saenz_1999 ========== A\. Saenz and P\. Lambropoulos, "Theoretical two\-, three\- and four\-photon ionization cross sections of helium in the XUV range", J\. Phys\. B\: At\. Mol\. Opt\. Phys\. 32, 5629\-5637 \(1999\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1088/0953\-4075/32/23/316 `__ **Source kind:** theory **Recorded technique:** Lowest\-order perturbation theory \(LOPT\) for N = 2, 3, 4\: the \(N \- 1\)\-fold sum over intermediate states of Eq\. \(1\), with no pole in it below the threshold \('only the last photon reaches into the continuum', p\. 5630\), evaluated in a discretised basis of two\-electron \(antisymmetrised, LS\-coupled, singlet, L = 0 to N\) eigenstates of He built from B\-splines in a radial box \(Lambropoulos, Maragakis and Zhang 1998\)\: 'converged results were obtained when employing R\_max ~ 300 au and 602 B\-splines of order nine for all orbital angular momenta l = 0\-6' \(p\. 5631\)\. The discretised continuum states are renormalised to final states by two methods, from the density of states and from matching to the asymptotic solution, which 'gave results that deviate from each other by less than 1%' \(p\. 5631\); to resolve sharp resonances, 'R\_max has been varied in between 280 and 320 au in steps of 2\.5 au' and the resulting spectra added \(p\. 5631\)\. The number of configurations is limited, a compromise that costs the accuracy of the ground\-state energy \(pp\. 5631\-5632\)\. Gauges\: length and velocity \(Fig\. 4\); a separate complex\-scaled CI calculation in Slater\-type orbitals, length gauge, at 13\.5 eV \(p\. 5634\)\. **Materials in structured entries:** He **Dataset quantities:** multiphoton\_ionization\_cross\_section **Contents:** 15 datasets; 0 formulas. Curation notes -------------- Journal pp\. 5629\-5637 = PDF pages 2\-10 \(PDF page 1 is the IOPscience download banner, its only image\); journal page = PDF page \+ 5627\. Identity checked on the first page \(PDF p\. 2\)\: J\. Phys\. B\: At\. Mol\. Opt\. Phys\. 32 \(1999\) 5629\-5637, Alejandro Saenz and P Lambropoulos, Max\-Planck\-Institut fur Quantenoptik, Garching, and FORTH\-IESL, Heraklion; received 23 August 1999 \(checked 2026\-10\-08\)\. THE CROSS SECTION \(p\. 5632, Eq\. 2\)\: 'Given the photon flux F in photons cm^\-2 s^\-1, the quantity W\_N = sigma^\(N\) F^N gives the transition probability per second for N\-photon ionization', sigma^\(N\) in cm^2N s^\(N\-1\); the definition of uiterwaal\_1998 and mckenna\_2004\. The cross sections are partial, by the symmetry of the final continuum\: two\-photon 1S and 1D \(Fig\. 1\), three\-photon 1P and 1F \(Fig\. 2\), four\-photon 1S, 1D and 1G \(Fig\. 3\)\. No total is drawn\. The polarisation is not named; it is linear\: the 1S, 1P and lower\-L final states the figures draw are reachable only by linearly polarised light \(circular polarisation reaches L = N alone\)\. THE PHOTON\-ENERGY AXES ARE SHIFTED BY THE AUTHORS, differently for each order \(p\. 5632\)\: 'Since our ground\-state energy is off by about 0\.69 eV, one way to present results of generalized cross sections sigma^\(N\) as a function of photon energy is to scale the latter so as to reach the same energy above the ionization threshold\. This is what has been done in figures 1\-4\. Needless to say, this scaling changes with the order N of the process in the calculation\.' The paper gives neither the computed ground\-state energy nor the formula of the scaling; read literally, N times the plotted photon energy minus the true threshold equals N times the computed photon energy minus the computed threshold, so the axes put the N\-photon threshold at its true value\. The values here are on the axes as drawn; nothing is shifted back or again\. The text's values at 13\.5 eV \(below\) are taken to be on the same axis \(they agree with Figs\. 