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Double focusing mass spectrometers of second order
219 International Jozunal of Mass Spectrometry and Ion Physiq, 14 (1974) 219-233 Q Elsevier Scientific Publishing Cdmpany, Amsterdam - Printed in The Netherlands
DOUBLE
FOCUSING
MASS
SPECTROMETERS
OF
SECOND
ORDER
H. MATSUDA Insritute of Physics, CoHege of Ger?eralEducation, Osaka University, Toyonaka (Japan)
(First received 4 January 1974; in final form 25 January 1974)
ABSTRACT
The possibilities of correcting for second order image aberrations of a double focusing mass spectrometer are investigated. Six coefficients of the second order aberration, A,, Az6, Add, A,,,,, A,,, AfiB, are calculated considering the influence of the fringing field and the conditions for complete second order focusing (ALI:= Ad = A, = AZ8 = Add = A, = A,,.B= A,, = 0) are sought. The results of the calculations and values for the parameters of the various arrangements are described. It would be possible to construct a complete second order double focusing mass spectrometer consisting of a toroidal electric field and a homogeneous magnetic field or of a cylindrical electric field plus an electrostatic quadrupole lens and a homogeneous magnetic field. The latter configuration would be advantageous because the vertical focusing is better. Straight boundaries of the magnetic field are adopted for practical reasons.
INTRODUCTION
The possibilities of correcting for second-order image aberrations of a double focusing mass spectrometer were discussed in a_previous paper [l] and the importance of the in3ueke of fringing fields was emphasized. The magnitude of the spread of the image due to the second order abekrations is given by the expression
(1) where r, is the mean radius in the magnetic field, ccand fi respectively the-radial ind the axial inclination of the incident be_am,y-the hzilf h&i&t df -tie_ object slit, _ and S - the.)el&ve eqergy‘ deviatibn.
220 consisting of a cylindrical electric field and a homogeneous magnetic field with normal incidence and exit, it has been shown that the aberration coefficients for the ion beams away from the median plane, B,, B,, and BBB,are very large, though the coefficients for beams on the median plane, B,, & and Baa, can be reduced to zero by choosing suitable parameters for the apparatus [l, 21. Such an effect is caused by the presence of the fringing fields and therefore it is quite important for the calculation of aberration coefficients to consider the influence of the fringing fields correctly. When one wants to design a mass spectrometer of very high resolution and of very high sensitivity, it is desirable that each of the above six aberration coefficients be as small as possible. For instance, if one assumes that the magnitudes of a, j?, S and y/r, are all of the order of l/100, the necessary resolution is 100.000 and the mass dispersion is equal to r,, all six aberration coefficients Bij should be smaller than 0.1, that is, an apparatus with complete second order double focusing is necessary. In this paper the possibility of obtaining complete second order focusing will be discussed and some examples of favourable design will be shown.
OF FRINGING
INFLUENCE
FIELD
As described in previous papers [l-3], the aberration coefficients are calculated by the matrix method. Figure 1 shows the arrangement of the fields and the profile planes. The final position of an ion is given by the equations: x9
=
A,x,+A,a,+Ad~+a,x~+A,,xoao+A,axo6+A,,a~+A~a,5
+
+Ada6*+A,yy~+ApSyoBo+A88B~
(2)
Y9 = A,Yo+AfiPo-
Fig. 1. Arrangement
bf field? and parameters.
(3)
Profile
planesare indicated by’n_+&
S9.
221. The influence of the fringing fields are taken into account through the t&sformation at the ideal boundary. For the electric fringing field, the values of the following definite integrals should be known [S] b I *a =
E,’
E(v) dv drr-46
SS3
_ (4
ss s b
I,,
= E,’
(5)
E(tt)2dt7d~--4,~b--3d
3
b
I 4s
=E02
Wd2dvzb
(6)
3
I,,
=
bE(,j)3dq - qt, f il
Eo -'
where E(q)/E,
is a distribution function of the electric fringing field along the main path and r,q is the distance from the ideal field boundary_ These integrals can be evaluated using the function E(q) gi ven by Herzog [4].The numerical values calculated are given in Table 1 for the case of a thin Herzog plate, where 2k is the gap distance between condenser electrodes and 2b the slit width. The distance, D, between the Herzog piate and condenser electrodes is chosen so that the ideal field
TABLE VALUES
1 OF
IN-I-JSGRAIS
FOR
THE
EIJXTRlC
FRINGING
FIELD
radius, 2k = gap distance, 26 = slit width, D = distance between Herzog p!ate and condenser electrodes. Ideal field boundary coiqcides with the end of condenser electrodes.
r,
=
mean
Uk
D/K
II,
0.15
0.5155
x (r,lkS
L,
x (r&W
I,
x (r&l
Ii,
x (r.lk)
IS x Wrc)
17 x (r&l
0.0892
0.1177
-0.2276
-0.3639
0.7878
0.0263
0.20 0.25 0.30
0.5076 -0.4973 0.4846
0.0916 0.0946 0.0982
0.1187 0.1201 0.1217
-00.2311 -0.2355 -0.2404
-0.3677 -0.3726 -0.3781
0.7641 0.7415 0.7200
0.0277 0.0295 0.0317
0.35 0.40 0.45 0.50
0.4693 0.4513 0.4305 0.4065
0.1025 0.1074 0.1130 0.1191
0.1238 0.1263 0.1291 0.1324
-02460
-0.2521 -0.2586 -00.2657
-0.3845 -0.3916 -0.3994 -0.4081
0.6994 0.6796
0.0341 0.0368
0.6605 0.6421
0.0398 0.0430
0.3791 0.1261 0.3479 0.1336 0.3123 0.1419 b-2716 __ O.lSll - 0.2248_ 0.1609 -_
g.1359 - 0.1401 0.1446 :0.1&98 0.1557
-c&2733 -0.2813 -0.2898 ~0~2989 -_0.3086
-0.4171 -0.4268 -0.4374 -o-4487_ ~0.4610
0.55 _0.60 0.65 0.70 0.75
0.6243 0.0464 0.6068 0.0500 0.5896 -_ . 0.0539 0.0579 1 0.5726 a_0.0621 - 0.5554
-
222
boundary coincides with the end of the condenser electrodes. It should be noted that values of the integrals depend on the ratio, k/l;. In all calculations described in this paper, the electrode structures assumed are as follows; b/k = 0.72, D/k = 0.2537, k/r, = 0.032. Then integrals are evaluated to be I,, = 0.0001587, I,, = 0.0001557, I,, = -00.009688, &, = -00.01451, I5 = 17.679 and I7 = 0.001907. For the magnetic fringing field, the values of integrals
11 =
ss s b
B,’
B(d dvdt7-+I:
(10)
D
b
1, = B;’
a
B(tl)‘dt7 - tlb
(11)
are necessary [6]. From the measured field distribution of a magnet and assuming that the gap distance is equal to O.O6r,, one obtains I, = 0.002276 and 1, = -0.03099. This magnet has round pole edges with a radius of 0_03r, to avoid saturation effects and a magnetic shunt with a hole of 0.15r, is placed at a distance O.l5r, from the pole. The values of integrals given above are different from the values used in the previous papers [l-3] because a wider slit width and gap distance than before was chosen in order to allow transmission of broad ion beams.
APPARATUS
WITH
CYLINDRICAL
ELECTRIC
FIELD
A double focusing mass spectrometer consisting of a cylindrical electric field and a homogeneous magnetic field will be discussed here. For a given 4, it is in many cases possible to find a set of parameters which satisfies the conditions A, = Ad = A,, = A, = A,, = 0, that is, the conditions for the second order double focusing on the median plane. Table 2 shows examples of these parameters and the values of A,, A,, A,, A,, A,, and A, in the c&e of += = 90”. The values in the parentheses of the first row are those calculated using the integral values given in ref. 1. The difference between the values in the first row and those in the second row shows the effect due to the difference between the fringing field distributions. The behaviour of the values of aberration coefficients Ay,,, A, and A,, is shown in Fig. 2a when the incident angle e’ to the magnetic field is changed. Five different cases are compared with each other. Here Ql is defined as the ratio r,/q’ where q’ is the radius of curvature of the ideal boundary at the entrance of electric field. The front face of the condenser ele :trodes is flat if Ql is zero and it is concave . _ if Ql has a negative value. - It is seen from the figure that the aberration coefficients are very large if the front fke of the condenser is flat and this causes a curved image. On the contrary,
.
-
__
-
.” -
-
-
0.803 0.607 0.425 0.255 0,110
0.855 0.894 0.930 0.971 1.014
-15 -15 -15 -15 -15
20 25 30 35 40
-6 -G -6 -G -6
5
1,279 I.245 1,199 1,143 1,081
lOBa 112.5 116.1 II9,5 122.2
-10 -10
1a227 0,839 0 460 0.287 0.133
0,858 0.986 1.125 1.207 1.301
I.328 1,321 1.260 1,213 I.156
95,G 102.6 108.7 111.1 112.8
-10
10 20 30 35 40
-4 -4 -4 -4 -4
4
87,O 93.4 99,3 101.9 104,l 105,9
-10 -10
0 10 20 25 30 35
-10 -10
-10 -10 -IO
-10
0 0
-2 -2 -2 -2 -2 -2
0
10
3
15
20 25 30
81.4 8G,5 90.9 92,5 93,5
0
0 0
10
_.
