Physics Research International

Physics Research International / 2009 / Article

Research Letter | Open Access

Volume 2009 |Article ID 524957 | 4 pages | https://doi.org/10.1155/2009/524957

On “GSI Oscillations” as Interference of Two Closely Spaced Ground Mass Eigenstates of H-Like Mother Ions

Academic Editor: Peter Blunden
Received26 Jun 2009
Accepted04 Aug 2009
Published30 Aug 2009

Abstract

We analyse the hypothesis that the “GSI oscillations” of the K-shell electron capture decay (EC) rates of the H-like heavy ions are caused by quantum beats from a coherent state of two closely spaced ground mass-eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© of decaying H-like heavy ions. We apply this mechanism to the calculation of the 𝛽+-decay rates of the H-like heavy ions and discuss the dynamics of the production of the H-like heavy ions with two closely spaced ground mass-eigenstates at GSI experiments. We show that such a mechanism cannot describe simultaneously the experimental data on both the EC-decay and 𝛽+-decay rates of the H-like heavy ions, measured at GSI.

1. Introduction

Recently Litvinov et al. [1] have observed that the K-shell electron capture (EC) decay rates of H-like 140Pr58+ and 142Pm60+ ions 140Pr58+⟶140Ce58++𝜈𝑒,142Pm60+⟶142Nd60++𝜈𝑒(1) have an unexpected periodic time modulation of exponential decay curves. The rates of the number 𝑁EC𝑑 of daughter ions 140Ce58+and 142Nd60+

𝑑𝑁EC𝑑(𝑡)𝑑𝑡=𝜆EC(𝑡)𝑁𝑚(𝑡),(2) where 𝑁𝑚(𝑡) is the number of the H-like mother ions 140Pr58+ or 142Pm60+[1] and 𝜆(H)EC(𝑡) is the EC-decay rate, are periodic functions, caused by a periodic time dependence of the EC-decay rates 𝜆EC(𝑡)=𝜆EC1+ğ‘ŽEC𝜔cosEC𝑡+𝜙EC(3) with a period 𝑇EC=2𝜋/𝜔EC≃7seconds, an amplitude ğ‘ŽEC≃0.20, and a phase 𝜙EC.

In the articles [2–4] we have proposed an explanation of the periodic time dependence of the EC-decay rates as an interference of two neutrino mass eigenstates 𝜈1 and 𝜈2 with masses 𝑚1 and 𝑚2, respectively. The period 𝑇EC of the time dependence has been related to the difference Δ𝑚221=𝑚22−𝑚21 of the squared neutrino masses 𝑚2 and 𝑚1 as follows: 𝜔EC=2𝜋𝑇EC=Δ𝑚2212𝛾𝑀𝑚,(4) where 𝑀𝑚 is the mass of the mother ion and 𝛾=1.43 is a Lorentz factor [1]. In a subsequent analysis we also showed that the 𝛽+branches of the decaying H-like heavy ions do not show time modulation, because of the broad energy spectrum of the neutrinos in the corresponding three-body decays and proposed a test of such a behaviour [4].

According to atomic quantum beat experiments [5–7], the explanation of the “GSI oscillations,” proposed in [2], bears similarity with quantum beats of atomic transitions, when an excited atomic eigenstate decays into a coherent state of two (or several) lower lying atomic eigenstates. In the case of the EC-decay one deals with a transition from the initial state |𝑚⟩ to the final state |𝑑𝜈𝑒⟩, where the electron neutrino is a coherent superposition of two neutrino mass eigenstates with the energy difference equal to 𝜔21=Δ𝑚221/2𝑀𝑚 related to 𝜔EC as 𝜔EC=𝜔21/𝛾.

Another mechanism of the “GSI oscillations” has been proposed by Giunti [8] and Kienert et al. [9]. The authors [8, 9] assume the existence of two closely spaced ground mass eigenstates of the mother of the H-like heavy ion in the initial state of the EC-decay and describe the initial state of the mother ion by the coherent superposition ||𝑚|𝑚⟩=cosğœƒî…žî¬||𝑚+sinğœƒî…žî…žî¬(5) of the wave functions of two mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© with masses 𝑀𝑚′ and 𝑀𝑚′′, respectively, and mass splitting of order Δ𝐸𝑚′𝑚′′=𝑀𝑚′−𝑀𝑚′′∼10−15eV; 𝜃 is a mixing angle.

