vanderhart_2005#
H. W. van der Hart and P. Bingham, “Two- and three-photon ionization of He between 10^13 and 10^14 W cm^-2”, J. Phys. B: At. Mol. Opt. Phys. 38, 207-221 (2005)
Source record · Catalogue · Review register
DOI: 10.1088/0953-4075/38/3/006
Source kind: theory
Recorded technique: R-matrix Floquet theory (Burke, Francken and Joachain 1991), non-perturbative and time-independent, with a B-spline basis in the inner region (p. 208): radius 20 au, 32 B-splines on a mixed exponential/linear knot set, three He+ target states (1s, 2s, 2p); ‘This basis set does not lead to a highly accurate position for the ground state of He, and we have thus shifted the He ground state energy to its experimental position.’ Two-photon regime: six absorption and two emission Floquet blocks, six total angular momenta (five at 1e13 W/cm2); three-photon regime: six absorption and two emission blocks (eight and three at 1e14 W/cm2), seven total angular momenta at 1 and 2 x 10^13 W/cm2, eight at 5e13 and 1e14; the R-matrix propagated to 80 au (to 240 au near threshold) before matching (p. 208). The result is the total ionisation rate of the ground state at 1, 2, 5 and 10 x 10^13 W/cm2, including above-threshold and higher-order ionisation. The 3l nl’ doubly excited states are not in the calculation (no n = 3 target state, p. 211). The polarisation is not named; it is linear (the two-photon 1sns 1S resonances of Figs. 7-8 are reachable only so, and the comparison is with Saenz and Lambropoulos’s linear-polarisation cross sections). The gauge is not stated. Non-relativistic, LS coupling.
Materials in structured entries: He
Dataset quantities: multiphoton_ionization_cross_section
Contents: 4 datasets; 0 formulas.
Curation notes#
Journal pp. 207-221 = PDF pages 2-16 (PDF page 1 is the IOPscience download banner, its only image); journal page = PDF page + 205. Identity checked on the first page (PDF p. 2: ‘J. Phys. B: At. Mol. Opt. Phys. 38 (2005) 207-221’, Hugo W van der Hart and Peter Bingham, Queen’s University Belfast; received 21 September 2004, published 24 January 2005) on 2026-10-08. The library’s identity is VALIDATED_WITH_WARNINGS (Crossref lists the first author as ‘Hart, Hugo W van der’); the first page prints the title, authors and locator; the copy is the version of record.
THE CROSS SECTIONS (Figs. 2 and 9, pp. 211, 219). ‘Our generalized cross sections are obtained through transformation of the total ionization rates obtained at an intensity of 10^13 W cm^-2’ (captions), ‘the lowest intensity used’; ‘we can only obtain approximate cross sections: higher-order processes are directly included in the approach’ (p. 211). The paper does not print the transformation. It is Gamma = sigma^(N) Phi^N, Phi = I / (hbar omega) the photon flux in cm^-2 s^-1 (the definition of uiterwaal_1998, mckenna_2004 and saenz_1999): Fig. 1’s 1e13 rate so converted (CODATA constants) is 0.3 to 0.8 % above Fig. 2’s line from 0.47 to 0.70 au, and Fig. 7’s 0.9 to 2.4 % above Fig. 9’s from 0.31 to 0.41 au, within the two readings (0.7 to 1.1 % each); the sign is that of the rounded 3.5e16 W/cm2 for the atomic unit of field that mckenna_2004’s Eq. 3 (the same group) prints, 0.5 % (N = 2) and 0.8 % (N = 3), which is not established here. The cross sections therefore carry everything in the 1e13 W/cm2 rate: the solid line of Fig. 2 has spikes near the 1s2p resonance (drawn at 0.769, 0.781 and 0.785 au, up to 3.7e-50 cm4 s) from the three-photon excitation of the autoionising 2l nl’ 1Po, 1Fo states: ‘These peaks are absent in the true generalized two-photon cross sections’ (caption); and at the 1s2p resonance itself the ground state’s mixing with 1s2p at 1e13 W/cm2 broadens and lowers the peak, ‘the difference [with perturbation theory] amounts to three orders of magnitude’ (p. 211; the drawn perturbative peak, 7.4e-48 at 0.7804 au, is 1230 times the R-matrix Floquet line there). The intensity dependence (Fig. 1, not taken): off resonance the rates scale as I^2 (three-photon: I^3 ‘and a ponderomotive shift’, p. 218), the authors expecting perturbation theory to hold to 10^15 W/cm2 for two-photon and 3 x 10^14 W/cm2 for three-photon ionisation off resonance (p. 220).
