bachau_1990 =========== H\. Bachau, P\. Lambropoulos and X\. Tang, "Method for the calculation of multiphoton and above\-threshold ionization with discretized bases in two\-electron atoms", Phys\. Rev\. A 42, 5801\-5804 \(1990\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/PhysRevA\.42\.5801 `__ **Source kind:** theory **Recorded technique:** Lowest\-order perturbation theory with an L^2 discretised basis \(pp\. 5801\-5803\)\: the intermediate states \|v'\> in the static\-exchange approximation, one electron frozen in the hydrogenic 1s orbital, the other expanded in Slater\-type functions \(Eqs\. 3\-4, Macias and Riera's basis\), 40 functions; the ground state a separate variational \(Slater\-type orbital\) calculation, E\_g = \-2\.8917 au against Pekeris's \-2\.90372 \(p\. 5802\)\. Above the one\-photon threshold \(two\-photon ATI\) the pole of the intermediate sum is handled by tuning a nonlinear basis parameter until one basis state \(the 30th of 40\) sits at E\_g \+ hbar omega, removing it from the principal part and adding the imaginary part with the density of states rho^2 = 2 / \|E\_\(k\-1\) \- E\_\(k\+1\)\| \(Eq\. 2, p\. 5803\)\. 'For ATI, we have obtained convergence only in the velocity gauge' \(p\. 5802\)\. The polarisation is not named\: the intermediate states are 1P with M = 0 \(Eq\. 3\) and the final states 'L = 0, 2' \(p\. 5802\), linear polarisation\. The comparison column of Table I is H\. Rudolph's single\-channel quantum\-defect Green's\-function calculation \(acknowledgements\)\. Non\-relativistic, LS coupling\. **Materials in structured entries:** He **Dataset quantities:** multiphoton\_ionization\_cross\_section **Contents:** 5 datasets; 0 formulas. Curation notes -------------- Phys\. Rev\. A 42, 5801\-5804 \(1990\), a Rapid Communication\: PDF pages 1\-4 = journal pp\. 5801\-5804\. Identity checked on the first page \(PDF p\. 1\: 'PHYSICAL REVIEW A VOLUME 42, NUMBER 9 1 NOVEMBER 1990', 'RAPID COMMUNICATIONS', H\. Bachau \(Bordeaux\), P\. Lambropoulos \(USC and FORTH/Crete\), X\. Tang \(USC\); received 6 August 1990\) on 2026\-10\-09\. The library's identity is VALIDATED\_WITH\_WARNINGS \(the PDF's date hints 2011, the download\); the copy is the version of record\. The PDF is a scan \(one 5060 x 6898 image per page, about 600 dpi\) with a text layer\. THE CROSS SECTIONS\. Table I \(p\. 5803\)\: 'Two\-photon ATI generalized cross sections sigma\_2 \(cm^4 s\) of He calculated with the L^2 basis and with the single\-channel quantum defect Green's function', 'multiplied by 10^\-53', at hbar omega = 0\.5, 1\.0, 1\.1, 1\.2 and 1\.3 au; 'The first entry \(at hbar omega = 0\.5\) does not involve ATI' \(caption\)\. The definition of sigma\_2 is not printed; its unit, cm^4 s, is that of W = sigma\_2 Phi^2 with Phi the photon flux in cm^\-2 s^\-1, and Fig\. 1\(a\)'s W / I^2 agrees with it at 1 Ry \(below\)\. Every cell of Table I checked against the table image \(all agree\)\. Fig\. 1\(a\) \(p\. 5802\)\: 'two\-photon ionization of He', W / I^2 in 10^\-16 s^\-1 W^\-2 cm^4 against the photon energy in Ry \(0\.6 to 2\.5\), 'long\-dashed line, our results', with Ritchie's \(short dashes, the 1s2p and 1s3p resonances\), L'Huillier, Jonsson and Wendin's \(solid\), Victor's \(line with solid circles\) and Kulander's \(plus signs\)\. Dotted lines mark '2ph expt\.' at 0\.9 Ry and '1s^\-1 expt\.' at 1\.81 Ry, the experimental two\- and one\-photon thresholds; the long\-dashed line starts at the first\. Whether the calculation's energies were shifted to the experimental thresholds is not stated \(its ground state is \-2\.8917 au, above Pekeris's \-2\.90372\)\. THE ENERGY AXES are the photon energy\: hbar omega in au \(Table I\) and in Ry \(Fig\. 