bebb_1966#

H. B. Bebb and A. Gold, “Multiphoton ionization of hydrogen and rare-gas atoms”, Phys. Rev. 143, 1-24 (1966)

Source record · Catalogue · Review register

Source kind: theory

Recorded technique: Lowest-order perturbation theory of N-photon ionisation (evolution operator, quantised single-mode field, dipole approximation, light linearly polarised along z: Eq. 23). The N-th order amplitude (Eq. 24) is reduced by the average-energy (closure) approximation: the sums over intermediate states are ‘collapsed’ with an average frequency (Eqs. 25-28), K = <f|z^N|g> / prod (omega_v - nu omega + i gamma/2), except in the order(s) holding a near resonance, whose states are summed explicitly, the rest of that sum being taken at a second average energy hbar Omega = 1.2 hbar omega_I (Eq. 42). hbar omega_v is the first excitation energy (10.2 eV for H, where direct sums, Appendix B and Table II, support it). Hydrogen: hydrogenic bound and Coulomb continuum states, N = 2 to 12, photon energies 1.14 to 13.4 eV (Figs. 2-13). Rare gases (Sec. IV): ‘rather crude’ scaled hydrogenic wave functions (the outer orbit’s Bohr radius scaled to the atomic radius through an effective Z, Table I) and scaled Coulomb continua, Eq. (43), only near the ruby photon energy 1.785 eV: Xe 7, Kr 8, Ar 9, Ne 13 and He 14 photons (Figs. 15-19), with level energies from Moore’s tables (Ref. 37). Phenomenological widths gamma/2 enter the denominators only at resonances.

Materials in structured entries: Xe

Dataset quantities: multiphoton_ionization_cross_section

Contents: 1 dataset; 0 formulas.

Curation notes#

Journal pp. 1-24 = PDF pages 1-24. Identity checked on the first page (PDF p. 1: The Physical Review, Second Series, Vol. 143, No. 1, 4 March 1966; ‘Multiphoton Ionization of Hydrogen and Rare-Gas Atoms’, H. Barry Bebb and Albert Gold, Institute of Optics, University of Rochester; received 8 October 1965) on 2026-10-09. The PDF is a scan: every page is one embedded image, 4810 x 6457 pixels, with an OCR text layer; every figure is raster (‘reproduced directly from the digital plotter output’, Fig. 2 caption). Library identity VALIDATED_WITH_WARNINGS (unmatched captions only).

THE QUANTITY. Eqs. (14)-(18) and (39): the rate per atom w^(N) = (m/hbar)/(2 pi)^2 (2 pi alpha F omega)^N |<k|tau^N|g>|^2 k (integrated over the electron’s direction), with ‘F (No. photons cm^-2 sec^-1) ... the photon flux’, the occupation number of the one field mode replaced through F (p. 4); Gaussian units. The figures plot w/F^N (‘the label W/(F^N) on the ordinate should be read as w_fg^(N)/F^N’, p. 10), i.e. the generalised cross section sigma_N in cm^(2N) s^(N-1), W = sigma_N F^N. This is lambropoulos_1976’s Eq. (20) for sigma_N (same prefactor (2 pi alpha)^N / (4 pi^2) (m K / hbar) omega^N). Eq. (43) is its numerical form for the scaled rare gases, w = Z^-3 (1.412e12) [0.84564e-33 Z^-2 F omega_eV]^N |<k|tau^N|g>|^2 k (matrix elements in (a0/Z)^(N+3/2), k in 1/a0).

PHOTON STATISTICS (pp. 21-22). The rates are for a single-mode field, <F^N> = <F>^N (Eq. 44); for multimode light ‘similar to a thermal source (Gaussian light)’, <F^N> = N! <F>^N (Eq. 45): ‘the transition rate is increased by N! over that produced by a single-mode laser with the same average power’, and the effective flux may exceed the measured average ‘by as much as (N!)^(1/N)’.