-
---
1.327 1.415 1,466 1.483 1.500 1.521
1.300 1,380 1,411 1,406 1,383
-
0.851 1.064 1,345 I a538 1,797 2.169
0,809 0,991 1,202 1,321 1,453 1,581 1.251 0.880 0.691 0.506 0.323
I.672 1.366 1.011 0,829 0.645
1.684 1.552 1.404 1,247
0
1.039 1.230 1.498 1,934
0
I.386 1.467 1.556 I.671
79.3 81.2 82.8 83.8
0 0 0 0
5
1,700
0
1.018
1,384
d
0 0
-2 -2 -2 -2 -2
1
I’,
rc
79,l
4,
----
-
_ -
0.581 0,602 0.624 0,647 0,669
0,764 0,808 0.856 0.908 0.966
__ I_
0.555 0.586 0,622 0.641 0.661
0.490 0.488 0.479 0.467 0.448 0,416
0.581 0.609 0.649 0.677 0.718
0.505 0.494 0.473 0.434
0,510
A,
0.798 0,914 1,051 1,135 1,241
0.698 0,803 0.912 0.969 1,027 1.089
0.921 1.116 1.361 1.52f.i I.751
0.937 1.035 1.138 1.252
0.940
I”,,,
A6 = A,, = s-&j = &,j = 0 AND
0
E”
=
0
6’
I,0
(0
Ql
=90°,Q2=0,r,,=
2
I
CASE OF $‘,
, BXAMPLEG OF PARAh!BTERS WHICH SA1ISFY THC CONDITIONS A,
TABLE 2
I,10 0.89 0.66 0.40
1.09
-. -
0,23 -0,I 1 -0.51 -0.96 - 1.47
0.78 on21 -0.52 -0,97 -I,50
1.22 0,78 On23 - 0,09 - 0.46 -0.87
1,lO 0,65 0.07 -0,29 - 0.74
A, P_
7.37 5.63 3.33 1.93 0.30
7.76 7.04 6.16 5,08
7.66
A,,
3.96 2687 I.78 OAB - 0,42
5.98 3.86 1.50 0,23 -1.15
7896 6.25 4.03 2A8 I .09 -0,90
4
VALUES OF A,,
A,,,,,
0,32 0.41 0.44 0,43 0.33
-0.15 0.03 0.05 -0,04 -0.21
-0,67 -0.42 -0,31 -0,32 -0,39 -0.54
-0.65 -0.44 -0,37 -0.41 -0.53
-0.91 -0.81 -0.73 -0.69
- 0.90
AYY
--
/i/j,
_
6,O 3.9 3.1 3.3 4.0
9.4 6.7 5.0 4.7 4.9 6.1
9s 7,l 5.9 6,O 6.7
IO,4 9,4 8,7 8,6
. __
- 3.1 - 2,3 - 1.8 - 1.7 - 1.8 f
-
-
-
-
-lo,3
-
_..
lo,3 7,9 6,2 5,1 4,5
- 17,7 -11,6 - 8,2 L. 786 - 768
-27sl -20.0 - 14.7 -13-3 -* 13,3 - 15,G
-26.7 -20,5 -16.2 -15,4 -15,9
- 30.2 -28.0 -26.8 -2746
-29.8)
_-
ACP
AND tfpp IN TIM
AYb
dyp
k _w
224 aberration coefficients are considerably reduced as the curvature and the incident angle e’ increase. Moreover, with such a condition, the magnitude of A, is also decreased (Table 2) and therefore the vertical focusing is improved. - The variation of the aberration coefficients with respect to the change of $, and the parameters of the second order double focusing on the median plane are
if the front face is concave the
given in Fig. 2b and Table
3. An example
of the effect due to the curved surface
at the exit of the electric field is given in the lower part of Table 3 and it is seen that this effect is small. The reason for this is considered to arise from the fact that the exit face of the condenser is situated near the intermediate image and therefore the change in angle of the ion beam there does not cause serious focusing changes at the final image position.
& 7a IO”
iv-
30”
6a 75” So” 85” 90” 9!i”qe .-
40” iE’
-
-aI -7b
- -5
-8c - -10
7bb
A
- -15
-
7c
-29
- -25 6c a
1
-30
b
Fig. 2. Behaviour gf the aberration coelikients of mass spectrometers consisting‘of a cylindrical ektrk field and a homogeneousmagneticfield-The coefficients A,, A,,# and AB8 are indicated by sUhePfi_ a, b and c respectively.Values of parameters are given in Table 2_and Table 3 and conditions A, = Ad = ;4, = A d = Aa = 0 are satisfied. (a) Variation with E’: 1: Ql = 0, E” = 0; 2: Ql = -2.0, E” = 0; 3: Ql = -2.0, s” = -10”; 4: Ql = ~4.0, E’J-k -10”; 5: - Ql_=_ -6.0, E” = -15”. @) Variation with #=e: 6: Ql = Q2 = d = d’ 7 0; 7: Ql = -2.0, E’ = 200, E’* = -16”; ‘8: Ql = -4.0, E’_= 35”, ,‘* = _-loo. -
90 95
ii
=Q2=6’=6”=0
6
7411 76,4 79.3 82,6
All
1.520 1.449 1.386 1.331
rc
_.
..
90 * 95
.
95,4 9880
1.101 1,160
Ql" -4, 42 = 0,r,,JR,; = -0,5, 6’ =
-10’
0, 6’ = 35’, E“ = -10” lOI, 1.297 105,l 1.223 10885 1.200 111.1 1.213 112,5 1.252
Ql = -4, Q2 = -2, 6’ = 35”,6” = 90 112,4 1.212
Ql = -4, Q2 = 8 75 80 85 90 95
Ql = -2, 42 = 0,e’ = 20°,E” = -10” 7 75 91,7 1.606 ,8O 9447 1.539 85 9783 1.493 90 99.3 1.466 NO,8 1.452 / g5
Ql
46e
In all cads f, = 1.0.
_
35”,6” 0,936 0,939
1.232
1,372 1,274 1,227 1.207 1,200
1.848 1,649 1,484 1,345 1.218
1.625 1a295 1.039 0.834
I’,
=
0.457 0.411
- 10’
0.261
0.798 0.553 0.392 0.287 0.223
1.409 1.179 1.004 0.880 0.794
_
.
1.653 1,466
1.115
1.278 1.267 1.214 1.136 1.055
1.000 0.976 0.946 0.912 0.875
0.880
1.029 0.988 0.937
1““I
.
0,847 0,776
0.633
0.612 0.647 0,653 0,641 0,620
0.400 0.430 0,457 0,479 0,497
0.401 0.453 0.505 0,556
4
.
-1.09 -1,ot
-0.98
-0.98 -0699 -0.97 -0.94
-0.93
0.26 0.25 0.24 0.23 0.23
.
on50 0.40
0,22
-0,08 on13 0.22 0,23 0.17
4,35 4.20 4.10 4.03 3,98
8.66 8.17 7,7G 7.42
4J
VALUES OF Ax,
1.11 1.10 1,lO 1.10
AY
A, = Ad = A,, = Aord= Aad = 0 AND
1.947 1.817 1.684 1.558
J
i!XAMPLBs OF PARAhlBTERS WHICH SATlSrY THE CONDITIONS
TABLE 3
_..
-.
0.29 0.17
-0,05
-0*02 0,Ol 0,oo -0.04 -0.09
-0.31 -0.31 -0,31 -0,31 -0,31
-0,90 -0,90 -0,91 -0.92
A YY
-
-
3.8 3.5 3.3 3.3 3.4
5,7 5.4 5.1 5.0 4,8
-
- 2.6 - 2.7
- 3.3
-
-
-11.6 -11,o - 10.4 -10.0
A,/1
-
A,, Ap, a-&,,, /iv/j ANb App
-
- 6,2 - 6J
- 7.7
- 9.9 - 8,5 -7.8 - 7,6 - 7,8
-19*2 -17.1 -15.7 -14.7 - 14,o
-33.5 -3082 -27.6
-37.9
4/J
,
1
’
226
..
.
0
0.1
0.2
0.3
0.4
0.5
. 0.6
-30 1 0
0.5
1.0
1 1.5
KSJ Fig; 3 (left). Variation of A,,-A,,J and APB with c = r,lR, when second order double focusing conditions ‘A; = Aa = A, = Ad = Aad = 0 are satisfied. Values of parameters are given in Table 4. .. _. __ Fig. 4 (right]. Vanatlon of A,, A,.8 and APB with K0 when second order double focusing conditions A, = Ad = A,, = AZ6 = Aal = 0 are satisfied. Values of parameters are given in Table 5. C
An example of the effect due to the curved boundary of the magnetic field is also shown in Table 3. If the entrance boundary of the magnet pole is concave (radius of curvature x) some improvement is observed but it is not very much. In all calculations .described above and throughout this paper, the mean radius in the magnetic field, rm is chosen to be 1.0 so. that the coefficients Arj in eqn; (2) coincide with the coefficients Bij in eqn; (1) which are usuahy defined as the coefficients of second order. .aberration. Nonetheless, jn a combinktion of -a cylindrical electric field and a homogeneous magnetic field, the limit of improvement is cqwicie~ed tb be when- A, z 0, A ya =-2, A,, x :5,and it seems d%icult to realize complete second order double focusing by simply changing parameters.
