The correction of image aberrations of second order for mass or energy separators consisting of one single magnetic sector field

NUCLEAR

INSTRUMENTS

AND METHODS

53 (I967) I97-215; © N O R T H - H O L L A N D

PUBLISHING

CO.

T H E C O R R E C T I O N OF IMAGE ABERRATIONS O F SECOND ORDER FOR MASS OR ENERGY SEPARATORS CONSISTING OF ONE SINGLE MAGNETIC S E C T O R FIELD H. WOLLNIK

IL Physikalisches Institut, Justus Liebig-Unioersitiit, Giessen, Germany Received 6 March 1967 Image aberrations of second order are presented for mass separators consisting of one homogeneous or inhomogeneous x-a) magnetic sector field, whereby the influence of the fringing fields 4, 5) are included in these expressions. Using a graphical method similar to linear programming techniques the derived aberrations can be corrected by proper choice of the curvatures of the entrance and exit bound.aries of the magnet. As an example finally such a correction is carried out for a stigmatic focusing symmetric mass separat6r using a magnetic field with the inhomogeneity factor n = 0.5 and with perpendicular entry and exit of the ion beam into and out of the sector field.

1. Properties of mass separators 1.1. GEOMETRICAL RELATIONS Using conical or toroidal pole faces 2' a) (fig. 1) a magnetic sector field can be constructed in which a particle that has the charge eeo (e = charge number, - e o = charge of one electron), energy Uo, and mass m o can move in a circular main path of radius rmo, along which the constant field strength Be is present. Such a sector field with radially decreasing field strength has decreased focusing power in the radial direction (Le. in the middle plane of deflection), increased focusing power in the axial direction (i.e. perpendicular to the middle plane of deflection), and increased mass and energy dispersion compared to a homogeneous magnetic sector field. Conical or toroidal pole faces have an axis of rotation and thus imply to introduce a cylinder coordinate system r, z, 0 or equivalent dimensionless coordinates radial coordinate: u = ( r - rmo)/rm o, axial coordinate: v = Z/rmo, azimuthal coordinate: w = 0. Because of the rotational symmetry the pole gap width b is independent of w and depends, for w = const., only on the radial coordinate u: b = boil + n u + ( n 2 +½~)u 2 + . . . I (1) where be describes the pole gap width at r = rm0 the location of the main path. n is called the inhomogeneity factor which is responsible for the distribution of the field strength B. From refs. 2'3) we know B, = B e [ - n o - ~uv + . . . ] , (2a) B v = Boil - n u - ½~u 2 + ½(~ + n)v 2 + . . . ] .

Fig. 1. Conical polefaces are shown which cause a radially inhomogeneous magnetic sector field.

197

(2b)

198

H. WOLLNIK

For all the following calculations v~e will use the same notation as in ref. 3) with u. rmo and v" rmo, where u ,f 1, < 1, describing the radial and axial deviations of a paraxial trajectory from the main path, and ~ and fl, where ,~ 1, fl 4 1, denoting its radial and axial angles of inclination relative to the main path, [i.e. angles that are ineluded by the main path and the projections of the paraxial trajectory first on the plane of symmetry v = 0 and second on a plane which is perpendicular to the plane v = 0 and which includes the tangent on the main path], "and m = ?no (1 +y) and U = Uo (1 +6), where y ,~ 1, 6 .~ 1, describing the mass and energy of a particle of charge eeo moving along this paraxial trajectory. Further ~m shall denote the deflecting angle of the sector field between the profile planes II6). e' and ~" shall denote the angles between the main path and the normals to the ideal field boundaries (ill[. 2) [i.e. the angles under which the main path enters and leaves the sector field], R'r" o and R"r'~o are the radii of curvature of the pole face boundaries at the entrance and exit side of the sector field~' 6) and l~rmo and l'~rmo the distances between the entrance and exit slits and the profile planes I (fig. 2) at the entrance and the exit side of the sector fieldS). 1.2. GENERAL ION OPTICAL PROPERTIES IN A SECOND ORDER APPROXIMATION

•Using the transfer matrice.~ of ref. 6) we can easily derive the position (us,vs) in the exit slit of a particle that has moved along a paraxial trajectory. For this calculation we must know only the point (uo,vo) where, and the angles of inclination (=o,flo) relative to the main path under which the particle under consideration moved through the entrance slit, and also its relative mass and energy deviation (y,6) from a main path particle. Using the notation of ref. 6) k,o = _ (Uo,~o,~,6,Uo,Uo~o,Uo~,Uo6,~o,~o~,~o6,r 2 2 2,~6,6o,Vo,Vo#o,#o 2 2 2.... ), k~o- (v0,#o .... ) we know: kr 5 =-=2 O.(l.,)'mA,r" tt 2 2 t O.(lm)'k.o, (3a) k:5

2

. 2 . D:(Im)'mA:r 2D~(lm)'k~o,

(3b)

where the matrices D and A are given in ref. 6). After carrying out four matrix multiplications these equations yield in a second order approximation us = uoA. + ~oA. + yA~ + 6A6 + u2 A.. + Uo~oA.. + uoy A.~ + uo6 A.6 + ~2A=~+ 2 2 2 2 + ¢xo~A=~+ O~o6A~6+ ~, A ~ + y6A~ + 6 A~ + veAl. + VofloA~p+ ~oApB (4a)

and in a first order approximation (4b)

v5 = voAu + floAp, with A. = ~ + lmv., A= = la. + lmv=+ l'(It. + I~V.), A~ = A~ = p~ + l~v~, Av = lzo+ lmV., ap =/~B + l~,vp+ l'(g~ + l~v~), A.. = p . . + l"mv.., A.= =/~.=+/~v..+2l'(/~..+ l;v..),

I" v A~ = /~.. q_ lmV=" q- Imvua)q-lm(12.u+lmv..) ,, ,2 ,, " q_ lm(/~u" , ' A., = A=8 = #., + l~v=r + l'(/~., + l~v.,), A~ = A66 = #yy+ lmV~,7, A)6 =/z~6 + l:,v~6, Aop = #op + l~vvp+ 21m(/l~u+ l'~vu~), It

l

It

l 1

ADp =/.tpp + Imvpp+ Im(/Z.p+ Imv.B)+ lm(l~.. + ImV.,), Ap~ = Ap~ =/zp~ + lmvpy+ Im(/a.~+/~vvr ).

THE CORRECTION OF IMAGE ABERRATIONS

199

Since axial image aberrations fall in the direction of the slit length a first order approximation for v5 is sufficient. One of the axial second order terms (Ap~ = Aps), however, is included in the listing oft.he A u because it is necessary for the determination of to: in eq. (10b). The g and v are defined in eq. (23) of ref. 6) as functions of R', R", g,8", and of the/7,~ given in eqs. (12) of this article. Eqs. (12) are derived from eqs. (14,18,19) of ref. 3). 1.3. FOCUSING AND DISPERSINGPROPERTIES Postulating that the apparatus must refocus particles which emerge from a point of the entrance slit under radial and axial angles of inclination a o and Po, we have to design it so that the equations A~ -- 0 (radial focusing condition),

(5a)

A# -- 0 (axial focusing condition),

(51~)

are fulfilled. Using the above mentioned g,v and calling lm and 1~ now l:m, t'r'm and i'm, l'.', if they are related to radial, or axial lens equations, eqs. (5) easily can be transformed to (tim --fr 4.'h~m) (li'm --fr + h i ' ) = f ~,

and (Ym - f z + h'm) (/~m--f~ 4- h~m) ---f~, or to the familiar form of thin lens equations (gin 4- h;m)-I 4- (gin 4- hlm)-I _- (f,) -1 (radial),

(6a)

(l; m-4-h;m)-t + (/~m4. h~m)-1 = (f~)-I (axial).

