Configuration-interaction and relativistic effects

PHYSICAL REVIEW A 68, 022714 共2003兲

Photoionization of the ground and first two metastable levels for Sc III: Configuration-interaction and relativistic effects Jiaolong Zeng,1,2 Gang Zhao,1 and Jianmin Yuan2 1

National Astronomical Observatories, Chinese Academy of Sciences, Datun Road 20A, Chaoyang, Beijing 100012, People’s Republic of China 2 Department of Applied Physics, National University of Defense Technology, Changsha 410073, People’s Republic of China 共Received 21 April 2003; published 28 August 2003兲 The photoionization cross section of the ground ( 关 Ne兴 3s 2 3p 6 3d 2 D 3/2) and first two metastable ( 关 Ne兴 3s 2 3p 6 3d 2 D 5/2 and 关 Ne兴 3s 2 3p 6 4s 2 S 1/2) levels of potassiumlike Sc III have been studied using fully relativistic R-matrix method. To show the effects of configuration interaction on the cross section, we first perform a nonrelativistic R-matrix calculation. The detailed structures and resonances of the cross section obtained by the fully relativistic method in the region of 3p→3d and 3p→4s resonances are described and analyzed in some detail. The positions, autoionization widths, and oscillator strengths of some prominent resonances are determined. The calculated cross section is compared with a recent experiment 关S. Schippers et al., Phys. Rev. Lett. 89, 193002 共2002兲兴. The resonance energies and intensities predicted by the fully relativistic calculations agree well with the experiment, especially for the 3p→4s resonances in the region of lower photon energy, while it is not so satisfactory for the nonrelativistic calculations. Our results show that not only adequate configuration interaction, particularly the 3p 3 →3d 3 core-valence correlations, but also relativistic effects should be carefully included in the calculation to obtain a good agreement with the experiment. DOI: 10.1103/PhysRevA.68.022714

PACS number共s兲: 32.80.Fb, 32.80.Hd, 32.80.Dz, 32.70.Cs

I. INTRODUCTION

In the past two decades the photoionization 共PI兲 of atoms and ions had been of increased interest. This is mainly due to the urgent need for the radiative data for astrophysical applications such as in the opacity project 关1,2兴, the iron project 关3兴, fusion research, and plasma diagnostics. The fundamental interest in the PI processes may also be a reason of this increase. Recently, some high-resolution quantitative measurements of the PI 关4 – 6兴 provide accurate data on ionic structure, and guidance to the development of theoretical models. To explain these high-resolution experiments, both elaborate configuration interaction 共CI兲 and relativistic effects should be considered. During the past few years the PI process of free scandium atoms has attracted special attention of both experimentalists 关7–11兴 and theorists 关12–15兴. This may partly be due to the existence of the 3p→3d giant resonances of this atom. On the other hand, scandium is the simplest atom to have only one 3d electron. This makes detailed studies of it particularly attractive both experimentally and theoretically. However, doubly ionized scandium, which also exists in the 3p→3d giant resonances, is much less investigated, either experimentally or theoretically. To the best of our knowledge, there is only one theoretical paper 关12兴 and only one experimental paper 关16兴 which deal with the PI of the doubly ionized scandium. Altun and Manson 关12兴 calculated the PI cross section of the ground state for Sc III using the multiconfiguration Hartree-Fock augmented many-body perturbation theory plus channel coupling method. LS coupling is assumed in their paper, and thus relativistic effects were neglected completely. Furthermore, CI was treated only approximately in their calculation. There is a large deviation between the theory 关12兴 and experiment 关16兴. Schippers 1050-2947/2003/68共2兲/022714共7兲/$20.00

