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 |m′〉 and |m″〉 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
have an unexpected periodic time modulation of exponential decay curves. The rates of the number NdEC of daughter ions 140Ce58+and 142Nd60+

dNdEC(t)dt=λEC(t)Nm(t),
where Nm(t) is the number of the H-like mother ions 140Pr58+ or 142Pm60+[1] and λEC(H)(t) is the EC-decay rate, are periodic functions, caused by a periodic time dependence of the EC-decay rates
λEC(t)=λEC(1+aECcos(ωECt+ϕEC))
with a period TEC=2π/ωEC≃7seconds, an amplitude aEC≃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 Δm212=m22-m12 of the squared neutrino masses m2 and m1 as follows:
ωEC=2πTEC=Δm2122γMm,
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=Δm212/2Mm 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〉=cosθ|m′〉+sinθ|m′′〉
of the wave functions of two mass eigenstates |m′〉 and |m′′〉 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 |m′〉 and |m′′〉 should be orthogonal 〈m′∣m′′〉=0.

2. EC- and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M51"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">β</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>-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]
λEC(m)(t)=λEC(1+sin2θcos(ΔEm′m′′t)),
where λEC is the EC-decay constant [2, 3, 10] and ΔEm′m′′ is the energy difference of the ground mass eigenstates |m′〉 and |m′′〉. This shows a periodic dependence of the EC-decay rate with a period inversely proportional to ΔEm′m′′

TEC=2πγΔEm′m′′.
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 aEC≃0.20. Since aEC=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+sin2θcos(ΔEm′m′′t)),
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 |m′〉 and |m′′〉 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 |m̃〉=-sinθ|m′〉+cosθ|m′′〉. Due to statistical equivalence and indistinguishability of the coherent states |m〉=cosθ|m′〉+sinθ|m′′〉 and |m̃〉=-sinθ|m′〉+cosθ|m′′〉 the probabilities Pm and Pm̃ of the production of the coherent states |m〉 and |m̃〉, related by Pm+Pm̃=1, should be equal Pm=Pm̃=1/2.

The decay rates λEC(m̃)(t) and λβ+(m̃)(t) of the EC and β+ decays of the H-like heavy ions from the coherent state |m̃〉 are equal to
λEC(m̃)(t)=λEC(1-sin2θcos(ΔEm′m′′t)),λβ+(m̃)(t)=λβ+(1-sin2θcos(ΔEm′m′′t)).
The total EC-decay and β+-decay rates of the H-like heavy ions from the coherent states |m〉 and |m̃〉 are defined by
λEC(t)=PmλEC(m)(t)+Pm̃λEC(m̃)(t)=λEC(1+sin2θ(Pm-Pm̃)cos(ΔEm′m′′t))=λEC,λβ+(t)=Pmλβ+(m)(t)+Pm̃λβ+(m̃)(t)=λβ+(1+sin2θ(Pm-Pm̃)cos(ΔEm′m′′t))=λβ+.
Since Pm=Pm̃=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 |m′〉 and |m′′〉 and, correspondingly, the coherent states |m〉 and |m̃〉 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)++⋯,
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],
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 |m′〉 and |m′′〉 of the H-like ion AX(Z-1)+ with masses Mm′ and Mm′′ and the mass-difference |Mm′-Mm′′′|~10-15eV, produced in the reaction (11). As a result coherent states |m〉=cosθ|m′〉+sinθ|m′′〉 and |m̃〉=-sinθ|m′〉+cosθ|m′′〉 should be created with equal probabilities Pm=Pm̃=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 AXF=3/2(Z-1)+→AXF=1/2(Z-1)++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 AXF=1/2(Z-1)+ 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|m′〉+sinθ|m′′〉, 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 |m′〉 and |m′′〉. Of course, the statistical population of the states |m′〉 and |m′′〉 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 |m′〉 and |m′′〉 and their coherent superpositions |m〉 and |m̃〉, 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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