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 · Catalogue · 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#

{
  "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#

{
  "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#

{
  "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#

{
  "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#

{
  "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#

{
  "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#

{
  "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"
  }
}

tong_2005/potential_Ne#

{
  "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#

{
  "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#

{
  "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"
  }
}

tong_2005/potential_Ar#

{
  "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+#

{
  "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+#

{
  "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+#

{
  "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"
  }
}

tong_2005/alpha_Ar+#

{
  "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+#

{
  "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+#

{
  "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"
  }
}

tong_2005/alpha_H2+#

{
  "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 review register for the checks and decisions.