Doubly-differential cross section calculations for $K$-shell vacancy production in lithium by fast O$^{8+}$ ion impact

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by  M. D. Śpiewanowski

Doubly-differential cross section calculations for K-shell vacancy production in
lithium by fast O8+ ion impact
M. D. S´piewanowski,1,∗ L. Guly´as,2 M. Horbatsch,1 and T. Kirchner1,†
1Department of Physics and Astronomy, York University, Toronto, Ontario, Canada M3J 1P3
2Institute for Nuclear Research, Hungarian Academy of Sciences (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary
(Dated: January 8, 2016)
Inner-shell vacancy production for the O8+-Li collision system at 1.5 MeV/amu is studied theo-
retically. Thetheorycombinessingle-electronamplitudesforeachelectroninthesystemtoextract
multielectroninformationaboutthecollisionprocess. Doubly-differentialcrosssectionsobtainedin
this way are then compared with the recent experimental data by LaForge et al. [J. Phys. B 46,
6 031001 (2013)] yielding good resemblance, especially for low outgoing electron energy. A careful
1 analysisoftheprocessesthatcontributetoinner-shellvacancyproductionshowsthattheimprove-
0 mentoftheresultsascomparedtosingle-active-electroncalculationscanbeattributedtotheleading
2 role of two-electron excitation-ionization processes.

n
PACSnumbers: 34.50.Fa
a
J

6 I. INTRODUCTION of all electrons are important already at the level of the

SDCS [11].
]
h Single,aswellasmultipleionizationprocessesofatoms Here, we study cross sections differential in energy
p by fast bare ion impact have been of great interest for of the outgoing electron and in the transverse momen-
m- decades [1] since they provide basic information about tum transfer by applying a novel strategy of combining
mechanismsforfew-bodybreakupprocesses. Experimen- single-electron amplitudes. In this way we incorporate
o
tally, the development of cold-target recoil-ion momen- the multielectron nature of the collision process into the
t
a tum spectroscopy (COLTRIMS) [2] allows to study the theoretical model. We apply the two-center basis gen-
s. collision processes in great detail. Yet, due to the limi- erator method (TC-BGM) [13] to calculate amplitudes
c tations posed by the supersonic gas jet target technique, forelectrontransitionswithinthetargetatom,andcom-
i
s the targets of interest are typically limited to rare gas bine them with continuum distorted wave eikonal initial
y atoms and molecules [3]. state (CDW-EIS) [14] single-electron ionization ampli-
h In recent experiments, laser cooling in a magneto- tudes. Thecombinedamplitudesarethenamendedwith
p
optical trap was combined with a reaction microscope aphasefactorthataccountsforthenucleus-nucleusinter-
[
(MOTReMi) [4, 5] to study single and multiple ioniza- action,andaresubsequentlyFouriertransformedinorder

1 tion of alkali metal atoms by ion impact. With this toobtaintheDDCSsdifferentialintheoutgoing-electron
v new approach the first kinematically complete experi- energy and in the transverse momentum transfer. When
0
ment for single ionization of lithium was achieved and compared with experimental data our results show im-
4
experiments were performed for the H+-Li collision sys- provements relative to previous attempts and lead to a
3
1 tem at 6 MeV/amu [6], the O8+-Li collision system at better understanding of the underlying processes that
0 1.5 MeV/amu [6, 7], and the Li2+-Li collision system at take place during inner-shell vacancy production for this
. 2.29 MeV/amu [8]. This triggered theoretical efforts for collision system.
