Analysis of magnetization, AC loss, and deff for various internal-Sn based Nb3Sn multifilamentary strands with and without subelement splitting

Cryogenics 44 (2004) 711–725 www.elsevier.com/locate/cryogenics

Analysis of magnetization, AC loss, and deff for various internal-Sn based Nb3Sn multifilamentary strands with and without subelement splitting M.D. Sumption *, X. Peng, E. Lee, X. Wu, E.W. Collings LASM, MSE, The Ohio State University, 2041 College Rd., Columbus, OH 43210, USA Received 27 March 2003; received in revised form 16 March 2004; accepted 18 March 2004

Abstract Magnetization, AC loss, vDC and deff were measured for several designs of rod-in-tube based internal-Sn Nb3 Sn type superconducting multifilamentary strands. Two kinds of subelement geometries were used in strand construction. The first had the standard annular Nb/Cu ring surrounding a Sn source; the second was similar but included an internal split intended to reduce magnetization and loss. Strands with 18 and 36 subelements were measured, at strand diameters of 0.5–0.8 mm. Optical, SEM, and EDS measurements were performed on these samples; average radii are reported and physical barrier integrity is found to be good. The magnetizations of these structures were analyzed in terms of a deff parameter, in this case calculated for annular structures. Analytical and numerical results of these calculations are presented. It was found that in general annular structures should be expected to have deff values somewhat larger than the subelement diameter; the value of this enhancement is reported. Also, the effect of subelement splitting on deff and magnetization was calculated. The results of these calculations are compared to the experimentally measured results. Reductions in deff due to subelement splitting are compared to direct, low-field susceptibility measurements. Magnetization values are seen to be nearly uniformly lower in the split subelement strands, and this leads in some but not all cases to significantly lower deff values. Possible reasons for these discrepancies are discussed. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Magnetization; Internal Sn; Nb3 Sn; deff ; Subelement

1. Introduction Magnetization, AC loss, low-field susceptibility (vDC ), and effective filament diameter (deff ) for three different strand types were measured in this study. All three were made from two slightly different subelement structures fabricated by Supergenics in conjunction with Outokumpu Advanced Superconductors. The two basic subelement structures are denoted B6 and B7, and are based on the rod-in-tube (RIT) subelement structure developed by Gregory and Pyon at IGC (now Outokumpu Advanced Superconductors) [1–3]. The strands were designed for high energy physics dipole magnets, and thus had relatively large values of critical current

*

Corresponding author. Tel.: +1-614-688-3684; fax: +1-614-6883677. E-mail address: [email protected] (M.D. Sumption). 0011-2275/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2004.03.021

density, Jc . As a consequence of the high Nb and Sn content used to maximize Jc , the magnetization and loss values were also large. In the past, large scale development of Nb3 Sn was driven by the special requirements of the ITER (International Tokomak Experimental Reactor) program. Strand manufacture and RHT protocol focussed on delivering material with specific Jc properties combined with hysteretic-loss constraints, e.g. a Jc (4.2 K,12 T) greater than 700 A/mm2 in association with a
3 T hysteretic loss less than 200 mJ/cm3 [3]. However, it has become clear that in order to be of use for future HEP magnets, higher current densities are needed. This has been achieved at the expense of relaxing the AC-loss (hysteretic) constraint of ITER; once this was removed, Jc (4.2 K,12 T)s, quickly rose beyond 2000 A/mm2 . To further this, a National Nb3 Sn Conductor Development Program was initiated by DOE’s Division of High Energy Physics [4]. Among the strand properties initially

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specified was a non-Cu Jc (4.2 K,12 T) of 3000 A/mm2 accompanied by an effective filament diameter (deff ) of 40 lm or less. Although Jc s of about 2600–2900 A/mm2 have been obtained so far [5,6], the deff of internal-Sn strand still higher than desired (frequently around 80– 120 lm [7]). However, some recent progress in achieving decreases in deff for high Jc strand has been reported. OS-ST wires using a RIT approach have resulted in Jc s of 2800–3000 A/mm2 with an associated deff of 40–60 lm, achieved by using 61 instead of 37 subelements. On the other hand, Supergenics has reported success with a split-barrier approach, where the individual subelements are subdivided into segments, or ‘‘split’’, by Ta–Nb fins [8]. This work focuses on the measurement of this latter type of split subelement strand. Magnetization loop measurements at 4.2 K and
9 T are used to extract magnetization, loss, and deff . Optical and SEM measurements are used to determine the subelement size and the integrity of the barrier. These results were compared to the standard expressions for deff [9–11], as well as calculations of deff specifically made for split and unsplit subelement geometries. Measurement and calculation of deff are compared and interpreted in terms of the effectiveness of the subelement splitting. This is in turn compared to direct measurements of vDC . It is generally found that; (i) unsplit subelements have a deff larger than the nominal subelement diameter, d0 , (ii) split barriers, in theory, can be effective, and (iii) some of the strands measured achieved deff reductions from the splitting, while others did not. These points are elaborated below.

2. Experimental 2.1. Strand fabrication and specifications Three different strand types were measured in this study. The two basic subelement structures, denoted B6 and B7, are described in Table 1. Here we can see that the main difference in the strands is the presence of subelement splits or fins within B6 which had been inserted in an attempt to reduce the resulting magnetization. The split material was Ta–40 wt.% Nb. We do note, however, that B6 had a 1.2% lower Sn:Nb ratio. Additionally, the number of filaments is slightly greater for B6. The subelements were restacked into a number of structures, three of which are examined in this work; a 19 (18+1) subelement restack of B6, a 19 (18+1) sub-

Fig. 1. Strand S4-HT2 after reaction heat treatment.