Unlike our analysis [2–4], the authors [8, 9] draw an analogy of the “GSI oscillations” with quantum beats of atomic transitions [7], when an atom, excited into a state of a coherent superposition of two closely spaced energy eigenstates, decays into a lower lying energy eigenstate. According to [7], the intensity of radiation, caused by a transition from such a coherent state into a lower energy eigenstate, has a periodic time dependent term with a period inversely proportional to the energy-difference Δ𝐸𝑚′𝑚′′ between two closely spaced energy eigenstates.

In this paper we apply the mechanism, proposed in [8, 9], to the analysis of the time modulation of the 𝛽+-decay rates of the H-like heavy ions. We analyse also the dynamics of the production of the H-like heavy ions with two closely spaced ground mass eigenstates at GSI experiments.

The mass splitting Δ𝐸𝑚′𝑚′′=𝑀𝑚′−𝑀𝑚′′∼10−15eV can be attributed either to the nucleus or to the energy level of the bound electron of the H-like mother ion. If the mass splitting is related to the energy level of the bound electron, one can show that in this case the coherent state |𝑚⟩, normalised to unity, reduces to the wave function of the unperturbed state of the H-like mother ion with a time dependent phase, which leads to no time modulation for the EC-decay rate of the H-like mother ion. Thus, we analyse below only the case, when the mass splitting is related to the nucleus of the H-like mother ion. By definition of the mass eigenstates, the mass eigenstates of the H-like mother ion |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© should be orthogonal âŸ¨ğ‘šî…žâˆ£ğ‘šî…žî…žâŸ©=0.

2. EC- and 𝜷+-Decay Rates, Caused by theDoubling of the Ground State of the Nuclei

The EC-decay rate of the mother ion from the |𝑚⟩ state is equal to [3] 𝜆(𝑚)EC(𝑡)=𝜆EC1+sin2𝜃cosΔ𝐸𝑚′𝑚′′𝑡,(6) where 𝜆EC is the EC-decay constant [2, 3, 10] and Δ𝐸𝑚′𝑚′′ is the energy difference of the ground mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ©. This shows a periodic dependence of the EC-decay rate with a period inversely proportional to Δ𝐸𝑚′𝑚′′

𝑇EC=2𝜋𝛾Δ𝐸𝑚′𝑚′′.(7) For 𝑇EC=7.06(8)seconds this gives Δ𝐸𝑚′𝑚′′=8.38(9)×10−16eV. According to the experimental data [1], the amplitude of the time modulated term is equal to ğ‘ŽEC≃0.20. Since ğ‘ŽEC=sin2𝜃, this gives 𝜃≃5.80.

However, the H-like heavy ions, subjected to the EC-decays, are unstable also under 𝛽+-decays [1]: 𝑚→𝑑+𝑒++𝜈𝑒. Following the standard procedure for the calculation of the 𝛽+-decay rates [3, 4, 10] one gets 𝜆𝛽(𝑚)+(𝑡)=𝜆𝛽+1+sin2𝜃cosΔ𝐸𝑚′𝑚′′𝑡,(8) where the 𝛽+-decay constant 𝜆𝛽+ has been calculated in [10]. Hence, according to [8, 9], the 𝛽+-decay rates of the H-like heavy ions should have the same periodic time dependence as the EC-decay rates. This contradicts the experimental data on the time dependence of the 𝛽+-decay rates of the H-like heavy 142Pm60+ ions at GSI [11–13], which indicate no time modulation. Of course, these experimental data are preliminary and one can wait for either the confirmation or rejection of them.

Nevertheless, this does not take away all problems. The point is that it seems that the doubling of the ground state of the nuclei of the H-like heavy ions, proposed in [8, 9], is unable to generate time dependence of both EC-decay rates and 𝛽+-decay rates of the H-like heavy ions at all. Indeed, the ground mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© of the mother H-like heavy ions, injected into the Experimental Storage Ring (ESR), should be statistically populated by the fast projectile fragmentation (see (11) and discussions below). Such a process populates also statistically the system of the mother H-like heavy ions with coherent states |𝑚⟩=−sin𝜃|ğ‘šî…žâŸ©+cos𝜃|ğ‘šî…žî…žâŸ©. Due to statistical equivalence and indistinguishability of the coherent states |𝑚⟩=cos𝜃|ğ‘šî…žâŸ©+sin𝜃|ğ‘šî…žî…žâŸ© and |𝑚⟩=−sin𝜃|ğ‘šî…žâŸ©+cos𝜃|ğ‘šî…žî…žâŸ© the probabilities 𝑃𝑚 and 𝑃𝑚 of the production of the coherent states |𝑚⟩ and |𝑚⟩, related by 𝑃𝑚+𝑃𝑚=1, should be equal 𝑃𝑚=𝑃𝑚=1/2.