1
0 the interpretation of the experimental data [8–12]. The paper is organized as follows. The novel method-
6 The’hydrogen-like’structureofthelithiumatomwith ology to extract multielectron information from single-
1 its large energy spacing between the K- and L-shells al- particle amplitudes is given in Sec. II. In Sec. III we
: lows for efficient laser cooling in a MOT and, hence, for present DDCSs for the O8+-Li system at 1.5 MeV/amu
v
i accurate studies of single-ionization processes. At the togetherwithacarefulanalysisofthecontributionsthat
X
levelofthesingly differentialcrosssection(SDCS)outer- constitute the overall DDCS. We draw conclusions in
r shell ionization has proven to be nearly a single-electron Sec. IV. We use atomic units ((cid:126) = m = e = 4π(cid:15) = 1)
a e 0
process with a marginal influence of the K-shell elec- throughout the paper unless stated otherwise.
trons [11]. Yet, doubly differential cross sections (DD-
CSs) for outer-shell ionization show that the inner-shell
electrons do play a role in the process, e.g., via core po-
II. METHODOLOGY
larization by the projectile [12]. In K-shell vacancy pro-
duction, on the other hand, the effects of the presence
We use the framework of the semiclassical approxima-
tion, in which the projectile follows a straight-line tra-
jectory R=ρ+vt with impact parameter vector ρ and
∗ [email protected] constant velocity v. The cross section differential in the
† [email protected] outgoing-electron energy E and in the transverse mo-
e
2
mentum transfer η is given as the detached electron having a well-defined momentum
k )isobtainedbypluggingEq.(7)intoEq.(3)andwork-
e
1 2π 2π
dσ (cid:90) (cid:90) (cid:90) ingthroughEqs.(2)and(1)foreverypossiblefinalstate.
dE dη =keη d(cosθe) dφe dφη|Rike(η)|2, (1) The result can be expressed as follows:
e
−1 0 0 dσvac = (cid:88) (cid:18)dσ1s↑f↓ke↑ + dσf↑1s↓ke↑ + dσ1s↑ke↓f↑(cid:19).
with the transition matrix element R (η) for the tran-
ike dEedη dEedη dEedη dEedη
sition from the initial state to a state that includes an f(cid:54)=1s
(8)
unbound electron with momentum k
e Here dσα1α2α3 istheDDCSfortheprocesswiththefinal
1 (cid:90) dEedη
Rike(η)= 2π dρeiη·ρaike(ρ), (2) sptraotjeecitniodnicsa,taenddbtyheαsium(ir=un1s,o2v,e3r)thweitthartgheetbporoupnedr-sstpatine
manifold. After some simplifications that are outlined in
and
the Appendix we arrive at
a (ρ)=eiδ(ρ)A (ρ). (3)
ike ike dσvac = dσexcl +(cid:88) dσfEI1 + (cid:88) dσfEI2
Here Aike(ρ) is the impact-parameter-dependent multi- dEedη dEedη dEedη dEedη
electronamplitudecalculatedwithouttheinfluenceofthe f(cid:54)=1s,2s f(cid:54)=1s
internuclear interaction with details given below. The + (cid:88) dσfex +∆ , (9)
phase accumulated during the collision process due to dE dη Ee,η
e
theprojectileinteractionwiththetargetnucleus,δ(ρ),is f(cid:54)=1s
given by (see, e.g., Ref.[15])
wherewecandistinguishthefollowingprocessesandam-
plitudes associated with them:
∞
(cid:90)
(i) Exclusive ionization, where one 1s electron is re-
δ(ρ)=− dtV (R(t)), (4)
NN moved, whereas the two electrons that are left behind
−∞ remain unaffected
with the nucleus-nucleus (NN) potential √
Aexcl = 2A A A ; (10)
1s→1s 1s→ke 2s→2s
Z Z
V (R)= P T. (5)
NN R (ii) Excitation-ionization (EI1), with one-electron re-
moval from the inner shell, one-electron excitation from
Within the independent-electron (IEL) model the full
the outer shell, and the residual inner-shell electron re-
three-electronHamiltonianofthecollisionproblemisre-
maining unaffected
placed by a sum of single-particle operators