et al. 关16兴 measured the PI cross sections for the ground and for two metastable levels of Sc III. In addition to the ground level ( 关 Ne兴 )3s 2 3 p 6 3d 2 D 3/2 , two metastable levels 3s 2 3 p 6 3d 2 D 5/2 and 3s 2 3 p 6 4s 2 S 1/2 also exist in their experiment. They determined the fine-structure composition of the ion beams by comparing Sc III PI cross sections with the experimental Sc IV photorecombination cross section. The experimental energy resolution is high (⬃25 meV) and the fine-structure splitting of the autoionization states are clearly visible. The theoretical calculation carried out by Altun and Manson 关12兴 does not reproduce the experiment well, either for the resonance positions or the intensities. There is obviously a need to carry out a more detailed study which includes both elaborate CI and fully relativistic effects in the calculation. The present paper considers the PI cross sections for the ground and two lowest metastable levels in the region of 3 p→3d and 3p→4s in an attempt to interpret and enlighten the nature of properties of the recent experimental results 关16兴. To show the CI and relativistic effects on the cross section, we first carry out two sets of nonrelativistic R-matrix calculations which include different CI with up to 3p 2 →3d 2 core-valence electron correlations in one and further 3p 3 →3d 3 correlations in another. Then, we perform detailed fully relativistic R-matrix calculations. In this way, one can see clearly the effects of elaborate CI and relativistic effects on the cross sections. II. THEORETICAL METHODS

The R-matrix method for electron-atom and photon-atom interactions has been discussed in great detail by Burke et al. 关17兴. The present calculations have been carried out by using the latest Belfast atomic R-matrix codes 共RMATRIX1兲 关18兴 and the fully relativistic R-matrix code DARC 关19,20兴. The

68 022714-1

©2003 The American Physical Society

PHYSICAL REVIEW A 68, 022714 共2003兲

ZENG, ZHAO, AND YUAN

R-matrix method is very effective in considering the resonance structures 关21–25兴, which is suitable for the present case. In an R-matrix calculation, the wave function of the N⫹1 electron system is given the form

TABLE I. Calculated nonrelativistic 共cases A and B兲 and experimental energy levels 关28兴 共in rydbergs兲 for the target Sc IV ion relative to the ground state. Configuration

⌿ k 共 X 1 •••X N⫹1 兲 ⫽Aˆ

2

兺i j c i jk ⌽ i共 X 1 •••X N rˆN⫹1 ␴ N⫹1 兲 u i j 共 r N⫹1 兲

⫹

兺j d jk ␾ j 共 X 1 •••X N⫹1 兲 ,

共1兲

where Aˆ is the antisymmetrization operator to take the exchange effect between the target electrons and the free electron into account. X i stands for the spatial (ri ) and the spin ( ␴ i ) coordinates of the ith electron. The functions u i j (r) under the first sum construct the basis sets for the continuum wave functions of the free electron, and ⌽ i is the coupling between the target wave function of a specific term S i L i ␲ i in the nonrelativistic calculation or level J i ␲ i in the relativistic calculation and the angular and spin part of the free electron. The correlation functions ␾ j in the second sum are constructed by the square-integrable orbitals to account for the correlation effects not adequately considered because of the cutoff in the first sum. In nonrelativistic calculations, the square-integrable orbitals are cast as linear combinations of Slater-type orbitals, P nl ⫽

兺j C jnl r I

jnl

exp共 ⫺ ␰ jnl r 兲 .

共2兲

The parameters ␰ jnl and coefficients C jnl are determined by a variational optimization on the energy of a particular state, while the powers of r(I jnl ) remain fixed. For the present calculation, we included eight real and one pseudo-orbitals (1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, and 4d). The pertinent parameters ␰ jnl and coefficients C jnl for all orbitals are obtained by using the CIV3 computer code 关26兴 according to the following rules. The 1s, 2s, 2p, 3s, and 3p orbitals are taken from those of the Hartree-Fock orbitals given by Clementi and Roetti 关27兴 for the Sc IV ground state 3s 2 3 p 6 1 S, and the 3d, 4s, and 4p orbitals are obtained by optimizing on the 3s 2 3p 5 3d 1 P 0 , 3s 2 3p 5 4s 1 P 0 , and 3s 2 3p 5 4 p 3 P states, respectively. The pseudo-orbital 4d is obtained by optimizing on the 3s 2 3p 5 3d 1 P o state. To show the effects of the CI on the PI cross section, two sets of calculations are carried out. In the first case 共case A), valence electron and 3p 2 →3d 2 core-valence electron correlations are included in the wave function expansion of the target and N⫹1 electron systems. In the second one 共case B), in addition to the CI included in case A, further 3p 3 →3d 3 core-valence electron correlations are also included. Explicitly, we take account of the CI among the single- and double-electron excitations among the included orbitals from the basic configuration 3s 2 3p 6 in the expansion of the target wave function for case A. For case B, further CI among the single-electron excita-

