Abstract

Photoabsorption and photoionization cross sections are reported from 3d24s4p D32o, 3d24s4p G33,4o and 3d34p F55o initial states of titanium, reaching final-state energies from −500 to 8000 cm−1 relative to the first ionization threshold 3d24s4F3/2. The nearly ab initio calculations use the eigenchannel R-matrix method, the multichannel quantum-defect theory, and the LSjj relativistic recoupling frame transformation. The last two steps use experimental energies of Ti+ levels. Radial orbitals needed to calculate short-range interaction parameters are obtained in a multiconfiguration Hartree–Fock approximation that directly generates natural orbitals for the target states. This method bypasses an intermediate step followed in earlier eigenchannel R-matrix studies, in which a set of primitive orbitals was used. Theoretical cross sections are compared with experimental data, in particular near the two lowest ionization thresholds 3d24s4F and 3d3 4F. Our results, which at these energies are accurate to within errors of ∼0.03 in the quantum defects, account for most of the experimental features. Predictions are also made for the spectra at higher energies, where overlapping Rydberg series seriously complicate the photoabsorption pattern.

© 1996 Optical Society of America

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1994 (1)

W. Stachowska, M. Fabiszisky, and J. Dembczynski, Z. Phys. D 32, 27 (1994).
[CrossRef]

1993 (6)

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4429 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4441 (1993).
[CrossRef] [PubMed]

D. J. Armstrong and F. Robicheaux, Phys. Rev. A 48, 4450 (1993).
[CrossRef] [PubMed]

A. K. Pradhan and K. A. Berrington, J. Phys. B 26, 157 (1993).
[CrossRef]

F. Robicheaux, Phys. Rev. A 48, 4162 (1993).
[CrossRef] [PubMed]

R. P. Wood, C. H. Greene, and D. Armstrong, Phys. Rev. A 47, 229 (1993).
[CrossRef] [PubMed]

1992 (3)

F. Robicheaux and C. H. Greene, Phys. Rev. A 46, 3821 (1992).
[CrossRef] [PubMed]

R. H. Reid, K. Bartschat, and P. G. Burke, J. Phys. B 25, 3175 (1992).
[CrossRef]

K. W. McLaughlin and D. W. Duquette, J. Opt. Soc. Am. B 9, 1953 (1992).
[CrossRef]

1991 (3)

Q. Wang and C. H. Greene, Phys. Rev. A 44, 1874 (1991).
[CrossRef] [PubMed]

C. FroeseFischer, Comp. Phys. Commun. 64, 369 (1991).
[CrossRef]

C. H. Greene and M. Aymar, Phys. Rev. A 44, 1773 (1991).
[CrossRef] [PubMed]

1990 (3)

1985 (2)

J. Sugar and C. Corliss, J. Phys. Chem. Ref. Data,  14 Suppl. 2, 147 (1985).

C. H. Greene, Phys. Rev. A 32, 1880 (1985).
[CrossRef] [PubMed]

1983 (1)

M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983).
[CrossRef]

1955 (1)

P.-O. Löwdin, Phys. Rev. 97, 1474 (1955).
[CrossRef]

1932 (1)

L. Brillouin, J. Phys. (Paris) 3, 373 (1932).

Armstrong, D.

R. P. Wood, C. H. Greene, and D. Armstrong, Phys. Rev. A 47, 229 (1993).
[CrossRef] [PubMed]

Armstrong, D. J.

D. J. Armstrong and F. Robicheaux, Phys. Rev. A 48, 4450 (1993).
[CrossRef] [PubMed]

Aymar, M.

C. H. Greene and M. Aymar, Phys. Rev. A 44, 1773 (1991).
[CrossRef] [PubMed]

M. Aymar, J. Phys. B 23, 2697 (1990).
[CrossRef]

M. Aymar, C. H. Green, and E. Luc-Koenig, “Multichannel spectroscopy of complex atoms,” Rev. Mod. Phys. (to be published); G. Miecznik, C. H. Greene, and F. Robicheaux, Phys. Rev. A 51, 513 (1995).
[CrossRef] [PubMed]

Bartschat, K.

