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Unified theory of ion optics
International Journal of Mass Spectrometry and Ion Physics, 52 (1983) 319-336 Elsevier Science Publishers 3.V., Amsterdam - Printed in The Netherlands
UNIFIED
319
TEIEZORY OF ION OPTICS
H. NAKABUSHI Department
T. SAKURAI
of Physics, Faculty of Science, Osaka Uhiversity, Toyonaka 560 (Japan)
and H. MATSUDA
Institute of Physics, Coilege of General Education, Osaka University, Toyonaka 560 (Japan) (Received 3 February 1983)
ABSTRACT A unified theory of ion optics is developed, which can be applied to 4 ion-optical devices used currently in particle spectrometers. The second-order trajectory calculation has been performed in unified fields consisting of crossed electric and magnetic fields with arbitrary distributions. Nine parameters (in the second-order approximation) are necessary to designate respective ion-optical devices as well as their arbitrary superpositions. Correction for the relativistic effect on particle mass is included, which is necessary for relativistically fast particles.
INTRODUCTION
In modern particle spectrometers, as well as beam transport systems, several kinds of electric and magnetic ion-optical devices are used for focusing, dispersing and aberration-compensating purposes. They are classified as: (a) electric cylindrical, spherical or toroidal sector; (b) magnetic homogeneous or inhomogeneous sector; (c) electric quadrupole, hexapole or multipole lens; (d) magnetic quadrupole, hexapole or multipole lens; (e) achromatic quadrupole lens; (f) Wien velocity filter; and (g) crossed field analyzer. In addition, drift space (h) is necessary to connect them. They are arranged in simple or complex ion-optical systems to meet specific purposes and realize required performances. Recently, more complex systems have been constructed incorporating several sector fields and additional focus-adjusting and aberration-compensating devices [I]. In designing such systems and 002k738 l/83/$03.00
Q 1983 Elsevier Science Publishers 3-V.
320
calculating their image aberrations, the method of matrix transformation is convenient and used extensively [2-41. Transfer matrices for ion-optical devices (a)-(g) have been calculated in a second- or third-order approximation by many authors [5- 181, and different matrix expressions are given for the respective devices. It would be advantageous to have a single transfer matrix which could describe all ion-optical devices (a)-(h) simultaneously. In order to establish such a convenient transfer matrix, a unified treatment is indispensable for combining them. This treatment necessitates unified understanding of their ion-optical characteristics. It is well known that those properties such as dispersion, focusing and aberration are closely related to multipole components in the power series expansion of ion-optical fields, that is: (i) energy or momentum dispersion (electric or magnetic dipole componen t) ; (ii) focusing or defocusing function (quadrupole component); and (iii) aberration-producing or -compensating function (hexapole or higherorder component). Any ion-optical field can be expressed in the form of a multipole expansion. If the unified expression of a multipole expansion is derived for any ion-optical field, it is possible to yield a single and unified transfer matrix by calculating particle trajectories in the unified field. Calculations with this intention have already been made but in an incomplete manner. The particle trajectories in electric and magnetic multipole lenses have been calculated respectively in refs. 11 and 14, the results of which are applied to pure quadrupole, hexapole or octapole lenses as well as their arbitrary superpositions, covering (c), (d) and (h). Similar calculations for crossed fields (g) have been performed in ref. 17 and relativistically in ref. 18, the results of which are applicable to pure electric or magnetic sectors (a) and (b) and also to a Wien filter (f). The former calculations are for ion-optical devices with non-dipole field components, the latter ones being for those with dipole field components. In the present paper we propose a unified theory of ion optics which describes all ion-optical devices (a)-(h) in a single trajectory expression and consequently in a single transfer matrix. The second-order particle trajectory has been calculated in general crossed electric and magnetic fields which are assumed to have dipole field components as variable parameters. Dispersion is expressed in terms of mass and energy, which is necessary if electric and magnetic fields are combined in a system. For completeness, correction for the relativistic effect is included, which is important when kinetic energies of particles to be analyzed increase up to values that cannot be neglected when compared with their rest masses. The calculation is limited to second order, but the procedure can be extended with relative ease to a higher-order
321
calculation. The present paper only concerns idealized main fields: the effect of frjnging fields is not included. DESCRIPTION
OF UNIFIED
FIELD
Following the normal procedure in ion optics, we choose a convenient circular main path of radius pO as an optical axis, along which moves a reference particle of rest mass m o, charge e, and kinetic energy U,. A curvilinear coordinate system relative to the optical axis is used as shown in Fig. I. The trajectory of an arbitrary particle is described by the coordinates x(z) and y(z) as functions of z, where z is the path length of the reference particle along the optical axis (z = pow with the angle o of deflection). All the expressions should be written in such a way that they give correct results at the limit of p,, + 00. Unified fields to be considered here consist of crossed electric and magnetic fields. Both electric and magnetic fields are assumed to be mirror-symmetric with respect to the median plane (y = 0 plane) and have no z-dependence. Thus the electric and magnetic fields on the median plane are defined as
Ex(x,0)=E,(h,+X,x+h,x2+
...)
B,(x,o)=B,(~,+~Ix+~*x2+
._.)
9
(1)
arbitrary
trajectory
Fig. 1. Curvilinear coordinate system used in describing particle trajectories in unified fields. The optical axis is a circular arc of radius pc, to whitih the coordinate system refers. In the case of a straight optical axis, h, ( = l/pa) = 0, it becomes a rectangular coordinate system. The median plane is the x- z plane (y = 0), being a bending or dispersive plane. The trajectory of an arbitrary particle is described by the coordinates X(Z) and v(z) as functions of z, where.2 is the path length of a reference particle moving along the optical axis.
322 where the other components vanish. The quantities A, and cl0 are dimensionless parameters, being 1 or 0 depending on whether dipole field components exist or not, and E, and B0 are the respective field strengths along the optical axis when dipole fields exist. Quadrupole and hexapole field components are generally defined as E
A
0
aExb,Y)
=
q?k
and B,I_L~=
x_o
ax
1
Y>
dx
y-o
x=0 y-o
(quadrupole components)
E,X,
=
1
a2E,(x,Y)
2
ax2
(2)
1 X=o and Bop2 = 7 y=o
a26,(x9 u) f3x2
x=0 y-_o
(hexapole components)
(3)
The electric and magnetic fields off the median plane are derived using Maxwell’s equations, and their second-order expressions are E,b,y)=E,[&+~
,x + x,x2 - f(2h,
E,{x,Y)
+ ho&&~ -
= E,[ - &
(2x2
+ &A, +
hoA,
-
hZ,Ao)v]
(4
&(x,y)=O
Rb~
- h;h,)y2]
Y) = &h,Y
+ aJ2w)
B,(x,y)=B,[~o+~1x+112x2--4(2~2+ho~l)y2] &(&Y)
= 0
(5)
with ho = l/po. The electric potential $B(x, y) and the magnetic vector potential component A,( X, j), which yield eqns. (4) and (5) by the relationships E = -grad + and B = rot A respectively, are given by $~(_x,JI>= -E,[h,x++A,~~-~(~,+h~&,)y~ + fX2x3
- +(2A,
+ h,h,
- h;Xo)xy2]
(6)
and A,(xA=
-B,[P +
i(G2
ox + 2(P, - hoPo)X2-
hop.,
+
3J&o)x3
-
hY2 ~2~~1
(7)
where +(O, 0) = 0 and A,(& 0) = 0 on the optical axis are assumed_ The usual assumption of A,(O, 0) = - B,~opo/2 (see ref. 8) is not adopted because A,( x, y} ---+ 60 when p. + ‘co. However, both assumptions of A,(O, 0) give the
323
same result in the final trajectory equation, although the expressions for A, ( X, y) have different forms to each other. Field expressions are obtained, for instance, for crossed -fields by putting A, = pO = 1, for a Wien filter by putting A, = p0 = 1 and h, = 0, and for electric and magnetic multipole lenses by putting A, = p,-, = h, = .O. In total there are nine parameters which are necessary to designate ion-optical devices (a)-(h) M=
l/PO)9 KAJ~
&3, A,, A*, 4DLl)~
PO, PIT
P2
as shown in Table 1, where h, and hZmare used instead of E, and B, (see eqn. (20)). In practical use, these parameters are initially set to zero and then appropriate values are given to the parameters necessary to designate the
TABLE
1
Parameters necessary to designate ion-optical devices for second-order approximation Ion-optical device
Electric field
(a) electric cylindrical sector electric spherical sector electric toroidal sector (b) magnetic homogeneous sector magnetic inhomogeneous sector (c) electric quadrupole lens electric hexapole lens electric multipole lens (d) magnetic quadrupole lens magnetic hexapole lens magnetic multipole lens (e) achromatic quadrupole lens (f) homogeneous Wien filter inhomogeneous Wien filter (g) crossed field analyzer (h) drift space
l/P* l/PO l/PO l/P,
l/PO 0 0 0 0 0 0 0 0 0 l/p,hla 0
Magnetic field
k
k3
A,
A,
hiTI rcL0 PI
ho h* h,la
1, 1
-ho -2h,
h; 3h; a
0
0
0
0
0 coc “00 =oc 0 0 0 =oe
0
0
0 0 c =0000 0 0 0 0 0 g = 0
0 0 0 0 0000 h, 1.0 h, 1 0000 0 0
0 0 0
0 0 0
f
1 1
0 s
0
0
0
f
I-L2
0 0
0 0
b
0 b
0
0
dOdO d OOd d 0 d eOeO f 1 0 f 1 g h lbb 0000
d ‘0 *
a A, = -(I + c)h,, and A, = (I+ c + +c” - +c’)hi using c and c’ from ref. 6. b p, = n,ho and p2 = n,hz using n, and n2 from ref. 8. and k,,, = h,h, with k,,, and k,,, being ion-optical parameters of electric = k,S = h,h, quadrupole and hexapole field components, respectively [ 111. being ion-optical parameters of magnetic d k4.m = h,p, and k,, = h,-,-,p:!with k*, and k,, quadrupole and hexapole field components, respectively [ 141. and k,,, = -2k,,, [15]. = k,,, = he&, k,., = h,p, f he= -h,. g A,=
_&=I rr? h h,=h,+h,.
