tong_2005 ========= X\. M\. Tong and C\. D\. Lin, "Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier\-suppression regime", J\. Phys\. B 38, 2593 \(2005\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1088/0953\-4075/38/15/001 `__ **Source kind:** theory **Recorded technique:** An empirical correction of the tunnelling \(ADK, MO\-ADK\) static\-field rate into the barrier\-suppression regime, Eq\. \(2\), with one constant alpha fitted per target to static rates computed by complex scaling in single\-active\-electron model potentials \(Eq\. \(4\), Table 1\) for H, He, Ne, Ar, Rb, Ne\+ and Ar\+, and checked on H2\+ and on correlated He rates \(Scrinzi et al\. 1999, scrinzi\_1999\)\. **Materials in structured entries:** Ar, Ar\+, H, H2\+, He, Ne, Ne\+ **Dataset quantities:** ionization\_energy, ionization\_model\_parameter **Contents:** 18 datasets; 0 formulas. Curation notes -------------- Journal pages 2593\-2600 = PDF pages 2\-9 \(PDF page 1 is the publisher's cover sheet\)\. solprop's BSI\_TONG\_LIN\_ALPHA \(He 7\.0; Ne, Ar, Kr, Xe 9\.0\) is this paper's alpha\. EXTRACTED\. Table 2 \(p\. 2596\)\: alpha and I\_p \(au\) of He, Ne, Ar and the ions Ne\+ and Ar\+ \(Rb is not curated\: alpha 6\.0, I\_p 0\.154, F\_b 0\.006; H's alpha and I\_p are, since atomic hydrogen became a material, TC\-833\)\. Table 1 \(p\. 2596\)\: the model\-potential parameters of the same five targets \(H\: Z\_c = 1, all a\_i = 0; Rb\: 1\.0, 24\.023, 11\.107, 115\.200, 6\.629, 11\.977, 1\.245\)\. Text \(pp\. 2596\-2597\)\: alpha = 6 for H2\+ \(inline\)\. THE FORMULAS \(p\. 2595\)\. Eq\. \(1\)\: the critical field F\_b = kappa^4/\(16 Z\_c\), 'the unperturbed atomic energy level lies on the top of the potential barrier'; kappa = sqrt\(2 I\_p\)\. Eq\. \(2\)\: W\_TBI\(F\) = W\_TI\(F\) exp\(\-alpha \(Z\_c^2/I\_p\)\(F/kappa^3\)\)\. Eq\. \(3\)\: W\_TI\(F\) = C\_l^2/\(2^\|m\| \|m\|\!\) \(2l \+ 1\)\(l \+ \|m\|\)\!/\(2 \(l \- \|m\|\)\!\) kappa^\-\(2Z\_c/kappa \- 1\) \(2 kappa^3/F\)^\(2Z\_c/kappa \- \|m\| \- 1\) exp\(\-2 kappa^3/3F\), 'the usual ADK rates for atoms', with C\_l 'derived from the ground\-state wavefunction calculated from the model potential, see table II of \(Tong et al 2002\)' \(tong\_2002\); for molecules W\_TI is the MO\-ADK rate\. Eq\. \(4\)\: V\(r\) = \-\(Z\_c \+ a\_1 exp\(\-a\_2 r\) \+ a\_3 r exp\(\-a\_4 r\) \+ a\_5 exp\(\-a\_6 r\)\)/r\. Eq\. \(5\)\: Error = \(W\_m \- W\_exact\)/W\_exact\. Eq\. \(6\)\: P = 1 \- exp\(\-int W\_m\(\|F\(t\)\|\) dt\)\. All in atomic units; static rates\. Rates are for the single active electron\: 'Static ionization rates per electron' for He \(Fig\. 4 caption\), and no occupation factor appears in Eqs\. \(2\)\-\(3\)\. F\_b CHECK\: Eq\. \(1\) with Table 2's I\_p and Z\_c \(1 for atoms, 2 for the ions, Table 1\) gives Rb 0\.00593, H 0\.0625, He 0\.2043, Ar\+ 0\.1290, Ar 0\.0838, Ne 0\.1572 \(printed 0\.006, 0\.063, 0\.204, 0\.129, 0\.084, 0\.157\) and Ne\+ 0\.2835 against the printed 0\.2845\: the Ne\+ entry is 0\.001 high \(a misprint, or F\_b from a more precise I\_p; not a dataset, so no erratum\)\. Table 1's Z\_c \+ a\_1 \+ a\_5 is the nuclear charge for every target \(He 2, Ne 10, Ar 18, Rb 37, Ne\+ 10, Ar\+ 18\: checked 2026\-09\-26\)\. THE RECOMMENDATION \(p\. 2596\)\: 'we further note that the fitted parameter alpha ranges from 6\.0 to 9\.0, covering binding energies from 0\.15 to 1\.5 au\. Based on table 2, we suggest that one chooses alpha = 6\.0 for electrons ionized from the s\-wave and alpha = 9\.0 for ionization from the higher partial waves\. With such choices the TBI rates agree with the numerically calculated ionization rates within 50%\.' This is the origin of solprop's 9\.0 for Kr and Xe, which the paper did not fit; for He \(an s electron\) the recommendation would give 6\.0 where Table 2's fit is 7\.0, and solprop uses the fitted 7\.0\. For atomic hydrogen alone alpha = 6 fits over a large range of F/F\_b \(p\. 