kulander_1987#
K. C. Kulander, “Time-dependent Hartree-Fock theory of multiphoton ionization: Helium”, Phys. Rev. A 36, 2726-2738 (1987)
Source record · Catalogue · Review register
Source kind: theory
Recorded technique: Time-dependent Hartree-Fock (TDHF, time-dependent self-consistent field): the helium ground state as one doubly occupied spatial orbital, propagated on a two-dimensional finite-difference grid in cylindrical coordinates (rho, z), spacing 0.2 a0 (259 x 100 points, boundaries at z = +-25.8 a0 and rho = 20.1 a0 for the 1.064 um case), Peaceman-Rachford alternating-direction propagator, 200-400 steps per optical period, an absorbing imaginary potential at the boundary (pp. 2727-2728). The field: linear polarisation, dipole approximation, length form, V_I = -e E(t) z sin(omega t) (Eq. 2), its envelope rising linearly over five to ten periods and then constant (Eq. 10); a rate is taken from the exponential decay during the constant part. Two models: TDHF, in which the orbital’s binding grows as the atom ionises, so that the rate falls with time and is extrapolated back to the static Hartree-Fock orbital energy (Fig. 12, p. 2733); and a frozen one-electron model, one electron frozen in its ground-state orbital and the other moving in the field of the nucleus and the frozen electron. A renormalised variant (RTDHF) is used at 9, 28 and 30 eV. The initial state, relaxed in imaginary time on the grid: total energy -75.95 eV (Hartree-Fock limit -77.87, exact -78.98), orbital energy -24.28 eV (Hartree-Fock -24.98; the true ionisation potential 24.58 eV; p. 2728).
Materials in structured entries: He
Dataset quantities: ionization_rate, multiphoton_ionization_cross_section, photoionization_cross_section
Contents: 8 datasets; 0 formulas.
Curation notes#
Phys. Rev. A 36, 2726-2738 (1987) = PDF pages 1-13 (journal page = PDF page + 2725). Identity checked on the first page (‘PHYSICAL REVIEW A VOLUME 36, NUMBER 6 SEPTEMBER 15, 1987’; Kenneth C. Kulander, Lawrence Livermore National Laboratory; received 16 March 1987) on 2026-10-09. The PDF is a scan (one 5169 x 7129 image per page) with an OCR text layer. The library’s identity is VALIDATED_WITH_WARNINGS; the copy is the version of record.
WHAT IS TAKEN (tools/extract/kulander_1987.py), all from Table I (p. 2732, tbl_8ef576c11381; every cell checked against the table image), all use = check: (a) ‘Helium ionization rates (s^-1)’ at 1.064 and 0.532 um (peak intensities 5e14, 1e15, 2e15 W/cm2) and at 9.0 eV (1e15 W/cm2), each in the columns ‘1s^2’ and “1s1s’”; (b) the TDHF ‘(generalized) cross sections’: two-photon, in cm4/W2 s, at 13.6 and 17.7 eV, and one-photon, in cm2, at 30.0 eV. Every cell prints its own power of ten; the files hold mantissa e power, as printed.
THE COLUMNS OF TABLE I(a) (the record’s reading; the table does not name them). The text compares the TDHF rate, ‘obtained by extrapolating the linear falloff back to the correct orbital energy’, with ‘the frozen, one-electron results in Table I for several wavelengths and peak intensities’ (p. 2733). ‘1s^2’ is read as TDHF (its one doubly occupied orbital) and “1s1s’” as the frozen model (one electron frozen in 1s, the other active): the frozen rate is the larger in every row, as the text says for 9 eV (the TDHF decay ‘slightly slower’, p. 2734). The text calls the long-wavelength agreement ‘excellent’; the table’s frozen/TDHF ratios are 1.4-2.2. At 9 eV the TDHF decay is not exponential at late times (Rabi oscillations, Fig. 14); how the 1s^2 value was taken there is not stated. RTDHF rates at the longer wavelengths ‘agree exactly with the extrapolated rates shown in Table I’ (p. 2735).
