Theoretical determination of two-electron one-photon transition characteristics for low- Z K -shell hollow atoms

Abstract

Studying K-shell hollow atom spectra broadens our knowledge on femtosecond phenomena in atomic physics, chemistry, and biology. Recent synchrotron measurements of the two-electron one-photon (TEOP) transitions of low-Z atoms have shown discrepancies between experiment and theoretical predictions of the TEOP relative intensities and their linewidths. The discrepancies seem to be a result of an incomplete description of an atomic response to the strong perturbation due to the K-shell double photoionization (DPI). A theoretical attempt, based on the multiconfiguration Dirac-Fock relativistic configuration interaction method, is presented for Mg, Al, and Si atoms. It is demonstrated that both the branching ratios and the TEOP linewidths can be closely reproduced by taking into account the influences of the core and valence electron correlations, open-shell valence configuration, and the outer-shell ionization and excitation processes following the K-shell DPI.

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PHYSICAL REVIEW A 96, 031402(R) (2017)
Theoretical determination of two-electron one-photon transition characteristics
for low- ZK-shell hollow atoms
Karol Kozioł*and Jacek Rzadkiewicz†
Narodowe Centrum Bada´
nJ˛
adrowych (NCBJ), Andrzeja Sołtana 7, 05-400 Otwock-´
Swierk, Poland
(Received 22 June 2017; published 14 September 2017)
Studying K-shell hollow atom spectra broadens our knowledge on femtosecond phenomena in atomic physics,
chemistry, and biology. Recent synchrotron measurements of the two-electron one-photon (TEOP) transitions
of low-Zatoms have shown discrepancies between experiment and theoretical predictions of the TEOP relative
intensities and their linewidths. The discrepancies seem to be a result of an incomplete description of an atomic
response to the strong perturbation due to the K-shell double photoionization (DPI). A theoretical attempt, based
on the multiconfiguration Dirac-Fock relativistic configuration interaction method, is presented for Mg, Al, and
Si atoms. It is demonstrated that both the branching ratios and the TEOP linewidths can be closely reproduced by
taking into account the influences of the core and valence electron correlations, open-shell valence configuration,
and the outer-shell ionization and excitation processes following the K-shell DPI.
DOI: 10.1103/PhysRevA.96.031402
K-shell hollow atoms, i.e., atoms in which electrons fill
the outer shells while the innermost shell is entirely empty,
constitute an attractive environment for studies of the nature
of exotic atomic states and of mechanisms leading to their
production. Such atoms can be produced in many physical
processes, including nuclear decays and ion-atom collisions.
K-shell hollow atoms can also be produced by the K-shell
absorption of a single photon followed by a purely quan-
tum mechanical shake-off or by a (quasi)classical knockout
process [1–3]. Another mechanism responsible for producing
K-shell hollow atoms is a sequential multiphoton absorption
on a time scale comparable with its decay time [4,5]. The latter
requires very short, intense x-ray pulses induced, e.g., by a free
electron laser [6,7].
K-shell hollow atoms decay by nonradiative Auger or
radiative transitions. The radiative transitions can occur via
one-electron one-photon (OEOP) or via much less probable
two-electron one-photon (TEOP) transitions (see Fig. 1).
The OEOP process in which an electron jumps from the
2psubshell to the empty Kshell (1s−2→1s−12p−1)is
accompanied by a single x-ray photon emission. In the case
of the TEOP process, the empty Kshell is completely filled
by simultaneous transitions of two electrons from the Lshell
(1s−2→2s−12p−1) also with the emission of a single x-ray
photon. Both OEOP and TEOP transitions are very sensitive
to the Breit interaction, quantum electrodynamics (QED), and
electron correlations [8–10]. Moreover, the natural widths of
the corresponding Khαand Kααhlines give direct information
on the K-shell hollow atoms lifetimes, which are the shortest
lifetimes of any known bound atomic state [3,11–13]. Thus,
the TEOP and OEOP transitions permit exploring both the
fundamentals of atomic physics and the nature of the K-shell
double photoionization (DPI) processes.
The TEOP transitions were predicted for the first time by
Heisenberg [14] in 1925. In 1931 Goudsmit and Gropper [15]
formulated the corresponding selection rules (i.e., n1,n2
can change arbitrarily, while l1=±1, l2=0,±2). In
*Karol.Koziol@ncbj.gov.pl
†Jacek.Rzadkiewicz@ncbj.gov.pl
1975 Wölfli et al. reported the first experimental observation
of TEOP transitions in Fe and Ni atoms [16]. Since then,
the TEOP transitions in K-shell hollow atoms have been
investigated in many experiments [17–24].
