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The trapping condition and a new instability of the ion motion in the ion cyclotron resonance trap
ELSEVIER
International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
and Ion Processes
The trapping condition and a new instability of the ion motion in the ion cyclotron resonance trap L. S c h w e i k h a r d a'*, J. Ziegler a, H. B o p p a, K. Lfitzenkirchen b alnstitut ffir Physik, Johannes Gutenberg-Universit~'t, D-55099 Mainz, Germany blnstitut ffir Kernchemie, Johannes Gutenberg-Universitdt, D-55099 Mainz, Germany Received 5 August 1994; accepted 4 October 1994
Abstract
In analogy to the critical mass, merit, a critical voltage, gcrit, (and a general trapping parameter, 7rtrap ) is defined, above which the ion motion in an ion cyclotron resonance (ICR) trap is unstable and the ions are lost from the trap. The theoretical values for the critical voltage are confirmed by experimental results. Singly charged gold cluster ions, Au2, of several sizes, n = 50, 60, 76, 100, 110, and 145 (the latter corresponding to an ion mass of 28 560 u), were injected into an ICR trap, stored, and detected by axial ejection and single ion counting using a microchannel plate detector. During the storage period the trapping voltage, U, was varied for extended durations (from a few to over one hundred milliseconds), to probe its effect on the trapping efficiency. In addition to the expected instability above the critical trapping condition (m/merit = U~ g c r i t = 7rtrap = 1), a new instability was observed below the critical value. This is explained in terms of a resonance effect occurring for a trapping parameter 7rtrap = g/gcrit = m/merit = 8/9, which corresponds to the smallest possible integer ratio of the eigenfrequencies of ion motion, i.e. co+ = w: = 2w (where co+, w~ and co stand for the reduced cyclotron, the trapping and the magnetron frequency, respectively).
Keywords." ICR; Penning trap; Critical mass; Trapping parameter; Instability of ion orbit
1. Introduction
In ion cyclotron resonance (ICR) traps, charged particles are stored via a superposition of static magnetic and electric fields. These traps are used, for example, in high resolution Fourier transform-(ICR) (FTICR). (For reviews on ICR traps see Refs. [1]-[7], and references cited therein.) There are two restrictions on the ions that can be stored: (i) their electric charge is determined * Corresponding author.
by the polarity of the electric field; (ii) the ions that can be stored cover the range from "zero" up to a maximum mass-over-charge ratio. This upper mass limit is also called the "critical mass", for which a charge state of unity, i.e. one elementary charge, is assumed. In the light of recent developments in the trapping of large biomolecules and cluster ions, the critical mass has gained considerable importance. In this work the ion motion in the region of the critical mass is studied in detail. The measurements reported here were performed using an ICR trap with hyperbolically
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L. Sehweikhard et al./lnternational Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
shaped electrodes, usually called a Penning trap. The experimental setup [8] was designed for the capture [9] and investigation of metal cluster ions [10-15]. Since these experiments may involve high-mass ions of low charge states, questions concerning the stability of the ion motions are of considerable interest. Therefore, the high-mass behavior of the Penning trap was probed by the storage of singlycharged gold cluster anions, Aun-, as a function of the trapping voltage. In this paper we report a comparison of the theoretical mass limit for an ideal Penning trap with the experimental results. Further, we report the observation of a new instability occurring in a mass region that had previously been considered to be stable. The next section reviews the ion motion in a Penning trap with emphasis on the interrelation between the eigenfrequencies, followed by a short description of the experimental setup. Finally, the experimental results are described and discussed.
2. Ion
where P0 and z0 are the shortest distances from the electrode surfaces to the trap center. Other electrode shapes, e.g. cylindrical, cubic, or elongated, that may be preferable for specific experimental purposes, also lead to an approximately quadrupolar field in the center of the trap. For ease of discussion and calculation we therefore only consider in this paper an ideal Penning trap (if not otherwise stated explicitly). With electrode geometries given by Eqs. (3a) and (3b), the trap dimension, d is represented by [1] d = l(p02 q-- 222) 1/2 (4) (An ideal Penning trap field is also created by a "two-electrode trap" [16], where d has to be redefined as in Ref. [17].) With the above fields the Lorentz force leads to the following set of differential equations for the ion motion: d2x
The ideal Penning trap consisting of a homogeneous magnetic field
d2y dx m-~+qB-~t t
~-~y = 0
(1)
and an axial quadrupolar electric field (2)
dt 2
(3a)
and two endcap electrodes with the surface equation p2 _72 = _72q_.~_ (3b)
dy -We dt
-
dZY q- aJc dx dt 2 dt dZz
serves as the paradigmatic model of ICR traps. The electric field is obtained by applying a potential difference, U, between a ring electrode obeying the surface equation z 2 = l(p2 _ p2)
qU
(5a) (5b) (5c)
These may be rewritten as dZx
U /~ = 2-~ (x, y, -2z)
qU
2d---5x = 0
d2z qU rn~+~-z = 0
trapping
B = (0, 0, B)
dy
m - ~ - qB dt
wz x 2
--
=
0
2
,~z y = 0 2
(6a)
(6b)
2
dt 2 ~-:VzZ= 0
(6c)
where the cyclotron frequency (in the absence of an electric field) wc = q B m and the (axial) trapping frequency
(7)
qU ) 1/2 wz=
~-~
have been introduced.
(8a)
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
79
2.1. Stable ion motion
Eq. (6c) obviously describes a simple harmonic oscillator of (angular) eigenfrequency COz, decoupled from the "radial" ion m o t i o n in the xy dimensions. Based on the conditions CO++ CO- = COc
(9)
CO+CO_= co2/2
(10a)
the frequencies CO+ and CO_ are defined: COc CO+ =5--4-
X
(lla)
and Eqs. (6a) and (6b) m a y be further rewritten as d2x
dt 2
(co+ + co_)
~t
_.~Y
- co+CO_x = 0
(12a)
dZy + (CO+ dx ~dt + CO_)~ - CO+CO_y= 0
(12b)
F r o m Eqs. (12a) and (12b) follows a radial ion trajectory given by x = p+ sin(CO+t + ~+) + p_ sin(CO_t + ~_) y = p+ cos(co+t + ~+) + p_ cos(co_t + ~_) (13a) According to these equations the ions move in epicycles, where the (angular) eigenfrequencies co+ and CO_ are given by the trap parameters (B, U, d), and the radii p+ and p_, and phases ~+ and ~p_ are given by the initial conditions'. The ion m o t i o n is a superposition of the axial harmonic trapping m o t i o n with the trapping frequency, COz, and two radial (i.e. in the xyplane) circular motions with the reduced (or modified) cyclotron frequency, CO+, and the m a g n e t r o n frequency, co_. The cyclotron and m a g n e t r o n modes of the ion m o t i o n are decoupled and their amplitudes are independent of each other; for example, note that p+ > p_ or p+ < p_. However, the corresponding frequencies are not independent.
Fig. 1. Ion motion for w+ = 4COz,wz = 8w_, and p_ = 4 Z a m p = 8p+: ---, magnetron motion only;..., combined magnetron and trapping motion; - - , full ion motion, i.e. combined magnetron, cyclotron, and trapping motion.
Rewriting Eq. (10a) we find that the ratio between the reduced cyclotron frequency and the trapping frequency is half the ratio between the trapping and m a g n e t r o n frequencies: co+ _ 1 COz coz 2 co_
(10b)
Fig. 1 shows the ion motions for co+ = 4coz, COz= 8CO_, and p_ = 4Zam p = 8 p + (where Zamp is the amplitude of the trapping motion). It is useful to introduce another combination of frequencies, i.e. that of the oscillations of the ions to and from the z-axis: COp= CO+ - co_
(14a)
The index " p " stands for "parametric," since a modulation of the trapping voltage at Wp leads to an excitation of the radial ion m o t i o n [16,18] with both p+ and p_ increasing exponentially with time [19]. (Analogously, FTI C R detection m a y be performed with signals at COp [17,19,20].) 2.2. The trapping parameter 7ftrap
So far it has been tacitly assumed that the argument of the square root of Eq. ( l l a ) is
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L. Schweikhard et al./International Journal o f Mass Spectrometry and Ion Processes 141 (1995) 77 90
1,0
non-negative, i.e. co2/4 ~> co2/2
(15)
0,9
Inserting expressions (7) and (8), this may be rewritten as
0,8
d2 B2
d 2 qB 2
2
(16b)
m
where (m/q)cri t is the well-known critical massover-charge (usually expressed as critical mass, mcrit, with the assumption of a charge state of one) and Ucrit is the critical trapping voltage analogously defined for easier comparison With the experimental results (see below). Eq. (1 la) may now be rewritten as co± = ~- 1 +
1
(mT-q-)-crit.]
