toma_2002

E. S. Toma and H. G. Muller, “Calculation of matrix elements for mixed extreme-ultraviolet-infrared two-photon above-threshold ionization of argon”, J. Phys. B: At. Mol. Opt. Phys. 35, 3435-3442 (2002)

Source record · Catalogue · Review register

DOI: 10.1088/0953-4075/35/16/306

Source kind: theory

Recorded technique: Lowest-order (second-order) perturbation theory of two-colour two-photon above-threshold ionisation of argon 3p by one odd harmonic of 800 nm and one 800 nm photon, linear parallel polarisations, length gauge (D_j = F_j . r, eq. 1), atomic units. Single-active-electron model: the configuration-averaged model potential V(r) = (1 + 5.4 e^-r + 11.6 e^-3.682r)/r (eq. 20, p. 3440), no spin-orbit coupling. The sum over intermediate states by the Dalgarno-Lewis method (eqs. 6-8), the outgoing-wave boundary condition giving the principal-value (real) and on-shell (imaginary) parts (eqs. 3-4); the asymptotic amplitude and phase from a WKB-type invariant (eqs. 9-14); the divergent continuum-continuum integrals regularised by moving-average filters (eqs. 15-19, order n = 6-8). Both time orders are summed into each radial element (p. 3440).

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 0 formulas.

Curation notes

Context only, no datasets. Identity checked on the rendered first page of the article (PDF p. 2): Toma and Muller, J. Phys. B: At. Mol. Opt. Phys. 35 (2002) 3435-3442, FOM-Institute AMOLF, received 15 March 2002, final form 3 July 2002, published 7 August 2002. PDF p. 1 is IOPscience’s download cover; PDF page n = journal page 3433 + n. Both tables were read on their images and on the rendered pages 3440 and 3442; the library’s cell text matches every printed string.

WHAT IS PRINTED. Table 1 (p. 3440), ‘The m-independent radial parts (l_i l’ l_f) of the matrix elements’, for the three pathways p->s->p, p->d->p and p->d->f (columns (psp), (pdp), (pdf)) and eight rows: 11 + IR, 13 + IR, 15 + IR, 17 + IR (harmonic q absorbed, one 800 nm photon absorbed) and 13 - IR, 15 - IR, 17 - IR, 19 - IR (harmonic q absorbed, one 800 nm photon emitted). Each entry is printed as ‘a - i pi b’ (or ‘-a + i pi b’ for (pdf)), the real part the principal-value sum and the imaginary part -pi times the on-shell product of one-photon elements (eq. 4, p. 3436); e.g. 11 + IR: ‘218.61 - i pi 12’, ‘98.85 - i pi 131.71’, ‘-105.92 + i pi 300.93’; 19 - IR: ‘80.78 - i pi 34.07’, ‘179.56 - i pi 38.23’, ‘-4.73 + i pi 52.10’. (‘12’ is printed without decimals, ‘145.4’ and ‘-11.3’ with one; all as on the image.) Table 2 (p. 3442): the radial phase shifts of the final continuum, delta_p and delta_f in rad, for the sidebands 12IR, 14IR, 16IR and 18IR (-6.17/-3.25, -6.12/-2.66, -6.10/-2.30, -6.10/-2.05), the asymptotic phase with the linear and logarithmic Coulomb terms removed (p. 3438); they are not included in Table 1’s elements, ‘because these were calculated with real wavefunctions’ (p. 3442).

THE DEFINITION AND NORMALISATION. M_if = <f| D_2 (E - H)^-1 D_1 |i>, E = E_0 + omega_1 (eq. 1). The tabulated radial parts combine with the angular factors c^m_{l-1,l} = sqrt((l^2 - m^2)/(4 l^2 - 1)) of Fig. 1 (p. 3441: m = 0, p->s and s->p (1/3)^1/2, p->d and d->p (4/15)^1/2, d->f (9/35)^1/2; m = +-1, p->d and d->p (1/5)^1/2, d->f (8/35)^1/2) into M~_if = sum_l’ c c (l_i l’ l_f), six processes (i, f) = (l_i m, l_f m); the physical element is M_if = M~_if F_2 F_1, the two fields factored out (p. 3440). Units: atomic units ‘throughout’ (p. 3436). The paper does not state the normalisation of the final continuum state, nor the model’s 3p binding energy, nor the photon energies in eV (800 nm, harmonics 11-19; 800 nm in air or vacuum not said). The IR-first time order is included in each element and not separated; ‘absorbing the infrared photon first does not bring the system into the continuum, and thus does not contribute to the imaginary part’ and is ‘small anyway’ (p. 3440).