6

3s 3p 2s 2 3p 5 3d 2s 2 3p 5 3d 2s 2 3p 5 3d 2s 2 3p 5 3d 2s 2 3p 5 3d 2s 2 3p 5 4s 2s 2 3p 5 4s 2s 2 3p 5 3d 2s 2 3p 5 4p 2s 2 3p 5 4p 2s 2 3p 5 4p 2s 2 3p 5 4p 2s 2 3p 5 4p 2s3p 6 3d 2s3p 6 3d

Term 1

S P0 3 0 F 1 0 D 3 0 D 1 0 F 3 0 P 1 0 P 1 0 P 3 S 3 D 3 P 1 P 1 D 3 D 1 D 3

Expt. 关28兴

Case A

Case B

0.0000 2.1972 2.2947 2.4370 2.4494 2.4700 3.0436 3.0754 3.1439 3.3875 3.4501 3.4933 3.4950 3.5053 3.9177 4.0282

0.0000 2.3600 2.4889 2.6329 2.6406 2.6572 3.2464 3.2790 3.4705 3.5897 3.6574 3.7025 3.6976 3.6846 4.0809 4.1907

0.0000 2.2008 2.3208 2.4807 2.4853 2.5011 3.0569 3.0840 3.2022 3.4018 3.4733 3.5142 3.5164 3.7169 3.9447 4.0708

tions from the basic configuration 3s 2 3 p 4 3d 2 is included as well. The appropriate R-matrix wave-function expansion was performed by including 17LS states of Sc IV which are listed in Table I, in which we also give the present calculated 共cases A and B) and experimental energy levels 关28兴 of these states relative to the ground state. It can easily be seen that the calculated energy levels in case B agree very well with the experiment, while those in case A do not. The difference is obviously caused by the different CI included in cases A and B. As mentioned above, only up to 3p 2 →3d 2 electron correlations are taken into account in case A, whereas further 3 p 3 →3d 3 correlations are considered in case B. For the calculations of PI cross section in both cases A and B, R-matrix boundary was chosen to be 12.5 a.u. to ensure that the wave function is completely wrapped within the R-matrix sphere. For each angular momentum, the continuum orbitals are expressed as a linear combination of 50 numerical basis functions. In relativistic R-matrix calculations, the orbitals are obtained from multiconfiguration Dirac-Fock code GRASP 关29兴 by using an extended average level method. The Sc IV target wave functions are described by a closed core 1s 2 2s 2 2p 6 and ten one-electron orbitals: 3s 1/2 , 3 p 1/2 , 3 p 3/2 , 3d 3/2 , 3d 5/2 , 4s 1/2 , 4p 1/2 , 4p 3/2 , 4d 3/2 , and 4d 5/2 . In performing the relativistic R-matrix calculation, 17 lowest levels of Sc IV are included in the expansion of wave function and the theoretical energy levels are shown in Table II along with the experimental values 关28兴. With the experience of the nonrelativistic calculations shown above, CI is included up to 3 p 3 →3d 3 core-valence electron correlations. R-matrix boundary was chosen to be the same as in the nonrelativistic calculations. For each angular momentum, the continuum orbitals are expressed as a linear combination of 25 numerical basis functions.

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PHOTOIONIZATION OF THE GROUND AND FIRST TWO . . .

PHYSICAL REVIEW A 68, 022714 共2003兲

TABLE II. Calculated relativistic and experimental energy levels 关28兴 共in rydbergs兲 for the target Sc IV ion relative to the ground state.