R. H. Reid, K. Bartschat, and P. G. Burke, J. Phys. B 25, 3175 (1992).
[CrossRef]

Berrington, K. A.

A. K. Pradhan and K. A. Berrington, J. Phys. B 26, 157 (1993).
[CrossRef]

Brillouin, L.

L. Brillouin, J. Phys. (Paris) 3, 373 (1932).

Burke, P. G.

R. H. Reid, K. Bartschat, and P. G. Burke, J. Phys. B 25, 3175 (1992).
[CrossRef]

Corliss, C.

J. Sugar and C. Corliss, J. Phys. Chem. Ref. Data,  14 Suppl. 2, 147 (1985).

Cowan, R. D.

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981), p. 180.

Dembczynski, J.

W. Stachowska, M. Fabiszisky, and J. Dembczynski, Z. Phys. D 32, 27 (1994).
[CrossRef]

Duquette, D. W.

Fabiszisky, M.

W. Stachowska, M. Fabiszisky, and J. Dembczynski, Z. Phys. D 32, 27 (1994).
[CrossRef]

Froese Fischer, C.

C. Froese Fischer, The Hartree-Fock Method for Atoms (Wiley, New York, 1977).

FroeseFischer, C.

C. FroeseFischer, Comp. Phys. Commun. 64, 369 (1991).
[CrossRef]

Green, C. H.

M. Aymar, C. H. Green, and E. Luc-Koenig, “Multichannel spectroscopy of complex atoms,” Rev. Mod. Phys. (to be published); G. Miecznik, C. H. Greene, and F. Robicheaux, Phys. Rev. A 51, 513 (1995).
[CrossRef] [PubMed]

Greene, C. H.

R. P. Wood, C. H. Greene, and D. Armstrong, Phys. Rev. A 47, 229 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4429 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4441 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 46, 3821 (1992).
[CrossRef] [PubMed]

Q. Wang and C. H. Greene, Phys. Rev. A 44, 1874 (1991).
[CrossRef] [PubMed]

C. H. Greene and M. Aymar, Phys. Rev. A 44, 1773 (1991).
[CrossRef] [PubMed]

C. H. Greene, Phys. Rev. A 32, 1880 (1985).
[CrossRef] [PubMed]

C. H. Greene, Fundamental Processes of Atomic Dynamics (Plenum, New York, 1988), p. 105; C. H. Greene and L. Kim, Phys. Rev. A 38, 5953 (1988).
[CrossRef] [PubMed]

Gudeman, C. S.

Knight, R. D.

Löwdin, P.-O.

P.-O. Löwdin, Phys. Rev. 97, 1474 (1955).
[CrossRef]

Luc-Koenig, E.

M. Aymar, C. H. Green, and E. Luc-Koenig, “Multichannel spectroscopy of complex atoms,” Rev. Mod. Phys. (to be published); G. Miecznik, C. H. Greene, and F. Robicheaux, Phys. Rev. A 51, 513 (1995).
[CrossRef] [PubMed]

McLaughlin, K. W.

Page, R. H.

Pradhan, A. K.

A. K. Pradhan and K. A. Berrington, J. Phys. B 26, 157 (1993).
[CrossRef]

Reid, R. H.

R. H. Reid, K. Bartschat, and P. G. Burke, J. Phys. B 25, 3175 (1992).
[CrossRef]

Robicheaux, F.

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4441 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4429 (1993).
[CrossRef] [PubMed]

D. J. Armstrong and F. Robicheaux, Phys. Rev. A 48, 4450 (1993).
[CrossRef] [PubMed]

F. Robicheaux, Phys. Rev. A 48, 4162 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 46, 3821 (1992).
[CrossRef] [PubMed]

Seaton, M. J.

M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983).
[CrossRef]

Sohl, J. E.

Stachowska, W.

W. Stachowska, M. Fabiszisky, and J. Dembczynski, Z. Phys. D 32, 27 (1994).
[CrossRef]

Sugar, J.

J. Sugar and C. Corliss, J. Phys. Chem. Ref. Data,  14 Suppl. 2, 147 (1985).

Wang, Q.

Q. Wang and C. H. Greene, Phys. Rev. A 44, 1874 (1991).
[CrossRef] [PubMed]

Wood, R. P.

