# Atomic Cascade Computations

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## Abstract

**:**

## 1. Introduction

## 2. Inner-Shell Transitions Revisited

#### 2.1. Following Atomic Decay Lines Basic Notations

#### 2.2. Role of Many-Electron Amplitudes

#### 2.3. Key Elements for Building Atomic Cascades

## 3. Modeling of Atomic Cascades

#### 3.1. Spectroscopic Observations

#### 3.2. Cascade Schemes

#### 3.3. Cascade Approaches

- (i)
**Average single-configuration approach (AverageSCA):**this is likely the simplest way to model a cascade in terms of its fine-structure levels and transition amplitudes. Here, the representation of ASF levels is significantly simplified, as they are (all) approximated by single configuration state function (CSF), and they are based on a common set of orbitals, as generated for the initial levels. Indeed, this approach neglectss all configuration mixing between the bound-state levels and, in addition, restricts the computations (by default) to the Coulomb interaction among the electrons, the electric-dipole (E1) transition amplitudes as well as to just a single set of continuum orbitals for each step of the cascade. However, the AverageSCA approach appears to be feasible for (almost) all atoms and ions from the periodic table, although (much) further work is necessary to understand how well this (very) simple approximation is able to describe the underlying relaxation of the system.- (ii)
**Single-configuration approach (SCA):**this approach still uses a common set of orbitals for all electron configurations (cascade blocks) from the same ionization stage, but includes configuration mixing within each block (i.e., by just taking into account the intermediate coupling effects, but not the so-called interaction of configurations). While most other limitations are quite similar to the AverageSCA approach above, the continuum orbitals are generated here for the correct fine-structure (transition) energies, although without the exchange interaction with the bound electrons [48].- (iii)
**Multiple-configuration approach (UserMCA):**this approach facilitates the incorporation of configuration mixings (electron–electron correlations) between user-selected configurations. Apart from some obvious rules for combining inner-shell configurations, the user is encouraged here to explicitly group different configurations together, based on physical insight into the cascade process [49,50]. The arrangement of the bound-state configurations into different groups is usually done either by means of their mean energy, a maximum size (number of CSF) of any individually selected cascade block, or need to be based upon some prior knowledge about strongly interacting configurations. However, care has to be taken by the user to ensure that each physical level (uniquely) belongs to just one group, so that “double counting” of rates, etc., does not occur in the subsequent simulations.- (iv)
**Multiple-configuration-shake approach (ShakeMCA):**this has been designed so far only but will help incorporate also shake-down and shake-up configurations, which do nominally not arise in any standard autoionization schemes of the atom or ion. These shake configurations can then be treated either as individual block of configurations or together with other configurations of the decay cascade. The UserMCA and ShakeMCA approaches will both enable one in the future to incorporate all major electron–electron correlations into the cascade computation by choosing proper “groups” of configurations, while the admixtures from other groups are still neglected.

#### 3.4. Cascade Computations

- Specification of the cascade tree, which means of all those configurations, whose levels are relevant for the observed spectra. Apart from a simple re-occupation of electron shells, this specification often requires insight by the user into important shake processes, as well as the mixing with other configurations that are energetically nearby.
- The set-up of the cascade blocks and generation of a self-consistent field, or, at least, a proper set of bound-state orbitals. In more detail, this refers to the selection and arrangement of configurations into some many-electron (CSF) basis, in which the atomic states are represented by diagonalizing the associated Hamiltonian matrix.
- The determination of the cascades steps and computation of the associated transition amplitudes and rates. A separate list of such bound-state transitions, i.e., amplitudes and rates, is then compiled for each atomic process of the cascade, such as photoionization, photoemission, autoionization, or others.
- Output of these (line) lists to some interface, either to disc or some Julia dictionary. In practice, all of these lists of transition data are subtypes of a common data structure, from which the desired information can be extracted subsequently.

#### 3.5. Cascade Simulations

## 4. Implementation of Atomic Cascades

#### 4.1. The Jac Toolbox

`Atomic.Computation`’s are based on explicitly specified electron configurations and (may) provide level energies, the representation of ASF, or selected level properties. These (atomic) computations also help to evaluate the transition amplitudes (and rates) for all of the processes mentioned above. These features make the Jac toolbox altogether well suited for studying atomic cascades.

