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Calculation of the parameters for the Schwarz-Christoffel transformation of ion slit lens systems
International Journal of Mass Spectrometry
1
and ion Processes, 61 (1984)
Elsevier Science Publishers B.V., Amsterdam
l-13 Printed in The Netherlands -
CALCULATION OF THE PARAMETERS FOR THE SCHWARZ-CHRISTOFFEL TRANSFORMATION OF ION SLIT LENS SYSTEMS
A.J.H.
BOERBOOM
and CHEN-HE-NENG
*
FOM - Institute for Atomic and Molecular Physics, 1098 SJ Amsterdam (The Netherlands)
(Received 24 January 1984)
ABSTRACT Simple series expansions and rapidly converging iteration methods are given for the calculation of the transformation parameters for conformal mapping of configurations of many parallel slitted electrodes used as slit lens systems, accelerators, or decelerators in ion optics. An example is given of an electrode system consisting of twelve electrodes with constant inter-electrode distances and slit widths equal to this distance. The precision obtained with series expansions amounts to +2%. The iterative method used allows a high degree of precision in the calculations. The case in which the slit width is greater than the inter-electrode separation is also treated.
INTRODUCTION
To compute the potential distribution in an electrode assembly, Laplace’s equation AV= 0 is to be solved with the boundary conditions set by the potentials at the electrodes and the field strength at infinity. If all electrodes are of infinite length in a certain direction, the problem can be considered to be two-dimensional in the plane perpendicular to tliat direction. If the cross-section of the electrodes with this plane has the shape of a (possibly degenerate) polygon, the Schwarz-Christoffel method of conformal mapping provides a fundamental solution of the problem. The method is well described in ref. 1. Application of the theory to practical cases has been very limited up to now, however, because the determination of the required parameters was a serious impediment.
* Present address: Republic of China. 0168-1176/84/$03.00
Factory
of Scientific
Instruments,
P-0.
Box 2724,
Q 1984 Elsevier Science Publishers B-V.
Beijing,
People’s
2 (~)
(-iEGG)
-----*---,
,..-__
:
,
__ .__ 4-
:
,a3_,______.’ ___.# 2.52 ._ - c..- _ _ __.
‘--___
_-__,
‘___.-r_
1
\ .-. _ --‘_ :
‘- --*---
_*_._.‘.&
ml
:
i ___.,2s,s)m-1____. ____i ----_ . __.._______
.----
I
r91__-.-___--
m
\
\.__-__ L___.. -- _ _ .---., I
2s N-1
+3-_------; L-_--__-_,,
I’\ %-2-.____ v+l ~._'__.)2*~,~ul__r_-____~~ ,___
_--
)__.
__-t__-_._ \
i
\
,
Av -0
\
\
\
a+ 27
-.
3% -0 +by2-
/
/
/
._
__
--
a/ -+a$ 3U
,’
‘.
AVv-0
I
2%
=o
Y’
Fig. 1. Conformal mapping of a slit system in the z plane on to the upper half of the w plane.
The case of ion slit lens systems has been discussed by Boerboom [2]. In a series of papers, he treats a configuration of an arbitrary number, N, of parallel electrodes with co-linear slits (Fig. I), The slits are assumed to be of infinite length perpendicular to the plane z = x + iv, and the electrodes are assumed to extend infinitely and to be of negligible thickness. Laplace’s equation is solved by first performing a conformal mapping of the positive half-plane w = u + iu, u 3 0 onto the area bounded by the electrode configuration in the plane z = x + iy. Laplace’s equation is invariant through this transformation and it is solved with ease in the w-plane. The transformation is produced by the inverse of the function z=
+“-~;)(~‘-~;)
. . . (S2-afN-3)
(SZ-u:d&S
(1)
WO
has been shown in [2(a)]. This expression contains 2N - 1 parameters ai, which are determined by the N slit widths and the N - 1 electrode distances. In accordance with [2(a)], the slit widths are denoted by 2s,, where n = 1,. . . N, and the electrode distances by or,.,, where n = 1,. . . N - 1. In [2(c)] the equations for the transformation parameters a, are given *. as
* In ref. 2(c), Eq. (l), a typographic error appears, which has, however, no consequences the further calculations. In the above equations this error has been corrected.
in
3
sn
=
a2 a2
+
+r,ln
a2n-1
+azn
G-1
-
-I-
5ln
+ _
QZn-1
2 2 ala3--sa2n--3 a2ff4.. 2 2
a2n--1 a2n_,
+
Q4 +
___
r,lna,
+ _
a2n--l a2n_-1
+rN_,lna2n-l
a2n
QZn-1
2 a2n--1
2 a2~+l~~~a2N-l
.a2n-2 2
a2nv 2
a2n-2 +
. . - +r,_Jy.+
+
a2N-2
-
a2N-2
+ _
a2n-1 LE2n_-1
2
(2)
- -a2N-2 2
where n = l,...N. r, =
where n = 1 , . . . N - 1. These 2N - 1 equations are to be solved for the unknown parameters a, (1 < i < 2N - 1). The physical meaning of the problem implies that all ui are real and that a, > CZ~+~ > 0 (1 < i -C2iV - 1). In [2@)] and [2(c)] these equations tie solved by series expansions, viz.
