Can Hyperfine Excitation explain the Observed Oscillation-Puzzle of Nuclear Orbital Electron Capture of Hydrogen-like Ions?
Nicolas Winckler, Katarzyna Siegien-Iwaniuk, Fritz Bosch, Hans Geissel, Yuri A. Litvinov, Zygmunt Patyk
aa r X i v : . [ nu c l - t h ] J u l Can Hyperfine Excitation explain the Observed Oscillation-Puzzle of Nuclear OrbitalElectron Capture of Hydrogen-like Ions?
Nicolas Winckler, Katarzyna Siegie´n-Iwaniuk, Fritz Bosch, Hans Geissel, Yuri Litvinov, ∗ and Zygmunt Patyk Gesellschaft f¨ur Schwerionenforschung (GSI), Planckstrasse 1, D-64291 Darmstadt, Germany Soltan Institute for Nuclear Studies, Hoza 69, PL-00-681 Warsaw, Poland Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
Modulated in time orbital electron capture (EC) decays have been observed recently in storedH-like Pr and Pm ions. Although, the experimental results are extensively discussed inliterature, a firm interpretation has still to be established. Periodic transitions between the hyperfinestates could possible lead to the observed effect. Both selected nuclides decay to stable daughternuclei via allowed Gamow-Teller transitions. Due to the conservation of total angular momentum,the allowed EC decay can only proceed from the hyperfine ground state of parent ions. In this workwe argue that periodic transitions to the excited hyperfine state (sterile) in respect to the allowedEC decay ground state cannot explain the observed decay pattern. PACS numbers: 23.40.-s, 31.30.Gs, 32.10.Fn
Experiments studying electron capture decay of storedhighly-charged
Pm and
Pm ions have been per-formed recently at GSI, Darmstadt. In these experi-ments, the highly-charged radioactive ions were producedvia fragmentation of about 500 MeV/u
Sm projectiles.The mono-isotopic beams in a selected ionic charge-statewere separated in-flight by the fragment separator FRS[1] and stored in the storage-cooler ring ESR [2]. In theESR, the ions were cooled employing first the stochas-tic [3] and then the electron cooling [4]. The cooledions were circulating in the ESR with revolution frequen-cies of about 2 MHz. Their decay properties have beenmeasured with time-resolved Schottky Mass Spectrome-try (SMS) [5, 6, 7]. Each ion at each revolution induceda mirror charge on a pair of capacitive pick-up platesinstalled inside the ring aperture. The subsequent fastFourier transform yielded revolution frequency spectra.While the frequencies in such spectra reflect the mass-over-charge ratios of the ions [7, 8], the areas of thefrequency peaks are proportional to the correspondingnumber of stored ions and to the square of their atomiccharge [7, 9].Two methods have been developed for the decay stud-ies [9]. In the first method, the number of stored parentand daughter ions has been monitored in time. A few tento a few hundred parent
Pm and
Pm ions have beenstored as fully-ionized, hydrogen- (H-like) or helium-like(He-like) ions [10, 11]. Constants of the three-body β + , λ β + , and orbital electron capture (EC), λ EC , decays aswell as the atomic-loss constant, λ loss , have been deter-mined, where the latter is mainly due to collisions withthe rest gas atoms and atomic capture of the electrons inthe cooler. In Table I we summarize the decay constantsobtained in Refs. [10, 11]. In the following we will referto this method as the many-ion (MI) spectroscopy.The second method has been applied in Ref. [12]and employs the sensitivity of SMS to single stored ions[5, 6, 7]. At maximum three parent ions were simultane- ously injected into the ESR and their frequencies, thatis the masses, were monitored in time. In EC decay themass of the ion changes by the corresponding Q EC valuewhile the atomic charge-state is preserved. Therefore,the EC decays were unambiguously identified by observ-ing sudden changes in the revolution frequency of theions. Several thousands EC decays have been measuredin this way. We refer to this method as the single-ion (SI)spectroscopy. Surprisingly, it has been observed that thenumber of EC decays per time unit deviates from theexpected exponential decay. The data were described byadding a modulation term (1 + 0 . × cos(0 . · ∆ t )),where t is the time after the injection of the ions into theESR, superimposed on the exponential decay [12]. Thefit parameters taken from Ref. [12] are given in Table II.The interpretation of the observation above has at-tracted the attention of many physicists working in differ-ent fields. Quantum beat phenomenon due to the emittedneutrinos, which are flavour - but not mass eigenstates ofthe weak-interaction hamiltonian, has been suggested by TABLE I: Measured β + , EC and atomic-loss decay constantsfor fully-ionized, H-like (in bold), and He-like Pr [10] and
