nikolopoulos_2001#
L. A. A. Nikolopoulos and P. Lambropoulos, “Multichannel theory of two-photon single and double ionization of helium”, J. Phys. B: At. Mol. Opt. Phys. 34, 545-564 (2001)
Source record · Catalogue · Review register
DOI: 10.1088/0953-4075/34/4/304
Source kind: theory
Recorded technique: Lowest non-vanishing order of perturbation theory (LOPT) for two photons, with multichannel two-electron wavefunctions of He in a discretised basis: LS-coupled antisymmetrised products of one-electron He+ (Z = 2) target functions expanded in B-splines in a spherical box, the continuum channels obtained by the free-boundary method and normalised by matching each channel to its asymptotic Coulomb solutions (effective charge 1 for single, 2 for double ejection; pp. 548-553); the intermediate states the L = 1 manifold, with the pole of the above-threshold sum treated by Eq. (3.2) (p. 554). ‘a box of radius 40 au and 62 B-splines’ (p. 556); with R = 40 au ‘five bound intermediate states’, and calculations with R = 100 au ‘do not change significantly’ (p. 554). The ground-state energy is -78.467 eV, ‘to be compared with the exact value E = 79.002 eV calculated by Pekeris (1962)’ (p. 553). Single ionisation sums the final channels with the ion bound (He+ 1s, 2s, 2p, ...), double ionisation those with both electrons in the continuum (Eqs. A.9, A.16). Linearly polarised light: final states L = 0 and 2 (p. 553). The dipole operator is defined in the velocity and the length gauge (p. 550); which one the figures use is not stated, and ‘tests on the accuracy of the results through calculations in other gauges’ are left for future work (p. 560).
Materials in structured entries: He
Dataset quantities: multiphoton_ionization_cross_section
Contents: 6 datasets; 0 formulas.
Curation notes#
Journal pp. 545-564 = PDF pages 2-21 (PDF page 1 is the IOPscience download banner, its only image); journal page = PDF page + 543. Identity checked on the first page (PDF p. 2): J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 545-564, L A A Nikolopoulos and P Lambropoulos, FORTH-IESL and University of Crete, Heraklion, and Max-Planck-Institut fur Quantenoptik, Garching; received 3 July 2000, in final form 15 November 2000 (checked 2026-10-08). The paper begins on p. 545, as cited.
THE CROSS SECTION (p. 562, Eq. A.9): sigma_N^+ = 2 pi (2 pi alpha)^N omega^N sum_mu |D^(N)(mu)|^2, the generalised N-photon cross section in atomic units, summed over the final channels of single ionisation; Eq. A.16 the same over the double-ionisation channels. The figures give it in cm4 s, and the text turns it into yields as Y_2 ~ sigma_2 (I_0 / (2 pi alpha hbar omega))^2 tau_2 (p. 558): the rate is sigma_2 F^2 with F the photon flux (photons cm^-2 s^-1), as uiterwaal_1998 and saenz_1999 define it. ‘He+’ and ‘He++’ in the figures are the final ion of He’s two-photon single and double ionisation, not ions irradiated: every dataset here is the neutral atom’s (material He).
FIG. 4’S UNIT IS MISPRINTED: its y axis reads ‘Twophoton cross section (cm^2 s)’ (p. 557). The values are cm4 s: the paper defines the generalised two-photon cross section in cm4 s (p. 558), Fig. 3 (the D wave, same page) and Fig. 5 label their axes ‘(cm^4 s)’, and Fig. 4’s D wave lies within 30 % of Fig. 3’s on the same decades (1e-52 to 1e-51). The record states the unit cm4 s; no printed value is wrong, so there is no erratum.
THE ENERGY AXES. The photon energy as drawn; the paper says nothing of shifting it. Its ground state is 0.535 eV above Pekeris’s (p. 553). Fig. 4’s two curves peak at the 1s2p 1P resonance at 20.950 eV, Fig. 3’s D wave at 21.200 eV, and saenz_1999’s (on its shifted axis) at 21.25 eV: Fig. 3 lies 0.25 eV above Fig. 4, about half the ground-state error (0.27 eV, the shift that puts a two-photon threshold right), within the grids’ 0.1 eV. Whether Fig. 3 was shifted to be compared with Saenz and Lambropoulos, or computed differently, the paper does not say; nothing is shifted here.