R. P. Wood, C. H. Greene, and D. Armstrong, Phys. Rev. A 47, 229 (1993).
[CrossRef] [PubMed]

Zhu, Y.

Comp. Phys. Commun. (1)

C. FroeseFischer, Comp. Phys. Commun. 64, 369 (1991).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. (Paris) (1)

L. Brillouin, J. Phys. (Paris) 3, 373 (1932).

J. Phys. B (3)

M. Aymar, J. Phys. B 23, 2697 (1990).
[CrossRef]

R. H. Reid, K. Bartschat, and P. G. Burke, J. Phys. B 25, 3175 (1992).
[CrossRef]

A. K. Pradhan and K. A. Berrington, J. Phys. B 26, 157 (1993).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

J. Sugar and C. Corliss, J. Phys. Chem. Ref. Data,  14 Suppl. 2, 147 (1985).

Phys. Rev. (1)

P.-O. Löwdin, Phys. Rev. 97, 1474 (1955).
[CrossRef]

Phys. Rev. A (9)

F. Robicheaux and C. H. Greene, Phys. Rev. A 46, 3821 (1992).
[CrossRef] [PubMed]

Q. Wang and C. H. Greene, Phys. Rev. A 44, 1874 (1991).
[CrossRef] [PubMed]

F. Robicheaux, Phys. Rev. A 48, 4162 (1993).
[CrossRef] [PubMed]

R. P. Wood, C. H. Greene, and D. Armstrong, Phys. Rev. A 47, 229 (1993).
[CrossRef] [PubMed]

C. H. Greene and M. Aymar, Phys. Rev. A 44, 1773 (1991).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4429 (1993).
[CrossRef] [PubMed]

F. Robicheaux and C. H. Greene, Phys. Rev. A 48, 4441 (1993).
[CrossRef] [PubMed]

D. J. Armstrong and F. Robicheaux, Phys. Rev. A 48, 4450 (1993).
[CrossRef] [PubMed]

C. H. Greene, Phys. Rev. A 32, 1880 (1985).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983).
[CrossRef]

Z. Phys. D (1)

W. Stachowska, M. Fabiszisky, and J. Dembczynski, Z. Phys. D 32, 27 (1994).
[CrossRef]

Other (4)

M. Aymar, C. H. Green, and E. Luc-Koenig, “Multichannel spectroscopy of complex atoms,” Rev. Mod. Phys. (to be published); G. Miecznik, C. H. Greene, and F. Robicheaux, Phys. Rev. A 51, 513 (1995).
[CrossRef] [PubMed]

C. Froese Fischer, The Hartree-Fock Method for Atoms (Wiley, New York, 1977).

C. H. Greene, Fundamental Processes of Atomic Dynamics (Plenum, New York, 1988), p. 105; C. H. Greene and L. Kim, Phys. Rev. A 38, 5953 (1988).
[CrossRef] [PubMed]

R. D. Cowan, The Theory of Atomic Structure and Spectra (U. California Press, Berkeley, Calif., 1981), p. 180.

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Figures (15)

Fig. 1
Fig. 1

Positions of experimental and theoretical LS-term values of Ti+ relative to the ground term 3d24s4F. Only states below 25 000 cm−1 (even-parity states only) are included in this figure. Numerical values are compared in Table 1.

Fig. 2
Fig. 2

Radial Hartree–Fock 3d and 4s orbitals plotted for 3d24s (solid curves), 3d3 (dotted curve) and 3d4s2 (dashed curves) configurations in Ti+. The scale on the horizontal axis is evenly distributed in r.

Fig. 3
Fig. 3

Quantum defects μd in 3d3(4F)d3F channels illustrating effects of the target correlation (on the scale of outer-electron energy). The solid curve is for a single-configuration Hartree–Fock target, the dashed curve is for a calculation that incudes 3d3, 3d2(3P)4d, and 3d2(3F)4d components in the target CI expansion, and the dotted curve represents a CI calculation with additional 3d4d2, 3d4p2, and 3d4f2 configurations in the target.