element restack of B7, and a 37 (36+1) subelement restack of B6. A typical cross section is shown in Fig. 1. The strand had the bulk of the Cu stabilizer on the outside of the filamentary array. Unlike past strands developed by Gregory et al. these strands had Nb barriers surrounding each individual subelement. Further strand specifications are given in Table 2. The strands were drawn down to sizes from 0.50 to 0.85 mm in diameter for heat treating and testing, leading to five distinct sample types (as a combination of subelement type, restack number, and final strand size), denoted S1-5 (see Table 2). These samples were distributed to three laboratories for HT, and four different HT schedules were applied as follows: HT1: The Ohio State University (OSU): 25 °C/h to 185 °C/100 h + 50 °C/h to 340 °C/48 h + 50 °C/ h to 700 °C/40 h. HT2: Fermi Accelerator National Laboratory (FANL): 25 °C/h to 210 °C/48 h + 50 °C/h to 340 °C/48 h + 75 °C/h to 700 °C/20 h. HT3: Lawrence Berkeley National Laboratory (LBNL): 16 °C/h to 210 °C/110 h + 10 °C/h to 340 °C/48 h + 25 °C/h to 650 °C/48 h. HT3B: Lawrence Berkeley National Laboratory (LBNL): 16 °C/h to 210 °C/110 h + 10 °C/h to 340 °C/48 h + 25 °C/h to 650 °C/96 h. Both loss and transport Jc samples were wound and heat treated (HT). The transport samples were wound onto ITER barrels in the standard way before HT. When the samples were wound onto barrels, about 1–2 ft extra was present on each end the barrel and this was

Table 1 B6 and B7 subelement specifications Subelement type

Fil. no.

Split?

Nb at.%

Sn at.%

Cu at.%

L=R

Nb area %

B6 B7

1095 1044

Y N

41.4 39.1

14.7 15.6

44.0 45.3

0.73 0.73

45.0 42.5

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

713

Table 2 Strand and sample specifications Sample

Name

S. Dia., mm

No. fil.

Split

% Non-Cu

L, cm

VnonCu 103 cm3

S1-HT1 S2-HT1 S3-HT1 S4-HT1 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2 S1-HT3 S2-HT3 S3-HT3B S4-HT3B

253.1/BAZ7-1014-5B 253.2/BAZ6-IR19 253.3/BAZ6-6101 253.4/BAZ7-1014-5C 254.1/BAZ7-1014-5B 254.2/BAZ6-IR19 254.3/BAZ6-6101 254.4/BAZ7-1014-5C 254.5/BAZ6-6102 255.1/BAZ7-1014-5B 255.2/BAZ6-IR19 255.3/BAZ6101 255.4/BAZ7-1014-5C

0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.80 0.53 0.50 0.815 0.85

18 18 18 18 18 18 18 18 36 18 18 18 18

N Y Y N N Y Y N Y N Y Y N

47 47 47 47 47 47 47 47 45 47 47 47 47

27.86 32.05 38.93 28.51 34.89 34.57 39.33 38.42 36.54 38.25 40.66 35.40 12.78

28.87 29.56 91.9 75.99 36.16 31.88 92.88 102.4 82.61 39.64 37.50 85.01 34.05

sealed by crimping or torch melting. After HT this section was removed. These barrels were subsequently tested at LBNL and FANL. The loss samples were HT at the same time as the transport samples at each respective laboratory and were then sent to OSU for magnetization and loss measurement. 2.2. Experimental measurement procedures 2.2.1. VSM measurement All magnetization and loss data were obtained using a bipolar
9 T vibrating sample magnetometer setup. The system was constructed from an EG&G M4500 controller in conjunction with a VSM head mounted on a 9 T Oxford dewar containing a NbTi based 9 T magnet. The solenoidal pickup coils were wound inhouse, and calibrations were performed using the saturation magnetization of Ni. All measurements were performed at 4.2 K. Samples up to 3.8 cm in transverse dimension and/or 3 cm along the solenoidal axis can be accommodated in this machine. However, for the present studies all of the samples were solenoidal, about 2.5 cm long, and approximately 5 mm in diameter. The strands were twisted before being wound into solenoids, and the specifications are listed in Table 2. This sample arrangement is similar to that used by Goldfarb and Itoh [12] in their study which showed the length dependence of the bridging effect in Nb3 Sn, a geometry which has become an informal standard for loss measurements in Nb3 Sn strands. Initial permeability measurements were made after zero field cooling. 2.2.2. Transport data All transport data were provided by LBNL and FNAL. The measurements were made using procedures and protocols developed during ITER benchmark testing [13], modified for HEP conductor needs. The holders were based on ITER barrel designs. FNAL used the standard ITER barrels with pressure contacts [14] and LBNL used soldered contacts [15].

2.2.3. Optical, SEM, and EDS measurements The SEM studies were made with a Phillips XL-30 ESEM with EDS attachment. Optical analysis was performed using standard light microscopy, and the subelement dimensions were based on area measurements using UTHSCSA image tool software. The subelements were approximated to be annular with cylindrical symmetry.

3. Calculation 3.1. The meaning of deff for internal-Sn strands with hollow subelement geometries The ‘‘standard’’ expression for deff is [9–11] DM ¼

4 deff Jc 3p

ð1Þ

Here DM is the height of the magnetization loop, and the units are SI (to obtain cgs-practical units, divide the RHS by 10). The normalizing area can be either that of the whole strand, or that of some subsection (e.g., the non-Cu) as long as both DM and Jc are normalized to the same area (although it has been reasonably argued by Barzi and Boffo [16,17] that it is best to remove the temptation to use different areas by writing them in explicitly). In using this expression we implicitly model the strand as consisting of a number of solid, round filaments. This allows an effective filament diameter to be extracted, which serves as a useful engineering parameter. In the case of ‘‘fully coupled’’ or ‘‘fully bridged’’ strands (frequent) deff is typically comparable to the subelement diameter. However, Goldfarb and Itoh [12] showed that this was a saturation of a more general effect, and that in its non-saturated form it could have sample length or sample twist pitch functional dependencies. Sumption [18] developed a model to describe these effects in the unsaturated regime.

