekin, Current transfer in multifilamentary superconductors. I. Theory,

/. Appl Phys. 49, 3406 (1978).

4. F. Voelker, Particle Accelerators 1, 205 (1970).

5. L. F. Goodrich and F. R. Fickett, Critical current measurements: A compendium

of experimental results, Cryogenics 22, 225 (1982).

6. J. W. Ekin, Irregularity in Nb-Ti filament area and electric field versus current

characteristics, Cryogenics 27, 603 (1987).

47
Kim mode! for magnetization of type-II superconductors

D.-X. Chena) and R. B. Goldfarb

Electromagnetic Technology Division, National Institute ofStandards and Technology, b) Boulder, Colorado 80303

(Received 31 January 1989; accepted for publication 18 May 1989)

We have calculated the initial magnetization curves and complete hysteresis loops for hard

type-II superconductors. The critical-current density Jc is assumed to be a function of the

internal magnetic field H, according to Kim’s model, Jc (//, ) = k /{H0 + \Hi \ ), where k and

H0 are constants. As is the case for other critical-state models, additional assumptions are that

bulk supercurrent densities are equal to Jc , and that the lower critical field is zero. Our analytic

solution is for an infinite orthorhombic specimen with finite rectangular cross section, laXlb

(a^b), in which a uniform field H is applied parallel to the infinite axis. Assuming equal flux penetration from the sides, we reduced the two-dimensional problem to a one-dimensional

calculation. The calculated curves are functions of b /a, a dimensionless parameter

p = (2ka) ],2 /H0, and the maximum applied field Hm . The field for full penetration is Hp = H0 [ ( 1 + p

2

) 1/2 — 1 ]. A related parameter is H* = H0 [ ( 1 +2p

2 )' 12 — 1 ]. Hysteresis

loops were calculated for the different ranges of Hm :Hm <Hp ,Hp <Hm <H*, and H * <Hm . The equations for an infinite cylindrical specimen of radius a are the same as those for a specimen with square cross section, a = b. In the limit p4 1 and a = b, our results reduce to those of the Bean model ( Jc independent of H, ) for cylindrical geometry. Similarly, in the

limit p4 1 and 6— oo, the results are the same as those for a slab in the Bean model. For

H> 1 . 5 Hp

, or H > 0 when p 4 1 , the width of the hysteresis loop AAf may be used to deduce Jc as a function of H. Jc (H) = NM{H)/[a( 1 — a/lb)].

I. INTRODUCTION

To derive magnetic properties of hard type-II superconductors, Bean 1

: 2 and London3 introduced what has come to be known as the critical-state model.4-5 The model assumes

that penetrated supercurrents flow with a density equal to the critical-current density JC (H,), where H, is the local internal field. The flux vortex array is stable and there is no

flux creep. The lower critical field is zero. In Bean’s model,

Jc (//, ) was considered to be a constant independent of //, . Since then, several different Jc (Hi ) functions have been proposed. Kim, Hempstead, and Stmad4-5 assumed that

Jc (Hi )=k/{H0 +\Hi \) , (1)

where k and H0 are positive constants ( Kim’s model ) . Watson

6 considered a simple linear function,

Je (H,) =A-C\H,\ , (2)

where A and C are positive constants (linear model). Irie and Yamafuji7 and Green and Hlawiczka8 proposed a power-law model:

Jc (Hl ) = k x \Hl \~\ (3)

where k , and q are positive constants. Fietz et al. 9 and Karasik, Vasil’ev, and Ershov 10 proposed an exponential-law

model:

JC {H,) =A, exp( - \H,\/CX ) , (4)

in which A , and C, are positive constants. In principle, the initial magnetization curve and hyster-

“ ’ Permanent address: Department of Solid State Physics, Royal Institute of Technology, S100-44 Stockholm, Sweden.

bl Formerly the National Bureau of Standards.

2489 J. Appl. Phys. 66 (6), 15 September 1989

esis loops of superconductors can be derived for every model

mentioned above. Kim et al.

5 obtained two sections of the

high-field loop for cylinders. Hulbert" solved for the initial curve and high-field loop for cylinders. Fietz et al. 9 derived

the initial curve and high-field loop for an infinite slab using

the Kim model and a nonzero lower critical field. Watson6

derived the initial curve for a cylindrical sample and calculated the loop for low fields. Irie and Yamafuji7 derived the

high-field loop for a slab. Ohmer and Heinrich 1

2

and Wollan

and Ohmer 13 derived the initial curve and the low-,

medium-, and high-field loops for a cylinder, for H0 = 0 in Kim’s model, and q = 1 in the power-law model. Karasik et

al.

]0 and Ravi Kumar and Chaddah 14 gave analytic solutions for the initial curves for cylinder and slab samples,

respectively, using the exponential-law model. The latter also gave numerical solutions for the hysteresis loops. In this paper we use Kim’s model to analytically derive

both the initial magnetization curve and the hysteresis loops

for an orthorhombic sample. The equations are somewhat

complicated because there are two constants in the expression for Jc {H, ) and the sample shape is not simple. The

loops may be of three types, depending on the value of the

maximum applied field. In addition, each curve has several

stages.

One of the motivations for using Kim’s model for the

derivation is that, of the models listed above, this one is quite

general. It subsumes the linear model when H0 ^>Hif and

Bean’s model when both k and H0 become infinite in such a way that k /H0

is a constant. It becomes a power-law model

for q = 1 if H0 = 0. A practical motivation is that, by using

Kim’s model, we can more accurately predict the magnetic

properties of superconductors for a realistic orthorhombic

2489

48
geometry. The physical significance of H0

is discussed by

Hulbert. 1

1

The paper is organized as follows. Section II gives the

general equation of magnetization for the orthorhombic ge- ometry. Section III derives the magnetization as a function

of field for different field ranges. Expressions are obtained

for the initial curves and the hysteresis loops. Section IV uses the equations to generate hysteresis loops for several cases. In Sec. V, the formulas are simplified for a general orthorhombic Bean model. In Sec. VI, it is shown how/,. ( H) may

be obtained from the width of the hysteresis loop.

II. GENERAL EXPRESSIONS FOR MAGNETIZATION

A. Orthorhombic geometry

We consider an infinitely long orthorhombic sample

with cross section 2ax2b {b^a). The boundaries of the

sample are at x = ±a and y = + b. An external field H is applied along the z axis. The configuration of the sample and

field direction are shown in Fig. 1(a). In this configuration,

the magnetic quantities have only z components, and the

supercurrents have only x and y components.

The critical-state model involves only macroscopic su- percurrent, magnetization, flux density, and field. The lower

critical field is assumed to be zero. The local internal field H,

is defined as H, = B //u0 , (5)

where B is the macroscopic local flux density and // 0is per- meability of free space. If H is the applied field and M, is the

local magnetization, that is, the field produced by bulk su- percurrents, we have

Mi = Ht — H

.

( 6 ) The total magnetization M is the average of M, over the

sample cross section.

