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End effects in first-order theory of quadrupole lenses
N U C L E A R I N S T R U M E N T S AND METHODS 76 0 9 6 9 ) 305-316; © N O R T H - H O L L A N D P U B L I S H I N G CO.
END EFFECTS IN FIRST-ORDER
THEORY
OF QUADRUPOLE
LENSES
G. E. LEE-WHITING
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 9 June 1969 Rapidly convergent series for the transverse position coordinates of a charged particle moving through an end zone ofa quadrupole lens are derived by iterating integral equations obtained from the first-order differential equations. Not more than three terms of the series appear to be needed for lens strengths normally used. The series contain form-factors formed by integrating the
field gradient along the axis with various weight functions. The focal lengths and the principal-plane coordinates of the whole lens are expressed in terms of numerically well behaved functions of the form-factors. Formulae are also derived for the effective lengths and strengths of the lens.
1. I n t r o d u c t i o n
F o r electrostatic q u a d r u p o l e lenses e q u a t i o n s o f the f o r m (1) c a n also be used. T h e f u n c t i o n k(z) is t h e n p r o p o r t i o n a l to
I n the first-order t h e o r y 1'2) o f the m o t i o n o f a c h a r g e d particle in a q u a d r u p o l e lens the transverse m o t i o n is f o u n d b y solving the differential e q u a t i o n s : d2x
--
dz 2 d2y dz 2
+ k(z)x = 0,
k(z)y = 0.
(la)
(lb)
T h e quantities x, y a n d z are o r t h o g o n a l C a r t e s i a n c o o r d i n a t e s o f the particle. T h e z-axis is a l o n g the fourfold s y m m e t r y axis o f the lens; the x- a n d y-axes lie in two o r t h o g o n a l planes o f s y m m e t r y . F o r a m a g n e t i c lens the f u n c t i o n k(z) is directly p r o p o r t i o n a l to the p a r t i c l e charge, e, a n d to the field g r a d i e n t on the axis, g, a n d is inversely p r o p o r t i o n a l to the particle m o m e n t u m , p. k = k(z) = eg/p, (2)
c3Br g=g(z)=-~x x=r=o
?)E_.__~ ox I x=y=0
(4)
i n s t e a d o f to g, b u t the d e p e n d e n c e on z is similar. W h i l e in describing the analysis it is c o n v e n i e n t to refer to m a g n e t s , the results a p p l y equally well to electrostatic q u a d r u p o l e lenses.
Pi
-z2 -Z~l
k(z)
0 (a)
k(z)
IP2
=Z
IZl
Z2
-Z2
k z)
!oll 0
(c)
.Z
Z2
k(z)
(3)
I n a static m a g n e t i c field p is a c o n s t a n t o f the m o t i o n . T h u s k has the s a m e z - d e p e n d e n c e as g. N e a r the centre o f the lens there is usually a r e g i o n in which g is effectively i n d e p e n d e n t o f z. T o w a r d the edges o f the lens g begins to fall off, a p p r o a c h i n g zero r a p i d l y at large distances f r o m the lens. U s u a l l y the designer o f q u a d r u p o l e m a g n e t s a t t e m p t s to m a k e the g r a p h o f k(z) r e s e m b l e p a r t (a) o f fig. 1; he tries to m a k e the central region in which the curve is a l m o s t flat l o n g c o m p a r e d to the e n d zones in which k(z) is r a p i d l y varying. H o w e v e r , in s o m e a p p l i c a t i o n s , such as the q u a d r u p o l e m a g n e t s inside the drift-tubes o f a lowenergy p r o t o n linear accelerator, l a c k o f space m a k e s this impossible. T h e n the g r a p h o f k(z) is m o r e like fig. l b . I n all cases we shall a s s u m e t h a t g(z) is effectively zero outside the interval - z 2 < z < z 2.
r,-77--I---V,,-], r -Z2
0
(b)
Z2
-Z2
-Zi
0
Zl
Z2
(d)
Fig. 1. Four different possibilities for the variation of k(z) with z are shown in the four parts of the figure. The value of k at z = 0 is k0. All but part (b) have a region stretching from - z l to zl in which k(z) is uniform. The value of k(z) falls from ko to 0 in the intervals zl to z2 and - z ~ to -z~; in part (c) z~=z2. In part (a) an arbitrary smooth curve has been used in the end zones; the parameter q is positive when the principal planes P1 and Pa lie between - z a and z~ as shown. Part (b) represents the bell-shape model with fl = 0.25 and zl = 0; the ratio of k(z2) to ko is 1/289. In part (c) we illustrate the rectangular model, also called the hard-edge approximation. The linear-ramp model in part (d) is drawn with the parameters of case II; the dashed lines show the corresponding effective rectangular model for the ( x - z) plane of motion. 305
306
G. E. LEE-WHITING
The simplest way to treat the first-order motion of a particle in a quadrupole lens is to approximate k(z) by the function shown in fig. l c - i . e , by a function which is zero outside the lens and uniform inside. This is the hard-edge approximation of Steffen a) or the rectangular model of Septier2). The eqs. (1) then have simple and well known solutions. In his review article 2) Septier mentions other work in which analytic solutions to the equations (1) are found for a "bell-shaped" model. Apart from these special cases the only method of calculation presently available appears to be the step-by-step numerical integration of the trajectory equations. The original purpose of the work reported in this paper was to provide a simpler method of treating lenses of the type shown in fig. la. In such cases most of the effect of the lens occurs in the central zone where the hard-edge approximation is appropriate. It did not seem too much to hope that an analytic calculation might be possible for the relatively small effects of the end zones. Such a formulat!on would also be useful in understanding how the lens parameters depend on the shape of the curve of k(z) versus z. It turned out that the desired formulation is indeed possible. Moreover, the analytic processes involved are sufficiently convergent that the method even applies to lens with no central z o n e - f i g , l b - f o r values of lens strength commonly used. The method of calculation to be used in sec. 2 involves an infinite sequence of approximate solutions. Why one may expect the sequence to be rapidly convergent is most easily explained for the hard-edge model of fig. lc. It suffices to consider equation (la); assume ko > 0 and put k o = z 2. Let us consider the transfer matrix, M, acting between the centre of the lens and z = zz. This matrix is well known for the hardedge m o d e l - s e e page 12 of ref. ~), for example. With the definition
= z2,
(s)
one finds (
cosq5
x-lsinq 5)
M =
.
\-tcsinq~
(6)
cos~b
The method of calculation to be discussed in sec. 2 is analogous to expanding the elements of M in powers of ~b. Recall that COS~ = 1--½t~2 + 2-~t~ 4 -- v~-~b 1 6 +...,
(7a)
and that 1 a,4 -5--~-o-w 1 . 7 . 6t ..... tk-lsinq ~ = 1-16tk 2 + 1-~'e
(7b)
Though it is true that a large number ot terms would be needed in these series if ~b were much greater than unity, this is not usually the case. In most applications the quantity 2zz2, which represents the strength 6) of the whole lens, is less than ½n. For ~b < ¼n ~ 0.785 the series (7) converge exceedingly rapidly. The fourth term in the expansion of cos q~ is less than 3.3 x 10-4; the fourth term in the series (7b) is even smaller. When dealing with eq. (lb) we would have to expand cosh ~b and sinh ~b instead of cos ~b and sin ~b. Though the advantage of the alternating signs in (7) would then be lost, the magnitude of the terms decrease at the same rate. It appears unlikely that one would over need more than three terms in (7); one or two terms will often suffice. If three terms of (7) give excellent results for the hard-edge model, then it is certainly desirable to look for an analogous method of calculation applicable to the cases illustrated by figs. la and b. One would expect to find the powers of 4)2 replaced by products of integrals (evaluated over the end zones) of k(z) multiplied by various weight functions. As a first step in finding such an expansion it seems natural to replace (1) by integral equations. 2. The integral equation
Assume that at z = z 1 it is known that x = xl and that x' = x~. We wish to find values of x and x' in the finite interval z 1 < z < Zz. It is not difficult to show that
k(z')(z-z')x(z')dz';
(8)
zl
the upper sign refers to (la), the lower to (lb). We shall find an approximate solution to the integral eq. (8) by an iterative procedure called the "Method of Successive Approximations" by Ince 3) and Picard's method by Piaggio4). For the zero-order approximation to the solution take
x°(z) = xl + x'l ( z - Zl);
(9)
this approximation ignores completely the effect of the field on the particle as it moves through the end zone. The first-order approximation is found by substituting x°(z ') for x(z') in the integral on the right in (8); the integrand is then composed entirely of known functions. The second-order approximation is then found by substituting the first-order approximation for x(z') under the integral sign in (8). In principle the method can be carried to any desired order. It is convenient to give the results in the form of the transfer matrix, M, operating between z~ and z. The elements of M may be expressed as series of integrals of
END EFFECTS IN FIRST-ORDER THEORY OF QUADRUPOLE LENSES progressively higher multiplicity. The series may be treated as power series in the parameter ~ = z6 = k~o(z2-zl), with coefficients which depend upon the shape of the function k(z) in the end zones. The terms analogous to three terms of the expansions (7) are
~ = (1-~-,~2~2"-}-o~2~4"}-...
mation (10) will be called the augmented twiceiterated solution. The determinant of I*1 is exactly equal to unity. When the determinant of the approximate matrix (10) is calculated, one finds that it departs from unity by
K'-l~[I-T-~¢12~2"~-o~'12~4-~...]/ "
/ The upper (la) (i.e. a explaining venient to
sign in the -T- in (10) refers to the equation convergent lens), the lower sign to (lb). In the remaining symbols in (10) it is conmake use of a new variable
= ( z - z~)/(z2 - z,),
(12)
The quantities o¢~ are then obtained by making the indicated replacement for q~(() in J, =
•,12
~x2
f'
0
h(~)q~x(~)d~;
use ~ ( ~ ) =
(13)
1 (1 - { ) ~(1 --~). ,4'=
J~ = f'oh(Od¢ f¢oh(¢')((-(')a'2((,¢')d¢' ;
(16)
o~1 + f 2 =~,~ -Jt-,ffJ12--Jl,~2.
(17)
Usually quadrupole magnets are designed to have a transverse plane of symmetry at z = 0. It is useful to have an expression for the transfer matrix applicable to the portion of the axis which is the mirror image of zl to z - i . e , to - z to - Z l . Let M' be this transfer matrix. It is possible to express M' in terms of the same integrals in which !,1 is expressed in (10). We obtain a formula for M' by substituting - z for Zl and - z, for z in (10) and by making use of k ( - z ) = k(z).
t. (,8) / (14)
for
J2
=J
(
Similarly
o,¢
Jl+J2 and
h(~) = k(z)/ k o.