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M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

If we now wish to apply this to internal-Sn strands where the coupled sections will be annular (rather than isolated, bridged strand pairs), we need to take into account both the geometry of the strand as well as the standard way in which conductor parameters are normalized. In this way, we can find a close connection between the calculated deff and the physical sample parameters. We can start with the expression for the magnetization of hollow cylinders (internal and external radii RI and RO , respectively) [19,20] see Appendix A: "  3 # 4 RI JcA15 d0 1  DM ¼ ð2Þ 3p RO However, we note that  2  RO  R2I Jc;nonCu ¼ JcA15 R2O

ð3Þ

Combining this with Eq. (2) leads to 2  3 3 RI 1  RO 4 6 7 d0 Jc;nonCu 4 DM ¼  2 5 3p 1  RROI

ð4Þ

Comparing Eqs. (1) and (4) above leads to   1  g3 deff ¼ d0 1  g2

ð5Þ

where g ¼ RI =RO . The brackets on the RHS will always be greater than 1, see Fig. 2. So, generally we should expect the deff values of internal-Sn based wires with hollow structures to be larger than the actual diameter of the subelement, at least when the subelement is fully bridged, and no ‘‘splits’’ are present. This will be true as long as the same areas are used to normalize magnetization and Jc (whether implicitly or explicitly [16,17]). This is just as it should be, as long as deff is meant to be a parameter representing the magnetization, rather than a real d occurring somewhere in the sample. That is, from a magnet engineering point of view, what matters is the

magnetization, normalized by the whole strand volume, as compared to the engineering critical current density, Je . If we ask for this to be parameterized by a deff value for the sake of simplicity, then the result of this is that deff has a less direct connection to geometrical parameters of the strand. The benefit of such a treatment, however, is that it gives us a simple, portable, parameterization of the magnetization. 3.2. The influence of subelement splitting The enhancement of deff relative to d0 can be seen as the result of the shielding of the hollow core, an effect that tends to go away with subdivision (splitting) of the cylinder. However, the reduction with splitting is in fact greater than this would suggest, because the flux can penetrate a split subelement semi-annular region both from the inside of the cylinder as well as the outside. The depth of penetration varies around the now-split annulus, and is a function of the angle of the field with respect to the split direction. Below we calculate this effect explicitly. Since the splits are randomly oriented, we will need to make some assumption about how the field is aligned with respect to them on average; the simplest and most reasonable assumption is that the alignment is perfectly random, and that we can average over all field orientations. This would require extensive work if done analytically, thus it will be done numerically below. However, we will also calculate an approximate result based on an average of the extreme orientations. In all cases we will assume that the subelements are round and uniform, as shown in Fig. 3. On the left we have an orientation which we can define as h ¼ 0, where the field is applied along the subelement splitting direction. On the right we have the case where h ¼ p=2, and the field is applied perpendicular the subelement splitting orientation. Values for deff are typically extracted at 12 T, where demagnetization effects are minimal, since B  Bp . In

1.6 θ=0

θ=π/2

1.5 1.4 1

deff / do

1

RO

1.3 2

RI

1.2

2 RI

R0

1.1 1.0 0.9 0.0

0.2

0.4

η

0.6

Fig. 2. deff =d0 vs g.

0.8

1.0

Fig. 3. Split subelement schematic at h ¼ 0 and h ¼ p=2.

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

this case, we can treat the magnetization as being the summation of the magnetization of a series of sections (‘‘planks’’) aligned (along the z-axis) perpendicular to the applied field (oriented along the y-axis) [21]. Then P P m Mi Ai i mi M¼ ¼ ¼ i ð6Þ V Vtot Atot Here mi is the moment of each individual section along the y-axis (and Mi is its magnetization), while Ai is the area of the section perpendicular to the z-axis, and Atot is the total area of the subelement, perpendicular to the z-axis. This leads to P Mi Ai JcA15 X w2i ðyÞDy ð7Þ DMnonCu ¼ i 2 ¼ 2 pRO pR2O i where RO is the outer radius of the subelement. The individual segment widths (along the x-axis) are given by wi , and the thickness by Dy. However, Ai ¼ wi Dy, and so Z 4  JcA15 4 d JcA15 ¼ DMnonCu ¼ w2 ðyÞ dy ð8Þ 3 eff pR2O 2 one quadrant  where deff is a new measure of the effective filament size.  In this particular calculation, deff includes the influence of the split, however, its difference from deff is more  general than this. In particular, in defining deff we normalize the magnetization to the non-Cu volume, but normalize Jc to the A15 area. This leads to  deff ¼ d0 ð1  g3 Þ, which has a distinct and well defined meaning for both split and unsplit subelements. This parameter is less useful than deff for magnet engineering purposes, but has a slightly closer connection to strand geometry. In particular, it allows a more straightforward and intuitive description of the magnetization of different subelement shapes (e.g., it has the convenient quality that as the annular wall gets thinner, this parameter grows sensibly smaller). It should be noted that if we wish to know how the standard deff is influenced we  should substitute deff ¼ deff ð1  g2 Þ in the LHS of Eq. (8). However, proceeding with our current definitions, we obtain for h ¼ 0,   Z RI Z RO 3 2  2 deff ¼ 2 ðx0  xI Þ dy þ x0 dy ð9Þ 2RO 0 RI

where xO and xI are the x components of RO and RI . Since x2O ¼ R2O  y 2 and x2I ¼ R2I  y 2 ,  Z RI  3 R2O þ R2I  2y 2 2R2O 0 Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 R2O  y 2 R2I  y 2 dy þ

 deff ¼

RO



R2O  y 2





 dy

RI

ð10Þ

715

leading to  ¼ deff

 3 2 3 2 3 p 2 R þ R  R RO 2R2O 3 I 3 O 2 I  2 ð1=2; 1=2; 2; ðRI =RO Þ Þ

2

F1 ð11Þ

Note that the last term is contains a hypergeometric function which in the limit of RI ! RO approaches 0.85. It can be expanded into the form f ðgÞ ¼