0
0 0 0 0 0
-2.0 -2.0 -2.0
-2.0 -2.5 -2.5
-2.0 -2.5 -3.q -3.5
-2.5 -3.0 -3.5
80 80 80
85 85 85
90 x 90 90 90
95 95 95
-4.0 -4.5 -4.5
-4.0 -4.5 -5.0
-5.0
105 105 ios
110 110 ’ 110
112
100 -3.0 >lOO -3,5 100, -4.0
0
0 0 0
0.5 0 0,5
-0.5 0 1.0
-0,5 0 0
-0.5 0 0,5 1.5
0 0 0,5
0 0 0
0 0 0 0 0
0
90
, 90 90 90 I 90 ’ 90
#e
Q2
’ Ql
In all cases
I#0 1.0 1.0 ia0 1.0 1.0
O,l 0,2 0.3 0.4 0,5 0.6
088 1.0 0,8 0,4
0.59
-.. ~. -
0
0,525 0 0.55 O&2 0,575 0,2
0.55 0.2 0.6 Oa6 0.625 Oa2
0.5 Oa8 0.55 0.8 0.625 042
0.525 1.0 0,55 1-o 0.55 OS8
0,55 0.525 0.612 0.675
0,525 1.0 0,625 1.0 0.575 1,o
0.575 1.0 0,55 1.0 086 1.0
Rpcx c
c 0 0 0 0 0 0 30 30 35 30 30 30 35 30 32 31 30 30 30 30 30 30 30 30 30 30 30 30 32
77.2 74.9 72,5 69,Y G7.2 G4,4 8487 92.9 86.1 88.8 9080 90,7 85,6 go,9 9080 93.9 90,o 92,l 99,G 9382 9386 94,l 96,8 91,o 91,o 960 95,6 93,l 90.0
-
-1085
-10 -10 -10
-10 -10 -10
-10 -10 -10
-10 -10 -I5
-10 -10 -10,l -10
-10 -13.2 -10
-
/ior =
0 0 0 0 0 0 -10 -15 -10
err
... -. _.. -
E’
4,
A~,J
SATISFY THE CONDITIONS
A,, A,, Ap , A,,),, A,,p AND r, = 1.0.
WHICH
OF PAHAMETLRS
VALULS
OF
EXAMPLES
TABLE 4
--
0,395 0,396 0.365 0.299
0.764 0.734 0.710 0.665
_ _
0.691 0,662 0.720
0.739 0.417 0.709 0,458 O.G70 0.409
.
0.643 0.753 0,755
0.683 0,632 0.497
0.878 0.800 0.612
1,417 1.471 1,547
1,466 1,635 1,670
1.414 1.485 1.623
1.425 1,456 1,024
1.696 1,397 1,577 1,687
1.361 1.128 1.379
1.345 0,998 1,507
0.985 I.036 1,089 1,148 1,209 1.275
I”,,,
I,25
1.00 1.03 1.08
I a00 1.05 1.13
0.79 0.87 1.08
AP
WITH
-2.3 -2.2 -2.9 -4.2
-1.9 -1.9 -2.3
-1,6 -1.4 -1.8
-4,4
-3.1 -3.6 -3,5
-3.4 -387 -4.0
-2.5 -3.0 -4.1
-2.0
-1.2 -1.7 -1.5
-1.5 -1.7 - 1.8
-1.3 -1.G -2.0
-1.5 -1.7 -1.4
-2.2 -1.5 -2.1 -2.5
-1.5 -1.6 -1.9
-1.5 -1.2 -2,G
0,48 6.5 5.2 -0.18 -1.0 4.0 -1.8 2.6 I.1 -2.5 -0.4 -3.4
A,,
0870 -2,3 0.76 -2.7 O.GO -2.3
0,73 0,66 0.80 1.01
0.55 0.49 0.61
0.49 0.39 0,54
0.50 0.50 0.50 0.49 0.49 0.48
A,
FOR THE APPARATUS
O.G90 1.769
0.816 0.756 0.689
0.844 0.793 0,804
1,002 0.876 0.955
1.105 1,033 0,960 0.731
1,251 1.539 1,140
1,706 1.586 1.464
1,766 1,861 1.965 2,086 2,220 2.371
d
0.935 0.862 0.935
0.930 0.985 0,826 0.676
1.246 1.179 1,137
1.060 1.058 0.958 0.935 0.958 0.816 0.664
1.423 1,588 1.410
1.041 1.042 1.043 1.044 1,050 1.059
I’,
1.143 1.223 1.070
1.317 1.249 1.182 1.116 1.052 0.989
r,
A8 - A,, = Aaa = A~J = 0
-0.03
-0.19 -0.13 -0.09
-0,06 -0.13 -0.20
-0.11 -0.13 -0.11
-0.23 -0.18 -0.22
-0.16 -0.12 -0.15 -0.08
Oa96 0*99 0.59 0.50
- on73 .!_ OS54 - 0027
- 0,82 - 0.72 f 0:58
1
- 1.29 - 0.51 - 0,94
- 0*94 - 1.23 - 1.18
-21 -14 - 8.4 - 4,l - 1.4 - 0.36
A/v
FIELD AND
0.10
0,27 0.14 0.23
- 0,07
- 0,28 - 0.26 - 0.16
On26 - OS35 , 0.19 - 0425 0.22 - 0.14
0.35 ’ 0.25 0,27
0.35 0.29 0.36
0,38 0.46 0.41 0.27
0,46 0.42 0.44
0.87 0.49 OS0
-0.20 -0.17 -0.14 --0. I4 -0.44 -0.17
-5.1 -1.0 1.7 2,8 2.3 -0,35
AYiJ
ELECTRIC
-0.33 -0.08 -0.22 -0.80 -1.93 -3.73
4,
TOROIDAL
5
(
228. .
_: :
:
in
of the equipotential surface the vertical direction and,R: = dR,/dr,: F&rue 3- ’ shows an example of the behaviour of the coefficients A,, _AXfl. and .AA,,when .C is changed keeping the-conditions A, = A, = A,,=.A&= ;A,, = 0.In the .&se4, = 90°, and- normal entry and exit from the magnet&zfield, an optimum con-. dition is obtained at c w 0.5 and the values of A,, are. reduced to about: l/20 of : -_ the value for cylindrical &se c = 0. _ By changing other parameters R:, Ql,_Q2, a’, E”, etc., much better improvement can,be obtained. In Tabl.6 4 examples of a favourable set of parameters and values of ApY,.ArPzind.ABBare given together with .the values of A,, and A, which indicate the nature of vertical focusing. As can be seen from the table, the values of the aberration coefficients decrease With an increase of the value of 4, until. .q%, = 112”, where all the aberration coefficients have values less than. 0.1. The combination. of parameters shown in the last. row of Table 4 (c$== 112”, 4, -= 909 c = 0.59, Ql = -4.0, E’ = 32”; a:’ = l&Y) has excellent focusing properties. The second order aberration coefficients are all quite small and -the parameters of the apparatus are all suitable for practical construction. As for .vertical-focusing; ions are once focused in the magnetic field and diverge again after passing through the magnet. This may be unfavourable from the view point .of getting a small image s&e but it is favourable to pass as many ions as possible through the narrow magnet gap.
APPARATUS
WITH CYLlIiDRICAL
ELECTRIC FIELD AND ELECTROSTATIC
QUADRUPOLELENS
A remarkable improvement can be obtained ‘by introducing an electrostatic quadrupole lens between the electric.and magnetic fields: The Q-lens is used to defocus the ion beam on the,median plane. In this dase four additional profile planes are necessary. The planes ql.and q2 are respectively.the front and back side of an ideal boundary at the entrance to the Q-lens and th_eplanes-q3 and q4 are those-at -the exit. The iufh.Ace of the fringing field is given by .a transformation between ql. and q2 and that between q3 and- q4 in a tiay similar. to that of the magx&d or electric field These transform&ions. and the transform&ion. which describes an id&l -Q-lens of length L and field.constant K,, are given by the follow.ifig-equatio&
[7]_
:
.EntranCe fringing field: jc,Z .=
: .. ‘Y _..
.. 1
.[l+K,2l,]x,l._t[~~~I~]tanag,-~K~I,]x,,6 : : :
-:. ‘..
‘.
: :
:. _.
-.;..:. .. ..;(f2>
I-
:
‘.
:.::
_:
.:
:
-. . . _.
-.
Jdeal el&&ost&ic
q&rtip~le
leti: (defocusiiig ‘on”the’&dian
&&ie$
:
.229
.:. '-
..
-y
l.. -.[g& + sin l&o .. .: tan &,a =. [K. -. $
yq3
=
J-%3 = J%fringing
L-&
s-t.. m
G K&:
sin /X0 L]x,, . .
[sin hKo&+KoLcos
[co&.Ko
-.
..
:
Gn h& L -_3L cos :hKo .....
(161.
+ [cos .hKo L] tan cxq2 :. hK,~]x,,d-[~Kj~~sin
K, L]
-wo sinK* L-j$,+ + [cosK,
hICoLjaq;6
(17)
(18)
pq2
p)
L]~qz.
field:
_xq4 =-Cl-K~I,]x,,+[2K$I,] tan
:.
tan aq3+[K~11]xq38
a,, 2 [--K&]x~~+~+K&]
-
tan aq,-[CKiI,]ccq36
(20) (21)
Yq4
=
Yq3
(W
P q4
=
893.
