1. Magnetization and decay effects in Nb3Sn, Bi:2212, and YBCO relevant to field errors
Center for Superconducting & Magnetic Materials (CSMM), The Ohio State University, Columbus, OH, USA
M. D. Sumption, C. Myers, M. Majoros, E. W. Collings
We thank Advanced Conductor Technologies and University of Colorado, Department of Physics, Boulder, CO, USA
D. van der Laan
W. Goldacker
We thank OST, Supramagnetics for Nb3Sn and Bi:2212
This work was funded by the U.S. Dept. of Energy, Division of High Energy Physics, under Grant No. under Grants No. DE-SC & DE-SC (OSU) (University Program)

2. Motivation – error fields and drift
While near-term (luminosity) LHC upgrades will require only NbTi and/or Nb3Sn, the subsequent energy upgrades of the LHC are expected to require HTS/LTS hybrids
In addition, numerous and various superconducting magnets needed for a muon-collider will require YBCO or Bi:2212 (and Nb3Sn)
The fact that strand and cable magnetization lead to field errors is well known, but up to date characterization is needed
This includes YBCO, Bi:2212, and ongoing Nb3Sn assessment
Below we will start with NbTi dipole magnet characteristics, and explore the relative magnetic properties of Nb3Sn, YBCO, and Bi:2212
To key collider issues will be, what is the size of b3 (the sextupole error field), and what might the use of HTS do to drift in b3

3. Field Errors at collision and injection
In accelerator operation, there is
(1) a low field injection phase, where the dipole magnets operate for some 20 minutes or more at a nominal 1 T injection “porch”
(2) An energy ramp, coast (beam collision)
(3) An energy dump, where the magnets are then cycled back to near zero, followed by a rise to the beam injection field to repeat the cycle
Any strand magnetization leads to deviations from the pure dipole field which tend to defocus the beam.
Such errors are described in terms of the high order multipoles of the field, a good measure is the sextupole component, b3
“Superconducting Accelerator Magnets” by P Ferracin, ETodesco, S O. Prestemon and H Felice, January 2012
The above process leads to a partial compensation of the error fields at injection, a (hopefully small) negative value usually dominated by the sextupole component, b3.

4. Magnetization Error Criteria
The magnet designers criteria for magnet quality is terms of the multipole error fields, a good error to key in on for dipoles in the sextupole component, b3.
b3 should be about “1 unit” or less
1 unit is 0.01% of the main field
The correlation between strand magnetization and error fields is magnet design dependent
cos theta dipoles and block magnets of different designs magnet cross section will have different magnetization to error field correlations
Amemiya, IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 20, NO. 3, JUNE 2010

5. NbTi Mag as a compare to 0.5 T, 15 mT => 12000 A/m
IEEE Trans
“Superconducting Accelerator Magnets” by P Ferracin, ETodesco, S O. Prestemon and H Felice, January 2012
0.5 T, 15 mT => A/m
From df and Jc, at 0.54, the values Mcoup,LHC = kA/m and Msh,inj,LHC = 10.3 kA/m, may be regarded as bench-marks against which the corresponding properties of Nb3Sn cables can be compared.

6. Let’s look at a LARP Cable/Stand
PPMS-measured M-B loop at 1.9 K to ± 14 T
MT23: 3PO3H-03: Extracted Strand Magnetizations of an HQ Type Nb3Sn Rutherford Cable and Estimation of Transport Corrections at Operating and Injection Fields, Edward W. Collings, Mike A. Susner, Mike D. Sumption, and Daniel R. Dietderich

7. Magnetization at Injection, Nb3Sn
At typical injection fields of 1 T, we have reached in this case the rather large value of Msh,cable,inj,1T = 170 kA/m – more than 10 times (about 17 times) the present day NbTi based values for the total Mparasitic
If we did a simple minded correlation of 10 kA/m to a b3 of 3 units, then -- This should lead to b3 values of ten-twenty times or so (30-40 units) for Nb3Sn
That this needs to be reduced is well known. But, what now about that for HTS and also the drift

8. First, let’s look at strand magnetization for some candidate HTS strands
OST Bi:2212
NbTi
Minj,NbTi – 10 kA/m
MinjBi kA/m
Factor of 2 greater – contributors?
deff = 100 m/7 m 15 times greater
Je = 10 kA/mm2 NbTi at 0.5 T/1.9 K
Je = 1000 A/mm2 (OST 2212) 1T 4 K
Factor of 10 less in Jc
Resultant factor  1.5
Minj=-20 kA/m