2

  g2 g4 5g6 F1  1=2; 1=2; 2; g2 ¼ 1    8 64 1024 ð12Þ

Thus  deff h¼0

  d0 f ðgÞ 2 g 1 þ g3  2 0:85 2

ð13Þ

Since f ðgÞ varies slowly for g values between 0.3 (0.96) and 1.0 (0.85) we can approximate this last equation by setting f ðgÞ ¼ 0:85. However, the full form of Eq. (13) is plotted in Fig. 4. A similar calculation for the h ¼ p=2 case leads to   3 3 f ðgÞ 2 3  g  g ð14Þ deff h¼p=2 d0 1 þ g  4 0:85 4 Directly averaging the h ¼ 0 and h ¼ p=2 cases leads to  deffhhi ¼

  d0 f ðgÞ 2 g 6 þ 5g3  3g  8 0:85 8

ð15Þ

for the angle-averaged value. This expression is plotted in Fig. 4 along with the numerical computation, where the two-angle approximation can be seen to be fairly good. The numerical calculation is detailed in Appendix B; here the averaging was done at 90 intervals over the region from h ¼ 0 to p=2. The agreement is acceptable for the area of g which is of interest. If we wished to know the response of the standard deff , we could then divide Eq. (15) by ð1  g2 Þ. In Fig. 5 we have plotted    deff split =deff unsplit vs g (note that the ratio deff split =  deff unsplit ¼ deff split =deff unsplit , both being equal to the same function of g). It can be noted that the magnetization for the split case is significantly smaller than for the unsplit case in the region of interest. This is because the lack of continuity of the subelement in the split case allows flux to penetrate into the inner regions of the subelement. The unsplit subelement shields and traps field within the subelement center (Sn-core), enhancing its magnetization, while the split subelement does not. On the contrary, the flux penetration into the split subelement makes the field at the internal boundary of the subelement close to the applied field, thus reducing flux shielding and trapping within the superconducting annulus itself. These two effects both tend to enhance the ratio of the unsplit subelement magnetization relative to

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M.D. Sumption et al. / Cryogenics 44 (2004) 711–725 Split (numerical caclculation) Split at θ = 0 Split at θ = π /2 Two angle average Unsplit

1.0

0.8

*

deff /d 0

0.6

0.4

0.2

0.0 0.0

0.2

0.4

η

0.6

0.8

1.0

 Fig. 4. deff =d0 vs g for a split subelement as compared to an unsplit subelement. Analytic calculations for the unsplit, split at h ¼ 0, and split at h ¼ p=2 cases are shown. An average of the h ¼ 0 and p=2 calculations is then compared to a numerical calculation averaging values at every degree within the h ¼ 0 to p=2 range.

0.8 Split/Unsplit (numerical) Spit/Unsplit (two angle)

deff*split /deff*unsplit

0.6

w L t

0.4

Outer shell

0.2

0.0 0.0

Fig. 6. Schematic of subelement re-coupling via weak links. 0.2

0.4

η

0.6

0.8

1.0

Fig. 5. dsplit =dunsplit vs g calculated using simple approximation as compared to numerical method.

that of the split subelement. This effect for the simple case of a slab is worked out heuristically and analytically in Appendix C. 3.3. Split subelement magnetization in the presence of weak-links crossing the splits The split geometry of the B6 type subelement is intended to reduce the magnetization by separating the half-annular segments of the structure, see Figs. 6. If no superconducting link exists between the half segments, the magnetization will be as calculated above. However, if a weak link does exist, the magnetization will be larger than this. If the weak link is strong enough, it will be as if the splitting is not present. This link might be in the form of a partially reacted outer Nb barrier, a growth of a superconducting link around the barrier on the inside of the annulus, a penetration though (breakage of) the

internal barrier, or a partial superconducting conversion of the Ta–Nb barrier. Whatever, the physical source of the weak link, we calculate below the criterion for it to effectively re-couple the annular structure. If we treat the subelement as a generalized 3-D cylindrical conductor with average Jc s, then, in an applied field, then the shielding (or trapping) current flowing through one of the half segments along the zaxis (see Fig. 6) may transfer over to the other halfannular segment by passing through, e.g, a weak skin reaction layer, if present. In particular, the current within the half segment given (flowing along the z-axis) is given by d0 ws JcA15 , and will require a matching current tLJweak . Here ws ¼ RO  RI , t is the thickness of the weakly superconducting path, and Jweak is the weak Jc . This leads to a condition for re-coupling or ‘‘healing’’ of the splits given by Jweak d0 ws > JcA15 L t

ð16Þ

We can expect that d0 =L will be about 1/100 for a twisted sample with an Lp of 1 cm. The value of t is difficult to

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

717

can be inferred. We can calculate the apparent deff and d0 for the unsplit subelement strand (where we have weighted for the influence of the volume of the magnetizing region [22]). Then " # 3 N X 8 Jc 3 1  ðRI;i =RO;i Þ DMnonCu ¼ R ð18Þ 3p N hRO i2 i O 1  ðRI;i =RO;i Þ2 However, we note that g is fairly constant for all subelements (the direct average is listed in Table 3), such that the term in square brackets can be brought outside the summation. If we then compare the standard form for deff , we find that   N 2 1  g3 X hdeff iunsplit size;ave ¼ R3O;i ð19Þ 2 N hRO i 1  g2 i

Fig. 7. Backscatter SEM of internal split for sample S3-HT1.

estimate, however, if it is 1/10th of the total diameter, then Jweak would need to be greater than 1/10th of JcA15 for re-coupling. If, on the other hand, we assume that the Ta–Nb split barrier itself is partially or weakly superconducting, then the weak current becomes ws LJweak , and then the criterion for re-coupling is Jweak =JcA15 > d0 =L (Fig. 7). 4. Results 4.1. Optical, SEM, and EDS measurements Light microscopy was used in conjunction with optical analysis to determine the outer and inner radii of the subelements within a representative cross section of the various strands. A cylindrically symmetric, annular geometry was assumed, and the results are displayed in Table 3. We note that, by design, there are three (somewhat) different sizes of subelements within these strands, in order to facilitate strand fabrication (a typical design strategy). This variation leads to the need to define an average d0 and deff (as measured from the magnetic response). In order to properly calculate this average, we must weight the radii with their magnetization values [22]. Using Eq. (4) from [22] the expression n X 1 DMnonCu ¼ DMi;nonCu Vi;nonCu ð17Þ Vtot;nonCu i

