168
M. F. Ciappina et al.
Acknowledgements We are deeply indebted to S. V. Popruzhenko for useful comment and suggestions. Authors acknowledge fruitful discussions with D. Batheja. Supported by the project High
Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449) from European Regional Development Fund
(HIFI) and by the project Advanced research using high intensity laser produced photons and particles (CZ.02.1.01/0.0/0.0/16_019/0000789) from European Regional Development Fund (ADONIS). The results of the project LQ1606 were obtained with the financial support of the Ministry
of Education, Youth and Sports as part of targeted support from the National Programme of Sustainability II. M. F. C acknowledges Prof. K. Yamanouchi for the invitation to participate in this
project and to M. Abe for her exceptional assistance. The text of the following chapter is partially
reproduced from [34], with kind permission from the American Physical Society.
Appendix A. Ionization Cascades
Starting from a given ground state of an atom or an ion, ionization can proceed
along different pathways. Clearly, for multi electron atoms the total number of such
paths grows very quickly with the atomic number. Most of them do not give any
considerable contribution into the production of ionic states owing to the structure
of the tunneling rate. The tunneling exponent in (8.9a) is maximal for the outermost
electron which has the minimal ionization potential. This makes ionization of inner
shells highly improbable before the outer shells have been stripped out, so that
such ionization pathways can be safely discarded. However, when electrons are
being removed from the same shell, ionization of lower s-levels may proceed with
comparable or even higher probability than that of p-levels with lower ionization
potentials. The reasons for that is (a) a smaller value of the asymptotic coefficients
C
2
νl in (8.9b) and (b) the factor F
|m| in (8.9a), which is small for nonzero magnetic
quantum numbers, owing to the condition F 1. As an example, for Kr
26+ with
the ground state configuration 1s
2 2s
2 2 p
6 , ionization of a p-electron with I p ≈ 2929
eV proceeds with probability comparable to that for an s-electron whose ionization
potential is by I p ≈ 235 eV higher [44]. Indeed, taking F F
∗
= 0.05, one obtains
that the tunneling exponent for the p-electron is exp((I p /I p F) ≈ 5 times higher than
that for the s-electron. At the same time, for the latter C
2
ν0 = 1.088, B 00 = 1, while
for the p-electron C
2
ν1 = 0.315 and B 10 = 3, B 1±1 = 3/2, and finally the factor F
|m|
gives 0.05 for m = ±1. Thus, the p-state rate averaged over the magnetic quantum
number appears only 5 times greater than that of the s-state. For partially stripped
p-shells the difference in ionization potentials appears to be even smaller, so that for
configurations 2s
2 2 p or 2s
2 2 p
2 , the s- and p-rates are almost equal. These estimates
show that the described sub-manifold of pathways may play an essential role in the
ionization dynamics.
An example of the structure of levels is shown on Fig. 8.9 for the initial configuration 1s
2 2s
2 2 p
6 , which corresponds to the neutral neon, Ar
10+ , Kr
26+ , etc.
For the initial configuration shown on Fig. 8.9 the total number of pathways is
equal to 28. For the 1s
2 2s
2 2 p
2 configuration only 5 relevant pathways left; starting
from the 1s
2 2s
2 configuration ionization proceeds along the unique pathway.
M. F. Ciappina et al.
Acknowledgements We are deeply indebted to S. V. Popruzhenko for useful comment and suggestions. Authors acknowledge fruitful discussions with D. Batheja. Supported by the project High
Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449) from European Regional Development Fund
(HIFI) and by the project Advanced research using high intensity laser produced photons and particles (CZ.02.1.01/0.0/0.0/16_019/0000789) from European Regional Development Fund (ADONIS). The results of the project LQ1606 were obtained with the financial support of the Ministry
of Education, Youth and Sports as part of targeted support from the National Programme of Sustainability II. M. F. C acknowledges Prof. K. Yamanouchi for the invitation to participate in this
project and to M. Abe for her exceptional assistance. The text of the following chapter is partially
reproduced from [34], with kind permission from the American Physical Society.
Appendix A. Ionization Cascades
Starting from a given ground state of an atom or an ion, ionization can proceed
along different pathways. Clearly, for multi electron atoms the total number of such
paths grows very quickly with the atomic number. Most of them do not give any
considerable contribution into the production of ionic states owing to the structure
of the tunneling rate. The tunneling exponent in (8.9a) is maximal for the outermost
electron which has the minimal ionization potential. This makes ionization of inner
shells highly improbable before the outer shells have been stripped out, so that
such ionization pathways can be safely discarded. However, when electrons are
being removed from the same shell, ionization of lower s-levels may proceed with
comparable or even higher probability than that of p-levels with lower ionization
potentials. The reasons for that is (a) a smaller value of the asymptotic coefficients
C
2
νl in (8.9b) and (b) the factor F
|m| in (8.9a), which is small for nonzero magnetic
quantum numbers, owing to the condition F 1. As an example, for Kr
26+ with
the ground state configuration 1s
2 2s
2 2 p
6 , ionization of a p-electron with I p ≈ 2929
eV proceeds with probability comparable to that for an s-electron whose ionization
potential is by I p ≈ 235 eV higher [44]. Indeed, taking F F
∗
= 0.05, one obtains
that the tunneling exponent for the p-electron is exp((I p /I p F) ≈ 5 times higher than
that for the s-electron. At the same time, for the latter C
2
ν0 = 1.088, B 00 = 1, while
for the p-electron C
2
ν1 = 0.315 and B 10 = 3, B 1±1 = 3/2, and finally the factor F
|m|
gives 0.05 for m = ±1. Thus, the p-state rate averaged over the magnetic quantum
number appears only 5 times greater than that of the s-state. For partially stripped
p-shells the difference in ionization potentials appears to be even smaller, so that for
configurations 2s
2 2 p or 2s
2 2 p
2 , the s- and p-rates are almost equal. These estimates
show that the described sub-manifold of pathways may play an essential role in the
ionization dynamics.
An example of the structure of levels is shown on Fig. 8.9 for the initial configuration 1s
2 2s
2 2 p
6 , which corresponds to the neutral neon, Ar
10+ , Kr
26+ , etc.
For the initial configuration shown on Fig. 8.9 the total number of pathways is
equal to 28. For the 1s
2 2s
2 2 p
2 configuration only 5 relevant pathways left; starting
from the 1s
2 2s
2 configuration ionization proceeds along the unique pathway.