]
1--gcrit/
(llc)
The critical condition may also be expressed in terms of the magnetic field, B, or the trap dimension, d, with all other parameters kept constant• For a general discussion, including the simultaneous variation of several trap and ion parameters, we introduce the dimensionless trapping parameter _
_
2Um d2B2q
(16c)
which now gives an expression for the radial eigenfrequencies of co+ -----211 + (1
- 71trap) 1/2]
(lld)
for the radial parametric frequency of cop = coc( 1 -
"a-trap) 1/2
Vz
LL 0,3 0,2
o1[ 0,2
0,4
0,6
0,8
1,0
TRAPPING PARAMETER, 7Ztrap
J
71.trap=2~ - U m/q coc U--~rit (m/q)crit
0,6 >0 0,5 Z LLI D O' 0,4 UJ
0,0 0,0
or
co:~=-~- 14-
.'~
(16a)
or
-
Vp
0,7
m/q <~ (m/q)crit - 2 U
U ~ Ucrit
V+
(14b)
Fig. 2. Eigenfrequencies of ion motion ~+, ~o~, and ~o_, and parametric frequency ~op (normalized to the cyclotron frequency) as a function of the trapping parameter 7"rtrap;this may be expressed as the ion mass-to-charge ratio, m/q, normalized to the critical ion mass-to-charge ratio, (m/q)crit, or equivalently as the trapping voltage, U, normalized to the critical trapping
voltage Ucrit.
and for the trapping frequency of COz = COc(71"trap/2) 1/2
(8b)
Fig. 2 shows the eigenfrequencies of the three modes of ion motion (co+, co_, coz) and the radial parametric frequency (COp) for values of the trapping parameter 0 ~< 7rtrap ~ 1. Note that for negative values of 71-trap the axial ion motion is unstable. This corresponds to the fact that for static electric trapping fields either positive or negative ions may be trapped depending on the polarity of the trapping voltage, but never both species simultaneously (see Eq. 8). To illustrate the behavior close to the
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L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
m a x i m u m possible trapping parameter for stable ion orbits, let us now consider a special set of initial conditions under which the ions are stationary at a given distance, Pini, from the trap center (from the z-axis, to be exact; a possible additional trapping m o t i o n is not relevant for this consideration): x = 0, y = Pini, dx/dt = dy/dt = 0. This leads immediately to the phases and radii of the ion motion: ~_ = 0 ,
g~+ = ~-
CO_
P+ =
-
Pini
cop -
(°r
P-
co+ =
~-p Pini
3o
Vp=V_/8~
...
p++p~/
--.=_ 25 -= z ) 20 ~) 15
7 ~
10
U
P± = 1 ( 1 ( 1 - - rCtrap)l/2 ) 17rtrap) / ~ Pini
1-1/10
1-1/100
1-1/1000
TRAPPING PARAMETER, ntrap and Eq. (13a) read for this case Fig. 3. Cyclotron and magnetron radii, and their sum, p+ + p , as a function of the trapping parameter, 7rtrap, for the following initial conditions: zero velocity in the x and y coordinates; position x = 0, y = Pini
X = Pini co+ sin(co_t) cop co -- Pini--Y- COS(CO+t) cop co+ Y = Pini
-
-
cop
for 0 ~< 7"ftrap ~ 1 COS(CO_t)
0.)_
+ P i n i - sin(co+t) cop
(17a) The radii are shown in Fig. 3 as functions of the trapping parameter; note that as the trapping parameter approaches unity, the radii increase rapidly. Fig. 4 shows the radial motions for the initial condition as discussed above for selected ratios between the cyclotron and magnetron frequency (co+/co_ = 10, 5, 2, 3/ 2, and 5/4, corresponding to the trapping parameters 71-trap= 40/121, 5/9, 8/9, 24/25, and 80/ 81, respectively). It is immediately clear that for a real Penning trap with a finite ring electrode of m i n i m u m radius, P0, only ions with P+ + D- = coc/°"p Pim = Pini/( 1
-
-
for example, ions are created or captured with zero radial kinetic energy at a radial distance of 1 m m from the center of the trap, their orbit will extend to 3 m m (distance from the axis) for 7rtrap = 8/9 and to 1 cm for 7rtrap : 1 - 1/100. In the limit 71-trap--+ 1 (corresponding to co+ = co- = coc/2), the amplitude of the radial m o t i o n approaches infinity linearly as a function of time: X = Pini sincoCt
2 coct
coct
-- Pini--~- COS "--~'-
Y -- Pini COS
COct
for 71"trap ~--- 1
coct. coct
q- Pini ~ - sln-~-
7rtrap) 1/2 < /90
remain trapped. All the others hit the trap electrode (within one period of the cyclotron motion) and are lost. Fig. 3 shows (p+ + p_)/ Pini as a function of the trapping parameter. If,
(17b) F o r trapping parameters above the critical v a l u e , 7rtrap > 1, the radii approach infinity
82
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
even faster:
where coo is defined as
cosh(co0t) sin c°ct
X = Pini
2
~)C
•
smh(c°°t) cos Wet
-- Pini ~ 0
T
0d c
f o r 71trap
cosh(co0t) cos 2 t
Y ~-- Pini
COct
+ Pini T
.
= ~ - (Trtrap -- 1)1/2
> 1
f o r 71-trap ~> 1
(18)
The general solution for the radial equations of motion (6a, 6b) may be written as
~cl
sinh(co0t) sin T
x = p+ sin(u;+t + 9~+) (17c)
/
+ p_ sin(a;_ t + ~_) for 0 ~< 7rtrap ~ 1
y = p+ cos(a;+t + 9~+) + p_ cos(w_t + ~_)
(3
03 =,s,
~+/~.=10 . . . .
I
I
. . . .
(13a)
'
'
'" '
i
i
i
X = Pconst s i n ~ - -
-F Ptime ~ -
--}- qOconst
ct.s l n (Wet \ 2 +
ggtime
/a~ct Y o~+/ m_-- 3/2 '
I
. . . .
I
,Oconstc o s ~ -
)
\
[ for
-~- ggconst )
71"trap z
1
]
wct /co c t \ I q- Ptime T C O S ~ T q- 99time) J
. . . .
(13b) where the subscripts " c o n s t " and " t i m e " indicate the terms of constant and time dependent Table 1 Some critical tune values, the corresponding trap parameters and the duration of an overall period (in units of the period of the magnetron motion, I/u_, and one period of the cyclotron motion in the absence of a trapping field, 1/Uc) m+ / m_ = 5/4 ,
I
. . . .
I
~z/&_ . . . .
I
i
i
i
i
I
i
i
i
i
60p/¢0
7rtrap
Toveral
I [l/v_]
Zoverall
I
Fig. 4. Radial ion motions (projection of ion motion in the xyplane) for initial conditions as in Fig. 3 for a selected number of ratios between the cyclotron and magnetron frequency (w+/w = 10, 5, 2, 3/2, and 5/4, corresponding to trapping parameters 7rtrap= 40/121, 5/9, 8/9, 24/25, 80/81, respectively). The central cross indicates the position of the trap axis (x = y = 0). The distance between the minor ticks on the axis corresponds to the minimum distance of the ions from the trap axis, and is equal to the y-coordinate value for the initial condition.
2
3 4 5 6 7 8 9 10
1
3.5 7 11.5 17 23.5 31 39.5 49
0.88889
1
3
0.59504 0.39506 0.27435 0.19945 0.15071 0.11754 0.09406 0.07689
2 1 2 1 2 1 2 1
11 9 27 19 51 33 83 52
[1/vc]
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 7~90
83
amplitudes, respectively, and
x = PchCOSh(a~0t) sin(\ c°d " /C°ct 2 + ~ch ) -I- Psh sinh(aJ0t) smt, ~ - + ~sh ) Y = Pch c°sh(c°0t) COS\ 2
+ Cpch) +
Psh
I
sinh(~0t) cos (Tc°ct + ~)sh) /
where "ch" and "sh" indicate the terms with cosh- and sinh-like time dependent amplitudes, respectively. Eqs. (13a)-(13c)describe the ion motion in an ICR cell in the limit of a perfect Penning trap. The stability of the ion orbits is given uniquely by the single trapping p a r a m e t e r 7rtrap.
2.3. A new instability In reality, no ICR trap is a perfect Penning trap. There are always distortions of the trapping forces which may be more or less severe, depending on the spatial homogeneity of the magnetic field and on the shapes of the trap electrodes. The fields may be shimmed with correction coils and electrodes, eliminating the effects of e.g. electrode segmentation and holes for ion injection and ejection. Other deviations from the perfect forces may even be impossible to correct for, such as the effect of image charges induced in the trap electrodes by the ions themselves [21,22] or variations in the relativistic mass of the ions [23] due to the variation in their kinetic energy along their trajectory. Any of these distortions of the trapping forces from the ideal Penning trap lead to perturbations and will have an influence on the ion orbit. Depending on the trapping conditions, the repeated perturbations may lead to an additional kind of instability of the ion motion and the ions may be lost from the trap. This will only happen when the perturbations do not cancel each other out, but add up due to periodic orbits. This periodicity may include all modes of the ion motion, the 'overall periodicity' (period Toverall) , and means that
for 71-trap) 1
(13c)
the ions return to the same position in the trap to undergo the same perturbation. This occurs if the relevant frequencies are integers or simple ratios of each other. The effect is expected to increase with a decrease in the overall period, i.e. a decrease in the integer numbers p and q of the simple ratios p/q of the frequencies of interest (see Table 1). The phenomenon discussed above is wellknown from accelerator and storage rings for ions and electrons. (For an introduction to this subject see Refs. [24] and [25].) While the particles go around in turns they perform 'betatron' oscillations, i.e. lateral (horizontal and vertical) displacements from their equilibrium orbit. One of the most critical design parameters for a storage ring is its 'tune' (Qh, Qv), i.e. the number of (horizontal or vertical, respectively) betatron oscillations during one turn. A ring can only be successfully operated if betatron resonance effects are avoided, i.e. if both Qh and Qv resemble neither integers nor simple fractions (the so-called stopbands). These values must be avoided for both the horizontal and the vertical tune; otherwise, effects due to magnet imperfections build up turn after turn and such a resonance results in loss of the beam. For accelerator and storage rings, Qh and Qv are generally independent of each other. We now consider the Penning trap and an ion orbit with large magnetron radius, p_, and moderate cyclotron radius, p+, and amplitude of z-motion, Zamp. For this case the "z-tune", or "trapping tune", may be defined as coz/a~_, the number of trapping oscillations, and the "cyclotron tune" may be defined as aJp/a~ ,
84
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90 !
