Record field in a 10 mm-period bulk high-temperature superconducting undulator

A 10 mm-period, high-temperature superconducting (HTS) undulator consisting of 20 staggered-array GdBa2Cu3O7−x (GdBCO) bulk superconductors has been fabricated and tested successfully. Each GdBCO disk was machined into a half-moon shape with micro-meter accuracy and shrink-fitted into a slotted oxygen-free copper disk which provided pre-stress and effective conduction-cooling. The HTS undulator prototype, consisting of GdBCO disks, copper disks, and CoFe poles fitted in a long copper shell, was field-cooled magnetized in fields of up to 10 T at 10 K. An undulator field of 2.1 T in a 4 mm magnetic gap was obtained. This field is the largest reported yet for the same gap and period length and exceeds the target value of 2 T for the meter-long HTS undulator scheduled for the hard x-ray I-TOMCAT beamline in the Swiss Light Source 2.0. We have demonstrated that bulk superconductor based undulators can provide significantly improved performance over alternative technologies.

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Introduction
Superconducting undulators (SCUs) exhibit excellent tunability of radiation wavelength and durable operation without demagnetization induced by electron beam losses. SCUs have been successfully developed and commissioned for user operation in the Karlsruhe Institute of Technology synchrotron and the Advanced Photon Source (APS) storage ring in the past decade [1][2][3]. Compared to permanent magnet undulators (PMUs) [4], SCUs provide unparalleled photon flux in the high energy part of the x-ray spectrum. However, when the undulator period, λ u , is below 15 mm a standard NbTi SCU does not provide a higher on-axis undulator field, B 0 , than the best cryogenic PMUs for the same magnetic gap, g [5]. The reduction in both λ u and B 0 results in a low deflection parameter K = 0.934λ u [cm]B 0 [T] below which no high harmonic photons can be effectively utilized in synchrotron light sources. Nb 3 Sn conductor with higher engineering current density J e has been selected for the development of several SCU prototypes for the APS storage ring and the Linac Coherent Light Source II [6,7], however, its superiority has not yet been validated for a period less than 15 mm.
Short undulator prototypes consisting of high-temperature superconducting (HTS) coils have been widely studied but have not yet delivered performance superior to NbTi or Nb 3 Sn undulators [8][9][10][11]. A promising approach for achieving high B 0 for a short period λ u is to employ a staggered-array bulk HTS undulator (BHTSU) [12][13][14]. The BHTSU consists of a series of staggered-array ReBa 2 Cu 3 O 7−x (ReBCO) bulks which are field-cooled (FC) magnetized by a superconducting solenoid. Experimental studies on 10 mm-period BHTSU prototypes demonstrate that the on-axis field B 0 can reach 0.85 T for g = 6 mm and 1.54 T for g = 4 mm, respectively [14,15]. Importantly, these results were obtained at 10 K which could be achieved with reduced cooling cost in comparison to ∼4 K required for NbTi/Nb 3 Sn undulators.
In this letter, we report an undulator field of 2.1 T obtained in a BHTSU prototype made of industrial GdBCO/Ag bulks with a designed period of 10 mm and magnetic gap of 4 mm.

Experiments
Quenches occurred in previous short undulator prototypes made of GdBCO/Ag bulks. These were ascribed to three possible influences: (a) poor assembly which resulted in relative motion between disks during magnetization; (b) poor cooling capability due to the gaps being filled with epoxy resin and (c) large Lorentz forces and associated tensile stresses in the GdB-CO/Ag bulks during magnetization. To address these issues we designed a new 10-period BHTSU prototype, as shown in figure 1. It consists of commercial GdBCO bulks from Nippon Steel Corporation and exploits the shrink-fit assembly technique which helps to constrain the GdBCO disks, provide better cooling conditions and reduce their tensile stresses [16,17]. Measured results of the mechanical strains in a previous half-period bulk YBCO undulator suggest the superconducting disk is well compressed by the copper disk after cool down to 77 K. More details about the strain measurement can be found in the appendix. To predict how much the applied prestress by the copper disk and the aluminum shell could compensate the Lorentz forces, a one-quarter 3D electromagneticmechanical coupled finite element analysis model was created in ANSYS to simulate the FC magnetization from 8 T with implementation of the E-J power law (J c = 1 × 10 10 A m −2 , n = 20) and compare the peak tensile stress in the GdBCO disk with and without being pre-stressed [18]. As shown in figure 2, during magnetization the peak 1st principal stress (maximum tensile stress) in the pre-stressed GdBCO disk reaches 39 MPa, significantly lower than the 82 MPa experienced by an unconstrained GdBCO disk. A direct comparison of the stress patterns for ∆B s = 8 T is provided on the right hand side of figure 2. It can be observed that the majority of the pre-stressed GdBCO disk experiences compressive stress which is unproblematic in ceramic-like materials, while the rest experiences only modest tensile stress.
The undulator design incorporates a period length of 10 mm, magnetic gap of 4 mm and mechanical gap of 3.4 mm. The half-moon shaped GdBCO disks, long copper shell, slotted oxygen-free copper disks and CoFe poles are machined with micro-meter accuracy utilizing electrical discharge machining wire erosion. The assembly was carried out as follows (i) The slotted copper disks were first heated up to 200 • C; (ii) The GdBCO disks and CoFe poles (expected to enhance the undulator field B 0 by 10%-15%) were then pressed into the copper disks;  (iii) All GdBCO-Cu and CoFe-Cu pieces were finally clamped by a long copper shell as shown in figure 1.
After assembly, the undulator sample was connected to the vertical magnetic field measurement system and installed in a variable temperature insert in the 12 T superconducting solenoid in the Bulk Superconductivity Group at the University of Cambridge. The undulator is cooled directly by flowing helium gas and its working temperature is controlled by a heater wrapped on the outer copper shell. A 3 mm-diameter x3yz-probe shown in figure 3, supported by a meter-long carbon fiber reinforced plastic tube and controlled by a motorized linear stage outside the cryostat, is employed to characterize the on-axis magnetic field in three orthogonal directions. The undulator field B y is measured at three different y positions, with one on-axis and the other two off-axis of ±0.1 mm. For other details regarding the experimental setup, please refer to [14].