~
(10)
terms of order ~6; this is still true when JF~ 4 and J12~ 4 are neglected. In showing that terms proportional to ~2 and to ~4 are absent one employs the useful identities:
(11)
and of the function
for
307
use ~ 2 ( ( , ( ' ) = 1
(1-O ¢'(1-O.
Finally ~ " = f 0l h(~)d~ f¢o h(~')(~- (')d~' f i ' h(~") (if' - ~") d~".
05) The once-iterated solution is obtained from (10) by discarding all terms beyond those containing J~, and the twice-iterated solution by discarding those beyond J ~ . The only term shown in (10) which belongs to the thrice-iterated solution is y{-~4 in M21. The approxi-
Note that M and M' have the same off-diagonal terms, but that the diagonal terms are interchanged. M' may also be calculated from M by means of a formula provided by BliamptisS). The positive dimensionless quantities J r , J ~ , oFv... act as form-factors for the variation of k(z) in the end zones. It is important to know how they depend on the shape of the function h(0. In table 1 their values are given for three different assumptions about h(~) in 0 < ~ < 1. The uniform case, or h(0 = 1, is of no practical importance; it is included merely to facilitate a test of the theory. Linear ramp means h(~) = 1 - ( ; it corresponds to fig. ld. The bell-shape m o d e l - d i s cussed on page 104 of ref. 2 ) - h a s h(0=/~'(/~2+¢2) -2,
~>__0.
(19)
The parameter fl is related to the width of the bellshaped curve; at ~ = fl h(0 has fallen to ¼ its value at = 0. If the model is to be of any use, h( 1) = fl#(1 + f12)- 2 must be small; h(1) is equal to 10 -2 and to 10 -3 forfl equal to & and to 0.1... respectively.
308
G. E. LEE-WHITING TABLE 1 Form-factors for different models. Uniform h(() = l
Linear ramp h(() = 1--(
Bell-shaped... f o r / / = 0.1 h(0 =//,(//2 + ¢2)- 2
7~
1
1
1 //2 2 1+//2
1
1 //tan- 1 / / = 0.078 507 2
1
1 //4 2 1 + 1/2 - 0.004 950 5
2
// 1
J~
1 / / - ~//tan - 1//= 0.073 556
1
E 1
1
J12
6
1-2
1
1
J
6
3--0
1
1
J'~
2-4
180
1
1
/2
2-4
7-2
1
1
J12
120
504
1
l
120
1440
If one substitutes the form-factors for uniform h(~) (given in table 1) into (10) with the upper sign, one recovers the expansion (7) for the elements of (6). With the lower sign one gets the expansion of the elements of the transfer matrix for eq. (lb). Thus the approximate solution generated by iterating the integral equation is indeed a generalization of the expansion (7). Note that the form-factors for the linear-ramp model of an end zone are smaller than those for uniform h((). The values for the bell-shaped model are smaller still. To show what this means for the convergence of the series for the elements of Ivl we have rearranged the data of table 1 in table 2. Each group of three rows refers to one element of M; the numbers are the ratios of the coefficients of the power of ¢ in the first, second and third terms to the first coefficient; for the bell shape only the leading term i n / / i s used. Convergence appears to be faster for the linear ramp than for the uniform case, and faster still for the bell shape; note, however, that for the bell shape the series for
1//3 (2 -- t a n - l / /
+g [(3
n 2) 4]
= 4.2644 x 10 -3
+~ 1 6
+... = 1.926 x 10-* n //7...
= 6.43
x 10 - 6
= 1.569x 10 -4
K ] + ~1 //6 _ 7[ j ~ _ ~_~//s ~//7 + . . . . 4.93×10-6 ~-~ 1 - - ~
]
//s + . . . . 1.7×10-7
Mal appears to converge considerably more slowly than the series for the other elements. If ¢ is held constant the rate of convergence increase as we move to the right in table 2. However, holding ~ constant does not necessarily make a fair comparison. If the end zones account for a large part of the focusing action of the lens, then ¢ will have to be increased as we move to the right in table 2 to maintain the same total effect. We return to this point near the end of sec. 4. As a check on the accuracy of the theory we have compared its results with values obtained by numerical integration. Three cases were used. Two of them are given in table 1-3 (page 62) of Steffen's bookX). In both these cases k(z) has the linear-ramp shape illustrated in fig. ld. Their parameters are given by: I II
2zl=0.92216m 2z 1=0.394 m
z2-zt=0.12092m z2-zl=0.266 m.
The calculations were carried out for three values of
END EFFECTS IN F I R S T - O R D E R THEORY OF Q U A D R U P O L E LENSES
309
TABLE 2
Rate of convergence of form-factor sequences. Belt-shaped model ... with 8 = 0.1 (approx.)
Rectangular model
Linear ramp
1
1
1
1
J2
0.5
0.33
n8
= 0.08
J2
0.042
0.014
n 83 i-6
=0.02×10_ 2
1
1
1
1
J12
0.17
0.083
~8
J12
0.0083
0.0020
0.067 84 = 0.7 x 10-5
1
1
1
1
0.17
0.067
~8
0.0083
0.0014
0.022 84 = 0.2 x 10- s
1
1
1
0.5
0.17
1 f12
0.042
0.0056
0.067 84 = 0.7 x 10-s
1
J~
k o - 0 . 2 5 , 1 and 4 m -2. The transfer matrix between - z 2 and z 2, T, was computed from the formula T = M H M'.
(20)
Here H is the matrix for the central region in which
k(z) is assumed to be perfectly flat. It is convenient to use a formula for H which applies to either a convergent (eq. la) or a divergent (eq. lb) lens.
H =
.
~ xS
(21)
C
Again the upper choice in an ambiguity such as + or -Trefers to a convergent lens, the lower to a divergent lens. The same principle applies to C and S: cos C=
sin (~b),
cosh
S=
(~b); sinh
(22)
1
1
2
z
= 0.005
= 0.002
= 0.005
here q~ = 2xz 1. A comparison of results obtained by the two methods is made in table 3. The elements 7"11, T12 and 7"21 should be equal to Steffen's C, S and C' respectively; T2z = Txl because the lens is symmetric; Steffen uses a subscript z for the convergent case, x for the divergent case. The results in rows opposite N are those obtained by numerical integration; those opposite 0, 1 and 2 correspond to zero, one and two iterations of the integral equation. Numbers opposite 2 + were obtained with the augmented twice-iterated approximate solution, in which the triple integral occurs in M2x ; this term has no effect on 7"12. The rows opposite F a r e discussed in the next section. Agreement between the numerical and the augmented twiceiterated results is excellent; the error is at most one unit in the fifth position after the decimal point. The ordinary twice-iterated results are almost as good; the worst error, 2 in the fourth place, occurs in 7"21 for the divergent lens. Only the results for the largest value of
310
G. E. L E E - W H I T I N G
TABLE3 Comparison of analytical and numerical results. Linear ramp, k0 = 4 m-2. Matrix element
Order etc. 0
T21 m-i
2 2+ N F
1.790 84 2.463 93 2.475 12 2.475 17 2.475 18 2.475 18
0 1 2 2+ N F
-1.925 65 -1.730 75 - 1.731 55 -1.731 55 - 1.731 55 -1.731 55
6.165 66 7.959 48 7.966 62 7.966 63 7.966 63 7.966 63
-1.417 89 - 1.901 96 -1.893 75 -1.893 81 -1.893 80 -1.893 81
1.74424 3.51448 3.549 34 3.549 53 3.54954 3.549 53
0.387 93 0.37120 0.371 24 0.371 24 0.37124
2.415 36 2.496 11 2.496 30 2.496 30 2.496 30
0.62935 0.526 89 0.528 03 0.538 03 0.528 02
1.265 37 1.44224 1.44426 1.44427 1.44427
TABLE4 Comparison of analytical and numerical results. Case III: bell shape, Zl = 0, fl = 0.1. Matrix element
Order etc.
Tll
1 2 2+ N F
-- 3.747 --3.397 --3.403 -3.407 -3.409
7.558 8.138 8.147 8.155 8.154
1 2 2+ N F
-9.290 -8.556 -8.583 -8.579 --8.583
13.320 14.599 14.637 14.638 14.637
1
-
1.395 - 1.232 - 1.237 - 1.237
4.208 4.46"/ 4.474 4.474
T12 m
2 N F
Div.
0.328 11 -0.010 53 -0.00488 -0.00491 -0.00491 -0.00491
ko are shown in table 3; for the smaller values the a g r e e m e n t is, as one w o u l d expect, even closer. F o r all values of k o the once-iterated results are p r o b a b l y good e n o u g h for most practical purposes. The agreement is better for case I t h a n for case II because ~ is smaller - 0.24 instead o f 0.53. F o r the third c o m p a r i s o n o f analytical a n d n u m e r i c a l
7"21 m-1
Case II Con.
3.986 52 4.568 13 4.57024 4.57024 4.57024 4.57024
0 1 2 N F
m
Div.
-0.50298 -0.598 03 -0.59764 -0.59764 -0.59764 -0.59764
1
Tll
Case I Con.
Case III Con. Div.
results a case was chosen with n o fiat central region. The f u n c t i o n h(() was t a k e n f r o m the bell-shape m o d e l discussed earlier. The p a r a m e t e r s for this case are: III z l = 0 z2=0.5m k o = 1 4 4 m -2 f l = 0 . 1 . T a b l e 4 for case III is similar to table 3, except that we have omitted the zeroth order iteration. Because case III has a m u c h larger value of ~ - 6 - t h a n the other cases, a n d because the increased rate of convergence of the form factors is insufficient c o m p e n s a t i o n , the agreem e n t in table 4 is n o t as good as it was in table 3. Even so the a u g m e n t e d twice-iterated (or perhaps the ordin a r y twice-iterated) results are close e n o u g h to be useful for m o s t practical purposes. The agreement improves rapidly as ~ is decreased. The chief cause of discrepancy in table 4 is the slowness of the convergence of the series for M l l m e n t i o n e d earlier. Better results could be o b t a i n e d by calculating M l l from the condition det(l*l) = 1. This is the chief reason for the closer agreement of the results in rows opposite the s y m b o l F (discussed in the next section). The f o r m u l a (10) m a y be used to calculate x a n d x' (or y a n d y ' ) at all points in the interval z I < z < z 2. These are exactly the functions needed i n calculating the third-order a b e r r a t i o n c o e f f i c i e n t s - s e e , for example, page 53 of ref. 1). Indeed, the f o r m u l a e were originally derived for this p u r p o s e ; it is i n t e n d e d that a n a c c o u n t of the higher-order work should appear later. F o r the rest of this paper, however, we shall be
END EFFECTS IN F I R S T - O R D E R concerned with the case z = z 2 - i . e , matrices for the whole end zone.
with transfer
3. Lens parameters If k(z) is effectively zero outside a finite interval of the z-axis, the action of the lens is conveniently described at points outside this interval by the parameters of a Gaussian thick lens. In general these would consist of a focal length and of the distances from two fiducial marks to the two principal planes. The action of the quadrupole magnet on motions in the x - z and y - z planes will usually be described by different thick lenses. Most often in practice the quadrupole magnet is designed to have a transverse plane of symmetry at z = 0. In this section we shall assume this to be the case. Analogous formulae could be derived for asymmetric lenses, but they would be a bit more complicated. L e t f b e the focal length of the lens. For the fiducial marks needed in specifying the positions of the principal planes let us use the points z = ___z 2 at which k(z) is assumed to go to zero. Because of the symmetry only one parameter, q, is needed to fix the position of the principal planes; q is to be positive when the principal planes lie between the fiducial marks (fig. la). Let T be the transfer matrix acting between z = - z 2 and z = + z2, as in sec. 2. Blewett 6) has shown how to o b t a i n f a n d q from T; the account in the appendix to ref. 7) might, however, be more accessible to the reader. f - 1 = _ T21, (23) q = (1 - T l l ) / ( - T21 ).