(23)
The field
constantKo
is given
by
K;=ek,
(24)
..2u,
where k. is the field gradient in the Q-lens and U, the eneygy of. amain path particle. The quantities II, I2 and I, are definite ‘integrals which-depend on the fringing field distribution and are given by I;
= k,:
P
(25)
where k(z)/ko is a distribution fun&n of $jc &eId gradient-along the Z-&S, the origin.- df which is on the -ideal bound&y;- Using :the distribtitiori cu% which ii - m&&&i w&h a mi@i&Q-J&s hid &su&i&t& &iks :of .fhe insciibed--crrcle ‘. &he &e&odes- t6 .&e0;07r,, #I&+-followitig &II& &e”&taine& iI .+~O.d;lO% . _-~~r2:.~~~()_O&02~34 andI ~-&o()OO1&3. ;_I :I.: ;l:..:-.- . ...-::..:-::-_ ~_... ;... .;.I : -._ _-
230 If the electrostatic quadrupole lens has a defocusing action on the median above, vertical focusing can be attained at the same time. An example of the effect of such a Q-lens is shown in Fig. 4 where the variation of the aberration coefficients A,,,,, A,,fl and A, are plotted against the value of K,,. In this case the length of the Q-lens, L, is chosen to be 0.22r, and the conditions A,=A,=A,= A,,= A,,= 0 are of course satisfied. The other parameters together with the values of A,,A, and A, are given in the first six rows of Table 5. As can be seen in Fig. 4 very small values of aberration coefficients are obtained for the most suitable focusing power of the Q-lens (k,z 1.85, L = 0.22). By adjusting the parameters of the apparatus, many favourable designs are found and examples of them are listed in Table 5. As compared with the apparatus with a toroidal electric field (Table 4), a significant feature of these instruments is that the incident angle to the magnetic field is small and negative and the exit angle is zero. Normal entry and exit is obtained by the apparatus with parameters in the line marked by * 1. The apparatus marked by *2 has quite small aberration coefficients. Ail six coefficients are smaller than 0.01 and therefore complete second order double focusing can be achieved with this arrangement. The change of each aberration coefficients is shown in Fig. 5 when one of the parameters is changed while the other constant. In this case the first order double focusing condition is satisfied by adjusting the values of dd,,the distance between the Q-lens and the magnetic
plane as mentioned
04
0 J -a1 z ;: -0.2 r: 2 -0.3 00 6 -04 4
-25
-20
L92
Lpd
L%K,
-1.5 ()2
Fig. 5. Variation of aberrationcoefficients of the apparatus Ati; 3: Aad; 4:-A,,,.; 5: Ass; 6: AB8. _-
marked by *2 in Table 5. 1: A,;
2:
0 -2.0 0 -2,o 0
-3,5 -3.5 -4.0 -4.0 -4.0
-3,5 -4,o -4,o -14.5 -4,o
-4.0 -4.0
-4.0
85 85 85 85 85
‘90 95
85
80 80 80
s80
80
0 0 0
-3s -3,5 -4.0
75 75 75
1.65 1.91
-_-
---..--..
1.915 1.92
1.875 1.91 I.91 1.933 1.91
1.86 1.885 1.895 1.94 1.895
1.848 1.85 1.885
1.855 1,84 1.83
0 0.5 1.0 1.5 1.8 2.0
&
-285
0 0
0 0 -2.5 0 -2,5
0 0 0
-3s -3s -3,5
70 70 70
0 0 0 0 0 0
Q2
-4,o -4,o -4,o -4,o -4,o -4,o
Qi
80 80 80 80 80 80
I_-_-
dc
70.5 73.8
73,7 74.1
71.9 72,5 70,7 73.3 70.4
=
0.869 0.935 1.108 1.304 1.384 1.194
--_-
I',
1.173 1.048
1,276 1.270
-II -10.5
-18 -21
--_------_..-
1,319 1,293
I.431 1.290 1.303 1.216 1.272
1.300 1,272 1.290 1.250 1.299
1.300 1,299
1.564 1.550 1.442 1.403 1.410
1.314 1.317 1.280 1.265 1.287
-10 -4 -9 - 1.8 -11 -13.5 -15 - 9.5 -14 - 10.5
1.757 1.680 1.539
1.322 1.336 I.299
0.15
0.15 0.15
0,l 0.15 0.15 0.15 0.2
0815 0815 0.15 0.15 0.2
0015 0.25 0.25
-_-
1,035 1,049
1,064 1,036
1,lGO 1.129 1,094 1,105 1,071
1.188 1,169 1.176 1.147 1,154
1,271 1,230 1,227
0815 1.418 OS21 1,401 0830 1.350
0.48 0.44
0843 0.42
0.42 0.44 0.49 0.47 0.49
0,43 0,50 0.47 0.58 0.46
0.46 0,43 0047
0851 0.47 0,45
0.58 0.57 0056 0.57 0,62 0,72
4x
.._--_--a--
0.957 1.111
0.787 0.686
0.941 0.894 I.153 0.926 1.130
1,083 1.439 I.127 I.634 1.058
I.369 1,195 1,246
1.793 1.573 1,415
1.046 1,065 1.144 1,349 1,612 1,998
4,,, -k
0.15 1.762 0,15 1.667 0,15 1.440 0,15 1.178 0.15 1.004 Oa15 0,887
dcq
AYp AND
1.325 “I.979 1.343 1.934 1.357 1.858
1.306 1.303 1.299 1.285 1.253 I.365
i-
A,,, np, I$,,,
- 3.5 -8 -7
3,5 0 - 3.5
_-
8'
-_._-
70.8 67.9 71,l 66.9 7181
68.8 G8.7 68,7
65.3 65.8 65.9
73.9 74.2 74,4 73.2 71.5 68.8
d,,
AND ELECTROSTATIC Q-LENS AND VALUES OF A,,
EXAMPLES Or PARAMETERS WHICH SATISFY THE CONDITIONS nrx = .+
--1.7 1.8
-1.5 -I,4
- I& -1.7 -1.8 -1.7 -1.7
--1.7 -2,0 -1,8 -2,3 -I,8
-1.9 -1.8 -1.9
-283 -2.1 - 1.9
1.1 0.9 0.2 -0.9 -I,8 -206
Av
-1.9 -1.7
-1.0 -0.G
-1.4 -1.4 -2.0 -1.5 -1.9
-168 -2.9 -2,l -3.5 -1.8
-2.8 -2.3 -2.5
-4.5 -3.8 -3,l
7,7 7,O 4,s I,2 -188 -482
_____^
A,J ---
-
0.17 0,25
_----.-._
-0,lO -0.11
0.04 0.09
-
0.14 0.13 0,lO 0.12 0,lO
-0.20 -0.20 - 0,09 -on15 -0,07 -0,19 -0.20
- 0,lO ’ - 0.01 - 0,07 0,oi *2 - 0.10
0,14 0.03 0.10 O,tM 0.12 -0.14 - 0.08 -0.12 -0,Ol -0.13
- 0808 0,OG *4 ‘5
0,17 0,20
0,08 0.04 *3 0,03 0.05 0,07
- 0.10 - 0,15 - 0,lO
- 0.03 - 0,Ol ‘1 - 0,06
-28.2 -23.6 -12.7 - 2,70 - 0,85 - 1892
%P
0.10 0,08 0.07
0,03, 0.03 0,09
-9.09 -6,8l - I,93 1.29 0.68 -0.85
AYP
- 0.09 -0.12 -0*09
-IO,OG - 0,OY - 0.09
-0.31 -0.05 0.43 0.57 0.33 0.08
A,
A,, = Ana = A&j = 0 FOR THE APPARATUS WlTH CYLlNDRlChL LLECTIUC FIELD App In all casts except for lust two rows, L = 0,22, h” = 0, rn, = 1.0. For the row marked by *4L = 0,3, E” = 0, r,,, = 1,Oand by *5 L = 0.22, .z” = -5”, I‘y = 1.0, --_ ---_-1-m
TABLE 5
z
232 field, and I’,. It is seen from Fig. 5 that the change is not very rapid and therefore_ the condition for second order focusing is not very critical. The only undesirable property of this arrangement is that the vertical focusing point is in the magnetic field. After passing through the magnet ions diverge again and at the image position their vertical position is given by eqn. (3) with Ap = -2.3, A, = -3.5. This focusing nature may be better for a small instrument because the vertical spread of ion beam-is narrowest in the magnet gap. For a big instrument, however, the spread of ions after the magnet would be serious if the aperture angle in this di, rection is large. The vertical focusing tends to be improved with increasing 4, or 4,. However, the aberration coefficients unfortunately increase with @,. The optimum condition for both requirements would be in a region c$, - 85”. For a large instrument, however, the ratio y,/r,,, would be small and therefore a small increase in the values of App and A, would be tolerable. From these considerations the instrument marked by *3 in Table 5 would be suitable for a big instrument such as a high resolution on-line mass spectrograph. The schematic drawing and the vertical trajectory of an ion in such an instrument is shown in Fig. 6. Examples of an apparatus having an oblique exit E” and different lengths of Q-lens L are shown in the lower part of Table 5 (*4 and *5).
-a
I
SCURCE
Fig. 6. Schematicdrawing of a double focusing mass spectrometer of second order with a cyiindriCal electric field- and an electrostatic quadrupole lens (*3 in TabIe 5). Vertical ion kajzctory is shown in the lower part. _
CONCLUSION -_ _
x!-w+d be possible to construct A complete se&d &let double focusing ,masS-~trometer.consi~~g of a toroidal elekic field _@d a; h6mogt+&,ti~g_ -_-, -,
233 netic field or consisting of a cylindrical electric-field ~1% electrostitic quad.xupole lens and a homogeneous magnetic field--The latter-design would be adaptable to a large instrument because the vertical focusing is better and machining would be easier. Straight boundaries of the magnetic field are adopted for practical rekons. - In the calculations described above the influence of fringing fields tire-taken into consideration with an accuracy effectively of third order.- The values of parameters given in this paper will be changed slightly if the distribution of the fringing field is different from that assumed here. Therefore, the parameters which satisfy the condition for second order focusing should be calculated using the distribution function for a respective apparatus. The expected change in parameters, however, are relatively small as shown in Table 2.
REFERENCES
1 H. Matsuda and T. Matsuo. ht. J. Maw Spectrom. Ion Ph>s_, 6 (1971) 385. 2 H. Matsuda and T. Matsuo, Atomic Masses and Fur&mental Constants, 4 (1972) 192. 3 H. Matsuda, Mass Spectrosc. (Tokyo), 21 (1973) 15. 4 R. He-g, Phys. Z., 41 (1940) 23. 5 H. Matsuda. Nucl. Instrum. Methods, 91 (1971) 637. 6 H. Matsuda and H. .Wollnik, Nuci. Instrum. Methock, 77 (1970) 40,2.+?3. 7 H. Matsuda and H. Wollnik, Nucl. Instrum Methonk, 103 (1972) 117.