(6b)

Heref,(fffi) denotes the radial (axial) focal length, I"m and l"s (/~m and l~') denote the radial (axial) object and image distances, while h"m and h~'m(h'm and h"m) denote the distances between the ideal field boundaries and the radial (axial) principle planes at the entrance and the exit side, respectively. After inserting the proper/t u and v u in eqs. (5) the coefficients of eqs. (6) read

f, = [ p s * - c*(t' + t")-(s*t't"/p)]-i, him = [1 - c* - (s*t"/p)']fr,

h~'m = I-1 - c*--(s*t'/p)]f,, f , = [qs t + ct(T' + T " ) - (stT'T"/q)]-', h'~m = [1 -- c t 4- (stT"/q)]f~, h"m = [1 - c t 4- (s~T'/q)]L, wherep = (1 - n ) *, q = n*,s * = sin (p~bm), c* = cos (p~b,,), s t = sin (q~bm), c t = cos (q~bm), t' = tan g, t" = tan ~", T' = t'-I~(1 +2tn)/c ', T" = t " - I~'(1 4-2t"~)/c ". 1~ and I~ are small quantities which depend on the distribution of the fringing field¢). If as a rough approximation a triangular fringing field distribution is assumed extending over the distance/?rmo and 5"rmo at the entrance and exit side respectively, I~ = ~5' and I ~ = ~ " . This approximation of I' and I" usually is sufficient For a more precise determination eq.(14b) of ref. 4) must be used. If eqs. (6) are fulfilled the radial and axial magnification factors A, and A~ respectively, using again the g and v of ref. ~) are:

A~ = c* 4- (s *t'/p)-- l" [ps* - c *(t' + t")-- (s *t't"[p)],

(7a)

A v = c t - (s t t'/q) - lm[qS t 4- Ct (t' 4- t")-- (St t't"/q)],

(7b)

while the mass- and energy-dispersions are: A r = A, = {1/(2p~)} {1 - c* + Im[s*p + t"(1 - c*)] }.

(8)

That means that particles of the masses me and m0(1 + ~), both having the energy Uo, or particles of the energies Uo and Uo(l + 6), both having the mass me, passing through the entrance slit of length 2Vormo and width 2Uormo,

200

H. W O L L N I K

will be refocused in two slit images,of length 2%Aor,,o and width 2uoA,rmo which are separated by the distance TAyrmo measured perpendicular to the main path in the plane of symmetry. Three coefficients of image aberrations of eqs. (4) must not necessarily cause an image broadening and thus a decrease in resolving power. These coefficients are A.~y,Ap~, and Au,. Ave causes a curvature of the slit image. This property must be taken into account by a curved exit and straight entrance slit or equivalently by a straight exit and a curved entrance slit. The radius of curvature Rsti, usually is qui~e large; 2 Rstitrmo = - { A o / ( 2 A o o ) ) r m o .

(9)

0

In eq. (9) R~,it is positive if its center of curvature lies in the direction towards the center of curvature of the main path. A~y and Ap~ cause an image broadening for particles of different masses if the image is assumed to lie in a plane perpendicular to the main path. This image broadening, however, vanishes if the image is allowed to lie in an inclined focusing plane. Because 0fthe symmetry e r a magnet the radial images of the entrance slit formed by particles of different mass lie in a plane perpendicular to the middle plane of deflection. The line of intersection of this plane with the middle plane of deflection, which will be referred to as radial focusing line, includes some angle to, with the main path which usually is different from 90° (fig. 2). The tangent of this angle can be taken from ref. 7). It nevertheless can be derived easily by calculating l~ once from etA, = 0 and then from ~tA~+ ctTA~ = 0, if ct ~ 0, and by using further the radial distance ~Ayrmo of the paraxial image point. Thus we get tg to, = -- ( v~ + I ' v , ) A , / A~,,

(10a)

where v= and vu are to be takeri from eq. (23) of ref. 6) and A~ and A~ are defined in eqs. (4). Similar as a radial focusing line also an axial focusing line in the plane of symmetry can be defined along which the axial slit images are located. This line includes the angle to~ with the main path. By calculating l'_m ' from/~A a = 0 and from flAa +~ffTAa~= 0, if ~ff# 0, and by using also TAy we obtain tg to~ = - (v a + l'v~)A~/Aa~.

(10b)

For stigmatic focusing mass separators (A~ = Aa = 0) that shall focus simultaneously different mass lines it must be postulated that co, = to,. This postulate is equivalent to (v. +

=

(,,

(11)

+

For the special case of homogeneous magnetic sector fields this formula is given in s). idiat fiel.d boundaries

E"~

¢'

MAIN PATH

"

Fig. 2. A magnetic sector field is shown with all its ion optical characteristics. The corrected ideal main path and a corrected idea] paraxial trajectory are shown passing through the sector field from the entrance to the exit slit.

THE CORRECTION

201

OF I M A G E A B E R R A T I O N S

1.4. IMAGE ABERRATIONS OF SECOND ORDER

The image aberrations given here differ from the ones givenin refs. 1, ~) since here also tlae fringing field effects are included up to terms of second order. In the thin lens approximation of eqs. (5,6) only slight changes due to I~ and 1~ can be noticed for the axial focusing power as compared to refs. 1. ~). For the coefficients of image aberrations, however, large changes are present for Avo,A~#,A##,A#r The other coefficients are independent of fringing field terms. Though it has been mentioned in refs. 4,s) it should be emphasized here again that the fringing field terms of the just mentioned four coefficients are all independent of the special fringing field distribution or extension. These terms are only due to the fact that a fringing field is present which does fulfill Maxwell's equations contrary to an abruptly cut off field. 1.4.1. The general coefficients of the image aberrations For numerical calculations the image aberrations in eqs. (4) can be presented in a more convenient manner. The coefficients given in eqs. (4) are nonlinear in ~b=, 8', e", n, and also in l~ and l~. The first order coefficients A,, A,, A~, A~, Ao, and A# further are independent of the curvatures of the pole shoe boundaries 1/R' and l/R", while the second order coefficients A,,, A~,, A,~, A,6, Aw A,~, A~6, Aye, Ars, A~s, A~, A~#, A##, and A#~ are linear functions of 1/R' and l/R". Thus we can write for ij = uu, u~, u~, u~, ~ , ~ , ~6, y~, ~6, 6~, or, v~, ~ , ~ A U = oAU + IAiflR' + 2 A u / R "