(24)
Using the notation (22) combined with the abbreviations Ivl = (adb)one can show that
T21 = 2Ccd + x - 1S(c2 -~ ~2d2)
(25)
and, with the help of det (M) = 1, that Tll = C(1 + 2bc) + ~- 1S(ac T- x2bd).
(26)
Substituting the expressions for the elements of I*! from (10) we get f - 1=_+~{S[(I .-T-~'I~ 2 .-~,~ 1~4)2.-T-~2(o.~ ~o~¢2.-~-,~4) 2] .-[+ 2C{(1 T J x { 2 + J , { 4 ) (j_T_j{2 + j(-~4)}
(27)
and
q =f{( 1 - C) +_S{[(1 g J 2 { 2 + J 2 {4) ( j T j ~ 2 +~f-{4) + + (1 T-~¢1~2 +Jl{')(1-7-~¢, 2~ 2 + j , 2 { 4 ) ] +
+-2C{Z( J T j ¢
2 +)U{4)( 1 T-J,2{ a + J , 2 { ' ) } .
(28)
The coefficient of C in (27) is known with essentially
THEORY OF Q U A D R U P O L E
LENSES
311
the same accuracy as is an element of the matrix M. If ~2d is not too close to u n i t y - a n d it will not be in any practical c a s e - t h e same can be said for the coeffficient of S. I f we insist, as is almost always the case, that ~b=2xzl be less than ½n, C and S are both positive. Then the two major terms inside the braces in (27) are both positive, and there can be no cancellation of their effects. Hence the error infcalculated from (27) is essentially the same as the error in the elements of !*! when calculated from (10). Inspection of (28) shows that an analogous statement can be made for q; all one has to do is to note that
(sin
t 2 1 - - C = _+2\sinhl (xzl).
(29)
As ~ 0 the formulae (27) and (28) approach the corresponding formulae for the hard-edge model, as expected. When there is no central z o n e - i.e. when z 1 = 0 - the formulae simplify considerably, because S = 0 and C = 1 ; note that in this case ~ = xz2. Even when there is a flat central zone we are allowed to put 21 = 0, if the resulting value of ~ gives an adequate speed of convergence. The simplification of (27) when z 1 = 0 is obvious, but that of (28) merits more attention. From (25) and (26) one can see that
q = b/d =b(1 -T-Ja2~2 +o,¢12¢4 + . . . ) / ( 1 -T-JI~2 "~-Ji~ 4 "{-...). (30) When ~ 0 (still with z 1 = 0) the transfer matrix calculated from (27) and (30) approaches the matrix appropriate for free drift through a distance 26, as one would expect. The transfer matrix for any two stations outside the interval - z 2 to +z2 can be calculated from f and q derived from (27) and (28) or (30). Of the various methods considered in this paper this one gives, in the opinion of the author, the best combination of simplicity and accuracy; it leads directly to the input data needed for the method of beam matching described in 7). The numbers in rows opposite the symbol F in table 3 were obtained from f and q. They agree with the results obtained by numerical integration to within one unit in the fifth decimal place. The results opposite F in table 4 were obtained using (27) and (30). They agree with the numerical results to better than 1 part in 103; the discrepancy is believed to be consistent with the magnitude of the omitted terms of the third order in k(z). The " F " rows agree better with the " N " rows in table 4 than do the " 2 + " rows, because the substitution
312
G. E. LEE-WHITING ad = 1 + bc,
used in deriving (26) eliminates the slowly convergent element a from the theory (for the case z 1 = 0). In tables 3 and 4 we have deliberately worked to more figures than are likely to be needed in practice, in order to study the convergence of the method. Formulae (27) and (28) or (30) combined with table 1 make it easy to estimate how many terms are needed for a desired accuracy. The formulae (27) and (28) or (30) could be reduced to polynomials in ~ by multiplying factors, expanding denominators, and neglecting higher-order terms comparable to those known to be lacking. But this would be an undesirable step. In the case zl = 0 it would be equivalent to applying the iterative method to the integral equation over the whole length of the lens. The value of the parameters analogous to ~ would be twice as great; the error terms would be increased by the factor 2 6 . Though the expansion of the formulae is not recommended for calculation, it does provide some indication of how the variation of k(z) in the end zones affects the lens parameters. For zl ¢ 0 we quote results to the first order in ~: f - ~ ~ ___x(S+2Co¢¢},
(31)
q ~ x - ~ tan tanh(XZ~) + 6 + [1 ___2C(C - 1)S- 2]J6. (32) The first term in each of these formulae is the contribution from the central region of uniform k(z). The final term is proportional to J , which is simply related to the average value of k(z) in the end zone. The quantity q - c5 is the distance of a principal plane from one of the positions z = _ zl. Thus, if we use the ends of the central zones as the fiducial marks, the amounts by which the lens parameters differ from those for the central zone alone are proportional to J . When z~ = 0 the first-order term survives in (31) but disappears in (32). Then we calculate q to the second order in ¢: q ~ 6 _+ (J1 -,-ff12)6~ 2.
(33)
Now (q--6) is the distance of a principal plane from the plane z = 0. Since D~1 >J~12, (33) says that a convergent lens has its first principal plane to the right of the plane of symmetry and its second principal plane to the left; the opposite is true for a divergent lens. This is a generalization of a known result for a hardedged lens. The Gaussian parameters for compound lenses may be calculated 6, a) from the f and q of each element, if the regions of appreciable g(z) do not overlap. When overlap occurs, one should construct the transfer
matrix for the whole lens, using (10) for the regions of varying g(z).
4. Effective lengths and strengths A transfer matrix has four elements. When one remembers that the determinant of the matrix is always unity, one sees that only three of the elements are independent. If the lens has a transverse plane of symmetry at z = 0 , T22 = 7"11. Thus the matrix for a symmetric lens has only two independent elements. The rectangular model for the function k(z) also has two independent p a r a m e t e r s - a length and a value of k o. Thus it is not unreasonable to hope that we might be able to find a rectangular model which has exactly the same transfer matrix as a given quadrupole lens. The length of this rectangular m o d e l - s e e fig. l d - we shall call the effective length; let us represent it by the symbol/*. The constant value of the function k(z) for the rectangular model will be represented by k*. Following Steffen 1) we shall call k* the effective quadrupole strength; for particles of fixed momentum it is a measure of the field gradient at the centre of the lens. It is convenient to introduce a parameter x* analogous to x by x* = (k*) ~. (34) Then we define the effective lens strength by the formula = :t*, (35) which is equivalent to the ct of Blewett6). The effective lens strength is a measure of the total focusing (or defocusing) action of the quadrupole lens. It is not difficult to derive a pair of simultaneous equations for I* and ~*. In Steffen's book 1) it is demonstrated that these equations may be combined into a single equation for tk*, which, in the present notation, becomes C* + ½c~*S* = T l l - z2T2t. (36) C* and S* are the functions (22) evaluated for the argument ~b*. T 11 and Tz~ are elements of the transfer matrix acting between z = - z 2 and z = +Zz. If (36) has a solution, values of l* and x* may be found from the easily derived relation -T-x ' S * = Tzl,
(37)
with the aid of (34) and (35). It is convenient to calculate T 2 ~ from the formulae for focal length given in the preceding section. Thus x* = +_f- ~/S*,
(38)
/* = ~b*/x*.
(39)
and
END
EFFECTS
IN
FIRST-ORDER
THEORY
Steffen 1) has p r o d u c e d curves o f quantities closely related to 1", tk* and k* against k o for the cases I and II considered in sec. 2. Septier 2) also discusses effective length, but he is m o r e concerned with trying to represent higher-order properties o f the lens; because he puts k* equal to k o, he c a n n o t get an exact description o f the first-order properties. In this section we shall be chiefly concerned with finding an analytic solution to (36), expressed in terms o f the form-factors defined in sec. 2. The first step is to expand the right side o f (36) in powers o f tp. Such an expansion is c o n t r a r y to the principle discussed near the beginning o f the last paragraph o f sec. 3, but it is apparently necessary. The f o r m o f the expansion depends on whether z~ # 0 or z 1 = 0. F o r z 1 # 0 one finds 1 o0 Tll-z2T21 = C+½d~S - - ~ n~0 e.q~4+2". (40) The coefficients e. are functions of the form-factors and o f the ratio p - ~ / z 1. Exact f o r m u l a r for e o and e 1 and an approximate formula for e 2 (containing all terms corresponding to the augmented thrice-iterated solution) have been derived, n o t without considerable labour. Because the formulae are lengthy, they are listed in the appendix. The formulae in the appendix have been checked by the numerical evaluation o f b o t h sides o f (40) or (45). A s s u m i n g the f o r m q~* = A~b[1 + dlt~ 2 --[-d2q~4 + . . . ] ,
(41)
one can show that A = (1 + Co)*,
(42)
d I = ¼el A - 4 __+6~-6(A2 - A - 4 ) ,
(43)
OF
QUADRUPOLE
313
LENSES
TABLE5 Effective lengths and strengths. Terms
~b*
1"
k*
m
m -2
Case II con. 1 1.370 62 2 1.366 80 3 1.366 52 N 1.366 50
0.709 28 0.706 76 0.706 57 0.706 56
3.734 2 3.740 0 3.740 4 3.740 4
div. 1.37062 1.374 44 1.374 16 1.374 18
0.711 24 0.716 32 0.715 95 0.715 98
3.713 7 3.681 6 3.683 9 3.683 7
1 2 3 N
Case III con. 1 1.352 9 2 1.337 7 3 1.335 9 N 1.3365
0.153 9 0.151 6 0.151 4 0.151 5
77.27 77.81 77.87 77.79
div. 1.352 9 1.368 2 1.366 5 1.367 3
0.166 8 0.171 7 0.171 1 0.171 4
65.75 63.50 63.75 63.65
1 2 3 N
1" was obtained f r o m (39) and k* f r o m the square o f (38). The numbers opposite N came from a direct numerical solution o f (36). The agreement is again excellent; the three-term results do not differ by m o r e than 1 in 104 from the true values. Calculations carried out for case I gave similar results, in spite o f the largeness o f ~b*. W h e n zl = 0 the right side o f (36) m a y be reduced to
and d 2 = ¼e2 A - 4 ++_l~A2dx - ½d 2 - (.44 - .4-4)/2240.
