Spatial Correlations in Finite Samples Revealed by Coulomb Explosion

We demonstrate that fast removal of many electrons uncovers initial correlations of atoms in a finite sample through a pronounced peak in the kinetic-energy spectrum of the exploding ions. This maximum is the result of an intricate interplay between the composition of the system from discrete particles and its boundary. The formation of the peak can be described analytically, accounting for correlations beyond a mean-field reference model. It can be experimentally detected with short and intense light pulses from 4th-generation light sources.

Figure 1

Ion-energy spectra as obtained from an exploding LJ cluster with 1000 particles for various pulse durations $T$ given in units of the expansion time $t_{\star }$, cf. Eq. (1). Energy is given in terms of $E_{\star }$ the highest energy for homogeneously charged sphere, cf. Eq. (3b). The dashed line marks the surface energy of a sphere, which is charged homogeneously according to a Gaussian pulse (see text).

Figure 2

Peak height of kinetic-energy spectra for ions shown in Fig. 1 for different pulse lengths $T/t_{\star }$ (a). The two arrows indicate the pulse lengths $T=t_{\star }/20$ and $T=20t_{\star }$ for which the evolution of the spectra $dP/dE$ are shown in (b) and (c), respectively, as a function of energy and time $\xi \in [0,1]$. The time variable $\xi =E(t)/E(\infty )$ describes how far the Coulomb explosion has evolved in terms of the kinetic energy $E(t)$ of an expanding homogeneously charged sphere.

Figure 3

Spectra from numerical propagation of systems with $N=100$, 1000, and 10 000 ions, respectively, are shown by orange or gray-shaded areas (a)–(c). The mean-field model as given by Eq. (3) is shown with dashed lines. Considering a correlation hole in the mean-field description by using Eq. (8) in Eq. (2) yields the spectrum shown as solid line (b). Additionally the pair-correlation function $g(r)$ according to Eq. (4) is shown for the cluster with $N=1000$ individual ions (d).

Figure 4

Sketch of the integration assuming a correlation hole with radius $\delta$ around the test particle at distance $r$. We show two situations, where the correlation hole is inside the bulk ($r_1$) and at the surface ($r_2$), respectively. The test particle force is obtained by integration over all shells (indicated by green or gray lines) with some of them being “open” (thick green or gray line, see also inset) due to the correlation hole. The force on a test particle as a function of the distance $r$ (a) due to an “open” charged spherical shell with radius $r$ according to Eq. (6) and (b) integrated over all shells of the charged sphere of radius $R$ according to Eq. (7).

Figure 5

Energy spectra as obtained from an exploding cluster with 1000 particles for various cluster edges $a$ (implemented with Fermi distribution function, see text), which is given in units of the correlation hole radius $\delta$. Energy is given in terms of $E_{\star }$, cf. Eq. (3b).