Solutions for an infinitely wide slab or an infinitely long

cylinder involve only one variable of integration. We have to consider a two-dimensional problem for an orthorhombic

sample. Fortunately, this two-dimensional problem can be

simplified to a one-dimensional calculation. Because the

sample is infinitely long and is located in a uniform H, both

Ht and Jc (i/, ) along the sample surface must be the same on

each side. Furthermore, the supercurrents penetrate the same depth into the sample from each side. The supercurrent

path is the rectangular circuit shown in Fig. 1(a). For an

(a) (b)

FIG. 1. (a) Sample configuration, (b) Supercurrent path.

infinitely long sample, the supercurrent density J, H{ and

M

,

along a given circuit are uniform and are written as functions of x: /(at), Ht (at), and M, (x).

To obtain the total magnetization M, we have to inte- grate M,(oc) over the cross-sectional area. Because Mis an

average over the sample and because of symmetry, it is suffi- cient to take only the first quadrant (at > 0, y > 0) into consideration. In this case the area is ab, and the differential area element is (2ot' + b — a)dx' , as shown in Fig. 1(b). Thus,

M = —— f (2x' + b — a)Ml (x') dx'

.

(7)

ab Jo In this equation we use primes to denote the variable of inte- gration. In the remainder of the paper, the primes are omitted in the integrations.

B. Extension to cylinders and other geometries

In the limit b /a — 1 , Eq. ( 7 ) for the average magnetization applies to a specimen with square cross section:

M = -^ f xMt (x) dx . (7a)

a~ Jo The same expression applies to an infinite cylindrical specimen of radius a, where the area is ira

2 and the differential area element is 2nx dx. Consequently, when b is set equal to a, the M{H) curves derived below are for either square

specimens, with cross section 2aX2a, or cylindrical specimens, with radius a.'

5 The principle of equal supercurrent penetration from

each side allows us to extend these arguments to samples

with cross sections in the shape of triangles and polygons.

The only requirement is that all sides are tangent to a circle

of radius a. A technologically useful example is a regular

hexagon.

III. DERIVATION OF M(H) FROM J(x)

A. General expression for J(x)

To obtain M we have to first derive the supercurrent

density J(x). Using Ampere’s law and Eq. ( 1 ), we have

dHj_

dx

sgn (J)Jc(H,) - sgn {J)k

H0 + sgn (H,)H,

( 8 )

where sgn is the sign function, equal to + 1. From Eq. (8),

J

[H0 + sgn(Hl )H,]dHi = - Jsgn (J)kdx. (9)

After integration, the solution for the quadratic in H,, in a region where H, and J do not change their signs, is H, = - sgn (H,)H0 ± [Hi - sgn(JH,)2k(x + c )]

1/2

,

( 10 ) where c is an integration constant to be determined by the

boundary conditions. Multiplying Eq. (10) by sgn (//,), we

obtain

H0 + sgn {Ht ) Hi = ± sgn(i/,

) [H o — sgn(Ji/, )2k(x + c)

] 1/2 (11)

2490 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989 D. -X. Chen and R. B. Goldfarb 2490

49
We set ± sgn(//, ) = 1 on the right-hand side because, from

Eq. ( 1 ), the left-hand side is always positive. Using Eqs. ( 1

)

and ( 1 1 ) we obtain

J(x) = sgn (J)Jc (Hl ) = sgn(/)/c/[/70 + sgn (//,-)//,-] = sgn(J)k /[H 20 — sgn(JH, )2k(x + c)

] 1/2

. (12)

This is the general expression for J(x).

B. Initial M(H) curve and full-penetration field

1. Current densities and local fields

We start from the initial state, H — M = 0, and increase

H in the z direction. According to Lenz’s law, the supercurrent J (of negative sign) will penetrate from the surface

(x — a) inward. If the supercurrent penetrates until x = x0, H, in the sample will be H at x — a, decrease to 0 at x = x0, and remain 0 for x<x0

. Figures 2(a) and 2(b) show the

J(x) and H,(x) functions, represented schematically by

straight-line segments. If H increases further, x0 decreases.

When x0 = 0, the sample is completely penetrated [Figs.

2(c) and 2(d)]. The corresponding field is called the full- penetration field Hp

. In the complete penetration state, J(x)

and//, {x) have similar forms, shown in Figs. 2(e) and 2(f).

For the initial magnetization curve, where the field is

first applied to the sample, we will denote/(x) as./0(;c). We

derive the supercurrent density J0 (x) for x0 <x<a. The

boundary condition is J0 (a)= -JC (H) . (13)

Substituting Eqs. ( 1 ) and (12) into Eq. ( 1 3 ) , we have

[Hq + 2k(a + c)]' 12 = H0 + H

,

(14)

from which

2kc = (H0 -f H) 2 — Hi — 2ka . (15)

Substituting Eq. (15) into Eq. ( 12), we obtain

J0 (x) = — k /[(H0 + H) 2 — 2k(a — x)] U2

( x0 <x<a).

(16)

We consider the magnetization for two stages:

0 <H<Hp (stage I) and Hp <H (stage II).

J* J

(a) (c) (e)

Hj H, Hi

(b) (d) (f)

FIG. 2. Schematic supercurrent density J and local internal field H

,

as functions of x for the initial magnetizing process. For purposes of illustra- tion, J and H, are sketched as straight line segments.

2491 J. Appl Phys., Vol. 66, No. 6, 15 September 1989

2. Stage / (0<H<Hp) In this range, x0 decreases from a to 0, and

Hj(x) — 0 (0<x<xo ) , (17a)

H,(x) = // + J

J0 (x) dx (x0 <x<a). (17b)

From Eqs. ( 17a), ( 17b), and (6) we obtain

Mi (x)=—H (0<x<xo ) , (18a)

M,(x) = J

J0 (x) dx (x0 <x<a) . (18b)

From Eqs. (17b) and (16), pc0

is determined: x0 = a-[(H0 + H) 2 -H20 ]/2k, (19)

using the boundary condition

H,(x0 ) = 0. (20)

Substituting Eqs. (18a), (18b), and (16) into Eq. (7), and

using Eq. ( 19), we obtain the final result: M(H) = — Hx0 (b — a + x0 )/ab — S{ (b + Xq) ( a - x0 )/ab

+ (S]-H i0 )Ql + 2H i0 (a-x0 )/5 kab

(0 <H<Hp ), (21)

where

S^Ho + H, (22a)

£?, = [5k(a + b) -2S]]/\5abk 2

. (22b)

For the case a = b, applicable to cylinders of radius a, Eq. (21 ) reduces to Hulbert’s Eq. ( lOd), with appropriate

symbol substitutions of H for B." In the limit b-> oo, for

infinite slabs of thickness 2a, Eq. (21 ) reduces to the solution of Fietz et al., Eq. (b) in Table I of Ref. 9.

3. Full-penetration field Hp When x0 in Eq. (19) becomes 0, the sample is completely penetrated [ Figs. 2 ( c ) and 2(d)]. Setting x0 = 0 so that,

by definition, H = Hp in Eq. ( 19), we obtain

Hp = {Hi -f 2ka) V2 — H0 . (23)

4. Stage II (Hp <H)

In this range [Figs. 2(e) and 2(0], Eqs. (7), (16),

(17b), and (18b) are still valid with x0 replaced by 0. The

final result is M(H) = -St + R])Ql +2R ]/5 kb

(.

