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Traces on ion yields and electron spectra of Ar inner-shell hollow states with
Free-Electron Lasers
A. O. G. Wallis,1H. Banks,1and A. Emmanouilidou1
1Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom
We explore the formation by Free-Electron-Laser radiation of Ar hollow states with two or three
inner-shell holes. We find that even charged Ar ion states can be more populated than odd charged
Ar ion states. This depends on the pulse intensity and the number of energetically accessible inner-
shell holes. Fully accounting for fine structure, we demonstrate that one electron spectra bare the
imprints of Ar hollow states with two inner-shell holes. Moreover, we show how the Auger spectra
of these hollow states can be extracted from two-electron coincidence spectra.
PACS numbers: 32.80.Fb, 41.60.Cr, 42.50.Hz, 32.80.Rm
The advent of extreme ultraviolet and X-ray Free-
Electron Lasers (FELs) allows the exploration of novel
states of matter. One fascinating aspect of FELs is that
the laser boils away electrons from the inside out giving
rise to hollow atoms and molecules. To monitor the fem-
tosecond time-scale dynamics of these hollow states one
needs to identify the ionization pathways that lead to
their formation. Understanding the processes leading to
the formation of hollow states will allow these states to
be employed as the basis for a new type of spectroscopy
for chemical analysis [1–4]. It will also assist in achieving
atomic resolution in diffraction patterns from biological
molecules interacting with FEL-radiation [5, 6].
We consider Ar interacting with FEL radiation. For
each additional inner-shell hole that becomes energeti-
cally accessible, a link of a PCand an AVtransition is
added to the ionization pathways. Even charged Ar ion
states are primarily populated by chains of these links. P
stands for a single-photon ionization of an electron and A
for an Auger decay with an electron from a higher orbital,
denoted as the subscript in A, dropping to fill in a hole.
C and V stand for a core and a valence electron, respec-
tively. We show that when hollow states with two inner-
shell holes are formed, the ion yield of Ar2n+ is larger
than the ion yield of Ar(2n−1)+, with n = 1, ..., h and h
the number of holes. This is true for all intensities. How-
ever, when three inner-shell holes are energetically acces-
sible, additional transitions become available. These are
Coster-Kronig Auger (AC) transitions [7] where the hole
and the electron dropping in to fill the hole occupy sub-
shells with the same n and different l numbers. As a
result, we find that it is only for higher intensities that
the yield of the even charged Ar ion states is larger than
the yield of the odd charged Ar ion states.
Focusing on two inner-shell holes, we demonstrate how
to identify the formation of the Ar2+(2p−2) hollow state
[8–10]. We show that the yield of Ar4+, which bares the
imprint of Ar2+(2p−2), is not sufficient for identifying
this hollow state. The reason is that Ar4+ is populated by
competing ionization pathways, however, not all of these
pathways contribute to the formation of the hollow state.
Unlike in the ion yields, we find that in the one electron
spectra these competing pathways leave different traces
and we can thus discern the formation of Ar2+(2p−2).
We also show how to extract the Auger spectrum of the
hollow state from two-electron coincidence spectra.
