Sequential two-photon double ionization of noble gas atoms

A S Kheifets

Research School of Physical Sciences and Engineering, The Australian National University, Canberra ACT 0200, Australia

E-mail: A.Kheifets@.anu.edu.au

Received 21 September 2007, in final form 10 October 2007
Published 5 November 2007

In recent years, strong-field ionization of noble gas atoms has been explored extensively using the Free electron LASer at Hamburg FLASH). Wabnitz et al (2005) studied multiple ionization of Ar and Xe atoms at the photon energy of ω  =  13 eV. Moshammer et al (2007) reported few-photon multiple ionization of Ne and Ar at ω  =  38.8 eV. Sorokin et al (2007) measured direct, sequential and resonant multi-photon ionization and excitation processes in Ne at ω  =  42.8 eV. Using an alternative high harmonic generation technique, Benis et al (2006) observed two-photon double ionization (TPDI) of Ar and Kr atoms by superposition of harmonics. Besides the total ionization rate, the latest experiments are aiming to obtain various differential cross-sections. Very recently, Braune et al (2007) reported angular anisotropy parameters in sequential TPDI of Ne at 47.5 eV and Ar at 38 eV. Recoil ion momentum-resolved measurements of Moshammer (2007) on Ne at 45 eV can also be interpreted in terms of these parameters.

On the theoretical side, little has been done so far to support these angular-resolved measurements. The only related work is a calculation of Kazansky and Kabachnik (2006) who reported angular distributions of the Auger 2p and 3d electrons in Ar and Kr in coincidence with photoelectrons in femtosecond and attosecond pulse regimes.

In the meantime, a theoretical framework to evaluate parameters in sequential TPDI of noble gas atoms can be readily developed. In a sequential regime, the TPDI process takes place as subsequent one-photon single ionization (OPSI) of the neutral atom and the singly charged ion. A competing non-sequential TPDI process, driven by electron correlations, is negligible in femtosecond pulse regime (Moshammer 2007, Braune et al2007). The photoelectron angular distribution in OPSI can be evaluated for an arbitrary atomic target (Jacobs and Burke 1972, Dill et al1975). The only missing link with the present measurements of parameters in sequential TPDI of noble gases is coupling of the angular momenta of two photons, two photoelectrons and two residual ions. This coupling can be worked out using the graphical angular momentum summation technique outlined, e.g., by Varshalovich et al (1988).

In the present communication, we perform this task and derive the expressions for parameters in sequential TPDI of a closed-shell atomic target. We use these expressions to evaluate the angular anisotropy parameters for TPDI of the outer valence subshells of Ne and Ar in a wide photon energy range. We make a comparison of the calculated parameters with available experimental data at few selected photon energies.

We illustrate our graphical method by first deriving well-known expressions of the parameter in OPSI of an arbitrary atomic target. We write the matrix element of the dipole operator between the initial and final many-electron states using notations of Sobelman (1972) in which we omit the spin variables:

Here is the dipole operator, is the polarization vector and is a fractional parentage coefficient. The expression for the reduced dipole matrix element is given by Chen et al (2003):

where the one-electron reduced dipole matrix element is defined according to Amusia (1990):

Here l> is greater than l₀ and l. In equations (1) and (2) we use the standard notations for the 3j and 6j symbols. For a closed-shell system, the factor preceding the one-electron reduced dipole matrix element in equation (2) is equal to 1.

Using these notations, we can write the differential, with respect to the photoelectron momentum, cross-section as

Here μ  =  0, 1 is associated with the linear and circular polarizations, respectively. The hat symbol denotes . Following Jacobs and Burke (1972), we introduce in equation (4) a shortcut for the phase-modulated and normalized dipole matrix element

The product of two spherical harmonics in equation (4) can be transformed using the Clebsch–Gordan series given by equation (5.6.9) (equation (9) of section 5.6) of Varshalovich et al (1988)

Thus, equation (4) contains the sum of five 3j symbols running over six angular momentum projections which can be exhibited graphically by the left-hand side diagram of figure 1

The sum can be readily evaluated using equation (12.1.13) of (Varshalovich et al1988) which leads to the standard photoelectron angular distribution

with . The coefficients A_J are given for linear polarization (μ  =  0) by the following expression:

Equation (7) is equivalent to equations (3)–(4) by Jacobs and Burke (1972).^Note1

When the electron–ion interaction is isotropic, we can neglect the effect of the total spherical symmetry of the ionized atom on the photoelectron and factor out the explicit L-dependence of the dipole matrix elements from equations (2) and (5):

The sum over the total angular momenta L, L ' can now be carried over using successively equations (12.2.18) and (12.2.7) of Varshalovich et al (1988). This finally leads to

The asymmetry parameter derived from equation (8) does not contain atomic or ionic quantum numbers. The fractional parentage coefficient occurs both in A₀ and A₂ and cancels out. Thus the parameter can be expressed solely by the one-electron dipole matrix elements of equation (3). This result can be obtained straightforwardly if we consider the photoionization in the independent electron approximation and describe it by the angular momentum coupling scheme exhibited by the central diagram of figure 1. Explicit formulae for the 3j and 6j symbols can be used to transform equation (8) into a closed-form expression for (Amusia 1990).

