Weak decays of H like 140 Pr 58+ and He like 140 Pr 57+ ions

Transcription

1 Weak decays of H like 40 Pr 58+ and He like 40 Pr 57+ ions A. N. Ivanov a M. Faber a R. Reda c P. Kienle bc arxiv:07.384v [nucl-th] 0 Feb 008 December 03 a Atominstitut der Österreichischen Universitäten Technische Universität Wien Wiedner Hauptstrasse 8-0 A-040 Wien Österreich b Physik Department Technische Universität München D Garching Germany c Stefan Meyer Institut für subatomare Physik Österreichische Akademie der Wissenschaften Boltzmanngasse 3 A-090 Wien Österreich Abstract The nuclear K shell electron capture (EC) and positron (β + ) decay constants λ EC and λ β + of H like 40 Pr 58+ and He like 40 Pr 57+ ions measured recently in the ESR ion storage ring at GSI were calculated using standard weak interaction theory. The calculated ratios R = λ EC /λ β + of the decay constants agree with the experimental values within an accuracy better than 3%. PACS:.5.Ff 3.5.+g 3.40.Bw t E mail: ivanov@kph.tuwien.ac.at Tel.: Fax: E mail: faber@kph.tuwien.ac.at Tel.: Fax: E mail: Paul.Kienle@ph.tum.de

2 Introduction The neutral 40 Pr 0+ atoms decay with a 99.4 % branch to the ground state of 40 Ce 0+ via a pure Gamow Teller transition []. The K shell EC to β + ratio was measured as R EC/β + = 0.74(3) by Biryukov and Shimanskaya [] and R EC/β + = 0.90(8) by Evans et al. [3]. Because of the discrepancy of these results the experiment was repeated by Campbell et al. who found R EC/β + = 0.73(3) and claimed a disagreement of about 5 % with the theoretical value by Bambynek et al. [5]. This problem came up again when Litvinov et al. succeeded in measuring the β + and orbital electron capture decay rates in fully ionised H like 40 Pr 58+ and He like 40 Pr 57+ ions in the Experimental Storage Ring (ESR) at GSI in Darmstadt [6]. For the He like 40 Pr 57+ ion a ratio R EC/β + = 0.96(8) agrees well with R EC/β + = 0.90(8) by Evans et al. [3] but deviates by about.5 standard deviations from the values of [] and [4]. In addition Litvinov et al. [6] found that the H like 40 Pr 58+ ion with one electron in the K shell decays.49(8) times faster than the He like one with two electrons in the K shell. In this work we show that using standard weak interaction theory one can reproduce the experimental values of the ratios of the EC and β + decay constants of the H like 40 Pr 58+ and He like 40 Pr 57+ ions with an accuracy better than 3 %. The paper is organised as follows. In Section we give the results of the calculation of the weak decay constants of the H-like 40 Pr 58+ and the He like 40 Pr 57+ ions and their ratios. We compare the theoretical results with the experimental data. In the Conclusion we discuss the obtained results. In Appendices A and B we adduce the detailed calculations of the weak decay constants of the H-like 40 Pr 58+ and the He like 40 Pr 57+ ions. Weak decay constants For the calculation of the weak decay constants of H-like and He like ions we use the Hamiltonian of the weak interaction taken in the standard form [7] H W (x) = G F V ud [ ψ n (x)γ µ ( g A γ 5 ) ψ p (x)] [ ψ νe (x)γ µ ( γ 5 )ψ e (x)] + h.c. () where G F =.66 0 MeV is the Fermi constant V ud = ± 0005 is the CKM matrix element g A =.695 ± is the axial coupling constant [8] ψ n (x) ψ p (x) ψ νe (x) and ψ e (x) are the operators of the neutron proton neutrino and electron/positron fields respectively. The detailed calculations of the weak decay constants of the H like 40 Pr 58+ and He like 40 Pr 57+ ions are given in Appendices A and B. Since the EC decay of the H like 40 Pr 58+ ion from the hyperfine state 40 Pr 58+ with F = 3 is suppressed (see Appendix F= 3 B and [6 9]) we take into account that the H like 40 Pr 58+ ion decays from the hyperfine ground state 40 Pr 58+ with F =. F= Using the Hamiltonian of the weak interaction () and following [5] for the EC decay constants of the H like 40 Pr 58+ and the He like 40 Pr 57+ F= I= ions in the ground states we

