2 Ion expansion in the inner zone

Initially, all the ions are uniformly distributed inside a sphere of radius $R$ . As the charge density is uniform, the radial electric field is linear. Therefore, there will be a uniform expansion for both slow and fast ions, and consequently the charge density will remain spatially constant inside the sphere $S_{s}$ , producing again a linear behaviour of the electric field. From this consideration, one can infer that the electric field inside $S_{s}$ remains a linear function of the radius $r$ at any time:

(2.1) $$\begin{eqnarray}E(r,t)=A(t)r\quad \text{for }r\leqslant R_{s}(t).\end{eqnarray}$$

Under this assumption, the equations of motion for the fast and the slow ions inside $S_{s}$ can be written in a simple way. The uniform expansion can be described by introducing two functions, ${\it\xi}_{s}(t)$ and ${\it\xi}_{f}(t)$ , such that slow ions and fast ions initially at $r=r_{0}$ are moved at time $t$ to $r=r_{0}{\it\xi}_{s}(t)$ and $r=r_{0}{\it\xi}_{f}(t)$ , respectively. Moreover, the initial densities of the two species, $n_{s,0}$ and $n_{f,0}$ , evolve in time as

(2.2a,b ) $$\begin{eqnarray}n_{s}(t)={\displaystyle \frac{n_{s,0}}{{\it\xi}_{s}^{3}(t)}},\quad n_{f}(t)={\displaystyle \frac{n_{f,0}}{{\it\xi}_{f}^{3}(t)}}.\end{eqnarray}$$

Using Gauss’s law, the electric field inside the sphere $S_{s}$ can be written as:

(2.3) $$\begin{eqnarray}E(r,t)=\frac{4{\rm\pi}}{3}\left(\frac{q_{f}n_{f,0}}{{\it\xi}_{f}^{3}(t)}+\frac{q_{s}n_{s,0}}{{\it\xi}_{s}^{3}(t)}\right)r,\end{eqnarray}$$

which has the correct dependence on $r$ , as in (2.1). After introducing the expression (2.3) into the equations of motion for the fast and slow ions, by writing the acceleration of the ions of the two species as $r_{0}\,\text{d}^{2}{\it\xi}_{f}/\text{d}t^{2}$ and $r_{0}\,\text{d}^{2}{\it\xi}_{s}/\text{d}t^{2}$ , one finally obtains

(2.4) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle {\displaystyle \frac{\text{d}^{2}{\it\xi}_{f}}{\text{d}t^{2}}}={\displaystyle \frac{4{\rm\pi}}{3}}{\displaystyle \frac{q_{f}}{m_{f}}}\left({\displaystyle \frac{q_{f}n_{f,0}}{{\it\xi}_{f}^{3}}}+{\displaystyle \frac{q_{s}n_{s,0}}{{\it\xi}_{s}^{3}}}\right){\it\xi}_{f},\\ \displaystyle {\displaystyle \frac{\text{d}^{2}{\it\xi}_{s}}{\text{d}t^{2}}}={\displaystyle \frac{4{\rm\pi}}{3}}{\displaystyle \frac{q_{s}}{m_{s}}}\left({\displaystyle \frac{q_{f}n_{f,0}}{{\it\xi}_{f}^{3}}}+{\displaystyle \frac{q_{s}n_{s,0}}{{\it\xi}_{s}^{3}}}\right){\it\xi}_{s},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

in which $r_{0}$ does not appear.

Equations (2.4) can be rewritten in a more compact way, as

(2.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}ccc@{}}{\displaystyle \frac{\text{d}^{2}{\it\xi}_{f}}{\text{d}t^{2}}}={\it\nu}^{2}\left({\displaystyle \frac{{\it\alpha}}{{\it\xi}_{f}^{3}}}+{\displaystyle \frac{{\it\beta}(1-{\it\alpha})}{{\it\xi}_{s}^{3}}}\right){\it\xi}_{f}, & {\it\xi}_{f}(0)=1, & {\displaystyle \frac{\text{d}{\it\xi}_{f}}{\text{d}t}}(0)=0\\ {\displaystyle \frac{\text{d}^{2}{\it\xi}_{s}}{\text{d}t^{2}}}={\it\nu}^{2}{\it\gamma}\left({\displaystyle \frac{{\it\alpha}}{{\it\xi}_{f}^{3}}}+{\displaystyle \frac{{\it\beta}(1-{\it\alpha})}{{\it\xi}_{s}^{3}}}\right){\it\xi}_{s}, & {\it\xi}_{s}(0)=1, & {\displaystyle \frac{\text{d}{\it\xi}_{s}}{\text{d}t}}(0)=0\end{array}\right\} & & \displaystyle\end{eqnarray}$$