r,”
p, = - 1 and p2 = r m 2
with TVand r, in ref. 16.
324
ion-optical device in question. It is noted here that these parameters are of different dimension
ION-OPTICAL
REFRACTIVE
INDEX
Following the trajectory of an arbitrary particle of rest mass m (note that m is not the relativistic mass), charge e and kinetic energy U (at + = 0), the relativistic relation of energy conservation in unified fields is written as
+ f5$$
e+(x,
y) = n-w2+ U
(8)
with 2) and c being respectively the particle velocity and the velocity of light. The particle trajectory will be derived [ 191 from Fermat’s variational principle nds=O
(9)
for time independent fields, where ds is the path length element taken on the particle trajectory in question and n is the ion-optical refractive index given bY
which is equal to the generalized momentum in the tangential direction of the trajectory. The components of the vector dr are [dx, d y, (1 + h,x)dz], and those of the vector potential A are [0, 0, A,] in the present case. Assuming z to be an independent variable and dividing n by [ mU(2 + wmc 2)I 1/2 to form a dimensionless quantity F, eqn. (9) is transformed to 6/=‘F dz = 0 =0
(11)
with
F(x,y,x’,y’)=
r
U
2+-
I i +
mU
(1 f
mc2
lp1/2j
)I
mv
(x’2 +y’2
1 /m
hox)2)“2 + e(1 + h,x)A,
02) 1
325
In the above equation and throughout this paper, the prime denotes d/dz. Equation (11) can also be written in the form of the Euler-Lagrange equations (13) In order to obtain second-order trajectories by solving the above equations,
F must be expanded in terms of x, x’, y and y’ to third order. Substitution of eqns. (6)-(S) into eqn. (12) yields F = F,, + F,+
+ Fz,x2
+ Fbz y2 + g,( x’2 + y”)
+ Fsox3 + F12xy2 + g2x(x’2
-t y”)
04)
where the coefficients are F,, = 1 F,, = h, - poM
- X,N
Fzo = - +[(/A, + hopO)M+ 52
= f[/W+
Fso=
-f +
+ (1,k2)x”oN2]
(A, + h,kdN]
[(P,
+
kda4+
(3,‘2c2)(A,
gz= - ;(h,,
(iI, + 2h,X,)N
(A2
+ h,A,)X,N2
+
tkJ,w
+ (3/‘2~*)h3,N~]
+ X,N)
(15)
with e=
1+(u/mc2)
M = (eB,/~)(1/\/1-1-E) N=
- (eE,/U)(c/(l
TRAJECTORY
+ c))
(16)
EQUATION
For most cases they coordinate is less important than the x coordinate, so the trajectory equations are derived to second order for the x direction and to first order for the y direction. Substitution of eqn. (14) into eqn. (13) yields x “ = F,, + 2F2,x + 3F3,x2 + F12y* - g,(2xx” y ” = 2F,,y
+ xr2 - y”) 07)
326
In solving eqn (17), fractional mass and energy deviations are introduced, defined as m/e
= (m,/e,)(l
+ y) and U/e = (U,/e,)(l
+ S)
(18)
Equation ( 16) is expanded in powers of y and 8 to second order as c = Eg+ (Co - l)(S - y - y6 + y2)
I
1-k
iw=h,
+ (24
N=h,
0
+ l)P}
(Y + Eo6)+
1
l-
1
(3y2 + 2(2e, - 1)ys 2(1 + QO)2
- 1)y +
(e; + 1)s)
i 1
+ 41
(2(E,-
l)y2+2&-
1)2y6+E0(f;+3)6Z)
+ coJ2
(191
with ‘0 = 1 + (U,/m,c*) h, = (%B,/~G-K)(
he = -
l//-=G)
(eo~o/WW(l
+ %JN
(20)
where co is a relativistic correction, and the non-relativistic limit is obtained by putting Eo= 1. The term Jz,(h.) describes the inverse of the radius of the optical axis when the magnetic (electric) dipole field alone is applied. The simple relation
(20
ho = &Jo + k&o
holds as the zeroth-order solution of eqn. ( 17). The above equation gives the relationship between the strengths and directions of the electric and magnetic dipole fields on the optical axis such that a centrifugal force is compensated, also showing that non-dipole fields do not cause any particle deflection. In a Wien filter the relationship h,, = 0 (and A, = p. = 1) holds, and therefore h, = -h,. Substituting eqn. (19) into eqn. (15), explicit forms of the trajectory equations ( 17) are written as x” + k,x = CJ + C,S -t DXXx2 + D,,xy +D,,y’
+ D,,xb
-I- Daaa2
+ D,&y6 + Da8aV2 + Dyy y2 + D,&32
(22)
327
y”
(2%
+ k,y = 0
with (Y and #3 being the angles of inclination relative to the optical axis for the x and y directions, respectively; a = X’ and j3 = y’ in a first-order approximation. The coefficients of eqns. (22) and (23) are given by k,=h,
(l/&)htXz,
+h,(h,A,+h,)+
k, = -h,
- h,h,Xo
(24) D xx =-h,
- +ho(2h,
f h,h,)
- (3/2c;)h&(X,
+ h,A,
Vh -
w%~, +kd%-nPa)
+ h,&,>
- (hl +h,%dkx
Duv=
-
1 241
+ Q>’ 1
Drs = -
%Cl + %I2 Dss=
-
1 2(1 + E0)2
( 2 CEg{2(4
l)2h,k,
+ 3POO
+ G, (2~ -
1)&p,)
+ (2G + 1knPcJ
D,,y=h2++h,(h,+h,A,)+-&h&,(h,+h,h,) 0
q3/3= - DdlQ
(25)
with h, = 4% h2
=&X2
+ k&u, -+k#2
(26)
where h, and h, are the ion-optical parameters of quadrupole and hexapole
328
fields, respectively. It is noted here that, among the coefficients of eqn, (25) which are related to y and 6, the following sums hold CY+ C,=h, Dx-, + Ox6 = 2k, - h, I&
(27)
-I- Dys -I- D66 = -ho
SECOND-ORDER
TRAJECTORY
The first-order solutions of eqns. (22) and (23) are easily obtained. The coordinates (x, y) and the angles ( LY,j?) of inclination, at z = z, are expressed to first order as X = cx, + Sa, + (l/k,)(l CX= -k,Sx,
t CLX,+ s&y
- C)(CJ
+ C&S)
+ c,s)
Y = CyYo + S,Po P = -k,$JYo
cw
+ cpo
where (x,,, yO) and ( aO, &,) are the initial values at z = 0 of the coordinates and the angles of inclination relative to the optical axis. S, C, S,, and Cu are defined in Table 2. It should be noted that the definitions of S and Sv are different from those usually used, such as S = sin(Jk,z) and Sv = sin(gz).
