Quadrupole Magnet Misalignment TolerancesPhilip F. Meads Jr. Citation: Review of Scientific Instruments 38, 1752 (1967); doi: 10.1063/1.1720663 View online: http://dx.doi.org/10.1063/1.1720663 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/38/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Kinetic equilibrium of space charge dominated beams in a misaligned quadrupole focusing channel Phys. Plasmas 20, 073107 (2013); 10.1063/1.4813251 The extended wedge method: Atomic force microscope friction calibration for improved tolerance toinstrument misalignments, tip offset, and blunt probes Rev. Sci. Instrum. 84, 055108 (2013); 10.1063/1.4804163 Tolerance Analysis of Misalignment for 2DMEMS FreeSpace Optical CrossConnect AIP Conf. Proc. 992, 594 (2008); 10.1063/1.2926934 Simulation of the particle misalignment of hard magnets after compaction J. Appl. Phys. 85, 5672 (1999); 10.1063/1.369836 Collapse of cycloidal electron flows induced by misalignments in a magnetically insulated diode Phys. Plasmas 5, 2447 (1998); 10.1063/1.872921

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1752 P. J. EBERT AND A. F. LAUZON

tional units, a dose rate of 20 Rlh of 1.25 MeV gamma rays. Since the sensitivity is independent of gamma-ray energy and since no temperature or pressure corrections are necessary, the vacuum diode would be useful for cross calibration of gamma-ray sources.

THE REVIEW l)F SCIENTIFIC INSTRUMENTS

ACKNOWLEDGMENTS

The interest and helpful advice of W. C. Dickinson are gratefully acknowledged. We also wish to thank H. Catron and M. Griffin for their assistance in performing many of the detector measurements.

VOLUME 38. NUMBER 12 DECEMBER 1967

Quadrupole Magnet Misalignment Tolerances*

PmLIP F. MEADS, JR.t Midwestern Universities Research Association, Stoughton, Wisconsin 53589

(Received 4 April 1966; and in final form, 7 August 1967)

The field produced by a quadrupole magnet is antisymmetric with respect to reflection through either of two orthogonal planes whose intersection is the optical axis of the magnet. Failure to perfectly align these planes for a given quadrupole magnet with the cardinal planes of the beam system incorporating that magnet induces an error field that perturbs the beam. The effects of misalignments of a number of quadrupoles are derived by including the induced error fields in the equation of motion. The results are expressed in terms of the known linear optical properties of the beam system and are sufficient for determining the tolerances required on each constituent quad­pole magnet.

1. INTRODUCTION

W E treat rigorously the optical effects of misaligning a number of quadrupole magnets in an arbitrary

beam system. Each quadrupole magnet is assumed to be well-constructed; that is, its magnetic field possesses the desired symmetry properties. Effects smaller than the in­herent third-order aberrations are assumed to be of no interest and are therefore dropped. The entire results are expressed in terms of the known linear properties of the beam system.

We first obtain a general expression for the magneto­static scalar potential for the misaligned magnet. From this scalar potential, we derive expressions for the error fields. The error field expressions are inserted into the equations of motion, which are then solved by a method of successive approximations. Integral equations are obtained by means of Green's functions. The complete solution is then written as the sum of integrals over each misaligned magnet. These integrals are solved in terms of the known linear solutions and the transfer matrix that carries the unperturbed beam from the center of a misaligned magnet to the point of ob­servation .. The displacements and change in slopes of a trajectory as seen by the observer downstream are shown to be equivalent to a displacement and change in slope occurring at the center of the misaligned magnet. The treatment is completely valid for thick lenses in contrast to earlier treatments.

t Present address: William M. Brobeck & Associates, Berkeley, Calif. 94710.

* Work supported by the U. S. Atomic Energy Commission.

2. GEOMETRY

2.1 Notation

We shall refer to a Cartesian coordinate system in which the Z axis is chosen to coincide with the optical axis. The two orthogonal axes are chosen to lie along the two planes of reflection antisymmetry for the ideal (perfectly aligned) quadrupole magnets. Trajectories starting in these planes always remain in them (ideal system). The optical axis runs between an object plane (initial conditions) denoted by the subscript" 0" (zo) and an observer at Z (arbitrary). Primed coordinates will be referred to the magnetic center of a particular misaligned magnet with z' = 0 at its mid­point. Unprimed coordinates refer to the beam system as a whole.

2.2 Misalignment Parameters

We consider the misalignment of a particular magnet such as that centered at Z= Zj. This magnet is assumed to be rotated about the optic axis by an angle Wi, tilted with respect to the optical axis by an angle if;j in the x direction and by an angle Tj in the y direction. Finally, the magnet is displaced normal to the optical axis by 5xj and 5Yj. The misalignment just described is illustrated in Fig. 1 (a), 1 (b), and 1 (c).