resonances. The continuum cross section is only 2.4 Mb near the threshold, but the cross section can be more than several hundred Mb in the region of resonances. From the observation of Fig. 1, one can see that the positions of the resonances are very sensitive to the CI included in the calculation. Taking into account of the core-valence electron correlations of 3p 3 →3d 3 共case B) can effectively lower the positions of the resonances. Take the giant 3p →3d resonance as an example. Case A predicts the position of this resonance at 40.75 eV, whereas at 38.52 eV for case B, which shifts to lower photon energy by 2.23 eV. Even so, case B predicts the position of this resonance higher than the experiment by more than 1 eV. The experimental result obtained by Schippers et al. 关16兴 is shown in Fig. 1 with a dotted line. In their experiment, in addition to the ground level 3s 2 3 p 6 3d 2 D 3/2 共accounting for 20.7%兲, two metastable levels 3s 2 3 p 6 3d 2 D 5/2 共54.6%兲 and 3s 2 3p 6 4s 2 S 1/2 共24.7%兲 also existed in the ion beam. Even though the significant deviation from the statistical populations of the finestructure levels and the contribution from the 3s 2 3p 6 4s 2 S 1/2 level are not taken into account in Fig. 1, there is large difference between the theory 共for both cases兲 and experiment. The experiment observed more structures than the theory. The structures are mainly caused by the relativistic splitting, as will be demonstrated in the following section. The width of the giant resonance predicted by the theory is smaller than the observed, the main reason might be the neglect of relativistic splitting of this resonance. Nevertheless, the strong resonance around 41 eV is caused by 3s 2 3p 6 4s 2 S 1/2 level, as can later be seen from the relativistic result. The only nonrelativistic theoretical result obtained by Altun and Manson 关12兴 predicts the giant 3p→3d resonance at 40.5 eV, which is very close to our result of 40.75 eV obtained by theory 共case A). There is further optimization space for the CI wave function used in our theory 共case A) and that obtained by Altun and Manson 关12兴. To obtain correct result of PI cross section of Sc III, one should at least include up to 3p 3 →3d 3 core-valence electron correlations. Relativistic effects are also important to predict the correct structures of resonances. In the following section, we turn to the fully relativistic R-matrix calculations which take up to 3p 3 →3d 3 correlations into account.

Configuration 2

6

Term 1

3s 3p 2s 2 3p 5 3d

3

S P0

2s 2 3p 5 3d

3

F0

2s 2 3p 5 3d 2s 2 3p 5 3d

1

D0 D0

2s 2 3p 5 3d 2s 2 3p 5 4s

2s 2 3p 5 4s 2s 2 3p 5 3d

3

1 3

1 1

F0 P0

P0 P0

J

Expt. 关28兴

Theory

0 0 1 2 4 3 2 2 3 1 2 3 2 1 0 1 1

0.0000 2.1845 2.1907 2.2036 2.2846 2.2974 2.3092 2.4370 2.4425 2.4552 2.4555 2.4700 3.0353 3.0473 3.0741 3.0754 3.1439

0.0000 2.2058 2.2125 2.2260 2.3283 2.3413 2.3533 2.4984 2.5005 2.5108 2.5133 2.5273 3.0690 3.0865 3.1097 3.1296 3.2585

III. RESULTS AND DISCUSSIONS A. Nonrelativistic R-matrix calculations

Figure 1 shows the PI cross section of the ground level 3s 2 3p 6 3d 2 D 3/2 of Sc III in the region of 3p→3d and 3p →4s resonances for cases A and B. The cross section near the ionization threshold region obtained in case B is shown in the inset. Good agreement is obtained between the length and velocity forms, therefore, only the length form is given throughout the paper. It can easily be seen that the continuum PI cross section is very small compared with that of the

B. Relativistic R-matrix calculations

FIG. 1. The photoionization cross section of the ground state 3s 2 2p 6 3d 2 D of Sc III in the region of giant 3p-3d resonance obtained by the nonrelativistic R-matrix method for cases A 共dashed line兲 and B 共solid line兲. The dotted line represents the experimental result obtained by Schippers et al. 关16兴. The cross section near the ionization threshold region obtained in case B is shown in the inset.

Figure 2 shows the PI cross section of the ground level 3s 2 3 p 6 3d 2 D 3/2 , with Figs. 2共a兲–共c兲 referring to the J ⫽5/2, 3/2, and 1/2 partial and Fig. 2共d兲 to the total cross section. The total cross section near the ionization threshold is shown in the inset in Fig. 1共d兲. The calculated ionization potential 共IP兲 is 24.518 eV, in rather good agreement with the experimental value 24.757 eV 关28兴. It can be seen that the complex resonances dominate the cross sections and J⫽5/2 partial cross section contributes most to the total cross section in the region of the shown photon energy. The continuum 共direct兲 cross section is very small compared to the indirect cross section around the resonances. The line shapes for the resonances above the giant resonance indicate strong interference between the direct and indirect PI channels.