ACCURACY. ‘The calculated ionization rates changed by less than 20% for reasonable variations of the parameters’ (grid spacing, size, time step; p. 2728): taken as relative 0.2, level not stated, for the rates. At the longest wavelength and highest intensities the grid lies inside the ponderomotive barrier, so those rates ‘are expected to be upper bounds’ (p. 2737): at 532 nm and 1e15 W/cm2 doubling the z extent lowered the rate by 15 % (p. 2732); ‘may be higher in other cases’. Because of the grid’s ionisation potential the 1.064 um process needs 22 photons, not 21 (p. 2729). The cross sections are given to one significant figure (3e-16 is 2.5-3.5e-16).
THE PHOTON ENERGIES are as printed: wavelengths in um (medium not stated) for (a), hbar omega in eV for the 9 eV row and for (b). Nothing is shifted: the calculation’s threshold is its own grid orbital energy, 24.28 eV (the two-photon threshold 12.14 eV), 0.3 eV below the true 24.58 eV. The 9.0 eV dataset’s wavelength_nm, 137.76, is hc / 9.0 eV (the record’s conversion; the parameter photon_energy_ev keeps the printed value).
THE TWO-PHOTON DEFINITION is not printed; the unit, cm4 W^-2 s^-1, is that of W / I^2, W the rate and I the intensity in W/cm2 (held as printed, ‘s-1 cm4/W2’, as bachau_1990’s Fig. 1(a)). As sigma_2 = (W / I^2) (hbar omega)^2 for W = sigma_2 Phi^2, Phi the photon flux (the script’s report): TDHF 1.424e-51 cm4 s at 13.6 eV and 8.04e-52 at 17.7 eV; the PT column 1.187e-51 and 8.85e-52. The intensity of the TDHF cross-section runs is not stated. Fig. 10 (p. 2733) shows the one- and two-photon ionising wave functions for 30 and 15 eV photons; the table’s two-photon energies are 13.6 and 17.7 eV.
COMPARISON with the curated two-photon cross sections of He (sigma_2 as above; each curve interpolated in log on its record’s own axis; ratio Kulander / other). TDHF at 13.6 eV: mihelic_2021 (TC-874, L = 0 + 2, 1.301e-51) 1.10, vanderhart_2005 (TC-904, 1.347e-51) 1.06, saenz_1999 (TC-873, 1S + 1D, 1.481e-51) 0.96; at 17.7 eV: 0.91 (8.80e-52), 0.92 (8.74e-52), 0.94 (8.57e-52). The PT column: 0.91, 0.88, 0.80 at 13.6 eV and 1.01, 1.01, 1.03 at 17.7 eV. bachau_1990’s Fig. 1(a) draws Kulander’s value as a ‘+’ at 1.0 Ry on its long-dashed line, which reads 2.81e-16 s^-1 W^-2 cm^4 there (the printed TDHF 3e-16). Within the one printed figure, TDHF agrees with all three. The one-photon 4e-18 cm2 at 30.0 eV is 0.74 of samson_1994’s measured 5.38 Mb.
NOT TAKEN. Table I(b)’s PT column, ‘Refs. 25-27’ (G. A. Victor, Proc. Phys. Soc. 91, 825 (1967); B. Ritchie, Phys. Rev. A 16, 2080 (1977), ritchie_1977; A. L’Huillier, L. Jonsson and G. Wendin, Phys. Rev. A 33, 3938 (1986), lhuillier_1986) and ‘Refs. 28’ (J. W. Cooper, Phys. Rev. 128, 681 (1962)): other papers’ values, and the row does not say which reference gives which number: 2.5e-16 and 1.1e-16 cm4/W2 s, 5e-18 cm2. The figures (expectation values, norms and projections against time, rates against orbital energy, wave-function density plots; Figs. 1-20): no quantity holds them, and the rates they give are Table I’s. The text’s statements without a number in a quantity: no excitation at 0.532 um and 1e14 W/cm2 within the numerical accuracy (the method needs rates of at least 1e11 s^-1, p. 2737); the observed saturation intensity 2e13 W/cm2 for a 50 ps pulse (Lompre et al., Ref. 24, quoted); the 28 eV Rabi oscillations; less than a tenth of a percent left on the grid at 1e17 W/cm2.
