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Fringing-field integrals of clamped quadrupoles
Nuclear Instruments and Methods in Physics Research A323 (1992) 580-582 North-Holland
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Fringing-field integrals of clamped quadrupoles H. Irnich, B. Pfreundtner, B. Hartmann and H. Wollnik II. Physikalisches Institut, Universität Giessen, D-W-6300 Giessen, Germany
Received 3 June 1992 Using the method of surface charges, the fringing field of electrostatic quadrupole lenses is investigated and the result is compared to measured field maps of magnetic quadrupoles of low flux densities. For different field clamp geometries third-order fringing-field integrals are calculated . 1. Introduction The investigation of the fringing fields of quadrupole lenses requires precisely measured field distributions or quite exact three-dimensional calculations . We have chosen to do calculations by the surface charge method [1,2] which has been known [3,4] for quite some time and recently has also been used for fringing-field calculations [5,6]. For our calculations we used the program IEQ3 of ref. [5] implemented [7] by a three-dimensional automatic mesh generator. In detail we calculated and measured the fringing-field distribution of a magnetic quadrupole that was driven by Co-Sm permanent magnets [8]. Such a device we had chosen for comparison since it can be better approximated by calculations of electrostatic potentials than a magnetic quadrupole driven by large coils. The cross section of this quadrupole is shown in fig. 1 together with the mesh of charge points chosen for corresponding potential calculations. Assuming no fringing-field clamps, the calculated go(z) =aB(O,O,z)g/ax in the fringing
field is shown in fig. 2 together with the measured distribution for a quadrupole . As one can see the agreement is quite good. Additionally, in fig. 2 also the calculated distribution go(z) of the gradient of the fringing-field distribution is shown for electrodes of circular cross section, i.e . for the case that the 120° segments of fig. 1 were replaced by 360° full circles. As one can see, the agreement in the main fringe-field region is quite good, however, there is a distinct disagreement in the tail of the fringing-field distribution . From this comparison between theoretical calculations and a measurement distribution we felt confident that our calculations reproduce the real fringing-field distribution quite accurately .
2. The fringing-Geld integrals Originally the method of fringing-field integrals was introduced for sector fields [8-13] only later it was
Fig. 1 . The cross section of a permanent-magnet driven quadrupole is shown together with the 120° electrode form and charge mesh used for a potential calculation by the surface-charge method. 0168-9002/92/$0_5.00 © 1992 - Elsevier Science Publishers B.V . All rights reserved
581
H. Irmch et al. / Fringing field of clamped quadrupoles
0.8 0.6
C
04 0.2
Fig. 2. The measured (stars) and the calculated (circles for full electrodes) distribution of the field gradient g(z) along the optic axis of the quadrupole shown in fig. 1. The solid line was calculated for the shown 120° electrodes . extended to the case of quadrupole lenses [14] . The basic idea here is that one defines an "effective field" that rises and falls abruptly at some "effective-field boundary" B where the position of this field boundary is chosen such that the integral along the optic axis over the "real field gradient g(z)" and over the "effective field gradient g*(z) [i .e . g *(z_
Table 1 fringing-field integrals
11, 72 ,
13,