We have

x' = x-5xj+ywj- (z- zj)if;j+ ... ; (1)

y'=y-5Yj-XWj- (z-zj)Tj+'" ;

z' = (z-zj)+xif;j+yTj+' . '.

(2)

(3)

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MAGNET TOLERANCES 1753

FIG. 1. Coordinate system describing misalignment.

optic oxis (out)

Y Y'

X'

(0)

The terms dropped as a result of the approximation of the trigonometric functions of w, T, and if; contribute to effects smaller than the inherent aberrations of the quad­rupole magnets.

2.3 Scalar Potential

A well-constructed quadrupole magnet is characterized by several symmetry properties. The magnetic field of an ideal quadrupole is antisymmetric with respect to reflection through both the x=O and the y=O planes; it is also sym­metric with respect to reflection through the x= y plane and with respect to reflection through the x= -y plane. The most general magnetostatic scalar potential that satis­fies Laplace's equation, satisfies these symmetry proper­ties, and is dominated by the linear quadrupole field is

eV(x',y',z') = Pc[ -x'y't/>(z')

+ l2 X'y'(X'2+ y'2) cf2t/>]+O(6). dZ'2

(4)

Here t/> is an analytic function of z', vanishing well outside of the magnet and assuming a nearly constant value within the central region of the quadrupole. This continuous func­tion is usually approximated by a step function t/>s(z'), which vanishes for Z'j> L j /2 and for z'i<Lj/2 where Li is known as the effective length of the magnet. Within the magnet

e dBy' e dBx' t/>~t/>.=k2= ___ =_--,

pc dX' pc dY'

where k is real for a magnet converging in the x-z plane and pure imaginary for a lens converging in the y-z plane. The effects of the difference function t/>-t/>s have been treated l ; they are already of the order of the inherent aberrations and thus can be dropped in this treatment.

2.4 Error Fields

When referred back to the unprimed coordinate system, the above scalar potential [Eq. (4)J yields the normal

X' X Y' Y

(b) (c)

quadrupole field and the following error fields which are complete through terms yielding effects of the order of the inherent aberrations.

(6)

and

e!!.Bz = pct/>(z)[ -yif;j-xTjJ+' . '. (7)

It is easily shown that the error field component in the z direction contributes only effects smaller than the in­herent aberrations.

In this discussion we have assumed that the error fields induced as a result of magnet misalignment are at least one order of magnitude smaller than the dominant quad­rupole field. All terms that yield perturbations down to the magnitude of the inherent third-order aberrations of the quadrupole magnets have been retained. The treatment is thus rigorous and fails only for grossly misaligned magnets.

3. METHOD OF SOLUTION

The equations of motion satisfied by the trajectories in the misaligned quadrupole field are l

cf2x (d:)2 (d82

dy dx -+t/>x=-!x - t/>-1X - t/>+~--tP dz2 dz dz dz dz

dy dt/> cf2t/> e +xy--_+(ixy2+l2x3)---!!.By+···, (8)

dz dz dz2 pc

cf2y (d02

(d:)2 dx dy --t/>y=!y - t/>+b - t/>-x---tP dz2 dz dz dz dz

dx dt/> cf2t/> e -xy--- (ix2y+l2y3)_+-!!.B,,+·· '. (9)

dz dz dz2 pc

1 P. F. Meads, Jr., "The Theory of Aberrations of Quadrupole Focusing Arrays" (Ph.D. Thesis, Univ. of California), available as Lawrence Radiation Laboratory Report UCRL-10807 (1963).

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1754 PHILIP F. MEADS, JR.

All of the misalignment contributions are contained in the terms t:.B" and t:.By. The remaining terms on the right hand side of these equations yield the inherent third-order aberrations of the beam system and are not treated here. Setting the right-hand sides to zero yields the linearized equations that determine the transfer matrices. The error fields t:.B" and t:.By are due to a sum of error fields produced by individual misaligned quadrupoles. We shall show that these equations may be treated separately over each quad­rupole so only the error field due to one quadrupole need be considered at any given time.

3.1 Formal Solution

We can write x(z)· and y(z) at the point of observation as a sum of (1) a linear contribution, obtained from the transfer matrices, (2) an inherent aberration contribution, and (3) the misalignment contributions t:.x(z) and t:.y(z). The differential equations may be turned into integral equations through the use of Green's functions, thus

t:.x(Z) = _~ jZ G,,(z,r)t:.By(x,y,t)dt; pc zo

t:.y(z) = ~ jZ Gy (z,r)t:.B" (x,y,r)dt, pc zo

where the Green's functions satisfy

d2

and

--G,,(Z,t)+q, (z)G" (z,t) = -o(z-t) dz2

d2

--Gy(z,t)-q,(z)Gy(z,t) = -o(z- t), dz2

respectively.

