Ion optics of multipole devices. I. Theory of the dodecapole

and Ion Processes, 63 (1985) 17-28 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

International Journal of Mass Spectrometry

ION OPTICS OF MULTIPOLE DODECAPOLE

A.J.H. BOERBOOM Chemistry

Department,

*, DOUGLAS

DEVICES.

B. STAUFFER

I. THEORY

17

OF THE

and F-W. McLAFFERTY

Cornell University, Ithaca, NY 14853

(U. S. A .)

(First received 20 March 1984; in final form 16 July 1984)

ABSTRACT Properties are established for an ion-optical device consisting of twelve parallel rods placed equidistantly on a circle. In comparison with conventional devices, this dodecapole features improved deflection and quadrupole focusing of ion beams, two-directional focusing, correction of second- and third-order aberrations, and the simultaneous application of combinations of these properties.

INTRODUCTION

In ion optical instruments (direction of ion propagation, z) it is well known [l] that any z-independent electrostatic or magnetic field can be considered as a superposition of multipole fields. In conventional mass spectrometry, the magnetic sector field and, if present, the electrostatic analyser are dipoles and thus deflect the ion beam. This makes it possible that magnetic and electrostatic fields can produce, respectively, a momentum and an energy dispersion. Deflection is a zero-order effect, i.e. its magnitude is, to a large extent, independent of the initial position or direction of the ions in a beam. The focusing effect of a magnetic sector field, realized in all conventional mass spectrometers, is mainly caused by the field boundaries; a magnetic slab with parallel entrance and exit planes has no focusing effect, as has been shown by Takeshita [2], All multipoles other than dipoles produce no deflection of the main ion trajectory, which thus shows no dispersion. Ion focusing can be realized by quadrupole lenses with stationary fields (i.e. not mass filters with high frequency fields). Giese 131 introduced a quadrupole doublet (two quadrupole lenses in sequence) into a mass spectrometer ion source in order to * Visiting Professor. Permanent address: FOM-Institute, The Netherlands. 0168-1176/85,‘$03.30

Kruislaan 407, 1098 SJ Amsterdam,

0 1985 Elsevier Science Publishers B.V.

18

shape the ion beam. Takeshita [2] used two quadrupole lenses as the focusing elements in a magnetic mass spectrometer. Matsuda [4] used a quadrupole lens to increase the transmittance and to reduce the magnitude of secondorder aberrations. Tuithof and Boerboom [5] installed a magnetic quadrupole in the first field-free region of a single focusing mass spectrometer and an electric one in the second field-free region. In this way, they succeeded in varying the dispersion of the magnetic analyser by a factor of 60 and in orienting the focal plane. A quadrupole field produces focusing properties in the median plane and defocusing in the perpendicular plane (X and y axes, respectively). Quadrupole focusing is a first-order effect, i.e. the focusing of ions emerging from an ion-optical object is approximately independent of their original direction. Magnetic field boundaries resemble quadrupoles, having different focusing properties in the median plane and in the perpendicular plane. Adding auxiliary electrodes on either side of each round quadrupole electrode can give a more accurate quadrupole field for either focusing [6] or mass filter [7] applications. The most common other type of multipole is the hexapole, introduced in mass-spectrometry by Boerboom [8]. It corrects second-order imaging aberrations, e.g. image curvature and opening aberrations. Octupoles, which were introduced by JEOL [9], correct for third-order aberrations. Matsuo, et al. [lo], describe a 12-pole device with unsymmetrical pole locations such that rod combinations representing quadrupole, hexapole, and octupole configurations can be selected. This study examines the properties of a dodecapole device (Fig. 1) consisting of parallel cylindrical electrodes whose ends are equally spaced on the circumference of the two circles .corresponding to the ends of a right cylinder; the cross-section of centers corresponds to a regular dodecagon. By proper application of potentials on four or six equally spaced rods, either a quadrupole or a hexapole lens, respectively, can be produced. However, the twelve potentials of the rods are twelve independent variables, which should in turn provide capabilities for twelve ion-optical functions. Of these, it is shown that useful applications can be expected w-hen the device is in one, or

000 0

0

0 0

0 0 OOO

Fig. 1. The dodecapole

electrode configuration

studied.