√
AEI1 = 2A A A ; (11)
H (t)=(cid:88)3 (cid:18)−1∆ (t)+V (cid:0)ri(cid:1)− Zp(cid:19). (6) f 1s→1s 1s→ke 2s→f
e 2 i t ri (iii) Excitation-ionization (EI2), with one-electron re-
i=1 p
moval from the outer shell, one-electron excitation from
The last two terms in Eq. (6) describe the interaction the inner shell, and the residual inner-shell electron re-
of the ith electron with the target and the projectile, maining unaffected
respectively; ri and ri are the distances between the ith
t p √
electron and the target and the projectile, respectively, AEI2 = 2A A A . (12)
and Z is the projectile charge. f 1s→1s 1s→f 2s→ke
p
TheTDSEwiththeHamiltoniangivenbyEq.(6)sep- The last two terms on the right-hand side of Eq. (9),
arates into a set of single-particle equations whose so- dσex
correspond to exchange processes, f , and an anti-
lutions are used to calculate electronic transition am- dEedη
symmetry correction term ∆ , that contains all the
plitudes for the three-electron system in a completely Ee,η
cross terms. The origin of these terms is explained in
specified final state. In order to do so we assemble
theAppendix. Theyturnouttobenegligiblysmalland,
the single-particle solutions to form a Slater determi-
henceforth, will be omitted [11].
nant (|ψ ψ ψ (ρ,t)(cid:105)), where the subscripts indi-
1s↑ 1s↓ 2s↑ (iv) Shake off. In addition to the above, we introduce
cate the initial single-electron states. We project it at
this correlated two-electron process that has proven to
t = t onto the three-electron final-state Slater determi-
f be of importance for the creation of low-energy electrons
nant (|α α α (cid:105)), where α (i = 1,2,3) are the final
1↑ 2↓ 3↑ i during inner-shell vacancy production [11]. One 1s elec-
states of interest [16]
tron is excited while the 2s electron is shaken off end-
A (ρ)=(cid:104)α α α |ψ ψ ψ (ρ,t )(cid:105). (7) ingupinthecontinuum,whereastheresidualinner-shell
α1↑α2↓α3↑ 1↑ 2↓ 3↑ 1s↑ 1s↓ 2s↑ f electron remains unaffected
The overall DDCS for the process of interest to this √
work,namely,vacancyproductionintheinnershell(with Ashake = 2A A Aoverlap. (13)
f 1s→1s 1s→f 2s
3
Following[11]weemployedtheindependentevent(IEV)
model for the EI2 process where, in our calculation, ion-
-1
10
ization takes place from the valence shell of the target in
theground-stateconfiguration,andtheelasticscattering
-2
and excitation processes take place after the inner elec- 10
trons rearrange to the Li+ ground-state configuration.
The amplitudes on the right-hand side of Eqs. (10)- -3
√ 10
s
(13) are single-particle transition amplitudes and 2 ac- u
ul
counts for the indistinguishability of the electrons in the d -4
o10
K-shell (see Appendix). In Eq. (13) Aoverlap is the over- m
2s
lap between the outer-shell electron orbital and a con-
-5
10
tinuum state of a modified Hamiltonian with the proper
outgoing-electron energy [11].
-6
Transition amplitudes for electron excitations for pro- 10
cesses (i)-(iv) are calculated using the two-center ba-
sis generator method (TC-BGM) [13]. The interactions 0 10 20 30 40
ρ (a.u.)
within the target atom [see Eq. (6)] have been approxi-
mated by the exchange-only optimized potential method
FIG. 1. (Color online) Modulus of the 1s → 2p single-
(OPM) of density functional theory [17]. Single-particle 0
particle transition amplitude as a function of impact param-
TDSEs are solved using a basis that consists of 1s−4f
eter. Solid line (magenta) - TC-BGM; dashed line (black) -
target states, and 1s−4f projectile states together with
fit.
