photoionization of al-like si using the r-matrix method

Photoionization of Al-like Si using the R-matrixmethod

Jagjit Singh, Sunny Aggarwal, A.K.S. Jha, A.K. Singh, and M. Mohan

Abstract: Relativistic calculations are made for photoionization of the ground state 3s23p (2Po1=2) and the lowest six excitedstates 3s23p (2Po3=2), 3s3p

2 (4P1/2,3/2,5/2), and 3s3p2 (2D3/2,5/2) of Al-like Si, using the BreitPauli Hamiltonian within the R-matrix method. Cross sections are determined by the Rydberg series of autoionizing resonances converging to various ionicstates. Relativistic effects as well as all the important physical effects like exchange, channel coupling, and short-range cor-relations have been considered in the present calculations. The present relativistic calculations for this ion using the lowest20 target levels of Si III in the LSJ coupling scheme will enhance the database sufficiently for practical applications of pho-toionization cross sections of Si II.

PACS Nos: 32.80.Fb, 32.10.Hq, 32.10.Fn

Rsum : Nous compltons des calculs relativistes de photo-ionisation pour le fondamental 3s23p (2Po1=2) et les six premierstats excits 3s23p (2Po3=2), 3s3p

2 (4P1/2,3/2,5/2) et 3s3p2 (2D3/2,5/2) du Si de type Al. Nous utilisons le Hamiltonien de BreitPauli lintrieur de la mthode de la matrice R. Les sections efficaces sont dtermines par les sries de Rydberg des rso-nances auto-ionisantes convergeant vers diffrents tats ioniques. Le prsent calcul tient compte des effets relativistes, maisaussi dimportants effets physiques comme lchange, les canaux coupls et les corrlations courte porte. Lactuel calculrelativiste pour cet ion, qui utilise les 20 premiers niveaux cible du Si III en couplage LSJ, va amliorer raisonnablement labase de donnes pour les applications pratiques de photo-ionisation du Si II.

[Traduit par la Rdaction]

1. Introduction

The absolute measurement of the photoionization crosssection of atomic ions started almost two decades ago [1].There has been great interest in photoionization studies ofopen-shell atoms and ions because of the importance of elec-tron correlation and interchannel coupling effects. Recently,there has been a major expansion in experimental work, withmerged beam experiments set up in Japan at Photon Factory,in Denmark at the ASTRID storage ring, and in the USA atthe Advanced Light Source (ALS) at Lawrence Berkley Na-tional Laboratory. All this work has been inspired by the im-portance of positive ions for solar physics and upperatmosphere physics, which require absolute cross section andspectroscopic data for modelling a stellar atmosphere andEarths ionosphere.Accurate theoretical calculations are needed as experimen-

tal data are available from various groups all around theworld. Because of increasing computing power, on the theo-retical side, the close-coupling R-matrix method [25] hasbeen the most accurate tool with which to study low energyelectron collision processes involving atoms and ions, as itconsiders all the important physical effects such as exchange,channel coupling, and short-range correlations.The Al isoelectronic sequence has received a great deal of

attention because of its various applications in astrophysics,

plasma physics, and spectroscopy. The study of Al-like Sihas also attracted considerable interest in astrophysics, asstrong emission lines from this ion have been observed insolar corona and solar flares [6, 7]. Si II lines have beenidentified in solar spectra in various observations [8, 9]. Si IIlines were obseved in the far-ultraviolet (FUV) andextreme-ultraviolet (EUV) in the spectral atlas of the Sun be-tween 670 and 1609 with the solar ultraviolet measure-ment of emitted radiation (SUMER) spectrograph on thesolar and heliospheric observatory (SOHO) [10] and between914 and 1177 by Feldman and Doschek [11]. In the upperchromosphere and lower transition region in the sun and late-type stars, line ratios among the Si II 1814 multiplet andthe intercombination (3s23p 2Po 3s3p2 4P) mutiplet near2335 are potentially useful density diagnostics. Radiativelifetimes of some of the levels of Si II, lying between950 and 1350 , were measured by Smith [12] with the Co-pernicus satellite and by various other authors [13, 14].The importance of Si II autoionization to explain the UV

opacity in the atmosphere of Ap-Si stars was pointed out byJamar et al. [15], followed by a quantitative investigation byArtru et al. [16]. Both cool and hot stars have strong Si lines.Ap-Si 4200 stars belong to a homogeneous group of stars.Their surfaces are those of dwarfs, and their effective temper-atures lie between 12 000 and 15 000 K. In the analysis ofthe international ultaviolet explorer (IUE) spectra of A and B

Received 4 July 2011. Accepted 10 September 2011. Published at www.nrcresearchpress.com/cjp on 20 October 2011.

