Hyperfine Interactions 127 (2000) 271–276
271
Testing atomic structure theories with high-accuracy mass
measurements on highly charged ions
H. Häffner a , N. Hermanspahn b , P. Indelicato c , H.-J. Kluge a , E. Lindroth d ,
V. Natarajan e , W. Quint a , S. Stahl a , J. Verdú b and G. Werth b
a
GSI Darmstadt, Planckstr. 1, D-64291 Darmstadt, Germany
Institut für Physik, Universität Mainz, D-55099 Mainz, Germany
c
Laboratoire Kastler-Brossel, Unité Mixte de Recherche du CNRS no. C8552, École Normale
Supérieure et Université Pierre et Marie Curie, Case 74, 4 place Jussieu, F-75252 Paris Cedex 05,
France
d
Department of Atomic Physics, Stockholm University, S-10405 Stockholm, Sweden
e
Department of Physics, Indian Institute of Science, Bangalore 560012, India
b
The mass of a highly charged ion is the sum of the mass of the nucleus, the mass of
the electrons and the electronic binding energies. High-accuracy mass measurements on
highly charged ions in a sequence of different charge states yield informations on atomic
binding energies, i.e., the ionisation potentials. In our contribution we discuss the possibility
of determining atomic binding energies of highly charged ions to better than 20 eV via
cyclotron frequency measurements in a Penning trap. At this level of accuracy different
contributions to the binding energies, like relativistic corrections, Breit corrections and QED
corrections, can be measured.
Keywords: atomic structure, mass measurements, Penning trap
1.
Motivation
Many-body problems occur in different fields of physics. In the case of atomic
physics theoretical calculations of the electronic configurations of few- and multielectron systems can be performed to high accuracy [1]. Theoretical evaluation methods like relativistic many-body perturbation theory can be tested by experimentally
determining the binding energy of a given electronic configuration.
The masses of atomic or molecular ions can be measured to extremely high accuracy using Penning trap mass spectrometers [2,3]. The presently achieved experimental
accuracy is sufficient to determine electronic binding energies for highly charged ions.
High-accuracy mass measurements on highly charged ions in a Penning trap are performed by the Seattle group [2] and by the SMILETRAP group in Stockholm [4]. In
this contribution we report on the measurement of the cyclotron frequency of a highly
charged ion (C5+ ) in our Penning trap mass spectrometer with a fractional resolution
of a few parts in 10−10 . We plan to extend these measurements in the future to heavy
highly charged ions.
 J.C. Baltzer AG, Science Publishers
272
2.
H. Häffner et al. / High-accuracy mass measurements
Atomic theory: electronic binding energies
In recent theoretical studies on cesium (Z = 55) in several charge states the
contributions to the total atomic binding energies were investigated with two different
computational schemes. The total binding energy of the neutral cesium atom, e.g., is
more than 200 keV. The main part of the binding energy (about 99.5%) can be obtained
from a single-configuration Dirac–Fock (i.e., relativistic Hartree–Fock) calculation.
Beyond the single-configuration Dirac–Fock approximation, several contributions
to the atomic binding energies are to be included: correlation energies due to the
Coulomb interaction between the electrons, the Breit interaction (magnetic interaction
between the electrons and retardation effects of the electron–electron interaction) and
the radiative QED effects (self-energy and vacuum polarisation). These three effects are
of the same order of magnitude. The radiative effects play a significant role only for the
innermost shells; about 90% come from the 1s-electrons and about 10% come from the
2s-electrons. The correlation energy is proportional to the number of electrons and is
of particular importance for the outer shells. The Breit interaction is also a two-particle
interaction and scales with the number of electrons, but it has, in addition, a strong
Z-dependence and the largest contributions are shifted towards inner electrons compared to the Coulomb interaction. In the comparison between binding energies of intermediately charged ions the Coulomb correlation is thus the most important ingredient.
Since the Coulomb correlation contributes with around 100 eV to the total binding
energy of an element as Cs, experiments which would test the atomic binding energy
calculations should be accurate to at least 10 to 20 eV, which means an experimental
accuracy of the mass measurement of about 1 part in 10−10 . At this level of accuracy
there are discrepancies between different computational schemes, and measurements
on this level would constitute a real challenge to theory. For heavy and highly charged
ions, i.e., ions with up to three electrons, the radiative effects are important contributions to the binding energy and, e.g., a measurement with an accuracy of 5 parts in
10−11 of hydrogen-like uranium (Z = 92) would compete in accuracy with the most
accurate experimental data available today.