Recently, the TEOP transitions following single-photon K-
shell DPI have been observed in a highly accurate synchrotron
experiment for Mg, Al, and Si by Hoszowska et al. [3]. In the
experiment, the TEOP transition energies, branching ratios
of the OEOP to TEOP transitions, and the TEOP linewidths
were precisely measured. So far, this single-photon impact
data provides the most reliable experimental results, which
can rigorously test the most advanced atomic modeling.
A comparison of the experimental values and theoretical
predictions has shown a good agreement only for the TEOP
energies. The experimental branching ratios of the OEOP to
TEOP transitions and the TEOP linewidths are rather poorly
reproduced by theory. The calculations based on second-
order perturbation theory using single-electron screened hy-
drogenic wave functions [25] as well as the Hartree-Fock
calculations based on a so-called “shake-down” model [26]
and the multiconfiguration Dirac-Fock (MCDF) [27] method
underestimate the experimental branching ratios [3]bya
factor of 2 to 3. It is only the employment of the relativistic
configuration interaction (RCI) formalism that allows reducing
the discrepancies to the level of 15%–30% [10]. In the case of
TEOP linewidths, it was found that the measured values are
larger by a factor of ∼2(or∼1.6 with the KK correction)
than the theoretical estimates based on the sum of the initial
and final state widths (for details, see [3]). Therefore, it is clear
that accurate calculations of the TEOP intensities for low-Z
elements are called for [28].
In this work, we report a comprehensive theoretical attempt
to reproduce the Kααhlinewidths and branching ratios of the
OEOP to TEOP transitions for Mg, Al, and Si measured in
Ref. [3]. The fully relativistic calculations of KK atomic level
widths and of L1L2,3widths are also presented.
The calculations of the radiative transition energies and
rates were carried out by means of the GRASP2K v1.1 [29] code,
based on the MCDF method. The methodology of the MCDF
calculations performed in the presented research is similar
to that published earlier, in many papers (see, e.g., [30,31]).
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1s−2
1s−12p−1
1/2
1s−12p−1
3/2
2s−12p−1
1/2
2s−12p−1
3/2
Khα2Khα1
OEOP
Khα2α3Khα1α3
TEOP
FIG. 1. Level scheme (not to scale) showing the decay of K-shell
hollow atoms via OEOP (blue arrows) and TEOP transitions (red
arrows).
The wave function for an N-electron system is given as an
atomic state function (ASF), which is a linear combination
of configuration state functions (CSFs). In the calculations,
the Breit interaction, self-energy, and vacuum polarization
QED corrections have been taken into account [32–34]. The
radiative transition rates were calculated in both the velocity
(Coulomb) [35] and length (Babushkin) [36] gauges. The
accuracy of the wave function depends on the CSFs included
in its expansion [37]. The accuracy can be improved by
extending the CSF set by including the CSFs originating from
substitutions from orbitals occupied in the reference CSFs to
unfilled orbitals of the active orbital set [active space (AS)].
The RCI method makes it possible to include a major part
of the electron correlation’s contribution to the energy of the
atomic levels and transition strengths [38,39]. The difference
between the transition rates calculated in the length and
velocity gauges is a common way to test the quality of a wave
function obtained in self-consistent calculations. Kadrekar and
Natarajan found [10] that the discrepancies between the Kααh
transition rates calculated in the length and velocity gauges
may be reduced by using the RCI approach. They also found
that the transition rates are very sensitive to the choice of the
orbital set.
Our studies, performed on a large number of test cases,
showed that using substitutions for “hole” orbitals in the RCI
expansion is crucial for producing reliable results that take into
account important correlation effects. Hence, the {ns}(n=
1–3) active space was used for the initial states of the Kααh
transitions (1s−2) (containing the CSFs produced by 1s–2sand
1s–3ssubstitutions) and the {ns,np}(n=1–3) active space
was used for the final states (2s−12p−1) (containing CSFs
produced by substitutions for “hole” 2sand 2porbitals). The
RCI expansion was created by using single (S) and double
(D) substitutions from a multireference set. In this way the
length-velocity transition rate ratio, Ilen/Ivel , may be reduced
from 3.58–3.64 to 1.02–1.09. In the next stages, we extended
the CSF set to {ns,np}(n=1−nmax,nmax =4,5) active
spaces, excluding only 1s–2psubstitutions for the initial states
(because of convergence issues).