By using the definitions of S, C, Sv and C,, in Table 2, eqn. (28)
becomes valid for positive, negative and near-zero values of k, and k,. Substituting the first-order soIutions of eqn. (28) into the right-hand side
TABLE
2
Definitions of S, C, S,, and C, a kx7 k,
Positive
S
(l/K)
c
-o/k,z)
cosh(,/?+)
SY
U/fi)W~z)
(l/,/q)
cy
-(/qz)
Negative sin(&)
(l/d-)
cosh( \i-ky
Near-zero sirWJTz)
z(l - ikxz2+&k,2z4) l-
+k,z2+&kzz4
sinh(dqr)
z(l-
g)
1 --‘k z2 +‘kzz4 24 Y ZY
ik,z2+
+iik:z4)
a S, C, SY and C, are analytical functions of k, and k,. k,S2 + C2 = 1 and k,,S; + C; = 1. LapIace transformations of each of S, C, S,, and CY have the same form for positive, negative and near-zero values of k, and k,. C and C, are dimensionless, but S and SY have a dimension of [L].
329
of eqn. (22)and then using the Laplace transformation r?(x)=/e=+)
defined as
dz
the second-order x coordinate is obtained. The procedure is the same as in ref. 18 and accordingly the same definition and notation are used here. Of special note is that the Laplace transformations of each of S, C, S’ and Cv have the same form, while each of these symbols is defined in different forms for positive, negative and near-zero values of k, and k, as shown in Table 2. Therefore, the second-order solution derived in this section is valid for any values of k, and k,. The final form of the second-order x coordinate may be expressed as X = cx, + Sa, + (l/k,)(l +Lx,2
+ %W%J
- c)( cyy + c,s) + K+Wf
+ &%S
+ H&4
+ QQY
+H,GaOS + Wy,y2 + H,,yS + HssS2 + Nyyuo’ + H,BY,P, + E-r,,&?
(29)
where the second-order coefficients rr,, = D__g(x2/xZ)
+ D,,g(aZ/x2)
H,, = &,g(
+ D,,g(
K,
x’/xa)
= %,g(x2/4
a2/xa)
are
+ h,S
+o,,s(~‘/~Y)
+%ygW4
w,, = D,,g(x2/xa)
+ D,,g(a2/4
+&gWx)
4,
+ D,,g(a2/a2)
= 4,g(x2/a2)
*aY = D,,g(x2/ary)+D,,g(a2/ay)+D,,g(x/a)
H,,=D,,g(x2/a6)+D,,gIu2/as)+D,b9(X/a) H,,=D,,g(~*/Y*)+D,,g(a*/Y*)+D,,g(~/Y)+D,*g(1/1) H,,=D,,g(x2/Y~)+D,,g(a2/Y6)+D,,g(x/G)+D,sg(x/Y)+Dygg(1/1) w,,=D,,g(~*/8~)+D,,g(~*/~~)+D,,g(x/~)+D,,g(l/l) H,,=D,,g(yz/Y2)+~~~g(P2/Y2) ~yle=Dyyg(~z/~~)+DlB8g(P2/~~) H,,=D,,g(Y2/~2)+~~~g(~2/82) The terms g( i/j)
and g( ij/kl), which are defined in the Appendix
(30) to ref. 18,
k,S;l
+z4(1 - $k,z2 t +,kfz4)
(C,/k;)[
C + k,S2)] (C,/k;)[rk,S-{(l-Ctk,S2)]
(1/3k;)[2(1
gtx2/a2)
&y!z6(1 - &k,z2 t &k;z4) &c~c8z6(1 - &k,z* + &kfz’) &C,Zz6(1- &k,z* t &k,2z4) t2k,)z2 t &(kf+2k,k, iz2[1 +(k,
(C;/3k;)[4(1-
KY ‘/s’>
(2,‘(k,
gtu 2/Y/R - S)
-4k,NS; -(2/kxXlC)-tk,S] a
- zC) a
(l/k;)[2(1-
(l/(k,
(Vk,)(S
-4k,W,C,
-4k,))Kl-(2k,/k,))(l(1/4k,)[2( 1 - C)t zk,S] ’
(l/(k,
C)l
C)-
(2C,C,/3k;)[4(1- C)-3zk,S + k,S*] (C,2/3k;)[4(1- C)-3zk,S t k,S*]
g(x2/S2) dY2/Y2)
dX2/Y~>
gtx2/v2)
t8k;)z4]
--
.
.-
(l/(k,-4k,))[S;-z2(l+kxz2t&,kfz4)lb
+z’[l-&(k,t4k,))z2+&(k,2
(2/(k,-4k,))[S,C,-~(l-~k,z~+$&z~)]~
-
.-
--
t4k,k,t16k;)z41
+z3[l-~(k,t4ky)z2t&(k~t4k,ky+16k;)z41
(l/(k,-4k,))[(k,-2k,)(z2/2)(l-$k,z2+~k~z4)-kyS~l’
- +k,z* t &k;t4)
.(Cs/3k;)(S-3zC+2SC),
l?tx2/4 C)-3zk,S
1- fk,z* t &k;z4) +$yz5(
i3(X2bY
&z5(1
t &k;z4)
- &k,z2 + &k;z4)
(C,./3k,2)(S-3zCt2SC)
k,S’]
+c,z4(1
t &k;z4)
ak,z* + &,k;z4) $k,z*
1
-C)-
zk,S - $( 1 -
C)
t k,S*]
+C,z4(1 - *k,z*
(2/3k,)S(l-
;z*(l jz'(l
-
dx2/x4 ldX2/XY > s(x2/x~)
C + k,S*)
(1/3kJl-
&*/x2)
- &k,z2 t $,k;z4)
&Ax/Q
&z4(l
C - fzk,S)
- C - fzk,S)
(C,/k;)(l
dX/Y
(C.Jk;)(l-
iz3(1 - hk,z* t &,k,2z4) $C,z4(1 - &k,z* t &k;z4)
(1/2k,)(S - zC)
gtx/a)
1
iz2(1- {k,z2 + &k;z4)
+ &,k,2z4)
fz*(l - +k,z2
tl/k,Ml- 0
jzs
l/l)
g(x/x)
gt
Expression for small k, and k, limits
Normal expression
Explicit expressions of g-functions
TABLE 3
= C,‘g(x2/a2)
= -2k&g(x2/a2)
g(a2/x8)
g(P2/v2) = ~:g(u2/~2) g(P2M) = - ~~g(Yz/yp) g(P2/B2) = g(u /v2)
’ In the case of k, = 4k,. b In the case of k, - 0, but not k, - 0. For the case of k, - 0, but not k, - 0, the normal expressions can be used without problem by substituting the expanded forms of Table 1 into 5” and Cy.
g(a*/a*)= g(x2/x2) g(02/ay) = Cyg(x2/4 g(a’/aQ = Csg(x2/xu)
g(a2/S2)
s@*/Y4= 2C&g(x2/a2)
= -2k,C,g(x*/c?)