71 BGM pseudostates [13]. Electron capture has been
shown to be very weak in this collision system [11], but
the projectile states are included nevertheless to ensure
consistency with the calculation of Ref. [11]. All basis
states are endowed with electron translation factors to
ensure Galilean invariance. Amplitudes for ionization, -2
on the other hand are obtained from CDW-EIS, due to
the difficulties posed by the extraction of these ampli- πof )
tudes from TC-BGM. We note that the ionization prob- s -4
abilities calculated from both methods are in very good nit
u
agreement (see Fig. 1 in Ref. [11]). e (
s
For the Fourier transform of Eq. (2) to be fully con- a -6
h
p
vergedoneneedsthethree-electronamplitudes(10)-(13)
forimpactparametersuptoafewhundredatomicunits.
Unlike the CDW-EIS, the TC-BGM calculations give re- -8
liable results only for a limited range of impact parame-
ters. It is, therefore, necessary to extrapolate the excita-
tion amplitudes from TC-BGM calculations, both their 0 10 20 30 40
ρ (a.u.)
moduli and phases, to avoid numerical issues within the
Fourier transform procedure. In Figs. 1 and 2 we depict
FIG. 2. (Color online) Phase of the 1s→2p single-particle
0
anexampleoftheabsolutevalueandphase,respectively, transitionamplitudeasafunctionofimpactparameter. Solid
of a transition amplitude from TC-BGM together with line (magenta) - TC-BGM; dashed line (black) - fit.
a fit for a range of impact parameters. For the purpose
ofthisworksuchfitswerecarriedoutforeachexcitation
channel. theprojectilewiththenucleusorthepassiveelectronsin
The sums in Eq. (9) run over the target bound-state the single-particle transition amplitude. Therefore, one
manifold. Due to the restricted basis in the TC-BGM should amend it by a phase factor that incorporates a
calculations we truncate these sums at n=4, where n is potential, V , that accounts for these interactions, i.e.,
NN
theprincipalquantumnumber. Evenafterthetruncation VNN = ZPRZT + Vscr(R) + Vpol(R), where the last two
the procedure is still very demanding numerically with terms on the right-hand side are the screening potential
the main challenge posed by the 2D Fourier transform of the passive electrons and a polarization potential, re-
of three-electron amplitudes in Eq. (2) that has to be spectively, see, e.g., [12]. It has been shown that one
calculated for each channel. hastoproperlytakeintoaccountthephaseaccumulated
We note that in most of the previous calculations for via V for a complete understanding of the outer-shell
NN
the lithium MOTReMI experiments the single-active- ionization process [12]. In our IEL model it would be
electron(SAE)modelwasused. Itisimportanttostress inconsistent to include screening and polarization terms
that in the SAE one does not account for interactions of in V (R), since all electrons are active. V (R) is in-
NN scr
4
10 10
u.) u.)
a. 1 a. 1
S ( S (
C C
D D
D D
0.1 0.1
0 0.5 11.5 22.5 3 0 0.5 11.5 22.5 3
η (a.u.) η (a.u.)
FIG. 3. (Color online) DDCS for the 1.5 MeV/amu O8+-Li FIG. 4. (Color online) DDCS for the 1.5 MeV/amu O8+-Li
collision system as a function of impact parameter for three collision system as a function of impact parameter for three
outgoing-electronenergies: E =2eV(orange),10eV(blue), outgoing-electronenergies: E =2eV(orange),10eV(blue),
e e
20eV(black). Solidlines-presentoverallresults,dashedlines 20eV(black). Squares-experimentaldatafromRef.[6],solid
- CDW-EIS in SAE approximation. lines - present overall results.
herently accounted for in this framework [15] while po- The original experimental data from Ref. [6] are rela-
larization would have to be modeled in terms of a time- tive only. In Ref. [4] the SDCSs were normalized using
dependenteffectivepotentialinthesingle-particleHamil- SAE CDW-EIS for the 2s ionization channel for which
tonian. the multielectron effects were found to be unimportant.