In Table Iwe show the transition rates of the Kααh
transitions for Mg. As one can see in the table, the Kααh
transition rates in length (AL) and velocity (AV)forms
significantly differ when the virtual orbital contributions are
neglected (AS0). Next, using only S substitutions to virtual
orbitals does not improve the ratio Ilen/Ivel . In order to get
convergence and an agreement between ALand AV, one has
to extend the calculations to the SD substitutions with an active
setatleastupton=4 (AS2). Therefore, to ensure that our
calculations take a reasonable time and obtain a reasonable
accuracy, we kept the calculations to the AS3 stage for Mg
and Al, and the AS2 stage for Si (for which the convergence
is more difficult to achieve). The lower part of Table Ipresents
the predictions for the 3ssatellites of the “pure” TEOP that,
TABLE I. Total transition rates of Kααhtransitions for Mg for various active spaces.
Active space for No. of CSFsaALAV
initial/final states 1s−22s−12p−1(1010 s−1)AL/AV
AS0 2s22p63s2/1s22s12p53s21 2 5.109 1.429 3.575
S substitutions only
AS1 {ns}/{ns,np}(n=1–3) 3 20 8.646 2.129 4.062
AS2 {ns,np}(n=1–4)b9 48 8.463 2.082 4.064
AS3 {ns,np}(n=1–5)b13 76 8.481 2.086 4.065
SD substitutions
AS1 {ns}/{ns,np}(n=1–3) 6 200 2.168 1.992 1.088
AS2 {ns,np}(n=1–4)b101 954 2.078 1.865 1.114
AS3 {ns,np}(n=1–5)b233 2278 2.083 1.862 1.119
Ref. [10] (limited RCI) 0.004 2.574 0.0015
Ref. [10] (large RCI) 2.262 2.169 1.043
Ref. [27]5.11
3s satellite (SD substitutions)
AS0 2s22p63s1/1s22s12p53s11 6 5.871 1.641 3.578
AS1 {ns}/{ns,np}(n=1–3) 8 493 2.258 2.228 1.014
AS2 {ns,np}(n=1–4)b213 2184 2.404 2.220 1.083
AS3 {ns,np}(n=1–5)b500 5094 2.459 2.216 1.109
aStates involved in Kααhtransitions only.
bExcluding 1s–2psubstitutions for initial states because of convergence issues.
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TABLE II. Total transition rates of Kααhtransitions (in 1010 s−1)
per 1s−2state.
Present Ref. [10]
ALAVALAVRef. [40]Ref.[27]
Mg diag. 2.083 1.862 2.262 2.169 5.11
3ssat. 2.459 2.216
Al diag. 2.535 2.382 3.248 2.854 9.69 6.61
3ssat. 3.033 2.689
3psat. 3.029 2.463
Si diag. 2.989 2.780 3.407 3.367 8.37
3ssat. 3.840 3.199
3psat. 3.503 3.047
due to the outer-shell ionization and excitation (OIE [12])
processes, can noticeably modify the effective (observed in
an experiment) transition rates and the corresponding TEOP
linewidths, as in the case of OEOP transitions [12]. Similar
calculations have been performed for OEOP transitions in Mg
and for OEOP and TEOP transitions for Al and Si atoms.
The final values of the transition strengths are presented in
Table II together with other theoretical predictions obtained by
means of the relativistic MCDF [27,40] and the MCDF-RCI
formalism [10], respectively. A good agreement between AL
and AVindicates the quality of the ASF representations [41].
One can also see that the transition rates for the 3sand 3p
satellites of the Kααhtransitions are about 20%–30% higher
than those for the diagram ones. It was also found that in the
case of the 3sand 3psatellites of OEOP (Khα) transitions,
the change in the corresponding transition rates is significantly
lower (up to 1%). Thus, it is clear that 3sand 3pOIE processes
can modify the branching ratios.
The Khαand Kααhbranching ratios were calculated by
using the expression
BR =I(Khα)
I(Kααh)=ij Aij
ik Aik
,(1)
where Aij and Aik are the rates for transitions between the
ith 1s−2initial states and the jth 1s−12p−1(Khαtransitions)
or kth 2s−12p−1final states (Kααhtransitions), respectively.
AsshowninTableII, OIE (shake) processes that change the
electronic configuration of deexcited K-shell hollow atoms,
modify their radiative transition rates. In order to take into
account this effect on the branching ratio, we used the equation
BR =I0(Khα)+sIs(Khα)Is
I0
I0(Kααh)+sIs(Kααh)Is
I0
,(2)
where Is/I0is the ratio of the intensity of the “diagram” Khα
and Kααhline (i.e., without additional spectator 3sor 3p
vacancies), I0, to the intensity of the its nl-shell satellite, Inl
s.