THE FREQUENCY AXIS is the photon energy in atomic units (hartree), not the total energy. The ground state is at its experimental energy (shifted; p. 208); at 1e13 W/cm2 it is dressed (-2.90416 au at 0.700 au, Table 2; -2.90803 at 1e14 W/cm2: ‘the binding energy of the He ground state is increased by roughly 0.004 au at a frequency of 0.7 au and an intensity of 10^14’, 40 % of it ponderomotive, p. 213). The autoionising states lie within 0.0044 au of the accurate field-free positions (Table 1: 2s2p 1Po -0.68869 au against Rost et al.’s -0.69313; LS-coupled, non-relativistic). The dashed lines (Saenz and Lambropoulos) are on their own axis, on which the authors moved each order’s photon energies so that their calculated threshold falls at the true one (saenz_1999, TC-873): the dashed lines here agree with saenz_1999’s partial curves, summed and placed at frequency x 27.2114 eV, within 0.6 % at 16 frequencies (checked by the script). Fig. 2’s frame and both lines run to 0.840 au (the text: ‘between 0.45 and 0.82 au’).
EXTRACTED (tools/extract/vanderhart_2005.py). The cross sections are only in Figs. 2 and 9 (no table, no supplement; the authors’ numbers went to Sato et al., below), vector drawings read from the PDF by data rule 8 (TC-699): every vertex of each drawn line, in drawing order. The four lines are told apart by their dash patterns (solid: R-matrix Floquet; dashed: Saenz and Lambropoulos). Axes calibrated from the drawing’s own ticks (every tick on the fitted scale, linear in frequency, logarithmic in sigma; the labels are text objects, checked against the page’s text): worst residuals 0.042/0.059 pt (Fig. 2, x/y), 0.048/0.000 (Fig. 9; only its five labelled y ticks), against a quantum of about 0.105 pt. Each line is one path of a few subpaths, each starting where the last ended, never stepping back in frequency, wholly inside the frame; the lines end on the frame’s right edge, which cuts them. Read for the checks only: Fig. 1 (rates, two-photon regime) and Fig. 7 (rates, three-photon regime).
THE CHECKS (the script fails if one is missed), against every number the text prints for these curves: ‘Differences [RMF against perturbation theory] are generally less than 10%’ off resonance (p. 211): drawn at most 9.1 % from 0.465 to 0.70 au; ‘Differences are generally less than 15%’ in the three-photon regime (p. 219): at most 15.8 % from 0.325 to 0.36 au (14 to 16 % from 0.34 to 0.37 au); resonance strengths, the integrals from 0.71 to 0.82 au (two-photon) and 0.355 to 0.405 au (three-photon), perturbative over R-matrix Floquet, ‘a factor 30’ (p. 211) and ‘a factor of 5.6’ (p. 220): drawn 29.6 and 5.75; near the 1s2s resonance ‘a factor of 10 difference’ (p. 219): the drawn peaks 1.48e-79 (0.3797 au) and 1.97e-80 (0.3800 au), 7.5 (reported). ‘An I^2-scaling ... shows very close agreement’ (p. 210): Fig. 1’s 2e13, 5e13 and 1e14 lines over I^2 times the 1e13 line, 0.47 to 0.70 au, +0.1 to -0.7 %, -0.5 to -2.3 % and -1.3 to -4.0 %. Fig. 1 and Fig. 7 against Figs. 2 and 9 (above). The extracted values, mapped back through the calibrations and overlaid on the rendered pages, follow every drawn line, its spikes and dashes (inspected 2026-10-08, again 2026-10-09).
DIGITISING UNCERTAINTY: half the quantum and the tick residuals are 0.6 % of a value on Fig. 2’s axis (36.3 pt per decade) and 1.0 % on Fig. 9’s (22.1 pt per decade): 1 % and 1.5 % are taken. In frequency the quantum is 0.0002 au (Fig. 2, 559 pt per au) and 0.0001 au (Fig. 9): on a resonance’s flank the value at a given frequency is uncertain by the slope times that. The spikes’ heights are those drawn through the calculation’s frequency grid, which the paper does not give. The paper states no uncertainty of the calculation; Table 1’s widths differ from more complete calculations by 5 to 25 % (a factor 5 for the narrowest state, p. 212).