EXTRACTED (tools/extract/nikolopoulos_2001.py). The paper prints no table and no supplement exists: the cross sections are in figures, all vector drawings, read from the PDF by data rule 8 (TC-699): Fig. 3 (p. 557), the D wave from 12.75 to 23.70 eV, ‘He+, this work, R = 40 a.u.’ (100 vertices); Fig. 4 (p. 557), the S and D waves from 12.35 to 38.00 eV (236 and 225 vertices); Fig. 5 (p. 558), the total single- and double-ionisation cross sections from 42.73 to 46.54 eV (17 markers each). Each curve is one stroked polyline; every vertex is taken, less two of Fig. 4’s D wave on the frame’s bottom edge, where the dip after the 1s2p resonance is clipped (21.17 and 21.18 eV). Fig. 5’s markers are the calculation’s points (the joining lines pass through their centres, checked); they lie 0.238 eV apart, not at round energies, as drawn. Axes calibrated from the drawing’s own tick marks (every tick on the fitted scale): worst residuals 0.074/0.085 pt (Fig. 3, x/y), 0.001/0.083 (Fig. 4), 0.063/0.076 (Fig. 5), the drawing quantised in steps of about 0.15 pt. The extracted values, mapped back through the calibrations and overlaid on the rendered pages, follow every drawn line and marker (inspected 2026-10-08).
FIG. 3 IS NOT FIG. 4’S D WAVE DRAWN AGAIN: the energy grids differ, the resonance is 0.25 eV higher (above), and off resonance Fig. 4’s D wave is 0.74 to 1.30 times Fig. 3’s at Fig. 3’s vertices (1.06 at 13 eV, 0.76 at 19 eV, 0.75 at 22.5 eV). Both are taken.
THE CHECKS (the script fails if one is missed): ‘for the two-photon He+ ionization cross section we obtain sigma_2(He+) ~ 1.2 x 10^-53 cm4 s’ at 45 eV (p. 558): Fig. 5 reads 1.267e-53 there (the markers either side 1.271 and 1.264e-53; the text’s figure is cut, not rounded, or approximate); ‘the He2+ cross section is about 10% of the He+ cross section’ (p. 558): 5.0 to 13.5 % across Fig. 5. The text’s 1.2e-53 is also taken as text (dataset text_sigma2_single_45eV).
SATO ET AL. (2011) QUOTE THIS CALCULATION (Table 2, ‘Theory (LOPT) [12]’, ref. [12] this paper) at 61.4, 58.4 and 56.0 nm (20.19, 21.23, 22.14 eV) as 8.0, 360 and 2.7 x 10^-52 cm4 s (Sato et al., J. Phys. B 44, 161001 (2011), in the library as Sato_2011_DeterminationAbsoluteTwo; checked on its table; a secondary quotation, not taken). They are Fig. 3’s D wave: it reads 1.043, 0.971 and 1.011 times them; Fig. 4’s S + D reads 0.96, 0.027 and 0.78 times them (on the 1s2p resonance at 21.23 eV Fig. 4’s curve is already 0.28 eV past its peak).
DIGITISING UNCERTAINTY: the tick residuals (up to 0.085 pt) and half the drawing’s quantum (0.075 pt) are each about 0.7 % of a value at 24 pt per decade (Figs. 3 and 4; Fig. 5 73 pt per decade), so 1 % is taken for the value at a drawn vertex or marker; in photon energy 0.009 eV (Fig. 4, 9.2 pt per eV), 0.004 eV (Fig. 3), 0.0015 eV (Fig. 5; a circle’s box, from its Bezier outline, is centred 0.08 pt left of the line’s vertex, 0.0013 eV). On a resonance’s flank the value at a given energy is uncertain by the slope times that, and the calculation’s grid (about 0.11 eV in Figs. 3 and 4) does not resolve the resonances’ shapes or heights. The paper states no uncertainty; it expects the calculation ‘to be reliable for a total kinetic energy of the two electrons of greater than 2 eV’ (p. 554) and calls Fig. 5’s range ‘the energy range for which we have converged results’ (p. 558); the artificial (pseudo-) resonances of the discretised continuum, the (+) points of Fig. 5, ‘have been simply replaced through linear interpolation over energies contributing to the main curve’ (p. 558).
