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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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