being

(2.6a−c ) $$\begin{eqnarray}{\it\nu}=\left[{\displaystyle \frac{4{\rm\pi}q_{f}^{2}(n_{f,0}+n_{s,0})}{3m_{f}}}\right]^{1/2},\quad {\it\alpha}={\displaystyle \frac{n_{f,0}}{n_{f,0}+n_{s,0}}},\quad {\it\beta}={\displaystyle \frac{q_{s}}{q_{f}}}.\end{eqnarray}$$

The quantity ${\it\nu}$ represents a characteristic frequency for the fast ion expansion, while ${\it\alpha}\in (0,1)$ is the fraction of fast ions in the cluster.

3 Expansion in the outer zone and shock shells

The system of equations (2.5) describes the dynamics of slow ions during the whole explosion and the motion of fast ions as long as they are inside $S_{s}$ . A fast ion, initially at $r=r_{0}\leqslant R$ , reaches the border of the inner region when $r_{0}{\it\xi}_{f}(t)=R{\it\xi}_{s}(t)$ . The time the ion crosses the sphere $S_{s}$ , which is indicated in the following as ${\it\tau}\!\left(r_{0}\right)$ , is obtained by solving the equation

(3.1) $$\begin{eqnarray}\frac{{\it\xi}_{s}({\it\tau})}{{\it\xi}_{f}({\it\tau})}=\frac{r_{0}}{R}.\end{eqnarray}$$

Considering a fast ion initially at $r_{0}$ , the electric field for $r\geqslant R_{s}$ can be written as $Q(r,t)/r^{2}$ , where $Q(r,t)$ is the charge in the sphere of radius $r$ at time $t$ ; as long as overtakings between fast ions do not occur, $Q(r,t)$ is the sum of the total charge of slow ions and the charge of the fast ions initially inside a sphere of radius $r_{0}$ :

(3.2) $$\begin{eqnarray}Q(r,t)=Q(r_{0},t)=\frac{4{\rm\pi}}{3}(q_{s}n_{s,0}R^{3}+q_{f}n_{f,0}r_{0}^{3}).\end{eqnarray}$$

Consequently, for $t>{\it\tau}(r_{0})$ , the equation of motion for a fast ion outside $S_{s}$ is

(3.3a−c ) $$\begin{eqnarray}m_{f}\frac{\text{d}^{2}r}{\text{d}t^{2}}=-\frac{\partial }{\partial r}\left(\frac{q_{f}Q(r_{0})}{r}\right),\quad r({\it\tau})={\it\xi}_{f}({\it\tau})r_{0},\quad \frac{\text{d}r}{\text{d}t}({\it\tau})=\frac{\text{d}{\it\xi}_{f}}{\text{d}t}({\it\tau})r_{0}.\end{eqnarray}$$

Equation (3.3) can be integrated in order to obtain the asymptotic kinetic energy of the fast ions, ${\it\epsilon}_{\infty }$ , as

(3.4) $$\begin{eqnarray}{\it\epsilon}_{\infty }=\frac{1}{2}m_{f}\left\{\frac{\text{d}{\it\xi}_{f}}{\text{d}t}({\it\tau}(r_{0}))r_{0}\right\}^{2}+\frac{q_{f}Q(r_{0})}{{\it\xi}_{f}({\it\tau})r_{0}}.\end{eqnarray}$$

The energy spectrum of the fast ions, ${\it\rho}_{{\it\epsilon}}$ , can be written as

(3.5) $$\begin{eqnarray}{\it\rho}_{{\it\epsilon}}={\displaystyle \frac{1}{N_{f}}}{\displaystyle \frac{\text{d}N_{f}}{\text{d}{\it\epsilon}_{\infty }}},\end{eqnarray}$$

where $N_{f}$ is the total number of fast ions, while $\text{d}N_{f}$ represents the number of ions having asymptotic energy in the interval $\text{d}{\it\epsilon}_{\infty }$ . Making use of (3.1) and (3.4), the spectrum can be expressed parametrically as a function of ${\it\tau}$ , as:

(3.6) $$\begin{eqnarray}{\it\rho}_{{\it\epsilon}}({\it\epsilon}_{\infty }({\it\tau}))={\displaystyle \frac{4{\rm\pi}r_{0}^{2}({\it\tau})n_{f,0}}{N_{f}}}{\displaystyle \frac{\text{d}r_{0}/\text{d}{\it\tau}}{\text{d}{\it\epsilon}_{\infty }/\text{d}{\it\tau}}}.\end{eqnarray}$$

As pointed out in a previous work (Li et al. Reference Li, Liu, Wang, Ni, Kim, Li and Xu2007), shock shells arise when the fraction of fast ions, ${\it\alpha}$ , is smaller than a critical value, ${\it\alpha}_{crit}$ . Typical phase-space evolutions of the fast ions are showed in figure 1 for ${\it\alpha}>{\it\alpha}_{crit}$ and in figure 2 for ${\it\alpha}<{\it\alpha}_{crit}$ . In particular, the curves in figure 2 show the typical wave-breaking behaviour that indicates the presence of a shock (Kaplan et al. Reference Kaplan, Dubetsky and Shkolnikov2003).

Figure 1.

Time evolution of the phase space distribution of fast ions in a mixture $\text{C}^{+}\text{H}^{+}$ with ${\it\alpha}=0.4$ , for $t$ in the range 15–43 fs. A cluster with ion density $n_{f,0}+n_{s,0}=10^{23}~\text{cm}^{-3}$ and $R=6.5$  nm has been considered.

Figure 2.

Time evolution of the phase space distribution of fast ions in a mixture $\text{C}^{+}\text{H}^{+}$ with ${\it\alpha}=0.2$ , for $t$ in the range 12–57 fs. A cluster with ion density $n_{f,0}+n_{s,0}=10^{23}~\text{cm}^{-3}$ and $R=6.5$  nm has been considered.

In fact, there is a simple way to discriminate Coulomb explosions without shocks from the ones in which this phenomenon appears without requiring a full numerical simulation of the explosion. Starting from (3.4), which in principle is valid only when shocks are absent, the derivative of ${\it\epsilon}_{\infty }$ with respect to $r_{0}$ is calculated. If the condition

(3.7) $$\begin{eqnarray}{\displaystyle \frac{\text{d}{\it\epsilon}_{\infty }}{\text{d}r_{0}}}>0\end{eqnarray}$$

is verified for every $r_{0}\in [0,R]$ , inner ions take a slower velocity respect to the ones in the outer zones and cannot reach them. In this case, no overtaking is present. Instead, if $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ is negative for some $r_{0}$ , in principle there will be overtakings. Of course, (3.4) is no longer valid in this case and it cannot be used to calculate the energy spectrum of fast ions, whose dynamics can be solved only numerically. Using this criterion, the existence of a limit value for ${\it\alpha}$ can be readily verified numerically. In general, if ${\it\alpha}\in ({\it\alpha}_{crit},1)$ the derivative $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ remains positive and no shock is formed; for ${\it\alpha}\in (0,{\it\alpha}_{crit})$ , $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ changes its sign, meaning that the fast particles overtake each other, giving rise to shock shells. The calculations show that ${\it\alpha}_{crit}$ depends only on the ratio between the ions charges and it is independent of their masses (Li et al. Reference Li, Liu, Wang, Ni, Kim, Li and Xu2007). In particular, ${\it\alpha}_{crit}=1/3$ for mixtures HD, HT, DT and  $\text{HC}^{+}$ .

