mckenna_2004#
C. McKenna and H. W. van der Hart, “Multiphoton ionization cross sections of neon and argon”, J. Phys. B: At. Mol. Opt. Phys. 37, 457-470 (2004)
Source record · Catalogue · Review register
DOI: 10.1088/0953-4075/37/2/013
Source kind: theory
Recorded technique: R-matrix Floquet theory (Burke et al. 1991, Dorr et al. 1992), non-perturbative and time-independent: an inner region of radius a holding all N+1 electrons, described by an antisymmetrised configuration-interaction expansion with full exchange and correlation, and an outer region with one electron outside, by a close-coupling expansion; the Floquet-Fourier expansion of the wavefunction in harmonics of the ‘monochromatic, linearly polarized laser field’ (Eqs. 1-2), the inner region in the length gauge, the R-matrix transformed to the velocity gauge and propagated outwards, then matched to an asymptotic expansion, which gives the quasi-energy and the total ionisation rate Gamma of the ground state (p. 458). Basis (p. 459): that of Burke and Taylor (1975), ‘a two-state model’ of the ion: Ne+ 1s2 2s2 2p5 2Po and 1s2 2s 2p6 2Se (1s closed; single excitations from 2s and 2p to 3s, 3p and 3d-bar pseudo-orbitals, double to 3d-bar); Ar+ 3s2 3p5 2Po and 3s 3p6 2Se (1s, 2s, 2p closed; single excitations from 3s and 3p to 4s-bar, 4p-bar and 3d pseudo-orbitals, double to 3d-bar); continuum electrons up to l = 7; the ground state energy adjusted to the experimental binding energies, 21.6 eV (Ne) and 15.8 eV (Ar). Checked against Burke and Taylor’s single-photon cross sections: ‘The agreement is excellent.’ Floquet blocks (p. 459): two-photon, ‘five terms in the Floquet expansion: three for absorption and one for emission’ (and the field-free block), L up to 5, intensity 10^12 W/cm2; three-photon, ‘two additional absorption blocks’ and L up to 6, intensity 10^13 W/cm2, ‘required to ensure that we achieve ionization rates of a reasonable magnitude from which we can hope to generate accurate cross sections’. Non-relativistic, LS coupling, no spin-orbit interaction, ‘This ... could result in a shift from experiment of the energy position of resonances in the spectra’ (p. 459). No convergence test with respect to the basis or the Floquet blocks is reported.
Materials in structured entries: Ar, Ne
Dataset quantities: multiphoton_ionization_cross_section
Contents: 7 datasets; 0 formulas.
Curation notes#
Journal pp. 457-470 = PDF pages 2-15 (PDF page 1 is the IOPscience download banner, its only image); journal page = PDF page + 455. Identity checked on the first page (PDF p. 2: ‘J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 457-470’, Claire McKenna and Hugo W van der Hart, Queen’s University Belfast; received 15 September 2003, published 17 December 2003) on 2026-10-06. The library folder’s year 2003 is the online publication; the volume is 2004’s, and the record is named by it.
THE CROSS SECTION (p. 458, Eq. 3): ‘From the total ionization rate Gamma in atomic units (au), we obtain the generalized cross section sigma^(N) in units of cm^2N s^(N-1) for N-photon ionization from the relation sigma^(N) = (8 pi alpha)^N (3.5 x 10^16 / I)^N omega^N Gamma a0^2N t0^(N-1)’, I the intensity in W/cm2, omega the photon energy in au. With 3.5 x 10^16 W/cm2 the intensity of a field of 1 au, (I/3.5e16)/(8 pi alpha) is the cycle-averaged intensity in atomic units and that over omega the photon flux Phi: Eq. 3 is Gamma = sigma^(N) Phi^N, Phi in photons cm^-2 s^-1, the definition of Uiterwaal et al. (uiterwaal_1998). The rounded 3.5e16 (for 3.509e16) makes sigma^(N) 0.5 % (N = 2) and 0.8 % (N = 3) smaller than with the exact constant; to recover the computed rate, use Eq. 3 as printed. The rates are those of the ground state at one intensity, 10^12 W/cm2 for N = 2 and 10^13 W/cm2 for N = 3: ‘Since the ionization rates are affected by non-linear effects, the cross sections obtained at high intensity may not necessarily coincide with cross sections from perturbation theory’ (pp. 458-459). Off resonance the authors find lowest-order behaviour (Fig. 4: the ground-state rate scales as I^3 between 10^12 and 10^13 W/cm2 near 10.53 eV, p. 464); at resonances the positions shift with the intensity (16.6 meV in the three-photon energy between the two, 11.6 meV ponderomotive, p. 464) and higher-order (four-photon) resonances appear.