Finally, we note that a similar approach has been em- The SDCS is obtained by integrating the experimental
ployed earlier to calculate SDCSs where, however, prob- DDCS for the lowest E =2 eV over the transverse mo-
e
abilities, instead of amplitudes, for different transitions mentum transfer. It is then normalized to match the
have been combined. In that case it was not necessary experimental value of the SDCS from Ref. [4]. This pro-
to carry out the 2D Fourier transform. The approach cedure fixes the normalization of all channels (in K-shell
allowed to explain SDCSs for the O8+-Li collision sys- ionization for E =2, 10, 20 eV).
e
tem at 1.5 MeV/amu [11]. However, it did not agree Finally, we carefully study the contributions to the
well with the experimental observations for the case of overall DDCS from the processes (i)-(iv) introduced in
Li2+-Li collisions at 2.29 MeV/amu [8]. the previous section.
III. RESULTS A. DDCSs
The method just outlined gives access to calculating In Fig. 3 we compare our overall results for electron
DDCSs differential in the outgoing-electron energy E energies E =2, 10, 20 eV (solid lines) with SAE CDW-
e e
and in the transverse momentum transfer η. This is not EISresults(dashedlines). WhencomparingwithRef.[6]
possible when having transition probabilities as a start- we note that our own SAE CDW-EIS results agree very
ing point (see Ref. [11]). The SDCSs can be calculated well in shape but we do not apply any arbitrary factors
by integrating the DDCSs of Eq. (1) over the transverse to change the spacing between the curves for all three
momentumtransferforeveryE . Toconnectourworkto energies.
e
the previous findings we cross-checked our overall SDCS The SDCS differential in the energy of the outgoing
resultsaswellasresultsfortheSDCSsforthecontribut- electron was shown to have a very weak dependence on
ing processes with those of [11] and concluded that the the electron energy in the range of interest when calcu-
agreementbetweenthetwoapproachesisverygood(not latedintheSAEmodel[11]. Thistrendisalsoseeninthe
shown). case of the DDCSs, i.e., the dashed lines are very close
Now we turn our attention to the results for inner- to each other. The SAE CDW-EIS theory predicts that
shell vacancy production obtained from the procedure forlowtransversemomentumtransfers(whichclassically
outlined in the previous section. We compare the results correspond to large impact parameters, i.e., distant col-
ofthepresentcalculationswiththeSAEresultsfromthe lisions) it is more likely to produce low-energy electrons,
CDW-EIS model and with the experimental data. and for high transverse momentum transfers one obtains
5
10 10
(i) (i)
(ii) (ii)
(iii) (iii)
(iv) (iv)
overall overall
SAE SAE
1 1
u.) u.)
a. a.
S ( S (
C C
D D
D D
0.1 0.1
0.01 0.01
0 0.5 11.5 22.5 3 0 0.5 11.5 22.5 3
η (a.u.) η (a.u.)
FIG. 5. (Color online) Partial contributions to the DDCS FIG. 6. (Color online) Same as Fig. 5 but for outgoing-
for processes (i)-(iv), the overall DDCS, and SAE CDW-EIS electron energy E =10 eV.
e
results for the O8+-Li system at 1.5 MeV/amu collision for
outgoing-electron energy E =2 eV.
e
The agreement between theory and experiment is less
satisfying for the two higher outgoing-electron energies
more high-energy electrons. The orange (2 eV) and blue
(E = 10 and 20 eV). While the general trend of the
e
(10 eV) dashed curves cross at transverse momentum
experimental results appears to be reproduced the ex-
transfer η ≈ 0.67 a.u., and the blue (10 eV) and black
tended plateaus between η = 0.25 a.u. to η = 0.75 a.u.
