Yields of X-ray transitions in muonic and pionic hydrogen and corresponding distributions of kinetic energies.

The intensities of the X-ray yields in the $\pi ^{-}p$ and $\mu ^{-}p$ atoms calculated with the Monte Carlo code [62,56] are shown in Table 3. The cascade model used is in good agreement with the experimental data on the $\mu ^{-}p$ X-ray yields [63] and in fair agreement with $\pi ^{-}p$ data [56,64].

Table 3: Calculated absolute yields (%) of muonic and pionic hydrogen $K$ -transition.

Transition	$1bar$		$15bar$
	$\pi ^{-}p$	$\mu ^{-}p$	$\pi ^{-}p$	$\mu ^{-}p$
$2\rightarrow 1$	6.5	47.6	3.4	55.4
$3\rightarrow 1$	2.7	21.0	2.9	36.6
$4\rightarrow 1$	3.5	22.1	2.0	7.0
$rest\rightarrow 1$	4.4	9.3	0.6	1.0
Total yield	17	100	8.9	99.9

Important for the proposed experiment is the fact that the muonic $2\rightarrow 1$ and $3\rightarrow 1$ transitions are about an order of magnitude stronger than the corresponding pionic transitions. This compensates the lower stop efficiency for muons. An extension of the measurement to the $4\rightarrow 1$ transition can also be considered; the intensity, however, decreases significantly for higher pressures. In addition the Bragg angles are becoming small and make this transition unfavorable. Rate estimates based on the yields of Table 3 are given in chapter 3.6.

In Figures 1 -4 calculated distributions of kinetic energies are shown for a pressure of 15 $bar$ for the exotic hydrogen atoms at the instant of the $2\rightarrow 1$ and the $3\rightarrow 1$ transition. (We always give the values for the pressures at a temperature of 293K.) The influence of these distributions on the form of the X-ray lines is depicted in Fig. 8-9.

**Figure 1:** The calculated spectrum of the kinetic energy of the muonic hydrogen atom at the instant of the $2\rightarrow 1$ transition. The pressure is $15bar$ , the total number of $\mu ^{-}p$ atoms is $10^{5}$ .
$\resizebox* {0.8\textwidth}{0.3\textheight}{\includegraphics{m215.eps}}$

**Figure 2:** The calculated spectrum of the kinetic energy of the muonic hydrogen atom at the instant of the $3\rightarrow 1$ transition. The pressure is $15bar$ , the total number of $\mu ^{-}p$ atoms is $10^{5}$ .
$\resizebox* {0.8\textwidth}{0.3\textheight}{\includegraphics{m315.eps}}$

**Figure 3:** The calculated spectrum of the kinetic energy of the pionic hydrogen atom at the instant of the $2\rightarrow 1$ transition. The pressure is $15bar$ , the total number of $\pi ^{-}p$ atoms is $10^{6}$ .
$\resizebox* {0.8\textwidth}{0.3\textheight}{\includegraphics{pip215.eps}}$

**Figure 4:** The calculated spectrum of the kinetic energy of the pionic hydrogen atom at the instant of the $3\rightarrow 1$ transition. The pressure is $15bar$ , the total number of $\pi ^{-}p$ atoms is $10^{6}$ .
$\resizebox* {0.8\textwidth}{0.3\textheight}{\includegraphics{pip315.eps}}$