#### Abstract

We analyse the hypothesis that the โGSI oscillationsโ 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 and of decaying H-like heavy ions. We apply this mechanism to the calculation of the -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 -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 and ions have an unexpected periodic time modulation of exponential decay curves. The rates of the number of daughter ions and

where is the number of the H-like mother ions or [1] and is the EC-decay rate, are periodic functions, caused by a periodic time dependence of the EC-decay rates with a period seconds, an amplitude and a phase .

In the articles [2โ4] we have proposed an explanation of the periodic time dependence of the EC-decay rates as an interference of two neutrino mass eigenstates and with masses and , respectively. The period of the time dependence has been related to the difference of the squared neutrino masses and as follows: where is the mass of the mother ion and is a Lorentz factor [1]. In a subsequent analysis we also showed that the 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 [5โ7], the explanation of the โGSI oscillations,โ 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 to the final state , where the electron neutrino is a coherent superposition of two neutrino mass eigenstates with the energy difference equal to related to as .

Another mechanism of the โGSI oscillationsโ 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 of the wave functions of two mass eigenstates and with masses and , respectively, and mass splitting of order ; is a mixing angle.

Unlike our analysis [2โ4], the authors [8, 9] draw an analogy of the โGSI oscillationsโ 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 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 -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 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 , 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 and should be orthogonal .

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

The EC-decay rate of the mother ion from the state is equal to [3] where is the EC-decay constant [2, 3, 10] and is the energy difference of the ground mass eigenstates and . This shows a periodic dependence of the EC-decay rate with a period inversely proportional to

For seconds this gives . According to the experimental data [1], the amplitude of the time modulated term is equal to . Since , this gives .

However, the H-like heavy ions, subjected to the EC-decays, are unstable also under -decays [1]: . Following the standard procedure for the calculation of the -decay rates [3, 4, 10] one gets
where the -decay constant has been calculated in [10]. Hence, according to [8, 9], the -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 -decay rates of the H-like heavy ions at GSI [11โ13], 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 -decay rates of the H-like heavy ions at all. Indeed, the ground mass eigenstates and 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 . Due to statistical equivalence and indistinguishability of the coherent states and the probabilities and of the production of the coherent states and , related by , should be equal .

The decay rates and of the EC and decays of the H-like heavy ions from the coherent state are equal to The total EC-decay and -decay rates of the H-like heavy ions from the coherent states and are defined by Since , no interference terms and time dependence appear in the EC-decay and -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 -decay rates of the H-like heavy ions, measured at GSI [1, 11โ13].

#### 3. Dynamics of Statistical Population

What is the dynamics of a statistical population of the ground mass eigenstates and and, correspondingly, the coherent states and of the mother H-like heavy ions in experiments at GSI?

At the GSI experiments the H-like heavy ions are produced in the reaction [11โ13]

where the incident ions with 500โ600โ kinetic energy per nucleon produce on a beryllium target the fragments of the highly ionised states like the H-like ions , , and so on, which are injected then with a kinetic energy of per nucleon into the ESR [11โ13].

According to the theory of high-energy nucleus-nucleus (or ion-ion) collisions [14], in the reactions (11) heavy nuclei are produced in excited states with excitation energies . The excited energy levels of the nucleus are distributed statistically with an energy level density . According to the theory of nuclear energy level density and the Bethe theorem [14โ22], a nuclear energy level density is a continuous function of , which can be deduced from a statistical analysis. The Bethe theorem gives the following general expression for the nuclear energy level density [14โ22] where has the meaning of nuclear temperature and is the entropy of the Fermi system of nucleons with a given number [14โ22].

Let, following [8, 9], the ground state of heavy nucleus be doubled with masses and and the massโsplitting . According to [14โ22], a transition of the nucleus from the excited states with a nuclear energy level distribution to the less excited states and finally to the ground states of the nucleus with quantum numbers [10] should have a statistical character [18]. As a result the ground states of the nucleus with quantum numbers and masses and , produced in the reaction (11), are populated statistically.

A statistical population of the ground states of the nucleus entails a statistical population of the mass eigenstates and of the H-like ion with masses and and the mass-difference , produced in the reaction (11). As a result coherent states and should be created with equal probabilities , that prohibits any time dependence of both the total EC-decay rates and the -decay rates of H-like heavy ions.

We would like to notice that in reaction (11) the H-like ions are produced both in the ground hyperfine states with atomic spin and in the excited hyperfine state with atomic spin , which decays into the ground hyperfine state with the lifetime of order of [10]. Of course, such transitions should replenish statistically the system of the mother H-like heavy ions with the ground hyperfine states with masses and and a mass splitting .

#### 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 . We have applied this mechanism to the calculation of the time modulation of the -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 , the -decay rates of the H-like heavy ions, decaying from the coherent state , should have the same period of the time dependence as the EC-decay rates. This contradicts recent experimental data at GSI [11โ13].

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 [14โ22], the system of the H-like heavy ions, injected into the ESR with an kinetic energy of about per nucleon, should be statistically populated with two closely spaced ground mass eigenstates and . Of course, the statistical population of the states and is not only determined by the energy-level density of the states in the nucleus produced in reaction (11), but also by transitions, defined by the -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 and and their coherent superpositions and , the EC and 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 -decay rates, measured at GSI [1, 11โ13].

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 , 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 decay rates the H-like mother ion.

#### Acknowledgment

The authors acknowledge fruitful discussions with T. Ericson.