kulander_1988
=============

K\. C\. Kulander, "Time\-dependent theory of multiphoton ionization of xenon", Phys\. Rev\. A 38, 778\-787 \(1988\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/kulander_1988.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevA\.38\.778 <https://doi.org/10.1103/PhysRevA.38.778>`__

**Source kind:** theory

**Recorded technique:** The time\-dependent Schrodinger equation of one active valence electron of xenon, solved on a finite\-difference grid \(cylindrical rho\-z, spacing 0\.3 a0, Peaceman\-Rachford propagator, 200 or 400 steps per optical period, an absorbing imaginary potential over the last 10\-20 grid points; pp\. 779\-780\)\. The other electrons are frozen in their ground\-state orbitals\: a local Hartree\-Slater potential \(Slater's exchange with the Herman\-Skillman correction, its strength fitted to the ionisation energy 12\.13 eV\) made into a single p\-wave effective core potential; no spin\-orbit interaction, so one continuum only \(the 2P1/2 limit at 13\.4 eV is ignored\)\. Grid orbital energies\: 5p\_0 12\.32 eV, 5p\_1 12\.23 eV, 5s 22\.3 eV \(experiment 23\.4\)\. Linear polarisation along z, the dipole approximation, length form, V\_I = \-sum e E\(t\) z\_i sin\(omega t\) \(Eq\. 2\); the field rises linearly over a chosen number of cycles and is then held constant \(Eq\. 7\), and the rate is the exponential decay of the initial orbital \(or of the norm\) at constant intensity\. The 5p\_0 and 5p\_\+\-1 orbitals are run separately; 'the total rate for the p shell' is the 5p\_0 rate \(two electrons\) plus the 5p\_\+\-1 rate \(four electrons\) \(Fig\. 1's caption and p\. 781\)\. A cross section is the rate over the photon flux to the power N at an intensity low enough for the rate's power law \('we check the linearity \.\.\. at two intensities', p\. 780\)\.

**Materials in structured entries:** Xe

**Dataset quantities:** ionization\_rate, multiphoton\_ionization\_cross\_section, photoionization\_cross\_section

**Contents:** 6 datasets; 0 formulas.

Curation notes
--------------

Phys\. Rev\. A 38, 778\-787 \(1988\) = PDF pages 1\-10 \(journal page = PDF page \+ 777\)\. Identity checked on the first page \('PHYSICAL REVIEW A VOLUME 38, NUMBER 2 JULY 15, 1988'; K\. C\. Kulander, Lawrence Livermore National Laboratory; received 25 January 1988\) on 2026\-10\-09\. The PDF is a scan \(one 5060 x 6898 image per page, about 610 dpi\) with an OCR text layer\. The library's identity is VALIDATED\_WITH\_WARNINGS; the copy is the version of record\. Every value of the text taken was read on the page image\.

WHAT IS TAKEN, all use = check\. \(1\) From the text \(inline points; pp\. 780\-781\)\: the total one\-photon photoionisation cross section, 46 Mb at 14\.4 eV and 29 Mb at 15\.63 eV; the two\-photon generalised cross section at 193 nm, 4 x 10^\-50 cm4 s; the three\-photon ones, 1 x 10^\-82 cm6 s2 at 293 nm \(the I^3 line fitted to Fig\. 1's rates\) and 2\.5 x 10^\-82 cm6 s2 at 266 nm\. \(2\) From the figures \(p\. 781; tools/extract/kulander\_1988\.py, read from the page image, the method in the script's docstring\)\: the rate of the p shell against the peak intensity at 293 nm \(Fig\. 1's total, 1e13 to 2e14 W/cm2, eight markers\), 586 nm \(Fig\. 2, 1e13 to 7\.5e13, seven\) and 1064 nm \(Fig\. 2, 2e13 to 8e13, seven\)\. The paper prints no rate in a table\.

THE ENERGY AXES\. Wavelengths in nm as printed \(the medium not stated\) and photon energies in eV \(one\-photon\)\. The model's threshold is the fitted 12\.13 eV \(single continuum, no 2P1/2 limit\)\: 193 nm \(6\.424 eV\) is two\-photon above it and below the real 2P1/2 limit; 266 nm 'produces electrons with energies above the second ionization threshold' \(p\. 781\), which the model does not have\. Nothing is shifted\.

