IC/94/295

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INNER- AND OUTER-SHELL EXCITATIONIN LITHIUM ISOELECTRONIC SEQUENCE

S.N. Tiwary

and

P. Kumar

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MIRAMARE-TRIESTE

IC/94/295

International Atomic Energy Agencyand

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

INNER- AND OUTER-SHELL EXCITATIONIN LITHIUM ISOELECTRONIC SEQUENCE l

S.N. Tiwary 2

International Centre for Theoretical Physics, Trieste, Italy

and

P. KumarDepartment of Physics, L.S. College BRA Bihar University,

Muzaffarpur, India.

ABSTRACT

The inner-shell excitation Is22s 2Se —* Is2s2 2Se, which leads to autoionization (Augertransition), as well as the outer-shell excitation Is22s 2Se —* Is22p 2P° transitions havebeen investigated in Li, Be+,B2+ and C3+ ions of the lithium isolectronic sequence em-ploying the configuration interaction wave functions for both the initial and final statesinvolved in the transition matrix element. Results are compared with other available the-oretical predictions and experimental observations. Comparison shows that our presenttheoretical results are encouraging.

MIRAMARE - TRIESTE

September 1994

:A part if this work was done while the author was Research Director and Professor,CNRS Laboratory, University of Paris-Sud, Paris and Observatoire de Paris, Meudon,Paris, France.

2Permanent address: Department of Physics, L.S. College BRA Bihar University,Muzaffarpur, India.

1 Introduction

Beyond the ionization limit of atomic systems with several electrons there

generally exist excitation levels corresponding to the excitation not of the

least bound electron but of an inner electron. These core- excited or multiply-

excited levels are degenerate with continuum states of the singly excited

system that is an ionic core in the ground state or low excitation state plus

a free electron. Thus the high lying multiply-excited state can decay easily

by autoionization except for the selection rules which result from parity or

the spin conservation or other reasons.

As a test case for the various effects apparent in the structure of excited sys-

tem, a fairly simple system ought to be chosen. The three- electron systems

provide some neat examples. The lightest alkali-metal atom, i.e., lithium

(Li) and Li-like ions are the best candidates for the study. There has been

growing interest in the inner-shell excitation of atoms and ions from both

experimentalists1'4 and theorists5'25, because inner-shell excitation, in

general, leads to autoionization which plays a very important role in ex-

plaining the structure observed in the total ionization cross section curve

for electron impact.

Accurate absorption optical oscillator strengths for atoms, molecules and

ions in the discrete and continuum regions provide valuable quantitative

information for understanding the high- precision electronic structure of

matter and its interaction with electromagnetic radiation. This information

is of importance in areas of application such as radiation-induced decompo-

sition, biophysics, testing and development of theoretical methods, lithog-

raphy, aeronomy, space chemistry and physics, radiation biology, dosimetry,

health physics, medical physics, radiation protection, astrophysics, atmo-

spheric physics, laser physics, radiation physics, plasma physics, gas dis-

1

i

charge, mass spectroscopy, space research, fusion research, etc. The oscilla-

tor strengths and cross sections provide a sensitive test for atomic structure

calculations. Reliable values of the oscillator strengths and cross sections

are also crucial for the development and evaluation of quantum mechanical

theoretical methods and for the modelling procedures used for various phe-

nomena involving electronic transitions induced energetic radiations.

Consequently, much of current theoretical as well as experimental work is

devoted to evaluate the accurate oscillator strengths of the inner-shell exci-

tation process in atoms and ions. From theoretical point of view, accurate

oscillator strengths can be expected only if accurate wave functions are em-

ployed in the transition matrix elements. If the wave functions are exact,

the length and velocity forms of the oscillator strengths (jx and jv) will

be equal. In general, the correct description of a transition depends on

(a) the form of the dipole operator (b) correlations and (c) choice of basis

orbitals used in the configuration-interaction calculations. Tiwary and his

co — workers5'25 have extensively investigated the excitation energies and

oscillator strengths in heavy alkali-metal atoms.

In continuation of our earlier work5"25 on heavy alkali-metal atoms, in this

paper we have extended our calculation to the lightest alkali-metal atom

(Li). The excitation energies have been calculated for the inner-shell exci-

tation Is22s 2Se -* Is2s2 2Se optically forbidden transition, which leads to

autoionization, in Li, Be+, B2+ and C3+ ions of the lithium isoelectronic

sequence employing the configuration interaction wave functions for both

the initial and final states involved in the transition matrix elements. We

have also performed calculations of the threshold energy and dimensionless

oscillator strengths, of both the length and velocity forms, for the electric

dipole allowed resonance excitation Is2 2s 2S£ —> Is22p 2P° transition in the

ions mentioned above using the configuration interaction wave functions for

the 2Se and 2P° states.

