Resonant Structures of Overlapping Autoionization ...

Journal of the Korean Physical Society, Vol. 51, No. 1, July 2007, pp. 312∼317

Resonant Structures of Overlapping Autoionization Rydberg Series in Photoionization of the N3+ Ion Dae-Soung Kim∗ e-Business Department, Kyonggi Institute of Technology, Siheung 429-450

Young Soon Kim† Department of Physics, Myongji University, Yongin 449-728 (Received 31 January 2007) The resonant structures of absolute photoionization cross sections for N3+ ion are calculated and compared with the recent experimental measurements and the previous calculations performed in the Opacity Project. We have used a noniterative eigenchannel R-matrix method, which is different from the R-matrix used in the Opacity Project. Comparisons with experimental results for the photoionization of the 1 S e ground state and the 1,3 P o excited states of N3+ in the photon energy range 62 – 90 eV show overall good agreement with the present theoretical results. The lower members of the autoionizing Rydberg series converging to each of the N4+ 2p threshold are identified and discussed. PACS numbers: 32.80.Fb Keywords: Photoionization, Photoelectron, Autoionization, Rydberg series, Atomic structure, R-matrix

of the earlier theoretical and experimental work has been limited to photoionization from the ground state. So far, the most extensive calculation beyond the ground state for Be sequence has been performed by the Opacity Project (OP) [11,12], in which an iterative R-matrix formulation of the close-coupling approximation was used and that formulation is different from our present noniterative eigenchannel R-matrix method. In this paper, we do not intend to present an exhaustive comparison with other earlier work, except for the previous theoretical calculation of the Opacity Project [11,12] and the recent experimental work by Bizau et al. [10].

I. INTRODUCTION The photoionization process plays an important role in many astrophysical objects, and it is an important task for atomic physics to provide appropriate data. Among the most relevant species for astrophysics are the element Be and Be-like ions. The present calculations of the photoionization cross sections for the N3+ ion have been carried out in response to the need to extend previous photoionization calculations for atomic Be [1–4] and for ionic Be-like B+ [5–7] and C2+ ions [8]. For these calculations, we employ a noniterative variational R-matrix method [9]. A brief description of our non-iterative techniques is given in the following section. In the photon energy region 62 – 90 eV, our calculations for the mixed photoionizations of the 1 S e ground state and the 1,3 P o excited states of N3+ ion are compared with the previous experimental results performed by Bizau et al. [10]. The associated autoionizing Rydberg series of the 1 S e ground and the 1,3 P o excited states of N3+ ion in the photon energy region below the N4+ 2p thresholds are identified and discussed. Although the neutral Be and Be-like ionic sequence have been the object of much research in the past, most ∗ E-mail: † E-mail:

II. METHOD Most of the present methods are the same as those used for a previous description of the Be and Be-like B+ and C2+ [1–8]. As the basic theory and computational techniques have been presented in the literature [9], we will give only a brief description of these methods. The major dynamics and numerical approximations used for the present calculations are contained in the noniterative eigenchannel R-matrix method [13]. In this method, the normal derivative of the basis funcP β tions, ψβ = i ci yi , at a given energy is defined by

[email protected]; Fax: +82-31-496-4637 [email protected]; Associate member of KIAS

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Resonant Structures of Overlapping Autoionization Rydberg· · · – Dae-Soung Kim and Young Soon Kim

∂ψβ /∂r = −bβ ψβ on the surface of the R-matrix volume. The R-matrix volume is defined by ri ≤ r0 , where all electrons are confined initially to radii less than r0 . The generalized matrix equation for the coefficients, cβi , of the basis functions is Γcβ = bβ Λcβ ,