THE FIGURES\. Both log\-log\. The calculated intensities are round \(every marker lies within 2 % of 1, 2, 3, 3\.5, 4, 5, 6, 7, 7\.5, 8, 10, 15 or 20 x 10^13 W/cm2; the text names 3, 3\.5, 5, 10, 15 and 20 x 10^13\), which checks the intensity scale; the data files keep the intensity read, and each row is named by the round value\. Read \(Fig\. 1\)\: the total's circles at all eight intensities \(two, at 3\.5e13 and 5e13, merge with the 5p\_0 triangle below and are centred from their top\)\. Fig\. 2\: 586 nm \(solid circles\) from 1e13 to 7\.5e13 and 1064 nm \(solid triangles\) from 2e13 to 8e13; from 1e14 up the two curves and their markers lie on one another \(and at 1\.5e14 on the 293 nm square\), so they are not read\. Fig\. 2's 293 nm squares are not taken \(Fig\. 1's total is the same calculation, drawn larger\); at 1e13 the 293 nm line starts at Perry et al\.'s open square, with no solid marker\. CHECKS \(the script raises if one fails\)\: Fig\. 2's open square reads 3\.99e11 s^\-1 against the text's '4 x 10^11 s^\-1 for a peak intensity of 1 x 10^13 W/cm^2' \(Perry et al\.\); the open circle and the 1064 nm open triangle read 4\.18e13 and 2\.78e13 W/cm2, where the text puts them \(4\.2e13, 2\.8e13\); the right frame lines read 2\.00e14 and 2\.04e14 W/cm2; Fig\. 1's total is 0\.96\-1\.02 of its 5p\_0 plus 5p\_1 markers at seven intensities; Fig\. 2's 293 nm squares are 0\.93\-1\.03 of Fig\. 1's total at six; Fig\. 1's dashed I^3 line has slope 2\.97 and gives sigma\_3 = 1\.05e\-82 cm6 s2 \(rate / \(I / hbar omega\)^3, 293 nm\), the printed 1 x 10^\-82\. The overlays of the values on the page image were inspected \(2026\-10\-09\)\: every value on its marker\.

ACCURACY\. Digitising\: 5 % in the rate \(a marker's centre within 4 pixels of where its connecting lines cross it\: 2\.2 % in Fig\. 1, 3\.4 % in Fig\. 2; the scale exact at the ticks\), and the intensity read within 2 % of the round value, which at the local slope \(3 at 293 nm, up to about 11 at 1064 nm\) moves a rate compared at the read intensity by more\. The paper\: for 293 nm above 1e14 W/cm2 'the decay is not purely exponential \.\.\. The theoretical error bars on these rates are the largest of all those rates presented \(probably no more than a factor of 2\), except perhaps for those in Fig\. 2 below 10^10 s^\-1 where the accuracy of the calculation suffers from numerical instabilities' \(p\. 785\)\: Fig\. 2's 586 nm at 1e13 \(6\.4e9\) and 1064 nm at 2e13 \(6\.7e9 s^\-1\)\. No uncertainty is stated for the cross sections \(one or two significant figures\)\.