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ZENG, ZHAO, AND YUAN

where ␴ a and ␴ b stand for constants related to the background of the resonance cross sections, respectively, ␧ ⫽2(E⫺E 0 )/⌫, with E being the photon energy, E 0 the resonance energy, ⌫ the resonance width, and the parameters q describes the shape of the resonance. For 3d 2 ( 3 F) 2 F 5/2 state, the resonance position and width are, respectively, 37.91 eV and 910 meV. This resonance has strong interference with nearby resonances, leading to very narrow widths for nearby resonances. On the left of this giant resonance, there are three resonances which are identified as 3d( 1 D)4s 2 D 5/2 , 3d( 1 F)4s 2 F 5/2 , and 3d( 3 D)4s 2 D 5/2 . Their widths are determined to be 0.54, 0.40, and 0.27 meV, respectively. On the right, the resonance located at 40.73 eV should be identified to be 3d 2 ( 3 F) 2 D 5/2 , with the width of 0.36 meV. It is clear that the widths of the four resonances in the vicinity of the giant resonance are more than four orders of magnitude smaller than 910 meV. To describe correctly and clearly these narrow resonances, a very fine energy grid (10⫺6 Ry) is taken in the present work. The positions and widths of the identified resonances are given in Table III. The oscillator strengths of the resonance transitions can be obtained from the PI cross sections,

f 共 J i ␲ i →J j ␲ j 兲 ⫽

FIG. 2. The partial and total photoionization cross section of the ground level 3s 2 3p 6 3d 2 D 3/2 of Sc III obtained by relativistic R-matrix code DARC: 共a兲 J⫽5/2 partial, 共b兲 J⫽3/2 partial, 共c兲 J ⫽1/2 partial, and 共d兲 the total. The total cross section near the ionization threshold is shown in the inset in 共d兲. In the designation of the resonances, 1s 2 2s 2 2p 6 3s 2 3p 5 has been omitted for convenience.

We have identified most of the prominent resonances using the fully relativistic MCDF code GRASP 关29兴. The levels of the relativistic configuration have been replaced by nonrelativistic configuration with term and level splitting. For example, the giant resonance near 38 eV is 2 2 3 3p 1/2 3p 3/2 (J⫽3)3d 3/2(J⫽2)3d 5/2(J⫽5/2) caused by 3s 1/2 autoionization state in relativistic designation. Actually, this 2 2 3 3p 1/2 3p 3/2 (J⫽3)3d 3/2(J level is strongly mixed with 3s 1/2 2 2 3 ⫽3)3d 5/2(J⫽5/2) and 3s 1/23p 1/23p 3/2(J⫽3)3d 3/2(J ⫽1)3d 5/2(J⫽5/2) levels. Using the label of nonrelativistic configuration with term and level splitting, it corresponds to ( 关 Ne兴 3s 2 3p 5 )3d 2 ( 3 F) 2 F 5/2 autoionization state. In the designation of the resonances shown in Fig. 2, 关 Ne兴 3s 2 3 p 5 has been omitted for convenience. It can easily be seen that the most striking feature in the total PI cross section is contributed by the this resonance. The position and width can be obtained by fitting the resonance a Fano profile,