1 and 4 there within 4 %, where the curves are flat\)\. The axes are photon energies \(not the total N\-photon energy\)\. Resonance positions in the figures are those of the LS\-coupled calculation on these axes \(drawn peaks\: two\-photon 21\.25 and 23\.06\-23\.10 eV; three\-photon 10\.333, 11\.468, 11\.842 eV \(1P\) and 11\.539, 11\.867 eV \(1F\); four\-photon 7\.08, 7\.70 eV \(1S, 1D\) and 7\.913, 8\.014 eV \(1G\)\); the 1S two\-photon cross section passes through zero near 18\.53 eV and 22\.55 eV, the three\-photon 1P near 10\.98 eV\. EXTRACTED \(tools/extract/saenz\_1999\.py\)\. 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\(a\), \(b\) \(p\. 5633\), Fig\. 2\(a\), \(b\) \(p\. 5634\), Fig\. 3\(a\), \(b\) \(p\. 5635\) and \(c\) \(p\. 5636\), Fig\. 4 \(p\. 5636\)\. Every path is a straight stroked segment of two vertices \(frames, ticks, curves and the label glyphs\); the curve of each panel is a chain of segments in drawing order, broken where the frame clips it\. Axes calibrated from the drawing's own tick marks \(every tick, short and long, on the fitted scale, linear in photon energy, logarithmic in the value\)\: the worst residuals are 0\.017/0\.017 pt \(Fig\. 1\(a\), x/y\), 0\.015/0\.016 \(1\(b\)\), 0\.016/0\.016 \(2\(a\)\), 0\.016/0\.017 \(2\(b\)\), 0\.015/0\.018 \(3\(a\)\), 0\.001/0\.019 \(3\(b\)\), 0\.012/0\.013 \(3\(c\)\), 0\.015/0\.016 \(4\)\. The panels' scales differ slightly\: Fig\. 1\(a\) runs from 12\.651 to 23\.799 eV, Fig\. 1\(b\) and Fig\. 4 to 23\.884 eV\. Every vertex of each drawn line is taken, less those on the frame's top or bottom edge \(where the frame clips a resonance peak or a zero\: there the value is only bounded\); a peak cut by the frame \(two\-photon 21\.25 eV in Fig\. 1, three\-photon 11\.47 eV in Fig\. 2\(a\), four\-photon 7\.08 eV in Fig\. 3\) is drawn to the frame and its height is not given\. Between about 8\.8 and 9\.1 eV \(Fig\. 2\(a\)\) and 6\.6 and 6\.7 eV \(Fig\. 3\(a\), \(b\)\) the drawn line is jagged on a scale of a few per cent, an artefact of the summed boxes \(not smoothed here\)\. Above the second resonance of each figure the calculation's energy grid is sparse \(straight segments of up to 0\.2 eV in Fig\. 1\)\: the curve between vertices is the drawing's interpolation, not the calculation's\. GAUGES\. Fig\. 4 draws the 1D two\-photon cross section in the length \(dashed\) and velocity \(solid\) gauges\: 'The smallness of the discrepancy between the two is rather reassuring' \(pp\. 5632\-5633\)\. The gauge of Figs\. 1\-3 is not stated\. Fig\. 1 is the velocity gauge\: Fig\. 1\(b\) and the solid line of Fig\. 4 agree within 0\.32 % from 13 to 22 eV \(one calculation drawn twice\), and Fig\. 1\(a\) at 13\.5 eV reads 0\.192e\-51 cm4 s, the text's velocity value 0\.2e\-51 \(\-4 %\), not its length value 0\.15e\-51 \(\+28 %\)\. Figs\. 2 and 3 are presumably the velocity gauge too \(not established\)\. THE CHECKS \(the script fails if one is missed\), against every number the text prints for these curves\: at 13\.5 eV, 1D velocity 1\.3e\-51 cm4 s \(Fig\. 4 solid 1\.295e\-51, \-0\.4 %; Fig\. 1\(b\) 1\.294e\-51, \-0\.4 %\), 1D length 1\.35e\-51 \(Fig\. 4 dashed, nearest vertex, 1\.340e\-51, \-0\.7 %\), 1S velocity 0\.2e\-51 \(Fig\. 1\(a\) 0\.192e\-51, \-4 %; the text gives one figure\); four\-photon\: 'According to figure 3 the generalized cross section is sigma^\(4\) ~ 10^\-118 cm8 s3, or perhaps larger by a factor of about two' at 7\.5 eV \(p\. 5635\)\: the sum of Fig\. 3\(a\), \(b\), \(c\) at 7\.5 eV is 2\.65e\-118 \(1S 0\.24e\-118, 1D 1\.70e\-118, 1G 0\.72e\-118\), consistent\. Reported, not a pass/fail check\: 'From figure 1 \.\.\. the partial wave of D symmetry dominates \.\.\. yielding the value sigma^\(2\) ~ 10^\-51 cm4 s' \(p\. 5632\), at the ninth harmonic of Ti\:sapphire, '13\.5 eV' \(p\. 5630\)\: Fig\. 1\(a\) \+ 1\(b\) at 13\.5 eV is 1\.49e\-51\. The extracted values, mapped back through the calibrations and overlaid on the rendered pages, follow every drawn line, its resonances, zeros and the dashes of Fig\. 