1\(a\)\), as printed; nothing converted\. EXTRACTED \(tools/extract/bachau\_1990\.py\)\. Table I, both columns\. Fig\. 1\(a\)'s long\-dashed line where it stands clear of the other curves, 0\.902 to 1\.100 Ry\: 28 samples on its first three dashes, read from the page image at its native pixels \(data rule 8, TC\-699; no vector drawing\)\: the axes calibrated from the frame's inward tick marks \(every 0\.1 Ry from 0\.7 to 2\.4 less 0\.9 and 1\.8, which lie under the dotted lines; every 0\.5 from 0 to 6\.5 in W / I^2\), each found in the image; the scan's shear \(the top edge's ticks 6\.5 to 10 pixels left of the bottom ones\) removed along the tick lines\. The ticks are irregularly spaced in x \(77 to 87 pixels per 0\.1 Ry, alike on both edges, the drawing's\)\: the linear scale fits them within 8\.9 pixels \(0\.011 Ry; 2 pixels at 1\.0 Ry\); y within 4\.3 pixels \(0\.03\)\. The dashes are the three dark regions 50\-90 pixels high in the window right of the 0\.9 Ry dotted line \(the short dashes are under 30, the solid line with its circles one region of 180\); every fourth column sampled where it crosses the dash in a single run of 9\-15 pixels, which leaves out the dashes' ends and the columns of Kulander's '\+' \(on the second dash at 1\.0 Ry\)\. Beyond 1\.10 Ry the long dashes run into the solid line and Victor's circles and are not read\. The overlay of the samples on the image was inspected \(2026\-10\-09\)\: every sample on its dash\. THE CHECKS \(the script fails if one is missed\)\. The text's 'largest discrepancy of a factor of 3\.6 between the two results' \(p\. 5803\)\: 29\.46 / 8\.12 = 3\.63 at 1\.0 au\. Fig\. 1\(a\) at 1 Ry \(= 0\.5 au; under the '\+', a quadratic through the three dashes' samples\)\: 2\.806 x 10^\-16 s^\-1 W^\-2 cm^4; Table I's 130\.69e\-53 cm4 s as sigma\_2 / \(hbar omega\)^2\: 2\.750, the drawing 1\.020 times it \(within 5 %\)\. NIKOLOPOULOS AND LAMBROPOULOS \(2001\) draw Table I's L^2 column at 1\.0\-1\.3 au as the squares of their Fig\. 4 \('He\+, Bachau et al \(1990\)'; nikolopoulos\_2001, TC\-875, which does not take them\)\: read with nikolopoulos\_2001's calibrated vector axes, the squares are 0\.996 to 0\.999 times Table I, at 27\.217, 29\.933, 32\.650 and 35\.383 eV \(1\.0\-1\.3 au times 27\.2114 eV within 0\.01 eV\)\. They agree\. DIGITISING UNCERTAINTY \(Fig\. 1\(a\)\)\: the x scale's residual, up to 0\.011 Ry, on the line's slope \(about 8 per Ry\) is up to 0\.09, 3 to 4 %; the y scale's 0\.03 and the run centre's pixel \(0\.007\) are 1 to 1\.5 %\: 4 % is taken\. The paper states no uncertainty of either calculation\: the L^2 results are judged by their 'convergence and stability \.\.\. as a function of the number of basis functions' \(p\. 5803\); the factor of 3\.6 the authors 'attribute \.\.\. to the rather unreliable representation of the ground\-state wave function in a single\-channel quantum\-defect approximation'\. THREE\- AND FOUR\-PHOTON ATI \(p\. 