#### 4.2. Bulding a Cascade Model

`Cascade.Computation`that enables the user to specify a (cascade) scheme and approach, along with all further input parameters. Figure 3 shows the internal definition of this data structure, while Table 1 displays a number of frequently occuring cascade schemes that are (partly) implemented and supported by the Jac toolbox. All of these schemes are subtypes of an abstract data structure

`AbstractCascadeScheme`and they are used to set the input data that are specific for a given casade computation. Figure 4 displays the explicit internal definition of two of these subtypes (schemes), namely the

`StepwiseDecayScheme`and the

`PhotonIonizationScheme`.

#### 4.3. Stepwise Decay of a K-Shell Hole in Atomic Magnesium. An Explicit Example

`perform(..)`the computation, and where the optional parameter

`output=true`tells the program that the generated lists of transition data are to be returned as a Julia dictionary (

`Dict`). Any cascade computation returns one (or a few) such list that comprises all lines of the same type, but that may occur here in any order with regard to their transition energies, level specification, or even the charge state of the ions. These “line” lists are the expensive parts of any cascade computation and, hence, are often also stored on a disk; they form the input for all subsequent cascade simulations. In the example above, these final lists are an instance of

`Cascade.DecayData <:Cascade.AbstractData`that comprises the Auger and radiative (lists of) lines.

#### 4.4. Running Cascade Simulations

`data`, this distribution of the ionic charge states can be obtained by the following cascade simulations:

