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Higher order non-linear resonances in a Paul trap
ELSEVIER
International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
Mass Spectrometry and Ion Processes
Higher order non-linear resonances in a Paul trap R. Alheit a, S. Kleineidam a, F. Vedel b, M. Vedel b, G. W e r t h a'* aInstitut fi~r Physik, Universiti~t Mainz, D-55099 Mainz, Germany bUniversitd de Provence, PI2M, F-13397 Marseille Cedex 20, France
Received 5 December 1995; accepted 5 April 1996
Abstract
We report some detailed investigations on non-linear resonances of high order as they appear in a radio-frequency ion trap, whose potential deviates from the ideal quadrupolar shape. The strength and shape of these resonances have been measured throughout the first stable region of the trap. Moreover, space charge shifts and also effects of a modulation of the trapping voltage on the shape of the resonances have been observed. Some of the resonances cannot be explained by the general rule that the sums of multiples of the ion macrofrequencies nrwr ÷ nzwz (nr, nz are integers) are multiples of the trapping field frequency fL Keywords." Ion trap; Non-linear resonances; Paul trap
1. I n t r o d u c t i o n
Ion traps have been developed to become a standard experimental tool used for high resolution mass spectrometry [1], in collision studies with charged particles [2], in quantum optics [3] and in commercial mass spectrometers. Of particular importance among the various types of trap is the so-called "Paul trap", named after its inventor [4]. It operates by applying a high frequency voltage V = V0 cos ~t to three electrodes, which are hyperboloids of revolution. A variable d.c. voltage U0 may be superimposed to vary the shape and depth of the three-dimensional potential well inside the electrode structure. For an ideal trap geometry and a single stored particle we have a * Corresponding author.
time-averaged harmonic trapping potential ("pseudopotential"). The ion trajectories in a trap with perfect rotational symmetry around the z-axis can be calculated analytically by the solution of the normalized Mathieu differential equation [5]: d2u
dr 2 + (ar,z - 2qr, zCOS2r)u = 0 u = x , y , z,
r = x, y,
(1)
T = ½ ~2t
-16eU0 az = - 2 a r = m(r2 + 2z2)f~2
(2)
8eVo qz = - 2 q r = m(r~ + 2z02)~2
(3)
where r0 is the radius of the ring electrode and 2z0 the closest distance between the endcaps. Stable ion trajectories occur for certain values
0168-1176/96/$15.00 © 1995 Elsevier Science B.V. All rights reserved P H S0168-1176(96)04380-7
156
R. Alheit et al./International Journal o f Mass Spectrometry and Ion Processes 154 (1996) 155-169
of the stability parameters flr,~ which depend on the parameters ar, z and qr, z:
which using eq. (5) can be written as nrCOr + nzCOz = t,,f~
(8)
q2 fl2 = a +
q2 (2 + 3) 2 - a 4
q2 (4 + fl)2 __ a + - (4)
They are related to the free oscillation frequency co of the trapped ion in the time-averaged potential (macromotion) by
flr,z ~-~ COr,z -- 2
(5)
The first stable region occurs when 0 < 3r,~ < 1. Deviations from the pure quadrupolar trap geometry, caused by misalignments, nonhyperbolic shapes, truncated electrodes, holes in the electrodes to obtain access to the trapping fields inside, space charge potential of a large ion cloud and influence of potentials from outside the trap, make the ion motion unstable for certain values of the trapping parameters, for which in the ideal case stability exists. Assuming that the rotational symmetry is not destroyed, this instability can be understood from an analytical expansion of the trapping potential in a power series of spherical coordinates. If we assume mirror symmetry with respect to the midplane of the trap, the expansion will contain only even orders: oo
[ / r , ~ 2N
¢(r,O, qo) = ~,C2N~,~)
P2N(COS~)
(6)
¢2N describes the contribution of the 2Nth order to the potential and PZN is the spherical harmonic of order 2N. Wang and co-workers [6, 7] derived a condition for the appearance of instabilities nz
~ flr + -~ flr = U
(7)
with nr, nz = 0, 4-1, 4-2... and nr + nz = N; u = 1,2, .... A similar equation had been derived earlier for the two-dimensional case of a linear mass filter by von Busch and Paul [8] and by Dawson and Whetten [9]:
nxa;x + nyCOy =
~
(9)
As stated by Wang et al. [6], energy exchange between the different coordinates occurs when nr or nz assumes a negative value, while for positive values the ions may gain energy from the time-varying trapping field and may get lost from the trap. Some of these instabilities have been experimentally observed in the two-dimensional case by von Busch and Paul [8] and Dawson and Whetten [9]. For the three-dimensional trap the first observation of a "black hole" was reported by Guidugli and Traldi [10] and several observations of unstable lines have been made by different groups. A high resolution mapping of a part of the stability diagram was performed by Eades et al. [11], who found several unstable lines. Of particular strength in all the traps under investigation was the resonance occurring for nr = 3, nz = 0 which arises from a hexapole contribution to the potential. Since the equation for a non-linear resonance contains the z / m ratio of the ion, it can be used to improve the performance of traps in high resolution mass spectrometry. Recently we reported very high resolution scans of the first stable region [12] and demonstrated that many unstable lines of reduced ion stability arising from higher order perturbation in the trapping potential can be observed. In this paper we will describe measurements which add further details to our previous observations. We used a different trap which possibly has different contributions of higher order perturbations. The main obvious difference is that we used a trap where the radius
R. Alheit et al./lnternational Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
I0 ."
copper, the other endcap was made of stainless steel with a mesh in the center to make the extraction of the ions possible. Since our trap was used simultaneously for an experiment in laser spectroscopy, we had two holes of 8 m m diameter drilled into the ring electrode opposite to each other in the trap's midplane. These holes served for the entrance and exit of a laser beam. The trap was driven by a 3 MHz oscillator. A.c. and d.c. amplitudes were controlled by a personal computer and could be varied in steps of 500 mV and 100 mV, respectively. The driving voltage was applied to the ring electrode while the endcaps were held at ground potential. Ions were created by ionization of the background gas which was typically at 10-9mbar after a three-day bakeout at 280°C. We used an electron beam from a filament placed below the lower endcap. The electrons entered the traps through a 1.5 m m diameter hole in the center of the lower endcap after being accelerated by 360 V. A cross-section of our trap is shown in Fig. 1.
•
"- .
r0
Fig. 1. Ion trap used in our experiment. The radius of the hyperbolic-shaped ring is 20 mm and the distance between the endcaps is 28.3 mm. The electrodes are modified by holes and grids.
was increased from 7 to 20 m m in the present experiment.
