trinh_2013#

V. H. Trinh, O. I. Tolstikhin, L. B. Madsen and T. Morishita, “First-order correction terms in the weak-field asymptotic theory of tunneling ionization”, Phys. Rev. A 87, 043426 (2013)

Source record · Catalogue · Review register

DOI: 10.1103/PhysRevA.87.043426

Source kind: theory

Recorded technique: The weak-field asymptotic theory (WFAT) of tunnelling ionisation in a static field, carried to first order in F, for one electron in a potential with a Coulomb tail (single-active-electron and frozen-nuclei approximations), in parabolic coordinates (Secs. II-III). Illustrated for H and for Ne(2p), Ar(3p), Kr(4p), Xe(5p) (m = 0) in the Green-Sellin-Zachor model potential: E0 and psi_0 from a Laguerre discrete-variable representation, alpha and psi_1 by spectral perturbation sums, g00, g10, a00 by fitting Eqs. (37)-(38) at large eta; the ‘exact’ rates for comparison by the Siegert-state method of batishchev_2010 (Sec. IV).

Materials in structured entries: Ar, Kr, Ne, Xe

Dataset quantities: ionization_model_parameter, tunnelling_structure_coefficient

Contents: 16 datasets; 0 formulas.

Curation notes#

Journal pp. 043426-1 to -14 = PDF pages 1-14 (pages cited as -n). Identity checked on the first page.

WHAT IS TABULATED AND WHICH QUANTITY (the question asked when the paper was requested). Table I (p. -9, tbl_44397cd5a3f8, image checked), ‘Characteristics of noble-gas atoms ... All values are given in atomic units’: N, u, v, E0, alpha, g00, g10, a00, A00, B00 per atom. - g00 and g10 ARE structure factors: the asymptotic coefficients g_nu of the parabolic channels nu = (0, 0) and (1, 0) (Eq. (37)), with G_nu = e^(-kappa mu_z) g_nu the WFAT structure factor (Eq. (50)) and mu_z = 0 for these atoms, so G00 = g00 (p. -10): recorded as tunnelling_structure_coefficient, model = wfat (wfat_<atom>). The row label is the parabolic channel (n_xi, m), not a partial wave l as for MO-ADK: the vocabulary’s ‘one per partial wave l’ does not fit WFAT exactly (for review). - a00, A00 and B00 are NOT structure factors: a00 is the first-order distortion coefficient of the asymptotic solution (Eqs. (34), (38)), A00 and B00 the first-order correction coefficients of the rate, Gamma = |G00|^2 W00(F) [1 + A00 F ln(F/(4 kappa^2)) + B00 F] (Eqs. (49), (53)): recorded as ionization_model_parameter, model = wfat (correction_<atom>). - N, u, v define the model potential (potential_<atom>); E0 and alpha are the model state’s energy and polarizability (state_<atom>): ionization_model_parameter. All use = check: properties of one-electron model states, which the paper uses to illustrate the theory.

CHECKS (computed 2026-09-27): (1) A00 from Eq. (40a) with beta^(0) = 1 - kappa/2 (Eq. (30a)), beta^(1) = -1/(2 kappa^2) (Eq. (31a)), gamma_0 = 1/4 (Eq. (16), D = 0), mu_z = 0: Ne 0.2456, Ar 0.1582, Kr 0.0415, Xe -0.2217, against 0.246, 0.158, 0.042, -0.222 printed (all agree to the printed digits; A00 = 0 at kappa = 1, E0 = -0.5, as p. -10 says). (2) B00 from Eq. (40b) with the printed E0, alpha and a00: Ne -2.62, Ar -7.58, Kr -10.42, Xe -16.48, against -2.6, -7.7, -10.5, -16.4: within the +-0.1 that a00’s one printed decimal allows (2 Re a00 enters B00), Ar by 0.12. (3) The model potential’s bound states, by finite differences on a 60 a0 radial grid: the lowest, second, third and fourth l = 1 levels with Trinh’s N, u, v are -0.7934 (Ne), -0.5791 (Ar), -0.5152 (Kr), -0.4460 (Xe), reproducing E0 to the printed three decimals. The same code gives Kr’s 4p at -0.4589 with Garvey et al.’s a = 1.351, b = 4.418 and -0.5139 with batishchev_2010’s modified a = 1.320, b = 4.355, reproducing that paper’s 0.459 and 0.514: Trinh’s Kr parameters (u = 1.340, v = 4.311) are a third set, and all four of Trinh’s sets are tuned so that E0 is minus the atom’s ionisation energy (batishchev_2010 quotes 0.793, 0.579, 0.515, 0.446 as ‘exact’).