(12)

and with k = 0,1,2, kAuu = klluu-F l" v,.,

kAu? = kAu6 = k/tu~ + l~akVuy, II

l

II

I ")

ii

kA.. = k/tu+ Imkv,,-F Im(k/t..+ IrakVu.)-FIm(k/tuu-FImkV..),

kA., = kA.6 = k/t.~+ l~kV., + l'm(k/t., + l'~kv.,), kA~ = kAy6 = k/t~ + l~kvyy, kAy6 = k/t~6+ l~kV~, kAoo

=

k/too+ l"mkV or,

kAy# = k/t.# + l~,kv,# + 21" (k/tw + kVv.), II

l

kA## = k/t## + Ira:## + l

II

+ ImkV°#)+

kA#~ = k/t#y + I/~kv#~+ Im(k/t.y +

! 2

+ kV.°),

lintY.y). ii

For the general case of inhomogeneous sector fields formed by conical or toroidal pole faces the k#U and kVu of eqs. (11) are given in eq. (23) of ref. 6). These terms are lengthy and complex functions of 8',8" and of the flu and ~ij which are the D o and D;j, respectively, of ref. a), evaluated for a magnetic sector field. In order to facilitate, however, numerical calculations, all flu and ~u of interest are quoted here again; ~ is defined in eq. (14) of ref. a) with ~ = - 2 n 2 for conical pole faces.

~, = 2 ~ = 2~'6 = s * / ( 1 - n ) "}, =

= (1-:)/20-n),

(13a) =

st/n ,,

~. = - s * ( 1 - n ) f, Vv =

--S?nf,

202

a. WOLLNI K /7.. = ~

1

otl-n )

[(~ + 5n - 3)s*'.+ (~ + 2n) (I - c*)].

S*

/7., = ~ [(~+5n-3)(1-c*)+3(1-n)], 3(l-n) ~

-/7"--....-~+ I---L--- (n+n+n2)s*4>,,,

/7"Y = / 7 " = ( 1 - - n )

-1

/7,.,,- (f_n)2

4(1-n) ~

[(n+5n_3)s,2_(2n+7n_3)(l_c,)], - 1

[ 2 ( n + 5n - 3)s*(1 - c * ) - 3 ( n + n + n 2) {s* - c*~m(1 - n)½}],

/7"r =/7"6 = 1 2 ( 1 - n) ~ /7,, = /7,, -

I

{(~ + 5n - 3)s .2 + 1"4(fi+ 2n) - 3(1 - n) 2] (1 -- c*) - 3(~ + n + n2)(1 -- n)½s*~bm}.

24(I--n) 3 1 -c*

/7~6 = 2/1~ -) 2 ( l - n ) ' 1 /7~ = 2(1 - 5n) (1 - n) [~(1 - n)s ) ' - {~(I - 3n) + n(1 --5n)} (1 - c*)]. --/i

/7"# = ( 1 - 5 n ) { n ( 1 - n ) } *

[(I

--

n)½stc t -- n½s*],

/7" = 2(l - 5~)ln(1 - n) In(1 - n)s t ' - ( 2 n +

'/

5n - l)n(1 - c*)],

(13b)

[-(n+n2+n)St~mn*]+

/Tv~ =/Tv, = a n ( l - n )

+ (1 -- n)~(1 -- 5n) [(1 -- 5n -- 2~)n½sts * + ~(1 -- n)½ct(1 -- c*)] . /7,;, = / T p , -

, { [--(~at'n2df'n)($t--ct~mnaz)]al-

4n~(l_n)

+ (1--n)½(12n - 5 n ) [(1-5n-2n){(l-n)½st-n~rcts*}+n(1-n)½st(l-c*)]} _

v~"

( n + 2 n ) s * (I +2c*). 6(1 - n) *

~+2n v" = 3(l-n)

(2s.~+c._

-1

v~Y = v"n = 1 2 ( 1 - n ) ~

1),

{2s.[E(~+En)c.+(~+5n_3)]_3(~+n+n2)[s.+(l_n)½c.q~m]},

$* 9,~ - - [2(~+2n)(1-c*)-3(1-n)], 6(1-n) ~

_

- 1

v,~ = ~,, - 4(1 _~)2

[2~,,(l_n)_(~+n+n2)( 1 _

n).l.s,~bm]

'

203

THE C O R R E C T I O N OF IMAGE A B E R R A T I O N S

1 v ~ = v " - 24(1 ~ -

[ 2 ( n + 2n)s*c* + (~ + 17n -- 6 n 2)s* -- 3(~ + n + n 2) (1 -- n)*c* ~bm],

n) 8"

~r, = 2 ~ r + 2(1 - - - - - - ~n) '1 { 2 n [ n ( 1 - n)]½stc t - [~(1 -- 3n) + n(1 -- 5 n ) ] s * } , 2(1-5n)(1-n)* ~"a = 1 - 5 n

( 2 s ) ' + c * _ 1), -1 (13b)

• vPa = 2(1 -- 5 n ) {n(1 - n)} "1 [ 2 n ( 1 - n)*stc + - ( 2 n + 5n -- 1)n½s*],

_

1

vv~ = ~, -- 4n½(l-n)

{_(n+n=+n)(c)dPmn,+st)+2nst+

-t (1 - n)'l(1 - 5 n ) [" - ~ n ( 1 -- n)½st(1 + c*) + ( i f ( 1 - 3 n ) + n(1 - 5 n ) ) n'~cts * ]

_ _ 1 { 2 v,, = vp, = 4 n ( 1 - n) - (n + n 2 + n)stdpm n½ • (1 - n)½(1-5n)

,

r(1 - n)½n~c)(1 - c * ) -

- n½ s ) s * ] } .

1.4.2. The subcoefficients /tfj, /7,j, v,j and ~ij for three special cases of mass separators 1. The/7 u and ~u For three special cases of mass separators the quantities #u and vu characterising the ion optical properties in the main deflecting field without fringing field effects which must be inserted into eqs. (22) of ref. 6) are considerably simplified as compared to eqs. (13)• A. An inhomogeneous magnetic sector field which has equal radial and axial focusing powers In = ½, eq. (1)], and which is formed by conical pole faces 1~ = - 2 n 2 = -½, ref. a)-I is characterized by the following/Tu and vu: /7. = v~ = / 7 v = vp = c*, /7~ =/Tp = 2 ~ = 2% = - 2~. = - 2~v = s*~/2,

/7~= /7,=1 - c *, /7=, = -2/7.. = ~., = ~p = - } ( I - c * - 2 s * ' ) , /7.. = 4~.. = - 2~.. = - 2~pB = ( } x / 2 ) s * ( 1 /7., = / 7 . ,

= --}[1

+ 2c*),

- c*-- 2s*'] + s*q~m[ ( 4 ~ / 2 ) ,

/7,, =/7,, = ,/218s*(1 + 3(s*- C*m / 42)I /7,, = / 7 , , = @~z['5(1- c * ) - 4s*' - 3s*~bm] ~/2], /7,, = -~[11(1 - c * ) -

4s*'-

3S*~bm/ X / 2 ] ,

/7.v = -~;[2(1 - c * ) - s * ' - I , /7.B = - 2~ov = ( ~ x / 2 ) s * ( 1 - c*),

/7.