FURTHER TWO-PHOTON DATA CITED, not in the library: G. A. Victor, Proc. Phys. Soc. 91, 825 (1967) (He, coupled Hartree-Fock).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
kulander_1987/tbl1a_rate_1064_1s2#
{
"id": "tbl1a_rate_1064_1s2",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_1064_1s2.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 1064,
"polarization": "linear",
"method": "time-dependent Hartree-Fock, one doubly occupied orbital, finite-difference grid (0.2 a0); rate extrapolated to the static Hartree-Fock orbital energy (Fig. 12)",
"column": "1s^2"
}
}
Table I(a) (p. 2732), 1.064 um, column ‘1s^2’ (TDHF, extrapolated to the static Hartree-Fock orbital energy; the record’s reading): 4.7e11, 1.5e13, 1.2e14 s-1 at 5e14, 1e15, 2e15 W/cm2. Tunnelling regime (p. 2729). 20 %: the paper’s sensitivity to the numerical parameters; the highest intensity may be an upper bound (the record’s notes).
kulander_1987/tbl1a_rate_1064_1s1sp#
{
"id": "tbl1a_rate_1064_1s1sp",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_1064_1s1sp.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 1064,
"polarization": "linear",
"method": "frozen one-electron model: one electron frozen in its static Hartree-Fock orbital, the other propagated on the finite-difference grid (0.2 a0)",
"column": "1s1s'"
}
}
Table I(a) (p. 2732), 1.064 um, column “1s1s’” (the frozen one-electron model; the record’s reading): 7.1e11, 2.1e13, 2.6e14 s-1 at 5e14, 1e15, 2e15 W/cm2. 20 %: the paper’s sensitivity to the numerical parameters; the highest intensity may be an upper bound (the record’s notes).
kulander_1987/tbl1a_rate_532_1s2#
{
"id": "tbl1a_rate_532_1s2",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_532_1s2.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 532,
"polarization": "linear",
"method": "time-dependent Hartree-Fock, one doubly occupied orbital, finite-difference grid (0.2 a0); rate extrapolated to the static Hartree-Fock orbital energy",
"column": "1s^2"
}
}
Table I(a) (p. 2732), 0.532 um, column ‘1s^2’ (TDHF, extrapolated; the record’s reading): 9.0e11, 1.9e13, 1.4e14 s-1 at 5e14, 1e15, 2e15 W/cm2. At 1e15 a grid twice as long in z lowered the rate by 15 % (p. 2732). 20 %: the paper’s sensitivity to the numerical parameters (the record’s notes).
kulander_1987/tbl1a_rate_532_1s1sp#
{
"id": "tbl1a_rate_532_1s1sp",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_532_1s1sp.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 532,
"polarization": "linear",
"method": "frozen one-electron model: one electron frozen in its static Hartree-Fock orbital, the other propagated on the finite-difference grid (0.2 a0)",
"column": "1s1s'"
}
}
Table I(a) (p. 2732), 0.532 um, column “1s1s’” (the frozen one-electron model; the record’s reading): 1.4e12, 2.8e13, 2.7e14 s-1 at 5e14, 1e15, 2e15 W/cm2. 20 %: the paper’s sensitivity to the numerical parameters (the record’s notes).
kulander_1987/tbl1a_rate_9eV_1s2#
{
"id": "tbl1a_rate_9eV_1s2",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_9eV_1s2.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 137.76,
"photon_energy_ev": 9.0,
"wavelength_definition": "hc / 9.0 eV, the printed photon energy (the record's conversion)",
"polarization": "linear",
"method": "time-dependent Hartree-Fock, one doubly occupied orbital, finite-difference grid (0.2 a0)",
"column": "1s^2"
}
}
Table I(a) (p. 2732), 9.0 eV photons, column ‘1s^2’ (TDHF; the record’s reading): 2.6e13 s-1 at 1e15 W/cm2, three-photon ionisation. The TDHF decay turns non-exponential at late times (Fig. 14); how this value was taken is not stated. 20 % (the record’s notes).