(4) with Vzn-1
= s, &
n
and v2n=
S -
I?+1
4%
where l<
n G N-l
(5)
It was shown that many coefficients are equal, e.g. A,;i = - 4/3 and PI,,;~~ = 86/45 for 0 -Ei -c n - 2, and that the coefficients can be relatively large, e.g. 52/9, 65/9. Obviously, the latter peculiarity badly influences convergence, but the appearance of equal coefficients suggests that simpler series can be constructed. It is the purpose of this paper to derive comparable expressions in higher approximations. Moreover, a powerful quickly converging iteration method is constructed which is efficient also for cases where the former methods are not applicable. Firstly, we will treat the much simpler case of configurations of equidistant electrodes with constant slit widths. ELECTRODE WIDTHS
CONFIGURATIONS
WITH
CONSTANT
DISTANCES
AND
SLIT
Often, slit systems occur with equidistant electrodes and with constant slit widths. Obviously, all ratios Ui (1 Q i < 2N - 2) are equal and the series become very simple. A derivation of the relevant expressions for ,this special
4 case
more insight
gives
solutions_ For an electrode
into
the structure
of the series and leads to better
far from the first and from the last electrodes
(1 -C n -=x
N ), we have r, =
=
(
-aZ,)(a:-aZ,)...(a~,-,-af,)
(f
-a:,)(a:-a:,)...(aI,_,-uf,)
(&-,--a:,) 2%
2 2 ( u2u4. 2 2a2,u~,a~,. - ‘a2n-_2
. . . (I-4&L-J
x
. . . (I-
a,‘n/aL-21
-. . I-
&_,/a&)
x
2 u,a,
2
22 a2a4
rn
~~-&+~/4J~~
..
(1 -
a:,-,/a5,)
a;,-,/a,‘,)
-.
(1 + CP.,“)
-“2n-2 2 2 ...a2n-3a2n--lu2n+l
2
2 2 2 u2a4...u2n-2u2n
Now, r, = r,+I rn-cl -=
4,/~L)
- &/a:).
( 1 -a~,+2/akJ--~
. . .a
2 2 ala3 Yn+1=
-
(
. . *( 1 -
rn=
(1 -
uQu,‘)(l
. .u~,u~,)(l
2 a,,--,u2,+,
i
l+P
n+l,iU2i 1
and, as far as end effects can be neglected, 2
u22,2a2,+2
x
2 u2?2+1
2a2n p=
=
Pni =
Pn+l,i, so
1
a2nu2n+2
a;,-,
and, finally, we find u~,,+~/u~~+~ = a 2n+l/u2n (approximately,
for 1 -+z n +z
N)w2n
Defining ~2,+2/a2.+, = w~,..+~ and a2n+l/a2n = w,,, we may put w~,,+~ = = w when end effects may be neglected. For Eq. (2) we may write
s, =S= With s
. . . +r,_,ln
1+
1+w 1+w + r,_,ln- 1 _ w + G*n- l_w+r,+I
w3
l--w3
ri = r and expanding
the logarithms
-~~vw++w3+~w5+~w7+~~9+~~~l+~w~3+~~~5+ 4r
* In this series all odd powers of w
In 1+ w3 1_w3+
..*
in power series, we find ___
*
(6)
appear. The coefficients are fractions: the denominator
equals the power, the numerator equals the sum of all divisors of the denominator, 1 and the denominator itself, e.g. 13 = 1 + 3 + 9, 24 = 1-C 3 + 5 + 15.
including
5
The convergence of this series can be improved (1 -w*)-‘, which yields .U=
by isolating
w(
1 - w*
1 +~w’-~w”-~w~+~~~-~w~~-~w~~+~~~~-
a factor . . . (7)
Inversion of Eq. (6) gives a badly converging series in u*, viz. kV=LJ(l - $2 +
34
-
16.7870~+ 77.487~’ - . , ,
(8)
and splitting off a factor (1 - u2) or (1 + u*)-* does not improve the convergence very much. Much better is to consider Eq. (7) as a quadratic equation in w, viz. vw2+(1+c)w-u=o where (1 + C) stands for the series expansion in Eq. (7). This equation can readily be solved to give w=
- (1+ c) + J(1 f c)’
+ 4u2
2v
(9)
(Only the positive root applies, as both w and v should be positive.) Iteration of this equation, with starting value wl= v2, leads to a rapid solution. In the example, it appears that this solution is a very good approximation even when only a limited number of terms is used iti Eq. (7). Equation (9) makes clear why thk series expansions in Eq. (4) as well as those appearing in the next section converge badly: they all run parallel to series expansion of the square root in Eq. (9), which is only valid if 4v2 is smaller than about 1. Rapid convergence only occurs if v* +z 0.25. In the limiting case mentioned intuitively in [2(a)] and [2(c)] (slit widths equal to the interelectrode distances), v = 0.3927 and v* = 0.1542. Equation (9) also applies for slit systems where the slit widths are considerably larger than the electrode distances (u z+ 1, w ( 1). DERIVATION
OF THE SERIES EXPANSIONS
In Eq. (4) of the present paper, as well as in the tables in refs. 2(a) and (c), series expansions are given which relate the unknown parameter a, to the first slit width (2~~). The repeated occurrence of coefficients of equal magnitude in these tables, however, indicates that an improvement will be obtained if the parameter di, is related to the immediately foregoing one, a,_,. In Eq. (18) in ref. 2(c), this has already been shown in first-order series expansions. Still better expressions are obtained by substituting the series expansion for the natural logarithm into Eq. (2) and writing both Eqs. (12)
6
and (13) [2( c>Iin terms of ui. We obtain up to the second order inclusive. a2n
XL
=A2”
e
s "~~2,+1=~2n+1=-----2P,+1
with a:'a:,--* &-3
n=l,...hJ-1
a2n-2
a,a,...aL-2 A
2 2 a2a4...a 2n+1=
2 2n-3a2n-l
Q, = 1 - u;,,_r - vi, + f~;,_~v;,_r + 4v2 + 102 Uz 3
3
2n
2n
2n+l
-
+ +J;,,_, + ~v;,,_~u;~
third order
Pnt1 = 1 - ;z?,“, - $v;n+l + v;,_rf& +
$$v&+~
+
1
n =O,...N-
a,a,...a 2 2 2
v~,+~v~,+~
+ gv;,
-third
+ $v;nv;n+l
order
These series show a much better behaviour: there are only two terms of order v2 and 5 of order v4, whereas in Eq. (4) there are n, and about *n2 terms, respectively; moreover, the coefficients are somewhat smaller in absolute value. Again, a reduction of the magnitude of the coefficients and an improved convergence appear when we introduce “adapted” slit widths and electrode distances, defined by S, =
Sn{l
+
3Uf,_2
4 4 -TTv2n--1
-5 $V,",_,
-V~,_lV~n+
+3&,_3&2
-U~n_3U~,_2-
ji~~,_4~~,_3~~,_2+
+&_3&_2&-l+
+i15v;,_2u;n_l
+
&U24,-2-
g&,-l
+V~,_2V~,_1
$V~,_3V~,_2
&';,,_2
+
+u;,_*u;n__lu;n
+
+';,_2&,_,
3v24,_1& + su;/J;,
+ Iv2 3 2n_-1v~nv~n+l- _. . + fourth.order) R, = r,(l
-14
+ u;,_~ + v;, - f~;~-~v;,_~
3v2n
-+Li;"u;n+l
12 -Vv2n-2v;,_1
- $J;,_~ - ~v;,,_~v&
u;,_&_2v~"_~
+&J;,_2u;,_1
+u~,-~~~~_~v~~+~~~~_~
+6~,2,_~&+
&_l&~&,+~ 2
+&%l+lV2n+2-
+
a*. +
+&&,
00)
+ -+&&,+1
fourth order}
6&+& +&&&n+~
00)
8
is taken and brought to the left-hand side
remains. The same procedure a2n+a2n--1
side of the equation.