=

- c * + s*'),

/7~r =/7~, = v~ = v~, = - ¼s*~bm[ ~/2 - 3(1 - c* - 2s* 2), /Tp, = / T p , = -- (¼~/2) { s * - c*q~m/ x / 2 } + ( } ~ / 2 ) s * ( 1 - c*),

(14)

204

H. WOLLNIK ~., = ~.e = ~-,x/218s*(1 - c*) + 3(s* + C*~m/42)], "v,, = "~,,~= ~-~x/21-2s*(2c* -- 5)-- 3c*~bm/ ~/2], v,e = ~-~x/2[-- 4s*(1 -- c*) -- 3C'4m / x/2],

(14)

vvr = "~v,= AX/2[S*( 5 + 4c*)-- 3c*q~m[ x/2],

~#,---- 9#, ----"--¼S*~bm/X/2+}(1 --C* +S'a). B. A homogeneous magnetic sector field which has only radial and no axial focusing power (n = 0) is characterized by even further simplified fi~ and 9~: ft. = ~. = c*,

fi, = - ~. = 29~ = 2i~ = s*,

ao=9~=1,

~a=¢m,

~=~.=½(1-:),

~.. = - ~ . , = - ~ . ~ - is *~, ~,, = - 2 ~ = - ½ ( 1 - c * - s*~),

~.~=s*c*,

fiv~ = flu = -- ~(1 - c* + s*'),

fiaa=-gar=-ga~=-½(1-c*),

Yu~, =

-

Yu~

-

= -

~

=

fi.r=fi.n--½s*(1-c*),

- T4v- ~ = - - ~ v-~ = - 2 v- v ~ = - vaa

=

(15)

is*,

¢. An inhomogeneous magnetic sector field which has only axial and no radial focusing power (n = 1, sometimes referred to as an 1/r field) is also characterized by very simple fi~j and ~ j ft. = ~,, = 1,

/.7,, = 2~y = 2~ 6 = (~m,

~. = 0,

fi~ = ~# =

c t,

fi# =

- ~ =*s,

fi,,:, = - 2~.. = q~m,

/7.. = ~.. = ~.. = ~.~ = ~.~ = 0, 3 A~2_L 1 ~4-

~. = ~ = --~'m T~'m, 4fi~. = - 4fi¢# = ~

fi~, = fi6 = ¼(~2,

~ = --~

= s~,

+ ~0~m,~ "

(16)

fiva = - ½ ( s t c ~ - dpm),

/7~, = / 7 ~ = -~m(S t + Ctq~m),

/7#r = ~#~ = ~l-St(1 + q92m)-- Cram-],

% = % = ff:(O£-

% = -- ~ . = ½ : d ,

1 ) + 3C*Om],

% = % = ~ O m ( 3 : - - dOm).

2. T h e / ~ j and v u For the special cases of perpendicular entry and exit of the particle beam into and out of the sector fields also the/~ and v of eqs. (23) of ref. 6) become very simple: Pi = fi~}~ for i = u, ~t, y, 6, v, fl, /

p* = pflp, O~dk! OVkZ

vv = pdp + vflp,

v~ = pflp,

fik~I- for k l = uu, ua, uy, u6, aa, a?, a6, ~?, ~6, 55, YkZ

!

(17)

THE CORRECTION

l['lu¢ =

l['lwt =

11a.. =

-

l/"lua =

1#~ =

1["1~ =

l[Jez~, =

l~"looJ =

205

OF I M A G E A B E R R A T I O N S

1~},}' :

libya =

lJJ&$ ~" 1~v,8 :

llJpp = O,

½~,

lvu. = _ lyvv = ½~¢,

2 P . . = 2P.,, = 2/~.~, = 2P,,.~ = 2/~,,,, = 2Y,,~, = 2/~,,a = 2/~,~, = 2/S,a = 21~aa = 2/~,,,, = 2/.t,,p = 2/~pp = 0, I

½p .,

2VVO = - - ½[12,

2. The. correction

of image

= p.p,,

=

2rung = - - ~tupp,

aberrations

= p..p,.

I

= ½PL

2Vpp = - - ½ff[~.

of second

(17)

J

order

By proper choice of the parameters ~m,e',e",n,lm,l'~,l/R' and 1/R" some of the coefficients A U of eqs. (4) can be put to zero exactly or at least approximately. This goal will be approached in two steps. 2.1. CHOICE OF THE FIRST ORDER PROPERTIESAND ROUGH CORRECTIONOF IMAGEABERRATIONS The first step is a rough correction. By keeping R' = R" = oo and choosing ~m,e',e",n,l~,l'~ properly, the geometry of an instrument is determined, for which at least the radial focusing condition I-eq. (5a)] is fulfilled and which has good dispersing properties and not too large image aberration subcoefficients oA u" 2.2. FINE CORRECTION OF IMAGE ABERRATIONS FOR AN INSTRUMENT WITH UNCHANGED FIRST ORDER PROPERTIES

For an instrument whose geometry has been determined as a second step a fine correction can be carried out. For this purpose a graph can be drawn as in fig. 3, with 1/R" and 1/R' being the ordinate and the abcissa, showing lines for which the coefficients A u listed in eqs. (4) or 0 2 ) vanish. Since the Aurare independent linear equations

~ I/R' 1 I.

o,s

I/R' o,s

I

o~

-4-I

Fig. 3. In this g r a p h lines are s h o w n in a coordinate system 1/R" a n d I/R" for which the different coefficients o f i m a g e aberrations take up the value 0 a n d two other values which h a v e a m a g n i t u d e so that the resultant aberration o f second order is o f t h e s a m e m a g n i t u d e as a c o r r e s p o n d i n g e s t i m a t e d aberration o f third order. T h e s h o w n e x a m p l e is characterized by ~m = 135 °, n = 0.5, e'= e', 1'= F. It is a s s u m e d that u0o = 5 x 10 -5, ~oo = 2 x 10 -2, re0 = 2 x 10 -3, fie0 = 5 x 10 -a, F = 0 , J = 2 x 10-5, a n d A ~ a -- 2. Uncritical aberrations are n o t ' s h o w n .

