- Email: [email protected]
A high precision Penning trap mass spectrometer
Nuclear Instruments North-Holland
and Methods
A HIGH PRECISION Ch. GERZ, Insiit~t fi Received
in Physics
Research
B47 (1990) 453-461
453
PENNING TRAP MASS SPECTROMETER
D. WILSDORF
and G. WERTH
Physik, U~iL~~sit~tMaim, D-6500 Maim, FRG 16 October
1989 and in revised form 23 February
1990
Penning ion traps are used as mass spectrometers for high precision determination of mass doublets. We present our experimental setup and discuss in detail possible sources of systematic uncert~nti~ from electric and magnetic fietd distortions, assignments and ion-ion interaction. Methods are discussed to deal with the associated line shifts. As a result, we demonstrate experimentally obtained resolutions of a few parts in 10v9 for light masses. The method allows very precise determinations of doublet mass differences of light ions.
1. Introduction A number of different methods exist for precision mass spectroscopy. Instruments like double focusing mass spectrometers, rf spectrometers and ion-cyclotron-resonance (ICR) spectrometer quote relative uncertainties in the 1O-8-1O-9 range [l]. A detailed comparison of these methods, however, shows that published results for the same particles by different authors using different instruments differ by several standard deviations. As an example, we refer to the most precise values of the mass differences of 3He and 3H as summarized in ref. [l]. The differences between the results beyond the quoted uncertainties indicate that systematic errors inherent in different methods might not be fully understood and are perhaps underestimated. However, this particular mass difference value is of special importance for neutrino mass determination from tritium beta decay. The result of an experiment, which quotes a nonzero neutrino mass 121, critically depends on the assumed value of the beta spectrum endpoint energy which is derived from the 3He/3H mass ratio [3]. The trapped ion method has been demonstrated as a technique of high resolution mass spectroscopy. Apart from the above mentioned ICR-technique, which is widely used in chemistry, Penning ion traps have been applied in the past to the most precise determination of the proton/electron mass ratio [4,5], the electron/ positron mass ratio [6], the proton atomic mass [7], to mass determinations of unstable particles [8] and most recently to molecular ions [9]. They use the measurement of eigenfrequencies of charged particles, confined in a parabolic electric well and a superimposed magnetic field. Since some of these frequencies depend on the trapped particles’ mass, their determination can be used for mass measurements. This technique offers some 0168-583X/90/$03.50
0 1990 - Ekvier
Science Publishers
advantages over other methods. In particular, the necessary knowledge of etectromagnetic fields is restricted to a small volume in space, the observation time of the particles can be made almost arbitrarily long to reduce transient time broadening of resonances, and the device can be operated with single particles to avoid space charge shifts. Moreover, a number of perturbations caused by trap imperfections have been treated in great detail and summarized by Brown and Gabrielse [lo] and by van Dyck et al. [ll]. In view of the potential of ion traps for high precision mass spectroscopy and of the importance of specific mass differences like 3He-3H we have set up a Penning ion trap, placed in a very homogeneous strong magnetic field, to investigate its possible limitations and appli~bility to the above mentioned problem. In this paper we present the experimental setup, give a theoretical description of the line shape and frequency shift, and demonstrate the achieved resolution.
2. Principle of operation Ions are created by electron-ionization of the background gas inside a static electric quadrupole ion trap with hyperbolic electrodes and a superimposed magnetic field directed along the axis of symmetry (fig. 1). They are stored for long times, limited by residual pressure. The motion of charged particles inside such a trap has been treated in much detail classically as well as quantum mechanically in the literature [10,12,13]. If we apply a voltage U to a perfect trap of a radius R and a distance 22 = @R between the endcaps, the electric potential between the electrodes is given by cp= +&2+y2-222)*
B.V. (North-Holland)
0)
454
Ch. Gerz et al. / Penning trap mass spectrometer
tion of resonant excitation of of is performed by a time of flight method [4]: At the end of the irradiation period, an electric pulse kicks the ions out of the trap through a narrow hole in the center of an endcap electrode. They travel along the magnetic field lines until they reach a microchannel plate (MCP) at the end of the solenoid. The output pulses of the MCP are fed via a discriminator into a multichannel scaler, which is active for only about 50 l~s. Thus no background problems arise. The ions’ time of flight is mainly determined by the force F acting upon them in the inhomogeneous part of the magnetic field outside the trap, F= grad(pl), where the orbital magnetic moment u is determined by the cyclotron radius rc:
Fig. 1. Penning ion trap.
j.k= +eocrc2 An ion inside with mass m oscillates along the axis of symmetry (z-axis) with a frequency w,
(2)
w, = (2eU/mR2)‘12.
In the x-y plane we have a superposition of two movements: the cyclotron orbit of the particle, whose frequency 0: is in the presence of an electric field slightly shifted away from the free particle value w, = (e/m)B 2 a;=-+
*c
2
WC-!!?
[ 4
2
2
1 l/2
’
and the magnetron motion which represents a drift of the cyclotron orbit center around the trap axis with frequency UC Om=T-
2 WC ---
[ 4
2 *z
2
1 l/2
.
Obviously in the ideal case o;+w,=o,. In practice, one has to deal with imperfections. One can measure 0: and w, precisely to calculate w, from a formula given by Brown and Gabrielse [lo], where most terms from perturbations cancel. Our apparatus, however, is not suited for an accurate determination of 0,. Therefore we restrict ourselves to mass doublets. To investigate them the precise knowledge of of is sufficient along with a less precise value of w, which we can measure: Am m
4
(5)
The numbers in brackets refer to the two masses. As w, does not depend on m in first order, we do not distinguish between w,(l) and o,(2). In our experiment we excite the frequencies by irradiation with the corresponding rf field. The detec-
(6)
and is a constant of motion. Excitation of the cyclotron motion increases r, and consequently shortens the time of flight to the detector. The initial energy spread of the stored ions along the z-axis and spurious electric fields in the drift region limit the sensitivity. While E, and AE, are typically of the order of 50 meV, we have to raise E, to about 0.2 eV in order to observe a decrease in the time of flight of about 20%. The change in magnetic moment from excitation of the magnetron orbit is too small to be detected by our method. We observe the resonance w, by an increase in the magnetron radius to a value which no longer allows the ions to escape through the central hole in the endcap electrode. Thus the decrease in detected ion number is our indication of the magnetron resonance. A complete cycle of ion creation, irradiation and ejection is needed to determine the time of flight corresponding to a given value of irradiation frequency. This cycle is repeated several times to improve the signal to noise ratio. Taking such a set of cycles for different values of the applied frequency, a resonance line is finally formed. It should be noted that similar devices have been built and described by van Dyck [14] and most recently by Cornell et al. [9]. The ions’ axial motion is detected via the image current induced in the endcaps. The detection of L$ resonance is performed by a coupling between radial and axial oscillations either by quadratic terms in the dc and rf fields [lo] or by the application of special rf coupling pulses [15].