19

a combination, of the two dipole modes, two quadrupole modes, or two hexapole modes, in the einzel lens mode, or in the dodecapole mode. The last four potential distributions are improper octupole and decapole modes apparently of lower utility. Only electrostatic multipoles will be treated here, but the theory is also applicable to magnetic multipoles. THEORY

The functions V, = r” cos n( 9 + $), where V is the electrostatic potential for n = 0, 1, 2, . . . . satisfy Laplace’s equation AV= 0; the radius r and the angle 8 are the polar coordinates, and + is an arbitrary phase angle. These functions form a complete set, i.e. any z-independent potential distribution V( r, 8) can unambiguously be represented by a linear combination of these functions. A potential distribution of the shape Vn(~, 6) = r” cos n(6 + +,) represents a 2n-pole field, so V, = r cos( 8 + $) is a dipole field, V, = r2 cos 2(8 + $J~) is a quadrupole field, etc. Electrodes are used to form or approximate such fields experimentally; for example, parallel plates produce a pure dipole field. Two parallel rods of equal length in the z direction and of opposite potentials (Fig. 2) produce a potential distribution which is not a pure dipole field, but can be described by a superposition of multipole fields, given by

E (4

J+, 0) =

cos no+

B,, sin n8>

n=l

0)

Symmetry with respect to the plane 8 = 0 gives V( r, 0) = - V( r, -S), and anti-symmetry with respect to the plane 8 = TIT/~gives V( I-, t?) = V(r, w - 0). Substitution of these identities in Eq. (1) gives B = 0 and n = odd, consequently V(r, 0) = E A,yn cos n6

n = 1,3,5 . . .

(2)

n=l

Thus the electrode configuration of Fig. 2 produces a field which can be

Fig. 2. Dipole

electrode configuration.

Fig. 3. Quadrupole

electrode configuration.

20

considered as a superposition of dipole, hexapole, decapole, 14-pole, etc. fields. The hexapole and higher fields can be viewed as the higher harmonics of the dipole field. However, the quadrupole, octupole, dodecapole, etc. fields are not the natural higher harmonics of the dipole field, as they possess a different type of symmetry. A quadrupole lens configuration as depicted in Fig. 3 produces a field which is the superposition of quadrupole, dodecapole, ‘20-pole, etc. fields. To produce a pure quadrupole field, the electrodes should have a hyperbolic cross-section [ll], i.e. the contributions of the higher fields will be zero. Also, by proper choice of the rod diameter (1.15 times the diameter of the inscribed circle [12]), the dodecapole field component vanishes, approximating the pure quadrupole field up to terms of the 20th order. The addition of auxiliary electrodes on either side of each quadrupole electrode [6,7] also can reduce this dodecapole component to zero. Another way to remove this component is to use the dodecapole device proposed here. However, this electrode configuration has many interesting additional properties, as will be shown below. THE DODECAPOLE

*

The potential field produced by an assembly of twelve parallel rods placed equidistantly on a circle has twelve degrees of freedom, i.e. the twelve potentials of the electrodes, so each field can be considered as the linear superposition of twelve elementary potential distributions. Selection of these (Table 1) was made analogously to the method used by Boerboom [8] for the hexapole configuration. As distribution I, a constant potential of + 1 unit of voltage on each rod was chosen. The potential field obtained possesses the highest possible symmetry: the field is invariant through a rotation over 30”, 60”, . . . 360”: cm ntl= cos n( 8 + 7r/6), which applies for n = 0, 12, 24, . . . So the field resembles closely a constant potential, as it exists inside a cylindrical electrode, only disturbed by a 24-, 48-, etc. pole component. Distribution II is obtained by giving the successive electrodes the potentials cos 8. The resulting field (Fig. 4) resembles closely the pure dipole field r cos 8; Eq. (2) is applicable because its symmetry requirements are fulfilled by the electrode geometry and potentials of distribution II. The function of and I = 1, such Eq- (2 sh ould take the values cos 8 for 8 = 30, 60,. . -360” that CA,

cos ~8 = cos 8 for 0 = 30°, 60°, . . _360°.

In &-der to find A n, a method analogous to Fourier analysis was used. IJyth sides were multiplied by cos 8 and added for all 12 values of 8. Now c cos ~&OS 8 becomes zero for n = 0, 2-10, 1 * Patent pending.

12, 14-22..

.,

and the higher

-l/2

0

x,2

l/2 l/2 o/2 0 1 - l/2

II

III IV

~0~8

c0s 28

IX

X

XI XII

sin 48

code

sin 58 cos 68

38

sin 38 cos 48

cos

V VI VII VIII

sin 28

sin 8

1

-1

l/2

a/2 -a/2

-fi/2 1

-&/2 l/2

-1 0 -l/2

m2

fi/2 - l/2

:,2 1

1

0

0

1

0 0

-1

-1

-1

-fi/2 1

- l/2

j3/2

-F2 0 -l/2

&/2 - l/2

8=120’

1

9=90°

4

I

8=60”

8=30°

3

Constant

2

1

Electrode

No.