(44)
F r o m the formulae in the Appendix one can see that the values o f e o and el are the same for eqs. (la) and (lb), while the values o f el for the two equations are opposite in sign. Hence the two equations have the same values o f A and d2, but have values o f d I with opposite signs. Thus there is one effective rectangular model for b o t h the x - z and the y - z motions only if one term o f (41) gives sufficient accuracy. When more terms are required, different rectangular models must be used for the two planes o f motion. The m e t h o d has been applied to the case II used earlier, with ko = 4 m -2. The n u m b e r s in rows opposite 1, 2 and 3 in the first c o l u m n o f table 5 were obtained using l, 2 and 3 terms respectively in (41);
= 1-
Tll-z2T2t
~ a,~ "+2".
(45)
n=0
The coefficients a n are functions o f the form-factors. F o r m u l a e for a o, a 1 and a 2 are listed in the appendix. We write q~* = B~[1 + bt~ 2 + b2~ 4 + . . . ] .
(46)
One finds that (47)
B = do*,
bl = ~ 1 B-" +-~ B2, 1_
n--4t
b 2 = ~-U2D
(48)
B,L 1 D2t31_2 "-I-T-fib O 1 - - ' ~ ' O 1 - - _ _
2240"
(49)
The formulae (47) etc. have been applied to case I I I o f sec. 2. The results appear in table 5. This time the
314
G.E.
LEE-WHITING
agreement is not so good, because of the larger value of 4. Even so, the three-term results agree with the exact results with an error less than 2 in 10 a, which ought to be good enough for most purposes. Recall that the fractional error in all the quantities calculated is proportional to 46 . If, as is often the case, the values of tk* in practical applications are smaller than in table 5, the errors may be very much smaller. In laying out a beam-transport system or a proton linear accelerator it is convenient to use the rectangular model for the quadrupole magnets. It would be useful to have some quick way to relate the model parameters to the function k(z), even if it is much less accurate than the three-term formulae (41) and (46). This is specially true when lack of space forces us to use a function k(z) resembling fig. lb, for which the parameters of the effective rectangular model are not very obvious. The first term in (41) or (46) suggests itself for use. Any more terms would require different rectangular models for the x - z and y - z planes of motion, and this is inconvenient and contrary to custom. The results of the one-term calculation, which we shall label with the subscript 0, are expressible in the form:
49~ = (tel)G,
(50a)
l* = IG 2,
(50b)
k* = k0G-2;
(50c)
l is the average length defined by
l - ko'
k ( z ) d z = 2(z, + J a ) .
(51)
~, - z 2
When
Z 1 > O,
G
;
(52)
J 1 2 ) ] ~ J -~.
(53)
= (1 + eo)¼(1 + J p ) - '
when z 1 = 0 G = [3(Jl
-
F r o m the formulae (50) one can see that
* * = Iko = fz2 loko
--g2
k(z)dz.
(54)
This means that the area under the curve k(z) versus z is equal to the same area for the effective rectangular model. Septier's 2) use of the concept of effective length in first-order problems is equivalent to replacing G by unity in (50). To show when this is justified we expand (52) in powers of p. G = 1+
~ ( 2 ~ 1 __ ~2)p2
+
. . . .
(52a)
F o r the linear-ramp model the coefficient of p2 in
(52a) is ~-~. The absence of a linear term in (52a) and the smallness of the coefficient of p2 provides an explanation of why Septier's approximation is sometimes successful. Next let us apply (53) to the models for which we have the form-factors in table 1. For uniform h ( 0 we have i = 2z 2 and G - - 1; the formulae then degenerate into l * = 2z 2 and k * = k o, which are exactly correct. Next consider the linear-ramp form of h(Q; since z 1 = 0, the plot of k(z) has the form of an isosoceles triangle. One finds l = z2,
(55a)
G = 24 ~ 1.189.
(55b)
and
For the bell-shape model the results are l = z2½rq~ + terms of order//4,
(56a)
and G=3~'2"n-~[l
fl x
3fl2 b terms of order fl 3] (56b)
2re2
The lowest-order lengths and strengths for cases I and II were calculated from the formula (50). They appear in table 6, together with values of l, p and G. The numbers in rows to the right of con. and div. are exact values for the planes in which the lens is convergent and divergent respectively. The approximate results are good to a few parts in 10 3 for case I, to about 1% for case II; the method is better for case l, because ~ is smaller. Septier's a p p r o x i m a t i o n - i . e . 1 " = I, k * = 4 - i s better for case I than for case II, because for the former case G is much closer to unity. But even here the discrepancy is greater than the departure of our one-term results from the con. or div. values. The formulae (56) were used with (50) to produce the first line of results for case III in table 6. The agreement with the exact results in the lines below is not as good as for the two other cases; the discrepance is about 6% for l* and 10% for k*. In all cases in table 6 the one-term result lies very close to the average of the values for the convergent and divergent planes of the lens. Hence the one-term result is probably the best approximation which uses the same effective parameters for the x - z and y - z planes. Finally, let us return to the question of the rate of convergence of the series for the elements of !*! considered in sec. 2 and in table 2. The only real difficulty occurs when the end zones account for a large part of the focusing action of the lens. Let us confine our attention to zl = 0. We wish to compare cases for which the k(z) functions have different forms but
315
END EFFECTS IN FIRST-ORDER THEORY OF QUADRUPOLE LENSES TABLE6 Lowest-order effective lengths and strengths.
Case
l-
p
G
lo*
m
I
II
III
1.0431
0.262
0.6600
1.350
0.0785
--
1.0033
¢ = 1E~,
(57)
e = (Ja)-'
(58)
The successive terms in the series actually differ by 42 ; hence E 2 is the relevant quantity. The values of E 2 for the uniform, linear-ramp, and bell-shape models are 1, 2~,~ 2.83, and (approximately( 0.74 fl-2 respectively; for fl = 0.1 the last becomes E z ~ 74. When ~b* is held constant the overall effect of the decrease in the formfactors and the increase in 42 in going from the uniform model to the linear r a m p is that the series for the elements of !*! converge somewhat more slowly, except for Mzz which is just about the same. Comparing the bell-shape model (for fl = 0.1) with the uniform model we find that the terms in M2a and 3//22 decrease at about the same rate, Ma2 somewhat more slowly, and Max very much more slowly. This reinforces our earlier conclusion that it is better to calculate Ma ~ from the condition d e t ( M ) = I. Appendix
In sec. 4 we make use of the expansion
~
e,q~"+2".
1.050 1.047 1.052
3.973 3.982 3.967
2.093 2.091 2.095
con. div.
0.712 0.707 0.716
3.71 3.74 3.68
1.371 1.366 1.374
con. div.
0.162 0.152 0.171
1.4356
with
(59)
n=0
The coefficients of the first three terms in the sum have been worked out; they are functions of the ratio
69.9 77.8 63.6
1.353 1.337 1.367
-m-e, = ¢~p + ( J , + ½j~)p2 + + [ J + l ( J , - ~ , 2 ) + 2 ~ , ] p 3+ + ~ [ 2 J , + ~ 2 + 2~¢J + 2 J ( ~ , - J , 2 ) ] p 4 +
-[-~[~fl - - ~ 1 2 +°I@'2--'~'l°'~/l 2 + + J ( 2 j + j , - J 2 ) + J(~¢, -o%)]p' + "~-~[J(6#~l --o~12) +Y(,3~ 1 --,~12)]p 6,
e2.~TA~Jp+ 4-L0-(0j 2
+ a-~0[ J , + 4 J - - J , 2 + l[j(j,
e o = 4 J p + 3 [ ( ~ "2 + 2 . f l ) p 2 + ( J 1 - - J 1 2 -[- 2 ° ¢ J l ) P 3 +
+J(Ja -J,E)P4],
(60)
(61)
+ 2Jl)p 2+ + 6JJ1]P 3+
_ j r 2) + J 2 + 2o#, + 2 J J ] p 4 +
+ 1[~¢,(j, - J , 2 ) + J , - J , 2 + + 2 y J - J j 2 - J Y 2 + 3 Jj , + + 3 J J 1 + 2~U]p 5 +
+~[Y(J,-J12)+J(J,-J,2)+ -~--~,¢16t¢1 - J - ~ " + l ~t~2]p6 + + 1-~[2,~1 o~1 --,~12 or~ , - - Y l o ~ l 2 -3t-J 2 -- 6ffo~2 "~-
+ J J , + (o¢ + 2 Y l ) ~ ] p 7 + + l~[J(J,
-J,
2)+~(Jt-Yx2)]P
s.
(62)
The expressions for e0 and e~ are exact, but the one for eu is approximate in that certain triple integrals that were not introduced in sec. 2 have been ignored. By an extension of our notation these integrals would be called -YF1, ~ 2 and ~"a2; they ought to contribute to the coefficients of p6, p7 and pS in e2. When z a =-0 we use the expansion
TII-Z2T21 =I-2~
p = 6/zl.
fro*
con. div. 1.0384
which have the same effective lens strength. The required value of the parameter 4 may be calculated from (50a) and (51); one finds
T,,-zzT2,=C+_I#~S - 1
ko* m -2
m
~ a"¢*+2"'
n=O
(63)
instead of (59). The first three coefficients in the sum are:
316
G. E. LEE-WHITING
ao = 4 8 J ( J l - J 1 2 ) ,
(64)
a~ = - T - 4 8 [ J ( J , - J ~ 2 ) + J ( J , - J , 2 ) ] ,
(65)
a2 = 48['J(~/'x --~'12) -~o~(o~ 1--~'12) -~-~"(J1 -- J12)]" (66) These expressions are exact. However, to be consistent with the degree of a p p r o x i m a t i o n used in early sections one would have to neglect the first term in the bracket in (66). T h e expressions (64) to ( 6 6 ) a r e sufficiently simple in f o r m that their extension to higher values of n is obvious.
References 1) K. G. Steffen, High energy beam optics (Interscience, New York, 1964). ~) A. Septier, Advan. Electron. Electron Phys. 14 0961) 85. a) E. L. Ince, Ordinary differential equations (Dover, New York, 1956). 4) H. T. H. Piaggio, Differential equations (Bell & Sons, London, 1945). s) E. E. Bliamptis, Rev. Sci. Instr. 35 (1964) 1521. 8)j. p. Blewett, BNL-JBB-13 (1959). 7) G. E. Lee-Whiting and N. Bezi/:, Nucl. Instr. and Meth. 71 (1969) 61. s) E. Regenstreif, in. Charged particle focusing (ed. A. Septier; Academic Press, New York, 1967).