## 5. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References and Note

- Mirakhmedov, M.N.; Parilis, E.S. Auger and X-ray cascades following inner-shell ionisation. J. Phys.
**1988**, B21, 795. [Google Scholar] [CrossRef] - Kucas, S.; Drabuzinskis, P.; Kynienė, A.; Masys, S.; Jonauskas, V. Evolution of radiative and Auger cascades following 2s vacancy creation in Fe
^{2+}. J. Phys. B**2019**, 52, 225001. [Google Scholar] [CrossRef] - Schippers, S.; Borovik, A.; Buhr, T.; Hellhund, K.; Holste, K.; Kilcoyne, A.L.D.; Klumpp, S.; Martins, M.; Müller, A.; Ricz, S.; et al. Stepwise contraction of the nf Rydberg shells in the 3d photoionization of multiply-charged xenon ions. J. Phys. B
**2015**, 48, 144003. [Google Scholar] [CrossRef] [Green Version] - Schippers, S.; Beerwerth, R.; Abrok, L.; Bari, S.; Buhr, T.; Martins, M.; Ricz, S.; Viefhaus, J.; Fritzsche, S.; Müller, A.; et al. Prominent role of multi-electron processes in K-shell double and triple photodetachment of oxygen anions. Phys. Rev. A
**2016**, 94, 041401. [Google Scholar] [CrossRef] [Green Version] - Khalal, M.A.; Lablanquie, P.; Andric, L.; Palaudoux, J.; Penent, F.; Bučar, K.; Žitnik, M.; Püttner, R.; Jänkälä, K.; Cubaynes, D.; et al. 4d-inner-shell ionization of Xe
^{+}ions and subsequent Auger decay. Phys. Rev. A**2017**, 96, 013412. [Google Scholar] [CrossRef] [Green Version] - Ma, X.; Stöhlker, T.; Bosch, F.; Brinzanescu, O.; Fritzsche, S.; Kozhuharov, C.; Ludziejewski, T.; Mokler, P.H.; Stachura, Z.; Warczak, A.; et al. State–selective electron capture in He–like U
^{90+}ions colliding with gaseous targets. Phys. Rev. A**2001**, 64, 012704. [Google Scholar] [CrossRef] - Ueda, K.; Shimizu, Y.; Chiba, H.; Kitajima, M.; Tanaka, H.; Fritzsche, S.; Kabachnik, N.M. Experimental and theoretical study of the Auger cascade following the 2p→4s photoexcitation in Ar. J. Phys. B
**2001**, 34, 107. [Google Scholar] [CrossRef] - Kabachnik, N.M.; Fritzsche, S.; Grum-Grzhimailo, A.N.; Meyer, M.; Ueda, K. Coherence and correlations in photoinduced Auger and fluorescence cascades in atoms. Phys. Rep.
**2007**, 451, 155. [Google Scholar] [CrossRef] - Le Bigot, E.O.; Boucard, S.; Covita, D.S.; Gotta, D.; Gruber, A.; Hirtl, A.; Fuhrmann, H.; Indelicato, P.; dos Santos, J.M.F.; Schlesser, S.; et al. High-precision X-ray spectroscopy in few-electron ions. Phys. Scr.
**2009**, T134, 014015. [Google Scholar] [CrossRef] - Gillaspy, J.D. Precision spectroscopy of trapped highly charged heavy elements: Pushing the limits of theory and experiment. Phys. Scr.
**2014**, 98, 1114004. [Google Scholar] [CrossRef] - Andersson, J.; Beerwerth, R.; Linusson, P.; Eland, J.H.D.; Zhaunerchyk, V.; Fritzsche, S.; Feifel, R. Triple ionization of atomic Cd involving 4p
^{−1}and 4s^{−1}inner-shell holes. Phys. Rev. A**2015**, 92, 023414. [Google Scholar] [CrossRef] [Green Version] - Wiese, W.L. Spectroscopic diagnostics of low temperature plasmas: Techniques and required data. Spectrochim. Acta B
**1991**, 46, 831–841. [Google Scholar] [CrossRef] - Saha, B.; Fritzsche, S. Influence of dense plasma on the low-lying transitions in Be-like ions: Relativistic multiconfiguration Dirac–Fock calculation. J. Phys. B
**2007**, 40, 259–270. [Google Scholar] [CrossRef] - Feldman, U.; Mandelbaum, P.; Seely, J.F.; Doschek, G.A.; Gursky, H. The potential for plasma diagnostics from stellar extreme-ultraviolet observations. Astrophy. J. Suppl.
**1992**, 81, 387–408. [Google Scholar] [CrossRef] - Dong, C.Z.; Fritzsche, S.; Fricke, B.; Sepp, W.-D. Branching ratios and lifetimes of the low–lying levels of Fe xx. Mon. Not. R. Astron. Soc.
**1999**, 307, 809–814. [Google Scholar] [CrossRef] [Green Version] - Battistini, C.; Bensby, T. The origin and evolution of r- and s-process elements in the Milky Way stellar disk. Astron. Astrophys.
**2016**, 586, A49. [Google Scholar] [CrossRef] [Green Version] - Popping, G.; Somerville, R.S.; Trager, S.C. Evolution of the atomic and molecular gas content of galaxies. Mon. Not. R. Astron. Soc.