9. Number of tapes and width vary Increasing I increases area,
Cable
Msat LF, 77 K (kA/m),
cable Vol
Minj, 4.2 K (kA/m), Cable Vol
N x t (mm)
Ic, A collision field
Minj, 4.2 K, kA/m, CV, per 3900 A
Minj,4.2 K kA/m Per strand vol, per 3900 A
M inj,4.2K kA/m per strand Vol, “any current”
CORC
100
500
39 x 4
3900a
1667
Roebel
120
600
10 x 2
4000a
750
Bi:2212 -- 16
240a 20
Nb3Sn
170
10,500b
 170
NbTi 10
 10
YBCO Cables
Similar at 77 K
Moving to 4 K, 1 T
X 10 x 1/2
Number of tapes and width vary
Increasing
I increases area,
M const
Norm to strand vol
Mag comparison for various strands
a 20 T, b 15 T
Iybco = 100 A/4 mm at 20 K
CORC Cable Total Vol =3.14*(7.34mm/2)2 = 42.3 mm2
CORC Cable Volume of Central non-SC region = 3.14*(4.28/2)2 =14.37 mm2
Volume of winding area = 27.93mm2
Strand Vol Danko =0.08*4*39=12.48mm2
CORC Fill factor = 30%
Roebel cable Vol = 5 x 0.08 mm x 5 = 5 x 0.4 = 2 mm2
Roebel Strand Vol = 4 *0.08*5=1.6 mm2
Roebel Fil factor = 80%
Nb3Sn 1500 A/mm2 at 15 T – assume 0.8 mm OD = 0.5 mm2 * 0.5 non-Cu = 375 * 28 = A

10. Comparing 1 T injection Magnetization
NbTi -> Nb3Sn, deff  x 9, x 2
NbTi -> Bi2212 deff  x 15 Jc  x 10
NbTi -> YBCO deff  x 500 Jc  1/2
So…
Do we take the magnetization per volume cable?
Or…
Do we take the magnetization per volume strand?
Or…
Do we take the magnetization per volume x the difference in Je – if so, the Je at what field?

11. Let’s consider it in terms of total moment of the winding
m = MV=MLA, but M  Jcd  m  JcdLA
Now, consider this for NbTi, then we replace with Nb3Sn in a magnet of same field
JcNb3Sn = J’ = JcNbTi = Jc, and deffNb3Sn = d’ = 10 dNbTi
Then m’ = Jc’D’L’A’, and but L’=L, and since B is
unchanged, I is unchanged,
And I = AJop, then A’/A = Jop/Jop’, and if Jop/Jop’ = J/J’, then m’/m = d’/d, as per expectation
But, how do we compare machines with different field targets? If B  I, then
B’/B = I’/I, then but now, I = AJop A=I/Jop
I’ = A’Jop’  A’=I’/Jop’
Then A’/A = (I’/I)(Jop/Jop’) = (B’/B)(Jop/Jop’)
Then m’/m = (Jcm’/Jcm)(d’/d)(B’/B)(Jop/Jop’)
𝑚 ′ 𝑚 = 𝐽 𝑐 ′ 𝑑 ′ 𝐿 ′ 𝐴′ 𝐽𝑐𝑑𝐿𝐴
𝑚 ′ 𝑚 = 𝐽 𝑐𝑚 ′ 𝑑 ′ 𝐿 ′ 𝐴′ 𝐽𝑐𝑚𝑑𝐿𝐴

12. A weighted comparison of magnetization
m’/m = (Jcm’/Jcm)(d’/d)(B’/B)(Jop/Jop’)
HTSC magnets are presumably to be hybrids, using, say Nb3Sn to reach 15 T, and HTS to get to 20T.
The final effective “average” contribution could be a 75%/25% mix – pushing YBCO/Nb3Sn hybrids to significantly lower mag –
However, the HTS will be presumably closer to the beam.
One final Note: While striating the YBCO will not influence Roebel cable Magnetization, it will reduce CORC cable magnetization.
From the above considerations, it seems that the striation target should be 100 filaments in a 4 mm wide strand

13. Drift on the injection Porch
Just as important as the absolute value of b3 is any change with time during the injection porch
It is possible to compensate for error fields with corrector coils, but the presence of drift makes this much more difficult
At right is shown the drift of the error fields as a function of time from zero to 1000 seconds for LHC magnets, followed by a snap-back once the energy ramp begins
The underlying mechanism for drift in NbTi magnets is the decay of coupling currents, (especially inhomogeneous and long length scale coupling currents) and their influence on the strand magnetization
Need to keep both b3 and its drift below 1 unit
For NbTi and Nb3Sn based magnets, this is possible

14. Important also to control drift
So, right now we are at several units of drift
Drift in LTS is due to influence of long range coupling current decay on strand magnetization
But HTS materials famously exhibit Giant Flux Creep (Y. Yeshurun and A. P. Malozemoff)
But, Creep goes like kT, so its not a problem at Low Temperatures, right?