2. Experimental apparatus Our trap of r0 = 20 m m has a ratio r o / z o = x/-2. The ring and one endcap were made of
[Pc].
lionl
cylinder
extrac~on-pulse
I multiplierl -15oov..-zsoov
I
genaator
J L
i'
: oov
..................... ,
ov.
drivingfrequency generator3MHz
I
ringeteclrode endcap
+lOOV
e - gunswitch ~
157
~.~,o
V
Fig. 2. Block diagram of the experimental setup.
ment
158
R. Alheit et aL/International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169 creation
waiting time
extraction
typ. 500 ms
typ. 300 ms
600 ns
detection
time e - gun
- -
+50V
',+50 V
- 350 V OV
OV
upper endcap
-150 V +150 V OV
OV
lower endcap
Fig. 3. Timing sequence of the experiment.
Variation of the filament current or the pulse length of the accelerating voltage between 10 and 2000ms allows one to vary the number of ions created by several orders of magnitude. After a delay time which could be varied between 10 ms and many seconds we applied a pulse of - 1 5 0 amplitude and 600 ns duration to the upper endcap and the same pulse with opposite polarity to the lower endcap. Both pulses had a fixed phase with respect to the trapping field oscillator. They drive the ions out of the trap. Fig. 2 shows the experimental setup and Fig. 3 the timing sequence. After passing through an ion optic the ions arrive at an ion multiplier placed 8 cm above the upper endcap. This distance is sufficient to separate ions of different masses in time by a few lOOns, mainly H +, H + and N + from the residual gas. In addition we observe H~- ions formed inside the trap by reaction of stored H~- with neutral H2: H~- + H2 ~ H + + H. The ion signals were amplified by the multiplier and digitized. By gating the analog-digital converter (ADC) we could choose a specific ion species for our investigation (Fig. 4). The measurements presented below were exclusively performed on an H + cloud because the corresponding signal was stronger than that of H + and its formation happened in a shorter time than for H~-. The storage time under normal trapping conditions outside a non-linear resonance was of the order of 10 s.
The main loss mechanism was the reaction with H 2 molecules mentioned above.
3. Scan of the stability diagram Fig. 5 shows a scan through almost the complete stability diagram of H~-. In this experiment we varied the values of az and qz in steps of 0.0019 and 0.00135, respectively, corresponding to voltage steps of AU0 = 0.7V and A V0 = 1 V. To improve the signal-to-noise ratio we added three signals for each of the 200 000 data points. The very high resolution
~4
-a,zt5 V?
~f L~
Fig, 4. Oscilloscope trace of an ion signal after ejection from the trap (H +, H~-, H~- from left to right). By gating the detection electronics we can choose any one of the signals for our experiments.
159
R. Alheit et al./International Journal o f Mass Spectrometry and Ion Processes 154 (1996) 155-169
0,0 0,1 0,15
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
0,15
0,10
0,10
0,05
0,05
0,00
0,00
az -0,05
43,05
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-0,10
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~,15
-0,20
-0,20
-0,25
~),25
-0,30
-0,30
-0,35
4), 35
-0,40 0,0 0,1
0,2
0,3
0,4 0,5
0,6
0,7
0,8
0,9
1,0
qz Fig. 5. High resolution scan of the major part of the first region of the stability diagram of our ion trap. The relative ion number is indicated in grey tones.
requires a total time of 120h to obtain the data. Care was taken to keep all the relevant experimental parameters constant during this time. The figure shows the existence of many sharp lines, indicating a reduced number of stored ions. Compared to our previous results [12] the width of the lines is smaller. We attribute this to a better alignment of the present trap. Double structures of some lines, as observed ealier [12] and possibly caused by violation of rotational symmetry (t3x ¢ flz) are hardly visible any more. We ascribe this to the fact that our present trap is larger in size by a factor of three in linear dimensions and that perturbations in the potential, caused by holes in the
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
qz Fig. 6. Calculated lines corresponding to the n,/2flr + nz/2~3z = 1 for different orders N = n r + nz.
relation:
electrodes and other imperfections, play a less significant role near the trap center. In addition, improved detection sensitivity allows us to observe lines associated with very high orders of the perturbation. The assignment of the observed lines follows from Fig. 6. Here we have plotted the theoretical lines of instabilities according to Eq (7) with u = 1 for perturbations of order N = 3 to N = 8. In a somewhat arbitrary manner we have drawn strong resonances by thick solid lines, weaker ones by thin solid lines and very weak ones by dashed lines. For clarity we have omitted lines of higher order in Fig. 6. However, in the experimental diagram of Fig. 5 one can observe lines belonging to orders up to N = 11. All lines belonging to a given order
160
R. Alheit et al./International Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
N cross at one point as indicated in Fig. 6. The assignment of the individual lines of any given order is such that the line corresponding to nz = 0, n, = N is parallel to the/3r boundary. Then, counterclockwise, the lines n, = 1, n, = N - 1; n, = 2, n, = N - 2 . . . follow. In a different way we identify the lines if we start at a node which appears at the fir = 0 border. All lines having the same n, value start here. The line o f n r = 0 is at the right side and then lines with increasing nr follow clockwise. In addition to these results we m a d e some observations which are not explained by the formalism given above (Refs [6,7]). First, we find coincidence with the experimentally observed and theoretically calculated lines only if we assume a shift o f the stability diagram, which corresponds to tSq = 0.046 at a = 0. Secondly, we see strong lines o f instabilities parallel to the boundaries o f the stability diagram at flz,/% = 0. At present we do not have a convincing explanation for these instabilities. While we observe a reduced ion number at our detector when we operate the trap at a non-linear resonance, it is not immediately obvious whether this is caused by a reduced trapping efficiency during ion creation, by a different spatial distribution, which could affect the ion extraction efficiency, or by a reduced stability during storage. In order to shed light on the mechanism we measured the storage time at individual operating parameters at a non-linear resonance. Fig. 7 shows a result for a dodecapole resonance at a, = 0.0136, qz = 0.586. If we plot the decay constants -r of an exponential ion loss when we vary q, in small steps around the resonance center we find the same shape as with the ion number. This indicates that it is in fact the instability of the ion trajectories which causes the decrease of the detected ion number. If we perform a very high resolution scan of the stability diagram, varying q, and a, in steps
1000
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,
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, / 0,0
qz Fig. 7. Ion number and time constant for exponential ion loss at a dodecapole resonance (az = 0.0136, qz = 0.586).