The total shake probabilities, i.e., shake-off and shake-up,
have been calculated by applying the sudden approximation
model [42] and using MCDF wave functions for two valence
ionization scenarios, namely, OIE1 and OIE2. The OIE1
corresponds to the ionization and/or excitation of the valence
(3sand 3p) electrons due to the sudden atomic potential
change resulting from the single K-shell vacancy. In the case of
TABLE III. Total shake probabilities (in percent per subshell) as
a result of single (OIE1) and double (OIE2) K-shell ionization.
Mg Al Si
Subshell 3s3s3p3s3p
OIE1 20.62 11.81 15.08 7.89 18.24
OIE2 49.21 33.65 37.09 24.71 44.62
the OIE2, the more pronounced potential change is caused by
two K-shell vacancies due to the quasisimultaneous removal
of two 1selectrons (for details, see [12]). The calculated values
of the total shake probabilities (in percent per subshell) for the
OIE1 and OIE2 scenarios presented in Table III were used for
the Is/I0factor calculations in Eq. (2), employing the binomial
distribution [43].
In Table IV the branching ratios for Mg, Al, and Si,
calculated using various approaches, including OIE1 and
OIE2, are presented and compared to the experimental data.
One can see that employing the RCI and OIE1 approach
improves distinctly the branching ratios over the simple
MCDF model. The branching ratio values calculated by using
the RCI+OIE2 model with the length gauge reproduces the
measured branching ratios better than any other theoretical
predictions published so far. Our calculations are also com-
pared with the experimental values and previous theoretical
predictions in Fig. 2. One can see that the inclusion of the
3sand 3pOIE2 contribution to the branching ratios reduces
the discrepancies between the experiment and theory. The
improvement is achieved at a relatively low cost related to the
increase of differences between the length and velocity gauge
branching ratio calculations within the RCI+OIE2 model
by up to 6%. This is a natural consequence of taking into
account the satellite TEOP transitions between states having
in general more open subshells than those for the “diagram”
ones. Nevertheless, only this approach can provide an atomic
model that is able to take into account the OIE processes that
can strongly affect TEOP transitions (see Table III).
In the next step of our studies, we made an attempt
to reproduce the effective linewidth of the TEOP (Kααh)
transitions. The Kααhlinewidth can be considered as the sum
of the initial and the final atomic level widths:
Kααh=KK +L1L2,3.(3)
TABLE IV. The Khαto Kααhbranching ratios for Mg, Al, and
Si, calculated by using various approaches.
Mg Al Si
MCDF len 909 1002 1105
vel 3102 3409 3807
RCI len 2276 2644 3096
vel 2363 2646 3148
RCI+OIE1 len 2200 2533 2964
vel 2284 2605 3076
RCI+OIE2 len 2122 2418 2827
vel 2205 2562 2998
Experiment: Ref. [3] 1838 ±258 2115 ±403 2610 ±370
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FIG. 2. Comparison between theoretical end experimental
branching ratios for Mg, Al, and Si. Experiment: Hoszowska et al. [3];
theory: ˚
Aberg et al. [26], Baptista [25], Saha et al. [27], “Kadrekar &
Natarajan” means compilation of data from Ref. [10]andRef.[44].
So far, the widths of the KK and L1L2,3levels have been
approximated by the expressions
KK 2K,(4a)
L1L2,3L1+L2,3.(4b)
As was shown in Ref. [3], the theoretical predictions of
TEOP linewidths based on Eqs. (4a) and (4b) underestimate
the experimental values. In order to reliably reproduce the
experimental values, one has to employ the fully relativistic
calculations of the KK and L1L2,3level widths for double
ionized states.
In our calculations of the KK level width we have
summed the partial radiative Khα1,2(KK-KL2,3) and Khβ1,3
(KK-KM2,3) and nonradiative KK-KLL and KK-KLM
Auger widths, respectively. For the L1L2,3level width we
have considered deexcitations of the 2s−12p−1state by the
following channels: (a) by filling the L1-shell hole by radia-
tive L1-L2,3and L1-M2,3transitions or by filling the L2,3-shell
hole by radiative L2,3-M1transitions (these transitions have
lesser intensities); and (b) by nonradiative LL-LLM and
LL-LMM Auger transitions. Calculations of nonradiative
transition rates have been carried out by means of the FAC
v1.1.4 [46] code, based on a modified Dirac-Hartree-Slater
method. It has been found that taking into account the
correlation effect by increasing the RCI expansion does not
substantially change the Auger transition rates (the reduction
was 0.7%–2.5% in the tested cases).