Following Eqs. (2) and (4) above, hd0 i is the same as Eq. (19) without the term in brackets. We have used this equation to calculate the apparent d0 values of Table 3. We note that in Table 3, the filaments are grouped into ‘‘big’’, ‘‘medium’’, and ‘‘small’’ in recognition of the fact that they were initially inserted as three different sizes. Nevertheless, there are two important points. The first is that the hd0 i appearing in Table 3 are a weighted average of the filamentary diameters––a way of including the fact that larger filaments will disproportionately skew the results to a larger hd0 i. The second is that this was done by individually summing the influence of each filament via Eq. (19). Thus, the averages shown for large and small filaments are for convenience and comparative purposes only (and these arbitrary groupings do not influence the extracted hd0 i values). 4.2. Magnetization loop measurement, AC loss, and deff Figs. 8–10 display the magnetization loops measured for various samples at 4.2 K. Fig. 8 shows the samples HT at LBNL, where the split subelement samples have a generally lower magnetization that that of the unsplit samples––although the subelement diameter variation is also important, with the 0.5 mm strands having lower magnetization than the 0.8 mm strands. Fig. 9 shows similar results for the OSU-HT samples, with again the unsplit samples lying below those of the split ones. Fig. 10 displays the results for the FNAL-HT samples, where we can see results similar to those of Figs. 8 and 9,

Table 3 Optical analysis of various strands (all R and d values given in lm) Strand type

R0;small

RI;small

R0;mid

RI;mid

R0;big

RI;big

g

d0;apparent

S1-HT1 S2-HT1 S3-HT1 S4-HT1 S5-HT2a

37.4
0.8 34.5
0.5 57.3
0.4 60.3
1.2 38.5
2.0

19.6
0.6 17.3
0.3 26.7
0.8 29.8
0.9 19.6
1.2

45.5
0.4 44.0
0.4 71.5
0.4 74.4
0.9 50.4
1.6

24.6
0.4 22.2
0.4 34.6
0.7 37.4
0.5 25.6
1.0

47.6
0.6 45.2
1.0 74.1
0.4 77.3
1.2 54.8
1.6

25.5
0.6 22.3
0.4 35.7
0.4 38.1
0.8 27.5
1.1

0.53 0.50 0.47 0.50 0.50

90 86 140 147 105

a

This dataset had more than average variability, leading to the apparently larger d0 .

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M.D. Sumption et al. / Cryogenics 44 (2004) 711–725 S1-HT3 (0.53 unsplit 18+1) S2-HT3 (0.50 split 18+1) S3-HT3B (0.815 split 18+1) S4-HT3B (0.85 unsplit 18+1)

1.5

3

Magnetization, M, (10 kA/m)

1.0

0.5

0.0

-0.5

-1.0

-1.5 -10

-8

-6

-4

-2

0

2

4

6

8

10

B, T

Fig. 8. Magnetization loops for LBNL-HT samples at 4.2 K.

S1-HT1 (0.53 unsplit 18+1) S2-HT1 (0.50 split 18+1) S3-HT1 (0.815 split 18+1) S4-HT1 (0.85 unsplit 18+1)

1.5

3

Magnetization, M, (10 kA/m)

1.0

0.5

0.0

-0.5

-1.0

-1.5 -10

-8

-6

-4

-2

0

2

4

6

8

10

B, T

Fig. 9. Magnetization loops for OSU-HT samples at 4.2 K.

S1-HT2 (0.53 unsplit 18+1) S2-HT2 (0.50 split 18+1) S3-HT2 (0.815 split 18+1) S4-HT2 (0.85 unsplit 18+ 1) S5-HT2 (0.80 split 36+1)

1.5

3

Magnetization, M, (10 kA/m)

1.0

0.5

0.0

-0.5

-1.0

-1.5 -10

-8

-6

-4

-2

0

2

4

6

8

10

B, T

Fig. 10. Magnetization loops for FANL-HT samples at 4.2 K.

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

4.5

∆M1/2H1/4/do1/2, (kA/m)1/2T1/4mm-1/2

UNSPLIT

S1-HT1 S2-HT1 S3-HT1 S4-HT1

4.0

3.5

3.0

2.5 SPLIT 2.0 2

4

6

8

10

12

B, T

Fig. 11. Subelement diameter normalized magnetization Kramer plot for OSU-HT samples at 4.2 K.

∆M1/2B1/4/d01/2, (kA/m)1/2T1/4mm-1/2

4.5 S1-HT3 S2-HT3 S3-HT3B S4-HT3B

UNSPLIT

4.0

3.5

UNSPLIT AND 36 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2

4 ∆M1/2H1/4/d0 (kA/m)1/2T1/4mm-1/2

however in this case we also have the 36 subelement sample, S5-HT2, which seems to have a relatively large magnetization, given the fact that it is both split, and has 36 subelements. Many of the magnetization curves show low-field flux jumping. This level may not be important for dipole magnet stability, but will influence the interpretation of the loss results, below. 1=2 Figs. 11–13 display curves of DM 1=2 B1=4 =d0 (a magnetic Jc Kramer plot). This allows us to compare something correlating to deff between strands with different subelement diameters, and additionally lets us do so at a variety of fields. In Fig. 12, we see the LBNL results, which essentially parallels the magnetization results of Fig. 8, with the results resolving into two groups, split and unsplit. Thus, for both 0.8 and 0.5 diameter strands, the magnetization is reduced in the strand with the subelement splits. In Fig. 11, we see quite similar results for the OSU-HT samples. Because

719

3

SPLIT 2

1 0

2

4

6

8

10

12

B, T

Fig. 13. Subelement diameter normalized magnetization Kramer plot for FANL-HT samples at 4.2 K.

