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Indian Journal of Pure & Applied Physics Vol. 3!J. January-February 2001. pp. 104-110

Accelerator magnet design for high field quality

P R Sarma

Variable Energy Cyclotron Centre. Dcparlment of Atomic Energy. Block-A F. Bid han Nagar. Calcull<l 700 OM Rccei vcd 7 November. 2000

Magnet~ of various types ;Jrc required in accelerators and hcam lines. Bending magnet~. switching magneh. analyzing magnets. quadrupolcs. scxtupolcs. Glascrs. etc. arc examples of accelerator magnets. In most of the accelerator <~pplicatiPns. cspcci;dly in high resolution hcam line~ and also in synchrotrons and storage rings the field quality of the magnets h;Jw to he very high. Tile l'icld quality in roomtcmpcralllrc iron magnets depends mainly on the shape of the pole. V;1rious mcthmls of determining the pole profile have hccn discussed.

I Introduction

Magnets or vanous types are required 111 accelerators and its beam lines. Bending magnets, switching magnets, analyzing magnets, quadrupoles, sextupoles. Glasers etc. are examples of accelerator magnets. Table I below lists some of the accelerator magnets under different classifications. Except for the cyclotron main magnet, all other magnets arc used for transporting the beam to the target or storing it for certain period of time. In most of the accelerator applications. especially in high resolution beam lines and also in synchrotrons and storage rings the field quality of the magnets have to be very high, in order that the beam quality does not deteriorate as the beam passes through the magnet either once (as in the beam lines) or many times (as in the synchrotrons). A typical example of high field quality magnets are the quadrupoles 111 PEP interaction region at LBL12 The sum of all higher order pole-field contributions is less than I ~ at the pole radius. i.e .. about 10·:1 at a distance of 0.6 times the half-aperture.

l.J Goal of magnet design

The aim of an accelerator magnet designer is to make a magnet which. first of all, has a field error within the allowed tolerance, has a simple electrical and mechanical design. is small in size and, or course. has a low cost. Some of these requirements are in conflict with the others and a designer makes a _judicious choice of the various aspects. There are other factors also involved in the design, like the reliability of performance and safety factor. One can always make a magnet of very big size (and large

aperture) and use only a small portion of the avai I able aperture where the field error is naturally small. But this is not what is wanted. One has to

make a magnet of minimum possible aperture and increase the so-called 'good field region'.

Tnhlc I - Examples of accclcralor mr~gnct~

Usc Type

Cyclotron magnet Dipole

Bending. switching. analyzing Dipole

Septum magnet Dipole

Focusing Quadrupole

Focusing Glaser

Second order correction Sextupolc

H ighcr order correct ion Octupolc

combined function Dipolc-rquadrupolc ..

Superconducting All types

This paper will deal with this aspect of magnet designing. The errors in a magnet can be understood in terms of the harmonic expansion of the potential (magnetic or electrostatic). For a dipole it is given by:

U(r.8) = C1r cos (8) + C1r1cos (38) + Cr'co-; ()8)

+C7r7 cos (78) + ... . .. (I) For a quadrupole the potential is:

(2)

SARMA: ACCELERATOR MAGNET DESIGN 105

U(r.8) = C:l..:>cos(28)+ C,/'cos (68)

+ C11,r1"cos (I 08) + C,~r14cos ( 148) + ··· ... (2) and so on. Here C s are the coefficients of various harmonics. The first terms in these equations are the ones that give the ideal potential and field. The other terms are the error harmonics. One should note that the error harmonics have rJ'-dependence and so the C1-term is the most dominant error harmonic for a dipole. Similarly, C.-term is the most dominant error harmonic in a quadrupole. In magnet designing, one always attempts to reduce these terms. In room temperature magnets this can be done by pole shaping. The authors will keep superconducting magnets out of their discussion.

