ritchie_1977
============

B\. Ritchie, "Calculation of cross sections for multiphoton ionization of complex atoms", Phys\. Rev\. A 16, 2080\-2088 \(1977\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/ritchie_1977.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevA\.16\.2080 <https://doi.org/10.1103/PhysRevA.16.2080>`__

**Source kind:** theory

**Recorded technique:** Lowest\-order perturbation theory for two\-photon ionisation of He 1 1S \(pp\. 2080\-2083\)\: an independent\-particle \('optical' electron\) model in which the first\-order wave function is the symmetrised product of the zeroth\-order 1s orbital and a first\-order function F^\(1\) \(Eq\. 3\); projection onto the passive electron gives an integro\-differential equation with the static potential and an exchange term \(Eq\. 4\), turned into an integral equation with the Coulomb Green's function and solved non\-iteratively \(Eqs\. 6, A1\-A8\)\. Unlike coupled Hartree\-Fock perturbation theory \(Dalgarno and Victor\) there is no coupling of the positive\- and negative\-frequency first\-order functions\. Final state\: the exchange approximation of e\-He\+ scattering \(Eq\. 7\), l = 0 and 2\. Zeroth\-order ground state\: the 12\-term Roothaan\-Sachs\-Weiss Hartree\-Fock function, epsilon^\(0\) = \-0\.918 au \(Table I\), or the one\-parameter variational function 2 zeta^\(3/2\) exp\(\-zeta r\), zeta = 27/16, epsilon^\(0\) = \-0\.896 au \(Table II\)\. 'E' \(exchange\) keeps the exchange term and the factor sqrt\(2\) of Eq\. 4; 'NE' \(nonexchange\) drops both \(the amplitude still uses the symmetrised forms, p\. 2085\)\. Dipole\-velocity form \('for linearly polarized light', Eq\. 9a\); one value in the dipole\-length form \(p\. 2084\)\. No dynamic Stark shift or broadening\. Numerical accuracy 'within about 5 %' \(p\. 2088\)\.

**Materials in structured entries:** He

**Dataset quantities:** multiphoton\_ionization\_cross\_section

**Contents:** 9 datasets; 0 formulas.

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

Phys\. Rev\. A 16, 2080\-2088 \(1977\), NUMBER 5, November 1977\: PDF pages 1\-9 = journal pp\. 2080\-2088\. Identity checked on the first page \(PDF p\. 1\: 'PHYSICAL REVIEW A VOLUME 16, NUMBER 5 NOVEMBER 1977', 'Calculation of cross sections for multiphoton ionization of complex atoms', Burke Ritchie, U\. S\. Army Missile Command, Redstone Arsenal; received 10 June 1976, revised 21 January 1977\) on 2026\-10\-09\. The library's identity is VALIDATED\_WITH\_WARNINGS; the copy is the version of record\. The PDF is a scan \(pdfimages\: one 4938 x 6686 bilevel JBIG2 image per page, 612 dpi; pdffonts\: only a non\-embedded Times\-Roman text layer\)\.

HELIUM ONLY\. Ritchie's tables are the only numbers\: two\-photon ionisation of He 1 1S with linear polarisation, below the one\-photon threshold \(E\_p 0\.456 to 0\.868 au, 12\.41 to 23\.62 eV\)\. No other gas and no other order of ionisation is calculated\: in particular NO KRYPTON \(nor xenon, argon or neon\) values of any kind\. The introduction cites calculations for other atoms \(Pindzola and Kelly's Ar, Bebb's alkalis, hydrogenic ions, metastable He; pp\. 2080\-2081\) and prints none of their numbers; the title's 'complex atoms' refers to the method\.