of the possibility for differing levels of reaction in the two subtly different subelements, we will need to compare these results to actual deff calculations, below. Fig. 13, which displays the FNAL-HT samples shows a similar division into split and unsplit groups. However, the 36 subelement strand lies in the unsplit group, suggesting that the splitting is not effective for this strand. Table 4 displays the loss results for all samples. Here we see that the loss reduction for the split samples is much smaller than would be expected from the magnetization results. This probably due to a low-field magnetization ‘‘cap’’ imposed on the unsplit, and hence more strongly flux jumping [23] strand. It may also have some influence coming from demagnetization modifications at low fields. We have used Kramer-plot-like extrapolations of the magnetization to determine the magnetization at 12 T, and this is displayed in Table 5. We then have used the standard expression for deff (Eq. (1)) along with transport Jc (measured at LBNL and FNAL) to determine deff , and the results are displayed in Table 5, along with the optically measured and weighted average d0 values. Table 6 displays deff measured, along with deff expected for unsplit samples, and two columns commenting on whether or not subelement-to-subelement bridging was observed, as well as the presence of broken subelement rings. These findings are discussed below.

3.0

4.3. Initial permeability and low field susceptibility

2.5 SPLIT 2.0

1.5 2

4

6

8

10

12

B, T

Fig. 12. Subelement diameter normalized magnetization Kramer plot for LBNL-HT samples at 4.2 K.

Initial permeability curves were taken, and the lowfield susceptibility values were extracted for all samples, with the results displayed in Table 7. Here we can see another measure of the effectiveness of the splits. In the Meissner regime we expect full exclusion from the superconducting annulus volume if the barriers are effective and from the whole of the subelement (the total

720

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

Table 4 AC loss for various samples EG name

OSU name

Split (Y/N)

ds , mm

L=Lp

Q, mJ/cm3 non-Cu

BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ6

S1-HT3 S2-HT3 S3-HT3B S4-HT3B S1-HT1 S2-HT1 S3-HT1 S4-HT1 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2

N Y Y N N Y Y N N Y Y N Y

0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.80

30.1 32.0 27.9 10.1 21.9 25.2 30.7 22.4 27.5 27.2 31.0 30.3 28.8

11,213 9103 13,488 19,530 12,755 9796 12,248 18,005 11,929 6778 12,343 14,510 13,229

(18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (36)

Lp ¼ 1:27 cm.

Table 5 12 T Magnetization and deff for various B6 and B7 structures EG name

OSU name

Split (Y/N)

ds , mm

DM 1=2 B1=4 (103 kA/m)1=2 T1=4

(3p=4ÞDM 106 A/m

12 T Jc 109 A/m2

deff ; lm

d0 ; lm

BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ6

S1-HT3 S2-HT3 S3-HT3B S4-HT3B S1-HT1 S2-HT1 S3-HT1 S4-HT1 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2

N Y Y N N Y Y N N Y Y N Y

0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.80

0.6576 0.4543 0.6504 0.9122 0.6718 0.4548 0.5638 0.8329 0.6682 0.3649 0.6802 0.7676 0.6782

0.2940 0.1403 0.2876 0.5657 0.3068 0.1406 0.2161 0.4716 0.3035 0.0905 0.3145 0.4006 0.3127

– 1.74 1.86 2.66 – – – – – 1.54 1.94 2.35 2.23

–

90 86 140 147 90 86 140 147 90 86 140 147 105

(18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (36)

81 155 213 – – – – – 59 163 170 140

Table 6 d0 and deff comparisons for various B6 and B7 structures EG name

OSU name

Split (Y/N)

ds , mm

d0 , lm

deff , expecteda , lm

deff , lm

Sub–sub bridging (Y/N)

Broken sub rings

BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ6

S1-HT3 S2-HT3 S3-HT3B S4-HT3B S1-HT1 S2-HT1 S3-HT1 S4-HT1 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2

N Y Y N N Y Y N N Y Y N Y

0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.53 0.50 0.815 0.85 0.80

90 86 140 147 90 86 140 147 90 86 140 147 105

107 100 162 170 107 100 162 170 107 100 162 170 123

–

Slight Y

N N N Y Y N N N Y N N N N

a

(18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (36)

(42) (68)

(42) (68)

(42) (68) (55)

81 155 213 – – – – – 59 163 170 140

N N N N N N N N Y

Split corrected value in parenthesis.

non-Cu volume) is the barriers are ineffective. This fact is not altered by the presence of the Nb barrier because the Ta–Nb split has a slight portion parallel with the Nb barrier (see Fig. 14) were it prevents the diffusion of Sn

into the barrier, and thus its superconducting conversion (in this area). Of course below the Bc2 of the Nb, the susceptibilities will be consistent with the non-Cu volume without regard to splitting. But this is only to about

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

721

Table 7 Low field vDC and split efficiency factor EG name

OSU name

Split (Y/N)

vunsplit

vsplit a

vmeasured

cb

v=vmax

HT

BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ7 BAZ6 BAZ6 BAZ7 BAZ6

S1-HT3 S2-HT3 S3-HT3B S4-HT3B S1-HT1 S2-HT1 S3-HT1 S4-HT1 S1-HT2 S2-HT2 S3-HT2 S4-HT2 S5-HT2

N Y Y N N Y Y N N Y Y N Y

0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159 0.159

[0.115] 0.119 0.124 [0.119] [0.115] 0.119 0.124 [0.119] [0.115] 0.119 0.124 [0.119] 0.119

0.116 0.119 0.121 0.143 0.131 0.151 0.113 0.137 0.121 0.087 0.106 0.110 0.125

– 1.0 1.0 – – 0.2 1.0 – – 1.0 1.0 – 0.55

0.73 – – 0.89 0.82 – – 0.86 0.76 – – 0.69 –

650 650 650 650 700 700 700 700 700 700 700 700 700

a b

(18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (18) (36)

°C/48 °C/48 °C/96 °C/96 °C/40 °C/40 °C/40 °C/40 °C/20 °C/20 °C/20 °C/20 °C/20

h h h h h h h h h h h h h

Brackets denote what the value for split would be for reference in this unsplit case. Assuming complete reaction.

vsplit ¼ vunsplit ð1  g2 Þ

ð20Þ

where the term in the parenthesis is the ratio of core area to non-Cu area. The initial slope values (excluding, however, the region below 2 kOe), along with the expectations for totally ineffective as well as totally effective splits are displayed in Table 7, along with an efficiency factor, defined as the fraction of the expected slope suppression achieved, viz., c¼

Fig. 14. Close-up of internal split near edge (backscatter SEM of S3HT1). Inset shows lack of Sn in Ta-split shielded barrier.