>'
80
U.JZ
60
I--
40
i /
/
(9
Z 12. rl
20
n," I,--
0
<
1000
2000
4000
3000
CYCLOTRON TUNE, vp / v !
I
i !
J~
!.J~.,
Z
_z n 12_
<
n," I-
1
i
0
4
CYCLOTRON TUNE, Vp / v_
Fig. 5. Working diagram for the Penning trap, where the abscissa gives the cyclotron tune ( ~ p / ~ ) and the ordinate the z-tune (wz/~_) as related by Eq. (10c).
the number of oscillations of the ion to and from the trap axis during one period of the magnetron motion. Note that according to Eq. (10b) these ratios are not independent of each other: =
½
-
1
(10c)
An integer value for the z-tune leads automatically to an integer or half-integer value for the cyclotron tune (but not vice versa). Fig. 5 shows the working diagram for the Penning trap, where the abscissa gives the cyclotron tune and the ordinate the z-tune as related by Eq. (10c). The lower figure shows the section of the working diagram for low tune values, with integer and half-integer values of the tunes indicated by solid and dashed lines, respectively. Table 1 lists some
critical tune values, the corresponding trap parameter, 7rtrap , and the length of an overall period, ToveralI (in periods of the magnetron motion l/u, which itself is a function of 7rtrap and, for comparison with an absolute unit, in periods of the cyclotron motion in the absence of the electric field, 1/Uc). Note that the overall period increases with increasing ratio of the eigenfrequencies of ion motion, i.e. for low trapping parameters (low mass, low trapping voltage, etc.). Under these conditions it takes a longer time for the ions to experience the same perturbation, and for standard experiments they will not be affected significantly. However, for high trapping parameters (small frequency ratios) the overall period may be very small. Fig. 6 shows some examples of ion orbits for small ratios of eigenfrequencies (for z-tunes, aJz/~_, of 5, 4, 3, and 2). The most extreme case is a~+ = aL, = 2aJ_; during one period of the magnetron motion the ion performs two oscillations in the axial direction and approaches the z-axis once (a~p = a~_). For even values of the z-tune the ion orbits close after one period of the magnetron motion. For odd values of the z-tune an additional period is needed for the closure.
3. Experimental procedure The experimental setup is described in detail elsewhere [8]. Fig. 7 gives an overview of the experimental sequence. The measurements are performed with gold cluster ions produced from the external source via laser vaporization ( T R I G G E R LASER) of a gold wire in a helium gas jet [26]. Negatively charged ions are guided by several ion optical elements to a Penning trap with a magnetic field of B = 5 T and electrode parameters of P0 = 20 m m and z 0 = po/21/2. The transfer section between ion source and trap allows differential pumping to obtain U H V conditions in the trap region. A single ion bunch is
L. Schweikhard et al./lnternational Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
85
y ~
-
.
.
.
.
.
.
.
.
.
.
.
.
×
Fig. 6. Some examples of ion orbits for small ratios of eigenfrequencies (for durations of 1/u ). Lines as in Fig. 1. In addition, the projection onto the xy-plane is shown below the full ion motion. Top left: ~:/co_ = 5; top right: ~ z / ~ = 4; bottom left: w:/a~_ = 3; bottom right: co=/~ = 2.
captured in flight (CAPTURE PULSE) [27] at a low trapping voltage, U = 3 V. Using this production and capture procedure a preselection of cluster sizes is already achieved [15]. In addition, all unwanted ions are ejected by resonant excitation of their cyclotron
motion (SELECTION) [8]. Only the cluster ions of the size (and mass) of interest remain in the trap. The trapping voltage is then changed to probe the trapping efficiency for a given value (TRAPPING VOLTAGE, Ux). After the trapping voltage is re-established at
86
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
>~ 1.2
I
TRIGGER LASER
CP
CAPTURE
o~ 0.8
z
PULSE
SELECTION
1.0
0.6 [3
!............. i T :
TRAPPING VOLTAGE Ux
Z
0.4
EJECTION
<
0.2
m
E- 0.0
TRANSIENTRECORDER 0
50
100
2
4 6 8 10 12 TRAPPING VOLTAGE [V]
14
150
t [ms]
Fig. 7. Timing sequence of the experiment. See text for details.
U = 3 V, the ions are ejected axially (EJECTION) and detected by a microchannel plate detector for time-of-flight analysis (TRANSIE N T R E C O R D E R ) . This sequence is typically repeated several hundred times, to increase the statistical significance of the results. After each sequence where the trapping voltage is pulsed to a particular value under consideration, an additional sequence is performed without pulsing the trapping voltage (i.e. the trapping voltage is kept constant at U = 3 V) and the resulting data are stored separately. These latter measurements are used to calibrate the trapping efficiencies of the pulsed mode, eliminating, for example, fluctuations in the cluster production. We define the experimental trapping efficiency, e(Ux), as the ratio of the number of ions counted after a sequence during which the trapping voltage has been changed from 3 V to the value of interest, Ux, to the number of ions counted after a sequence during which no such change in trapping voltage occurred.
Fig. 8. Measured trapping efficiency for gold cluster anions, Aul00, and trapping periods of 40 ms as a function of trapping voltage.
(i) most values of the trapping efficiencies are slightly less than unity, indicating that there is some ion loss during the change of the trapping potential; (ii) the trapping efficiency decreases very slowly from about e ~ 0.9 to e ~ 0.75 over a wide range of the trapping voltage, up to Ux ~ 11.5 V (for comparison, Ucrit ~ 12.25 V), and drops to zero for higher values; (iii) there is an outlyer at U ~ 9 V; systematic studies confirmed that this was due to statistical fluctuations. (iv) there is a resonance-like structure, consisting of several data points, at U 10.5 V; this structure has been observed repeatedly.
~
1,00
i
0,98 -
~ 0,96> O 0,94Z0,92 < r~ 0,90N.-
4. Experimental results
0,88. U
Z < 0,86 -
Fig. 8 shows the results of a typical series of measurements. After gold cluster anions (AuT00 with a mass of 19 697 u) were captured and selected, the trapping efficiency, as defined above, was determined as a function of the trapping voltage. The figure shows several features:
n- 0,84 O Z
410
610
8I 0
I 100
I 120
J 140
160
SIZE OF GOLD CLUSTER ION, n
Fig. 9. Experimental critical trapping voltage, (E]), and trapping voltage of new instability (O), normalized to the theoretical value of the critical trapping voltage as a function of gold cluster size. O, trapping voltage of new instability normalized to the experimental value of the critical trapping voltage.
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
The data shown were taken with a storage duration of 40 ms at the trapping voltage under consideration. Measurements with storage durations of 4 and 146 ms yield the same results [28]. The investigations have also been extended to other cluster sizes, n = 50, 60, 76, 120, and 145 (corresponding to m = 9848, 11818, 14969, 21666, and 28560 u). The results are summarized in Fig. 9. The open squares indicate the experimental values of the critical trapping voltage. They were determined from the measurements in the same way as in Fig. 8, where the data points were taken in the region of Ucrit. A straight line is fitted to the data n e a r Ucrit with trapping efficiency values between 20 and 80% of the value for small trapping voltage. This line is extrapolated to the abscissa and the corresponding trapping voltage is defined as the experimental critical trapping voltage. Its values as a function of cluster size are shown in Fig. 9, normalized to the values expected from the parameters of the Penning trap and the ion species under investigation. The open cycles indicate the centre of the above-discussed resonance-like structure, again normalized with respect to the expected critical trapping voltage. Note that these values deviate only slightly from each other, i.e. they show a slow decrease as a function of cluster mass. Further, when normalized with respect to the experimental values of Ucrit (solid circles), they are independent of the cluster mass and compatible with a value of 8/9 (indicated by the dashed line). As stated above, the eigenfrequencies of ion motion are at the lowest possible integer ratios of each other for this value of the trapping parameter and the motions are therefore expected to be unstable. Therefore, in addition to the instability of the ion motion for trapping parameters exceeding the upper limit a new kind of instability is observed for trapping parameters that so far have not been considered to be "critical".
87
5. Discussion and conclusion
The concept of critical mass has been generalized to the critical trapping parameter, 71-trap, and the occurrence of closed ion orbits, in particular those with a small overall period, has been examined. Using large gold cluster ions the dependence of the trapping efficiency on the trapping voltage has been studied. From the data an experimental critical voltage is determined which is slightly below the theoretical value. This deviation is not surprising; even though the ions have little kinetic energy after their capture in the trap, they may still have a finite initial distance from the trap axis. As has been shown above, this leads to very large amplitudes of the radial modes of motion for trapping parameters close to the upper limit. Hence, while the ion orbits are still stable (i.e. do not approach infinity with time) the ions may collide with the electrodes due to the finite trap dimensions. (Alternatively, they may not be able to leave the trap through the exit hole of 5 m m diameter during the axial ejection for time-of-flight detection.) Note that the initial position determines the experimental trapping efficiencies. Therefore, by a series of measurements of the trapping efficiency as a function of trapping voltage, information on the radial distribution of stored ions may be obtained. This method, however, is outside the scope of the present paper. At a trapping voltage of approximately 8/9 Ucrit a new instability was observed which seems to be associated with a closed ion orbit of small period. The origin of this instability has yet to be studied in more detail. Since only a few ions (< 10) were trapped at any one time in the present investigations, the instability is hardly a space-charge (collective) effect, but rather is due to the interaction of single ions with the electromagnetic trapping field. The various ratios of eigenfrequencies of the different modes of the ion motion are not
88
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
__
~
Y
,r"
×
i
Fig. 10. For the case a~+ = a~. = 2cv_, where the experiments indicate an instability o f the ion motion, a few orbits are shown (for details see captions of Figs. 1 and 6). Starting with the top left figure and going clockwise the phase of the cyclotron mode is incremented by 7r/8 for each figure.
