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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

In Table 8 the energies and natural line widths are given for He-like atoms for transitions falling in the energy range covered by our spectrometer. The values are taken from [74] and agree well with calculations using the MCDF code [75,76,77,78]. The error in the energies is on the level of some 10 meV. It can be noted that the natural line widths in some cases are very small. Also some transition pairs are suited for the test of the two-crystal arrangement.

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

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Pionic Hydrogen Collaboration