**2014**, 442, 2398–2418. [Google Scholar] [CrossRef] [Green Version] - Jakas, M.M. Many-body effects in atomic-collision cascades. Phys. Rev. Lett.
**1985**, 55, 1782. [Google Scholar] [CrossRef] [PubMed] - Schippers, S.; Martins, M.; Beerwerth, R.; Bari, S.; Holste, K.; Schubert, K.; Viefhaus, J.; Savin, D.W.; Fritzsche, S.; Müller, A. Near L-edge single and multiple photoionization of singly charged iron ions. Astrophys. J.
**2017**, 849, 5. [Google Scholar] [CrossRef] [Green Version] - Beerwerth, R.; Buhr, T.; Perry-Sassmannshausen, A.; Stock, S.O.; Bari, S.; Holste, K.; Kilcoyne, A.L.D.; Reinwardt, S.; Ricz, S.; Savin, D.W.; et al. Near L-edge single and multiple photoionization of triply charged iron ions. Astrophys. J.
**2019**, 887, 189. [Google Scholar] [CrossRef] [Green Version] - Fritzsche, S. A fresh computational approach to atomic structures, processes and cascades. Comput. Phys. Commun.
**2019**, 240, 1–14. [Google Scholar] [CrossRef] - Fritzsche, S. JAC: User Guide, Compendium & Theoretical Background. Available online: https://github.com/OpenJAC/JAC.jl (accessed on 10 February 2021).
- Fritzsche, S. Ratip—A toolbox for studying the properties of open—Shell atoms and ions. J. Electr. Spec. Rel. Phenon.
**2001**, 114–116, 1155–1164. [Google Scholar] [CrossRef] - Fritzsche, S. The Ratip program for relativistic calculations of atomic transition, ionization and recombination properties. Comput. Phys. Commun.
**2012**, 183, 1525–1559. [Google Scholar] [CrossRef] - Barnes, J.; Kasen, D. Effect of a high opacity on the light curves of radioactively powered transients from compact object mergers. Astrophys. J.
**2013**, 775, 18. [Google Scholar] [CrossRef] - Kravtsov, A.V.; Borgani, S. Formation of Galaxy Clusters. Annu. Rev. Astrononmy Astrophys.
**2012**, 50, 353–409. [Google Scholar] [CrossRef] [Green Version] - Cowan, R.D. The Theory of Atomic Structure and Spectra; University of California Press: Berkeley, CA, USA, 1981. [Google Scholar]
- Fritzsche, S. Large-scale accurate structure calculations for open-shell atoms. Phys. Scr.
**2002**, T100, 37. [Google Scholar] [CrossRef] - Perry-Sassmannshausen, A.; Buhr, T.; Borovik, A., Jr.; Martins, M.; Reinwardt, S.; Ricz, S.; Stock, S.O.; Trinter, F.; Müller, A.; Fritzsche, S.; et al. Multiple photodetachment of carbon anions via single and double core-hole creation. Phys. Rev. Lett.
**2020**, 124, 083203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Andersson, J.; Beerwerth, R.; Roos, A.H.; Squibb, J.J.; Singh, R.; Zagorodskikh, S.; Talaee, O.; Koulentianos, D.; Eland, J.H.D.; Fritzsche, S.; et al. Auger decay of 4d inner-shell holes in atomic Hg leading to triple ionization. Phys. Rev. A
**2017**, 96, 012505. [Google Scholar] [CrossRef] [Green Version] - Grant, I.P. Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Eliav, E.; Fritzsche, S.; Kaldor, U. Electronic structure theory of the superheavy elements. Nucl. Phys. A
**2015**, 944, 518–550. [Google Scholar] [CrossRef] - Surzhykov, A.; Fritzsche, S.; Stöhlker, T.; Tachenov, S. Polarization studies on the radiative recombination of highly charged bare ions. Phys. Rev. A
**2003**, 68, 022710. [Google Scholar] [CrossRef] [Green Version] - Surzhykov, A.; Indelicato, P.; Santos, J.P.; Amaro, P.; Fritzsche, S. Two-photon absorption of few-electron heavy ions. Phys. Rev. A
**2011**, 84, 022511. [Google Scholar] [CrossRef] - Fritzsche, S.; Grum-Grzhimailo, A.N.; Gryzlova, E.V.; Kabachnik, N.M. Angular distributions and angular correlations in sequential two-photon double ionization of atoms. J. Phys. B
**2008**, 41, 165601. [Google Scholar] [CrossRef] - Li, Y.; Ullrich, C.A. Time-dependent transition density matrix. Chem. Phys.
**2011**, 391, 157–163. [Google Scholar] [CrossRef] - Charlwood, F.C.; Billowes, J.; Campbell, P.; Cheal, B.; Eronen, T.; Forest, D.H.; Fritzsche, S.; Honma, M.; Jokinen, A.; Moore, I.D.; et al. Ground state properties of manganese isotopes across the N=28 shell closure. Phys. Lett. B
**2010**, 690, 346–351. [Google Scholar] [CrossRef] - Yordanov, D.T.; Balabanski, D.L.; Bieroń, J.; Bissell, M.L.; Blaum, K.; Budinčević, I.; Fritzsche, S.; Frömmgen, N.; Georgiev, G.; Geppert, C.; et al. Spins, Electromagnetic moments and isomers of
^{107-129}Cd. Phys. Rev. Lett.**2013**, 110, 192501. [Google Scholar] [CrossRef] [Green Version] - Cheal, B.; Cocolios, T.E.; Fritzsche, S. Laser spectroscopy of radioactive isotopes: Role and limitations of accurate isotope-shift calculations. Phys. Rev. A
