Calculation of the photoionization cross sections for the ions of the cesium isoelectronic sequence by the Dirac-Fock method

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Calculation of the photoionization cross sections for the ions of the cesium isoelectronic sequence by the Dirac-Fock method

The photoionization cross sections for the ground and same excited states of 10 ions cesium isoelectronic sequence from Ba⁺ through Tb¹⁰⁺ are calculated by the relativistic Dirac-Fock self-consistent field method. The energy dependence of the photoionization cross sections and the changes in the threhold values along the isoelectronic sequence are considered. Along with the total photoionization cross sections, the asymmetry parameter in the angular distribution of photoelectrons and the degree of their polarization ara also colculated.

Since the investigation of diverse physical processes involving multiply charged ions is becoming increasingly extensive, there is growing interest in studying the photoionization cross sections of such ions [1]. There are currently few experimental data for such cross sections, so theoretical methods of calculation become important. For the purpose of interpolation or extrapolation, it is expedient to study the characteristics of ions along isoelectronic sequences. The study of the behavior of photoionization cross sections along the isoelectronic sequence is of interest also because the photoionization cross sections in the near-threshold energy region are associated with the corresponding oscillator strengths.

For neutral atoms, in particular, for atoms whose outer shell is closed or possesses several valence electrons, the interaction of a photoelectrons with the other electrons of this shell plays a significant role. A number of methods have been developed that more or less take into account these correlations. These methods include the method of random phase approximation with exchange [2], the many-body perturbation theory [3], the R-matrix theory [4], the model potential method [5], etc.

Results and Discussion

In this paper, photoionization of Cs-like ions is investigated. These ions have a relatively simple configuration - one optical electron above the filled shells. The electrons of closed shells are relatively strongly coupled, and the interaction between the photoelectron and the electrons of the core is relatively weak. In addition, the influence of the correlation and polarization effects with increasing charge of the ion q decreases. But as the ion charge increases, relativistic effects increase noticeably. Taking this into account, in this paper, in the calculation of photoionization cross sections, the relativistic Dirac-Fock self-consistent field method (DF) was used [6].

In [7], the procedure for calculating the photoionization cross sections based on the DF method with the exchange effect correctly taken into account was described. In the present paper we calculate the photoionization cross sections for the ground and certain excited states of ions of the isoelectronic sequence of cesium (In this paper, the photoionization cross sections are measured in Mb (1 Mb = 10-18 cm2 ) and the photon energy щ and the photoelectron energy E in Ry). It should be noted that, unlike, for example, the isoelectronic sequence of lithium, the ground state varies several times along the isoelectronic sequence of cesium. The Ba⁺ ion, as well as the neutral Cs, has a 6s ²S_1/2 configuration in the ground state. During photoionization, transitions to the continuous spectrum 6s_1/2  > Ep_1/2 and Ep_3/2 are allowed for the valence electron. For La⁺⁺, the ground state of 5d ²D_3/2 and the electron from the 5d_3/2 level can go over into the continuous spectrum by Ep_1/2, Ep_3/2 and Ef_5/2, Starting with the Ce³⁺ ion, the configuration is 4f ²F_5/2 and 4f_5/2 > Ed_3/2 , Ed_5/2 and Eg_7/2  transitions are allowed. This complicates calculations. As is well known, contributions to the total cross section of photoionization yield all allowed transitions.

Considered were relatively small photon energies, for which the ionization of the electrons of the inner shells has not yet occurred. Autoionization and processes involving internal electrons were not taken into account. The exchange and retardation effects are accurately taken into account. All calculations were carried out for two values of the gauge constant: G = 0 (Coulomb gauge) and G = v2 (Babushkin gauge) [7]. In the nonrelativistic theory, the Coulomb gauge corresponds to using the matrix element in the form of “velocity,” and Babushkin's gauge, in the form of “length”; therefore, the corresponding cross-section values are labeled by the indices “V” and “L”. We also considered the influence of the choice of gauge on the calculated values of the cross sections, i.e. the difference between у_L and у_V . One would expect that with the growth of the ion charge this difference will decrease, since usually the ions become more hydrogen-like.