3 obtain the following expressions (see Appendix A) EC = 3 F + M ψ (Z) λ (He) EC = 3 s I + M ψ (Z ) s Q H π ψ (Z) (s) Q He π () where F = / and I = are the total angular momenta of the H like 40 Pr 58+ and the He like 40 Pr 57+ ions respectively Q H = (3348 ± 6) kev and Q He = (335 ± 6) kev are the Q values of the decays 40 Pr Ce ν e and 40 Pr Ce ν e M is the nuclear matrix element of the Gamow Teller transition [0] ψ (Z) s and ψ (Z ) s are the wave functions of the H like ions 40 Pr 58+ and 40 Ce 57+ respectively Z = 59 is the electric charge of the mother nucleus 40 Pr 59+ and ψ (Z) (s) is the wave function of the He like ion 40 Pr 57+ ψ (Z) s and ψ (Z ) s ψ (Z) (s) are defined by d ψ (Z) 3 xψ (Z) s ( r )ρ( r ) s = d3 xρ( r ) d 3 x ψ (Z ) s ψ (Z) d 3 x ψ (Z ) s ( r )ψ (Z) (s) ( r (s) = r )ρ( r ) (3) d3 x ρ( r ) where ρ( r ) is the nuclear density [0]. Using the Woods Saxon shape for the nuclear density ρ(r) and the Dirac wave functions for the H like ion and the He like ion [ ] one gets ψ (Z) s = ψ (Z ) s ψ (Z) (s) =.66/ πa 3 B where a B = /Zαm e = 897 fm is the Bohr radius of the H like ion 40 Pr 58+ ; m e = 0.5 MeV is electron mass and α = / is the fine structure constant [8]. The β + decay constants of the H like 40 Pr 58+ and the He like 40 Pr 57+ ions are defined by (see Appendix A) β + = λ (He) β + = M f(q H F + 4π 3 β + Z ) 3 M f(q He I + 4π 3 β + Z ). (4) Since the Q values of the decays 40 Pr Ce 58+ +e + +ν e and 40 Pr Ce e + +ν e are equal Q H β = Q He + β = Q + β + = (3396±6) kev the Fermi integral f(q β + Z ) = (.±0.03) MeV 5 is defined by the phase volume of the final states of the decays and the Fermi function describing the Coulomb repulsion between the positron and the nucleus 40 Ce 58+ for Z = 59 [7]. The theoretical values of the weak decay constants are defined up to the unknown nuclear matrix element M [0] of the Gamow Teller transition which cancels in the ratios. The theoretical ratios of the weak decay constants are given by EC/β = 3π Q H ψ(z) s + f(q β + Z ) =.40(4) R (H)th In the ratio 3 of the EC decay constant λ(h) EC the factors 3 and are caused by the hyperfine structure of the ground state of the H like 40 Pr 58+ ion and the phase volume of the final state of the F= decay respectively (see Appendix A). 3

4 R (He)th ψ (Z) EC/β = π Q He ψ(z ) s (s) = 0.94(3) + f(q β + Z ) R (H/He)th EC/EC = I + F + ψ (Z) s ψ (Z ) s ψ (Z) Q H (s) Q He =.50(4) (5) calculated for F = constants are [6] and I =. The experimental data on the ratios of the weak decay R (H)exp EC/β + =.36(9) R (He)exp EC/β + = 0.96(8) R (H/He)exp EC/EC =.49(8). (6) The theoretical values agree well with the experimental data [6]. 3 Conclusion We have calculated the EC and β + decay constants of the H like and He-like ions 40 Pr 58+ and 40 Pr 57+ respectively. Following the standard theory of weak decays of heavy nuclei [5] we have expressed the decay constants in terms of the nuclear matrix element M of the Gamow Teller transition. We have shown that the complete set of experimental data on the ratios of weak decay constants of the ions 40 Pr 58+ and 40 Pr 57+ can be explained within the standard theory of weak interactions of heavy ions. Our theoretical values of the ratios of the decay constants for H like 40 Pr 58+ and He like 40 Pr 57+ ions agree with the experimental data within an accuracy better than 3 %. This high precision of the theoretical analysis of the weak decays of the H like 40 Pr 58+ and He like 40 Pr 57+ ions is due to the small number of electrons the behaviour of which can be described by the solution of the Dirac equation. The dependence of the ratio R EC/β + on the electron structure of the decaying system is confirmed for both the experimental data and our theoretical analysis of the He like ion 40 Pr 57+. Indeed for the He like 40 Pr 57+ ion the ratio R (He)exp EC/β = 0.96(8) is smaller than for the H like 40 Pr ion R (H)exp EC/β =.36(9). Of course such a regularity should be confirmed experimentally + by the measurements of EC and β + decays of the Li like 40 Pr 56+ ions. According to the hyperfine structure of the H like ion 40 Pr 58+ [3] the bound electron can be in two states with a total angular momentum F = and F = 3 with the energy splitting equal to E = E sf= E sf= 3 = α(αz) 3 µ m { e ( δ)( ε) µ N m p ( + γ)( + γ) + x rad (7) where γ = (αz) µ = +.5 µ N is the magnetic moment of the nucleus 40 Pr 59+ [6] µ N = e/m p is the nuclear magneton m p = MeV/c is the proton mass δ is the nuclear charge distribution correction ε is the nuclear magnetisation distribution correction (the Bohr Weisskopf correction [4]) x rad denotes the radiative correction calculated to lowest order in α and αz [5]. Numerical estimates carried our for different 4