4 \(inspected 2026\-10\-08\)\. DIGITISING UNCERTAINTY\: the drawing's coordinates are given to about 0\.001 pt and the tick residuals are at most 0\.019 pt, against 21 to 62 pt per decade\: 0\.2 % of a value at worst\. The same calculation drawn twice \(Figs\. 1\(b\), 4\) agrees within 0\.32 %\. 0\.5 % is taken for the value at a drawn vertex; in photon energy 0\.001 eV \(Fig\. 1, 24\.5 pt per eV\), less in Figs\. 2 and 3; on a resonance's flank the value at a given energy is uncertain by the slope times that\. The paper states no uncertainty of the calculation; the two normalisations agree within 1 %, the gauges within the difference of Fig\. 4 \(the length gauge within 5 % of the velocity gauge below 16 eV, 10 % above it at 17 eV, 25 % at 20 eV\), and the complex\-scaling calculation gives 1\.15e\-51 \(1D\) and 0\.1e\-51 \(1S\) at 13\.5 eV against the B\-spline 1\.3 and 1\.35e\-51 and 0\.2 and 0\.15e\-51\. THE VALUES AT 13\.5 eV \(p\. 5634\) are taken as text \(datasets text\_\*\)\: the B\-spline values in each gauge and the complex\-scaling values\. The complex\-scaling calculation is the authors' own \('a separate calculation of the two\-photon ionization cross section has been performed that is based on a completely different approach \.\.\. a complex\-scaled CI calculation in a basis of Slater\-type orbitals', the cross section '\(in the length gauge\) \.\.\. extracted from the corresponding complex\-scaled resolvent in a similar way as has been described for the case of atomic hydrogen by Froelich and Flores\-Riveros \(1986\)', pp\. 5633\-5634\)\: Froelich and Flores\-Riveros, J\. Chem\. Phys\. 84, 6732 \(1986\) is the method's reference for hydrogen, not the source of the values, so they are not reproduced\_from\. NOT TAKEN\: the total cross sections \(not drawn; the sum of the partial ones is not the authors' figure\); the ion\-yield estimates \(Y\_2 ~ 85 ions per shot at 115 GW/cm2 for Kobayashi et al\.'s experiment, p\. 5632; about 3e12 W/cm2 for one ion per pulse in four\-photon ionisation at 7\.5 eV, p\. 5636\), which no quantity holds and which depend on assumed volumes and densities\. The paper cites the experiments of Kobayashi, Sekikawa, Nabekawa and Watanabe, Opt\. Lett\. 23, 64 \(1998\) \(He two\-photon at the ninth harmonic\) and Xenakis et al\., J\. Phys\. B 29, L457 \(1996\) \(Ar two\-photon at 15 eV\), and Watanabe 1999 \(private communication, four\-photon\), whose values it does not print\. mihelic\_2021 and nikolopoulos\_2001 reproduce these curves \(Fig\. 3 in each\) as comparisons; those copies are not taken there\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- saenz\_1999/fig1a\_sigma2\_1S ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig1a_sigma2_1S", "material": "He", "file": "He/saenz_1999__fig1a_sigma2_1S_He.csv", "table": "fig_c3ccd1aa8c7b", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1S (L = 0)", "gauge": "velocity (not stated; established by Fig. 4 and the text, the record's notes)", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated two-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 1\(a\) \(p\. 5633\)\: the generalised two\-photon ionisation cross section of the He ground state to the 1S continuum \(W = sigma F^2\), cm4 s \(the axis prints 10^\-52 cm4 sec\), against the photon energy on the authors' shifted axis \(the record's notes\), 12\.651 to 23\.799 eV\: every vertex of the drawn line, 1832 vertices\. Through zero near 18\.53 and 22\.55 eV \(drawn to the frame's bottom, 1e\-56 to 2e\-57 at the last vertices\); the 1s2p resonance peak at 21\.25 eV runs off the frame\. 0\.192e\-51 at 13\.5 eV, the text's velocity value 0\.2e\-51\. Velocity gauge \(established, the record's notes\)\. 0\.5 %\: the digitising uncertainty of a drawn vertex \(the record's notes\); the paper states none\. saenz\_1999/fig1b\_sigma2\_1D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig1b_sigma2_1D", "material": "He", "file": "He/saenz_1999__fig1b_sigma2_1D_He.csv", "table": "fig_c3ccd1aa8c7b", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "velocity (not stated; established by Fig. 4 and the text, the record's notes)", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated two-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 1\(b\) \(p\. 