DOUBLE
FOCUSING
MASS
SPECTROMETERS
OF
SECOND
ORDER
H. MATSUDA Insritute of Physics, CoHege of Ger?eralEducation, Osaka University, Toyonaka (Japan)
(First received 4 January 1974; in final form 25 January 1974)
ABSTRACT
The possibilities of correcting for second order image aberrations of a double focusing mass spectrometer are investigated. Six coefficients of the second order aberration, A,, Az6, Add, A,,,,, A,,, AfiB, are calculated considering the influence of the fringing field and the conditions for complete second order focusing (ALI:= Ad = A, = AZ8 = Add = A, = A,,.B= A,, = 0) are sought. The results of the calculations and values for the parameters of the various arrangements are described. It would be possible to construct a complete second order double focusing mass spectrometer consisting of a toroidal electric field and a homogeneous magnetic field or of a cylindrical electric field plus an electrostatic quadrupole lens and a homogeneous magnetic field. The latter configuration would be advantageous because the vertical focusing is better. Straight boundaries of the magnetic field are adopted for practical reasons.
INTRODUCTION
The possibilities of correcting for second-order image aberrations of a double focusing mass spectrometer were discussed in a_previous paper [l] and the importance of the in3ueke of fringing fields was emphasized. The magnitude of the spread of the image due to the second order abekrations is given by the expression
(1) where r, is the mean radius in the magnetic field, ccand fi respectively the-radial ind the axial inclination of the incident be_am,y-the hzilf h&i&t df -tie_ object slit, _ and S - the.)el&ve eqergy‘ deviatibn.
220 consisting of a cylindrical electric field and a homogeneous magnetic field with normal incidence and exit, it has been shown that the aberration coefficients for the ion beams away from the median plane, B,, B,, and BBB,are very large, though the coefficients for beams on the median plane, B,, & and Baa, can be reduced to zero by choosing suitable parameters for the apparatus [l, 21. Such an effect is caused by the presence of the fringing fields and therefore it is quite important for the calculation of aberration coefficients to consider the influence of the fringing fields correctly. When one wants to design a mass spectrometer of very high resolution and of very high sensitivity, it is desirable that each of the above six aberration coefficients be as small as possible. For instance, if one assumes that the magnitudes of a, j?, S and y/r, are all of the order of l/100, the necessary resolution is 100.000 and the mass dispersion is equal to r,, all six aberration coefficients Bij should be smaller than 0.1, that is, an apparatus with complete second order double focusing is necessary. In this paper the possibility of obtaining complete second order focusing will be discussed and some examples of favourable design will be shown.
OF FRINGING
INFLUENCE
FIELD
As described in previous papers [l-3], the aberration coefficients are calculated by the matrix method. Figure 1 shows the arrangement of the fields and the profile planes. The final position of an ion is given by the equations: x9
=
A,x,+A,a,+Ad~+a,x~+A,,xoao+A,axo6+A,,a~+A~a,5
+
+Ada6*+A,yy~+ApSyoBo+A88B~
(2)
Y9 = A,Yo+AfiPo-
Fig. 1. Arrangement
bf field? and parameters.
(3)
Profile
planesare indicated by’n_+&
S9.
221. The influence of the fringing fields are taken into account through the t&sformation at the ideal boundary. For the electric fringing field, the values of the following definite integrals should be known [S] b I *a =
E,’
E(v) dv drr-46
SS3
_ (4
ss s b
I,,
= E,’
(5)
E(tt)2dt7d~--4,~b--3d
3
b
I 4s
=E02
Wd2dvzb
(6)
3
I,,
=
bE(,j)3dq - qt, f il
Eo -'
where E(q)/E,
is a distribution function of the electric fringing field along the main path and r,q is the distance from the ideal field boundary_ These integrals can be evaluated using the function E(q) gi ven by Herzog [4].The numerical values calculated are given in Table 1 for the case of a thin Herzog plate, where 2k is the gap distance between condenser electrodes and 2b the slit width. The distance, D, between the Herzog piate and condenser electrodes is chosen so that the ideal field
TABLE VALUES
1 OF
IN-I-JSGRAIS
FOR
THE
EIJXTRlC
FRINGING
FIELD
radius, 2k = gap distance, 26 = slit width, D = distance between Herzog p!ate and condenser electrodes. Ideal field boundary coiqcides with the end of condenser electrodes.
r,
=
mean
Uk
D/K
II,
0.15
0.5155
x (r,lkS
L,
x (r&W
I,
x (r&l
Ii,
x (r.lk)
IS x Wrc)
17 x (r&l
0.0892
0.1177
-0.2276
-0.3639
0.7878
0.0263
0.20 0.25 0.30
0.5076 -0.4973 0.4846
0.0916 0.0946 0.0982
0.1187 0.1201 0.1217
-00.2311 -0.2355 -0.2404
-0.3677 -0.3726 -0.3781
0.7641 0.7415 0.7200
0.0277 0.0295 0.0317
0.35 0.40 0.45 0.50
0.4693 0.4513 0.4305 0.4065
0.1025 0.1074 0.1130 0.1191
0.1238 0.1263 0.1291 0.1324
-02460
-0.2521 -0.2586 -00.2657
-0.3845 -0.3916 -0.3994 -0.4081
0.6994 0.6796
0.0341 0.0368
0.6605 0.6421
0.0398 0.0430
0.3791 0.1261 0.3479 0.1336 0.3123 0.1419 b-2716 __ O.lSll - 0.2248_ 0.1609 -_
g.1359 - 0.1401 0.1446 :0.1&98 0.1557
-c&2733 -0.2813 -0.2898 ~0~2989 -_0.3086
-0.4171 -0.4268 -0.4374 -o-4487_ ~0.4610
0.55 _0.60 0.65 0.70 0.75
0.6243 0.0464 0.6068 0.0500 0.5896 -_ . 0.0539 0.0579 1 0.5726 a_0.0621 - 0.5554
-
222
boundary coincides with the end of the condenser electrodes. It should be noted that values of the integrals depend on the ratio, k/l;. In all calculations described in this paper, the electrode structures assumed are as follows; b/k = 0.72, D/k = 0.2537, k/r, = 0.032. Then integrals are evaluated to be I,, = 0.0001587, I,, = 0.0001557, I,, = -00.009688, &, = -00.01451, I5 = 17.679 and I7 = 0.001907. For the magnetic fringing field, the values of integrals
11 =
ss s b
B,’
B(d dvdt7-+I:
(10)
D
b
1, = B;’
a
B(tl)‘dt7 - tlb
(11)
are necessary [6]. From the measured field distribution of a magnet and assuming that the gap distance is equal to O.O6r,, one obtains I, = 0.002276 and 1, = -0.03099. This magnet has round pole edges with a radius of 0_03r, to avoid saturation effects and a magnetic shunt with a hole of 0.15r, is placed at a distance O.l5r, from the pole. The values of integrals given above are different from the values used in the previous papers [l-3] because a wider slit width and gap distance than before was chosen in order to allow transmission of broad ion beams.
APPARATUS
WITH
CYLINDRICAL
ELECTRIC
FIELD
A double focusing mass spectrometer consisting of a cylindrical electric field and a homogeneous magnetic field will be discussed here. For a given 4, it is in many cases possible to find a set of parameters which satisfies the conditions A, = Ad = A,, = A, = A,, = 0, that is, the conditions for the second order double focusing on the median plane. Table 2 shows examples of these parameters and the values of A,, A,, A,, A,, A,, and A, in the c&e of += = 90”. The values in the parentheses of the first row are those calculated using the integral values given in ref. 1. The difference between the values in the first row and those in the second row shows the effect due to the difference between the fringing field distributions. The behaviour of the values of aberration coefficients Ay,,, A, and A,, is shown in Fig. 2a when the incident angle e’ to the magnetic field is changed. Five different cases are compared with each other. Here Ql is defined as the ratio r,/q’ where q’ is the radius of curvature of the ideal boundary at the entrance of electric field. The front face of the condenser ele :trodes is flat if Ql is zero and it is concave . _ if Ql has a negative value. - It is seen from the figure that the aberration coefficients are very large if the front fke of the condenser is flat and this causes a curved image. On the contrary,
.
-
__
-
.” -
-
-
0.803 0.607 0.425 0.255 0,110
0.855 0.894 0.930 0.971 1.014
-15 -15 -15 -15 -15
20 25 30 35 40
-6 -G -6 -G -6
5
1,279 I.245 1,199 1,143 1,081
lOBa 112.5 116.1 II9,5 122.2
-10 -10
1a227 0,839 0 460 0.287 0.133
0,858 0.986 1.125 1.207 1.301
I.328 1,321 1.260 1,213 I.156
95,G 102.6 108.7 111.1 112.8
-10
10 20 30 35 40
-4 -4 -4 -4 -4
4
87,O 93.4 99,3 101.9 104,l 105,9
-10 -10
0 10 20 25 30 35
-10 -10
-10 -10 -IO
-10
0 0
-2 -2 -2 -2 -2 -2
0
10
3
15
20 25 30
81.4 8G,5 90.9 92,5 93,5
0
0 0
10
_.