**2012**, 86, 042501. [Google Scholar] [CrossRef] - Ferrer, R.; Barzakh, A.; Bastin, B.; Beerwerth, R.; Block, M.; Creemers, P.; Grawe, H.; de Groote, R.; Delahaye, P.; Flechard, X.; et al. Towards high-resolution laser ionization spectroscopy of the heaviest elements in supersonic gas jet expansion. Nat. Commun.
**2017**, 8, 14520. [Google Scholar] [CrossRef] [Green Version] - Surzhykov, A.; Jentschura, U.D.; Stöhlker, T.; Gumberidze, A.; Fritzsche, S. Alignment of heavy few-electron ions following excitation by relativistic Coulomb collisions. Phys. Rev.
**2008**, A77, 042722. [Google Scholar] [CrossRef] - Åberg, T.; Howat, G. Theory of the Auger Effect. In Corpuscles and Radiation in Matter I; Encyclopedia of Physics Vol. XXXI; Mehlhorn, W., Ed.; Springer: Berlin/Heidelberg, Germany, 1982; p. 469. [Google Scholar]
- Verma, H.R. A study of radiative Auger emission, satellites and hypersatellites in photon-induced K x-ray spectra of some elements in the range 20≤Z≤32. J. Phys. B
**2000**, 33, 3407. [Google Scholar] [CrossRef] - Brieger, M.; Schuessler, H.A. What is quantum interference in atoms? Eur. Lett.
**1996**, 35, 1. [Google Scholar] [CrossRef] - Fritzsche, S.; Nikkinen, J.; Huttula, S.-M.; Aksela, H.; Huttula, M.; Aksela, S. Interferences in the 3p
^{4}nl satellite emission following the excitation of argon across the 2p1/254s and 2p3/253dJ=1 resonances. Phys. Rev. A**2007**, 75, 012501. [Google Scholar] [CrossRef] - Shimizu, Y.; Yoshida, H.; Okada, K.; Muramatsu, Y.; Saito, N.; Okashi, H.; Tamenori, Y.; Fritzsche, S.; Kabachnik, N.M.; Tanaka, H.; et al. High resolution angle-resolved measurements of Auger emission from the photo-excited 1s
^{−1}3p state of Ne. J. Phys. B At. Mol. Opt. Phys.**2000**, 33, L685. [Google Scholar] [CrossRef] - Kitajima, M.; Okamoto, M.; Shimizu, Y.; Chiba, H.; Fritzsche, S.; Kabachnik, N.M.; Sazhina, I.P.; Koike, F.; Hayaishi, T.; Tanaka, H.; et al. Experimental and theoretical study of the Auger cascade following 3d→5p photoexcitation in krypton. J. Phys. B
**2001**, 34, 3829. [Google Scholar] [CrossRef] - Fritzsche, S.; Fricke, B.; Sepp, W.-D. Reduced L
_{1}level-width and Coster-Kronig yields by relaxation and continuum interactions in atomic zinc. Phys. Rev. A**1992**, 45, 1465. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dong, C.Z.; Fritzsche, S. Relativistic, relaxation and correlation effects in spectra of Cu II. Phys. Rev. A
**2005**, 72, 012507. [Google Scholar] [CrossRef] - Fritzsche, S.; Froese Fischer, C.; Fricke, B. Calculated level energies, transition probabilities and lifetimes for phosphorus-like ions of the iron group in the 3s3p
^{4}and 3s^{2}3p^{2}3d configurations. At. Data Nucl. Data Tables**1998**, 68, 149–179. [Google Scholar] [CrossRef] [Green Version] - Available online: https://docs.julialang.org/ (accessed on 10 February 2021).
- Bezanson, J.; Chen, J.; Chung, B.; Karpinski, S.; Shah, V.B.; Vitek, J.; Zoubritzky, J. Julia: Dynamism and performance reconciled by design. Proc. ACM Program. Lang.
**2018**, 2, 120. [Google Scholar] [CrossRef] [Green Version] - Julia Comes with a Full-Featured Interactive and Command-Line REPL (Read-Eval-Print Loop) that is built into the Executable of the Language.
- Schippers, S.; Beerwerth, R.; Bari, S.; Buhr, T.; Holste, K.; Kilcoyne, A.D.; Perry-Sassmannshausen, A.; Phaneuf, R.A.; Reinwardt, S.; Savin, D.W.; et al. Near L-edge single and multiple photoionization of doubly charged iron ions. Astrophys. J.
**2021**, 908, 52. [Google Scholar] [CrossRef] - Sharma, L.; Surzhykov, A.; Srivastava, R.; Fritzsche, S. Electron-impact excitation of singly charged metal ions. Phys. Rev. A
**2011**, 83, 062701. [Google Scholar] [CrossRef]

