Synchrotron radiation Mössbauer spectra of a rotating absorber with implications for testing velocity and acceleration time dilation

Many Mössbauer spectroscopy (MS) experiments have used a rotating absorber in order to measure the second-order transverse Doppler (TD) shift, and to test the validity of the Einstein time dilation theory. From these experiments, one may also test the clock hypothesis (CH) and the time dilation caused by acceleration. In such experiments the absorption curves must be obtained, since it cannot be assumed that there is no broadening of the curve during the rotation. For technical reasons, it is very complicated to keep the balance of a fast rotating disk if there are moving parts on it. Thus, the Mössbauer source on a transducer should be outside the disk. Friedman and Nowik have already predicted that the X-ray beam finite size dramatically affects the MS absorption line and causes its broadening. We provide here explicit formulas to evaluate this broadening for a synchrotron Mössbauer source (SMS) beam. The broadening is linearly proportional to the rotation frequency and to the SMS beam width at the rotation axis. In addition, it is shown that the TD shift and the MS line broadening are affected by an additional factor assigned as the alignment shift which is proportional to the frequency of rotation and to the distance between the X-ray beam center and the rotation axis. This new shift helps to align the disk's axis of rotation to the X-ray beam's center. To minimize the broadening, one must focus the X-ray on the axis of the rotating disk and/or to add a slit positioned at the center, to block the rays distant from the rotation axis of the disk. Our experiment, using the ⁵⁷Fe SMS, currently available at the Nuclear Resonance beamline (ID18) at the ESRF, with a rotating stainless steel foil, confirmed our predictions. With a slit installed at the rotation axis (reducing the effective beam width from 15.6 µm to 5.4 µm), one can measure a statistically meaningful absorption spectrum up to 300 Hz, while, without a slit, such spectra could be obtained up to 100 Hz only. Thus, both the broadening and the alignment shift are very significant and must be taken into consideration in any rotating absorber experiment. Here a method is offered to measure accurately the TD shift and to test the CH.