ACCURACY: no uncertainty is stated for Table I; g00, g10, a00 are printed to one decimal. The WFAT(1) rates agree with the exact Siegert-state rates up to about F_c, the field where the WFAT(1) rate turns zero: F_c ~ 0.3, 0.12, 0.095, 0.065 for Ne, Ar, Kr, Xe (Eq. (62) gives 0.4, 0.18, 0.13, 0.09); second-order perturbation theory for the energy holds up to F ~ 0.1, 0.05, 0.05, 0.03 (pp. -10, -11). For H(1s) the paper compares with the exact rates and with higher-order expansions (Figs. 1-3, pp. -7 and -8).

FIGURES (not digitised): Fig. 1, H(1s) Gamma/W00 against F; Fig. 2, excited H states; Figs. 3, 8-11, transverse momentum distributions; Figs. 4-7, the energy, the rate and its ratio to the field factor for Ne(2p), Ar(3p), Kr(4p), Xe(5p) (exact, WFAT(0), WFAT(1)): the exact SAE static rates of these model atoms are in figures only.

FOR REVIEW: solprop uses ADK/PPT and MO-ADK, not WFAT; these are the atoms’ weak-field asymptotic coefficients the roadmap asks for (Trinh et al.), in the WFAT’s own normalisation (the parameters’ normalisation field), with the first-order corrections.

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

Datasets#

trinh_2013/wfat_Ne#

{
  "id": "wfat_Ne",
  "material": "Ne",
  "file": "Ne/trinh_2013__wfat_Ne.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "tunnelling_structure_coefficient",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "orbital": "2p",
    "m": 0,
    "normalisation": "Trinh et al. 2013 Eq. (37): g_nu = eta^(1/2 - beta_nu^(0)/kappa) e^(kappa eta/2) <Phi_nu^(0)|psi_0(r)> as eta -> inf, psi_0 the real normalised bound state, Phi_nu^(0) the parabolic channel functions of Eq. (30b), beta_nu^(0) = Z - kappa(n_xi + (|m| + 1)/2), kappa = (2|E0|)^(1/2); G_nu = e^(-kappa mu_z) g_nu (Eq. (50)); the rate Gamma = (2 - delta_m0)|G_0m|^2 W_0m(F)(1 + A_0m F ln(F/(4 kappa^2)) + B_0m F) (Eq. (53)) with W_nu(F) = (kappa/2)(4 kappa^2/F)^(2Z/kappa - 2 n_xi - |m| - 1) exp(-2 kappa^3/(3F)) (Eq. (51)), Z = 1"
  }
}

Table I, p. 043426-9: g00 and g10, the asymptotic coefficients of the parabolic channels nu = (n_xi, m) = (0, 0) and (1, 0) of the 2p (m = 0) orbital in the model potential of Eqs. (63)-(64) (potential_Ne); the rows are the channel, not a partial wave l. G_00 = g00 is the WFAT structure factor of the dominant channel (mu_z = 0); g10 enters the transverse momentum distribution. Printed to two digits. Values of the SAE model state, not of the many-electron atom: use = check (record notes).

trinh_2013/correction_Ne#

{
  "id": "correction_Ne",
  "material": "Ne",
  "file": "Ne/trinh_2013__correction_Ne.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "parameter": "a00, A00, B00 of Eqs. (38), (40), the row label"
  }
}