is followed
At the right-hand
with the term
.
r,h a2n-a2n--1
We rewrite, however, 2r,a,, with the aid of the equation for r,,, to obtain a factor a,,_,(1 - a&Ja’,,_ ]). If desired, the first term cz2n_-1and the last one in Eq. (2), both containing a factor a2n_-1, can be brought to the left-hand side. S, is brought to the right-hand side to join the remaining logarithmic terms. In this way we obtain
+I+2
2 2 2 a1 '~~a2n-3a2n+l'~.a2N-1
2
2 2 a2n-2a2n.'.
2 a2N-2
a2...
a
-
In 2n ’
+a2n i
a2n-I-a2n a2n--1+a2??+2
_
2a2n-1
=sn+rn_,
-In
a2n--2+azn-1
a2*-2 I
a2n--2-a,,-,
{
- f-n_2 In
a2n4+a2*-1 a2n--4
_
...
-a2n-1
__.
-rn+dn a2n-1
-a2n+2
In the first and last of these equations (n = 1 and n = N, respectively) one logarithmic term should be brought td the left-hand side. The equation for r,, is transformed into
a2n=
(af-a&,)-.- (a:n-l-afn)
(a:~-ah+l)bbs (ak?-agN-l)
2 ( a2-a2n
(a:,-&+,)...
2
)_..
CW
only
a&
In both sides of the former and in the right-hand side of the latter equation as many terms and factors are included as to obtain the desired accuracy. All terms and factors can be included to obtain the exact equations. These equations can be used for the iterative solution. They show, however, only a slow convergence_ It is well known from the theory of iteration that the iterative equation = 0 at the point where x = f(x). If this is x R-t1 = f( xn) converges best if f’(x) not the case, we may add a term - x f; to both sides of this equation [where fd = f’(x) is taken near the solution of the equation]. In this way we obtain
9
the alternative iterative equation x = [f(x) - xfd]/[l - f,‘]. In our case, we have a good approximation for x,,, so by adding the derivative term we can accelerate the convergence considerably. Depending on the number of electrodes and the ratios between slit widths and neighbouring electrode distances, the convergence can amount to one or several digits per iteration. Also, in the derivative as many terms are included as desired. Here, however, the ultimate accuracy is not influenced by neglect of minor terms. If, during the iteration process, a large change in ai occurs somewhere, it may be advisable to vary the subsequent ui (i < j < 2N - I) proportionally. play such an important role in the This is because the ratios a/a,+, equations that large deviations of (a’ - a,?+l) in m or ( ai - a,+l) in s,,, or even these factors becoming negative, should be avoided. The iteration method produces rapid convergence, even if the condition of slit widths smaller than the distance to the neighbouring electrodes (2s, -C “‘n-1; 2s, -=zark) is violated. In the latter case, a starting value for a,/a,_, between 0.5 and 1 should be taken. EXAMPLE
As an example we take the post-acceleration section in the tandem mass spectrometer of Tuithof et al. [3]. It consists of twelve electrodes with 20 mm slits 20 mm apart. So the slit widths equal the electrode distances, the Iirniting case of the series expansions. We have 23 parameters a1 _. . az3, N = 12, s, = 10, r, = 6.3662 and v, = 0.392699. The purpose of the device is the simultaneous acceleration of a beam of fragment ions of considerable varying masses and energies (up to a ratio of 1: 12 for both quantities) without a lens action in the median plane and with limitation of the divergence for all ions in the perpendicular plane. Table 1 shows the results of various approximation methods. The first column gives the computer-calculated values obtained by applying a Fletcher-Powell program. The parameters are evenly distributed over more than ten orders of magnitude. And, indeed, LZ,/LZ~+~ is constant: within 1: lo3 between a3 and azl, so in the region between the second and last electrode but one, and within 1 : lo6 for 6 G i < 17 (more than three electrodes from either end). The second column gives the zeroth approximation s
a, =-,aitl
2
=
s
j.9
(4yz>
Though a, is reasonably well approximated, the values for ai (i > 1) gradually deviate, eventually rising to an error of a factor of more than 30 for az3. Obviously, the continued application of a formula with a constant error
-2
-3 -3 -4 -4
2.143 513
7.206 975 2.423 149 8,147 182 2.739269
9.210046 3.096 628
6 7 8
9 10
11 12
-5 -5
-1 -2
-1
1.896236 6.375 305
4 5
3
5.168056 1.683723 5.641962
43.608 17.125
4.669 18.337 7.201 28.278 11.105 7.909 2.618
1.991 6.590 2.181 7.219 2.390
1.817 6.015
3.028
5.257 1.659 5.491
First approx.
