Abstract

The properties of entire spectra—specifically, the line positions and transition probabilities of perturbed series and of their adjoining continua as well as autoionization line profiles and related collision cross sections—have been interpreted in terms of a few parameters, such as quantum defects and oscillator-strength densities, which vary slowly with the excitation energy. By use of Seaton’s quantum-defect theory, the concept of separate configurations is replaced by the more comprehensive concept of channels. Alternative sets of channels are considered, related by a frame transformation whose construction is a main goal of the analysis of each spectrum. Concepts and procedures employed for these purposes since 1969 are summarized and restated here. Reference is made to extension of the treatment to negative ions, and to the direct ab initio calculation of the channel parameters to be compared with experimental results. Illustrative results are presented, as well as a guide to relevant literature.

© 1975 Optical Society of America

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References

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  1. M. J. Seaton, Proc. Phys. Soc. (Lond.) 88, 801 (1966).
    [Crossref]
  2. U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
    [Crossref]
  3. U. Fano, Phys. Rev. A 2, 353 (1970).
    [Crossref]
  4. G. Herzberg, Phys. Rev. Lett. 25, 1081 (1969).
    [Crossref]
  5. G. Herzberg and Ch. Jungen, J. Molec. Spectrosc. 41, 425 (1972).
    [Crossref]
  6. O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
    [Crossref]
  7. D. Dill, Phys. Rev. A 6, 160 (1972).
    [Crossref]
  8. D. Dill, E. S. Chang, and U. Fano, in Electronic and Atomic Collisions, VIII ICPEAC, Beograd, Yugoslavia, 1973, edited by B. C. Čobić and M. V. Kurepa (Institute of Physics, Beograd, Yugoslavia), p. 536.
  9. O. Atabek and Ch. Jungen, private communication.
  10. E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).
    [Crossref]
  11. R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972); E. S. Chang, Phys. Rev. Lett. 33, 1644 (1974).
    [Crossref]
  12. K. T. Lu, Phys. Rev. A 4, 579 (1971).
    [Crossref]
  13. K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).
    [Crossref]
  14. D. Dill, Phys. Rev. A 7, 1976 (1973).
    [Crossref]
  15. J. A. R. Samson and J. L. Gardner, Phys. Rev. Lett. 31, 1327 (1973).
    [Crossref]
  16. A. F. Starace, J. Phys. B 6, 76 (1973).
    [Crossref]
  17. C. M. Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).
    [Crossref]
  18. J. Geiger, in Vacuum Ultraviolet Radiation Physics, edited by E. Koch, R. Haensel, and C. Kunz (Vieweg, Braunschweig, 1974), p. 28.
  19. C. D. Lin, Astrophys. J. 187, 385 (1974).
    [Crossref]
  20. K. T. Lu, J. Opt. Soc. Am. 64, 706 (1974).
    [Crossref]
  21. K. T. Lu, in Ref. 18, p. 23.
  22. C. M. Brown, S. G. Tilford, and M. L. Ginter, J. Opt. Soc. Am. 65, 385 (1975).
    [Crossref]
  23. A. R. P. Rau and U. Fano, Phys. Rev. A 4, 1751 (1971).
    [Crossref]
  24. A. R. P. Rau, in Electron and Photon Interaction with Atoms, edited by H. Kleinpoppen and M. R. C. McDowell (Plenum, New York, 1975).
  25. C. M. Lee, Phys. Rev. A 11, 1692 (1975).
    [Crossref]
  26. See, e.g., D. L. Moores, Proc. Phys. Soc. (Lond.) 91, 830 (1967); H. E. Saraph and M. J. Seaton, Phil. Trans. R. Soc. Lond. A 271, 1 (1971).
    [Crossref]
  27. U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973); C. M. Lee, Phys. Rev. A 10, 584 (1974).
    [Crossref]
  28. C. M. Lee, Phys. Rev. A 10, 1598 (1974).
    [Crossref]

1975 (2)

1974 (4)

