The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Hybrid Permanent Magnets for Accelerator Applications
Kuninori ENDO and Mika MASUZAWA
Accelerator Laboratory, High Energy Accelerator Research Organization
1-1 Oho, Tsukuba-shi, lbaraki-ken 305-0801, JAPANAbstract

To shape the precision field distribution in the magnet
aperture, the hybrid permanent magnets can be used. The
permanent magnets in the magnetic circuit provide themagnetomotive force instead of the excitation coil. If thepole pieces are made of iron which is shaped to give adesired field distribution and lined with the permanentbricks, the precise hybrid permanent magnet of theaccelerator grade is obtained. The field strength dependence
on the material and volume of the permanent magnet areinvestigated.

without disturbing it because they have a permeability close
to unity. These magnets are not hybrid but composed ofpure SmCo or NdFeB blocks ensuring a uniform high field
gradient to converge both electron and positron beams at thecrossing point.
In the present study we focus on the performance of thehybrid permanent magnets so as to apply them to thestorage ring or beam transport line. The use of thepermanent magnets benefits in the reduction of thefabrication and running costs. We select two kinds ofpermanent materials, Sr ferrite and SmCo, for the evaluationof the magnet performance.

1 Introduction

2 Permanent Magnet Material and Properties

Recent development of the high performance permanent
magnet, high remanent field, large coercive force and highenergy product, makes possible its application to the fieldof the high energy accelerators. Their magnets require thehighly homogeneous magnetic field over the beam aperture
to guide the beam without causing instabilities, otherwise
the beam will be lost before reaching the maximum energy.
In the accelerator application the deviation of the field
distribution must stay within the limit imposed by thebeam optics. In a circular accelerator such as a storage
electron or (anti-) proton ring at the fixed energy the beam
circulates many times in the same magnets, so the field
quality is essentially important. This is the reason why thehybrid permanent magnet with machined iron poles is used.
There are many product series of permanent magnets
ranging from alnico to rare-earth magnets and they have awide range of applications such as motor, speaker, relay andso on in the industrial products. The first large scale
application is embodied at the Recycler antiproton storage
ring of Fermilab between Booster and Main Injector (MI)after an elaborate hybrid permanent magnet R&D program
which include the counter measure against the temperaturedependent field variation and the protection against theleakage field from the nearby pulse magnets. Recycler ring
recycles the anti-proton beam at 8 GeV from the Tevatronthrough MI to get the luminosity gain of the Tevatron pp
collider by a factor of two.
A small scale application is found at SLAC. In this casea part of the injection line magnets and the chromaticity
correction magnets of the dumping ring use the permanentmagnets [1]. Most recently PEP-II has adopted the highgradient permanent quadrupoles made of samarium cobalt(SmCo) at the interaction region [2] and CESR plans toreplace the present SmCo magnets with ones made ofneodymium iron boron (NdFeB) for the phase-III upgrade[3]. They are installed close to an interaction point in adetector symmetrically and suffer from the solenoidal field

The basic properties of the permanent magnet isexpressed by the demagnetization curve at the second
quadrant which is obtained when its magnetic hyteresis
curve is measured under the sinusoidal external magnetic
field giving a saturation at the extreme fields. Most
important properties are the residual induction and thecoercive force. The larger both values, the higher themagnetic field available. An operating point of thepermanent magnet is given by the point (H, B) on thedemagnetization curve which is determined by the magnetic
circuit. As an available energy is HB !81r, its maximum(HB)max is given by the point where the demagnetizing
curve and the hyperbola HB = const are tangent. (HB)max isthe maximum energy product expressed by a unit of Megagauss Oersted (MG·Oe). Magnetic properties are summarized
for typical permanent magnets in Table 1 [4].
Permanent magnet materials are grouped into three
categories, metallic alloy, ferrite and rare earth. Alnico is themetallic alloy with 1.1-11 MG·Oe and has an excellent
magnetic stability against temperature variation, but it israther expensive because it contains Co.
Ferrite is a double oxide MOFe 20 3, iron oxide Fez03
being a main component and M is Co (cobalt ferrite), Ba(barium ferrite), Sr (strontium ferrite), and etc. Among them
Sr ferrite is most popular because it is cheep and chemicallystable besides it has large coercive force and large resistivity.
Its maximum energy product is 2.0-4.8 MG·Oe andmagnetic properties suffer from the temperature variation.
SmCo5 and Sm 2Co7 have a vary large energy product(16-32 MG·Oe). As it is very expensive, about ten times
more expensive than Sr ferrite, its application is limited to asmall scale. Cheaper neodymium iron boron NdzFe 14B has
the largest energy product (23-49 MG·Oe) at present andwill find various applications including a high energy
accelerator.
Irreversible and reversible thermal demagnetization of thepermanent magnet are observed but the former is a one-time-296-