(10)

(11)

(12)

(13)

We now separate these integrals into a sum of separate integrals, each of which is taken over a single misaligned quadrupole magnet. To make the separation complete, we assume that q,(z) vanishes at some point between each magnet; this is normally the case. With this separation, the error field for the jth magnet appears only in the jth integral. We thus have

t:.x(z) = - ~ {f:~~~: q,(t)G,,(z,t)

XC-OXj+2Wjy-(r-Zj)¥-'j]dr}, (14)

(sum over magnets)

t:.y(z) = ~ {f:~~:: q,(t)Gy(z,r)

XC-OYj-2WjX-(r-Zj)TJdr}, (15)

(sum over magnets)

where (16)

3.2 Solution in Terms of Linear Functions

We can now replace q,(z) by kl without altering the solution within the accuracy desired. The linear solutions within the jth magnet are

X= aj coskj(z-zj)+bj sinkj(z-zj); (17)

y=Cj coshkj(z-zj)+dj sinhk/(z-zj). (18)

At this point let us note that we can make the transfor­mation from any equation in x to the corresponding equa­tion in y by replacing (x, y, q" k, w, T, and ¥-') by (y, x, -q"

ik, -W, ¥-" and T); henceforth we shall write only the equa­tions for x.

The integrals can be expressed entirely in terms of known functions once we define the transfer matrix that carries x and dx/dz at Zj to x and dx/dz at z, the point of observation; this transfer matrix is evaluated for a per­fectly aligned beam system. We write

(19)

We now express the Green's function in terms of this transfer matrix as follows:

G(Z,r) = W 12 (Z,Zj) cosk(r-zj)

-k-1W ll (z,Zj) sink (t-Zj). (20)

Furthermore, any linear trajectory within the jth magnet satisfies

x(r)=Xj cosk(t-zj)+k-1dx/dzlj sink (r-Zj), (21)

where Xj and dx/ dz I j are the displacement and slope of the trajectory at the center of the jth magnet in the absence of any misalignment. The corresponding expressions for yare obtained upon application of the rule outlined above. If the jth magnet is diverging in the x-z plane, k is pure imaginary; however, the expressions hold in this case also.

Upon approximating x and y in the integrands by Eq. (21), which is a valid approximation within the accu­racy desired, we obtain explicit integrals in terms of known linear functions. For example, if the only misaligned magnet is that at Zh then, upon replacing t-Zj by t,

tL;

x(z) = -f CW12(Z,Zj) coskr- W ll(z,zj)k-1 sinkr] -lL;

xC -OXj+ 2Wj(Yi coshkr+k-1dy/ dz I j Xsinhkr)-Nj]dr. (22)

Recalling the significance of the transfer matrix W(z,Zj), it is apparent that the displacement and change in slope at the point of observation z appears to the observer at z to be the result of increments in x and dx/dz (and y,dy/dz)

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MAGNET TOLERANCES 1755

occuring at Zj, thus we can write

[

.1X] [ .1Xj 1 .1dx =W(z,Zj) .1dX/.

dz z dz ,

(23)

3.3 Result

The expressions for the apparent displacements and change of slopes are:

.1Xj= 2krl(sintOj coshtOj- costOj sinhtOj)wjdy / dz I j+ 2krl(tOj costOj- sintOj)1fj;

.1dx/ dz I j= - 2kj(sintOj coshtOj+ costOj sinhtOj)w,Yj+ 2kj(sintOj)oxj;

(24)

(25)

(26)

(27)

.1yj= -2krl(costOj sinhtOj-sintOj coshtO,)wjdx/dzl j+2krl(tOj coshtOj-sinhtOj)Tj ;

In these expressions, OJ=kjL}. We have in Eqs. (24)-(27) the exact optical effects,

ignoring only effects smaller than the inherent aberrations, of misaligning the magnet centered at Zh as seen by an ob­server at a completely arbitrary point z, downsteam from the misaligned magnet. These effects depend upon the optical properties of the portion of the beam system beyond Zj only through the transfer matrix W (z,Zj) and upon the optical properties of the portion of the beam system pre­ceding the misaligned magnet only through Xj and dx/ dz \ j.

The expressions are more complicated than the usual expressions associated with the misalignments of quad­rupole magnets because they apply to thick lenses and are not the result of a thin lens approximation. (We nowhere assume that OJ is small.) The thin lens approximation is obtained by keeping the focal length fixed while letting OJ go to zero, i.e., k jOj ---+ I-I (fixed) as OJ ---+ O. At this limit, we obtain

.1X·"'----- -------+ 0 1 Wj dYI 1 1fj

, 6 k/P dz j 12 klP , (28)

and

dX/ 2 ox} .1- '" --y,"Wj+-.

dz j 1 1 (29)

The corresponding expressions for .1yj and .1 (dy/dz) I j are easily obtained upon application of the rule outlined pre­viously. Thus in the thin lens approximation there is no apparent displacement, but only a change in the apparent slope, which is as we would expect.