-
---
1.327 1.415 1,466 1.483 1.500 1.521
1.300 1,380 1,411 1,406 1,383
-
0.851 1.064 1,345 I a538 1,797 2.169
0,809 0,991 1,202 1,321 1,453 1,581 1.251 0.880 0.691 0.506 0.323
I.672 1.366 1.011 0,829 0.645
1.684 1.552 1.404 1,247
0
1.039 1.230 1.498 1,934
0
I.386 1.467 1.556 I.671
79.3 81.2 82.8 83.8
0 0 0 0
5
1,700
0
1.018
1,384
d
0 0
-2 -2 -2 -2 -2
1
I’,
rc
79,l
4,
----
-
_ -
0.581 0,602 0.624 0,647 0,669
0,764 0,808 0.856 0.908 0.966
__ I_
0.555 0.586 0,622 0.641 0.661
0.490 0.488 0.479 0.467 0.448 0,416
0.581 0.609 0.649 0.677 0.718
0.505 0.494 0.473 0.434
0,510
A,
0.798 0,914 1,051 1,135 1,241
0.698 0,803 0.912 0.969 1,027 1.089
0.921 1.116 1.361 1.52f.i I.751
0.937 1.035 1.138 1.252
0.940
I”,,,
A6 = A,, = s-&j = &,j = 0 AND
0
E”
=
0
6’
I,0
(0
Ql
=90°,Q2=0,r,,=
2
I
CASE OF $‘,
, BXAMPLEG OF PARAh!BTERS WHICH SA1ISFY THC CONDITIONS A,
TABLE 2
I,10 0.89 0.66 0.40
1.09
-. -
0,23 -0,I 1 -0.51 -0.96 - 1.47
0.78 on21 -0.52 -0,97 -I,50
1.22 0,78 On23 - 0,09 - 0.46 -0.87
1,lO 0,65 0.07 -0,29 - 0.74
A, P_
7.37 5.63 3.33 1.93 0.30
7.76 7.04 6.16 5,08
7.66
A,,
3.96 2687 I.78 OAB - 0,42
5.98 3.86 1.50 0,23 -1.15
7896 6.25 4.03 2A8 I .09 -0,90
4
VALUES OF A,,
A,,,,,
0,32 0.41 0.44 0,43 0.33
-0.15 0.03 0.05 -0,04 -0.21
-0,67 -0.42 -0,31 -0,32 -0,39 -0.54
-0.65 -0.44 -0,37 -0.41 -0.53
-0.91 -0.81 -0.73 -0.69
- 0.90
AYY
--
/i/j,
_
6,O 3.9 3.1 3.3 4.0
9.4 6.7 5.0 4.7 4.9 6.1
9s 7,l 5.9 6,O 6.7
IO,4 9,4 8,7 8,6
. __
- 3.1 - 2,3 - 1.8 - 1.7 - 1.8 f
-
-
-
-
-lo,3
-
_..
lo,3 7,9 6,2 5,1 4,5
- 17,7 -11,6 - 8,2 L. 786 - 768
-27sl -20.0 - 14.7 -13-3 -* 13,3 - 15,G
-26.7 -20,5 -16.2 -15,4 -15,9
- 30.2 -28.0 -26.8 -2746
-29.8)
_-
ACP
AND tfpp IN TIM
AYb
dyp
k _w
224 aberration coefficients are considerably reduced as the curvature and the incident angle e’ increase. Moreover, with such a condition, the magnitude of A, is also decreased (Table 2) and therefore the vertical focusing is improved. - The variation of the aberration coefficients with respect to the change of $, and the parameters of the second order double focusing on the median plane are
if the front face is concave the
given in Fig. 2b and Table
3. An example
of the effect due to the curved surface
at the exit of the electric field is given in the lower part of Table 3 and it is seen that this effect is small. The reason for this is considered to arise from the fact that the exit face of the condenser is situated near the intermediate image and therefore the change in angle of the ion beam there does not cause serious focusing changes at the final image position.
& 7a IO”
iv-
30”
6a 75” So” 85” 90” 9!i”qe .-
40” iE’
-
-aI -7b
- -5
-8c - -10
7bb
A
- -15
-
7c
-29
- -25 6c a
1
-30
b
Fig. 2. Behaviour gf the aberration coelikients of mass spectrometers consisting‘of a cylindrical ektrk field and a homogeneousmagneticfield-The coefficients A,, A,,# and AB8 are indicated by sUhePfi_ a, b and c respectively.Values of parameters are given in Table 2_and Table 3 and conditions A, = Ad = ;4, = A d = Aa = 0 are satisfied. (a) Variation with E’: 1: Ql = 0, E” = 0; 2: Ql = -2.0, E” = 0; 3: Ql = -2.0, s” = -10”; 4: Ql = ~4.0, E’J-k -10”; 5: - Ql_=_ -6.0, E” = -15”. @) Variation with #=e: 6: Ql = Q2 = d = d’ 7 0; 7: Ql = -2.0, E’ = 200, E’* = -16”; ‘8: Ql = -4.0, E’_= 35”, ,‘* = _-loo. -
90 95
ii
=Q2=6’=6”=0
6
7411 76,4 79.3 82,6
All
1.520 1.449 1.386 1.331
rc
_.
..
90 * 95
.
95,4 9880
1.101 1,160
Ql" -4, 42 = 0,r,,JR,; = -0,5, 6’ =
-10’
0, 6’ = 35’, E“ = -10” lOI, 1.297 105,l 1.223 10885 1.200 111.1 1.213 112,5 1.252
Ql = -4, Q2 = -2, 6’ = 35”,6” = 90 112,4 1.212
Ql = -4, Q2 = 8 75 80 85 90 95
Ql = -2, 42 = 0,e’ = 20°,E” = -10” 7 75 91,7 1.606 ,8O 9447 1.539 85 9783 1.493 90 99.3 1.466 NO,8 1.452 / g5
Ql
46e
In all cads f, = 1.0.
_
35”,6” 0,936 0,939
1.232
1,372 1,274 1,227 1.207 1,200
1.848 1,649 1,484 1,345 1.218
1.625 1a295 1.039 0.834
I’,
=
0.457 0.411
- 10’
0.261
0.798 0.553 0.392 0.287 0.223
1.409 1.179 1.004 0.880 0.794
_
.
1.653 1,466
1.115
1.278 1.267 1.214 1.136 1.055
1.000 0.976 0.946 0.912 0.875
0.880
1.029 0.988 0.937
1““I
.
0,847 0,776
0.633
0.612 0.647 0,653 0,641 0,620
0.400 0.430 0,457 0,479 0,497
0.401 0.453 0.505 0,556
4
.
-1.09 -1,ot
-0.98
-0.98 -0699 -0.97 -0.94
-0.93
0.26 0.25 0.24 0.23 0.23
.
on50 0.40
0,22
-0,08 on13 0.22 0,23 0.17
4,35 4.20 4.10 4.03 3,98
8.66 8.17 7,7G 7.42
4J
VALUES OF Ax,
1.11 1.10 1,lO 1.10
AY
A, = Ad = A,, = Aord= Aad = 0 AND
1.947 1.817 1.684 1.558
J
i!XAMPLBs OF PARAhlBTERS WHICH SATlSrY THE CONDITIONS
TABLE 3
_..
-.
0.29 0.17
-0,05
-0*02 0,Ol 0,oo -0.04 -0.09
-0.31 -0.31 -0,31 -0,31 -0,31
-0,90 -0,90 -0,91 -0.92
A YY
-
-
3.8 3.5 3.3 3.3 3.4
5,7 5.4 5.1 5.0 4,8
-
- 2.6 - 2.7
- 3.3
-
-
-11.6 -11,o - 10.4 -10.0
A,/1
-
A,, Ap, a-&,,, /iv/j ANb App
-
- 6,2 - 6J
- 7.7
- 9.9 - 8,5 -7.8 - 7,6 - 7,8
-19*2 -17.1 -15.7 -14.7 - 14,o
-33.5 -3082 -27.6
-37.9
4/J
,
1
’
226
..
.
0
0.1
0.2
0.3
0.4
0.5
. 0.6
-30 1 0
0.5
1.0
1 1.5
KSJ Fig; 3 (left). Variation of A,,-A,,J and APB with c = r,lR, when second order double focusing conditions ‘A; = Aa = A, = Ad = Aad = 0 are satisfied. Values of parameters are given in Table 4. .. _. __ Fig. 4 (right]. Vanatlon of A,, A,.8 and APB with K0 when second order double focusing conditions A, = Ad = A,, = AZ6 = Aal = 0 are satisfied. Values of parameters are given in Table 5. C
An example of the effect due to the curved boundary of the magnetic field is also shown in Table 3. If the entrance boundary of the magnet pole is concave (radius of curvature x) some improvement is observed but it is not very much. In all calculations .described above and throughout this paper, the mean radius in the magnetic field, rm is chosen to be 1.0 so. that the coefficients Arj in eqn; (2) coincide with the coefficients Bij in eqn; (1) which are usuahy defined as the coefficients of second order. .aberration. Nonetheless, jn a combinktion of -a cylindrical electric field and a homogeneous magnetic field, the limit of improvement is cqwicie~ed tb be when- A, z 0, A ya =-2, A,, x :5,and it seems d%icult to realize complete second order double focusing by simply changing parameters.
