Time-reversal Studies in Photorecombination and
Photoionization Experiments with Ion Beams
A. Müller , S. Schippers, A. Aguilar† , I. Alvarez‡ , M. E. Bannister§ , J. Bozek†,
C. Cisneros‡ , A. M. Covington , G. H. Dunn¶ , M. F. Gharaibeh , G. Hinojosa ‡ ,
S. Ricz, A. S. Schlachter† and R. A. Phaneuf
Institut für Kernphysik, Strahlenzentrum, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
†
Advanced Light Source, Lawrence Berkeley National Laboratory, MS 7-100, Berkeley, CA 94720, USA
Department of Physics, MS 220, University of Nevada, Reno, NV 89557-0058, USA
‡
Centro de Ciencias Físicas, UNAM, Apartado Postal 6-96, Cuernavaca 62131, México
§
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
¶
JILA, University of Colorado, Boulder, CO 80309-0440, USA
Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), H-4001 Debrecen, Hungary
Abstract. The principle of detailed balance relates the cross sections σPR for photorecombination (PR) and σPI for
photoionization (PI) of ions on a state-to-state level. Measuring one or the other of the two cross sections provides direct
information about the time-reversed process. Measurements carried out at the Advanced Light Source, Berkeley, for PI of
C2 and Sc2 are compared with experimental results from the heavy-ion storage ring TSR, Heidelberg, on PR of C3 and
Sc3 , respectively. From that comparison, state selective cross sections can be inferred both for PR and PI. Both experimental
approaches provide possibilities for high-resolution spectroscopy of multiply excited states.
INTRODUCTION
formation obtained in PI and PR studies a new level of
insight into the mechanisms of both processes becomes
available.
Photoabsorption in the interstellar medium modifies the
radiation spectrum of distant objects in the universe and
thus complicates the interpretation of astrophysical observations. Photoionization of ions is an important mechanism for the production of highly charged ions in plasmas exposed to hot sources of radiation. Ionization in
such plasmas is usually balanced by low-energy electronion recombination [1]. Because of their applied importance photon-ion and electron-ion collision processes
have received long-standing interest by the plasma and
astrophysics communities [2].
With the advent of heavy-ion storage rings [3] a prime
source of experimental information on electron-ion recombination became available and the field of recombination studies rapidly matured [4]. Now, with the development of suitable experimental facilities at thirdgeneration synchrotron radiation facilities, research on
photon-ion interactions is also rapidly evolving. The
state of the art in photoionization experiments with ion
targets has recently been reviewed [5].
The work described in this paper relates measurements
on photorecombination (PR) obtained at a heavy ion storage ring with photoionization (PI) experiments carried
out at a synchrotron light source. From the combined in-
DETAILED BALANCE
Photoionization of an ion A q from a given initial state i
to a final state f
hν Aqi
A f e
q 1
(1)
is the time-reversed (photo-)recombination of the ion
Aq1 from state f to state i of the ion A q
Aq1 f e
hν A i
q
(2)
provided the energy hν of the photon and the energy E cm
in the electron-ion center-of-mass (c. m.) frame are the
same in both equations and are related by
hν Ii Ecm
(3)
where Ii is the ionization potential for state i of the ion
A q .
Both reactions can proceed directly or in a multi-step
fashion involving intermediate production of multiply
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
191
2
55
2
1s 2pnl'
6
5
50
1s 2p
AA
DC
RD
n=4
Energy ( eV )
well defined, parent beams of a given charge state may
contain ions in long lived excited states. In present storage ring PR experiments the parent ion is generally in its
ground state, however, the final state of the recombination process is not specified since only the new charge
state of the recombined ion is detected. Hence, when
comparing measured PI and PR cross sections on the
basis of Eq. 4, one has to be aware of the fact that the
number of PR channels starting from (almost generally
ground) state f is usually much larger than the number
of PI channels ending in state f . This is illustrated in
Fig. 1 for the case of PR of C 3 and PI of C2 ions.