5633\)\: the generalised two\-photon ionisation cross section of the He ground state to the 1D continuum \(W = sigma F^2\), cm4 s, against the photon energy on the authors' shifted axis \(the record's notes\), 12\.650 to 23\.884 eV\: every vertex of the drawn line, 1729 vertices\. 1\.2 to 1\.3e\-51 from 13 to 15 eV, 0\.73e\-51 at 20 eV; the 1s2p \(drawn peak 21\.245 eV, 6\.7e\-49\) and 1s3p \(23\.104 eV\) resonances with the asymmetric dips after them \(21\.424 eV\)\. 1\.294e\-51 at 13\.5 eV, the text's velocity value 1\.3e\-51 \(\-0\.4 %\)\. The same calculation as Fig\. 4's solid \(velocity\) line, within 0\.32 %\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig2a\_sigma3\_1P ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig2a_sigma3_1P", "material": "He", "file": "He/saenz_1999__fig2a_sigma3_1P_He.csv", "table": "fig_4538c5a2639e", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm6 s2", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 3, "polarization": "linear", "final_symmetry": "1P (L = 1)", "gauge": "not stated", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_3 = sigma^(3) F^3 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated three-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 2\(a\) \(p\. 5634\)\: the generalised three\-photon ionisation cross section of the He ground state to the 1P continuum \(W = sigma F^3\), cm6 s2 \(the axis prints 10^\-84 cm6 sec2\), against the photon energy on the authors' shifted axis \(the record's notes\), 8\.496 to 11\.944 eV\: every vertex of the drawn line, 1044 vertices\. Two\-photon resonances at 10\.333 eV \(1\.5e\-79\) and 11\.468 eV \(runs off the frame\), 11\.842 eV; zero near 10\.98 eV; jagged by a few per cent from 8\.8 to 9\.1 eV \(the record's notes\)\. The text prints no value of this curve\. Gauge not stated\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig2b\_sigma3\_1F ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig2b_sigma3_1F", "material": "He", "file": "He/saenz_1999__fig2b_sigma3_1F_He.csv", "table": "fig_4538c5a2639e", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm6 s2", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 3, "polarization": "linear", "final_symmetry": "1F (L = 3)", "gauge": "not stated", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_3 = sigma^(3) F^3 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated three-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 2\(b\) \(p\. 5634\)\: the generalised three\-photon ionisation cross section of the He ground state to the 1F continuum \(W = sigma F^3\), cm6 s2, against the photon energy on the authors' shifted axis \(the record's notes\), 8\.496 to 11\.953 eV\: every vertex of the drawn line, 1034 vertices\. Flat at 0\.13 to 0\.29e\-84 below 11\.3 eV; resonances at 11\.539 and 11\.867 eV\. The text prints no value of this curve\. Gauge not stated\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig3a\_sigma4\_1S ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig3a_sigma4_1S", "material": "He", "file": "He/saenz_1999__fig3a_sigma4_1S_He.csv", "table": "fig_f558c85055c3", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm8 s3", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 4, "polarization": "linear", "final_symmetry": "1S (L = 0)", "gauge": "not stated", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_4 = sigma^(4) F^4 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated four-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 3\(a\) \(p\. 5635\)\: the generalised four\-photon ionisation cross section of the He ground state to the 1S continuum \(W = sigma F^4\), cm8 s3 \(the axis prints 10^\-116 cm8 sec3\), against the photon energy on the authors' shifted axis \(the record's notes\), 6\.343 to 8\.140 eV\: every vertex of the drawn line, 634 vertices\. Resonances at 7\.08 eV \(runs off the frame\) and 7\.699 eV, a zero near 7\.64 eV, structure above 7\.9 eV\. 