research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Journal logoJOURNAL OF
SYNCHROTRON
RADIATION
ISSN: 1600-5775

Elaborations on Mössbauer rotor experiments with synchrotron radiation and with usual resonant sources

aDepartment of Physics, Belarus State University, 4 Nezavisimocti Avenue, Minsk 220030, Belarus, bDepartment of Engineering, Okan University, Akfirat, Istanbul, Turkey, cSavronik, Eskisehir, Turkey, dIstanbul University, Istanbul, Turkey, and eBogazici University, Istanbul, Turkey
*Correspondence e-mail: alkholmetskii@gmail.com

Edited by D. A. Reis, SLAC National Accelerator Laboratory, USA (Received 21 February 2017; accepted 21 August 2018; online 28 September 2018)

A comparative analysis of Mössbauer experiments in a rotating system between a recent application using synchrotron radiation [Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]). Eur. Phys. Lett. 114, 50010; Friedman et al. (2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]). J. Synchrotron Rad. 24, 661–666] and usual sources of resonant radiation is carried out. The principal methodological difference between these experiments can be related to the fact that in the former set of experiments the source of the resonant radiation rests in a laboratory frame whereas for the latter set of experiments the source is attached to a rotating system. It is concluded that the utilization of ordinary Mössbauer sources remains the most promising path for further research appertaining to the Mössbauer effect in rotating systems.

1. Introduction

It is known that the majority of Mössbauer experiments in a rotating system had been carried out by the early 1960s, soon after the discovery of the Mössbauer effect (Hay et al.. 1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]; Champeney & Moon, 1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]; Hay, 1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]; Granshaw & Hay, 1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]; Champeney et al., 1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]; Kündig, 1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]). In these experiments either an absorber orbited around a source of resonant radiation or vice versa. The goal was to verify the relativistic time dilation for a moving resonant absorber, which induces a relative energy shift between emission and absorption lines by the widely acclaimed value

[\Delta E/E = -u^{2}/2c^{2},\eqno(1)]

where u is the tangential velocity of the absorber and c is the velocity of light in a vacuum. For sub-sound speeds of u ≃ 300 m s−1, the value of ΔE/E is of the order of 10−12, which can be reliably measured using iron-57 Mössbauer spectroscopy, because this provides a relative energy resolution of resonant γ-quanta of about 10−14 and higher. Correspondingly, all the authors of the mentioned papers reported a confirmation of the classical relativistic expression (1)[link] with an accuracy of about 1%. Later, the relativistic time dilation had been tested – though not in rotation but in uniform translational motion – with a much better precision (10−8–10−9) in experiments on ion beams and free muons undergoing an inertial motion (Bailey et al., 1977[Bailey, J., Borer, K., Combley, F., Drumm, H., Krienen, F., Lange, F., Picasso, E., von Ruden, W., Farley, F. J. M., Field, J. H., Flegel, W. & Hattersley, P. M. (1977). Nature (London), 268, 301-305.]; McGowan et al., 1993[McGowan, R. W., Giltner, D. M., Sternberg, S. J. & Lee, S. A. (1993). Phys. Rev. Lett. 70, 251-254.]), whereafter these achievements deprived scientists of further interest in the repetition of such-like Mössbauer rotor setups.

Nonetheless, we recently paid closer attention to the experiments in question in order to verify the prediction by Yarman et al. on the additional variation of the time rate in bound systems (e.g. Yarman, 2004[Yarman, T. (2004). Ann. Fond. de Broglie, 29, 459-492.], 2006[Yarman, T. (2006). Found. Phys. Lett. 19, 675-693.]). Initially, we devoted our focus on the renowned experiment conducted by Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]), since Kündig was the only one to apply a first-order Doppler modulation of energy of γ-quanta on a special rotor setup which provided an oscillating motion of the source along the radius of the rotor, and which hence facilitated the accumulation of Mössbauer spectra of the resonant absorber at each fixed angular velocity. This allowed Kündig to record the shape and the position of the resonant line on the energy scale versus the rotational speed. As it so happens, if the vibrations are random, they do not affect the position of the resonant line, so that the measured energy shift between emission and absorption resonant lines becomes practically insensitive to the presence of such mechanical vibrations in the rotor system.

In contrast, others (Hay et al., 1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]; Champeney & Moon, 1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]; Hay, 1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]; Granshaw & Hay, 1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]; Champeney et al., 1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]) measured only the count rate of detected γ-quanta at each fixed angular velocity, and their results were not protected from the distortions of resonant lines due to vibrations. This explains why Kündig's experiment is much more informative and reliable than the other experiments undertaken on this subject matter. Even so, we regrettably revealed a number of errors in Kündig's data processing calculations (Kholmetskii et al., 2008[Kholmetskii, A. L., Yarman, T. & Missevitch, O. V. (2008). Phys. Scr. 77, 035302.]). The raw data available in the paper by Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]) allowed us to derive the correct experimental values for the relative energy shift between emission and absorption lines versus the rotational frequency, which manifested as

[\Delta E/E = -ku^{2}/c^{2},\eqno(2)]

with

[k=0.596\pm0.006.\eqno(3)]

Thus, this result substantially differs from the relativistic prediction k = 1/2, and the revealed difference exceeds the measurement uncertainty by more than ten times.