We first describe the rate equations we use to obtain
our results [11, 12]. We account for the general case when
multiple states lead to state j, for example, i →j→k and
i0→j→k. To compute the contribution of the state i to
the yield I(q−1)
j(i) of the ion state j with charge q −1 we
solve the rate equations:
d
dtI(q−1)
j(i) (t) =(σi→jJ(t) + Γi→j)I(q−2)
i(t) (1)
−X
k0
(σj→k0J(t) + Γj→k0)I(q−1)
j(i) (t)
d
dtP(q)
j(i)→k=σj→kJ(t)I(q−1)
j(i) (t)
d
dtA(q)
j(i)→k=Γj→kI(q−1)
j(i) (t),
where σi→jand Γi→jare the single-photon absorption
cross section and the Auger decay rate from the initial
state i to the final state j, respectively. J(t) is the photon
flux, which is modeled with a Gaussian function. Atomic
units are used in this work. For details on how we com-
pute Γi→j, see [11]. The first term in Eq. (1) accounts for
the formation of the state j with charge q −1 through the
single-photon ionization and the Auger decay of the state
i with charge q −2. The second term in Eq. (1) accounts
for the depletion of state j by single-photon ionization
and Auger decay to the state k0with charge q. In ad-
dition, we compute the photo-ionization P(q)
j(i)→kand the
Auger A(q)
j(i)→kyields, with q the charge of the final state
k. These yields provide the probability for observing two
electrons with energies corresponding to the transitions
i→j and j →k. Using these yields, we obtain the coin-
cidence two-electron spectra. The one electron spectra,
that is, the transition yields from an initial state j with
charge q −1 to a final state k with charge q and the ion
yields of the state j and of all the states with charge q −1
arXiv:1502.00907v1 [physics.atom-ph] 3 Feb 2015
2
FIG. 1. Ion yields of Arn+ for a pulse of 5×1015 W cm−2
intensity, 10 fs duration and different photon energies. For
each photon energy, the number of accessible inner-shell holes
is different: (a) 200 eV, no inner-shell holes; (b) 260 eV, a sin-
gle 2p inner-shell hole; (c) 315 eV, two 2p inner-shell holes.
(d) 360 eV, three 2p and a combination of two 2p and one
2s inner-shell holes. Highlighted in red is the contribution of
Coster-Kronig Auger transitions. (e) for 315 eV and (f) for
360 eV show the contribution of pathways that are differenti-
ated by the maximum number of core holes in any state along
each pathway: light grey corresponds to zero maximum num-
ber of core holes, grey to one, black to two and striped black
lines to three.
are given by
P(q)
j→k=X
iP(q)
j(i)→kA(q)
j→k=X
iA(q)
j(i)→k(2)
I(q−1)
j=X
iI(q−1)
j(i) I(q−1) =X
jI(q−1)
j.(3)
In Fig. 1 we compute the yields of the Arn+ ion states
for four photon energies and for a high pulse intensity
of 5×1015 W cm−2. We do so accounting only for the
electronic configuration of the ion states in the rate equa-
tions and without including fine structure [11]. We first
consider a photon energy sufficiently low, 200 eV, that
single-photon ionization events do not lead to the for-
mation of inner-shell holes. In Fig. 1(a), we show that
the Arn+ ion states are populated in descending order.
For 260 eV, a single 2p inner-shell hole is accessible by
a PCprocess from neutral Ar. A PCis a much more
likely transition than a PVone. As a result, for all inten-
sities, the population going through Ar+(2p−1) is much
FIG. 2. Ionization pathways between different electronic con-
figurations of Ar accessible with P (red lines) and A (green
lines) events (a) up to Ar4+ for ~ω= 315 eV and (b) up
to Ar3+ for ~ω= 360 eV. The labels 2s−a2p−b3v
−cstand
for the electronic configuration (2s2−a2p6−b3s2−d3p6−e), with
d + e = c the number of valence holes. In (a) PCAVPCAV
(blue arrows) and PCPCAVAV(brown arrows) are the path-
ways which contribute the most to the ion yield of Ar4+. In
(b) the blue arrows indicate the pathway PCACAVthat in-
volves a Coster-Kronig ACtransition and populates Ar3+.
larger than the population ending up in or going through
Ar+(3v−1). In addition, since the PCphoto-ionization is
followed by an AVdecay—PCAVpathway—the ion yield
of Ar2+ is higher than the ion yield of Ar+, see Fig. 1(b).
For 315 eV, two 2p inner-shell holes are accessible by
two PCevents, see Fig. 2(a). As for 260 eV, Ar2+ has
a larger population than Ar+. In addition, PCAVPCAV
and PCPCAVAVare now energetically allowed pathways
that populate Ar4+, see Fig. 2(a). Since, Ar3+ is pop-
ulated by pathways involving at least one PVprocess,
Ar4+ has a larger population than Ar3+, see Fig. 1(c).