WHAT IS TAKEN: the one rare-gas generalised cross section printed as a number, Xe seven-photon at the ruby line (p. 22, Sec. V): Voronov and Delone ‘find a “cross section” w/F^7, of ~5 x 10^-206, in good agreement with the value 4 x 10^-206 predicted from theory’. The theory value is the dataset sigma7_Xe. CHECKED against Fig. 19(b) (page image at native resolution, during review; not a dataset): the five y ticks 1e-201 to 1e-205 lie on one log scale, 238.2 pixels per decade (residuals under 0.003 decade), the x ticks 1.75-1.78 eV on 301.1 pixels per 0.01 eV; the curve at 1.785 eV is at 10^-205.32 = 4.8e-206 (line 7 pixels, +-0.015 decade), so the printed 4e-206 agrees with the drawing to 0.08 decade.

NOT TAKEN. (a) Voronov and Delone’s measured ~5 x 10^-206 (G. S. Voronov and N. B. Delone, Zh. Eksp. Teor. Fiz. Pis’ma Red. 1, 66 (1965) [JETP Lett. 1, 66 (1965)]): reproduced, the original not in the library (data rule 6). (b) Hydrogen (Figs. 2-13, 2-12 photons from 1s; Fig. 13, two-photon ionisation of 2S H against Zernik’s exact calculation; the text’s 8-photon 1s rate at 1.82 eV, 2.5 x 10^-235 F^8 s^-1 for gamma/2 = 1e-5 eV, p. 11; the abstract’s ~10^-244 F^8 at 1.78 eV): H is not a material. (c) The rare gases’ dispersion curves w/F^N near 1.785 eV (Figs. 15(b), 16(b), 17(b), 18(b), 19(b)) and threshold fluxes (panels c): raster only, of a method that lambropoulos_1976 (p. 12) calls ‘an order of magnitude calculation’, at seven to fourteen photons; not digitised (no printed number but Xe’s to check them against). (d) Table I (tbl_59a6439e8eba, p. 14; image checked): per gas N, hbar omega_I, hbar omega_v, atomic radius, Z, the order nu of the near resonance and the levels summed explicitly with their energies: He 14, 24.580, 21.0 eV, 0.93 A, 0.57 (nu = 13: 1s3p 1P1 23.082, 4p 1P1 23.737, 4f 1F3 23.732, 5f 1F3 24.038; nu = 12: 1s2s 1S0 20.611); Ne 13, 21.559, 17.0, 1.12, 0.96 (12: 10p[1/2]0 21.4038, 11p[3/2]2 21.4294; 11: 4s’[1/2]1 19.7749); Ar 9, 15.755, 11.8, 1.54, 1.03 (8: 5p[1/2]1 14.46, 5p[3/2]2 14.51); Kr 8, 13.996, 11.3, 1.69, 1.25 (7: 6s[3/2]2 12.3494, 6s[3/2]1 12.3827, 4d[5/2]2 12.2552, 4d[5/2]3 12.2816, 4d[3/2]1 12.3518); Xe 7, 12.127, 9.6, 1.90, 1.4 (6: 7p[1/2]1 10.8995). The energies are Moore’s (reproduced; NIST’s lists are curated, TC-983); hbar omega_v and Z are inputs chosen for this model, not fitted constants (not ionization_model_parameter) nor properties of the gas. (e) The initiation fluxes (one electron from 1e20 atoms cm^-3 in 1e-8 cm^3 in 10 ns, i.e. a rate of 1e-4 s^-1 per atom): no quantity, and they depend on the assumed focal volume and pulse. (f) Table II (average energies of hydrogen) and the Appendix A table (angular bookkeeping).