3. Theory of line shape and frequency shifts 3.1. Line shape in the ideal trap The time of flight of the ions is a function of their orbital magnetic moment and their initial velocity paral-
Ch. Germ et al. / Penning trap mass spectrometer
lel to the symmetry axis when ejected out of the trap. Small changes in time of flight can be assumed to depend linearly on the magnetic moment and thus on the cyclotron energy EC. This approximation will be made throughout this paper. The equation of motion with an additional oscillating homogeneous electric field (amplitude E) shows that the cyclotron energy of an ion initially at rest is after a time t given by w = e2E2 sin2(Awt/2) c !jrn (Aw/~)~
455
z,Z-r2
2 +c21d,
SW, = -
R2
R2
2z,Z - 2r: - rz
+ +d,q,, +“d8
r2
+ +C2u;C
R2
6 wm
3204- 62: (2r:
+ rm)’ + 3r: + 6rzrz
+ rz
X R4 (11) r2
’
+ JIc2w,c R2
where Aw denotes the difference frequency and the eigenfrequency.
between
the applied
3.2. Field inhomogeneities
(8) (the trap radius R is used to make the expansion coefficients dimensionless) and the electric potential up to the dodecapole:
= i
d&l’,
cos 0.
I-1
The 1= 2 term corresponds to the unperturbed potential, d, = 1. Using classical methods [16] to solve the nonlinear equation of motion, one finds that the new eigenfrequencies depend on the cyclotron radius rc, the magnetron radius r,,, and the amplitude z,, of the z-oscillation. The deviations from the unperturbed values are, to linear order in the expansion coefficients and neglecting in the result o, against wi: SW; = tc,o;
zo’- r,’ - ri R2
$d,q,,-
- +d,o,
2
rm - rc R2
3z,2( ri - r,Z)-rz+r4
= . (12) R4 As the coupling between the degrees of freedom is very small even with 2 or 3 ions in the trap instead of a single one, rc is determined by the radial energy, r, by the initial coordinates and zc by the initial position as well as by the initial kinetic energy in z-direction. The z-amplitude dominates over r, and r,,,, so only the terms with zz and z,$ will be kept. Since E, is proportional to zi, we can write down the general form of any shift +“d4
A real trap suffers from imperfections. The electrodes do not extend to infinity, in our experiment they have holes in the center of the endcaps. The nonvanishing susceptibility of their metal causes the originally very homogeneous magnetic field to be distorted. These perturbations preserve rotation symmetry, but not necessarily inversion symmetry with respect to the z = 0 plane (in our apparatus, the holes are of unequal size, and there is a mount on one side of the trap). To understand the experimental observations, the magnetic field has to be expanded at least to quadratic terms
;
2
+
w 6m
Sw=AE,+BE,Z.
(13) Since we use an ensemble of ions, in which the probability of finding E, is given by a distribution (dp/d E,)E,, the observed line is the sum over each ion’s shifted resonance: K(Am) e2E2
=-/, &?I
c=sin’[(Aw+Sw(E,))t/2]
[( Aw + So( EZ))/212
=
(14) Since the interaction between the ions is very small, dp/dE, is determined by the initial potential and kinetic energy. The ionization is uniform along the z-axis, so the potential energy distribution is easily calculated. If the velocity of the particle does not change during ionization (which is a reasonable approximation for non-dissociative processes), the kinetic energies E, are thermally distributed, and the convolution integral gives:
2zi - r,’ - 212 R2
,e(E,-E,)B(E,-(E,-E,))
- yd,w,
@GFzY
x 3~04- 6zz( r,’ + 2rz)
dp dE dE*.
+ r,* + 6&i
+ 3rA
R4 GO)
dE
k
Ch. Gerz et al. / Penning trap mass spectrometer
456
E,, denotes the energy of an ion oscillating from endcap to endcap, and I,, is the modified Bessel function of the first kind. Introducing a = AEo/27r, b = BE$/2n, q = E,/kT and Av = Aw/2a, WC can be cast in the form
.A
of the trap relative to the magnetic field
Ref. [lo] gives a set of equations for the eigenfrequencies if there is an angle B between trap and field axis: --r2-2-2 UC w%l-
_
d 6 4%9
T 5 2500-
+++
t tt
t
t
tt
t tt
zooo-
g
1500-
:: 2 a
IOOO5000-t’. 0.0
+
+t
(1,
+ * *t
t+
+*+*t,
+ + f
l
*+
’ , 0.2
, 0.4
,
, 0.6
frequency
,
, 0.8
,
, 1.0
,,
1.2
,
,) 1.4
shift [Hz]
Fig. 2. Calculated probability distribution of frequency shifts from ion-ion interaction. The ions are assumed to homoge-
neously fill a cylindrical volume of 300 pm diameter, reaching from one endcap to the other.
(17) where the bar denotes the eigenfrequencies in the misaligned trap and wC and o, are as defined in section 2. Assuming small frequency changes, an approximative solution of this set of equations leads to:
3.4. Ion-ion
+
3000-
r; 2
3.3. Misalignment
tt+tt
3500-
interaction
Since we confine several ions simultaneously, and our detection method is not only sensitive to the center of mass motion, we also have to take into account the interaction between the particles. In the general case, this can be done only by numerical calculations, but the problem of two ions of the same mass can be treated analytically and gives unexpected results. To solve it, one starts with the equations of motion and transforms to a frame rotating with 63,. This eliminates the electric field and modifies the strength of the magnetic field. The center of mass motion can be separated, and it behaves like a single particle. The remaining problem an ion in a homogenous magnetic field and an axially symmetric electric field - is very similar to that of the normal trap, except that the r-dependence of the electric field has changed. Guided by this analogy, one can remove the electric field by transforming to a rotating frame again. The field disappears only at a certain value r. of r which is the mean distance between the two ions. Now an approximation has to be made, the validity of which will be discussed later: the distance between the ions does not deviate much from its mean. Then the
interaction potential can be expanded into a series in (r - ro). Keeping only the quadratic terms and neglecting the azimuthal curvature of the resulting potential valley, the ion moves in an ellipse with calculable frequency. The approximation is valid as long as either radius of the ellipse (which is of the order of the cyclotron radius) is small compared to the distance of the ions. The ions oscillate in the z-direction with relative amplitude 5. The mutual force has to be averaged over that oscillation. If one further assumes 1 X= r,, the interaction potential becomes logarithmic. This has a simple interpretation: for one ion, the other appears smeared out to a line charge. Referencing all frequencies to a coordinate frame at rest finally results in SW; = 0,
The eigenfrequency wf is insensitive to the second ion! In an ensemble of ion pairs, r, and 5 vary, and the resonance is broadened. Fig. 2 shows a computer generated distribution function of frequency shifts according to eq. (19). The ions were assumed to homogenously fill a cylindrical volume. For measurements see section 5. Van Dyck et al. Ill] discussed the number dependency of the ions’ eigenfrequency by the image charges induced in the trap electrodes. They obtain a relative shift of wf proportional to the ion number n and inversely proportional to B2 and R’. For our experimental parameters this shift amounts to at most 10e2 ppb and thus is negligible. In contrast to our observation, however, they found a zero number dependency on 0,’ + @,, while their resonance c.$ was shifted linear with n, mainly due to the fact that their detection scheme is not sensitive to the relative motion of the ions
Ch. Gem et al. / Penning trap mass spectrometer
0
2
4
I
10
12
14
457
18
II)
20
time /h Fig. 3. Temporal drift of the magnetic field measured by the cyclotron resonance of the stored protons. The measured drift amounts to 2.47(4)x 10-9/h.