2 Pole Shaping in Quadrupoles

There are a number of ways in which the field error can he reduced in quadrupoles. Hinterberger and others-' used a pure hyperbolic profile terminated by shims at the coil windows. Others use hyperbolic poles truncated with straight lines at the outer ed!!es"A More com pi icated shapes have also been

L d h 7-lJ

used'''. Another way used by Danby an ot ers IS

to make poles consisting entirely of plane surfaces.

2.1 Circular profile

A simple way of improving the field quality in quadrupoles is to use a circular profile and to optimize the radius of the circle in order to minimize the C,,-term'"-'~. A numerical technique is used for evaluating the harmonic coefficients in the potential. A number of points (10-15) (r,,8,), (r~,8~),

etc. are chosen on the pole face and these coordinates are put in Eq. (2). Assuming that the potential is constant (say

n

on the pole face, a set of equations is obtained.

Cr,"cos (28) + C.r,"cos (68) + Cor,'"cos (108) + C~r;'~cos (148) + ... =I

i= I ,2,... . .. (3)

The ahrive set of simultaneous equations is solved to obtain the harmonic coefficients. This is done for a number of radii. The radius for which Cr.

is zero is the optimum radius. This comes out to be R = 1.145 (in terms of the half-aperture) ... (4)

With this simple design the error field at r = 0.6 is 1.5xlo-~. This suffices for many applications. The quadrupoles used in the beam lines of VEC, Calcutta, are of this design.

2.2 A new method of finding the profile

If higher field quality is required, one has to use different profiles as indicated earlier. Sarma and Bhandari 1~ have recently shown an ingenious method of determining the pole profile. In the earlier methods based on Poisson calculation or other numerical methods, one first chooses a pole profile and then calculates the field and ultimately finds the harmonic components. If the dominant harmonic coefficients are not small, the geometry of the pole is changed and the calculation is repeated.

This process is continued until the field errors are small. In the new method, the authors first choose the harmonic coefficients and then determine the profile, which generates the chosen coefficients. In this way the field quality of the magnet is chosen o priori. Since a better quality than the circular profile is desired. the authors take Cr.= Co= 0 and so the pole profile is determined by:

C~r cos (28) + C4r1~ cos ( 148)

+Cxr1xcos(l88) + ... = I ... (5)

The pole profile is determined numerically hy putting values for r in the above equation and solving for

e .

The deviation of the profile from the ideal one obviously depends on the value of

c j

chosen. In the numerical procedure one starts with a small value of C14 and then varies the higher harmonics to get a profile. which gives a finite and constant pole width. Fig. I compares the new profile with an optimized circular profile and the hyperbolic profile. Fig. 2 shows how the pole width increases with better field qualities. It should he noted that the fabrication of the circular profile i~

easy but in the fabrication of the new profile, numerically controlled machine is required. This. however, is not a problem in laminar magnets once a die is made.

2.3 Conformal mapping method

Conformal mapping technique is often used,_, in the design of magnets. Design of quadrupole magnets and sextupole magnets becomes easier'' it.

the geometry is conformally mapped onto dipole geometry. The author has used this technique to optimize the pole profile of Danby type quadrupolcs

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106 INDIAN .I PURE & APPL PHYS. VOL 39, JANUARY-FEBRUARY 2001

which consist entirely of plane surfaces. Some of the geometries suggested by Danby are quite complicated. Some simpler geometries have been investigated. Since the pole is polygonal one can use Schwarz-Christoffel transformation technique of conformal mapping. Fig. 3 shows one of the pole geometries and the mappings. The transformation from the 2-plane to an intermediate t-plane is:

,

'

' '

HYPERBOLIC

'

Fig. I - Pole rrolilc of a quadrurolc obtained with the new method for C14

=

.00004. For comrarison. an ideal hyperbolic rrolilc and an optimized circular rrolile are also shown

I

2 = re'H =P

J

()

w = U(r,8) + iV(r,8) = (2/rt) sin-1(t/a)

... (6)

... (7) where U(r,8) is the potential, and P, a and hare the parameters of the Schwarz-Christoffel transformation. The boundary conditions, which determine these parameters, are:

(i) z.(t =a)= I is the half-aperture EF in Fig. 3

(ii) ::.(t=b)- 2U=a) =length FG xi

(iii) 2U= I)-2U=h) =length GH (sin 8 + i cos 8)

The harmonic coefficients can be found out by putting the expansion of t in terms of 2 in Eq.(7).

