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Double photoexcitation processes at the near K-edge region of Ne, Na and Ar
V.L. Sukhorukov, A.N. Hopersky, I.D. Petrov, V.A. Yavna, V.F. Demekhin
To cite this version:
V.L. Sukhorukov, A.N. Hopersky, I.D. Petrov, V.A. Yavna, V.F. Demekhin. Double photoexcitation processes at the near K-edge region of Ne, Na and Ar. Journal de Physique, 1987, 48 (10), pp.1677- 1683. �10.1051/jphys:0198700480100167700�. �jpa-00210607�
Double
photoexcitation
processes at the nearK-edge region
of
Ne,
Na and ArV. L. Sukhorukov, A. N. Hopersky, I. D. Petrov, V. A. Yavna and V. F. Demekhin
Rostov Railway Engineers Institute, Chair of Physics, 344017 Rostov-on-Don, U.S.S.R.
(Reçu le 22 janvier 1987, révisé le 12 mai 1987, accepté le 20 mai 1987)
Résumé. 2014 Nous avons calculé la structure fine du spectre de photoabsorption en couche K de Ne, Na et Ar
dans le domaine de la photoexcitation simple et photoionisation double. Pour calculer les sections efficaces d’ionisation, nous utilisons la théorie des orbitales non orthogonales, et pour calculer les énergies nous
utilisons la méthode de l’interaction de configuration. Nous avons obtenu un bon accord entre spectres calculés
et mesurés. On a montré que pour une bonne description de la probabilité de l’effet photoélectrique en couche interne, il suffit de tenir compte du réarrangement monopolaire des couches électroniques. L’interprétation de
la structure fine des spectres étudiés impose de prendre en considération une corrélation angulaire des
électrons du c0153ur et des électrons excités pendant le mouvement. De même, il faut tenir compte du fait que les voies d’ionisation simples et doubles possèdent des seuils différents.
Abstract. 2014 K-absorption structures were calculated for Ne, Na and Ar within the region of photo-double
excitation/ionization. Ionization cross section were calculated using the theory of non-orthogonal orbitals and the energies were obtained via configuration interaction method. Calculated spectra are in good agreement with the experiment. It is shown that to describe photoionization probabilities it suffices to take into account the monopole rearrangement of electron shells. In order to interpret the fine structure of the experimental spectra one must consider angular correlations in the movement of both core and excited electrons as well as
the fact that the single and multiple ionization channels open at specific energies.
Classification
Physics Abstracts
32.80F
1. Introduction.
Methods of studying the structure of matter by
extended X-ray absorption fine structure (EXAFS)
have been developed intensively during the last
decade. These methods are based on a simple
relation [1-2] of the geometrical structure of matter
with the oscillations of the absorption coefficient above the edge. The statement of such a relation was possible because a fast moving photoelectron is weakly bound to core electrons of the absorbing
atom and, therefore, the EXAFS can be described
satisfactorily within a one-electron approximation.
The statement of a simple relation between the structural parameters of matter and the X-ray ab- sorption near edge structure (XANES) is more
difficult because of at least two reasons. Firstly, the photoelectron forming a XANES is slow-moving.
Thus, many-electron correlations can be significant
even when describing single ionization. Secondly,
the near edge structure may contain additional
« white lines » corresponding to the processes of
photo-double excitation/ionization.
The first difficulty may be overcome by exploiting
the XANES of inner shells. As has been shown in [3- 5] in this case, the main multi-electron effect is the
monopole rearrangement of electron shells which may be easily taken into account using the methods
of the theory of non-orthogonal orbitals.
One should approach the second problem (the investigation of two-electron processes in photoab- sorption) by investigating the most simple systems, e.g. atoms. Double ionization of free atoms has been studied experimentally in [6-8] for Ne, Na and Ar, correspondingly. In those papers K-absorption spec- tra in the region of single and double ionizations
were obtained using synchrotron radiation, and a preliminary assignment of spectra was given. The present work aims mainly to give a more detailed assignment of K-spectra in the region of double ionization, and to distinguish the principal multi-
electron effects which must be taken into account in
making the assignment.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480100167700
1678
2. Wavefunctions and energies.
Atomic radial orbitals (AO) of the initial state are
obtained by solving the Hartree-Fock (HF) equa- tions for the atom’s ground state. The energies of
final state are calculated as follows. AO of core and excited electrons have been obtained solving the HF equations averaged over the configurations 1s-1 nl III 1 n2 12 n3 l3. One should notice that the core’s AO practically do not depend on configura-
tions of the excited electrons, and AO of the excited
states having the same symmetry are practically orthogonal. This allowed us to calculate AO of the continuous spectrum using the frozen core approxi-
mation by solving the HF equation for the El- electron of configuration 1s-1 n, 11 1 El.
