Linewidth and lifetime of atomic levels and the time evolution of spectra and coincidence spectra

The time-dependent theory of an atomic cascade decay process is discussed. We show that it is useful to measure the spectra of the emitted particles (electrons or photons) as a function of time and introduce time-dependent spectra and time-dependent coincidence spectra. The analysis reveals that there are several timescales involved in the atomic cascades and in the respective spectra. The work prepares the ground for the discussion of molecules, clusters, and solids, where the nuclear dynamics strongly participate in the cascade decay process making it much more complex.

Figure 1

Schematic representation of the two-step cascade. Shown are the four energy levels involved. The initial state $i$ is excited to the decaying state $d_1$, which decays to $d_2$ and this finally to $f$. In the first decay step an electron with energy $E_1$ and in the second decay step an electron with energy $E_2$ are emitted. Equations (1, 2, 3) govern the time evolution of the whole cascade process.

Figure 2

Spectra of a model example. (Top) Spectrum of the electrons emitted in the first step of the decay compared to the spectrum, which one would obtain in the absence of the second decay step (i.e., in the absence of a cascade). The latter spectrum has been scaled to let the maxima of both spectra be the same. (Bottom) The two decay spectra, that of the first and that of the second electron. Parameters used: ${\Gamma }_{d_1}=0.2$ eV, ${\Gamma }_{d_2}=0.1$ eV and $E_{d_1}-E_{d_2}=100$ eV, $E_{d_2}-E_f=10$ eV.

Figure 3

Coincidence spectrum of the first and the second electrons of the cascade. Relative intensity as a function of the energies $E_1$ and $E_2$ of the two emitted electrons. The relative intensity is provided by the scale on the right-hand side of the figure. The parameters used are the same as those in Fig. 2.

Figure 4

Time evolution of the coincidence spectrum at 10 fs. The spectrum intensity is scaled such that the final spectrum [i.e., after a long time, see Eq. (5)] has a maximum intensity of one. Parameters used: ${\Gamma }_{d_1}=0.03$ eV, ${\Gamma }_{d_2}=0.3$ eV and $E_{d_1}-E_{d_2}=100$ eV, $E_{d_2}-E_f=10$ eV. As expected, this coincidence spectrum is centered at the energy grid point $(E_1,E_2)=(100\hspace{0.5em}\text{eV},10\hspace{0.5em}\text{eV})$. However, the spectrum spreads over a large energy range that changes in time.

Figure 5

Top view of the time evolution of the coincidence spectrum. The spectrum is becoming narrower as time increases. This can be explained by the uncertainty principle. The parameters used are the same as in Fig. 4.

Figure 6

Top view of the coincidence spectrum at 300 fs. The whole decay process is completed after about 300 fs and thus the depicted spectrum represents the final coincidence spectrum. The relative intensity is scaled and its corresponding color bar is given in the figure. The parameters used are the same as those in Figs. 4 and 5.

Figure 7

Two different timescales of the same process. (Top) The population of the first decaying level, $|a_{d_1}(t)|{}^2$, decreases in time exponentially (note the logarithmic scale). Accordingly, the decay of the first level is completed in less than 20 fs. (Bottom) The time evolution of the spectrum of the electron emitted in the first decay step calculated by Eq. (10). As one can see, the spectrum develops strongly at times much longer than 20 fs and reaches its final form only after about 140 fs. (See text for the numerical parameters used.)

Figure 8

Time evolution of the spectrum of the electron emitted in the first step of the cascade and its breakdown into its contributions. The top panel shows the time evolution of the spectrum of the first emitted electron computed by Eq. (12). One can see that the spectrum is almost completely finished at 20 fs (green solid circle). The final spectrum (solid black line) is computed by Eqs. (5) and (6). The middle panel shows how the population of $d_2$ level, $|a_{d_2}(E_1,t)|{}^2$, decreases in time. Finally the whole population will vanish when the second decay process is completed (i.e., when the whole cascade is completed). The lower panel shows the “leak” of $|a_{d_2}(E_1,t)|{}^2$ calculated via the time evolution of the coincidence spectrum (i.e., $\int \text{d}{E}_2|a_f(E_1,E_2,t)|{}^2$). This intensity grows with time and nicely keeps balance with the intensity from $|a_{d_2}(E_1,t)|{}^2$ so that the total spectrum is not influenced by the partial intensity changes after 20 fs. The parameters used are the same as in Fig. 7.