Total recombination cross sections, that are usually measured in merged-beams electron-ion storage-ring experiments [3, 4], therefore, cannot be directly compared to
PI cross sections obtained in present photon-ion mergedbeam measurements. In spite of this apparent complication, the combined results of PI and PR experiments can
provide a new level of understanding. This will be illustrated below with two examples.
2
P1/2,3/2
2
2
45
1s 2snl'
5
n=4
PE
3
2
1s 2s2p
2e1γ
5
P0,1,2
2
1s 2s
S1/2
RR
DPI
C
3+
P2E
0
1
2
S0
1s 2s
C
2
2+
FIGURE 1. Schematic energy level diagram for PI of C2
and PR of C3 . For C2 two series of Rydberg states 1s2 2snl
and 1s2 2pnl are indicated. Note the break in the energy axis
because of which states with n 3 are not shown. PI can
proceed by direct removal of an outer-shell electron (DPI =
direct photoionization) from the initial state of the C2 ion
or by excitation of the ion to an autoionizing intermediate
state which can then decay by emission of an electron (AA
= autoionization). In the present energy range the formation
of a doubly excited intermediate state requires photo double
excitation of the ground state C2 ion (P2E = photo excitation
of 2 electrons) or single excitation (PE = photo excitation of
1 electron) of a singly excited metastable ion. PE and P2E
are almost exclusively restricted to dipole transitions. In the
storage ring recombination experiments the parent ions are
strictly in their ground state C3 1s2 2s. Recombination can
directly lead to bound states (RR = radiative recombination)
which may be excited. Alternative routes involve capture with
excitation of the valence electron (DC = dielectronic capture)
and subsequent emission of radiation (RD = radiative decay).
Decay directly to the ground state is only possible by a twoelectron transition (2e1γ = two-electron–one-photon decay).
RESULTS
The techniques applied in PI and PR experiments have
been described in detail previously [6, 7]. In particular,
measurements for PI of C 2 [8] and for PR of C 3
[9, 10] have already been published. The two data sets
are compared in Fig. 2 which shows a narrow energy
range of the measured cross section function for PI of
C2 together with a calculated PI cross section obtained
from PR data by strictly applying Eq. 4 to the available
PR cross sections. At about 48.5 eV a quite prominent
and broad feature with an asymmetric resonance shape
dominates the PI cross section in that energy range.
The feature is associated with the doubly excited state
1s2 2p4d 1 P that can be reached from the ground state of
C2 by a dipole allowed transition involving excitation
of two electrons.
The thick solid line is the result of a model calculation of direct photoionization for 60 % ground state and
40 % metastable C2 ions present in the parent beam of
the experiment [8] and includes a Fano-Voigt fit of the
1s2 2p4d 1 P resonance found at at hν 48486 eV. The
level energy and the total natural width of the 1s 2 2p4d 1 P
state in the fit were basically taken from previous investigations of PR of C3 ions [9, 10]. The PR study
yielded a resonance energy E cm 0586 eV, slightly
shifted (by 13 meV) with respect to the PI experiment.
The strength of the PI resonance was found to be S PI 76 1022 cm2 eV (for the 60% fraction of ground state
C2 ions in the parent beam). This can be compared with
SDR 16 1020 cm2 eV found for dielectronic recombination of C3 via the 2p4d 1 P resonance. This large
excited autoionizing states (see Fig. 1). The indirect (resonant) PR channel is also called dielectronic recombination (DR). The principle of detailed balance, based
on time-reversal symmetry, relates the cross sections σ PI
and σPR of PI and PR, respectively, on a state-to-state
level. For nonrelativistic photon energies hν m e c2 one
obtains
PR
σ f i Ecm hν 2
gi PI
σ hν 2me c2 Ecm g f i f
(4)
where the quantities g i and g f are the statistical weights
of the ionic initial and final states, respectively.