The photoionization cross sections for the 6s_1/2-levels of Cs and Ba⁺ near the ionization threshold have a Cooper minimum, so the absolute values of у(E) are small. In this case, the choice of gauge is strongly influenced and the values of у_L(E) and у_V(E) differ significantly. But as can be seen from Fig. 1, the difference between у_L(0)  and у_V(0) decreases with increasing charge of the ion q. For levels 5d_3/2  of La⁺⁺ and Ce³⁺ ions, the value of у(E) is larger and the effect of calibration is relatively smaller.

Fig. 1 - Change in the total threshold photoionization cross sections along the isoelectronic sequence of cesium for 6s_1/2 states

photoionization calculating dependence

The solid line is у_V(E), the dashed line is у_L(E), and q is the ion charge.

Fig. 2 - Photoionization cross sections for the 5d_3/2 state of the Ce³⁺ ion

The solid line is the total cross section у(E), the dashed line is the cross sections for the channel 5d_3/2 > Ef_5/2.

In Fig. 2 shows the total photoionization cross section for the 5d_3/2 Ce³⁺ level. In the same place, у(E) is depicted for the channel 5d_3/2 > Ef_5/2. It is seen that the channel 5d_3/2> Ef_5/2  determines the total contribution, and the remaining (5d_3/2 > Ep_1/2  and Ep_3/2 ) only about 6%. A similar situation occurs when the 4f ²F^o_5/2  levels of Ce³⁺  and Pr⁴⁺.  ions are photoionized. There the channel 4f_5/2 > Eg_7/2  is about 98% of the total cross section - see Fig. 3. Using the well-known relationship between the photoionization cross sections and the oscillator strengths, the values of ф(E_n) (Fig. 2 and Fig. 3) are determined using the values of the oscillator strengths for the 5d_3/2> 8f_5/2  and 4f_5/2 > 8g_7/2  transitions are plotted in the discrete spectrum, respectively.

Fig. 3 - Photoionization cross sections for the 4f_5/2 state of the Pr⁴⁺ ion

The solid line is the total cross section у(E), the dashed line is the cross section for the 4f_5/2 > Eg_7/2 channel.

The calculated total photoionization cross sections for у_L(E) and у_V(E) of the ground state 5d_3/2  of the La⁺⁺ ion are listed in the table.

Table 1 - The total photoionization cross sections у(E) for the ground state 5d_3/2  of the La⁺⁺ ion. щ is the photon energy, E is the photoelectron energy

Calculations of the photoionization cross sections were carried out on the basis of the DF method with correct exchange allowance. In [8], the influence of exchange accounting on the calculated values of the cross sections was studied. The obtained results showed that a correct exchange allowance leads to a significant convergence of the values of у_L and у_V. Therefore, allowance for exchange for photoionization is essential.

In studying the photoionization of the 6s electrons of the Cs-like ions, wie also determined the values of the asymmetry parameter in the angular distribution of photoelectrons в(E) and the degree of polarization of photoelectrons p(E) [2], [7]. According to the Cooper-Zare theorem, in the nonrelativistic approximation, the asymmetry parameter for the ns-electron does not depend on the energy and в(E)=2. In the relativistic theory, the interference between the ns_1/2 > Ep_1/2 and ns_1/2 > Ep_3/2 channels causes в(E) to vary from -1 to 2 in the vicinity of the Cooper minimum [9]. For Ba⁺, the Cooper minimum is near the threshold and в_L(0)=-0.972 and в_L(5)= 1.91 were obtained. In [10], the dependence of в(E) on energy and on the choice of gauge for a neutral Cs atom was studied. We also calculated the degree of the polarization of photoelectrons p(E), where p(E) is the degree of polarization of photoelectrons under photoionization by circularly polarized light (the Fano effect). For Ba⁺, the large value of p(E) has p_L(0) = 0.379 for the ionization threshold, and p_L(1) = -0.409. The Cs-like ions differ by this property from Li-like ions, for which (except for Li) p(E) is practically zero [7].

Conclusion

Unfortunately, direct comparison with experimental findings is imposible since experimental results for the ions in question are unavailable. The photoionization cross sections for the 6s_1/2 levels near the ionization threshold have a Cooper minimum; therefore, the absolute values of у(E) are small. In this case, the choice of calibration strongly affects and the values of у_L(E) and у_V(E) differ significantly. Correlation corrections should be taken into account here. It can be expected that the calculated values of photoionization cross sections for the 5d_3/2 and 4f_5/2 levels will be more accurate, since the corresponding у_L(E) and у_V(E) differ less.

List of references

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