Third-order particle motion through the fringing field of a homogeneous bending magnet

Nuclear Instruments and Methods in Physics Research A 344 (1994) 278-285 North-Holland

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A

Third-order particle motion through the fringing field of a homogeneous bending magnet B. Hartmann *, H. Wollnik

II. Physikalisches Institut der Justus-Liebig-Universität, 35392 Giessen, Germany

(Received 13 December 1993) The ion trajectories through the extended fringing field of a homogeneous magnet are approximated by analytic formulas including the effects of inclined and curved field boundaries . The results are expressed by transfer matrices . All expressions are compared to numerical integrations through typical field distributions. 1. Introduction The global ion-optical properties of a magnet can be described by a nonlinear map relating final phasespace coordinates rf to initial ones r, of individual charged particles: For most application this map is represented by a transfer matrix to a prespecified order. In case of the main field direct analytical formulas can be found. In case of an extended fringing field no analytical expressions exist. Therefore the matrix elements can only be obtained by time consuming numerical integrations [1] or approximated by analytical formulas [2,3]. The results in refs . [2,3] do not include the longitudinal effects and comparing the results of ref. [1] with refs . [2,3] there are some disagreements for elements that include the effects of a curved field boundary . Therefore, in this article the analytical expressions are newly calculated and represented in an effective third-order approximation. Assuming the length of the fringing field to be of the same size of the half gap Go of the magnets, the shift and bending of the main path is expressed in a polynomial up to third order in G, and the nth-order matrix elements are expressed in polynomials of (3 - n) order. 2. Determining the main path of a reference particle It is desirable to distinguish between a real main path and an ideal main path, which both should coin* Corresponding author .

cide well inside the magnet [4]. The ideal main path describes the trajectory of a reference particle in a field that changes its magnitude at the effective field boundary [4] abruptly from zero to a constant value. This ideal main path is described by a straight line in the field-free region and a circle of radius po inside the magnet (Fig. 1) . In case of an extended fringing field the bending curvature h = 1/p of the main path increases smoothly from zero at su to h o = 1/p, t at s h . If

Fig. 1. The trajectory of the reference particle as it passes through the fringing field region under the influence of an ideal field (ideal main path) and under the influence of a real field distribution (real main path). Both trajectories coincide inside the magnet, while the real main path is shifted by 0 X and tilted by 00 relative to the ideal main path at the effective field boundary of the magnet .

0168-9002/94/$07 .00 C 1994 - Elsevier Science B.V. All rights reserved SSDI0168-9002(93)E1334-T

279

B. Hartmann, H. Wollnik /Nucl. Instr. and Meth. to Phys. Res A 344 (1994) 278-285

cos (P and using the initial conditions fi. =X = 0 and Za = sQ one finds for the final coordinates : 1

eb = -fh du, CO

xn

C;

ffh d 2 u,

)2 1 1 1 Zb = -u - - s h du du . Co 2 c o f (f

Fig. 2. The field distribution of B Y (u) is assumed to be known from experiments or calculations along the dashed line . A particle at the position P = (Z, 0) at a distance d = Z cos E from the effective field boundary is deflected by the flux density By (d) = BY(u) .

both main paths must coincide at sb, the real main path must be shifted by AX and bent by the angle AO when passing the effective field boundary. The magnitude of AX and A(P can be determined by solving a4p as

-h(s),

ax as-

az

- sin (P, as =cos 0.

e = fsbh(c,Z) dZ = 1f Sbc ')h(u) du sa

Co

To simplify Eqs. (4) all integration boundaries are left off. This system of equations describes to first order the motion of the reference particle in the fringing field of a magnet that has a straight field boundary . In case of a field boundary that is curved with a radius R the distance d between the reference particle that moves along the real main path and the field boundary is defined by

where Z, = R cos E and Xo = -R sin e are the coordinates of the center of the circle that describes the curvature of the effective field boundary . Expanding Eq . (5) and using c = 1 /R and s o = sin E the distance d(X, Z) is approximated to third order by d(X, Z)=co Z-so X-

SaCO

where c o = cos E, u = c o Z and h = h(u) = BY(u)IX. with BY (u) being the field distribution along the straight line. Inserting Eq . (3) into Eq . (2), expanding sin 0,

Zc(c0X 2 +2soco XZ+s0Z 2)

+ 2C2(s~,c2X3 + Co (3sô -

1)X2Z

+s o(1-3cô ) XZ 2 -sôc o Z3 ) .

