Ⅱ-9. Instrumentation

RIKEN Accel. Prog. Rep. 47 (2014)

Extraction of 3D field maps of magnetic multipoles from 2D surface
measurements†
H. Takeda,∗1 T. Kubo,∗1 K. Kusaka,∗1 H. Suzuki,∗1 N. Inabe,∗1 and J. A. Nolen∗2

†
∗1
∗2

Condensed from the article in Nucl. Instr. Meth. B317, 798809 (2013)
RIKEN Nishina Center
Argonne National Laboratory

Br ( r ,θ , z ) , Bθ ( r , θ , z ) , Bz ( r , θ , z )measurements on the surface of a cylinder

Fourier analysis with respect toBr , n ( r , z ) , Bθ , n ( r , z )Fourier transform with respect toBr , n ( k ) , Bθ , n ( k )Eqs.(5), (6)bn ,0 ( k )inverse Fourier transform

bn ,0 ( z )Eq.(4)

Eq.(2)

bn , m ( k )inverse Fourier transform

bn , m ( z )Eqs.(3a), (3b), (3c)Full 3D field maps of multipoles

Fig. 1. Diagram of full 3D ﬁeld map extraction from 2D
surface measurements. The process of extracting theleading term bn,0 (z) and the pseudo terms bn,m (z) fromthe 2D measurements of the surface of a cylinder are
shown step-by-step. The equation numbers shown by
the arrows indicate those used for the corresponding
processes described in the original paper† .
1.6
1.4
1.2
10.8
0.6
0.4
0.2
0-0.2-800

b2,0b2,1b2,2b2,m (T)In large-aperture, short-length magnets with strong
magnetic ﬁelds, such as superconducting triplet
quadrupole (STQ)1) magnets in the BigRIPS2) , thefringing ﬁeld region is generally very large, and theshape and eﬀective length of the magnetic ﬁeld distribution change with the excitation current due to saturation of the iron core. Further, higher-order pseudo
terms become relatively large in these magnets compared to those in small-aperture, long-length magnets
because they originate from the changes of the magnetic ﬁeld in the direction along the beam axis. It isindispensable to correctly extract pseudo quadrupole
components as well as ﬁrst-order quadrupole components from measured 3D ﬁeld maps even for ﬁrst-orderion-optical simulations.
Pseudo terms have the same azimuthal angle dependence as that for the leading term, such as cos 2θ fora quadrupole, but have a higher-order radial dependence, such as r3 rather than r for a quadrupole. Atﬁrst glance, it appears that ﬁeld map data measured at
diﬀerent radii are required to solve the r dependence.
However, we present a practical numerical method thateliminates the need for this data. In this method, themeasurement data for one radius of one component
in the cylindrical coordinates, i.e., 2D ﬁeld measurements on the surface of a cylinder, are suﬃcient todetermine the full 3D magnetic multipole ﬁeld in thecylinder. Using this novel method, we can extract thedistributions along the beam axis for the coeﬃcient of
the ﬁrst-order 2n-pole component bn,0 (z), which is theleading term of the 2n-pole components in the multipole expansion of magnetic ﬁelds. Higher-order pseudo
components bn,m>0 (z) can be deduced from the leading
term via recursion relations. The full 3D ﬁeld map of
2n-pole is completely described by these components.
See the original paper† for details about the formalism and procedure of the method. Steps of the processof extraction of full 3D ﬁeld maps of magnetic multipoles from 2D surface measurements are summarized
in Fig. 1.
The proposed method was applied to large-aperture
STQ magnets in the BigRIPS fragment separator at
the RIKEN Nishina Center RI Beam Factory. Figure 2 shows the leading term b2,0 (z) together with thepseudo terms b2,1···4 (z), which were obtained from themeasurement result of Bθ,2 at 100 A. Here b2,1 is relatively large, showing that the pseudo terms cannot beb2,3b2,4-400

0400800z position (mm)Fig. 2. Examples of b2,0···4 (z). b2,0 was extracted fromBθ,2 , which was measured at a radius of 107 mm and at
an excitation current of 100 A for a Q500 quadrupole
magnet in STQ24. Pseudo terms b2,1···4 were calculated
from b2,0 with the diﬀerential recursion relation (2) inthe original paper† .

ignored.
The obtained b2,0 (z) distributions were parametrizedusing the Enge functions to ﬁt the fringe ﬁeld shapes
at all excitation current values, so that unmeasured
values are interpolated. We implemented these parameters in the ion-optical calculation code COSY INFINITY3) and realized a ﬁrst-order calculation that incorporates the eﬀect of large and varying fringe ﬁelds moreaccurately.
References
1) K. Kusaka et al.: IEEE Trans. Appl. Supercond. 14,310 (2004).
2) T. Kubo: Nucl. Instr. Meth. B204, 97 (2003).
3) K. Makino, M. Berz: Nucl. Instr. Meth. A558, 346(2006).

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