2 Method

We have performed our configuration interaction (CI) calculation using the

computer program CIV3 of Hibbertm in the same way as in our earlier

work5"25. The CI wave function is written as:M

tftZS) = 2>i*.-(a,-£S) (1)tsl

Each of the M single-configuration functions $; is constructed from one-

electron functions, whose orbital and spin momenta are coupled to form the

common total angular- momentum quantum numbers L and S according to

a prescription denoted in (1) by a,.

We express the radial parts of the one-electron functions in analytical form

as a sum of Slater-type orbitals, following Clementi and Roetti27:

The parameters in (2) can be varied to optimize the energy of any state,

subject to the orthonormality conditions

Pni{r)Pn>dT)dr = 6nn, (3)

Once wave functions of form (1) have been determined in this way , they

can be used to obtain the optical oscillator strengths, /L and fy, for tran-

sitions between initial and final states *£' and ty? with energies E' and E?

respectively:

and

where AE = Ef - E' and gl = {2Ll + 1) (25; + 1) is the statistical weight of

the lower state $ ' . For the exact wave functions , equations (4) and (5) give

identical results. For approximate wave functions the two equations may

yield different results. The reliability of either depends on the closeness of

II and fy. The parameters for the basis orbitals used in the present calcu-

lation are shown in Tables 1-4.

3 Results and discussion

Table 5 displays our present non-relativistic theoretical excitation thresh-

old (A£(in eV)) of the lowest lying autoioiiizing level generated due to the

inner-shell optically forbidden excitation Is22s 2Se —> \s2s2 2Se transition

in Li, Z?e+, B2+ and C3 + ions of the lithium isoelectronic sequence obtained

using the configuration-interaction (CI) wave functions for both the initial

and final states involved in the transition matrix elements. Table 5 also

exhibits the excitation energy and the opticaJ oscillator strengths , of both

the length and velocity forms (fi and fv respectively), for the optically al-

lowed excitation Is22s 2Se —f Is22p 2P° resonance transition in exactly the

same ions as mentioned above obtained employing the CI wave functions

for both 2Se and 2P° states. In the case of Be+, we have also reported the

Hartree-Fock (HF) excita.tion energy and oscillator strengths for the reso-

nance transition in order to test the validity of such description. For the

physically meaningful comparison, we have also presented other available

relevant theoretical results and experimental data in the same Table.

Several features of importance emerge from Table 5. In the case of inner-

shell excitation Is22s 2 5 e —> Is2s2 2Se optically forbidden transition, which

leads to autoionization, we have compared our excitation threshold energy

with the experimental results of Rodbro et al28 and theoretical results of

Crandall et a/29. For Li, our theoretical excitation energy is closer to exper-

imental result than the theoretical prediction of Crandall et al, which reflects

the accuracy of our present calculation. For 5 e + , our result is very close

to the theoretical result of Crandall et al. There is no experimental result

available in this case. For B2+, our result is in better agreement with the

experiment than the other theory. Finally, for C'3+, the situation is similar

to Li and B2+. In general, our present theoretical result tends to lie closer

to the experiment compare to other theoretical results in all ions of present

consideration.

In the case of outer-shell excitation Is22s lSf — Is22p iP° optically allowed

transition, we have compared our theoretical excitation threshold energy and

optical oscillator strengths, of both the length {JL) and velocity (fv) forms,

with the experiment of Martin and Wiese30 and Anderson et a/31 as well as

other theoretical results of Weiss32 and Martinson33. For Li, our theoret-

ical excitation energy and oscillator strengths are closer to the experiment

of Martin and Wiese compai'e to the other theoretical results of Martinson.

For Be+, we have shown our Hartree-Fock (1IF) //, and fv as well as CI

JL and fv in the Table. It is clear from the Table that the HF ft and

fv are not in good agreement which indicates that the HF description is

not adequate. The CI ft a i ld fv are in good agreement which reflects that

the correlation plays a crucial role in order to obtain reliable results. For

B2+, other theoretical predictions and experimental observations are not

available. However, there is a good agreement between the length and ve-

locity forms of the oscillator strengths which provides the confidence about

the reliability of our results. For C3 + , our theoretical transition energy and

oscillator strengths are in good agreement with the experimental data of

Martin and Wiese. Thus, we have achieved encouraging results.

4. Conclusion

Our present non-relativistic theoretical investigation of the excitation en-

ergy as well as both the length and velocity forms of the optical oscillator

strengths demonstrates the importance of correlations in the lithium iso-

electronic sequence. The present configuration interaction wave functions

may be of use for accurate calculation of the cross sections, which are very

sensitive to the wave functions, by photon and election impact employing

the R-matrix method. We hope that this work will stimulate experimental

and other elaborate theoretical investigations.