g@/xy)
!e2/Y2)=C,‘g~x2/cu2)
= kig( X2/4x2) k,g(x2/xa)
g(a2/xcw) = -
g( a’/x’)
332
are functions of z and are given explicitly in Table 3. However, Dij is not a function of z, as explicitly expressed in eqn. (25). The angle Q of inclination is @ven by 1 (y= 1 +&x
dx = xF - hOXXl dz
to second order and may be written as cu= - k,Sx,
+ Ca,
+ G,,x$
+ S( C,,y + C,s)
+ GxaxO~,-, + G~+,Y
+ Gayctoy + G,s~oG + GYYy2
+ G,,a;
+ G,,Y~ + Gad2
+ G,,x,2
+ Gyyd
(31)
+ GY~~o&+G&
where the second-order coefficients are Gxx = o,,g’(
x2/x2)
-t D,,g’(
a2,‘x2)
+ h,k,SC
GXQ = D,,g’(
x2/w)
+ D,,g’(
a2/xa)
- h,( 1 - C - 2k,S2)
G X-y= D,,g’(x2/xY)
+ QX,g’(~2/xY)
+ D,,g’(x/x)
+ bw
- 2C)Cy
GXs = D,,g’(
x2/x6)
+ D,,g’(
a2/x8)
+ D,,g’(
+ h,S(
1 - 2C) C,
G aa = o,,g’(
x2/a”)
-t D,,g’(
a2,‘cy2) - h,SC
G ay = 4,g’(x2/~Y) + (h,/k,)(l
G as = D*,g’(x2/d3) + (h,/k,)(l G yy =D_yXg’(x2/Y2) -
&Jk,)S(l
G+ = Lg’(x’/YS) +%g’Wu) GM = 4,g’(x2P2) - &‘k,)S(l
+ Qu,d(~2/~Y)
x/x)
+D,,g'W4
- C-2k,S2)C, + D,,g’(a2/cd)
+ D,,g’(x/a)
- C-2k,S2)C, +D,,g’(eY2)
-
+D,yg’WY)
+o,.g’O/~)
C,C,’
+ D,ag’(az/Y~) + D,,g’O/l) + D,,g’(a2/62) - C)C,z
G yy = o,,9’(Y2/Y2)
+qi&3g’(B2/Y2)
G yJ3= o,,8’(Y2/YP)
+ q9pS’(B2/YP)
GpB = Dyyg’(Y2/P2)
+D,s,sg’(P2/P2)
+ O,,g’(x/S) - (2hcJkJW + R,g’(x/G)
- C>C,C, + &g’(W)
333
The terms g’(i/j) and g’( q/H) denote the derivatives of g-functions with regard to z and are explicitly given in Table 4. CASES
WITH
NON-FOCUSING
ACTION
There are several ion-optical devices with zero or near-zero values of k, = 0),a multipole lens with very and/or k,,such as a hexapole lens (k,= k,, small value of k, and k,, an r- * magnetic field (k, = 0),a homogeneous magnetic field (k,= 0),a cylindrical electric field (k, = 0),and a drift space = 0). Furthermore, in fitting work for designing complex ion-optical W,=k, systems, there often arise cases in which k, or k, change continuously from positive to negative values, or vice versa, through zero. In such cases of kX - 0 and k, - 0,as already described in the previous section, the expanded forms in terms of k, and k, are used for the symbols S, C, SY and CY. In the usual procedure of the definition of these terms such as S = sin(Ez)
for
k, > 0 and S = sinh(,/qz)
for k, -C0, S is irregular at k, = 0 and the Taylor expansion cannot be applied to S. However, using the definitions given in Table 2, S and C( SY and C,) and their derivatives with regard to k,(k,,) become continuous at k,(k,,) = 0,thus making it possible to express them using the Taylor expansion of powers of k,(k,,). Another advantage of the definitions used here is that Laplace transformations of each of S, C, S,, and CY have the same form, while each symbol is defined in different forms for positive, negative and near-zero values of k, or k,_ Accordingly, the second-order solutions of eqns. (29) and (31) are applicable to the cases of kX - 0 and k, - 0, by substituting the expanded forms of S, C, Su and C” into g( i/j) and g( ij/kl)and their derivatives g’( i/j) and g’( ij/kZ). These expanded forms of g- and g’- functions are also given explicitly in the third columns of Tables 3 and 4, respectively. SUMMARY
A unified theory of ion optics has been developed, which can be applied to all ion-optical devices (a)-(h) used widely in particle spectrometers and beam transport systems. The trajectory calculations have been performed to second order for the radial x direction and to first’ order for the axial y direction, in unified fields consisting of crossed electric and magnetic fields with arbitrary distributions. The necessary parameters to designate respective ion-optical devices and their arbitrary superpositions are summarized in Table 1, which also shows their relationships with quantities conveniently used in the respective devices. For slow particles (2, e c), putting co = 1 yields nonrelativistic
results.
g’tr2/b2)
g’(v‘/ra>
hk,z’
. -
-.
(2/(k, -4kJXSyCy - S) (l/k,)@ - zC) a
(Mk, -4k,Mk, -2k,P i(3S + ZC) B (2,/(/c,-4k,))(l-C-2kyS_) ZS”
-
-
-2k,S,C’l
;z3[1--+,(k,
z’[l-+(k,
t4k,)z2
t&k;
+4k,k,
t 16k;)z4J
t4k,k,+16k;)z4]
t&(k,2 t2k,k,+8k~)z4J t4k,)z2+&(k;
z[l -+(k, t2k,)z2
+ &k,2z4)
f +k,2t4)
&C~Z5(l - fk,z2
(2C&/3k;)(S t2SC -3zC) (C;/3k,z)(St2SC-3zC)
- fk,z2
t #z4)
&C,zz’(l - fk,z2
(C;/3kf)(S+2SC-3iC)
&C;z5(1
t &k;z4) - +k,z’
:C,z4(l
fC,z’(l- $k,z2 +$k;z4) +z3(1- +k,z2 t &k;z4) aCyz4(1- ik,z”+ &k;z4)
+ &k,2z4)
i- &,k,Zz4)
+C,z’(l -
z2(1-
(2/3k,)S(lC> (C,/3k,2)(2(1- C)+3zk,S -4k,S2] (C,/3k;)[2(1C)+3zk,S -4k,S2]
(C,/2k,)(S i ZC) fS(l t 2C) (2/3k,)(C - 1t 2k,S2) (C,/3k,)(St3zC-4SC) (C,/3k,)(S +3tC -4X)
Pq2US-
jZS
zc>
z(1 -ik,Z2 t &jky) z(1 -+k,z2 t &k,2z4) fz2(1 - &4,22 t &kZz4) gyzyl - hk,z2 t &k,2z4) &z’(l - +,k,z’ -I-&,k,2z4) z(1 - +k,z* f $$k;z4)
s :(s + ZC)
Expression for small k, and k, limits
Normal expression
Explicit expressions of g’-functions
TABLE 4
S’(P2/Y2) = k,2s’(Y2/~2~
= g’(x2/x2)
= C,g’(x’/xa)
= C,g’(x’/xa)
g’(a2/az)
g’(a’/ay)
g’(a*/as)
’ In the case of k, = 4k,. In the case of only k, - 0 or k, - 0 in g’( y2/y *), g’( y */yfi) and g’( y ‘//3’), the normal expressions can be used without problem by substituting the expanded forms of Table 2 into S, C, S,, and CY.
sU2/B2)= g’(Y2/Y2)
8’(P2/YS) = - k,d(Y2/YB)
= C,‘g’(x2/a2)
g’(a2/S2)
fb2/XY) = -2k,C,g’(x2/a2) g’( a2/xq = -2kxC~g'(x2/a2)
= C,‘g’(x2/a2) = 2C,c~g’(x2/a2)
g’(a’/y’) g’(a2/y8)
g’(a2/x2)= k,2g’(x2/a2) g’( a2/xa)= - k,g’( x ‘/xa)
336
The unified transfer matrix is easily constructed using the results herein. The coefficients in eqns. (29) and (31) give the corresponding elements of the first and second rows of the transfer matrix. The other non-vanishing elements are calculated from these first- and second-row elements by forming appropriate products. REFERENCES 1 H.A. Enge, Nucl. Instrum. Methods, 186 (198 1) 413. 2 K.L. Brown, R. Belbeoch and P. Bounin, Rev. Sci. Instrum., 35 (1964) 481; K.L. Brown and SK. Howrg, SLAC-91 (1970). 3 H. Wollnik, Nucl. Instrum. Methods, 52 (1967) 250. 4 T. Matsuo, H. Matsuda, Y. Fujita and H. Wollnik, Mass Spectrosc. (Jpn), 24 (1976) 19. 5 H. Hintenberger and L.A. Kiinig, 2. Naturforsch. Teil A, 12 (1957) 773. 6 T. Matsuo, H. Matsuda and H. Wollnik, Nucl. Instrum. Methods, 103 (1972) 5 15. 7 R. Ludwig, Z. Naturforsch. Teil A, 22 (1967) 553. 8 T. Matsuo and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 6 (1971) 361. 9 Y. Fujita and H. Matsuda, Nucl. Instrum. Methods, 123 (1975) 495. 10 S. Taya and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 9 (1972) 235. 1 I T. Matsuo, H. Matsuda, H. Nakabushi, Y. Fujita and A.J.H. Boerboom, Int. J. Mass Spectrom. Ion Phys., 42 (1982) 217. 12 G.E. Lee-Whiting, Nucl. Instrum. Methods, 83 (1970) 232. 13 K.L. Brown, SLAC-75 (1967). 14 H. Nakabushi and T. Matsuo, Nucl. Instrum. Methods, 198 (1982) 207. 15 S.Ya. Yavor, A.D. Dymnikov and L.P. Ovsyannikova, Nucl. Instrum. Methods, 26 (1964) 13. 16 D. Ioanoviciu, Int. J. Mass Spectrom. Ion Phys., 11 (1973) 169. 17 D. Ioanoviciu. Int. J. Mass Spectrom. Ion Phys., 15 (1974) 89. 18 H. Nakabushi, T. Sakurai and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 49 (1983) 89. I9 W. Glaser, in S. Fltigge (Ed.), Handbuch der Physik, 33, Berlin, 1956, p. 123.