J. Singh, S. Aggarwal, A.K.S. Jha, and M. Mohan. Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India.A.K. Singh. Department of Physics, D.D.U. College, University of Delhi, Delhi 110015, India.

Corresponding author: J. Singh (e-mail: [email protected]).

1119

Can. J. Phys. 89: 11191126 (2011) doi:10.1139/P11-106 Published by NRC Research Press

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stars, Artru and Lanz [17] explained the broad flux depres-sion at 140 nm as the effect of a single, strong Si II autoion-ization resonance. Some of the resonances of Si II in stellarspectra were reported by Lanz [9].To the best of our knowledge, no theoretical work in the

LSJ coupling scheme has appeared in the literature for photo-ionization of Si II, and this is the first calculation for the pho-toionization cross section of ground state 3s23p (2Po1=2) andthe lowest six excited states 3s23p (2Po3=2), 3s3p2 (4P1/2,3/2,5/2),and 3s3p2 (2D3/2,5/2) of Si II, by taking the lowest 20 targetstates of Si III using the relativistic R-matrix method.

2. Calculation method

2.1. Target eigenstatesThe Mg-like Si III target states have been represented us-

ing a configurationinteraction (CI) type wave function [18]

FJ XMi1

aiFiaiLSJ 1

where {Fi} represents a set of configuration state functionsthat posseses the same angular symmetries as, and in the pre-sent calculations are constructed from, the set of one-electronorbitals whose radial parts are written as a linear combinationof Slater-type orbitals (STOs)

Pnlr Xkj1

Cjnlcjnlr 2

which satisfy the following orthonormality condition:Z10

PnlrPn0lrdr dnn0 l < n0 n 3

The 1s, 2s, and 2p orbitals are those calculated by Clem-entti and Roetti [19], while the 3s, 3p, 3d, 4s, 4p, 4d, and 4fwere obtained variationally using Hibberts CIV3 code [20]by optimizing on 3s2 (1S), 3p2 (1S), 3p3d (1Po), 3p4s (1Po),3s4p (1Po), 3s4d (3D), and 3s4f (1Fo), respectively.The energies of 20 LSJ states of Si III in the BreitPauli

approximation are given by

hFiJjHNBPjFiJ 0i ENi djj0 4We include the mass-correction, Darwin, and spinorbit

interaction terms of the BreitPauli approximation where

HNBP HNNR HNmass HND HNSO 5awith

HNmass 1

8a2XNi1

r4i 5b

HND 1

8a2Z

XNi1

r2i1

ri

5c

HNSO 1

2a2Z

XNi1

li Sir3i

5d

These energies for the lowest 20 target sates in the LSJcoupling scheme were calculated using the RECUPD moduleof R-matrix code and are presented in Table 1. Configura-tions used in CI expansion of the Si III target states arepresented in Table 2. Configurations having negligible corre-lation with target states are excluded from the present calcu-lations. It can be seen from Table 1 that the LSJ excitationenergies (relative to ground state) are reasonably close toNIST [21] values.

2.2. R-matrix calculationThe R-matrix method developed by Berrington et al. [24],

Burke and Robb [5], and Seaton [22] has been used in thepresent calculation. In the internal region [23], the total wavefunction is expanded in terms of antisymmetrized energy-dependent R-matrix basis states jk

Table 1. Comparison of calculated and experimental energies ofSi III.

Key Config. Term J Present (Rydberg) NIST (Rydberg)1 3s2 1S 0 0.000 000 0 0.000 000 02 3s3p 3Po 0 0.469 907 3 0.480 462 73 3s3p 3Po 1 0.470 793 4 0.481 634 54 3s3p 3Po 2 0.472 580 3 0.484 019 65 3s3p 1Po 1 0.773 110 5 0.755 298 36 3p2 1D 2 1.105 106 0 1.113 700 77 3p2 3P 0 1.195 427 4 1.181 990 48 3p2 3P 1 1.196 337 3 1.183 207 19 3p2 3P 2 1.198 131 0 1.185 563 210 3s3d 3D 3 1.311 158 3 1.302 599 211 3s3d 3D 2 1.311 178 6 1.302 618 312 3s3d 3D 1 1.311 209 1 1.302 640 313 3s4s 3S 1 1.440 049 1 1.397 674 514 3p2 1S 0 1.496 139 6 1.398 286 715 3s4s 1S 0 1.518 845 2 1.449 548 916 3s3d 1D 2 1.602 507 4 1.510 561 817 3s4p 3Po 0 1.728 009 0 1.596 813 318 3s4p 3Po 1 1.728 212 3 1.597 114 919 3s4p 3Po 2 1.728 650 0 1.597 781 620 3s4p 1Po 1 1.736 662 0 1.608 269 6

Table 2. Configurations used in CI expansion ofSi III target state.