(2)

These equations of motion of the reference particle can be solved, if the actual curvature h(s) of the trajectory is known at every position s. Assuming a reference particle of charge qo and momentum po, the curvature h(s) equals BY(s)/Xm, where X m = po/qo is the particle's magnetic rigidity . In many cases the flux density By is known along a straight line perpendicular to the effective field boundary (dashed line in Fig. 2). The coordinates 0(s), X(s), Z(s) can be determined iteratively. At a first step one assumes a straight field boundary and that the reference particle is not deflected at all. Integrating then the first relation of Eq . (2) from sa to sb, the overall deflection is found to first order as :

(4)

(6)

Using Eq. (4) the distance d can be expressed by d = u + Ad, where Ad contains only terms of integrals over the known curvature h(u) . The curvature h(s) along the real main path can be approximated by expanding h(u) : h(s) =h +h'Ad + 2'-h"( Ad )2 =h+h'(-2

,2

2

f(fhdu l du+ t ö ffhd 2 u

to~ +cCou ffh d 2 u - zctâu2c~

+'h"(cô

+

âcztoua

ZCZt0U3

(ffh d2 U~ Z -c C6 u 2 ff h d2u (7)

with the abbreviations t o = tan E, h = h(u), h' = ah(u)/au and h" = a2h(u)/aU2. Inserting Eq . (7) into

B . Hartmann, H. Wollnik / Nucl. Instr. and Meth. in Phys . Res . A 344 (1994) 278-285

280

Eq . (2), the final location of the reference particle is : Xb

e

hd2u d4u - ô ffh'ffh o fc

1 tô

+2

c2

Eq . (11) over u yielding : fi b = ~fh du-

cffh' U 2 d2u,

h d3u +

+2 c fh'ff

1 1 1 2 Zb = -u - - s f (fil du) du . CO 2 c,

3

(8)

h'u2

- 2 co cfh'f_

z

c3 fh' ffh' ff h dsu d3 u + 2 co

cf d3 u h'uffh

z - Zc t~

2 t ° fh'u 3 du h'u2 du - Zc 2 c Co f

Solving the integrals by partial integration and comparing the coordinates with the coordinates of the ideal main path, the difference can be written as :

+2

t(2 1 3 to OX= -2holia+ c 2 c 3h°I3 ° -3 2ch°I2 °' o o o AZ=

2

z ôfh' f(fh du) d2 u

1 1 co hpi3a

- Zc

-2

co fh 3

co



h du (ff d2u)2

fh"u2 ffh d3u

t 3 to4 h"u4 o toz with zholia+ +ac2c f_ c3hol,a 2 o o o d2u Il . _ ho (sbco)2/2) = 1(ffh tô 1 1 - 3 ch 2 1 +--h0313a . (12) 2 2a co 2 co I2a =h o '(f (tif h du) du - (sbco)3/3) GO, Also here the occurring integrals can be solved by partial integration and the difference between the real 2 and ideal main path can be expressed by 73a =ho 2 (f (fh du) du - (sbco) 3 /3) ^' Go* (10 ) If the final coordinates of both paths coincide at sb, the start position of the real main path must be shifted slightly by terms of second and third order and therefore h(s) must be modified to : h(s) = h +

z

to fhdu) du + - ffhd2u h'(-2c~f( C

+ cô ffh' ffh +c

C O u ff h

d 4u

d2u

-

-

2 c~ cff h'u2

ZCt2U2

d2u

- zc2t2u3 -

co

2 t°

CO

ch ° h a -ch2

t3 zl3a+ 2( 213a - I2b) 4 co c t

t2 + 3c2h,) 3 12a , o t1 X =

1

3 to 2 tâ hoIia - 2 -h 03a + 3cho co I2a, co co

AZ=

1 1 2 C 3h°13a

(13 )

with the additional integral h°ha

1 1 t 3 t2 3 ch ol2a +- zhôl3a +~Il0 3a-3 2 c° CO 2 c°

I2b = ho 2 (fh 2u2 du - (sbco) 3/3) - Goo . In case of a straight boundary AO vanishes, showing that in this case the real main path is only shifted by AX and AZ and not bent at the effective field boundary.