206

r~. WOLLNI~

in 1/J~' and 1/R" it can not be expected that a point can be found for which more than two coefficients vanish or are approximately zero simultaneohsly. This is even more so if additionally it is required that the values of 1/R' and I/R" are not to be larger than 1 or 1.5". A correction of image aberrations of second order, however, in reality does not require the above given coefficients A is to be exactly zero. It is sufficient if the sum of the aberrations of second order is smaller than or equal to the sum of the aberrations of third order which are still •present. This is approximately achieved if all aberrations of second order ioJohij = ioio(oAij + 1Aij/R'-F 2A~j/R"), with .ioJo = UoVo,%,/3o,T,~ are smaller than the largest aberration of third order 2See A 3 .... where See is assumed to be the largest possible quantity of all uo,vo,~o,/3o,T,& Since all Asss are unknown we can only estimate this quantity to be in the order of magnitude of 1 or I0. Recent third order calculations s) seem to confirm that this assumption is realistic. Thus all aberrations ioJoA o of second order are allowed to be as large as 2S]oAs~ ~[with 1 < A~ s < 101, allowing Aij to be smaller or equal to 2Bs~S3oo/(iooJoo) where ioo and Joe characterize the maximum values of all io and Jo that are allowed by the diaphragms of the instrument. The equations Ais = +__2s~oAs~/(iooJoo) represent each time two parallel lines to Aaj = 0 in fig. 3, marking in each case a region.in which each point specifies an instrument that is "corrected" for one special aberration. If for 1/R' < 1.5 and I/R" < 1.5" an area can be found which is included by all the regions each of which is limited by two so constructed parallel lines, we can say that each point of this area represents an instrument that is "corrected" for all image aberrations of second order. Usually several of the aberrations are quite uncritical and thus can be neglected in such a graph. In fig. 3 as an example this drawing is made for an instrument with ~bm = 135 °, n = 0.5, 5' = 8" = 0, and l ' r = lm,. Further it was assumed that Uoo = 5 x 10-S,~oo = 2 x 10-2,roe = 2 x 10-2,8oo = 5 x 10 - 3 , ~ = 0, ~ = 2 × 10 -5. Under these conditions only the aberration coefficients Au,Avp, and App are of importance while the others (Aup,A~,A~6,A66) following the above given argument are so uncritical that their corresponding allowed regions cover the whole area of the diagram of fig. 3. The coefficients Av~,A~, and Apt need not to be corrected since they cause only an image curvature or inclinations of the focusing lines relative to the main path [eqs. (9) and (10)]. 2.3. OPTIMAL RESOLUTION AND LUMINOSITY FOR A CALCULATED INSTRUMENT The usual question to be solved by the layout of a newly designed instrument is to achieve maximal beam intensity for a postulated resolution by proper choice of Uoo,~oo,Ooo and/~oo t. The resolution R = m/Am is here defined as the distance between two lines of mass m o and 2me divided by the width of one line: R = Ar[2Uoo lAp J+ 26 IA61 + Uo2olAp, I +2Uoo~oo lAp, I+ 2Uoo6 lAp6 i + ~ o IA~ I +2~oo6 IA~6 [+ + 6z IA66] + 2Ooo/~oolApp I + / ~ o IApp I + third order terms'I- ~. (18) All mixed aberrations are counted twice since the two parameters can have equal and different sign. Assuming the density of particles emitted from the ion source to be constant over the area of the entrance slit and constant also for different parts of the illuminated solid angle 2~oo2~oo, the recorded separated particle intensity is the product of the number of particles that leave the entrance slit per unit area and unit solid angle and of the lumi2 2 multiplied by the transmission T,i.e. the from the nosity L. This quantity is the entrance slit area 2Uoo2Voormo spectrometer accepted solid angle 2~oo2~ffoo divided by the illuminated solid angle 2~oo2]~oo, provided that ~oo > %o and/~oo >/~oo. If one of these relations is not valid either %0 or ~ffoomust be replaced by the corresponding ~oo or/~oo. This luminosity is

L = {(UooOoo~oo1~oo)/(~too~oo)}r~o.

(19)

Thus the above presented problem is identical to finding the proper values of Uoo, nee, ~oo, and/~oo, which make L in eq. (19) to a maximum while giving R in eq. (18) the postulated value. R - z as well as L depend on Ooo linearly. For increasing roe the luminosity L, however, increases much faster than the resolution decreases. Thus %o should be made as large as possible limited only by the pole gap width. Because of the linear term ~ IA 6 I, ~ must be extremely small for any high resolution so that 61A 61 < Uoo I Ap I. This is a severe postulate for an ion source. * In refs. 4, 5) the terms ~zS/R 9 or ~za/R ~ (with 8~,~, and R being not dimensionless quantities) were neglected as third orderterms. Thus, if the coefficients of all third order aberrations are assumed to have the same order of magnitude, 8~S/RZ also must be of the same order of magnitude as ~z~, forcing R to be larger or only negligibly smaller than rmo. t In section 2.2 it was assumed that uoo,~o,v0o and ~0o were known. A relatively rough knowledge, however, was sufficient there.

THE

CORRECTION

OF IMAGE

207

ABERRATIONS

For this reason we can drop the even smaller second order terms 2 Uoo6 IA.6[, 2~oo 6 IA~e I, and 6 2 IA 66l completely. For a high resolution also Uoo must be very small. Therefore we can neglect in this case similarly the second order terms 2Uoo~ooI A.~I and Uo2oIA.I compared to the other aberrations. Due to these simplifications Voo has a certain value and thus the luminosity L is a simple function of Uoo, ~oo, and floo only, while the resolution is

R = a~[2Uoo Ia. i + 26 Ia, I+ ~o~oIAu I+ 20oo/~oo IA~p [+ P~o lapp [+ 2~o3oIA . . I + other third order terms]-*, if one aberration of third order is included. For many high current mass separators it is postulated now that no diaphragms are to be met by parts of the beam. So if the ion source determines ~oo = Zoo and floo = ]~oo only the width Uoormo of the entrance slit or for a plasma-source the virtual source width can be adjusted. By proper operation of a high current plasma ion source Uoo usiJally can be made very small without loosing beam intensity1°). For a separator which can use diaphragmg to limit ~oo and floo the resolution can be increased further by reducing the most important aberrations. If Voo and Uoo are defined by the effective ion source area, and R has a constant postulated value, we can eliminate floo from the eqs. (18,19) obtaining L as some function of ~oo- L(%o) is defined only for positive %0 depending on the postulated resolution R. Since an image formed by a lens system that has aberrations can not be smaller than the ion image formed by a lens system that has no aberrations, the postulated resolution R must be chosen to be smaller than Ay/(Uoo IAs I ). The amount, it has been chosen smaller, is assigned to the aberrations. The maximum of this function L(~oo) is obtained as a resolution of the function E~L(0~oo) / ~0~o0]~op t = 0 =

2

6

5

2

4

25Apl~A~=O~opt + 20Apl~AafAa~O~op t + 4AB~A~opt + 2

2

- {8vooA.p+ IOA,,[Ar(R

-1

3

- 2 6 ) - 2uooA.] } A ~ . p t +

2 2 + 4AppEA~(R -,_26)_2uooA,]}A,a2op,+ - {3vooA,p _ EA~(R-l_26)_2uooA.]A2.pv~o + EAy(R-1--26)--2uooA,] 2App,

(20)

where for all A u their absolute value is to be used. Generally this sixth order equation must be solved yielding the optimal ~op, and with the help of eq. (19) the optimal ~opt. For A~, or A~,~ equal to zero, however, eq. (20) reduces to a second order equation. For separators which do not use a plasma ion source and whose separated ion beam intensity increases with Uoo the maximal luminosity is obtained in a similar procedure. For a postulated resolution Uoo is eliminated from 10

[111

9 8 7 6 5 l, 3 2 1 0

40° 50° 60° 70° 00o 90° 100° 110° 120° 130° 1/.00150° 160° 170° 180° 190°

F i g . 4. I n this f i g u r e lmr is presented f o r s t i g m a t i c f o c u s i n g s y m m e t r i c mass s e p a r a t o r s w i t h n = 0.5, 8' = e" = 0 a n d d i f f e r e n t values o f the d e f l e c t i n g angle ~m. F u r t h e r l'm~, the a x i a l f o c u s i n g distance is s h o w n i f lp = 1/60.