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

loss depending only on the lowest temperature for Sr ferriteor the highest temperature for SmCo to which the magnet
has been exposed. Freezing Sr ferrite to condition canremove the irreversible demagnetization [5]. This kind ofstabilization is called a thermal seasoning.
An intrinsic temperature coefficient of the residual
inductance is -0.02--0.03%1°C for alnico, -0.03--0.05 for
Fe-Cr-Co, -0.18--0.20 for ferrite, -0.02--0.043 for SmCoand -0.09--0.13 for NdFeB. A large temperature effect on
the field strength is expected for the Sr ferrite magnet,
requiring a measure to compensate the temperature effect. At
Fermilab several packages of 30%Ni-70%Fe alloy strips,
having a low Curie temperature ( -50-70°C) andpermeability with a strong function of temperature,interspersed longitudinally between Sr ferrite bricks subtract
flux in a fashion to null out the overall thermal dependenceof the integrated field strength to less than 0.01%1°C [5,6].
Table 1
CHARACTERISTICS OF TYPICAL PERMANENTMAGNETS.
Residual Coercive Max. energy Curie
Material
product temperature
Induction force(oC)(MQ·Oe)
(kQe)
(kG)
850
1.1-11
0.5-1.6
5.0-14.0
alnico670
1.1-7.0Fe-Cr-Co 8.0-14.5 0.4-0.8
2.0-4.8 450-460
3.0-4.5 1.6-3.95
Sr ferrite
16-32 710-820
8.0-12.0 4.0-10.7
SmCo31023-42
2.0-14.1
9.8-14,5
NdFeB

Usually e"" 1.3 in spite of L; I h"" 10 and /1; I J.lo > 1000
for accelerator magnets assuming A8 = AP. The flux leakage
inevitable in the pole gap is defined by the leakage factor(/>1) as follows,(5)f = BPAP I BgAg.
Inserting (4) and (5) into (3), the slope (s) defining theoperating point at the second quadrant of the permanent
magnet is given by
Li~---------~

f--"------.-.------.-----.!r--1-2df-------1-------'---____L__!_____lll_TI
I,_________ !
Li

q=2(d+p+h)~ Permanent magnet

Fig.l Geometry of the hybrid bending magnet.

0.5

6..

0.3

~

0.2

£

~

3 Simple Formulae of Magnet Designo.•

Assuming that the average half length of the iron yoke
is L; , the half length of the permanent magnet d, and thehalf gap height has shown in Fig.1, the following relation
is obtained according to the Ampere's law,(1)
H8 h+H;L;+HPd= O,
where H8 , H; and HP are the field strength of the pole gap,
iron yoke and permanent magnet, respectively. Let BP theflux density of the permanent magnet at its operating point,(2)
BPAP = J.l;H;A; = J1 0 H8 A8 ,where AP, A; and A8 are crosssections of the permanentmagnet, yoke and pole gap, respectively. J.l; is thepermeability of iron and J.lo that of air (close to thepermeability of vacuum).
From (1) and (2),(B8 A8 )(B8 h)(1 + J1 0 L;A8 I J.l;hA;)
H B =II A dp pp(3)1""0is derived, where B8 is the flux density of the pole gap. Theterm (1 + J.loL;A8 I J.l;hA;) corresponds to the reluctancecoefficient ( t: > 1) of the magnetomotive force (MMF)
which is the correction factor corresponding to the MMF
loss in the iron yoke. Another factor contributing to thisloss is the flux leakage from the iron yoke given by f3,e = HPd I H8 h = 1 + J10 L;A8 I J.l;hA; + [3. (4)__L