EXTRACTED (tools/extract/mckenna_2004.py). The paper prints no table and no supplement exists: the cross sections are only in figures, all vector drawings, read from the PDF by data rule 8 (TC-699): Fig. 1 (Ne, N = 2, p. 461), its enlargement Fig. 2 (p. 462), Fig. 3 (Ne, N = 3, p. 463), Fig. 5 (Ar, N = 2, p. 465), Fig. 7 (Ar, N = 2, 14.25-14.65 eV, p. 467) and Fig. 8 with its inset (Ar, N = 3, p. 468). Each curve is one line, the paper’s only calculation; every vertex of the drawn polyline is taken, in drawing order. Axes calibrated from the drawing’s own tick marks (short and long, every one checked on the fitted scale, linear in photon energy, logarithmic in decades in sigma, the inset linear): the worst tick residuals are 0.06/0.07 pt (Fig. 1, x/y), 0.06/0.07 (Fig. 2), 0.001/0.07 (Fig. 3), 0.06/0.07 (Fig. 5), 0.12/0.07 (Fig. 7), 0.06/0.07 (Fig. 8), 0.04/0.05 (inset), against a drawing quantised in steps of about 0.13 pt. The curve of each figure is drawn as many polyline pieces in no order; the script joins them end to end and checks that the result never steps back in photon energy. Vertices where the bottom of the frame cuts the threshold’s rise (one each in Figs. 1 and 5) are dropped. Not taken: Figs. 4 and 6, ionisation rates of the ground and resonant states against photon energy at one intensity (the anatomy of the 5f and 4d/6s resonances; Fig. 6(c) is the authors’ interpretation of the Ar ground-state rate between 14.08 and 14.33 eV, the gap between Figs. 5 and 7), which no quantity of the vocabulary holds.
THE CHECKS (the script fails if one is missed), against every number the text prints for these curves: Ar N = 3 ‘rise up to 1.0 x 10^-82 cm6 s2 from a minimum value of 6.75 x 10^-84’ between 5.5 and 6.5 eV (p. 468): the drawn minimum 6.735e-84 (at 5.53 eV; -0.2 %), and the curve reaches 1.0e-82 at 6.490 eV; Ar N = 2 ‘rises to 3.6 x 10^-48 cm4 s’ at 13.9 eV (p. 466): drawn peak 3.72e-48 (+3.4 %); Ar N = 2 ‘within 8% of 1.2 x 10^-50’ below 11.5 eV (p. 465): 1.119e-50 to 1.268e-50 from 8.0 to 11.4 eV (-6.7 to +5.7 %; the 4s 1Se peak ‘at approximately 12.77 eV’, p. 465, drawn at 12.773 eV), the wing of the 4s 1Po resonance reaching 1.08 times 1.2e-50 at 11.43 eV and 1.40e-50 at 11.5 eV; Ne N = 2 ‘within 10% of 1.65 x 10^-51’ between 11 and 16 eV (p. 460): 1.505e-51 to 1.821e-51 (-8.8 to +10.4 %); the interference minima of Fig. 2 ‘at 20.74 eV’, ‘20.08 eV’ and ‘21.07 eV’ (p. 462): drawn at 20.743, 20.087 (a flat bottom 0.007 eV wide) and 21.069 eV. The same calculation drawn twice agrees: Fig. 1 over Fig. 2 off resonance (19.55 to 20.9 eV) within 0.7 %, Fig. 8 over its inset (7.122 to 7.138 eV) within 0.7 %. The inset’s 4d 1De peak is 1.16 times the dip after it; the text: the cross section ‘is enhanced by nearly 20%’ by the 4d 1De and ‘by 2%’ by the 6s 1Se four-photon resonances (p. 469). The extracted values, mapped back through the calibrations and overlaid on the rendered pages, follow every drawn line, its resonances and the inset (inspected 2026-10-06).