H

p< H<Hm ), (24)

where

R, = (S] -2ka) U2 (25)

C. Hysteresis loops for the low-Hm case (Hm <HP ) To obtain hysteresis loops we have to derive reverse M(H) curves from a given maximum field Hm on the initial curve. The reverse M(H) curve starts from ( Hm ,Mm ) and

ends at ( — Hm , — Mm ), forming the descending branch of

the hysteresis loop. The ascending branch will then be

D. -X. Chen and R. B Goldfarb 2491

50
— — H). The equations governing the shape of the hysteresis loops depend on Hm . The first case is for Hm <Hp

, where the specimen is never fully penetrated. The second

case is for H * < Hm , when the reverse supercurrent penetrates to the center of the specimen before His cycled back to

zero. (The expression for /f * is derived in Sec. Ill D 2.)

The third case is intermediate, Hp < Hm <H*.

1. Current densities and local fields

The low-Hm reverse M(H) curve starts from a point on

stage I of the initial curve, Hm <Hp

. Figures 3 ( a

)—3 (j

)

show the J{x) and H,(x) functions developing with decreasing H. Figures 3(a) and 3(b), similar to Figs. 2(a) and

2(b), correspond to the starting point with H = Hm and a negative supercurrent penetrating to xm . When H decreases

from Hm , the induced supercurrent with positive J will penetrate from the sample surface tox = x„ and the corresponding J(x) and H,(x) are shown in Figs. 3(c) and 3(d). At

this point, /(x) forxm <x<x, [denoted as Jm (x) ] remains

the same as the J0 (x) for the initial magnetization when

H = Hm . J(x) is /,(x) for x, <x<c. This arrangement is maintained until H = 0, as shown in Fig. 3(e) and 3(f).

Further decreasing H to a negative value complicates the

situation. As shown in Figs. 3(g) and 3(h), the expression

for J(x ) has to be divided into three parts: Jm (x),J2 (x), and

y3 (x), and the corresponding H, changes sign at x = x3. This situation lasts until H = — Hm , when both Jm and J2 are removed. Figures 3(i) and 3(j) show/(x) and H, (x) for

Xm «

(a)

J J

1

Ji

Xm n Xm

X

1

1 x 0

]•

XCl Xn

Hi. Hi Hm - -y^

Xm « X

(b)

(c)

_Z!L

Xm a x

(d)

Jk J 3

Xm ^3

x, a

1

Jm

(e)

Xm a X

(f)

Xm a

(g) (i) H,

*5*3- 0 -H m

Xm a

(h) (j)

FIG. 3. Schematic supercurrent density and local internal field as functions of x for the reverse magnetizing process from Hm < Hp

.

H = — Hm ; they are opposite to the case shown in Figs.

3(a) and 3(b).

We give the expressions for Jm (x), /, (x), J2 (x), and

73 (x). Because Jm (x) =/0 (x) when H = Hm , we obtain

from Eq. ( 16):

Jm (x) = -k/[(H0 + Hm ) 2 -2k(a-x)]' n

(xm < X < X , )

. The boundary condition for7,(x) is

(26)

J\(a) = Jc (H) . (27)

Substituting Eqs. ( 1 ) and (12) into Eq. (27) we have

[Hi -2k(a + c)] U2 = H0 + H,

from which

(28)

2kc = - (//0 + H) 2 + H20 - 2ka .

Substituting Eq. (29) into Eq. (11) we obtain

(29)

y,(x) = k/[ (H0 -f H) 2 + 2k(a - x)] 1/2 (x, <x<a) .

(30)

The boundary conditions for/2 (x) and/3 (x) are y2 (x3 ) = Jc (0)

and

(31a)

y3 (a) = Jc (H) . By a similar derivation, we obtain(31b)

y,(x) = k /[Hi 4- 2k(x3 — x) ]

1/2 (x2 <x<x3 ) and

(32a)

y3 (x) = k/[(H0 - H) 2 — 2k(a — x)] 1/2 (x3 <x<a) . (32b)

The magnetization process can be divided into two

stages. Stage I corresponds to 0 < H < Hm , and stage II cor- responds to — Hm < H < 0.

2. Stage i (0<H<Hm) In stage I, H, (x) is

H,(x) = 0 (0 <x <xm ) , (33a)

//,(x) = Hm + /m (x) dx (xm < x < x | ) , (33b)

Ht (x) = H + f°y,(x) dx (x ! < x < u ) . (33c)

From Eqs. (33a), (33b), and (33c) and Eq. (6), Af,(x) is obtained as M( (x)=—H (0 <x <xm ) , (34a)

3/,(x) =Hm -H +

^

J

m {x)dx (xm <x <x, ) , (34b)

Mt (x) = J

Jl (x)dx (x, <x <a) . (34c)

xm can be determined by replacing H in Eq. (19) with Hm : x„ =a- [(Ha + Hm )

2 -H 20 ]/2k. (35a)

Using the continuity of H, at x = x,, and from Eqs. (33b)

and (33c), we obtain

2492 J. Appl. Phys., Vol. 66. No. 6. 1 5 September 1 989 D. -X. Chen and R. B. Goldfarb 2492

51
xt =a- [(Hn + Hm ) 2 -(H0 + H) 2 ]/4k. (35b)

Substituting Eqs. (34a), (34b), (34c), (26), and (30) into

Eq. (7), and using xm and x, defined by Eqs. (35a) and

(35b) we obtain the final result: M(H) — — Hxm (b — a + xm )/ab

-S'1 [o6+ (a-b)xm -x2m ]/ab

-(S\-R\)Q2 +{R\-Hl)Q, + 2[(x, —a)R \ + (x, - xm )H 30 ]/5kab

(0 <H<Hm ), (36)

where

S2 = H0 + Hm , (37a)

R2 = [Si + 2k(a — x,)] 1/2 (37b)

Q2 = [5k(a + b) + 2S]]/I5abk 2

, (37c)

Q? = [ 10A’(a + b) +3S? - IS

\

] /30abk 2 (37d)

3. Stage //(-Hm <H<0)

In stage II, H, (x) and M, (x) are

H, (x) = 0 (0 <x <xm ) , (38a)

H,(x)=Hm +

J

Jm (x)dx (

x

m < x <

x

2 ) , (38b)

Hi (x)=H + J2 (x)dx+ f J3 (x)dx (x2 <*<*3 ) » J X X 4 (38c)

H,(x) =H +

J J3 (x)dx (x3 <x<a) , (38d)

and

M, (x) = -H (0 <x <xm ) , (39a)

M,(x) =Hm --// + J

Jm (x)dx (xm < x <

x

2 ) , (39b)

M, (x) = J

J2 (x)dx + ^

J3 (x)dx (x2 < x <

x

3 ) ,

(39c)

Af,(x)=J J3 (x)dx (x3 <x<a) . (39d)

xm is given in Eq. (35a), and x3 and x2 can be derived using

the continuity conditions at x = x3 and x2 from Eqs. (38d),

(38c), and (38b):

x3 = a- [(HQ -H)2 -H20 ]/2k, (40a)

x2 — Q \(H0 — H) 2

-\- (H0 + Hm ) 2 — 2H\ ]/4k . (40b)

The final result is obtained in the standard way used above:

2493 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989

M(H) =- Hxm ( b-a + xm )/ab + S3 (a - x3 )(b + x3 )/ab

S, (x3 x2 ) [ 2 (b — a x2 ) + x3 + xm ]/ab

-(Si - Hl)Q4 + (Rl - Hl)(Qs + Q6 ) + 2 [ (x3 — a)H o - (x3 -x2 )(^ l -H30 )]/5kab

(~Hm <H< 0) , (4i)

where

S3 = Hq — H , (42a)

^3 = [^o + 2k(x3 -x2 )]

1/2

, (42b)

Q4 = [5k(a + b)-2Sl]/\5abk 2

, (42c)

05 = [5A:(fl + 6 ) -55^ +1H 20 ]/I5abk 2

, (42d)

Qb = [\0k(a + b) -35^ -1S\ +6H 2o ]/30abk 2

. (42e)

D. Hysteresis loops for the high-/ym case (H% <Hm )

1. Current densities and local fields

For Hm > HP we have to consider two cases. In the highHm case, the reverse supercurrent completely penetrates to the sample center before H has decreased to 0. This corre- sponds to Hm >H*m — (H 20 + 4ka)' /2 — H0 as shown below. The second case is for medium Hm , where the reverse supercurrent is not completely penetrated when H = 0. This

will be discussed in Sec. Ill E. J(x) and H, (x) for the high-//m case are shown in Figs.