of 0.00068 and 0.00136, respectively (corresponding to A U 0 = A V 0 = 0 . 5 V in our case; Fig. 8(a)), we observe additional weak lines, which were not visible in Fig. 5. The assignment of these lines to the orders of the perturbation is made in Fig. 8(b). We note in particular that weak lines corresponding to iso/~r lines fir = 2/7, 2/8, 2/9 and 2/10 are visible. A three-dimensional plot of the data in a small part o f the stability diagram (Fig. 9) improves visibility o f the structure of ion number variation and justifies the term "black canyon" (Ref. (10)) for the lines of instabilities. A n even higher scan resolution o f Aa, = 0.00027, Aq, = 0 . 0 0 0 6 8 (Fig. 10) reveals a line, starting near az = 0 and crossing the node at az = - 0 . 0 1 8 , qz = 0.80, where the ion number is slightly increased (by about 5%). This line corresponds to flz = v7213r, for which the depth of the trapping potential is identical in both r- and z-directions. Some o f the weak lines appearing in Fig. 8 can only be explained by the resonance condition°of Eq. (7) if we set u = 2 nr
nz
~-/% + ~-f12 = 2
(10)
They can arise if the trapping voltage is not perfectly sinusoidal but contains harmonics
R. Alheit et al./lnternational Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
161
0,045
0,030 0,015
a z o,ooo -0,015
-0,030 -0,045
(a)
0,50
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u, f b
u,13u
qz (N-3)13r/2 + 313_/2 = 1
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5
4
n -o1,t ra
0,045 0,030 0,015
a z o,ooo -0,015
-0,030 -0,045
(b) (N-6)13r/2 +( Fig. 8. (a) Very high resolution scan of a part of the stability diagram, showing many weak non-linear resonances. Resolution:
Aqz = 0.000 68, Aaz = 0.001 36. (b) Theoretical non-linear resonances of the part of the stability diagram shown in Fig. 8(a).
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
162
rotational symmetry. The experimental observation of these resonances indicates that this is not the case in our trap. Similarly resonances of odd N should appear only if the mirror symmetry of the trap with respect to the z = 0 plane is violated as seems to be the case in our trap. 0,02 0,01
•,~ IN
4. Shape and amplitude of non-linear resonances
o,oo -0,01
Fig. 11 shows a high resolution scan through a non-linear resonance where we added many ion ejection pulses to obtain a good signal-tonoise ratio. The example shown corresponds to N = 4, r/r ~ 1, r/z ----3. The half width is tSq~ = 0.007 which is a typical number also for other resonances. F o r mass spectrometric use this would correspond to a mass resolution o f m / 6 m = q / 6 q = 110. As in all other observed resonances the shape is slightly asymmetric, showing a larger slope at lower q values. A possible reason for such an asymmetry could be deviation from rotational symmetry of our trap as mentioned previously. This would cause non-linear resonances for the x- and ycoordinates to appear at slightly different q values. If the strength of these nearby resonances is not the same and if they are not
-0,02 -0,03 -0,04 -0,05 0,65 0,66 0,67 0,68 0,69 0,70 0,71 0,72 u,13
qz-aXis Fig. 9. Three-dimensional picture of a small part of the stability diagram, showing different "black canyons" of reduced ion number.
at 2fL F o r even numbers n r and nz they coincide with fundamental non-linear resonances. The appearance of m a n y lines of high order can be related to some properties of our trap. As stated by Wang et al. [6] resonances of odd nr and odd nz should vanish if the trap has perfect
az
0,50
0,55
0,60
0,65
0,70
0,75
0,80
qz Fig. 10. Very high resolution scan of a part of the stability diagram (Aqz = 0.000 68, Aa z = 0.000 28) showing a slightly increased ion number along a line (starting in the upper left corner and crossing the different nodes) where fir = x/~13zand the shape of the trap's timeaveraged potential is spheroid.
R. A lheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169 I
¢0
30000
"~
25000
I
I
I
÷
tO
20000 15000
0
10000
..Q
5000
E
~ e--
resonance center and outside the resonances. As expected, the resonance intensity decreases with increasing order. In general the strength of the resonances with r/r or n z 0 are very weak and often not visible at all. We should note, however, that Fig. 13 gives only a qualitative picture, since the relative strength o f the different resonances, as they occur in our experiment, depend on the delay time between ion creation and extraction, as is evident from the storage time variation at a resonance as shown in Fig. 7. Weak resonances become more visible only after a long delay time, since their reduction in storage time is small. =
÷
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÷
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.~÷ ~-÷
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0
~
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qz Fig. 11. Shape o f a n o n - l i n e a r o c t o p o l e resonance ( N = 4, n r = 1, n z = 3).
resolved, their sum would appear as one asymmetric line. The strength o f the resonances, that is the fraction reduction in ion number, is quite different throughout the stability diagram. Fig. 12 shows a scan for constant a (az = 0.0136) along the qz-axis. Following the identification of the resonances from those in the complete stability diagram we have assigned the numbers N / n r / n z to individual lines. In Fig. 13 we have plotted the amplitudes o f the different resonances; we define amplitude as the relative difference of the detected ion numbers at
5. Space charge effects We have investigated the appearance of the non-linear resonances at different ion numbers. As an example we discuss Fig. 14(a), where we plotted the resonance N = 4 , n r =- 1, n z ---- 3 for different detected ion numbers, which vary between 150 and 250 000. The active volume of our trap is about 1 cm 3. While the relative strength of the resonances stays I
8oo
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.
_~ 5oo
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~ 300 N I n r l nz : ¢" ~200 =
7/0/7 7/2/5
7•3/4
7/5/2
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T I
6/2/4 100
5/1/4
" i
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nr'13r/2 + nz*~z/2 = 1 ] s,
0 0,2
n r + n z = N, I , 0,3
'2
2N: order of perturbation potential r I i 0,4
0,5
0,6
~./2/2
' I 0,7
4/1/3
I 0,8
3/0/3
ll ~ 13/1/2 ~ 0,9
, 1,0
qz Fig. 12. Detected ion n u m b e r vs. q= for a c o n s t a n t value of az (0.0136). The a s s i g n m e n t o f the different n o n - l i n e a r r e s o n a n c e s to the p o t e n t i a l o f o r d e r 2 N at the integers nr, nz is indicated.
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
164
1,0
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N=3
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nr
0,0
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Fig. 13. Strengths of the observed resonances for different orders of the potential. The resonance intensity is defined as (no - n)/no, where n is the detected ion number outside a resonance and no that at resonance center.
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
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0,002 0,001
t,,,.,,.
t,--
0,000 100
1000
10000
max. number of detected ions Fig. 14. (a) Effect of increasing ion number on the position and on the shape of a non-linear resonance (N = 4, nT = 1, nz = 3). The stored ion numbers are obtained by multiplying the detected ion numbers by roughly a factor of 10 to account for the estimated detection efficiency. (b) Shift of the resonance center of Fig. 14(a) for different detected ion numbers.
166
R. Alheit et al./International Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
approximately constant, the width increases by a factor of 2. The non-linearity of the space charge potential causes a distribution of the motional eigenfrequencies with increasing width if the ion number becomes larger and consequently the condition for non-linear resonances is covered by a distribution of a and q values. At the same time the position of minimum ion number is shifted to higher q values with the ion number as expected. The value of/3 decreases with space charge and consequently the q value corresponding to the same/3 value without and with space charge should increase. From Fig. 14(b) we see that this shift becomes significant above a detected ion number of several thousand. If we assume that the overall detection efficiency is about 10%, this number is in accordance with
600
i
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i
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'--"
~ I
predictions that space charge effects in general can be neglected below an ion number of 104 [13].