DIGITISING UNCERTAINTY: half the drawing’s quantum (0.07 of 0.13 pt) and the tick residuals (up to 0.07 pt in y) are each 0.3 to 0.6 % of a value on the log axes (39 to 48 pt per decade; Fig. 3 27 pt), so 2 % is taken for the value at a drawn vertex. In photon energy the quantum is 0.007 eV in Fig. 1 (19 pt/eV), 0.004 eV in Fig. 5, 0.002 eV in Figs. 3 and 8, 0.0009 eV in Fig. 2, 0.0002 eV in Fig. 7 and 0.00004 eV in the inset: on a resonance’s flank the value at a given energy is uncertain by the slope times that (in Fig. 1 the resonances above 19.5 eV are better read from Fig. 2). The peak heights are those drawn, through the calculation’s energy grid, which the paper does not give; in the narrowest resonances several vertices share one energy (the line runs up and down in one 0.13 pt column). The paper states no uncertainty of the calculation.
RESONANCES AND LABELS. The two-photon spectra are dominated by one-photon resonances with 1Po Rydberg states (Ne ns; Ar 4s, 3d, 5s) and, in Ar, by two-photon doubly excited 3s 3p6 ns 1Se and nd 1De states below the 3s 3p6 2S threshold (29.2 eV); the three-photon spectra by two-photon np 1Se, 1De resonances, with nf 1De and four-photon 1Ge series above 10.4 eV (Ne) and 7.5 eV (Ar). At the ns 1Po resonances the ground state and the Rydberg state exchange character (laser-induced degenerate states); ‘at the resonance position, we have not shown the adiabatic change from one state to another, but it is to be understood that this change takes place at the maximum peak of each ns resonance’ (pp. 460-462), and the assignment of the ground state ‘is thus not always unambiguous’ where the nd and nf states interfere (pp. 462, 464, 468). Without spin-orbit coupling the resonance positions may differ from experiment (p. 459). The captions of Figs. 1 and 2 and the text (p. 460) name the Ne Rydberg series ‘3s2 3p5 ns 1Po’ and Fig. 3’s resonances ‘3p 1De’, ‘3p 1Se’: for Ne the core is 2s2 2p5 (the ns series of Ne+ 2p5); a slip in the labels, no effect on the values.
NOT TAKEN, other papers’ values the text compares with (no other values are drawn in the figures): Moccia, Rahman and Rizzo, J. Phys. B 16, 2737 (1983) (Ne, Ar two-photon, perturbative; Ne ‘up to a factor of 4’ below the present, ‘strongly gauge dependent’, p. 460); Pan, Gao and Starace, Phys. Rev. A 41, 6271 (1990) (Ar two-photon ‘within 20% of 0.85 x 10^-50 cm4 s’, about 30 % below the present, p. 465); McGuire, Phys. Rev. A 24, 835 (1981) (circular polarisation); Pindzola and Kelly, Phys. Rev. A 11, 1543 (1975); Starace and Jiang, Phys. Rev. A 36, 1705 (1987); Saenz and Lambropoulos, J. Phys. B 32, 5629 (1999) (He).
COVERAGE for the rate’s multiphoton limit: Ar three-photon at 193 nm (6.424 eV), on the rising wing of the 4p 1De two-photon resonance (6.667 eV in this LS-coupled calculation): 6.43e-83 cm6 s2 (log-interpolated between drawn vertices; the curve rises 3.3 decades per eV there). Nothing at 248.6 nm (4.99 eV): Ar three-photon starts at its threshold, 5.3 eV; the paper has no four-photon, Kr or He calculation.
Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
mckenna_2004/fig1_sigma2_Ne#
{
"id": "fig1_sigma2_Ne",
"material": "Ne",
"file": "Ne/mckenna_2004__fig1_sigma2_Ne.csv",
"table": "fig_f6941eec6a0c",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 2,
"intensity_w_cm2": 1000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ne+ target states (2s2 2p5 2Po, 2s 2p6 2Se; Burke and Taylor 1975 basis), L <= 5, Floquet blocks: one emission, three absorption",
"definition": "sigma^(2) = Gamma / Phi^2 (Eq. 3), Gamma the ground state's total ionisation rate at 1e12 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 1 (p. 461): the generalised two-photon ionisation cross section of the Ne ground state (Eq. 3; Gamma = sigma Phi^2, Phi in cm^-2 s^-1), cm4 s, against the photon energy, 10.80 to 21.45 eV: every vertex of the drawn line (the record’s notes). Resonances of the 2p5 ns 1Po series (drawn peaks: 3s at 17.12 eV, 1.78e-49; 4s at 19.80 eV ...) with nd 1Po, 1Fo between; off resonance 1.5 to 1.8e-51 between 11 and 16 eV. At 19 pt/eV the drawing resolves 0.007 eV: above 19.5 eV Fig. 2 is the better reading. 2 %: the digitising uncertainty of a drawn vertex (the record’s notes); the paper states none.
mckenna_2004/fig2_sigma2_Ne#
{
"id": "fig2_sigma2_Ne",
"material": "Ne",
"file": "Ne/mckenna_2004__fig2_sigma2_Ne.csv",
"table": "fig_4da3adbbfca0",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 2,
"intensity_w_cm2": 1000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ne+ target states (2s2 2p5 2Po, 2s 2p6 2Se; Burke and Taylor 1975 basis), L <= 5, Floquet blocks: one emission, three absorption",
"definition": "sigma^(2) = Gamma / Phi^2 (Eq. 3), Gamma the ground state's total ionisation rate at 1e12 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 2 (p. 462): the same calculation as Fig. 1, enlarged, 19.50 to 21.10 eV: the 4s, 5s, 6s 1Po resonances and the nd 1Po, 1Fo interferences, with minima at 20.08, 20.74 and 21.07 eV (the text; read at 20.087, 20.743, 21.069). cm4 s against photon energy, every vertex of the drawn line. Agrees with Fig. 1 off resonance within 0.7 %. 2 % (the record’s notes).
mckenna_2004/fig3_sigma3_Ne#
{
"id": "fig3_sigma3_Ne",
"material": "Ne",
"file": "Ne/mckenna_2004__fig3_sigma3_Ne.csv",
"table": "fig_0af720bedf15",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm6 s2",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 3,
"intensity_w_cm2": 10000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ne+ target states (2s2 2p5 2Po, 2s 2p6 2Se; Burke and Taylor 1975 basis), L <= 6, Floquet blocks: one emission, five absorption",
"definition": "sigma^(3) = Gamma / Phi^3 (Eq. 3), Gamma the ground state's total ionisation rate at 1e13 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 3 (p. 463): the generalised three-photon ionisation cross section of the Ne ground state (Eq. 3; Gamma = sigma Phi^3), cm6 s2, against the photon energy, 7.21 to 10.67 eV: every vertex of the drawn line. No resonance below 9 eV (0.6 to 1.05e-84 there); two-photon 2p5 np 1Se, 1De resonances above, nf 1De and four-photon 1Ge from 10.4 eV. The text prints no value of this curve. 2 % (the record’s notes; 25 pt per decade, the coarsest y axis).
mckenna_2004/fig5_sigma2_Ar#
{
"id": "fig5_sigma2_Ar",
"material": "Ar",
"file": "Ar/mckenna_2004__fig5_sigma2_Ar.csv",
"table": "fig_bdcfada5769c",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 2,
"intensity_w_cm2": 1000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ar+ target states (3s2 3p5 2Po, 3s 3p6 2Se; Burke and Taylor 1975 basis), L <= 5, Floquet blocks: one emission, three absorption",
"definition": "sigma^(2) = Gamma / Phi^2 (Eq. 3), Gamma the ground state's total ionisation rate at 1e12 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 5 (p. 465): the generalised two-photon ionisation cross section of the Ar ground state (Eq. 3; Gamma = sigma Phi^2), cm4 s, against the photon energy, 7.92 to 14.10 eV: every vertex of the drawn line. Flat to 11.4 eV within 8 % of 1.2e-50 (the text; read -6.7 to +5.7 %); the 4s 1Po one-photon resonance (drawn peak 11.97 eV), the doubly excited two-photon 4s 1Se ‘at approximately 12.77 eV’ (read 12.773) and 3d 1De / 5s 1Se at 13.9 eV (read 13.88), whose peak the text gives as 3.6e-48 (read 3.72e-48). 2 % (the record’s notes).
