#
Effects of the FEL Fluctuations on the 2s2p Li^{+} Auto-Ionization Lineshape

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## Abstract

**:**

## 1. Introduction

## 2. Electronic Structure and the Fano Picture of Resonant Ionization

#### 2.1. Atomic Structure Calculations

#### 2.2. Calculation of the Fano Parameters of the AIS

#### 2.2.1. Photoionization Cross Section and Position, width of the Autoionizing Resonance

**S**, ${I}_{0}={\langle S\rangle}_{t}=c{\u03f5}_{0}{E}_{0}^{2}/2$ and set ${\epsilon}_{0}=1/4\pi $, with ${\epsilon}_{0}$ the vacuum’s dielectric constant. This will give ${I}_{0}=c{E}_{0}^{2}/8\pi $. Then by use of standard perturbation theory the ionization yield is related with the photoionization cross section (all quantities in a.u.):

#### 2.2.2. Calculation of the Fano Parameters, ${q}_{a},{d}_{ga},{d}_{gc}$

## 3. The Density-Matrix EOMs in the Fano Representation

## 4. The Averaged Density-Matrix EOMs

#### 4.1. FEL Radiation as a Gaussian, Non-Stationary Stochastic Process

#### 4.2. Averaged Form of the EOMs

## 5. Results and Discussion

#### 5.1. The AIS Line Shape

#### 5.2. Effects of the Fluctuation’s Coherence Time

#### 5.3. Effects of the Pulse Duration

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Autocorrelation |

AIS | Autoionization State |

CI | Configuration Interaction |

EOMs | Equations of Motion |

FEL | Free Electron Laser |

FWHM | Full Width at Half Maximum |

MC | Monte-Carlo |

TISE | Time Independent Schrödinger Equation |

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**Figure 1.**Ionization scheme of Li${}^{+}$. $|g\rangle $$1{s}^{2}$ is the ground state, $|a\rangle $ 2s2p is the auto-ionizing state, $|c\rangle $ is the continuum above the ionized state of Li${}^{2+}$ 1s and the $|{c}^{\prime}\rangle $ is the continuum above the ionized state Li${}^{3+}$. The energies depicted on the right are not to scale but only give an idea of the values of the levels.

**Figure 2.**The effect of smoothing the familiar Fano-profile, caused by averaging the fluctuations can be clearly seen in this figure. The red-dashed curve with highest peak is obtained for a deterministic pulse whereas the black-solid curve is obtained for a stochastic pulse with coherence time of 1.3 fs (53.74 a.u.). The peak intensity is 10${}^{13}$ W/cm${}^{2}$ and the pulse duration ${\tau}_{p}=12.7$ fs (525.02 a.u.).

**Figure 3.**Effect of coherence time on the ionization yield is shown in this plot. As ${\tau}_{c}$ increases, the ionization yield also increases, near resonance. The peak intensity is 10${}^{13}$ W/cm${}^{2}$ and the pulse duration ${\tau}_{p}=12.7$ fs (525.02 a.u.).

**Figure 4.**Effect of pulse duration on the ionization yield is shown in this plot. The coherence time used is ${\tau}_{c}=3$ fs (124.02 a.u.) and the peak intensities are: 10${}^{13}$ W/cm${}^{2}$ for (

**a**), 10${}^{14}$ W/cm${}^{2}$ for (

**b**) and 10${}^{15}$ W/cm${}^{2}$ for (

**c**). For lower intensities, as pulse duration increases, the ionization yield also increases. But the pattern flips after certain peak intensity and for the highest peak intensity of 10${}^{15}$ W/cm${}^{2}$, the yield drops as pulse duration increases.

**Figure 5.**Effect of peak intensity on the population of Li${}^{2+}$. The peak intensities used are: 10${}^{13}$ W/cm${}^{2}$ for (

**a**), 10${}^{14}$ W/cm${}^{2}$ for (

**b**) and 10${}^{15}$ W/cm${}^{2}$ for (

**c**). Black-solid curve is when the further ionization channel is ignored (${\gamma}_{c}=0$) and the red-dashed curve is when the further ionization channel is considered. The pulse duration ${\tau}_{p}=$ 20 fs (826.8 a.u.) and the coherence time ${\tau}_{c}=$ 3 fs (124.02 a.u.). The red-dashed curve suggests that for (

**a**) and (

**b**) the population grows after the peak intensity at t = 0 fs, but for (

**c**) it drops. This causes the change in the trend of ionization of Figure 4.

Parameters | Widths |
---|---|

$({E}_{a},{q}_{a})$ | (5.52, −2.2) |

${d}_{ga}$ | 0.01527 |

${\Gamma}_{a}$ | 0.00235 |

${\gamma}_{g}$ | 0.0819 |

${\gamma}_{c}$ | 0.0584 |

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**MDPI and ACS Style**

Katravulapally, T.; Nikolopoulos, L.A.A.
Effects of the FEL Fluctuations on the 2s2p Li^{+} Auto-Ionization Lineshape. *Atoms* **2020**, *8*, 35.
https://doi.org/10.3390/atoms8030035

**AMA Style**

Katravulapally T, Nikolopoulos LAA.
Effects of the FEL Fluctuations on the 2s2p Li^{+} Auto-Ionization Lineshape. *Atoms*. 2020; 8(3):35.
https://doi.org/10.3390/atoms8030035

**Chicago/Turabian Style**

Katravulapally, Tejaswi, and Lampros A. A. Nikolopoulos.
2020. "Effects of the FEL Fluctuations on the 2s2p Li^{+} Auto-Ionization Lineshape" *Atoms* 8, no. 3: 35.
https://doi.org/10.3390/atoms8030035