COMPARISONS \(Kulander's quotations against the curated originals, at 193 nm = 6\.4241 eV, the photon energy; each figure interpolated on its record's own axis\)\. Two\-photon Xe\: 'l'Huillier and Wendin \.\.\. Hartree\-Fock' 1\.5e\-49 cm4 s\: lhuillier\_1987 \(TC\-883\) Fig\. 6 HF reads 1\.49e\-49; their 'random\-phase approximation with exchange' 2e\-50\: lhuillier\_1987's RPAE reads 2\.58e\-50 and lhuillier\_1987b's \(TC\-916\) 2\.61e\-50 \(the quotation is 0\.77 of the drawn curves; neither paper prints a number at 193 nm\); McGuire 8e\-50\: mcguire\_1981 \(TC\-879\) reads 8\.36e\-50; Gangopadhyay et al\. 5e\-51\: gangopadhyay\_1986 \(TC\-914\) Fig\. 1 at the two\-photon total energy reads 5\.03e\-51; McCown et al\. 4e\-50 cm4 s\: mccown\_1982 \(TC\-884\), 4e\-32 cm4 W\-1, is 4\.12e\-50 cm4 s as alpha hbar omega\. Kulander's 4e\-50 is 0\.97 of McCown et al\.'s and 0\.27, 1\.55, 0\.48 and 7\.9 times the HF, RPAE, McGuire and MQDT curves\. lagutin\_2017 \(TC\-880\) starts at the 2P1/2 threshold \(6\.72 eV\) and has no value at 193 nm; its first CIPFCP vertex is 4\.49e\-50 cm4 s\. Three\-photon\: lhuillier\_1989 \(TC\-882\) quotes 'Kulander calculates a cross section equal to 2\.5 x 10^\-82 cm6 s2' at 266 nm\: this paper prints exactly that \(p\. 781\)\. FINDING\: lhuillier\_1989 sets 'Perry and Landen have measured sigma\_3 = 1\.2 x 10^\-82 cm6 s2' beside its 266 nm values; here that number is Perry et al\.'s 293 nm rate \(4e11 s^\-1 at 1e13 W/cm2, unpublished\) converted, 'This corresponds to a three\-photon cross section of 1\.2 x 10^\-82' \(p\. 781\)\: a 293 nm value\. FINDING\: Gangopadhyay et al\.'s three\-photon values, quoted here as 3 x 10^\-76 \(293 nm\) and 5 x 10^\-76 cm6 s2 \(266 nm\), carry 'an error in a conversion factor in their three\-photon calculations' found afterwards\: 'they now obtain results which are in good agreement with our result' \(p\. 781, citing P\. Lambropoulos, private communication\)\. gangopadhyay\_1986's Table II \(TC\-914\) at 4\.6667 eV \(265\.7 nm\) holds 2\.02e\-76 \+ 3\.15e\-76 = 5\.17e\-76 cm6 s2, the quoted 5e\-76\: those values are about 1e6 too large \(lhuillier\_1989's MQDT, from the same group, 6e\-82 at 266 nm\)\. One\-photon\: the 46 and 29 Mb are 0\.74 and 0\.50 of chan\_1992's \(TC\-68\) measured df/dE converted \(109\.76 Mb eV x df/dE\: 62\.4 Mb at 14\.4 eV, 58 Mb at 15\.6 eV\); the text quotes Marr and West's 60\.1 and 54\.5 Mb and an exact calculation in the same potential, 41 and 27 Mb\.

NOT TAKEN\. Fig\. 1's 5p\_0 and 5p\_1 curves \(the rates of one kind of orbital\: no quantity holds a partial rate; read only for the check of the total\)\. Fig\. 2's open symbols\: other papers' measured saturation intensities turned into rates by this paper's rule, 'three times the inverse of the pulse length' \(p\. 781; Perry et al\., Refs\. 24\-26; L'Huillier et al\., Phys\. Rev\. A 27, 2503 \(1983\); Freeman et al\., Phys\. Rev\. Lett\. 59, 1092 \(1987\)\)\. The 5s photoionisation cross section, 4\.5e\-20 cm2 at 24 eV \(exact 5e\-20; one subshell, not the total\), and the 5s rate at 293 nm and 1e14 W/cm2, 'less than 10^11 s^\-1' \(a bound\)\. The decay rates of the Stark\-shifted resonances \(9\.7e13 s^\-1 out of, and about 1\.4e12 s^\-1 into, the 4f\-like state at 586 nm and 3\.5e13 W/cm2; 5e13 and 4e13 s^\-1 for the d resonances at 293 nm and 2e14 W/cm2, pp\. 783\-785\)\: rates of a state, not of the atom\. The other papers' cross sections the text quotes \(above\)\. Figs\. 3\-12 \(wave functions, norms, level diagram\)\.

MISPRINTS IN THE TEXT \(no value affected\)\. Eq\. 7 prints f\(t\) = 1/t\_max for t \< t\_max\: the text says the field 'increases linearly', t/t\_max \(as Kulander 1987's Eq\. 10\)\. p\. 781\: 'The dotted line in Fig\. 2 represents a fitted I^3 rate'\: the dashed line is in Fig\. 1 \(its caption\)\. Ref\. 20 is printed 'Phys\. Rev\. A 26, 3318 \(1987\)'\: mccown\_1982, 1982\.