SATO ET AL. (2011) quote this calculation (sato_2011, TC-876, Table 2 ‘Theory (R-matrix Floquet method)’, ‘provided ... his theoretical data’): 8.5, 63, 3.5 and 6.9 x 10^-52 cm4 s at 61.4, 58.4, 56.0 and 53.4 nm (0.7421, 0.7802, 0.8136, 0.8532 au). Read here (log-interpolated): 8.74, 60.0 and 3.54 x 10^-52 from Fig. 2 (the quoted values 0.97, 1.05 and 0.99 times these; 58.4 nm is on the top of the 1s2p peak, where the line runs from 5.1 to 6.4 x 10^-51 within 0.001 au), and at 53.4 nm, beyond Fig. 2, 5.29 x 10^-52 from Fig. 1’s 1e13 rate converted as above: the quoted 6.9 is 1.30 times this. 53.4 nm is on the steep upper flank of the 1s3p resonance (drawn peak 0.8485 au): the line reaches 6.9e-52 at 0.8520 au, 0.0012 au (0.07 nm) below 53.4 nm, five times Fig. 1’s frequency quantum (0.00024 au). The disagreement at 53.4 nm is not resolved; the other three agree within the readings.
NOT TAKEN. Tables 1-3 (pp. 212, 213, 216): the field-free and dressed total energies (au, below the double-ionisation threshold) and widths of the six lowest odd autoionising states and the dressed ground state at 1, 2, 5, 10 x 10^13 W/cm2: no quantity holds a dressed or a total energy, or a width (level_energy is an energy above the ground level). Figs. 1, 4-8: total ionisation rates (au) against frequency at four intensities, which no quantity holds (ionization_rate’s abscissae are the intensity and the field); the rates at 2, 5 and 10 x 10^13 W/cm2 are the paper’s only intensity dependence (Sato et al.’s Fig. 3 converts them to cross sections). Fig. 3: dressed energies. The text’s rates of the 2p3d 1Po state and of the ground state at that resonance (6e-7 to 7e-6 au and 4.3e-7 to 1.1e-5 au from 1e13 to 1e14 W/cm2, p. 216; the resonance is drawn in Fig. 6 at 0.7856 au at 1e13 W/cm2, where Fig. 1’s 1e13 and 1e14 lines read 4.8e-7 and 1.0e-5 au, consistent; Table 3’s column gives 0.795 au as the frequency of the calculation), and the fractions of three- and four-photon ionisation (below 0.1 % at 0.7 au; 3.3 % at 0.35 au, 1e14 W/cm2; pp. 209-210, 218).
FURTHER TWO-PHOTON DATA CITED, not in the library: Purvis, Dorr, Terao-Dunseath, Joachain, Burke and Noble, Phys. Rev. Lett. 71, 3943 (1993) (R-matrix Floquet two-photon ionisation rates of He above 0.6 au at 1e12 W/cm2); Glass, Burke, van der Hart and Noble, J. Phys. B 30, 3801 (1997) (three-photon excitation of He’s autoionising states, 5e12 and 1e13 W/cm2); van Enk, Zhang and Lambropoulos, Z. Phys. D 37, 211 (1996) and J. Phys. B 29, 4041 (1996) (He third-order susceptibility near the doubly excited states); Lagmago Kamta and Starace, Phys. Rev. A 65, 053418 (2002) and Parker et al., J. Phys. B 36, L393 (2003) (time-dependent He).
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
vanderhart_2005/fig2_sigma2_rmf#
{
"id": "fig2_sigma2_rmf",
"material": "He",
"file": "He/vanderhart_2005__fig2_sigma2_rmf_He.csv",
"table": "fig_d93cfccc63b5",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "hartree",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"intensity_w_cm2": 10000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, non-perturbative; B-spline inner region (20 au), He+ 1s, 2s, 2p targets, ground state shifted to the experimental energy; five total angular momenta, six absorption and two emission Floquet blocks at 1e13 W/cm2",
"definition": "Gamma = sigma^(2) Phi^2, Gamma the ground state's total ionisation rate at 1e13 W/cm2 (higher orders included), Phi = I / (hbar omega) the photon flux in cm^-2 s^-1; the formula is not printed: Fig. 1's rate so converted reproduces this line within 0.8 % (the record's notes)"
}
}
Fig. 2 (p. 211), solid line: the generalised two-photon ionisation cross section of the He ground state from the R-matrix Floquet total ionisation rate at 1e13 W/cm2, cm4 s, against the photon energy in hartree (the frequency in au), 0.4529 to 0.8400 au: every vertex of the drawn line. 1.36e-51 at 0.453 au falling to 7.9e-52 at 0.70 au; the 1s2p one-photon resonance near 0.78 au with three-photon autoionising spikes (0.769, 0.781, 0.785 au, up to 3.7e-50) ‘absent in the true generalized two-photon cross sections’ (caption); minimum 3.5e-52 at 0.817 au. Within 9.1 % of the dashed perturbative line off resonance (0.465-0.70 au). 1 %: the digitising uncertainty of a drawn vertex (the record’s notes); the paper states none.
