Application of low temperature calorimeters for precise Lamb shift measurements on hydrogen-like very heavy ions

The precise determination of the 2 s_{1}/{2}, 2 p_{1}/{2}, 2 p_{3}/{2} → 1 s_{1}/{2} X-ray transitions in hydrogen-like very heavy ions such as ²⁰⁸Pb ⁸¹⁺ or ²³⁸U ⁹¹⁺ provides a sensitive test of QED, especially of higher order contributions to the self-energy which are not accessible by alternative methods. Such measurements became possible recently by X-ray spectroscopy using highly charged ions stored and cooled in heavy ion storage rings. A better energy resolution of the X-ray detector is most essential in order to improve the experimental accuracy, which is presently about one order of magnitude worse than the theoretical uncertainty. An energy resolution of ΔE = 30-50 eV for E_γ = 50-100 keV, and a photopeak efficiency above 30% may be provided by a low temperature calorimeter. It will be constructed on the basis of already existing arrays of silicon microcalorimeters, but with larger volume and high Z absorbers. Such a detector will allow a more precise determination of the 1s-Lamb shift and for the first time the direct investigation of the 2s-Lamb shift in hydrogen-like heavy ions. In the present contribution the experimental scenario for QED experiments at storage rings, the present status of experimental and theoretical investigations and the design of an appropriate low temperature detector are discussed.