208

H. WOLLNIK

eqs. (18,19) yielding L as a function of~oo and floo.OL/&too = 0 and OL/Oflo o = 0 then present two equations which determine the optimal Cropt and flopS.The corresponding Uopt can be obtained from eq. (19). 3. The correction of image aberrations of second order for a special case of a mass separator

As a simple example we will discuss here only one group of a much used type of mass separators for which r ~ t! i -R t 1,m = l;m -- 1,m,l:m, a n d R" a r e a r b i t r a r y . F o r s u c h i n s t r u m e n t s tl

8' a n d g ' a r e z e r o o r a l m o s t z e r o , n = 0 . 5 , l"m

o n l y t h e d e f l e c t i n g a n g l e q~m b e t w e e n t h e p r o f i l e p l a n e s I I ( f i g . 2 ) r e m a i n s a s a n a r b i t r a r y approximation.

parameter

T h u s t h e r o u g h c o r r e c t i o n is h e r e q u i t e s i m p l e , a l l o w i n g u s t o u s e a l l r e a s o n a b l e

The radial focusing condition

in a first order

~bm, i .e . 0 < ~bm <

yields the equal object and image distances simply as

lrm

= l , "m = l , m = 2 C t g

(¼~bm4

2 ),

(21)

while the axial focusing condition also is approximately fulfilled. The dependence of lrm and of l~m o n ~ m a r e shown in fig. 4 and the table 1. I'- is calculated taking into account the axial fringing field focusing caused by I~ and 1~ (eqs. 6). The radial and axial magnification factors and the mass and energy dispersion in this special case are constant for all ~bm: A . = A v = - I,

(22)

A~, = A6 = 2.

(23)

TABLE 1

In this table all d i m e n s i o n s of the investigated s t i g m a t i c focusing ma s s s e p a r a t o r s are presented. Ea c h s e p a r a t o r consists of a r a d i a l l y i n h o m o g e n e o u s m a g n e t i c sector field with the i n h o m o g e n i t y factor be i ng 0.5. F o r p e r p e n d i c u l a r e nt ry a n d exit of the particle be a m the axial focusing distan ce/'zm is different from

Irm. F o r

8' = e" = 0.48, however, this difference vanishes. The angles co, a n d ~oz between the

r adial a n d axial focusing lines a n d the m a i n p a t h as well as the c urva t ure s o f the exit or e n t r a n c e slit, respectively, are presented for the cases I a n d II (fig. 6) which are distingui s he d by t he pos t ul a t e s Aa~ =

l/R" =

0 a nd Aa~ = A v# = 0.

case I

~m

hm

l"zm

(R')-I

(R-)-x

case I I

for

¢,0z

(Rsllt) - 1

(R')-I

(R")-I

for

Coz

(Rsllt) - 1

- 0.006

30 °

7.552

10.085

-- 0.262

13.5 °

4.2 °

0.004

-0.219

-0.043

13.8 °

4.2 °

40 °

5.613

6.899

- 0.349

16.8 °

6.5 °

0.007

-0.2932

-0.056

17.3 °

6.4 °

- O.OI 1

50 °

4.437

5.203

-

0.435

19.4 °

9.0 °

0.008

- 0.368

- 0.067

20.2 °

8.9 °

-0.017

60*

3.644

4.144

- 0.521

0.007

- 0.443

- 0.078

22.5 °

11.6 °

- 0.025

3.068

3.415

- 0606

21.4 ° 23.00

12.0 °

70 °

15.4 °

0.002

-0.520

--0.086

24.3 °

14.8 °

- 0.035

80*

2.628

2.878

- 0.687

24.2 °

19.3 °

- 0.007

- 0.090

-- 0.090

25.7 °

18.4 °

- 0.050

90*

2.279

2.465

- 0.765

25.2 °

23.7 °

-0.023

-0.675

--0.09

26.7 °

22.5 °

- 0.064

100"

1.993

2.134

-0.835

26.1 °

28.8 °

- 0.047

--0.752

--0.083

27.4 °

27.3 °

-

1100

1.753

1.861

-0.893

26.9 °

34.5 °

-

0.080

- 0.827

- 0.066

27.9 °

33.0 °

-0.108

120 °

1.547

1.631

--0.933

27.7 °

40.8 °

-0.126

--0.897

--0.035

28.2 °

39.8 °

-0.139

1300

1.368

1.432

-0.941

28.6 °

47.5 °

-0.184

--0.958

0.017

28.4 °

48.0 °

-0.178

140 °

1.208

1.258

-0.899

29.6 °

54.4 °

-0.258

-

0.102

28.4 °

57.9 °

- 0.228

150 °

1.064

1.103

--0.773

30.7 °

61.4 °

-0.353

- 1.013

0.239

28.3 °

69.7 °

- 0.294

160*

0.934

0.963

-0.504

32.0 °

68.1 °

- 0.473

-0.966

0.462

28.0 °

83.3*

-0.382

170 °

0.813

0.836

0.021

33.5 °

74.5 °

-0.630

-0.809

0.830

27.5 °

97.9 °

- 0.504

180 °

0.701

0.717

1.020

35.3 °

80.6 °

-0.841

-0.441

1.461

26.7 °

112.4

°

- 0.678

190 °

0.595

0.607

2.963

37.3*

86.2 °

-1.130

0.361

2.601

25.5 °

125.7 °

-0.941

200 °

0.495

0.503

6.955

39.8 °

91.6 °

-- 1.593

2.130

4.825

23.9 °

137.3"

- 1.360

l

I

1.001

0.084

THE

CORRECTION

OF IMAGE

ABERRATIONS

209

Also the radial image aberrations of second order, described by eq. (4), can be simplified considerably. Using eq. (21) we get: oA.. = ½c*(1 +2c*), oAuat ~-

½(s'x/2 ) (3 ctg * ~ - 1),

oAu7

oA.6 = ~¢[15 + 7c* - 16c .2 + 3~bmctg* / x/El,

o A ~ = ~t(3 ctg *~ - 1),

oAot), oA~,~, oA~,~ oAvv

oA.n = ~;[3(s*x/2 + ~b,.)/(1 - c*) + 8s*~/2], oAu = ~ - ( - 9 - 1 1 c* + 8c * ~ - 3~bmctg* / x/2),

(24a)

~;( - 3 - 11 c* + 8c *~ - 3~bmctg* / x/2), ½c*(1 - c * ) - 1,

oat,# = - (ix/2) (2 + c*) ctg*,

oA## =

-~-(3 ctg*~ + 1),

oAar = /;[(x/2)s*(17 + 4 c * ) -

3 ~ b m ] / ( 1 -- c * ) ,

1A:. = - tAt,t, = ctg*/x/2, 1A,~ = - 1Aop = 2ctg .2,

(24b)

1A~ = - lApp = (x/2)ctg *~, xA,7 = 1A,6 = xA~r = tA~6 = tAyy = rAy6 = tAu = 1A#~ = 0, 2A.. -- - 2At,t, = c*' ctg*/x/2, 2Au~ = - - 2Av# = 2 c *

ctg *a,

2A,,,, = -2App = (x/2)ctg .3,

2A~y = 2Au = ½2Ay6 = s * ( 1 - c * ) / 4 2 , ] 2Auy

2Au6

s*c*x/2,

2A,,~

2A,,,

- 2 A p t = 2(1+c*),

(24c) J

where c* = cos(q~m/x/2),

s* = sin(q~m/x/2),

ctg* = ctg(~bm/2~/2 ).