(1)

where Γij = 2hyi |E − H|yj i − hyi |∂/∂r|yj iS and Λij = hyi |yj iS , the subscript S representing integration only over the surface of the R-matrix volume. In the eigenchannel R-matrix approach, the basis set is divided into open functions, which are nonzero at r = r0 , and closed functions, which are zero at r = r0 . In the present calculation, we use only two open orbitals for each l. In the present calculation, the Hamiltonian, H, does not refer to the full atomic system, but only the valence shells. This Hamiltonian works well as long as the energy of the calculational system is smaller than the energy to excite an inner-core 1s electron. With a closed 1s2 core of a Be-like N3+ ion, our task is to represent the wave function of the two valence electrons by a superposition of independent-particle wave functions, adequate to describe the correlated motion within the R-matrix volume V . Then, a non-relativistic model Hamiltonian can be used to effectively describe the two valence electrons of a Be-like N3+ ion: 1 1 1 H = − ∇21 − ∇22 + U (r1 ) + U (r2 ) + . (2) 2 2 r12 ~ = ~l1 + ~l2 For a pair of electrons coupled to a given L within V , we write the wave function as follows: X ψ = cn1 n2 yn1 n2 (3) n1 ,n2

= A

X

cn1 n2 φn1 l1 (r1 )φn2 l2 (r2 )Yl1 l2 LML (Ω1 , Ω2 ),

n1 ,n2

where A represents the antisymmetrization. More specifically, yn1 n2 can be represented by 1 Φn (r1 , Ω1 , Ω2 ) φn2 l2 (r2 ) yn1 ,n2 = √ [ 1 r2 2 Φn (r2 , Ω2 , Ω1 ) φn2 l2 (r1 )], +(−1)q 1 r1

(4)

where q ≡ l1 + l2 − L + S, and by Φn1 (r1 , Ω1 , Ω2 ) =

φn1 l1 (r1 ) Yl1 l2 LM (Ω1 , Ω2 ), r1

(5)

which connects more naturally with the quantum-defect description of the outer region. The channel functions Φn are the functions for the ionic states, including the angular wave functions of both electrons. Next, a major consideration of R-matrix calculations concerns an accurate description of the target-state functions. We describe some of the approximations for successful calculation of N3+ spectra. The N3+ ion can be

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very well described using a multiconfiguration HartreeFock (MCHF) method [14]. The first step consists of solving the MCHF equations for five interacting electrons moving in the model potential to determine the correlation effects of 2s, 2p, 3d, and 4f orbitals interacting with the 1s2 core wave function. Next, we construct local s and p potentials, which give the Hartree-Fock ns, np, nd and nf orbitals as solutions. For example, the 2snp configuration owing to the 1 S e ground state has been represented by products of φ2s (r1 ) and of the following twenty-two φnp (r2 ) orbitals. Twenty of these orbitals have been obtained by solving the oneelectron wave equation 1 d2 l(l + 1) c + + U (r) − Enl }φcnl (r) 2 2 2 dr 2r X = λnm φml (r),

{−

(6)

m

where the φml are the Hartree-Fock wave functions of c angular momentum l. The Enl and the λnm are chosen c c so that φnl (r0 ) = 0 and hφnl |φml i = 0. In addition, the lowest two positive energy solutions φonl for each l are generated by 1 d2 l(l + 1) o + + U (r) − Enl }φonl (r) = 0, (7) 2 2 dr 2r2 o o where Enl are chosen so that dφonl (r0 )/dr = 0 and Enl − U (r0 ) ≥ 0, and they are added to the basis set. It is noted that all closed orbitals are orthogonal to each other and that good convergence is achieved in the present calculation with an accuracy of 10−8 or better. Then, the proper twenty-one or twenty-two orbitals are calculated for any other nl in the 2pns and 2pnd configurations. Similarly, the 2pnp 1,3 S, 1,3 D, and 2pnf 1,3 D owing to the excited states 1,3 P are also constructed successfully. The matrix elements of the one electron Hamiltonian H(~r) are easily calculated because the basis functions in Eq. (4) are the eigenfunctions of H(~r1 ) + H(~r2 ). On the other hand, the evaluation of the matrix elements for 1/r12 is more complicated, but they need to be evaluated only once because our basis set is energy independent. The energy-normalized eigenstates ψα in each eigenchannel can be represented by a linear combination of the unnormalized eigenstates which, along with the ground state wave function ψ0 , are used to calculate X ψα = ψβ (I −1 )β,i Uiα cos(πµα ). (8) {−

