Laser spectroscopy of hyperfine structure in highly-charged ions: a test
  of QED at high fields
Laser spectroscopy of hyperfine structure in highly-charged ions:
a test of QED at high fields
D.F.A. Winters ∗, M. Vogel
GSI mbH, Planckstrasse 1, Darmstadt D-64291, Germany
D.M. Segal, R.C. Thompson
Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, United Kingdom
W. Nörtershäuser
Universität Mainz, Fritz-Strassmann-Weg 2, Mainz D-55099, Germany
GSI mbH, Planckstrasse 1, Darmstadt D-64291, Germany
Abstract
An overview is presented of laser spectroscopy experiments with cold, trapped, highly-charged ions, which will
be performed at the HITRAP facility at GSI in Darmstadt (Germany). These high-resolution measurements of
ground state hyperfine splittings will be three orders of magnitude more precise than previous measurements.
Moreover, from a comparison of measurements of the hyperfine splittings in hydrogen- and lithium-like ions of the
same isotope, QED effects at high electromagnetic fields can be determined within a few percent. Several candidate
ions suited for these laser spectroscopy studies are presented.
Key words: QED, highly-charged ions, hyperfine structure, laser spectroscopy, trapping and cooling
PACS: 12.20.Fv, 21.10.Ky, 32.10.Fn, 32.30.-r
1. Introduction
Quantum electrodynamics (QED) was the first
quantum field theory to be formulated and has suc-
cessfully passed every experimental test at low and
intermediate fields. A well-known example of QED
effects at low fields (∼ 109 V/cm) is the Lamb
shift in hydrogen [1]. At low fields, the QED effects
(self-energy and vacuum polarisation) can still be
∗ Corresponding author.
Email address: d.winters@gsi.de (D.F.A. Winters).
treated as a perturbation, only taking into account
lower order terms [2]. However, up to now QED
calculations have never been tested at high fields
(∼ 1015 V/cm) because such fields cannot be pro-
duced in a laboratory, nor by the strongest lasers
available. At high fields, perturbative QED is no
longer valid and higher order terms become impor-
tant as well [2]. Experiments carried out at high
fields therefore test different aspects of QED cal-
culations and are complementary to high-precision
tests of the lower order terms.
Heavy atoms that have been stripped of almost
Preprint submitted to Canadian Journal of Physics 4 November 2018
http://arxiv.org/abs/0704.0560v2
all their electrons, the so-called highly-charged ions
(HCI), are ideal ‘laboratories’ for tests of QED at
high fields. These ions have, for example, electric
field strengths of the order of 1015 V/cm close to
the nucleus [2] and can be produced at high veloc-
ities at the Gesellschaft für Schwerionenforschung
(GSI) in Darmstadt, Germany.
At the HITRAP facility, which is currently be-
ing built at GSI, ions coming from the experimen-
tal storage ring (ESR) with MeV energies will be
slowed down by linear and radiofrequency stages
to keV kinetic energies, trapped and cooled down
to sub-eV energies, and finally made available for
experiments. Within the HITRAP project, instru-
mentation is being developed for high-precision
measurements of atomic and nuclear properties,
mass and g-factor measurements and ion-atom
and ion-surface interaction studies [3,4,5].
2. Hydrogen- and lithium-like ions
Hydrogen- and lithium-like ions are the best can-
didates for our studies, since they have s-electrons
which are very close to the nucleus. The (higher
order) QED effects are most pronounced at the
high fields close to the nucleus, therefore the best
measurable quantity is the ground state hyperfine
splitting (HFS). Due to the simple electronic struc-
ture of H- and Li-like species, accurate (higher or-
der) calculations of ground state HFS can be done,
which will then be compared with accurate exper-
imental results.
As a first approximation, good within about 4%,
the energy of the (1s) 2S1/2 ground state HFS of
hydrogen-like ions is given by [2,6]:
EHFS = α(Zα)
2(2I + 1)
2As(1− δ)(1)
where α is the fine structure constant, gI =
µ/(µNI) is the nuclear g-factor (with µ the nuclear
magnetic moment and µN the nuclear magneton),
I the nuclear spin, me and mp are the electron and
proton mass, respectively, and c is the speed of
light. Equation (1) represents the normal ground
state HFS multiplied by a correction As for the
relativistic energy of the s-electron, and by a fac-
tor (1− δ), which takes the ‘Breit-Schawlow’ (BS)
effect into account. The BS effect is due to the spa-
tial distribution of the nuclear charge. It corrects
for the fact that we cannot assume a homogeneous
charge distribution over a spherical nucleus. The
values for δ were taken from [6], those for gI and
I from [7]. In principle eq.(1) should also contain
a correction for the finite nuclear mass, but since
this correction is very small it can be neglected
[2]. The energy of the (1s22s) 2S1/2 ground state
HFS of lithium-like ions only differs from eq.(1)
by a factor 1/n3 = 1/8 and by the As-value [2].