0
0 0 0 0 0
-2.0 -2.0 -2.0
-2.0 -2.5 -2.5
-2.0 -2.5 -3.q -3.5
-2.5 -3.0 -3.5
80 80 80
85 85 85
90 x 90 90 90
95 95 95
-4.0 -4.5 -4.5
-4.0 -4.5 -5.0
-5.0
105 105 ios
110 110 ’ 110
112
100 -3.0 >lOO -3,5 100, -4.0
0
0 0 0
0.5 0 0,5
-0.5 0 1.0
-0,5 0 0
-0.5 0 0,5 1.5
0 0 0,5
0 0 0
0 0 0 0 0
0
90
, 90 90 90 I 90 ’ 90
#e
Q2
’ Ql
In all cases
I#0 1.0 1.0 ia0 1.0 1.0
O,l 0,2 0.3 0.4 0,5 0.6
088 1.0 0,8 0,4
0.59
-.. ~. -
0
0,525 0 0.55 O&2 0,575 0,2
0.55 0.2 0.6 Oa6 0.625 Oa2
0.5 Oa8 0.55 0.8 0.625 042
0.525 1.0 0,55 1-o 0.55 OS8
0,55 0.525 0.612 0.675
0,525 1.0 0,625 1.0 0.575 1,o
0.575 1.0 0,55 1.0 086 1.0
Rpcx c
c 0 0 0 0 0 0 30 30 35 30 30 30 35 30 32 31 30 30 30 30 30 30 30 30 30 30 30 30 32
77.2 74.9 72,5 69,Y G7.2 G4,4 8487 92.9 86.1 88.8 9080 90,7 85,6 go,9 9080 93.9 90,o 92,l 99,G 9382 9386 94,l 96,8 91,o 91,o 960 95,6 93,l 90.0
-
-1085
-10 -10 -10
-10 -10 -10
-10 -10 -10
-10 -10 -I5
-10 -10 -10,l -10
-10 -13.2 -10
-
/ior =
0 0 0 0 0 0 -10 -15 -10
err
... -. _.. -
E’
4,
A~,J
SATISFY THE CONDITIONS
A,, A,, Ap , A,,),, A,,p AND r, = 1.0.
WHICH
OF PAHAMETLRS
VALULS
OF
EXAMPLES
TABLE 4
--
0,395 0,396 0.365 0.299
0.764 0.734 0.710 0.665
_ _
0.691 0,662 0.720
0.739 0.417 0.709 0,458 O.G70 0.409
.
0.643 0.753 0,755
0.683 0,632 0.497
0.878 0.800 0.612
1,417 1.471 1,547
1,466 1,635 1,670
1.414 1.485 1.623
1.425 1,456 1,024
1.696 1,397 1,577 1,687
1.361 1.128 1.379
1.345 0,998 1,507
0.985 I.036 1,089 1,148 1,209 1.275
I”,,,
I,25
1.00 1.03 1.08
I a00 1.05 1.13
0.79 0.87 1.08
AP
WITH
-2.3 -2.2 -2.9 -4.2
-1.9 -1.9 -2.3
-1,6 -1.4 -1.8
-4,4
-3.1 -3.6 -3,5
-3.4 -387 -4.0
-2.5 -3.0 -4.1
-2.0
-1.2 -1.7 -1.5
-1.5 -1.7 - 1.8
-1.3 -1.G -2.0
-1.5 -1.7 -1.4
-2.2 -1.5 -2.1 -2.5
-1.5 -1.6 -1.9
-1.5 -1.2 -2,G
0,48 6.5 5.2 -0.18 -1.0 4.0 -1.8 2.6 I.1 -2.5 -0.4 -3.4
A,,
0870 -2,3 0.76 -2.7 O.GO -2.3
0,73 0,66 0.80 1.01
0.55 0.49 0.61
0.49 0.39 0,54
0.50 0.50 0.50 0.49 0.49 0.48
A,
FOR THE APPARATUS
O.G90 1.769
0.816 0.756 0.689
0.844 0.793 0,804
1,002 0.876 0.955
1.105 1,033 0,960 0.731
1,251 1.539 1,140
1,706 1.586 1.464
1,766 1,861 1.965 2,086 2,220 2.371
d
0.935 0.862 0.935
0.930 0.985 0,826 0.676
1.246 1.179 1,137
1.060 1.058 0.958 0.935 0.958 0.816 0.664
1.423 1,588 1.410
1.041 1.042 1.043 1.044 1,050 1.059
I’,
1.143 1.223 1.070
1.317 1.249 1.182 1.116 1.052 0.989
r,
A8 - A,, = Aaa = A~J = 0
-0.03
-0.19 -0.13 -0.09
-0,06 -0.13 -0.20
-0.11 -0.13 -0.11
-0.23 -0.18 -0.22
-0.16 -0.12 -0.15 -0.08
Oa96 0*99 0.59 0.50
- on73 .!_ OS54 - 0027
- 0,82 - 0.72 f 0:58
1
- 1.29 - 0.51 - 0,94
- 0*94 - 1.23 - 1.18
-21 -14 - 8.4 - 4,l - 1.4 - 0.36
A/v
FIELD AND
0.10
0,27 0.14 0.23
- 0,07
- 0,28 - 0.26 - 0.16
On26 - OS35 , 0.19 - 0425 0.22 - 0.14
0.35 ’ 0.25 0,27
0.35 0.29 0.36
0,38 0.46 0.41 0.27
0,46 0.42 0.44
0.87 0.49 OS0
-0.20 -0.17 -0.14 --0. I4 -0.44 -0.17
-5.1 -1.0 1.7 2,8 2.3 -0,35
AYiJ
ELECTRIC
-0.33 -0.08 -0.22 -0.80 -1.93 -3.73
4,
TOROIDAL
5
(
228. .
_: :
:
in
of the equipotential surface the vertical direction and,R: = dR,/dr,: F&rue 3- ’ shows an example of the behaviour of the coefficients A,, _AXfl. and .AA,,when .C is changed keeping the-conditions A, = A, = A,,=.A&= ;A,, = 0.In the .&se4, = 90°, and- normal entry and exit from the magnet&zfield, an optimum con-. dition is obtained at c w 0.5 and the values of A,, are. reduced to about: l/20 of : -_ the value for cylindrical &se c = 0. _ By changing other parameters R:, Ql,_Q2, a’, E”, etc., much better improvement can,be obtained. In Tabl.6 4 examples of a favourable set of parameters and values of ApY,.ArPzind.ABBare given together with .the values of A,, and A, which indicate the nature of vertical focusing. As can be seen from the table, the values of the aberration coefficients decrease With an increase of the value of 4, until. .q%, = 112”, where all the aberration coefficients have values less than. 0.1. The combination. of parameters shown in the last. row of Table 4 (c$== 112”, 4, -= 909 c = 0.59, Ql = -4.0, E’ = 32”; a:’ = l&Y) has excellent focusing properties. The second order aberration coefficients are all quite small and -the parameters of the apparatus are all suitable for practical construction. As for .vertical-focusing; ions are once focused in the magnetic field and diverge again after passing through the magnet. This may be unfavourable from the view point .of getting a small image s&e but it is favourable to pass as many ions as possible through the narrow magnet gap.
APPARATUS
WITH CYLlIiDRICAL
ELECTRIC FIELD AND ELECTROSTATIC
QUADRUPOLELENS
A remarkable improvement can be obtained ‘by introducing an electrostatic quadrupole lens between the electric.and magnetic fields: The Q-lens is used to defocus the ion beam on the,median plane. In this dase four additional profile planes are necessary. The planes ql.and q2 are respectively.the front and back side of an ideal boundary at the entrance to the Q-lens and th_eplanes-q3 and q4 are those-at -the exit. The iufh.Ace of the fringing field is given by .a transformation between ql. and q2 and that between q3 and- q4 in a tiay similar. to that of the magx&d or electric field These transform&ions. and the transform&ion. which describes an id&l -Q-lens of length L and field.constant K,, are given by the follow.ifig-equatio&
[7]_
:
.EntranCe fringing field: jc,Z .=
: .. ‘Y _..
.. 1
.[l+K,2l,]x,l._t[~~~I~]tanag,-~K~I,]x,,6 : : :
-:. ‘..
‘.
: :
:. _.
-.;..:. .. ..;(f2>
I-
:
‘.
:.::
_:
.:
:
-. . . _.
-.
Jdeal el&&ost&ic
q&rtip~le
leti: (defocusiiig ‘on”the’&dian
&&ie$
:
.229
.:. '-
..
-y
l.. -.[g& + sin l&o .. .: tan &,a =. [K. -. $
yq3
=
J-%3 = J%fringing
L-&
s-t.. m
G K&:
sin /X0 L]x,, . .
[sin hKo&+KoLcos
[co&.Ko
-.
..
:
Gn h& L -_3L cos :hKo .....
(161.
+ [cos .hKo L] tan cxq2 :. hK,~]x,,d-[~Kj~~sin
K, L]
-wo sinK* L-j$,+ + [cosK,
hICoLjaq;6
(17)
(18)
pq2
p)
L]~qz.
field:
_xq4 =-Cl-K~I,]x,,+[2K$I,] tan
:.
tan aq3+[K~11]xq38
a,, 2 [--K&]x~~+~+K&]
-
tan aq,-[CKiI,]ccq36
(20) (21)
Yq4
=
Yq3
(W
P q4
=
893.