C. M. Lee, Phys. Rev. A 10, 1598 (1974).
[Crossref]

C. D. Lin, Astrophys. J. 187, 385 (1974).
[Crossref]

K. T. Lu, J. Opt. Soc. Am. 64, 706 (1974).
[Crossref]

O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
[Crossref]

1973 (5)

D. Dill, Phys. Rev. A 7, 1976 (1973).
[Crossref]

J. A. R. Samson and J. L. Gardner, Phys. Rev. Lett. 31, 1327 (1973).
[Crossref]

A. F. Starace, J. Phys. B 6, 76 (1973).
[Crossref]

C. M. Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).
[Crossref]

U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973); C. M. Lee, Phys. Rev. A 10, 584 (1974).
[Crossref]

1972 (4)

G. Herzberg and Ch. Jungen, J. Molec. Spectrosc. 41, 425 (1972).
[Crossref]

D. Dill, Phys. Rev. A 6, 160 (1972).
[Crossref]

E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).
[Crossref]

R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972); E. S. Chang, Phys. Rev. Lett. 33, 1644 (1974).
[Crossref]

1971 (2)

K. T. Lu, Phys. Rev. A 4, 579 (1971).
[Crossref]

A. R. P. Rau and U. Fano, Phys. Rev. A 4, 1751 (1971).
[Crossref]

1970 (2)

K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).
[Crossref]

U. Fano, Phys. Rev. A 2, 353 (1970).
[Crossref]

1969 (1)

G. Herzberg, Phys. Rev. Lett. 25, 1081 (1969).
[Crossref]

1968 (1)

U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

1967 (1)

See, e.g., D. L. Moores, Proc. Phys. Soc. (Lond.) 91, 830 (1967); H. E. Saraph and M. J. Seaton, Phil. Trans. R. Soc. Lond. A 271, 1 (1971).
[Crossref]

1966 (1)

M. J. Seaton, Proc. Phys. Soc. (Lond.) 88, 801 (1966).
[Crossref]

Atabek, O.

O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
[Crossref]

O. Atabek and Ch. Jungen, private communication.

Brown, C. M.

Chang, E. S.

E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).
[Crossref]

R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972); E. S. Chang, Phys. Rev. Lett. 33, 1644 (1974).
[Crossref]

D. Dill, E. S. Chang, and U. Fano, in Electronic and Atomic Collisions, VIII ICPEAC, Beograd, Yugoslavia, 1973, edited by B. C. Čobić and M. V. Kurepa (Institute of Physics, Beograd, Yugoslavia), p. 536.

Cooper, J. W.

U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

Dill, D.

O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
[Crossref]

D. Dill, Phys. Rev. A 7, 1976 (1973).
[Crossref]

D. Dill, Phys. Rev. A 6, 160 (1972).
[Crossref]

D. Dill, E. S. Chang, and U. Fano, in Electronic and Atomic Collisions, VIII ICPEAC, Beograd, Yugoslavia, 1973, edited by B. C. Čobić and M. V. Kurepa (Institute of Physics, Beograd, Yugoslavia), p. 536.

Fano, U.

U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973); C. M. Lee, Phys. Rev. A 10, 584 (1974).
[Crossref]

E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).
[Crossref]

A. R. P. Rau and U. Fano, Phys. Rev. A 4, 1751 (1971).
[Crossref]

U. Fano, Phys. Rev. A 2, 353 (1970).
[Crossref]

K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).
[Crossref]

U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

D. Dill, E. S. Chang, and U. Fano, in Electronic and Atomic Collisions, VIII ICPEAC, Beograd, Yugoslavia, 1973, edited by B. C. Čobić and M. V. Kurepa (Institute of Physics, Beograd, Yugoslavia), p. 536.

Gardner, J. L.

J. A. R. Samson and J. L. Gardner, Phys. Rev. Lett. 31, 1327 (1973).
[Crossref]

Geiger, J.

J. Geiger, in Vacuum Ultraviolet Radiation Physics, edited by E. Koch, R. Haensel, and C. Kunz (Vieweg, Braunschweig, 1974), p. 28.