Key Term Configurations used1 1Po 3s3p, 3s4p, 3p4s, 3p3d, 3p4d2 1Do 3p3d, 3p4d3 1Fo 3s4f, 3p3d, 3p4d4 3Po 3s3p, 3s4p, 3p4s, 3p3d, 3p4d5 3Do 3p3d, 3p4d6 3Fo 3s4f, 3p3d, 3p4d7 1S 3s2, 3p2, 3s4s, 3p4p8 1P 3p4p9 1D 3p2, 3s3d, 3s4d, 3p4p10 3S 3s4s, 3p4p11 3P 3p2, 3p4p12 3D 3s3d, 3s4d, 3p4p

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jk AXij

CijkFiX1X2::::::XN ;brN1; sN1uijrN1Xj

djkfjX1:::::XN1 6

where A is an antisymmetrization operator that accounts forelectron exchange between the target electrons and the freeelectron. The channel functions, Fi, are obtained by couplingthe target states with the angular and spin functions ofcontinuum electrons to form states of the total angular mo-

mentum and parity. The continuum orbitals, uij, represent themotion of the scattered electron.Here, we have used a pair-coupling scheme [24] because it

is expected to be realized in medium-sized atomic systems.

Ji l K K 12 J 7

where Ji is the total angular momentum of the target stateand J is the total angular momentum of the system. InR-matrix theory, configuration space is partitioned into two

Fig. 1. Calculated photoionization cross section from the ground state 3s23p (2Po1=2) of Si II as a function of photon energy up to 3 Rydberg.

Fig. 2. Calculated photoionization cross section from the 3s23p (2Po3=2) state of Si II as a function of photon energy up to 3 Rydberg.

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regions by a sphere of radius a in such a way as to enclosethe target electrons effectively [25]. Electron exchange be-tween the scattered electron and target is negligible in the ex-ternal region (r > a).The continuum orbitals, uij, in (6), which are eigenfunc-

tions of a zero-order nonrelativistic model Hamiltonian, sat-isfy the following boundary conditions:

uij0 0 8

a

uij

duij

dr

ra

b 9

A zero logarithmic derivative, b = 0, has been imposed atthe R-matrix boundary radius, a = 8.0 a.u., and retained 15continuum orbitals for each value 0 6 of continuumelectron. The coefficients Cijk and djk can be obtained by di-agonalizing the (N+1)-electron BreitPauli Hamiltonian ma-

Fig. 3. Calculated photoionization cross section from the 3s3p2 (4P1/2) state of Si II as a function of photon energy up to 3 Rydberg.


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trix in the inner region. The radial equations are solved in theouter region assuming purely a Coulombic asymptotic inter-action

s 43P2aa20

u

g

XL;lf

jLlf Ef jDLjij2 10

Finally, the boundfree photoionization cross section is ex-pressed [2] in the dipole length form where g is the statisticalweight factor of the initial state, and DL is the dipole opera-tor. More theoretical details are given by Seaton [22].

3. Result and discussion

In this section, we present a photoionization process that cor-responds to photoionization of Si II from its ground state 3s23p(2Po1=2) and the lowest six 3s23p (

2Po3=2), 3s3p2 (4P1/2,3/2,5/2), and3s3p2 (2D3/2,5/2) excited states using the lowest 20 targetstates of Si III. For correlation effects, we have included12 symmetries, namely 1Po, 1Do, 1Fo, 3Po, 3Do, 3Fo, 1S, 1P,1D, 3S, 3P, and 3D from all the contributing configurationsof Si III involving n = 3 and 4. However, many of the lev-els belonging to these symmetries do not appear up to the


Fig. 6. Calculated photoionization cross section from the 3s3p2 (2D3/2) state of Si II as a function of photon energy up to 3 Rydberg.