Fig. 4
Fig. 4

Quantum defects μd in 3d24s(4F)d3G channels illustrating effects of the target polarization. Solid curve, a single-configuration Hartree–Fock target; dashed curve, adding closed channels (3d4s4p + 3d24p)p3G and (3d4s4p + 3d24p)f3G; dotted curve, adding extra 3d4s4fp3G and 3d4s4ff3G closed channels. Lowering of the dashed and dotted curves at the threshold is caused by a short-range perturber 3d24p2 3G.

Fig. 5
Fig. 5

Experimental (upper curve) and total convolved theoretical (lower curves) cross sections for photoionization of the 3d24s4p G 3 4 o level, plotted on energy scale relative to the titanium ground level 3d24s2 3F2. The two theoretical lines are length- and velocity-gauge calculations.

Fig. 6
Fig. 6

Perturbation in the autoionizing Rydberg series in Fig. 5. The solid curves are the real part of quantum defects in the 3d24s(4F7/2)d3/2 and 3d24s(4F7/2)d5/2 channels for J = 5, obtained by approximate solution of Eq. (8). They are plotted on the scale of effective quantum numbers derived from the 3d24s 4F9/2 threshold. In (b) we also compare experimental (stars) and theoretical (diamonds) positions of resonances resolved in Fig. 5. The latter are obtained with the time-delay matrix analysis. Notice the broad perturber near ν9/2 = 24 associated with the 3d24s(2F)6d 1H5 resonance.

Fig. 7
Fig. 7

J = 5 partial cross sections for photoionization of 3d24s4p G 3 4 o compared between a 6-channel calculation [upper curves, including only 3d24s( F 4 J c)l channels] and a 15-channel calculation [lower curves, including additionally 3d3( F 4 J c)l and 3d24s( F 2 J c) channels]. The enhancement of the cross section in the lower traces and the change in resonance features is caused by the 3d24s(2F)6d 1H5 perturber. Solid curves, length-gauge calculations; dotted curves, velocity-gauge calculations.

Fig. 8
Fig. 8

Experimental (upper curve) and total convolved theoretical (lower curve) cross sections for photoionization of the 3d24s4p G 3 3 o level, plotted as a function of energy relative to the titanium ground level 3d24s 3F2.

Fig. 9
Fig. 9

Lu–Fano plot illustrating perturbation in the J = 4 partial photoionization cross section of the 3d24s4p G 3 3 o level in the energy range near 3d24s 4F5/2. The solid lines represent quantum defects in the 3d24s(4F3/2)d5/2 channel as a function of effective quantum numbers associated with the 3d24s 4F5/2 threshold. Diamonds show positions of resonances obtained through a search for maxima of the eigenphase sum; stars are experimental positions. The broad perturber near ν5/2 = 17 is 3d24s(2F)s 3F.

Fig. 10
Fig. 10

Experimental (upper curve) and convolved theoretical (lower curves) cross sections for photoionization of the 3d24s4p D 3 2 o level, plotted as a function of energy relative to the titanium ground level 3d24s2 3F2.

Fig. 11
Fig. 11

Experimental (upper curve) and theoretical (lower curve) fluorescence yield in photoionization of 3d34p F 5 5 o, plotted as a function of energy relative to the titanium ground level.

Fig. 12
Fig. 12

Comparison of autoionization probabilities of closed channels 3d24s(4F7/2)d (solid lines) and 3d3(4F9/2)d (dashed and dashed–dotted lines) for J = 5 and J = 6 partial waves, respectively. The energy is calculated relative to either 3d24s 4F7/2 (solid lines) or 3d3 4F9/2 (dashed and dashed–dotted lines) thresholds. 3d24s(4F7/2)d (J = 5) can autoionize into 3d24s(4F5/2)d channels, whereas 3d3(4F9/2)d (J = 6) autoionizes into 3d24s(4F7/2)d and, if allowed by energy conservation, into the 3d3(4F7/2)d continuum. Notice how opening of the 3d3(4F7/2)d5/2 channel, indicated by vertical arrows, increases the autoionization rate of the closed channel 3 d 3 ( F 9 / 2 4 ) d 3 / 2 .

