Quasi-sheet quadrupole triplets

ARTICLE IN PRESS

Nuclear Instruments and Methods in Physics Research A 524 (2004) 39–45

Quasi-sheet quadrupole triplets V.S. Skachkov*,1, A.N. Ermakov2, V.I. Shvedunov2 World Physics Technologies, Inc., 1105 Highland Circle, Blacksburg, VA 24060, USA Received 21 November 2002; received in revised form 15 December 2003; accepted 3 January 2004

Abstract We have constructed quasi-sheet quadrupole triplet magnets from homogeneously magnetized Nd–Fe–B prisms whose base angles and magnetizations we calculated to obtain the focusing gradients required by the reverse orbits of a recirculating electron accelerator. We describe here the triplet design, construction, tuning, measured parameters, and operation. r 2004 Elsevier B.V. All rights reserved. PACS: 75.50.W Keywords: Quadrupole magnets; Multipole magnets; Rare earth magnets; Magnet design

1. Introduction Our Quasi-Sheet Multipole (QSM) model [1] allows compact magnets to be constructed using a minimum of Rare-Earth Permanent Magnet (REPM) material. QSM magnets are distinguished by their magnetically soft iron yoke, which is essential to their function and only incidentally provides a magnetic shield. The simplest QSM designs, close analogs of ‘window frame’ dipole electromagnets [2] and Panofsky lenses [3], have working regions with large longitudinal-to-gap dimensions. QSM magnets have much in common with REPM split-pole multipoles [4,5] in that they: (i) have pure multi*Corresponding author. [email protected] (W.P. Trower) Fax:+1-540-953-2249. 1 Permanent address: Nuclear Physics Institute, Moscow State University, 119899 Moscow, Russia. 2 Permanent address: Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia.

pole fields excited by accurately arranging REPM material around a working region; (ii) do not require magnetically soft pole tips; (iii) provide desired magnetization distributions by approximating an ideal azimuth magnetization rotation using homogeneously magnetized REPM blocks, circular segments [4] or rods [5] in split-pole multipoles; (iv) allow any number of magnetic elements up to and including continuously magnetized blocks of complex shapes [6]; and (v) can have right cylindrical working regions. QSM dipoles and quadrupoles can be constructed entirely of homogeneously magnetized REPM blocks with triangular or rectangular bases, simplifying magnet assembly while reducing attendant safety hazards. Since there are no molecular volume currents, (i.e., the curl of the block magnetization, I; vanishes) homogeneously magnetized REPM magnets have only microscopic sheet currents on their surfaces. These currents, when precisely deployed close to the interior yoke

0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.01.051

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V.S. Skachkov et al. / Nuclear Instruments and Methods in Physics Research A 524 (2004) 39–45

surface, excite a pure multipole field in practically any arbitrarily shaped working region, a significant advantage of QSM magnets over split-pole magnets [6]. Conventional REPM magnets, including ‘box’ magnets, leak flux from the pole tips in the transverse cross-section both close to, and outside of, the working region and also in the inaccessible spaces between the pole tips and the yoke. QSM magnets, in contrast, deploy REPM material so that the molecular sheet currents exactly follow the yoke profile rendering pole tips redundant and utilizing almost all the generated magnetic flux. Magnet openings produce fringe fields and only there do QSM magnet fields deviate substantially from the desired multipolarity. However, accurate field source distributions even in short QSM quadrupoles provide linear integral fields, which a relativistic particle beam sees as an ideal lens with a longitudinal ‘step’ field. On this basis we developed two QSM quadrupoles, the so-called amagnets [7], one having fixed and the other variable gradient fields, that are operating in our RaceTrack Microtron (RTM) injection beam lines [8,9]. Only the entrance/exit slits and mechanical imperfections limited their field linearity to 2–3%. The finite quadrupole length effect of all short lenses is important for QSM magnets since their field tuning depends on their specific fringe fields. A multipole magnet field quality is usually described by a two-dimensional harmonic expansion of the magnetic field or potential. However, short multipole magnet fringe fields have no characteristic longitudinal flat top and so are not adequately described by cylindrical multipoles. Thus, our measure of field quality is the normalized transverse magnet field behavior. Since linearity guarantees good relativistic charged particle beam optics, this method is simple, reliable, and sufficient. Requiring no power supplies, and having comparable capital but much reduced operating costs, REPM quadrupoles enjoy decisive advantages over their electromagnetic counterparts for focusing and transporting charged particle beams as they produce stronger magnetic fields in more compact devices. QSM quadrupoles are attractive for drift tube accelerators that use highly compact