For 360 eV, three 2p inner-shell holes or a combination of
one 2s and two 2p inner-shell holes are accessible through
three PCevents, see Fig. 2(b). Pathways involving three
PCand three AVtransitions, such as PCAVPCAVPCAV,
are now energetically allowed and populate Ar6+ . For
200 eV, 260 eV and 315 eV the odd charged states are
populated only by pathways that include at least one
PVprocess. In contrast, for 360 eV, pathways that in-
clude Coster-Kronig Auger transitions between the 2s
and 2p sub-shells are energetically allowed. These path-
3
FIG. 3. As a function of pulse intensity: (a) for 315 eV and
10 fs, the ion yields of Ar4+ (black squares), of PCAVPCAV
(blue circles), of PCPCAVAV(brown diamonds) and of all the
other pathways contributing to Ar4+ (grey triangles); (b) for
360 eV and 10 fs, the ion yield of Ar6+ (black squares), of
PCPCPCAVAVAV(brown diamonds), of {3PC+ 3AV}(blue
circles) and of all the other pathways contributing to Ar6+
(grey triangles).
ways do not necessarily involve a PVevent. For instance,
in Fig. 2(b), we show the PCACAVpathway that includes
a Coster-Kronig transition (AC) and populates Ar3+.
When no Coster-Kronig transitions are present, the most
probable pathways populating the Ar(2n−1)+ states and
those populating the Ar(2n)+ states have the same num-
ber of P events, with n = 1, ..., h. Thus, for 260 eV and
for 315 eV, the yield of the Ar(2n−1)+ states is less than
the yield of the Ar(2n)+ states for all intensities. How-
ever, when a Coster-Kronig transition is present some
of the most probable pathways populating the Ar(2n−1)+
states have one P transition less than the most probable
pathways populating the Ar(2n)+ states. As a result, the
yield of the Ar(2n−1)+ states is larger/smaller than the
yield of the Ar(2n)+ states for low/high intensities. For
a high pulse intensity of 5×1015 W cm−2, in Fig. 1(d),
we show that the ion yields of Ar4+ and Ar6+ are larger
than the ion yields of Ar3+ and Ar5+, respectively.
For the results in Fig. 1, double ionization (DI) and
double Auger (DA) decays are not accounted for. These
are both processes where two electrons are ejected in one
step. Some of the pathways DI and DA give rise to have
one P process less compared to pathways where only one
electron is ejected at each ionization step. As a result, the
contribution of these two processes is less for high inten-
sities. Moreover, these two processes are significantly less
likely than the ionization processes we currently account
for in our calculations. For instance, the probability for
a DA decay from a 2p hole in Ar is roughly 10% of the
probability for a single Auger decay [13].
Focusing on hollow states with two inner-shell holes, we
look for observables with clear imprints of Ar2+(2p−2).
We first consider the ion yields. In Fig. 3(a), for 315 eV,
we plot as a function of intensity the ion yield of Ar4+.
We also plot the contributions to this latter yield of the
PCPCAVAVand of the PCAVPCAVpathways and the
contribution of all the remaining pathways. Ar2+(2p−2)
is formed from Ar by two sequential PCevents while it is
is depleted through two sequential AVevents. Thus, the
PCPCAVAVpathway bares the imprint of the formation
of Ar2+(2p−2), see Fig. 2(a). In Fig. 3(a), we also show
that the yield of all other pathways that involve a PV
transition is much smaller than the yields of PCPCAVAV
and PCAVPCAV. We choose a small pulse duration, 10
fs, since it favors the contribution of the PCPCAVAV
pathway. The reason is that for small pulse durations
high intensities are reached faster. This favors a PCPC
sequence rather than a PCAVsequence. However, even
for this small pulse duration, the yield of PCPCAVAV
is similar to the yield of PCAVPCAV. Fig. 1(e) also
shows that ion yields alone do not trace the formation
of Ar2+(2p−2). Indeed, pathways that go through the
two inner-shell hollow state contribute to the ion yield of
Ar4+ as much, if not less, as pathways that go through
hollow states with up to one inner-shell hole. For hollow
states with three inner-shell holes, it is even more diffi-
cult to discern the pathway that bares the imprint of the
hollow state. We show this to be the case for 360 eV
in Fig. 3(b) where we plot as a function of intensity the
yield of Ar6+. We also plot the contribution to this latter
yield of the PCPCPCAVAVAVpathway, the contribution
of the sum of the other {3PC+ 3AV}pathways that in-
volve three PCand three AVevents and the contribution
of all the remaining pathways. The PCPCPCAVAVAV
pathway bares the imprint of the Ar3+(2p−22s−1) and of
the Ar3+(2p−3) states. Its yield is smaller than the yield
of the other {3PC+ 3AV}pathways. Also, it is only
slightly larger than the sum of the yields of the pathways
that involve at least one PVprocess. Fig. 1(f) also shows
that, using ion yields alone, we can not discern the for-
mation of the states Ar3+(2p−22s−1) and Ar3+ (2p−3).