Moreover, we paid attention to a possible systematic error in the experiment by Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]), where parts of the piezotransducer – used to implement an oscillating motion to the resonant source – inevitably experience some centrifugal force, which ought to have led to the dependence of the piezoelectric constant on the rotational frequency. Although Kündig estimated this effect to be negligible, a convincing proof was not presented. At the same time, the presence of this effect can decrease the measured energy shift between emission and absorption lines, so that the actual value of the coefficient k must turn out to be even larger in comparison with our rectified estimation (3)[link]. Hence, instead of the equality (3)[link], we ended up with the inequality

[k\,\ge\,0.60.\eqno(4)]

In addition, Kündig observed an approximately exponential increase of the linewidth by up to 1.5 times under a variation of the rotational frequency from 183 rev s−1 to 517 rev s−1. Yet this does not mean that the same appreciable increase of linewidth took place for the rotor setups applied in the experiments described in the papers by Hay et al. (1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]), Champeney & Moon (1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]), Hay (1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]), Granshaw & Hay (1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]) and Champeney et al. (1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]). At the same time, it is rather difficult to believe that a line broadening was totally absent – as was tacitly assumed by the authors of the above-mentioned papers. Among these, the experiment by Champeney et al. (1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]) attracts attention on the basis of large amounts of experimental data obtained for different absorbers and the Mössbauer sources 57Co in two different matrices. However, the re-analysis of this experiment, presented by Kholmetskii et al. (2008[Kholmetskii, A. L., Yarman, T. & Missevitch, O. V. (2008). Phys. Scr. 77, 035302.]), has shown that, unlike what Champeney et al. had originally reported, their outcome too fits well the inequality (4)[link].

The result (4)[link] indicates that, next to the usual relativistic dilation of time for an orbiting resonant absorber, there is an additional component of time dilation arising in a rotating system, which induces an excess of ΔE/E in comparison with equation (1)[link] to a value much larger than the measured uncertainty.

This finding stimulated the performance of our own experiments on the given subject, though we did not apply a straight repetition of the experiment by Kündig so as to avoid the presence of possible systematic errors in the measurement of the energy shift between emission and absorption lines. Instead, and similarly to what had been effectuated with regards to the experiments reported by Hay et al. (1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]), Champeney & Moon (1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]), Hay (1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]), Granshaw & Hay (1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]) and Champeney et al. (1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]), we have decided to measure the count rate of resonant γ-quanta by a detector at different rotational frequencies of the resonant absorber. On the other hand, in contrast to the experiments of Hay et al. (1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]), Champeney & Moon (1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]), Hay (1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]), Granshaw & Hay (1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]) and Champeney et al. (1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]), we did evaluate the level of vibrations in the rotor system via the measurement of absorption curves for two different resonant absorbers, whose resonant lines are shifted on the energy scale with respect to each other approximately by their linewidth. The idea behind this method is based on the fact that the vibrations do broaden the resonant line but do not affect its area. In addition, if we assume that the vibrations are random, then, as we have mentioned above, they do not affect the position of the resonant line on the energy scale. Under these conditions, one can easily realize that an equal broadening of the lines of such absorbers due to vibrations implies a different level of resonant absorption with the change of rotational frequency. Therefore, carrying out a joint processing of the measured data obtained with both resonant absorbers, we were capable of separating the variation of the detector's count rate caused by the energy shift (1)[link] from the variation of the count rate due to vibrations, and in such a way to exclude the influence of vibrations on the measured value of k. The data processing algorithm for the realization of this method presented, for example, by Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.]), allows an unbiased estimation of k to be obtained, where the contribution of all possible instrumental factors are explicitly accounted for.

Applying this methodology, our team performed two experiments: first in 2008 (Kholmetskii et al., 2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.]) and later in 2014 (Kholmetskii et al., 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]; Yarman et al., 2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]), which differ from each other by the technical characteristics of the rotor systems. Accordingly, we arrived at the following results,

[k=0.66\pm0.03\eqno(5)]

(Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.]),

[k=0.69\pm0.02\eqno(6)]

(Kholmetskii et al., 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]; Yarman et al., 2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]).

The obtained results (4)[link]–(6)[link] are extraordinary and definitely require an explanation. To date, there have been three attempts to clarify them:

