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Energy Levels of Doubly Excited States in Argon.

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Bnnalen der Physik. 7. Folge, Rand 39, Heft 2, 1982, S. 107-111
J. A. Barth, Leipzig
Energy Levels of Doubly Excited States in Argon
By TH. M. EL-SHERBINI
and S. H. ALLAM
Physics Department, Faculty of Science, Cairo University, Cairo, A. R. Egypt
A b s t r a c t . Term energies of doubly excited states in argon and argon ions have been determined
using single-configuration Hartree-Fock calculations.
The energies calculated for KL3s3p6nZand KL393p4nZn’Z’states of Ar show a fairly good agreement
with the experimental results obtained by various techniques. However, for Ar+, Ar2+ and Ar3+ states
there are no experimental data available in the literature for comparison.
Energieniveaus der zweifach angeregten Zilstiinde des Argons
I n h a l t s u b e r s i c h t . Die Energieterme der zweifach angeregten Zustande des Argons und der
Argonionen wurden mittels der Hartree-Fock-Methode berechnet.
Fur die Zustande KL3~3p6nZund KL3s23p4nZn‘Z’ des Ar ergaben sich eine sehr gute Ubereinstimmung mit den nach verschiedenen Techniken ermittelten experimentellen Werten. Fur die Znstande in Ar+; Ar2+ und Ar3+ fehlen zum Vergleich die experimentellen Daten.
1. Introduction
When an atom has two electrons excited relative to its ground state configuration
then the atom is said to be in a doubly-excited state. If the two excited electrons
toget her have more energy than is needed to conipletely remove a single electron from
the ground state atom (the first ionization potential) the doubly-excited atom is energetically able to autoionize; that is to decay to the ground state of the ion with the
ejection of one of the excited electrons. States in which a single inner-shell electron is
excited (core excitation) with comparable energies to doubly-excited states can also
decay through autoionization.
A large niiniber of doubly-excited states and core-excited states for atoms has been
identified in electron-impact experiments [ 1, 21 and photo-absorption experiments [3, 41.
However, only recently doubly-excited states and core-excited states in ions have been
observed in beamfoil spectra [5, 61 and in ejected-electron spectra from ion-atom collision experiments [7, 81.
Theoretical calculations of doubly-excited and core-excited levels in rare-gas atoms
have been done by several methods. For He calculations were done with sufficient
accuracy both by the close-coupling method [9] and by quasi-variational procedures
[lo]. For Ne, the next simplest rare-gas atom, theoretical calculations have been carried
out for core-excited states by both frozen-core Hartree-Fock [ 111 and close-coupling
[121 methods. While doubly-excited states of Ne have been calculated by close-coupling
method [13] and by a frozen-core superposition of configurations [14]. To our knowledge
no calculations have been published for core-excited states or doubly-excited states of Ar.
We describe in this paper single Configuration Hartree-Fock calculations of coreexcited states and doubly-excited states of argon atom together with singly, doubly
TH. M. EL-SHERBINI
and S. H. ALLAM
108
and triply ionized argon. The aim of the work is to obtain theoretical energy values for
doubly-excited and core-excited states in argon and argon ions which might be useful
in the interpretation of the experimental data.
2. Method
The Hartree-Fock approach was used for obtaining approximate total wave functions. The steps which have been taken in order to obtain the total energy of the state
are as follows (see ref. [15]): i) a functional form is selected and defined in terms of
certain functions to be determined later ii) an expression of the total energy is derived
in terms of these functions iii) the variational principle is applied and equations derived
whose solutions are functions that leave the total energy stationary (i.e. t o be minimum).
Let Y be the total Hartree-Fock approximate wave function of the configuration
and be represented by a linear combination of trial functions @?:
Y=
2
ci @i
i
(1)
where the coefficients Ciare taken as those which minimize the total energy. The total
energy of the state is therefore:
where Hii, = ( Yil H 1 !Pi),Nii = (!Pi]
!Pi), and H is the non relativistic Schrodinger
Hainiltonian of N-electron atom which can be written, in atomic units, as
where Z is the atomic number, ri is the distance of the ith electron from the nucleus,
and rii is the distance between the ith and the jth electron. The condition for making
the energy a minimum leads to a system of linear homogeneous equations in the C?
the solution for which is given by the secular equation:
1
1
1
H..
- - E 3 . . = 0.