quadrupole lenses as their sole focusing element. Although QSM quadrupoles are somewhat larger than functionally equivalent split-pole magnets [4– 6], they are easier to construct as they use a small number of simple geometry magnetic elements and their iron yoke prevents the intrusion of external fields. The QSM triplets we describe here were designed for our 14-orbit 70 MeV RTM [9] with its rectangular accelerating structure that provides Radio-Frequency quadrupole horizontal focusing and vertical defocusing. We found that a single electromagnetic quadrupole located on the common RTM axis could not provide sufficient focusing so we placed quadrupole triplets with B2 m vertical focal lengths on the six evennumbered reverse orbits. Since reverse orbit beams are dispersive, and to avoid longitudinal and horizontal oscillation mixing, our triplets horizontally focus at infinity. The space into which the triplets must fit is restricted to 33.5 mm between orbits and 80 mm between end magnets. We obtained the required triplet optical parameters with 18-mm diameter aperture, 8-mm long outer lenses, Q1 and Q3 ; and a 16-mm long central lens, Q2 : The optical properties of these short triplets depend critically on the longitudinal field distribution especially since the overlap of neighboring triplet oppositely directed fields decrease their focusing strength. All our triplets had the same geometry, required special field tuning [1], only the mentioned demagnetization, and provided the design gradients to within B1%.

2. Triplet design Our QSM quadrupole triplets are seen in Fig. 1 and their principal parameters are listed in Table 1. The relativistic electrons energies in six triplet equipped orbits range from 9.8 to 57.8 MeV.Approximating the field gradient in each ‘step’ along the lens axis, seen in Fig. 2, and using conventional beam transport matrix methods, we found that Q2 could be twice as long as Q1 and Q3 so that all three lenses would have approximately the same gradients thus all REPM blocks could be the same size.

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easily installed directly on the 18-mm diameter beam pipes.

Yoke Q2/ 2 Q2/ 2

Q1

41

Q3 32

3. Triplet construction 18

8

8

8

18

8

80

(a) y

I Yoke 45° I REPM r0 x

(b) Fig. 1. Quadrupole triplet lens (a) arrangement and (b) quadrant cross-section.

Triangular-base REPM blocks excite the field in a volume bounded by an interior yoke surface seen in Fig. 1(b). For an infinitely long QSM quadrupole lens with minimal transverse dimensions, aperture radius r0 ; and block magnetization I; and angle c; the gradient is [1] G¼

m0 I sin 2c 2R0

where m0 is the free space permeability. We defined the molecular sheet current density by choosing a set of cs that was simple to construct and then we calculated the magnetization for each of our triplets. The requirement that yoke leakage flux minimally intrude on neighboring RTM orbits sets the 5-mm yoke thickness and the o1.0 T yoke field. All yokes are 22-mm on a side except for the 12th orbit, which is 26 mm. A soft iron shield surrounds and extends 6 mm beyond the physical triplet to reduce the leakage flux on adjacent orbits and to improve the fringe fields. Each Nd–Fe–B prismatic block is 8-mm long and has coercivity on magnetization of less than 1.5 T.3 Our triplets are 3

Each numerical I and H is multiplied by m0 :