inside the trap and that the spatial distribution is different from ours.
of ions
3.5. Magnetic field instabilities
The magnetic field of our Peking trap, produced by a superconducting solenoid, shows a temporal drift. We found that for time intervals of several hours, typical for a set of measurements, the shift can satisfactorily be described as linear. Fig. 3 gives an example of a measurement, taken on protons. Here the relative drift amounts to 2.47(4) x 10m9/h. During a period of several weeks it varied between 2 x 10m9/h and 4 x 10e9/h.
solenoid in a field of about 6 T. The field inhomogeneity is lo-‘/cm3 in the empty magnet and an order of magnitude worse inside the trap. On the field axis about 50 cm away from the trap there is a hot filament providing the electron beam, and at about 35 cm on the other side there are two microchannel plates in chevron configuration for ion detection. To avoid electric stray fields which decrease the sensitivity of the time of flight method, the whole drift region is shielded by a carefully polished copper tube. The complete experimental setup is at room temperature. If hydrogen is to be measured, it is introduced into the apparatus via a heated palladium leak. For other gases, a conventional leak valve is used.
4. Experimental setup 5. Experimental methods for reducing the line widths A scheme of the experimental setup is shown in fig. 4. A glass tube of 54 mm diameter, evacuated by an ion getter pump, served as a vacuum envelope. Typical operating pressure was 0.1 pPa. The central part of the apparatus is a quadrupole trap with carefully machined hyperbolic electrodes of OFHC copper. The electrodes have been gold plated to avoid surface charges on oxidized spots. The trap radius is 6 mm and the distance between the endcaps 8.4 mm. In addition, we have two guard electrodes located between the ring and endcaps as first described by van Dyck et al. [17]. A hole of 0.4 mm radius in one of the endcaps allows an electron beam to enter the trap for creating ions. A hole of 0.8 mm radius lets the ions pass when, at the end of the irradiation period, they are ejected from the trap by raising the ring voltage to the potential of the endcaps. The trap is located at the center of a superconducting
5.1. Reducing ion-ion interaction
The number of ions simultaneously present in the trap has to be reduced as much as possible. Our ion creation mechanism is not sensitive to the ion species, so as a first step all unwanted ions have to be removed. As the ions’ axial oscillation periods depend on their masses, this can be done by applying appropriate fre quencies to the endcaps. In this way, the z-amplitude is made large enough to drive the ions out of the trap. Fig. 5 shows as an example the preparation of 4He+. Even if an ion species contributes only 1% to the initial trap population, it can be isolated in this way. The larger the change in the time of flight for a given radial energy, the less ions are needed to detect an increase in the radial energy. Maximum sensitivity re-
458
Ch. Gets et al. / Penning trap mass spectrometer palladium superconducting
trap
with
solenoid
detecting
electron
system
eiectrical
gun
ion
ieakage
getter
pump
ionisation
gas
gauge
inlet
feedthroughs
Fig. 4. Schematic view of the experimental setup.
40
80
k 5 :
20 10
“0 30
160
200
240
200
ii40
Ltb) .1._ time
in 40 5 ._
120
quires the initial energy of the ejected ions parallel to the magnetic field to be as low as possible. This energy stems from the axial oscillation. The shallower the potential well, the more it is reduced. Therefore, we use a trapping potential of typicaly only 200 mV. Fig. 6 compares TOF spectra with and without applied cyclotron frequency, taken with one detected ion per cycle on the average. Estimating our detector efficiency as f to +, that corresponds to 2-3 trapped ions/cycle. Experiments with Penning traps as mass spectrometers use the sideband w: + w,,, [4,8] as well as wf [5,91 for determining mass ratios. As we are interested only in small mass differences, we prefer using 0: to get rid of the ion number dependence of the resonance frequency. Fig. 7 shows that wf is indeed independent of ion number within 1.3 standard deviations, whereas 0: + w,,, clearly varies as predicted from theory. The radial diameter of the ion “cloud” in the trap can be only roughly estimated, so the calculated shift is uncertam within a factor of about 3, but within this factor is coincides with the observed value. Using w: requires an approximate knowledge of w,,, to determine the mass difference according to eq. (5) given in section 2. Fig. 8 shows a typical o,,, resonance. The correct line shape could be calculated only numerically, because w,.,, depends on the distance between the ions which changes during irradiation with w,,,_ We take the half width of the resonance to be the upper limit of the error of the center frequency.
of
fright
C~SI
120
160
I
40
80
tine of CLight
[PSI
Fig. 5. Time of flight spectra (a) of all stored ions produced by electro-ionisationand (b) after removal of all ions except 4He+ by strong excitation of their r-amplitudes.
Ch. Germ et al. / Penning trap mass spectrometer
?
(a)
-0.00
fi
-0.10
$ a,
-0.20
mHz/lon
23(18)
t -0.30 0.0
40
80 time of
"0
30
k *
20
120
0.4
200
160
0.8
average
1.2
1.6
number
2.0
of
2.4
ions
flight Cpsl
Lb)
: 10
E;
374(31)
0.30.