independent, i.e. simple ratios of c0z/aJ_ yield simple ratios of a~+/a~ z (Eq. 10b); however, in general the opposite is not true. Therefore it is not clear which particular mode of the ion motion leads to the instability. One may speculate that the nature of the
magnetron motion has a significant influence. The magnetron motion of ions in an ICR trap is metastable [1]. In general, perturbations tend to increase the magnetron radius. For example, ion collisions with neutral atoms and molecules will have this effect [19,29-31],
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
leading to ion loss from the trap if counteracting measures [32,33] are not taken. Further, note that not all ions are lost from the trap under the resonance condition. The question of instability may also depend on the particular phases of the ion motion with respect to trap imperfections, and to the relative phases between the modes of ion motion (Fig. 10). The observed instability is analogous to betatron instabilities of particle accelerator and storage rings where simple ratios have to be avoided for the tunes of the betatron oscillations about the equilibrium orbit (stopbands). The instability may thus be due to a coupling of the radial and axial motions, corresponding to the coupling of horizontal and vertical betatron motions of a particle in a ring. But there are some differences between these latter systems and ion cyclotron resonance traps. For example, there is only one equilibrium orbit in rings while the amplitudes of magnetron, cyclotron and trapping motion in the trap are independent of each other and are not restricted, except for the finite trap electrode dimensions. There is another analogy worth considering. In celestial mechanics the orbits of satellites and planets can in principle be of any radius within a wide range, as for ion orbits in the ICR trap. However, for planets and satellites the periods of these orbits are functions of the radius. As is well-known, the perturbation of additional satellites or planets leads to voids, e.g. the regions between Saturn's rings or the distribution of asteroids in the solar system [34-36]. For example, a particle orbiting in the Cassini division would have half the orbital period of Saturn's m o o n Mimas. Likewise, the Kirkwood gaps of asteroids are explained by gravitational perturbations due to Jupiter. Again, simple fractions of the ratios of orbital frequencies play an important role. The observed instability of ion orbits may be called a "static resonance" [37]: a resonant
89
phenomenon that does not depend on the excitation by a time-varying signal of an external device (e.g. a frequency generator), but occurs for static magnetic and electric fields. This resonant phenomenon may provide another means for manipulating ions stored in an ICR trap. However, further experimental studies and theoretical considerations will be necessary to exploit the analogies indicated, to fully understand the nature of the new instability, and to use this effect for future applications.
Acknowledgments We thank H.-J. Kluge for his continuous support of the Mainz cluster project. A. Streun is acknowledged for helpful discussions concerning the analogies and differences of ion traps and storage rings. The experiments were supported by the Deutsche Forschungsgemeinschaft and by the Materialwissenschaftliches Forschungszentrum of the Johannes Gutenberg-Universitfit of Mainz.
References [1] L.S. Brown and G. Gabrielse, Rev. Mod. Phys., 58 (1986) 233. [2] B. Asamoto (Ed.), FT-ICR/MS Analytical Applications of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry, VCH, Weinheim, 1991. [3] G. Bollen and C. Carlberg (Eds.), Proc. Workshop on Physics with Penning Traps, Lertorpet, Sweden, 1991, Phys. Scr., RS 20 (1993). [4] A.G. Marshall and P.B. Grosshans, Anal. Chem., 63 (1991) 215A. [5] R. Blatt, P. Gill and R.C. Thompson (Eds.), Physics of trapped ions, Special Issues of J. Mod. Opt., 39(2) (1992). [6] A.G. Marshall and L. Schweikhard, Int. J. Mass Spectrom. Ion Processes, 118/119 (1992) 37. [7] J.M. D'Auria, D.R. Gill and A.I. Yavin (Eds.), Proc. Workshop Traps for Antimatter and Radioactive Nuclei, TRIUMF, Vancouver, BC, 1993, Hyperfine Interactions, 81 (1993). [8] St. Becker, K. Dasgupta, G. Dietrich, H.-J. Kluge, S. Knutnetzov, M. Lindinger, K. Lfitzenkirchen, L. Schweikhard and J. Ziegler, in preparation.
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L. Sehweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
[9] H.-U. Hasse, St. Becker, G. Dietrich, N. Klisch, H.-J. Kluge, M. Lindinger, K. Lfitzenkirchen, L. Schweikhard and J. Ziegler, Int. J. Mass Spectrom. Ion Processes, 132 (1994) 181. [10] St. Becker, G. Dietrich, H.-U. Hasse, N. Klisch, H.-J. Kluge, D. Kreisle, S. KriJckeberg, M. Lindinger, K. LiJtzenkirchen, L. Schweikhard, H. Weidele and J. Ziegler, Rapid Commun. Mass Spectrom., 8 (1994) 401; Comp. Mater. Sci., 2 (1994) 633; Z. Phys. D, 30 (1994) 341. [11] G. Dietrich, St. Becker, H.-U. Hasse, H.-J. Kluge, S. Kuznetsov, M. Lindinger, K. Ltitzenkirchen, L. Schweikhard and J. Ziegler, Aun+-induced decomposition of N20, Ber. Bunsenges. Phys. Chem., in press. G. Dietrich, K. Dasgupta, M. Lindinger, K. Liitzenkirchen, L. Schweikhard and J. Ziegler, Chemisorption of H 2 on vanadium clusters, in preparation. [12] J. Ziegler, St. Becker, G. Dietrich, H.-J. Kluge, M. Lindinger, K. Ltitzenkirchen, L. Schweikhard and C. Walther, Photofragrnentation of Stored Cluster Ions, SASP 94, Symp. on Atomic and Surface Physics, Innsbruck, 20-26 March 1994, in T.D. M/irk, R. Schrittwieser and D. Smith (Eds.), Maria Alm, Austria, book of abstracts, p. 293. [13] C. Walther, St. Becker, K. Dasgupta, G. Dietrich, H.-J. Kluge, M. Lindinger, K. Liitzenkirchen, L. Schweikhard and J. Ziegler, Photogragmentation of Stored Cluster Ions, in Proc. 7th Int. Symp. Resonance Ionization Spec., 3 8 July 1994, Bernkastel-Kues, Germany, in press. [14] H. Weidele, St. Becker, H.-J. Kluge, M. Lindinger, L. Schweikhard, C. Walther, J. Ziegler and D. Kreisle, Delayed Electron Emission of Negatively Charged Tungsten Clusters, Surf. Rev. Lett., submitted for publication. [15] L. Schweikhard, St. Becker, K. Dasgupta, G. Dietrich, H.J. Kluge, D. Kreisle, S. Kuznetsov, M. Lindinger, K. Ltitzenkirchen, B. Obst, H. Weidele, C. Walther and J. Ziegler, Trapped metal cluster ions, Phys. Scri., submitted for publication. [16] L. Schweikhard, M. Lindinger and H.-J. Kluge, Rev. Sci. Instrum., 61 (1990) 1055. [17] J. Marto, A.G. Marshall and L. Schweikhard, Int. J. Mass Spectrom. Ion Processes, in press. [18] S.-H. Lee, K.-P. Wanczek and H. Hartmann, Adv. Mass Spectrom., 8B (1980) 1645. [19] L. Schweikhard and A.G. Marshall, J. Am. Soc. Mass Spectrom., 4 (1993) 433.
[20] D.L. Rempel, E.B. Ledford, Jr., S.K. Huang and M.L. Gross, Anal. Chem., 59 (1987) 2527. [21] R.S. Van Dyck, Jr., F.L. Moore, D.L. Farnham and P.B. Schwinberg, Phys. Rev. A, 40 (1989) 6308. [22] X. Xiang, P.B. Grosshans and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 125 (1993) 33. [23] L. Schweikhard, G.M. Alber and A.G. Marshall, Phys. Scr., 46 (1992) 598. [24] K. Wille, Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen, Stuttgart, 1992. [25] E. Wilson, several contributions to the CERN Accelerator school: General Accelerator Physics, Gif-sur-Yvette, Paris, France, 3-14 September 1984, CERN 85-19, Vol. 1, 1985, and the CERN Accelerator school: Advanced Accelerator Physics, Queen's College, Oxford, 16 27 September 1985, Nonlinear resonances, CERN 87 03, Vol. 1, 1987. [26] H. Weidele, U. Frenzel, T. Leisner and D. Kreisle, Z. Phys. D, 20 (1991) 425. [27] H. Schnatz, G. Bollen, P. Dabkievicz, P. Egelhof, F. Kern, H. Kalinowsky, L. Schweikhard, H. Stolzenberg and H.-J. Kluge, The ISOLDE Collaboration, Nucl. Instrum. Methods A, 251 (1986) 17. [28] H. Bopp, Diploma thesis (in German, unpublished), Mainz, 1993. [29] T.E. Sharp, J.R. Eyler and E. Li, Int. J. Mass Spectrom. Ion Phys., 9 (1972) 421. [30] T.J. Francl, E.K. Fukuda and R.T. Mclver, Jr., Int. J. Mass Spectrom. Ion Phys., 50 (1983) 151. [31] R.C. Dunbar, J.H. Chen and J.D. Hays, Int. J. Mass Spectrom. Ion Processes, 57 (1984) 39. [32] G. Savard, St. Becker, G. Bollen, H.-J. Kluge, R.B. Moore, L. Schweikhard, H. Stolzenberg and U. Wiess, Phys. Lett. A, 158 (1991) 247. [33] L. Schweikhard, S. Guan and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 120 (1992) 71. [34] E.v.P. Smith and K.C. Jacobs, Introductory Astronomy and Astrophysics, W.B. Saunders Company, Philadelphia, PA, 1973. [35] J. Moser, Nachr, Akad. Wiss, Grttingen, IIa (1955) 87. [36] J. Wisdom, Proc. R. Soc. London, Ser. A, 413 (1987) 109. [37] L. Schweikhard, Rapid Commun. Mass Spectrom., 8 (1994) 10.