Indeed, the largest contribution to the ion yield of Ar6+
comes from pathways that go through states with up to
two inner-shell holes.
We now explore whether we can identify the for-
mation of Ar2+(2p−2) from one electron spectra. In
Fig. 4, for 315 eV, we compute the one electron photo-
ionization and Auger spectra for pulse parameters that
optimize the contribution of the PCPCAVAVpathway.
Unlike the ion yields, to accurately calculate the elec-
tron spectra we now fully account for the fine struc-
ture of the ion states in the rate equations. To obtain
these fine structure ion states we perform calculations
using the grasp2k [14] and ratip [15] packages, within
the relativistic Multi-Configuration Dirac-Hartree-Fock
(MCDHF) formalism, see [11] for details. Out of all
P and A transitions shown in Fig. 2(a), the transitions
that bare the imprint of Ar2+(2p−2) are the PCtran-
sition Ar+(2p−1)→Ar2+(2p−2) and the AVtransition
Ar2+(2p−2)→Ar3+ (2p−13v−2). Can we separate these
transitions from all others in one electron spectra?
For 315 eV, the single-photon ionized electrons from
an inner-shell (2s or 2p) escape with energies between
0 eV and 70 eV while those ionized from a valence
shell (3s or 3p) escape with energy between 214 eV
and 300 eV. The P(q)
j→kyields for valence shell elec-
4
trons are very small and not visible in Fig. 4. More-
over, the Auger electrons escape with energies between
150 eV and 240 eV. Thus, Auger electrons are well
separated from single-photon ionized electrons. The
most probable Auger and single-photon ionization tran-
sitions are depicted in Fig. 4. During the transi-
tions: i) Ar+(2p−1)→Ar2+ (3v−2) the first Auger elec-
tron is ejected along PCAVPCAVwith energy from 173
eV to 208 eV [14], ii) Ar2+ (2p−2)→Ar3+(2p−13v−2)
the first Auger electron is ejected along PCPCAVAV
with energy from 181 eV to 241 eV and iii)
Ar3+(2p−13v−2)→Ar4+ (3v−4) the second Auger elec-
tron is ejected along PCPCAVAVand PCAVPCAVwith
energy from 140 eV to 198 eV. Thus, the Auger tran-
sition ii) that bares the imprint of Ar2+(2p−2) can be
clearly discerned only for energies above 208 eV. For
smaller energies Auger transitions i) and ii) strongly over-
lap. During the transitions: iv) Ar →Ar+(2p−1) the
first photo-ionized electron is ejected along PCPCAVAV
and PCAVPCAVwith energy from 65 eV to 67.5 eV, v)
Ar+(2p−1)→Ar2+(2p−2) the second photo-ionized elec-
tron is ejected along PCPCAVAVwith energy from 1 eV
to 25 eV and vi) Ar2+(3v−2)→Ar3+ (2p−13v−2) the sec-
ond photo-ionized electron is ejected along PCAVPCAV
with energy from 22 eV to 41 eV. From 22 eV to 25 eV
there is an overlap between photo-ionization transitions
v) and vi). However, transition v) is orders of magnitude
larger than vi). Since transition v) bares the imprint of
Ar2+(2p−2), we can clearly identify the formation of the
hollow state from the one-electron spectra.
