Pionic Atoms

A pionic atom is formed when a negative pion is stopped in matter and is captured by an atom. The incident pion slows down by successive electromagnetic interactions with the electrons and nuclei, and when it reaches the typical velocity of atomic electrons, the pion is captured by ejecting a bound electron from its Bohr orbit (14). The typical principal quantum number of the captured pions is estimated as $n \sim (m_{\pi}/m_e)^{\frac{1}{2}} \sim 15$.

Subsequent stepwise de-excitation, mostly by Auger and $x$-ray emissions, occur until the captured pion is in a low-lying bound state. Auger emission dominates the de-excitation from the higher $n$-states. In the Auger process ($\Delta~n~=~1$, $\Delta~l~=~0$), as the pion de-excites the Auger electron escapes and carries off the excess energy. For lower $n$-states, the $x$-ray cascade ($E1$ electric dipole emission, $\Delta l = 1$) predominates.

Table 2.1: Energy eigenfunctions $u_{n l m}(r,\theta ,\phi )$ for a hydrogenic atom for values of $n$= 1, 2 . $n, l, m$ are the principal, orbital and magnetic quantum numbers respectively, and $r_o$ is the Bohr radius, given by $h^2\varepsilon_o/\pi m_{\pi} e^2$ for the pionic hydrogen atom.

Quantum numbers	Energy eigenfunctions for hydrogen atom
	$u_{n l m}(r,\theta ,\phi )$
$(n,l,m)$
$s$ states
$n=1$, $l=0$, $m=0$	$\frac{\displaystyle 1}{\displaystyle \sqrt{\pi r_{o}^3}} \exp(-r/r_o)$
$n=2$, $l=0$, $m=0$	$\frac{\displaystyle 1}{\displaystyle \sqrt{32 \pi r_{o}^3}} (2 - r/r_o) \exp(-r/2r_o)$
$p$ states
$n=2$, $l=1$, $m=0$	$\frac{\displaystyle 1}{\displaystyle \sqrt{32 \pi r_{o}^3}} (r/r_o) \exp(-r/2r_o) \cos\theta$
$n=2$, $l=1$, $m$ = $\pm 1$	$\frac{\displaystyle 1}{\displaystyle \sqrt{64 \pi r_{o}^3}} (r/r_o) \exp(-r/2r_o) \exp(\pm \phi)\sin\theta$

Following the atomic cascade process when size of the pionic orbit becomes smaller than the innermost electronic orbit around the nucleus, the pion is no longer screened by the remaining electrons of the atom, and the system becomes a hydrogen-like pionic atom. In hydrogen, when pion is captured replacing the atomic electron, the system is known as pionic hydrogen.

The hydrogen isotopes are unique in pion capture studies. A pionic hydrogen atom is neutral and so can pass freely through neighboring atoms where it experiences the strong electric field of the atomic interior. These fields perturb the pion cascade through Stark transitions ( $nl \rightarrow nl', l' \neq l$), forcing the $\pi ^-$ into $s$ orbits with high $n$. At the origin of the pionic atom $r$ = 0, the pionic wave function vanishes for the $l \geq 1$ orbitals, while for the $s$ states it is non-vanishing (Table 2.1). Thus the $s$ states have nonzero overlap over the nucleonic wave function at the origin resulting in the enhanced $nS$ absorption width. As shown in Table 2.2, the capture begins to be important at $n$=5 and is dominated by $n$=4, and 3 states (15).

Table: $nS$-state pion capture fractions in hydrogen (15)

Principal quantum number	$nS$-state capture fraction
$n$
7	0.003
6	0.013
5	0.09
4	0.44
3	0.39
2	0.04

Table 2.3: Pion capture fractions in light nuclei $^{6}$Li and $^{12}$C (16)

	$^{6}$Li		$^{12}$C
$n$	$l$ = 0	$l$ = 1	$l$ = 0	$l$ = 1
5	0.006 $\pm$ 0.004	0.005 $\pm$ 0.005	0.0035 $\pm$ 0.0015	0.038 $\pm$ 0.007
4	0.016 $\pm$ 0.006	0.035 $\pm$ 0.001	0.0025 $\pm$ 0.0015	0.082 $\pm$ 0.008
3	0.022 $\pm$ 0.007	0.130 $\pm$ 0.006	0.0015 $\pm$ 0.0008	0.145 $\pm$ 0.002
2	0.019 $\pm$ 0.005	0.430 $\pm$ 0.074	0.0030 $\pm$ 0.0010	0.630 $\pm$ 0.500
1	0.335 $\pm$ 0.065		0.0600 $\pm$ 0.0200

The general trend for light nuclei is that, only $s$ and $p$ orbit captures contribute significantly, and either $1S$, and/or $2P$ state capture (16) dominates. As seen from Table 2.3, the $p$ state capture rates ($2P$ 63%, $3P$ 15%, $4P$ 8%) dominate considerably over the $s$ state capture rate ($1S$ 6%) for pion capture in $^{12}$C nuclei.