vanderhart_2005/fig2_sigma2_saenz#
{
"id": "fig2_sigma2_saenz",
"material": "He",
"file": "He/vanderhart_2005__fig2_sigma2_saenz_He.csv",
"table": "fig_d93cfccc63b5",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "hartree",
"use": "check",
"reproduced_from": [
"saenz_1999"
],
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"method": "lowest-order perturbation theory, two-electron B-spline basis (Saenz and Lambropoulos 1999); the photon-energy axis scaled by those authors to put their two-photon threshold at the true one",
"definition": "W_2 = sigma^(2) F^2, F the photon flux in photons cm^-2 s^-1 (saenz_1999, Eq. 2); the sum of the 1S and 1D partial cross sections"
}
}
Fig. 2 (p. 211), dashed line: Saenz and Lambropoulos’s (1999) perturbative two-photon cross section, ‘in numerical form’ from A. Saenz (p. 220), cm4 s against the photon energy in hartree, 0.4649 to 0.8400 au: every vertex of the drawn line. The total over the 1S and 1D final states, which saenz_1999 draws only as partial curves: their sum, at frequency x 27.2114 eV on saenz_1999’s axis (its scaled photon energy, TC-873), agrees with this line within 0.6 % at eight frequencies from 0.47 to 0.82 au. The 1s2p peak drawn 7.4e-48 at 0.7804 au. Not this source’s calculation. 1 % (the record’s notes).
vanderhart_2005/fig9_sigma3_rmf#
{
"id": "fig9_sigma3_rmf",
"material": "He",
"file": "He/vanderhart_2005__fig9_sigma3_rmf_He.csv",
"table": "fig_f2b3c82b4132",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm6 s2",
"abscissa_unit": "hartree",
"use": "check",
"uncertainty": {
"relative": 0.015,
"level": "not stated"
},
"parameters": {
"photons": 3,
"intensity_w_cm2": 10000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, non-perturbative; B-spline inner region (20 au), He+ 1s, 2s, 2p targets, ground state shifted to the experimental energy; seven total angular momenta, six absorption and two emission Floquet blocks at 1e13 W/cm2",
"definition": "Gamma = sigma^(3) Phi^3, Gamma the ground state's total ionisation rate at 1e13 W/cm2 (higher orders included), Phi = I / (hbar omega) the photon flux in cm^-2 s^-1; the formula is not printed: Fig. 7's rate so converted reproduces this line within 2.4 % (the record's notes)"
}
}
Fig. 9 (p. 219), solid line: the generalised three-photon ionisation cross section of the He ground state from the R-matrix Floquet total ionisation rate at 1e13 W/cm2, cm6 s2, against the photon energy in hartree, 0.3024 to 0.4200 au: every vertex of the drawn line. 1.2e-85 at 0.302 au rising to the 1s2s two-photon resonance (drawn peak 1.97e-80 at 0.3800 au), the 1s3s/1s3d wing at 0.42 au. Within 15.8 % of the dashed perturbative line off resonance (0.325-0.36 au); a factor 7.5 below its peak at 1s2s. 1.5 %: the digitising uncertainty (22 pt per decade; the record’s notes); the paper states none.
vanderhart_2005/fig9_sigma3_saenz#
{
"id": "fig9_sigma3_saenz",
"material": "He",
"file": "He/vanderhart_2005__fig9_sigma3_saenz_He.csv",
"table": "fig_f2b3c82b4132",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm6 s2",
"abscissa_unit": "hartree",
"use": "check",
"reproduced_from": [
"saenz_1999"
],
"uncertainty": {
"relative": 0.015,
"level": "not stated"
},
"parameters": {
"photons": 3,
"polarization": "linear",
"method": "lowest-order perturbation theory, two-electron B-spline basis (Saenz and Lambropoulos 1999); the photon-energy axis scaled by those authors to put their three-photon threshold at the true one",
"definition": "W_3 = sigma^(3) F^3, F the photon flux in photons cm^-2 s^-1 (saenz_1999, Eq. 2); the sum of the 1P and 1F partial cross sections"
}
}
Fig. 9 (p. 219), dashed line: Saenz and Lambropoulos’s (1999) perturbative three-photon cross section, cm6 s2 against the photon energy in hartree, 0.3122 to 0.4200 au: every vertex of the drawn line. The total over the 1P and 1F final states: saenz_1999’s partial curves summed (its scaled axis, TC-873) agree with this line within 0.6 % at eight frequencies from 0.33 to 0.41 au. The 1s2s peak drawn 1.48e-79 at 0.3797 au. Not this source’s calculation. 1.5 % (the record’s notes).
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.