Our theoretical predictions for the widths of the KK and
L1L2,3atomic levels, as well as the fluorescence yields (ωKK)
and the lifetime of the K−2states (τKK), for Mg, Al, and Si,
are collected in Table V. One can see in Table Vthat the KK
and L1L2,3level widths obtained from the fully relativistic
calculations are significantly larger (by up to a factor of ∼2)
than the approximation based on the sum of the single hole
states [Eqs. (4a) and (4b)]. The newly improved theoretical
values of the KK and L1L2,3level widths significantly reduce
the discrepancy between the experimental and theoretical
values of the Kααhlinewidths [see column “Eq. (3)” in
Table V]. In order to further improve the predictions of the
Si Kααh (OVC)
Line profile (sum
of Lorentz profiles)
Fit (by one
Lorentz profile)
Si Kααh (OVC+OIE2)
Main line
3s satellite
3p satellite
Sum
Fit
Energy [eV]
3560 3565 3570 3575 3580
Intensity (arbitrary units)
FIG. 3. The “diagram” and OIE2-originating satellite contribu-
tions to the width of the Kααhline of Si.
Kααhlinewidth, it is necessary to extend the calculations
to open-shell valence configuration (OVC) and OIE effects.
The OVC effect is related to the open-shell atoms that have
many initial and final states that can additionally broaden x-ray
lines. The OVC broadening is a result of an overlap of many
component x-ray lines having slightly different energies and
widths. It has been previously shown that the OVC effect can
enlarge the Khαlinewidths of atoms with 20 Z30 [12].
Details concerning the evaluation procedure for effective
linewidths relating to the OVC effect can be found in Ref. [47].
Another reason for the broadening observed experimentally
for the x-ray lines can be attributed to the OIE effect [12,47].
Ionization and excitation processes following the K-shell DPI
lead to a larger contribution of the open-shell configurations
amongst the deexciting atoms and, in consequence, to a
broadening of the line. The evaluation procedure of the OIE
effect (OIE2 scenario) is illustrated for Si in Fig. 3.Itis
worth noticing that the OIE effect affects not only the total
TEOP linewidths but also the widths of the initial (KK) and
final (L1L2,3) levels. For example, the 3sand 3pOIE causes
a reduction of the L1L2,3atomic level by up to 40%–60%
(OIE2) because of the suppression of the LL-LLM and
LL-LMM deexcitation channels. In columns “OIE1” and
“OIE2” of Table V, the total TEOP linewidths are presented
for the OIE1 and OIE2 ionization scenarios, respectively.
As can be seen, only the theoretical predictions assuming
strong shake processes (OIE2 scenarios) can reproduce all
the experimental TEOP linewidths. This finding is consistent
with our results obtained for branching ratios and with results
in the previous studies [12] concerning Khα1,2hypersatellite
linewidths. In the case of Mg and Al, the final elucidation of
the OVC and OIE contributions to TEOP linewidths requires
the measurements with a higher accuracy. Such results should
ultimately verify the dominant role of the knockout character
of DPI in the region of the K-shell DPI cross-section maxima
that, in contrast to the pure shake-off mechanism, can initiate
strong OIE processes (for details, see [3,12]).
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TABLE V. Theoretical predictions for widths of KK and L1L2,3atomic levels and Kααhlinewidths.
Atom Width of atomic level (eV) τKK Natural linewidths (eV)
(conf. val.) KKaEq. (4a)[45]L1L2,3
aEq. (4b)[45]ωKK (10−16 s) Eq. (3) OVC OIE1bOIE2bExpt. [3]
Mg 3s20.901 0.66 1.032 0.49 0.0366 7.31 1.93 1.93 2.20 2.71 2.5±0.6
Al 3s23p10.998 0.74 1.184 0.82 0.0491 6.60 2.18 2.31 2.68 3.06 2.9±1.7
Si 3s23p21.094 0.86 1.342 0.95 0.0622 6.02 2.44 2.57 2.87 3.19 3.8±0.9
aAverage value per KK or L1L2,3atomic level.
bIncluding influence of OIE1/OIE2 effect on the width of L1L2,3atomic level.
In conclusion, we have shown that employing the MCDF-
RCI calculations with OVC and OIE corrections enables one to
reproduce the experimental TEOP linewidths and correspond-
ing branching ratios for Mg, Al, and Si. The obtained theoret-
ical values are in agreement within 7%–18% and 10%–14%
of the measured linewidths and branching ratios, respectively.
Thus, the results of our studies set theoretical limits for the
TEOP transitions in the low-Zatomic range. We also hope
that this work will guide future theoretical studies for higher-Z
elements and new experiments with a higher accuracy.
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