mckenna_2004/fig7_sigma2_Ar#
{
"id": "fig7_sigma2_Ar",
"material": "Ar",
"file": "Ar/mckenna_2004__fig7_sigma2_Ar.csv",
"table": "fig_c9dbe2d04620",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 2,
"intensity_w_cm2": 1000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ar+ target states (3s2 3p5 2Po, 3s 3p6 2Se; Burke and Taylor 1975 basis), L <= 5, Floquet blocks: one emission, three absorption",
"definition": "sigma^(2) = Gamma / Phi^2 (Eq. 3), Gamma the ground state's total ionisation rate at 1e12 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 7 (p. 467): the same as Fig. 5 from 14.26 to 14.64 eV, where the 3s 3p6 nd 1De (5d, 6d, 7d ...) and ns 1Se (7s, 8s ...) doubly excited Rydberg series dominate; cm4 s against photon energy, every vertex of the drawn line. Rydberg series of the 3s2 3p4 (3P) 4s/3d 2P thresholds, not in the calculation, ‘may already influence the spectrum’ (p. 468). The text prints no value of this curve. 2 % (the record’s notes).
mckenna_2004/fig8_sigma3_Ar#
{
"id": "fig8_sigma3_Ar",
"material": "Ar",
"file": "Ar/mckenna_2004__fig8_sigma3_Ar.csv",
"table": "fig_714c65223ddb",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm6 s2",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 3,
"intensity_w_cm2": 10000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ar+ target states (3s2 3p5 2Po, 3s 3p6 2Se; Burke and Taylor 1975 basis), L <= 6, Floquet blocks: one emission, five absorption",
"definition": "sigma^(3) = Gamma / Phi^3 (Eq. 3), Gamma the ground state's total ionisation rate at 1e13 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 8 (p. 468), main plot: the generalised three-photon ionisation cross section of the Ar ground state (Eq. 3; Gamma = sigma Phi^3), cm6 s2, against the photon energy, 5.30 to 7.79 eV: every vertex of the drawn line. A minimum of 6.75e-84 (the text; read 6.735e-84 at 5.53 eV), rising to 1.0e-82 by 6.5 eV (read at 6.490 eV) on the wing of the two-photon 4p 1De (drawn peak 6.667 eV) and 4p 1Se (6.785 eV) resonances; 5p 1De, 1Se above 7.3 eV, nf 1De and four-photon 1Ge from 7.5 eV. 193 nm (6.424 eV): 6.43e-83. 2 % (the record’s notes).
mckenna_2004/fig8_inset_sigma3_Ar#
{
"id": "fig8_inset_sigma3_Ar",
"material": "Ar",
"file": "Ar/mckenna_2004__fig8_inset_sigma3_Ar.csv",
"table": "fig_714c65223ddb",
"digitised": "vector",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm6 s2",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.02,
"level": "not stated"
},
"parameters": {
"photons": 3,
"intensity_w_cm2": 10000000000000.0,
"polarization": "linear",
"method": "R-matrix Floquet, length/velocity gauge, two Ar+ target states (3s2 3p5 2Po, 3s 3p6 2Se; Burke and Taylor 1975 basis), L <= 6, Floquet blocks: one emission, five absorption",
"definition": "sigma^(3) = Gamma / Phi^3 (Eq. 3), Gamma the ground state's total ionisation rate at 1e13 W/cm2, Phi the photon flux in cm^-2 s^-1, with 3.5e16 W/cm2 for the atomic unit of field"
}
}
Fig. 8 (p. 468), inset: the same calculation from 7.121 to 7.140 eV on a linear axis printed in units of 1e-84 cm6 s2 (converted: cm6 s2 here), against the photon energy: every vertex of the drawn line. The four-photon 4d 1De (drawn peak 7.1283 eV) and 6s 1Se features, which ‘in principle, do not contribute to the three-photon cross sections’ and enhance it by nearly 20 % and 2 % at 1e13 W/cm2 (p. 469). Agrees with the main plot within 0.7 %. 2 % (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.