FURTHER TWO\-PHOTON DATA CITED, not in the library\: W\. K\. Bischel, J\. Bokor, D\. J\. Kligler and C\. K\. Rhodes, IEEE J\. Quantum Electron\. QE\-15, 380 \(1979\) and R\. V\. Hodges, L\. C\. Lee and J\. T\. Moseley, Int\. J\. Mass Spectrom\. Ion Phys\. 39, 133 \(1981\) \(Xe at 193 nm, measured; mccown\_1982 holds the values it quotes from them\); P\. Lambropoulos, Phys\. Rev\. Lett\. 55, 2141 \(1985\) \(Xe, 1\.6e\-49 cm4 s at 193 nm, quoted\)\.

**Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets
--------

kulander\_1988/text\_sigma1\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "text_sigma1_Xe",
     "material": "Xe",
     "table": "text",
     "quantity": "photoionization_cross_section",
     "unit": "Mb",
     "abscissa_unit": "eV",
     "use": "check",
     "uncertainty": {
       "level": "not stated"
     },
     "parameters": {
       "method": "single active electron, frozen Hartree-Slater core (effective core potential), finite-difference time-dependent Schrodinger equation, the rate over the photon flux"
     }
   }

2 inline points in the source record.

p\. 780\: 'a total cross section of 46 Mb at a photon energy of 14\.4 eV' and 29 Mb at 15\.63 eV, the frozen single\-active\-electron model \(5p\_0 and 5p\_\+\-1 together\), from the decay rate at an intensity low enough to be linear\. The exact values for the same potential are 41 and 27 Mb; measured \(Marr and West\) 60\.1 and 54\.5 Mb, quoted; 0\.74 and 0\.50 of chan\_1992 \(the record's notes\)\. No uncertainty stated\.

kulander\_1988/text\_sigma2\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "text_sigma2_Xe",
     "material": "Xe",
     "table": "text",
     "scale": 1e-50,
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm4 s",
     "abscissa_unit": "nm",
     "use": "check",
     "uncertainty": {
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "wavelength_nm": 193,
       "polarization": "linear",
       "method": "single active electron, frozen Hartree-Slater core (effective core potential fitted to 12.13 eV, no spin-orbit), finite-difference time-dependent Schrodinger equation",
       "definition": "the generalised cross section in cm4 s, sigma_2 = W / Phi^2 with W the decay rate and Phi the photon flux in cm^-2 s^-1 (the formula is not printed; the unit implies it)",
       "gauge": "length (V_I = -sum e E z_i sin(omega t), Eq. 2)",
       "energy_axis": "the wavelength as printed; the model's threshold is 12.13 eV (single continuum), not shifted"
     }
   }

1 inline points in the source record.

p\. 780\: 'We find the two\-photon generalized cross section at this wavelength \[193 nm\] to be 4 x 10^\-50 cm4 s', one significant figure\. Two photons \(12\.85 eV\) above the model's single threshold \(12\.13 eV\), below the real 2P1/2 limit; the model has no spin\-orbit interaction\. 0\.97 of mccown\_1982's measured value \(the record's notes\)\. No uncertainty stated; the wavelength medium not stated\.

kulander\_1988/text\_sigma3\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "text_sigma3_Xe",
     "material": "Xe",
     "table": "text",
     "scale": 1e-82,
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "cm6 s2",
     "abscissa_unit": "nm",
     "use": "check",
     "uncertainty": {
       "level": "not stated"
     },
     "parameters": {
       "photons": 3,
       "polarization": "linear",
       "method": "single active electron, frozen Hartree-Slater core (effective core potential fitted to 12.13 eV, no spin-orbit), finite-difference time-dependent Schrodinger equation; at 293 nm an I^3 law fitted to the rates from 1e13 W/cm2",
       "definition": "the generalised cross section in cm6 s2, sigma_3 = W / Phi^3 with W the rate and Phi the photon flux in cm^-2 s^-1 (the formula is not printed; Fig. 1's dashed line so converted gives the printed value)",
       "gauge": "length (V_I = -sum e E z_i sin(omega t), Eq. 2)"
     }
   }

2 inline points in the source record.

p\. 781\: at 293 nm 'The generalized cross section corresponding to this line \[the I^3 fit to the rates of Fig\. 1, which depart from I^3 above 5e13 W/cm2\] is 1 x 10^\-82 cm6 s2', 'within about 20% with the Green's\-function results of McGuire'; at 266 nm 'we obtain a three\-photon cross section of 2\.5 x 10^\-82 cm6 s2', the value lhuillier\_1989 quotes\. Fig\. 1's dashed line read by the script gives 1\.05e\-82 at 293 nm \(the record's notes\)\. No uncertainty stated; the wavelength medium not stated\.