6. Modulation of the trapping voltage The treatment of the inhomogeneous Mathieu equation to determine the stability of ion trajectories in a potential with higher order contributions by different authors [6-9] implicitly assumes that the electric a.c. trapping field has a perfectly sinusoidal shape. This might not necessarly be the case in an actual experiment since high voltage amplifiers may show phase and amplitude noise, sidebands or higher harmonics of the fundamental frequency. We have started to investigate the effect of such imperfections by modulating the
i
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O 300
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at a z = 0,0136
excitation amplitude at lower endcap
0,560
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" 250 mV
0,570
0,575
0,580
•
Vex = 3000 kHz
--x--
Vex = 3030 kHz
--o--
Vex = 3060 kHz
,D
Vex = 3100 kHz
0,585
0,590
0,595
qz Fig. 15. Sidebands at the non-linear octupole resonance (N = 4, n r 1, n~ = 3) for different excitation frequencies added to the trap driving frequency of 3000 kHz. The arrows indicate the position of the sidebands for the different modulation frequencies. =
167
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
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Voe×= 9o0o mv
I
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,
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i
0,600
qz Fig. 16. Higher order sidebands to the non-linear resonance of Fig. 15 at large amplitudes of the additional excitation voltage.
0,05 0,04 0,03 0,02 0,01 a z
0,00
-0,01 -0,02 -0,03 -0,04 -0,05 0,50
0,55
0,60
0,65
0,70
0,75
0,80
qz Fig. 17. Part of the stability diagram with additional excitation voltage at 30kHz frequency difference from the trapping field, showing sidebands occurring at some of the non-linear resonances.
168
R. Alheit et al./lnternational Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
trapping voltage and looking for its influence on a non-linear resonance. We added a second frequency, slightly different from the 3 MHz driving field frequency, to the trap electrodes. When we take the octupole resonance N = 4, nr = 1, nz = 3 as an example (Fig. 15) we see sidebands occurring at the frequency difference between the driving field and the additional voltage. Although the amplitude of the additional voltage is small compared to that of the trapping field (250 mV compared to 400 V in the example of Fig. 15) the sidebands are quite pronounced. If we increase the amplitude of the additional voltage the total number of stored ions is decreased, the central non-linear resonance almost disappears and additional sidebands at multiples of the difference frequency show up (Fig. 16). The indication of a double structure in this resonance is maintained also in the sidebands. We note, however, that those sidebands do not appear at all resonances. Fig. 17 shows a high resolution scan of a part of the stability diagram, when the additional voltage was applied. Sidebands are clearly observed at some resonances, in particular the hexapole resonance 3 r = 2/3 (N = 3, nr = 3, nz = 0), while at other resonances they do not occur or are too weak to be observed.
7. Conclusion We have added a number of detailed investigations to our previously [12] reported observation of non-linear resonances in the first stable region of a Paul trap. Numerous sharp resonances arising from higher order contributions to the ideal quadrupole potential are observed, the highest order being N = 11. Additional resonances which follow the condition (nr/2)3r + (nz/2)3z = 2 are observed for the first time. Furthermore, we observe strong resonances near the boundary of the stability diagram which have not been explained so far. Broadening and shifts of the
resonances due to space change have been observed and sidebands are created when we modulate the a.c. driving field. While the general structure of the observations seems to be well understood there are some questions which require further investigation. This concerns a confirmation of the origin of the strong resonances near the boundary of the stability region as well as the different behavior of different resonances when we modulate the trap's driving frequency. It would be of interest to continue the measurements with the possibility of changing the trap's geometry (mirror symmetry, rotational symmetry) in a controlled way and to investigate other trap geometries which are in use in commercial mass spectrometers. Another important parameter which has not yet been changed is the background pressure. Collisions with background atoms whose mass is larger than that of the confined ion lead to rapid ion loss. For this reason our experiments on H~- were performed under ultrahigh vacuum conditions to minimize collisions with heavy background molecules. If we perform our measurements on heavy ions we could vary the background gas by introducing light gases such as H2 or He into our apparatus. Collisions of the heavy stored ions with these gases would reduce the ions' kinetic energy. It remains to be seen what influence the collisions have on the appearance of the non-linear resonances. The established existence of non-linear resonances may have a significant influence on the result in experiments where the mass of the confined ion is changed by chemical reactions, electro- or photo-dissociation, or other processes. If the parent molecule is stored in a stable part of the stability area, it may well happen that a daughter ion appears at or near a non-linear resonance. This would change the observed number of daughter ions significantly. Even for small values of the trapping parameters a and q, where only high orders of a perturbing potential play a role in
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
the non-linear resonances and their strength is small, a change in signal height by 10-15% can easily be observed.
Acknowledgements Our experiment, performed at the University of Mainz, was supported by the Deutsche Forschungsgemeinschaft. PI2M is Unit6 de Recherche Associ6e au C.N.R.S. 773.
References [1] R.E. March and J.F.J. Todd (Eds.), Practical Aspects of Ion Trap Mass Spectrometry. CRC Press, Cleveland, OH, 1995. [2] D.A. Church, Phys. Rep., 228 (1993) 254. [3] P.E. Toschek, in B. Kramer (Ed.), The Art of Measurement, VCH, Weinheim, 1988, pp. 139-160.
169
[4] W. Paul, O. Osberghaus and E. Fischer, Forschungsber. Wirtsch. Verkehrsminist. Nordrhein-Westfalen, (1958) 415. [5] J. Meixner and F.W. Sch~ifke, Mathieusche Funktionen und Sph~iroidfunktionen, Springer, Berlin, 1954. [6] Y. Wang, J. Franzen and K.P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 124 (1993) 125. [7] (a) Y. Wang and J. Franzen, Int. J. Mass Spectrom. Ion Processes, 132 (1994) 155. (b) J. Franzen, R.-H. Gabling, M. Schubert and Y. Wang, in (R.E. March and J.F.J. Todd (Eds.), Practical Aspects of Ion Trap Mass Spectrometry, Vol. 1, CRC Press, Cleveland, OH, 1995, pp. 49-167. [8] F. von Busch and W. Paul, Z. Phys., 164 (1961) 581. [9] P.H. Dawson and N.R. Whetten, Int. J. Mass Spectrom. Ion Phys., 2 (1969) 45. [10] (a) F. Guidugli and P. Traldi, Rapid Commun. Mass Spectrom., 5 (1991) 343. (b) F. Guidugli, P. Traldi, A.M. Franklin, M.L. Langford, J. Murell and J.F.J. Todd, Rapid Commun. Mass Spectrom., 6 (1992) 229. [11] D.M. Eades, J.V. Johnsen and R.A. Yost, J. Am. Soc. Mass Spectrom., 4 (1993) 917. [12] R. Alheit, C. Hennig, R. Morgenstern, F. Vedel and G. Werth, Appl. Phys., B, 61 (1995) 277. [13] F. Vedel, Int. J. Mass Spectrom. Ion Processes, 106 (1991) 33.