Table I, p. 043426-9: a00 (Eq. (38), the coefficient of the first-order distortion of the orbital in the asymptotic expansion, Eq. (34)) and the first-order correction coefficients A00 and B00 (Eqs. (40)) of the dominant channel (0, 0), for the 2p (m = 0) SAE state. Units: atomic units of inverse field (they multiply F). A00 reproduced from E0 (record notes).

trinh_2013/potential_Ne#

{
  "id": "potential_Ne",
  "material": "Ne",
  "file": "Ne/trinh_2013__potential_Ne.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "N, u, v of Eqs. (63)-(64), the row label"
  }
}

Table I, p. 043426-9: N (the nuclear charge, Z_eff(0)), u and v of the effective charge Z_eff(r) = N - (N - 1){1 - [(v/u)(e^(u r) - 1) + 1]^(-1)} (Eq. (64)) of the SAE potential V = -Z_eff(r)/r (Eq. (63)), ‘the recommended values’ (p. -9, citing Green et al. 1969 and Garvey et al. 1975); u and v in inverse bohr. They reproduce E0 (checked, record notes).

trinh_2013/state_Ne#

{
  "id": "state_Ne",
  "material": "Ne",
  "file": "Ne/trinh_2013__state_Ne.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "E0, alpha of the model's active electron, the row label"
  }
}

Table I, p. 043426-9: E0, the unperturbed energy of the 2p (m = 0) state in the model potential, and alpha, ‘its polarizability’ (alpha_zz of the active electron in the model, which enters B00 through Eq. (40b)). Properties of the SAE model state, not of the atom: E0 equals minus the atom’s first ionisation energy to the printed digits (by the choice of u, v), but alpha is the one-electron model’s, not the atom’s dipole polarizability, and must not be read as one.

trinh_2013/wfat_Ar#

{
  "id": "wfat_Ar",
  "material": "Ar",
  "file": "Ar/trinh_2013__wfat_Ar.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "tunnelling_structure_coefficient",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "orbital": "3p",
    "m": 0,
    "normalisation": "Trinh et al. 2013 Eq. (37): g_nu = eta^(1/2 - beta_nu^(0)/kappa) e^(kappa eta/2) <Phi_nu^(0)|psi_0(r)> as eta -> inf, psi_0 the real normalised bound state, Phi_nu^(0) the parabolic channel functions of Eq. (30b), beta_nu^(0) = Z - kappa(n_xi + (|m| + 1)/2), kappa = (2|E0|)^(1/2); G_nu = e^(-kappa mu_z) g_nu (Eq. (50)); the rate Gamma = (2 - delta_m0)|G_0m|^2 W_0m(F)(1 + A_0m F ln(F/(4 kappa^2)) + B_0m F) (Eq. (53)) with W_nu(F) = (kappa/2)(4 kappa^2/F)^(2Z/kappa - 2 n_xi - |m| - 1) exp(-2 kappa^3/(3F)) (Eq. (51)), Z = 1"
  }
}

Table I, p. 043426-9: g00 and g10, the asymptotic coefficients of the parabolic channels nu = (n_xi, m) = (0, 0) and (1, 0) of the 3p (m = 0) orbital in the model potential of Eqs. (63)-(64) (potential_Ar); the rows are the channel, not a partial wave l. G_00 = g00 is the WFAT structure factor of the dominant channel (mu_z = 0); g10 enters the transverse momentum distribution. Printed to two digits. Values of the SAE model state, not of the many-electron atom: use = check (record notes).

trinh_2013/correction_Ar#

{
  "id": "correction_Ar",
  "material": "Ar",
  "file": "Ar/trinh_2013__correction_Ar.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "parameter": "a00, A00, B00 of Eqs. (38), (40), the row label"
  }
}