International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
and Ion Processes
The trapping condition and a new instability of the ion motion in the ion cyclotron resonance trap L. S c h w e i k h a r d a'*, J. Ziegler a, H. B o p p a, K. Lfitzenkirchen b alnstitut ffir Physik, Johannes Gutenberg-Universit~'t, D-55099 Mainz, Germany blnstitut ffir Kernchemie, Johannes Gutenberg-Universitdt, D-55099 Mainz, Germany Received 5 August 1994; accepted 4 October 1994
Abstract
In analogy to the critical mass, merit, a critical voltage, gcrit, (and a general trapping parameter, 7rtrap ) is defined, above which the ion motion in an ion cyclotron resonance (ICR) trap is unstable and the ions are lost from the trap. The theoretical values for the critical voltage are confirmed by experimental results. Singly charged gold cluster ions, Au2, of several sizes, n = 50, 60, 76, 100, 110, and 145 (the latter corresponding to an ion mass of 28 560 u), were injected into an ICR trap, stored, and detected by axial ejection and single ion counting using a microchannel plate detector. During the storage period the trapping voltage, U, was varied for extended durations (from a few to over one hundred milliseconds), to probe its effect on the trapping efficiency. In addition to the expected instability above the critical trapping condition (m/merit = U~ g c r i t = 7rtrap = 1), a new instability was observed below the critical value. This is explained in terms of a resonance effect occurring for a trapping parameter 7rtrap = g/gcrit = m/merit = 8/9, which corresponds to the smallest possible integer ratio of the eigenfrequencies of ion motion, i.e. co+ = w: = 2w (where co+, w~ and co stand for the reduced cyclotron, the trapping and the magnetron frequency, respectively).
Keywords." ICR; Penning trap; Critical mass; Trapping parameter; Instability of ion orbit
1. Introduction
In ion cyclotron resonance (ICR) traps, charged particles are stored via a superposition of static magnetic and electric fields. These traps are used, for example, in high resolution Fourier transform-(ICR) (FTICR). (For reviews on ICR traps see Refs. [1]-[7], and references cited therein.) There are two restrictions on the ions that can be stored: (i) their electric charge is determined * Corresponding author.
by the polarity of the electric field; (ii) the ions that can be stored cover the range from "zero" up to a maximum mass-over-charge ratio. This upper mass limit is also called the "critical mass", for which a charge state of unity, i.e. one elementary charge, is assumed. In the light of recent developments in the trapping of large biomolecules and cluster ions, the critical mass has gained considerable importance. In this work the ion motion in the region of the critical mass is studied in detail. The measurements reported here were performed using an ICR trap with hyperbolically
0168-1176/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0168-1 176(94)04092-3
78
L. Sehweikhard et al./lnternational Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
shaped electrodes, usually called a Penning trap. The experimental setup [8] was designed for the capture [9] and investigation of metal cluster ions [10-15]. Since these experiments may involve high-mass ions of low charge states, questions concerning the stability of the ion motions are of considerable interest. Therefore, the high-mass behavior of the Penning trap was probed by the storage of singlycharged gold cluster anions, Aun-, as a function of the trapping voltage. In this paper we report a comparison of the theoretical mass limit for an ideal Penning trap with the experimental results. Further, we report the observation of a new instability occurring in a mass region that had previously been considered to be stable. The next section reviews the ion motion in a Penning trap with emphasis on the interrelation between the eigenfrequencies, followed by a short description of the experimental setup. Finally, the experimental results are described and discussed.
2. Ion
where P0 and z0 are the shortest distances from the electrode surfaces to the trap center. Other electrode shapes, e.g. cylindrical, cubic, or elongated, that may be preferable for specific experimental purposes, also lead to an approximately quadrupolar field in the center of the trap. For ease of discussion and calculation we therefore only consider in this paper an ideal Penning trap (if not otherwise stated explicitly). With electrode geometries given by Eqs. (3a) and (3b), the trap dimension, d is represented by [1] d = l(p02 q-- 222) 1/2 (4) (An ideal Penning trap field is also created by a "two-electrode trap" [16], where d has to be redefined as in Ref. [17].) With the above fields the Lorentz force leads to the following set of differential equations for the ion motion: d2x
The ideal Penning trap consisting of a homogeneous magnetic field
d2y dx m-~+qB-~t t
~-~y = 0
(1)
and an axial quadrupolar electric field (2)
dt 2
(3a)
and two endcap electrodes with the surface equation p2 _72 = _72q_.~_ (3b)
dy -We dt
-
dZY q- aJc dx dt 2 dt dZz
serves as the paradigmatic model of ICR traps. The electric field is obtained by applying a potential difference, U, between a ring electrode obeying the surface equation z 2 = l(p2 _ p2)
qU
(5a) (5b) (5c)
These may be rewritten as dZx
U /~ = 2-~ (x, y, -2z)
qU
2d---5x = 0
d2z qU rn~+~-z = 0
trapping
B = (0, 0, B)
dy
m - ~ - qB dt
wz x 2
--
=
0
2
,~z y = 0 2
(6a)
(6b)
2
dt 2 ~-:VzZ= 0
(6c)
where the cyclotron frequency (in the absence of an electric field) wc = q B m and the (axial) trapping frequency
(7)
qU ) 1/2 wz=
~-~
have been introduced.
(8a)
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
79
2.1. Stable ion motion
Eq. (6c) obviously describes a simple harmonic oscillator of (angular) eigenfrequency COz, decoupled from the "radial" ion m o t i o n in the xy dimensions. Based on the conditions CO++ CO- = COc
(9)
CO+CO_= co2/2
(10a)
the frequencies CO+ and CO_ are defined: COc CO+ =5--4-
X
(lla)
and Eqs. (6a) and (6b) m a y be further rewritten as d2x
dt 2
(co+ + co_)
~t
_.~Y
- co+CO_x = 0
(12a)
dZy + (CO+ dx ~dt + CO_)~ - CO+CO_y= 0
(12b)
F r o m Eqs. (12a) and (12b) follows a radial ion trajectory given by x = p+ sin(CO+t + ~+) + p_ sin(CO_t + ~_) y = p+ cos(co+t + ~+) + p_ cos(co_t + ~_) (13a) According to these equations the ions move in epicycles, where the (angular) eigenfrequencies co+ and CO_ are given by the trap parameters (B, U, d), and the radii p+ and p_, and phases ~+ and ~p_ are given by the initial conditions'. The ion m o t i o n is a superposition of the axial harmonic trapping m o t i o n with the trapping frequency, COz, and two radial (i.e. in the xyplane) circular motions with the reduced (or modified) cyclotron frequency, CO+, and the m a g n e t r o n frequency, co_. The cyclotron and m a g n e t r o n modes of the ion m o t i o n are decoupled and their amplitudes are independent of each other; for example, note that p+ > p_ or p+ < p_. However, the corresponding frequencies are not independent.
Fig. 1. Ion motion for w+ = 4COz,wz = 8w_, and p_ = 4 Z a m p = 8p+: ---, magnetron motion only;..., combined magnetron and trapping motion; - - , full ion motion, i.e. combined magnetron, cyclotron, and trapping motion.
Rewriting Eq. (10a) we find that the ratio between the reduced cyclotron frequency and the trapping frequency is half the ratio between the trapping and m a g n e t r o n frequencies: co+ _ 1 COz coz 2 co_
(10b)
Fig. 1 shows the ion motions for co+ = 4coz, COz= 8CO_, and p_ = 4Zam p = 8 p + (where Zamp is the amplitude of the trapping motion). It is useful to introduce another combination of frequencies, i.e. that of the oscillations of the ions to and from the z-axis: COp= CO+ - co_
(14a)
The index " p " stands for "parametric," since a modulation of the trapping voltage at Wp leads to an excitation of the radial ion m o t i o n [16,18] with both p+ and p_ increasing exponentially with time [19]. (Analogously, FTI C R detection m a y be performed with signals at COp [17,19,20].) 2.2. The trapping parameter 7ftrap
So far it has been tacitly assumed that the argument of the square root of Eq. ( l l a ) is
80
L. Schweikhard et al./International Journal o f Mass Spectrometry and Ion Processes 141 (1995) 77 90
1,0
non-negative, i.e. co2/4 ~> co2/2
(15)
0,9
Inserting expressions (7) and (8), this may be rewritten as
0,8
d2 B2
d 2 qB 2
2
(16b)
m
where (m/q)cri t is the well-known critical massover-charge (usually expressed as critical mass, mcrit, with the assumption of a charge state of one) and Ucrit is the critical trapping voltage analogously defined for easier comparison With the experimental results (see below). Eq. (1 la) may now be rewritten as co± = ~- 1 +
1
(mT-q-)-crit.]