β,i

We need to find a new set of solutions ψi () with i = 1,· · ·,No from ψα . These solutions should remain well behaved as r −→ ∞. The No ×No open-channel reaction matrix is written as [15] oo Kphys = K oo − K oc [K cc + tan πn∗ ]−1 K co ,

(9)

where the effective quantum number in the closed channels is given by Z −3 n∗ = p , (10) 2(It − ω)

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Journal of the Korean Physical Society, Vol. 51, No. 1, July 2007

Table 1. Onset thresholds (in eV) for each of the 1 S e ground state, the metastable 3 P o state, and the excited 1 P o state of ionic N3+ . Initial State 1 e S 3 o P 1 o P

Present 77.6130 68.9845 60.8429

Opacity Project (OP) [16] 77.3312 68.9708 60.8700

Table 2. Energy differences (in eV) between the low lying states: 2s2 2s2p 3 P − 2s2p 1 P in the ionic N3+ . Transition 2s2 1 S − 2s2p 3 P 2s2 1 S − 2s2p 1 P 2s2p 3 P − 2s2p 1 P

Present 8.6285 16.7701 8.1416

OP [16] 8.3604 16.4611 8.1008

with It representing the ionization energy of target state. The eigenphase in each channel i is then defined as δi = tan−1 λi ,

i = 1, · · · , No ,

(11)

oo where λi is the eigenvalue of Kphys . Other detailed theoretical descriptions are given in the previous calculations [2].

III. RESULT AND DISCUSSION In Table 1, we compare the energies of the onset thresholds for the 1 S e ground state and the 1,3 P o excited states of ionic N3+ . The present ionization thresholds relative to the 1 S e ground and the 3 P o metastable state of ionic N3+ are 77.6130 eV and 68.9845 eV, respectively. The corresponding previous experimental results of Bizau et al. are 77.1 eV and 69.96 eV, which are in good agreement with our present results. The previous theoretical calculations of the Opacity Project are also in good agreement with our present results. We believe that the investigation of these energies should be continued, because there exist not enough theoretical and experimental data yet for these thresholds. In Table 2, the current excitation energies of the Belike N3+ ions are compared with the OP [16], the multiconfiguration Dirac-Fock (MCDF) [17], the multiconfiguration relativistic random-phase approximation (MCRRPA) [19], and the NIST data [20]. For the MCRRPA result, we have quoted the results calculated with core excitation channels [19].

1. The Autoionization Rydberg Series from the Excited States 2s2p 1,3 P o of a Be-like N3+ Ion

The photoionization cross sections for ionization to the Li-like N4+ state from the 2s2p 1,3 P o excited states are

1

Experiment. [10] 77.1 69.96

S

−

2s2p

MCDF [17] 8.5439 17.7299 9.1970

3

P , 2s2

1

S

MCRRPA [19] 8.5043 16.7504 8.2440

−

2s2p

1

P and

NIST [20] 8.3407 16.2041 7.8634

shown in Fig. 1 (a) and (c), along with the related previous OP results in Fig. 1 (b) and (d) for the singlet and triplet cases, respectively. The length and velocity results are displayed by the solid and dotted curves, respectively. The vertical dotted line represents the N4+ 2p thresholds for the photoionization from the 2s2p 1,3 P o states. As Fig. 1 shows, the features of the photoionization spectra for the singlet and the triplet processes differ in several aspects, including the onset threshold values, the resonance positions and, hence, the shapes of the interfering resonances. In the energy region below the N4+ 2p threshold, we have considered three Rydberg series, 2pnp 1,3 S e , 1,3 De and 2pnf 1,3 De , for the singlet and the triplet manifolds, each converging to the N4+ 2p threshold. A resonance position, Er , is calculated from the maximum derivative of the eigenphase sum δ, which is the sum over δi in Eq. (11), and the width is

 Γ=2

dδ dE

−1 .