(20 eV) dashed curves cross at around η ≈1.10 a.u..
and η = 0.25 a.u. to η = 1.0 a.u. for E = 10 eV and
e
After the multielectron processes have been incorpo-
E =20eV,respectively, arenotaccountedforproperly.
e
rated the SDCS decreases much more rapidly with in-
Insteadoftheaforementionedplateausthetheoreticalre-
creasing outgoing-electron energy [11]. At the level of
sults display narrower maxima followed by ’knees’ that
the DDCS this leads to the cross sections being more
widen the distribution over the momentum transfer as
separated from each other, as seen in Fig. 3 (solid lines).
compared to the SAE CDW-EIS results.
Moreover,thelinesnevercrossandtendtobeparallelat
We elaborate more on the origin of the maxima and
η >2.5 a.u.. Hence, slow electrons are more likely to be
knees in the next subsection. For momentum transfers
released than the fast ones for the whole range of η.
betweenthemaximaandkneesthereisaregionforwhich
Our results are compared with the experimental data
a large discrepancy between theory and experiment is
in Fig. 4. The similarity in shape of the theoretical and
observed. Moreover, similarly to the case of the lowest
experimental results for E =2 eV up to η ≈2.2 a.u. is
e outgoing-electronenergythetheoryoverestimatestheex-
striking. The general trend of the theoretical curve lying
perimentaldataforhighmomentumtransfers,especially
below the experimental points is attributed to the fact
for E = 10 eV. Nevertheless, there is significant im-
e
that only a limited number of excitation channels have
provement as compared to the SAE CDW-EIS results
beenaccountedfor(upton=4). InthecaseoftheSDCS
allowingustoconcludethatwequalitativelyunderstand
incorporation of all excitation channels in an approxi-
the experimental data.
mate way increased the cross section by 20% [11]. This
To elucidate the role of the multielectron processes for
fact, we believe, explains the small difference between
this collision system we now turn to a discussion of each
theory and experiment for low momentum transfers. An
process individually.
analogous approximation on the level of amplitudes is
not possible without introducing arbitrary phases of ex-
citationamplitudescausingunpredictablechangesinthe
DDCSs. Therefore, we refrain from extrapolating the B. Contributions
excitation amplitudes for n>4.
ThesuddendecreaseofDDCSsataroundη =2a.u. at Each process discussed in Sec. II has a different de-
smallelectronenergiesseenin[6]isnowdeemedtobean pendenceonthetransversemomentumtransferη fordif-
artifact caused by the limited experimental momentum ferentoutgoing-electronenergyE . Thisresultsinadis-
e
acceptance [18]. Therefore, we have removed the three tinctvariationofthemagnitudeoftheDDCSfordifferent
experimentalpointsforE =2and10eVforthehighest E andindifferentshapesofeachDDCSasseeninFigs.3
e e
transverse momentum transfer (cf. Fig. 4 and Fig. 3b and4. ThecontributionsaredepictedinFigs.5,6,and7
from [6]). for E =2, 10, and 20 eV, respectively.
e
6
The DDCSs for processes with the inner-shell electron
detachmentastheionizationchannel,aswellastheshake 10
(i)
process [(i),(ii), and (iv)] resemble the SAE results, i.e., (ii)
(iii)
they have narrow maxima at low values of the momen-
overall
tumtransferanddecreaserapidlywithincreasingη. The SAE
EI2 DDCS, on the other hand, peaks at larger η and de-
1
creaseslessrapidlyasηincreases. Thisbehaviorisinher- u.)
ited from the outer-shell ionization [see Figs. 1(c)-3(c) a.
of Ref. [12] for the valence-shell ionization DDCSs] by S (
C
combining the outer-shell ionization amplitude with am- D
D
plitudes for inner-shell excitations and elastic scattering. 0.1
While all the processes from (i) to (iv) play an impor-
tant role in increasing the overall DDCS to match the
experimental values at low η, it is the EI2 process that
is responsible for the broadening of the DDCS such that
0.01
it agrees well with the experimentat E =2 eV (Fig. 5).