(1) A venture by Corda (2015[Corda, Ch. (2015). Ann. Phys. 355, 360-366.], 2016[Corda, Ch. (2016). Ann. Phys. 368, 258-266.]) to introduce, under the framework of the general theory of relativity, an additional effect of clock synchronization between the spinning resonant source and the stationary detector placed outside of the rotor system. In particular, the author claims that for this measurement configuration there appears an additional energy shift between the source and the detector with the relative value (ΔE/E)synch = u2/6c2, which should be added to the shift due to the time dilation effect (ΔE/E)dilation = u2/2c2. Correspondingly, the coefficient k in equation (2)[link] should be determined via the summation of (ΔE/E)synch and (ΔE/E)dilation, which yields k = 2/3, in perfect agreement with equations (5)[link] and (6)[link]. However, we had already pointed out (Kholmetskii et al., 2015b[Kholmetskii, A. L., Yarman, T. & Arik, M. (2015b). Ann. Phys. 363, 556-558.]) that the component of energy shift (ΔE/E)dilation emerges between the source and the absorber, whereas the component of energy shift (ΔE/E)synch exists between the source and the detector. At the same time, for usual detectors of resonant γ-quanta (used in all Mössbauer rotor experiments performed up to date), their energy resolution should be increased by about ten (!) orders of magnitude, in order to be sensitive to the shift (ΔE/E)synch = u2/6c2. Therefore, this component of energy shift remains strictly not tracked by the detector (which only counts resonant γ-quanta, passing across the resonant absorber) and should be totally ignored in the evaluation of k. Further attempts by Corda to defend his approach, via the claim (Corda, 2016[Corda, Ch. (2016). Ann. Phys. 368, 258-266.]) that an observer attached to the orbiting absorber and an observer attached to the resting detector measure different energy shifts of resonant lines, directly contradicts classical causality, as we have shown (Kholmetskii et al., 2016[Kholmetskii, A. L., Yarman, T., Yarman, O. & Arik, M. (2016). Ann. Phys. 374, 247-254.]).

Thus, the origin of the extra-energy shift between emission and absorption lines in rotating systems still awaits a consistent explanation in the general theory of relativity.

(2) The explanation suggested by Friedman et al. on the basis of their generalization of special relativity (Friedman & Semon, 2005[Friedman, Y. & Semon, M. D. (2005). Phys. Rev. E, 72, 026603.]; Friedman & Gofman, 2010[Friedman, Y. & Gofman, Yu. (2010). Phys. Scr. 82, 015004.]; Friedman, 2011[Friedman, Y. (2011). Ann. Phys. 523, 408-416.]) with the negation of the `clock hypothesis' by Einstein (1953[Einstein, A. (1953). The Meaning of Relativity. Princeton University Press.]). They assumed the presence of a hypothetical maximal acceleration in nature denoted by am, which allegedly affects, in general, the time rate of an orbiting absorber. This then induces an additional energy shift between emission and absorption lines (next to the usual relativistic dilation of time), which, for coefficient k to assume 0.6, leads to the value of am ≃ 1019 m s−2.

(3) The explanation contingent upon the novel framework of the Yarman–Arik–Kholmetskii (YARK) theory of gravity (e.g. Yarman & Kholmetskii, 2013[Yarman, T. & Kholmetskii, A. L. (2013). Eur. Phys. J. Plus, 128, 8-17.]; Yarman et al., 2015[Yarman, T., Kholmetskii, A. L. & Arik, M. (2015). Eur. Phys. J. Plus, 130, 191-206.], 2016b[Yarman, T., Kholmetskii, A. L., Arik, M. & Yarman, O. (2016b). Can. J. Phys. 94, 558-562.]), which had, for the past few years, been developed on the basis of Yarman's earlier approach appertaining to any ponderable gravitational interaction scenario (Yarman, 2004[Yarman, T. (2004). Ann. Fond. de Broglie, 29, 459-492.], 2006[Yarman, T. (2006). Found. Phys. Lett. 19, 675-693.]). In the case of the Mössbauer rotor experiments, this theory predicts the exact equality k = 2/3 (Yarman et al., 2015[Yarman, T., Kholmetskii, A. L. & Arik, M. (2015). Eur. Phys. J. Plus, 130, 191-206.]), which agrees with the results of the latest measurements (3)[link] and (4)[link].

Now, the entire problem remains open to scientific debate, and new Mössbauer experiments in a rotating system with increased precision are required. In this respect, recent papers by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]) that report the results of the application, apparently for the first time, of synchrotron radiation to measure the Mössbauer effect in a rotating system look to be topical.

Friedman et al. emphasized that the principal advantage of their experiments, in comparison with the previous Mössbauer rotor experiments where ordinary sources of resonant radiation had been applied (Hay et al., 1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]; Champeney & Moon, 1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]; Hay, 1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]; Granshaw & Hay, 1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]; Champeney et al., 1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]; Kholmetskii et al., 2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]; Yarman et al., 2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]), is the possibility to measure the entire Mössbauer spectra of an orbiting resonant absorber at different tangential velocities, just like in Kündig's experiment (Kündig, 1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]) but in a simpler technical way. Therefore, in the opinion of Friedman et al., much more detailed information can be obtained about the energy shift between a resting source and a rotating absorber, as well as with respect to the influence of various instrumental factors on the measurement results. At the same time, Friedman et al. still did not consider the goal of estimating the coefficient k in equation (2)[link]; in particular, the paper by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) has a methodological character and focuses on the estimation of distortions in Mössbauer measurements caused, first of all, by vibrations in the rotor system.

Based on this analysis, Friedman et al. present their recommendations for the improved performance of Mössbauer experiments in a rotating system, which are considered to be universal, i.e. equally applicable to both kinds of measurements, either with synchrotron radiation or with ordinary sources. Based on their recommendations, Friedman et al. expressed their doubts on the correctness of our results, equations (5)[link] and (6)[link], implying that a further increase of measurement precision should be closely linked to progress in the performance of Mössbauer rotor experiments with a synchrotron source of resonant radiation.