THE QUANTITY\. Tables I and II print 'Values of sigma^\(2\) x I\_0/2I in a\_0^2' \(the captions\), Fig\. 1's axis 'sigma^\(2\) \(2I/I\_0 a\_0^2\)'\. Eq\. 1 defines sigma^\(N\) through d sigma^\(N\) / d Omega = \(alpha a\_0^2 / 2 pi\)\(k / E\_p\)\(I / I\_0 E\_p^2\)^\(N\-1\) \|\.\.\.\|^2 with 'I the light intensity in W/cm^2, I\_0 = 14\.038 x 10^16 W/cm^2'\: sigma^\(2\) is a cross section in cm^2 proportional to I, for the rate sigma^\(2\) Phi \(Phi the photon flux\)\. For a printed value v the generalised cross section of Gamma = sigma\_2 Phi^2 \(Phi = I / hbar omega\) is therefore sigma\_2 = 2 v a\_0^2 E\_p E\_h / I\_0, and W / I^2 = sigma\_2 / \(hbar omega\)^2 = 9\.151 x 10^\-17 v / E\_p \(au\) s^\-1 W^\-2 cm^4\. Not converted in the data \(the values are as printed, unit a0^2, the definition in each dataset's parameters\); the conversion is confirmed by lhuillier\_1986's Fig\. 13, which draws Table I's E column as W / I^2\: its dashed line is 0\.93, 0\.95 and 1\.01 times the converted values at 0\.506, 0\.570 and 0\.651 au \(lhuillier\_1986's checks; a factor\-2 misreading of 2I / I\_0 would give 0\.5 or 2\)\. At 0\.456 au, v = 1\.267 is sigma\_2 = 1\.005 x 10^\-51 cm^4 s\.

THE ENERGY AXIS is the photon energy E\_p in au as printed \(Table I prints the wavelength in angstrom beside it, kept in 'printed'; the abscissa is E\_p\)\. The final state's energy uses the accurate ionisation potential, 'N E\_p = k^2/2 \+ V\_I' with V\_I = 0\.904 au \(p\. 2081\), but the intermediate \(Green's\-function\) spectrum is the model's\: the poles sit at the orbital energy, so the 1s2p and 1s3p resonances are displaced\. Table I's first resonance lies near 0\.786\-0\.802 au \(unshifted pole 0\.793 au\), against the accurate 0\.780 au; Table II's near 0\.763\-0\.782 \(unshifted 0\.771\)\. The author applied no shift \('We recommend use of a semiempirical modification \.\.\. in the vicinity of resonances', p\. 2085, not done here\)\. LS coupling, non\-relativistic\.

EXTRACTED \(tools/extract/ritchie\_1977\.py\)\: Tables I and II, every column, the plain values and the parenthesised ones \(the latter 'obtained ignoring final\-state, perturbed\-initial\-state nonorthogonality', i\.e\. with phi\_0 instead of the Schmidt\-orthogonalised phi\-hat\_0 of Eq\. 10\)\: eight datasets\. The p\. 2084 dipole\-length value at 1000 angstrom as an inline point\. Every cell of both tables checked against the table images and the page image at its native pixels\: all agree\. The specks over the decimal points of five of Table II's E cells \('1\:469', '1/361', '0\:8283', '0\:6248', '\(0\:6199\)' in a low\-resolution view\) are dust on the scan; the digits are unambiguous\.

THE CHECKS \(the script fails if one is missed\)\: every wavelength of Table I against its E\_p \(hc / E\_h = 455\.634 angstrom au; within the roundings of both\: 0\.760 au at 600 angstrom is the largest miss, 455\.634 / 600 = 0\.7594, within 0\.0011\); the unshifted first resonance \|epsilon^\(0\)\| \- 1/8 = 0\.793 \(\-0\.918\) and 0\.771 \(\-0\.896\) au \(pp\. 2084\-2085\: nu = 2 in Eq\. A1\); epsilon^\(0\) = zeta^2/2 \- 2 zeta \+ 5 zeta/8 = \-0\.8965 for zeta = 27/16 \('\-0\.896'\); the accurate V\_I = 0\.904 au and resonance 0\.780 au \(nist\_asd\_2026's He IE 24\.587389 eV = 0\.90357 au and 1s2p 1P1 171134\.89 cm\-1 = 0\.77975 au\); 'about 20 %' dynamical error at threshold\: 1 \- E / 2 NE at 0\.456 au = 0\.199 \(Table I\) and 0\.200 \(Table II\)\.

