Multiple core-hole formation by free-electron laser radiation in molecular nitrogen

We obtain the bound molecular and bound atomic orbital wavefunctions using Molpro [21]. We employed the Hartree–Fock approximation with a correlation-consistent polarized triple-zeta basis set. For FEL-driven N₂ we treat the nuclei as fixed, with the distance between the nuclei set equal to the equilibrium distance, i.e. 2.08 a.u. This internuclear distance is used to compute the molecular orbitals. We use the single-center expansion method [22] to calculate the molecular continuum wavefunctions and Hartree–Fock Slater [23] calculations to compute the atomic continuum wavefunctions [17]. We then use these wavefunctions to obtain single-photon ionization cross sections and auger rates, see [24]. The dissociation of the molecule is treated through terms in the rate equations. Specifically, we assume that ${{\rm{N}}}_{2}^{4+}$ and ${{\rm{N}}}_{2}^{3+}$ molecular ion states with no core holes dissociate instantaneously into ${{\rm{N}}}^{2+}+{{\rm{N}}}^{2+}$ and ${{\rm{N}}}^{2+}+{{\rm{N}}}^{+},$ respectively. We also assume that all states of ${{\rm{N}}}_{2}^{2+}$ dissociate with a lifetime of 100 fs [25]. The final fragments of this latter dissociation are ${{\rm{N}}}^{+}+{{\rm{N}}}^{+}$ and ${{\rm{N}}}^{2+}+{\rm{N}}$ with 74% and 26% probability, respectively [15, 16]. Starting from the ground state of molecular nitrogen with all orbitals fully occupied ($1{\sigma }_{{g}}^{2},1{\sigma }_{{u}}^{2},2{\sigma }_{{g}}^{2},2{\sigma }_{{u}}^{2},1{\pi }_{{ux}}^{2},1{\pi }_{{uy}}^{2},3{\sigma }_{{g}}^{2}$), we solve the rate equations to obtain the final atomic ion yields. Moreover, by solving additional rate equations, we compute all the possible pathways which contribute to these final atomic ion states. By pathway, we refer to the series of molecular and atomic states (electronic configurations) that are accessed before the formation of a given atomic ion yield. In what follows, we compute the contribution of pathways that include the formation of a SSDCH, TSDCH or TCH molecular state.

As mentioned above, we use molecular orbitals to obtain the final atomic ion yields and the populations of the pathways from the rate equations. We note that the use of molecular bound state orbitals is important for obtaining electron spectra. Indeed, it has been shown that with high-resolution electron spectroscopy one can observe the energy splitting of the molecular core-hole states 1σ_g and 1σ_u [26–29]. In order to determine whether a pathway accesses a TSDCH molecular state or a SSDCH molecular state during the interaction of N₂ with an FEL pulse, it is necessary at each time step of the propagation to project the delocalized inner-shell molecular orbitals onto inner-shell orbitals localized on atomic sites. We denote the DCH molecular states that involve the inner-shell molecular orbitals 1σ_g and 1σ_u by $| 1{\sigma }_{g/u}1{\sigma }_{g/u}\rangle$.

The delocalized molecular orbitals are expressed in terms of orbitals localized on atomic sites by $| 1{\sigma }_{g}\rangle \,=\tfrac{1}{\sqrt{2}}(| 1{s}_{a}\rangle +| 1{s}_{b}\rangle )$ and $| 1{\sigma }_{u}\rangle =\tfrac{1}{\sqrt{2}}(| 1{s}_{a}\rangle -| 1{s}_{b}\rangle )$ where $| 1{s}_{a}\rangle$ and $| 1{s}_{b}\rangle$ are $| 1s\rangle$ orbitals localized on the atomic sites a and b of N₂, respectively. At every time step, we check whether a DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{g}\rangle ,| 1{\sigma }_{g}1{\sigma }_{u}\rangle$ or $| 1{\sigma }_{u}1{\sigma }_{u}\rangle$ has been accessed. Expressing the DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{g}\rangle$ in terms of orbitals localized on atomic sites, we obtain the following:

$$\begin{eqnarray}| 1{\sigma }_{g}1{\sigma }_{g}\rangle & = & \displaystyle \frac{1}{2}[| 1{s}_{a}\rangle | 1{s}_{a}\rangle +| 1{s}_{a}\rangle | 1{s}_{b}\rangle \\ & & +| 1{s}_{b}\rangle | 1{s}_{a}\rangle +| 1{s}_{b}\rangle | 1{s}_{b}\rangle ],\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}| \langle 1{s}_{a}1{s}_{a}| 1{\sigma }_{g}1{\sigma }_{g}\rangle {| }^{2} & = & | \langle 1{s}_{a}1{s}_{b}| 1{\sigma }_{g}1{\sigma }_{g}\rangle {| }^{2}\\ =\,| \langle 1{s}_{b}1{s}_{a}| 1{\sigma }_{g}1{\sigma }_{g}\rangle {| }^{2} & = & | \langle 1{s}_{b}1{s}_{b}| 1{\sigma }_{g}1{\sigma }_{g}\rangle {| }^{2}=\displaystyle \frac{1}{4}.\end{eqnarray} \tag{ 2 }$$

Thus, the formation of a DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{g}\rangle$ corresponds to 50% probability of accessing a SSDCH molecular state, i.e. the $| 1{s}_{a}1{s}_{a}\rangle$ or the $| 1{s}_{b}1{s}_{b}\rangle$ state, and 50% probability of accessing a TSDCH molecular state, i.e. the $| 1{s}_{a}1{s}_{b}\rangle$ or the $| 1{s}_{b}1{s}_{a}\rangle$ state. One can show that the same probabilities are obtained for the DCH molecular state $| 1{\sigma }_{u}1{\sigma }_{u}\rangle$.