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lowest 20 target states but are needed for correlation effects.The target includes singlets and triplets, which allows the(N + 1) configuration to be doublet or quartet. Forphotoionization of Si II 3s23p 2Po, J = 1/2 ground state, SiIII plus electron will combine to give symmetries of 2S, 2P,and 2D. We have considered transitions J = 1/2 1/2, 3/2;J = 3/2o 1/2, 3/2, 5/2; J = 1/2 1/2o, 3/2o; J = 3/2 1/2o, 3/2o, 5/2o; and J = 5/2 3/2o, 5/2o, 7/2o. The follow-ing possible transitions have been included for ground statephotoionization: 2So1=2 2S1/2,, 2P1/2,3/2;

2Po1=2 2S1/2, 2P1/2,3/2,2D3/2; 4Po1=2 4S3/2,, 4P1/2,3/2, 4D1/2,3/2;

4Do1=2 4F3/2,, 4P1/2,3/2,4D1/2,3/2 by considering dipole selection rules. Pair-couplingschemes have been used for photoionization cross sectioncalculation of ground state 3s23p (2Po1=2) and excited states3s23p (2Po3=2) and 3s3p2 (4P1/2,3/2,5/2,2). Processes for groundstate 3s23p (2Po1=2) photoionization cross section calculationsare reported in the following list:

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6 3S2(1S0), 3P2(3P0), 3P2(1S0), 3s4s (1S0))+ e(s)J = 1/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6 3S2(1S0), 3P2(3P0), 3P2(1S0), 3s4s (1S0)) + e(d)J = 3/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p63s3p, 3s4p (3Po0)) + e(p)J = 1/2, 3/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p63s3p, 3s4p (3Po1,

1Po1)) + e(p)J = 1/2 Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6

3s3p, 3s4p (3Po1,1Po1)) + e(p, f)J = 3/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p63s3p, 3s4p (3Po2)) + e(p, f)J = 1/2, 3/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6 3p2(3P2, 1D2), 3s3d (3D2, 1D2)) + e(d)J = 1/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6 3p2(3P2,1D2), 3s3d (3D2,1D2)) + e(s,d,g)J = 3/2

Fig. 7. Calculated photoionization cross section from the 3s3p2 (2D5/2) state of Si II as a function of photon energy up to 3 Rydberg.

Table 3. Photoionization cross section.

Cross section (Mb)

Energy(Rydberg)

3s23p(2Po1=2)

3s23p(2Po3=2)

3.00 0.9659 1.76803.10 0.9380 1.74203.20 0.8856 1.68503.30 0.8213 1.58103.40 0.7540 1.43703.50 0.6905 1.28103.60 0.6400 1.13803.70 0.6091 1.02703.80 0.5796 0.95343.90 0.5506 0.91574.00 0.5485 0.9089

Table 4. Photoionization cross section.

Cross section (Mb)

Energy(Rydberg)

3s3p2(4P1/2)

3s3p2(4P3/2)

3s3p2(4P5/2)

3s3p2(2D3/2)

3s3p2(2D5/2)

2.60 1.1070 1.0770 0.3085 1.4600 0.65872.70 1.0770 1.0390 0.2726 1.3820 0.63032.80 1.0340 0.9955 0.2398 1.2930 0.61602.90 0.9915 0.9635 0.2140 1.2010 0.60823.00 0.9484 0.9430 0.1968 1.1060 0.59733.10 0.8911 0.9195 0.1845 1.0120 0.57393.20 0.8191 0.8845 0.1737 0.9256 0.53593.30 0.7457 0.8413 0.1645 0.8563 0.49533.40 0.6828 0.7997 0.1587 0.8141 0.46383.50 0.6377 0.7644 0.1568 0.8037 0.4436

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Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p6 3p2(3P1), 3s3d (3D1), 3s4s (3S1)) + e(s,d)J = 1/2, 3/2

Si II (1s22s22p63s23p 2Po1=2) + hn Si III (1s22s22p63s3d (3D3)) + e(d,g)J = 1/2, 3/2In atomic or ionic photoionization one has an electronion

system with an electron in continuum. A Rydberg series ofresonances arises from the strong Coulomb potential betweenthe electron and the ion. Resonances occur because of super-position or interaction of bound state wave functions andcontinuum state wave functions. From basic scattering theory,they correspond to a rapid change in phase shift by P. Thenthe reactance matrix, K, leads to a sharp change (rise andfall) in the corresponding scattering matrix or cross section.Sharp resonances or peaks that are associated with an energythreshold and lie at an energy below the threshold are Fes-bach resonances [26].Figure 1 presents the photoionization cross section of the