END EFFECTS IN FIRST-ORDER
THEORY
OF QUADRUPOLE
LENSES
G. E. LEE-WHITING
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 9 June 1969 Rapidly convergent series for the transverse position coordinates of a charged particle moving through an end zone ofa quadrupole lens are derived by iterating integral equations obtained from the first-order differential equations. Not more than three terms of the series appear to be needed for lens strengths normally used. The series contain form-factors formed by integrating the
field gradient along the axis with various weight functions. The focal lengths and the principal-plane coordinates of the whole lens are expressed in terms of numerically well behaved functions of the form-factors. Formulae are also derived for the effective lengths and strengths of the lens.
1. I n t r o d u c t i o n
F o r electrostatic q u a d r u p o l e lenses e q u a t i o n s o f the f o r m (1) c a n also be used. T h e f u n c t i o n k(z) is t h e n p r o p o r t i o n a l to
I n the first-order t h e o r y 1'2) o f the m o t i o n o f a c h a r g e d particle in a q u a d r u p o l e lens the transverse m o t i o n is f o u n d b y solving the differential e q u a t i o n s : d2x
--
dz 2 d2y dz 2
+ k(z)x = 0,
k(z)y = 0.
(la)
(lb)
T h e quantities x, y a n d z are o r t h o g o n a l C a r t e s i a n c o o r d i n a t e s o f the particle. T h e z-axis is a l o n g the fourfold s y m m e t r y axis o f the lens; the x- a n d y-axes lie in two o r t h o g o n a l planes o f s y m m e t r y . F o r a m a g n e t i c lens the f u n c t i o n k(z) is directly p r o p o r t i o n a l to the p a r t i c l e charge, e, a n d to the field g r a d i e n t on the axis, g, a n d is inversely p r o p o r t i o n a l to the particle m o m e n t u m , p. k = k(z) = eg/p, (2)
c3Br g=g(z)=-~x x=r=o
?)E_.__~ ox I x=y=0
(4)
i n s t e a d o f to g, b u t the d e p e n d e n c e on z is similar. W h i l e in describing the analysis it is c o n v e n i e n t to refer to m a g n e t s , the results a p p l y equally well to electrostatic q u a d r u p o l e lenses.
Pi
-z2 -Z~l
k(z)
0 (a)
k(z)
IP2
=Z
IZl
Z2
-Z2
k z)
!oll 0
(c)
.Z
Z2
k(z)
(3)
I n a static m a g n e t i c field p is a c o n s t a n t o f the m o t i o n . T h u s k has the s a m e z - d e p e n d e n c e as g. N e a r the centre o f the lens there is usually a r e g i o n in which g is effectively i n d e p e n d e n t o f z. T o w a r d the edges o f the lens g begins to fall off, a p p r o a c h i n g zero r a p i d l y at large distances f r o m the lens. U s u a l l y the designer o f q u a d r u p o l e m a g n e t s a t t e m p t s to m a k e the g r a p h o f k(z) r e s e m b l e p a r t (a) o f fig. 1; he tries to m a k e the central region in which the curve is a l m o s t flat l o n g c o m p a r e d to the e n d zones in which k(z) is r a p i d l y varying. H o w e v e r , in s o m e a p p l i c a t i o n s , such as the q u a d r u p o l e m a g n e t s inside the drift-tubes o f a lowenergy p r o t o n linear accelerator, l a c k o f space m a k e s this impossible. T h e n the g r a p h o f k(z) is m o r e like fig. l b . I n all cases we shall a s s u m e t h a t g(z) is effectively zero outside the interval - z 2 < z < z 2.
r,-77--I---V,,-], r -Z2
0
(b)
Z2
-Z2
-Zi
0
Zl
Z2
(d)
Fig. 1. Four different possibilities for the variation of k(z) with z are shown in the four parts of the figure. The value of k at z = 0 is k0. All but part (b) have a region stretching from - z l to zl in which k(z) is uniform. The value of k(z) falls from ko to 0 in the intervals zl to z2 and - z ~ to -z~; in part (c) z~=z2. In part (a) an arbitrary smooth curve has been used in the end zones; the parameter q is positive when the principal planes P1 and Pa lie between - z a and z~ as shown. Part (b) represents the bell-shape model with fl = 0.25 and zl = 0; the ratio of k(z2) to ko is 1/289. In part (c) we illustrate the rectangular model, also called the hard-edge approximation. The linear-ramp model in part (d) is drawn with the parameters of case II; the dashed lines show the corresponding effective rectangular model for the ( x - z) plane of motion. 305
306
G. E. LEE-WHITING
The simplest way to treat the first-order motion of a particle in a quadrupole lens is to approximate k(z) by the function shown in fig. l c - i . e , by a function which is zero outside the lens and uniform inside. This is the hard-edge approximation of Steffen a) or the rectangular model of Septier2). The eqs. (1) then have simple and well known solutions. In his review article 2) Septier mentions other work in which analytic solutions to the equations (1) are found for a "bell-shaped" model. Apart from these special cases the only method of calculation presently available appears to be the step-by-step numerical integration of the trajectory equations. The original purpose of the work reported in this paper was to provide a simpler method of treating lenses of the type shown in fig. la. In such cases most of the effect of the lens occurs in the central zone where the hard-edge approximation is appropriate. It did not seem too much to hope that an analytic calculation might be possible for the relatively small effects of the end zones. Such a formulat!on would also be useful in understanding how the lens parameters depend on the shape of the curve of k(z) versus z. It turned out that the desired formulation is indeed possible. Moreover, the analytic processes involved are sufficiently convergent that the method even applies to lens with no central z o n e - f i g , l b - f o r values of lens strength commonly used. The method of calculation to be used in sec. 2 involves an infinite sequence of approximate solutions. Why one may expect the sequence to be rapidly convergent is most easily explained for the hard-edge model of fig. lc. It suffices to consider equation (la); assume ko > 0 and put k o = z 2. Let us consider the transfer matrix, M, acting between the centre of the lens and z = zz. This matrix is well known for the hardedge m o d e l - s e e page 12 of ref. ~), for example. With the definition
= z2,
(s)
one finds (
cosq5
x-lsinq 5)
M =
.
\-tcsinq~
(6)
cos~b
The method of calculation to be discussed in sec. 2 is analogous to expanding the elements of M in powers of ~b. Recall that COS~ = 1--½t~2 + 2-~t~ 4 -- v~-~b 1 6 +...,
(7a)
and that 1 a,4 -5--~-o-w 1 . 7 . 6t ..... tk-lsinq ~ = 1-16tk 2 + 1-~'e
(7b)
Though it is true that a large number ot terms would be needed in these series if ~b were much greater than unity, this is not usually the case. In most applications the quantity 2zz2, which represents the strength 6) of the whole lens, is less than ½n. For ~b < ¼n ~ 0.785 the series (7) converge exceedingly rapidly. The fourth term in the expansion of cos q~ is less than 3.3 x 10-4; the fourth term in the series (7b) is even smaller. When dealing with eq. (lb) we would have to expand cosh ~b and sinh ~b instead of cos ~b and sin ~b. Though the advantage of the alternating signs in (7) would then be lost, the magnitude of the terms decrease at the same rate. It appears unlikely that one would over need more than three terms in (7); one or two terms will often suffice. If three terms of (7) give excellent results for the hard-edge model, then it is certainly desirable to look for an analogous method of calculation applicable to the cases illustrated by figs. la and b. One would expect to find the powers of 4)2 replaced by products of integrals (evaluated over the end zones) of k(z) multiplied by various weight functions. As a first step in finding such an expansion it seems natural to replace (1) by integral equations. 2. The integral equation
Assume that at z = z 1 it is known that x = xl and that x' = x~. We wish to find values of x and x' in the finite interval z 1 < z < Zz. It is not difficult to show that
k(z')(z-z')x(z')dz';
(8)
zl
the upper sign refers to (la), the lower to (lb). We shall find an approximate solution to the integral eq. (8) by an iterative procedure called the "Method of Successive Approximations" by Ince 3) and Picard's method by Piaggio4). For the zero-order approximation to the solution take
x°(z) = xl + x'l ( z - Zl);
(9)
this approximation ignores completely the effect of the field on the particle as it moves through the end zone. The first-order approximation is found by substituting x°(z ') for x(z') in the integral on the right in (8); the integrand is then composed entirely of known functions. The second-order approximation is then found by substituting the first-order approximation for x(z') under the integral sign in (8). In principle the method can be carried to any desired order. It is convenient to give the results in the form of the transfer matrix, M, operating between z~ and z. The elements of M may be expressed as series of integrals of
END EFFECTS IN FIRST-ORDER THEORY OF QUADRUPOLE LENSES progressively higher multiplicity. The series may be treated as power series in the parameter ~ = z6 = k~o(z2-zl), with coefficients which depend upon the shape of the function k(z) in the end zones. The terms analogous to three terms of the expansions (7) are
~ = (1-~-,~2~2"-}-o~2~4"}-...
mation (10) will be called the augmented twiceiterated solution. The determinant of I*1 is exactly equal to unity. When the determinant of the approximate matrix (10) is calculated, one finds that it departs from unity by
K'-l~[I-T-~¢12~2"~-o~'12~4-~...]/ "
/ The upper (la) (i.e. a explaining venient to
sign in the -T- in (10) refers to the equation convergent lens), the lower sign to (lb). In the remaining symbols in (10) it is conmake use of a new variable
= ( z - z~)/(z2 - z,),
(12)
The quantities o¢~ are then obtained by making the indicated replacement for q~(() in J, =
•,12
~x2
f'
0
h(~)q~x(~)d~;
use ~ ( ~ ) =
(13)
1 (1 - { ) ~(1 --~). ,4'=
J~ = f'oh(Od¢ f¢oh(¢')((-(')a'2((,¢')d¢' ;
(16)
o~1 + f 2 =~,~ -Jt-,ffJ12--Jl,~2.
(17)
Usually quadrupole magnets are designed to have a transverse plane of symmetry at z = 0. It is useful to have an expression for the transfer matrix applicable to the portion of the axis which is the mirror image of zl to z - i . e , to - z to - Z l . Let M' be this transfer matrix. It is possible to express M' in terms of the same integrals in which !,1 is expressed in (10). We obtain a formula for M' by substituting - z for Zl and - z, for z in (10) and by making use of k ( - z ) = k(z).
t. (,8) / (14)
for
J2
=J
(
Similarly
o,¢
Jl+J2 and
h(~) = k(z)/ k o.
~
(10)
terms of order ~6; this is still true when JF~ 4 and J12~ 4 are neglected. In showing that terms proportional to ~2 and to ~4 are absent one employs the useful identities:
(11)
and of the function
for
307
use ~ 2 ( ( , ( ' ) = 1
(1-O ¢'(1-O.