Next, we show that the DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{u}\rangle$ corresponds to different probabilities of accessing a SSDCH molecular state versus a TSDCH molecular state depending on the spin of the state. Denoting by S and T the spatial part of a singlet or triplet DCH molecular state, respectively, we obtain

$$\begin{eqnarray}&&| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S/T}=\displaystyle \frac{1}{\sqrt{2}}(| 1{\sigma }_{g}\rangle | 1{\sigma }_{u}\rangle \pm | 1{\sigma }_{u}\rangle | 1{\sigma }_{g}\rangle ).\end{eqnarray} \tag{ 3 }$$

Expressing the spatial part of the DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{T}$ in terms of orbitals localized on the atomic sites, we obtain

$$\begin{eqnarray}&&| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{T}=\displaystyle \frac{1}{\sqrt{2}}(| 1{s}_{b}\rangle | 1{s}_{a}\rangle -| 1{s}_{a}\rangle | 1{s}_{b}\rangle ),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&| \langle 1{s}_{a}1{s}_{b}{| }^{T}| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{T}{| }^{2}=1.\end{eqnarray} \tag{ 5 }$$

and, thus, this state corresponds to 100% probability of occupying a TSDCH molecular state. Similarly, expressing the DCH molecular state $| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S}$ in terms of orbitals localized on the atomic sites, we obtain:

$$\begin{eqnarray}&&| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S}=\displaystyle \frac{1}{\sqrt{2}}(| 1{s}_{a}\rangle | 1{s}_{a}\rangle -| 1{s}_{b}\rangle | 1{s}_{b}\rangle ),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&| \langle 1{s}_{a}1{s}_{a}| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S}{| }^{2}=| \langle 1{s}_{b}1{s}_{b}| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S}{| }^{2}=\displaystyle \frac{1}{2}\end{eqnarray} \tag{ 7 }$$

and, thus, this state corresponds to 100% probability of occupying a SSDCH molecular state. Taking into account that a DCH molecular state has 75% probability to be in the $| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{T}$ state and 25% to be in the $| 1{\sigma }_{g}1{\sigma }_{u}{\rangle }^{S}$ state, based on the multiplicities of the singlet and triplet states [30], it follows that the DCH state $| 1{\sigma }_{g}1{\sigma }_{u}\rangle$ has 75% probability to access a TSDCH molecular state and 25% probability to access a SSDCH molecular state. The above probabilities are incorporated at every time step of our computations in order to calculate the probability of pathways that access TSDCH and SSDCH molecular states during the interaction of N₂ with an FEL laser pulse.

Using the molecular and atomic auger and photo-ionization yields, we plot in figures 1(a), (b) the electron spectra generated by an FEL pulse with 525 eV (1100 eV) photon energy, with a full width at half maximum (FWHM) pulse duration of 4 fs and with a peak intensity of 10¹⁷ W cm⁻² (10¹⁸ W cm⁻²). Our calculations, which treat nuclear motion only through rate equations, are more accurate for short duration and high intensity laser pulses, since the molecular transitions for these parameters take place at small internuclear distances [12]. Moreover, a pulse duration of 4 fs is accessible experimentally [31] and peak intensities of 10¹⁷ W cm⁻² (10¹⁸ W cm⁻²) result in large contributions of DCH states to electron spectra. We find that for both FEL pulses the transitions that involve a $| 1{\sigma }_{g}1{\sigma }_{u}\rangle$ state or a TCH molecular state can not be easily discerned. The reason for this, is that these latter transitions have very small probability or that their energy is very close to the energies corresponding to other transitions. In contrast, we find that the transition involving a $| 1{\sigma }_{g}1{\sigma }_{g}\rangle$ or a $| 1{\sigma }_{u}1{\sigma }_{u}\rangle$ state can be easily discerned at 55 eV kinetic energy for the 525 eV FEL pulse and at 630 eV for the 1100 eV FEL pulse. However, the transition involving a $| 1{\sigma }_{g}1{\sigma }_{g}\rangle$ or a $| 1{\sigma }_{u}1{\sigma }_{u}\rangle$ state has 50% probability to involve a TSDCH molecular and a SSDCH molecular state. Therefore, in the current formulation, we can not determine from the electron spectra whether a TSDCH and/or a SSDCH molecular state is formed.