**Figure 1.**Important building blocks of all cascade computations. Each cascade is formed by the—list of energetically sorted—levels of electron configurations (blue lines in yellow boxes) from neighbored charge states. These levels are connected to each by fine-structure transitions (red and green arrows, often called lines below) of different kind, and where the number of these transitions rapidly increases $\propto \phantom{\rule{0.222222em}{0ex}}{n}_{i}\times {n}_{f}$ with the number of the initial- and final-state levels of the associated configurations. A pathway (green arrows) decribes a possible decay path of an atom and it connects three or more fine-structure levels by different excitation and/or decay processes. Moreover, one or several electron configurations can be comprised together into a single cascade block (gray frames) to properly deal with inter-electronic correlations, but to ensure that each level just belongs to one block. Apart from those configurations of an atomic cascade, which are readily derived from the atomic shell model, further shake configurations can also be considered, although this incorporation typically results in a sizeable increase of the overall complexity. Finally, pairs of cascade blocks give rise to one or several cascade steps (wide light-red arrow) in order to deal with the different atomic processes, and that occur during the relaxation of the atom or ion. The formal decomposition of a cascade into these building blocks suggests various cascade approaches, as explained in the text below.

**Figure 2.**Simplified view of frequently occuring cascade schemes. Apart from (

**a**) the stepwise decay of an initial N-electron hole-state configuration of an ion via two or more intermediate cascade blocks, including bound levels with less than (the initially) N electrons, these schemes also help account for the excitation of the atoms or ions. Such an excitation may arise from the (

**b**) photoexcitation or photoionization, (

**c**) electron-impact excitation, (

**d**) radiative or dielectronic capture of electrons, (

**e**) the formation of hollow ions via multiple electron capture, or (

**f**) by a muon capture.

**Figure 3.**Definition of the data structure

`Cascade.Computation`in Jac that helps to specify all relevant data about a cascade computation as described in Section 3.4. This structure enables the user to select both, a cascade

`scheme::Cascade.AbstractCascadeScheme`and (cascade)

`approach::Cascade.AbstractCascadeApproach`in order to distinguish different computational models for a cascade.

**Figure 4.**Definition of the data structures

`Cascade.StepwiseDecayScheme`(upper panel) and

`Cascade.PhotonIonizationScheme`(lower panel) in Jac. For each of these structures, different information must be given to control the corresponding cascade computations. In particular, the subfield

`processes::Array{Basics.AbstractProcess,1`} allow the user to specify the processes, such as

`Auger(), Radiative(), ...`, and which are taken into account to determine the cascade blocks.

**Figure 5.**Selected printout from the example in Section 4.3. Apart from the initial levels (multiplet), this printout lists the cascade tree, the generated blocks, and steps of the cascade, as well as a summary of all generated continuum orbitals and transition amplitudes. This printout can indeed be quite long, but it enables the user to reconstruct the individual calculations and check for inconsistencies or warning that are issued during the execution.

**Figure 6.**(

**a**) Level structure of the $1s\to 3p$ excited neutral and $1s$ ionized atomic magnesium. (

**b**) Comparison of the final ion distribution for the $1s$ hole-state levels as obtained in the averaged single-configuration approach

`(AverageSCA)`. Relative ion distribution are shown for an initially occupied $1s\phantom{\rule{0.166667em}{0ex}}2{s}^{2}2{p}^{6}3{s}^{2}3p\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.222222em}{0ex}}}^{1}{P}_{1}$ level of neutral Mg (red bars), a statistical distribution of all four $1s\phantom{\rule{0.166667em}{0ex}}2{s}^{2}2{p}^{6}3{s}^{2}3p\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.222222em}{0ex}}}^{1,3}{P}_{J}$ levels (blue bars) for the $1s$ ionized $1s\phantom{\rule{0.166667em}{0ex}}2{s}^{2}2{p}^{6}3{s}^{2}\phantom{\rule{0.277778em}{0ex}}{\phantom{\rule{0.222222em}{0ex}}}^{2}{S}_{1/2}$ level of Mg${}^{+}$ (orange bars). The small difference between the red and blue bars arises from the contributions of different decay paths.

**Table 1.**Cascade schemes, as (partly) implemented and supported by the Jac toolbox. These schemes help to distinguish cascades of different context and complexity. They internally refer to separate kinds of cascade computations, cf. Section 3.3 and the data structures below.