Results
The BHTSU prototype made of Nippon GdBCO bulks was first FC magnetized from 10 T to 0 T at 10 K with the onaxis field measured every 1 T and then subcooled to 7.5 K to minimize the flux creep. The field ramp rate was kept at ∼−1 T h −1 during magnetization. By analyzing the measured field profiles from one on-axis InAs sensor and the other two off-axis of ±0.1 mm we observe that there is an offset of ∼0.1 mm between the undulator axis and the axis of the 3 mmdiameter x3yz-probe. The measurement field profile with the smallest amplitude is thus considered as the practical on-axis undulator field. Figure 4 summarizes the measured on-axis magnetic field profiles during the full FC magnetization process. Higher field amplitude was observed at both ends because the periodicity stops and the sizes of the end bulks were not optimized in this stage. To evaluate the performance under various ∆B s , we have compared the mean undulator field B 0 (the two end peaks are excluded) and the field inhomogeneity σ B /B 0 as shown in figure 5. σ B is the modified standard deviation and estimated by     the field inhomogeneity can be further minimized and a root mean square (RMS) phase error of only a few degrees can be achieved [19].
To minimize the flux creep-induced field decay, the BHTSU prototype was subcooled to 7.5 K after FC magnetization from 10 T to zero. The undulator field was measured immediately after subcooling and during the initial field decay over 120 h. The measured data points of B 0 were fitted by the 'interpolation formula', the time dependence of the persistent current j(t) in the collective creep model suggested in [20]. It is expected that B −µ 0 (µ = 1) is proportional to lnt with further increase in the waiting time as shown in figure 6(a). In the meantime, the field inhomogeneity increases over time but its change, σ B /B 0 −σ i B /B 0 , is very small and it seems to stop with an exponential trend, i.e. proportional to 1−e −t/τ where τ = 2.2 d as shown in figure 6(b).

Conclusion
A HTS undulator prototype made of industrial GdBCO/Ag bulks has been fabricated with new machining and assembling techniques. We have achieved a mean B 0 of up to 2.1 T with a 10 mm-period and 4 mm-magnetic gap. This undulator field of 2.1 T is the highest achieved to date for such a short period and exceeds the target value required for the I-TOMCAT beamline in the SLS 2.0. The quenches that hindered our previous undulator prototypes have been eliminated in this experiment thanks to new undulator fabrication techniques. The field inhomogeneity caused by J c variation among GdBCO disks is within 2.7%. It is observed that the flux creep is not negligible after subcooling from 10 K to 7.5 K while the field homogeneity reduces slightly over time.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Acknowledgments
This work was performed under the auspices and with support from the Swiss Accelerator Research and Technology (CHART) program (www.chart.ch), the LEAPS-INNOV WP6 (www.leaps-innov.eu/) and Cambridge Royce facilities grant EP/P024947/1. The work of M D Ainslie was carried out while affiliated with the Bulk Superconductivity Group, Department of Engineering, University of Cambridge, and supported by the EPSRC Early Career Fellowship, EP/P020313/1. The authors would like to acknowledge the RIKEN SPring-8 Center for the constant support to our development, specifically for the procurement of the ReBCO crystals used for the preparation of the undulator sample and to thank L A C. Huber from Paul Scherrer Institute for technical support in preparing the sample. All data are provided in full in the results section of this paper.

Appendix
To validate the shrink-fit assembly technique, we first assembled a half-period undulator prototype consisting of a 4 mm-thick, half-moon shaped YBCO disk, slotted copper disk and aluminum shell, as shown in figure 7(a). To quantify the pre-stress contributions, eight strain gauges were mounted on the YBCO disk and one strain gauge was mounted on a small stress-free YBCO block for thermal compensation, as shown in figure 7(b). An inverse measurement of the mechanical strains was carried out as follows Taking the final stress-free YBCO disk as the reference, the total measured mechanical strain, as shown in figure 7(e), consists of four contributions: (a) 'S RT YC − S RT Y ' due to shrink-fitting with the copper disk; (b) 'S RT YCA − S RT YC ' due to shrink-fitting with the aluminum shell; (c) 'S 77K YC − S 77K Y ' due to the compressive force provided by the copper disk upon cooling and (d) 'S 77K YCA − S 77K YC ' due to the compressive force provided by the aluminum shell upon cooling. The strain data for No. 2 and No. 8 are missing because of broken wires during the measurements. It can be observed from the plot that the major compressive strain comes from the compressive force provided by the copper disk when cooled to 77 K. Both shrink-fit assemblies contribute little to compressing the YBCO disk. Most of the compressive force provided by the aluminum shell at 77 K is absorbed by the copper disk and the residual force transmitted to the YBCO disk is limited. This suggests that the shrinkfit assembly with aluminum shell is not necessarily required in the bulk HTS undulator.