z*)=go]" are equal. Then trajectories are traced from outside the quadrupole into the main field for both field distributions where the trajectories traced through the effective field are bent and shifted slightly when they cross the effective field boundary . These bends and shifts here must be chosen such that the trajectories are identical in the main field, independent of which fringing-field distribution was used . In any case it should be noted that the focusing effects of quadrupole lenses are mainly due to the strength and length of the main field region while the image aberration are mainly due to the existence of the fringing fields [15] . The advantage of the method of fringing-field integrals lies in the fact that the largest integrals can be solved exactly as long as Laplace's equation is assumed to be correct throughout the fringing field. Consequently it is sufficient to solve the fringing-field integrals in some approximation for which the fringing field distribution must be known only roughly. The result of a calculation that uses fringing-field integrals is thus usually more accurate than a ray tracing procedure in which the field distribution is calculated by some finite element method that fulfills Laplace's equation only to perhaps three decimals . For third-
14
G
D
B
1,
I2
1,
0.0147 0.0202 0.0285 0.0386 0.0591 0.0759
14 -0 .1587 -0 .1823 -0 .2149 -0.2495 -0 .3089 -0.3494
0.50
0.25 0 .50 0.75 1.00 1.50 2.00
-0 .0728 0.0932 0.2126 0.2981 0.4044 0.4606
0.0531 0.0649 0.0828 0.1047 0.1524 0.1959
0.0036 0.0046 0.0052 0.0047 -0 .0018 -0 .0156
0.0175 0.0230 0.0310 0.0407 0.0604 0.0767
-0 .1802 -0.2003 -0 .2283 -0.2589 -0 .3132 -0 .3515
0.75
0.25 0.50 0.75 too 1 .50 2.00
-0 .0098 0.1365 0.2405 0.3160 0.4121 0.4642
0.0713 0.0815 0.0974 0.1171 0.1608 0.2014
0 .0018 0 .0024 0.0025 0.0015 -0 .0055 -0 .0193
0.0231 0 .0281 0.0355 0.0444 0.0628 0.0782
-0 .2098 -0.2252 -0 .2475 -0.2729 -0 .3202 -0.3549
1 .00
0.25 0.50 0.75 1 .00 1.50 2.00
0.0940 0.1978 0.2782 0.3399 0.4225 0.4691
0.0960 0.1034 0.1165 0.1334 0.1721 0.2089
-0 .0023 -0 .0018 -0 .0021 -0 .0035 -0 .0110 -0 .0246
0.0315 0.0354 0.0418 0.0496 0.0661 0.0801
-0 .2427 -0 .2529 -0 .2696 -0 .2896 -0 .3291 -0 .3595
0.25
0.25 0.50 0.75 1 .00 1.50 2.00
-0.1047 0.0690 0.1963 0.2875 03997 0.4585
0.0425 0.0548 0.0739 0.0972 0.1474 0.1926
0.0043 0.0055 0.0065 0.0064 0.0004 -0 .0135
582
H Irnich et al. / Fringing field of clamped quadrupoles of the fringing-field clamps shown in fig. 3. Assuming a quadrupole of aperture 2G o the fringing-field integrals are shown in table 1 for different values of the distance
D between the clamps and the end of the full circle
electrodes and for different clamp apertures 2G . The values of table 1 agree resonably to those of ref [6),
though there are main noticable differences when comparing the values of the effective field boundary .
References
Fig. 3. Section of quadrupole lens equipped with fringing-field clamps . This geometry was used for the calculation of the integrals in table 1.
order calculations the fringing-field integrals [14] I ' I2=
1
Zb
=
ko f= a f =a 1
zn r zl
kof=a
1
I4=
1
kz0 f
k(Z ) dz
zb
az- 1/2zb,
z f =
k(z) dzl dz -1/3zb, 2
dz - 113z
3
dnk(z) Zdz-z b ,
must be determined accurately for different geometries
[1] E. Munro and H.C. Chu, Optik 61 (1982) 1 . [2] P.W . Hawkes and E. Kasper, Principles of Electron Optics, Vol. I (Academic Press, 1989). [3] A.B . Birtles, B.J . Mayo and A.W . Bennet, Proc . IEEE 120 (1973) 213. [4] M. Eupper, Thesis, Tiibmgen (1985). [5] M. Str6er, Optik 81 (1) (1988) 12. [6] M. Ishihara and T. Matsuo, Nucl . Instr. and Meth . B70 (1992) 445. [7] H. Irnich . Thesis Giessen (1991). [8] H. Wollnik, Thesis München (1964). [9] H Wollnik and H. Ewald Nucl . Instr. and Meth 36
(1965) 93 . [10] H. Matsuda and H. Wollnik Nucl . Instr. and Meth . 77 (1970) 40, 283. [111 H. Matsuda, Nucl . Instr. and Meth . 91 (1971) 637. [12] T. Sakurai, T. Matsuo and H. Matsuda, Int. Mass Spectr . and Ion Phys . 91 (1989) . [13] Sagalovsky, Nucl . Instr. and Meth . 298 (1990) [14] H. Matsuda and H. Wollnik, Nucl . Instr. and Meth . 103 (1972) 117. [15] H. Wollnik, Nucl . Instr. and Meth . B56/57 (1991) 1096 .