Finally ~ " = f 0l h(~)d~ f¢o h(~')(~- (')d~' f i ' h(~") (if' - ~") d~".
05) The once-iterated solution is obtained from (10) by discarding all terms beyond those containing J~, and the twice-iterated solution by discarding those beyond J ~ . The only term shown in (10) which belongs to the thrice-iterated solution is y{-~4 in M21. The approxi-
Note that M and M' have the same off-diagonal terms, but that the diagonal terms are interchanged. M' may also be calculated from M by means of a formula provided by BliamptisS). The positive dimensionless quantities J r , J ~ , oFv... act as form-factors for the variation of k(z) in the end zones. It is important to know how they depend on the shape of the function h(0. In table 1 their values are given for three different assumptions about h(~) in 0 < ~ < 1. The uniform case, or h(0 = 1, is of no practical importance; it is included merely to facilitate a test of the theory. Linear ramp means h(~) = 1 - ( ; it corresponds to fig. ld. The bell-shape m o d e l - d i s cussed on page 104 of ref. 2 ) - h a s h(0=/~'(/~2+¢2) -2,
~>__0.
(19)
The parameter fl is related to the width of the bellshaped curve; at ~ = fl h(0 has fallen to ¼ its value at = 0. If the model is to be of any use, h( 1) = fl#(1 + f12)- 2 must be small; h(1) is equal to 10 -2 and to 10 -3 forfl equal to & and to 0.1... respectively.
308
G. E. LEE-WHITING TABLE 1 Form-factors for different models. Uniform h(() = l
Linear ramp h(() = 1--(
Bell-shaped... f o r / / = 0.1 h(0 =//,(//2 + ¢2)- 2
7~
1
1
1 //2 2 1+//2
1
1 //tan- 1 / / = 0.078 507 2
1
1 //4 2 1 + 1/2 - 0.004 950 5
2
// 1
J~
1 / / - ~//tan - 1//= 0.073 556
1
E 1
1
J12
6
1-2
1
1
J
6
3--0
1
1
J'~
2-4
180
1
1
/2
2-4
7-2
1
1
J12
120
504
1
l
120
1440
If one substitutes the form-factors for uniform h(~) (given in table 1) into (10) with the upper sign, one recovers the expansion (7) for the elements of (6). With the lower sign one gets the expansion of the elements of the transfer matrix for eq. (lb). Thus the approximate solution generated by iterating the integral equation is indeed a generalization of the expansion (7). Note that the form-factors for the linear-ramp model of an end zone are smaller than those for uniform h((). The values for the bell-shaped model are smaller still. To show what this means for the convergence of the series for the elements of Ivl we have rearranged the data of table 1 in table 2. Each group of three rows refers to one element of M; the numbers are the ratios of the coefficients of the power of ¢ in the first, second and third terms to the first coefficient; for the bell shape only the leading term i n / / i s used. Convergence appears to be faster for the linear ramp than for the uniform case, and faster still for the bell shape; note, however, that for the bell shape the series for
1//3 (2 -- t a n - l / /
+g [(3
n 2) 4]
= 4.2644 x 10 -3
+~ 1 6
+... = 1.926 x 10-* n //7...
= 6.43
x 10 - 6
= 1.569x 10 -4
K ] + ~1 //6 _ 7[ j ~ _ ~_~//s ~//7 + . . . . 4.93×10-6 ~-~ 1 - - ~
]
//s + . . . . 1.7×10-7
Mal appears to converge considerably more slowly than the series for the other elements. If ¢ is held constant the rate of convergence increase as we move to the right in table 2. However, holding ~ constant does not necessarily make a fair comparison. If the end zones account for a large part of the focusing action of the lens, then ¢ will have to be increased as we move to the right in table 2 to maintain the same total effect. We return to this point near the end of sec. 4. As a check on the accuracy of the theory we have compared its results with values obtained by numerical integration. Three cases were used. Two of them are given in table 1-3 (page 62) of Steffen's bookX). In both these cases k(z) has the linear-ramp shape illustrated in fig. ld. Their parameters are given by: I II
2zl=0.92216m 2z 1=0.394 m
z2-zt=0.12092m z2-zl=0.266 m.
The calculations were carried out for three values of
END EFFECTS IN F I R S T - O R D E R THEORY OF Q U A D R U P O L E LENSES
309
TABLE 2
Rate of convergence of form-factor sequences. Belt-shaped model ... with 8 = 0.1 (approx.)
Rectangular model
Linear ramp
1
1
1
1
J2
0.5
0.33
n8
= 0.08
J2
0.042
0.014
n 83 i-6
=0.02×10_ 2
1
1
1
1
J12
0.17
0.083
~8
J12
0.0083
0.0020
0.067 84 = 0.7 x 10-5
1
1
1
1
0.17
0.067
~8
0.0083
0.0014
0.022 84 = 0.2 x 10- s
1
1
1
0.5
0.17
1 f12
0.042
0.0056
0.067 84 = 0.7 x 10-s
1
J~
k o - 0 . 2 5 , 1 and 4 m -2. The transfer matrix between - z 2 and z 2, T, was computed from the formula T = M H M'.
(20)
Here H is the matrix for the central region in which
k(z) is assumed to be perfectly flat. It is convenient to use a formula for H which applies to either a convergent (eq. la) or a divergent (eq. lb) lens.
H =
.
~ xS
(21)
C
Again the upper choice in an ambiguity such as + or -Trefers to a convergent lens, the lower to a divergent lens. The same principle applies to C and S: cos C=
sin (~b),
cosh
S=
(~b); sinh
(22)
1
1
2
z
= 0.005
= 0.002
= 0.005
here q~ = 2xz 1. A comparison of results obtained by the two methods is made in table 3. The elements 7"11, T12 and 7"21 should be equal to Steffen's C, S and C' respectively; T2z = Txl because the lens is symmetric; Steffen uses a subscript z for the convergent case, x for the divergent case. The results in rows opposite N are those obtained by numerical integration; those opposite 0, 1 and 2 correspond to zero, one and two iterations of the integral equation. Numbers opposite 2 + were obtained with the augmented twice-iterated approximate solution, in which the triple integral occurs in M2x ; this term has no effect on 7"12. The rows opposite F a r e discussed in the next section. Agreement between the numerical and the augmented twiceiterated results is excellent; the error is at most one unit in the fifth position after the decimal point. The ordinary twice-iterated results are almost as good; the worst error, 2 in the fourth place, occurs in 7"21 for the divergent lens. Only the results for the largest value of
310
G. E. L E E - W H I T I N G
TABLE3 Comparison of analytical and numerical results. Linear ramp, k0 = 4 m-2. Matrix element
Order etc. 0
T21 m-i
2 2+ N F
1.790 84 2.463 93 2.475 12 2.475 17 2.475 18 2.475 18
0 1 2 2+ N F
-1.925 65 -1.730 75 - 1.731 55 -1.731 55 - 1.731 55 -1.731 55
6.165 66 7.959 48 7.966 62 7.966 63 7.966 63 7.966 63
-1.417 89 - 1.901 96 -1.893 75 -1.893 81 -1.893 80 -1.893 81
1.74424 3.51448 3.549 34 3.549 53 3.54954 3.549 53
0.387 93 0.37120 0.371 24 0.371 24 0.37124
2.415 36 2.496 11 2.496 30 2.496 30 2.496 30
0.62935 0.526 89 0.528 03 0.538 03 0.528 02
1.265 37 1.44224 1.44426 1.44427 1.44427
TABLE4 Comparison of analytical and numerical results. Case III: bell shape, Zl = 0, fl = 0.1. Matrix element
Order etc.
Tll
1 2 2+ N F
-- 3.747 --3.397 --3.403 -3.407 -3.409
7.558 8.138 8.147 8.155 8.154
1 2 2+ N F
-9.290 -8.556 -8.583 -8.579 --8.583
13.320 14.599 14.637 14.638 14.637
1
-
1.395 - 1.232 - 1.237 - 1.237
4.208 4.46"/ 4.474 4.474
T12 m
2 N F
Div.
0.328 11 -0.010 53 -0.00488 -0.00491 -0.00491 -0.00491
ko are shown in table 3; for the smaller values the a g r e e m e n t is, as one w o u l d expect, even closer. F o r all values of k o the once-iterated results are p r o b a b l y good e n o u g h for most practical purposes. The agreement is better for case I t h a n for case II because ~ is smaller - 0.24 instead o f 0.53. F o r the third c o m p a r i s o n o f analytical a n d n u m e r i c a l
7"21 m-1
Case II Con.
3.986 52 4.568 13 4.57024 4.57024 4.57024 4.57024
0 1 2 N F
m
Div.
-0.50298 -0.598 03 -0.59764 -0.59764 -0.59764 -0.59764
1
Tll
Case I Con.
Case III Con. Div.
results a case was chosen with n o fiat central region. The f u n c t i o n h(() was t a k e n f r o m the bell-shape m o d e l discussed earlier. The p a r a m e t e r s for this case are: III z l = 0 z2=0.5m k o = 1 4 4 m -2 f l = 0 . 1 . T a b l e 4 for case III is similar to table 3, except that we have omitted the zeroth order iteration. Because case III has a m u c h larger value of ~ - 6 - t h a n the other cases, a n d because the increased rate of convergence of the form factors is insufficient c o m p e n s a t i o n , the agreem e n t in table 4 is n o t as good as it was in table 3. Even so the a u g m e n t e d twice-iterated (or perhaps the ordin a r y twice-iterated) results are close e n o u g h to be useful for m o s t practical purposes. The agreement improves rapidly as ~ is decreased. The chief cause of discrepancy in table 4 is the slowness of the convergence of the series for M l l m e n t i o n e d earlier. Better results could be o b t a i n e d by calculating M l l from the condition det(l*l) = 1. This is the chief reason for the closer agreement of the results in rows opposite the s y m b o l F (discussed in the next section). The f o r m u l a (10) m a y be used to calculate x a n d x' (or y a n d y ' ) at all points in the interval z I < z < z 2. These are exactly the functions needed i n calculating the third-order a b e r r a t i o n c o e f f i c i e n t s - s e e , for example, page 53 of ref. 1). Indeed, the f o r m u l a e were originally derived for this p u r p o s e ; it is i n t e n d e d that a n a c c o u n t of the higher-order work should appear later. F o r the rest of this paper, however, we shall be
END EFFECTS IN F I R S T - O R D E R concerned with the case z = z 2 - i . e , matrices for the whole end zone.
with transfer
3. Lens parameters If k(z) is effectively zero outside a finite interval of the z-axis, the action of the lens is conveniently described at points outside this interval by the parameters of a Gaussian thick lens. In general these would consist of a focal length and of the distances from two fiducial marks to the two principal planes. The action of the quadrupole magnet on motions in the x - z and y - z planes will usually be described by different thick lenses. Most often in practice the quadrupole magnet is designed to have a transverse plane of symmetry at z = 0. In this section we shall assume this to be the case. Analogous formulae could be derived for asymmetric lenses, but they would be a bit more complicated. L e t f b e the focal length of the lens. For the fiducial marks needed in specifying the positions of the principal planes let us use the points z = ___z 2 at which k(z) is assumed to go to zero. Because of the symmetry only one parameter, q, is needed to fix the position of the principal planes; q is to be positive when the principal planes lie between the fiducial marks (fig. la). Let T be the transfer matrix acting between z = - z 2 and z = + z2, as in sec. 2. Blewett 6) has shown how to o b t a i n f a n d q from T; the account in the appendix to ref. 7) might, however, be more accessible to the reader. f - 1 = _ T21, (23) q = (1 - T l l ) / ( - T21 ).