Distribution function

-1

l/2

G/2

-a/2

-m2 0 1 -l/2

l/2 l/2

-x,2

8=150°

5

0 1

0 1

-1

0

-1O 0 1

-:

8=180”

6

a/2 -l/2 -1

o/2

l/2 a/2 0 -1 -l/2

-l/2

-b/2

e=210”

7

Twelve independent potential distributions on the dodecapole electrode configuration

TABLE 1

F/2

-m2 -l/2

0 -l/2

(3/2

-fi/2 -l/2

-l/2

1

e=240°

8

-1 -1

0

0

0 0 1 1

-1 -1

0

1

e=27O”

9

F/2

l/2

a/2

-o/2 -1 0 -l/2

l/2 -o/2 - l/2

1

e=JOO”

10

-l/2 -1

-/x/2

-o/2

-o/2 0 -1 -l/2

-l/2 l/2

;,2

e=330°

11

0 1

1

0

0 1 0 1

0 1

:

e=360°

12

22

I Cylindrical

fl

Vl

VIII

X

Dipoles

rrr

tbxlpolor

VKI

0ctupolss

Decapo1ar

XI

Fig. 4. The twelve potential distributions for the dodecapole device of Fig. 1.

23

order terms of Eq. (2) drop out except for n = 12-fold &- 1. Thus the field produced is much more a “pure” dipole field than that produced by the two electrodes of Fig. 1. The reason that the eleventh and thirteenth order terms do not drop out is illustrated graphically in Fig. 5. -The values of cos 9, cos 110, cos 138, etc., become equal at the 12 values of 8 = 30°, 60°, 9o”, _. . 36OO. This is true because cos 8 - cos 118 = 2 sin 68 sin 58, cos 8 cos 136 = 2 sin 78 sin 68, and sin 68 = 0 for & = 30”, 600,. . -360”. Thus the deviation of distribution II from the pure dipole distribution has the shape of a combination of 22-, 26-, 46-, 50-, etc. pole fields. The magnitude of the coefficients A,, depends on the relative diameter and shape of the dodecapole electrodes, but the higher-order contribution is very small because it is proportional to r” -I. In the region Y c R/2 (R = radius of the circle of electrode centers) this contribution is less than 0.1% (OS”), going exponentially to zero as r goes to zero. Potential distribution III (Fig. 4) produces exactly the same field as distribution II, except that it is rotated by 8 = 90°. Linear superposition of weighted distributions II and III can produce a dipole field in any direction. Distribution IV has quadrupole symmetry (Fig. 4) so only the 12-, 20-, etc. higher harmonics are possible. However, the voltages applied on the electrodes in between, numbers I, 2; 4, 5; 7, 8; 10, 11, eliminate the dodecapole field component. The 20-pole field component remains, but has only a minor influence on the quadrupole field (proportional to Yn-2 = r8). In ordinary

Fig. 5. The functions cos 8, cos 118, 90,...360”.

and cos 138 which produce equal values for 8 = 30, 60,

24

quadrupoles choosing the radius of the rods, R,, as 1.15 (R - R,), reduces the dodecapole field component nearly to zero [12]. In the dodecapole electrode configuration, the dodecapole field component is eliminated irrespective of the radius of the rods, but their radius might be chosen to minimize the 20-pole field component also. Distribution V arises when distribution IV is rotated through 45” (Fig. 4), one-eighth of a full turn. Due to the fact that the number of electrodes is not divisible by 8, other potentials must be applied. Thus the maximum and minimum voltages, + 1 and - 1, respectively, are not achieved in this distribution. Whereas in the case of the dipole distributions II and III, the field remains exactly the same after 90° rotation, now there is this minor change, but it only appears in the 20th harmonic. A linear combination of IV and V gives a quadrupole field in any desired direction, for which the magnitude of the 20th-order term will be changed. Distribution VI does not essentially differ from the field in an ordinary hexapole electrode configuration. Because in the latter a dodecapole field component is essentially absent, the extra rods have no appreciable effect. Indeed, they are at zero potential. These extra rods could shield the hexapole field from penetration by outside fields. Rotation through 30” yields distribution VII. Rods are present at the proper positions so the field remains exactly the same. Distributions VIII and IX impose octupole fields on the twelve electrodes. Now, the disturbing component is the 16pole fieId which is superimposed on the octupole field. The intensity is proportional to r4, so for rmax= R/2 the contribution is less than 6%. Distributions X and XI are decapole fields which are still more seriously disturbed by 14-pole field components. Distribution XII is a pure dodecapole field with a 36-pole field as the first higher harmonic, which is completely negligible as it depends on YI2 . However, this field cannot be rotated; sin 68 = 0 for all rod positions. Distributions II-XII all have the property that the sum of the charges on the rods is zero, as for any linear combination of these distributions. In order to represent any type of potential distribution on the rods, at least one should have a net charge. This is the case with distribution I. It remains to be shown that distributions I-XII are linearly independent. This follows from the fact that the determinant D, of the matrix of Table 1 equals 93312 = 12 x 65 and so is not zero. APPLICATIONS