In figs. 5a and 5b these coefficients of second order aberrations are presented graphically as functions of ~b=. It there can be seen that only very small differences exist in the second order properties o f different instruments for ~bm > 120 °. By curving the entrance and exit field boundaries by proper values~of the'curvature_'radii R' the second order aberrations can be reduced using the m e t h o d described in section 2.2. Hereby it was assumed that Zoo = 3 x 10 -2, floe = 0.5 x 10 -2, %0 = 2 × 10 -2, and Uoo = 5 × 10 -5. The example shown there in fig. 3 is the instrument having an angle of deflection ~bm = 135 °. F o r q5m larger than 170 ° corrections of image aberrations are impossible if the R' and R" are not to be too small. The largest aberration generally is 0to2oA,,. Thus one always should postulate A u to be very small. App and At,p are the two other coefficients of importance (section 2.2). F r o m the dependence of A,, and App on R' and R" it can be seen that almost no diminishing of App can be accomplished if A,, must be small. If we make now drawings like in fig. 3 for all ~bm we find two especially advantageous solutions for all cases. One referred to as I is characterized by an uncurved exit field boundary (R" = oo) showing relative small At,p and App if A,, is put to zero by proper choice of R'. The other referred to as II postulates A,~ and At,p to be zero simultaneously by proper choice of R' and R" leaving App still small enough. The R' and R" thus obtained are shown in fig. 6 and table 1. With these R' and R" now residual aberration coefficients result, which are plotted in fig. 7a for case I and in fig. 7b for case II. F r o m these coefficients A u the actual aberrations are obtained by multiplying each one of them by the corresponding iooJoo. Using the quantities Uoo = 5 x 10 -5, Ctoo= 2 x 10 -2, roe = 2 × 10 -2, floe = 5 x 10 -5,

210

H. WOLLN~K

12

o Aij

",

11

\

10

o%,~

-

OA(XOt

,,

........

50=

~-£

-

I A

o: . . .uu ...

...........

/.0°

60" 30"

80

90"

--..2~....... _ Z ~ 100" ~.

~2-~--::[g{--[~

25~°33 '

169""2'

- ~ ~ ~'

T

I

_ ....

--"""-'-~--.~--

__ . . . .

- - ~ ---- - - - - - - . . . . . . . . . . . . . . .

L.-- ~ a - --F - ~ ; . : . ~ - ~ : . . ~ . . _ - 7 ........ .;...... ;

-~-_.7-7--J-f-~T--i--~F--{---F--~-,

I ~'~' . . - - ~ - - - - - l " 7 _ . - b - - d

:

,~" ,?o" ,oo" ,9o" zoo' 21o" zzo' 2ao" z~o" z5o'

-3

." I)o" ~o' ~o" ~o" "~o'

/ -5 /

-6 -7

//

/

/ !

/

-9

,,

/

40

'~ /,,opp

/

-11 -13

(a)

12!

\

11'

',

lO 9. 6

2Av o /\~ \ ~." ..x,,,\ \

7 6

\ ".. ",,\

5 4

,~

%.

3 2 1 o

~ - ~ 1.0°

7..77~ 50o

60" 70"

-J 80"

90.

100" 110. 120" 130" 1/.0' 150" 1 6 0 ' ~ 1 "

1BO" 190"--200"~-~""2m.

Sm 230" 240" 250"

(b) Fig. Sa,b. The coefficientsof the image aberrations of second order as given in eqs. (24) arc presented graphically for ~z < ~ m < ~zv'2.

THE C O R R E C T I O N

211

OF I M A G E A B E R R A T I O N S

= 0, and 6 = 2 x 10 -s, these actual aberrations are plotted in figs. 8a and 8b. Here, however, vooAv,,,otvA,,y , and flvApr are left off since they cause only an image curvature [eq. (9)] and angles co,and coz[eqs. (10)], formed by the main path and the lines of radial and axial focusing (fig. 2). The dependence of this image curvature on ~bm is shown in fig. 9 whilethe dependence of co, and co= on ~bm is shown in fig. 10. To fig. 8 it should be mentioned that quite a few of the aberrations are too small to be seen on this scale. On the other hand A~, is not assumed to be zero, though it is corrected, because of unprecise machining and adjustment the values of n and of ~bm deviate by small amounts An and A~bm from the calculated values and thus give rise to a small AA~,. Naturally suchAn and A~bm cause also a small change in I'm and l;'m but this small change easily is detected by looking for the best focused mass line. The quantity AA,,,, we derive from eq. (24) assuming that 1" is also changed so that/~m =/'m and the instrument again is symmetric, (if I"m is left at its old value and only l~'mis adjusted so that good focusing is obtained, the derived formula should be almost correct): AA~ = { -ctg(~bm/2x/2)/sin 2(q~m/2X/2)} [X/2 + ½{(R')-t + ( R " ) - 1}ctg(~bm/2x/2)] [I d ~bmI+ q~mId n I].

(25)

The resultant rest aberration OtooAA,,~, is presented in fig. 8 whereby it is assumed that Atkm= -I- 0.1 ° and An = __+0.005 which both should be achievable with some but not too much effort. As a simple and without any reason chosen assumption finally A,~, is assumed to be equal to oA,, if 0A,~ is larger than 2 and A,~, is put to 2 if oA,, is smaller than 2. The resultant fictitious aberration of third order%aoA,,, is also indicated in fig. 8. Thus all ion optical properties of stigmatic focusing symmetric spectrometers with n = 0.5 are presented graphically to give a survey over these very often used instruments. It should be mentioned at this point that the discussed instruments were considered to have ideal field boundaries 2) which for e' = e" = 0 coincide with the profile planes II and thus include the deflecting angle ~m with each other and whose fringing fields are limited by shielding diaphragms. For unshielded instruments these calculations, however, should not yield too unsatisfying results if the position of the ideal field boundaries has been determined accurately defining ~bm,d, and e". A finite curvature even for uncurved pole shoe ends then also should be taken into account 11). These calculations all can be made without any special knowledge of the fringing-field distribution of an instrument because the aberrations of second order are influenced only by the existence of a fringing field and not ~/~

?

VRi

6 5

4.