3”
11.891
5.000 1.963 7.711
Zeroth approx.
Computer value
Exp.
2’
lb
1 2
(1, n
0”
TABLE 1
9.420 3.175
2.165 7.297 2.460 8.291 2.795
6.422
1.905
5.106 1.688 5.652
Second approx.
4c
9.344 3.146
2.161 7.275 2.449 8.245 2.776
6.419
1.907
5.215 1.686 5.670
d
d
f
d I
d r
d
I
1
d
d
6.375 5
1.8962
5.168 9 1.6843 5.6409
Es. (9) corr.
7d
I
f
f
1.896
4.989 1.677 5.639
Es. (9)
Third approx.
6
5”
6.886 2.254
1.829 5.989 1.961 6.420 2.103
5.585
1.706
5.093 1.597 5,196
iter.
First
ge
1.041157 3.500608 1.176984 3.957290 1.330531 4.473 547 1.504101 5.056915 1.699605 5.695 180 1.855457
-5 -6 -6 -7 -7 -8 -8 -9 -9 -10 -10 - 3.3% 31.6 x
6.725 26.409 10.371 40.726 15.993 62.804 24.663 96.852 38.034 149.358 58.653 +1.7% -30 %
8.664 2.868 9.492 3.142 1.040 3.442 1.139 3.771 1.248 4.131 1.304 -1.2% + 6.4%
3.608 1.216 4.099 1.382 4.657 1.570 5.291 1.784 5.973 1.974
1,070
-I-0.9% + 2.2%
1.059 3.565 1.200 4.041 1.360 4.580 1.542 5.190 1.745 5.869 1.897
’ Parameter number. b Computer value with lo-exponent for colums l-8. ’ .Using Eqs. (10) and (11). d Eq. (9) with correction for end effects; u6-u,s, identical with column 1. ’ First iteration, starting from column 2. After 5 iterations a precision of 1: lo5 is obtained. ’ See column 1.
‘23
exact
Accuracy: 01 exact
13 14 15 16 17 18 19 20 21 22 i3
+ 3.6%
+ 3.6%
5.057 1.700 5.717 1.922 0.0% 0.0%
1.5044 5.0570 1.7003 5.695 7 1.8559
d
d
d
d
d
d
- 1.5% -42%
0.738 2.417 0.791 2.592 0.849 2.782 0.915 3.005 1.014 3.350 1.075
12
leads to large deviations if the number of electrodes increases. But, already, the first approximation, which includes only the u* terms in S, and R, [Eq. (5)], gives 6 fair set of values with ati accuracy ranging from + 1.7% for a, to - 30% for a23. The second and third approximations (including the v4 and u6 terms, respectively) give better values, but demand expressions of increasing complexity and show only a modest improvement. Column 6 gives the results when applying Eq. (9). This small formula with its extremely simple derivation gives astonishingly good results. In the central range (6 c i -=E17), the ratio a,+Ja, agrees better than 1: lo9 (computer precision) with the exact ratios. To obtain the results, terms up to NJ*’ in Eq. (7) have been taken into account. For 3 5 i -=z20 the precision is better than O.l%, taking only the w3 terms in C(w). Equation (9) only gives ratios. One of the r equations in (3), however, suffices to give absolute values. Equation (9) is derived by neglecting end effects and therefore at both ends deviations appear. A correction for these effects can be made by taking deviating ratios of w at the begining and end. This has been done by introducing wl, w2, w3 and wzo, wzl, wz2 only. In column 7 the almost exact results are shown. Unfortunately, Eq. (9) is only valid for slit systems with equal slit widths and equidistant electrodes. Column 8 gives the results of the iteration method. For the starting values, the zeroth approximation (column 2) was taken: moderately good values of a, for small n but strongly deviating values for large PZ.After only one iteration (column 8) a dramatic increase in precision occurs: a23 was previously wrong by a factor 31.6, but appears now to be only 42% too small. After five iterations a precision of better than 1 : lo5 for all a, is obtained. This example shows the power of the various methods. With a small and not necessarily fast computer the complicated system of transcendent equations can be solved with a very high precision and in this way a simple method of finding the exact potential distribution in complicated electrode systems is obtained. CONCLUSION
The parameters governing the Schwarz-Christoffel transformation of an electrode configuration of the shape described in the introduction, can be calculated with ease. In slit systems with equidistant electrodes and constant slit widths Eq. (9) gives an excellent value for the ratios of the parameters. End effects can easily be taken into account. In configurations with electrode distances larger than the slit widths in the
13
adjacent electrodes, the series expansions (10) give a solution (11) with an ac?uracy of 1% or better. For the general case, the iteration method provides a solution with any degree of precision. As starting values one may take a, = s1/2, a,, 1 = viai for vi2< 1, with vi defined in (5). For vi >, 0.5, the ratio a,+Ja, approaches 1; a reasonable value is ui/(l + u,.). In the intermediate region, ui = 0.4, the average of both proposed values gives an excellent starting value. The iteration method makes an easy calculation of the Schwarz-Christoffel parameters possible for any slit system (also non-symmetric ones). As far as the authors are aware, a general solution of the problem has never been published. ACKNOWLEDGEMENTS
One of the authors (C.H.) is indebted to the Academica Sinica for a leave of absence. The authors wish to express their thanks to Dr. W.E. v.d. Kaay for performing the computer calculations. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research). REFERENCES 1 P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I, McGraw-Hill,
New York, 1953, p. 445. 2 A.J.H. Boerboom, Z. Naturforsch. Teil A, (a) 14 (1959) 809; (b) 15 (1960) 244; (c) 15 (1960) 253. 3 F-W. McLafferty (Ed.), Tandem Mass Spectrometry, Wiley, New York, 1983, Chap. 11.