We individually demagnetized each block before assembling the magnet. We discuss this wellknown widely used practice here as we applied it to our triangle-base blocks. We were able to use this procedure because our blocks were small in both their longitudinal and transverse dimensions. Thus any elemental block volumes demagnetization uncertainty did not substantially alter the field linearity. When we tuned our triplets we found no significant demagnetizing induced magnetization non-uniformities and our measured normalized fields were transversely linear. Local magnetization corrections would have been required only if the magnet had been several times longer than its aperture radius. We did not use local re-magnetization. To obtain the required magnetizations for each block type, seen in Fig. 3(a) for c ¼ 15 , we saturated the block and then selectively applied an external demagnetizing field so that the working point moved from the limit demagnetization curve onto the lower magnetization hysteresis loop, as seen in Fig. 3(b). For some demagnetizing fields the working point was at A and all the block working points lay on the return curve 1. For stronger demagnetizing fields we reach a lower point C and the corresponding curve 2 had lower remnant magnetization. Because our quadrupoles have relatively small fields their block magnetic operating point lay in the 2nd, not the 3rd, demagnetization loop quadrant. The block magnetization was achieved in a small number of tuning iterations and was relatively undisturbed by insertion in the triplet assembly. We used demagnetization regimes that passed through the third demagnetization loop quadrant and found appropriate individual demagnetization factors that produced the required block magnetic moment. Thus we had equal average block magnetization for each quadrupole lens [1]. Operationally we placed a demagnetized block in a pulse-coil with its magnetization perpendicu-

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Table 1 Triplet parameters RTM Orbit Beam energy (MeV) Gradient (T/m)

Q1; Q3 Q2

c IðTÞ

Q1; Q3 Q2

2

4

6

8

10

12

9.8 9.4 10.0 15 0.40 0.43

19.4 18.2 19.2 15 0.78 0.82

29.0 27.0 28.4 30 0.59 0.62

38.6 35.8 37.7 30 0.79 0.83

48.2 44.4 47.0 30 0.98 1.03

57.8 53.3 56.2 45 0.98 1.03

25 Gw

G [ T/m]

20 15 10 5 0 -20

-15

-10

-5

0

5

10

15

20

z [mm]

Fig. 2. Quadrupole gradient curve and ‘step’ approximation. Fig. 4. QSM quadrupole triplets.

I A

C 15°

90°

120°

Holder

45° B

Pulse coil B

11 mm

I

(a)

REPM

1.2 0.9

2

0.6

C

Hall probe

µ 0 I [T]

1 A

0.3

µ 0 H CI

0.0 - 1.5

(b)

- 1.2

-0.9

-0.6

µ 0 H [T]

-0.3

0.0

(c)

Fig. 3. Block (a) geometry (b) demagnetization scheme (c) demagnetization set up.

lar to the triangular side, AB, and parallel to coil axis, as seen in Fig. 3(c). After each demagnetizing pulse we measured the block field B5 mm from its surface with a Hall probe. We adjusted a block magnetic moment in three or four iterations to better than B1% of the value required by our QSM model. This did not guarantee that every elemental block volume was uniformly demagnetized since a variety of working point trajectories

along different demagnetization curves can produce different magnetization changes. With our small blocks we obtained B1% field linearity. For larger blocks and higher accuracy, when our simple demagnetization technique cannot sufficiently suppress the undesired block magnetization non-uniformity, the blocks can be divided into smaller component blocks and/or local demagnetization can be used [1,7]. After assembling each triplet, seen in Fig. 4, we measured the gradient at its center and, if necessary, adjusted the block moments by repeating the above procedure.

