Cascade Mechanisms

A brief summary of the cascade processes in the exotic atoms with

is given in Table 1. The radiative de-excitation and the nuclear absorption (in hadronic atoms) do not depend on experimental conditions directly. All other processes occur in collisions with surrounding atoms and their rates are proportional to the hydrogen density and usually depend on energy.

At least tree cascade mechanisms are essential for the basic understanding of the atomic cascade [1]: the radiative transitions, the external Auger effect, and the Stark mixing. In this paper, the cascade models, which include these three mechanisms only, will be called the minimal cascade model¹ (MCM).

Table: Cascade processes in exotic atoms with

and their energy dependence.

Mechanism	Example	-dependence	Refs.
Radiative	$(\mu p)_i \to (\mu p)_f + \gamma$	none	see [1]
External Auger effect	$(\mu p)_i +{\rm H}_2 \to (\mu p)_f + e^- +{\rm H}_2^+$	weak	[1,5]
Stark mixing	$(\mu p)_{nl} + {\rm H} \to (\mu p)_{nl^\prime} + {\rm H}$	moderate	[1,6,7,8,9,10,11]
Elastic scattering	$(\mu p)_{n} + {\rm H} \to (\mu p)_{n} + {\rm H}$	strong	[11,12,13,14,15]
Coulomb transitions	$(\mu p)_{n_i} + p \to (\mu p)_{n_f} + p$ ,	strong	[16,17,18,19,20,21,22]
Transfer (isotope exchange)	$(\mu p)_{n} + d \to (\mu d)_{n} + p$	strong	[23,24,25,26]
Absorption	$(\pi^-p)_{nS} \to \pi^0+n,\ \gamma+n$	none	see [1]

**Figure:** The rates of (a) radiative and (b) Auger de-excitation (LHD) in muonic hydrogen.
$\begin{figure} \mbox{\hspace{20mm} (a) \hspace{60mm} (b)}\\ [-\baselineskip] \mb... ...pRLegoBW.eps}}%\mbox{\epsfysize =6cm\epsffile {mupALegoBW.eps}} } \end{figure}$

Figure 1 demonstrates the

-dependence of the total radiative and Auger de-excitation rates for muonic hydrogen. The main features of these de-excitation mechanisms were discussed in [1,2].

The Auger rates calculated in the Born approximation (Fig. 1b) are energy independent. The eikonal approximation [5] predicts a rather weak energy dependence, with the results being very close to the ones in the Born approximation for $n\leq 6$ and for a kinetic energy of the order of 1 eV. The initial and final state interactions in the Auger transitions were discussed in [17], however, no detailed calculations have been done.

The Stark mixing corresponds to transitions among the

-sublevels with the same

. It is a very fast collisional process because the exotic atoms with

are small and electroneutral and have no electrons, so that they can easily pass through the regions of the strong electric field inside ordinary atoms. When the Stark mixing rate is much larger than all other transition rates, the statistical population of the

-sublevels is determined by the principle of detailed balance.

**Figure:** The effective (statistical average) rates for (a) muonic hydrogen [4] and (b) pionic hydrogen [29] in liquid hydrogen. The *effective* absorption rates for the $\pi ^-p$ defined as in [1] and the deceleration rates correspond to the kinetic energy $E=2\;$ eV (solid line) and $E=30\;$ eV (dotted line). The Coulomb de-excitation rates for $E=2\;$ eV (solid line) and $E=0.04\;$ eV (dashed line) are from [20,21].
$\begin{figure} \mbox{\hspace{20mm} (a) \hspace{60mm} (b)}\\ [-\baselineskip] \mb... ...mupRASBW.eps}}%\mbox{\epsfysize =6cm\epsffile {pipRASDCBW.eps}} } \end{figure}$

The relative importance of the various cascade processes in muonic and pionic hydrogen is demonstrated in Figs. 2a,b. The cascade models of the exotic atoms with

are listed in Table 2; they are of two types. In one group, there are various implementations of the MCM [1,7,8,10] where the kinetic energy is assumed to be constant through the whole cascade. The other group [4,29,30,31] consists of detailed kinetics models which take the energy evolution during cascade into account by explicit treatment of acceleration and deceleration mechanisms.

Table: The cascade processes included in various models of the lightest exotic atoms in addition to the MCM containing the radiative and Auger de-excitation and the Stark mixing.

Model	Systems	Coulomb	Elastic	-evolution
Leon, Bethe (1962) [1]	$\pi ^-p$ ,
Borie, Leon (1980) [7]	$\mu p$ , $\pi ^-p$ , , $\bar{p}p$
Markushin (1981) [8]	$\mu p$ , $\mu d$
Reinfenröter et al. (1988) [28]	$\bar{p}p$
Czaplinski et al. (1990) [27]	$\mu p$ , $\mu d$
Markushin (1994) [4]	$\mu p$ , $\mu d$
Czaplinski et al. (1994) [26]	$\mu p$ , $\mu d$
Aschenauer et al. (1995) [29,30]	$\pi ^-p$
Aschenauer, Markushin (1997) [31]	$\mu p$ , $\mu d$
Terada, Hayano (1997) [10]	$\pi ^-p$ , , $\bar{p}p$