Equilibrium charge-state distributions for heavy ions exiting carbon foils

Abstract

It has been recently shown that equilibrium charge-state distributions (ECSD's) of heavy ions exiting carbon foils are well fitted by the ${\ensuremath{\chi}}^{2}$ model, the Gaussian model, and the reduced ${\ensuremath{\chi}}^{2}$ model in the low-, intermediate-, and high-velocity regions, respectively. A qualitative interpretation of these statistical distributions is given in the present work, and it is pointed out that the reduced ${\ensuremath{\chi}}^{2}$ model is mathematically identical to the ${\ensuremath{\chi}}^{2}$ model if the charge state $i$ is replaced by $Z\ensuremath{-}i$($Z$ is the projectile atomic number) in the probability density function. This equivalence is physically explained by the existence of a charge-state limitation in these two velocity regions. The author discusses also the validity of semiempirical relations giving the mean charge $\overline{i}$ and the standard deviation of the charge $s$ in the different velocity domains. At sufficiently high velocities $\frac{E}{M}\ensuremath{\gtrsim}100$ keV ${\mathrm{amu}}^{\ensuremath{-}1}$, $Z\ensuremath{\lesssim}20$, experimental data available seem to indicate that $\overline{i}$ and $s$ depend monotonically on $Z$ and $v$, but this is not the case at lower velocities where shell effects play a major role. Finally, the author analyzes ECSD's observed for heavy ions, $59\ensuremath{\lesssim}Z\ensuremath{\lesssim}82$, at low energies $\frac{E}{M}\ensuremath{\lesssim}7$ keV ${\mathrm{amu}}^{\ensuremath{-}1}$) and finds that these ECSD's are also very well fitted by the ${\ensuremath{\chi}}^{2}$ model proposed previously for lighter ions, $Z\ensuremath{\lesssim}26$, at $\frac{E}{M}\ensuremath{\lesssim}20$ keV ${\mathrm{amu}}^{\ensuremath{-}1}$. However, for these heavy ions, ECSD's are already Gaussian for $\frac{E}{M}\ensuremath{\gtrsim}5$ keV ${\mathrm{amu}}^{\ensuremath{-}1}$.