Z 40
80
120
160
Fig. 6. Time of flight spectra of a mixture of 4He and D, (a) with and (b) without applied cyclotron frequency for D;. showing the shift in time of flight. The average number of trapped ions per cycle is 2-3.
with field
0.2
200
tine of fl;ght :psl
5.2. Dealing
, , ,
0.20-, 0.0
-
imperfections
As stated above, our trap contains correction electrodes which mainly generate an electric cctapole potential. E$. (10) shows that by adjusting d, (which corresponds to adjusting the voltage at the correction electrodes), the contributions to wf proportional to z,, and r,,, can be made to cancel, leaving a small r, term and higher orders in the amplitudes. However, if the shifts of the other observable frequencies are to be reduced simultaneously, electric and magnetic fields must be compensated separately. To trim the magnetic field, a correction coil has been built whose field varies as z2 along its axis and which fits into the bore of the superconducting solenoid. The current in the coil is chosen to give a minimum linewidth at the sideband predominantly on magnetic 0; + 0,. which depends imperfections. Subsequently, the voltage at the correction electrodes is varied to minimize the width of the wi resonance. This procedure results in a narrowing of the sideband by a factor of 3 and c.$ by an order of magnitude. The limit is given by perturbations of higher
0.4
average Fig. 7. Dependence
0.6
mHz/lon 0.8
number
1.0
,
of
1.2
,
1.4
,)
ions
of the (a) of and (b) (WA+ a,,,) resonance on trapped ion number.
order. Since the dominant terms in these perturbations are proportional to high powers of zo, further reduction of the linewidth requires that the oscillation amplitude be diminished. This is done by raising the trapping potential “adiabatically”, that is to say so slowly that, in a quantum mechanical view, the occupation number does not change. The z-energy of the particle is then proportional to oz. From that one deduces that z.
22 E 20 s 18 ;
16
L 14 :
12
E lo =
8 6 4 , ; , , , 24 , 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.6 6.0 6.2
frequency
[Hz]
+
7200
Fig. 8. q,, resonance of Hz.
>
460
Ch. Germ et al. / Penning trap mass spectrometer
+6
V
endcap
voltage
electron
gun
- u.
ring
0
1
2
3
45
67
voltage
8
Fig. 9. Time sequence of ring and endcap voltages. Switching points: 0: start of cycle, 1-2 (0.2 s): ion creation, 1-3 (0.4 s): removal of unwanted ions by wobbled oscillator, 4-5 (0.1 s): trapping voltage increased to 10 V reference, 5-6 (0.5-10 s): excitation of wf, 6-7 (0.1 s): decreasing trapping voltage, 8: end of cycle and ion ejection.
varies as U-‘/4. When we create ions at 150 mV and change the trapping voltage to 10 V, the sideband of + W, becomes narrower by an order of magnitude
(the limit being given by ion-ion interaction) and wg by nearly two orders of magnitude. Fig. 9 shows the complete sequence of a single measurement, including ion creation, the removal of unwanted ions, the adiabative potential change, the irradiation of the ions by rf fields and the ejection of ions.
6. Results
o.io
.-I
o.ko
frequency
o.bo
0.60
[Hz]
t
0.60
0.70
'
44714959
1650 160155. y
150-
-
145.
= .? =
140-
f
135-
';, 130.
He+
QJ 125.&
120115llO-
--.0.0
0.1
0:2
frequency
0:3
0:4
[Hz]
0.5
0.6
0.7
0.6
.
+ 22511020
Fig. 10. Narrowest resonances for H; and 4He+ in our experiment. Experimental points are fitted by least squares to the lineshape given in eq. (16) (solid line). The relative linewidths are 2 X 10m9 and 4 X 10e9, respectively. The uncertainty of the center frequency is 2X10-to and 3x10-lo, respectively, for fixed distortion parameters a and b of eq. (16).
and summary
Fig. 10 shows the achieved minimum linewidth for two light ions, Hl and 4He+. The widths are 2 X 10m9 and 4 x 10P9, respectively, and the unshifted frequencies v, of eq. (16) are determined to relative uncertainties of 2 X 10e9 and 1.0 X 10w9. The mass separation in a doublet will be known even more precisely, as the uncertainty in v,, includes the uncertainty to which residual field imperfections can be derived from the lineshape. As the ions are compared in the same trap with the same fields, these imperfections can be fixed once they have been determined. Regarding them no longer as variable, the unshifted frequency v,, is fitted to 2 x lo-” and 3 X lo-“, respectively, for the sample curves of fig. 9. In summary, we have built an apparatus which gives very narrow mass resonances for light particles. Their width and shape can be explained by two factors, field inhomogeneities together with the spread of the axial energies on one hand and ion-ion interaction on the other. The guard electrodes, a correction coil and an adiabatic compression of the volume which the ions occupy cope with the field imperfections. Restriction to mass doublets not only excludes errors due to trap misalignments, but also allows the use of the wd frequency which is, at small ion numbers, the least sensitive to ion-ion interaction. Removing wrong masses
Ch. Germ et al. / Penning trap mass spectrometer w, and using only 2-3 particles further diminishes this interaction. We have applied the method described above to a determination of the 4He-D, mass difference and obtained an overall uncertainty of 5 x 10e9 amu. Details of this measurement will be published elsewhere [18]. via
This experiment was supported by the Deutsche Forschungsgemeinschaft.
References 111 G. Audi, R.L. Graham and J.S. Geiger, 2. Phys. A321
(1985) 533.
PI S. Boris et al., Phys. Rev. Lett. 58 (1987) 2019. [31J.J. Simpson,Proc. 7th Int. Conf. on Atomic Masses and Fundamental Constants, AMCO-7, Darmstadt (1985). and J. Traut, Z. Phys. A297 [41G. Graff, H. Kalinowsky (1980) 35. [51 R.S. van Dyck, Jr., F.L. Moore, D.L. Farnham and P.B. Schwinberg, Int. J. Mass Spectr. and Ion Proc. 66 (1985) 327.
461
[6] P.B. Schwinberg, R.S. van Dyck, Jr. and H.G. Dehmelt, Phys. Rev. Lett. 811 (1981) 119. [7] R.S. van Dyck, Jr., F.L. Moore, D.L. Famham and P.B. Schwinberg, in: Frequency Standards on Metrology, ed. A. de Marchi (Springer, Heidelberg, 1989) p. 349. [8] G. Bollen et al., Hyperfine Interactions 38 (1987) 793. [9] E.A. Cornell et al., Phys. Rev. Lett. 63 (1989) 1674. [lo] L.S. Brown and G. Gabrielse, Rev. Mod. Phys. 58 (1986) 233. (111 R.S. van Dyck, Jr., F.L. Moore, D.L. Famham and P.B. Schwinberg, Phys. Rev. A40 (1989) 6308. [12] G. Graff, E. Klempt and G. Werth, Z. Phys. 222~ (1979) 201. [13] D.J. Wineland, W.M. Itano and R.S. van Dyck, Jr., Adv. At. Mol. Spectr. 19 (1983) 135. [14] R.S. van Dyck, Jr., Phys. Scripta T22 (1988) 228. 1151 E.A. Cornell et al., Phys. Rev. A41 (1990) 312. [16] D. Hagedorn, Nichtlineare Schwingungen (Akad. Verlagsgesellschaft, Berlin, 1978). [17] R.S. van Dyck, Jr. and P.B. Schwinberg, Phys. Rev. Lett. 47 (1981) 395. [18] Ch. Get-z, D. Wilsdorf and G. Werth, to be published.
and Methods
A HIGH PRECISION Ch. GERZ, Insiit~t fi Received
in Physics
Research
B47 (1990) 453-461
453
PENNING TRAP MASS SPECTROMETER
D. WILSDORF
and G. WERTH
Physik, U~iL~~sit~tMaim, D-6500 Maim, FRG 16 October
1989 and in revised form 23 February
1990
Penning ion traps are used as mass spectrometers for high precision determination of mass doublets. We present our experimental setup and discuss in detail possible sources of systematic uncert~nti~ from electric and magnetic fietd distortions, assignments and ion-ion interaction. Methods are discussed to deal with the associated line shifts. As a result, we demonstrate experimentally obtained resolutions of a few parts in 10v9 for light masses. The method allows very precise determinations of doublet mass differences of light ions.