+zh"(cô(ffh d2u) 2 -c Ôu c 2ff h d2 u + âc 2 tôu4 1 .

0O=

( 11 )

The overall bend then can be expressed by integrating

3. Determining the path of an arbitrary particle To calculate the trajectory of an arbitrary particle throughout the fringing-field region one advanta-

B. Hartmann, H. Wollnik/Nucl. Instr. and Meth . in Phys. Res. A 344 (1994) 278-285

281

geously uses canonical particle-optical coordinates [1,5] r l =x, r2 = a = PXlpo, r3 = y , r4 = b

=

P,,lpo,

r5=1="o(t-ta), rb =s K

=

Klq - Kol go Kolgo

,

rnlq - molgo

(14)

molgo

where x is the distance to the real main path in the plane of deflection and y the distance to this plane, while po, co, KIt/q o , m11/q, and t o denote momentum, velocity, kinetic energy-to-charge ratio, mass-to-charge ratio and time of flight of the reference particle, whereas p, v, K/q, m/q and t stand for the same quantities of the particle under consideration . Within a magnetic sector the equations of motion become : ax ay Po a1 Vo P _ = Po _ -_ - I as a _, as = P as = b ^' P P

Fig. 3. The position Pk of an arbitrary particle relative to the position P of the reference particle which is moving along the real main path is expressed in the coordinate system (z, x) which is rotated by an angle (P .

aa

From Eq . (17) one finds the components of B to third order as :

ab

av BY(s, x, y) - -~ ax)

1 P =-(bpBz-B,,)(I+hz)+h , as Po Xm p

as

1 Xm \

p -aPBz+B, (1+hx) .

(15)

If we use the abbreviations :

1

B Y (s, x, y) =

P = PV I - (Po/P) 2(a z + b z ) , c=L,

Po

5k)

(I

+

2V0,3Y 2- 6V3 ,1x 3

)

- VO,IX

(1+

2 7ISK+ S,n 1+271

X

,,i=o

,

-

2

Vi" 1XY

2 V2,1 x y

)'

(16)

Here the factor vl =Ko /2m o u2 is used for relativistic corrections with o, denoting the speed of light. The flux density B in Eq . (15) must be determined from a magnetic potential V in the rotated coordinate system (z, x, y), where z is parallel to the reference-particle trajectory and its origin is placed to the position s of this particle which is moving along its path (Fig . 3) : V,,(s) i i )1

2 7V1,3xy 2 ,

B,(s, x, y) = _(a5

__1

L'

V(s, x, Y) _

- (ay) 1

77 sK + s m) , 1 +71

~ 76 K %8 m (1+sK)(1+ o= 1+,~

3

- vo ' , - V,lx - 2V2,1x

I-(Po/P)2(a2+bz)

V (l +

-V1,lY-Vz,IxY-zV3,Ix2Y

(17)

_

1

6V0,3

y3 .

(18)

Since the magnetic potential V must fulfill the Laplace equation AV= 0, the coefficients Vo, 3 and V1 , 3 can be expressed by others : _ Vz,l, Va .3 = Vo,l (19) V1,3 = - V1"1 - V3,1 ,

Thus B depends only on the coefficients V, ,1 which can be expressed by expanding the coefficients of BY (s, 0, 0) along the real main path : 1 i.

V(k) 1

az k+ ' - az kax ,By(s, az k + ` =Xm

0, 0)

k ~ h(s, 0, 0) = XmhzkX , . azax

(20)

282

B. Hartmann, H. Wollnik/Nucl. Instr. and Meth. i n Phys. Res . A 344 (1994) 278-285

Using Eqs. (20) and (18), the Eqs. (15) can be expanded to third order (neglecting the terms including 5 k and B.) yielding :

ax = az

allas =1+hx+(a 2 +b 2 +hxa 2 +hxb 2)/2, aa/as = ( -h x -h 2 )x + ( -hxx/ 2 - hh x )x 2 -ha 2 /2 +(h, +h xx )y 2 /2+h'yb -hb2/2 + ( - hxx/6 - hhxxl2) x3

+ ( h zzx + hh 2z + hxxx + hhxx)xy2/2 +(hzx - hhz)xyb,

ab/as = + hx y + (h xx

+

hh x )xy - hzay

+(hxxxl2 + hhxx)xxy + ( - hzx - hhz)xay + ( - hzzx - hxxx)y3/6 .

ax

1 1

ax - 1

2 C(2

dx

0

+c

h xzz = hddxz , + hdd(dzzdx + d xz d z) + 3hddddxd2

-ct 2 u fh du +

Dz - 1

2 Cô

aZ_ ax

(22)

+x k sin(O) +ZP ,

+Xp .