The decay rates 𝜆(𝑚)EC(𝑡) and 𝜆(𝑚)𝛽+(𝑡) of the EC and 𝛽+ decays of the H-like heavy ions from the coherent state |𝑚⟩ are equal to 𝜆(𝑚)EC(𝑡)=𝜆EC1−sin2𝜃cosΔ𝐸𝑚′𝑚′′𝑡,𝜆(𝑚)𝛽+(𝑡)=𝜆𝛽+1−sin2𝜃cosΔ𝐸𝑚′𝑚′′𝑡.(9) The total EC-decay and 𝛽+-decay rates of the H-like heavy ions from the coherent states |𝑚⟩ and |𝑚⟩ are defined by 𝜆EC(𝑡)=𝑃𝑚𝜆(𝑚)EC𝜆(𝑡)+𝑃𝑚(𝑚)EC(𝑡)=𝜆EC𝑃1+sin2𝜃𝑚−𝑃𝑚cosΔ𝐸𝑚′𝑚′′𝑡=𝜆EC,𝜆𝛽+(𝑡)=𝑃𝑚𝜆𝛽(𝑚)+𝜆(𝑡)+𝑃𝑚(𝑚)𝛽+(𝑡)=𝜆𝛽+𝑃1+sin2𝜃𝑚−𝑃𝑚cosΔ𝐸𝑚′𝑚′′𝑡=𝜆𝛽+.(10) Since 𝑃𝑚=𝑃𝑚=1/2, no interference terms and time dependence appear in the EC-decay and 𝛽+-decay rates of the H-like heavy ions.

This implies that the mechanism of two closely spaced ground mass eigenstates of the nuclei of the mother H-like heavy ions is unable to provide a correct simultaneous description of the EC-decay and 𝛽+-decay rates of the H-like heavy ions, measured at GSI [1, 11–13].

3. Dynamics of Statistical Population

What is the dynamics of a statistical population of the ground mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© and, correspondingly, the coherent states |𝑚⟩ and |𝑚⟩ of the mother H-like heavy ions in experiments at GSI?

At the GSI experiments the H-like heavy ions 𝐴𝑋(𝑍−1)+ are produced in the reaction [11–13]

152Sm+9Be⟶𝐴𝑋(𝑍−1)++⋯,(11) where the incident ions 152Sm with 500–600 MeV kinetic energy per nucleon produce on a beryllium target 9Be the fragments of the highly ionised states 𝐴𝑋(𝑍−1)+like the H-like ions 140Pr58+, 142Pr60+, 122I52+, and so on, which are injected then with a kinetic energy of 400MeV per nucleon into the ESR [11–13].

According to the theory of high-energy nucleus-nucleus (or ion-ion) collisions [14], in the reactions (11) heavy nuclei 𝐴𝑋∗𝑍+ are produced in excited states with excitation energies 𝐸∗. The excited energy levels of the nucleus 𝐴𝑋∗𝑍+ are distributed statistically with an energy level density 𝜌(𝐸∗). According to the theory of nuclear energy level density and the Bethe theorem [14–22], a nuclear energy level density 𝜌(𝐸∗) is a continuous function of 𝐸∗, which can be deduced from a statistical analysis. The Bethe theorem gives the following general expression for the nuclear energy level density 𝜌(𝐸∗) [14–22] 𝜌𝐸∗∼𝑒𝑆[𝐸∗,𝑇],(12) where 𝑇 has the meaning of nuclear temperature and 𝑆[𝐸∗,𝑇] is the entropy of the Fermi system 𝐴𝑋∗𝑍+ of nucleons with a given number 𝐴 [14–22].

Let, following [8, 9], the ground state of heavy nucleus 𝐴𝑋𝑍+ be doubled with masses 𝑀1 and 𝑀2 and the mass–splitting |𝑀1−𝑀2|∼10−15eV. According to [14–22], a transition of the nucleus 𝐴𝑋∗𝑍+ from the excited states with a nuclear energy level distribution 𝜌(𝐸∗) to the less excited states and finally to the ground states of the nucleus 𝐴𝑋𝑍+ with quantum numbers 𝐽𝜋=1+ [10] should have a statistical character [18]. As a result the ground states of the nucleus 𝐴𝑋𝑍+ with quantum numbers 𝐽𝜋=1+ and masses 𝑀1 and 𝑀2, produced in the reaction (11), are populated statistically.