5803\), taken as inline points\: at the photon energy of the four\-photon resonance with the 1S0e\(1\) doubly excited state, 'four photons of energy 14\.38 eV each', the ATI channels competing with it, 'three\- and four\-photon ionization into the continua 1skp\(L=1\) and 1sks\(L=0\), with one and two ATI absorptions'\: 'W^\(3\) = 500 I^3 a\.u\. and W^\(4\) = 5\.083 x 10^3 I^4 a\.u\.' The rate and the intensity are in atomic units; which atomic unit of intensity is meant is not stated \(the paper gives the Rabi frequency's conversion, \-3\.5 I^2 au = \-9\.02e\-18 I^2 rad/s with I in W/cm2, not the rates'\)\. Kept as printed \(unit au\), not converted\. NOT TAKEN\. The four\-photon Rabi frequency Omega^\(4\)\[1s^2 \-\> 1S0e\(1\)\] = \-3\.5 I^2 au = \-9\.02e\-18 I^2 rad/s \(I in W/cm2\), and Be\+'s \-4\.5e\-3 I^2 au = 11\.7e\-21 I^2 rad/s \(sign as printed, p\. 5803\)\: no quantity holds a multiphoton Rabi frequency\. Fig\. 1\(a\)'s other curves and markers \(other authors' calculations\: B\. Ritchie, Phys\. Rev\. A 16, 2080 \(1977\), since curated as ritchie\_1977; A\. L'Huillier, L\. Jonsson and G\. Wendin, Phys\. Rev\. A 33, 3938 \(1986\), lhuillier\_1986; G\. A\. Victor, Proc\. Phys\. Soc\. 91, 825 \(1967\), not in the library; K\. C\. Kulander, Phys\. Rev\. A 36, 2726 \(1987\), kulander\_1987\)\. Fig\. 1\(b\)\: the photoelectron angular distribution of Mg's five\-photon ATI at 532 nm \(with Kim et al\.'s data\), no quantity\. The long\-dashed line beyond 1\.10 Ry \(not separable from the solid line and the circles\)\. FURTHER TWO\-PHOTON DATA CITED\: the four papers above \(He two\-photon ionisation below the one\-photon threshold; three since curated, Victor's not in the library\); not in the library\: M\. Edwards, X\. Tang, P\. Lambropoulos and R\. Shakeshaft, Phys\. Rev\. A 33, 4444 \(1986\) and M\. Edwards, X\. Tang and R\. Shakeshaft, Phys\. Rev\. A 35, 3758 \(1987\) \(hydrogen, Green's function\)\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- bachau\_1990/tbl1\_sigma2\_L2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "tbl1_sigma2_L2", "material": "He", "file": "He/bachau_1990__tbl1_sigma2_L2_He.csv", "table": "tbl_c78f5248bff6", "scale": 1e-53, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "hartree", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "gauge": "velocity", "method": "lowest-order perturbation theory, L^2 discretised basis (40 static-exchange 1P functions, Slater-type; the 30th placed at the pole above the threshold), ground state variational, E_g = -2.8917 au", "definition": "sigma_2 in cm4 s, the definition not printed: by the unit W = sigma_2 Phi^2, Phi the photon flux in cm^-2 s^-1 (Fig. 1(a)'s W / I^2 agrees at 0.5 au, the record's notes)" } } Table I \(p\. 5803\), 'L^2 basis' column\: the generalised two\-photon ionisation cross section of He, x 10^\-53 cm4 s, at hbar omega = 0\.5 au \(below the one\-photon threshold\) and 1\.0\-1\.3 au \(two\-photon ATI\)\: 130\.69, 29\.46, 17\.89, 11\.77, 7\.72\. The total over the final 1S and 1D states \(not resolved\)\. Fig\. 1\(a\) at 1 Ry agrees with the 0\.5 au entry within 2 % \(the record's notes\)\. No uncertainty stated\. bachau\_1990/tbl1\_sigma2\_qd ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "tbl1_sigma2_qd", "material": "He", "file": "He/bachau_1990__tbl1_sigma2_qd_He.csv", "table": "tbl_c78f5248bff6", "scale": 1e-53, "quantity": "multiphoton_ionization_cross_section", "unit": "cm4 s", "abscissa_unit": "hartree", "use": "check", "parameters": { "photons": 2, "polarization": "linear", "method": "single-channel quantum-defect Green's-function method (H. Rudolph, printed here; not described)", "definition": "sigma_2 in cm4 s, as the L^2 column (the definition not printed)" } } Table I \(p\. 