kulander_1987/tbl1a_rate_9eV_1s1sp#
{
"id": "tbl1a_rate_9eV_1s1sp",
"material": "He",
"file": "He/kulander_1987__tbl1a_rate_9eV_1s1sp.csv",
"table": "tbl_8ef576c11381",
"quantity": "ionization_rate",
"unit": "s-1",
"abscissa": "intensity",
"abscissa_unit": "W/cm2",
"use": "check",
"uncertainty": {
"relative": 0.2,
"level": "not stated"
},
"parameters": {
"wavelength_nm": 137.76,
"photon_energy_ev": 9.0,
"wavelength_definition": "hc / 9.0 eV, the printed photon energy (the record's conversion)",
"polarization": "linear",
"method": "frozen one-electron model: one electron frozen in its static Hartree-Fock orbital, the other propagated on the finite-difference grid (0.2 a0)",
"column": "1s1s'"
}
}
Table I(a) (p. 2732), 9.0 eV photons, column “1s1s’” (the frozen one-electron model; the record’s reading): 5.8e13 s-1 at 1e15 W/cm2; exponential decay, direct ionisation (Fig. 13). 20 % (the record’s notes).
kulander_1987/tbl1b_sigma2_tdhf#
{
"id": "tbl1b_sigma2_tdhf",
"material": "He",
"file": "He/kulander_1987__tbl1b_sigma2_tdhf.csv",
"table": "tbl_8ef576c11381",
"quantity": "multiphoton_ionization_cross_section",
"unit": "s-1 cm4/W2",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"method": "time-dependent Hartree-Fock (equivalent to RPAE in the short-time limit, p. 2728), finite-difference grid",
"definition": "W / I^2, W the ionisation rate (s^-1) and I the intensity (W/cm2), as the unit cm4/W2 s implies (the definition is not printed); sigma_2 = (W / I^2)(hbar omega)^2 for W = sigma_2 Phi^2, Phi the photon flux",
"gauge": "length (V_I = -e E z sin(omega t), Eq. 2)",
"energy_axis": "photon energy as printed; the calculation's ionisation threshold is its grid orbital energy, 24.28 eV (two-photon threshold 12.14 eV), not shifted"
}
}
Table I(b) (p. 2732), column TDHF: the generalised two-photon cross section of He, 3e-16 at 13.6 eV and 1e-16 cm4/W2 s at 17.7 eV (photon energies, below the one-photon threshold), one significant figure. As sigma_2 = (W / I^2)(hbar omega)^2: 1.42e-51 and 8.04e-52 cm4 s, within 10 % of mihelic_2021, vanderhart_2005 and saenz_1999 (the record’s notes). ‘Our calculated one- and two-photon cross sections agree very well with perturbation results’ (p. 2732). No uncertainty stated.
kulander_1987/tbl1b_sigma1_tdhf#
{
"id": "tbl1b_sigma1_tdhf",
"material": "He",
"file": "He/kulander_1987__tbl1b_sigma1_tdhf.csv",
"table": "tbl_8ef576c11381",
"quantity": "photoionization_cross_section",
"unit": "cm2",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"level": "not stated"
},
"parameters": {
"method": "time-dependent Hartree-Fock, finite-difference grid; rate over intensity (the intensity not stated)"
}
}
Table I(b) (p. 2732), column TDHF: the one-photon photoionisation cross section of He, 4e-18 cm2 at 30.0 eV, one significant figure (Cooper’s 5e-18, the PT column, not taken). 0.74 of samson_1994’s measured 5.38 Mb at 30 eV. No uncertainty stated.
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.