2 (fhdu) ,

t 1 -fît du+ 0 fh'ffhd 3u Co c2 -

toz t2 c -fh' u 2 du + c o h âh a , 2 c co

0

to CO

(1 -3CÔ)u 2 ,

f

25

hx = hddx

(23)

The derivatives of the Z and X to z and x can be obtained by expanding sin eh, cos (P and using Eq . (12) : 1 1

ZC2

Finally h x can be approximated by :

In Fig. (3) the position P of the reference particle and the position Pk of an arbitrary particle are shown. P denotes the origin of the rotated coordinate system (z, x) . Thus the position of Pk in this system is given by zk, x k and in the coordinate system (Z, X) by :

aZ

ffh d2u - chIla - ctocou

o.

h xxx = h d d xxx + 3hdddxxdx + hddddz

+xk COs(P)

2

d xxx = 3c2tac3

2 h, = hdd  +hdddZ,

sin(O)

co (fh

du)

dxx = -cc, - 2cto co h du + C Z (3toCÔ - 1)u,

hxx = hddxx + hdddx,

-zk

_toto + 2

+ ctôh0

h x =h d d x ,

Xk =

(24)

= fhdu+ c-0 fh"'fhdu- 2ctofh'u2 du

In Eqs. (21) only the curvature h occurs and its derivatives with respect to the coordinates z and x of the rotated coordinate system. Knowing h(d) and d(x, z), thus the derivatives can be obtained by using the chain rule several times:

Zk =zk COs(O)

(fhdu) .

The distance drk between Pk and the effective field boundary ~ can be obtained by replacing Z by Zk and X by `~ k in Eq . (6). The expansion coefficients of d are calculated as derivatives at x = z = 0. In x-direction the derivatives are found up to the needed approximation order from :

(21)

h z = hddZ,

fhdu

Co

2 to2 -fh' ho + Ç2 c u2 du - c t0 CO o

ax/as = a + hax + a3/2 + ab 2/2,

ay/as = a + hax + a2b/2 + b3/2,

t0 1 fhdu- fh'f

Co

"dx Ad + zh"'dx(Ad )2 = h'dx + h t --h' fh du + -0 h' fh' h d3u - 1 ct 2h'fh'u2 du Co

ff

t0 +ct2 h'h o h Q -h't,c o + 1 h'(fh du) 2 cl o + ch'

2

ffh d2u - ch'hI, a - ct,coh'u

- Ct 2h'uf h du 2 t ltc0h" (1-3cô)h'u 2 + 2 f (f h du) du +zc2c 0

-

t~h

0

h d 2 u - -°t3 h h,ff h d4 u ff ff

+ 1 t4ch"

ff h' U2 d2u -2 Ct 2h"uffh d2u

+ Zctôcoh"u2

283

B . Hartmann, H. Wollnik I Nuct. Instr. and Meth. m Phys . Res . A 344 (1994) 278-285

+ cz t,~c o h"u 3 +

t Co

f

t0 (Y, B)=2h ocz l, a

h"Ih du fh dZ u

0

(B,Y)_ - Zctôh"ufhdu+tôh"h o l la - t0ho - h0 ,4a -1tâh_ , (IIhd 2 co

Zu \z fI

+Zct~h"'u z ffhd Z u

- Hc Ztocoh"'u4, c and analogously one obtains

- c2hOIl a

3h'"d x d = h'd,x, + 3h "d,d xx + xx Ad + h"r(dx)3 . +h(dx ) 3 0d + Zh(dx)3(od)2 (28)

The real particle trajectory now can be determined by inserting Eqs. (26)-(28) into Eq . (21) and using the method of successive approximation. The difference of the particle trajectory under the influence of an ideal and a real field distribution is best expressed in a matrix representation . For the entrance fringing field the matrix elements have the following form (with X = x, A = a, Y= y, B = b, L =1 and P = (p -po)lpo) 1st order: to (X, X) = 1-ch oI a - (2 + 3t2) (A to (X, A) = -2h,,-l,° 1 P) = (X, Zha co

