Absorption spectrum of singly-ionized aluminum

J. Quant.Spectrosc. Radiat. TransferVol.45, No. 2, pp. 121-125,1991 Printed in Great Britain.All rightsreserved

0022-4073/91 $3.00+0.00 Copyright© 1991PergamonPressplc

A B S O R P T I O N SPECTRUM OF SINGLY-IONIZED ALUMINUM Jos~ MANUEL P. SERR.~O Centro de Fisica Molecular das Universidades de Lisboa, INIC, Complexo I, Av. Rovisco Pais, 1000 Lisboa, Portugal (Received 29 June 1989; receivedfor publication 18 May 1990) Abstract--Energy terms, dipole oscillator strengths and photoionization cross-sections from the ground state have been calculated for singly-ionized aluminum. Autoionization Ip0 state-transition energies and resonance line widths in the continuum are also obtained. The configuration interaction method was used for initial and final states. The ionic orbitals are generated through angular-momentum-dependent, scaled Thomas-Fermi-Dirac-Amaldi potentials.

1. I N T R O D U C T I O N Since singly-ionized aluminum AI II is the first member of the magnesium-like isoeleetronic sequence, we are concerned with the relevance of intereleetronic correlation interactions for alkaline-earth atoms of the type already studied for neutral magnesium. 1 These effects play a very important role. We shall investigate to what extent the ionization state of the atom acts to modify these effects. Also, the A1 II ion has been shown to be important in astrophysical applications 2-6 and lasers. 7 We calculate energies for terms of singly- and doubly-excited states and oscillator strengths for singly-excited states from the ground state through dipole electronic transitions. We present the photoionization cross-sections for the background and autoionizing (3pns)~P °, n = 5-6, resonance structures in the continuum. The corresponding calculated autoionization line widths are also provided. We use the configuration mixing method 8-~° with the electronic orbitals generated by means of Thomas-Fermi-Dirac-Amaldi (TFDA) potentials," which are scaled through angular-momentum-dependent parameters) ~,13 The free states are approximated by a single continuum. All calculations are done in the Russel-Saunders coupling scheme. A summary of the theory is presented in See. 2 since details were provided in a previous paper. ~° 2. T H E O R Y Configuration interaction bound-state wave functions Wmassociated with the eigen-energies Era, are expressed as linear combinations of configuration-state basis functions ¢ , , viz. W,, = ~ a,,,@,.

(1)

n

Here, the @, are Slater determinants, which are made up of one-electron orbitals. These bound-state orbitals are generated by solving the Schr6dinger equation, which contains the V(2t, r) T F D A potentials, where 2t is an angular-momentum-dependent scaling parameter, and r is the electron coordinate. In Eq. (1), we have assumed that contributions from the continuum configuration-state basis functions are negligible. This is often a very good approximation. The a,,, coefficients (eigenveetor values) are found through diagonalization of the total Hamiltonian matrix. Eissner and Nussbaumer's ~3 computer code is used to obtain all of the bound-state wave functions. The WE continuum state wave functions associated with eigen-energies E, for the range of the ionic continuum states, is expressed as WE = ~ croW,, + f c , @ , de, d m

121

(2)

122

Jos# MANUEL P. S~Ra~,o Table 1. Eigenvector contributions to the (3s2)1S ground state and to the (3snp)~P ° excited states of AI II.