4(a)-4(n). Figures 4(a) and 4(b) correspond to the start- ing point when H — Hm and the sample is completely penetrated by negative supercurrent with density Jm (x). Decreasing H from Hm induces a positive supercurrent with

density J\(x). As can be seen in Figs. 4(c) and 4(d), the

reverse positive supercurrent penetrates to x — x,, at which

point H = Hprh , the full reverse penetration field for the

high -Hm case, defined below. This is stage I. Stage II starts when x, becomes 0, shown in Figs. 4(e)

and 4(f), and ends when H reaches 0, shown in Figs. 4(g)

and 4(h). After that comes stage III, in which the local in- ternal fields at the center and the surface of the sample have

different signs, corresponding to different functions for J2 (x) and J3 (x), bounded by x = x3 , where H, = 0. This

situation is shown in Figs. 4 ( i ) and 4 (j ) . Further decreasing

H results in the situation of Figs. 4(k) and 4(1), when the

whole sample has negative H,, and the magnetization process enters stage IV. In stage IV, 7(x) keeps the form of

y3 (x) until H— — Hm , when the process ends. The final J(x) and H, (x) are shown in Figs. 4(m) and 4(n).

The expressions for Jm (x), /,(x), J2 (x), and J3 (x)

have already been given in Eqs. (26), (30), (32a), and

( 32b ) , and we need only change the regions ofx, referring to the figures as described above.

2. Stage / (Hprh <H<

H

m) In stage I, H, (x) and A/, (x) are

H,(x) = Hm 4- Jm (x)dx (0 <x <x,

)

, (43a)

D. -X. Chen and R. B. Goldfarb 2493

52
J

FIG. 4. Same as Fig. 3, from H,„ >//*,.

H,(x)=H+[ J\{x)dx ( x,<x<a ), (43b)

and

M,(x) = Hm - H + J

Jm (x)dx ( 0 <x<x,),

(44a)

A/,(x)=J J, (x)dx (x,<x<a). (44b)

x, is given in Eq. (35b), and the final result is M(H) = -Sl -(S] -R 32 )Q2 +{R 32 -Rl)Q3 + 2[ (x, — a) R\ -f x,R 4 \/5kab

(Hprh <H<Hm ) , (45)

in which

R4 = (S\ -2ka)' 12

, (46)

and Hprh is the reverse full-penetration field for the high -Hm

case. It can be determined by taking x, = 0 in Eq. (35b):

Hprlt = [(H0 + Hm f-4ka] U2 -H0

. (47)

The boundary between the high- and medium-//^ case can

also be determined from Eq. (35b) by taking x, = 0 and

H = 0:

H* = (

H

20 +4ka) l,2 -H0

. (48)

In the limit b—oo, Eq. (45) reduces to Eq. (d) in Table I, Ref. 9, with the typographic correction

“...{(•v/2/2) [

... + 22? £ ]

3. Stage // (0<H<Hprtt) In stage II, the H( (x) and M, (x) are

i/,(x)=^+J /1 (x)dx (0 <x<a), (49)

3/,(x) = J

Jt (x)dx (0 <x<a). (50)

x, is given in Eq. (35b), and the final result is M{H) = — S, — (S] — R l )Q2 — 2R l/5kb

(0 <H<Hprh ), (51)

where

R5 =(S] +2ka) ] ' 2. (52)

For the case a = b, applicable to cylinders of radius a, Eq. (51) reduces to Kim’s solution for M(H) in the first quadrant, Eq. ( 14) in Ref. 5, and to Hulbert’s solution, Eq.

(10b) in Ref. 11. In the limit b— oo, for infinite slabs of

thickness 2a, Eq. (51) reduces to the solution of Fietz et al., Eq. (e) in Table I of Ref. 9.

4. Stage III (—Hp <H<0)

In stage III, the H, (x) and M, (x) are H

j

(x) =//-)-

J* y2 (x)^x +

J

Ji(x)dx (0 <x <x3 ) ,

and

,(x)=//-tJ

J^x)dx (x3 <x<o),

(53a)

(53b)

J2 {x)dx + f J2 {x)dx (0<x<x3 ), (54a)

M,(x) =

J Jy {x)dx (x3 <x <a) . (54b)

x3 is given in Eq. (40a), and the final result is M{H) = S3 (a — x2 ){b + x2 )/ab — 5l x3 (6 — a + x2 )/ab - (Si - H30 )Q4 - (Hi - Rl)Q5 - 2[ (a —x^Hl + x3 R l]/5kab

(~Hp <H< 0) , (55)

where

Rb =(H 20 +2kx3 )' 12

. (56)

M,(x) -f

The lower H boundary in this stage is — Hp

, as can be seen by taking x3 = Oin Eq. (40a). For the case a = b, applicable

to cylinders of radius a, Eq. (55) reduces to Hulbert’s Eq.

( 1 0c ) , with appropriate symbol substitutions ofH for B.

11 In the limit b-+ oo, Eq. (55) reduces to Eq. (g) in Table I, Ref.

9, with the typographic correction

“-(B„+ H.) + ...{[... - ( B0 - H0 ) 2 + ...]

3 ' 2 - ...} + ... .”

2494 J. Appl. Phys., Vol. 66, No. 6, 15 September 1989 D. -X. Chen and R. B. Goldfarb 2494

53
5. Stage IV (-Hm <H<-Hp) In stage IV, the //, (x) and Mi (x) are

H,(x) = // + J* Ji (x)dx (0 <x<a) (57)

M,(x) = (58) JJi (x)dx (0 <x<a),

and the final result is M(H) = S3 — (5 3 — R])Q4 — 2R ]/5kb

{-Hm <H< -Hp

) , (59)

where

R,= (S\ -2ka) x ' 2 (60)

To generate the complete hysteresis loop, stages I-IV

are reflected about the origin onto the third and fourth quadrants, as indicated at the beginning of Sec. Ill C. For example, Eq. (59) in the second quadrant becomes

M(H) = -St + (S\ -R\)Qt + 2R ]/5kb

( HD <H<Hm ) (59a)

in the fourth quadrant, in which H and M(H) are replaced

by — H and — M(H). Equation (59a) is the same as Eq.