1. Introduction A number of different methods exist for precision mass spectroscopy. Instruments like double focusing mass spectrometers, rf spectrometers and ion-cyclotron-resonance (ICR) spectrometer quote relative uncertainties in the 1O-8-1O-9 range [l]. A detailed comparison of these methods, however, shows that published results for the same particles by different authors using different instruments differ by several standard deviations. As an example, we refer to the most precise values of the mass differences of 3He and 3H as summarized in ref. [l]. The differences between the results beyond the quoted uncertainties indicate that systematic errors inherent in different methods might not be fully understood and are perhaps underestimated. However, this particular mass difference value is of special importance for neutrino mass determination from tritium beta decay. The result of an experiment, which quotes a nonzero neutrino mass 121, critically depends on the assumed value of the beta spectrum endpoint energy which is derived from the 3He/3H mass ratio [3]. The trapped ion method has been demonstrated as a technique of high resolution mass spectroscopy. Apart from the above mentioned ICR-technique, which is widely used in chemistry, Penning ion traps have been applied in the past to the most precise determination of the proton/electron mass ratio [4,5], the electron/ positron mass ratio [6], the proton atomic mass [7], to mass determinations of unstable particles [8] and most recently to molecular ions [9]. They use the measurement of eigenfrequencies of charged particles, confined in a parabolic electric well and a superimposed magnetic field. Since some of these frequencies depend on the trapped particles’ mass, their determination can be used for mass measurements. This technique offers some 0168-583X/90/$03.50
0 1990 - Ekvier
Science Publishers
advantages over other methods. In particular, the necessary knowledge of etectromagnetic fields is restricted to a small volume in space, the observation time of the particles can be made almost arbitrarily long to reduce transient time broadening of resonances, and the device can be operated with single particles to avoid space charge shifts. Moreover, a number of perturbations caused by trap imperfections have been treated in great detail and summarized by Brown and Gabrielse [lo] and by van Dyck et al. [ll]. In view of the potential of ion traps for high precision mass spectroscopy and of the importance of specific mass differences like 3He-3H we have set up a Penning ion trap, placed in a very homogeneous strong magnetic field, to investigate its possible limitations and appli~bility to the above mentioned problem. In this paper we present the experimental setup, give a theoretical description of the line shape and frequency shift, and demonstrate the achieved resolution.
2. Principle of operation Ions are created by electron-ionization of the background gas inside a static electric quadrupole ion trap with hyperbolic electrodes and a superimposed magnetic field directed along the axis of symmetry (fig. 1). They are stored for long times, limited by residual pressure. The motion of charged particles inside such a trap has been treated in much detail classically as well as quantum mechanically in the literature [10,12,13]. If we apply a voltage U to a perfect trap of a radius R and a distance 22 = @R between the endcaps, the electric potential between the electrodes is given by cp= +&2+y2-222)*
B.V. (North-Holland)
0)
454
Ch. Gerz et al. / Penning trap mass spectrometer
tion of resonant excitation of of is performed by a time of flight method [4]: At the end of the irradiation period, an electric pulse kicks the ions out of the trap through a narrow hole in the center of an endcap electrode. They travel along the magnetic field lines until they reach a microchannel plate (MCP) at the end of the solenoid. The output pulses of the MCP are fed via a discriminator into a multichannel scaler, which is active for only about 50 l~s. Thus no background problems arise. The ions’ time of flight is mainly determined by the force F acting upon them in the inhomogeneous part of the magnetic field outside the trap, F= grad(pl), where the orbital magnetic moment u is determined by the cyclotron radius rc:
Fig. 1. Penning ion trap.
j.k= +eocrc2 An ion inside with mass m oscillates along the axis of symmetry (z-axis) with a frequency w,
(2)
w, = (2eU/mR2)‘12.
In the x-y plane we have a superposition of two movements: the cyclotron orbit of the particle, whose frequency 0: is in the presence of an electric field slightly shifted away from the free particle value w, = (e/m)B 2 a;=-+
*c
2
WC-!!?
[ 4
2
2
1 l/2
’
and the magnetron motion which represents a drift of the cyclotron orbit center around the trap axis with frequency UC Om=T-
2 WC ---
[ 4
2 *z
2
1 l/2
.
Obviously in the ideal case o;+w,=o,. In practice, one has to deal with imperfections. One can measure 0: and w, precisely to calculate w, from a formula given by Brown and Gabrielse [lo], where most terms from perturbations cancel. Our apparatus, however, is not suited for an accurate determination of 0,. Therefore we restrict ourselves to mass doublets. To investigate them the precise knowledge of of is sufficient along with a less precise value of w, which we can measure: Am m
4
(5)
The numbers in brackets refer to the two masses. As w, does not depend on m in first order, we do not distinguish between w,(l) and o,(2). In our experiment we excite the frequencies by irradiation with the corresponding rf field. The detec-
(6)
and is a constant of motion. Excitation of the cyclotron motion increases r, and consequently shortens the time of flight to the detector. The initial energy spread of the stored ions along the z-axis and spurious electric fields in the drift region limit the sensitivity. While E, and AE, are typically of the order of 50 meV, we have to raise E, to about 0.2 eV in order to observe a decrease in the time of flight of about 20%. The change in magnetic moment from excitation of the magnetron orbit is too small to be detected by our method. We observe the resonance w, by an increase in the magnetron radius to a value which no longer allows the ions to escape through the central hole in the endcap electrode. Thus the decrease in detected ion number is our indication of the magnetron resonance. A complete cycle of ion creation, irradiation and ejection is needed to determine the time of flight corresponding to a given value of irradiation frequency. This cycle is repeated several times to improve the signal to noise ratio. Taking such a set of cycles for different values of the applied frequency, a resonance line is finally formed. It should be noted that similar devices have been built and described by van Dyck [14] and most recently by Cornell et al. [9]. The ions’ axial motion is detected via the image current induced in the endcaps. The detection of L$ resonance is performed by a coupling between radial and axial oscillations either by quadratic terms in the dc and rf fields [lo] or by the application of special rf coupling pulses [15].