A ck now le dge me nt s

The author would like to thank Professor Abdus Salam, the International

Atomic Energy Agency and UNESCO for hospitality at the International

Centre for Theoretical Physics, Trieste, Italy, and the Swedish Agency for

Research Cooperation with Developing Countries (SAREC) for financial

support during his visit at the ICTP under the Associateship scheme. He

is also grateful to Profs. Connerade, Faisal, Persico, Barut and Denardo for

stimulating discussions and encouragement. He would like to express his

thanks to BRAB University, India, for leave and also the UGC, New Delhi,

for Major Research Project.

6

References

1. B. Peart, J. G. Stevenson and K. Dolder, J. Phys. B: 6, 146 (1973).

2. B. Peart and K. Dolder, J. Phys. B: 8, 56 (1975).

3. K. J. Nygaard, Phys. Rev. A 11, 1475 (1975).

4. V. Pejcev and K. J. Ross, J. Phys. B: 10, 291 (1977).

.5. S. N. Tiwary, Cheni. Phys. Lett. 93, -17 (1982).

6. A. Hibbert, A. E. Kingston and S. N. Tiwary, J. Phys. B: 15, L643 (1982).

7. S. N. Tiwary, Chem. Phys. Lett. 96. 333 (1983).

8. S. N. Tiwary, A. E. Kingston and A. Hibbert, J. Phys. B: 16, 2457 (1983).

9. S. N. Tiwary, Astrophys. Journal 269, 803 (1983).

10. S. N. Tiwary, Astrophys. Journal 272, 781 (1983).

11. S. N. Tiwary, Proc. Ind. Acad. Sci. 93, 1345 (1984).

12. S. N. Tiwary, Invited Talk, Book-World Book Publisher, New York, 1987.

13. A. E. Kingston, A. Hibbert and S. N. Tiwary, J. Phys. B: 20, 3907 (1987).

14. S. N. Tiwary,A. P. Singh, D. D. Singh and R. J. Sharma, Can. J. Phys. 66, 405(1988).

15. S. N. Tiwary, Praman-J. Phys. 35, 89 (1990).

16. S. N. Tiwary, Fizika, 22, 577 (1990).

17. S. N. Tiwary and P. Kumar, Acta Phys. Pol. 80, 23 (1991).

18. S. N. Tiwary, Int. J. Theor. Phys. 30, 825 (1991).

19. S. N. Tiwary, Int. J. Theor. Phys. 32, 2047 (1993).

20. S. N. Tiwary, Fizika 23, 27 (1991).

21. S. N. Tiwary, Nuovo Cimento D, 13. 1073 (1991).

22. S. N. Tiwary and D. D. Singh, Nuovo Cimento D 14, 739 (1992).

23. S. N. Tiwary, M. Kumar, D. D. Singh and P. Kumar, Nuovo Cimento D 15, 77 (1993).

24. S. N. Tiwary, P. Kandpal and A. Kumar, Nuovo Cimento D 15, 1181 (1993).

25. S. N. Tiwary and P. Kandpal, Nuovo Cimento D 16, 339 (1994).

26. A. Hibbert, Comput. Phys. Commun. 9, 141 (1975).

27. E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974).

28. M. Rodbro, R. Bruch and P. Bisgaard, J. Phys. B: 12, 2413 (1979).

29. D. H. Crandall, R. A. Phaneuf, D. C. Gregory, A. M. Howald,

D. W. Mueller, T. J. Morgan, D. H. Dunn, D. C. Griffin and R. J. W. Henry,

Phys. Rev. A 34, 1757 (1986).

30. G. A. Martin and W. L. Wiese, J. Phys. Chem. Ref. Data 5, 537 (1976).

31. T. Anderson, K. A. Jessen and G. Soren.sen, Phys, Rev. 188, 76 (1969).

32. A. W. Weiss, Astrophys. J. 138, 1262 (1962).

33- I. Martinson, Private Communication, 1990.

Table 1

Parameters for the bound orbitals used in the. present calculation for

Li. Each orbital is a sum of Slater-type orbitals.

Orbital

I s

2s

3s

4s

2 p

3 p

4 p

3d

4 d

4f

Coefficient

6.9868746

2.2795897

-0.0003546

0.0358937

0.0003165

-1.1297426

-0.3198818

0.2463431

-0.3749575

0.2323533

0.6558707

-0.2119038

0.0248425

2.5948057

-1.9430932

0.3337925

-0.0000702

0.2327960

0.1358806

-0.0106379

0.0927230

-0.0169656

0.0002743

0.0090078

0.0069834

-0.0004127

0.0003606

Power of r

1

1

2

2

2

1

1

2

2

2

1

2

3

1

2

3

2

2

3

2

3

4

3

3

4

4

Exponent

2.4758301

4.6894197

0.7650000

1.7530003

0.6271100

2.4758301

4.6894197

0.7650000

1.7530003

0.6271100

2.2952986

0.5243351

0.3972201

2.0023003

1.0011997

0.8421000

0.2103000

0.5269900

0.4608071

0.3195900

0.3503345

0.3173317

0.2298303

0.3332464

0.2748408

0.2462557

0.2789979

- m «>• — — « ••<—-

Table 2

Parameters for the bound orbitals used in the

Be+. Each orbital is a sum of Slater-

present calculation for

type orbitals.