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Fringing-field integrals of clamped quadrupoles H. Irnich, B. Pfreundtner, B. Hartmann and H. Wollnik II. Physikalisches Institut, Universität Giessen, D-W-6300 Giessen, Germany
Received 3 June 1992 Using the method of surface charges, the fringing field of electrostatic quadrupole lenses is investigated and the result is compared to measured field maps of magnetic quadrupoles of low flux densities. For different field clamp geometries third-order fringing-field integrals are calculated . 1. Introduction The investigation of the fringing fields of quadrupole lenses requires precisely measured field distributions or quite exact three-dimensional calculations . We have chosen to do calculations by the surface charge method [1,2] which has been known [3,4] for quite some time and recently has also been used for fringing-field calculations [5,6]. For our calculations we used the program IEQ3 of ref. [5] implemented [7] by a three-dimensional automatic mesh generator. In detail we calculated and measured the fringing-field distribution of a magnetic quadrupole that was driven by Co-Sm permanent magnets [8]. Such a device we had chosen for comparison since it can be better approximated by calculations of electrostatic potentials than a magnetic quadrupole driven by large coils. The cross section of this quadrupole is shown in fig. 1 together with the mesh of charge points chosen for corresponding potential calculations. Assuming no fringing-field clamps, the calculated go(z) =aB(O,O,z)g/ax in the fringing
field is shown in fig. 2 together with the measured distribution for a quadrupole . As one can see the agreement is quite good. Additionally, in fig. 2 also the calculated distribution go(z) of the gradient of the fringing-field distribution is shown for electrodes of circular cross section, i.e . for the case that the 120° segments of fig. 1 were replaced by 360° full circles. As one can see, the agreement in the main fringe-field region is quite good, however, there is a distinct disagreement in the tail of the fringing-field distribution . From this comparison between theoretical calculations and a measurement distribution we felt confident that our calculations reproduce the real fringing-field distribution quite accurately .
2. The fringing-Geld integrals Originally the method of fringing-field integrals was introduced for sector fields [8-13] only later it was
Fig. 1 . The cross section of a permanent-magnet driven quadrupole is shown together with the 120° electrode form and charge mesh used for a potential calculation by the surface-charge method. 0168-9002/92/$0_5.00 © 1992 - Elsevier Science Publishers B.V . All rights reserved
581
H. Irmch et al. / Fringing field of clamped quadrupoles
0.8 0.6
C
04 0.2
Fig. 2. The measured (stars) and the calculated (circles for full electrodes) distribution of the field gradient g(z) along the optic axis of the quadrupole shown in fig. 1. The solid line was calculated for the shown 120° electrodes . extended to the case of quadrupole lenses [14] . The basic idea here is that one defines an "effective field" that rises and falls abruptly at some "effective-field boundary" B where the position of this field boundary is chosen such that the integral along the optic axis over the "real field gradient g(z)" and over the "effective field gradient g*(z) [i .e . g *(z
Table 1 fringing-field integrals
11, 72 ,
13,