]
1--gcrit/
(llc)
The critical condition may also be expressed in terms of the magnetic field, B, or the trap dimension, d, with all other parameters kept constant• For a general discussion, including the simultaneous variation of several trap and ion parameters, we introduce the dimensionless trapping parameter _
_
2Um d2B2q
(16c)
which now gives an expression for the radial eigenfrequencies of co+ -----211 + (1
- 71trap) 1/2]
(lld)
for the radial parametric frequency of cop = coc( 1 -
"a-trap) 1/2
Vz
LL 0,3 0,2
o1[ 0,2
0,4
0,6
0,8
1,0
TRAPPING PARAMETER, 7Ztrap
J
71.trap=2~ - U m/q coc U--~rit (m/q)crit
0,6 >0 0,5 Z LLI D O' 0,4 UJ
0,0 0,0
or
co:~=-~- 14-
.'~
(16a)
or
-
Vp
0,7
m/q <~ (m/q)crit - 2 U
U ~ Ucrit
V+
(14b)
Fig. 2. Eigenfrequencies of ion motion ~+, ~o~, and ~o_, and parametric frequency ~op (normalized to the cyclotron frequency) as a function of the trapping parameter 7"rtrap;this may be expressed as the ion mass-to-charge ratio, m/q, normalized to the critical ion mass-to-charge ratio, (m/q)crit, or equivalently as the trapping voltage, U, normalized to the critical trapping
voltage Ucrit.
and for the trapping frequency of COz = COc(71"trap/2) 1/2
(8b)
Fig. 2 shows the eigenfrequencies of the three modes of ion motion (co+, co_, coz) and the radial parametric frequency (COp) for values of the trapping parameter 0 ~< 7rtrap ~ 1. Note that for negative values of 71-trap the axial ion motion is unstable. This corresponds to the fact that for static electric trapping fields either positive or negative ions may be trapped depending on the polarity of the trapping voltage, but never both species simultaneously (see Eq. 8). To illustrate the behavior close to the
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L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
m a x i m u m possible trapping parameter for stable ion orbits, let us now consider a special set of initial conditions under which the ions are stationary at a given distance, Pini, from the trap center (from the z-axis, to be exact; a possible additional trapping m o t i o n is not relevant for this consideration): x = 0, y = Pini, dx/dt = dy/dt = 0. This leads immediately to the phases and radii of the ion motion: ~_ = 0 ,
g~+ = ~-
CO_
P+ =
-
Pini
cop -
(°r
P-
co+ =
~-p Pini
3o
Vp=V_/8~
...
p++p~/
--.=_ 25 -= z ) 20 ~) 15
7 ~
10
U
P± = 1 ( 1 ( 1 - - rCtrap)l/2 ) 17rtrap) / ~ Pini
1-1/10
1-1/100
1-1/1000
TRAPPING PARAMETER, ntrap and Eq. (13a) read for this case Fig. 3. Cyclotron and magnetron radii, and their sum, p+ + p , as a function of the trapping parameter, 7rtrap, for the following initial conditions: zero velocity in the x and y coordinates; position x = 0, y = Pini
X = Pini co+ sin(co_t) cop co -- Pini--Y- COS(CO+t) cop co+ Y = Pini
-
-
cop
for 0 ~< 7"ftrap ~ 1 COS(CO_t)
0.)_
+ P i n i - sin(co+t) cop
(17a) The radii are shown in Fig. 3 as functions of the trapping parameter; note that as the trapping parameter approaches unity, the radii increase rapidly. Fig. 4 shows the radial motions for the initial condition as discussed above for selected ratios between the cyclotron and magnetron frequency (co+/co_ = 10, 5, 2, 3/ 2, and 5/4, corresponding to the trapping parameters 71-trap= 40/121, 5/9, 8/9, 24/25, and 80/ 81, respectively). It is immediately clear that for a real Penning trap with a finite ring electrode of m i n i m u m radius, P0, only ions with P+ + D- = coc/°"p Pim = Pini/( 1
-
-
for example, ions are created or captured with zero radial kinetic energy at a radial distance of 1 m m from the center of the trap, their orbit will extend to 3 m m (distance from the axis) for 7rtrap = 8/9 and to 1 cm for 7rtrap : 1 - 1/100. In the limit 71-trap--+ 1 (corresponding to co+ = co- = coc/2), the amplitude of the radial m o t i o n approaches infinity linearly as a function of time: X = Pini sincoCt
2 coct
coct
-- Pini--~- COS "--~'-
Y -- Pini COS
COct
for 71"trap ~--- 1
coct. coct
q- Pini ~ - sln-~-
7rtrap) 1/2 < /90
remain trapped. All the others hit the trap electrode (within one period of the cyclotron motion) and are lost. Fig. 3 shows (p+ + p_)/ Pini as a function of the trapping parameter. If,
(17b) F o r trapping parameters above the critical v a l u e , 7rtrap > 1, the radii approach infinity
82
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
even faster:
where coo is defined as
cosh(co0t) sin c°ct
X = Pini
2
~)C
•
smh(c°°t) cos Wet
-- Pini ~ 0
T
0d c
f o r 71trap
cosh(co0t) cos 2 t
Y ~-- Pini
COct
+ Pini T
.
= ~ - (Trtrap -- 1)1/2
> 1
f o r 71-trap ~> 1
(18)
The general solution for the radial equations of motion (6a, 6b) may be written as
~cl
sinh(co0t) sin T
x = p+ sin(u;+t + 9~+) (17c)
/
+ p_ sin(a;_ t + ~_) for 0 ~< 7rtrap ~ 1
y = p+ cos(a;+t + 9~+) + p_ cos(w_t + ~_)
(3
03 =,s,
~+/~.=10 . . . .
I
I
. . . .
(13a)
'
'
'" '
i
i
i
X = Pconst s i n ~ - -
-F Ptime ~ -
--}- qOconst
ct.s l n (Wet \ 2 +
ggtime
/a~ct Y o~+/ m_-- 3/2 '
I
. . . .
I
,Oconstc o s ~ -
)
\
[ for
-~- ggconst )
71"trap z
1
]
wct /co c t \ I q- Ptime T C O S ~ T q- 99time) J
. . . .
(13b) where the subscripts " c o n s t " and " t i m e " indicate the terms of constant and time dependent Table 1 Some critical tune values, the corresponding trap parameters and the duration of an overall period (in units of the period of the magnetron motion, I/u_, and one period of the cyclotron motion in the absence of a trapping field, 1/Uc) m+ / m_ = 5/4 ,
I
. . . .
I
~z/&_ . . . .
I
i
i
i
i
I
i
i
i
i
60p/¢0
7rtrap
Toveral
I [l/v_]
Zoverall
I
Fig. 4. Radial ion motions (projection of ion motion in the xyplane) for initial conditions as in Fig. 3 for a selected number of ratios between the cyclotron and magnetron frequency (w+/w = 10, 5, 2, 3/2, and 5/4, corresponding to trapping parameters 7rtrap= 40/121, 5/9, 8/9, 24/25, 80/81, respectively). The central cross indicates the position of the trap axis (x = y = 0). The distance between the minor ticks on the axis corresponds to the minimum distance of the ions from the trap axis, and is equal to the y-coordinate value for the initial condition.
2
3 4 5 6 7 8 9 10
1
3.5 7 11.5 17 23.5 31 39.5 49
0.88889
1
3
0.59504 0.39506 0.27435 0.19945 0.15071 0.11754 0.09406 0.07689
2 1 2 1 2 1 2 1
11 9 27 19 51 33 83 52
[1/vc]
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 7~90
83
amplitudes, respectively, and
x = PchCOSh(a~0t) sin(\ c°d " /C°ct 2 + ~ch ) -I- Psh sinh(aJ0t) smt, ~ - + ~sh ) Y = Pch c°sh(c°0t) COS\ 2
+ Cpch) +
Psh
I
sinh(~0t) cos (Tc°ct + ~)sh) /
where "ch" and "sh" indicate the terms with cosh- and sinh-like time dependent amplitudes, respectively. Eqs. (13a)-(13c)describe the ion motion in an ICR cell in the limit of a perfect Penning trap. The stability of the ion orbits is given uniquely by the single trapping p a r a m e t e r 7rtrap.
2.3. A new instability In reality, no ICR trap is a perfect Penning trap. There are always distortions of the trapping forces which may be more or less severe, depending on the spatial homogeneity of the magnetic field and on the shapes of the trap electrodes. The fields may be shimmed with correction coils and electrodes, eliminating the effects of e.g. electrode segmentation and holes for ion injection and ejection. Other deviations from the perfect forces may even be impossible to correct for, such as the effect of image charges induced in the trap electrodes by the ions themselves [21,22] or variations in the relativistic mass of the ions [23] due to the variation in their kinetic energy along their trajectory. Any of these distortions of the trapping forces from the ideal Penning trap lead to perturbations and will have an influence on the ion orbit. Depending on the trapping conditions, the repeated perturbations may lead to an additional kind of instability of the ion motion and the ions may be lost from the trap. This will only happen when the perturbations do not cancel each other out, but add up due to periodic orbits. This periodicity may include all modes of the ion motion, the 'overall periodicity' (period Toverall) , and means that
for 71-trap) 1
(13c)
the ions return to the same position in the trap to undergo the same perturbation. This occurs if the relevant frequencies are integers or simple ratios of each other. The effect is expected to increase with a decrease in the overall period, i.e. a decrease in the integer numbers p and q of the simple ratios p/q of the frequencies of interest (see Table 1). The phenomenon discussed above is wellknown from accelerator and storage rings for ions and electrons. (For an introduction to this subject see Refs. [24] and [25].) While the particles go around in turns they perform 'betatron' oscillations, i.e. lateral (horizontal and vertical) displacements from their equilibrium orbit. One of the most critical design parameters for a storage ring is its 'tune' (Qh, Qv), i.e. the number of (horizontal or vertical, respectively) betatron oscillations during one turn. A ring can only be successfully operated if betatron resonance effects are avoided, i.e. if both Qh and Qv resemble neither integers nor simple fractions (the so-called stopbands). These values must be avoided for both the horizontal and the vertical tune; otherwise, effects due to magnet imperfections build up turn after turn and such a resonance results in loss of the beam. For accelerator and storage rings, Qh and Qv are generally independent of each other. We now consider the Penning trap and an ion orbit with large magnetron radius, p_, and moderate cyclotron radius, p+, and amplitude of z-motion, Zamp. For this case the "z-tune", or "trapping tune", may be defined as coz/a~_, the number of trapping oscillations, and the "cyclotron tune" may be defined as aJp/a~ ,
84
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90 !
