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 · Catalogue · 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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
{
"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#
[
{
"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 review register for the checks and decisions.