(23)
The field
constantKo
is given
by
K;=ek,
(24)
..2u,
where k. is the field gradient in the Q-lens and U, the eneygy of. amain path particle. The quantities II, I2 and I, are definite ‘integrals which-depend on the fringing field distribution and are given by I;
= k,:
P
(25)
where k(z)/ko is a distribution fun&n of $jc &eId gradient-along the Z-&S, the origin.- df which is on the -ideal bound&y;- Using :the distribtitiori cu% which ii - m&&&i w&h a mi@i&Q-J&s hid &su&i&t& &iks :of .fhe insciibed--crrcle ‘. &he &e&odes- t6 .&e0;07r,, #I&+-followitig &II& &e”&taine& iI .+~O.d;lO% . _-~~r2:.~~~()_O&02~34 andI ~-&o()OO1&3. ;_I :I.: ;l:..:-.- . ...-::..:-::-_ ~_... ;... .;.I : -._ _-
230 If the electrostatic quadrupole lens has a defocusing action on the median above, vertical focusing can be attained at the same time. An example of the effect of such a Q-lens is shown in Fig. 4 where the variation of the aberration coefficients A,,,,, A,,fl and A, are plotted against the value of K,,. In this case the length of the Q-lens, L, is chosen to be 0.22r, and the conditions A,=A,=A,= A,,= A,,= 0 are of course satisfied. The other parameters together with the values of A,,A, and A, are given in the first six rows of Table 5. As can be seen in Fig. 4 very small values of aberration coefficients are obtained for the most suitable focusing power of the Q-lens (k,z 1.85, L = 0.22). By adjusting the parameters of the apparatus, many favourable designs are found and examples of them are listed in Table 5. As compared with the apparatus with a toroidal electric field (Table 4), a significant feature of these instruments is that the incident angle to the magnetic field is small and negative and the exit angle is zero. Normal entry and exit is obtained by the apparatus with parameters in the line marked by * 1. The apparatus marked by *2 has quite small aberration coefficients. Ail six coefficients are smaller than 0.01 and therefore complete second order double focusing can be achieved with this arrangement. The change of each aberration coefficients is shown in Fig. 5 when one of the parameters is changed while the other constant. In this case the first order double focusing condition is satisfied by adjusting the values of dd,,the distance between the Q-lens and the magnetic
plane as mentioned
04
0 J -a1 z ;: -0.2 r: 2 -0.3 00 6 -04 4
-25
-20
L92
Lpd
L%K,
-1.5 ()2
Fig. 5. Variation of aberrationcoefficients of the apparatus Ati; 3: Aad; 4:-A,,,.; 5: Ass; 6: AB8. _-
marked by *2 in Table 5. 1: A,;
2:
0 -2.0 0 -2,o 0
-3,5 -3.5 -4.0 -4.0 -4.0
-3,5 -4,o -4,o -14.5 -4,o
-4.0 -4.0
-4.0
85 85 85 85 85
‘90 95
85
80 80 80
s80
80
0 0 0
-3s -3,5 -4.0
75 75 75
1.65 1.91
-_-
---..--..
1.915 1.92
1.875 1.91 I.91 1.933 1.91
1.86 1.885 1.895 1.94 1.895
1.848 1.85 1.885
1.855 1,84 1.83
0 0.5 1.0 1.5 1.8 2.0
&
-285
0 0
0 0 -2.5 0 -2,5
0 0 0
-3s -3s -3,5
70 70 70
0 0 0 0 0 0
Q2
-4,o -4,o -4,o -4,o -4,o -4,o
Qi
80 80 80 80 80 80
I_-_-
dc
70.5 73.8
73,7 74.1
71.9 72,5 70,7 73.3 70.4
=
0.869 0.935 1.108 1.304 1.384 1.194
--_-
I',
1.173 1.048
1,276 1.270
-II -10.5
-18 -21
--_------_..-
1,319 1,293
I.431 1.290 1.303 1.216 1.272
1.300 1,272 1.290 1.250 1.299
1.300 1,299
1.564 1.550 1.442 1.403 1.410
1.314 1.317 1.280 1.265 1.287
-10 -4 -9 - 1.8 -11 -13.5 -15 - 9.5 -14 - 10.5
1.757 1.680 1.539
1.322 1.336 I.299
0.15
0.15 0.15
0,l 0.15 0.15 0.15 0.2
0815 0815 0.15 0.15 0.2
0015 0.25 0.25
-_-
1,035 1,049
1,064 1,036
1,lGO 1.129 1,094 1,105 1,071
1.188 1,169 1.176 1.147 1,154
1,271 1,230 1,227
0815 1.418 OS21 1,401 0830 1.350
0.48 0.44
0843 0.42
0.42 0.44 0.49 0.47 0.49
0,43 0,50 0.47 0.58 0.46
0.46 0,43 0047
0851 0.47 0,45
0.58 0.57 0056 0.57 0,62 0,72
4x
.._--_--a--
0.957 1.111
0.787 0.686
0.941 0.894 I.153 0.926 1.130
1,083 1.439 I.127 I.634 1.058
I.369 1,195 1,246
1.793 1.573 1,415
1.046 1,065 1.144 1,349 1,612 1,998
4,,, -k
0.15 1.762 0,15 1.667 0,15 1.440 0,15 1.178 0.15 1.004 Oa15 0,887
dcq
AYp AND
1.325 “I.979 1.343 1.934 1.357 1.858
1.306 1.303 1.299 1.285 1.253 I.365
i-
A,,, np, I$,,,
- 3.5 -8 -7
3,5 0 - 3.5
_-
8'
-_._-
70.8 67.9 71,l 66.9 7181
68.8 G8.7 68,7
65.3 65.8 65.9
73.9 74.2 74,4 73.2 71.5 68.8
d,,
AND ELECTROSTATIC Q-LENS AND VALUES OF A,,
EXAMPLES Or PARAMETERS WHICH SATISFY THE CONDITIONS nrx = .+
--1.7 1.8
-1.5 -I,4
- I& -1.7 -1.8 -1.7 -1.7
--1.7 -2,0 -1,8 -2,3 -I,8
-1.9 -1.8 -1.9
-283 -2.1 - 1.9
1.1 0.9 0.2 -0.9 -I,8 -206
Av
-1.9 -1.7
-1.0 -0.G
-1.4 -1.4 -2.0 -1.5 -1.9
-168 -2.9 -2,l -3.5 -1.8
-2.8 -2.3 -2.5
-4.5 -3.8 -3,l
7,7 7,O 4,s I,2 -188 -482
_____^
A,J ---
-
0.17 0,25
_----.-._
-0,lO -0.11
0.04 0.09
-
0.14 0.13 0,lO 0.12 0,lO
-0.20 -0.20 - 0,09 -on15 -0,07 -0,19 -0.20
- 0,lO ’ - 0.01 - 0,07 0,oi *2 - 0.10
0,14 0.03 0.10 O,tM 0.12 -0.14 - 0.08 -0.12 -0,Ol -0.13
- 0808 0,OG *4 ‘5
0,17 0,20
0,08 0.04 *3 0,03 0.05 0,07
- 0.10 - 0,15 - 0,lO
- 0.03 - 0,Ol ‘1 - 0,06
-28.2 -23.6 -12.7 - 2,70 - 0,85 - 1892
%P
0.10 0,08 0.07
0,03, 0.03 0,09
-9.09 -6,8l - I,93 1.29 0.68 -0.85
AYP
- 0.09 -0.12 -0*09
-IO,OG - 0,OY - 0.09
-0.31 -0.05 0.43 0.57 0.33 0.08
A,
A,, = Ana = A&j = 0 FOR THE APPARATUS WlTH CYLlNDRlChL LLECTIUC FIELD App In all casts except for lust two rows, L = 0,22, h” = 0, rn, = 1.0. For the row marked by *4L = 0,3, E” = 0, r,,, = 1,Oand by *5 L = 0.22, .z” = -5”, I‘y = 1.0, --_ ---_-1-m
TABLE 5
z
232 field, and I’,. It is seen from Fig. 5 that the change is not very rapid and therefore_ the condition for second order focusing is not very critical. The only undesirable property of this arrangement is that the vertical focusing point is in the magnetic field. After passing through the magnet ions diverge again and at the image position their vertical position is given by eqn. (3) with Ap = -2.3, A, = -3.5. This focusing nature may be better for a small instrument because the vertical spread of ion beam-is narrowest in the magnet gap. For a big instrument, however, the spread of ions after the magnet would be serious if the aperture angle in this di, rection is large. The vertical focusing tends to be improved with increasing 4, or 4,. However, the aberration coefficients unfortunately increase with @,. The optimum condition for both requirements would be in a region c$, - 85”. For a large instrument, however, the ratio y,/r,,, would be small and therefore a small increase in the values of App and A, would be tolerable. From these considerations the instrument marked by *3 in Table 5 would be suitable for a big instrument such as a high resolution on-line mass spectrograph. The schematic drawing and the vertical trajectory of an ion in such an instrument is shown in Fig. 6. Examples of an apparatus having an oblique exit E” and different lengths of Q-lens L are shown in the lower part of Table 5 (*4 and *5).
-a
I
SCURCE
Fig. 6. Schematicdrawing of a double focusing mass spectrometer of second order with a cyiindriCal electric field- and an electrostatic quadrupole lens (*3 in TabIe 5). Vertical ion kajzctory is shown in the lower part. _
CONCLUSION -_ _
x!-w+d be possible to construct A complete se&d &let double focusing ,masS-~trometer.consi~~g of a toroidal elekic field _@d a; h6mogt+&,ti~g_ -_-, -,
233 netic field or consisting of a cylindrical electric-field ~1% electrostitic quad.xupole lens and a homogeneous magnetic field--The latter-design would be adaptable to a large instrument because the vertical focusing is better and machining would be easier. Straight boundaries of the magnetic field are adopted for practical rekons. - In the calculations described above the influence of fringing fields tire-taken into consideration with an accuracy effectively of third order.- The values of parameters given in this paper will be changed slightly if the distribution of the fringing field is different from that assumed here. Therefore, the parameters which satisfy the condition for second order focusing should be calculated using the distribution function for a respective apparatus. The expected change in parameters, however, are relatively small as shown in Table 2.
REFERENCES
1 H. Matsuda and T. Matsuo. ht. J. Maw Spectrom. Ion Ph>s_, 6 (1971) 385. 2 H. Matsuda and T. Matsuo, Atomic Masses and Fur&mental Constants, 4 (1972) 192. 3 H. Matsuda, Mass Spectrosc. (Tokyo), 21 (1973) 15. 4 R. He-g, Phys. Z., 41 (1940) 23. 5 H. Matsuda. Nucl. Instrum. Methods, 91 (1971) 637. 6 H. Matsuda and H. .Wollnik, Nuci. Instrum. Methock, 77 (1970) 40,2.+?3. 7 H. Matsuda and H. Wollnik, Nucl. Instrum Methonk, 103 (1972) 117.