>'
80
U.JZ
60
I--
40
i /
/
(9
Z 12. rl
20
n," I,--
0
<
1000
2000
4000
3000
CYCLOTRON TUNE, vp / v !
I
i !
J~
!.J~.,
Z
_z n 12_
<
n," I-
1
i
0
4
CYCLOTRON TUNE, Vp / v_
Fig. 5. Working diagram for the Penning trap, where the abscissa gives the cyclotron tune ( ~ p / ~ ) and the ordinate the z-tune (wz/~_) as related by Eq. (10c).
the number of oscillations of the ion to and from the trap axis during one period of the magnetron motion. Note that according to Eq. (10b) these ratios are not independent of each other: =
½
-
1
(10c)
An integer value for the z-tune leads automatically to an integer or half-integer value for the cyclotron tune (but not vice versa). Fig. 5 shows the working diagram for the Penning trap, where the abscissa gives the cyclotron tune and the ordinate the z-tune as related by Eq. (10c). The lower figure shows the section of the working diagram for low tune values, with integer and half-integer values of the tunes indicated by solid and dashed lines, respectively. Table 1 lists some
critical tune values, the corresponding trap parameter, 7rtrap , and the length of an overall period, ToveralI (in periods of the magnetron motion l/u, which itself is a function of 7rtrap and, for comparison with an absolute unit, in periods of the cyclotron motion in the absence of the electric field, 1/Uc). Note that the overall period increases with increasing ratio of the eigenfrequencies of ion motion, i.e. for low trapping parameters (low mass, low trapping voltage, etc.). Under these conditions it takes a longer time for the ions to experience the same perturbation, and for standard experiments they will not be affected significantly. However, for high trapping parameters (small frequency ratios) the overall period may be very small. Fig. 6 shows some examples of ion orbits for small ratios of eigenfrequencies (for z-tunes, aJz/~_, of 5, 4, 3, and 2). The most extreme case is a~+ = aL, = 2aJ_; during one period of the magnetron motion the ion performs two oscillations in the axial direction and approaches the z-axis once (a~p = a~_). For even values of the z-tune the ion orbits close after one period of the magnetron motion. For odd values of the z-tune an additional period is needed for the closure.
3. Experimental procedure The experimental setup is described in detail elsewhere [8]. Fig. 7 gives an overview of the experimental sequence. The measurements are performed with gold cluster ions produced from the external source via laser vaporization ( T R I G G E R LASER) of a gold wire in a helium gas jet [26]. Negatively charged ions are guided by several ion optical elements to a Penning trap with a magnetic field of B = 5 T and electrode parameters of P0 = 20 m m and z 0 = po/21/2. The transfer section between ion source and trap allows differential pumping to obtain U H V conditions in the trap region. A single ion bunch is
L. Schweikhard et al./lnternational Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
85
y ~
-
.
.
.
.
.
.
.
.
.
.
.
.
×
Fig. 6. Some examples of ion orbits for small ratios of eigenfrequencies (for durations of 1/u ). Lines as in Fig. 1. In addition, the projection onto the xy-plane is shown below the full ion motion. Top left: ~:/co_ = 5; top right: ~ z / ~ = 4; bottom left: w:/a~_ = 3; bottom right: co=/~ = 2.
captured in flight (CAPTURE PULSE) [27] at a low trapping voltage, U = 3 V. Using this production and capture procedure a preselection of cluster sizes is already achieved [15]. In addition, all unwanted ions are ejected by resonant excitation of their cyclotron
motion (SELECTION) [8]. Only the cluster ions of the size (and mass) of interest remain in the trap. The trapping voltage is then changed to probe the trapping efficiency for a given value (TRAPPING VOLTAGE, Ux). After the trapping voltage is re-established at
86
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
>~ 1.2
I
TRIGGER LASER
CP
CAPTURE
o~ 0.8
z
PULSE
SELECTION
1.0
0.6 [3
!............. i T :
TRAPPING VOLTAGE Ux
Z
0.4
EJECTION
<
0.2
m
E- 0.0
TRANSIENTRECORDER 0
50
100
2
4 6 8 10 12 TRAPPING VOLTAGE [V]
14
150
t [ms]
Fig. 7. Timing sequence of the experiment. See text for details.
U = 3 V, the ions are ejected axially (EJECTION) and detected by a microchannel plate detector for time-of-flight analysis (TRANSIE N T R E C O R D E R ) . This sequence is typically repeated several hundred times, to increase the statistical significance of the results. After each sequence where the trapping voltage is pulsed to a particular value under consideration, an additional sequence is performed without pulsing the trapping voltage (i.e. the trapping voltage is kept constant at U = 3 V) and the resulting data are stored separately. These latter measurements are used to calibrate the trapping efficiencies of the pulsed mode, eliminating, for example, fluctuations in the cluster production. We define the experimental trapping efficiency, e(Ux), as the ratio of the number of ions counted after a sequence during which the trapping voltage has been changed from 3 V to the value of interest, Ux, to the number of ions counted after a sequence during which no such change in trapping voltage occurred.
Fig. 8. Measured trapping efficiency for gold cluster anions, Aul00, and trapping periods of 40 ms as a function of trapping voltage.
(i) most values of the trapping efficiencies are slightly less than unity, indicating that there is some ion loss during the change of the trapping potential; (ii) the trapping efficiency decreases very slowly from about e ~ 0.9 to e ~ 0.75 over a wide range of the trapping voltage, up to Ux ~ 11.5 V (for comparison, Ucrit ~ 12.25 V), and drops to zero for higher values; (iii) there is an outlyer at U ~ 9 V; systematic studies confirmed that this was due to statistical fluctuations. (iv) there is a resonance-like structure, consisting of several data points, at U 10.5 V; this structure has been observed repeatedly.
~
1,00
i
0,98 -
~ 0,96> O 0,94Z0,92 < r~ 0,90N.-
4. Experimental results
0,88. U
Z < 0,86 -
Fig. 8 shows the results of a typical series of measurements. After gold cluster anions (AuT00 with a mass of 19 697 u) were captured and selected, the trapping efficiency, as defined above, was determined as a function of the trapping voltage. The figure shows several features:
n- 0,84 O Z
410
610
8I 0
I 100
I 120
J 140
160
SIZE OF GOLD CLUSTER ION, n
Fig. 9. Experimental critical trapping voltage, (E]), and trapping voltage of new instability (O), normalized to the theoretical value of the critical trapping voltage as a function of gold cluster size. O, trapping voltage of new instability normalized to the experimental value of the critical trapping voltage.
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
The data shown were taken with a storage duration of 40 ms at the trapping voltage under consideration. Measurements with storage durations of 4 and 146 ms yield the same results [28]. The investigations have also been extended to other cluster sizes, n = 50, 60, 76, 120, and 145 (corresponding to m = 9848, 11818, 14969, 21666, and 28560 u). The results are summarized in Fig. 9. The open squares indicate the experimental values of the critical trapping voltage. They were determined from the measurements in the same way as in Fig. 8, where the data points were taken in the region of Ucrit. A straight line is fitted to the data n e a r Ucrit with trapping efficiency values between 20 and 80% of the value for small trapping voltage. This line is extrapolated to the abscissa and the corresponding trapping voltage is defined as the experimental critical trapping voltage. Its values as a function of cluster size are shown in Fig. 9, normalized to the values expected from the parameters of the Penning trap and the ion species under investigation. The open cycles indicate the centre of the above-discussed resonance-like structure, again normalized with respect to the expected critical trapping voltage. Note that these values deviate only slightly from each other, i.e. they show a slow decrease as a function of cluster mass. Further, when normalized with respect to the experimental values of Ucrit (solid circles), they are independent of the cluster mass and compatible with a value of 8/9 (indicated by the dashed line). As stated above, the eigenfrequencies of ion motion are at the lowest possible integer ratios of each other for this value of the trapping parameter and the motions are therefore expected to be unstable. Therefore, in addition to the instability of the ion motion for trapping parameters exceeding the upper limit a new kind of instability is observed for trapping parameters that so far have not been considered to be "critical".