International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
Mass Spectrometry and Ion Processes
Higher order non-linear resonances in a Paul trap R. Alheit a, S. Kleineidam a, F. Vedel b, M. Vedel b, G. W e r t h a'* aInstitut fi~r Physik, Universiti~t Mainz, D-55099 Mainz, Germany bUniversitd de Provence, PI2M, F-13397 Marseille Cedex 20, France
Received 5 December 1995; accepted 5 April 1996
Abstract
We report some detailed investigations on non-linear resonances of high order as they appear in a radio-frequency ion trap, whose potential deviates from the ideal quadrupolar shape. The strength and shape of these resonances have been measured throughout the first stable region of the trap. Moreover, space charge shifts and also effects of a modulation of the trapping voltage on the shape of the resonances have been observed. Some of the resonances cannot be explained by the general rule that the sums of multiples of the ion macrofrequencies nrwr ÷ nzwz (nr, nz are integers) are multiples of the trapping field frequency fL Keywords." Ion trap; Non-linear resonances; Paul trap
1. I n t r o d u c t i o n
Ion traps have been developed to become a standard experimental tool used for high resolution mass spectrometry [1], in collision studies with charged particles [2], in quantum optics [3] and in commercial mass spectrometers. Of particular importance among the various types of trap is the so-called "Paul trap", named after its inventor [4]. It operates by applying a high frequency voltage V = V0 cos ~t to three electrodes, which are hyperboloids of revolution. A variable d.c. voltage U0 may be superimposed to vary the shape and depth of the three-dimensional potential well inside the electrode structure. For an ideal trap geometry and a single stored particle we have a * Corresponding author.
time-averaged harmonic trapping potential ("pseudopotential"). The ion trajectories in a trap with perfect rotational symmetry around the z-axis can be calculated analytically by the solution of the normalized Mathieu differential equation [5]: d2u
dr 2 + (ar,z - 2qr, zCOS2r)u = 0 u = x , y , z,
r = x, y,
(1)
T = ½ ~2t
-16eU0 az = - 2 a r = m(r2 + 2z2)f~2
(2)
8eVo qz = - 2 q r = m(r~ + 2z02)~2
(3)
where r0 is the radius of the ring electrode and 2z0 the closest distance between the endcaps. Stable ion trajectories occur for certain values
0168-1176/96/$15.00 © 1995 Elsevier Science B.V. All rights reserved P H S0168-1176(96)04380-7
156
R. Alheit et al./International Journal o f Mass Spectrometry and Ion Processes 154 (1996) 155-169
of the stability parameters flr,~ which depend on the parameters ar, z and qr, z:
which using eq. (5) can be written as nrCOr + nzCOz = t,,f~
(8)
q2 fl2 = a +
q2 (2 + 3) 2 - a 4
q2 (4 + fl)2 __ a + - (4)
They are related to the free oscillation frequency co of the trapped ion in the time-averaged potential (macromotion) by
flr,z ~-~ COr,z -- 2
(5)
The first stable region occurs when 0 < 3r,~ < 1. Deviations from the pure quadrupolar trap geometry, caused by misalignments, nonhyperbolic shapes, truncated electrodes, holes in the electrodes to obtain access to the trapping fields inside, space charge potential of a large ion cloud and influence of potentials from outside the trap, make the ion motion unstable for certain values of the trapping parameters, for which in the ideal case stability exists. Assuming that the rotational symmetry is not destroyed, this instability can be understood from an analytical expansion of the trapping potential in a power series of spherical coordinates. If we assume mirror symmetry with respect to the midplane of the trap, the expansion will contain only even orders: oo
[ / r , ~ 2N
¢(r,O, qo) = ~,C2N~,~)
P2N(COS~)
(6)
¢2N describes the contribution of the 2Nth order to the potential and PZN is the spherical harmonic of order 2N. Wang and co-workers [6, 7] derived a condition for the appearance of instabilities nz
~ flr + -~ flr = U
(7)
with nr, nz = 0, 4-1, 4-2... and nr + nz = N; u = 1,2, .... A similar equation had been derived earlier for the two-dimensional case of a linear mass filter by von Busch and Paul [8] and by Dawson and Whetten [9]:
nxa;x + nyCOy =
~
(9)
As stated by Wang et al. [6], energy exchange between the different coordinates occurs when nr or nz assumes a negative value, while for positive values the ions may gain energy from the time-varying trapping field and may get lost from the trap. Some of these instabilities have been experimentally observed in the two-dimensional case by von Busch and Paul [8] and Dawson and Whetten [9]. For the three-dimensional trap the first observation of a "black hole" was reported by Guidugli and Traldi [10] and several observations of unstable lines have been made by different groups. A high resolution mapping of a part of the stability diagram was performed by Eades et al. [11], who found several unstable lines. Of particular strength in all the traps under investigation was the resonance occurring for nr = 3, nz = 0 which arises from a hexapole contribution to the potential. Since the equation for a non-linear resonance contains the z / m ratio of the ion, it can be used to improve the performance of traps in high resolution mass spectrometry. Recently we reported very high resolution scans of the first stable region [12] and demonstrated that many unstable lines of reduced ion stability arising from higher order perturbation in the trapping potential can be observed. In this paper we will describe measurements which add further details to our previous observations. We used a different trap which possibly has different contributions of higher order perturbations. The main obvious difference is that we used a trap where the radius
R. Alheit et al./lnternational Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
I0 ."
copper, the other endcap was made of stainless steel with a mesh in the center to make the extraction of the ions possible. Since our trap was used simultaneously for an experiment in laser spectroscopy, we had two holes of 8 m m diameter drilled into the ring electrode opposite to each other in the trap's midplane. These holes served for the entrance and exit of a laser beam. The trap was driven by a 3 MHz oscillator. A.c. and d.c. amplitudes were controlled by a personal computer and could be varied in steps of 500 mV and 100 mV, respectively. The driving voltage was applied to the ring electrode while the endcaps were held at ground potential. Ions were created by ionization of the background gas which was typically at 10-9mbar after a three-day bakeout at 280°C. We used an electron beam from a filament placed below the lower endcap. The electrons entered the traps through a 1.5 m m diameter hole in the center of the lower endcap after being accelerated by 360 V. A cross-section of our trap is shown in Fig. 1.
•
"- .
r0
Fig. 1. Ion trap used in our experiment. The radius of the hyperbolic-shaped ring is 20 mm and the distance between the endcaps is 28.3 mm. The electrodes are modified by holes and grids.
was increased from 7 to 20 m m in the present experiment.