However, we now highlight two principal facts that make an essential difference for the methodologies of both kinds of Mössbauer rotor experiments:

(i) For a synchrotron source which rests in a laboratory frame, the linear Doppler effect between the source and the rotating absorber turns out to be essential even for a very narrow synchrotron beam (a few micrometres in width); whereas, in contrast, for an ordinary resonant source which rests in a rotating frame, the linear Doppler shift between emission and absorption lines does not emerge, and this would be so regardless of the particular configuration of the measurement geometry and divergence of the γ-beam (§2[link]).

(ii) For a synchrotron source, rotor vibrations, on the whole, affect the measurement results; whereas, in contradiction, when a resonant source and a resonant absorber are both rigidly fixed on a rotor, it is essential to take into account only relative vibrations between them, which, in general, are much smaller in magnitude than the absolute vibrations of the rotor itself (§3[link]).

Below, we will show that these facts, when considered together, substantially diminish the practical significance of the methodological recommendations made by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) with regards to the realization of Mössbauer rotor experiments where ordinary sources are utilized. We finally conclude in §4[link] with the conviction that the latter kind of experiments still remain the most promising path for future research on the Mössbauer effect in rotating systems.

2. Mössbauer rotor experiments and the linear Doppler effect

In this section, we consider separately the configuration of Mössbauer rotor experiments with an ordinary point-like source of resonant radiation and with a synchrotron source, and determine the contribution of the linear Doppler effect to the energy shift between emission and absorption lines.

2.1. Mössbauer rotor experiments with usual sources of resonant radiation

In this configuration, a source of resonant radiation [57Co for the experiments described by Hay et al. (1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]), Champeney & Moon (1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]), Hay (1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]), Granshaw & Hay (1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]), Champeney et al. (1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]), Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]), Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]) and Yarman et al. (2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.])] is rigidly attached to the rotor and spins on its axis. Let us show that, in the idealized case where no vibrations are present in the rotor system, the linear Doppler effect between the source and the absorber is totally absent irrespective of their sizes and regardless of the divergence of the resonant beam. In other words, the entire energy shift between emission and absorption lines is proportional to the ratio u2/c2 in the lowest order in (u/c).

In order to prove this statement, it is sufficient to show that, for two arbitrary points A and B on the rotor surface (see Fig. 1[link], where A stands for the point-like source and B for the point-like absorber), the relative energy shift between the emission line and the absorption line does not contain linear terms of the order of (u/c).

[Figure 1]
Figure 1
Diagram for the calculation of the frequency (energy) shift between a point-like source A and point-like absorber B, with both items rigidly fixed on the rotor surface.

For a laboratory observer, the frequency of emitted γ-quanta is equal to (Møller, 1973[Møller, C. (1973). The Theory of Relativity. Oxford: Clarendon Press.])

[\nu_{\rm{em}} = {{ \nu_0\left(1-u_A^2/c^2\right)^{1/2} }\over{ \left[1-\left({\bf{n}}\cdot{\bf{u}}_A\right)/c\right] }},]

where ν0 is the proper frequency of the γ-quanta, uA is the tangential velocity of point A at the emission time moment, and n is the unit vector along the direction of propagation of the γ-quanta as they are emitted from point A and absorbed at point B.

Correspondingly, the frequency of the absorbed radiation in the laboratory frame is

[\eqalignno{ \nu_{\rm{ab}} & = {{ \nu_{\rm{em}}\left[1-\left({\bf{n}}\cdot{\bf{u}}_B\right)/c\right] }\over{ \left(1-u_B^2/c^2\right)^{1/2} }} \cr& = {{ \nu_0\left(1-u_A^2/c^2\right)^{1/2} \,\left[1-\left({\bf{n}}\cdot{\bf{u}}_B\right)/c\right] }\over{ \left(1-u_B^2/c^2\right)^{1/2} \,\left[1-\left({\bf{n}}\cdot{\bf{u}}_A\right)/c\right] }}, &(7)}]

where uB is the tangential velocity of point B at the absorption time moment.

In order to calculate the frequency (7)[link], we designate rA and θA as the radial and angular coordinates of the point A at the moment of emission of a γ-quantum, and, correspondingly, rB and θB as the radial and angular coordinates of point B at the moment of absorption of the γ-quantum the way it can be seen in Fig. 1[link]. With these designations, we have the following components,

[\eqalignno{ {n_x} & = {{{{\left({{r_{AB}}} \right)}_x}} \over {{r_{AB}}}} = {{{r_B}\cos {\theta _B} - {r_A}\cos {\theta _A}} \over {{r_{AB\vphantom{\big{|}}}}}} , &(8b) \cr {n_y} & = {{{{\left({{r_{AB}}} \right)}_y}} \over {{r_{AB}}}} = {{{r_B}\sin {\theta _B} - {r_A}\sin {\theta _A}} \over {{r_{AB}}}} , &(8b)}]

[\eqalignno{ {u_{Bx}} & = \omega\, {r_B}\sin {\theta _B}, \quad (9a), \qquad {u_{By}} = - \omega\, {r_B}\cos {\theta _B}, & (9b) \cr {u_{Ax}} & = \omega\, {r_A}\sin {\theta _A}, \quad (9c), \qquad {u_{Ay}} = - \omega\, {r_A}\cos {\theta _A}, &(9d) }]

where rAB is the distance between point A at the instant of emission and point B at the instant of absorption.