2595, Fig\. 1\)\. FIGURES \(not digitised\)\: Fig\. 1 H static rates against F/F\_b \(ADK, TBI, complex scaling\); Fig\. 2 the TI and TBI errors against F/F\_b for Rb, He, Ne, Ar, Ne\+, Ar\+; Fig\. 3 H2\+ static rates; Fig\. 4 He rates per electron against Scrinzi et al\.'s correlated results; Fig\. 5 H ionisation probability in 25 fs pulses at 800 and 1064 nm \(TDSE against TI and TBI; saturation 2\.75 and 4\.40 x 10^14 W/cm^2 for TI and TBI at 98 %\)\. **Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- tong\_2005/alpha\_H ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_H", "material": "H", "file": "H/tong_2005__alpha_H.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\) for atomic hydrogen, fitted to its exact static rates \(the bare Coulomb potential, Table 1's row H\: Z\_c = 1, every a\_i = 0\)\. Curated for the comparisons of optmat's rate with the one\-electron calculations of hydrogen \(TC\-827, TC\-833\)\. tong\_2005/ip\_H ~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_H", "material": "H", "file": "H/tong_2005__ip_H.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: hydrogen's 1s energy in atomic units with infinite nuclear mass, 0\.5\. tong\_2005/alpha\_He ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_He", "material": "He", "file": "He/tong_2005__alpha_He.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\), fitted to single\-active\-electron complex\-scaling static rates computed in the Eq\. \(4\) potential \(potential\_He\); 'the TBI rates agree with the numerically calculated ionization rates within 50%' \(p\. 2596, for the recommended choice\)\. solprop's BSI\_TONG\_LIN\_ALPHA for this gas\: 7\.0\. tong\_2005/ip\_He ~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_He", "material": "He", "file": "He/tong_2005__ip_He.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: the ionisation energy the model potential was tuned to \('We further fine tune a\_3 to make the ionization potential calculated from the model potential in good agreement with the measured one', p\. 2596\); the measurement is not cited\. Adiabatic and vertical coincide for an atom or atomic ion\. tong\_2005/potential\_He ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "potential_He", "material": "He", "file": "He/tong_2005__potential_He.csv", "table": "tbl_3240f8c18932", "quantity": "ionization_model_parameter", "unit": "au", "abscissa": "label", "use": "check", "parameters": { "model": "tong_lin_2005_potential", "parameter": "Z_c, a_1 ... a_6 of Eq. (4), the row label" } } Table 1 \(p\. 2596\)\: Z\_c and a\_1 \.\.\. a\_6 of the single\-active\-electron potential Eq\. \(4\) of the 1s electron, 'obtained by fitting the numerical potential calculated from the self\-interaction free density functional theory \(Tong and Chu 1997a\)' with a\_3 fine\-tuned to the measured I\_p; a\_1, a\_3, a\_5 and Z\_c are charges \(a\_3 per bohr\), a\_2, a\_4, a\_6 inverse lengths, in atomic units\. Z\_c \+ a\_1 \+ a\_5 is the nuclear charge \(checked\)\. tong\_2005/alpha\_Ne ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ne", "material": "Ne", "file": "Ne/tong_2005__alpha_Ne.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\), fitted to single\-active\-electron complex\-scaling static rates computed in the Eq\. \(4\) potential \(potential\_Ne\); 'the TBI rates agree with the numerically calculated ionization rates within 50%' \(p\. 2596, for the recommended choice\)\. solprop's BSI\_TONG\_LIN\_ALPHA for this gas\: 9\.0\. tong\_2005/ip\_Ne ~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_Ne", "material": "Ne", "file": "Ne/tong_2005__ip_Ne.