e 0 0.5 11.5 22.5 3
As the energy of the outgoing electron E increases, η (a.u.)
e
the partial DDCS shapes for all the aforementioned pro-
cesses, except the EI2 process, remain practically un- FIG. 7. (Color online) Same as Fig. 5 but for outgoing-
changed and only the overall strength diminishes. In electron energy Ee = 20 eV. The shake-off process has been
removed due the negligible influence on the overall DDCS.
particular the shake off process becomes negligible at
E =20eV.TheDDCSfortheEI2process,ontheother
e
hand, is clearlysuppressed, changingtheshapemostno-
Multielectron processes become less important as the
ticeably (see Figs. 5, 6, and 7): the maximum moves to
perturbation parameter (β = Z /v) decreases. Hence,
higher momentum transfers as E increases. This trend p
e
it would be interesting to see experimental results for
allows us to explain the broadening of the experimen-
systems with perturbation parameter between β = 1.03
tal DDCS, as well as the appearance of knees described
for the O8+-Li collision system at 1.5 MeV/amu [6] and
in the previous subsection. Hence, it can be concluded
β =0.21forLi2+-Licollisionsat2.29MeV/amu[8]. This
that the shift of the maximum in the EI2 DDCS with
would allow us to trace how the role of the multielectron
increasingoutgoing-electronenergyisresponsibleforthe
processes scales with the perturbation parameter.
broadening of the overall DDCS. This feature improves
theagreementwiththeexperimentaldataformostofthe
transverse momentum transfer range.
Finally,wenotethatcalculationsfortheH+-Lisystem ACKNOWLEDGEMENTS
at 6 MeV/amu have shown that for weak perturbations
the role of multielectron processes becomes negligible. A This work was supported by the Natural Sciences and
similar observation has been made for the Li2+-Li sys- EngineeringResearchCouncilofCanada(NSERC)under
tem at 2.29 MeV/amu [8]. In the latter case multielec- Grant No. RGPIN-2014-03611 and by the Hungarian
tron processes amounted to less than 10% of the over- Scientific Research Fund (OTKA Grant No. K 109440).
all SDCS whereas for the H+-Li system at 6 MeV/amu We thank Dr. D. Fischer for providing details about the
these contributions are less than 1%. The cause of the experimental data.
observeddiscrepancybetweentheoryandexperimentfor
these systems remains unknown, but can perhaps be at-
tributed to electron correlations that are not accounted APPENDIX
for in our model.
We show the validity of Eq. (9) by exploiting the ap-
proach given by Eq. (7). The three-electron amplitude
IV. CONCLUSIONS for a given set of spin projections is written as
(cid:12) (cid:12)
We presented a method to analyze multielectron pro- (cid:12)(cid:104)α1↑|ψ1s↑(cid:105) 0 (cid:104)α1↑|ψ2s↑(cid:105)(cid:12)
(cid:12) (cid:12)
cessesbycombiningTC-BGMresultsforexcitationwith Aα1↑α2↓α3↑ =(cid:12)(cid:12) 0 (cid:104)α2↓|ψ1s↓(cid:105) 0 (cid:12)(cid:12),
CDW-EIS results for ionization at the level of ampli- (cid:12)(cid:104)α3↑|ψ1s↑(cid:105) 0 (cid:104)α3↑|ψ2s↑(cid:105)(cid:12)
tudes. This approach allowed us to explain the ex- (A1)
perimental DDCSs for the O8+-Li collision system at where we made use of the fact that due to the lack of
1.5 MeV/amu. We attribute the leading role in broad- spin dependence of the Hamiltonian projections of the
ening the DDCSs as compared to SAE CDW-EIS results final single-electron wave functions onto final states with
to excitation-ionization processes, in particular, the EI2 an opposite spin projection are zero and, henceforth, we
process. omit the spin projections for brevity with the exception
7
of the subscripts in the amplitudes that follow to specify
the final-state configuration.