The expansion is:

t = ((J?rraz~)(z~/2' + (CJ"tf()2")(z"l6!)

The parameters a, h and

8

are optimized to minimize the coefficients

Cr.,

C1o and

C

1• The parameter Pis used to normalize the half-aperture to I. The table below compares the field qualities of a conformally optimized quadrupole with that of a circular quadrupole and a quadrupole of new profile.

Table 2- Harmonic coefficients of potential in varinu~

quadrupoles Profile

Circular

Plane surface New prolile

w 1.5 a: ::J

...

a: w

~

.

1.0 u. .J

<(

~

0.5

...

0 ~

C,

<105

<10''

0

c111 cl4 error at r =

o .c,

-0.0024X 0.0002X l.) xiO''

<In··' <l<r' 2.4 xltr5

() .00004 l.l x 1rr''

4 E : ; . . - - - 20 12

0.0 ,__ _ _ ....___...:._..._ _ _ _,_ _ _ ...J...__J

0.5 1.0 1.5 2.0 2.5

HEIGHT/HALF-APERTURE

Fig. 2- Variation of the width of the new pole profile for various values of c l4

3 Design of Sextupoles and Octupoles

A large number of sextupoles are used in synchrotrons and storage rings for correction to some of the beam characteristics and control the growth of beam emittance. All modern synchrotrons and many spectrometers employ sextupolcs for canceling the effect of chromaticity which results from the fact that the focal lengths of quadrupoles

(4)

SARMA: ACCELERATOR MAGNET DESIGN 107

A

i

s • • • • • • • • •

A1 81 c1 01 E1 F1 G1 H1 11 8

N N

A2 12

82 H2

s

c2 G2

E2

02 F2

Fig. 3 _ The pole configuration of a plane surface quadrupole. The intermediate and the final mappings are also shown

and hence the tune of a machine depend on the deviation of a particle's momentum from the central momentum. To give some examples, the European Synchrotron Radiation Facility (ESRF) at Grenoble'r. uses 224 sextupoles and the 8 GeV storage ring at RIKEN, Tokyo17 employs a total number of 336 sextupole magnets. Similarly, the ELETTRA storage ring at Trieste'~. the synchrotron light source ROSY at Dresden'') and many other synchrotrons"""' require many sextupoles for the correction of not only the natural chromaticity but also the sextupole contribution from the dipoles"~.

The basic design procedures of the sextupoles are the same as the quadrupoles. The most dominant error harmonic in a sextupole is the r'J-term and so the good field region is larger compared to a quadrupole. In an octupole it is still better. The simplest pole profile in these cases is once aga111 circular with optimum radii"'

R = 0.56 (for a sextupole) and R = 0.37 (for an ocrupole)

Other complicated pole shapes can be employed here also''-The new method also can be used21'.

3.1 Combined profile magnets

In many applications one needs combined profile magnets where one deliberately introduce~

other harmonic fields to the main field. A cilpole magnet having field gradient is an example of such a magnet. Such magnets bend and focus the beam simultaneously. The analyzing magnet at YEC, Kolkata is a double focusing magnet where the beam is analyzed and also focused in both the trans verse planes equally. Com~,i ned profile magnets can easily be designed with the help of the new method described earlier17.