4. Triplet field tuning To study our QSM quadrupole field we measured the longitudinal gradient, GðzÞ; in a single lens with length l; and then numerical integrated an approximate ‘step’ gradient, seen in Fig. 2, Z 1 þN Gw ¼ GðzÞ dz: l N

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Our 8-mm lens integral gradient was 63% larger than that measured at the lens center. For 16-mm lens it was 12% larger. This excess was caused by the superposition of the fields from the two other lenses in the triplet. An individual lens magnetic field gradient at its center, G0 ; is reduced as the lens is shortened. For example, a REPM split-pole quadrupole with the minimal outer diameter has G0 =Gw E0:6  l=r0 when l5r0 : We did not calculate the field distribution in determining the desired gradient integral. Instead we used a general QSM magnet theorem, which states that the effective and geometrical lens lengths are equal. So the gradient integral prediction accuracy depends principally on our block magnetization uncertainty, which in practice we reduced to less than 1%. Thus for a QSM quadrupole triplet we only needed to demonstrate

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the field linearity of our three finite length component lens. Since overlapping neighbor lens fields, seen in Fig. 5(a), reduce the triplet strength causing part of the integral to vanish, a secondorder approach is required. Here we divided the integration range into three parts where the gradient seen in Fig. 5(b) conserves its sign. Each part we associated with a corresponding quadrupole and measured the gradient of the first assembled triplet to determine the correct block moments. Using this technique we created compact, unified high-gradient triplets with less than optimal quality REPM material. Our short quadrupole transverse field, seen in Fig. 5(a), was not linear with distance nor was its gradient, seen in Fig. 6(a). This was particularly troubling in the central transverse lens planes where the gradient varied the most.

0.3

40

0.2

20

-0.1 x [mm]

-0.2 -6

6

0

B [T]

0.0

-20 x [mm] -6-15

-40

-0.3

-60

40 0

20

40

20

0 -20

-20 -40

G [T/m]

0.1

-40

z [mm]

z [mm]

-60

-60

(a)

(a) 0.3

45

0.2 2 B norm [ T ]

G [T/m]

30 15 0

-15 -30 -60

(b)

0.1 0.0 -0.1

3

-0.2 1

-0.3 -40

-20

0 z [mm]

20

40

60

Fig. 5. 10th orbit median plane triplet (a) field and (b) z-axis gradient.

-6

(b)

-4

-2

0 x [mm]

2

4

6

Fig. 6. 10th orbit median plane triplet (a) gradient and (b) normalized fields.

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magnetization is a shared property of QSM and split-pole multipole magnets [4–6].

The normalized magnetic field is Bnorm ðxÞ ¼

1 l

Z

z2

By ðx; 0; zÞ dz

z1

5. Triplet operation

3

3

2

2

1

1

0 -1 -2

0 -1 -2

-3

-3 0

5

10

15

(a)

20 z [m]

25

30

35

40

3

3

2

2

1

1 y [mm]

x [mm]

Uncertainty in our accelerating structure RF focusing and possible synchronous phase shifts can both be compensated for by using a commonaxis quadrupole. However, to do so decreases the vertical focusing causing the beam diameter to expand as seen in the calculated envelops of Fig. 7(a). Although REPM triplets give up some of their vertical focusing and substantially reduce the common axis quadrupole strength, they can be used to adjust the horizontal focusing while simultaneously minimizing the vertical beam growth as seen in Fig. 7(b). Our installed triplets have increased the beam transmission through the RTM by B50%.

y [mm]

x [mm]

where the vertical median plane field component, By ; is integrated over the triplet length in all three regions, [zi ; ziþ1 ] (i ¼ 1; 2; 3) in which the gradient sign is conserved. This field was linear, as seen in Fig. 6(b). Typically all three lens gradients were initially about equal as required but were several percent lower than those given in Table 1. After correcting the triplet block magnetizations, we measured normalized field gradient and glued the blocks to the yoke. The measured normalized field linearity of a single finite length QSM quadrupole is an intrinsic property of all such quadrupoles excited by twodimensional magnetization distributions. This independence of both the block dimensions and

0

-1

-2

-2

10

15

20 z [m]

0

5

10

15

20 z [m]

25

30

35

40

-3 0

(b)

5

0

-1

-3

0

5

10

15

20 z [m]

25

30

35

40

Fig. 7. Transverse beam envelopes (a) with and (b) without triplets.