B) = 1 - 2hôl,a-`2 + hOIlb 2 (1+2 t0) co co

(B,

(L,

x) _ -ch,

z t-" CO

co

(2t0 + 3t~)

Ila

1 (L, A) = -h,Ila 2 . CO

2nd order:

t2 (X , XX) = - 7'h (X,

to 6t2) ) = zho2 + zhô I4a (5 + c co

1 (A, XX) = ,';cho3 , XA) = hot2

(A, YY) = Zhôt (1 + 2t() - zcho

0

z (A,A)=1-2hôt21,a+chol,ato(2+3tô) CO

tz 0 P)=hô 0h t a -ch o co c o ha

(Y,Y)=1-hôl, b 2(1+2tô)+choIla 1(2to+3tô) co col

1 3co

toz +ihôI4a -(7+ lOt2) c0

t 11_ 2 + 6t2 ) 4a, - 2O cl-(5 o

co

tz -ch0I, a o (2+3t()

(A,

- ch,I,Q

t0 X)=t,h o +c zh oI, a 2(2+3t2 )

c0

C

Co

Co

c

t

2O (2+3 tô ) z

(27)

h ,xx = hddxx, + 3hddd,xdx +hdddd3

(A,

(1+2t,2 )

- chOI,a t0 (2 + 3t,ß ) + choI,b t0 (5 + 6 0 c0 0

h xx = hddxx + hdddX + h (dx) z(Ad) z z

o

+h'I,, 2(1+2tô)+h'I9 t °(7+lOt~ t 1) c0 c0 (26)

=1t'dxx+hlldxxAd +h"(dx)2+h'rr(dx)ZOd

1

(A, YB) _ - h0to -hôl4a C , (1 + 2tO ) (Y,

z 1 3 XY)=t0ho+ h0214 . -(to+2t0) co

1 (B, AT) = -ch,)-s - chôl4a (5to + llt3 +6 t5) 0

(B,XB) = -tôho - hôl4a

1 (t0+2tô) CO

284

B. Hartmann, H. Wollnik/Nucl. Instr. and Meth.

(B, AY) _ -h,,- -hôl4a -(St o +6t3) CO c CO

(L, YY) =h0,4a

~

1 1 2co

t)~

(Y,

Co

(A, XXA)

= Zcho

to z h oc 2 c 4

_ -I

0

t (B, XXB) _ - ;hoc 3

(X, XXA) = -hotô (B, to (X, XYY) _ - zh0(t4 +2 t4) + zhoc 3 co (B, (X, XYB)=h o(t o +2tO) (X XXP) = (B, zhotn to (B, (X, AYY) =h o co (B, 1 (X, YYP) = - zh0 z co (B, to _ Xy ~y ,czh (A, ) 03 to

YYY) = lh) 3 (1 +6 t )2 ) + 2h2h, -(1+212) co o

(B, XXY)

to = - Zchoco

c

Res. A 344 (1994) 278-285

(Y, YYB) _ _ zho tco

3rd order: (X,XXX)

m Phys.

o

Co)

t XAY) _ -3hoc 3 ci XAB)=h o(-t o -2t3) t0 AAY) = - h0 z XBP) = t2 h, AYP) =h,-, c ~) YYY)

+'-,hôl )0 ( t ; (18+28 t 2 ) co to (1+2ti~ - 0

to (A, XYY) = 'Th)2 c 3 (1 + 6t4) - -hocz C CO c0 O t3

(A,

XYB) =hôt 2 (1 +2 t02) -3ho c(

(A,

XBB) _ - t3()h,

(A,

t0 AYY)=''h2tô(5+612)-zhoc 3

Co

- ~h)c ls (1 + 6tC) - 3hôc1~

+ 2hocz t4

)

CO

(B, YYB) = 2ho c ° t - zh `- ic c0

= Z h o c t° c0

1 CO

(1+2 tô)

1 -2h()2 I62 (1 + 2t(, ) c(

CO

XXY)

cz

1

(A, XAP) = -h Otzo

(Y,

t2)

+ -`,hot 0(4 + jOt2 +6 t4)

(A, XAA)=h ot3

(A,YYP)=hô(-Zt o -tô) 2 (A, YBP) =h0t

1

h

ci)

(A, AYB)=h o (-t -2to;)