Ginter, M. L.

Henry, R. J. W.

R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972); E. S. Chang, Phys. Rev. Lett. 33, 1644 (1974).
[Crossref]

Herzberg, G.

G. Herzberg and Ch. Jungen, J. Molec. Spectrosc. 41, 425 (1972).
[Crossref]

G. Herzberg, Phys. Rev. Lett. 25, 1081 (1969).
[Crossref]

Jungen, Ch.

O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
[Crossref]

G. Herzberg and Ch. Jungen, J. Molec. Spectrosc. 41, 425 (1972).
[Crossref]

O. Atabek and Ch. Jungen, private communication.

Lee, C. M.

C. M. Lee, Phys. Rev. A 11, 1692 (1975).
[Crossref]

C. M. Lee, Phys. Rev. A 10, 1598 (1974).
[Crossref]

U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973); C. M. Lee, Phys. Rev. A 10, 584 (1974).
[Crossref]

C. M. Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).
[Crossref]

Lin, C. D.

C. D. Lin, Astrophys. J. 187, 385 (1974).
[Crossref]

Lu, K. T.

K. T. Lu, J. Opt. Soc. Am. 64, 706 (1974).
[Crossref]

C. M. Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).
[Crossref]

K. T. Lu, Phys. Rev. A 4, 579 (1971).
[Crossref]

K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).
[Crossref]

K. T. Lu, in Ref. 18, p. 23.

Moores, D. L.

See, e.g., D. L. Moores, Proc. Phys. Soc. (Lond.) 91, 830 (1967); H. E. Saraph and M. J. Seaton, Phil. Trans. R. Soc. Lond. A 271, 1 (1971).
[Crossref]

Rau, A. R. P.

A. R. P. Rau and U. Fano, Phys. Rev. A 4, 1751 (1971).
[Crossref]

A. R. P. Rau, in Electron and Photon Interaction with Atoms, edited by H. Kleinpoppen and M. R. C. McDowell (Plenum, New York, 1975).

Samson, J. A. R.

J. A. R. Samson and J. L. Gardner, Phys. Rev. Lett. 31, 1327 (1973).
[Crossref]

Seaton, M. J.

M. J. Seaton, Proc. Phys. Soc. (Lond.) 88, 801 (1966).
[Crossref]

Starace, A. F.

A. F. Starace, J. Phys. B 6, 76 (1973).
[Crossref]

Tilford, S. G.

Astrophys. J. (1)

C. D. Lin, Astrophys. J. 187, 385 (1974).
[Crossref]

J. Molec. Spectrosc. (1)

G. Herzberg and Ch. Jungen, J. Molec. Spectrosc. 41, 425 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

J. Phys. B (1)

A. F. Starace, J. Phys. B 6, 76 (1973).
[Crossref]

Phys. Rev. A (11)

C. M. Lee and K. T. Lu, Phys. Rev. A 8, 1241 (1973).
[Crossref]

E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).
[Crossref]

R. J. W. Henry and E. S. Chang, Phys. Rev. A 5, 276 (1972); E. S. Chang, Phys. Rev. Lett. 33, 1644 (1974).
[Crossref]

K. T. Lu, Phys. Rev. A 4, 579 (1971).
[Crossref]

K. T. Lu and U. Fano, Phys. Rev. A 2, 81 (1970).
[Crossref]

D. Dill, Phys. Rev. A 7, 1976 (1973).
[Crossref]

D. Dill, Phys. Rev. A 6, 160 (1972).
[Crossref]

U. Fano, Phys. Rev. A 2, 353 (1970).
[Crossref]

A. R. P. Rau and U. Fano, Phys. Rev. A 4, 1751 (1971).
[Crossref]

C. M. Lee, Phys. Rev. A 11, 1692 (1975).
[Crossref]

C. M. Lee, Phys. Rev. A 10, 1598 (1974).
[Crossref]

Phys. Rev. Lett. (4)