UNIFIED
319
TEIEZORY OF ION OPTICS
H. NAKABUSHI Department
T. SAKURAI
of Physics, Faculty of Science, Osaka Uhiversity, Toyonaka 560 (Japan)
and H. MATSUDA
Institute of Physics, Coilege of General Education, Osaka University, Toyonaka 560 (Japan) (Received 3 February 1983)
ABSTRACT A unified theory of ion optics is developed, which can be applied to 4 ion-optical devices used currently in particle spectrometers. The second-order trajectory calculation has been performed in unified fields consisting of crossed electric and magnetic fields with arbitrary distributions. Nine parameters (in the second-order approximation) are necessary to designate respective ion-optical devices as well as their arbitrary superpositions. Correction for the relativistic effect on particle mass is included, which is necessary for relativistically fast particles.
INTRODUCTION
In modern particle spectrometers, as well as beam transport systems, several kinds of electric and magnetic ion-optical devices are used for focusing, dispersing and aberration-compensating purposes. They are classified as: (a) electric cylindrical, spherical or toroidal sector; (b) magnetic homogeneous or inhomogeneous sector; (c) electric quadrupole, hexapole or multipole lens; (d) magnetic quadrupole, hexapole or multipole lens; (e) achromatic quadrupole lens; (f) Wien velocity filter; and (g) crossed field analyzer. In addition, drift space (h) is necessary to connect them. They are arranged in simple or complex ion-optical systems to meet specific purposes and realize required performances. Recently, more complex systems have been constructed incorporating several sector fields and additional focus-adjusting and aberration-compensating devices [I]. In designing such systems and 002k738 l/83/$03.00
Q 1983 Elsevier Science Publishers 3-V.
320
calculating their image aberrations, the method of matrix transformation is convenient and used extensively [2-41. Transfer matrices for ion-optical devices (a)-(g) have been calculated in a second- or third-order approximation by many authors [5- 181, and different matrix expressions are given for the respective devices. It would be advantageous to have a single transfer matrix which could describe all ion-optical devices (a)-(h) simultaneously. In order to establish such a convenient transfer matrix, a unified treatment is indispensable for combining them. This treatment necessitates unified understanding of their ion-optical characteristics. It is well known that those properties such as dispersion, focusing and aberration are closely related to multipole components in the power series expansion of ion-optical fields, that is: (i) energy or momentum dispersion (electric or magnetic dipole componen t) ; (ii) focusing or defocusing function (quadrupole component); and (iii) aberration-producing or -compensating function (hexapole or higherorder component). Any ion-optical field can be expressed in the form of a multipole expansion. If the unified expression of a multipole expansion is derived for any ion-optical field, it is possible to yield a single and unified transfer matrix by calculating particle trajectories in the unified field. Calculations with this intention have already been made but in an incomplete manner. The particle trajectories in electric and magnetic multipole lenses have been calculated respectively in refs. 11 and 14, the results of which are applied to pure quadrupole, hexapole or octapole lenses as well as their arbitrary superpositions, covering (c), (d) and (h). Similar calculations for crossed fields (g) have been performed in ref. 17 and relativistically in ref. 18, the results of which are applicable to pure electric or magnetic sectors (a) and (b) and also to a Wien filter (f). The former calculations are for ion-optical devices with non-dipole field components, the latter ones being for those with dipole field components. In the present paper we propose a unified theory of ion optics which describes all ion-optical devices (a)-(h) in a single trajectory expression and consequently in a single transfer matrix. The second-order particle trajectory has been calculated in general crossed electric and magnetic fields which are assumed to have dipole field components as variable parameters. Dispersion is expressed in terms of mass and energy, which is necessary if electric and magnetic fields are combined in a system. For completeness, correction for the relativistic effect is included, which is important when kinetic energies of particles to be analyzed increase up to values that cannot be neglected when compared with their rest masses. The calculation is limited to second order, but the procedure can be extended with relative ease to a higher-order
321
calculation. The present paper only concerns idealized main fields: the effect of frjnging fields is not included. DESCRIPTION
OF UNIFIED
FIELD
Following the normal procedure in ion optics, we choose a convenient circular main path of radius pO as an optical axis, along which moves a reference particle of rest mass m o, charge e, and kinetic energy U,. A curvilinear coordinate system relative to the optical axis is used as shown in Fig. I. The trajectory of an arbitrary particle is described by the coordinates x(z) and y(z) as functions of z, where z is the path length of the reference particle along the optical axis (z = pow with the angle o of deflection). All the expressions should be written in such a way that they give correct results at the limit of p,, + 00. Unified fields to be considered here consist of crossed electric and magnetic fields. Both electric and magnetic fields are assumed to be mirror-symmetric with respect to the median plane (y = 0 plane) and have no z-dependence. Thus the electric and magnetic fields on the median plane are defined as
Ex(x,0)=E,(h,+X,x+h,x2+
...)
B,(x,o)=B,(~,+~Ix+~*x2+
._.)
9
(1)
arbitrary
trajectory
Fig. 1. Curvilinear coordinate system used in describing particle trajectories in unified fields. The optical axis is a circular arc of radius pc, to whitih the coordinate system refers. In the case of a straight optical axis, h, ( = l/pa) = 0, it becomes a rectangular coordinate system. The median plane is the x- z plane (y = 0), being a bending or dispersive plane. The trajectory of an arbitrary particle is described by the coordinates X(Z) and v(z) as functions of z, where.2 is the path length of a reference particle moving along the optical axis.
322 where the other components vanish. The quantities A, and cl0 are dimensionless parameters, being 1 or 0 depending on whether dipole field components exist or not, and E, and B0 are the respective field strengths along the optical axis when dipole fields exist. Quadrupole and hexapole field components are generally defined as E
A
0
aExb,Y)
=
q?k
and B,I_L~=
x_o
ax
1
Y>
dx
y-o
x=0 y-o
(quadrupole components)
E,X,
=
1
a2E,(x,Y)
2
ax2
(2)
1 X=o and Bop2 = 7 y=o
a26,(x9 u) f3x2
x=0 y-_o
(hexapole components)
(3)
The electric and magnetic fields off the median plane are derived using Maxwell’s equations, and their second-order expressions are E,b,y)=E,[&+~
,x + x,x2 - f(2h,
E,{x,Y)
+ ho&&~ -
= E,[ - &
(2x2
+ &A, +
hoA,
-
hZ,Ao)v]
(4
&(x,y)=O
Rb~
- h;h,)y2]
Y) = &h,Y
+ aJ2w)
B,(x,y)=B,[~o+~1x+112x2--4(2~2+ho~l)y2] &(&Y)
= 0
(5)
with ho = l/po. The electric potential $B(x, y) and the magnetic vector potential component A,( X, j), which yield eqns. (4) and (5) by the relationships E = -grad + and B = rot A respectively, are given by $~(_x,JI>= -E,[h,x++A,~~-~(~,+h~&,)y~ + fX2x3
- +(2A,
+ h,h,
- h;Xo)xy2]
(6)
and A,(xA=
-B,[P +
i(G2
ox + 2(P, - hoPo)X2-
hop.,
+
3J&o)x3
-
hY2 ~2~~1
(7)
where +(O, 0) = 0 and A,(& 0) = 0 on the optical axis are assumed_ The usual assumption of A,(O, 0) = - B,~opo/2 (see ref. 8) is not adopted because A,( x, y} ---+ 60 when p. + ‘co. However, both assumptions of A,(O, 0) give the
323
same result in the final trajectory equation, although the expressions for A, ( X, y) have different forms to each other. Field expressions are obtained, for instance, for crossed -fields by putting A, = pO = 1, for a Wien filter by putting A, = p0 = 1 and h, = 0, and for electric and magnetic multipole lenses by putting A, = p,-, = h, = .O. In total there are nine parameters which are necessary to designate ion-optical devices (a)-(h) M=
l/PO)9 KAJ~
&3, A,, A*, 4DLl)~
PO, PIT
P2
as shown in Table 1, where h, and hZmare used instead of E, and B, (see eqn. (20)). In practical use, these parameters are initially set to zero and then appropriate values are given to the parameters necessary to designate the
TABLE
1
Parameters necessary to designate ion-optical devices for second-order approximation Ion-optical device
Electric field
(a) electric cylindrical sector electric spherical sector electric toroidal sector (b) magnetic homogeneous sector magnetic inhomogeneous sector (c) electric quadrupole lens electric hexapole lens electric multipole lens (d) magnetic quadrupole lens magnetic hexapole lens magnetic multipole lens (e) achromatic quadrupole lens (f) homogeneous Wien filter inhomogeneous Wien filter (g) crossed field analyzer (h) drift space
l/P* l/PO l/PO l/P,
l/PO 0 0 0 0 0 0 0 0 0 l/p,hla 0
Magnetic field
k
k3
A,
A,
hiTI rcL0 PI
ho h* h,la
1, 1
-ho -2h,
h; 3h; a
0
0
0
0
0 coc “00 =oc 0 0 0 =oe
0
0
0 0 c =0000 0 0 0 0 0 g = 0
0 0 0 0 0000 h, 1.0 h, 1 0000 0 0
0 0 0
0 0 0
f
1 1
0 s
0
0
0
f
I-L2
0 0
0 0
b
0 b
0
0
dOdO d OOd d 0 d eOeO f 1 0 f 1 g h lbb 0000
d ‘0 *
a A, = -(I + c)h,, and A, = (I+ c + +c” - +c’)hi using c and c’ from ref. 6. b p, = n,ho and p2 = n,hz using n, and n2 from ref. 8. and k,,, = h,h, with k,,, and k,,, being ion-optical parameters of electric = k,S = h,h, quadrupole and hexapole field components, respectively [ 111. being ion-optical parameters of magnetic d k4.m = h,p, and k,, = h,-,-,p:!with k*, and k,, quadrupole and hexapole field components, respectively [ 141. and k,,, = -2k,,, [15]. = k,,, = he&, k,., = h,p, f he= -h,. g A,=
_&=I rr? h h,=h,+h,.