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: the ionisation energy the model potential was tuned to \('We further fine tune a\_3 to make the ionization potential calculated from the model potential in good agreement with the measured one', p\. 2596\); the measurement is not cited\. Adiabatic and vertical coincide for an atom or atomic ion\. tong\_2005/potential\_Ne ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "potential_Ne", "material": "Ne", "file": "Ne/tong_2005__potential_Ne.csv", "table": "tbl_3240f8c18932", "quantity": "ionization_model_parameter", "unit": "au", "abscissa": "label", "use": "check", "parameters": { "model": "tong_lin_2005_potential", "parameter": "Z_c, a_1 ... a_6 of Eq. (4), the row label" } } Table 1 \(p\. 2596\)\: Z\_c and a\_1 \.\.\. a\_6 of the single\-active\-electron potential Eq\. \(4\) of the 2p electron, 'obtained by fitting the numerical potential calculated from the self\-interaction free density functional theory \(Tong and Chu 1997a\)' with a\_3 fine\-tuned to the measured I\_p; a\_1, a\_3, a\_5 and Z\_c are charges \(a\_3 per bohr\), a\_2, a\_4, a\_6 inverse lengths, in atomic units\. Z\_c \+ a\_1 \+ a\_5 is the nuclear charge \(checked\)\. tong\_2005/alpha\_Ar ~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ar", "material": "Ar", "file": "Ar/tong_2005__alpha_Ar.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\), fitted to single\-active\-electron complex\-scaling static rates computed in the Eq\. \(4\) potential \(potential\_Ar\); 'the TBI rates agree with the numerically calculated ionization rates within 50%' \(p\. 2596, for the recommended choice\)\. solprop's BSI\_TONG\_LIN\_ALPHA for this gas\: 9\.0\. tong\_2005/ip\_Ar ~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_Ar", "material": "Ar", "file": "Ar/tong_2005__ip_Ar.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: the ionisation energy the model potential was tuned to \('We further fine tune a\_3 to make the ionization potential calculated from the model potential in good agreement with the measured one', p\. 2596\); the measurement is not cited\. Adiabatic and vertical coincide for an atom or atomic ion\. tong\_2005/potential\_Ar ~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "potential_Ar", "material": "Ar", "file": "Ar/tong_2005__potential_Ar.csv", "table": "tbl_3240f8c18932", "quantity": "ionization_model_parameter", "unit": "au", "abscissa": "label", "use": "check", "parameters": { "model": "tong_lin_2005_potential", "parameter": "Z_c, a_1 ... a_6 of Eq. (4), the row label" } } Table 1 \(p\. 2596\)\: Z\_c and a\_1 \.\.\. a\_6 of the single\-active\-electron potential Eq\. \(4\) of the 3p electron, 'obtained by fitting the numerical potential calculated from the self\-interaction free density functional theory \(Tong and Chu 1997a\)' with a\_3 fine\-tuned to the measured I\_p; a\_1, a\_3, a\_5 and Z\_c are charges \(a\_3 per bohr\), a\_2, a\_4, a\_6 inverse lengths, in atomic units\. Z\_c \+ a\_1 \+ a\_5 is the nuclear charge \(checked\)\. tong\_2005/alpha\_Ne\+ ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ne+", "material": "Ne+", "file": "Ne+/tong_2005__alpha_Ne+.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\), fitted to single\-active\-electron complex\-scaling static rates computed in the Eq\. \(4\) potential \(potential\_Ne\+\); 'the TBI rates agree with the numerically calculated ionization rates within 50%' \(p\. 2596, for the recommended choice\)\. A cation\: the rate of Ne\+ \-\> Ne2\+ \(Z\_c = 2\)\. tong\_2005/ip\_Ne\+ ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_Ne+", "material": "Ne+", "file": "Ne+/tong_2005__ip_Ne+.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: the ionisation energy the model potential was tuned to \('We further fine tune a\_3 to make the ionization potential calculated from the model potential in good agreement with the measured one', p\. 