1
and ion Processes, 61 (1984)
Elsevier Science Publishers B.V., Amsterdam
l-13 Printed in The Netherlands -
CALCULATION OF THE PARAMETERS FOR THE SCHWARZ-CHRISTOFFEL TRANSFORMATION OF ION SLIT LENS SYSTEMS
A.J.H.
BOERBOOM
and CHEN-HE-NENG
*
FOM - Institute for Atomic and Molecular Physics, 1098 SJ Amsterdam (The Netherlands)
(Received 24 January 1984)
ABSTRACT Simple series expansions and rapidly converging iteration methods are given for the calculation of the transformation parameters for conformal mapping of configurations of many parallel slitted electrodes used as slit lens systems, accelerators, or decelerators in ion optics. An example is given of an electrode system consisting of twelve electrodes with constant inter-electrode distances and slit widths equal to this distance. The precision obtained with series expansions amounts to +2%. The iterative method used allows a high degree of precision in the calculations. The case in which the slit width is greater than the inter-electrode separation is also treated.
INTRODUCTION
To compute the potential distribution in an electrode assembly, Laplace’s equation AV= 0 is to be solved with the boundary conditions set by the potentials at the electrodes and the field strength at infinity. If all electrodes are of infinite length in a certain direction, the problem can be considered to be two-dimensional in the plane perpendicular to tliat direction. If the cross-section of the electrodes with this plane has the shape of a (possibly degenerate) polygon, the Schwarz-Christoffel method of conformal mapping provides a fundamental solution of the problem. The method is well described in ref. 1. Application of the theory to practical cases has been very limited up to now, however, because the determination of the required parameters was a serious impediment.
* Present address: Republic of China. 0168-1176/84/$03.00
Factory
of Scientific
Instruments,
P-0.
Box 2724,
Q 1984 Elsevier Science Publishers B-V.
Beijing,
People’s
2 (~)
(-iEGG)
-----*---,
,..-__
:
,
__ .__ 4-
:
,a3_,______.’ ___.# 2.52 ._ - c..- _ _ __.
‘--___
_-__,
‘___.-r_
1
\ .-. _ --‘_ :
‘- --*---
_*_._.‘.&
ml
:
i ___.,2s,s)m-1____. ____i ----_ . __.._______
.----
I
r91__-.-___--
m
\
\.__-__ L___.. -- _ _ .---., I
2s N-1
+3-_------; L-_--__-_,,
I’\ %-2-.____ v+l ~._'__.)2*~,~ul__r_-____~~ ,___
_--
)__.
__-t__-_._ \
i
\
,
Av -0
\
\
\
a+ 27
-.
3% -0 +by2-
/
/
/
._
__
--
a/ -+a$ 3U
,’
‘.
AVv-0
I
2%
=o
Y’
Fig. 1. Conformal mapping of a slit system in the z plane on to the upper half of the w plane.
The case of ion slit lens systems has been discussed by Boerboom [2]. In a series of papers, he treats a configuration of an arbitrary number, N, of parallel electrodes with co-linear slits (Fig. I), The slits are assumed to be of infinite length perpendicular to the plane z = x + iv, and the electrodes are assumed to extend infinitely and to be of negligible thickness. Laplace’s equation is solved by first performing a conformal mapping of the positive half-plane w = u + iu, u 3 0 onto the area bounded by the electrode configuration in the plane z = x + iy. Laplace’s equation is invariant through this transformation and it is solved with ease in the w-plane. The transformation is produced by the inverse of the function z=
+“-~;)(~‘-~;)
. . . (S2-afN-3)
(SZ-u:d&S
(1)
WO
has been shown in [2(a)]. This expression contains 2N - 1 parameters ai, which are determined by the N slit widths and the N - 1 electrode distances. In accordance with [2(a)], the slit widths are denoted by 2s,, where n = 1,. . . N, and the electrode distances by or,.,, where n = 1,. . . N - 1. In [2(c)] the equations for the transformation parameters a, are given *. as
* In ref. 2(c), Eq. (l), a typographic error appears, which has, however, no consequences the further calculations. In the above equations this error has been corrected.
in
3
sn
=
a2 a2
+
+r,ln
a2n-1
+azn
G-1
-
-I-
5ln
+ _
QZn-1
2 2 ala3--sa2n--3 a2ff4.. 2 2
a2n--1 a2n_,
+
Q4 +
___
r,lna,
+ _
a2n--l a2n_-1
+rN_,lna2n-l
a2n
QZn-1
2 a2n--1
2 a2~+l~~~a2N-l
.a2n-2 2
a2nv 2
a2n-2 +
. . - +r,_Jy.+
+
a2N-2
-
a2N-2
+ _
a2n-1 LE2n_-1
2
(2)
- -a2N-2 2
where n = l,...N. r, =
where n = 1 , . . . N - 1. These 2N - 1 equations are to be solved for the unknown parameters a, (1 < i < 2N - 1). The physical meaning of the problem implies that all ui are real and that a, > CZ~+~ > 0 (1 < i -C2iV - 1). In [2@)] and [2(c)] these equations tie solved by series expansions, viz.