␴ ⫽ ␴ a⫹ ␴ b

共 q⫹␧ 兲 2

1⫹␧ 2

,

共3兲

冉

1 4 ␲ ␣ a 20 2

冊冕

⌬E r

␴ PI 共 ⑀ ;J i ␲ i →J j ␲ j 兲 d ⑀ , 共4兲

where ␣ and a 0 are the fine-structure constant and the Bohr radius, J i ␲ i and J j ␲ j are total angular momenta and parities of the initial bound level and the final continuum state, governed by the dipole selection rules. The final result for the transition 3s 2 3 p 6 3d 2 D 3/2→3d 2 ( 3 F) 2 F 5/2 is 1.150, which shows strong photoabsorption. The resonance oscillator strengths of some other resonances are determined and given in Table III as well. Figures 3 and 4 show the PI cross sections of the two metastable levels 3s 2 3 p 6 3d 2 D 5/2 and 3s 2 3 p 6 4s 2 S 1/2 . Figures 3共a兲–共c兲 refer to the J⫽7/2, 5/2, and 3/2 partial cross sections and Fig. 3共d兲 to the total cross sections. Figures 4共a兲 and 4共b兲 refer to the J⫽3/2 and 1/2 partial cross sections and Fig. 4共c兲 to the total cross sections. The calculated IP is 24.490 and 21.365 eV, respectively, for the two metastable levels, in rather good agreement with corresponding experimental values of 24.732 and 21.590 eV. Complex resonance structures dominate the PI cross sections while the direct cross section is very small, which is similar to that shown in Fig. 2. In Fig. 3, the J⫽7/2 partial cross section contributes most to the total cross section. The giant 3p→3d resonance is caused by 3s 2 3 p 6 3d 2 D 5/2→3d 2 ( 3 F) 2 F 7/2 transition. The width of this resonance is determined to be 920 meV, which is nearly equal to that of the 3s 2 3 p 6 3d 2 D 3/2→3d 2 ( 3 F) 2 F 5/2 resonance shown in Fig. 2共a兲 共910 meV兲. On the left of this giant resonance, a resonance should be identified as 3d( 1 F)4s 2 F 7/2 with the width of 0.4 meV. On the right 共photon energy less than 43 eV兲, no resonance exists. This is

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PHOTOIONIZATION OF THE GROUND AND FIRST TWO . . . TABLE III. Resonance positions 共in eV兲, widths 共in meV兲, and oscillator strengths f for the (1s 2 2s 2 2p 6 3s 2 3p 5 )3d 2 and (1s 2 2s 2 2p 6 3s 2 3p 5 )3d4s resonances. In the designation of the resonances, 1s 2 2s 2 2p 6 3s 2 3p 6 has been omitted in the case of 2 D 3/2 , 2 D 5/2 , and 2 S 1/2 initial states and 1s 2 2s 2 2p 6 3s 2 3p 5 in the case of doubly excited intermediate states. Lower level 2

D 3/2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 3/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 D 5/2 2 S 1/2 2 S 1/2 2 S 1/2 2 S 1/2 2 S 1/2 2

Upper level

Energy

Widths

f

3d 2 ( 1 D) 2 F 5/2 3d( 3 F)4s 2 F 5/2 3d( 1 D)4s 2 D 5/2 3d( 1 F)4s 2 F 5/2 3d( 3 D)4s 2 D 5/2 3d 2 ( 3 F) 2 F 5/2 3d 2 ( 3 F) 2 D 5/2 3d( 3 P)4s 2 D 5/2 3d( 3 P)4s 2 F 5/2 3d( 3 P)4s 2 P 3/2 3d 2 ( 1 D) 2 D 3/2 3d( 3 D)4s 2 D 3/2 3d( 1 D)4s 2 D 3/2 3d 2 ( 3 P) 2 P 3/2 3d 2 ( 3 F) 2 D 3/2 3d( 3 P)4d 2 D 3/2 3d( 3 P)4s 2 P 1/2 3d 2 ( 3 P) 2 P 1/2 3d 2 ( 1 D) 2 F 7/2 3d 2 ( 1 G) 2 F 7/2 3d( 3 F)4s 2 F 7/2 3d( 1 F)4s 2 F 7/2 3d 2 ( 3 F) 2 F 7/2 3d( 3 F)4s 2 F 5/2 3d( 1 D)4s 2 D 5/2 3d( 1 F)4s 2 F 5/2 3d 2 ( 3 F) 2 F 5/2 3d 2 ( 3 F) 2 D 5/2 3d( 3 P)4s 2 P 3/2 3d 2 ( 3 F) 2 D 3/2 3d 2 ( 3 P) 2 P 3/2 4s 2 ( 1 S) 2 P 3/2 3d( 1 P)4s 2 P 3/2 4s 2 ( 1 S) 2 P 1/2 3d( 1 P)4s 2 P 1/2