A statistical population of the ground states of the nucleus 𝐴𝑋𝑍+ entails a statistical population of the mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© of the H-like ion 𝐴𝑋(𝑍−1)+ with masses 𝑀𝑚′ and 𝑀𝑚′′ and the mass-difference |ğ‘€ğ‘šâ€²âˆ’ğ‘€ğ‘šâ€²â€²î…ž|∼10−15eV, produced in the reaction (11). As a result coherent states |𝑚⟩=cos𝜃|ğ‘šî…žâŸ©+sin𝜃|ğ‘šî…žî…žâŸ© and |𝑚⟩=−sin𝜃|ğ‘šî…žâŸ©+cos𝜃|ğ‘šî…žî…žâŸ© should be created with equal probabilities 𝑃𝑚=𝑃𝑚=1/2, that prohibits any time dependence of both the total EC-decay rates and the 𝛽+-decay rates of H-like heavy 𝐴𝑋(𝑍−1)+ ions.

We would like to notice that in reaction (11) the H-like ions 𝐴𝑋(𝑍−1)+ are produced both in the ground hyperfine states with atomic spin 𝐹=1/2 and in the excited hyperfine state with atomic spin 𝐹=3/2, which decays into the ground hyperfine state 𝐴𝑋(𝑍−1)+𝐹=3/2→𝐴𝑋(𝑍−1)+𝐹=1/2+𝛾 with the lifetime of order of 𝜏∼10−2s [10]. Of course, such transitions should replenish statistically the system of the mother H-like heavy ions with the ground hyperfine states 𝐴𝑋(𝑍−1)+𝐹=1/2 with masses 𝑀𝑚′ and 𝑀𝑚′′ and a mass splitting |𝑀𝑚′−𝑀𝑚′′|∼10−15eV.

4. Conclusion

We have analysed the mechanism of two closely space mass eigenstates of the H-like heavy ions with a mass splitting of order of 10−15eV. We have applied this mechanism to the calculation of the time modulation of the 𝛽+-decay rates of the H-like heavy ions and analysed the dynamics of the production of the H-like heavy ions at GSI experiments.

We have shown that in case the nuclei of the H-like heavy ions have the ground states splitted with a mass-difference of order of 10−15eV, the 𝛽+-decay rates of the H-like heavy ions, decaying from the coherent state |𝑚⟩=cos𝜃|ğ‘šî…žâŸ©+sin𝜃|ğ‘šî…žî…žâŸ©, should have the same period of the time dependence as the EC-decay rates. This contradicts recent experimental data at GSI [11–13].

We have analysed the dynamics of the production of the H-like heavy ions with two closely spaced mass eigenstates at GSI experiments. We have shown that, according to the theory of high-energy ion-ion collisions [14–22], the system of the H-like heavy ions, injected into the ESR with an kinetic energy of about 400MeV per nucleon, should be statistically populated with two closely spaced ground mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ©. Of course, the statistical population of the states |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© is not only determined by the energy-level density of the states in the nucleus produced in reaction (11), but also by 𝛾 transitions, defined by the 𝛾-strength functions [23], that lead to de-excitation of the excited states populated directly in the nuclear reactions. Since such a statistical population of the nuclear states leads to a statistical equivalence of both two closely spaced ground mass eigenstates |ğ‘šî…žâŸ© and |ğ‘šî…žî…žâŸ© and their coherent superpositions |𝑚⟩ and |𝑚⟩, the EC and 𝛽+ decay rates of the H-like heavy ions do not depend on time at all. Thus, we can conclude that such a hypothesis of two closely spaced ground mass eigenstates of heavy nuclei is unable to explain correctly the experimental data on the time modulation of both the EC-decay rates and 𝛽+-decay rates, measured at GSI [1, 11–13].

As we have mentioned above the mass splitting of the H-like mother ion can be attributed to the splitting of the energy level of the bound electron. However, since in this case the coherent state |𝑚⟩, normalised to unity, reduces to the wave function of the unperturbed state of the H-like mother ion with a common time dependent phase, the splitting of the energy level of the bound electron of the H-like mother ion leads to no time modulation for the EC and the 𝛽+ decay rates the H-like mother ion.

Acknowledgment

The authors acknowledge fruitful discussions with T. Ericson.

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