2 kOe––and above this, as long as the Sn is kept from the Ta protected regions, the Nb barriers are themselves expected to be secondary, thin annular segments which do not significantly influence the magnetization, deff , or low-field susceptibility. Since the subelements are cylindrical and in a transverse field, their demagnetization factor is 1/2. That of the split subelement is much more difficult to determine, for simplicity we will assume that it is also 1/2. For the present purposes, this approximation is sufficient to distinguish between splits which are effective, and those which are not effective. We will start by normalizing v to the non-Cu volume. For unsplit subelements, or subelements where the splitting was not effective, flux will be excluded from the whole subelement (the non-Cu region) and we would expect v ¼ 1=2. On the other hand if the splits are effective, flux will be excluded only from the annular region of the subelement, and the absolute value of v will be smaller. The resulting v for splits which are 100% efficient is given by

vunsplit  vmeasured vunsplit  vsplit

ð21Þ

Also listed in Table 7 are the fractions of measured susceptibility to maximum possible, in the case of unsplit strands. These should have some general correlation to reaction completeness or aggressiveness (Fig. 15).

Fig. 15. Backscatter SEM for S3-HT1 showing degree of reaction completeness.

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M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

5. Discussion While reductions in DM are seen for most split samples, it is not fully clear if the reductions are due to lower Jc in the B6 subelements, or lower effective subelement widths (splitting). In particular, somewhat lower Jc values for B6 are to be expected, because of the 1.2% lower Sn:Nb ratio, as well as the fact that the barriers will tend to absorb some of the Sn. Thus, we should be hesitant in drawing too much from the magnetization reduction values themselves. Nevertheless, the observation of reduced deff for several strands argues that in some cases the splits are lowering the loss. However, it is also important to ask whether the splits would be effective in a high Sn environment, because at higher Sn concentrations there may be more of a tendency to react through or around the internal barriers (splits). The observations above are also potentially complicated by the fact that in some cases the subelements have been drawn to different sizes, and the efficiency of the splits may be degraded at smaller sizes (see the 37 subelement results). Clearly, further experiments are needed to clarify this. Below we outline some detailed observations. 1. DM for 19 subelement based split samples (see column 6 of Table 5) is roughly 1/2 that of their unsplit counterparts. 2. DM=d0 plots show curves which fall into two groups, split and unsplit, with the unsplit curves lying above those of the split ones. The 36 split subelement sample falls in with the unsplit curves. 3. Calculated deff for unsplit samples is larger than d0 by approximately 18%. 4. Calculations show that dsplit should be lower than dunsplit by a factor of about 0.42–0.44. This seems to be close to the experimental DM results. 5. On the other hand, deff reductions (which include the variation in Jc ) seem to be present is some cases, and not in others. 6. In some cases, subelement splitting seems to be effective, in some cases it seems to be only partially so. It was most efficient for sample S2-HT2, one of the samples with the least aggressive HT (700 °C/20 h). Splitting was partially effective for S2-HT3 which had a 650 °C/48 h HT. However, it was ineffective for S3HTB3, with the most aggressive HT of 650 °C/96 h. 7. On the other hand, two samples with less aggressive HTs had no improvement with splitting, S5-HT2 and S3-HT2. In the case of S5-HT2 (the 36 subelement strand) the low efficiency may be due subelement-to-subelement bridging, since this has been observed optically. In the case of S3-HT2, it may be due to Jc degradation associated with subelement breakage. 8. vDC results for the unsplit samples show that the less aggressive HTs are not excluding flux from all of the

calculated non-Cu regions. This suggests a less than maximal reaction in some cases––in line with the small unreacted regions seen in SEM. 9. In general the vDC values are lessened by subelement splitting. This implies that either vDC is a less sensitive measure of bridging or that Jc limitation of one form or other (measurement limitations or subelement breakage) is the limiting factor in reaching the desired deff . 6. Summary We have measured the magnetization, AC loss, vDC , and deff for several strands with split and unsplit subelement designs, at strand diameters of 0.8 and 0.5 mm. The strands have been characterized optically to determine subelement sizes, areas, and the variations in these quantities. Optical measurements have also allowed us to characterize subelement integrity and check for the presence of subelement-to-subelement bridging. Backscatter SEM has been used to confirm the mechanical integrity of the internal barrier. Calculations have been made to determine expected deff values for unsplit and split annular subelement geometries. In the latter case a simple, approximate, analytical approach to the calculation was compared to results from a numerical method. Our results show that deff for unsplit annular subelements with typical geometries should be about 18% higher than the outer radius of the annulus, an effect which stems in part from the standard definition of deff . It is also shown that, optimally, deff should be about 45% of the unsplit value. The conditions for weak-link re-coupling of the split subelements are calculated, as well as the influence on deff of variations in subelement size. Experimental and theoretical results are compared, and we see that the splits may be having an influence, but to varying levels of efficiency. Interpretation of the results is made more difficult by the fact that the Sn contents in B6 is less than in B7 (1.2% nominally). Additionally, the behavior that would be found for similar strands with high levels of Sn (for Jc optimization) is unclear. At present some factors which may be suppressing the effectiveness of the split are (in one case) subelement-to-subelement bridging, and local variations in Jc (defects and low Sn in some cases). This said, subelement splitting does seem to be reducing magnetization and deff in some cases. However, a full explanation of the factors affecting the efficiency of the subelement splits awaits more experimentation. Acknowledgements We thank Eric Gregory and Bruce Zeitlin (Supergenics) for providing the subelements and samples. We

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

also thank Dan Dietderich (LBNL) and Emanuela Barzi (FNAL) for transport Jc data and the reaction of some of the Nb3 Sn loss samples. This work was supported by the US Department of Energy under Grant DE-FG0295ER40900.