00·········~·..,0.5dlh

Fig.2 Field strength obtained by the simple formula andPandira code. Hc=3.35k0e, Br=0.35T for Sr ferrite and
Hc=l0.5k0e, Br=ll.OT for SmCo. Subscripts "pand"and "p" mean results by Pandira code and analyticformula, respectively.
(6)s =BPI HP = -j10 A8 df I APhe.
The permeance ( P8 ) of the rectangular pole gap isapproximately defined as~ ""J1 0 [.ewl2h+0.264 (2.e+2w)+4x0 .077x2h],
where .e and w are the pole length and pole width(A8 = .ew), respectively. Here we neglected contributions
from other parts of the yoke. Then, the leakage coefficient isgiven byf"" 1+ 1.056(hl .e + hlw)+ 1.232h 2 I .ew.
Considering h2 I .ew << 1 and hi .e << 1 for accelerator
magnets, an approximate leakage coefficient isf "" 1+ 1.056h I w.
Considering 0.1 < hI w < 0.2 for accelerator magnets,
1.1 < f < 1.2. If A8 = AP, the operating point of thepermanent magnet is given by
BPI J.loHp = --D.885d I h,-297-The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

and the field strength by
B8 =:BpI f.
Detailed simulation was perfonmed with the Pandira code andcompared with the results obtained by above simple formulain Fig.2.
For an evaluation of the hybrid quadrupole magnet ofFig.3, the similar relations are obtained as follows. The 118
quad can be transformed into 114 dipole in new coordinatesu , v as in Fig.4 through the conformal transformation [7],(7)w = z 2 12R,magnet,
quadrupole
theofradiusboreaisR
wherez = x + iy and w = u + iv.

In the followings Y,pBzpd::: y wpBwpdw is assumed for thesake of simplicity. If there is no leakage flux,Bw8 (2 -li32)R = B,P (2 -114 )R = 2GR 2 ,where G is the field gradient in the z-plane.
The leakage factor is now defined as
f = Bz/2 -114)RI2GR 2 = 7B,P 18GR,(9)

and the reluctance coefficient is(10)t: = 2H,Pd I y 0 Bw8 R.
Assuming the same values for f and E,
B,P I H,P = -9 fd 14y0 t:R = -1.99d I y 0 R,and(11)Iron pass length
Li=Li1+Li2In Fig.5 the field gradient estimated from (11)compared with the Pandira results.

Fe-8- Gp/SrFerr (T/m)... - -[]- · Gpand/SrFerr (T /m)

2R-+- Gp/SmCo (T/m)- ·X · Gpand/SmCo (T /m):IS:::::J::·······-~·-·........... :···················+··

Fig.3 Geometry of 1/4 hybrid quadrupole magnet.
w-planed/R

r 0 =1 .0

Fig.5 Comparison between the relation (11) and Pandiraresult. H c =3.35k0e, H, =0.35T for Sr ferrite and
He =10.5k0e, H,=ll.OT for SmCo. Subscripts "pand"and "p" mean results by Pandira code and analytic
formula, respectively.

Return yokemagnetization
PermanentReferences

Air gapuFig.4 Transformed 1/8 quadrupole in thew-plane for the bore
radius of R = 1 , d = 0.4 and LY = 0.4.
In the transformed geometry the half gap height of thedipole is R 12 and the Ampere's law for the closed
integration loop in Fig.4 becomesfH,dz = frwBww'*dw I w' = fy wBwz*dw I z = 0, (8)v,lw'*IIBwl)andrz<x,y,IB,i) = Yw(u,r = lf/1,w' = dw I dz. An asterisk means the complex conjugate andsubscripts z and w stand for z- and w-plane, respectively.
Taking the integration on the v-axis (u=O) in the gap,
w * I w = -1 without losing generality. Relation (8) is also
expressed asfH,dz = frwBwz*dw I z = fr wBw~w· lwdw = 0.
where[1] M. Baltay et al, "Comparison of SmCo and NdFeB in
PM Multipoles," Proc. 1987 PAC, 1987, pp.l431-3.
[2] Conceptual Design Report, "An Asymmetric B
Factory Based on PEP," LBL PUB-5303, 1991.
[3] W. Lou et al, "Stability considerations of permanent
quadrupoles for CESR phase-III upgrade," PRST-AB,
1' 0224011 (1998).
[4] Permanent magnets Vol.l (Tokin, Ltd.), Ferrite andRare earth magnet (Hitachi metals, Ltd) and Permanentmagnet (Sumitomo Special Metals Co., Ltd).
[5] H.D. Glass et al, "Permanent Dipole Magnets for the 8
GeV Transfer Line at FNAL," PAC97, 1997, pp.32579.
[6] R. Yamada et al, "Recycler Model Magnet Test on
Temperature Compensation for Strontium Ferrite,"Internal Report, TM-1950, 1995.
[7] K. Halbach, "Application of Conformal Mapping toEvaluation and Design of Magnets Containing Iron
with Nonlinear B(H) Characteristics," NIM, 64 (1968)
278-284.

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