NOT TAKEN: the curve of Saenz and Lambropoulos in Fig. 3 (red dotted; saenz_1999’s own Fig. 1(b) is taken there); the markers of Bachau, Lambropoulos and Tang, Phys. Rev. A 42, R5801 (1990) and Proulx and Shakeshaft, J. Phys. B 26, L7 (1993) in Fig. 4 (He two-photon single ionisation, other papers’, neither in the library); the (+) pseudoresonance points of Fig. 5 (the authors’ rejected values); Figs. 1 and 2 (single-photon double and single ionisation, tests of the basis against Lambropoulos et al 1998, J. Phys. B 28, L571 (1995) and Venuti et al 1996); the one-photon cross section ‘sigma_1(He+) ~ 1.1 x 10^-19 cm2’ at 45 eV and the yields (p. 558), outside this record’s two-photon scope; the comparisons the text quotes from other papers (p. 559): Crance and Aymar, J. Phys. B 18, 3529 (1985), double ionisation 2.05, 1.74, 1.73 x 10^-55 cm4 s at 43.5, 45, 46.25 eV; Kornberg and Lambropoulos, J. Phys. B 32, L603 (1999), 3.25e-17 cm4 W^-2 s at 45 eV, converted by the authors to 3.54e-54 cm4 s; Pindzola and Robicheaux, J. Phys. B 31, L823 (1998), yields at 45 eV. None of these is in the library.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
nikolopoulos_2001/fig3_sigma2_D#
{
"id": "fig3_sigma2_D",
"material": "He",
"file": "He/nikolopoulos_2001__fig3_sigma2_D_He.csv",
"table": "fig_9a66eb16b035",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "1D (L = 2)",
"final_state": "He+ (single ionisation, the ion in any bound state)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel",
"definition": "sigma_2 = 2 pi (2 pi alpha)^2 omega^2 sum_mu |D^(2)(mu)|^2 (Eq. A.9, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1",
"energy_axis": "photon energy as drawn; no shift stated; the 1s2p peak 0.25 eV above Fig. 4's (the record's notes); ground state 0.535 eV above Pekeris's"
}
}
Fig. 3 (p. 557), black line (‘He+, this work, R = 40 a.u.’): the generalised two-photon single-ionisation cross section of He to the 1D (L = 2) final states, cm4 s (the axis prints sigma_2^+ (cm^4 s)), against the photon energy as drawn, 12.75 to 23.70 eV, below the N = 2 threshold: every vertex of the drawn line, 100 vertices (the dip after the resonance, 2.5e-55 at 21.42 eV, lies above the frame’s bottom). 1.2e-51 at 14 eV, 0.79e-51 at 19 eV; the 1s2p resonance peak at 21.200 eV (1.5e-49). Not Fig. 4’s D wave drawn again: 0.25 eV higher in its resonance and up to a third higher off it (the record’s notes). Sato et al. (2011)’s values for this calculation agree with it within 4 %. 1 %: the digitising uncertainty (the record’s notes); the paper states none.
nikolopoulos_2001/fig4_sigma2_S#
{
"id": "fig4_sigma2_S",
"material": "He",
"file": "He/nikolopoulos_2001__fig4_sigma2_S_He.csv",
"table": "fig_521c07c70805",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "1S (L = 0)",
"final_state": "He+ (single ionisation, the ion in any bound state)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel",
"definition": "sigma_2 = 2 pi (2 pi alpha)^2 omega^2 sum_mu |D^(2)(mu)|^2 (Eq. A.9, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1",
"energy_axis": "photon energy as drawn; no shift stated; the 1s2p peak at 20.95 eV, 0.25 eV below Fig. 3's (the record's notes); ground state 0.535 eV above Pekeris's"
}
}
Fig. 4 (p. 557), black line (‘He+, S-wave’): the generalised two-photon single-ionisation cross section of He to the 1S (L = 0) final states, cm4 s (the axis misprints cm^2 s; the record’s notes), against the photon energy as drawn, 12.35 to 38.00 eV: every vertex of the drawn line, 236 vertices. A minimum of 3.8e-54 at 18.0 eV; the 1s2p resonance at 20.950 eV; doubly excited (2l2l’) resonances from 22 to 25 eV (drawn peak 3.4e-48 at 24.0 eV) and above 29 eV; core-excited channels (He+ 2s, 2p) above 32 eV (p. 556). The text prints no value of this curve. 1 % (the record’s notes).
nikolopoulos_2001/fig4_sigma2_D#
{
"id": "fig4_sigma2_D",
"material": "He",
"file": "He/nikolopoulos_2001__fig4_sigma2_D_He.csv",
"table": "fig_521c07c70805",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "1D (L = 2)",
"final_state": "He+ (single ionisation, the ion in any bound state)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel",
"definition": "sigma_2 = 2 pi (2 pi alpha)^2 omega^2 sum_mu |D^(2)(mu)|^2 (Eq. A.9, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1",
"energy_axis": "photon energy as drawn; no shift stated; the 1s2p peak at 20.95 eV, 0.25 eV below Fig. 3's (the record's notes); ground state 0.535 eV above Pekeris's"
}
}
Fig. 4 (p. 557), red line (‘He+, D-wave’): the generalised two-photon single-ionisation cross section of He to the 1D (L = 2) final states, cm4 s (the axis misprints cm^2 s; the record’s notes), against the photon energy as drawn, 12.35 to 38.00 eV: every vertex of the drawn line, 225 vertices (two on the frame’s bottom edge, at the dip after the 1s2p resonance, 21.17 and 21.18 eV, dropped). 1.2e-51 at 13 eV, 0.6e-51 at 19 eV; the 1s2p resonance at 20.950 eV; the 1s2p peak of Fig. 3 at 21.200 eV, which this curve does not reproduce off resonance either (0.74 to 1.30 times Fig. 3’s; the record’s notes). At 29.9 eV the ‘disagreement ... with the datum given by Bachau et al is not surprising’ (p. 556; Bachau’s static-exchange final state). 1 % (the record’s notes).