These AO were then used to construct a full set of basic wave-functions of the type :
(a is for the set of intermediate angular momenta
and configurations). Such a scheme of angular
momentum summation is chosen because the non-
spherical parts of 1s-ni 11 and n2 12-n3 13 Coulomb
interaction make the greatest contribution to the
diagonal matrix elements (a LS I V I a LS) where
and Hee and V HF (K) are respectively the Coulomb
interactions operator and the potential averaged over
the configuration K. After calculating both diagonal
and non-diagonal matrix elements of the operator V via the technique of reference [9] and solving equa- tions of the type
the energies of stationary states and their wavefunc- tions
were found. Equation (2) contains the full HF
energy averaged over the configuration K. In cal-
culating this energy, the relativistic correction to the energy of the 1s-level was taken into account ; the correction taken from [10] for Ne and Na and from
[11] for Ar and is equal to 1.8, 2.4 and 10.0 eV, correspondingly. The energies of transitions from
ground state to state (3) are calculated through the
formula
Eo being the full HF energy of the ground state.
Formula (4) does not involve the correlation energy which is equal to 1-3 eV for each electron [12]. This figure provides an estimation of the accuracy of the calculated absolute energies of the absorption fea-
tures.
3. Transition probabilities.
It has been shown [3-5] that the probabilities of single ionization may be accurately described by considering the monopole rearrangement of elec- tronic shells during photoionization. It has also been shown that the easiest way to take the rearrangement
into account is by the use of non-orthogonal or-
bitals [9]. Therefore, in the present work, the theory
of non-orthogonal orbitals was used to obtain the expressions for the amplitudes of double excitations.
The amplitudes of such processes were calculated in
[13] considering that a change of HF potential leads only to monopole excitations. In the cases of interest
they are
where N is a product of overlap integrals for the AO
of the electrons not involved in a transition, and 0) is the ground state wavefunction. Within (5)-(7) l max = max (/1’ l2 ) ;
bra-AO are the functions of final, and ket-AO - that of the initial state ; according to [3-5, 13], the expression for the matrix element of operator d considering the first order terms is as follows :
summation (8) being performed over all the occupied
states of l2-symmetry, where F is the Fermi level.
The formulae (5) and (6) allow us to calculate the
amplitudes of lsnp-nl pn2 p and lsns-nl pn2 s transi- tions in Ne and Ar, and formula (7) - that of ls3s-
npn’s transitions in Na. The amplitude of the transi- tion from ground state to the state (3) is
Amplitudes (9) are related to the oscillator strengths
for the corresponding transitions via the expression
while the photon energy w is determined in (4).
Oscillator strengths are related to the area and amplitude of the Lorentian curve via
In (11)-(12) w, a and r are measured in atomic units while the measure of U ELS is determined by
that of a2
If one of the electrons is excited into the continu-
ous spectrum then (10)-(11) give the value of the ionization cross section. When both electrons are
excited into the continuous spectrum, then the
expression for the ionization cross section is
where integration is over the surface E + E’ =
cd - IP 12 ; s and E’ are the energies of electrons in the continuum, and IP12 is the atomic double ionization potential.
One should note finally, that, as has been shown in [3-5, 13], using (5-7) the requirement of orthogon- ality for complete wavefunctions of excited states to the functions of low lying states of the same sym- metry is important. For instance, satisfying this requirement leads to the disappearance of the second
terms in (6-7).
4. Double photoexcitation spectrum of Ne.
Before describing the results of the calculation we now comment on how the spectra of photo-double
excitation/ionization were extracted from full exper- imental spectra [6-8]. To solve this problem the spectra of single ionization were calculated consider-
ing the rearrangement of electronic shells through
the methods [3-5], calculated spectra were applied to
the experimental ones within the region of double ionization, and the double photoexcitation spectra
were taken as the differences between the exper- imental and calculated spectra of single ionization.