r,”
p, = - 1 and p2 = r m 2
with TVand r, in ref. 16.
324
ion-optical device in question. It is noted here that these parameters are of different dimension
ION-OPTICAL
REFRACTIVE
INDEX
Following the trajectory of an arbitrary particle of rest mass m (note that m is not the relativistic mass), charge e and kinetic energy U (at + = 0), the relativistic relation of energy conservation in unified fields is written as
+ f5$$
e+(x,
y) = n-w2+ U
(8)
with 2) and c being respectively the particle velocity and the velocity of light. The particle trajectory will be derived [ 191 from Fermat’s variational principle nds=O
(9)
for time independent fields, where ds is the path length element taken on the particle trajectory in question and n is the ion-optical refractive index given bY
which is equal to the generalized momentum in the tangential direction of the trajectory. The components of the vector dr are [dx, d y, (1 + h,x)dz], and those of the vector potential A are [0, 0, A,] in the present case. Assuming z to be an independent variable and dividing n by [ mU(2 + wmc 2)I 1/2 to form a dimensionless quantity F, eqn. (9) is transformed to 6/=‘F dz = 0 =0
(11)
with
F(x,y,x’,y’)=
r
U
2+-
I i +
mU
(1 f
mc2
lp1/2j
)I
mv
(x’2 +y’2
1 /m
hox)2)“2 + e(1 + h,x)A,
02) 1
325
In the above equation and throughout this paper, the prime denotes d/dz. Equation (11) can also be written in the form of the Euler-Lagrange equations (13) In order to obtain second-order trajectories by solving the above equations,
F must be expanded in terms of x, x’, y and y’ to third order. Substitution of eqns. (6)-(S) into eqn. (12) yields F = F,, + F,+
+ Fz,x2
+ Fbz y2 + g,( x’2 + y”)
+ Fsox3 + F12xy2 + g2x(x’2
-t y”)
04)
where the coefficients are F,, = 1 F,, = h, - poM
- X,N
Fzo = - +[(/A, + hopO)M+ 52
= f[/W+
Fso=
-f +
+ (1,k2)x”oN2]
(A, + h,kdN]
[(P,
+
kda4+
(3,‘2c2)(A,
gz= - ;(h,,
(iI, + 2h,X,)N
(A2
+ h,A,)X,N2
+
tkJ,w
+ (3/‘2~*)h3,N~]
+ X,N)
(15)
with e=
1+(u/mc2)
M = (eB,/~)(1/\/1-1-E) N=
- (eE,/U)(c/(l
TRAJECTORY
+ c))
(16)
EQUATION
For most cases they coordinate is less important than the x coordinate, so the trajectory equations are derived to second order for the x direction and to first order for the y direction. Substitution of eqn. (14) into eqn. (13) yields x “ = F,, + 2F2,x + 3F3,x2 + F12y* - g,(2xx” y ” = 2F,,y
+ xr2 - y”) 07)
326
In solving eqn (17), fractional mass and energy deviations are introduced, defined as m/e
= (m,/e,)(l
+ y) and U/e = (U,/e,)(l
+ S)
(18)
Equation ( 16) is expanded in powers of y and 8 to second order as c = Eg+ (Co - l)(S - y - y6 + y2)
I
1-k
iw=h,
+ (24
N=h,
0
+ l)P}
(Y + Eo6)+
1
l-
1
(3y2 + 2(2e, - 1)ys 2(1 + QO)2
- 1)y +
(e; + 1)s)
i 1
+ 41
(2(E,-
l)y2+2&-
1)2y6+E0(f;+3)6Z)
+ coJ2
(191
with ‘0 = 1 + (U,/m,c*) h, = (%B,/~G-K)(
he = -
l//-=G)
(eo~o/WW(l
+ %JN
(20)
where co is a relativistic correction, and the non-relativistic limit is obtained by putting Eo= 1. The term Jz,(h.) describes the inverse of the radius of the optical axis when the magnetic (electric) dipole field alone is applied. The simple relation
(20
ho = &Jo + k&o
holds as the zeroth-order solution of eqn. ( 17). The above equation gives the relationship between the strengths and directions of the electric and magnetic dipole fields on the optical axis such that a centrifugal force is compensated, also showing that non-dipole fields do not cause any particle deflection. In a Wien filter the relationship h,, = 0 (and A, = p. = 1) holds, and therefore h, = -h,. Substituting eqn. (19) into eqn. (15), explicit forms of the trajectory equations ( 17) are written as x” + k,x = CJ + C,S -t DXXx2 + D,,xy +D,,y’
+ D,,xb
-I- Daaa2
+ D,&y6 + Da8aV2 + Dyy y2 + D,&32
(22)
327
y”
(2%
+ k,y = 0
with (Y and #3 being the angles of inclination relative to the optical axis for the x and y directions, respectively; a = X’ and j3 = y’ in a first-order approximation. The coefficients of eqns. (22) and (23) are given by k,=h,
(l/&)htXz,
+h,(h,A,+h,)+
k, = -h,
- h,h,Xo
(24) D xx =-h,
- +ho(2h,
f h,h,)
- (3/2c;)h&(X,
+ h,A,
Vh -
w%~, +kd%-nPa)
+ h,&,>
- (hl +h,%dkx
Duv=
-
1 241
+ Q>’ 1
Drs = -
%Cl + %I2 Dss=
-
1 2(1 + E0)2
( 2 CEg{2(4
l)2h,k,
+ 3POO
+ G, (2~ -
1)&p,)
+ (2G + 1knPcJ
D,,y=h2++h,(h,+h,A,)+-&h&,(h,+h,h,) 0
q3/3= - DdlQ
(25)
with h, = 4% h2
=&X2
+ k&u, -+k#2
(26)
where h, and h, are the ion-optical parameters of quadrupole and hexapole
328
fields, respectively. It is noted here that, among the coefficients of eqn, (25) which are related to y and 6, the following sums hold CY+ C,=h, Dx-, + Ox6 = 2k, - h, I&
(27)
-I- Dys -I- D66 = -ho
SECOND-ORDER
TRAJECTORY
The first-order solutions of eqns. (22) and (23) are easily obtained. The coordinates (x, y) and the angles ( LY,j?) of inclination, at z = z, are expressed to first order as X = cx, + Sa, + (l/k,)(l CX= -k,Sx,
t CLX,+ s&y
- C)(CJ
+ C&S)
+ c,s)
Y = CyYo + S,Po P = -k,$JYo
cw
+ cpo
where (x,,, yO) and ( aO, &,) are the initial values at z = 0 of the coordinates and the angles of inclination relative to the optical axis. S, C, S,, and Cu are defined in Table 2. It should be noted that the definitions of S and Sv are different from those usually used, such as S = sin(Jk,z) and Sv = sin(gz).