Appendix A. Magnetization for hollow tubes The magnetization of a hollow element can be calculated as follows. We start with the general expression for magnetization in SI units (see for instance Carr [24]) M¼

 Z  1 1 ~ rx~ J ð~ rÞ ds V 2

ðA:1Þ

723

Appendix B. Numerical calculation of split subelement magnetization In order to calculate a parameter associated with the average width (suitable for use with the R Bean slab model) we must find the integral I ¼ w2i dy, corresponding to each rotation angle. Fig. 17 shows the parameter definitions, where the split width is c and AB and CD define the split edges; their expression in the original coordinate system is y0 ¼ c. We must calculate I for all angles a (for ease of calculation a ¼ h þ p=2Þ. In order to do so we define a set of new coordinate systems using the rotation matrix   cos a sin a R¼ ðB:1Þ  sin a cos a After rotation, the split edge is defined by the line

1=2

where r ¼ ðq2 þ z2 Þ , as depicted in Fig. 16. Taking into account the current crossing over at the ends (giving a factor of two), and also integrating out z and h, we get    Z R 20 M¼ q2 Jcz ðqÞ dq ðA:2Þ 5pR 0 If we take RO to be the outer diameter of the tube, and RI to be the inner diameter of the tube, then   R 20 1 JcA15 q3 RO M¼ 2 I 5pR 3    3  2 RO  R3I ¼ d0 JcA15 ðA:3Þ 3p R3O leading to the expressions of [19,20], and Eq. (2) of the text.

c cos a

ðB:2Þ

and the points A, B, C, and D are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yA ðRO ; a; gÞ ¼  sin a  R2O  c2 þ c  cos a

ðB:3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2I  c2 þ c  cos a

ðB:4Þ

yB ðRO ; a; gÞ ¼  sin a  yC ðRO ; a; gÞ ¼ sin a 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2I  c2 þ c  cos a

ðB:5Þ

yD ðRO ; a; gÞ ¼ sin a 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2O  c2 þ c  cos a

ðB:6Þ

where g ¼ RI =RO . We then can calculate I from I¼

J 1X ðS1 ðak Þ þ S2 ðak Þ þ S3 ðak ÞÞ J k¼1

ðB:7Þ

where ak runs from 0 to p=2 as k runs from 1 to J , and where S1 is the lengths in region 1 and is given by 2 Z yB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2O  y 2  j xj dy ðB:8Þ S1 ¼

J ξ

y 0 ¼ x tan a þ

r

yA

r

y

y0

ρ

α

A

B

C

D x0

2c

x

Fig. 16. Basic subelement cylindrical geometry and parameter definitions.

Fig. 17. End-on view of split subelement geometry with associated definitions.

724

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

S2 defines the lengths in region 2 and is given by 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ffi 2 R RO  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R > 2 2 2 2  2 2 < RI 2 R  y  y dy þ 2 R  y dy R I O O y RI S2 ¼ R B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 > : yC R2O  y 2  R2I  y 2 dy yB

9 = a ¼ 0>

if

ðB:9Þ

> ;

otherwise

and S3 defines the lengths in region 3 and is given by

S3 ¼

8 R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 yD > 2 2  jxj dy þ RO 2 >  y R2o  y 2 dy R > O yC yD > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > > R yD 2 2  jxj dy >  y R > O y > > C  > 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2  > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > R RI 2 2 < R2  y 2  R2  y 2 dy R  y  jxj þ O

yC

if if

I

O

2 2 R yD pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2 2 þ jxj dy þ RO 2 2 > þ  y R  y dy R > O O R y D > > I   >   2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 > > R yD 2 2 2 2 2 2 > RO  y  R I  y RO  y  jxj þ dy > > yC > > >    > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 RR > : þ R RI 2  R2  y 2  R2  y 2 dy þ O 2 R2  y 2 dy yD

I

O

 p c c a < 6 sin RO RI p 2  c a < 0 6 sin 2 RO

RI

The results of these calculations are shown in Fig. 4 of the main text, along with those of the approximate treatment.

Appendix C. Slab calculation of the influence of magnetization of the central region Let us compare the magnetization of a ‘‘split’’ (central flux assumed not present) and ‘‘unsplit’’ (central flux assumed present) slab model, see Fig. 18. In this case, the field within the superconducting regions is given by BðxÞ ¼ C1 Jc ðRO  RI Þ þ lH on the shielding branch, where C1 depends on the system of units. Then the average value of B  lH normalized over the whole sample (split case) is

O

if

ðB:10Þ

> > > > > > > > > > > > > > > > > ;

jyD j P RI

otherwise

h B  lH isplit ¼

9 > > > > > > > > > > > > > > > > > =

i 1 h C1 Jc ðRO  RI Þ2 wL 2AL

ðC:1Þ

For the unsplit case, this is " # 2 C 1 J c ð RO  RI Þ þ RI ð RO  RI Þ h B  lH i ¼ 2 2RO

ðC:2Þ

Since d / DM / B  lH , and using A ¼ wRO =2, we see that dsplit 1 h i ¼ dunsplit 1 þ 2g 1g 2 1g

ðC:3Þ

Normalized versions of Eqs. (C.1) and (C.2) are plotted in Fig. 19. 1.0

Split Unsplit

0.8

deff*/deff*(η = 0)

w

y x

L

0.6

0.4

0.2

RO RI

Fig. 18. Simple slab geometry and associated parameter definitions.

0.0 0.0

0.2

0.4

η

0.6

0.8

1.0

  Fig. 19. deff =deff ðg ¼ 0Þ in split and unsplit cases for simple slab geometry.

M.D. Sumption et al. / Cryogenics 44 (2004) 711–725

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