5 ions by Shabaev et al. [3] show that for the calculation of E with an accuracy better than % one can drop the contributions of the corrections δ ε and x rad and get E = ev. The lifetime τ F= 3 of the hyperfine state of the H like 40 Pr 58+ ion is defined F= 3 by the radiative transition 40 Pr Pr γ only. It is equal to τ F= 3 F= F= 3 = sec [6]. Since the lifetime τ F= 3 = sec is much shorter than the cooling time ions decay into the hyperfine ground states 40 Pr 58+. F= of about s [6] all H like 40 Pr 58+ F= 3 In our calculation of the EC decay of the H like 40 Pr 58+ ion from the hyperfine ground state 40 Pr 58+ F= the hyperfine structure is taken into account in the spinorial wave function of the bound electron (see Appendices A and B). The influence of the hyperfine structure on the probabilities of weak decays of heavy ions at finite temperature has been investigated by Folan and Tsifrinovich [9]. As has been mentioned by Folan and Tsifrinovich [9] the EC decay of the H like ion 40 Pr 58+ F= 3 from the hyperfine state with F = 3 is forbidden by a conservation of angular momentum. This agrees with our analysis (see Appendix B). Our result for the ratio R (H/He)th EC/EC = 3 agrees quantitatively well with that obtained by Patyk et al. [6] who also accounted for the hyperfine structure of the H like 40 Pr 58+ ion and the estimate carried out by Litvinov et al. [6]. According to Patyk et al. [6] and Litvinov et al. [6] the ratio R (H/He)th EC/EC is fully caused by the statistical factors = 3 R (H/He) EC/EC = I + F + = 3 (8) calculated for I = and F =. Unfortunately this assertion is not completely correct. The result R (H/He)th EC/EC = 3 appears only at the neglect of the electron screening of the electric charge of the nucleus Z in the He like 40 Pr 57+ ion. Having neglected the electron screening of the electric charge of the nucleus Z one can represent the wave function of the bound (s) state of the He like ion in the form of the product of the one electron Dirac wave functions. In this case ψ (Z) s = ψ (Z ) s ψ (Z) (s) and the ratio R (H/He)th EC/EC given by Eq.(5) reduces to Eq.(8). Unlike our analysis of the weak decays of the H like and He like ions Patyk et al. [6] as well as Litvinov et al. [6] did not analyse the contributions of the Coulomb wave functions of the bound electrons averaged over the nuclear density. Nevertheless such contributions are important for the correct calculation of both the ratio R (H/He) EC/EC for ions with small electric charges Z and the ratios R(H) EC/β and R (He) + EC/β + (see Eq.(5)) which were calculated by neither Patyk et al. [6] nor Litvinov et al. [6]. Acknowledgement We acknowledge many fruitful discussions with M. Kleber F. Bosch and Yu. A. Litvinov in the course of this work. One of the authors (A. Ivanov) is greatful to N. I. Troitskaya and V. M. Shabaev for the discussions of properties of heavy ions. 5

