shekhtman_2020 ============== D\. Shekhtman, M\. A\. Mustafa and N\. J\. Parziale, "Two\-photon cross\-section calculations for krypton in the 190\-220 nm range", Appl\. Opt\. 59, 10826 \(2020\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1364/AO\.410806 `__ **Source kind:** theory **Recorded technique:** Lowest\-order perturbation theory for the two\-photon excitation of krypton from its 4p6 1S0 ground state to eight 5p, 5p' and 6p levels, one per excitation line \(192\.749 to 216\.667 nm\), in linear polarisation only \(q = 0, p\. 5\)\. The two\-photon matrix element M\_fg^\(2\) \(Eq\. 12\) is a sum over a finite basis of intermediate states \(the 'matrix mechanics' form D G D, Eq\. 19\), truncated at n \<= 7 by a Debye\-length argument \(Eq\. 13\)\. The dipole matrix elements are evaluated with quantum\-defect radial functions \(Eq\. 33\) calibrated on NIST level energies and Einstein coefficients \(the integer I\_m chosen to reproduce the NIST A values, Table 4\), the ground state's Hartree\-Fock 4p orbital of Clementi \(Eq\. 39\), and, in the 'with oscillator strengths' variant, the NIST transition probabilities where they exist \(Eq\. 41\), with the sign taken from the quantum\-defect radial integral\. Three bases \(Table 3's states\)\: set 1, \|g\> and \|1\> to \|20\> \(5s, 6s, 7s intermediates\); set 2, also \|21\> \(4d\); set 3, also \|22\> to \|24\> \(5d, 6d\)\. The single\-path approximation of Table 9 keeps one intermediate state per line\. The cross section is sigma\_o^\(2\) = \(2 pi\)^3 alpha^2 omega\_L^2 \|M\_fg^\(2\)\|^2 a\_o^4 \(Eq\. 8\), with W\_fg = sigma^\(2\) F^2 \(Eq\. 6\), F the photon flux, and sigma^\(2\) = sigma\_o^\(2\) g\(2 omega\_L\) \(Eq\. 7\), g a Gaussian line shape whose peak is Eq\. \(9\)\. The experimental part \(relative fluorescence at 212\.56, 214\.77, 216\.67 nm, 7 ns, 1350 MHz dye laser, 1 torr of 1 % Kr in N2\) is a check of the relative values only\. **Materials in structured entries:** Kr **Dataset quantities:** two\_photon\_excitation\_cross\_section **Contents:** 7 datasets; 0 formulas. Curation notes -------------- Appl\. Opt\. 59\(34\), 10826\-10837 \(1 December 2020\) = PDF pages 1\-12 \(journal page = PDF page \+ 10825\); identity checked against the first page \(journal, volume, pages, authors\: D\. Shekhtman, M\. A\. Mustafa, N\. J\. Parziale\)\. Version of record\. WHAT IS TAKEN \(tools/extract/shekhtman\_2020\.py\)\. sigma\_o^\(2\), cm4, of Tables 5, 6 and 7 \(basis sets 1, 2, 3; PDF pp\. 10\-11, journal pp\. 10835\-10836\), quantum\-defect \(QD\) and quantum\-defect with oscillator strengths \(QD\+OS\) columns, and of Table 9 \(p\. 11/10836, single path\)\: seven datasets of eight lines\. The row is the final level in Racah notation \(5p\[3/2\]2; primed for the 2P\_1/2 core\), taken from Table 3 \(p\. 9/10834\) by its lambda\_L; the abscissa is the printed lambda\_L, the exciting photon's vacuum wavelength \(Table 3 note c\: 'as measured in vacuum'; checked\: 2 h c / \(E \- E\_G\) from Table 3's own energies reproduces every lambda\_L to 0\.0005 nm, while standard\-air values would be 0\.07 nm shorter\)\. The authors' result is basis set 3 with oscillator strengths \(Fig\. 3, and the value the text compares with experiment, 'In Table 7, the calculated cross\-section for 214\.769 nm excitation is 4\.18 x 10^\-35 cm4', p\. 