Hence, substituting (8)[link] and (9)[link] into (7)[link], we derive

[\eqalignno{ \nu_{\rm{ab}} & = {{ \nu_0\left(1-u_A^2/c^2\right)^{1/2} }\over{ \left(1-u_B^2/c^2\right)^{1/2} }} {{ \left[1-\left({\bf{n}}\cdot{\bf{u}}_B\right)/c\right] }\over{ \left[1-\left({\bf{n}}\cdot{\bf{u}}_A\right)/c\right] }} \cr & = {{ \nu_0\left(1-u_A^2/c^2\right)^{1/2} }\over{ \left(1-u_B^2/c^2\right)^{1/2} }} {{ \left[1-\left(n_xu_{Bx}+n_yu_{By}\right)/c\right] }\over{ \left[1-\left(n_xu_{Ax}+n_yu_{Ay}\right)/c\right] }} \cr& = {{ \nu_0\left(1-u_A^2/c^2\right)^{1/2} \big\{1-\left[\omega{r_A}{r_B}\sin\left(\theta_A-\theta_B\right)\right]/r_{AB}\,c\big\} }\over{ \left(1-u_B^2/c^2\right)^{1/2} \big\{1-\left[\omega{r_B}{r_A}\sin\left(\theta_A-\theta_B\right)\right]/r_{AB}\,c\big\} }} \cr & = {{ \nu_0 \left(1-u_A^2/c^2\right)^{1/2} }\over{ \left(1-u_B^2/c^2\right)^{1/2} }}.&(10) }]

Thus, the terms of the numerator and the denominator, which contain linear terms in (u/c), mutually cancel each other, so that no contribution of the linear Doppler effect to the energy shift between emission and absorption lines emerges. Therefore, the frequency (energy) shift is determined by the second-order Doppler shift (or time dilation effect) alone, as long as the extra energy shift revealed in the experiments presented by Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]) and Yarman et al. (2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]) is not taken into account at this time.

Since equation (10)[link] has been derived for two arbitrary points A and B on the rotor surface, it also remains in force for any spatially extended source and absorber fixed on the rotor surface, and does not depend on the divergence of the γ-beam. The only point is that, for such a spatially extended source centered on the rotational axis, the tangential velocity uA at the edge of the source and the tangential velocity at the center evidently differ from each other, and this can cause a broadening of the emitting resonance line. However, for a source of resonant radiation sufficiently compact and bearing a typical configuration of customary Mössbauer rotor experiments, this effect is quite negligible as numerically estimated by Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]). Thus, for practical purposes, we can put uA = 0 in the case of a compact source, so that the relative energy shift becomes

[{{\Delta{E}}\over{E}} = {{ v_0-\nu_{\rm{ab}} }\over{ \nu_0 }} = 1 - {{ 1 }\over{ \left(1-u_B^2/c^2\right)^{1/2} }} \,\simeq - {{ u_B^2 }\over{ 2c^2 }} \eqno(11)]

[where we temporarily exempt ourselves from dealing with the extra energy shift revealed in experiments detailed by Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]) and Yarman et al. (2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.])].

Therefore, we conclude that the broadening of the resonant line observed in Kündig's experiment (and definitely taking place in all other experiments that feature an ordinary resonant source fixed on a rotor system) should be attributed to rotor vibrations only, as Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]) assumed.

2.2. Mössbauer rotor experiment with a synchrotron source

In this section, we will consider the experiments presented by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]), where a source of synchrotron radiation is applied to measure the energy shift of the resonant line in an orbiting absorber. In these experiments, the authors used a rotor with a 50 mm radius capable of rotating with a frequency of up to 1 kHz. A semicircular-shaped single-line absorber [an enriched 57Fe(95%)K4Fe(CN)6·3H2O single-line material] was placed on the rim of the disk, and the synchrotron Mössbauer source (SMS) of the European Synchrotron Radiation Facility (providing 57Fe resonant radiation at 14.4 keV within a bandwidth of 15 neV) was focused by Kirkpatrick–Baez optics to a 10 µm × 5 µm spot size. Hence, the beam from the SMS first hits the rotating absorber, whereafter it is detected by the detector diametrically opposed to the SMS. This beam crosses the rotor along its diameter. During rotation, the radial acceleration of the absorber alternately assumes parallel [denoted as state (b)] and anti-parallel [denoted as state (a)] orientations to the photon wavevector [see Fig. 1[link] of Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.])]. According to Friedman et al., this can yield opposite signs for the additional frequency shift due to the aforesaid acceleration as postulated by Friedman & Semon (2005[Friedman, Y. & Semon, M. D. (2005). Phys. Rev. E, 72, 026603.]), Friedman & Gofman (2010[Friedman, Y. & Gofman, Yu. (2010). Phys. Scr. 82, 015004.]) and Friedman (2011[Friedman, Y. (2011). Ann. Phys. 523, 408-416.]). There are some technical improvements in the latest experiment (Friedman et al., 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]) in comparison with the earlier experiment (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]), which, however, do not affect the principal methodology of both experiments and thus are not commented on here; separate attention to the experiment (Friedman et al., 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]) is given in our recent paper (Kholmetskii et al., 2018[Kholmetskii, A. L., Yarman, T., Yarman, O. & Arik, M. (2018). Eur. Phys. J. Plus, 133, 261.]).