Table I, p. 043426-9: a00 (Eq. (38), the coefficient of the first-order distortion of the orbital in the asymptotic expansion, Eq. (34)) and the first-order correction coefficients A00 and B00 (Eqs. (40)) of the dominant channel (0, 0), for the 3p (m = 0) SAE state. Units: atomic units of inverse field (they multiply F). A00 reproduced from E0 (record notes).

trinh_2013/potential_Ar#

{
  "id": "potential_Ar",
  "material": "Ar",
  "file": "Ar/trinh_2013__potential_Ar.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "N, u, v of Eqs. (63)-(64), the row label"
  }
}

trinh_2013/state_Ar#

{
  "id": "state_Ar",
  "material": "Ar",
  "file": "Ar/trinh_2013__state_Ar.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "E0, alpha of the model's active electron, the row label"
  }
}

Table I, p. 043426-9: E0, the unperturbed energy of the 3p (m = 0) state in the model potential, and alpha, ‘its polarizability’ (alpha_zz of the active electron in the model, which enters B00 through Eq. (40b)). Properties of the SAE model state, not of the atom: E0 equals minus the atom’s first ionisation energy to the printed digits (by the choice of u, v), but alpha is the one-electron model’s, not the atom’s dipole polarizability, and must not be read as one.

trinh_2013/wfat_Kr#

{
  "id": "wfat_Kr",
  "material": "Kr",
  "file": "Kr/trinh_2013__wfat_Kr.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "tunnelling_structure_coefficient",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "orbital": "4p",
    "m": 0,
    "normalisation": "Trinh et al. 2013 Eq. (37): g_nu = eta^(1/2 - beta_nu^(0)/kappa) e^(kappa eta/2) <Phi_nu^(0)|psi_0(r)> as eta -> inf, psi_0 the real normalised bound state, Phi_nu^(0) the parabolic channel functions of Eq. (30b), beta_nu^(0) = Z - kappa(n_xi + (|m| + 1)/2), kappa = (2|E0|)^(1/2); G_nu = e^(-kappa mu_z) g_nu (Eq. (50)); the rate Gamma = (2 - delta_m0)|G_0m|^2 W_0m(F)(1 + A_0m F ln(F/(4 kappa^2)) + B_0m F) (Eq. (53)) with W_nu(F) = (kappa/2)(4 kappa^2/F)^(2Z/kappa - 2 n_xi - |m| - 1) exp(-2 kappa^3/(3F)) (Eq. (51)), Z = 1"
  }
}

Table I, p. 043426-9: g00 and g10, the asymptotic coefficients of the parabolic channels nu = (n_xi, m) = (0, 0) and (1, 0) of the 4p (m = 0) orbital in the model potential of Eqs. (63)-(64) (potential_Kr); the rows are the channel, not a partial wave l. G_00 = g00 is the WFAT structure factor of the dominant channel (mu_z = 0); g10 enters the transverse momentum distribution. Printed to two digits. Values of the SAE model state, not of the many-electron atom: use = check (record notes).

trinh_2013/correction_Kr#

{
  "id": "correction_Kr",
  "material": "Kr",
  "file": "Kr/trinh_2013__correction_Kr.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "parameter": "a00, A00, B00 of Eqs. (38), (40), the row label"
  }
}

Table I, p. 043426-9: a00 (Eq. (38), the coefficient of the first-order distortion of the orbital in the asymptotic expansion, Eq. (34)) and the first-order correction coefficients A00 and B00 (Eqs. (40)) of the dominant channel (0, 0), for the 4p (m = 0) SAE state. Units: atomic units of inverse field (they multiply F). A00 reproduced from E0 (record notes).

trinh_2013/potential_Kr#

{
  "id": "potential_Kr",
  "material": "Kr",
  "file": "Kr/trinh_2013__potential_Kr.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "N, u, v of Eqs. (63)-(64), the row label"
  }
}

trinh_2013/state_Kr#

{
  "id": "state_Kr",
  "material": "Kr",
  "file": "Kr/trinh_2013__state_Kr.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "E0, alpha of the model's active electron, the row label"
  }
}