(24)
Using the notation (22) combined with the abbreviations Ivl = (adb)one can show that
T21 = 2Ccd + x - 1S(c2 -~ ~2d2)
(25)
and, with the help of det (M) = 1, that Tll = C(1 + 2bc) + ~- 1S(ac T- x2bd).
(26)
Substituting the expressions for the elements of I*! from (10) we get f - 1=_+~{S[(I .-T-~'I~ 2 .-~,~ 1~4)2.-T-~2(o.~ ~o~¢2.-~-,~4) 2] .-[+ 2C{(1 T J x { 2 + J , { 4 ) (j_T_j{2 + j(-~4)}
(27)
and
q =f{( 1 - C) +_S{[(1 g J 2 { 2 + J 2 {4) ( j T j ~ 2 +~f-{4) + + (1 T-~¢1~2 +Jl{')(1-7-~¢, 2~ 2 + j , 2 { 4 ) ] +
+-2C{Z( J T j ¢
2 +)U{4)( 1 T-J,2{ a + J , 2 { ' ) } .
(28)
The coefficient of C in (27) is known with essentially
THEORY OF Q U A D R U P O L E
LENSES
311
the same accuracy as is an element of the matrix M. If ~2d is not too close to u n i t y - a n d it will not be in any practical c a s e - t h e same can be said for the coeffficient of S. I f we insist, as is almost always the case, that ~b=2xzl be less than ½n, C and S are both positive. Then the two major terms inside the braces in (27) are both positive, and there can be no cancellation of their effects. Hence the error infcalculated from (27) is essentially the same as the error in the elements of !*! when calculated from (10). Inspection of (28) shows that an analogous statement can be made for q; all one has to do is to note that
(sin
t 2 1 - - C = _+2\sinhl (xzl).
(29)
As ~ 0 the formulae (27) and (28) approach the corresponding formulae for the hard-edge model, as expected. When there is no central z o n e - i.e. when z 1 = 0 - the formulae simplify considerably, because S = 0 and C = 1 ; note that in this case ~ = xz2. Even when there is a flat central zone we are allowed to put 21 = 0, if the resulting value of ~ gives an adequate speed of convergence. The simplification of (27) when z 1 = 0 is obvious, but that of (28) merits more attention. From (25) and (26) one can see that
q = b/d =b(1 -T-Ja2~2 +o,¢12¢4 + . . . ) / ( 1 -T-JI~2 "~-Ji~ 4 "{-...). (30) When ~ 0 (still with z 1 = 0) the transfer matrix calculated from (27) and (30) approaches the matrix appropriate for free drift through a distance 26, as one would expect. The transfer matrix for any two stations outside the interval - z 2 to +z2 can be calculated from f and q derived from (27) and (28) or (30). Of the various methods considered in this paper this one gives, in the opinion of the author, the best combination of simplicity and accuracy; it leads directly to the input data needed for the method of beam matching described in 7). The numbers in rows opposite the symbol F in table 3 were obtained from f and q. They agree with the results obtained by numerical integration to within one unit in the fifth decimal place. The results opposite F in table 4 were obtained using (27) and (30). They agree with the numerical results to better than 1 part in 103; the discrepancy is believed to be consistent with the magnitude of the omitted terms of the third order in k(z). The " F " rows agree better with the " N " rows in table 4 than do the " 2 + " rows, because the substitution
312
G. E. LEE-WHITING ad = 1 + bc,
used in deriving (26) eliminates the slowly convergent element a from the theory (for the case z 1 = 0). In tables 3 and 4 we have deliberately worked to more figures than are likely to be needed in practice, in order to study the convergence of the method. Formulae (27) and (28) or (30) combined with table 1 make it easy to estimate how many terms are needed for a desired accuracy. The formulae (27) and (28) or (30) could be reduced to polynomials in ~ by multiplying factors, expanding denominators, and neglecting higher-order terms comparable to those known to be lacking. But this would be an undesirable step. In the case zl = 0 it would be equivalent to applying the iterative method to the integral equation over the whole length of the lens. The value of the parameters analogous to ~ would be twice as great; the error terms would be increased by the factor 2 6 . Though the expansion of the formulae is not recommended for calculation, it does provide some indication of how the variation of k(z) in the end zones affects the lens parameters. For zl ¢ 0 we quote results to the first order in ~: f - ~ ~ ___x(S+2Co¢¢},
(31)
q ~ x - ~ tan tanh(XZ~) + 6 + [1 ___2C(C - 1)S- 2]J6. (32) The first term in each of these formulae is the contribution from the central region of uniform k(z). The final term is proportional to J , which is simply related to the average value of k(z) in the end zone. The quantity q - c5 is the distance of a principal plane from one of the positions z = _ zl. Thus, if we use the ends of the central zones as the fiducial marks, the amounts by which the lens parameters differ from those for the central zone alone are proportional to J . When z~ = 0 the first-order term survives in (31) but disappears in (32). Then we calculate q to the second order in ¢: q ~ 6 _+ (J1 -,-ff12)6~ 2.
(33)
Now (q--6) is the distance of a principal plane from the plane z = 0. Since D~1 >J~12, (33) says that a convergent lens has its first principal plane to the right of the plane of symmetry and its second principal plane to the left; the opposite is true for a divergent lens. This is a generalization of a known result for a hardedged lens. The Gaussian parameters for compound lenses may be calculated 6, a) from the f and q of each element, if the regions of appreciable g(z) do not overlap. When overlap occurs, one should construct the transfer
matrix for the whole lens, using (10) for the regions of varying g(z).
4. Effective lengths and strengths A transfer matrix has four elements. When one remembers that the determinant of the matrix is always unity, one sees that only three of the elements are independent. If the lens has a transverse plane of symmetry at z = 0 , T22 = 7"11. Thus the matrix for a symmetric lens has only two independent elements. The rectangular model for the function k(z) also has two independent p a r a m e t e r s - a length and a value of k o. Thus it is not unreasonable to hope that we might be able to find a rectangular model which has exactly the same transfer matrix as a given quadrupole lens. The length of this rectangular m o d e l - s e e fig. l d - we shall call the effective length; let us represent it by the symbol/*. The constant value of the function k(z) for the rectangular model will be represented by k*. Following Steffen 1) we shall call k* the effective quadrupole strength; for particles of fixed momentum it is a measure of the field gradient at the centre of the lens. It is convenient to introduce a parameter x* analogous to x by x* = (k*) ~. (34) Then we define the effective lens strength by the formula = :t*, (35) which is equivalent to the ct of Blewett6). The effective lens strength is a measure of the total focusing (or defocusing) action of the quadrupole lens. It is not difficult to derive a pair of simultaneous equations for I* and ~*. In Steffen's book 1) it is demonstrated that these equations may be combined into a single equation for tk*, which, in the present notation, becomes C* + ½c~*S* = T l l - z2T2t. (36) C* and S* are the functions (22) evaluated for the argument ~b*. T 11 and Tz~ are elements of the transfer matrix acting between z = - z 2 and z = +Zz. If (36) has a solution, values of l* and x* may be found from the easily derived relation -T-x ' S * = Tzl,
(37)
with the aid of (34) and (35). It is convenient to calculate T 2 ~ from the formulae for focal length given in the preceding section. Thus x* = +_f- ~/S*,
(38)
/* = ~b*/x*.
(39)
and
END
EFFECTS
IN
FIRST-ORDER
THEORY
Steffen 1) has p r o d u c e d curves o f quantities closely related to 1", tk* and k* against k o for the cases I and II considered in sec. 2. Septier 2) also discusses effective length, but he is m o r e concerned with trying to represent higher-order properties o f the lens; because he puts k* equal to k o, he c a n n o t get an exact description o f the first-order properties. In this section we shall be chiefly concerned with finding an analytic solution to (36), expressed in terms o f the form-factors defined in sec. 2. The first step is to expand the right side o f (36) in powers o f tp. Such an expansion is c o n t r a r y to the principle discussed near the beginning o f the last paragraph o f sec. 3, but it is apparently necessary. The f o r m o f the expansion depends on whether z~ # 0 or z 1 = 0. F o r z 1 # 0 one finds 1 o0 Tll-z2T21 = C+½d~S - - ~ n~0 e.q~4+2". (40) The coefficients e. are functions of the form-factors and o f the ratio p - ~ / z 1. Exact f o r m u l a r for e o and e 1 and an approximate formula for e 2 (containing all terms corresponding to the augmented thrice-iterated solution) have been derived, n o t without considerable labour. Because the formulae are lengthy, they are listed in the appendix. The formulae in the appendix have been checked by the numerical evaluation o f b o t h sides o f (40) or (45). A s s u m i n g the f o r m q~* = A~b[1 + dlt~ 2 --[-d2q~4 + . . . ] ,
(41)
one can show that A = (1 + Co)*,
(42)
d I = ¼el A - 4 __+6~-6(A2 - A - 4 ) ,
(43)
OF
QUADRUPOLE
313
LENSES
TABLE5 Effective lengths and strengths. Terms
~b*
1"
k*
m
m -2
Case II con. 1 1.370 62 2 1.366 80 3 1.366 52 N 1.366 50
0.709 28 0.706 76 0.706 57 0.706 56
3.734 2 3.740 0 3.740 4 3.740 4
div. 1.37062 1.374 44 1.374 16 1.374 18
0.711 24 0.716 32 0.715 95 0.715 98
3.713 7 3.681 6 3.683 9 3.683 7
1 2 3 N
Case III con. 1 1.352 9 2 1.337 7 3 1.335 9 N 1.3365
0.153 9 0.151 6 0.151 4 0.151 5
77.27 77.81 77.87 77.79
div. 1.352 9 1.368 2 1.366 5 1.367 3
0.166 8 0.171 7 0.171 1 0.171 4
65.75 63.50 63.75 63.65
1 2 3 N
1" was obtained f r o m (39) and k* f r o m the square o f (38). The numbers opposite N came from a direct numerical solution o f (36). The agreement is again excellent; the three-term results do not differ by m o r e than 1 in 104 from the true values. Calculations carried out for case I gave similar results, in spite o f the largeness o f ~b*. W h e n zl = 0 the right side o f (36) m a y be reduced to
and d 2 = ¼e2 A - 4 ++_l~A2dx - ½d 2 - (.44 - .4-4)/2240.