4 Reduction of Aberration in Glaser Magnets Generally quadrupole magnets are used for focusing the beam in a beam line. Another type of focusing magnet, which is sometimes used in low energy beam lines is the Glaser magnet. These arc basically solenoid magnets with iron yoke and pole pieces. It can be visualized as a cylindrical dipole ma"net with a central bore for allowing the beam to

o . I

pass. The Glaser magnet generates a strong ax 1a magnetic field and the particles arc focused hy the action of this field. The advantage of a Glaser magnet over a quadrupole is that it is cylindrically symmetric and so it focuses the beam in a II the meridian planes. Obviously, a Glaser magnet does

(5)

lOX INDIAN J PURE & APPL PHYS, VOL 39, JANUARY-FEBRUARY 2001

nol have any astigmatic error. Spherical aberration however, is quite large in this magnet. Recently, the author worked on a scheme of improving the field quality and reducing the spherical aberration11'1 of Glaser magnets hy shaping the profile of the pole pieces. A Glaser magnet with tapered pole pieces has been designed and fabricated (Fig.4). The field in a cylindrically symmetric system has the general form:

%% COIL

CONVENTIONAL NEW DESIGN

Fig. 4- Pole prolllc or an improved Glaser magnet compared to that or the conventional magnet with flat poles

8,(r,-:.) = 8(::.)-(r /4)8" (z:)

+ (r4/64)81v(z.) + ... . .. (9) where 8(z) is the field along the axial distance .::.

The equation of motion in this field takes the form311: r .. = -(e8J2m)!l r-.5 (B"(z:) B(z))r'+ ... ] ... ( 10) and so there will be aberration due to the presence of r' term and higher order terms. This leads to the emittance growth also. The coefficient of spherical aberration C is defined11 by:

l'!r = C,r'

where l'!r is the aberration in the image plane and r is the height of the ray in the aperture plane, and is calculated to be:

~ ~

C= ( 1/2)

f

18'(z:)l1d:.l

f B~ (z: )d z:l

... ( I I )

From Eq. (I 0) it is seen that the aberration can he reduced if one reduces the term 8"(z), i.e. the longitudinal field is made more uniform in the median plane. This can be achieved by increasing

the pole gap, but this reduces the focal power also. It has been shown that using tapered poles the aberration can be reduced (Fig. 5). Fig. 6 shows that the spherical aberration coefficient decreases as the taper angle increases, while the focal length remains essentially constant.

294

27

202

56

t

140

Fig. 5- Mechanical drawing or the Glaser magnet. The dimensions shown arc in rnm

5 End-shaping of Magnets

An ideal magnet of very high field quality has to have an infinite length (along the direction of the beam). But any practical magnet can have only a finite length. This also introduces harmonic errors.

In a short magnet the effects of these error harmonics are very severe. End shaping can reduce this effect. The potential due to a magnet of finite length has the following solution:

U(r,8) = L.,e"'lcos (m8) ./,.(ar) +cos Om8)

J.,"'(ar) + ...

I .. . (

12)

(6)

SARMA: ACCELERATOR MAGNET DESIGN I ()LJ

68 0 FOCAL LENGTH

:c

0.45

I -C) 0 SPH. ABERR. COEFF.

z w 2- u.: u..

w 0

(..)

z 0

!;;;:

a: a:

m w

<(

::r:::

0...

en

0.40

0.35

0.30

0 10 20 30 40

66

64

62

60 50 TAPER ANGLE (DEG.)

Fig. 6 - Plot of the rocallcngth (for a 15 keY deuteron) and the spherical aberration coefficient versus the taper angle

Due to the presence of the e"' factors the end errors come in. ror a magnet of accuracy I o-J in the field a simple cut across the pole ends is sufficient.

But for higher accuracy special end-design has to be adopted. If a cut is made, the angle and extent of cut can be adjusted to minimize the total integrated error clue to the most dominant harmonicJ"· Another method is to contour the pole in the middle of the magnet. At CERN this has often been adopted". It should be noted that the contribution of the end- effects can he made smaller if the length of the

ma~net is increased. But this is not always possible

L '

due to the optics requirements in the beam line.

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I I 0 INDIAN .I PURE & APPL PHYS. VOL 39, .JANUARY-FEBRUARY 2001

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