However, eq.(1) requires two further important
corrections, the one of most interest to us being
that which corrects for the QED effects. The other
correction takes the ‘Bohr-Weisskopf’ (BW) effect
into account [8]. The BW effect is due to the spa-
tial distribution of the nuclear magnetisation and
is only known with an accuracy of 20-30 %, which
is mainly due to the single-particle model used for
its calculation [9]. Unfortunately, the QED effects
are of the same order of magnitude as the uncer-
tainty in the BW effect [10]. Thus, from a HFS
measurement of a single species (i.e. H- or Li-like)
the QED effects cannot be determined accurately.
Equation (1) can also be written as E1sHFS =
C1s +E1sQED, where the constant C
1s includes ev-
erything except the QED effects. Since the equa-
tions for the (1s) and (2s) states are so similar, it
is possible to write the difference between the two
HFS as ∆EHFS = E
HFS−ξE
HFS = Enon−QED+
EQED [10]. The factor ξ only contains non-QED
terms and can be calculated to a high precision [10].
From the difference between the HFS measure-
ments of H- and Li-like ions of the same isotope,
the QED effects can thus be determined within a
few percent. However, this requires measurements
of transitionwavelengthswith an experimental res-
olution of the order of 10−6.
The transition lifetime t is defined as t = A−1
(see e.g. [11]). The transition probability A, for
an M1 transition from the excited to the lower
hyperfine state, is given by [2]
4α(2πν)3~2I (2κ+ 1)
27m2ec
4 (2I + 1)
where ~ is Planck’s constant divided by 2π and κ
Table 1
Calculated HFS transition wavelengths (λ) and lifetimes (t)
of the most interesting ion species for systematic studies.
Also shown are the nuclear spin (I) and magnetic moment
(µ), taken from [7]. The half-lives of these species are longer
than 10 minutes. (The values listed are truncated and the
QED and BW effects are not included.)
element ion type λ (nm) t (ms) I µ (µN )
lead 207Pb81+ H-like 973 45 1/2 0.59
bismuth 209Bi82+ H-like 239 0.38 9/2 4.11
209Bi80+ Li-like 1469 87
protactinium 231Pa90+ H-like 262 0.64 3/2 2.01
231Pa88+ Li-like 1511 123
lead [12] 207Pb+ P3/2 - P1/2 710 41 1/2 0.59
chlorine [13] 35Cl+ 3P2 -
1D2 858 - 3/2 0.82
3P1 -
1D2 913 -
argon [14] 37Ar2+ 3P2 -
1D2 714 - 7/2 1.3
3P1 -
1D2 775 -
is related to the electron’s angular momentum [2].
From eq.(2) it can be seen that A scales with the
transition frequency as ν3, whereas ν is propor-
tional to Z3, see eq.(1). Therefore, the transition
lifetime scales with Z as t ∝ Z−9 and is roughly of
the order of milliseconds for Z > 70.
In table 1 the calculated transition wavelengths
(λ) and lifetimes (t), together with their corre-
sponding I and µ values, of the most interesting
species for our laser spectroscopy studies are listed.
(The QED and BW effects are not taken into ac-
count.) The half-lives of these species exceed 10
minutes, which corresponds to the minimum time
required for a measurement. Although the wave-
lengths span a broad range, roughly from 200 to
1600 nm, these transitions are still accessible with
standard laser systems. The three species (Pb [12],
Cl [13] and Ar [14]) at the bottom of the table
are considered as candidates for pilot experiments.
They are singly charged ions, which are easily pro-
duced, have M1 transitions at convenient wave-
lengths, and can be used to test the laser spec-
troscopy part of the experiment. A measurement
of the HFS in 207Pb+ is of special interest, be-
cause it will be possible to extract the value of µ.
Currently two different values exist, which unfor-
tunately leads to a 2% difference in the HFS cal-
culations [15].
In principle, similar experiments could be car-
ried out with metastable hafnium (180Hf, level
energy 1141 keV, half-life 5.5 h [7]). For H-like
hafnium, the transition values are λ = 217 nm and
t = 0.25 ms. For the Li-like ion, λ = 1434 nm and
t = 72 ms are obtained. The difficulty with this
isotope is that its nucleus is in an excited state,
which is difficult to produce.
Figure 1 shows the calculated transition wave-
lengths of all H-like lead, and all H- and Li-like
bismuth isotopes with half-lives exceeding 10 min-
utes. (The QED and BW effects are not included.)