Comparing the values of cross sections in the region
before the threshold of double ionization we had to
change the theoretical values for Ne, Na and Ar by
+ 5 %, - 2 % and - 5 % correspondingly.
The basis for the calculation of double photoexci-
tation spectra of Ne contained the configurations
Excitation-ionization processes were taken into ac- count by including the configurations 1 s-12p-13p Ep
and ls-’2p-’4pep, and the processes of double ionization - by including the configuration
ls-’2p-’Eps’p. Channels ls2p-3pEp, ls2p-4pEp and ls2p-sps ’p open at the energies 906.8 eV, 911.7 eV
and 916.6 eV, correspondingly. Hereafter, the mix- ing of channels of excitation-ionization and double ionization with the doubly excited states was not
taken into account.
The configurations of the first column in (14)
determine the integral intensities of double photo-
excitation features since they are the final states for
the direct transitions from the ground state. The
remainder of configurations were considered because the AO with equal main quantum numbers overlap significantly and therefore must allow one to describe angular correlations in the movement of the excited electrons.
Calculations have been performed including the
states (14) step by step to watch the influence of correlations of different types. The results of the
first-stage calculation are shown in figure la and
table Ia. In this calculation only ls2p-npn’p excita-
tions were considered and the non-spherical part of the Coulomb operator was not taken into account in calculating the energies of 1s-1Zp-lnpn’p con-
1680
figurations. This approximation corresponds to con- sidering rearrangement of the electronic shells and
neglecting angular correlations in the movement of excited electrons completely.
One should notice a strong mutual cancellation of the first and the second terms in (5) for the terms
with odd values of L + S when calculating the probabilities of ls2p-npn’p transitions. This has led to the fact that the excitation cross section of the
terms 1S, 3p, ’D (arising from npn’p) is by two orders
of magnitude greater than that of the terms 3S,
1P, 3D. ,
Angular correlations are partly taken into account if the multiplet splitting of 1s-IZp-Inpn’p is con-
sidered. Calculated values of the operator V (1)
matrix elements of Coulomb interaction are reduced
by a factor 1.5, which corresponds to the inclusion of multi-electron correlations [14]. The Coulomb inter-
Fig. 1. - NeK-absorption spectrum at the region of
double photoexcitation : ... experiment [6] ; theory (this work) ; --- partial excitation-ionization and double ionization ls2p-3psp, 4pep, Eps’p cross sec- tions ; 2013’2013’2013’2013 double excitation spectrum obtained as
a sum of Lorentian curves with r = 0.65 eV. Approxi-
mations a, b, c described in the paper. Lorentian ampli-
tudes are calculated via (12) with r = 0.65 eV [6]. Nota-
tion of final states labelled here with numbers is given in
table I.
action 1s-n’p and 2p-n’p was not considered because it is small compared to ls-2p, 1s-np and np-n’p
interactions. The spectrum calculated with the inclu- sion of multiplet splitting is shown in figure 1b, and
the wavefunctions of the most intense components
are listed in table Ib. Figure 1b demonstrates that consideration of multiplet splitting makes the agree- ment between theory and experiment better, while
the origin of some details in structure of the first
white lines remains unclear. Consideration of angu- lar correlations by inclusion of the whole basic
configurations (14) makes the agreement between calculated and experimental spectra in the low-ener- gy region quite satisfactory (see Fig.1c and Tab. Ic).
There are still some discrepancies present on the
high energy side which could be caused by the fact
that the basic set (14) is not wide enough. The results of calculations of all the stages are given in more
detail in [15].
5. Double photoexcitation spectrum of Na.
The calculation of the Na spectrum is somewhat different from Ne. The difference is that the main contribution to the spectrum is given by the excita- tion of the optical 3s - electron, and, therefore, one
may confine the calculation to the inclusion of
configurations with three unfilled shells. The wave-
functions of excited states are calculated within the
following scheme of summation
and are listed in table II. The effective value of Coulomb interaction within the states (15) was
calculated via the methods [16]. Also listed there are
the energies of the states obtained by solving (2),
functions (3) and oscillator strengths (f) for double
excitation. Figure 2 illustrates a good agreement of
Fig. 2. - NaK-absorption spectrum at the region of
double photoexcitation : ... experiment [7] ; - theory (this work) ; --- partial excitation-ionization
ls3s-3pEs, 4sEp cross sections. Lorentian amplitudes are
calculated via (12) with r = 0.55 eV [7]. Notation of final states labelled here with numbers is given in table II.