3. Theory of line shape and frequency shifts 3.1. Line shape in the ideal trap The time of flight of the ions is a function of their orbital magnetic moment and their initial velocity paral-
Ch. Germ et al. / Penning trap mass spectrometer
lel to the symmetry axis when ejected out of the trap. Small changes in time of flight can be assumed to depend linearly on the magnetic moment and thus on the cyclotron energy EC. This approximation will be made throughout this paper. The equation of motion with an additional oscillating homogeneous electric field (amplitude E) shows that the cyclotron energy of an ion initially at rest is after a time t given by w = e2E2 sin2(Awt/2) c !jrn (Aw/~)~
455
z,Z-r2
2 +c21d,
SW, = -
R2
R2
2z,Z - 2r: - rz
+ +d,q,, +“d8
r2
+ +C2u;C
R2
6 wm
3204- 62: (2r:
+ rm)’ + 3r: + 6rzrz
+ rz
X R4 (11) r2
’
+ JIc2w,c R2
where Aw denotes the difference frequency and the eigenfrequency.
between
the applied
3.2. Field inhomogeneities
(8) (the trap radius R is used to make the expansion coefficients dimensionless) and the electric potential up to the dodecapole:
= i
d&l’,
cos 0.
I-1
The 1= 2 term corresponds to the unperturbed potential, d, = 1. Using classical methods [16] to solve the nonlinear equation of motion, one finds that the new eigenfrequencies depend on the cyclotron radius rc, the magnetron radius r,,, and the amplitude z,, of the z-oscillation. The deviations from the unperturbed values are, to linear order in the expansion coefficients and neglecting in the result o, against wi: SW; = tc,o;
zo’- r,’ - ri R2
$d,q,,-
- +d,o,
2
rm - rc R2
3z,2( ri - r,Z)-rz+r4
= . (12) R4 As the coupling between the degrees of freedom is very small even with 2 or 3 ions in the trap instead of a single one, rc is determined by the radial energy, r, by the initial coordinates and zc by the initial position as well as by the initial kinetic energy in z-direction. The z-amplitude dominates over r, and r,,,, so only the terms with zz and z,$ will be kept. Since E, is proportional to zi, we can write down the general form of any shift +“d4
A real trap suffers from imperfections. The electrodes do not extend to infinity, in our experiment they have holes in the center of the endcaps. The nonvanishing susceptibility of their metal causes the originally very homogeneous magnetic field to be distorted. These perturbations preserve rotation symmetry, but not necessarily inversion symmetry with respect to the z = 0 plane (in our apparatus, the holes are of unequal size, and there is a mount on one side of the trap). To understand the experimental observations, the magnetic field has to be expanded at least to quadratic terms
;
2
+
w 6m
Sw=AE,+BE,Z.
(13) Since we use an ensemble of ions, in which the probability of finding E, is given by a distribution (dp/d E,)E,, the observed line is the sum over each ion’s shifted resonance: K(Am) e2E2
=-/, &?I
c=sin’[(Aw+Sw(E,))t/2]
[( Aw + So( EZ))/212
=
(14) Since the interaction between the ions is very small, dp/dE, is determined by the initial potential and kinetic energy. The ionization is uniform along the z-axis, so the potential energy distribution is easily calculated. If the velocity of the particle does not change during ionization (which is a reasonable approximation for non-dissociative processes), the kinetic energies E, are thermally distributed, and the convolution integral gives:
2zi - r,’ - 212 R2
,e(E,-E,)B(E,-(E,-E,))
- yd,w,
@GFzY
x 3~04- 6zz( r,’ + 2rz)
dp dE dE*.
+ r,* + 6&i
+ 3rA
R4 GO)
dE
k
Ch. Gerz et al. / Penning trap mass spectrometer
456
E,, denotes the energy of an ion oscillating from endcap to endcap, and I,, is the modified Bessel function of the first kind. Introducing a = AEo/27r, b = BE$/2n, q = E,/kT and Av = Aw/2a, WC can be cast in the form
.A
of the trap relative to the magnetic field
Ref. [lo] gives a set of equations for the eigenfrequencies if there is an angle B between trap and field axis: --r2-2-2 UC w%l-
_
d 6 4%9
T 5 2500-
+++
t tt
t
t
tt
t tt
zooo-
g
1500-
:: 2 a
IOOO5000-t’. 0.0
+
+t
(1,
+ * *t
t+
+*+*t,
+ + f
l
*+
’ , 0.2
, 0.4
,
, 0.6
frequency
,
, 0.8
,
, 1.0
,,
1.2
,
,) 1.4
shift [Hz]
Fig. 2. Calculated probability distribution of frequency shifts from ion-ion interaction. The ions are assumed to homoge-
neously fill a cylindrical volume of 300 pm diameter, reaching from one endcap to the other.
(17) where the bar denotes the eigenfrequencies in the misaligned trap and wC and o, are as defined in section 2. Assuming small frequency changes, an approximative solution of this set of equations leads to:
3.4. Ion-ion
+
3000-
r; 2
3.3. Misalignment
tt+tt
3500-
interaction
Since we confine several ions simultaneously, and our detection method is not only sensitive to the center of mass motion, we also have to take into account the interaction between the particles. In the general case, this can be done only by numerical calculations, but the problem of two ions of the same mass can be treated analytically and gives unexpected results. To solve it, one starts with the equations of motion and transforms to a frame rotating with 63,. This eliminates the electric field and modifies the strength of the magnetic field. The center of mass motion can be separated, and it behaves like a single particle. The remaining problem an ion in a homogenous magnetic field and an axially symmetric electric field - is very similar to that of the normal trap, except that the r-dependence of the electric field has changed. Guided by this analogy, one can remove the electric field by transforming to a rotating frame again. The field disappears only at a certain value r. of r which is the mean distance between the two ions. Now an approximation has to be made, the validity of which will be discussed later: the distance between the ions does not deviate much from its mean. Then the
interaction potential can be expanded into a series in (r - ro). Keeping only the quadratic terms and neglecting the azimuthal curvature of the resulting potential valley, the ion moves in an ellipse with calculable frequency. The approximation is valid as long as either radius of the ellipse (which is of the order of the cyclotron radius) is small compared to the distance of the ions. The ions oscillate in the z-direction with relative amplitude 5. The mutual force has to be averaged over that oscillation. If one further assumes 1 X= r,, the interaction potential becomes logarithmic. This has a simple interpretation: for one ion, the other appears smeared out to a line charge. Referencing all frequencies to a coordinate frame at rest finally results in SW; = 0,
The eigenfrequency wf is insensitive to the second ion! In an ensemble of ion pairs, r, and 5 vary, and the resonance is broadened. Fig. 2 shows a computer generated distribution function of frequency shifts according to eq. (19). The ions were assumed to homogenously fill a cylindrical volume. For measurements see section 5. Van Dyck et al. Ill] discussed the number dependency of the ions’ eigenfrequency by the image charges induced in the trap electrodes. They obtain a relative shift of wf proportional to the ion number n and inversely proportional to B2 and R’. For our experimental parameters this shift amounts to at most 10e2 ppb and thus is negligible. In contrast to our observation, however, they found a zero number dependency on 0,’ + @,, while their resonance c.$ was shifted linear with n, mainly due to the fact that their detection scheme is not sensitive to the relative motion of the ions
Ch. Gem et al. / Penning trap mass spectrometer
0
2
4
I
10
12
14
457
18
II)
20
time /h Fig. 3. Temporal drift of the magnetic field measured by the cyclotron resonance of the stored protons. The measured drift amounts to 2.47(4)x 10-9/h.