32.62 34.69 36.07 36.31 36.59 37.91 40.73 43.42 43.72 33.06 36.11 36.31 36.59 40.55 40.76 43.23 32.87 40.47 32.23 33.85 34.45 36.56 38.20 34.65 36.03 36.28 37.89 40.91 33.03 40.72 37.73 39.29 42.5 39.89 42.63

117 51 0.54 0.40 0.27 910 0.36 12.0 119 45 0.27 0.31 0.35 0.68 0.31 0.27 46 37 117 0.34 51 0.40 920 51 0.54 0.40 910 0.36 44 3.22 0.68 4.0 143 4.0 143

0.025 0.248 0.017 0.030 0.006 1.150 0.102 0.042 0.029 0.012 0.007 0.061 0.056 1.063 0.015 0.073 0.060 0.542 0.004 0.018 0.202 0.088 1.108 0.012 0.033 0.148 0.046 1.101 0.001 0.070 0.027 0.232 2.276 0.108 1.233

different from the situation of the J⫽5/2 partial cross section from the initial level 3s 2 3p 6 3d 2 D 3/2 . It is understandable because the resonances near the giant resonance of 3d 2 ( 3 F) 2 F 5/2 in Fig. 2 are 3d( 1 D)4s 2 D 5/2 and 3d 2 ( 3 F) 2 D 5/2 levels. There is no J⫽7/2 level for the 2 D term. On the other hand, the resonance transition 3s 2 3p 6 3d 2 D 5/2 →3d 2 ( 3 F) 2 F 5/2 is very weak, almost invisible in Fig. 3共b兲. This is quite different from the case of the giant resonance shown in Fig. 2共a兲. Near the broad weak resonance, there are a few resonances with very narrow widths, which is similar to the case shown in Fig. 2共a兲 in the vicinity of the giant resonance. The resonance structures of the cross section from the

FIG. 3. The same as Fig. 2, but for the metastable level 3s 2 3p 6 3d 2 D 5/2 : 共a兲 J⫽7/2 partial, 共b兲 J⫽5/2 partial, 共c兲 J⫽3/2 partial, and 共d兲 the total cross section.

metastable level 3s 2 3 p 6 4s 2 S 1/2 shown in Fig. 4 is simpler than those in Figs. 2 and 3. The main contributions come from a few resonances: 4s 2 2 P 3/2 , 3d( 1 P)4s 2 P 3/2 , 4s 2 2 P 1/2 , and 3d( 1 P)4s 2 P 1/2 , among which the strongest is 3d( 1 P)4s 2 P 3/2 . The oscillator strengths of the transitions 3s 2 3 p 6 4s 2 S 1/2-3d( 1 P)4s 2 P 3/2 and 3s 2 3 p 6 4s 2 S 1/2-3d( 1 P)4s 2 P 1/2 are determined to be 2.276 and 1.233, respectively. The former is in good agreement with the experimental result 2.1⫾0.4 关16兴. The positions, widths, and oscillator strengths of the resonances shown in Figs. 3 and 4 are also given in Table III. It is worth noting that most work carried out on the oscillator strengths of Sc III is limited to the transitions between the valence-shell excited states 关30–34兴. Work is very scarce on the oscillator strengths of the inner-shell transitions. Few studies have been carried out on the oscillator strengths of the 3p→3d transitions using nonrelativistic method 关35,36兴. To the best of our knowledge, there are no relativistic studies on the oscillator strengths or autoionization widths of the 3p→3d inner-shell excitations. In the experiment carried out by Schippers et al. 关16兴, 20.7% of the ground level 3s 2 3 p 6 3d 2 D 3/2 and 54.6% and 24.7% of the metastable levels 3s 2 3 p 6 3d 2 D 5/2 and 3s 2 3 p 6 4s 2 S 1/2 existed in the ion beam. Figure 5 shows the result representing a linear combination of the products of the PI cross sections shown in Figs. 2– 4 times the ion fraction for the ground and two metastable levels. This spectrum