(12)

E=Er

The calculated resonance parameters, such as the resonance positions Er , the effective quantum numbers n∗ , and the widths Γ, for lower members of the Rydberg series are listed in Table 3. The 2pnp 1,3 S e , 1,3 De and the 2pnf 1,3 De states for n ≤ 4 are bound so that those lie below the onset threshold and are not listed in Table 3, and all of those Rydberg series for n ≥ 4 display smooth regular behaviors with approximately constant quantum defects. It is noted that the 2pnp 1,3 S resonances are wider than the corresponding 2pnp 1,3 D resonances. However, the 2pnf 1,3 D resonances are very weak and exhibit very narrow peaks. Next, the background cross sections above the N4+ 2p threshold are larger than those below the threshold for both singlet and triplet states.

Resonant Structures of Overlapping Autoionization Rydberg· · · – Dae-Soung Kim and Young Soon Kim

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Fig. 1. Photoionization cross sections for ionization to the Li-like N4+ state from the 2s2p 1 P o excited states of the N3+ ion as a function of the photon energy: (a) present results for 1 P o (solid curve, present length results; dotted curve, present velocity results) (b) OP results for 1 P o (c) present results for 3 P o (solid curve, present length results; dotted curve, present velocity results) and (d) OP results for 3 P o . The vertical dotted line represents the N4+ 2p thresholds for photoionizations. Table 3. Position (Er in eV), effective quantum number (n∗ ) and width (Γ in eV) of resonances converging to the N4+ (2p) threshold from the photoionization of 2s2p 1 P o excited states. n 5 6 7 8 9 10 n 5 6 7 8 9 10 n 5 6 7 8 9 10

2pnp 1 S e Er 62.5994 65.3635 67.0116 68.0705 68.7935 69.3096 2pnp 1 De Er 62.2142 65.1708 66.8918 67.9904 68.7371 69.2686 2pnf 1 De Er 62.5268 65.3955 67.0256 68.0768 68.7975 69.3128

∗

n 4.9449 5.9550 6.9625 7.9646 8.9646 9.9647

Γ 0.5438 0.3175 0.2062 0.1411 0.1003 0.0736

n∗ 4.8413 5.8636 6.8714 7.8732 8.8728 9.8722

Γ 0.0056 0.0037 0.0012 0.0003

n∗ Γ 4.9249 0.0023 5.9706 0.0010 6.9734 0.0003 7.9719 8.9713 9.9712

2pnp 3 P e Er 70.1836 73.2266 74.9819 76.0980 76.8554 77.3935 2pnp 3 De Er 70.0330 73.1516 74.9349 76.0666 76.8334 77.3776 2pnf 3 De Er 70.6343 73.5168 75.1528 76.2085 76.9372 77.4493

n∗ 4.7977 5.8243 6.8334 7.8354 8.8356 9.8354

Γ 0.2030 0.1304 0.0873 0.0597 0.0421 0.0307

n∗ Γ 4.7593 0.0023 5.7906 0.0009 6.7992 0.0002 7.8008 8.8009 9.8009 n∗ Γ 4.9155 0.0012 5.9601 0.0007 6.9622 0.0001 7.9603 8.9591 9.9594

Fig. 2. Photoionization cross sections for ionization to the Li-like N4+ state from the 2s2 1 S e state of N3+ ion as a function of the photon energy: (a) present results for 1 S e (solid curve, present length results; dotted curve, present velocity results), and (b) OP results for 1 S e . The vertical dotted line represents the N4+ 2p thresholds for photoionization of the 2s2 1 S e ground state.