(44)
F r o m the formulae in the Appendix one can see that the values o f e o and el are the same for eqs. (la) and (lb), while the values o f el for the two equations are opposite in sign. Hence the two equations have the same values o f A and d2, but have values o f d I with opposite signs. Thus there is one effective rectangular model for b o t h the x - z and the y - z motions only if one term o f (41) gives sufficient accuracy. When more terms are required, different rectangular models must be used for the two planes o f motion. The m e t h o d has been applied to the case II used earlier, with ko = 4 m -2. The n u m b e r s in rows opposite 1, 2 and 3 in the first c o l u m n o f table 5 were obtained using l, 2 and 3 terms respectively in (41);
= 1-
Tll-z2T2t
~ a,~ "+2".
(45)
n=0
The coefficients a n are functions o f the form-factors. F o r m u l a e for a o, a 1 and a 2 are listed in the appendix. We write q~* = B~[1 + bt~ 2 + b2~ 4 + . . . ] .
(46)
One finds that (47)
B = do*,
bl = ~ 1 B-" +-~ B2, 1_
n--4t
b 2 = ~-U2D
(48)
B,L 1 D2t31_2 "-I-T-fib O 1 - - ' ~ ' O 1 - - _ _
2240"
(49)
The formulae (47) etc. have been applied to case I I I o f sec. 2. The results appear in table 5. This time the
314
G.E.
LEE-WHITING
agreement is not so good, because of the larger value of 4. Even so, the three-term results agree with the exact results with an error less than 2 in 10 a, which ought to be good enough for most purposes. Recall that the fractional error in all the quantities calculated is proportional to 46 . If, as is often the case, the values of tk* in practical applications are smaller than in table 5, the errors may be very much smaller. In laying out a beam-transport system or a proton linear accelerator it is convenient to use the rectangular model for the quadrupole magnets. It would be useful to have some quick way to relate the model parameters to the function k(z), even if it is much less accurate than the three-term formulae (41) and (46). This is specially true when lack of space forces us to use a function k(z) resembling fig. lb, for which the parameters of the effective rectangular model are not very obvious. The first term in (41) or (46) suggests itself for use. Any more terms would require different rectangular models for the x - z and y - z planes of motion, and this is inconvenient and contrary to custom. The results of the one-term calculation, which we shall label with the subscript 0, are expressible in the form:
49~ = (tel)G,
(50a)
l* = IG 2,
(50b)
k* = k0G-2;
(50c)
l is the average length defined by
l - ko'
k ( z ) d z = 2(z, + J a ) .
(51)
~, - z 2
When
Z 1 > O,
G
;
(52)
J 1 2 ) ] ~ J -~.
(53)
= (1 + eo)¼(1 + J p ) - '
when z 1 = 0 G = [3(Jl
-
F r o m the formulae (50) one can see that
* * = Iko = fz2 loko
--g2
k(z)dz.
(54)
This means that the area under the curve k(z) versus z is equal to the same area for the effective rectangular model. Septier's 2) use of the concept of effective length in first-order problems is equivalent to replacing G by unity in (50). To show when this is justified we expand (52) in powers of p. G = 1+
~ ( 2 ~ 1 __ ~2)p2
+
. . . .
(52a)
F o r the linear-ramp model the coefficient of p2 in
(52a) is ~-~. The absence of a linear term in (52a) and the smallness of the coefficient of p2 provides an explanation of why Septier's approximation is sometimes successful. Next let us apply (53) to the models for which we have the form-factors in table 1. For uniform h ( 0 we have i = 2z 2 and G - - 1; the formulae then degenerate into l * = 2z 2 and k * = k o, which are exactly correct. Next consider the linear-ramp form of h(Q; since z 1 = 0, the plot of k(z) has the form of an isosoceles triangle. One finds l = z2,
(55a)
G = 24 ~ 1.189.
(55b)
and
For the bell-shape model the results are l = z2½rq~ + terms of order//4,
(56a)
and G=3~'2"n-~[l
fl x
3fl2 b terms of order fl 3] (56b)
2re2
The lowest-order lengths and strengths for cases I and II were calculated from the formula (50). They appear in table 6, together with values of l, p and G. The numbers in rows to the right of con. and div. are exact values for the planes in which the lens is convergent and divergent respectively. The approximate results are good to a few parts in 10 3 for case I, to about 1% for case II; the method is better for case l, because ~ is smaller. Septier's a p p r o x i m a t i o n - i . e . 1 " = I, k * = 4 - i s better for case I than for case II, because for the former case G is much closer to unity. But even here the discrepancy is greater than the departure of our one-term results from the con. or div. values. The formulae (56) were used with (50) to produce the first line of results for case III in table 6. The agreement with the exact results in the lines below is not as good as for the two other cases; the discrepance is about 6% for l* and 10% for k*. In all cases in table 6 the one-term result lies very close to the average of the values for the convergent and divergent planes of the lens. Hence the one-term result is probably the best approximation which uses the same effective parameters for the x - z and y - z planes. Finally, let us return to the question of the rate of convergence of the series for the elements of !*! considered in sec. 2 and in table 2. The only real difficulty occurs when the end zones account for a large part of the focusing action of the lens. Let us confine our attention to zl = 0. We wish to compare cases for which the k(z) functions have different forms but
315
END EFFECTS IN FIRST-ORDER THEORY OF QUADRUPOLE LENSES TABLE6 Lowest-order effective lengths and strengths.
Case
l-
p
G
lo*
m
I
II
III
1.0431
0.262
0.6600
1.350
0.0785
--
1.0033
¢ = 1E~,
(57)
e = (Ja)-'
(58)
The successive terms in the series actually differ by 42 ; hence E 2 is the relevant quantity. The values of E 2 for the uniform, linear-ramp, and bell-shape models are 1, 2~,~ 2.83, and (approximately( 0.74 fl-2 respectively; for fl = 0.1 the last becomes E z ~ 74. When ~b* is held constant the overall effect of the decrease in the formfactors and the increase in 42 in going from the uniform model to the linear r a m p is that the series for the elements of !*! converge somewhat more slowly, except for Mzz which is just about the same. Comparing the bell-shape model (for fl = 0.1) with the uniform model we find that the terms in M2a and 3//22 decrease at about the same rate, Ma2 somewhat more slowly, and Max very much more slowly. This reinforces our earlier conclusion that it is better to calculate Ma ~ from the condition d e t ( M ) = I. Appendix
In sec. 4 we make use of the expansion
~
e,q~"+2".
1.050 1.047 1.052
3.973 3.982 3.967
2.093 2.091 2.095
con. div.
0.712 0.707 0.716
3.71 3.74 3.68
1.371 1.366 1.374
con. div.
0.162 0.152 0.171
1.4356
with
(59)
n=0
The coefficients of the first three terms in the sum have been worked out; they are functions of the ratio
69.9 77.8 63.6
1.353 1.337 1.367
-m-e, = ¢~p + ( J , + ½j~)p2 + + [ J + l ( J , - ~ , 2 ) + 2 ~ , ] p 3+ + ~ [ 2 J , + ~ 2 + 2~¢J + 2 J ( ~ , - J , 2 ) ] p 4 +
-[-~[~fl - - ~ 1 2 +°I@'2--'~'l°'~/l 2 + + J ( 2 j + j , - J 2 ) + J(~¢, -o%)]p' + "~-~[J(6#~l --o~12) +Y(,3~ 1 --,~12)]p 6,
e2.~TA~Jp+ 4-L0-(0j 2
+ a-~0[ J , + 4 J - - J , 2 + l[j(j,
e o = 4 J p + 3 [ ( ~ "2 + 2 . f l ) p 2 + ( J 1 - - J 1 2 -[- 2 ° ¢ J l ) P 3 +
+J(Ja -J,E)P4],
(60)
(61)
+ 2Jl)p 2+ + 6JJ1]P 3+
_ j r 2) + J 2 + 2o#, + 2 J J ] p 4 +
+ 1[~¢,(j, - J , 2 ) + J , - J , 2 + + 2 y J - J j 2 - J Y 2 + 3 Jj , + + 3 J J 1 + 2~U]p 5 +
+~[Y(J,-J12)+J(J,-J,2)+ -~--~,¢16t¢1 - J - ~ " + l ~t~2]p6 + + 1-~[2,~1 o~1 --,~12 or~ , - - Y l o ~ l 2 -3t-J 2 -- 6ffo~2 "~-
+ J J , + (o¢ + 2 Y l ) ~ ] p 7 + + l~[J(J,
-J,
2)+~(Jt-Yx2)]P
s.
(62)
The expressions for e0 and e~ are exact, but the one for eu is approximate in that certain triple integrals that were not introduced in sec. 2 have been ignored. By an extension of our notation these integrals would be called -YF1, ~ 2 and ~"a2; they ought to contribute to the coefficients of p6, p7 and pS in e2. When z a =-0 we use the expansion
TII-Z2T21 =I-2~
p = 6/zl.
fro*
con. div. 1.0384
which have the same effective lens strength. The required value of the parameter 4 may be calculated from (50a) and (51); one finds
T,,-zzT2,=C+_I#~S - 1
ko* m -2
m
~ a"¢*+2"'
n=O
(63)
instead of (59). The first three coefficients in the sum are:
316
G. E. LEE-WHITING
ao = 4 8 J ( J l - J 1 2 ) ,
(64)
a~ = - T - 4 8 [ J ( J , - J ~ 2 ) + J ( J , - J , 2 ) ] ,
(65)
a2 = 48['J(~/'x --~'12) -~o~(o~ 1--~'12) -~-~"(J1 -- J12)]" (66) These expressions are exact. However, to be consistent with the degree of a p p r o x i m a t i o n used in early sections one would have to neglect the first term in the bracket in (66). T h e expressions (64) to ( 6 6 ) a r e sufficiently simple in f o r m that their extension to higher values of n is obvious.
References 1) K. G. Steffen, High energy beam optics (Interscience, New York, 1964). ~) A. Septier, Advan. Electron. Electron Phys. 14 0961) 85. a) E. L. Ince, Ordinary differential equations (Dover, New York, 1956). 4) H. T. H. Piaggio, Differential equations (Bell & Sons, London, 1945). s) E. E. Bliamptis, Rev. Sci. Instr. 35 (1964) 1521. 8)j. p. Blewett, BNL-JBB-13 (1959). 7) G. E. Lee-Whiting and N. Bezi/:, Nucl. Instr. and Meth. 71 (1969) 61. s) E. Regenstreif, in. Charged particle focusing (ed. A. Septier; Academic Press, New York, 1967).