9/10834\)\: that dataset is primary, the others \(smaller bases, QD alone, single path\) are checks\. No uncertainty is stated; the QD Einstein coefficients of Table 4 differ from NIST's by 2 % to 541 %, and the bases differ by up to three orders of magnitude for the 6p lines \(192\-194 nm\), which the text calls 'predictions'\. For the 5p and 5p' lines \(202\-217 nm\) the paper reports 'good agreement' of the relative values with measured relative excitation spectra \(Table 8\), 'especially using basis sets 2 and 3' \(p\. 7/10832\)\. THE LINE PROFILE IS NORMALISED IN ANGULAR FREQUENCY \(frequency\_variable\)\. The tables' second column, sigma^\(2\) = sigma\_o^\(2\) g\(2 omega\_L\) in cm4 s, divided by sigma\_o^\(2\) is Eq\. \(9\)'s Gaussian peak g = 2 sqrt\(ln 2 / pi\) / sqrt\(2 Delta omega\_L^2 \+ Delta omega\_T^2\) with Delta omega\_L = 2 pi x 1350 MHz and the Doppler width \(Eq\. 10\) of Kr at 298 K at the two\-photon frequency 2 omega\_L, both angular\: 1\.0045 to 1\.0125 times it \(mean 1\.008\) in all 52 regular cells; the mean is 1\.000 at 292 K, so the authors' temperature \(or mass\) may differ slightly from 298 K\. With g normalised in ordinary frequency the ratio would be 2 pi smaller\. So W = sigma\_o^\(2\) g Phi^2 with g normalised in the two\-photon angular frequency, the convention of kroll\_1990 and drag\_2021; Eq\. \(8\) is drag\_2021's Eq\. \(9\), 8 pi^3 alpha^2 omega^2 \|M\|^2\. NOT TAKEN\. The profile\-dependent sigma^\(2\) \(cm4 s\), the peak for the authors' line shape and laser width, which the vocabulary does not hold \(it is sigma\_o^\(2\) times a number fixed above\); used as the check above\. The normalised columns sigma^\(2\) / max sigma^\(2\) \(checked\)\. Table 3's level energies \(NIST's, below the ionisation limit\) and quantum\-defect parameters, Table 4's Einstein coefficients \(NIST's and the quantum\-defect ones\: no quantity of the vocabulary holds an A coefficient\), Table 8's relative excitation signals \(Richardson et al\., Grib et al\. and this work\: relative fluorescence, no quantity\), Tables 1 and 2 \(angular\-momentum bookkeeping\)\. The photoionisation cross section of Eq\. \(5\) is Khambatta et al\.'s formula, not evaluated\. PRINTED INCONSISTENCIES \(the script's checks list them; every other cell agrees with its row\)\: \(1\) Table 6, QD, 193\.494 nm\: sigma\_o^\(2\) printed 9\.85 x 10^\-35, the 204\.196 nm cell two rows below repeated; its sigma^\(2\) \(7\.42e\-46\) and normalised ratio \(0\.084\) give 2\.26e\-35\: erratum\. \(2\) Table 9, 214\.769 nm\: sigma\_o^\(2\) printed 8\.54 x 10^\-35; its sigma^\(2\) \(3\.05e\-46\) and ratio \(0\.318\) give 8\.5e\-36\: erratum \(the power of ten\)\. \(3\) Table 5, QD, 214\.769 and 216\.667 nm\: sigma^\(2\) printed 1\.98 x 10^\-46 and 2\.24 x 10^\-46, ten times smaller than their sigma\_o^\(2\) g and their printed ratios 0\.404 and 0\.455 imply\: the power of ten; not a taken column\. \(4\) Table 7, QD, 193\.494 nm\: the normalised ratio printed 0\.198, against 1\.74e\-45 / 1\.03e\-44 = 0\.169\: not a taken column\. The PDF's text layer has the same numbers as the table images in all four cases\. \(5\) The text \(p\. 9/10834\) quotes dakka\_2018's sigma\_o^\(2\) as '5\.2 \+\- 2\.2 x 10^\-35 cm4'; the library's copy of Dakka et al\. \(the submitted version\) prints 5\.2 \+\- 2\.1\. REFERENCES CITED FOR TWO\-PHOTON DATA, NOT IN THE LIBRARY\: N\. M\. Khambatta, J\. A\. Oertel, R\. Silk, L\. J\. Radziemski and J\. M\. Mack, J\. Appl\. Phys\. 