(4) with Vzn-1
= s, &
n
and v2n=
S -
I?+1
4%
where l<
n G N-l
(5)
It was shown that many coefficients are equal, e.g. A,;i = - 4/3 and PI,,;~~ = 86/45 for 0 -Ei -c n - 2, and that the coefficients can be relatively large, e.g. 52/9, 65/9. Obviously, the latter peculiarity badly influences convergence, but the appearance of equal coefficients suggests that simpler series can be constructed. It is the purpose of this paper to derive comparable expressions in higher approximations. Moreover, a powerful quickly converging iteration method is constructed which is efficient also for cases where the former methods are not applicable. Firstly, we will treat the much simpler case of configurations of equidistant electrodes with constant slit widths. ELECTRODE WIDTHS
CONFIGURATIONS
WITH
CONSTANT
DISTANCES
AND
SLIT
Often, slit systems occur with equidistant electrodes and with constant slit widths. Obviously, all ratios Ui (1 Q i < 2N - 2) are equal and the series become very simple. A derivation of the relevant expressions for ,this special
4 case
more insight
gives
solutions_ For an electrode
into
the structure
of the series and leads to better
far from the first and from the last electrodes
(1 -C n -=x
N ), we have r, =
=
(
-aZ,)(a:-aZ,)...(a~,-,-af,)
(f
-a:,)(a:-a:,)...(aI,_,-uf,)
(&-,--a:,) 2%
2 2 ( u2u4. 2 2a2,u~,a~,. - ‘a2n-_2
. . . (I-4&L-J
x
. . . (I-
a,‘n/aL-21
-. . I-
&_,/a&)
x
2 u,a,
2
22 a2a4
rn
~~-&+~/4J~~
..
(1 -
a:,-,/a5,)
a;,-,/a,‘,)
-.
(1 + CP.,“)
-“2n-2 2 2 ...a2n-3a2n--lu2n+l
2
2 2 2 u2a4...u2n-2u2n
Now, r, = r,+I rn-cl -=
4,/~L)
- &/a:).
( 1 -a~,+2/akJ--~
. . .a
2 2 ala3 Yn+1=
-
(
. . *( 1 -
rn=
(1 -
uQu,‘)(l
. .u~,u~,)(l
2 a,,--,u2,+,
i
l+P
n+l,iU2i 1
and, as far as end effects can be neglected, 2
u22,2a2,+2
x
2 u2?2+1
2a2n p=
=
Pni =
Pn+l,i, so
1
a2nu2n+2
a;,-,
and, finally, we find u~,,+~/u~~+~ = a 2n+l/u2n (approximately,
for 1 -+z n +z
N)w2n
Defining ~2,+2/a2.+, = w~,..+~ and a2n+l/a2n = w,,, we may put w~,,+~ = = w when end effects may be neglected. For Eq. (2) we may write
s, =S= With s
. . . +r,_,ln
1+
1+w 1+w + r,_,ln- 1 _ w + G*n- l_w+r,+I
w3
l--w3
ri = r and expanding
the logarithms
-~~vw++w3+~w5+~w7+~~9+~~~l+~w~3+~~~5+ 4r
* In this series all odd powers of w
In 1+ w3 1_w3+
..*
in power series, we find ___
*
(6)
appear. The coefficients are fractions: the denominator
equals the power, the numerator equals the sum of all divisors of the denominator, 1 and the denominator itself, e.g. 13 = 1 + 3 + 9, 24 = 1-C 3 + 5 + 15.
including
5
The convergence of this series can be improved (1 -w*)-‘, which yields .U=
by isolating
w(
1 - w*
1 +~w’-~w”-~w~+~~~-~w~~-~w~~+~~~~-
a factor . . . (7)
Inversion of Eq. (6) gives a badly converging series in u*, viz. kV=LJ(l - $2 +
34
-
16.7870~+ 77.487~’ - . , ,
(8)
and splitting off a factor (1 - u2) or (1 + u*)-* does not improve the convergence very much. Much better is to consider Eq. (7) as a quadratic equation in w, viz. vw2+(1+c)w-u=o where (1 + C) stands for the series expansion in Eq. (7). This equation can readily be solved to give w=
- (1+ c) + J(1 f c)’
+ 4u2
2v
(9)
(Only the positive root applies, as both w and v should be positive.) Iteration of this equation, with starting value wl= v2, leads to a rapid solution. In the example, it appears that this solution is a very good approximation even when only a limited number of terms is used iti Eq. (7). Equation (9) makes clear why thk series expansions in Eq. (4) as well as those appearing in the next section converge badly: they all run parallel to series expansion of the square root in Eq. (9), which is only valid if 4v2 is smaller than about 1. Rapid convergence only occurs if v* +z 0.25. In the limiting case mentioned intuitively in [2(a)] and [2(c)] (slit widths equal to the interelectrode distances), v = 0.3927 and v* = 0.1542. Equation (9) also applies for slit systems where the slit widths are considerably larger than the electrode distances (u z+ 1, w ( 1). DERIVATION
OF THE SERIES EXPANSIONS
In Eq. (4) of the present paper, as well as in the tables in refs. 2(a) and (c), series expansions are given which relate the unknown parameter a, to the first slit width (2~~). The repeated occurrence of coefficients of equal magnitude in these tables, however, indicates that an improvement will be obtained if the parameter di, is related to the immediately foregoing one, a,_,. In Eq. (18) in ref. 2(c), this has already been shown in first-order series expansions. Still better expressions are obtained by substituting the series expansion for the natural logarithm into Eq. (2) and writing both Eqs. (12)
6
and (13) [2( c>Iin terms of ui. We obtain up to the second order inclusive. a2n
XL
=A2”
e
s "~~2,+1=~2n+1=-----2P,+1
with a:'a:,--* &-3
n=l,...hJ-1
a2n-2
a,a,...aL-2 A
2 2 a2a4...a 2n+1=
2 2n-3a2n-l
Q, = 1 - u;,,_r - vi, + f~;,_~v;,_r + 4v2 + 102 Uz 3
3
2n
2n
2n+l
-
+ +J;,,_, + ~v;,,_~u;~
third order
Pnt1 = 1 - ;z?,“, - $v;n+l + v;,_rf& +
$$v&+~
+
1
n =O,...N-
a,a,...a 2 2 2
v~,+~v~,+~
+ gv;,
-third
+ $v;nv;n+l
order
These series show a much better behaviour: there are only two terms of order v2 and 5 of order v4, whereas in Eq. (4) there are n, and about *n2 terms, respectively; moreover, the coefficients are somewhat smaller in absolute value. Again, a reduction of the magnitude of the coefficients and an improved convergence appear when we introduce “adapted” slit widths and electrode distances, defined by S, =
Sn{l
+
3Uf,_2
4 4 -TTv2n--1
-5 $V,",_,
-V~,_lV~n+
+3&,_3&2
-U~n_3U~,_2-
ji~~,_4~~,_3~~,_2+
+&_3&_2&-l+
+i15v;,_2u;n_l
+
&U24,-2-
g&,-l
+V~,_2V~,_1
$V~,_3V~,_2
&';,,_2
+
+u;,_*u;n__lu;n
+
+';,_2&,_,
3v24,_1& + su;/J;,
+ Iv2 3 2n_-1v~nv~n+l- _. . + fourth.order) R, = r,(l
-14
+ u;,_~ + v;, - f~;~-~v;,_~
3v2n
-+Li;"u;n+l
12 -Vv2n-2v;,_1
- $J;,_~ - ~v;,,_~v&
u;,_&_2v~"_~
+&J;,_2u;,_1
+u~,-~~~~_~v~~+~~~~_~
+6~,2,_~&+
&_l&~&,+~ 2
+&%l+lV2n+2-
+
a*. +
+&&,
00)
+ -+&&,+1
fourth order}
6&+& +&&&n+~
00)
8
is taken and brought to the left-hand side
remains. The same procedure a2n+a2n--1
side of the equation.