COMPARISON with the curated He calculations \(Table I's E column as sigma\_2, interpolated in log sigma; reported by the script, not checked\)\: Ritchie over mihelic\_2021 \(L = 0 \+ 2; OEP 6's He model, TC\-943, the same\) 0\.77, 0\.81, 0\.84, 0\.85, 0\.89 at E\_p = 0\.456, 0\.506, 0\.570, 0\.651, 0\.701 au \(12\.41\-19\.08 eV\); over saenz\_1999 \(1S \+ 1D, velocity\) 0\.71, 0\.79, 0\.87, 0\.96 \(from 0\.506 au\); over nikolopoulos\_2001 \(S \+ D\) 0\.62, 0\.77, 0\.90, 1\.03, 1\.18; over bachau\_1990's Table I L^2 value at 0\.5 au \(Table I interpolated to 0\.5 au, 1\.036 x 10^\-51 cm^4 s\) 0\.79\. The independent\-particle Hartree\-Fock values lie 10\-30 % below the correlated calculations near threshold, closer at 18\-19 eV\. lhuillier\_1986 \(LDA\-RPA\) calls its own curve and Ritchie's 'in very good agreement' for 0\.90 \< omega \< 1\.40 Ry; its Fig\. 13 draws Ritchie's about 20 % below its own there \(1\.98 against 2\.51 at 1 Ry, both read from the page images\)\.

UNCERTAINTY\. 'We estimate that the use of these procedures provides cross sections accurate to within about 5 %' \(p\. 2088\)\: the numerical accuracy of the solution of Eq\. 6, taken as relative 0\.05 \(level not stated\)\. It does not include the model's error \(an independent\-particle Hartree\-Fock model, no correlation; the length and velocity forms differ by 11 % at 1000 angstrom, 1\.403 against 1\.267\), nor the error of the resonance positions\.

NOT TAKEN\. Fig\. 1 \(p\. 2084\)\: Table I's E column plotted with a hand\-drawn line through the points \(the points are the table\)\. Figs\. 2\-4\: radial response functions u\_1^\(1\)\(r\) at 585, 568 and 555 angstrom, no quantity\. The comparison with Victor's dipole\-length result 'in excellent agreement' at 1000 angstrom\: graphical in Victor's paper, no number printed here\.

FURTHER TWO\-PHOTON DATA CITED, not in the library\: G\. A\. Victor, Proc\. Phys\. Soc\. 91, 825 \(1967\) \(He, coupled Hartree\-Fock, length form, graphical\); M\. S\. Pindzola and H\. P\. Kelly, Phys\. Rev\. A 11, 1543 \(1975\) \(Ar, many\-body perturbation theory; in the library as Pindzola\_1975, record pindzola\_1975\)\.

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

Datasets
--------

ritchie\_1977/tbl1\_E
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl1_E",
     "material": "He",
     "file": "He/ritchie_1977__tbl1_E_He.csv",
     "table": "tbl_2491aff218c3",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "independent-particle lowest-order perturbation theory, Coulomb Green's function, static-exchange first-order equation (exchange, 'E'), Roothaan-Sachs-Weiss Hartree-Fock ground state, exchange-approximation final state",
       "definition": "sigma^(2) x I_0 / 2I in a0^2 (the caption): sigma^(2) the intensity-dependent cross section of Eq. 1 (rate sigma^(2) Phi), I in W/cm2, I_0 = 14.038e16 W/cm2; for Gamma = sigma_2 Phi^2, sigma_2 = 2 v a0^2 E_p E_h / I_0"
     }
   }