64, 4809 \(1988\) \(Kr two\- and three\-photon excitation cross sections, 6p levels\); N\. M\. Khambatta, L\. J\. Radziemski and S\. N\. Dixit, Phys\. Rev\. A 39, 3842 \(1989\) \(Ar three\-photon\); J\. Bokor, J\. Y\. Zavelovich and C\. K\. Rhodes, Phys\. Rev\. A 21, 1453 \(1980\) \(Kr 6p, single path\); M\. A\. Mustafa, D\. Shekhtman and N\. J\. Parziale, Phys\. Rev\. Appl\. 11, 064013 \(2019\) \(single path, 212\.556 nm\); and the relative excitation spectra of D\. R\. Richardson et al\., Opt\. Lett\. 42, 3498 \(2017\) and S\. W\. Grib et al\., Appl\. Opt\. 58, 7621 \(2019\)\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- shekhtman\_2020/qd\_basis1 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qd_basis1", "material": "Kr", "file": "Kr/shekhtman_2020__qd_basis1.csv", "table": "tbl_795599a25fb2", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect", "basis": "set 1: |g> and |1> to |20> of Table 3 (5s, 6s, 7s intermediates)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 5 \(p\. 10/10835\), Quantum\-Defect sigma\_o^\(2\) column; basis set 1\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. This column's sigma^\(2\) at 214\.769 and 216\.667 nm is misprinted by a power of ten \(record notes\), not this one\. shekhtman\_2020/qdos\_basis1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qdos_basis1", "material": "Kr", "file": "Kr/shekhtman_2020__qdos_basis1.csv", "table": "tbl_795599a25fb2", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect_with_oscillator_strengths", "basis": "set 1: |g> and |1> to |20> of Table 3 (5s, 6s, 7s intermediates)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 5 \(p\. 10/10835\), Quantum\-Defect with Oscillator Strengths sigma\_o^\(2\) column; basis set 1\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. shekhtman\_2020/qd\_basis2 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qd_basis2", "material": "Kr", "file": "Kr/shekhtman_2020__qd_basis2.csv", "table": "tbl_c5a835601f3b", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect", "basis": "set 2: |g> and |1> to |21> of Table 3 (5s, 6s, 7s and one 4d intermediate)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 6 \(p\. 10/10835\), Quantum\-Defect sigma\_o^\(2\) column; basis set 2\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. The 193\.494 nm cell \(printed 9\.85e\-35\) is corrected by an erratum\. shekhtman\_2020/qdos\_basis2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qdos_basis2", "material": "Kr", "file": "Kr/shekhtman_2020__qdos_basis2.csv", "table": "tbl_c5a835601f3b", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect_with_oscillator_strengths", "basis": "set 2: |g> and |1> to |21> of Table 3 (5s, 6s, 7s and one 4d intermediate)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 6 \(p\. 10/10835\), Quantum\-Defect with Oscillator Strengths sigma\_o^\(2\) column; basis set 2\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. shekhtman\_2020/qd\_basis3 ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qd_basis3", "material": "Kr", "file": "Kr/shekhtman_2020__qd_basis3.csv", "table": "tbl_9e3a7f52b1d8", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect", "basis": "set 3: |g> and |1> to |24> of Table 3 (5s, 6s, 7s, 4d, 5d, 6d intermediates)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 7 \(p\. 