KRYPTON (OEP 6, Q2): the paper gives no two-photon ionisation value of any rare gas, neither table nor figure. Its only Kr result is eight-photon, Fig. 18(b) (and the threshold flux, 18(c)), raster, 1.76 to 1.80 eV, w/F^8 between 1e-234 and 1e-229, with seventh-order resonances on the 6s and 4d levels of the 2P3/2 core near 12.35 eV. (pindzola_1975’s notes describe this paper as ‘H and the rare gases, 2-12 photons’: the 2-12 photons are hydrogen’s; the rare gases are 7 to 14 photons at 1.785 eV.)

INCONSISTENCIES, none touching the dataset. (1) Kr (p. 15): ‘the lower energy (hbar omega = 1.784) peak arising from the two nearly degenerate levels 6s[3/2]2 and 4d[3/2]1 and the higher energy (hbar omega ~ 1.766) peak arising from the 6s[3/2]1 level’; Fig. 18(b) draws the two peaks near 1.764 and 1.769 eV, as Table I’s energies over seven give (12.3494/7 = 1.7642, 12.3518/7 = 1.7645, 12.3827/7 = 1.7690 eV): ‘1.784’ reads as 1.764 and ‘~1.766’ as ~1.769. (2) Fig. 18(a) labels the Kr limit 13.966 eV; Table I has 13.996. (3) The abstract gives initiation fluxes ‘~10^29 cm^-2 sec^-1 for Xe, Kr, and Ar’; Sec. IV.2.c (p. 14) says ‘comparable initiation fluxes (~5 x 10^28 photons cm^-2 sec^-1)’.

OTHER CALCULATIONS AND MEASUREMENTS CITED (not in the library): A. Gold and H. B. Bebb, Phys. Rev. Lett. 14, 60 (1965) (the earlier rare-gas estimate, with hbar omega_v at the ionisation energy, ‘somewhat too small’, p. 7); W. Zernik, Phys. Rev. 135, A51 (1964) (two-photon ionisation of 2S H); G. S. Voronov and N. B. Delone (above, Xe at the ruby line).

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets#

bebb_1966/sigma7_Xe#

{
  "id": "sigma7_Xe",
  "material": "Xe",
  "table": "text",
  "scale": 1e-206,
  "quantity": "multiphoton_ionization_cross_section",
  "unit": "cm14 s6",
  "abscissa_unit": "eV",
  "use": "check",
  "parameters": {
    "photons": 7,
    "polarization": "linear",
    "method": "lowest-order perturbation theory, Bebb and Gold's average-energy (closure) approximation: the sixth-order sum over the near-resonant 7p[1/2]1 level (10.8995 eV) done explicitly, the lower orders collapsed at hbar omega_v = 9.6 eV, the rest at hbar Omega = 1.2 hbar omega_I; scaled hydrogenic orbitals (Z = 1.4, atomic radius 1.90 A) and a scaled Coulomb continuum (Eqs. 42-43, Table I); 'computed in several different approximations ... the different situations lead to consistent answers' (p. 21)",
    "definition": "w = sigma F^7: w the ionisation rate per atom (s^-1), F the photon flux (photons cm^-2 s^-1) of a single-mode field, <F^N> = <F>^N (Eqs. 17-18, 39, 44); for chaotic (Gaussian) light the rate is 7! times larger at the same average flux (Eq. 45)",
    "photon_statistics": "single mode (coherent)",
    "ionization_energy_ev": 12.127
  }
}

1 inline points in the source record.

p. 22 (Sec. V): ‘the value 4 x 10^-206 predicted from theory’ for the seven-photon ‘cross section’ w/F^7 of Xe at the ruby line, 1.785 eV (Sec. IV; Fig. 19(b) marks ‘Ruby 6943 A line’, an air wavelength). Agrees with Fig. 19(b) read at 1.785 eV, 4.8e-206 (the record’s notes). No uncertainty is stated; the method is an order-of-magnitude one (lambropoulos_1976, p. 12). The near resonance 6 hbar omega ~ 7p[1/2]1 (10.8995 eV) lies at 1.8166 eV, above the ruby line.

Formulas#

No formulas are recorded.

Corrections#

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.