U. Fano and C. M. Lee, Phys. Rev. Lett. 31, 1573 (1973); C. M. Lee, Phys. Rev. A 10, 584 (1974).
[Crossref]

G. Herzberg, Phys. Rev. Lett. 25, 1081 (1969).
[Crossref]

O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974).
[Crossref]

J. A. R. Samson and J. L. Gardner, Phys. Rev. Lett. 31, 1327 (1973).
[Crossref]

Proc. Phys. Soc. (Lond.) (2)

M. J. Seaton, Proc. Phys. Soc. (Lond.) 88, 801 (1966).
[Crossref]

See, e.g., D. L. Moores, Proc. Phys. Soc. (Lond.) 91, 830 (1967); H. E. Saraph and M. J. Seaton, Phil. Trans. R. Soc. Lond. A 271, 1 (1971).
[Crossref]

Rev. Mod. Phys. (1)

U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968).
[Crossref]

Other (5)

D. Dill, E. S. Chang, and U. Fano, in Electronic and Atomic Collisions, VIII ICPEAC, Beograd, Yugoslavia, 1973, edited by B. C. Čobić and M. V. Kurepa (Institute of Physics, Beograd, Yugoslavia), p. 536.

O. Atabek and Ch. Jungen, private communication.

J. Geiger, in Vacuum Ultraviolet Radiation Physics, edited by E. Koch, R. Haensel, and C. Kunz (Vieweg, Braunschweig, 1974), p. 28.

K. T. Lu, in Ref. 18, p. 23.

A. R. P. Rau, in Electron and Photon Interaction with Atoms, edited by H. Kleinpoppen and M. R. C. McDowell (Plenum, New York, 1975).

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Figures (9)

FIG. 1
FIG. 1

Block diagram of alternative procedures for including effect of electron, or other, internal correlations in the theory of Rydberg levels.

FIG. 2
FIG. 2

Radial wave function of an excited electron in the outer, Coulomb field, region of an atom. - - - - -, Coulomb field extends into inner core; —, non-Coulomb field in inner core.

FIG. 3
FIG. 3

Traditional diagram of upper Rydberg levels with J ≤ 1 of para H2, showing ionization thresholds that correspond to two allowed rotational levels of para H 2 + (Ref. 5). Levels above N = 0 threshold are greatly broadened by autoionization (see Fig. 4). Vertical hatching above the thresholds indicates continuous spectrum. Actual mixing of σ and π character of levels is not shown here.

FIG. 4
FIG. 4

Absorption spectrum of para H2 near the ionization thresholds. I0 to the right and I2 farther to the left. The quantum numbers assigned to the lines do not coincide with the parameters ν0 and ν2 of the present paper. Note the autoionization profiles to the left of the first threshold and the intense perturbing lines labeled with vibrational indices v ≠ 0 (Ref. 5).

FIG. 5
FIG. 5

Diagram analogous to that of Fig. 3 showing the mixing of σ and π channels induced by rotational uncoupling of the Rydberg orbit in different ranges of the spectrum. Each level, n, is represented by a vector A n, whose components in the σ and π direction are coefficients of the σ and π components of the level’s eigenfunction. The intensity of the nth absorption line is proportional to ( A n · D ) 2; the excitation dipole D is nearly parallel to the N = 0 axis according to Ref. 5. The vectors A n remain parallel to the σ or π axis in the lower spectrum but spiral around in the higher spectrum, causing alternations of the intensity factor ( A n · D ) 2. The spiralling becomes continuous, at a nonuniform rate, in the autoionization region, causing the alternation of intensity shown in Fig. 4. The intensity in the two continuum channels above I2 is determined by shifting the phases of the dipole components Dσ and Dπ by πμσ and πμπ and projecting onto the axis N = 0 or N = 2. Note that the vertical scale is linear in ν2. (Courtesy C. E. Theodosiou.)