kulander\_1988/fig1\_rate\_293\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "fig1_rate_293_Xe",
     "material": "Xe",
     "file": "Xe/kulander_1988__fig1_rate_293_Xe.csv",
     "table": "fig_b14628317562",
     "digitised": "raster",
     "quantity": "ionization_rate",
     "unit": "s-1",
     "abscissa": "intensity",
     "abscissa_unit": "W/cm2",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "wavelength_nm": 293,
       "polarization": "linear",
       "method": "single active electron, frozen Hartree-Slater core (effective core potential fitted to 12.13 eV, no spin-orbit), finite-difference time-dependent Schrodinger equation; the 5p_0 (two electrons) and 5p_+-1 (four electrons) rates summed"
     }
   }

Fig\. 1 \(p\. 781\), circles on the heavy line\: the total rate of the 5p shell at 293 nm \(5p\_0 plus 5p\_\+\-1\), 4\.1e11 s\-1 at 1e13 to 2\.3e14 at 2e14 W/cm2, read from the page image \(the record's notes\)\. Three\-photon; I^3 up to about 5e13 W/cm2, then saturating; above 1e14 a Stark\-shifted d resonance makes the decay non\-exponential \(a factor of 2, p\. 785\)\. 5 %\: the digitising uncertainty \(the record's notes\); the paper states none below 1e14 W/cm2\.

kulander\_1988/fig2\_rate\_586\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "fig2_rate_586_Xe",
     "material": "Xe",
     "file": "Xe/kulander_1988__fig2_rate_586_Xe.csv",
     "table": "fig_bfb3456aaf40",
     "digitised": "raster",
     "quantity": "ionization_rate",
     "unit": "s-1",
     "abscissa": "intensity",
     "abscissa_unit": "W/cm2",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "wavelength_nm": 586,
       "polarization": "linear",
       "method": "single active electron, frozen Hartree-Slater core (effective core potential fitted to 12.13 eV, no spin-orbit), finite-difference time-dependent Schrodinger equation; the 5p_0 and 5p_+-1 rates summed"
     }
   }

Fig\. 2 \(p\. 781\), solid circles\: the total rate of the 5p shell at 586 nm, 6\.4e9 s\-1 at 1e13 to 4\.3e13 at 7\.5e13 W/cm2, read from the page image \(the record's notes\); above 7\.5e13 the markers lie on the 1064 nm ones and are not read\. Six\-photon in lowest order; a jump just above 3e13 W/cm2 from a Stark\-shifted 4f\-like resonance \(5p\_0 only, pp\. 782\-783\)\. The 1e13 value is below 1e10 s\-1, where 'the accuracy of the calculation suffers from numerical instabilities' \(p\. 785\)\. 5 %\: the digitising uncertainty \(the record's notes\)\.

kulander\_1988/fig2\_rate\_1064\_Xe
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "fig2_rate_1064_Xe",
     "material": "Xe",
     "file": "Xe/kulander_1988__fig2_rate_1064_Xe.csv",
     "table": "fig_bfb3456aaf40",
     "digitised": "raster",
     "quantity": "ionization_rate",
     "unit": "s-1",
     "abscissa": "intensity",
     "abscissa_unit": "W/cm2",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "wavelength_nm": 1064,
       "polarization": "linear",
       "method": "single active electron, frozen Hartree-Slater core (effective core potential fitted to 12.13 eV, no spin-orbit), finite-difference time-dependent Schrodinger equation; the 5p_0 and 5p_+-1 rates summed"
     }
   }

Fig\. 2 \(p\. 781\), solid triangles\: the total rate of the 5p shell at 1064 nm, 6\.7e9 s\-1 at 2e13 to 2\.3e13 at 8e13 W/cm2, read from the page image \(the record's notes\); from 1e14 the markers lie under the 586 nm circles and are not read\. Eleven\-photon in lowest order; no resonance effects, the outgoing flux collimated along the field, 'good evidence that the tunneling mechanism is in effect' at 1e14 \(p\. 785\); per electron the 5p\_0 rate exceeds the others' by more than 20 times \(p\. 786\)\. The 2e13 value is below 1e10 s\-1, where the accuracy suffers from numerical instabilities \(p\. 785\)\. 5 %\: the digitising uncertainty \(the record's notes\)\.

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 :doc:`review register </register>` for the checks and decisions.