is followed
At the right-hand
with the term
.
r,h a2n-a2n--1
We rewrite, however, 2r,a,, with the aid of the equation for r,,, to obtain a factor a,,_,(1 - a&Ja’,,_ ]). If desired, the first term cz2n_-1and the last one in Eq. (2), both containing a factor a2n_-1, can be brought to the left-hand side. S, is brought to the right-hand side to join the remaining logarithmic terms. In this way we obtain
+I+2
2 2 2 a1 '~~a2n-3a2n+l'~.a2N-1
2
2 2 a2n-2a2n.'.
2 a2N-2
a2...
a
-
In 2n ’
+a2n i
a2n-I-a2n a2n--1+a2??+2
_
2a2n-1
=sn+rn_,
-In
a2n--2+azn-1
a2*-2 I
a2n--2-a,,-,
{
- f-n_2 In
a2n4+a2*-1 a2n--4
_
...
-a2n-1
__.
-rn+dn a2n-1
-a2n+2
In the first and last of these equations (n = 1 and n = N, respectively) one logarithmic term should be brought td the left-hand side. The equation for r,, is transformed into
a2n=
(af-a&,)-.- (a:n-l-afn)
(a:~-ah+l)bbs (ak?-agN-l)
2 ( a2-a2n
(a:,-&+,)...
2
)_..
CW
only
a&
In both sides of the former and in the right-hand side of the latter equation as many terms and factors are included as to obtain the desired accuracy. All terms and factors can be included to obtain the exact equations. These equations can be used for the iterative solution. They show, however, only a slow convergence_ It is well known from the theory of iteration that the iterative equation = 0 at the point where x = f(x). If this is x R-t1 = f( xn) converges best if f’(x) not the case, we may add a term - x f; to both sides of this equation [where fd = f’(x) is taken near the solution of the equation]. In this way we obtain
9
the alternative iterative equation x = [f(x) - xfd]/[l - f,‘]. In our case, we have a good approximation for x,,, so by adding the derivative term we can accelerate the convergence considerably. Depending on the number of electrodes and the ratios between slit widths and neighbouring electrode distances, the convergence can amount to one or several digits per iteration. Also, in the derivative as many terms are included as desired. Here, however, the ultimate accuracy is not influenced by neglect of minor terms. If, during the iteration process, a large change in ai occurs somewhere, it may be advisable to vary the subsequent ui (i < j < 2N - I) proportionally. play such an important role in the This is because the ratios a/a,+, equations that large deviations of (a’ - a,?+l) in m or ( ai - a,+l) in s,,, or even these factors becoming negative, should be avoided. The iteration method produces rapid convergence, even if the condition of slit widths smaller than the distance to the neighbouring electrodes (2s, -C “‘n-1; 2s, -=zark) is violated. In the latter case, a starting value for a,/a,_, between 0.5 and 1 should be taken. EXAMPLE
As an example we take the post-acceleration section in the tandem mass spectrometer of Tuithof et al. [3]. It consists of twelve electrodes with 20 mm slits 20 mm apart. So the slit widths equal the electrode distances, the Iirniting case of the series expansions. We have 23 parameters a1 _. . az3, N = 12, s, = 10, r, = 6.3662 and v, = 0.392699. The purpose of the device is the simultaneous acceleration of a beam of fragment ions of considerable varying masses and energies (up to a ratio of 1: 12 for both quantities) without a lens action in the median plane and with limitation of the divergence for all ions in the perpendicular plane. Table 1 shows the results of various approximation methods. The first column gives the computer-calculated values obtained by applying a Fletcher-Powell program. The parameters are evenly distributed over more than ten orders of magnitude. And, indeed, LZ,/LZ~+~ is constant: within 1: lo3 between a3 and azl, so in the region between the second and last electrode but one, and within 1 : lo6 for 6 G i < 17 (more than three electrodes from either end). The second column gives the zeroth approximation s
a, =-,aitl
2
=
s
j.9
(4yz>
Though a, is reasonably well approximated, the values for ai (i > 1) gradually deviate, eventually rising to an error of a factor of more than 30 for az3. Obviously, the continued application of a formula with a constant error
-2
-3 -3 -4 -4
2.143 513
7.206 975 2.423 149 8,147 182 2.739269
9.210046 3.096 628
6 7 8
9 10
11 12
-5 -5
-1 -2
-1
1.896236 6.375 305
4 5
3
5.168056 1.683723 5.641962
43.608 17.125
4.669 18.337 7.201 28.278 11.105 7.909 2.618
1.991 6.590 2.181 7.219 2.390
1.817 6.015
3.028
5.257 1.659 5.491
First approx.
