Charging and ion ejection dynamics of large helium nanodroplets exposed to intense femtosecond soft X-ray pulses

Abstract

Ion ejection from charged helium nanodroplets exposed to intense femtosecond soft X-ray pulses is studied by single-pulse ion time-of-flight (TOF) spectroscopy in coincidence with small-angle X-ray scattering. Scattering images encode the droplet size and absolute photon flux incident on each droplet, while ion TOF spectra are used to determine the maximum ion kinetic energy, $E_{\text {kin}}$, of $\hbox {He}_{j}^{+}$ fragments (j = 1–4). Measurements span $\hbox {He}_N$ droplet sizes between $N\sim 10^{7}$ and $\sim 10^{10}$ (radii $R_0$ = 78–578 nm), and droplet charges between $\sim 9\times 10^{-5}$ and $\sim 3\times 10^{-3}$ e/atom. Conditions encompass a wide range of ionization and expansion regimes, from departure of all photoelectrons from the droplet, leading to pure Coulomb explosion, to substantial electron trapping by the electrostatic potential of the charged droplet, indicating the onset of hydrodynamic expansion. The unique combination of absolute X-ray intensities, droplet sizes, and ion $E_{\text {kin}}$ on an event-by-event basis reveals a detailed picture of the correlations between the ionization conditions and the ejection dynamics of the ionic fragments. The maximum $E_{\text {kin}}$ of He$^{+}$ is found to be governed by Coulomb repulsion from unscreened cations across all expansion regimes. The impact of ion-atom interactions resulting from the relatively low charge densities is increasingly relevant with less electron trapping. The findings are consistent with the emergence of a charged spherical shell around a quasineutral plasma core as the degree of ionization increases. The results demonstrate a complex relationship between measured ion $E_{\text {kin}}$ and droplet ionization conditions that can only be disentangled through the use of coincident single-pulse TOF and scattering data.

Appendices

Appendix A: Simulation of cluster charge for partially frustrated ionization

As described in Sect. 4.1 of the main text, once the Coulomb potential (Eq. 4 of the main text) around $r = 0$ becomes sufficiently deep (i.e., the potential energy reaches about 813 eV) at some time $t = t_0$ during the X-ray-pulse, photoelectrons start to get trapped near the droplet center [17]. For $t > t_0$, continued ionization further deepens the potential and extends the range of frustrated ionization to larger radii $r_{\text {frust}}(t) > 0$. At any time $t = t_i > t_0$ during the X-ray pulse, electron trapping within the region $r \le r_{\text {frust}}(t_i)$ leads to partial screening of the ionic background, which in turn affects the trapping potential for the remainder of the pulse, i.e, for all $t > t_i$. This mutual dependence and dynamic evolution of the droplet charge state and electron trapping potential requires a time-dependent modeling of the charging/trapping dynamics to estimate the total number of trapped electrons as a result of partial frustration. To this end, an iterative, finite time step algorithm is implemented to track the evolution of the Coulomb potential within the droplet as ionization proceeds. In this model, it is assumed that trapped electrons concentrate in the deepest part of the Coulomb potential near the droplet center and efficiently screen ions generated in this region throughout the X-ray pulse. This description leads to charge distribution corresponding to a homogeneously charged spherical shell surrounding a quasi-neutral core [17]. As such, with the onset of frustration, it becomes more accurate to model the cluster Coulomb potential as that of a charged spherical shell of finite thickness:

$$\begin{aligned} \begin{matrix} V(r)= \left\{ {\begin{array}{llllll} -\frac{\rho _{\text {He}^+}}{2\epsilon _0}(R_0^2 - R_{el}^2) ,&{}\quad r \le R_{el}\\ -\frac{\rho _{\text {He}^+}}{2\epsilon _0}(R_0^2 - \frac{2R_{el}^3}{3r}-\frac{r^2}{3}) ,&{}\quad R_{el}< r < R_0 \\ -\frac{e}{4\pi \epsilon _0} \frac{N_{\text {eff}}}{r} ,&{}\quad r\ge R_0 \\ \end{array} } \right. \end{matrix} \end{aligned}$$

(12)

Here, e is the elementary charge, $\epsilon _0$ is the permittivity of free space, r corresponds to the distance from the droplet’s center, $R_0$ and $R_{\text {el}}$ are the radii of the droplet and quasineutral core, respectively, $\rho _{\text {He}^+}$ is the charge density of ions in the charged spherical shell, and $N_{\text {eff}}$ is the net cluster charge. Since we assume a homogeneous distribution of ionization events, the ion charge density is taken to be $\rho _{\text {He}^+}= e\cdot N_{\text {ion}}/(\frac{4}{3}\pi R_0^3)$. With continued frustrated ionization, the quasi-neutral core region expands while the thickness of the charged shell is reduced.