Further, we emphasize that, in the experiments by Friedman et al., the source of resonant radiation rests in the laboratory frame; and hence the contribution of the linear Doppler effect to the energy shift between resonant lines of the source and the rotating absorber does not, in general, vanish. In particular, one can realize that the finite projection of the tangential velocity of the resonant absorber onto the axis of the synchrotron beam emerges due to two circumstances: the finite width of the synchrotron beam, denoted hereinafter as δ (see Fig. 2[link]), and the finite distance between the synchrotron beam and the axis x of Fig. 2[link], denoted hereinafter as h. Due to the finite values of δ and h, the expression for the energy shift between emission and absorption lines should contain linear terms in (u/c), which are responsible for the corresponding linear Doppler effect contribution.

[Figure 2]
Figure 2
The linear Doppler effect contribution due to the final width δ of the synchrotron beam, and also due to the finite distance h between the axis x and the axis of the synchrotron beam. In both cases the tangential velocity of the resonant absorber has a finite projection onto the axis of the synchrotron beam.

The finite value of the width of the synchrotron beam δ leads to the broadening of the resonance line, insofar as γ-quanta on the periphery of the beam pass the point of the resonant absorber where the absorber's tangential velocity has a finite projection on the propagation direction of such γ-quanta (see Fig. 2[link]), with the tangential velocity being equal to

[\delta u \simeq \left({{\delta/{2r}}}\right)\,u = {{\delta\omega}/2}, \eqno(12)]

where r is the rotor radius and ω is the angular velocity. Hence, at δ = 5 µm and given the rotational frequency ν = 300 rev s−1 [the maximum value used in the experiment (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]), corresponding to the tangential velocity of the absorber of about 100 m s−1], equation (12)[link] yields

[\delta u \simeq 4.7\,\,{\rm{mm\,\,s}}^{-1}, \eqno(13)]

which substantially exceeds the proper width of the resonant line (about 0.3 mm s−1 for 57Fe).

Therefore, at the maximal tangential velocity near 300 m s−1 (applied in the experiments with ordinary sources performed up to now), the width of the resonant line would be three times larger, i.e. about 14 mm s−1, which exceeds the natural linewidth by approximately 50 times! Thus, the sensitivity to a relative energy shift between emission and absorption lines decreases by the same amount of times in comparison with rotor experiments using ordinary sources, even in the idealized case where any vibrations in the rotor are fully suppressed.

The Mössbauer spectrum of the resonant absorber measured at ν = 300 rev s−1 (u ≃ 100 m s−1) is shown in the lower part of Fig. 3 of the paper by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]). Though Friedman et al. do not explicitly indicate the width of their resonant line, it can be qualitatively estimated to be near 9–10 mm s−1, which approximately twice exceeds the value (13)[link]. Thus, we can conclude that the observed broadening of the resonant line happens not only due to the final width of the synchrotron beam but due to vibrations too. Unfortunately, the authors (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) do not present any estimation of the relative contribution of these effects to the measured linewidth. Under these conditions it is reasonable to assume that, at the given rotational frequency ν = 300 rev s−1 (corresponding to u = 100 m s−1), the contribution of rotor vibrations to the line broadening has approximately the same value (13)[link], and its further analysis is given in the next section.

Further, when the synchrotron beam as a whole is shifted a distance h along the axis y of Fig. 2[link], this induces a linear Doppler shift of the measured resonant line of the orbiting absorber, which is named by Friedman et al. as the alignment shift (AS). In this case, similar to equation (13)[link], the projection of the tangential velocity of the absorber to the propagation direction of the γ-quantum at the point of contact is defined at a small h as

[\Delta {u_{\rm AS}} \simeq \left({{h/r}}\right)\,u = h\omega. \eqno(14)]

By the same token, Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]) find important the fact that the alignment shift has the same values in the state (a) (when the radial acceleration of the absorber is anti-parallel to the photon wavevector) and in the state (b) (when the radial acceleration of the absorber is parallel to the photon wavevector); and hence the relative AS between these states, in the idealized case of the absence of any vibrations, should vanish, so that the influence of acceleration on the relative energy shift of the resonant line can be investigated in a pure form.

In a real situation, the vibrations in the rotor system are always present, and, in the case where they are not random (and which, in general, is actually the case), the relative AS between the states (a) and (b) is not vanishing (see §3[link]).

3. Mössbauer rotor experiments and vibrations

In the general case, vibrations in the rotor system lead to the broadening of the resonant line; in addition, when a non-random component of such vibrations is present, it may affect the position of the resonant line on the energy scale. It is then obvious that, for the case where a source of resonant radiation rests in a laboratory frame, like in the Mössbauer rotor experiments with synchrotron radiation (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]), absolute rotor vibrations, which are measured in the laboratory frame, directly affect the resonant absorption. Contrariwise, where the source of resonant radiation is rigidly fixed to the rotor (like for Mössbauer rotor experiments with ordinary resonant sources), only relative vibrations between the source and the absorber may distort the resonant line. It is obvious that, under any reasonable experimental conditions, the relative vibrations are always less than the absolute vibrations of the rotor, and, thus, from this point of view, the experiments with ordinary sources look much more attractive than the experiments with synchrotron radiation.