11/10836\), Quantum\-Defect sigma\_o^\(2\) column; basis set 3\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. This column's normalised ratio at 193\.494 nm is misprinted \(record notes\), not sigma\_o^\(2\)\. shekhtman\_2020/qdos\_basis3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "qdos_basis3", "material": "Kr", "file": "Kr/shekhtman_2020__qdos_basis3.csv", "table": "tbl_9e3a7f52b1d8", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "primary", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "quantum_defect_with_oscillator_strengths", "basis": "set 3: |g> and |1> to |24> of Table 3 (5s, 6s, 7s, 4d, 5d, 6d intermediates)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 7 \(p\. 11/10836\), Quantum\-Defect with Oscillator Strengths sigma\_o^\(2\) column; basis set 3, the authors' result \(Fig\. 3; the 214\.769 nm value, 4\.18e\-35 cm4, is the one the text compares with dakka\_2018\)\. Row\: the final level \(Table 3\); x\: the laser's vacuum wavelength\. No uncertainty is stated\. shekhtman\_2020/single\_path ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "single_path", "material": "Kr", "file": "Kr/shekhtman_2020__single_path.csv", "table": "tbl_305221c17bc2", "quantity": "two_photon_excitation_cross_section", "unit": "cm4", "abscissa_unit": "nm", "use": "check", "parameters": { "frequency_variable": "angular", "polarization": "linear", "photon_statistics": "corrected (a calculation: lowest-order perturbation theory for a monochromatic field, no G2(0))", "method": "single_path", "basis": "one intermediate state per line (Table 9's |k>)", "definition": "W_fg = sigma^(2) F^2 (Eq. 6), F the photon flux; sigma^(2) = sigma_o^(2) g(2 omega_L) (Eq. 7), g normalised in the two-photon angular frequency (checked, record notes); sigma_o^(2) = (2 pi)^3 alpha^2 omega_L^2 |M_fg^(2)|^2 a_o^4 (Eq. 8)" } } Table 9 \(p\. 11/10836\), single\-path approximation\: one intermediate state, \|1\> \(5s\[3/2\]1\) or \|3\> \(5s'\[1/2\]1\) as the table names it, with the NIST oscillator strengths\. Row\: the final level \(Table 3, and Table 9's own \|f\> column, which agrees\); x\: the laser's vacuum wavelength\. The 214\.769 nm cell \(printed 8\.54e\-35\) is corrected by an erratum\. Formulas -------- No formulas are recorded. Corrections ----------- .. code-block:: text [ { "dataset": "qd_basis2", "row": "6p[3/2]2", "column": "value", "printed": "9.85e-35", "corrected": "2.26e-35", "reason": "Table 6 prints the 204.196 nm cell (9.85 x 10^-35, two rows below) at 193.494 nm: the row's own sigma^(2) = 7.42 x 10^-46 cm4 s and normalised ratio 0.084 (= 7.42e-46 / 8.86e-45) give sigma_o^(2) = sigma^(2) / g(193.494 nm) = 2.26e-35, where g is Eq. (9)'s Gaussian peak times the mean ratio sigma^(2) / (sigma_o^(2) g) = 1.008 of the 52 regular cells of Tables 5-7 and 9 (their spread, 1.0045 to 1.0125, makes the correction good to +-0.01 in the last figure); the printed value gives a ratio of 0.232.", "evidence": "Table 6 image and the PDF's text layer both read 9.85 x 10^-35 in the 193.494 and the 204.196 nm rows of the Quantum-Defect column; checked by tools/extract/shekhtman_2020.py (checks)." }, { "dataset": "single_path", "row": "5p[3/2]2", "column": "value", "printed": "8.54e-35", "corrected": "8.54e-36", "reason": "The row's own sigma^(2) = 3.05 x 10^-46 cm4 s (and its normalised ratio 0.318 = 3.05e-46 / 9.57e-46) gives sigma^(2) / (sigma_o^(2) g) = 0.101 with the printed sigma_o^(2), and 1.010 with 8.54 x 10^-36, inside the 1.0045 to 1.0125 of the regular cells: the power of ten is misprinted.", "evidence": "Table 9 image and the PDF's text layer both read 8.54 x 10^-35; checked by tools/extract/shekhtman_2020.py (checks)." } ] 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.