1S

3s2

3s4s

3s5s

3s6s

3p2

3s2

0.971

0.037

0.014

0.008

0.219

1pO

3s3p

3s4p

3s5p

3s6p

3p4s

3p4p

3p5p

3p6p

3d2

-0.072 -0.036-0.023 3p5s

3p6s

0.056

0.024

3d5d

-0.012-0.009-0.006

3p3d I

3d4d

3p4d

3p5d

4s2

4p2

-0.010-0.013

3d4p

3d5p

4s4p

4d2 -0.004 4p4d

I

3s3p

0.970 -0.049 -0.020 -0.012

0.0141-0.2001-0.088 -0.052 -0.041

-0.019

-0.019 -0.019

3s4p

0.045

0.978

0.019

0.008 -0.162

-0.038 -0.020 -0.036 ~ -0.071 -0.042 -0.012

-0.006

-0.016 -0.044

3s5p

0.020 -0.055

0.975

0.061 -0.187

-0.061 -0.021 -0.035

-0.005 -0.026 -0.003

-0.003

-0.043

0.000

3s6p

0.012 -0.036 -0.103 -0.976 -0.173

-0.039 -0.026 -0.033

-0.004

0.000

0.035

0.002

0.001

0.002

where the q~c are Slater determinants containing one free orbital and are also generated by solving the Schr6dinger equation with the same V(2~, r) as for the bound-state orbitals. The W,~ are autoionizing state functions. The energy-dependent eigenvector coefficients Cmand c, are obtained by diagonalization of the total Hamiltonian matrix ( 'EIHI E) = E

(3)

(E -- E'),

The method of diagonalization for continuum states is based on earlier work by Rice,s Fano,7 and AItick and Moore.9 Equation (2) has been applied to the case of a single channel. With energy-normalized continuum state wave functions 7rE, the photoionization cross-section a~ in the length formulation is given in megabarns (I0 -'s cm2) by

(4)

~v(E) = 2.69 hv E,I(~,01RI~, )1=' g

Table 2. Transition energies in Rydberg of AI II. Transition

Term

A

B

C

D

3s2 3s2.3s3p -3s4s -3s4p -3s3d -3s5s -3s4d -3s5p -3s6s -3s5d -3s6p -3p4s -3soop

lS 1po 1S 1pO 1D IS 1D 1pO IS 1D 1pO IS Ipo

0.0000 0.5647 0.8694 0.9785 1.0277 1.1173 1.1566 1.1537 1.2194 1.2374 1.2366 1.3724 1.3916

0.0000 0.6243 0.8694 1.0038 1.0247 1.1173 1.1566 1.1643 1.2194 1.2375 1.2433

0.0000 0.5570 0.8628 0.9614 0.9949 1.1066 1.1378 1.1476 1.2106 1.2210 1.2301

0.0000 0.5457 0.8694 0.9748 1.0037

1.3916

1.3846

Autoionizing states -3s3d -3pSs -3p4d -3p6s -3p5d

lpO 1pO 1pO 1pO lpo

1.4982 1.7243 1.6607 1.7243 1.7515

A, calculated energies that best fit the experimental data of lowest terms; B, autoionizing states have not been included in the calculations. All atomic orbitals are as in column A; C, Tayal and Hibbert; ~5 D, experimental energy terms from Bashkin and Stoner) 6

Absorption spectrum o f singly-ionized a l u m i n u m

123

where R = ~ r~ is the dipole length operator for the N atomic electrons, and ~0 stands for the initial state as defined by Eq. (1); hv is the photon energy, which equals the threshold ionization energy I0 plus the kinetic energy k 2 of the ejected electron in Rydberg, and E' means summation over the finite final quantum states. The autoionization line widths FE, are defined (in atomic units) by

FE~=

2n

~E

~"tn

,

(5)

where r 0 is the inter-electronic length. 3. R E S U L T S A N D

DISCUSSION

3. I. Bound-states energy In Table 1, we display the configuration basis sets and also their configuration mixing eigenvector values for the ground state and for the ip0 excited states. In the notation for configuration basis states, we indicate only the two outer electrons and omit the core electron orbitals (ls22s22pr). Inspection of these eigenvector values shows that doubly-excited configuration states (3pnp)lS, n = 3-6, make a very important contribution to the ground state wave function. Doubly-excited (3pns)lP ° configuration basis states also play an important contribution to (comparable, or sometimes greater than) singly-excited basis states in the (3snp)lP ° series. The configuration basis states (3dnd)lS and (3pnd)lp ° are seen to play important roles as well. Thus, in A1 II, as well as for neutral magnesium, I the electronic correlations are very important. The calculated ground state energy is -483.194 a.u., which agrees with the value -483.335 obtained by Clementi and Roetti 14 using the analytic Hartree-Fock method. Transition energies from the ground state are given in Table 2; they show good agreement with measured values, 16 specially for upper states for which agreement within 0.5% is obtained for the first ionization energy. Comparisons of calculated term energies given in Table 2 in columns A and B (B does not include doubly-excited states in the calculations while the electronic orbitals have been frozen from column A) shows the relevance of these configuration basis states in electronic correlation effects.