As applications, we consider only the use of the dodecapole electrode configuration as a time-independent device to influence ion beams traversing

25

it in the axial direction and in the neighborhood of its axis. Distribution I produces, to a high approximation, a constant potential. When the dodecapole is of finite length and situated in a cylindrical space of different potential, it resembles very much an einzel lens, so it will show focusing effects with almost the quality of a genuine tubular lens. Like all einzel lenses, it acts as a positive lens, adjusting the focal distance only if it is too long. But this action can be combined with each of the following effects or with a linear combination of them. The effect on ion trajectories of distribution II, which is almost a pure dipole field, is clear: this distribution produces an almost homogeneous electric field and all ions are deflected in the x direction over essentially equal angles. Theoretically, a uniform dipole field such as this is produced by two flat and parallel electrodes, These electrodes, however, should be very long in the y direction in comparison with their x spacing to avoid penetration from the sides. Obviously, the dimensions of electrodes are limited by the diameter of the ion flight tube, giving the dodecapole electrode configuration in mode II a substantial advantage over parallel plates. The former provides a nearly ( > 99.9%) homogeneous field for the cross-sectional area of diameter R (the tube diameter is - 2R plus the electrode diameter). However, the length of parallel plates should be at least five times their spacing [13]; thus, plates inside such a tube can be only - 0.4R apart. Another advantage of the dodecapole over the parallel plates appears in the case of mechanical maladjustment, i.e. when the symmetry plane of the device does not coincide with the median plane of the main instrument but is tilted over an angle At?. Although for parallel plates only a cumbersome correction is possible, the dodecapole electrode configuration allows an elegant solution simply by applying potentials cos( 8 - A@) on the electrodes. When A0 becomes 90”, the potentials are those of distribution III, producing y axis deflection. For the quadrupole modes IV or V, similar considerations can be made. In comparison with a quadrupole lens consisting of four cylindrical electrodes, the dodecapole configuration is better because it compensates for the dodecapole field component. For the lens with four hyperbolic electrodes, a perfect quadrupole field is only produced by extending the hyperbolic surface infinitely in the x-y plane, limiting the cross-section in which a high quality quadrupole field can be produced. Further, the hyperbolic surfaces of such electrodes are much more difficult to make precisely than are the cylindrical electrodes of the dodecapole configuration. Again, the dodecapole configuration makes possible the rotation of the quadrupole field, such as for rotational alignment correction, with a continuous transition to the oblique quadrupole mode of distribution V. Also, a correction is possible if the main trajectory of the ion beam does not coincide with the axis of the

26

dodecapole configuration, while in an ordinary quadrupole this would cause the ion beam to be deflected. In the dodecapole device, a potential distribution V = r2 eos 28 can be applied where r is the distance of the rods from the actual axis, so that the rods are no longer considered to lie on the circle. For a hexapole field, the dodecapole configurations do not constitute an essential improvement except for unlimited rotation of the hexapole field (such as to correct rotational maladjustment) and to compensate for misplacement of the ion beam axis. Possibly the most important property of the dodecapole configuration, however, is the unique feature that it can combine its excellent quadrupole and hexapole (and other) functions. Thus, the quadrupole field can be used for focusing and the hexapole field can be used simultaneously for the correction of image curvature and opening aberrations. Experimental tests of such a dodecapole configuration will be described in a subsequent paper in this series, as will the combined use of dodecapole configurations before and after a magnet. Further, an octupole component can be superimposed on these quadrupole and/or hexapole fields using mode VIII of the dodecapole configuration. fn applications that require only low octupole fields, such as correction of third-order aberrations [9], the l&pole component produced by the dodecapole will be less serious. However, a genuine octupole configuration employed as a separate device is undoubtedly better. DECOMPOSITION OF AN STANDARD DISTRIBUTION