0\.24e\-118 at 7\.5 eV \(the record's notes\: the text's sigma^\(4\) ~ 1e\-118\)\. Gauge not stated\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig3b\_sigma4\_1D ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig3b_sigma4_1D", "material": "He", "file": "He/saenz_1999__fig3b_sigma4_1D_He.csv", "table": "fig_f558c85055c3", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm8 s3", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 4, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "not stated", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_4 = sigma^(4) F^4 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated four-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 3\(b\) \(p\. 5635\)\: the generalised four\-photon ionisation cross section of the He ground state to the 1D continuum \(W = sigma F^4\), cm8 s3, against the photon energy on the authors' shifted axis \(the record's notes\), 6\.343 to 8\.139 eV\: every vertex of the drawn line, 640 vertices\. Resonances at 7\.081 eV \(runs off the frame\) and 7\.696 eV, a zero near 7\.67 eV\. 1\.70e\-118 at 7\.5 eV, the largest of the three partial cross sections there\. Gauge not stated\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig3c\_sigma4\_1G ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig3c_sigma4_1G", "material": "He", "file": "He/saenz_1999__fig3c_sigma4_1G_He.csv", "table": "fig_38efdb3d55c0", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm8 s3", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 4, "polarization": "linear", "final_symmetry": "1G (L = 4)", "gauge": "not stated", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_4 = sigma^(4) F^4 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated four-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 3\(c\) \(p\. 5636, 'Figure 3\. Continued\.'\)\: the generalised four\-photon ionisation cross section of the He ground state to the 1G continuum \(W = sigma F^4\), cm8 s3, against the photon energy on the authors' shifted axis \(the record's notes\), 6\.343 to 8\.141 eV\: every vertex of the drawn line, 637 vertices\. Smooth from 1\.3e\-119 to 8e\-119 below 7\.85 eV; three\-photon resonances at 7\.913 and 8\.014 eV\. 0\.72e\-118 at 7\.5 eV\. Gauge not stated\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig4\_sigma2\_1D\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig4_sigma2_1D_velocity", "material": "He", "file": "He/saenz_1999__fig4_sigma2_1D_velocity_He.csv", "table": "fig_f06112f44e83", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "velocity", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated two-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 4 \(p\. 5636\), solid line\: the 1D two\-photon cross section in the velocity gauge \(caption\), cm4 s, against the photon energy on the authors' shifted axis, 12\.650 to 23\.884 eV\: every vertex of the drawn line, 1733 vertices \(one chain\)\. The same calculation as Fig\. 1\(b\) within 0\.32 %\. 1\.295e\-51 at 13\.5 eV, the text's 1\.3e\-51 \(\-0\.4 %\)\. 0\.5 % \(the record's notes\)\. saenz\_1999/fig4\_sigma2\_1D\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig4_sigma2_1D_length", "material": "He", "file": "He/saenz_1999__fig4_sigma2_1D_length_He.csv", "table": "fig_f06112f44e83", "digitised": "vector", "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "uncertainty": { "relative": 0.005, "level": "not stated" }, "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "length", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1", "energy_axis": "photon energy, shifted by the authors so the calculated two-photon threshold falls at the true one (ground state 0.69 eV off; p. 5632); as drawn" } } Fig\. 4 \(p\. 