2s2

1

2. The Autoionization Rydberg Series from the S e Ground State of a Be-like N3+ Ion

In the energy region below 90 eV, the present and the OP photoionization cross sections for ionization to the Li-like N4+ state from the 2s2 1 S e state are shown in Figs. 2 (a) and (b), respectively. The shapes of the cross

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Journal of the Korean Physical Society, Vol. 51, No. 1, July 2007

Table 4. Position (Er in eV), effective quantum number (n∗ ) and width (Γ in eV) of resonances converging to the N4+ (2p) threshold from the photoionization of 2s2 1 S e ground state. 2pns 1 P o n Er 5 78.4530 6 81.6289 7 83.4409 8 84.6020 9 85.3898 10 85.9483

∗

n 4.7343 5.7315 6.7241 7.7189 8.7155 9.7133

Γ 0.3383 0.2361 0.1557 0.1039 0.0718 0.0514

2pnd 1 P o Er 79.4378 82.1822 83.8083 84.8528 85.5672 86.0778

n∗ 4.9687 5.9869 6.9963 7.9983 8.9984 9.9979

Γ 0.1131 0.0682 0.0455 0.0314 0.0224 0.0164

sections in the two calculations show good agreement. However, the resonance structure in the OP calculation [16] is shifted by about 0.90 eV to the lower energy side. There are two Rydberg series of autoionizing resonances converging to the N4+ 2p threshold. These two Rydberg series of doubly excited states, a broad 2pns 1 P o and a narrow 2pnd 1 P o , show strong interference in Fig. 2. The calculated resonance parameters, such as the resonance positions Er , the effective quantum numbers n∗ , and the widths Γ, for each Rydberg series are listed in Table 4. As in the case of 2s2p 1,3 P o excited states, the lowest autoionizing resonances for the 2pns 1 P and 2pnd 1 P series are the 2p5s 1 P and the 2p5d 1 P resonances at 78.45 and 79.44 eV, respectively. Since there are no series perturbations from resonances converging to higher thresholds, all of the autoionizing Rydberg series show regular patterns in this energy range.

3. Comparison with the Previous Experimental Results of Bizau et al. [10]

In the photon energy region 60 – 90 eV, our present results for the 1 S e ground state and the 1,3 P o excited states for Be-like N3+ are displayed in Fig. 3(a) together with the recent experimental results of Bizau et al. [10] in Fig. 3(b). We have assigned 10 % and 30 % for the cross sections of 1,3 P excited states from Fig. 1 (a) and (c), respectively, and 60 % for the cross sections of 1 S ground state from Fig. 2 (a), to attribute similar portions as in the result of Bizau et al. [10]. Overall good agreement between the experiment and the present calculations is shown, though the experimental spectra do not reveal the finely resolved details of each peak. The resonances appearing below the threshold for photoionization of 2s2 1 S ground-state ions at 77.62 eV are associated with the photoionization of 2s2p 1,3 P o excited states. Three autoionizing Rydberg series, the 2pnp 1,3 S, the 2pnp 1,3 D, and the 2pnf 1,3 D states converging to each of their respective threshold limits are discernible.

Fig. 3. (a) Present and (b) experimental photoionization cross sections of N3+ ions [10] as functions of photon energy (solid curve, present length results; dotted curve, present velocity results).

IV. CONCLUSIONS We have calculated the resonant structures of the photoionization spectra for the 1 S ground state and the 1,3 P excited states of the N3+ ion in the photon energy range between 60 and 90 eV. Recent experiments and previous calculations show good agreement with the present results. All lower members of the Rydberg series of autoionizing resonances each converging to the N4+ 2p threshold are identified using a methodology based on eigenphase sum gradients, and their positions and widths are given. Reasonable agreement between length and velocity results is found, and the present calculations show good overall agreement with the earlier Opacity Project calculations. However, significant quantative differences between the two results for the thresholds are noted. We believe that the investigation of these energies should be continued, becuase we don’t have enough previous theoretical and experimental data for these thresholds of the N3+ ion.

ACKNOWLEDGMENTS This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-312-C00103).

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