Fig. 13
Fig. 13

Theoretical total photoionization cross sections (unconvolved) of different initial levels for final-states energies reaching 8000 cm−1 relative to Ti+ ground level 3d24s 4F3/2. Vertical arrows mark energies of fine-structure ionization thresholds (Ref. 20). (a) 3d24s4p D 3 2 o, (b) 3d24s4p G 3 3 o, (c) 3d24s4p G 3 4 o, (d) 3d34p F 4 5 o.

Fig. 14
Fig. 14

Perturbations in the autoionizing Rydberg series 3d24s(2F)l in the photoionization of the 3d24s4p D 3 2 o initial level. The top curve shows the total cross section obtained with artificially opened 3d24s( F 4 J c)l and 3d3( F 4 J c)l channels (initially closed). For the lower curve we additionally opened 3d24s( F 2 J c)l channels. All perturbers in the lower cross section are associated with Ti+ states above 3d24s 4F7/2 and are classified in Table 5. Both cross sections are calculated here in the length gauge.

Fig. 15
Fig. 15

Perturbations in the photoionization cross section of 3d24s4p G 3 4 o. Cross sections in (a) and (b) (in length gauge) are obtained in the same way as the analogous cross sections in Fig. 14. Perturbers identified in this spectrum are classified in Table 5.

Tables (5)

Tables Icon

Table 1 Theoretical and Experimental Term-Average Energies (cm−1) of Even-Parity Ti+ States Relative to the Ground Level 3d24s 4F3/2

Tables Icon

Table 2 Dominant Components and Their Weights (square of the coefficient) in CI Expansions for the Four Lowest Target States

Tables Icon

Table 3 Theoretical Energies and Effective Quantum Numbers for the Initial Odd-Parity States Compared with Experimental Dataa

Tables Icon

Table 4 Theoretical and Experimental Energies (cm−1) and Effective Quantum Numbers of 3d24s(4F7/2) d Autoionizing Series in the Photoionization Cross Section of the 3d24s4p G 3 4 o Level

Tables Icon

Table 5 Classification of Some Prominent Perturbers Identified in Photoionization Cross Sections of 3d24s4p D 3 2 o and 3d24s4p G 3 4 o

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

[ 1 2 d 2 d r 2 + V eff + l ( l + 1 ) 2 r 2 ] P nl ( r ) = P nl ( r ) , V eff ( r ) = Z eff + ( Z Z eff ) exp ( α 1 l r ) r α 2 l exp ( α 3 l r ) + V pol ( r ) , V pol ( r ) = α d { 1 exp [ ( r / r c ) 3 ] } 2 2 r 4 .
Ψ HF ( P nl ) | H | Ψ HF ( P nl P n l ) = 0 .
[ 1 2 d 2 d r 2 + l ( l + 1 ) 2 r 2 Z r + V H ( r ) ] P nl ( r ) = E nl P nl ( r ) + n λ nn P n l ( r ) , V H ( r ) = n q n l [ 0 r 1 r > P n l 2 ( s ) d s ] ,
Ψ i ( r ) = 1 r j Φ j ( ω ) [ f j ( r ) δ ij g j ( r ) K ij ] ,
K phys = K oo K oc ( K cc + tan π ν ) 1 K co ,
P 1 , S 3 , P 3 , D 3 , P 5 , D 5 , F 5 J = 1 D 1 , P 3 , D 3 , F 3 , P 5 , D 5 , F 5 , G 5 J = 2 F 1 , D 3 , F 3 , G 3 , P 5 , D 5 , F 5 , G 5 , H 5 J = 3 G 1 , F 3 , G 3 , H 3 , D 5 , F 5 , G 5 , H 5 J = 4 H 1 , G 3 , H 3 , I 3 , F 5 , G 5 , H 5 J = 5 I 1 , H 3 , I 3 , K 3 , G 5 , H 5 J = 6
S 1 + i K 1 i K ,
S phys = S oo S oc [ S cc exp ( 2 i π ν ) ] 1 S co ,
det [ S cc exp ( 2 i π ν ) ] = 0
E E 7 / 2 1 2 ( n Re μ 7 / 2 ) 2 i 2 2 Im μ 7 / 2 ( n Re μ 7 / 2 ) 3 .
I = I 0 ( 1 σ W ћ ω ) .
1 r ij = k = 0 k = r < k r > k + 1 C i ( k ) · C j k .

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