FIG. 6
FIG. 6

Comparison of experimental levels of H2 absorption spectrum (v = 0 → v′ = 1 transitions) with Eqs. (6). The two sections of the rising sigmoid represent Eq. (6b) with μσ = 0.21 and μπ = −0.06. The descending straight segments represent the additional relationship between ν0 and ν2 established by Eq. (6a); different segments correspond to ranges of ν0 and ν2 with different integer parts (note that scales represent only the decimal part). The length of each segment represents an error bar, proportional to ν3. Experimental points should lie at intersections of the curve and segments except for errors due to disregarding vibrational coupling; estimated corrections shift the raw data points (triangles) to positions (circles) closer to the intersection (Ref. 4.)

FIG. 7
FIG. 7

Asymmetry parameter β of angular-distribution law for Xe photoelectrons, as a function of wavelength. (a) Experimental β value shown in relation to experimental photoabsorption resonances (dashed curve). (b) Theoretical β values (solid line) shown in relation to theoretical photoabsorption cross section (dashed curve) (Ref. 15.)

FIG. 8
FIG. 8

Upper: Oscillator strength densities df(ρ)/dE for the three separate eigenchannels of the autoionization spectrum of Ar, obtained by over-all fitting of the absorption spectrum. The observed photoabsorption is given by the sum of the ordinates of the three graphs for each value of ν1/2. Lower: Relative intensities obtained from observed lines classified as belonging to the three eigenchannels (Ref. 17.)

FIG. 9
FIG. 9

Intensity profiles of autoionization lines in the Ar spectrum. Upper: High resolution, relative measurements. Lower: - - - Low resolution, absolute measurements; — ab initio calculation (Ref. 27.)

Tables (4)

Tables Icon

TABLE I Experimental and theoretical energies of J = 1 levels of H2 (in units of cm−1) (Ref. 6). The upper portion gives npπ levels between the ionization limits v = 0 and v = 1; N = 1. The lower portion gives levels of the C Π 1 u - state.

Tables Icon

TABLE II Sucessive extension of quantum-defect equations for a given total angular momentum J.

Tables Icon

TABLE III Relative step heights of photodetachment cross section at various fine-structure thresholds (Ref. 23).

Tables Icon

TABLE IV Parameters for J = 1 states of Ar at lowest ionization threshold. Calculated (semi-empirical) values, from Ref. 27.

Equations (12)

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u ( r ) = f ( r ) cos π μ - g ( r ) sin π μ .
σ ph = π e 2 h m c d f d E ,
sin π [ ν ( E ) + μ ( E ) ] = 0 ,             E = I - 13.6 eV / ν 2 ( E ) .
E n = I - 13.6 eV / [ n - μ ( E n ) ] 2 .
f n = d f d E | E n d E n d n = d f d E | E n 27.2 eV [ n - μ ( E n ) ] 3
E = I 0 - 13.6 eV / ν 0 2 = I 2 - 13.6 eV / ν 2 2 ,
| cos β sin π ( μ σ + ν 0 ) sin β sin π ( μ π + ν 0 ) - sin β sin π ( μ σ + ν 2 ) cos β sin π ( μ π + ν 2 ) | = 0.
cos β exp ( i π μ σ ) D σ + sin β exp ( i π μ π ) D π 2 .
- sin β sin π ( μ σ + ν 2 ) A σ + cos β sin π ( μ π + ν 2 ) A π = 0 , E = I 2 - 13.6 eV / v 2 2 ,
| cos β sin ( π μ σ - δ 0 ) sin β sin ( π μ π - δ 0 ) - sin β sin π ( μ σ + ν 2 ) cos β sin π ( μ π + ν 2 ) | = 0 ,
α = 1 2 3 4 5 p 5 d 3 D 1 ° , p 5 d 1 P 1 ° p 5 d 3 P 1 ° , p 5 s 1 P 1 ° , p 5 s 3 P 1 ° ;
i = 1 2 3 4 5 p 5 ( P 2 3 / 2 ° ) d 5 / 2 , p 5 ( P 2 3 / 2 ° ) d 3 / 2 , p 5 ( P 2 3 / 2 ° ) s 1 / 2 , p 5 ( P 2 1 / 2 ° ) d 3 / 2 , p 5 ( P 2 1 / 2 ° ) s 1 / 2 .