25

30

35

40

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6. Conclusions We have described here QSM quadrupole triplets using homogeneously magnetized REPM prismatic blocks that have pure multipole fields in cylindrical working volumes. The technical advantage of our QSM quadrupoles is that only a few simple shaped blocks are required that are easily manufactured and safely assembled. Our short quadrupole normalized fields are linear in the transverse plane. Our experience in producing these quadrupoles and with their operation in relativistic electron beams further validated our expectations for QSM magnets.

Acknowledgements We thank O.S. Sergeeva for useful discussions, G.A. Novikov for assistance with magnetic measurements and calculations, and W.P. Trower for support and encouraging this work.

References [1] V.S. Skachkov, Nucl. Instr. and Meth. A 500 (2003) 43. [2] G. Parzen, BNL-50536 (1976). [3] L.N. Hand, W.K.H. Panofsky, Rev. Sci. Instrum. 30 (1959) 927. [4] K. Halbach, Nucl. Instr. and Meth. 169 (1980) 1. [5] N.V. Lazarev, V.S. Skachkov, in: R.L. Witkover (Ed.), Proceedings of the 1979 Linear Accelerator Conference, Montauk, New York, BNL-51134, 1979, p. 380.

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[6] V.S. Skachkov, A.V. Selin, I.M. Kapchinskiy, N.V. Lazarev, in: H. Henke, H. Homeyer, Ch. Petit-Jean-Genaz (Eds.), Proceedings of the 1992 European Particle Accelerator Conference, Vol. 2, Editions Frontieres, Berlin, 1992, p. 1400; I.M. Kapchinskiy, N.V. Lazarev, E.A. Levashova, A.P. Preobragenskiy, A.V. Selin, V.S. Skachkov, R.M. Vengrov, in: W. Lord (Ed.), Proceedings of the 12th International Conference on Magnet Technology, Vol. 28, IEEE, Piscataway, NJ, 1992, p. 531; V.S. Skachkov, in: S. Myers, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, J. Poole (Eds.) Proceedings of the 1996 European Particle Accelerator Conference, Vol. 3, Institute of Physics, Bristol, 1996, p. 2190. [7] V.S. Skachkov, A.N. Ermakov, V.I. Shvedunov, in: J.L. Laclare, W. Mitaroff, Ch. Petit-Jean-Genaz, J. Poole, M. Regler (Eds.) Proceedings of the 2000 European Particle Accelerator Conference, World Scientific, Singapore, 2000, p. 2125; V.S. Skachkov, in: J.L. Laclare, W. Mitaroff, Ch. PetitJean-Genaz, J. Poole, M. Regler (Eds.), Proceedings of the 2000 European Particle Accelerator Conference, World Scientific, Singapore, 2000, p. 2122. [8] E.A. Knapp, A.W. Saunders, V.I. Shvedunov, W.P. Trower, Nucl. Instr. and Meth. B 139 (1998) 517; V.I. Shvedunov, A.N. Ermakov, A.I. Karev, E.A. Knapp, N.P. Sobenin, W.P. Trower, in: P. Lucas, S. Webber (Eds.), Proceedings of the 2001 European Particle Accelerator Conference, Vol. 4, IEEE, Piscataway, NJ, 2001, p. 2595. [9] V.I. Shvedunov, R.A. Barday, V.P. Gorbachev, A.M. Gorokhov, E.A. Knapp, N.P. Sobenin, A.A. Sulimov, W.P. Trower, A. Vetrov, D.A. Zayarny, in: P. Lucas, S. Webber (Eds.), Proceedings of the 2001 European Particle Accelerator Conference, Vol. 2, IEEE, Piscataway, NJ, 2001, p. 1601.