3

2

-I

/

I

I 120° 130° 1/.0° 150° 160" 170° ~

190o 200°

-2

Fig. 6. The values o f I/R' and 1/R" are shown here which give a partial correction o f image, aberrations. F o r different deflecting angles. The curve with index I shows values obtained from the postulate that A~a (R'hR"~ = oo) = 0. The curves with index II show values for I/R' and I/R ~ obtained from the postulate Aa,,(R'II,R"n) = Av#(R'ir,R"n) = 0. These values can be taken also from table 1.

212

H. W O L L N I K

8~b A,m(D 7-

x

6.

\

5t,. 3.

/

Avv

-,'---~-

o _I~,-:-7 ...~

. . . .

//

.

~'"-,-~"

.

-"

~ .....

---".

A,,

.-'~-

•

200o

: : ! : : . L :,o.... , Y , . . ~ .=o,~o.,,oo,,o.,,o. ~. - - ~ - - .v :I"~ ~,o0., =.,,oo,o.~-~:~_

/ Fig. 7a. The residual coefficients of image aberrations are shown for case I corresponding to R'z as it may be taken from fig. 6 or table 1 and to .Kw = oo. Aa is equal to zero for all angles of deflection ~m. The scale here is enlarged as compared to fig. 5.

,J ',\ A~j(1[) 7 6

\

/

", \

j \\

/ %

5 //" ..d " ~

/, .... 3 2

./

A~y

/I /

Any

1 0 1 --

2"

-

3

to-~oo soo soo 7'0° ~

~

1~)o0 ~o01~zoo1~ 1~.oo1~oo l'6ool~/ool~oo-'~\~Pl 3

Fig. 7b. The residual coefficients of image aberrations are shown corresponding to R'H and R'~x as they may b¢ taken from fig. 6 or table 1. A== and Avp are zero for all deflection angles ~m. The scale here is enlarged as compared to fig. 5.

THE C O R R E C T I O N

213

OF I M A G E A B E R R A T I O N S

A

18

\

\

WS

\ \

\ \

IL

\

12 ~

\\\

lo - - ~ - -

'

\\\

~

Z '(R'I°0°°)

\\,

Z /

(a)

\ 18 16,

\\

I/,,

\\ \

12[\

\\\ .

o

.

I

.

.

.

.

.

.

.

.

.

.

.

.

-

.

s'lR=10000) .

.

.

.

.

.

.

.

.

.

.

.

.

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40° 50° 60O 70O 80° 90° 100° 110° 120° 130° lZ.0O150° 160° 1"/0° 180° 190° 200o -

(~m

(b) Fig. 8a,b. The residual aberration for case I and for case I! (fig. 6) are shown if uoo = 5 x 10-5, =oo = 2 x ]0 -2, voo -- 2 x 10-~, ~oo = 5 x 10-a,y = 0,6 = 2 x 10 -5. The rest aberration o~o02A~ is shown for the case that n and ~m are machined or adjusted a little wrong by An = _+ 0.005 a n d / [ ~ m = -F 0.], see eq. (25). The two cases are marked by u2(,4n) and =~(A~). Assuming these values o f / I n a n d / I ~ both to be present alsofloo2`4Bp and 2u0ofloo`4u# are shown marked f12 and off. Further the sum ofall aberrations o f second order is shown marked 27. Then a line is shown marked s"(R = 10000), characterizing the maximal width a mass line is allowed to have. Finally also an aberration o f third order 2=o03,4=== is shown, where it is assumed that ,4=== = o,4==if o,4== > 2 and '4=== = 2 if o,4==< 2. This assumption might be considerably wrong.

214

H. WOLLNIK

of its ~xtension*). The axial focusing properties, however, do depend on the special distribution of the fringing field because of I~ and I~ in eqs. (6), Assuming 4) I~ and I ; to be 1/60, the axial focusing power of the sector field is decreased yielding a larger l"~ as shown in fig. 4. Choosing, however, d = d' = 0 such a sector field still can be made to give a stigmatic focus. For I~ = I~ ~ tgd = tgs" or 8' = *" = 0.48 ° if I~ = I~ = 1/60. For ,' = d' = 0 also stigmatic focusing instruments can be obtained if n is slightly different from 0.5. These ~ or the differences of n from 0.5 are still so small that all the second order quantities presented in figs. 5-10 are essentially unchanged. Finally the luminosity of such instruments should be discussed assuming instruments equipped with a high -I Rsht - 1.75

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I00 ° II0° 120 ° 130° 140° I.~W~ ° 160 ° 170° I B ~

190° 20[~°

Fig. 9. The radius o f curvature Rs of the image according to eq. (10), is given in table 1 and s h o w n here for the cases I and II (fig. 6)

100J

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90o 100° 110° 12.0° 130° 140° 50 ° 1600170o 1~o 1 o

Fig. 10. The angles mr and oJz between the main path and the lines of radial and axial focusing are shown separately for the cases I and II (see fig. 6). Corresponding indices are attached to mr and cot. oJri are indePendent o f R". These va ues can he taken also f r o m table 1.

THE CORRECTION OF IMAGE ABERRATIONS

215

current plasma ion source for which the luminosity does not depend on Uoo, ref. 9) and assuming A,= = Avp = 0, we get f r o m eqs. (20) and (19) %p, = [ M R - * - 26 - Uoo)A~=]*,

(26a)

Uoo)Apa]½,

(26b)

]~op, = [s~(R - x - 2 6 -

and the luminosity f r o m eq. (18). These values are all a little too high since A,~ and A,,p are not exactly zero. The differences, however, should not exceed 10% very much. This example o f one type of mass separator is presented here in order to give some survey a b o u t its ion optical properties. Besides this, it should be regarded only as an example h o w such a procedure should be carried out. F o r the continuous interest in this work I want to thank here Prof. H. Ewald. Also I want to thank Mr. R. Amad.ori for most o f the numerical results o f the example given, which he obtained from a versatile computeiprogram.

References 1) L. Kfnig and H. Hintenberger, Z. Naturf. 12a (1957) 377. z) A. J. H. Boerboom, H. A. Tasman and H. Wachsmuth, Z. Naturf. 14a (1959), 121,816, 818; H. A. Tasman, thesis (Leiden). 3) H. Wollnik, Nucl. Instr. and Meth. 34 (1965) 213. 4) H. Wollnik and H. Ewald, Nucl. Instr. and Meth. 36 (1965) 93. 5) H. Wollnik, Nucl. Instr. and Meth. 38 (1965) 56. 6) H. Wolinik, Nucl. Instr. and Meth. 52 (1967) 250. 7) H. Liebl, Z. Naturf. 13a (1958) 490; H. Liebl, Optik 16 (1959) 19. 8) M. Camac, Rev. Sci. Instr. 22 (1951), 196; G. Andersson, B. Hedin and G. Rudstam, Nucl. Instr. and Meth. 28 (1964) 245. 9) R. Ludwig, Z. NaturL in preparation. 10) j. Chavet, Nuel. Instr. and Meth. 38 (1965) 37. 11) W. Ploch and W. Walcher, Z. Physik 127 (1950) 274; H. Enge, Rev. Sei. Instr. 35 (1964) 278.