To sketch the steps to get from Eq. (8) to Eq. (9) we
write the amplitudes of interest as
(cid:12) (cid:12)
(cid:12)(cid:104)1s|ψ1s(cid:105) 0 (cid:104)1s|ψ2s(cid:105)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
A1s↑f↓ke↑ =(cid:12)(cid:12) 0 (cid:104)f|ψ1s(cid:105) 0 (cid:12)(cid:12) (cid:12)(cid:12)(cid:104)1s|ψ1s(cid:105) 0 (cid:104)1s|ψ2s(cid:105)(cid:12)(cid:12)
(cid:12)(cid:104)ke|ψ1s(cid:105) 0 (cid:104)ke|ψ2s(cid:105)(cid:12) A1s↑ke↓f↑ =(cid:12)(cid:12) 0 (cid:104)ke|ψ1s(cid:105) 0 (cid:12)(cid:12)
=(cid:104)1s|ψ (cid:105)(cid:104)f|ψ (cid:105)(cid:104)k |ψ (cid:105) (A2) (cid:12)(cid:104)f|ψ1s(cid:105) 0 (cid:104)f|ψ2s(cid:105)(cid:12)
1s 1s e 2s
−(cid:104)ke|ψ1s(cid:105)(cid:104)f|ψ1s(cid:105)(cid:104)1s|ψ2s(cid:105), =(cid:104)1s|ψ1s(cid:105)(cid:104)ke|ψ1s(cid:105)(cid:104)f|ψ2s(cid:105) (A4)
−(cid:104)f|ψ (cid:105)(cid:104)k |ψ (cid:105)(cid:104)1s|ψ (cid:105).
1s e 1s 2s
(cid:12) (cid:12)
(cid:12)(cid:104)f|ψ1s(cid:105) 0 (cid:104)f|ψ2s(cid:105)(cid:12)
(cid:12) (cid:12)
Af↑1s↓ke↑ =(cid:12)(cid:12) 0 (cid:104)1s|ψ1s(cid:105) 0 (cid:12)(cid:12)
(cid:12)(cid:104)ke|ψ1s(cid:105) 0 (cid:104)ke|ψ2s(cid:105)(cid:12)
=(cid:104)f|ψ1s(cid:105)(cid:104)1s|ψ1s(cid:105)(cid:104)ke|ψ2s(cid:105) (A3) Using Eqs. (A2)-(A4), we can express Eq. (8) via
−(cid:104)k |ψ (cid:105)(cid:104)1s|ψ (cid:105)(cid:104)f|ψ (cid:105), Eqs. (1)-(3) as
e 1s 1s 2s
 
dσvac = (cid:88) dσ211sss→→→k1fse + dσ121sss→→→k1fse + dσ211sss→→→k1fse + dσ112sss→→→k1fse + dσ112sss→→→k1fse + dσ121sss→→→k1fse +∆f  (A5)
dEedη  dEedη dEedη dEedη dEedη dEedη dEedη Ee,η
f(cid:54)=1s
 
= (cid:88) 2dσ211sss→→→k1fse +2dσ121sss→→→k1fse +2dσ112sss→→→k1fse +∆f  (A6)
 dEedη dEedη dEedη Ee,η
f(cid:54)=1s
(cid:32) (cid:33)
= dσexcl +(cid:88) dσfEI1 + (cid:88) dσfEI2 + dσfex +∆f , (A7)
dEedη dEedη dEedη dEedη Ee,η
f(cid:54)=1s,2s f(cid:54)=1s
1s→α1 terms. Finally, the factors of two in Eq. (A6) are incor-
dσ1s→α2 √
where 2s→α3 are the DDCSs for the transitions to the porated into the respective amplitudes, resulting in 2
dEedη
|α α α (cid:105) final state and ∆f consists of all the cross factors in Eqs. (10)-(12).
1 2 3 Ee,η
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