(24) for stage II of the initial curve. For the case a = b, applicable to cylinders of radius a, Eq. (59a) reduces to Kim’s solution for M(H) in the fourth quadrant, Eq. ( 14) in Ref. 5, and to Hulbert’s solution, Eq. (10a) in Ref. 11. In the

limit 6-* oo, for infinite slabs of thickness 2a, Eq. (59a) re- duces to the solution of Fietz et al., Eq. (c) in Table I of

Ref. 9.

E. Hysteresis loops for the medium-//^ case (Hp <Hm <H*m )

1.

Current densities and local fields

For the medium-7/m case, the reverse magnetization

process is shown in Figs. 5 ( a )-5 ( n ) . The difference between

Figs. 5(a)-5(n) and Figs. 4(a)^l(n) is only for stage II. At

the end of stage I [ Figs. 5(e) and 5 ( f)],// = 0, but x , is still greater than 0. Therefore, in stage II, when H < 0, J(x) has

to be divided into three parts: Jm (x), J2 {x), and 73 (x),

shown in Figs. 5(g) and 5(h). The next stage starts from the

point where x2 = 0, as shown in Figs. 5(i) and 5(j). The

expressions for J(x) are the same as in the high-Hm case.

2.

Stage / (0<H<Hm) This stage is the same as for the high -Hm case except for the field interval, which is from Hm to 0 here.

3.

Stage // (Hprm <H<0)

In stage II, H, (x) and M(x) are

Hi(x) = Hm + j

Jm (x)dx (0 <x <x2 ) , (61a)

a x h

J

a x x, a x 1

(a) (c) (e) (g)

Hi. Hs H, Hi. Hj—

a x /i\. 0 x,a x

0 x, a x 0 i^Tx

a

(d) (f) (h)

•*2 4} J. Ji ^1

V

X 3 x2\ X ® { ’x 0 a

(i) (k) (m)

H,

a prm

ni ‘t

sX3 a

. n

a

. n

x ° -HP

\ ; x

0

-Hm -

(j) 0) (n)

FIG. 5. Same as Fig. 3, from Hm between H and H*„

£'/2 (x)^x + £

Ht (x) = H + I J2 (x)dx+ j J2 (x)dx (x2 <x<x3 ),

(61b)

H,(x)=H + j

J3 (x)dx (x3 <x <a) ,

and

(61c)

M, (x) = Hm — H +

J

Jm (x)dx (0 <x <x2 ) , (62a)

M,(x) — JJ2 (x)dx + J* J3 (x)dx (x2 <x <x3 ) , (62b)

M,(x) = JJ3 {x)dx (x3 <x < a) . (62c)

x2 and x3 are given in Eqs. (40b) and (40a), and the final result is M{H) = S3 (a — x3 )(6 -f x3 )/ab — Si [x3 + (b — a)x3 ]/ab - (S\ - H 30 )Q4 - (Hi - R\)QS + (R \ - R

34 )Q6 — 2[ (a — x3 )//q + (^3 — x2 )R 3 — x2R 4 ]/5 kab

(Hprm <H< 0), (63)

2495 J. Appl. Phys., Vol. 66, No. 6, 15 September 1989 D. -X. Chen and R. B Goldfarb 2495

54
where Hprm is the reverse full-penetration field for the medium-Hm case, which can be determined by taking x2 = 0 in Eq. (40b):

Hptm = H„-[4ka + 2Hi - (fl0 + Hm ) 3

]

1,2

. (64)

4. Stage lll(-Hp <H<Hpm)

This stage is the same as the high -Hm case except for the

interval of H, which is here from Hprm to - Hp

.

5. Stage /V(-Hm <H<-HP)

This stage is the same as the high -Hm case.

IV. COMPUTED M(H) CURVES

We have analytically verified, for each case in Sec. Ill, that the stages are continuous at their end points. To illus- trate the formulas in Sec. Ill, we give some computed M{H)

curves. To reduce the number of variables, we define a new

parameter, similar to one used by Kim5

: p= (2ka)' l2 /H0

. (65)

Equations (23) and (48) can be rewritten as Hp =H0 [(\+p2 )' l2 -\], (66)

H*m =H0 [(\+2p2 )' l2 -\]. (67)

The shapes of the M(H) curves are now determined by p

and b/a. Figures 6(a)-6(e) give the initial and hysteresis

M(H) curves for b/a = 1 and p = 0.3, 1, 3, 10, and 1000.

For each case, five M(H) loops are drawn for Hm — Hp /2, Hp

, ( Hp + H* )/2,//* , and AHp

. The curves in Figs. 7(a)

and 7(b) give the initial and hysteresis M(H) curves for p = 1 ,Hm = H* and 4Hp

, and b /a — 1 (smallest), 1.5, 2, 5, and 100 (largest). For all the curves, M and //are normalized to Hp

. We can observe from Figs. 6(a)-6(e) that the M(H)

curves derived from Kim’s model have a wide variety. The

curves shown in Fig. 6(a) are very similar to those derived

from Bean’s model. 2 Ifp were smaller than 0.1, there would

be virtually no difference between Kim’s and Bean’s models.

Figure 6(e) for p = 1000 is almost the same as for the limit- ing case p-+ oo, where Kim’s model reduces to Jc (H,)=k/H,.

The initial curves have minima except when p = 0. We

can see from Figs. 6(a)-6(e) that, with increasing p, the

field where the minimum is located decreases from Hp to

0.

56 Hp

, and the minimum M decreases from — 0.33 Hp to — 0.36 Hp

. The initial slopes of the initial curves are — 1. Forp = 1000, the initial curve is linear within 1 % up to 0.09

Hp

, despite the assumption of zero lower critical field in the

derivation. This linear region does not come from a Meissner

state, but is simply a reflection of a large Jc at low H,

.

The initial reverse slopes at the comers of the hysteresis

loops in Figs. 6(a)-6(e) are also — 1. This is a consequence

of shielding by surface supercurrent at the beginning of stage

1. Because Jc is lower, this 1 % linear region is smaller.

The second and the fourth loops correspond to the two

boundaries between the low-, medium-, and high -Hm cases. We can see from Figs. 6(a)-6(e) that, for the medium- and

high-Hm cases, the initial and the hysteresis curves merge

H/Hp

FIG. 6. Theoretical M-H curves, scaled by Hp

, for a sample with b/a — 1, for p = (a) 0.3, (b) 1, (c) 3, (d) 10, and (e) 1000. In each figure, loops are shown for H,n /Hp =0.5 (smallest), 1,4-+- //*,

/

2Hp

, H * /Hp

, and 4 (lar- gest).

2496 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989 D. -X. Chen and R. B. Goldfarb 2496

55
FIG. 7. Theoretical M-H curves, scaled by Hp

, for p= 1 and (a) Hm = H*,< (b) //,„ - 4Hp

. In each figure, curves are shown for samples

with aspect ratio b /a = 1 (smallest), 1.5, 2, 5, and 100 (largest).

when H^Hp

. The middle parts, around H — 0, are the same

for all the loops in the high-Hm case. With increasing b /a, the width of the normalized hysteresis loop increases asymptotically, as seen in Figs. 7(a)

and 7(b). One interesting feature in Figs. 7(a) and 7(b) is that, for each figure, the loops with the same p and Hm cross

at two points. This crossover effect is general for Hm >

For Hm — H * the crossing points are at H = ±Hp

, but for higher Hm the points are at higher fields. For a set of samples

with the same a and different b, when H — Hm , H,( 0)

should be the same for all the samples, because H, is a function only of x. When H is reduced from this Hm to a value

Hcro that equals this H,-( 0), x t will become a/2. Changing

the variable in Eq. (7) such that £ = x — a/2, we have

« /» 4- a/2 Mcro = —- (2£ + b)M,iS)di. (68)

ab J - a/2 Because A/,(£) is an even function of £, Eq. (68) can be

written as Mcto =-\ M.&di, (69) a J - a/2 independent of b. Since both Hcro and Mcro are independent

of b for constant a, the point ( Hcro ,Mcro ) must be a crossing

point, as illustrated in Figs. 7(a) and 7(b).