inside the trap and that the spatial distribution is different from ours.
of ions
3.5. Magnetic field instabilities
The magnetic field of our Peking trap, produced by a superconducting solenoid, shows a temporal drift. We found that for time intervals of several hours, typical for a set of measurements, the shift can satisfactorily be described as linear. Fig. 3 gives an example of a measurement, taken on protons. Here the relative drift amounts to 2.47(4) x 10m9/h. During a period of several weeks it varied between 2 x 10m9/h and 4 x 10e9/h.
solenoid in a field of about 6 T. The field inhomogeneity is lo-‘/cm3 in the empty magnet and an order of magnitude worse inside the trap. On the field axis about 50 cm away from the trap there is a hot filament providing the electron beam, and at about 35 cm on the other side there are two microchannel plates in chevron configuration for ion detection. To avoid electric stray fields which decrease the sensitivity of the time of flight method, the whole drift region is shielded by a carefully polished copper tube. The complete experimental setup is at room temperature. If hydrogen is to be measured, it is introduced into the apparatus via a heated palladium leak. For other gases, a conventional leak valve is used.
4. Experimental setup 5. Experimental methods for reducing the line widths A scheme of the experimental setup is shown in fig. 4. A glass tube of 54 mm diameter, evacuated by an ion getter pump, served as a vacuum envelope. Typical operating pressure was 0.1 pPa. The central part of the apparatus is a quadrupole trap with carefully machined hyperbolic electrodes of OFHC copper. The electrodes have been gold plated to avoid surface charges on oxidized spots. The trap radius is 6 mm and the distance between the endcaps 8.4 mm. In addition, we have two guard electrodes located between the ring and endcaps as first described by van Dyck et al. [17]. A hole of 0.4 mm radius in one of the endcaps allows an electron beam to enter the trap for creating ions. A hole of 0.8 mm radius lets the ions pass when, at the end of the irradiation period, they are ejected from the trap by raising the ring voltage to the potential of the endcaps. The trap is located at the center of a superconducting
5.1. Reducing ion-ion interaction
The number of ions simultaneously present in the trap has to be reduced as much as possible. Our ion creation mechanism is not sensitive to the ion species, so as a first step all unwanted ions have to be removed. As the ions’ axial oscillation periods depend on their masses, this can be done by applying appropriate fre quencies to the endcaps. In this way, the z-amplitude is made large enough to drive the ions out of the trap. Fig. 5 shows as an example the preparation of 4He+. Even if an ion species contributes only 1% to the initial trap population, it can be isolated in this way. The larger the change in the time of flight for a given radial energy, the less ions are needed to detect an increase in the radial energy. Maximum sensitivity re-
458
Ch. Gets et al. / Penning trap mass spectrometer palladium superconducting
trap
with
solenoid
detecting
electron
system
eiectrical
gun
ion
ieakage
getter
pump
ionisation
gas
gauge
inlet
feedthroughs
Fig. 4. Schematic view of the experimental setup.
40
80
k 5 :
20 10
“0 30
160
200
240
200
ii40
Ltb) .1._ time
in 40 5 ._
120
quires the initial energy of the ejected ions parallel to the magnetic field to be as low as possible. This energy stems from the axial oscillation. The shallower the potential well, the more it is reduced. Therefore, we use a trapping potential of typicaly only 200 mV. Fig. 6 compares TOF spectra with and without applied cyclotron frequency, taken with one detected ion per cycle on the average. Estimating our detector efficiency as f to +, that corresponds to 2-3 trapped ions/cycle. Experiments with Penning traps as mass spectrometers use the sideband w: + w,,, [4,8] as well as wf [5,91 for determining mass ratios. As we are interested only in small mass differences, we prefer using 0: to get rid of the ion number dependence of the resonance frequency. Fig. 7 shows that wf is indeed independent of ion number within 1.3 standard deviations, whereas 0: + w,,, clearly varies as predicted from theory. The radial diameter of the ion “cloud” in the trap can be only roughly estimated, so the calculated shift is uncertam within a factor of about 3, but within this factor is coincides with the observed value. Using w: requires an approximate knowledge of w,,, to determine the mass difference according to eq. (5) given in section 2. Fig. 8 shows a typical o,,, resonance. The correct line shape could be calculated only numerically, because w,.,, depends on the distance between the ions which changes during irradiation with w,,,_ We take the half width of the resonance to be the upper limit of the error of the center frequency.
of
fright
C~SI
120
160
I
40
80
tine of CLight
[PSI
Fig. 5. Time of flight spectra (a) of all stored ions produced by electro-ionisationand (b) after removal of all ions except 4He+ by strong excitation of their r-amplitudes.
Ch. Germ et al. / Penning trap mass spectrometer
?
(a)
-0.00
fi
-0.10
$ a,
-0.20
mHz/lon
23(18)
t -0.30 0.0
40
80 time of
"0
30
k *
20
120
0.4
200
160
0.8
average
1.2
1.6
number
2.0
of
2.4
ions
flight Cpsl
Lb)
: 10
E;
374(31)
0.30.