0 50 100 150 200
Energy (eV)
0.00
0.04
0.08
0.12
Ar→Ar+(2p−1)
Ar+(2p−1)→Ar2+(2p−2)
Ar2+(3v−2)→Ar3+(2p−13v−2)
Ar+(2p−1)→Ar2+(3v−2)
Ar2+(2p−2)→Ar3+(2p−13v−2)
Ar3+(2p−13v−2)→Ar4+(3v−4)
FIG. 4. One electron spectra, for a pulse of 5×1015 W cm−2
intensity, 10 fs duration and 315 eV photon energy.
Finally, we show how to extract the Auger spectrum of
Ar2+(2p−2) from two-electron coincidence spectra. Co-
incidence experiments have been performed extensively
with synchrotron radiation [16, 17]. It is expected that
coincidence experiments with FEL-radiation will take
place in the near future [10, 18]. In anticipation of these
experiments, in Fig. 5, we plot the coincidence spectra
of a single-photon ionized electron and an Auger elec-
tron. This choice of electrons is based on the fact that
single-photon ionized electrons are well separated in en-
ergy from Auger electrons, see Fig. 4. Moreover, we have
already shown that for energies of a single-photon ion-
ized electron (EP) up to 25 eV we can clearly discern
the second PCevent—previously denoted as transition
v)—in the PCPCAVAVpathway. Indeed, as shown in
Fig. 5, there is no trace of the PCAVPCAVpathway
for EP<25 eV. Focusing on EP<25 eV, the energies
of the Auger electron from 140 eV to 198 eV corre-
spond to the transition Ar3+(2p−13v−2)→Ar4+(3v−4),
while from 181 eV to 241 eV correspond to the tran-
sition Ar2+(2p−2)→Ar3+ (2p−13v−2). It is this latter
transition that corresponds to the Auger spectra of the
Ar2+(2p−2) hollow state. In more detail, the Auger spec-
trum of the 1S0fine structure state is the sum of the
spectra corresponding to EParound 1.4 eV and 3.6 eV.
These two energies correspond to the 2P3/2and 2P1/2
fine structure states of Ar+(2p−1). The Auger spectrum
of the 1D2fine structure state is the sum of the spec-
tra corresponding to EParound 13.5 eV and 15.6 eV.
Finally, it is more difficult to discern the Auger spectra
of the 3P0,1,2fine structure states in the interval 20.2
eV<EP<24.6 eV. It is also mainly the Auger spectra
of these 3P0,1,2states that overlaps in the energy inter-
val from 181 eV to 198 eV with the Auger transition
Ar3+(2p−13v−2)→Ar4+ (3v−4).
140 160 180 200 220 240
0
20
40
60
80
EA(eV)
EP(eV)
10−4
10−3
10−2
PCAPCA&PCPCAA
PCAPCA
PCPCAA
1S0{
1D2{
3P0,1,2{
FIG. 5. Coincidence spectra of an Auger and a photo-ionized
electron. The pulse parameters are 5×1015 W cm−2intensity,
10 fs duration and 315 eV photon energy.
In conclusion, we explored how Ar states with multi-
ple inner-shell holes affect the ion yields. We found that
the ion yields of even charged ion states are larger than
the ion yields of odd charged ion states either for all in-
tensities or only for higher ones. This depends on the
type of transitions that are energetically allowed. Our
results hold for two and three inner-shell holes in Ar. It
would be interesting to further explore how our results
are affected by an even larger number of inner-shell holes.
Finally, motivating future FEL coincidence experiments,
5
we demonstrated how two-electron spectra carry infor-
mation regarding the Auger spectra of hollow states.
A.E. acknowledges support from EPSRC under Grant
No. H0031771 and J0171831 and use of the Legion com-
putational resources at UCL.
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