V. M(H) FORMULAS FOR ORTHOHOMBIC SAMPLES IN THE BEAN LIMIT

Bean derived the M(H) formulas for cylindrical add

infinite-slab samples for Jc independent of H,. In this paper,

we have obtained the M(H) formulas for the orthorhombic

geometry using Kim’s model for Jc , Eq. ( 1 ). In this section,

2497 J. Appl. Phys., Vol 66, No. 6, 1 5 September 1 989

we reduce the orthorhombic Kim formulas to general orthorhombic Bean formulas. Finally, we reduce these to the simple cylinder and slab forms.

A. Bean’s model for orthorhombic samples

Bean’s model can be written as JC (H,)=JC , (70)

where Jc is constant. To modify Kim’s model, Eq. ( 1 ), for

the Bean limit, let k- oo, (71a)

H0

-> oo , (71b)

k/H0 = Jc

. (71c)

All the formulas for Kim’s model can be reduced to the

corresponding ones for Bean’s model. For some cases, namely Eqs. (74), (75b), (77a), and (77b) below, it was neces- sary to use binomial expansions before taking the limits in Eqs. (71a), (71b), and (71c). For the terms raised to the

power 3/2 in Eqs. ( 24 ) , ( 45 ) , and (51), expansions has to be

carried to third order.

The general expression for the supercurrent density can

be obtained from Eq. (12):

J(x) =sgn(7)7c . (72)

For each specific case we have

Jo = Jm= -Jc (73a)

and

J{ =J2 = Ji = Jc

. (73b)

We do not have to identify J2 , and J2 because we do not need to separate J(x) into several sections, since Jc is constant. For the same reason, for every M(H) curve, only one

or two stages have to be considered, and for the reverse curves, only two cases have to be considered.

The full-penetration field can be obtained from Eq. ( 23

)

as Hp ~ Jc a • (74)

The initial M(H) curve can be derived from Eqs. (21 ) and

(24) as M(H) = - H+ (H 2/2Jc )(\/a 4- \/b) - H

3/(3J 2e ab)

(0 <H<Hp ), (75a)

M(H)= -Jc a^-a/6b) (HP <H). (75b)

Equations (75a) and (75b) may be found in Ref. 15. The

reverse curve for the low-Hm (0 <Hm <Hp

) case can be

derived from Eq. (36) as

M(H)= -H + [(H 2m +2HHm -// 2 )/4/f ](l/a+ 1/6) - (3H 3m +3H2m H-3Hm H 2 + H 3 )/(l2J 2c ab)

(~Hm <H<Hm ). (76)

The reverse curve for the high -Hm (Hp < Hm ) case can be

derived from Eqs. (45) and (51 ) as

D. -X. Chen and R B. Goldfarb 2497

56
M(H) = -Jc a{\- a/6b) + Hm - H

-[(Hm -H)2/4Jc ](\/a+\/b)

+ (Hm -H?/(\2J 2

c ab)

(Hm — 2Hp <H<Hm ) , (77a)

M(H) = Jc a(\-a/6b) ( - Hm <H<Hm -2Hp

) . (77b)

B. Reduction to simple geometries

For the infinite slab, b /a-* oo, the above equations for

M(H) become, for the initial curve, M(H) = -H+ H2/2Jc a (0 <H<Hp ), (78a)

M(H) = — Jc a/2 (Hp <H) ; (78b)

for the low-Hm (Hm < Hp

) curve, M(H) = -H+ (H 2m +2HHm -H2 )/\Jc a

( — Hm <H<

H

m ) ; (79)

and for the high-Hm ( Hp <Hm ) curve, M(H) = - Jc a/2 + Hm -H-(Hm - H) 2/\Jc a (Hm -2Hp <H<Hm ), (80a)

M(H)=Jc a/2 ( — Hm <H <

H

m — 2Hp

) . (80b)

For the cylinder, b /a = 1, the corresponding equations

are, for the initial cruve, M(H) = — H + H2/Jc a — H3/3(Jc a) 2 (0 <H<Hp ),

(81a)

M(H) = -Jc a/3 (Hp <H) ; (81b)

for the low-Hm (Hm <Hp

) curve, M(H) = -H+ (H 2m + 2HHm -H2 )/2Jc a - (3Hi + 3H2m H- 3Hm H

2 + H*)/U(Jc a) 2 (-Hm <H<Hm

)- (82)

and for the high-Hm ( Hp <Hm ) curve, M(H) = -Jc a/3+Hm -H

- (Hm - H) 2/2Jc a + (Hm - H) 3/\2(Jc a) 7 (Hm -2Hp< H<Hm ), (83a)

M(H)=Jc a/3 ( — Hm <H<

H

m — 2Hp

) (83b)

Equations (78a), (78b), (79), (81a), (81b), and (82) for

slabs and cylinders are the same as Eqs. (2), ( 3 ) , and ( 5 ) in Ref. 2. Bean did not treat the high-Hm case, only the initial curves and the low-Hm case.

C. Hysteresis loss

Although this paper deals with magnetization curves

rather than hysteresis loss W, it is straightforward to calculate energy loss per unit volume per field cycle for orthorhombic samples in the Bean limit. Generally,

W= fj, 0 (j) HdM~fi0 (j) MdH . (84a)

For the complete low-Hm loop, from Eq. (76),

2498 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989

W= 2+IqH i [Jc (a + b)- Hm ]/(3J\ab) . (84b)

For the complete high-/fm loop, from Eqs. (77a) and (77b),

W= 2fi0Jc a[Jc a(a - 2b) + Hm (3b - a)]/3b . (84c)

VI. DETERMINATION OF JC (H) FROM THE WIDTH OF

THE HYSTERESIS LOOP

In this section we discuss two topics. First, we examine

the necessary conditions for using Bean’s model to determine Jc from hysteresis loop measurements assuming that

Jc (H, ) actually follows Eq. ( 1 ). Second, we offer a modification of the conventional cylinder and slab formulas for Jc determination for the general orthorhombic Bean model.

A. Prerequisites for Bean formulation

The traditional way to determine critical-current density of superconductors from magnetic measurements is based on Bean’s model, where Jc is considered constant. To

determine JC (H), a hysteresis loop is obtained, and the

width of the hysteresis loop at a given field, t^M(H), is measured. Bean’s model2 gives

Jc (H) = lbM(H)/2a (85a)

for cylinders of radius a, and

JC (H) = LM(H)/a (85b)

for slabs of thickness 2a. Note that a field-dependent Jc is contrary to the assumption used to derive the Bean equations.There are two requirements for using Eqs. (85a) and

(85b) for Jc determination if Jc (i/, ) is assumed to actually

follow Kim’s model. ( 1 ) The magnetization on ascending

and descending branches of the hysteresis loop at a given H

must correspond to fully penetrated states. (2) The maximum deviation of JC (H,) in the sample from Jc (H, = H

)

must be small. H is the uniform applied field, and H, is the

local internal field. The notation Jc ( H, = H) means Jc (H,

)

for Ht = H These conditions will be expressed in terms of

recommended values of Hm , H, and p.