Pm [11] ions obtained in MI measurements (see text). Thedecay constants are given in the rest frame of the ions. Thetotal decay constant λ = λ EC + λ β + + λ loss is presented inthe last column.Ion λ β + [ s − ] λ EC [ s − ] λ loss [ s − ] λ [ s − ] Pr + Pr Pm + Pm TABLE II: The decay parameters obtained in SI measure-ments (see text) for Pr (upper part) and Pm (lower part) [12]. In case a), the fits of the data points havebeen done assuming the pure exponential decay N ′ EC ( t ) = N (0) · λ EC · e − λt (the quantity N (0) · λ EC is constant), where λ = λ EC + λ β + + λ loss and the prime index denotes the timederivative. In case b), a modulated in time EC decay constanthas been assumed N ′ EC ( t ) = N (0) · λ EC · e − λt (1+ a × cos ( ωt + ϕ )). The decay constants λ for Pr have large uncertaintiesdue to a short time of the total observation [12].Fit parameters of Pr dataMethod λ [ s − ] a ω [ s − ]a) - -b) Pm dataMethod λ [ s − ] a ω [ s − ]a) - -b) several authors, see for instance Refs. [13, 14]. However,this possibility is strongly disputed, see for example Refs.[15, 16]. An alternative explanation has been proposedbased on the coupling of the electron and nuclear spinsto the rotation in the ring (Thomas precession) [17].In this Paper we investigate the hypothesis of periodictransfers between the hyperfine states (e.g. due to in-teractions with electromagnetic fields or with the coolingsystem) of H-like Pr and Pm ions. Due toconservation of the total angular momentum, the upperhyperfine state does not decay by allowed EC decay [10]which can cause the observed modulations.A detailed theoretical description of the EC decay ratesin H-like and He-like ions has been performed in Refs.[18, 19]. There, it has been proven that the ratio of theserates equal to λ H − like EC ≈ / · λ He − like EC to a few percent.Based on this fact, we will show that the measured de-cay constants in Refs. [10, 11] (see Table I) cannot acco-modate the hypothesis of periodic transfers between thehyperfine states.Moreover, we discuss the possibility of observing themodulated decays also in MI experiments, which, if suc-cessful, would be of a great advantage due to the smallstatistics presently achievable in SI measurements.Both investigated nuclei Pr and
Pm have spin I = 1 + and decay by allowed Gamow-Teller transitionto the ground state of stable daughter nuclei with spin0 + . The parent H-like ions can therefore have two hyper-fine states |−i and | + i each with total angular momenta F − = I − / F + = I + 1 /
2, respectively. Theorder of the hyperfine states depends on the sign of thecorresponding nuclear magnetic moment. For the mag-netic moment parallel to the nuclear spin the hyperfineground state has spin F − and in the opposite case it is TABLE III: Hyperfine splitting parameters. The ion and thenuclear transition are given in the first two columns. Nuclearmagnetic moments used in this work are given in the secondcolumn. The hyperfine splitting δE and the relaxation decayconstants λ hf are given in the third and fourth columns, re-spectively. The excitation constant b (see text) is given in thelast column.Ion Transition µ/µ N δE [eV] λ hf [ s − ] b [ s − ] Pr 1 + → + +2.5 1.26 38.2 7.8 Pm 1 + → + +2.5 1.12 26.2 5.4 F + . However, in the allowed Gamow-Teller EC transi-tions I → I ±
1, due to conservation of the total angularmomentum, only states can decay that have spins F ± .The energy splitting δE between the two states |−i and | + i can be estimated by using [20]: δE = 43 α ( αZ ) µµ N mm p I + 12 I A ( αZ ) mc , (1)where m , m p , µ , µ N , and α are the electron mass, protonmass, nuclear magnetic moment, nuclear magneton, andthe fine-structure constant, respectively. The relativisticfactor A ( αZ ) is defined as follows A ( αZ ) = 1(2 p − ( αZ ) − p − ( αZ ) . (2)We note, that the corrections for the nuclear charge dis-tribution, the nuclear magnetization distribution and theQED effects are not included in Eq. (1).The probability for a spontaneous transition from theexcited hyperfine state |±i with angular momentum F ± to the ground state |∓i is expressed by a simple for-mula [21, 22] λ hf = 4 α δE hm c F ∓ + 12 I + 1 . (3)The calculated energy splitting δE and the decay proba-bility λ hf are presented for H-like Pr and