2596\); the measurement is not cited\. The ion's own ionisation energy \(the atom's second ionisation energy\); the final state is not stated\. Adiabatic and vertical coincide for an atom or atomic ion\. tong\_2005/potential\_Ne\+ ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "potential_Ne+", "material": "Ne+", "file": "Ne+/tong_2005__potential_Ne+.csv", "table": "tbl_3240f8c18932", "quantity": "ionization_model_parameter", "unit": "au", "abscissa": "label", "use": "check", "parameters": { "model": "tong_lin_2005_potential", "parameter": "Z_c, a_1 ... a_6 of Eq. (4), the row label" } } Table 1 \(p\. 2596\)\: Z\_c and a\_1 \.\.\. a\_6 of the single\-active\-electron potential Eq\. \(4\) of the 2p electron, 'obtained by fitting the numerical potential calculated from the self\-interaction free density functional theory \(Tong and Chu 1997a\)' with a\_3 fine\-tuned to the measured I\_p; a\_1, a\_3, a\_5 and Z\_c are charges \(a\_3 per bohr\), a\_2, a\_4, a\_6 inverse lengths, in atomic units\. Z\_c \+ a\_1 \+ a\_5 is the nuclear charge \(checked\)\. tong\_2005/alpha\_Ar\+ ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_Ar+", "material": "Ar+", "file": "Ar+/tong_2005__alpha_Ar+.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } Table 2 \(p\. 2596\)\: alpha of Eq\. \(2\), fitted to single\-active\-electron complex\-scaling static rates computed in the Eq\. \(4\) potential \(potential\_Ar\+\); 'the TBI rates agree with the numerically calculated ionization rates within 50%' \(p\. 2596, for the recommended choice\)\. A cation\: the rate of Ar\+ \-\> Ar2\+ \(Z\_c = 2\)\. tong\_2005/ip\_Ar\+ ~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "ip_Ar+", "material": "Ar+", "file": "Ar+/tong_2005__ip_Ar+.csv", "table": "tbl_87ba350a98ce", "quantity": "ionization_energy", "unit": "hartree", "abscissa": "none", "use": "check", "parameters": { "kind": "adiabatic" } } Table 2, 'I\_p' \(au\)\: the ionisation energy the model potential was tuned to \('We further fine tune a\_3 to make the ionization potential calculated from the model potential in good agreement with the measured one', p\. 2596\); the measurement is not cited\. The ion's own ionisation energy \(the atom's second ionisation energy\); the final state is not stated\. Adiabatic and vertical coincide for an atom or atomic ion\. tong\_2005/potential\_Ar\+ ~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "potential_Ar+", "material": "Ar+", "file": "Ar+/tong_2005__potential_Ar+.csv", "table": "tbl_3240f8c18932", "quantity": "ionization_model_parameter", "unit": "au", "abscissa": "label", "use": "check", "parameters": { "model": "tong_lin_2005_potential", "parameter": "Z_c, a_1 ... a_6 of Eq. (4), the row label" } } Table 1 \(p\. 2596\)\: Z\_c and a\_1 \.\.\. a\_6 of the single\-active\-electron potential Eq\. \(4\) of the 3p electron, 'obtained by fitting the numerical potential calculated from the self\-interaction free density functional theory \(Tong and Chu 1997a\)' with a\_3 fine\-tuned to the measured I\_p; a\_1, a\_3, a\_5 and Z\_c are charges \(a\_3 per bohr\), a\_2, a\_4, a\_6 inverse lengths, in atomic units\. Z\_c \+ a\_1 \+ a\_5 is the nuclear charge \(checked\)\. tong\_2005/alpha\_H2\+ ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "id": "alpha_H2+", "material": "H2+", "table": "text", "quantity": "ionization_model_parameter", "unit": "1", "abscissa": "none", "use": "check", "parameters": { "model": "tong_lin_2005", "parameter": "alpha" } } 1 inline points in the source record. Text, pp\. 2596\-2597, Fig\. 3\: the static rate of the H2\+ ground state at the equilibrium distance, field parallel to the axis, computed exactly, 'shown to be well reproduced by the TBI model in the high field region with alpha = 6', the TI rate being MO\-ADK \(tong\_2002\)\. 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.