At the start of the simulation, the total charge is set such that the Coulomb potential is sufficiently deep to trap ionized electrons at the center of the droplet. At each step, the total charge of the droplet is increased by $+1e$, $\rho _{\text {He}^+}$, $R_{\text {el}}$ and $N_{\text {eff}}$ are updated, and the Coulomb potential is calculated as described in Eq. 12 in order to calculate the trapping threshold radius, $r_{\text {frust}}(t_i)$. The corresponding (fractional) increase in the trapped electrons at each step, $\varDelta N_{\text {trap}}$, is taken to be the product of the added charge density, $\rho ^+$, and the volume within the trapping threshold radius:

$$\begin{aligned} \varDelta N_{\text {trap}}(t_i)=\rho _+\frac{4}{3}\pi r_{\text {frust}}^3(t_i). \end{aligned}$$

(13)

The simulation progresses until the total charge equals $N_{\text {ion}}$, and the total number of trapped electrons is obtained by summing over the trapped electrons of all steps, leading to: $N_{\text {eff}}=N_{\text {ion}}-N_{\text {trap}}= N_{\text {ion}}-\sum _{i}^{}\varDelta N_{\text {trap}}(t_i)$.

The results of the simulation are also used to determine at which value of $\alpha$ full frustration is achieved. Note that $\alpha$ corresponds to the ratio of all photogenerated ions, $N_{\text {ion}}$, including both screened and unscreened ions, to the net charge $N_{\text {frust}}$ of unscreened ions needed for full frustration. Thus, full frustration does not correspond to $\alpha = 1$. Instead, $N_{\text {ion}}$ must be greater than $N_{\text {frust}}$, so that $N_{\text {eff}}$ = $N_{\text {frust}}$. Using the simulation, the net charge and surface Coulomb potential is calculated for each hit, based on the size and flux extracted from the corresponding scattering image. Within the dataset herein, $\alpha \sim 2.5$ is the threshold at which the surface Coulomb potential reaches 813.4 eV, the threshold for electron trapping throughout the droplet.

Fig. 7

Results from Fig. 4 in a log–log representation. The slope of a linear fit indicated by the blue line is 0.47, indicating that $E^{\max }_{\text {kin}}/U_{\text {Coul}} \propto \sqrt{\alpha }$

Appendix B: Scaling of $E^{\max }_{\text {kin}}/U_{\text {Coul}}$ with $\alpha$

Presenting $E^{\max }_{\text {kin}}/U_{\text {Coul}}$ plotted against $\alpha$ with log–log scaling elucidates a linear trend, with a slope 0.47, as shown in Fig. 7. This indicates that $E^{\max }_{\text {kin}}/U_{\text {Coul}} \propto \sqrt{\alpha }$. As such, we use this scaling to plot a trend line in Fig. 4 of the main text. Analysis of this relationship is beyond the scope of this work.

Fig. 8

Thickness of the shell of unscreened ions as a function of the frustration parameter

Appendix C: Shell Thickness of Unscreened Ions

Unscreened cations are expected to be homogeneously distributed below the onset of frustration $(\alpha < 0.67)$ and begin to localize towards the surface of the charged droplet with increasing frustration. As derived from Eq. 11 in the main text, the shell thickness $\delta _{\text {shell}}$ can be expressed as

$$\begin{aligned} \delta _{\text {shell}}=R_0[1-(1-\frac{N_{\text {eff}}}{N_{\text {ion}}})^{1/3}], \end{aligned}$$

(14)

where $R_0$ is the droplet radius, $N_{\text {ion}}$ is the number of ions from photoionization, and $N_{\text {eff}}$ is the net cluster charge as defined in Sect. 5.1 of the main text. In Fig. 8, the shell thickness occupied by unscreened ions is plotted as a function of the frustration parameter $\alpha$.