3.
Experimental principle
In the Penning trap mass spectrometer the cyclotron frequency ωc of a single
highly charged ion
Q
B
(1)
M
with mass M and charge Q is measured in a homogeneous magnetic field B. In
the comparison of the cyclotron frequencies ωc,1 and ωc,2 of two different ions the
magnetic field B cancels and the ratio of the cyclotron frequencies
ωc =
Q1 M2
ωc,1
=
ωc,2
Q2 M1
(2)
H. Häffner et al. / High-accuracy mass measurements
273
gives directly the mass ratio. The mass Mi of an atomic ion with i electrons is
composed of the mass of the bare nucleus Mnuc , the mass of the electrons i · me and
the total (negative) electronic binding energy EBtot
Mi c2 = Mnuc c2 + i · me c2 + EBtot .
(3)
The total electronic binding energy EBtot is the sum of the ionisation potentials
of the different charge states where i is the number of electrons. The ionisation
potential EBi of a highly charged ion, i.e., the binding energy of the outer electronic
shell, is experimentally determined in the comparison of the cyclotron frequencies ωc,i
and ωc,i−1 of the ion in two neighbouring charge states with i and i − 1 electrons,
given by
EBi
Qi Mi c2 − me c2 − EBi
ωc,i
=
.
ωc,i−1
Qi−1
Mi c2
Solving eq. (4) for the ionisation potential EBi yields
ωc,i Qi−1
i
EB =
− 1 Mi c2 + me c2 .
ωc,i−1 Qi
{z
} | {z } | {z }
|
(i)
(ii)
(4)
(5)
(iii)
With eq. (5) the ionisation potential EBi can be determined from the ratio of
measured cyclotron frequencies ωc,i /ωc,i−1 , the known charge ratio Qi−1 /Qi , the mass
Mi of the highly charged ion and the mass me of the electron. In the following we
discuss the three expressions (i)–(iii) in eq. (5) and their contributions to the uncertainty
in the determination of the ionisation potential EBi .
(i) The cyclotron frequency ωc of a single ion in a Penning trap can be measured
with very high accuracy. The principle of the Penning trap as a mass spectrometer
is described in chapter 4. In measurements on C4+ - and C5+ -ions, e.g., an
accuracy of 1.3 · 10−11 has been achieved [2]. With the assumption that the
cyclotron frequencies ωc,i and ωc,i−1 can be measured with an accuracy of 10−10
the ionisation potential EBi of a highly charged ion with mass A = 200, for
example, can be determined with an accuracy of about 20 eV.
(ii) Mass units have to be converted to energy units, i.e., the mass Mi c2 of the
highly charged ion has to be known in units of electron Volts (eV), in order to
compare the experimentally measured masses with the theoretically calculated
binding energy EBi . A systematic error arises from this conversion because the
relation between the atomic mass unit Mu and the energy unit eV is known only
with an uncertainty of 4 · 10−8 (Mu c2 = 931 494 013 (37) eV) [5]. However, the
contribution of this uncertainty to the total systematic error in the determination
of the binding energy EBi is in general negligible (<0.1 eV).
The mass Mi of the highly charged ion can be expressed in atomic mass units Mu
by comparing its cyclotron frequency to the cyclotron frequency of a carbon ion
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H. Häffner et al. / High-accuracy mass measurements
(e.g., C6+ or C5+ ) since the atomic mass unit is defined as 1/12 of the mass of the
neutral carbon atom 12 C. For this comparison of cyclotron frequencies an accuracy
of 10−6 is sufficient to reach an uncertainty of <1 eV in the determination of the
binding energy EBi because the expression (i) in eq. (5) is a small quantity of the
order of 10−5 .
(iii) The mass me c2 of the electron in units of eV is known with an uncertainty of
0.021 eV (me c2 = 510 998.902 (21) eV). The contribution of this uncertainty to
the total systematic error is negligible.
4.