3”
11.891
5.000 1.963 7.711
Zeroth approx.
Computer value
Exp.
2’
lb
1 2
(1, n
0”
TABLE 1
9.420 3.175
2.165 7.297 2.460 8.291 2.795
6.422
1.905
5.106 1.688 5.652
Second approx.
4c
9.344 3.146
2.161 7.275 2.449 8.245 2.776
6.419
1.907
5.215 1.686 5.670
d
d
f
d I
d r
d
I
1
d
d
6.375 5
1.8962
5.168 9 1.6843 5.6409
Es. (9) corr.
7d
I
f
f
1.896
4.989 1.677 5.639
Es. (9)
Third approx.
6
5”
6.886 2.254
1.829 5.989 1.961 6.420 2.103
5.585
1.706
5.093 1.597 5,196
iter.
First
ge
1.041157 3.500608 1.176984 3.957290 1.330531 4.473 547 1.504101 5.056915 1.699605 5.695 180 1.855457
-5 -6 -6 -7 -7 -8 -8 -9 -9 -10 -10 - 3.3% 31.6 x
6.725 26.409 10.371 40.726 15.993 62.804 24.663 96.852 38.034 149.358 58.653 +1.7% -30 %
8.664 2.868 9.492 3.142 1.040 3.442 1.139 3.771 1.248 4.131 1.304 -1.2% + 6.4%
3.608 1.216 4.099 1.382 4.657 1.570 5.291 1.784 5.973 1.974
1,070
-I-0.9% + 2.2%
1.059 3.565 1.200 4.041 1.360 4.580 1.542 5.190 1.745 5.869 1.897
’ Parameter number. b Computer value with lo-exponent for colums l-8. ’ .Using Eqs. (10) and (11). d Eq. (9) with correction for end effects; u6-u,s, identical with column 1. ’ First iteration, starting from column 2. After 5 iterations a precision of 1: lo5 is obtained. ’ See column 1.
‘23
exact
Accuracy: 01 exact
13 14 15 16 17 18 19 20 21 22 i3
+ 3.6%
+ 3.6%
5.057 1.700 5.717 1.922 0.0% 0.0%
1.5044 5.0570 1.7003 5.695 7 1.8559
d
d
d
d
d
d
- 1.5% -42%
0.738 2.417 0.791 2.592 0.849 2.782 0.915 3.005 1.014 3.350 1.075
12
leads to large deviations if the number of electrodes increases. But, already, the first approximation, which includes only the u* terms in S, and R, [Eq. (5)], gives 6 fair set of values with ati accuracy ranging from + 1.7% for a, to - 30% for a23. The second and third approximations (including the v4 and u6 terms, respectively) give better values, but demand expressions of increasing complexity and show only a modest improvement. Column 6 gives the results when applying Eq. (9). This small formula with its extremely simple derivation gives astonishingly good results. In the central range (6 c i -=E17), the ratio a,+Ja, agrees better than 1: lo9 (computer precision) with the exact ratios. To obtain the results, terms up to NJ*’ in Eq. (7) have been taken into account. For 3 5 i -=z20 the precision is better than O.l%, taking only the w3 terms in C(w). Equation (9) only gives ratios. One of the r equations in (3), however, suffices to give absolute values. Equation (9) is derived by neglecting end effects and therefore at both ends deviations appear. A correction for these effects can be made by taking deviating ratios of w at the begining and end. This has been done by introducing wl, w2, w3 and wzo, wzl, wz2 only. In column 7 the almost exact results are shown. Unfortunately, Eq. (9) is only valid for slit systems with equal slit widths and equidistant electrodes. Column 8 gives the results of the iteration method. For the starting values, the zeroth approximation (column 2) was taken: moderately good values of a, for small n but strongly deviating values for large PZ.After only one iteration (column 8) a dramatic increase in precision occurs: a23 was previously wrong by a factor 31.6, but appears now to be only 42% too small. After five iterations a precision of better than 1 : lo5 for all a, is obtained. This example shows the power of the various methods. With a small and not necessarily fast computer the complicated system of transcendent equations can be solved with a very high precision and in this way a simple method of finding the exact potential distribution in complicated electrode systems is obtained. CONCLUSION
The parameters governing the Schwarz-Christoffel transformation of an electrode configuration of the shape described in the introduction, can be calculated with ease. In slit systems with equidistant electrodes and constant slit widths Eq. (9) gives an excellent value for the ratios of the parameters. End effects can easily be taken into account. In configurations with electrode distances larger than the slit widths in the
13
adjacent electrodes, the series expansions (10) give a solution (11) with an ac?uracy of 1% or better. For the general case, the iteration method provides a solution with any degree of precision. As starting values one may take a, = s1/2, a,, 1 = viai for vi2< 1, with vi defined in (5). For vi >, 0.5, the ratio a,+Ja, approaches 1; a reasonable value is ui/(l + u,.). In the intermediate region, ui = 0.4, the average of both proposed values gives an excellent starting value. The iteration method makes an easy calculation of the Schwarz-Christoffel parameters possible for any slit system (also non-symmetric ones). As far as the authors are aware, a general solution of the problem has never been published. ACKNOWLEDGEMENTS
One of the authors (C.H.) is indebted to the Academica Sinica for a leave of absence. The authors wish to express their thanks to Dr. W.E. v.d. Kaay for performing the computer calculations. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research). REFERENCES 1 P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I, McGraw-Hill,
New York, 1953, p. 445. 2 A.J.H. Boerboom, Z. Naturforsch. Teil A, (a) 14 (1959) 809; (b) 15 (1960) 244; (c) 15 (1960) 253. 3 F-W. McLafferty (Ed.), Tandem Mass Spectrometry, Wiley, New York, 1983, Chap. 11.