5803\), 'Green's function' column\: H\. Rudolph's single\-channel quantum\-defect Green's\-function two\-photon cross section of He, x 10^\-53 cm4 s, at hbar omega = 0\.5 and 1\.0\-1\.3 au\: 57\.28, 8\.12, 6\.35, 5\.11, 4\.21; 2\.3 to 3\.6 times below the L^2 column, attributed by the authors to the single\-channel ground state\. No uncertainty stated\. bachau\_1990/fig1a\_W2\_L2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "fig1a_W2_L2", "material": "He", "file": "He/bachau_1990__fig1a_W2_L2_He.csv", "table": "fig_7d7bc8abfffe", "digitised": "raster", "scale": 1e-16, "quantity": "multiphoton_ionization_cross_section", "unit": "s-1 cm4/W2", "abscissa_unit": "rydberg", "use": "check", "uncertainty": { "relative": 0.04, "level": "not stated" }, "parameters": { "photons": 2, "polarization": "linear", "method": "lowest-order perturbation theory, L^2 discretised basis, as Table I", "definition": "W / I^2, W the two-photon ionisation rate (s^-1) and I the intensity (W/cm2), as the axis prints it; equal to sigma_2 / (hbar omega)^2 for W = sigma_2 Phi^2" } } Fig\. 1\(a\) \(p\. 5802\), long\-dashed line \('our results'\)\: W / I^2 of two\-photon ionisation of He, 10^\-16 s^\-1 W^\-2 cm^4, against the photon energy in Ry, read from the page image \(the record's notes\)\: 28 samples on the first three dashes, 0\.902 to 1\.100 Ry, 3\.71 falling to 2\.11\. Below the one\-photon threshold \(1\.81 Ry\); starts at the '2ph expt\.' line, 0\.9 Ry\. 4 %\: the digitising uncertainty \(the record's notes\); the paper states none\. bachau\_1990/text\_W3\_ATI ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_W3_ATI", "material": "He", "table": "text", "quantity": "multiphoton_ionization_cross_section", "unit": "au", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 3, "polarization": "linear", "gauge": "velocity", "channel": "1skp (L = 1)", "method": "lowest-order perturbation theory, L^2 discretised basis, two poles in the continuum", "definition": "W^(3) = c I^3, the transition probability per unit time, W and I in atomic units (the atomic unit of intensity not stated); c is the value" } } 1 inline points in the source record. p\. 5803\: 'W^\(3\) = 500 I^3 a\.u\.', three\-photon ionisation of He into 1skp \(L = 1\), one ATI absorption, at 14\.38 eV \('four photons of energy 14\.38 eV each' reach the 1S0e\(1\) resonance\)\. Rate and intensity both in atomic units; which atomic unit of intensity is not stated\. No uncertainty stated\. bachau\_1990/text\_W4\_ATI ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "text_W4_ATI", "material": "He", "table": "text", "scale": 1000.0, "quantity": "multiphoton_ionization_cross_section", "unit": "au", "abscissa_unit": "eV", "use": "check", "parameters": { "photons": 4, "polarization": "linear", "gauge": "velocity", "channel": "1sks (L = 0)", "method": "lowest-order perturbation theory, L^2 discretised basis, two poles in the continuum", "definition": "W^(4) = c I^4, the transition probability per unit time, W and I in atomic units (the atomic unit of intensity not stated); c is the value" } } 1 inline points in the source record. p\. 5803\: 'W^\(4\) = 5\.083 x 10^3 I^4 a\.u\.', four\-photon ionisation of He into 1sks \(L = 0\), two ATI absorptions, at 14\.38 eV\. Rate and intensity both in atomic units; which atomic unit of intensity is not stated\. 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 :doc:`review register ` for the checks and decisions.