3.2. Oscillator strengths Table 3 shows the dipole-length orbital integrals p,p = (3sir In?). we see that the absolute value of P3p is many times greater than for upper-state transitions. This result shows the dependence in Table 4 on P3p of the dipole oscillator strengths f(3s, np), n = 3-6, because of configuration mixing. We note that our calculated values agree well with the experimental data for f(3s, 3p) Table 4. Dipole length oscillator strengths for transitions from the ground state. Transition

C

Table 3. Radial integral dipole length values

3s - 3p

3s - 4p

3s - 5p

1.8960

0.0082

0.0001

1.9098

0.0125

1.871

0.0063

D

1.77

E

1.83

F

1.836

G

1.836

H

1.84

0.0064

P.p=(3s[rlnp).

n

3

4

5

6

Pnp

-2.606

0.229

0.053

0.045

A and B are calculations corresponding to ionic potentials of columns A and B of Table 2, respectively; C, Tayal and Hibbert; ~5 D, Victor et al; 17 E, Froese-Fischer and Godefroid; ~8 F, Weiss; 19 G, Zare; ~° H, experimental value from Wiese et al 2~ with accuracy within 10%.

124

Jo$1~ MANUELP. SERRiO

O'13 ( Mbl

o.

I 30

I 40

i 50

I 60

I 70

I 80

I 90 hi) (eV)

100

Fig. 1. Photoionization cross-sections a, in Mbarn without autoionization resonant states (background). ( - - ) our calculations; ( - - - ) calculated by Reilman and Manson.22

w h o s e a c c u r a c y is within 10%. 21 N o p u b l i s h e d d a t a are a v a i l a b l e for t r a n s i t i o n s involving u p p e r states.

3.3. Autoionization probabilities T h e c a l c u l a t e d a u t o i o n i z a t i o n p r o b a b i l i t i e s F(3pns)~P °, n = 5-6, in a t o m i c units for the two r e s o n a n c e s o f the (3pns)~P ° series are F(3p5s) = 4.776 x 103 a n d F(3p6s) = 2.495 x 103.

3.4. Photoionization T h e c a l c u l a t e d b a c k g r o u n d p h o t o i o n i z a t i o n cross-sections ( n o t i n c l u d i n g a u t o i o n i z i n g states in the basis set) are s h o w n in Fig. 1; these are c o m p a r e d with results c a l c u l a t e d by R e i l m a n a n d 2.

0-U (Mb)

O. 0.1

0.3

0.5

k21Ryd)

Fig. 2. Photoionization cross-section a, in Mbarn; k 2 is the photoelectron energy in Rydberg; ( - - ) resonance curve obtained with inclusion of the (3p, ns)lP°, n -- 5-6, autoionizing states; ( - - - ) autoionizing states have not been included in the calculations.

Absorption spectrum of singly-ionizedaluminum

125

Manson n who used the Herman-Skilman 23 ionic potentials. Near the ionization threshold, the results of Reilman and Manson n are some 30% greater than ours. Figure 2 shows the cross-sections near the ionization threshold. 4. C O N C L U S I O N S Electronic correlation effects have been shown to be important in studying configuration mixing for AI II. Comparisons of our calculated energy terms and oscillator strengths with theoretical values of other authors and with available experimental results show good agreement. Our calculated photoionization cross-sections have backgrounds that compare well with calculations of other authors. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

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