Nevertheless, the analysis of the influence of absolute rotor vibrations on the measurement results implemented by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) in the case of synchrotron radiation is still interesting. Carrying out this analysis, Friedman et al. applied the Jeffcott model for the description of rotor vibrations, and determined a non-random component throughout such vibrations. They anticipated that the presence of this non-random component induces an additional energy shift between emission and absorption lines, which is comparable with the proper width of the resonant line. Thereby, Friedman et al. concluded that the results (5)[link] and (6)[link] obtained by our team are not convincing, because our data processing procedure described in Kholmetskii et al. (2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]) and Yarman et al. (2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]) implies that the vibrations are random, and apparently remains in contrast with the observations by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]).

However, Friedman et al. did not realize that in these experiments (Kholmetskii et al., 2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]; Yarman et al., 2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]), which led to the results (5)[link] and (6)[link], the relative vibrations between the source and the absorber are indeed much smaller than the absolute rotor vibrations observed in their measurements; thus, the results obtained by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) cannot be straightforwardly extended to the case of the application of a usual source mounted on a rotor.

In the latter case, the relative vibrations between the source and the absorber induce a fluctuation of their relative velocity, which is defined as the difference – as seen by the laboratory observer – between the velocity of the source at the emission moment and the velocity of the absorber at the absorption moment. It is then obvious that, in being responsible for the linear component of the Doppler shift between emission and absorption lines, only the radial component of such relative velocity should be taken into account. In order to estimate such a radial component of relative velocity, we point out that the displacement of the rotor on the whole due to its vibration along the line joining the source and the absorber at the value δx (where we assume that this line is parallel to the x-axis at the considered time moment) during a short time interval δt induces a much smaller relative displacement δxsa between the source and the absorber, which is determined via the speed of sound vs in the rotor material, i.e.

[\delta {x_{\rm sa}} \simeq \delta x \sin ({{\omega r}/{{v_{\rm s}})}}. \eqno(15)]

In this equation we adopted the common fact that the main harmonic of the vibration frequency spectrum coincides with the rotational frequency. At ωr = 100 m s−1 [the maximal tangential velocity of the absorber in the experiment (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.])] and vs ≃ 6000 m s−1 (speed of sound in an aluminium alloy), the latter equation yields

[\delta {x_{\rm sa}} \simeq {{\delta x}/{60}}. \eqno(16)]

The same relationship (16)[link] holds for the radial component of relative velocity between the source and the absorber caused by rotor vibrations, and, thus, for known configurations of the Mössbauer rotor experiments with usual sources, the influence of rotor vibrations on the shape of the resonant line is almost two orders of magnitude smaller than that framed by the synchrotron experiment reported by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]). Therefore, the systematic component of the energy shift between emission and absorption lines due to vibrations estimated in the experiment by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) (amounting to about 0.60 mm s−1) is reduced to the value 0.60/60 = 1 × 10−2 mm s−1. This value is a few tens of times smaller than the proper width of the resonant line, and is quite negligible for Mössbauer rotor experiments using ordinary sources. Hence, the assumption about the random character of vibrations for the kind of experiments undertaken by Kündig and reported by Kündig (1963[Kündig, W. (1963). Phys. Rev. 129, 2371-2375.]) as well as by other authors of Mössbauer rotor experiments using ordinary resonant sources (Hay et al., 1960[Hay, H. J., Schiffer, J. P., Cranshaw, T. E. & Egelstaff, P. A. (1960). Phys. Rev. Lett. 4, 165-166.]; Champeney & Moon, 1961[Champeney, D. C. & Moon, P. B. (1961). Proc. Phys. Soc. 77, 350-352.]; Hay, 1962[Hay, H. J. (1962). Proceedings of the Second Conference on the Mössbauer effect, edited by A. Schoen and D. M. T. Compton, p. 225. New York: Wiley.]; Granshaw & Hay, 1963[Granshaw, T. E. & Hay, H. J. (1963). Proceedings of the International School of Physics `Enrico Fermi', p. 220. New York: Academic Press.]; Champeney et al., 1965[Champeney, D. C., Isaak, G. R. & Khan, A. M. (1965). Proc. Phys. Soc. 85, 583-593.]; Kholmetskii et al., 2009[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2009). Phys. Scr. 79, 065007.], 2011[Kholmetskii, A. L., Yarman, T., Missevitch, O. V. & Rogozev, B. I. (2011). Int. J. Phys. Sci. 6, 84-92.], 2015a[Kholmetskii, A. L., Yarman, T., Arik, M. & Missevitch, O. V. (2015a). AIP Conf. Proc. 1648, 510011-510014.]; Yarman et al., 2016a[Yarman, T., Kholmetskii, A. L., Arik, M., Akkuş, B., Öktem, Y., Susam, L. A. & Missevitch, O. V. (2016a). Can. J. Phys. 94, 780-789.]) is quite warranted in any practical situation, and makes fully reliable our recent results (5)[link], (6)[link] which were obtained on the basis of the given assumption.