Now we extend our formalism to the case of sequential TPDI. In analogy to equation (4), cross-section of this process can be written as

Here we adopt the following angular momentum coupling scheme:

By integrating equation (9) over and employing the Clebsch–Gordan series (6), we end up with the expression containing the sum of nine 3j symbols running over eleven angular momentum projections. This sum can be exhibited by a triangular diagram similar to the one on the left of figure 1 but with a larger number of `ladder steps'. Varshalovich et al (1988) do not provide a direct summation formula for this case. Nevertheless, the summation can be carried over using reduction formulae (12.1.10) and (12.1.11) which reduce the number of ladder steps by 1. As a result, the angular distribution of the photoelectron is given by the Legendre polynomial expansion , where the angular anisotropy coefficients can be derived from the following coefficients:

Here we denoted for brevity of notations and set μ  =  0.

For a closed atomic shell, L₀  =  0 and L_i  =  L '_i  =  l₀, L₁  =  L '₁  =  1. Further simplification of equation (10) can be achieved if we neglect the dependence of the wavefunction of the second photoelectron on the total angular momentum of the singly ionized atomic system L₂. Then the explicit L₂ dependence of the matrix element can be factored out using equation (7), and the summation over L₂, L '₂ can be carried over. This leads to the anisotropy parameters which can be extracted from the following coefficients:

As in the case of equation (8), the parameters extracted from equation (11) do not contain the ionic or atomic quantum numbers except for the total angular momentum of the doubly charged ion L_a. Equation (11) can be derived straightforwardly if we consider an independent electron TPDI process in which the angular momenta of the two holes couple to the total angular momentum L_a. This process is exhibited graphically by the right-hand side diagram of figure 1.

We use equation (11) to evaluate the parameters of TPDI of Ne 2p and Ar 3p valence subshells. Isotropy of the system containing the doubly charged ionic core and the photoelectron, which is essential for application of equation (11), can be judged, at least partly, from the properties of the discrete excited states of the singly charged ion. Inspection of the Ne II energy levels (Moore 1949) shows that 2p⁴[L_aS_a]nl}LS manifold is indeed governed by the quantum numbers L_aS_a and depends very weakly on LS. This is less so in Ar II. Therefore the present Ar results should be treated as qualitative rather than quantitative. A more accurate calculation using equation (10) is needed to get the Ar -parameters on the same level of accuracy as for Ne.

In practical computations, we obtain the one-electron dipole matrix elements for the neutral atom by running the RPAE computer code (Amusia and Chernysheva 1997). The random phase approximation with exchange (RPAE) provides a very accurate description of single-photon one-electron ionization processes in noble gas atoms (Amusia and Cherepkov 1975). The parameters derived from the RPAE dipole matrix elements using equation (8) are in excellent agreement with experimental data of Braune et al (2007). For Ne at 47.5 eV, as compared with the experimental value of 1.00. For Ar at 38 eV, as compared with the experimental value of 1.88.

In principle, the RPAE method can be modified to describe ionization of the singly charged ion with an open np⁵ shell (Cherepkov and Chernysheva 1977). This, however, would require an explicit account for anisotropy of the ionized system which we neglect when substituting equation (10) with equation (11). Therefore, we opted for a less computationally demanding Hartree–Fock method when calculating the one-electron dipole matrix elements for the singly charged ion. The radial orbitals and R_kl entering equation (3) are calculated using the self-consisted and frozen-core Hartree–Fock codes, respectively (Chernysheva et al1976, 1979).

Results of our computations are shown in figure 2 for Ne (left) and Ar (right). In the same figure we indicate the numerical values of the experimental anisotropy parameters for Ne at 47.5 eV and for Ar at 38 eV as reported by Braune et al (2007) for the ³P final state of the doubly charged ion. These results are in fair agreement with the present calculation. However, more experimental data across a wider photon energy range are clearly needed to gauge the quality of the present theoretical model and its numerical implementation.

Figure 2. The angular anisotropy -parameters for Ne (left) and Ar (right). The experimental data are from Braune et al (2007).

It is noteworthy that the parameters depend weakly on the symmetry of the doubly charged ion. Conversely, the parameters demonstrate a very strong dependence with very small absolute values for ¹D state and relatively large in magnitude and opposite in sign values for ³P and ¹S states. This behaviour can be understood from equation (11). The statistical average is proportional to δ_K0 which eliminates the coefficients A₄ and reduces A₀ to a single-photon one-electron value (8). Although the statistical average does not apply directly to the ratio A_J/A₀, one would expect the term average value of close to zero which is indeed the case for Ne for which shows little term dependence. For Ar, shows strong term dependence near the threshold, and the statistical average of deviates from zero in this photon energy range.

Acknowledgment

The author wishes to thank Markus Braune and Robert Moshammer for communicating their experimental result prior to publication

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Notes

Note1

 Notations of Jacobs and Burke (1972) are ambiguous as the angular momenta of the atom and the ion are both denoted by the same symbol. When unambiguous notations are employed, the equivalence of the original equations and the present equation (7) becomes evident.