ACTUAL POTENTIAL MODES I-XII

DISTRIBUTION

INTO

THE

As the twelve distribution modes form a fundamental set (determinant # 0), any actual potential distribution can be considered as a linear superposition of these standard modes. This decomposition can be realized by solving twelve equations with twelve unknowns of the form J$ = CAi,V,, but can more easily be accomplished using Table 2. This was obtained by inverting the matrix of Table 1 and mirroring the resulting matrix across its main diagonal. Considering the twelve potentials of the electrodes as a column vector, multiplication of this vector at the left-hand side by the matrix of Table 2 gives the contributions of each of the twelve standard modes and so facilitates the study of the effect of a particular potential distribution on an ion beam. The determinant D,t of the matrix of Table 2 is l/93312. The distributions I-XII ma be normalized by multiplying each of them by a constant J ck such that c c;V;.’ = 1. For distributions I and XII, this constant would i- 1

l/12 l/12

a/12 0

III IV

V VI

l/12

-l/12

- &/12

XI XII

J1;/12

X

IX

l/6 -l/12

G/l2

II

VII VIII

l/12

1

Electrode

I

Distribution

0

l/12

- fi/l2 l/I2

l/12

- &/I2

- l/12

D/l2 -l/6 0

l/6 -l/12

0

I/6 0

0 0 -l/6

l/6 --l/6

l/12

3

l/12

G/l2 - l/l2

2 l/12

l/12

-a/12 l/12

-l/12

a/12 l/12 -l/12

b/12

-h/12

l/6 -l/12

-&/12 0

-a/12 l/6 0 -l/12

l/12 -l/12

-&‘12

5

Wl2 -l/12

-l/12

4 l/12

l/12

0

-l/6

I/6 0

0

l/6 0 -l/6

0

-l/6

6

-l/12

-l/12

G/l2

e/12

- l/12

- l/6

fi/12 0

a/12 l/12

-&‘12 -l/12 -l/12

l/12 a/12 l/12

0 - l/6 -l/12

-l/12

-e/12

-l/6 -l/12

-e/12 0

-l/12 l/12

\/7/12

l/12

11

&/12

- &12

0 -l/12

-e/12 -l/6

0 0

l/12

l/12 q/12 -l/12

0

l/12

10

-l/6 - l/6

9

l/6 I/6 0

l/6 0 -l/12

a./12

-a/12 -l/12

-l/12 l/12

l/12 -l/12

l/12

8

-o/12

7

TABLE 2 Matrix to determine the contribution of distributions I-XII from the values of the twelve electrode potentials

l/12

0

l/6

0

l/6 0 l/6

l/6 0

0

l/6

l/12

12

28

be l/m and for the other distributions, l/ 6. In this case, the determinant D, would be 1. Calculation of the mirrored inverse of this normalized matrix I would show that it is invariant through this operation. CONCLUSION

Twelve electrodes in a dodecagon configuration constitute a valuable device for influencing ion beams. The most important applications are deflection in two perpendicular directions, quadrupole focusing, correction of second- and third-order aberrations, and two-directional focusing. A unique feature is that several of these applications can be effected simultaneously by this single device. Further studies on the ion orbits and experimental realization of the dodecapole configuration are in progress. ACKNOWLEDGEMENTS

A.J.H. Boerboom thanks the FOM Foundation of Fundamental Research on Matter for leave of absence and Cornell University for their invitation. Financial support was provided by the National Institutes of Health and the Army Research Office, Durham. REFERENCES 1 P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. 2 I. Takeshita, Mass Spectrom. (Jpn.), 23 (1975) 173. 3 C.F. Giese, Rev. Sci. Instrum., 30 (1959) 260. 4 H. Matsuda, Int. J. Mass Spectrom. Ion Phys., 14 (1974) 219. 5 H.H. Tuithof and A.J.H. Boerboom. Int. J. Mass Spectrom. Ion Phys., 20 (1976) 107. 6 H. Matsuda and T. Matsuo, Int, J. Mass Spectrom. Ion Phys., 24 (1977) 107. 7 R.D. Board, Hewlett-Packard, Palo Alto, CA, 1971. 8 A.J.H. Boerboom, Int. J. Mass Spectrom. Ion Phys., 8 (1972) 475. 9 DX300 Mass Spectrometer, JEOL Ltd., 1418 Nakagami Alcishima, Tokyo, Japan. 10 T. Matsuo, H. Matsuda, H. Nakabushi, Y. Fujita and A.J.H. Boerboom, Int. J. Mass Spectrom. Ion Phys., 42 (1982) 217. 11 W. Paul, H.P. Reinhard and H. von Zahn, Z. Phys., 152 (1958) 143. 12 I.E. Dayton, F.C. Shoemaker and R.F. Mozley, Rev. Sci. Instrum., 25 (1954) 485. 13 E. Durand, Electrostatique, Tome 2, Masson, Paris, 1966.