4 Theoretical derivation of ${\it\alpha}_{crit}$

Starting with the case of no shock, by indicating with $Q_{s}$ and $Q_{f}$ the total charge of slow and fast ions, the dynamics of a fast ion initially at $r=r_{0}$ is governed by the Hamiltonian

(4.1) $$\begin{eqnarray}\mathscr{H}(r,p_{r};r_{0})={\displaystyle \frac{p_{r}^{2}}{2m_{f}}}+{\displaystyle \frac{q_{f}Q_{f}}{r}}\left(\frac{r_{0}}{R}\right)^{3}+q_{f}{\it\Phi}_{s}(r,R_{s}(t)),\end{eqnarray}$$

where ${\it\Phi}_{s}(r,{\it\rho})$ is the electrostatic potential in $r$ due to fixed ions uniformly distributed in a sphere of radius ${\it\rho}$ :

(4.2) $$\begin{eqnarray}\displaystyle {\it\Phi}_{s}(r,{\it\rho})=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{3Q_{s}}{2{\it\rho}}}-{\displaystyle \frac{Q_{s}r^{2}}{2{\it\rho}^{3}}}\quad & \text{for }r\leqslant {\it\rho},\\ {\displaystyle \frac{Q_{s}}{r}}\quad & \text{for }r\geqslant {\it\rho}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

In the limit $m_{s}\rightarrow +\infty$ , $R_{s}(t)$ is constant and equal to $R$ . Consequently, as $\mathscr{H}$ does not depend explicitly on time, it is a constant of motion, and the asymptotic energy ${\it\epsilon}_{\infty }$ expressed as a function of $r_{0}$ can be written immediately as

(4.3) $$\begin{eqnarray}{\it\epsilon}_{\infty }(r_{0})=q_{f}\left[{\displaystyle \frac{Q_{f}-Q_{s}/2}{R^{3}}}r_{0}^{2}+{\displaystyle \frac{3Q_{s}}{2R}}\right]\!,\end{eqnarray}$$

from which the energy spectrum can be readily calculated (Li et al. Reference Li, Liu, Wang, Ni, Kim, Li and Xu2007; Murakami & Mima Reference Murakami and Mima2009).

Figure 3.

Energy spectra of a mixture with slow ions at rest ( $m_{s}\rightarrow +\infty$ ) and hydrogen ions, for different value of ${\it\alpha}$ in the range $[0.2,0.5]$ . The same initial density and radius as in figure 1 has been considered. Curves a–c represent mixtures with ${\it\alpha}>{\it\alpha}_{crit}$ ; in the absence of shocks the numerical spectra correspond to the analytic curves. The limit case of ${\it\alpha}={\it\alpha}_{crit}$ is shown in curve d, where the energy spectra is singular. Curves e–g display situations with ${\it\alpha}>{\it\alpha}_{crit}$ , for which the spectra have been computed by means of the shell method.

Two situations can occur, depending on the sign of $Q_{f}-Q_{s}/2$ . If $Q_{f}>Q_{s}/2$ the ion velocity is an increasing function of $r_{0}$ , so overtakings do not occur and the analytic spectrum is correct, as shown in figure 3, curves a–c. When $Q_{f}<Q_{s}/2$ , according to (4.3), ${\it\epsilon}_{\infty }$ is a decreasing function of $r_{0}$ , but this implies that the ions closer to the centre of the cluster are bound to overtake the ions close to the border of the sphere. In other words, the hypothesis of no overtakings, and consequently (4.3), is no longer valid. In figure 3, curves e–g, the energy spectra have been calculated correctly by using the shell method. The case $Q_{f}=Q_{s}/2$ (figure 3, curve d) can be considered as a limit for $Q_{f}-Q_{s}/2\rightarrow 0^{+}$ and, consequently, (4.3) can still be used. In the general case, where the mass of slow ions is finite, in the Hamiltonian (4.1) the potential due to slow ions, ${\it\Phi}_{s}$ , is a function of time, as it depends on the frontline $R_{s}(t)$ , and $\mathscr{H}$ is no longer a constant of motion:

(4.4) $$\begin{eqnarray}{\displaystyle \frac{\text{d}\mathscr{H}}{\text{d}t}}={\displaystyle \frac{\partial \mathscr{H}}{\partial t}}=q_{f}{\displaystyle \frac{\partial {\it\Phi}_{s}}{\partial R_{s}}}{\displaystyle \frac{\text{d}R_{s}}{\text{d}t}}.\end{eqnarray}$$

In principle, the value of the asymptotic energy can be obtained by integrating (4.4) in time:

(4.5) $$\begin{eqnarray}{\it\epsilon}_{\infty }=\mathscr{H}(t\rightarrow +\infty )=\mathscr{H}(t=0)+q_{f}\int _{0}^{{\it\tau}(r_{0})}{\displaystyle \frac{\partial {\it\Phi}_{s}}{\partial R_{s}}}(r(t),R_{s}(t)){\displaystyle \frac{\text{d}R_{s}}{\text{d}t}}\,\text{d}t,\end{eqnarray}$$

where the upper integration limit can be set equal to ${\it\tau}(r_{0})$ , as for $t>{\it\tau}$ the ion is outside $S_{s}(t)$ and the term $\partial {\it\Phi}_{s}/\partial R_{s}$ vanishes. Noting that $\mathscr{H}(t=0)$ is equal to the asymptotic energy for fixed slow ions, (4.1), and expressing $r(t)$ and $R_{s}(t)$ as functions of ${\it\xi}_{f}$ and ${\it\xi}_{s}$ , one finally obtains

(4.6) $$\begin{eqnarray}{\it\epsilon}_{\infty }=q_{f}\left[{\displaystyle \frac{Q_{f}-Q_{s}/2}{R^{3}}}r_{0}^{2}+{\displaystyle \frac{3Q_{s}}{2R}}\right]-{\displaystyle \frac{3Q_{s}q_{f}}{2R}}I(r_{0}),\end{eqnarray}$$

where $I(r_{0})$ is defined as

(4.7) $$\begin{eqnarray}I(r_{0})=\int _{0}^{{\it\tau}(r_{0})}\left[1-\left({\displaystyle \frac{r_{0}{\it\xi}_{f}(t)}{R{\it\xi}_{s}(t)}}\right)^{2}\right]{\displaystyle \frac{{\it\xi}_{s}^{\prime }(t)}{{\it\xi}_{s}^{2}(t)}}\,\text{d}t.\end{eqnarray}$$

The quantity $I$ is always non-negative, and in fact, the motion of slow ions takes some energy away from the fast ones.

The condition $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}>0$ for $r_{0}\in [0,R]$ is now investigated. Starting from (4.7), one obtains:

(4.8) $$\begin{eqnarray}{\displaystyle \frac{\text{d}I}{\text{d}r_{0}}}={\displaystyle \frac{{\it\xi}_{s}^{\prime }({\it\tau})}{{\it\xi}_{s}({\it\tau})^{2}}}\left(1-{\displaystyle \frac{r_{0}^{2}{\it\xi}_{f}({\it\tau})^{2}}{R^{2}{\it\xi}_{s}({\it\tau})^{2}}}\right){\displaystyle \frac{\text{d}{\it\tau}}{\text{d}r_{0}}}-\int _{0}^{{\it\tau}(r_{0})}{\displaystyle \frac{2r_{0}{\it\xi}_{f}^{2}{\it\xi}_{s}^{\prime }}{R^{2}{\it\xi}_{s}^{4}}}\,\text{d}t.\end{eqnarray}$$

From this formula, considering that $\text{d}{\it\tau}/\text{d}r_{0}<0$ , one has

(4.9a,b ) $$\begin{eqnarray}{\displaystyle \frac{\text{d}I}{\text{d}r_{0}}}(r_{0})\leqslant 0\quad \forall r_{0}\in [0,R],\quad {\displaystyle \frac{\text{d}I}{\text{d}r_{0}}}(R)=0.\end{eqnarray}$$

Being

(4.10) $$\begin{eqnarray}{\displaystyle \frac{\text{d}{\it\epsilon}_{\infty }}{\text{d}r_{0}}}=2q_{f}{\displaystyle \frac{Q_{f}-Q_{s}/2}{R^{3}}}r_{0}-{\displaystyle \frac{3Q_{s}q_{f}}{2R}}{\displaystyle \frac{\text{d}I}{\text{d}r_{0}}},\end{eqnarray}$$

if $Q_{f}>Q_{s}/2$ the derivative $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ is always non-negative, as it is the sum of two non-negative quantities. Instead, if $Q_{f}<Q_{s}/2$ , $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ is the sum of a negative term and a non-negative term and, in principle, nothing can be said about the sign of $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}$ for a generic $r_{0}$ . However, $\text{d}I/\text{d}r_{0}$ vanishes for $r_{0}=R$ , and therefore $\text{d}{\it\epsilon}_{\infty }/\text{d}r_{0}(R)$ must be negative; consequently, in this case there are overtakings.