21
4
21
(4)
The solution of equation (4)for the energy eigen values E for the states under consideration is equivalent to the diagonalization of the energy matrix derived from a nonorthogonal
basis set.
The basis set of the trial function @ is a determinantal product of one electron spinorbit functions :
1
@nlmlms (r, 0, @, 0 ) = - P(nl;r ) YZm,(@, @) xms
(5)
r
where r, 0,@, 0 are the spherical polar and spin coordinates of the electron, Yzml(O,@)
is a spherical harmonic, P(n1;r ) is the radial wave functions and Xms is the spin wave
function. The radial wave functions are chosen to be screen hydrogenic and are used
to determine the Hamiltonian matrix elements Hii (see ref. [16]).
3. Results and Discussion
The method explained in the previous section was used in computing the energies
of the following states:
i) Ar : 3s3p63d, 3s3p64s, 3s3p64p, 3s3p64d, 3p43d2,3p444s2, 3p43d4s, 3p43d4p, 3 ~ ~ 4 . ~ 4 2 ,
ii) Ar+: 3s3p53d, 3s3p54s, 3s3p54p, 3p33d2,3p34s2,3p34p2, 3p33d4s, 3p33d4p, 3p34s4p
109
Energy Levels of Doubly Excited States in Argon
iii) Ar2+: 3s3p43d, 3s3p44s, 3s3p44p, 3p23d2,3p24s2,3p24p2,3p23d4s, 3p23d4p, 3p24s4p
iv) Ar3+: 3s3p33d, 3s3p34s, 3s3p34p, 3p3d2, 3p4s2, 3p4p2, 3p3d4s, 3p3d4p, 3 ~ 4 ~ 4 2 ,
The energies of the core-excited states and doubly excited states of Ar atom with
respect to the 3s23p6ground state is listed in Table 1. The Table also includes the photoabsorption measurements of MADDENet a]. [17] and the electron impact data of TWEED
et al. [18] together with the results of JORGENSEN
et al. [19] from ion-atom collision
experiments. Although there is a deviation of about 1eV between our results and the
Table 1. Energies of doubly excited and core-excited states of Ar
State
3s 3p6(2S) 4s
3s 3p6(2S) 4p
3s 3p"ZS) 3d
3s 3p"2S) 4cl
3p4(3P) 4 9
3p4('D) 4s2
3p4(lS) 4.5-2
3p4(3P) 3d2
3p4(ID) 3d2
3p4('S) 3 d 2
3p4(3P)
3d 4s
3p4(1U)3d 4s
3p4(1S)3d 4s
3p4(3P)4s 4p
3p4(lD) 4s 4p
3p4(1S)4s 42,
3p4(3P)3d 4p
3p4(1U)3d 4p
3p4(1S) 3d 4p
Energy in (eV)
This work
MADDEN et al.
~171
26.12
27.42
28.61
29.43
26.24
28.08
30.85
31.84
34.12
34.77
28.23
30.21
33.18
27.63
29.49
32.27
29.86
31.84
34.81
26.61
TWEED
et al.
[181
JBRGENSEN
et al.
~191
25.17
26.60
27.54
28.35
25.16
26.55
27.50
26.95
28.66
31.74
33.47
34.01
29.04
31.24
33.52
30.72
32.86
34.99
28.11
29.96
30.69
32.54
Table 2. Energies of doubly excited and core-excited states of Ar+
State
Energy
(eV)
State
Energy
(eV)
3s ~ $ I ~ 3d
(~P)
3s 3p5(lP) 3d
3s 3 $ 1 ~ (4s
~p)
3s 3p5('P) 4s
3s 3p5(3P) 4p
3s 3p5(lP) 4p
3p3('8) 3d2
3p3(2U) 3d2
3p3(") 3d2
35.83
36.75
34.74
35.58
37.36
38.22
36.31
39.34
41.35
36.04
39.21
41.32
3p3(4S) 4p2
3p3(2D) 4p2
3p"ZP) 4p2
3p"4S) 3d 4s
3p3(2D)3d 4s
3p3(2P)
3d 4s
3~"~
3ds42,
)
3p"ZD) 3d 4p
3p3(2P)3d 4p
3p3(4S) 4s 4p
3p3(2D)4s 4p
3p3(2P)4s 4p
41.84
45.09
47.26
35.12
38.21
41.29
37.74
40.85
42.92
38.35
41.55
43.69
3p3(48)4s2
3p3(") 482
3p3(2P)482
TH.M. EL-SHERBINIand S. H. ALLAM
110
experimental ones for core-excited states, the energy values for doubly excited states
show fairly reasonable agreement with the experimental results. A reason for the deviation can be the interactions between the configurations of the core-excited states and
adjacent configurations having nearly the same energies [20]. I n our single-configuration
calculations configuration interaction effects on the level positions were not taken into
account. Multiconfiguration Hartree-Fock calculations for Ar is in progress which niight
remove the deviation between the experimental results and our values.
