LS Note 311: Achievable Magnetic Fields of Super-Ferric Helical Undulators for the ILC

ANL/APS/LS-311

Achievable Magnetic Fields of Super-Ferric Helical Undulators for the ILC

S.H. Kim

Advanced Photon Source, Argonne National Laboratory

April 13, 2006

Abstract – The magnetic fields on the beam axis of helical undulators for the proposed International Linear Collider (ILC) gamma-ray production were calculated for undulator periods of 10 mm and 12 mm. The calculation assumed the use of low-carbon steel for the magnetic poles and a beam chamber outer diameter of 6.3 mm. Using NbTi superconducting coils at 4.2 K, the on-axis field for a 10-mm-period undulator was 0.62 T at the critical current density. The field for a 12-mm undulator period was 0.95 T, which gives a K value of 1.06. The K value for an 11-mm undulator with Nb₃Sn superconducting coils was estimated to about 1.1.

A transverse periodic helical magnetic field may be generated on the axis of a double-helix coil with equal currents in opposite directions in each helix [1]. An ideal helical field may be expressed as

	(1)

where B₀ is the magnetic field modulus on the undulator axis, and k = 2π/λ with λ as the undulator magnetic period length along the electron-beam direction on the z-axis. The radiated photon energy e_n for the nth harmonic is given by

	(2)

with E as the electron beam energy, γ is the relativistic factor of the electron energy, and θ is the angle between the z-axis and the radiated photon beam direction. The deflection parameter K is defined as

K = 0.0934 λ(mm)B₀(T)	(3)

From Eq. (2), the photon energy may be calculated. Currently, the proposed International Linear Collider (ILC) lists λ= 10 mm and K = 1 for the helical undulator parameters [2]. For K = 1, the spectral power density of the photon beam is maximum at γθ =1 for a single electron [1]. This note calculates the fields on the undulator axis with a beam chamber outer diameter of 6.3 mm for periods of 10 mm and 12 mm to evaluate whether the listed K value could be achieved.

Figure 1 shows a double-helix model coil for the low-carbon steel poles and the beam chamber. A double-helix superconducting (SC) coil with equal currents in opposite directions in each helix was inserted in between the steel coils for the field calculations. A typical calculation for the on-axis fields corresponding to Eq. (1) is plotted in Fig. 2. The calculation used OPERA-3d [3]; the B(H) data of “low-carbon” steel included in the software were used for the steel poles shown in Fig. 1.

In Fig. 3, on-axis field modulus B₀ (right axis) calculated for periods λ = 10 mm and 12 mm, and the corresponding maximum fields in the coil (left axis) are plotted as a function of the average current density in the coil. The average critical current densities J_c(NbTi) and J_c(Nb₃Sn) at 4.2 K, for the NbTi and Nb₃Sn SC coils (bottom axis), respectively, are functions of the coil maximum field (left axis) and limit the coil current densities and the on-axis fields. The figure shows that, at J_c(NbTi) around 1.15 kA/mm², the on-axis fields are approximately 0.55 T (K = 0.51) and 0.95 T (K = 1.06) for λ = 10 mm and 12 mm, respectively. The J_c(Nb₃Sn) for λ = 10 mm is about 1.9 kA/mm², which gives an on-axis field of 0.85 T (K = 0.8). From the figure, one could estimate an achievable K value of about 1.1 for λ = 11 mm.

When the operating current density is close to the critical current density J_c(NbTi) (see Fig. 3), the stability margins of the device may become an issue and may be simulated with some form of heat loads from inside the beam chamber [4]. For the development of planar-type SC undulators at the Advanced Photon Source (APS), an average J_c(NbTi) up to 1.4 kA/mm² was achieved for a 14.5-mm-period short section. Considering the coil geometry for the helical undulator, J_c(NbTi) and J_c(Nb₃Sn) in Fig. 3 were reduced by 20% from that used in ref. [4]. The stability margin at the operating current density for λ = 12 mm may be enhanced by using an Nb₃Sn coil.

This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.

References

[1] B.M. Kincaid, J. Appl. Phys. 48, 2684 (1977).
[2] http://www.linearcollider.org/cms/
[3] OPERA-3d, Vector Fields Ltd., Oxford, England. The author does not imply that similar software by other vendors cannot perform the work.
[4] S.H. Kim et al., IEEE Trans. Appl. Supercond. 15, 1240 (2005); S.H. Kim et al., R&D of Short-Period NbTi and Nb3Sn Superconducting Undulators for the APS, Proc. 2005 PAC, 2410 (2005).

Fig. 1 A model of a double-helix coil for the low-carbon steel poles and beam chamber. A double-helix SC coil with equal currents in opposite directions in each helix is to be inserted between the steel coils.

Fig. 1. A model of a double-helix coil for the low-carbon steel poles and beam chamber. A double-helix SC coil with equal currents in opposite directions in each helix is to be inserted between the steel coils.

Fig. 2. Plots of calculated Bx and By, two components in Eq. (1), for one 12-mm period along the undulator axis. The on-axis field B0 was 0.88 T for an average coil current density of 1 kA/mm2 and the beam chamber outer diameter as in Fig. 1.

Fig. 2. Plots of calculated B_x and B_y, two components in Eq. (1), for one 12-mm period along the undulator axis. The on-axis field B₀ was 0.88 T for an average coil current density of 1 kA/mm² and the beam chamber outer diameter as in Fig. 1.

Fig. 3. On-axis field modulus B₀ (right axis) calculated for periods λ = 10 mm and 12 mm, and the corresponding maximum fields in the coil (left axis) are plotted as a function of the average current density in the coil. The average critical current densities J_c(NbTi) and J_c(Nb₃Sn) at 4.2 K, for the NbTi and Nb₃Sn SC coils (bottom axis), respectively, are functions of the coil maximum field (left axis) and limit the coil current densities and the on-axis fields.