Calculated energies are increased by 1.4 eV for applying
to the experiment.
Table I. - NeK-absorption spectrum features caused by ls2p - n1 11 n2 12 excitations (Fig. 1).
N 1 : Approximation.
N2 : Number of a component in figure 1.
(1) Table lists the intermediate quantum numbers in a wavefunction :
(2) Approximations are noted as in the paper.
Table II. - NaK-absorption spectrum features caused by ls3s - n1 11 n2 12 excitations (Fig. 2).
N : Number of a component in figure 2.
(1) Calculated energies are increased by 1.4 eV to be applied to the experiment.
(2) Table lists the intermediate quantum numbers in a wavefunction :
1682
calculated and measured spectra after ls3s-3pEs and ls3s-4ssp transitions were taken into account.
6. Double photoexcitation spectrum of Ar.
Calculation of the Ar spectrum in the region of
3 220 -- co -- 3 228 eV is identical to calculation of the Ne spectrum. Basic configurations included are
those listed in (14) with an exception that the main quantum numbers of outer p and excited electrons
are increased by 1. Calculating the spectrum at
w > 3 228 eV it was taken into account that the
channels ls3p-4pEp and ls3p-5pEp are open above
w = 3 228.5 eV and w = 3 232.4 eV, correspon-
dingly. The total excitation cross section was found
as a sum of partial ones (10)-(11) over LS. The multiplet splitting of 1s-13p- lnp was not considered,
not was the multiplet splitting of 1s-13p-1 in calculat-
ing the photo-double ionization ls3p-EpE’p cross
section using formula (13) which was summed over
LS.
Calculating ls3s-nsn’p double excitation the fol-
lowing configuration were included
Also considered was the dipole polarization of the 3p shell by 3s vacancy via 3p3p-3snd (n = 3, 4, 5)
excitation. Accurate calculation of energies of states
described by the configurations with 5 non-occupied
Fig. 3. - ArK-absorption spectrum at the region of
double photoexcitation : ... experiment [8] ; - theory (this work) ; --- partial excitation-ionization and double ionization 1s3p-4pep, 5pep, epe ’p cross sec- tions ; -.-.-.- double excitation spectrum obtained as
a sum of Lorentian curves with r = 0.69 eV. Lorentian
amplitudes are calculated via (12) with T = 0.69 eV [3].
Notation of final states labelled here with numbers is given
in table III.
Table III. - ArK-absorption spectrum features caused by ls3p - nl ll n2 l2 and ls3s - nl sn2 P excitations
(Fig. 3).
N : Number of a component in figure 3.
(1) Table lists the intermediate quantum numbers in a wavefunction :
shells seem too complex. Therefore the structure
connected with ls3s-nsn’p excitations was calculated
considering the non-spherical contribution for only 3p-3d interactions.
The results of the final calculation are shown in
figure 3 and table III. Figure 3 shows partial spectra
and total spectrum while table III lists energies,
oscillator strengths and wavefunctions of the most intense components. It can be seen that the calcu- lation describes most of the experimental features.
7. Conclusion.
Our investigation has allowed us to assign the main
features in the spectra of double photoexcitation of Ne, Na and Ar. The main effect one should take into account in calculating total excitation cross sections is a monopole rearrangement of electronic shells.
Good agreement between calculated and measured values of cross sections shows that the monopole approximation is sufficient. Within this approxi-
mation one core electron is excited by photon while
another one is excited by the change of Coulomb potential.
To describe the energies of the excited states one
needs to consider angular correlations in the move-
ment of the excited electrons, and in calculating ls3s-npn’s excitations the dipole polarization of 3p
shell by 3s vacancy must be considered. Some of the
spectral features in Ne, Na and Ar are caused by the opening of channels of single and double photo-
ionization. One should notice that the presence of open channels and discrete states within this channels
(for example,1s-13s- 1npn’s state lies in the continu-
ous spectrum ls- ’3p- np Ep) may lead to the appear-
ance of additional « Fano structure » in theoretical spectra. The trend of the development in the theory
of double photoexcitation X-ray spectra is towards
considering such an inter-channel mixing effects.
Acknowledgments.
The authors are grateful to the referees for valuable remarks.
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