2. Experimental apparatus Our trap of r0 = 20 m m has a ratio r o / z o = x/-2. The ring and one endcap were made of
[Pc].
lionl
cylinder
extrac~on-pulse
I multiplierl -15oov..-zsoov
I
genaator
J L
i'
: oov
..................... ,
ov.
drivingfrequency generator3MHz
I
ringeteclrode endcap
+lOOV
e - gunswitch ~
157
~.~,o
V
Fig. 2. Block diagram of the experimental setup.
ment
158
R. Alheit et aL/International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169 creation
waiting time
extraction
typ. 500 ms
typ. 300 ms
600 ns
detection
time e - gun
- -
+50V
',+50 V
- 350 V OV
OV
upper endcap
-150 V +150 V OV
OV
lower endcap
Fig. 3. Timing sequence of the experiment.
Variation of the filament current or the pulse length of the accelerating voltage between 10 and 2000ms allows one to vary the number of ions created by several orders of magnitude. After a delay time which could be varied between 10 ms and many seconds we applied a pulse of - 1 5 0 amplitude and 600 ns duration to the upper endcap and the same pulse with opposite polarity to the lower endcap. Both pulses had a fixed phase with respect to the trapping field oscillator. They drive the ions out of the trap. Fig. 2 shows the experimental setup and Fig. 3 the timing sequence. After passing through an ion optic the ions arrive at an ion multiplier placed 8 cm above the upper endcap. This distance is sufficient to separate ions of different masses in time by a few lOOns, mainly H +, H + and N + from the residual gas. In addition we observe H~- ions formed inside the trap by reaction of stored H~- with neutral H2: H~- + H2 ~ H + + H. The ion signals were amplified by the multiplier and digitized. By gating the analog-digital converter (ADC) we could choose a specific ion species for our investigation (Fig. 4). The measurements presented below were exclusively performed on an H + cloud because the corresponding signal was stronger than that of H + and its formation happened in a shorter time than for H~-. The storage time under normal trapping conditions outside a non-linear resonance was of the order of 10 s.
The main loss mechanism was the reaction with H 2 molecules mentioned above.
3. Scan of the stability diagram Fig. 5 shows a scan through almost the complete stability diagram of H~-. In this experiment we varied the values of az and qz in steps of 0.0019 and 0.00135, respectively, corresponding to voltage steps of AU0 = 0.7V and A V0 = 1 V. To improve the signal-to-noise ratio we added three signals for each of the 200 000 data points. The very high resolution
~4
-a,zt5 V?
~f L~
Fig, 4. Oscilloscope trace of an ion signal after ejection from the trap (H +, H~-, H~- from left to right). By gating the detection electronics we can choose any one of the signals for our experiments.
159
R. Alheit et al./International Journal o f Mass Spectrometry and Ion Processes 154 (1996) 155-169
0,0 0,1 0,15
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
0,15
0,10
0,10
0,05
0,05
0,00
0,00
az -0,05
43,05
-0,10
-0,10
-0,15
~,15
-0,20
-0,20
-0,25
~),25
-0,30
-0,30
-0,35
4), 35
-0,40 0,0 0,1
0,2
0,3
0,4 0,5
0,6
0,7
0,8
0,9
1,0
qz Fig. 5. High resolution scan of the major part of the first region of the stability diagram of our ion trap. The relative ion number is indicated in grey tones.
requires a total time of 120h to obtain the data. Care was taken to keep all the relevant experimental parameters constant during this time. The figure shows the existence of many sharp lines, indicating a reduced number of stored ions. Compared to our previous results [12] the width of the lines is smaller. We attribute this to a better alignment of the present trap. Double structures of some lines, as observed ealier [12] and possibly caused by violation of rotational symmetry (t3x ¢ flz) are hardly visible any more. We ascribe this to the fact that our present trap is larger in size by a factor of three in linear dimensions and that perturbations in the potential, caused by holes in the
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
qz Fig. 6. Calculated lines corresponding to the n,/2flr + nz/2~3z = 1 for different orders N = n r + nz.
relation:
electrodes and other imperfections, play a less significant role near the trap center. In addition, improved detection sensitivity allows us to observe lines associated with very high orders of the perturbation. The assignment of the observed lines follows from Fig. 6. Here we have plotted the theoretical lines of instabilities according to Eq (7) with u = 1 for perturbations of order N = 3 to N = 8. In a somewhat arbitrary manner we have drawn strong resonances by thick solid lines, weaker ones by thin solid lines and very weak ones by dashed lines. For clarity we have omitted lines of higher order in Fig. 6. However, in the experimental diagram of Fig. 5 one can observe lines belonging to orders up to N = 11. All lines belonging to a given order
160
R. Alheit et al./International Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
N cross at one point as indicated in Fig. 6. The assignment of the individual lines of any given order is such that the line corresponding to nz = 0, n, = N is parallel to the/3r boundary. Then, counterclockwise, the lines n, = 1, n, = N - 1; n, = 2, n, = N - 2 . . . follow. In a different way we identify the lines if we start at a node which appears at the fir = 0 border. All lines having the same n, value start here. The line o f n r = 0 is at the right side and then lines with increasing nr follow clockwise. In addition to these results we m a d e some observations which are not explained by the formalism given above (Refs [6,7]). First, we find coincidence with the experimentally observed and theoretically calculated lines only if we assume a shift o f the stability diagram, which corresponds to tSq = 0.046 at a = 0. Secondly, we see strong lines o f instabilities parallel to the boundaries o f the stability diagram at flz,/% = 0. At present we do not have a convincing explanation for these instabilities. While we observe a reduced ion number at our detector when we operate the trap at a non-linear resonance, it is not immediately obvious whether this is caused by a reduced trapping efficiency during ion creation, by a different spatial distribution, which could affect the ion extraction efficiency, or by a reduced stability during storage. In order to shed light on the mechanism we measured the storage time at individual operating parameters at a non-linear resonance. Fig. 7 shows a result for a dodecapole resonance at a, = 0.0136, qz = 0.586. If we plot the decay constants -r of an exponential ion loss when we vary q, in small steps around the resonance center we find the same shape as with the ion number. This indicates that it is in fact the instability of the ion trajectories which causes the decrease of the detected ion number. If we perform a very high resolution scan of the stability diagram, varying q, and a, in steps
1000
,
.'-2.
i
,
[
.
[
,
[
4,0
,
900
3,5
8oo '~ •g
700
3,0 o) ~ 'i
,
2,5 " ~ ~ 0 5 0 0 6 0. 0+ ,
,,
,,oo
,),/
"o
c~ E
300
t-
200
I 0,580
,
I 0,584
,
.
,
,']
/
.
~
100
~,'
I 0,588
2,0 ~ 1,5 ~ 1,o .~.