0

= hto ( B, YBB )- 0

CO

(L, XXX) = -'-h2t' b 0 0 (L, XXA)

_ - Zhotn

(L, XYY) = - ;hûtô (Y, XXB) =h o () (L, XYB) =h ot)~ (Y, XAY)=h0(t0+2tô) 1 (29) (L, AYY) Zh 3 . _ -töh o (Y, XYP) c o

=

t~) CO

(5 + 6t~;)

B Hartmann, H. Wollmk/Nucl Instr. and Meth. in Phys. Res A 344 (1994) 278-285

The listed integrals are: Il°

-h o I

I,

2

- h°_

(ffh (ff

d2ll

- (Sbco)Z/21

~ Go

h2 d2u - I4asbCo - (Sbco)2/2)

Go

I4a=ho2(fh2du-sbcol ~ Go

IS = h o

2

f (dh/du)

2

du - Go

I6=h~ 2 fu(dh/du) 2 du I,

=h~

I lo = h o

3(f

[h2

fît

t

G°

du l du - (Sbco)2/2) - Goo

3 f «dhldu ) 2

fil

du)

du - G,,«

In Eq . (29) the momentum dispersion is considered . In order to get the energy and mass dispersion, the elements (., . . . P) must be replaced by ( ., . . .D)=( ., .-P)Tk and ( ., . . .G)=( ., . . .P)7 , (30) with G = Sm and D = Sk . The terms 27k = 1 + r7/(1 + rl) and 2-r, = 1 - 77/(1 + 71) contain the relativistic correction . So far we have determined only the transfer matrix of the entrance fringing field. The also necessary transfer matrix of the exit fringing field is obtained as the reversed matrix of the entrance fringing field, i.e . the matrix of the entrance fringing field must be inverted and the signs of a, b and 1 must be exchanged. 4. Discussion The expressions of the matrix elements in Eq . (29) were compared with the results of numerical integrations [1], where the values of the shifts and bends of the real main path and of the matrix elements were calculated for different values of the half gap width Go . All elements agree in an effective third-order approximation . Comparing the expressions for the real main path to

285

the results of ref. [2] shows agreement for the shift AX . The bend De of Eq . (17) in ref. [2], however, should be replaced by our AO of Eq . (2). To compare the results of the present article to the matrix elements of refs . [2,3] the different coordinate systems must be taken into account, since there the coordinates tan a = =Prlpz and tan /3 p y lp z were used instead of the canonical normalized momentum a =p xlp o here used and b =pylp o [5]. In order to compare our Eq . (29) to Eq . (38) of ref. [2] and Eq . (27) of ref. [3] it was necessary to transform Eq . (29) into the coordinate system used in refs. [2,3]. After this transformation we found complete agreement except for higher-order terms in the first-order matrix elements f'(u, u), f'(tan a, u), f'(tan a, tan a), f'(tan a, y), f'(tan a, S) in Eq . (38) of ref. [2] as well as G, H, Hß in Eq . (27) of ref. [3]. We would recommend to modify these elements according to our Eq . (29) . The inaccuracies of these first-order coefficients of ref. [2] were mentioned already in ref. [6]. The corrected formulas in ref. [6] agree with our results. Also we compared our results to the expressions of ref. [7]. The author of ref. [7] listed the second order terms in Eq . (65) and mentioned some disagreement of the terms T233 and T234 to the corresponding terms of ref. [2], which in our notation are called (A, YY) and (A, YB). Our results show agreement to the formulas of ref. [2] thus indicating that the terms T233, T234 in ref. [7] need to be corrected.

References [1] B. Hartmann, M. Berz and H. Wollnik, Nucl . Instr. and Meth . A 297 (1990) 343. [2] H. Matsuda and H. Wollnik, Nucl . Instr. and Meth . 77 (1970) 283. [3] T. Sakurai, T. Matsuo and H. Matsuda, Int. J. Mass . Spectr. and Ion Phys . 91 (1989) 51 . [4] H. Wollnik and H. Ewald, Nucl . Instr. and Meth . 36 (1965) 93 . [5) H. Wollnik, Optics of Charged Particles (Academic Press, Orlando, Florida, 1987). [6] M.I . Yavor, Nucl . Instr. and Meth . A 337 (1993) 16 . [7] L. Sagalovsky, Nucl . Instr. and Meth A 298 (1990) 205.