The isotopes are labelled by their corresponding
atomic mass units (in u) and the stable isotopes
(207Pb and 209Bi) are indicated by the small ar-
rows. For Pb, only the H-like isotopes are acces-
sible with standard lasers, because the transition
wavelengths of Li-like isotopes are much longer
than 1600 nm. For Bi, many isotopes of both ion
species are accessible, although their transition
wavelengths differ considerably.
From Fig.1 it is clear that both elements offer
many candidates for laser spectroscopy measure-
ments of ground state HFS and that bismuth, in
particular, allows for a systematic study of the
(higher order) QED effects at high fields. Further-
more, a systematic study of different isotopes of
the same species, for example a study of the H-like
Pb isotopes, will make it possible to study trends
in nuclear properties across a range of isotopes.
There already exist two previous measurements
of the 2s ground state HFS in 209Bi. A direct mea-
surement [16] was carried out using the ESR at
GSI (Darmstadt), but unfortunately no resonance
was found at the predicted value of ≈ 1554 nm
[10]. An indirect measurement [17] was performed
in an electron beam ion trap (EBIT) and yielded
a value of ≈ 1512 nm, but the error in the mea-
surement was rather large (≈ 50 nm). In the ESR
the ions have relativistic velocities (≈ 200MeV/u),
H-like Li-like
H-like
Fig. 1. Calculated transition wavelengths for H-like Pb
and Bi isotopes (full circles), and Li-like Bi isotopes (open
circles). Only isotopes with half-lives exceeding 10 minutes
are shown. The small arrows indicate the stable isotopes,
the numbers are the masses in u. (The QED and BW effects
are not included.)
which are used to shift the transition wavelength
to a lower value (≈ 532 nm), and the transitions
are Doppler-broadened (≈ 40 GHz). In the EBIT
the ions have temperatures of several hundreds of
eV (∼ 106 K), which lead to considerable Doppler
broadening (≈ 10 GHz). The resolution obtained
in previous measurements at the ESR is of the or-
der of 10−4, whereas that of the EBIT measure-
ment is of the order of 10−2.
3. Experiment overview
A detailed description of the proposed experi-
ments, as well as a treatment of the techniques
used, can be found elsewhere [18,19]. Briefly, an
externally produced bunch of roughly 105 HCI at
an energy of a few keV is loaded into a cylindrical
open-endcap Penning trap [20] on axis, i.e. along
the magnetic field lines. Electron capture (neutral-
isation) by collisions is strongly reduced by operat-
ing the trap at cryogenic temperatures under UHV
conditions. The HCI are captured in flight, con-
fined, cooled by ‘resistive cooling’ [21] and radially
compressed by a ‘rotating wall’ [22] technique. Af-
ter these steps a cold and dense ion cloud is ob-
tained. The spectroscopy laser enters the trap ax-
ially through an open-endcap and will fully irradi-
ate the ion cloud. The fluorescence from the excited
HCI is detected perpendicular to the cooled axial
motion (trap axis) through segmented ring elec-
trodes, which are covered by a highly-transparent
copper mesh. (The ring is segmented for the rotat-
ing wall technique.)
The above mentioned transition lifetimes imply
that, for a detection efficiency of ∼ 10−3, accept-
able fluorescence rates, up to a few thousand counts
per second, from M1 transitions can be expected
from a (∼ 3mm diameter) cloud of 105 ions [18,19].
Confining the HCI in a trap, and cooling and com-
pressing the cloud, will thus enable fluorescence
detection and ensure long interrogation times by
the laser.
However, due to the high density of HCI in the
cloud, space charge effects will play a role and will
lead to shifts of the motional frequencies of the
trapped ions. We have studied this effect in detail
and understand the corresponding frequency shifts
well [23]. Since these shifts are fairly small, the (fre-
quency dependent) cooling and compression tech-
niques can still be applied.
The HCI also need to be strongly cooled to re-
duce Doppler broadening of the transitions. This
will be achieved by resistive cooling of the (axial)
ion motion in the trap. For example, for the F =
1 → F = 0 transition in 207Pb81+ at ν ≈ 3× 1014
Hz, the Doppler broadened linewidth at a temper-
ature of 4 K is ∆νD ≈ 3× 10
7 Hz. The anticipated
resolution is therefore of the order of 107/1014 =
10−7. This is three orders of magnitude better than
any previous measurement, see e.g. [15,24], and
good enough to measure the QED effects within a
few percent.
4. Acknowledgments
This work is supported by the European Com-
mission within the framework of the HITRAP
project (HPRI-CT-2001-50036). W.N. acknowl-
edges funding by the Helmholtz Association (VH-
NG-148).
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	Introduction
	Hydrogen- and lithium-like ions
	Experiment overview
	Acknowledgments
	References