By using the definitions of S, C, Sv and C,, in Table 2, eqn. (28)
becomes valid for positive, negative and near-zero values of k, and k,. Substituting the first-order soIutions of eqn. (28) into the right-hand side
TABLE
2
Definitions of S, C, S,, and C, a kx7 k,
Positive
S
(l/K)
c
-o/k,z)
cosh(,/?+)
SY
U/fi)W~z)
(l/,/q)
cy
-(/qz)
Negative sin(&)
(l/d-)
cosh( \i-ky
Near-zero sirWJTz)
z(l - ikxz2+&k,2z4) l-
+k,z2+&kzz4
sinh(dqr)
z(l-
g)
1 --‘k z2 +‘kzz4 24 Y ZY
ik,z2+
+iik:z4)
a S, C, SY and C, are analytical functions of k, and k,. k,S2 + C2 = 1 and k,,S; + C; = 1. LapIace transformations of each of S, C, S,, and CY have the same form for positive, negative and near-zero values of k, and k,. C and C, are dimensionless, but S and SY have a dimension of [L].
329
of eqn. (22)and then using the Laplace transformation r?(x)=/e=+)
defined as
dz
the second-order x coordinate is obtained. The procedure is the same as in ref. 18 and accordingly the same definition and notation are used here. Of special note is that the Laplace transformations of each of S, C, S’ and Cv have the same form, while each of these symbols is defined in different forms for positive, negative and near-zero values of k, and k, as shown in Table 2. Therefore, the second-order solution derived in this section is valid for any values of k, and k,. The final form of the second-order x coordinate may be expressed as X = cx, + Sa, + (l/k,)(l +Lx,2
+ %W%J
- c)( cyy + c,s) + K+Wf
+ &%S
+ H&4
+ QQY
+H,GaOS + Wy,y2 + H,,yS + HssS2 + Nyyuo’ + H,BY,P, + E-r,,&?
(29)
where the second-order coefficients rr,, = D__g(x2/xZ)
+ D,,g(aZ/x2)
H,, = &,g(
+ D,,g(
K,
x’/xa)
= %,g(x2/4
a2/xa)
are
+ h,S
+o,,s(~‘/~Y)
+%ygW4
w,, = D,,g(x2/xa)
+ D,,g(a2/4
+&gWx)
4,
+ D,,g(a2/a2)
= 4,g(x2/a2)
*aY = D,,g(x2/ary)+D,,g(a2/ay)+D,,g(x/a)
H,,=D,,g(x2/a6)+D,,gIu2/as)+D,b9(X/a) H,,=D,,g(~*/Y*)+D,,g(a*/Y*)+D,,g(~/Y)+D,*g(1/1) H,,=D,,g(x2/Y~)+D,,g(a2/Y6)+D,,g(x/G)+D,sg(x/Y)+Dygg(1/1) w,,=D,,g(~*/8~)+D,,g(~*/~~)+D,,g(x/~)+D,,g(l/l) H,,=D,,g(yz/Y2)+~~~g(P2/Y2) ~yle=Dyyg(~z/~~)+DlB8g(P2/~~) H,,=D,,g(Y2/~2)+~~~g(~2/82) The terms g( i/j)
and g( ij/kl), which are defined in the Appendix
(30) to ref. 18,
k,S;l
+z4(1 - $k,z2 t +,kfz4)
(C,/k;)[
C + k,S2)] (C,/k;)[rk,S-{(l-Ctk,S2)]
(1/3k;)[2(1
gtx2/a2)
&y!z6(1 - &k,z2 t &k;z4) &c~c8z6(1 - &k,z* + &kfz’) &C,Zz6(1- &k,z* t &k,2z4) t2k,)z2 t &(kf+2k,k, iz2[1 +(k,
(C;/3k;)[4(1-
KY ‘/s’>
(2,‘(k,
gtu 2/Y/R - S)
-4k,NS; -(2/kxXlC)-tk,S] a
- zC) a
(l/k;)[2(1-
(l/(k,
(Vk,)(S
-4k,W,C,
-4k,))Kl-(2k,/k,))(l(1/4k,)[2( 1 - C)t zk,S] ’
(l/(k,
C)l
C)-
(2C,C,/3k;)[4(1- C)-3zk,S + k,S*] (C,2/3k;)[4(1- C)-3zk,S t k,S*]
g(x2/S2) dY2/Y2)
dX2/Y~>
gtx2/v2)
t8k;)z4]
--
.
.-
(l/(k,-4k,))[S;-z2(l+kxz2t&,kfz4)lb
+z’[l-&(k,t4k,))z2+&(k,2
(2/(k,-4k,))[S,C,-~(l-~k,z~+$&z~)]~
-
.-
--
t4k,k,t16k;)z41
+z3[l-~(k,t4ky)z2t&(k~t4k,ky+16k;)z41
(l/(k,-4k,))[(k,-2k,)(z2/2)(l-$k,z2+~k~z4)-kyS~l’
- +k,z* t &k;t4)
.(Cs/3k;)(S-3zC+2SC),
l?tx2/4 C)-3zk,S
1- fk,z* t &k;z4) +$yz5(
i3(X2bY
&z5(1
t &k;z4)
- &k,z2 + &k;z4)
(C,./3k,2)(S-3zCt2SC)
k,S’]
+c,z4(1
t &k;z4)
ak,z* + &,k;z4) $k,z*
1
-C)-
zk,S - $( 1 -
C)
t k,S*]
+C,z4(1 - *k,z*
(2/3k,)S(l-
;z*(l jz'(l
-
dx2/x4 ldX2/XY > s(x2/x~)
C + k,S*)
(1/3kJl-
&*/x2)
- &k,z2 t $,k;z4)
&Ax/Q
&z4(l
C - fzk,S)
- C - fzk,S)
(C,/k;)(l
dX/Y
(C.Jk;)(l-
iz3(1 - hk,z* t &,k,2z4) $C,z4(1 - &k,z* t &k;z4)
(1/2k,)(S - zC)
gtx/a)
1
iz2(1- {k,z2 + &k;z4)
+ &,k,2z4)
fz*(l - +k,z2
tl/k,Ml- 0
jzs
l/l)
g(x/x)
gt
Expression for small k, and k, limits
Normal expression
Explicit expressions of g-functions
TABLE 3
= C,‘g(x2/a2)
= -2k&g(x2/a2)
g(a2/x8)
g(P2/v2) = ~:g(u2/~2) g(P2M) = - ~~g(Yz/yp) g(P2/B2) = g(u /v2)
’ In the case of k, = 4k,. b In the case of k, - 0, but not k, - 0. For the case of k, - 0, but not k, - 0, the normal expressions can be used without problem by substituting the expanded forms of Table 1 into 5” and Cy.
g(a*/a*)= g(x2/x2) g(02/ay) = Cyg(x2/4 g(a’/aQ = Csg(x2/xu)
g(a2/S2)
s@*/Y4= 2C&g(x2/a2)
= -2k,C,g(x*/c?)