Table I \(p\. 2083\), column E, the plain values\: sigma^\(2\) x I\_0/2I in a\_0^2 for two\-photon ionisation of He, Hartree\-Fock ground state \(epsilon^\(0\) = \-0\.918 au\), exchange, at 22 photon energies 0\.456\-0\.868 au \(1000\-525 angstrom\), below the one\-photon threshold, across the 1s2p \(near 0\.79 au\) and 1s3p \(near 0\.86 au\) resonances at the model's positions \(the record's notes\)\. 1\.267 at threshold, 0\.5877 at 0\.701 au, 2\.419 at 0\.786 au\. The author's preferred set \(plotted in Fig\. 1\)\. The total over the final 1S and 1D states\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl1\_E\_nonorth
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl1_E_nonorth",
     "material": "He",
     "file": "He/ritchie_1977__tbl1_E_nonorth_He.csv",
     "table": "tbl_2491aff218c3",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl1_E, the final state not orthogonalised to the perturbed initial state",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table I \(p\. 2083\), column E, the values in parentheses\: as tbl1\_E but 'obtained ignoring final\-state, perturbed\-initial\-state nonorthogonality' \(phi\_0 for phi\-hat\_0, Eq\. 10\)\. Within 4 % of tbl1\_E off resonance \(1\.320 against 1\.267 at threshold\), up to 7 % near the resonances\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl1\_NE
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl1_NE",
     "material": "He",
     "file": "He/ritchie_1977__tbl1_NE_He.csv",
     "table": "tbl_2491aff218c3",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl1_E without the exchange term and the sqrt(2) normalisation of the inhomogeneous term ('NE')",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table I \(p\. 2083\), column NE, the plain values\: the nonexchange model \(the exchange term and the sqrt\(2\) of Eq\. 4 dropped\), Hartree\-Fock ground state\. A factor 1\.6 below tbl1\_E at threshold \(0\.7911 against 1\.267\), mostly the missing normalisation factor of the symmetrised wave function \(p\. 2085\); its resonances lie elsewhere \(6\.289 at 0\.786 au\)\. Not a physical model\: a measure of the exchange effect\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl1\_NE\_nonorth
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl1_NE_nonorth",
     "material": "He",
     "file": "He/ritchie_1977__tbl1_NE_nonorth_He.csv",
     "table": "tbl_2491aff218c3",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl1_NE, the final state not orthogonalised to the perturbed initial state",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table I \(p\. 2083\), column NE, the values in parentheses\: the nonexchange model without the final\-state orthogonalisation\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl2\_E
~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl2_E",
     "material": "He",
     "file": "He/ritchie_1977__tbl2_E_He.csv",
     "table": "tbl_7b95e928b8ad",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl1_E with the one-parameter variational ground state 2 zeta^(3/2) exp(-zeta r), zeta = 27/16",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table II \(p\. 2083\), column E, the plain values\: as tbl1\_E with the one\-parameter variational ground state \(zeta = 27/16, epsilon^\(0\) = \-0\.896 au\), at ten photon energies 0\.456\-0\.799 au, through the first resonance only \(near 0\.776 au in this model, p\. 2085\)\. 7 % above tbl1\_E at threshold \(1\.359 against 1\.267\); 'we expect the Hartree\-Fock results to be of greater accuracy in the nonresonant region' \(p\. 2085\)\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl2\_E\_nonorth
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl2_E_nonorth",
     "material": "He",
     "file": "He/ritchie_1977__tbl2_E_nonorth_He.csv",
     "table": "tbl_7b95e928b8ad",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl2_E, the final state not orthogonalised to the perturbed initial state",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table II \(p\. 2083\), column E, the values in parentheses\: as tbl2\_E without the final\-state orthogonalisation\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl2\_NE
~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl2_NE",
     "material": "He",
     "file": "He/ritchie_1977__tbl2_NE_He.csv",
     "table": "tbl_7b95e928b8ad",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl2_E without the exchange term and the sqrt(2) normalisation ('NE')",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table II \(p\. 2083\), column NE, the plain values\: the nonexchange model with the one\-parameter variational ground state\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/tbl2\_NE\_nonorth
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "tbl2_NE_nonorth",
     "material": "He",
     "file": "He/ritchie_1977__tbl2_NE_nonorth_He.csv",
     "table": "tbl_7b95e928b8ad",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "uncertainty": {
       "relative": 0.05,
       "level": "not stated"
     },
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "velocity",
       "method": "as tbl2_NE, the final state not orthogonalised to the perturbed initial state",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

Table II \(p\. 2083\), column NE, the values in parentheses\: the nonexchange model without the final\-state orthogonalisation\. 5 %\: the numerical accuracy \(p\. 2088\)\.

ritchie\_1977/text\_length\_1000A
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "text_length_1000A",
     "material": "He",
     "table": "text",
     "quantity": "multiphoton_ionization_cross_section",
     "unit": "a0^2",
     "abscissa_unit": "hartree",
     "use": "check",
     "parameters": {
       "photons": 2,
       "polarization": "linear",
       "gauge": "length",
       "method": "as tbl1_E (implied, not restated), dipole-length form",
       "definition": "sigma^(2) x I_0 / 2I in a0^2, as tbl1_E"
     }
   }

1 inline points in the source record.

p\. 2084\: 'we did use the dipole length form at 1000 A, obtaining 1\.403 \(2I/I\_0\) a\_0^2', against the velocity form's 1\.267 \(tbl1\_E at 0\.456 au, the 1000 angstrom row\)\: 11 % higher\. Which zeroth\-order function and exchange treatment are not restated; the sentence follows Table I's discussion and is compared with Victor's Hartree\-Fock length result, so Table I's model with exchange is implied \(not stated\)\. The abscissa is Table I's E\_p for 1000 angstrom\. 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 </register>` for the checks and decisions.