Our rerults for the energies of doubly excited states and core-excited states of Ar+,
Ar2+ and Ar3+ relative to the ground state of each ion are listed in Tables 2, 3 and 4
respectively. There are no experimental data available from literature for comparison.
These results for doubly excited states in argon ions might be useful in identifying the
peaks in the ejected electron spectra from ion-atom collision experiments.
Table 3. Energies of doubly excited and core-excited states of Ar2+
State
Energy
(eV)
State
Energy
(eV)
State
Energy
(eV)
3s 3p4(4P) :k/
3s 3p4(2P) 3d
3s 3p4(") :Id
3s 3p4('5) 3rl
36.95
38.29
39.36
42.29
41.86
43.20
44.27
47.21
45.56
46.87
47.95
50.88
3p'(3P) 3d2
3p2('D) 3d2
3pZ('S) 3d2
3p2(3PP)4s'
3p*('D) 4s'
3p*('S) 452
Rp'(3P) 4p2
3p2(1D) 4p2
3p'('S) 4p2
42.89
44.99
48.13
50.14
52.36
55.67
57.44
59.78
G3.29
3p2(3P) 3d 4s
3p2('D) 3d 4s
3p2('S) 3d 4s
3p2(3P) 3d 4p
3p2('D) 3d 4p
3p*('S) 3d 4p
3p2(3P) 4s 4p
3p2('D) 4s 41,
3$('S) 4s 4p
45.08
47.23
50.45
48.64
50.83
54.12
53.27
55.53
58.91
3s 3p4(4P) 4h
3s 3p4(2P)4 s
3s 3p'(ZD) 4s
3s 3p4('S) 4s
3s 3p4(4P)4p
3s 3p"'P) 4p
3s 3p4(2D) 4p
3s 3p4('S) 4p
'JLble 1. Energies of doubly excited and core-excited states of Ar3+
~
State
~
~
Energy
(eV)
State
56.99
58.79
60.41
61.32
62.40
63.30
63.51
65.33
67.03
67.94
W.Oi
(8.98
3s 3p3(5S) 4p
8s 3p3(3S)4p
3s 3p3(30) 4p
3s 3p3('D) 4p
3s 3p3(3P) 41,
3s 3p3('P) 4p
3p(2P) 3d2
3p('P) 4s2
3 d 2 P )4P'
3p(2P) 3d 4s
Rp(2P) 3d 4p
3p (2P)
4s 4p
Energy
(eV)
ti7.9!J
G9.85
71.5ti
72.49
73.62
71.55
119.0
135.2
143.4
125.3
131.7
145.6
Energy Levels of Doubly Excited States in Argon
111
References
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R. J. TWEED,
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R. P. MADDEN,
and D. L. EDERER,
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1C. CODLING,
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T. QNDERSEN, 8. M. BENTZEN,
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[13] T. M. LUKE,J. Phys. B: 8 (1975) 1601.
[141 J. LANGLOIS
and J. M. SICHEL,J. Phys. B: 13 (1980) 881, 3109.
[15] Th. M. EL- SHERBINI,Atomkernenergie-Kerntechnik 39 (1981) 63.
[MI C. FROESE
FISCHER,Comput. Phys. Commun. 1 (1969) 151.
[l'i]R. P. MADDEN,
D. L. EDERER,
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[18] H. J. TWEED,F. GELEBART,
and J. PERESSE,
J. Phys. B: 9 (1976) 2643.
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[8]
[3]
[4]
[5]
[6]
[7]
Bei der Redaktion eingegangen am 15. April 1981.
Anschr. d. Verf.: TH. M. EL-SHERBINI
and S. H. ALLAM
Physics Department, Faculty of Science
Cairo University, Cairo, A.R. Egypt
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