---4)--
number of ions
" ~
storage time ~ l 0,5
,
I 0,592
,
I 0,596
, / 0,0
qz Fig. 7. Ion number and time constant for exponential ion loss at a dodecapole resonance (az = 0.0136, qz = 0.586).
of 0.00068 and 0.00136, respectively (corresponding to A U 0 = A V 0 = 0 . 5 V in our case; Fig. 8(a)), we observe additional weak lines, which were not visible in Fig. 5. The assignment of these lines to the orders of the perturbation is made in Fig. 8(b). We note in particular that weak lines corresponding to iso/~r lines fir = 2/7, 2/8, 2/9 and 2/10 are visible. A three-dimensional plot of the data in a small part o f the stability diagram (Fig. 9) improves visibility o f the structure of ion number variation and justifies the term "black canyon" (Ref. (10)) for the lines of instabilities. A n even higher scan resolution o f Aa, = 0.00027, Aq, = 0 . 0 0 0 6 8 (Fig. 10) reveals a line, starting near az = 0 and crossing the node at az = - 0 . 0 1 8 , qz = 0.80, where the ion number is slightly increased (by about 5%). This line corresponds to flz = v7213r, for which the depth of the trapping potential is identical in both r- and z-directions. Some o f the weak lines appearing in Fig. 8 can only be explained by the resonance condition°of Eq. (7) if we set u = 2 nr
nz
~-/% + ~-f12 = 2
(10)
They can arise if the trapping voltage is not perfectly sinusoidal but contains harmonics
R. Alheit et al./lnternational Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
161
0,045
0,030 0,015
a z o,ooo -0,015
-0,030 -0,045
(a)
0,50
0,55
0,60
0,65
U, [U
u, f b
u,13u
qz (N-3)13r/2 + 313_/2 = 1
N:
9
8
7
6
5
4
n -o1,t ra
0,045 0,030 0,015
a z o,ooo -0,015
-0,030 -0,045
(b) (N-6)13r/2 +( Fig. 8. (a) Very high resolution scan of a part of the stability diagram, showing many weak non-linear resonances. Resolution:
Aqz = 0.000 68, Aaz = 0.001 36. (b) Theoretical non-linear resonances of the part of the stability diagram shown in Fig. 8(a).
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
162
rotational symmetry. The experimental observation of these resonances indicates that this is not the case in our trap. Similarly resonances of odd N should appear only if the mirror symmetry of the trap with respect to the z = 0 plane is violated as seems to be the case in our trap. 0,02 0,01
•,~ IN
4. Shape and amplitude of non-linear resonances
o,oo -0,01
Fig. 11 shows a high resolution scan through a non-linear resonance where we added many ion ejection pulses to obtain a good signal-tonoise ratio. The example shown corresponds to N = 4, r/r ~ 1, r/z ----3. The half width is tSq~ = 0.007 which is a typical number also for other resonances. F o r mass spectrometric use this would correspond to a mass resolution o f m / 6 m = q / 6 q = 110. As in all other observed resonances the shape is slightly asymmetric, showing a larger slope at lower q values. A possible reason for such an asymmetry could be deviation from rotational symmetry of our trap as mentioned previously. This would cause non-linear resonances for the x- and ycoordinates to appear at slightly different q values. If the strength of these nearby resonances is not the same and if they are not
-0,02 -0,03 -0,04 -0,05 0,65 0,66 0,67 0,68 0,69 0,70 0,71 0,72 u,13
qz-aXis Fig. 9. Three-dimensional picture of a small part of the stability diagram, showing different "black canyons" of reduced ion number.
at 2fL F o r even numbers n r and nz they coincide with fundamental non-linear resonances. The appearance of m a n y lines of high order can be related to some properties of our trap. As stated by Wang et al. [6] resonances of odd nr and odd nz should vanish if the trap has perfect
az
0,50
0,55
0,60
0,65
0,70
0,75
0,80
qz Fig. 10. Very high resolution scan of a part of the stability diagram (Aqz = 0.000 68, Aa z = 0.000 28) showing a slightly increased ion number along a line (starting in the upper left corner and crossing the different nodes) where fir = x/~13zand the shape of the trap's timeaveraged potential is spheroid.
R. A lheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169 I
¢0
30000
"~
25000
I
I
I
÷
tO
20000 15000
0
10000
..Q
5000
E
~ e--
resonance center and outside the resonances. As expected, the resonance intensity decreases with increasing order. In general the strength of the resonances with r/r or n z 0 are very weak and often not visible at all. We should note, however, that Fig. 13 gives only a qualitative picture, since the relative strength o f the different resonances, as they occur in our experiment, depend on the delay time between ion creation and extraction, as is evident from the storage time variation at a resonance as shown in Fig. 7. Weak resonances become more visible only after a long delay time, since their reduction in storage time is small. =
÷
"O
163
÷
÷
÷÷ -Is
.~÷ ~-÷
I
0
~
0,770
I
~
0,775
I
~
0,780
I
0,785
qz Fig. 11. Shape o f a n o n - l i n e a r o c t o p o l e resonance ( N = 4, n r = 1, n z = 3).
resolved, their sum would appear as one asymmetric line. The strength o f the resonances, that is the fraction reduction in ion number, is quite different throughout the stability diagram. Fig. 12 shows a scan for constant a (az = 0.0136) along the qz-axis. Following the identification of the resonances from those in the complete stability diagram we have assigned the numbers N / n r / n z to individual lines. In Fig. 13 we have plotted the amplitudes o f the different resonances; we define amplitude as the relative difference of the detected ion numbers at
5. Space charge effects We have investigated the appearance of the non-linear resonances at different ion numbers. As an example we discuss Fig. 14(a), where we plotted the resonance N = 4 , n r =- 1, n z ---- 3 for different detected ion numbers, which vary between 150 and 250 000. The active volume of our trap is about 1 cm 3. While the relative strength of the resonances stays I
8oo
I
7
.
_~ 5oo
"-
'
•
~'~
5/4/1
~ 300 N I n r l nz : ¢" ~200 =
7/0/7 7/2/5
7•3/4
7/5/2
9/4/5 9/5/4 6/1/5 8/6/2 4/2/2 8/1/7 8/3/5 8/4/4 5/0/5 7/1/6 6/2/4 5/2/3
T I
6/2/4 100
5/1/4
" i
6•4/2
nr'13r/2 + nz*~z/2 = 1 ] s,
0 0,2
n r + n z = N, I , 0,3
'2
2N: order of perturbation potential r I i 0,4
0,5
0,6
~./2/2
' I 0,7
4/1/3
I 0,8
3/0/3
ll ~ 13/1/2 ~ 0,9
, 1,0
qz Fig. 12. Detected ion n u m b e r vs. q= for a c o n s t a n t value of az (0.0136). The a s s i g n m e n t o f the different n o n - l i n e a r r e s o n a n c e s to the p o t e n t i a l o f o r d e r 2 N at the integers nr, nz is indicated.