z*)=go]" are equal. Then trajectories are traced from outside the quadrupole into the main field for both field distributions where the trajectories traced through the effective field are bent and shifted slightly when they cross the effective field boundary . These bends and shifts here must be chosen such that the trajectories are identical in the main field, independent of which fringing-field distribution was used . In any case it should be noted that the focusing effects of quadrupole lenses are mainly due to the strength and length of the main field region while the image aberration are mainly due to the existence of the fringing fields [15] . The advantage of the method of fringing-field integrals lies in the fact that the largest integrals can be solved exactly as long as Laplace's equation is assumed to be correct throughout the fringing field. Consequently it is sufficient to solve the fringing-field integrals in some approximation for which the fringing field distribution must be known only roughly. The result of a calculation that uses fringing-field integrals is thus usually more accurate than a ray tracing procedure in which the field distribution is calculated by some finite element method that fulfills Laplace's equation only to perhaps three decimals . For third-
14
G
D
B
1,
I2
1,
0.0147 0.0202 0.0285 0.0386 0.0591 0.0759
14 -0 .1587 -0 .1823 -0 .2149 -0.2495 -0 .3089 -0.3494
0.50
0.25 0 .50 0.75 1.00 1.50 2.00
-0 .0728 0.0932 0.2126 0.2981 0.4044 0.4606
0.0531 0.0649 0.0828 0.1047 0.1524 0.1959
0.0036 0.0046 0.0052 0.0047 -0 .0018 -0 .0156
0.0175 0.0230 0.0310 0.0407 0.0604 0.0767
-0 .1802 -0.2003 -0 .2283 -0.2589 -0 .3132 -0 .3515
0.75
0.25 0.50 0.75 too 1 .50 2.00
-0 .0098 0.1365 0.2405 0.3160 0.4121 0.4642
0.0713 0.0815 0.0974 0.1171 0.1608 0.2014
0 .0018 0 .0024 0.0025 0.0015 -0 .0055 -0 .0193
0.0231 0 .0281 0.0355 0.0444 0.0628 0.0782
-0 .2098 -0.2252 -0 .2475 -0.2729 -0 .3202 -0.3549
1 .00
0.25 0.50 0.75 1 .00 1.50 2.00
0.0940 0.1978 0.2782 0.3399 0.4225 0.4691
0.0960 0.1034 0.1165 0.1334 0.1721 0.2089
-0 .0023 -0 .0018 -0 .0021 -0 .0035 -0 .0110 -0 .0246
0.0315 0.0354 0.0418 0.0496 0.0661 0.0801
-0 .2427 -0 .2529 -0 .2696 -0 .2896 -0 .3291 -0 .3595
0.25
0.25 0.50 0.75 1 .00 1.50 2.00
-0.1047 0.0690 0.1963 0.2875 03997 0.4585
0.0425 0.0548 0.0739 0.0972 0.1474 0.1926
0.0043 0.0055 0.0065 0.0064 0.0004 -0 .0135
582
H Irnich et al. / Fringing field of clamped quadrupoles of the fringing-field clamps shown in fig. 3. Assuming a quadrupole of aperture 2G o the fringing-field integrals are shown in table 1 for different values of the distance
D between the clamps and the end of the full circle
electrodes and for different clamp apertures 2G . The values of table 1 agree resonably to those of ref [6),
though there are main noticable differences when comparing the values of the effective field boundary .
References
Fig. 3. Section of quadrupole lens equipped with fringing-field clamps . This geometry was used for the calculation of the integrals in table 1.
order calculations the fringing-field integrals [14] I ' I2=
1
Zb
=
ko f= a f =a 1
zn r zl
kof=a
1
I4=
1
kz0 f
k(Z ) dz
zb
az- 1/2zb,
z f =
k(z) dzl dz -1/3zb, 2
dz - 113z
3
dnk(z) Zdz-z b ,
must be determined accurately for different geometries
[1] E. Munro and H.C. Chu, Optik 61 (1982) 1 . [2] P.W . Hawkes and E. Kasper, Principles of Electron Optics, Vol. I (Academic Press, 1989). [3] A.B . Birtles, B.J . Mayo and A.W . Bennet, Proc . IEEE 120 (1973) 213. [4] M. Eupper, Thesis, Tiibmgen (1985). [5] M. Str6er, Optik 81 (1) (1988) 12. [6] M. Ishihara and T. Matsuo, Nucl . Instr. and Meth . B70 (1992) 445. [7] H. Irnich . Thesis Giessen (1991). [8] H. Wollnik, Thesis München (1964). [9] H Wollnik and H. Ewald Nucl . Instr. and Meth 36
(1965) 93 . [10] H. Matsuda and H. Wollnik Nucl . Instr. and Meth . 77 (1970) 40, 283. [111 H. Matsuda, Nucl . Instr. and Meth . 91 (1971) 637. [12] T. Sakurai, T. Matsuo and H. Matsuda, Int. Mass Spectr . and Ion Phys . 91 (1989) . [13] Sagalovsky, Nucl . Instr. and Meth . 298 (1990) [14] H. Matsuda and H. Wollnik, Nucl . Instr. and Meth . 103 (1972) 117. [15] H. Wollnik, Nucl . Instr. and Meth . B56/57 (1991) 1096 .