87
5. Discussion and conclusion
The concept of critical mass has been generalized to the critical trapping parameter, 71-trap, and the occurrence of closed ion orbits, in particular those with a small overall period, has been examined. Using large gold cluster ions the dependence of the trapping efficiency on the trapping voltage has been studied. From the data an experimental critical voltage is determined which is slightly below the theoretical value. This deviation is not surprising; even though the ions have little kinetic energy after their capture in the trap, they may still have a finite initial distance from the trap axis. As has been shown above, this leads to very large amplitudes of the radial modes of motion for trapping parameters close to the upper limit. Hence, while the ion orbits are still stable (i.e. do not approach infinity with time) the ions may collide with the electrodes due to the finite trap dimensions. (Alternatively, they may not be able to leave the trap through the exit hole of 5 m m diameter during the axial ejection for time-of-flight detection.) Note that the initial position determines the experimental trapping efficiencies. Therefore, by a series of measurements of the trapping efficiency as a function of trapping voltage, information on the radial distribution of stored ions may be obtained. This method, however, is outside the scope of the present paper. At a trapping voltage of approximately 8/9 Ucrit a new instability was observed which seems to be associated with a closed ion orbit of small period. The origin of this instability has yet to be studied in more detail. Since only a few ions (< 10) were trapped at any one time in the present investigations, the instability is hardly a space-charge (collective) effect, but rather is due to the interaction of single ions with the electromagnetic trapping field. The various ratios of eigenfrequencies of the different modes of the ion motion are not
88
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77-90
__
~
Y
,r"
×
i
Fig. 10. For the case a~+ = a~. = 2cv_, where the experiments indicate an instability o f the ion motion, a few orbits are shown (for details see captions of Figs. 1 and 6). Starting with the top left figure and going clockwise the phase of the cyclotron mode is incremented by 7r/8 for each figure.
independent, i.e. simple ratios of c0z/aJ_ yield simple ratios of a~+/a~ z (Eq. 10b); however, in general the opposite is not true. Therefore it is not clear which particular mode of the ion motion leads to the instability. One may speculate that the nature of the
magnetron motion has a significant influence. The magnetron motion of ions in an ICR trap is metastable [1]. In general, perturbations tend to increase the magnetron radius. For example, ion collisions with neutral atoms and molecules will have this effect [19,29-31],
L. Schweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
leading to ion loss from the trap if counteracting measures [32,33] are not taken. Further, note that not all ions are lost from the trap under the resonance condition. The question of instability may also depend on the particular phases of the ion motion with respect to trap imperfections, and to the relative phases between the modes of ion motion (Fig. 10). The observed instability is analogous to betatron instabilities of particle accelerator and storage rings where simple ratios have to be avoided for the tunes of the betatron oscillations about the equilibrium orbit (stopbands). The instability may thus be due to a coupling of the radial and axial motions, corresponding to the coupling of horizontal and vertical betatron motions of a particle in a ring. But there are some differences between these latter systems and ion cyclotron resonance traps. For example, there is only one equilibrium orbit in rings while the amplitudes of magnetron, cyclotron and trapping motion in the trap are independent of each other and are not restricted, except for the finite trap electrode dimensions. There is another analogy worth considering. In celestial mechanics the orbits of satellites and planets can in principle be of any radius within a wide range, as for ion orbits in the ICR trap. However, for planets and satellites the periods of these orbits are functions of the radius. As is well-known, the perturbation of additional satellites or planets leads to voids, e.g. the regions between Saturn's rings or the distribution of asteroids in the solar system [34-36]. For example, a particle orbiting in the Cassini division would have half the orbital period of Saturn's m o o n Mimas. Likewise, the Kirkwood gaps of asteroids are explained by gravitational perturbations due to Jupiter. Again, simple fractions of the ratios of orbital frequencies play an important role. The observed instability of ion orbits may be called a "static resonance" [37]: a resonant
89
phenomenon that does not depend on the excitation by a time-varying signal of an external device (e.g. a frequency generator), but occurs for static magnetic and electric fields. This resonant phenomenon may provide another means for manipulating ions stored in an ICR trap. However, further experimental studies and theoretical considerations will be necessary to exploit the analogies indicated, to fully understand the nature of the new instability, and to use this effect for future applications.
Acknowledgments We thank H.-J. Kluge for his continuous support of the Mainz cluster project. A. Streun is acknowledged for helpful discussions concerning the analogies and differences of ion traps and storage rings. The experiments were supported by the Deutsche Forschungsgemeinschaft and by the Materialwissenschaftliches Forschungszentrum of the Johannes Gutenberg-Universitfit of Mainz.
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L. Sehweikhard et al./International Journal of Mass Spectrometry and Ion Processes 141 (1995) 77 90
[9] H.-U. Hasse, St. Becker, G. Dietrich, N. Klisch, H.-J. Kluge, M. Lindinger, K. Lfitzenkirchen, L. Schweikhard and J. Ziegler, Int. J. Mass Spectrom. Ion Processes, 132 (1994) 181. [10] St. Becker, G. Dietrich, H.-U. Hasse, N. Klisch, H.-J. Kluge, D. Kreisle, S. KriJckeberg, M. Lindinger, K. LiJtzenkirchen, L. Schweikhard, H. Weidele and J. Ziegler, Rapid Commun. Mass Spectrom., 8 (1994) 401; Comp. Mater. Sci., 2 (1994) 633; Z. Phys. D, 30 (1994) 341. [11] G. Dietrich, St. Becker, H.-U. Hasse, H.-J. Kluge, S. Kuznetsov, M. Lindinger, K. Ltitzenkirchen, L. Schweikhard and J. Ziegler, Aun+-induced decomposition of N20, Ber. Bunsenges. Phys. Chem., in press. G. Dietrich, K. Dasgupta, M. Lindinger, K. Liitzenkirchen, L. Schweikhard and J. Ziegler, Chemisorption of H 2 on vanadium clusters, in preparation. [12] J. Ziegler, St. Becker, G. Dietrich, H.-J. Kluge, M. Lindinger, K. Ltitzenkirchen, L. Schweikhard and C. Walther, Photofragrnentation of Stored Cluster Ions, SASP 94, Symp. on Atomic and Surface Physics, Innsbruck, 20-26 March 1994, in T.D. M/irk, R. Schrittwieser and D. Smith (Eds.), Maria Alm, Austria, book of abstracts, p. 293. [13] C. Walther, St. Becker, K. Dasgupta, G. Dietrich, H.-J. Kluge, M. Lindinger, K. Liitzenkirchen, L. Schweikhard and J. Ziegler, Photogragmentation of Stored Cluster Ions, in Proc. 7th Int. Symp. Resonance Ionization Spec., 3 8 July 1994, Bernkastel-Kues, Germany, in press. [14] H. Weidele, St. Becker, H.-J. Kluge, M. Lindinger, L. Schweikhard, C. Walther, J. Ziegler and D. Kreisle, Delayed Electron Emission of Negatively Charged Tungsten Clusters, Surf. Rev. Lett., submitted for publication. [15] L. Schweikhard, St. Becker, K. Dasgupta, G. Dietrich, H.J. Kluge, D. Kreisle, S. Kuznetsov, M. Lindinger, K. Ltitzenkirchen, B. Obst, H. Weidele, C. Walther and J. Ziegler, Trapped metal cluster ions, Phys. Scri., submitted for publication. [16] L. Schweikhard, M. Lindinger and H.-J. Kluge, Rev. Sci. Instrum., 61 (1990) 1055. [17] J. Marto, A.G. Marshall and L. Schweikhard, Int. J. Mass Spectrom. Ion Processes, in press. [18] S.-H. Lee, K.-P. Wanczek and H. Hartmann, Adv. Mass Spectrom., 8B (1980) 1645. [19] L. Schweikhard and A.G. Marshall, J. Am. Soc. Mass Spectrom., 4 (1993) 433.
[20] D.L. Rempel, E.B. Ledford, Jr., S.K. Huang and M.L. Gross, Anal. Chem., 59 (1987) 2527. [21] R.S. Van Dyck, Jr., F.L. Moore, D.L. Farnham and P.B. Schwinberg, Phys. Rev. A, 40 (1989) 6308. [22] X. Xiang, P.B. Grosshans and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 125 (1993) 33. [23] L. Schweikhard, G.M. Alber and A.G. Marshall, Phys. Scr., 46 (1992) 598. [24] K. Wille, Physik der Teilchenbeschleuniger und Synchrotronstrahlungsquellen, Stuttgart, 1992. [25] E. Wilson, several contributions to the CERN Accelerator school: General Accelerator Physics, Gif-sur-Yvette, Paris, France, 3-14 September 1984, CERN 85-19, Vol. 1, 1985, and the CERN Accelerator school: Advanced Accelerator Physics, Queen's College, Oxford, 16 27 September 1985, Nonlinear resonances, CERN 87 03, Vol. 1, 1987. [26] H. Weidele, U. Frenzel, T. Leisner and D. Kreisle, Z. Phys. D, 20 (1991) 425. [27] H. Schnatz, G. Bollen, P. Dabkievicz, P. Egelhof, F. Kern, H. Kalinowsky, L. Schweikhard, H. Stolzenberg and H.-J. Kluge, The ISOLDE Collaboration, Nucl. Instrum. Methods A, 251 (1986) 17. [28] H. Bopp, Diploma thesis (in German, unpublished), Mainz, 1993. [29] T.E. Sharp, J.R. Eyler and E. Li, Int. J. Mass Spectrom. Ion Phys., 9 (1972) 421. [30] T.J. Francl, E.K. Fukuda and R.T. Mclver, Jr., Int. J. Mass Spectrom. Ion Phys., 50 (1983) 151. [31] R.C. Dunbar, J.H. Chen and J.D. Hays, Int. J. Mass Spectrom. Ion Processes, 57 (1984) 39. [32] G. Savard, St. Becker, G. Bollen, H.-J. Kluge, R.B. Moore, L. Schweikhard, H. Stolzenberg and U. Wiess, Phys. Lett. A, 158 (1991) 247. [33] L. Schweikhard, S. Guan and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 120 (1992) 71. [34] E.v.P. Smith and K.C. Jacobs, Introductory Astronomy and Astrophysics, W.B. Saunders Company, Philadelphia, PA, 1973. [35] J. Moser, Nachr, Akad. Wiss, Grttingen, IIa (1955) 87. [36] J. Wisdom, Proc. R. Soc. London, Ser. A, 413 (1987) 109. [37] L. Schweikhard, Rapid Commun. Mass Spectrom., 8 (1994) 10.