Penning trap mass spectrometer
In a Penning trap a highly charged ion is stored in the combination of a homogeneous magnetic field and an electrostatic quadrupole field. The ion oscillates along the
magnetic field lines with axial frequency ωz due to the electric harmonic binding force.
In the plane perpendicular to the magnetic field lines the electric field transforms the
free cyclotron motion into two circular motions: the perturbed cyclotron motion (with
frequency ω+ < ωc ) and the magnetron motion (with frequency ω− ωc ) which is a
~ ×B
~ drift motion around the trap axis. The free cyclotron frequency ωc = (Q/M )B
E
is, therefore, not an eigenfrequency of the ion in the Penning trap. However, the frequency ωc can be derived from the three oscillation frequencies ω− , ωz and ω+ of the
stored ion by [6]
2
2
ωc2 = ω−
+ ωz2 + ω+
.
(6)
Due to the hierarchy of the eigenfrequencies (ω− ωz ω+ ) only the reduced
cyclotron frequency ω+ and the axial frequency ωz have to be measured with high
accuracy.
For a measurement of the cyclotron frequency of an ion in a Penning trap with
an accuracy of better than 10−9 , it is essential to perform the experiment on a single
ion because the Coulomb interaction between simultaneously trapped ions can substantially shift their oscillation frequencies [7]. Also deviations of the electrostatic trapping
field from the ideal quadrupole geometry (anharmonicities) and inhomogeneities of the
magnetic field shift the eigenfrequencies. The anharmonicities of the electric field can
be minimised by adjusting the voltage applied to special compensation electrodes.
The magnetic field homogeneity in the Penning trap is optimised by using only nonmagnetic material close to the trap center and by tuning the currents in special shim
coils of the magnet which provides the magnetic field. With state-of-the-art superconducting magnets temporal stabilities of better than ∆B/B < 10−9 in one hour can be
reached, which is sufficient to compare the masses of two ions with an accuracy of
better than 10−9 .
Our present experimental set-up is designed for the determination of the g-factor
of the bound electron in light hydrogen-like systems [9,10]. A single stored C5+ -ion
is nondestructively detected by picking up the image currents induced in the trap
H. Häffner et al. / High-accuracy mass measurements
275
Figure 1. Cyclotron-frequency spectrum of a single trapped 12 C5+ -ion. The line width of the signal
(∆ν/ν = 8 · 10−10 ) is due to magnetic field inhomogeneities. We, therefore, expect the frequency
resolution to increase with improved magnetic field homogeneity.
electrodes by the periodic ion motion. The image currents are transformed into a
voltage signal by electronic resonant circuits attached to the trap electrodes. During
the detection process the ion’s motional amplitudes are reduced, i.e., the ion is cooled,
due to a dissipative force produced by the interaction with the attached circuit. After
a few seconds the ion reaches thermal equilibrium with the resonant circuit.
A demonstration of a cyclotron frequency measurement on a single trapped
C5+ -ion is shown in figure 1. Using eq. (6) the cyclotron frequency of the single
ion is determined in this specific example with a precision of 8 · 10−10 . The ion moves
(with motional amplitudes of axial 50 µm, cyclotron 40 µm, magnetron 10 µm) in an
inhomogeneous magnetic field (linear term 60 µT/mm, quadratic term 8.2 µT/mm2 )
resulting in frequency shifts and line broadening. The inhomogeneity is mainly due to
a ferromagnetic ring which is needed to perform the g-factor measurement and which
is situated close to the trap. We expect to improve the precision by more than one
order of magnitude by removing the ferromagnetic ring and by tuning the currents in
the shim coils, which will allow us to reach accuracies of better than 10−10 .
5.
Conclusions and outlook
High-accuracy mass measurements on highly charged ions in different charge
states in a Penning trap make it possible to determine experimentally atomic binding
energies, i.e., the ionisation potentials, of highly charged ions with an accuracy of
better than 20 eV. Penning trap mass spectrometers are therefore an interesting tool
for the investigation of the atomic structure of highly charged heavy ions.
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H. Häffner et al. / High-accuracy mass measurements
Acknowledgements
We are grateful to Gerrit Marx and Johannes Schönfelder for helpful discussions.
V.N. acknowledges travel support by a DST-DAAD fellowship. J.V. acknowledges
financial support by the European Community (TMR Network CT-97-0144).
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