nikolopoulos_2001/fig5_sigma2_single#
{
"id": "fig5_sigma2_single",
"material": "He",
"file": "He/nikolopoulos_2001__fig5_sigma2_single_He.csv",
"table": "fig_47d9f579f9f5",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "total (L = 0 and 2)",
"final_state": "He+ (single ionisation, the ion in any bound state)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel; the above-threshold pole by Eq. (3.2)",
"definition": "sigma_2^+ = 2 pi (2 pi alpha)^2 omega^2 sum_mu |D^(2)(mu)|^2 over the single-ionisation channels (Eq. A.9, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1",
"energy_axis": "photon energy as drawn; no shift stated; ground state 0.535 eV above Pekeris's"
}
}
Fig. 5 (p. 558), black circles (‘He+’): the generalised two-photon single-ionisation cross section of He, all final symmetries (L = 0 and 2) and every bound state of the ion, cm4 s (the axis prints sigma_2^+, sigma_2^++ (cm^4 s)), against the photon energy as drawn, 42.73 to 46.54 eV: the 17 markers, the calculation’s points. 3.5e-53 at 42.73 eV falling to 1.24 to 1.28e-53 above 44.6 eV; 1.267e-53 at 45 eV, the text’s ‘~ 1.2 x 10^-53’ (p. 558; dataset text_sigma2_single_45eV). The paper’s range of ‘converged results’ (p. 558). 1 % (the record’s notes).
nikolopoulos_2001/fig5_sigma2_double#
{
"id": "fig5_sigma2_double",
"material": "He",
"file": "He/nikolopoulos_2001__fig5_sigma2_double_He.csv",
"table": "fig_47d9f579f9f5",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.01,
"level": "not stated"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "total (L = 0 and 2)",
"final_state": "He2+ (double ionisation)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel (effective charge 2 for double ejection); the above-threshold pole by Eq. (3.2)",
"definition": "sigma_2^++ = 2 pi (2 pi alpha)^2 omega^2 sum over the double-ionisation channels with 0 < epsilon < E/2 of |D^(2)(mu)|^2 (Eq. A.16, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1",
"energy_axis": "photon energy as drawn; no shift stated; ground state 0.535 eV above Pekeris's"
}
}
Fig. 5 (p. 558), red squares (‘He++’): the generalised two-photon double-ionisation cross section of He (both electrons ejected), cm4 s, against the photon energy as drawn, 42.73 to 46.54 eV: the 17 markers, the calculation’s points. 1.44 to 1.78e-54, ‘about 10% of the He+ cross section’ (p. 558; 5.0 to 13.5 % here). Correlation in the final state ‘tends to lower the value of the cross section’ (p. 559); the pseudoresonance points (+) were replaced by interpolation (p. 558) and are not taken. 1 % (the record’s notes).
nikolopoulos_2001/text_sigma2_single_45eV#
{
"id": "text_sigma2_single_45eV",
"material": "He",
"table": "text",
"scale": 1e-53,
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"parameters": {
"photons": 2,
"polarization": "linear",
"final_symmetry": "total (L = 0 and 2)",
"final_state": "He+ (single ionisation, the ion in any bound state)",
"gauge": "not stated",
"method": "LOPT, multichannel two-electron B-spline basis (box R = 40 au, 62 B-splines), free-boundary continuum normalised channel by channel; the above-threshold pole by Eq. (3.2)",
"definition": "sigma_2^+ = 2 pi (2 pi alpha)^2 omega^2 sum_mu |D^(2)(mu)|^2 over the single-ionisation channels (Eq. A.9, atomic units), in cm4 s; the rate sigma_2 F^2 with F the photon flux in photons cm^-2 s^-1"
}
}
1 inline points in the source record.
p. 558: ‘consider photons of energy omega = 45 eV ... for the two-photon He+ ionization cross section we obtain sigma_2(He+) ~ 1.2 x 10^-53 cm4 s’: He’s two-photon single ionisation (the ion He+ the product; material He, not the ion). Fig. 5 reads 1.267e-53 there (the record’s notes). No uncertainty is 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.