Testing and tuning of the bent crystals.

A first step of a calibration and understanding the functioning of the spectrometer is a measurement with X-rays of well understood line shape in the preparatory state of the experiment. Fluorescence X-rays are not useful for this purpose as their line shape is rather complicated because of satellite structures. Also their natural width is much broader than the resolution of the available crystals. X-ray transitions of one-electron atoms are better suited as their energies are known with a precision of typically $1meV$ in the energy region of interest. The line shape is also influenced by Doppler broadening depending on the temperature which is needed to ionize the atoms in the ECR source. A Maxwellian distribution of the velocities in the ECR plasma results in a Gaussian lineshape of the X-ray lines. The temperature (energy) of the ions in the ECR plasma is not a priori known and depends on the magnetic field geometry of the ECR source as well as on the high frequency parameters. A value between $1eV$ and $10eV$ per charge value is most probable [80]. For the following estimate a value of $5eV$ per charge unit is assumed. The Doppler width is expected to be a factor of 20 higher than the natural line width for the small Z and almost a factor of 10 higher for the higher Z. This is much more favorable than the conditions with fluorescence lines. The intensity of emitted X-rays is expected to be high enough to do calibration measurements in a time much shorter (less than one hour) compared to measurements with exotic atoms which last typically some days per measured energy. It is therefore planned to set up an ECR source and do calibration measurements well before the real measurement with pions and muons [81]. These calibration measurements will start with a diagnosis of the ECR source using survey measurements with CCD detectors and then continue with a high resolution X-ray spectroscopy with $Si$ crystals. In this way yield and Doppler width of the ECR source will be determined. In a final step the two-crystal set-up with $SiO_{2}$ crystals will be tested. In Table 7 some characteristics of one-electron lines and of fluorescence lines relevant for the required energy region are shown. At first it can be stated that no line really coincides with a muonic or pionic hydrogen transition. It can however be noted that the energies of the transitions from the one-electron atom of nuclear charge Z almost coincide with the fluorescence lines of Z+1 as one electron almost shields one nuclear charge. This offers the possibility to determine the peak positions of the fluorescence lines during the calibration measurement with $meV$ precision. This more technical fact may be of help during the real measurements to check the angular precision of the spectrometer as fluorescence lines are readily available.

Table 7: Energies and line widths (FWHM) are given both for one-electron MCDF code (fully relativistic, with QED corrections and finite nuclear size) [75,76,77,78] and fluorescence transitions [79] for different Z. The mass M is in nuclear mass units. The Gaussian width $\Gamma _{Max}$ (FWHM) caused by a Maxwellian velocity distribution of the ECR plasma is calculated for an ion temperature of $5eV$ per charge. The widths for the fluorescence lines are from ref [73].

Z	M	1 $e^{-}$ atoms				fluorescence lines
		$2p_{\frac{1}{2}}\rightarrow$ $1s$	$\Gamma _{2p_{\frac{1}{2},\frac{3}{2}}}$	$\Gamma _{Max}$	$2p_{\frac{3}{2}}\rightarrow 1s$	$K\alpha _{1}$	$\Gamma _{K\alpha _{2}}$	$K\alpha _{2}$	$\Gamma _{K\alpha _{2}}$
		$[eV]$	[ $meV$ ]	$[meV]$	$[eV]$	$[eV]$	$[meV]$	$[eV]$	[ $meV$ ]
13	26.974	1727.687	11.8, 11.8	200	1729.003	1486.27	430	1486.70	430
14	27.969	2004.325	15.9, 15.8	236	2006.095	1739.38	524	1739.98	539
15	30.966	2301.651	20.9, 20.8	269	2303.982	2012.27	570	2013.7	560
16	31.963	2619.701	27.1, 26.9	311	2622.720	2306.64	769	2307.84	722
17	34.960	2958.528	34.5, 34.3	347	2962.376	2620.8	925	2622.4	945
18	39.953	3313.179	43.4, 43.1	375	3323.019	2955.63	810	2957.70	800

Table 8: Transition energies and widths for He-like transitions in the energy region covered by the crystal spectrometer.

Z	1s2p 3P1		1s2p 1P1		1s2p 3P2		1s2s 3S1
	energy	width	energy	width	energy	width	energy	width
	[eV]	[meV]	[eV]	[meV]	[eV]	[meV]	[eV]	[meV]
13	1588.16	0.05	1598.33	18	1588.79	1.37E-05	1575.00	1.10E-07
14	1853.80	0.10	1865.04	25	1854.69	2.55E-05	1839.47	2.37E-07
15	2140.13	0.20	2152.47	33	2141.35	4.54E-05	2124.58	4.83E-07
16	2447.19	0.38	2460.69	43	2448.78	7.77E-05	2430.40	9.39E-07
17	2775.00	0.69	2789.67	56	2777.11	1.29E-04	2756.92	1.75E-06
18	3123.58	1.18	3139.64	70	3126.34	2.07E-04	3104.18	3.15E-06
19	3493.01	1.96	3510.51	88	3496.55	3.23E-04	3472.27	5.49E-06