Measuring one or the other of the two cross sections σPI and σPR provides direct information about the
time-reversed process. However, neither the usual storage ring PR experiments nor the present synchrotron
light source measurements of PI can immediately provide state-selective cross sections for which both the initial and the final state of the ion are known. In present
PI experiments the initial state of the ions is not always
192
0
n = 10
48.0
1
10
2p 4d P
1
2p 4f D
10
3
2
1,3
10
2p 4f F
3
2p 4d P
3
3
10
2p 4d D
Cross section ( Mb )
3
2p 4f G
sition occurs with a probability of only 79%.
The application of detailed balance to experiments on
PI of Sc2 [11] and PR of Sc 3 [12] exhibits some
unique features. Low energy PR of argon-like Sc 3 Ar
and PI of potassium-like Sc 2 Ar3d just above threshold involve resonances associated with intermediate autoionizing states with configurations Mg3p 5 3d 2 and
Mg3p5 3d4s, both based on a Mg-like core. Both configurations support states with very fast autoionization
decay and, hence, many of the associated resonances are
broad. Full radiative relaxation of both groups of autoionizing states requires emission of only one electron
which is different from most other autoionizing configurations, such as the C 2 1s2 2pnl resonances whose
dominant radiative decay is by sequential emission of
two photons.
Measured Sc2 PI and Sc3 PR cross sections are
shown in Fig. 3 together with state selective cross sections for PI and PR deduced from the experimental data
by applying the principle of detailed balance. According
to Eq. 3 the electron energy scale (panel c) ) has been
shifted with respect to the photon energy scale (panels
a) and b) ) by the ionization potential (24.757 eV) of the
Sc2 (3p6 3d 2 D32 ) ground state, such that the PI and PR
spectra can easily be compared. Apart from the lower energy resolution of the PR experiment there are obvious
similarities with respect to relative resonance positions
and strengths. The resonance at hν 418 eV in the PI
spectrum appears in the PR spectrum at an energy shifted
by about 3.1 eV. This shift corresponds to the 3d
4s
excitation energy and is therefore a clear indication for
the presence of metastable Sc 2 (3p6 4s parent ions in
the PI experiment. In addition to the fact that unknown
fractions of metastable components were present in the
primary beam the measured cross section for PI of Sc 2
was initially only relative. On the basis of an identification of most of the PI resonances in the energy range displayed in Fig. 3 and under the well founded assumption
that radiative decay of the intermediate resonant states
exclusively proceeds by 3d
3p transitions, application of the principle of detailed balance and comparison
with the absolute cross section for PR of ground state
Sc3 provided the fractions of the 3p 6 3d 2 D32 ground
state as well as of the 3p6 3d 2 D52 and 3p6 4s 2 S12 Sc2
metastable components in the parent beam used in the PI
experiments and hence also yielded absolute cross sections for PI of the three different initial Sc 2 states involved in that experiment (see Fig. 3, panel b) ). From
the time-reversal analysis final-state resolved contributions to PR of ground-state Sc 3 (see Fig. 3, panel c) )
could also be inferred.
Application of the principle of detailed balance to
state-of-the-art measurements at ion storage rings and
synchrotron light sources provides a powerful tool for
11
48.2
12
48.4
13
48.6
14
48.8
Photon energy ( eV )
FIGURE 2. Experimental cross sections for PI of C2 (open
circles) [8] are compared with C3 PR data converted by using
the principle of detailed balance (Eq. 4). The conversion was
carried out with statistical weights gf 2 and gi 1 of the
ground states 1s2 2s 2 S and 1s2 2s2 1 S of the investigated ions,
respectively. The ionization potential of ground state C2 is Ii =
47.888 eV. The solid circles with statistical error bars originate
from a previous PR experiment [10], the associated thin solid
line stems from many-body-perturbation-theory (MBPT) [9]
and was scaled by a factor 0.8 to match the experimental PR
data. The thick solid line in the lower portion of the figure
shows the 2p4d 1 P contribution to PI of ground-state C2
obtained by a Fano-Voigt fit to the PI data maintaining the
resonance energy and the resonance width from the MBPT
PR calculation. Contributions of Rydberg states to the PI cross
section resulting from PE excitations of metastable parent C2
ions are indicated by the shaded areas.
difference calls for an explanation.