g@/xy)
!e2/Y2)=C,‘g~x2/cu2)
= kig( X2/4x2) k,g(x2/xa)
g(a2/xcw) = -
g( a’/x’)
332
are functions of z and are given explicitly in Table 3. However, Dij is not a function of z, as explicitly expressed in eqn. (25). The angle Q of inclination is @ven by 1 (y= 1 +&x
dx = xF - hOXXl dz
to second order and may be written as cu= - k,Sx,
+ Ca,
+ G,,x$
+ S( C,,y + C,s)
+ GxaxO~,-, + G~+,Y
+ Gayctoy + G,s~oG + GYYy2
+ G,,a;
+ G,,Y~ + Gad2
+ G,,x,2
+ Gyyd
(31)
+ GY~~o&+G&
where the second-order coefficients are Gxx = o,,g’(
x2/x2)
-t D,,g’(
a2,‘x2)
+ h,k,SC
GXQ = D,,g’(
x2/w)
+ D,,g’(
a2/xa)
- h,( 1 - C - 2k,S2)
G X-y= D,,g’(x2/xY)
+ QX,g’(~2/xY)
+ D,,g’(x/x)
+ bw
- 2C)Cy
GXs = D,,g’(
x2/x6)
+ D,,g’(
a2/x8)
+ D,,g’(
+ h,S(
1 - 2C) C,
G aa = o,,g’(
x2/a”)
-t D,,g’(
a2,‘cy2) - h,SC
G ay = 4,g’(x2/~Y) + (h,/k,)(l
G as = D*,g’(x2/d3) + (h,/k,)(l G yy =D_yXg’(x2/Y2) -
&Jk,)S(l
G+ = Lg’(x’/YS) +%g’Wu) GM = 4,g’(x2P2) - &‘k,)S(l
+ Qu,d(~2/~Y)
x/x)
+D,,g'W4
- C-2k,S2)C, + D,,g’(a2/cd)
+ D,,g’(x/a)
- C-2k,S2)C, +D,,g’(eY2)
-
+D,yg’WY)
+o,.g’O/~)
C,C,’
+ D,ag’(az/Y~) + D,,g’O/l) + D,,g’(a2/62) - C)C,z
G yy = o,,9’(Y2/Y2)
+qi&3g’(B2/Y2)
G yJ3= o,,8’(Y2/YP)
+ q9pS’(B2/YP)
GpB = Dyyg’(Y2/P2)
+D,s,sg’(P2/P2)
+ O,,g’(x/S) - (2hcJkJW + R,g’(x/G)
- C>C,C, + &g’(W)
333
The terms g’(i/j) and g’( q/H) denote the derivatives of g-functions with regard to z and are explicitly given in Table 4. CASES
WITH
NON-FOCUSING
ACTION
There are several ion-optical devices with zero or near-zero values of k, = 0),a multipole lens with very and/or k,,such as a hexapole lens (k,= k,, small value of k, and k,, an r- * magnetic field (k, = 0),a homogeneous magnetic field (k,= 0),a cylindrical electric field (k, = 0),and a drift space = 0). Furthermore, in fitting work for designing complex ion-optical W,=k, systems, there often arise cases in which k, or k, change continuously from positive to negative values, or vice versa, through zero. In such cases of kX - 0 and k, - 0,as already described in the previous section, the expanded forms in terms of k, and k, are used for the symbols S, C, SY and CY. In the usual procedure of the definition of these terms such as S = sin(Ez)
for
k, > 0 and S = sinh(,/qz)
for k, -C0, S is irregular at k, = 0 and the Taylor expansion cannot be applied to S. However, using the definitions given in Table 2, S and C( SY and C,) and their derivatives with regard to k,(k,,) become continuous at k,(k,,) = 0,thus making it possible to express them using the Taylor expansion of powers of k,(k,,). Another advantage of the definitions used here is that Laplace transformations of each of S, C, S,, and CY have the same form, while each symbol is defined in different forms for positive, negative and near-zero values of k, or k,_ Accordingly, the second-order solutions of eqns. (29) and (31) are applicable to the cases of kX - 0 and k, - 0, by substituting the expanded forms of S, C, Su and C” into g( i/j) and g( ij/kl)and their derivatives g’( i/j) and g’( ij/kZ). These expanded forms of g- and g’- functions are also given explicitly in the third columns of Tables 3 and 4, respectively. SUMMARY
A unified theory of ion optics has been developed, which can be applied to all ion-optical devices (a)-(h) used widely in particle spectrometers and beam transport systems. The trajectory calculations have been performed to second order for the radial x direction and to first’ order for the axial y direction, in unified fields consisting of crossed electric and magnetic fields with arbitrary distributions. The necessary parameters to designate respective ion-optical devices and their arbitrary superpositions are summarized in Table 1, which also shows their relationships with quantities conveniently used in the respective devices. For slow particles (2, e c), putting co = 1 yields nonrelativistic
results.
g’tr2/b2)
g’(v‘/ra>
hk,z’
. -
-.
(2/(k, -4kJXSyCy - S) (l/k,)@ - zC) a
(Mk, -4k,Mk, -2k,P i(3S + ZC) B (2,/(/c,-4k,))(l-C-2kyS_) ZS”
-
-
-2k,S,C’l
;z3[1--+,(k,
z’[l-+(k,
t4k,)z2
t&k;
+4k,k,
t 16k;)z4J
t4k,k,+16k;)z4]
t&(k,2 t2k,k,+8k~)z4J t4k,)z2+&(k;
z[l -+(k, t2k,)z2
+ &k,2z4)
f +k,2t4)
&C~Z5(l - fk,z2
(2C&/3k;)(S t2SC -3zC) (C;/3k,z)(St2SC-3zC)
- fk,z2
t #z4)
&C,zz’(l - fk,z2
(C;/3kf)(S+2SC-3iC)
&C;z5(1
t &k;z4) - +k,z’
:C,z4(l
fC,z’(l- $k,z2 +$k;z4) +z3(1- +k,z2 t &k;z4) aCyz4(1- ik,z”+ &k;z4)
+ &k,2z4)
i- &,k,Zz4)
+C,z’(l -
z2(1-
(2/3k,)S(lC> (C,/3k,2)(2(1- C)+3zk,S -4k,S2] (C,/3k;)[2(1C)+3zk,S -4k,S2]
(C,/2k,)(S i ZC) fS(l t 2C) (2/3k,)(C - 1t 2k,S2) (C,/3k,)(St3zC-4SC) (C,/3k,)(S +3tC -4X)
Pq2US-
jZS
zc>
z(1 -ik,Z2 t &jky) z(1 -+k,z2 t &k,2z4) fz2(1 - &4,22 t &kZz4) gyzyl - hk,z2 t &k,2z4) &z’(l - +,k,z’ -I-&,k,2z4) z(1 - +k,z* f $$k;z4)
s :(s + ZC)
Expression for small k, and k, limits
Normal expression
Explicit expressions of g’-functions
TABLE 4
S’(P2/Y2) = k,2s’(Y2/~2~
= g’(x2/x2)
= C,g’(x’/xa)
= C,g’(x’/xa)
g’(a2/az)
g’(a’/ay)
g’(a*/as)
’ In the case of k, = 4k,. In the case of only k, - 0 or k, - 0 in g’( y2/y *), g’( y */yfi) and g’( y ‘//3’), the normal expressions can be used without problem by substituting the expanded forms of Table 2 into S, C, S,, and CY.
sU2/B2)= g’(Y2/Y2)
8’(P2/YS) = - k,d(Y2/YB)
= C,‘g’(x2/a2)
g’(a2/S2)
fb2/XY) = -2k,C,g’(x2/a2) g’( a2/xq = -2kxC~g'(x2/a2)
= C,‘g’(x2/a2) = 2C,c~g’(x2/a2)
g’(a’/y’) g’(a2/y8)
g’(a2/x2)= k,2g’(x2/a2) g’( a2/xa)= - k,g’( x ‘/xa)
336
The unified transfer matrix is easily constructed using the results herein. The coefficients in eqns. (29) and (31) give the corresponding elements of the first and second rows of the transfer matrix. The other non-vanishing elements are calculated from these first- and second-row elements by forming appropriate products. REFERENCES 1 H.A. Enge, Nucl. Instrum. Methods, 186 (198 1) 413. 2 K.L. Brown, R. Belbeoch and P. Bounin, Rev. Sci. Instrum., 35 (1964) 481; K.L. Brown and SK. Howrg, SLAC-91 (1970). 3 H. Wollnik, Nucl. Instrum. Methods, 52 (1967) 250. 4 T. Matsuo, H. Matsuda, Y. Fujita and H. Wollnik, Mass Spectrosc. (Jpn), 24 (1976) 19. 5 H. Hintenberger and L.A. Kiinig, 2. Naturforsch. Teil A, 12 (1957) 773. 6 T. Matsuo, H. Matsuda and H. Wollnik, Nucl. Instrum. Methods, 103 (1972) 5 15. 7 R. Ludwig, Z. Naturforsch. Teil A, 22 (1967) 553. 8 T. Matsuo and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 6 (1971) 361. 9 Y. Fujita and H. Matsuda, Nucl. Instrum. Methods, 123 (1975) 495. 10 S. Taya and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 9 (1972) 235. 1 I T. Matsuo, H. Matsuda, H. Nakabushi, Y. Fujita and A.J.H. Boerboom, Int. J. Mass Spectrom. Ion Phys., 42 (1982) 217. 12 G.E. Lee-Whiting, Nucl. Instrum. Methods, 83 (1970) 232. 13 K.L. Brown, SLAC-75 (1967). 14 H. Nakabushi and T. Matsuo, Nucl. Instrum. Methods, 198 (1982) 207. 15 S.Ya. Yavor, A.D. Dymnikov and L.P. Ovsyannikova, Nucl. Instrum. Methods, 26 (1964) 13. 16 D. Ioanoviciu, Int. J. Mass Spectrom. Ion Phys., 11 (1973) 169. 17 D. Ioanoviciu. Int. J. Mass Spectrom. Ion Phys., 15 (1974) 89. 18 H. Nakabushi, T. Sakurai and H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 49 (1983) 89. I9 W. Glaser, in S. Fltigge (Ed.), Handbuch der Physik, 33, Berlin, 1956, p. 123.