1. Fully penetrated states

For fully penetrated states, shielding currents circulate

in only one sense throughout the volume of the specimen for

the upper branch of the hysteresis loop and in the opposite

sense for the lower branch. The condition of full penetration

for both branches of the hysteresis loop is satisfied for the

high-Hm case (Sec. Ill D) when

Hm >H*m (86)

and

H<Hprh . (87)

If Hm = Hi, then Hprh = 0, and the condition is satisfied only for H = 0. To obtain a useful H interval, Eq. (86) is restricted to Hm >H*m . (86a)

As examples, we obtain J(H), using Eq. (85a), from the

major hysteresis loops in Figs. 6(a)-6(e) (for which

Hm 5>H * ). The symbols in Figs. 8(a)-8(e) give the/c (H)

D. -X. Chen and R. B. Goldfarb 2498

57
H/Hp, H,/Hp

FIG. 8. Comparison of Jc (//) [symbols, obtained from the major loops in Figs. 6(a)-6(e) using Eq. (85a)] with JC (H,) [smooth curve, obtained from Eq. ( 1 ) which was also used to compute the loops in Fig. 6] . Hand H,

are normalized to Hp

. Jc ( H) and JC (H,) are normalized to JC (H, = Hp).

2499 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989

thus obtained for p = 0.3, 1,3, 10, and 1000. For comparison, the smooth curves in Figs. 8(a)-8(e) give Je (H( ) derived from Eq. ( 1

) . The H scales are normalized to Hp

, and

the Jc scales are normalized to Jc (H, = Hp

). We can see from these figures that Jc (H) and Jc (//, ) overlap in a certain H interval, somewhat different for each case. The upper

limit of this correspondence interval is Hprh , Eq. (47), even

for hysteresis loops where Hm >4Hp

.

2. Uniform JC (H,)

Referring again to Figs. 8(a)-8(e), the lower limit of

overlap is about 1.5 Hp for most cases. For the case p 4 1 , Fig.

8(a), this limit is much smaller. The reason for the extended

overlap range is that Jc (H, ) is rather uniform for small p, so

that JC (H)~JC (H, =H).

We consider stages II and IV of the hysteresis loop (Sec.

HID). From Eqs. (51) and (59a), we obtain AM(H).

Dropping the higher-order terms and substituting into Eqs.

(85a) and (85b), we obtain

JC {H) =JC (H,=H){ 1 +5[<5i/,(i/)] 2 }, (88)

where

6Hi (H)=AHi (H)/(H0 + H) , (88a)

AH,(H)=aJc (Hl =H) , (88b)

and where s= l/4for6/a-» oo (infinite slab), and s = 1/20

for b /a — 1 (cylinder).

The field change relative to (H0 + H) in the sample is given by Eq. (88a). AH,(H) defined by Eq. (88b) is the

first-order difference between H,( 0) and H,{a). From Eq.

(88), JC (H) derived from AM(H) is always greater than

JC (H,=H). The difference between JC (H) and

Jc (H, = H) is determined by SH, (//), decreasing with decreasing a and Jc (Ht ) and increasing H0 and H. From the

factor s, the error in Jc ( H) is much smaller for a cylinder

than for a slab with the same a. When 0, we have to consider stages II and III of the

loop. The requirements for small errors are the same as for larger H except that Jc (H) is always less than Jc (H, = H).

When p4 1, H0is very large, and the error in Jc (H) is very

small, even at H = 0.

B. Formula for orthorhombic samples

AM{H) in Eqs. (85a) and (85b) is the vertical width of

the hysteresis loop. In Sec. IV, we showed that the vertical

width of the hysteresis loop increases with increasing b /a. If we use Eq. ( 85b) to calculate J(H) for the samples with the same a but different b, different Jc will be obtained even for the same superconductor material. Thus, we derive a for- mula for orthorhombic samples using Bean’s model. In the

penetrated state, using Eq. (18b) with x0 = 0 and

J(x ) = — Jc , we obtain

Mj(x) = — Jc {a — x) . (89)

Substituting Eq. (89) into Eq. (7), or simply subtracting

Eqs. (77b) and (75b), we have

&M = Jc a(\-a/3b). (90)

Rearranging, we obtain the general formula

D. -X. Chen and R. B. Goldfarb 2499

58
Jc (H) = AM(H)/[a(\-a/3b)] . (91)

It reduces to Eq. (85a) when b /a — 1, and Eq. (85b) when

b /a-* oo . Equation ( 85a ) applies to polygons as well as cylinders, following the argument of Sec. II B. Equation (91)

has also been derived by Clem. 16

VII. CONCLUSION

Since the original work of Bean and London, the criti- cal-state model has been used by many researchers to describe the magnetic response of hard type-II superconductors. The model has provided a simple, intuitive framework

in which data could be analyzed, despite the need for ap- proximations when applying the model to samples of finite dimensions. The refinements by Kim et a/, were an effort to incorporate the field dependence of critical-current density.

The results were magnetization curves that more closely re- sembled experimental data, particularly at low fields. Several other authors have attempted, with some success, to ex- tend the critical-state model for various applications. In this paper we have developed some useful equations

for the analysis of magnetization of type-II superconductors

within the construct of the critical-state theory. Using Kim’s

model for critical-current density, Eq. ( 1 ), we have analytically derived magnetization equations for the general case of an infinite superconductor with rectangular cross section.

Different equations apply to the various parts of the magnetic hysteresis loop. Section III D gave the equations for the

most useful case of large maximum applied field. Examples

of the possible variety of magnetization curves were given in the figures. If we take different dimensional limits, the solutions apply to infinite slabs, cylinders, and rods with polygonal cross sections.

The general Kim solution can be reduced to a general

Bean model for rectangular cross section by reducing the

Kim equation for critical-current density (Sec. V A). In the

appropriate dimensional limits, these equations become the

well known Bean solutions for slabs and cylinders (Sec.

2500 J. Appl. Phys., Vol. 66, No. 6, 1 5 September 1 989

V B). A simple formula was derived to relate the width of a measured hysteresis loop to the critical-current density as a function of applied field, for orthorhombic samples in the

Bean limit (Sec. VI B).

ACKNOWLEDGMENTS

We thank J. Nogues for help with computer programming, W. L. Bahn for drawing Figs. 1-5, and J. S. Munoz, R.

Puzniak, V. Skumryev, and K. V. Rao for helpful discussions. We thank M. A. R. LeBlanc for valuable comments

and making us aware of Ref. 11. macsyma, a symbolic manipulation program produced by Symbolics, Inc., was used

to compare the cylinder and slab limits of our equations with

the results in Refs. 5, 9, and 1 1 , and to verify the continuity of

the stages in Sec. III. This work was supported by the Swedish Board for Technical Development, STU, and the U.S.

Department of Energy, Division of High Energy

Physics.