Figure 4.

Mean value of the asymptotic energy for different mixtures of carbon and hydrogen, as a function of the fraction of fast ions.

Figure 5.

Standard deviation of the energy spectra as a function of the fraction of fast ions, for different mixtures of carbon and hydrogen.

In summary, the condition $Q_{f}=Q_{s}/2$ is valid in any case and can be used in order to discriminate between explosions with and without shocks. This is the rigorous proof of a property of the Coulomb explosions of mixtures that was found numerically in a previous work (Li et al. Reference Li, Liu, Wang, Ni, Kim, Li and Xu2007). The condition $Q_{f}=Q_{s}/2$ provides the critical value of ${\it\alpha}$ :

(4.11) $$\begin{eqnarray}{\it\alpha}_{crit}={\displaystyle \frac{{\it\beta}}{2+{\it\beta}}},\end{eqnarray}$$

which depends only on the charge ratio.

5 Energy spectrum of the fast ions

The analysis performed for the simplified situation of $m_{s}\rightarrow +\infty$ suggest that fast ion spectra could be particularly narrow when ${\it\alpha}$ approaches ${\it\alpha}_{crit}$ . To confirm this, a full set of calculations has been carried out for the mixtures $\text{C}^{+}\text{H}^{+}$ , $\text{C}^{2+}\text{H}^{+}$ , $\text{C}^{3+}\text{H}^{+}$ and $\text{C}^{4+}\text{H}^{+}$ . Clusters with ion density $n_{f,0}+n_{s,0}=10^{23}~\text{cm}^{-3}$ and $R=6.5$ nm (as in figures 1–3) have been considered. In figures 4 and 5 the mean value of the energy and the energy spread have been calculated for each type of mixture for ${\it\alpha}\in (0,1)$ . Moreover, in figures 6–9 the energy spectra are reported for each mixture for ${\it\alpha}$ close to ${\it\alpha}_{crit}$ . All the calculations have been performed with the shell method. For the mixture $\text{C}^{4+}\text{H}^{+}$ the results presented here are in excellent agreement with previously published calculations (in particular, with the results reported in figure 2 of the paper by Murakami & Mima (Reference Murakami and Mima2009)).

Figure 6.

Energy spectra of a mixture $\text{C}^{+}\text{H}^{+}$ with different values of ${\it\alpha}$ in the range $[0.2,0.4]$ . The thick curve indicates the distribution with the minimum value of ${\it\sigma}_{{\it\epsilon}}$ .

Figure 7.

Energy spectra of a mixture $\text{C}^{2+}\text{H}^{+}$ with different values of ${\it\alpha}$ in the range $[0.3,0.5]$ . The thick curve indicates the distribution with the minimum value of ${\it\sigma}_{{\it\epsilon}}$ .

The plots of the mean ion energy, $\langle {\it\epsilon}_{\infty }\rangle =\int _{0}^{+\infty }{\it\epsilon}{\it\rho}_{{\it\epsilon}}({\it\epsilon})\,\text{d}{\it\epsilon}$ , as function of ${\it\alpha}$ , show a linear behaviour. In fact, the total kinetic energy of the fast ions, $\mathscr{E}_{f}$ , can be written as the difference between the potential energy at $t=0$ and the total kinetic energy of the slow ions

(5.1) $$\begin{eqnarray}\mathscr{E}_{f}={\mathcal{U}}(t=0)-\mathscr{E}_{s}={\displaystyle \frac{3(Q_{s}+Q_{f})^{2}}{2R}}-{\displaystyle \frac{3}{5}}N_{s}{\displaystyle \frac{m_{s}}{2}}[R{\it\xi}_{s}^{\prime }(+\infty )]^{2}\end{eqnarray}$$

and, therefore, it does not depend on the possible presence of shocks.

Figure 8.

Energy spectra of a mixture $\text{C}^{3+}\text{H}^{+}$ with different values of ${\it\alpha}$ in the range $[0.45,0.65]$ . The thick curve indicates the distribution with the minimum value of ${\it\sigma}_{{\it\epsilon}}$ .

Figure 9.