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
164
1,0
1,01
N=4
N=3
0,8
0,8
0,6
0,6
0,4
0,4
0,2
0,2
X 0,0
X
2
0,0
3
1,0
0
1
2
X
X
3
4
1,0
cI1) "t"-
0,8
0,8
>
0,6
0,6
"~
0,4
N=5
N=6
o4L]III 0,2
0,2 0,0
0
1
2
3
4
5
1,0
0,0
1 ~0
N=7
0,8
0,6
0,6
0,4
0,4
0,2
o,2 2
3
4
5
6
1
2
3 ,
,
4 ,
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R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
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166
R. Alheit et al./International Journal of Mass Spectrometry and 1on Processes 154 (1996) 155-169
approximately constant, the width increases by a factor of 2. The non-linearity of the space charge potential causes a distribution of the motional eigenfrequencies with increasing width if the ion number becomes larger and consequently the condition for non-linear resonances is covered by a distribution of a and q values. At the same time the position of minimum ion number is shifted to higher q values with the ion number as expected. The value of/3 decreases with space charge and consequently the q value corresponding to the same/3 value without and with space charge should increase. From Fig. 14(b) we see that this shift becomes significant above a detected ion number of several thousand. If we assume that the overall detection efficiency is about 10%, this number is in accordance with
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predictions that space charge effects in general can be neglected below an ion number of 104 [13].
6. Modulation of the trapping voltage The treatment of the inhomogeneous Mathieu equation to determine the stability of ion trajectories in a potential with higher order contributions by different authors [6-9] implicitly assumes that the electric a.c. trapping field has a perfectly sinusoidal shape. This might not necessarly be the case in an actual experiment since high voltage amplifiers may show phase and amplitude noise, sidebands or higher harmonics of the fundamental frequency. We have started to investigate the effect of such imperfections by modulating the
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167
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
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168
R. Alheit et al./lnternational Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
trapping voltage and looking for its influence on a non-linear resonance. We added a second frequency, slightly different from the 3 MHz driving field frequency, to the trap electrodes. When we take the octupole resonance N = 4, nr = 1, nz = 3 as an example (Fig. 15) we see sidebands occurring at the frequency difference between the driving field and the additional voltage. Although the amplitude of the additional voltage is small compared to that of the trapping field (250 mV compared to 400 V in the example of Fig. 15) the sidebands are quite pronounced. If we increase the amplitude of the additional voltage the total number of stored ions is decreased, the central non-linear resonance almost disappears and additional sidebands at multiples of the difference frequency show up (Fig. 16). The indication of a double structure in this resonance is maintained also in the sidebands. We note, however, that those sidebands do not appear at all resonances. Fig. 17 shows a high resolution scan of a part of the stability diagram, when the additional voltage was applied. Sidebands are clearly observed at some resonances, in particular the hexapole resonance 3 r = 2/3 (N = 3, nr = 3, nz = 0), while at other resonances they do not occur or are too weak to be observed.
7. Conclusion We have added a number of detailed investigations to our previously [12] reported observation of non-linear resonances in the first stable region of a Paul trap. Numerous sharp resonances arising from higher order contributions to the ideal quadrupole potential are observed, the highest order being N = 11. Additional resonances which follow the condition (nr/2)3r + (nz/2)3z = 2 are observed for the first time. Furthermore, we observe strong resonances near the boundary of the stability diagram which have not been explained so far. Broadening and shifts of the
resonances due to space change have been observed and sidebands are created when we modulate the a.c. driving field. While the general structure of the observations seems to be well understood there are some questions which require further investigation. This concerns a confirmation of the origin of the strong resonances near the boundary of the stability region as well as the different behavior of different resonances when we modulate the trap's driving frequency. It would be of interest to continue the measurements with the possibility of changing the trap's geometry (mirror symmetry, rotational symmetry) in a controlled way and to investigate other trap geometries which are in use in commercial mass spectrometers. Another important parameter which has not yet been changed is the background pressure. Collisions with background atoms whose mass is larger than that of the confined ion lead to rapid ion loss. For this reason our experiments on H~- were performed under ultrahigh vacuum conditions to minimize collisions with heavy background molecules. If we perform our measurements on heavy ions we could vary the background gas by introducing light gases such as H2 or He into our apparatus. Collisions of the heavy stored ions with these gases would reduce the ions' kinetic energy. It remains to be seen what influence the collisions have on the appearance of the non-linear resonances. The established existence of non-linear resonances may have a significant influence on the result in experiments where the mass of the confined ion is changed by chemical reactions, electro- or photo-dissociation, or other processes. If the parent molecule is stored in a stable part of the stability area, it may well happen that a daughter ion appears at or near a non-linear resonance. This would change the observed number of daughter ions significantly. Even for small values of the trapping parameters a and q, where only high orders of a perturbing potential play a role in
R. Alheit et al./International Journal of Mass Spectrometry and Ion Processes 154 (1996) 155-169
the non-linear resonances and their strength is small, a change in signal height by 10-15% can easily be observed.
Acknowledgements Our experiment, performed at the University of Mainz, was supported by the Deutsche Forschungsgemeinschaft. PI2M is Unit6 de Recherche Associ6e au C.N.R.S. 773.
References [1] R.E. March and J.F.J. Todd (Eds.), Practical Aspects of Ion Trap Mass Spectrometry. CRC Press, Cleveland, OH, 1995. [2] D.A. Church, Phys. Rep., 228 (1993) 254. [3] P.E. Toschek, in B. Kramer (Ed.), The Art of Measurement, VCH, Weinheim, 1988, pp. 139-160.
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[4] W. Paul, O. Osberghaus and E. Fischer, Forschungsber. Wirtsch. Verkehrsminist. Nordrhein-Westfalen, (1958) 415. [5] J. Meixner and F.W. Sch~ifke, Mathieusche Funktionen und Sph~iroidfunktionen, Springer, Berlin, 1954. [6] Y. Wang, J. Franzen and K.P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 124 (1993) 125. [7] (a) Y. Wang and J. Franzen, Int. J. Mass Spectrom. Ion Processes, 132 (1994) 155. (b) J. Franzen, R.-H. Gabling, M. Schubert and Y. Wang, in (R.E. March and J.F.J. Todd (Eds.), Practical Aspects of Ion Trap Mass Spectrometry, Vol. 1, CRC Press, Cleveland, OH, 1995, pp. 49-167. [8] F. von Busch and W. Paul, Z. Phys., 164 (1961) 581. [9] P.H. Dawson and N.R. Whetten, Int. J. Mass Spectrom. Ion Phys., 2 (1969) 45. [10] (a) F. Guidugli and P. Traldi, Rapid Commun. Mass Spectrom., 5 (1991) 343. (b) F. Guidugli, P. Traldi, A.M. Franklin, M.L. Langford, J. Murell and J.F.J. Todd, Rapid Commun. Mass Spectrom., 6 (1992) 229. [11] D.M. Eades, J.V. Johnsen and R.A. Yost, J. Am. Soc. Mass Spectrom., 4 (1993) 917. [12] R. Alheit, C. Hennig, R. Morgenstern, F. Vedel and G. Werth, Appl. Phys., B, 61 (1995) 277. [13] F. Vedel, Int. J. Mass Spectrom. Ion Processes, 106 (1991) 33.