'C. P. Bean, Phys. Rev. Lett. 8. 250 ( 1962).

: C. P. Bean, Rev. Mod. Phys. 36, 31 ( 1964).

’H. London, Phys. Lett. 6, 162 ( 1963).

4 Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev. Lett 9, 306

(1962).

5 Y. B. Kim, C. F. Hempstead, and A. R. Stmad, Phys. Rev. 129, 528

(1963).

h

J. H. P. Watson, J. Appl. Phys. 39, 3406 ( 1968).

7 F. Irie and K. Yamafuji, J. Phys. Soc. Jpn. 23, 255 ( 1967).

K

I. M. Green and P. Hlawiczka, Proc. IEE 114. 1329 ( 1967). AV. A. Fietz, M R. Beasley, J. Silcox, and W. W. Webb, Phys. Rev. 136, A335 (1964).

"’V. R. Karasik, N. G. Vasil’ev, and V. G Ershov, Sov. Phys. JETP 32. 433

(1971) (Zh. Eksp. Teor. Fiz. 59, 790 (1970)].

"J. A. Hulbert, Brit. J. Appl. Phys. 16, 1657 ( 1965).

i: M. C. Ohmer and J. P. Heinrich, J. Appl. Phys. 44, 1804 ( 1973).

'\J. J. Wollan and M. C. Ohmer, Cryogenics 16, 271 ( 1976).

I4 G. Ravi Kumar and P. Chaddah, Phys. Rev. B 39, 4704 ( 1989).

i 5 A. M. Campbell and J. E. Evetts, Adv. Phys. 21, 199 ( 1972).

I6J. R. Clem, cited in: A. Umezawa, G. W. Crabtree, J. Z. Liu, H. W. Weber,

W. K. Kwok, L. H. Nunez, T. J. Moran, C. H. Sowers, and H. Claus,

Phys. Rev. B 36, 7151 ( 1987).

D. -X. Chen and R. B. Goldfarb 2500

59

U.S. GOVERNMENT PRINTING OFFICE: 1991-576-422/45,153 REGION NO. 8

UNITS FOR MAGNETIC PROPERTIES

Conversion

Quantity Symbol Gaussian & cgs emu

0 factor, C

b SI & rationalized mks c

Magnetic flux density,

magnetic induction

B gauss (G) d

io- 4 tesla (T), Wb/m2

Magnetic flux <t> maxwell (Mx), G-cm2 ur8 weber (Wb), volt second (V-s)

Magnetic potential difference,

magnetomotive force

U, F gilbert (Gb) 10/477 ampere (A)

Magnetic field strength,

magnetizing force

H oersted (Oe),

e Gb/cm 103 /4t7 A/m f

(Volume) magnetization 8 M emu/cm3 h 103 A/m

(Volume) magnetization 477M G 103 /4t7 A/m

Magnetic polarization,

intensity of magnetization

J, I emu/cm3 477 X 10

-4 T, Wb/m2 '

(Mass) magnetization <T, M emu/g

1477 X 10

—

7

A-m2 /kg

Wb-mAg

Magnetic moment m emu, erg/G 10- 3 A-m2

, joule per tesla (J/T)

Magnetic dipole moment j emu, erg/G 477X 10-'° Wb-m '

(Volume) susceptibility X- * dimensionless, emu/cm3 477

(4tt)2 x10-7 dimensionless

henry per meter (H/m), Wb/(A-m)

(Mass) susceptibility Kp cm3/g, emu/g

477 X 1 0

— 3

(477)

2 XlO-'°

m

3 Ag

H-m7kg

(Molar) susceptibility Xmol) Kmol cm3/mol, emu/mol

477 X 10~ 6

(477)

2 X 10- 13 m

3 /mol

H-m2 /mol

Permeability M dimensionless 4t7X10- 7 H/m, Wb/(A-m)

Relative permeability'' Pr not defined dimensionless

(Volume) energy density,

energy product k w erg/cm3 io-' J/m 3

Demagnetization factor D, N dimensionless 1/477 dimensionless

a. Gaussian units and cgs emu are the same for magnetic properties. The defining relation is B =// -\-AvM

.

b. Multiply a number in Gaussian units by C to convert it to SI (e.g., 1 Gx IO

-4 T/G= IO-4

T).

c. SI (Syst'eme International d'Unites) has been adopted by the National Bureau of Standards. Where two conversion factors are given, the upper one is recognized under, or consistent with, SI and is based on the definition B — y.o(H +M), where

/xo = 47r x 10~ 7 H/m. The lower one is not recognized under SI and is based on the definition B =/aoH +J, where the symbol

/ is often used in place of J

.

d. 1 gauss = 105 gamma (y).

e. Both oersted and gauss are expressed as cm _1/2

-g

l/2

-s -1

in terms of base units.

f. A/m was often expressed as “ampere-turn per meter” when used for magnetic field strength.

g. Magnetic moment per unit volume.

h. The designation “emu” is not a unit.

/. Recognized under SI, even though based on the definition B = y.oH -\-J

.

See footnote c.

j. Hr = ii/fio= 1 +x, all in SI. p. r is equal to Gaussian p.

k. B H and H have SI units J/m3

; M H and B H/4tt have Gaussian units erg/cm 3

.

R. B. Goldfarb and F. R. Fickett, U.S. Department of Commerce, National Bureau of Standards, Boulder, Colorado 80303, March 1985 NBS Special Publication 696 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

61

BL-114A U.S. DEPARTMENT OF COMMERCE

(5-90) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION OR REPORT NUMBER

NISTIR 3975

2. PERFORMING ORGANIZATION REPORT NUMBER

3. PUBLICATION DATE

August 1991

4. TITLE AND SUBTITLE

Magnetic Measurements for High Energy Physics Applications

5. AUTHOR(S)

R. B. Goldfarb, Editor

6. PERFORMING ORGANIZATION OF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BOULDER, COLORADO 80303-3328

7. CONTRACT/GRANT NUMBER

DE-AI05-85-ER40240

8. TYPE OF REPORT AND PERIOD COVERED

Final Report, 6/15/88-6/14/91

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

U.S. Department of Energy

Division of High Energy Physics

Washington, DC 20545

10.

SUPPLEMENTARY NOTES

11.

ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBLIOGRAPHY OR

LITERATURE SURVEY, MENTION IT HERE.)

This report is a collection of papers describing magnetic measurements on multifilamentary Nb-Ti

superconductor wires and cables as a function of magnetic field and time at liquid-helium temperatures. The

papers deal with ac losses and interfilament coupling by proximity effect and eddy currents. Flux creep was

investigated under different experimental conditions. A Hall-probe magnetometer, which was used to measure

magnetization and flux creep in the presence of a transport current, is described. A method for increasing the

critical current of superconducting cable by controlling twist pitch is demonstrated. A critical-state model for

the magnetization of superconductors was developed for samples with field-dependent critical current density

and rectangular cross section.

12.

KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITALIZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS) ac losses; critical current; critical state; eddy currents; flux creep; Hall sensors; hysteresis; magnetization;

magnetometry; multipole fields; niobium-titanium; proximity effect; Superconducting Super Collider;

superconductors.

13. AVAILABILITY 14. NUMBER OF PRINTED PAGES

72 X UNLIMITED FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIBJ), VA 22161.

IS. PRICE

X

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.