Si II ground state in the photon energy range from 1S0threshold to 3 Rydberg. The calculated ionization energy forthe LSJ target level (1.114 77 Rydberg) is in fair agreementwith experimental ionization energy (1.201 39 Rydberg). Alot of resonances are found in the energy range 1.1152.851 Rydberg due to different Rydberg series of resonancesconverging to various thresholds up to 3s4p (1Po1) of Si III.Various series of resonances are expected in the energy range1.1151.89 Rydberg due to following states: 3s3p (3Po0 {np[1]1/2,3/2}, 3Po1 {np [0]1/2, np [1]1/2,3/2, nf [2]3/2},

3Po2 {np [1]1/2,3/2,np [2]1/2,3/2}, 1Po1 {np [0]1/2, np [1]1/2,3/2, nf [2]3/2}).In the energy range 1.892.851 Rydberg, Rydberg reso-

nances are expected due to various even and odd states: 3p2(1D2 {nd [0]1/2, nd [1]1/2,3/2, nd [2]3/2, ns [2]3/2, ng [2]3/2},3P0 {nd [2]3/2, ns [0]1/2}, 3P1 {ns [1]1/2,3/2, nd [1]1/2,3/2, nd [2]3/2},3P2 {nd [0]1/2, nd [1]1/2,3/2, nd [2]3/2, ns [2]3/2, ng [2]3/2}, 1S0{nd [2]3/2, ns [0]1/2}), 3s3d (3D1 {ns [1]1/2,3/2, nd [1]1/2,3/2,nd [2]3/2}, 3D2 {nd [0]1/2, nd [1]1/2,3/2, nd [2]3/2, ns [2]3/2,ng [2]3/2}, 3D3 {nd [1]1/2,3/2, ng [1]1/2,3/2, nd [2]3/2, ng [2]3/2},1D2 {nd [0]1/2, nd [1]1/2,3/2, nd [2]3/2, ns [2]3/2, ng [2]3/2}),3s4s (3S1 {ns [1]1/2,3/2, nd [1]1/2,3/2, nd [2]3/2}, 1S0 {nd [2]3/2,ns [0]1/2}), 3s4p (3Po0 {np [1]1/2,3/2},

3Po1 {np [0]1/2, np [1]1/2,3/2,nf [2]3/2}, 3Po2 {np [1]1/2,3/2, np [2]1/2,3/2},

1Po1 {np [0]1/2,np [1]1/2,3/2, nf [2]3/2}). Here, we have used the JLK couplingscheme that is written in the form ntlt 2S1LPJt nl[K]J [27].The results of this calculation for the photoionization cross

section of Si II for 3s23p (2Po3=2) and 3s3p2 (4P1/2) states havebeen plotted in Figs. 2 and 3, respectively. The correspond-ing values of ionization enegies for these two states are1.112 61 and 0.789 112 Rydberg. Results for photoioniza-tion of 3s3p2 (4P3/2) are presented in Fig. 4, having ioniza-tion energy 0.788 335 Rydberg. In Fig. 5 we report thephotoionization cross section results for 3s3p2 (4P5/2) statewith ionization energy 0.787 052 Rydberg. Finally, in Figs. 6and 7 we have reported the photoionization cross sectionfor 3s3p2 (2D3/2) and 3s3p2 (2D5/2), having ionization ener-gies 0.629 915 and 0.629 855 Rydberg, respectively. Wehave also provided photoionization cross sections in tabu-lated form in Tables 3 and 4 at some energy points. To thebest of our knowledge, no theoretical or experimental datais available for photoionization of Si II in the LSJ couplingscheme to compare with the present results.

4. ConclusionIn summary, photoionization cross section calculations

have been reported for photoionization of Si II from itsground state 3s23p (2Po1=2) and the lowest six 3s23p (

2Po3=2),3s3p2 (4P1/2,3/2,5/2), and 3s3p2 (2D3/2,5/2) excited states, usingthe relativistic R-matrix method. To the best of our knowl-edge, these extensive calculations in the LSJ coupling schemefor photoionization of Si II are presented for the first time us-ing the lowest 20 target levels of Si III. We expect our resultsto be accurate, as all the important physical effects like ex-change, channel coupling, short-range correlation, and rela-tivistic effects have been taken into account and shouldenhance the database sufficiently for practical application ofthe photoionization cross section of Si II.

AcknowledgementJS is thankful to U.G.C. (India) for JRF and MM is thank-

ful to D.S.T. (India), U.G.C. (India), and University of Delhifor financial support.

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photoionization of al-like si using the r-matrix method

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