Pm ionsin Table III. The magnetic moment of
Pr has beenestimated in Ref. [23]. The
Pm nucleus has a similarstructure, therefore we applied the same magnetic mo-ment values as in the case of
Pr. Note, the hyperfinedecay constant λ hf depends on the ninth power of theatomic number Z . In our case, the ground state of H-likeions |−i has the spin F − and can decay via EC and β + decay. However, the excited state | + i can only decay by β + decay. We note, that ions in both hyperfine statescan be lost due to atomic interactions. We will denotethe ions as active or non-active to distinguish the ionswhich can or cannot decay by the allowed EC decay, re-spectively. The number of active ions (A) in the state |−i is N A ( t ) and the number of non-active (N) nuclei inthe state | + i is N N ( t )We simulate the time-modulation of the number of ECdecaying ions by introducing a simple mechanism: nucleiin the A-state |−i are periodically excited with a proba-bility b ×{ ωt + ϕ ) } to the N-state | + i . The excitedstate | + i decays spontaneously to the ground state |−i with decay constant λ hf (See Fig. 1). Experimentally,such periodic excitations could be due to motion in theelectromagnetic fields of the ESR or due to spin-flip re-actions in the cooler. The relevant hyperfine splittingparameters are summarized in Table III. The modula- F r ac t i on o f i on s active (upper)time [s] non-active (lower) total mother daughter FIG. 1: Fractions of EC active mother nuclei (upper solid),non-active mother nuclei (lower solid) and the total number ofmother nuclei (dot) and fraction of daughter nuclei (dashed-dotted) as a function of laboratory time. Note, the total num-bers of mother and daughter nuclei almost do not oscillate intime. tion of the EC decay constant is described by the set offour parameters { λ hf , b, ω, ϕ } . For the sake of simplicitywe put ϕ = 0. We can write three coupled differentialequations for the number of parent A-ions N A ( t ), N-ions N N ( t ), and for the number of daughter nuclei N D ( t ): N ′ A ( t ) = − ( λ EC + λ loss + λ β + ) N A ( t )+ λ hf N N ( t ) − b × { ωt ) } N A ( t ) , (4) N ′ N ( t ) = − ( λ loss + λ β + ) N N ( t ) − λ hf N N ( t ) + b × { ωt ) } N A ( t ) , (5) N ′ D ( t ) = − λ loss N D ( t ) + λ EC N A ( t ) , (6)where the prime index denotes the time derivative. Theseequations have been solved numerically using the Eulermethod. A time step of 0.00056 s has been chosen. In thebeginning we establish an approximate relation betweenthe constants λ hf and b . Let us define x = < N N ( t ) >< ( N N ( t ) + N A ( t )) > (7)which is the ratio of the average number of N-nuclei tothe total number of parent ions. An average number ofions decaying from the N-state to the A-state ( N → A ) a b/ hf x EC (a)/ EC (0)-1 FIG. 2: The ratio b/ λ hf (dotted), the renormalization factor λ EC ( a ) /λ EC (0) (solid) and the average occupancy of the ex-cited hyperfine state (dashed) as a function of the oscillationamplitude a . (The meaning of the vertical axis is given in thelegend.) approximately equals to xλ hf and it should be equal tothe number of ions b (1 − x ) being excited from the A-stateto the N-state ( A → N ) xλ hf ≈ b (1 − x ) ⇒ x ≈ b/λ hf b/λ hf . (8)For a fixed ratio x , the parameters λ hf and b are free.Using Eq. 7 the ratio x can be connected to the mod-ulation amplitude a : x ≈ a/ (1 + a ). Comparing thisrelation with Eq. 8 we find that a ≈ b/λ hf For theobserved a ≈ . x ≈ . − x fraction of par-ent nuclei is active. Thus, the effective decay constant λ ≈ (1 − x ) λ EC + λ β + λ loss is smaller since some ofthe ions are in ”sterile” N-state. Therefore, if the peri-odic transfer between the hyperfine states occurs in theMI experiments, the decay constants obtained in H-likeions shall be readjusted. Due to the absence of hyper-fine splitting, no such adjustment is needed in the caseof He-like ions. The measured ratio of decay constants λ H − likeEC /λ He − likeEC has to be corrected approximately by1 / (1 − x ) = 1 .