THE SAME TOTAL ENERGY, DIFFERENT SPLITS. Each sideband is reached by two rows, as the paper says (pp. 3440-3441): 11 + IR and 13 - IR (sideband 12), 13 + IR and 15 - IR (14), 15 + IR and 17 - IR (16), 17 + IR and 19 - IR (18). In OEP 8’s split variables (its eq. 3) with the IR frequency signed, omega_a = q omega and omega_b = +-omega at Omega = 2n omega, the pairs sit at nu = 5 omega and 7 omega for sideband 12, up to nu = 8 and 10 omega for sideband 18 (7.75 to 15.5 eV for omega = 1.55 eV); the ‘- IR’ row is absorption plus stimulated emission, an E-E+ term that OEP 8’s eq. 1 omits. Computed for this record from the printed values (not printed by the paper): the sum over the six processes of |M~|^2 is 2.45, 2.18, 2.06 and 2.03 times larger for the ‘+’ than for the ‘-’ row of sidebands 12, 14, 16 and 18.

CHECKED AGAINST haber_2009 (TC-1008), which computed its SOPT values from these tables. sum |M~(q + IR)|^2 / sum |M~(q - IR)|^2 from Table 1 and Fig. 1’s factors is 1.418 (q = 13) and 1.279 (q = 15), Haber et al.’s SOPT 1.42 and 1.28; the m = 0 to m = 1 ratios are 2.19, 2.62 (13 +, 13 -) and 2.57, 2.99 (15 +, 15 -), Haber et al.’s sigma_pz/sigma_px,y exactly. Haber et al.’s SOPT beta_2 and beta_4 were not reproduced from the angular formula of p. 3441 with Table 2’s phases (e.g. 13 - IR gives 2.08 and 0.50 against their 1.97 and 0.40, 13 + IR 2.72 and 1.18 against 1.68 and 0.54); the phase convention they used is not stated, and no dataset depends on it.

WHY NOTHING IS TAKEN. No absolute two-photon cross section, rate or generalised cross section is printed, so nothing fits multiphoton_ionization_cross_section; the ‘- IR’ rows are not two-photon absorption in any case. NOT TAKEN (no quantity): Table 1’s 24 complex radial elements. two_photon_transition_amplitude was approved for bound final levels only (TC-1040); a continuum amplitude awaits a small-split source. Its value is relative (continuum normalisation unstated). NOT TAKEN (no quantity): Table 2’s continuum phase shifts, which are not a photoionisation delay and are not proposed.

FOR OEP 8 (Q6) AND OEP 6 (Q9). The elements are nondegenerate, channel-resolved (l_i, m, l_f, and the intermediate l’) and complex, but every XUV photon (about 17-29 eV) ionises alone, far above the first intermediate resonance (Ar 4s[3/2]1 11.62 eV, NIST) and above the threshold (15.76 eV); the IR-first intermediate lies about 1.55 eV above the ground state, far below 4s. Nothing is below the first resonance per photon and nothing is at small splits (the smallest |nu| is 5 omega = 7.75 eV); like haber_2009 and vinbladh_2022 (TC-1008, TC-1009), these cannot test OEP 8’s eq. 4 (the curvature kappa in nu^2).

REFERENCES. The model potential is that of Muller, Phys. Rev. A 60, 1341 (1999) (ref. 8; in the library as Muller_1999_NumericalSimulationHigh, no record). Not in the library: P. M. Paul et al., Science 292, 1689 (2001) (ref. 1, the RABBIT experiment, whose table quotes these atomic phases); H. G. Muller, PhD thesis, Free University Amsterdam (1985) (ref. 5, the method).

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

Datasets

No datasets are recorded.

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.