Z 40
80
120
160
Fig. 6. Time of flight spectra of a mixture of 4He and D, (a) with and (b) without applied cyclotron frequency for D;. showing the shift in time of flight. The average number of trapped ions per cycle is 2-3.
with field
0.2
200
tine of fl;ght :psl
5.2. Dealing
, , ,
0.20-, 0.0
-
imperfections
As stated above, our trap contains correction electrodes which mainly generate an electric cctapole potential. E$. (10) shows that by adjusting d, (which corresponds to adjusting the voltage at the correction electrodes), the contributions to wf proportional to z,, and r,,, can be made to cancel, leaving a small r, term and higher orders in the amplitudes. However, if the shifts of the other observable frequencies are to be reduced simultaneously, electric and magnetic fields must be compensated separately. To trim the magnetic field, a correction coil has been built whose field varies as z2 along its axis and which fits into the bore of the superconducting solenoid. The current in the coil is chosen to give a minimum linewidth at the sideband predominantly on magnetic 0; + 0,. which depends imperfections. Subsequently, the voltage at the correction electrodes is varied to minimize the width of the wi resonance. This procedure results in a narrowing of the sideband by a factor of 3 and c.$ by an order of magnitude. The limit is given by perturbations of higher
0.4
average Fig. 7. Dependence
0.6
mHz/lon 0.8
number
1.0
,
of
1.2
,
1.4
,)
ions
of the (a) of and (b) (WA+ a,,,) resonance on trapped ion number.
order. Since the dominant terms in these perturbations are proportional to high powers of zo, further reduction of the linewidth requires that the oscillation amplitude be diminished. This is done by raising the trapping potential “adiabatically”, that is to say so slowly that, in a quantum mechanical view, the occupation number does not change. The z-energy of the particle is then proportional to oz. From that one deduces that z.
22 E 20 s 18 ;
16
L 14 :
12
E lo =
8 6 4 , ; , , , 24 , 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.6 6.0 6.2
frequency
[Hz]
+
7200
Fig. 8. q,, resonance of Hz.
>
460
Ch. Germ et al. / Penning trap mass spectrometer
+6
V
endcap
voltage
electron
gun
- u.
ring
0
1
2
3
45
67
voltage
8
Fig. 9. Time sequence of ring and endcap voltages. Switching points: 0: start of cycle, 1-2 (0.2 s): ion creation, 1-3 (0.4 s): removal of unwanted ions by wobbled oscillator, 4-5 (0.1 s): trapping voltage increased to 10 V reference, 5-6 (0.5-10 s): excitation of wf, 6-7 (0.1 s): decreasing trapping voltage, 8: end of cycle and ion ejection.
varies as U-‘/4. When we create ions at 150 mV and change the trapping voltage to 10 V, the sideband of + W, becomes narrower by an order of magnitude
(the limit being given by ion-ion interaction) and wg by nearly two orders of magnitude. Fig. 9 shows the complete sequence of a single measurement, including ion creation, the removal of unwanted ions, the adiabative potential change, the irradiation of the ions by rf fields and the ejection of ions.
6. Results
o.io
.-I
o.ko
frequency
o.bo
0.60
[Hz]
t
0.60
0.70
'
44714959
1650 160155. y
150-
-
145.
= .? =
140-
f
135-
';, 130.
He+
QJ 125.&
120115llO-
--.0.0
0.1
0:2
frequency
0:3
0:4
[Hz]
0.5
0.6
0.7
0.6
.
+ 22511020
Fig. 10. Narrowest resonances for H; and 4He+ in our experiment. Experimental points are fitted by least squares to the lineshape given in eq. (16) (solid line). The relative linewidths are 2 X 10m9 and 4 X 10e9, respectively. The uncertainty of the center frequency is 2X10-to and 3x10-lo, respectively, for fixed distortion parameters a and b of eq. (16).
and summary
Fig. 10 shows the achieved minimum linewidth for two light ions, Hl and 4He+. The widths are 2 X 10m9 and 4 x 10P9, respectively, and the unshifted frequencies v, of eq. (16) are determined to relative uncertainties of 2 X 10e9 and 1.0 X 10w9. The mass separation in a doublet will be known even more precisely, as the uncertainty in v,, includes the uncertainty to which residual field imperfections can be derived from the lineshape. As the ions are compared in the same trap with the same fields, these imperfections can be fixed once they have been determined. Regarding them no longer as variable, the unshifted frequency v,, is fitted to 2 x lo-” and 3 X lo-“, respectively, for the sample curves of fig. 9. In summary, we have built an apparatus which gives very narrow mass resonances for light particles. Their width and shape can be explained by two factors, field inhomogeneities together with the spread of the axial energies on one hand and ion-ion interaction on the other. The guard electrodes, a correction coil and an adiabatic compression of the volume which the ions occupy cope with the field imperfections. Restriction to mass doublets not only excludes errors due to trap misalignments, but also allows the use of the wd frequency which is, at small ion numbers, the least sensitive to ion-ion interaction. Removing wrong masses
Ch. Germ et al. / Penning trap mass spectrometer w, and using only 2-3 particles further diminishes this interaction. We have applied the method described above to a determination of the 4He-D, mass difference and obtained an overall uncertainty of 5 x 10e9 amu. Details of this measurement will be published elsewhere [18]. via
This experiment was supported by the Deutsche Forschungsgemeinschaft.
References 111 G. Audi, R.L. Graham and J.S. Geiger, 2. Phys. A321
(1985) 533.
PI S. Boris et al., Phys. Rev. Lett. 58 (1987) 2019. [31J.J. Simpson,Proc. 7th Int. Conf. on Atomic Masses and Fundamental Constants, AMCO-7, Darmstadt (1985). and J. Traut, Z. Phys. A297 [41G. Graff, H. Kalinowsky (1980) 35. [51 R.S. van Dyck, Jr., F.L. Moore, D.L. Farnham and P.B. Schwinberg, Int. J. Mass Spectr. and Ion Proc. 66 (1985) 327.
461
[6] P.B. Schwinberg, R.S. van Dyck, Jr. and H.G. Dehmelt, Phys. Rev. Lett. 811 (1981) 119. [7] R.S. van Dyck, Jr., F.L. Moore, D.L. Famham and P.B. Schwinberg, in: Frequency Standards on Metrology, ed. A. de Marchi (Springer, Heidelberg, 1989) p. 349. [8] G. Bollen et al., Hyperfine Interactions 38 (1987) 793. [9] E.A. Cornell et al., Phys. Rev. Lett. 63 (1989) 1674. [lo] L.S. Brown and G. Gabrielse, Rev. Mod. Phys. 58 (1986) 233. (111 R.S. van Dyck, Jr., F.L. Moore, D.L. Famham and P.B. Schwinberg, Phys. Rev. A40 (1989) 6308. [12] G. Graff, E. Klempt and G. Werth, Z. Phys. 222~ (1979) 201. [13] D.J. Wineland, W.M. Itano and R.S. van Dyck, Jr., Adv. At. Mol. Spectr. 19 (1983) 135. [14] R.S. van Dyck, Jr., Phys. Scripta T22 (1988) 228. 1151 E.A. Cornell et al., Phys. Rev. A41 (1990) 312. [16] D. Hagedorn, Nichtlineare Schwingungen (Akad. Verlagsgesellschaft, Berlin, 1978). [17] R.S. van Dyck, Jr. and P.B. Schwinberg, Phys. Rev. Lett. 47 (1981) 395. [18] Ch. Get-z, D. Wilsdorf and G. Werth, to be published.