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In figure 2, we compare our results for the final atomic ion yields when N₂ interacts with an FEL pulse with computations that use atomic instead of molecular orbitals [16] and with experimental results [31]. To compare with experiment, we take the FEL flux to be given by

$$\begin{eqnarray}&&{J}({x},{y},{t})=\rho ({x},{y}){{\rm{\Gamma }}}_{\mathrm{ph}}({t}),\end{eqnarray} \tag{ 8 }$$

where the transverse beam profile is given by

$$\begin{eqnarray}&&\rho ({x},{y})=\displaystyle \frac{4\mathrm{ln}2}{\pi {\rho }_{{x}}{\rho }_{{y}}}{{\rm{e}}}^{-4\mathrm{ln}2[{({x}/{\rho }_{{x}})}^{2}+{({y}/{\rho }_{{y}})}^{2}]}\end{eqnarray} \tag{ 9 }$$

with ρ_x = 2.2 μm and ρ_y = 1.2 μm in accord with the experimental parameters in [31]. The temporal profile for the rate of photons is given by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{ph}}({t})={{\rm{\Gamma }}}_{\mathrm{ph},0}{{\rm{e}}}^{-4\mathrm{ln}2{({t}/\tau )}^{2}},\end{eqnarray} \tag{ 10 }$$

where τ is the FWHM duration of the pulse and Γ_ph,0 is the maximum rate of photons

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{ph},0}=2\sqrt{\displaystyle \frac{\mathrm{ln}2}{\pi }}\displaystyle \frac{{{n}}_{\mathrm{ph}}}{\tau }.\end{eqnarray} \tag{ 11 }$$

${{n}}_{\mathrm{ph}}=\tfrac{{{E}}_{{P}}}{\omega }$ is the number of photons, each with energy ω, in a pulse with energy ${{E}}_{{P}}$. We note that, when we compare with experimental results [31], we also multiply the pulse energy by a factor which accounts for the photon beam transport losses.

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We obtain the final atomic ion yields by computing the photon flux at different grid points (x, y), solving the rate equations at each grid point and finally integrating over an area of 10 μm × 10 μm. We normalize so that the yields of the final atomic ions sum up to 1. When all pathways are accounted for, our results, displayed in figure 2 (red color) compare very well with the experimental results [31] (black color). We also compute the atomic ion yields when we account only for pathways that go through SCH molecular states or through SCH plus TSDCH molecular states. We find that in all these cases our results agree well with the theoretical results in [16]. As expected, the more pathways we exclude, the higher the yield is of the lower charged atomic ion. Indeed, when pathways that access solely SCH molecular states are accounted for, we find that the atomic ion yield for the ${{\rm{N}}}^{+}$ state is the highest. Moreover, as expected, we find that for longer FEL pulse durations the yields for higher charged atomic ions have larger values, since the number of inner-shell electrons ionizing by single-photon absorption increases.

In figure 3, for different FEL pulses, we compute the final atomic ion yields as well as the contribution of SCH, TSDCH, SSDCH and TCH molecular states to each of the final atomic ion yields. In figures 3(a) and (c), we find that for short duration of 4 fs FWHM and high intensity FEL pulses, 49% and 56% of all pathways contributing to all final atomic ion yields are pathways that have accessed TSDCH and TCH molecular states. Moreover, we find that the contribution of pathways that access TSDCH and TCH molecular states increases for higher charged atomic ion states. The reason for this, is that the contribution of pathways where two single-photon ionizations take place sequentially, i.e. before an auger process takes place following the first single-photon ionization, is larger for atomic ions N³⁺ and N⁴⁺ compared to N²⁺. For instance, the contribution of pathways that have accessed SSDCH, TSDCH and TCH molecular states account for roughly 90% of the N⁵⁺ yield for the short duration and intense FEL pulses, see figures 3(a) and (c). For the long duration of 80 fs FWHM and small intensity FEL pulses, we find that no more than 10% of all pathways contributing to the atomic ion yields are pathways that have accessed SSDCH, TSDCH and TCH molecular states. In addition, the atomic ion yields of the higher charged states have much larger values for the 80 fs pulse rather than for the 4 fs FWHM FEL pulse. This is reasonable since for the long duration FEL pulse more single-photon ionization processes take place leading to the formation of higher charged atomic ions.

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In figure 4, we plot the population of pathways that accesses SCH, SSDCH, TSDCH and TCH molecular states as a function of intensity for a 4 fs FEL pulse for 525 eV photon energy (a) and for 1100 eV photon energy (b). We find that for the 525 eV (1100 eV) FEL pulse most of the population accesses multiple-core-hole molecular states for intensities above 10¹⁷ W cm⁻² (10¹⁸ W cm⁻²). The intensity is higher for the higher photon energy pulse since the single-photon ionization cross sections are higher for the smaller photon energy FEL pulse. Moreover, we find that the contribution of TCH molecular states compared to TSDCH molecular states is higher for the 1100 eV rather than for the 525 eV FEL pulse for high intensities. The reason for this, is that more molecular states are energetically accessible with the 1100 eV photon energy FEL pulse.

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