Resonant PI of C2 1s2 2s2 1 S at around 48.5 eV proceeds via dipole excitation to the 1s 2 2p4d 1 P state. This
state can decay by the emission of radiation or by autoionization to the C 3 1s2 2s 2 S ground state. In the PI
experiment the latter channel is observed. The time reversed process is dielectronic capture of a 0.6 eV electron
by a C3 1s2 2s 2 S ion with subsequent relaxation of the
intermediate 1s2 2p4d 1 P resonance by a two-electron –
one photon transition to the C 2 1s2 2s2 1 S ground state.
In the PR experiment, however, the intermediate resonant state can also decay by numerous additional routes,
predominantly sequential emission of two photons (see
Fig. 1). From the comparison of measured resonance
strengths one can conclude that only about 79% of PR
via the 1s2 2p4d 1 P resonance is due to the time reversed
photoexcitation studied here, i.e. radiative stabilization
of the PR resonance by a two-electron – one-photon tran-
193
S1/2 → 3d( P)4s P1/2,3/2
2
1
2
2 3
2
2
2
2 3
3
2
2
D3/2, 5/2 → 3d ( F) F5/2, 7/2
D5/2 → 3d( F)4s F7/2
3
0.1
2
2
D3/2 → 3d( P)4s P1/2
We gratefully acknowledge financial support by
NATO Collaborative Linkage Grant CLG-976362,
by Deutsche Forschungsgemeinschaft under project
number MU 1068/10, by the US DOE under contracts DE-FG03-00ER14787, DE-A102-95ER54293,
DE-AC05-00OR22725, by CONACyT and DGAPA
(Mexico).
0.2
2
-15
PI cross section ( 10
ACKNOWLEDGMENTS
D3/2, 5/2→ 3d ( P) P1/2, 3/2
a)
2
cm )
recovering detailed information on both photoionization
and photorecombination processes.
0.0
35
40
45
b)
6
2
6
2
3p 4s S1/2
2
2
2
1
2
2
2 3
2
6
3p 3d D5/2
S1/2 → 3d( P)4s P1/2,3/2
D3/2, 5/2 → 3d ( F) F5/2, 7/2
2
0.0
2
3p 3d D3/2
2
2 3
D3/2 → 3d( F)4s F5/2
2
3
0.5
3
Ferland, G., and Savin, D. W., editors, Spectroscopic
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D5/2 → 3d( P)4s P3/2
1.
2
PI cross section ( 10
-15
REFERENCES
D3/2, 5/2→ 3d ( F) D3/2, 5/2
1.0
2
cm )
Photon energy ( eV )
35
40
45
2
5
2 3
2
3p 3d ( F) F
5
1
2
3p 3d( P)4s P
2 3
2
3
5
2
3
5
1.0
3p 3d( F)4s F
2.0
5
2
2
2
6
3p 4s S1/2
sum
2 3
3.0
3p 3d ( F) D
2
6
3p 3d D5/2
5
6
3p 3d D3/2
3p 3d ( P) P
c)
3p 3d( P)4s P
PR cross section ( 10
-19
2
cm )
Photon energy ( eV )
0.0
10
15
20
Electron energy ( eV )
FIGURE 3. Experimental cross sections for PI of Sc2 a)
[11] and PR of Sc3 ions c) [12]. Most of the detected resonances are identified at the state-to-state level as indicated in
the different panels. Fine structure doublets have been resolved
in the PI experiments. The assignments can be read from panels
a) and b) which both show PI results. The data displayed in
panel b) are absolute cross sections for 3 different initial states
of Sc2 , the weighted sum of which is the original PI measurement of panel a). The time-reversed PI processes of panel b)
make up for most of the PR cross section displayed in panel c)
as indicated by the sum line. The shaded resonance in b) is due
to PI of the 3p6 4s 2 S metastable component of the parent beam
and corresponds to the shaded peak in c) which is the resonance
3p5 3d 1 P4s 2 P decaying almost exclusively to the metastable
3p6 4s 2 S state.
194