20. The numerical solutions of Eqs. 4-6fitted to the experimental data [10] gives the correctionfactor of 1.23. However, the readjusted experimental ra-tios for
Pm and
Pr ions are 1.77(7) and 1.83(10)which disagree, respectively, by about 4 and 3 standarddeviations to the theoretical ratio of 1.5.The numerical solutions of Eqs. 4-6 for a fixed valueof the oscillation amplitude a were fitted to the MI ex-perimental mother and daughter populations [10]. As aresult we obtained the ratio b/λ hf , the average occupancyof the excited hyperfine state x and the EC decay con-stant λ EC ( a ). The results as a function of the amplitudea are plotted in Fig. 2.Let us analyze a different model and assume that in-deed the electron capture probability varies in time as1 + 0 . .
89 ∆ t + ϕ ), where ∆ t is the time intervalfrom the creation until the decay of the ion. The originof such modulations is still intensively discussed. Exper-iments with implanted neutral Pm and
Re atoms,performed in Berkeley [24] and in Garching [25], respec-tively, have not observed time-modulations of EC decays.The mathematical model with the time-modulated ECconstant λ EC has the following form: dN M ( t ) = − λ EC { a cos ( ωt + ϕ ) } N M ( t ) dt − ( λ loss + λ β + ) N M ( t ) dt, (9) dN D ( t ) = λ EC { a cos ( ωt + ϕ ) } N M ( t ) dt − λ loss N D ( t ) dt, . (10)The decay constants can be taken from Table I and thecorresponding time-modulated parameters a and ω canbe taken from Table II. The solution of Eq. 9 can bewritten as: N M ( t ) = N M (0) e − λ loss t − λ β + t − λ EC ( t − a sin ( ωt + ϕ ) /ω ) . The Eq. 10 has been integrated analytically and alsochecked numerically. The obtained results show that theassumption of a time-dependent decay probability ob-served in SI measurements is in perfect agreement withthe decay and growth curves extracted from the experi-mental data of MI experiments.However, we can construct a different approach andassume that all decay constants are time-modulated or,what is mathematically equivalent, that the time scalefluctuates. The numbers of mother N M ( t ) and daughter N D ( t ) nuclei are connected by the two following equa-tions: dN M ( t ) = − dt { a cos ( ωt + ϕ ) } × ( λ EC + λ loss + λ β + ) N M ( t ) , (11) dN D ( t ) = dt { a cos ( ωt + ϕ ) } × ( λ EC N M ( t ) − λ loss N D ( t )) . (12)Eqs 11-12 can be easily solved by introducing a fluctu-ating time scale t ′ connected with time t by the relation t ′ = t − a sin ( ωt + ϕ ) /ω . Thus, the numbers of motherand EC daughter ions can be expressed by analytic func-tions of time t ′ (see e.g. [10]) . For large t (comparedwith 1 /ω ) the ratio of both times is approaching one.We see that experimental populations in MI can be fit-ted with almost the same parameters for a = 0 and for a = 0. We have shown that if the modulation appears onlyfor H-like ions then the ratio λ H − likeEC /λ He − likeEC ≈ / λ H − likeEC /λ He − likeEC ratios from MI experiments are in excellent agreementwith the factor 3 /
2. This disagreement could be resolvedif the described modulation appears in He-like ions aswell. However, there are no hyperfine states in He-likeions and the discussed mechanism is physically disabled.We emphasize, that in order to describe consistently thepresent data of MI and SI experiments, the modulationphenomenon with similar amplitude has also to occur inHe-like ions. Therefore, it is indispensable to perform SIexperiments on He-like Pm or Pr ions. Fur-thermore, the ratio λ H − likeEC /λ He − likeEC turned out to bea sensitive probe and more accurate measurements are,therefore, required.We have investigated the consistency of the measureddata in respect to the modulated EC decay constant.We have shown that the 20 % modulation amplitudedetermined from SI experiments translates into a tiny,much less than a percent, modulation amplitude of decaycurves measured with the MI spectroscopy. Such accu-racy is presently out of reach for the MI experiments. ∗ Electronic address: [email protected]; On leave fromGSI Helmholtzzentrum f¨ur Schwerionenforschung[1] H. Geissel et al. , Nucl. Instr. Meth.
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