Orbital

I s

2s

3s

4s

2p

3p

4p

3d

4d

4f

Coefficient

12.0709343

2.6127710

0.0009156

0.1483740

-2.6770458

-0.5193034

1.9510384

-1.7176704

1.5172777

-0.9699644

0.2801738

0.9590698

-0.6502516

0.2950754

-0.0114662

1.3615055

0.0544544

-0.0093094

0.9339063

-0.1556458

0.0017858

0.1021642

0.0804749

-0.0083149

0.0063983

Power of r

1

1

2

2

1

1

2

2

1

2

3

1

2

3

4

2

2

3

2

3

4

3

3

4

4

Exponent

3.4977102

6.5022697

1.1838303

2.6277103

3.4977102

6.5022697

1.1838303

2.0277103

2.9252968

0.8292635

0.7417576

2.8583193

0.6659091

0.6620640

0,5133845

1.0681200

0.4608071

0.3195900

0.9371601

0.6317720

0.4114543

0.6669648

0.5688674

0.4888195

0.5286312

10

Table 3

Parameters for the bound orbitals used in the

B2+. Each orbital is a sum of Slater-

present calculation for

type orbitals.

Orbital

Is

2s

3s

4s

2p

3p

4p

3d

4d

4f

Coefficient

17.9168091

2.9601736

0.0027155

0.2605826

-4.5712872

-0.6459638

4.8968992

-4.3977871

2.5246229

-2.5788403

1.1002264

1.5875397

-1.8374224

0.9574825

-0.0591679

3.7150707

2.0801020

-0.6748603

1.3946943

-0.6423528

0.0465272

0.4233974

0.3298890

-0.0460600

0.0365160

Power of r

1

1

2

2

1

1

2

2

I

2

3

1

2

3

4

2

2

3

2

3

4

3

3

4

4

Exponent

4.5025797

8.2783804

1.6829901

3.4060802

4.5025797

8.2783804

1.6829901

3.4060802

3.4766245

1.1671171

1.0738478

3.2467966

1.0867414

0.9602834

0.7411374

1.5958757

1.2638054

1.0059681

1.1421089

0.9053420

0.7078838

1.0011911

0.8702564

0.7242978

0.7784781

11

Table 4

Parameters for the bound orbitals used in the

C3+ . Each orbital is a sum of Slater-

present calculation for

type orbitals.

Orbital

I s

2s

3s

4s

2 p

3 p

4 p

3 d

4d

4f

Coefficient

24.3502197

3.4065514

0.0067115

0.3366341

-6.7230558

-0.7653501

9.6030931

-8.6993093

3.6779690

-5.3328676

2.1234426

2.4205656

-3.9582319

3.0101938

-0.1761737

7.492S656

4.3530245

-1.3748379

2.9058151

-2.0661373

0.1596443

1.1590681

0.8983302

-0.2410561

0.1269981

Power of r

1

1

2

2

1

1

2

2

1

2

3

1

2

3

4

2

2

3

2

3

4

3

3

4

4

Exponent

5.4931202

9.9242697

2.1782598

4.1925802

5.4931202

9.9242697

2.1782598

4.1925802

4.0089560

i.6859655

1.3627119

3.7515182

3.3366337

1.2961988

0.9572796

2.1128635

1.8422756

1.2810955

1.4071512

1.2357731

0.9373122

1.3349876

1.0674610

0.9946850

0.9946850

12

Table 5.

Excitation energy (A£(eV)) and optical oscillator strengths for the

transitions Is22s 25e — \s2s2 2Se and Is22s -Se — lsa2p 2f° in Li

isoelectronic sequence

Systems

Present results

AE I /z. | fv

Other results

AE I fL | fv

Experimental results

AE I f

Li

-* Is2s2

-+ Is22p

56.021

1.844

0.0

0.763

0.0

0.771

56.954

1.873

0.02

0.768

0.0

0.791

55.899

1.849 0.753

Is22s -+ U2s2115.213

3.9784.021*

0.0

0.510

0.511'

0.0

0.5230.549*

115.2

3.995 0.505 0.521 3.964 0.52

B2+

U22s 193.152

5.996

0.0

0.3G2

0.0

0.374

194.1 192.8

C 3+

Is22s

U22p

293.401

8.011

0.0

0.283

0.0

0.291

293.0 291.7

7.997 0.286

Hartree-Fock (HF)

. « * - » * . , . . * • " « * • • ; * - ' ' * • •