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ZENG, ZHAO, AND YUAN

FIG. 4. The same as Fig. 2, but for the metastable level 3s 2 3p 6 4s 2 S 1/2 : 共a兲 J⫽3/2 partial, 共b兲 J⫽1/2 partial, and 共c兲 the total cross section.

is highly resolved and shows that many resonances contribute to the cross section. Some resonances are very broad and some very narrow. The curve shown in Fig. 5共a兲 is convoluted with a Gaussian profile with the experimental resolution 25 meV to get the final result shown in Fig. 5共b兲. A few narrow resonances coalesce after the broadening caused by the experimental instrument is taken into account. Reasonable good agreement is obtained between the theory and experiment. Nearly all structures observed in the experiment have been predicted by the theory. Furthermore, our identification of the resonances are consistent with those by Schippers et al. 关16兴, which can easily be seen from Figs. 2– 4. On the other hand, we have also determined some resonances which are not identified by Schippers et al. 关16兴. From the identification of Figs. 2– 4, we determined that the resonances observed on the left of the giant resonance are caused by 2 D 3/2,5/2→3d( 1 D)4s 2 D 3/2,5/2 and 2 D ,3/2,5/2 →3d( 1 F)4s 2 F 5/2,7/2 , whereas on the right by 2 S 1/2→4s 2 2 P 3/2 . The predicted positions and intensities agree excellently with the measurement for the 3p→4s resonances in the region of lower photon energy 共less than 37 eV兲. Moreover, the theoretical width of the giant resonance agrees much better with the experiment after the relativistic effects being considered. While for the strong 3p→3d resonances, the theoretical positions are systematically higher than the observation by about 1 eV. The intensity is larger than the observation by 32% and 25%, respectively, for the resonances located at 37.3 and 41.8 eV. The main reason may be due to the complexity of 3s 2 3p 5 3d 2 configuration, for which it is not easy to accurately calculate all its levels. More CI may improve the result, but this will result in too many levels and make the calculation intractable. For example, inclusion of further

FIG. 5. Comparison between the theoretical 共solid line兲 and experimental 共shaded line兲 photoionization cross section of Sc III in the photon energy range of 32.5– 43.5 eV: 共a兲 the instrumental broadening having not been considered and 共b兲 integrating curve 共a兲 using the experimental energy spread ⬃25 meV. Some unidentified resonances in the experiment have been determined.

CI with 3s3 p 3 3d 5 , for example, should, in general, lower the position and intensity of the 3p→3d resonances, but this configuration alone will split into 1337 levels. On the other hand, we predict a resonance which should be identified as 2 S 1/2→3d 2 ( 3 P) 2 P 3/2 superimposed on the giant 3p→3d resonance. It is very narrow 共only 0.68 meV兲 and rather weak 共the oscillator strength is 0.027兲 but not negligible. As mentioned above, the experimental resolution is ⬃25 meV, which is much larger than the width of this resonance. Therefore, it might escaped from the experimental scanning. In conclusion, close-coupling calculations are performed for the photoionization cross section of the ground and two lowest metastable levels of Sc III using both the relativistic and nonrelativistic R-matrix method. Complex resonances dominate the cross section over the 3p excitation region. The basic parameters of the resonances, such as positions, widths, and oscillator strengths, are determined and analyzed in some detail. The giant 3p→3d resonance from the ground and the first lowest metastable levels have broad widths 共910 and 920 meV兲 and strong absorption 共oscillator strengths are 1.150 and 1.108, respectively兲, whereas the strongest absorption is contributed by 3s 2 3 p 6 4s 2 S 1/2→3d( 1 P)4s 2 P 3/2 whose oscillator strength is 2.276. To obtain good agreement with the observation, elaborate configuration interaction and relativistic effects should be carefully considered.

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PHYSICAL REVIEW A 68, 022714 共2003兲

PHOTOIONIZATION OF THE GROUND AND FIRST TWO . . . ACKNOWLEDGMENTS

This work was supported by the National Science Fund for Distinguished Young Scholars under Grant No. 10025416, the National Natural Science Foundation of China

under Grant Nos. 10204024 and 19974075, the National High-Tech ICF Committee in China, and China Research Association of Atomic and Molecular Data. One of the authors 共Jiaolong Zeng兲 acknowledges the support from the CAS.

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