5636\), dashed line\: the 1D two\-photon cross section in the length gauge \(caption\), cm4 s, against the photon energy on the authors' shifted axis, 12\.650 to 23\.884 eV\: every vertex of the 80 drawn dashes \(1030 vertices; nothing between dashes, about 6 pt of line each, gaps as long; the printed cell names the dash\)\. Within 5 % of the velocity gauge below 16 eV, 10 % above it at 17 eV, 25 % at 20 eV \(nearest vertices\); the 1s2p peak drawn to 1\.27e\-48 at 21\.245 eV\. At 13\.5 eV the nearest vertex reads 1\.340e\-51, the text's 1\.35e\-51 \(\-0\.7 %\)\. 0\.5 % \(the record's notes\)\. saenz\_1999/text\_sigma2\_1D\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1D_velocity", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "velocity", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: 'the B\-spline approach yields for a photon energy of 13\.5 eV and transitions into the 1D continuum the cross sections sigma\_D^\(2\) = 1\.3 x 10^\-51 cm4 s \(velocity gauge\)'\. The calculation of Figs\. 1 and 4; 13\.5 eV on the authors' shifted axis \(the record's notes\)\. No uncertainty is stated\. saenz\_1999/text\_sigma2\_1D\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1D_length", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "length", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: '\.\.\. and 1\.35 x 10^\-51 cm4 s \(length gauge\)', the 1D continuum at 13\.5 eV, B\-spline basis\. No uncertainty is stated\. saenz\_1999/text\_sigma2\_1S\_velocity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1S_velocity", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1S (L = 0)", "gauge": "velocity", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: 'For transitions into the 1S continuum one finds at this energy with the B\-spline method sigma\_S^\(2\) = 0\.2 x 10^\-51 cm4 s \(velocity\)', at 13\.5 eV; printed to one figure\. No uncertainty is stated\. saenz\_1999/text\_sigma2\_1S\_length ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1S_length", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1S (L = 0)", "gauge": "length", "method": "LOPT, discretised two-electron B-spline basis (602 B-splines of order 9, l = 0-6, R_max 280-320 au, spectra added), singlet LS-coupled states", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: '\.\.\. and 0\.15 x 10^\-51 cm4 s \(length\)', the 1S continuum at 13\.5 eV, B\-spline basis\. No uncertainty is stated\. saenz\_1999/text\_sigma2\_1D\_complex\_scaling ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1D_complex_scaling", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1D (L = 2)", "gauge": "length", "method": "LOPT, complex-scaled CI in Slater-type orbitals, the cross section from the complex-scaled resolvent (as Froelich and Flores-Riveros 1986 for H)", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: 'the complex\-scaling approach gives sigma\_D^\(2\) = 1\.15 x 10^\-51 cm4 s', the 1D continuum at 13\.5 eV\: the authors' own separate calculation \(not reproduced from another paper; the record's notes\), length gauge\. The paper does not say whether its photon energy is shifted as the figures' are\. No uncertainty is stated\. saenz\_1999/text\_sigma2\_1S\_complex\_scaling ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_sigma2_1S_complex_scaling", "material": "He", "table": "text", "scale": 1e-51, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "final_symmetry": "1S (L = 0)", "gauge": "length", "method": "LOPT, complex-scaled CI in Slater-type orbitals, the cross section from the complex-scaled resolvent (as Froelich and Flores-Riveros 1986 for H)", "definition": "W_2 = sigma^(2) F^2 (Eq. 2), W the ionisation probability per second, F the photon flux in photons cm^-2 s^-1" } } 1 inline points in the source record. p\. 5634\: '\.\.\. compared to the value of 0\.1 x 10^\-51 cm4 s obtained with the complex\-scaling approach', the 1S continuum at 13\.5 eV\: the authors' own calculation, length gauge; printed to one figure\. 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 :doc:`review register ` for the checks and decisions.