Transition energies of pionic hydrogen and Bragg angles.

The energies of the muonic and pionic hydrogen Lyman X-rays vary between 1896 and 3244 eV. In Table 5 the transition energies for pionic and muonic hydrogen are given together with the relevant crystal plane and reflection angles.

Table 5: Pionic and muonic hydrogen X-rays are listed together with the Bragg angles of relevant single crystals.

Transition	Energy[eV]	$\Theta _{B}$ [ $^{0}$ ]
		$SiO_{2}$ 100	$SiO_{2}$ 10.1	$SiO_{2}$ 11.0	$Si$ 111
$\mu ^{-}p_{2\rightarrow 1}$	1896	50.22	78.22	-	-
$\mu ^{-}p_{3\rightarrow 1}$	2248	40.40	55.7	-	61.58
$\mu ^{-}p_{4\rightarrow 1}$	2371	37.88	51.52	-	56.51
$\pi ^{-}p_{2\rightarrow 1}$	2433	36.76	49.65	-	54.35
$\mu ^{-}p_{\infty \rightarrow 1}$	2528	35.17	47.31	87.29	51.46
$\pi ^{-}p_{3\rightarrow 1}$	2885	30.31	40.00	61.08	43.26
$\pi ^{-}p_{4\rightarrow 1}$	3042	28.60	37.60	56.11	40.54
$\pi ^{-}p_{5\rightarrow 1}$	3114	27.88	36.6	54.19	39.42
$\pi ^{-}p_{\infty \rightarrow 1}$	3244	26.67	34.9	51.11	37.56

Not all Bragg angles which are listed in Table 5 are suited for the experiment. Their choice is limited by the following considerations. First, the energy resolution as derived from the Bragg relation varies as $cot\Theta _{B}$ . The broadening (see Appendix 1) which is caused by the finite width of the crystal increases even with $(cot\Theta _{B})^{2}$ . Therefore an experiment should be performed at about $\theta _{B}$ $>$ $40^{^{0}}$ . A reasonable range for the Bragg angles is $40^{^{0}}<\theta _{B}<65^{^{0}}$ because the detector must be well shielded from the pion source which excludes higher Bragg angles. In order to shield the CCD detector an extended concrete shielding must be tailored around the path of the X-rays and especially around the CCD detector. Therefore not even this angular range is accessible without reworking the whole shielding.

Table 5 shows that there exist very close pairs of Bragg angles for pionic hydrogen and the corresponding muonic hydrogen transitions provided the $SiO_{2}$ 100 crystal is used for muons and the $SiO_{2}$ 10.1 crystal for pions. We plan to use this fact to simultaneously measure the muonic and pionic X-rays by using two suitably arranged crystals. In this way instabilities between a pion measurement and the reference measurement with muons cancel. The two crystals will be mounted one on top of each other symmetrically above and below a horizontal plane given by the center of the target and the CCD detector. Further implications of this approach are discussed in chapter 4.