15. Creep in YBCO and Bi:2212 at LT
But, what about present day strands?

16. OST, B = 1 T
Supramagnetics, B = 1 T
Supramagnetics, B = 12 T
OST, B = 12 T

17. Magnetization Change Bi:2212
A creep rate, Rc (=kT/U0) was extracted for each case assuming τ was negligible and M0 = b.
From this the effective pinning potential was extracted
In addition, the magnetization change over 20 minutes was measured, in terms of ratio and absolute difference
So if we associate 10 kA/m with 3 units, then we have about 1 unit of drift below

18. Measurements of 4 mm wide Superpower coated Conductor
So if we associate 10 kA/m with 3 units, then we have about units of drift below
FO 12 mm wide tape would give units
This would be mitigated by hybrid magnets or field parallel orientations
Measurements of 4 mm wide Superpower coated Conductor

19. Summary and Conclusions
Magnetization contributions for YBCO, Bi:2212, and Nb3Sn have been compared against NbTi
On a direct Moment/Vol, and not correcting for Jc potential, Bi:2212 was seen to be close to Nb3Sn, and YBCO was significantly larger than these ( times)
A weighted contribution assessment, taking into account how much winding is needed and the shape of the Jc curve, led to effective m’ contributions somewhat lower
A striation target was suggested : 100 filaments/4 mm width
Potential contributions to drift on the injection porch from creep (as opposed to the standard LTS mechanism) in Bi:2212 and YBCO were considered
10-20% changes in magnetization over a 20 minute time window were seen at 4K, for an 18 stack Bi:2212 with OK deff and moderate magnetization, this leads to 1 unit or so – but only if Bi:2212 is well filamented
For YBCO, tending to have higher Magnetization, units was seen, would be diluted by hybrid implementation or parallel field orientation
The above striation target, implemented in CORC cable, would make drift also OK for YBCO

20. Appendix

21. YBCO tape At 100 mT, M is 0.5 Am, (i.e., Moment per meter)
If we then divide by the area, (10 mm x 0.08 mm) = 0.8 mm2 = 0.8 x 10-6 m2, then M = .625x 106 = 625 kA/m at 77 K
Going to 4.2 K and 1 T, we get 5 x , leading to 3125 kA/m
This is a 12 mm wide strand, so slicing by 3 gives about 1000 kA/m
Advances pp 509
Resultant factor (Minj/Minj,NbTi) =

22. Now, let’s look at Cables!

23. CORC cable – M(H) loop in perp. field at 50 Hz
M(H) loop of cable #7 at 50 Hz and 77.3 K – 39 strands, 100 A per at 77 K SF
M determined considering volume occupied by tapes
The straight line (in red) represents an initial M(H) curve
Field of full penetration Bp ≈ 0.06 T
Here we have 125 mT = 100 kA/m as compared to the LHc’s 10 kA/m, and a Nb3Sn strands 170 kA/m
But, “what about 4.2 K and in field?”
Cable 7: 10 layers, 39 tapes, ID = 4.8 mm, OD = 7.34 mm, Lp = 34 mm

24. Jc (B) curves of YBCO CCs
A. Xu, J. J. Jaroszynski, F. Kametani, Z. Chen, D. C. Larbalestier, Y. L. Viouchkov, Y. Chen, Y. Xie and V. Selvamanickam SUST, vol. 23, p , 2010
Jc (B) curves of YBCO CCs
Dropping from 77 K SF to 4.2 K SF increases Jc by x 10.
Increasing from SF to 20 T decreases  by about 10
Increasing from SF to 1 T, about 2 times less
So, we expect
Minj  500 kA/m
M20 T  100 kA/m
This is expected for any field orientation, since cable is cylindrically symmetric

25. M-H Roebel Cable, B
Cable Type: Roebel, 10/2 (10 strands of 2 mm width)
Pitch length: 90 mm
Strand: AMSC coated conductor (Ni-5at%W substrate)
Ic (77K, self-field) ≈ 400 A
Sample Length: 50 cm
M = 150 mT = 120 kA/m
Dropping from 77 K SF to 4.2 K SF increases Jc by x 10. Increasing from SF to 20 T decreases J by about 10
Increasing from SF to 1 T, about 2 times less, so, we expect
Minj  600 kA/m
M20 T  120 kA/m
This is expected for B

26. Comparing 1 T injection Magnetization
NbTi -> Nb3Sn, deff  x 9, Jc x 2
NbTi -> Bi2212 deff  x 15 Jc  x 10
NbTi -> YBCO deff  x 500 Jc  1/2
P Lee
ASC/
NHMFL