We can return easily to the general case, where a number .of quadrupole magnets are misaligned, by recalling that we could separate the expression for the cumulative dis­placement .1x(z) and change in slope ,1{dx/dz\ z into a sum of .1x(z) and .1 (dx/dz) \ z due to misaligning each magnet

individually. Thus we have

[;Xl = ~ [we"~,;) r ;~ l] (30)

dz z l dz j

where the sum is over all quadrupole magnets, and .1Xj and .1 (dx/dz) I j are the apparent displacement and change in slope at Zh as seen by an observer at z, that result from the misalignment of only that magnet centered at Zj.

The fact that the beam system may contain magnets other than quadrupoles does not alter these results. The modifying effects of bending magnets, separators, and solenoid magnets are contained in the transfer matrices. We do not in this paper calculate the effects of misaligning other than quadrupole magnets.

4. INTERPRETATION

4.1 Significance of the Thick Lens Expressions

A thin lens analysis leads one to disregard the effects of tilts of the magnet axis with respect to the optical axis, whereas the preceding thick lens analysis correctly pre­dicts that beam loss can occur as a result of such tilts.

A second result is that the tolerance on displacement of a magnet orthogonal to the optical axis is most severe for a converging lens when 0=7r.

From Eqs. (24) and (26), we see there is no effective dis­placement due to rotation of a lens where tantOj= tanhtOj; this condition yields the most relaxed tolerance on rotation where there is a focus inside the lens. The most relaxed tolerance on rotation for a lens in which the beam is essen­tially parallel is obtained when tantO= -tanhtO; this is the condition that no effective change in slope occur as a result of lens rotation.

The primary purpose of this paper is to develop the ex-

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1756 PHILIP F. MEADS, JR.

pressions that relate the displacements and change in slopes at some point in a beam system to the misalignment of the quadrupole magnets within the system and to express the results in terms of the well-known linear optical properties of the system. This has been accomplished.

4.2 Aperture Requirements

The author has not applied these expressions to the problem of determining the tolerances on an actual beam system. A suggested method of application which uses linear programming is described in the remainder of this paper.

It is usual to assume that, apart from misalignment and. dispersive effects, the beam distribution in phase space is symmetric with respect to reflection through the origin in either the x- (dx/dz) plane or the y- (dy/dz) plane. Thus if Xm is the maximum displacement at the mth magnet in the absence of misalignments and Xm the available aper­ture at that magnet, the condition that no beam be lost at this point as a result of insufficient x aperture is

OJ 8j) dYj - cos- sinh- WJ-

2 2 dz j

!2(O. O· O,)! ] + k ~ cos ~ -sin; 1/;j

OJ OJ)! ! ( OJ)! } +cos2 sinhz wijh + 2k sinz ox; . (31)

A similar expression must hold if there is to be no loss of beam due to insufficient y aperture. An expression of this type exists for each aperture in the beam system. The

totality of these inequalities constitutes a triangular array of linear inequalities that must be satisfied by the mis­alignment parameters ox;, oy;, W;, 1/;;, and 'rj (which are here forced to be positive quantities) if no beam is to be lost as a result of misalignments.

In order to determine the tolerances that will guarantee no beam loss, we could proceed in a number of ways. A typical approach to this problem is to assume that the mis­alignments are randomly distributed within certain bounds, the tolerances. One then examines a large en­semble of systems, adjusting these bounds until a pre­determined (large) fraction of the ensemble of systems are loss-free. A slight modification is to adjust the bounds until the ensemble expectation value of the beam loss reaches. an acceptable limit.

A novel approach to this problem is to use the methods. of linear programming to locate the most sensitive adjust­ment and determine the tolerance required on that ad­justment. Let us first restate all of the misalignment parameters in terms of positive displacements Qi; for ~xample, the rotation of a given quadrupole can be ex­pressed in terms of the displacement of the magnet ends. We obtain a set of linear inequalities that must be satisfied by the Qi directly from Eq. (31). Let Q be positive and a lower bound to the set Qi. We then solve the linear pro­gramming problem of finding the maximum value for Q subject to Q being a lower bound to the Qi which are sub­ject to the linear inequalities (31). The solution to the problem, Q, is the magnitude of the tightest tolerance that must be met if we are to guarantee no beam loss. One or more of the Qi will be equal to Q; it is these alignments that must be accurate to within Q. To proceed further, we elimi­nate the tightest tolerances by setting the corresponding Qi to the maximum Q just found and then repeat the problem for the reduced set. If continued in this manner to its conclusion, this procedure will yield the magnitude of all tolerances on the quadrupole magnets. The solutions are the most liberal tolerances that guarantee no beam loss even if each element is misaligned by just the tolerance allowed.

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