Energy spectra of a mixture $\text{C}^{4+}\text{H}^{+}$ with different values of ${\it\alpha}$ in the range $[0.7,0.9]$ . The thick curve indicates the distribution with the minimum value of ${\it\sigma}_{{\it\epsilon}}$ .

From the numerical solution of the system (2.4), it has been found that $[{\it\xi}_{s}^{\prime }(+\infty )]^{2}$ depends linearly on ${\it\alpha}$ , and using this result the average kinetic energy $\langle {\it\epsilon}_{\infty }\rangle =\mathscr{E}_{f}/N_{f}$ becomes a linear function of ${\it\alpha}$ .

The standard deviation of the energy of fast ions, ${\it\sigma}_{{\it\epsilon}}=[\int _{0}^{+\infty }({\it\epsilon}-\langle {\it\epsilon}_{\infty }\rangle )^{2}{\it\rho}_{{\it\epsilon}}\,\text{d}{\it\epsilon}]^{1/2}$ , presents a minimum for a value of ${\it\alpha}$ which is close to ${\it\alpha}_{crit}$ when $(m_{s}/q_{s})/(m_{f}/q_{f})\gg 1$ . In fact, for ${\it\alpha}\simeq {\it\alpha}_{crit}$ , the spectrum has a large plateau and a sharp peak. Probably, the plateau gives a non-negligible contribution to ${\it\sigma}_{{\it\epsilon}}$ , while for practical applications the presence of the peak and its energy spread are more important.

6 Summary

In this paper the dynamics of particles during the Coulomb explosion of ion mixtures has been investigated, with particular interest in the energy spectrum of the component with the larger charge-to-mass ratio. Exploiting the hypothesis of no overtakings among ions of the same species, and the new approach based on the linear behaviour for the electric field inside the inner sphere, $S_{s}$ , containing both slow and fast ions, analytical formulas describing the acceleration both inside and outside $S_{s}$ (where only fast ions are present), have been derived together with the energy spectrum. For the first time (at least to the authors’ knowledge), a simple way to deduce if overtakings between fast ions will take place during the expansion has been presented and discussed, confirming the existence, empirically deduced in a previous work by using numerical simulations (Li et al. Reference Li, Liu, Wang, Ni, Kim, Li and Xu2007), of a limit value for the mixture composition, ${\it\alpha}_{crit}$ , above which no shocks form. The analysis here proposed is based on the calculation of the derivative of ${\it\epsilon}_{\infty }$ with respect to the initial ion position $r_{0}$ : when this quantity is negative for some $r_{0}$ , the hypothesis of no overtaking is no longer valid. Moreover, making use of the Hamiltonian of fast particles and starting with the simple case where the mass of the slow species is considered infinite, a rigorous proof of the existence of ${\it\alpha}_{crit}$ , which is valid for every value of $m_{f}$ and $m_{s}$ , has been obtained. In the limit situation of $m_{s}\rightarrow +\infty$ , when ${\it\alpha}>{\it\alpha}_{crit}$ approaches the critical value, the energy spectrum becomes increasingly narrow, meaning that the ions tend to acquire very similar velocities, and for ${\it\alpha}={\it\alpha}_{crit}$ the spectrum becomes monoenergetic. On the other hand, the condition ${\it\alpha}={\it\alpha}_{crit}$ is the limit situation for the rising of shock shells. As most of these considerations are valid also for the general case in which $m_{s}$ is finite, one can conclude that the rising of shock shells is induced by a very narrow energy spectrum, and not vice versa.

In conclusion, conditions to obtain nearly monoenergetic ions from Coulomb explosions of heteronuclear clusters have been individuated and supported by a rigorous theoretical analysis. The analysis has been carried out supposing that the electrons are completely expelled from the cluster; also in order to explain recent experimental results (Hohenberger et al. Reference Hohenberger, Symes, Madison, Sumeruk, Dyer, Edens, Grigsby, Hays, Teichmann and Ditmire2005; Iwan et al. Reference Iwan, Andreasson, Bergh, Schorb, Thomas, Rupp, Gorkhover, Adolph, Möller and Bostedt2012), future investigation should take into proper account also the electron dynamics and the laser–cluster interaction in order to have a clearer picture of the phenomenon.