Therefore, the explanation of the origin of the extra-energy shift between emission and absorption lines for Mössbauer experiments in a rotating system, as expressed by equations (5)[link] and (6)[link], remains in force as a very topical problem.

Looking closer at the contribution of vibrations to the measured shift of the resonant line in the experiment conducted by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]), we recall the disclosure of the previous section where we found that the contribution of rotor vibrations to the line broadening is comparable with the value (13)[link] at u = 100 m s−1. Insofar as the level of vibrations is proportional to u2, then at u ≃ 300 m s−1 (a typical maximal tangential velocity in Mössbauer experiments with ordinary sources) the line broadening due to vibrations is expected to be nine times larger in comparison with the value (13)[link], i.e. more than 40 mm s−1. This exceeds the natural width of the resonant line (≤0.3 mm s−1) by more than 100 times (!), which, in fact, leads to a drastic decrease of measurement sensitivity to the relative energy shifts between emission and absorption lines under the rotation of the absorber. This means that the uncertainty in the determination of the coefficient k in equation (2)[link] in the Mössbauer rotor experiments with synchrotron radiation should many times exceed the measurement uncertainty of the coefficient k in the experiments with ordinary sources of resonant radiation. This circumstance explains the fact that Friedman et al. did not even try to evaluate the coefficient k in both experiments (Friedman et al., 2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]).

Thus, Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) restricted their research to the measurement of the difference in AS between the state (a) (when the radial acceleration of the absorber is anti-parallel to the photon wavevector) and the state (b) (when the radial acceleration of the absorber is parallel to the photon wavevector). They claimed that, for run 39 of Friedman et al. (2916), the observed relative shift of the line between states (a) and (b) `is significantly larger than the calculated vibrational one'. In such a way, the authors presume to hint that their result indicates a violation of the clock hypothesis by Einstein, and that we are dealing with the presence of a maximal acceleration in nature.

However, their results are not convincing because the shift of the resonant line due to the non-random character of vibrations in the rotor system does not provide quantitative information about the corresponding relative AS. The reason is that the line broadening in the presence of vibrations is defined by the fluctuation of the velocity of the absorber along the synchrotron beam (the axis x in Fig. 2[link]), whereas the fluctuation of the AS is defined by the fluctuation of the distance h due to vibrations, which, in general, has different values in the states (a) and (b) for non-random vibrations, and cannot be estimated via the line broadening. The same remark remains in force with respect to the latest experiment by Friedman et al. (2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]) (see Kholmetskii et al., 2018[Kholmetskii, A. L., Yarman, T., Yarman, O. & Arik, M. (2018). Eur. Phys. J. Plus, 133, 261.]).

In fact, a correct way to determine the shift of resonant lines as a function of rotational frequency is to measure the coefficient k in equation (2)[link] as was done in all experiments with ordinary sources of resonant radiation performed up to now. However, this is still not the case for the synchrotron experiment reported by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.], 2017[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G. & Chumakov, A. I. (2017). J. Synchrotron Rad. 24, 661-666.]), where low measurement sensitivity to the relative energy shifts of resonant lines due to considerable broadening of the lines still does not allow the coefficient k in equation (2)[link] to be determined with the measurement precision comparable with the values obtained in other experiments on this subject, where ordinary sources of resonant radiation had been applied.

4. Conclusion

As an outcome of this comparative study, we conclude that the quotidian application of compact point-like ordinary sources of resonant radiation rigidly attached to a rotor remains the most efficient way to perform Mössbauer experiments in a rotating system. Experiments on this subject with a synchrotron source (which rests in a laboratory frame) have a number of principal shortcomings in comparison with the case of ordinary sources as disclosed in the present research. Correspondingly, among practical recommendations made in the concluding section of the paper by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]) with respect to improved performance of Mössbauer experiments in a rotating system, only the initial (and obviously trivial) suggestion to use `vibrationless rotor systems' is relevant for both kinds of experiments.

All other practical recommendations by Friedman et al. (2016[Friedman, Y., Nowik, I., Felner, I., Steiner, J. M., Yudkin, E., Livshitz, S., Wille, H.-C., Wortmann, G., Arogeti, S., Levy, R., Chumakov, A. I. & Rüffer, R. (2016). Europhys. Lett. 114, 50010.]), such as, in particular, being mindful of the non-random component of vibrations in the rotor system, are relevant only for further development of Mössbauer rotor experiments relying on a synchrotron source, and are not appreciably significant for experiments making use of ordinary sources; which, due to the absence of the linear Doppler shift of the resonant line, as well as a diminished level of vibrations between the source and the absorber, will remain the most promising path for further research on the Mössbauer effect in rotating systems in the foreseeable future. This fact is all the more accentuated in the case of other plausible Mössbauer setups with variations on the placement of the source and the absorber.

Perspectives for the performance of such experiments with improved precision have been recently analyzed by Kholmetskii et al. (2016[Kholmetskii, A. L., Yarman, T., Yarman, O. & Arik, M. (2016). Ann. Phys. 374, 247-254.]).

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