Excitation or DecayScheme & Brief Explanation |
---|

Stepwise decay scheme: this scheme starts from either one or a few excited (electron) configurations, or a set of initial levels, with inner-shell holes. These exited levels then decay by different user-selected atomic processes, such as autoionization, photon emission, and others, until a given number of electrons is released and/or the ions cannot further decay to any lower level. This scheme often extends a prior excitation scheme; cf. the data structure StepwiseDecayScheme. |

Photoionization scheme: enables one to model the initial photoionization of an atom or ion. It starts from its ground configuration and generates all of those electron configurations that can be reached by the given photon energies; cf. PhotonExcitationScheme. |

Photoexcitation scheme: enables one to model the initial photoexcitation of an atom or ion. It starts from some (ground) configuration and generates all those configurations that can be reached by photons from a given range of photon energies; cf. PhotonExcitationScheme. |

Electron-capture scheme: this scheme implements a dielectronic-capture process. It starts from some (ground) configuration of an atom or ions and generates all doubly-excited configurations with one additional electron for a range of free-electron energies; cf. ElectronCaptureScheme. |

Impact-excitation scheme: enables one to model the electron-impact excitation of an atom; not yet implemented. |

Hollow-ion scheme: enables one to start from any hollow-ion configuration whose decay is modeled. It expects a list of electron shells that can be temporarely populated during the decay; not yet implemented. |

Muon cascade: this scheme starts with a single muon that is initially captured into one or several $\phantom{\rule{0.222222em}{0ex}}(n\ell )\phantom{\rule{0.222222em}{0ex}}$ subshells of an atom, which then subsequently decays via photon and electron emission; not yet implemented. |

**Table 2.**Frequently required distributions in atomic cascade simulations. These distributions are (partly) implemented and supported by the Jac toolbox.

Distribution & Brief Explanation |
---|

Ion distribution, i.e., the distribution of ionic charge states after the (stepwise decay) cascade of some excited and initially occupied atomic level(s); cf. the data structure Cascade.IonDistribution. |

Final-level distribution, i.e., the distribution of the finally occupied levels, following the (stepwise decay) cascade of some excited and initially populated atomic level(s); cf. Cascade.FinalLevelDistribution. |

Photon spectrum from a cascade as function of the photon energy; cf. Cascade.PhotonIntensities. |

Electron spectrum from a cascade as function of the electron energy; cf. Cascade.ElectronIntensities. |

Absorption cross sections of atoms or ions in some given ground-state configuration as function of the incident photon energy; cf. Cascade.AbsorptionCrossSection. |

DR plasma rate coefficients for the dielectronic recombination of ions and for one or several electron temperatures. |

Muon x-ray spectrum following the capture of an muon into some shell or subshell of an atom or ion. |

**Table 3.**Selected data structures of the Jac toolbox that are relevant for cascade computations and simulations.

Struct & Brief Explanation |
---|

Cascade.AbstractCascadeApproach: defines an abstract type for dealing with the cascade approach that is applied to the generation and evaluation of all many-electron amplitudes. |

Cascade.AbstractCascadeScheme: defines an abstract data type to discriminate between different excitation, ionization and decay schemes of an atomic cascade; see also Figure 4. |

Cascade.AbstractData: defines an abstract type to distinguish the output data from different cascade computations, such as Cascade.DecayData, Cascade.PhotoIonData, Cascade.ExcitationData, and others. |

Cascade.AbstractSimulationProperty: defines an abstract type to specify the property or distribution that should be simulated, based on given cascade data; cf. Table 2. |

Cascade.Block: defines a data structure for an individual (cascade) block of configurations that is given by a list of configurations, and which gives rise to a common multiplet. Any individual level of the atom or ion must not belong to more than one of the blocks in order to avoid “double counting” in the cascade. |

Cascade.Computation: defines a data structure for the computation of a photoexcitation, photoionization, stepwise decay or several other cascade computations. |

Cascade.Level: defines a level specification for dealing with fine-structure transitions in cascade simulations. |

Cascade.Simulation: defines a structure to deal with simulations, based on given cascade data. |

Cascade.SimulationSettings: defines a data structure to specify all additional parameters for controling a cascade simulation. |

Cascade.Step: defines a structure for an individual cascade step that is determined by two blocks of the initial- and final-state configurations as well as by one atomic process, such as Auger, PhotoEmission, or others, and which relate the corresponding fine-structure levels to each other. |

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Fritzsche, S.; Palmeri, P.; Schippers, S.
Atomic Cascade Computations. *Symmetry* **2021**, *13*, 520.
https://doi.org/10.3390/sym13030520

**AMA Style**

Fritzsche S, Palmeri P, Schippers S.
Atomic Cascade Computations. *Symmetry*. 2021; 13(3):520.
https://doi.org/10.3390/sym13030520

**Chicago/Turabian Style**

Fritzsche, Stephan, Patrick Palmeri, and Stefan Schippers.
2021. "Atomic Cascade Computations" *Symmetry* 13, no. 3: 520.
https://doi.org/10.3390/sym13030520