Table I, p. 043426-9: E0, the unperturbed energy of the 4p (m = 0) state in the model potential, and alpha, ‘its polarizability’ (alpha_zz of the active electron in the model, which enters B00 through Eq. (40b)). Properties of the SAE model state, not of the atom: E0 equals minus the atom’s first ionisation energy to the printed digits (by the choice of u, v), but alpha is the one-electron model’s, not the atom’s dipole polarizability, and must not be read as one.

trinh_2013/wfat_Xe#

{
  "id": "wfat_Xe",
  "material": "Xe",
  "file": "Xe/trinh_2013__wfat_Xe.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "tunnelling_structure_coefficient",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "orbital": "5p",
    "m": 0,
    "normalisation": "Trinh et al. 2013 Eq. (37): g_nu = eta^(1/2 - beta_nu^(0)/kappa) e^(kappa eta/2) <Phi_nu^(0)|psi_0(r)> as eta -> inf, psi_0 the real normalised bound state, Phi_nu^(0) the parabolic channel functions of Eq. (30b), beta_nu^(0) = Z - kappa(n_xi + (|m| + 1)/2), kappa = (2|E0|)^(1/2); G_nu = e^(-kappa mu_z) g_nu (Eq. (50)); the rate Gamma = (2 - delta_m0)|G_0m|^2 W_0m(F)(1 + A_0m F ln(F/(4 kappa^2)) + B_0m F) (Eq. (53)) with W_nu(F) = (kappa/2)(4 kappa^2/F)^(2Z/kappa - 2 n_xi - |m| - 1) exp(-2 kappa^3/(3F)) (Eq. (51)), Z = 1"
  }
}

Table I, p. 043426-9: g00 and g10, the asymptotic coefficients of the parabolic channels nu = (n_xi, m) = (0, 0) and (1, 0) of the 5p (m = 0) orbital in the model potential of Eqs. (63)-(64) (potential_Xe); the rows are the channel, not a partial wave l. G_00 = g00 is the WFAT structure factor of the dominant channel (mu_z = 0); g10 enters the transverse momentum distribution. Printed to two digits. Values of the SAE model state, not of the many-electron atom: use = check (record notes).

trinh_2013/correction_Xe#

{
  "id": "correction_Xe",
  "material": "Xe",
  "file": "Xe/trinh_2013__correction_Xe.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "wfat",
    "parameter": "a00, A00, B00 of Eqs. (38), (40), the row label"
  }
}

Table I, p. 043426-9: a00 (Eq. (38), the coefficient of the first-order distortion of the orbital in the asymptotic expansion, Eq. (34)) and the first-order correction coefficients A00 and B00 (Eqs. (40)) of the dominant channel (0, 0), for the 5p (m = 0) SAE state. Units: atomic units of inverse field (they multiply F). A00 reproduced from E0 (record notes).

trinh_2013/potential_Xe#

{
  "id": "potential_Xe",
  "material": "Xe",
  "file": "Xe/trinh_2013__potential_Xe.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "1",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "N, u, v of Eqs. (63)-(64), the row label"
  }
}

trinh_2013/state_Xe#

{
  "id": "state_Xe",
  "material": "Xe",
  "file": "Xe/trinh_2013__state_Xe.csv",
  "table": "tbl_44397cd5a3f8",
  "quantity": "ionization_model_parameter",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "model": "gsz_sae_potential",
    "parameter": "E0, alpha of the model's active electron, the row label"
  }
}

Table I, p. 043426-9: E0, the unperturbed energy of the 5p (m = 0) state in the model potential, and alpha, ‘its polarizability’ (alpha_zz of the active electron in the model, which enters B00 through Eq. (40b)). Properties of the SAE model state, not of the atom: E0 equals minus the atom’s first ionisation energy to the printed digits (by the choice of u, v), but alpha is the one-electron model’s, not the atom’s dipole polarizability, and must not be read as one.

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.