6 Appendix A: Calculation of weak decay constants of H-like 40 Pr 58+ and He like 40 Pr 57+ ions In this Appendix we adduce the detailed calculations of the EC and β + decay constants of the H like 40 Pr 58+ ion in the hyperfine ground state 40 Pr 58+ and the He like 40 Pr 57+ F= I= ion in the ground (s) state. For the calculation of the weak decay constants of H-like and He like ions we follow [5] and use the Hamiltonian Eq.(). EC decay of the H like 40 Pr 58+ ion The K shell electron capture decay (the EC decay) 40 Pr 58+ F= 40 Ce 58+ I=0 +ν e describes a transition of the H like mother ion 40 Pr 58+ from the hyperfine ground state F M F with F = / and M F = ±/ into the daughter ion 40 Ce 58+ in the ground state I I z = 0 0. The EC decay constant of the H like 40 Pr 58+ ion is defined by EC = M m F + M F FM F (π) 4 δ (4) (p d + k p m ) d 3 p d (π) 3 E d d 3 k (π) 3 E ν (A-) where p m p d and k are 4 momenta of the mother ion the daughter ion and the neutrino respectively. According to [5] the amplitudes MF F of the EC decay are equal to = { M m E d E ν + 3 [ϕ σ ϕ n p+ ] [ϕ n σ ) σ ϕ ν ( e ] 6 [ϕ σ ϕ n+ p+ ϕ σ ϕ n p ] [ϕ n σ ) σ ϕ ν ( e+ ] = M m E d E ν { 6 [ϕ σ ϕ n+ p+ ϕ n 3 [ϕ n+ ] [ϕ n σ ) σ ϕ ν ( e+ ] ] [ϕ n σ ) σ ϕ ν ( e ] (A-) where we have used the non relativistic approximation for nucleons [5] /3 and /6 are the Clebsch Gordon coefficients of the spinorial wave function of the bound electron in the hyperfine state with F = caused by the spinorial wave functions of the nucleus 40 Pr 59+ with spin I = and the electron with spin s = n = k/e ν is a unit vector alone the 3 momentum of the neutrino and is the matrix element of the Gamow Teller transition defined by G F = g A V ud d 3 x Ψd( r )Ψ m ( r ) ψ (Z) s (r) (A-3) where Ψd ( r ) and Ψ m( r ) are the wave functions of the daughter 40 Ce 58+ and mother 40 Pr 59+ nuclei respectively and ψ (Z) s (r) is the radial wave function of the bound electron in the hyperfine ground state of the H like 40 Pr 58+ ion with electric charge Z. It is equal F= to [] ψ (Z) + γ ( r ) γ (r) = e r/a B (A-4) Γ(3 + γ) a B πa 3 B 6

7 where γ = α Z []. For numerical calculations we set Ψ d ( r )Ψ m( r ) ρ(r) where ρ(r) has the Woods Saxon shape [0]. For the subsequent analysis it is convenient to rewrite the matrix element Eq.(A-3) as follows where we have denoted M = g A G F V ud = M ψ (Z) s d 3 xρ(r) ψ (Z) s = (A-5) d 3 xρ(r)ψ (Z) s ( r ). (A-6) d 3 xρ(r) Since a neutrino is polarised anti parallel to its spin i.e. the spinorial wave function is the eigenfunction of the operator n σ with the eigenvalue (see [7 7]) n σ ϕ ν = ϕ ν (A-7) a non zero contribution to the EC decay constant comes from the amplitude { = Mm E d E ν M ψ (Z) 6 [ϕ σ ϕ n+ p+ ϕ n s 3 [ϕ n+ ] [ϕ ϕ ν σ e ] ] [ϕ ϕ ν σ e+ ] only (A-8) where we have taken into account Eq.(A-7). Since the spinorial matrix elements are equal to the amplitude [ϕ n+ σ ϕ p+ ϕ n is [ϕ ϕ n+ σ p ] [ϕ ϕ ν σ e+ ] = + = 3 M m E d E ν ψ(z) s. ] [ϕ ϕ ν σ e ] = (A-9) (A-0) Taking the squared absolute value of the amplitude Eq.(A-0) and substituting into Eq.(A-) we get EC = 3 F + M ψ (Z) s = F + 3 M ψ (Z) s Q H π δ (4) (p d + k p m ) d3 p d d 3 k 8π = (A-) where in the ratio 3 the factors 3 and are caused by the hyperfine structure of the ground state of the H like 40 Pr 58+ ion and the phase volume of the final state of the F= decay respectively. Thus the EC decay constant of the H like 40 Pr 58+ ion is EC = 3 F + M ψ (Z) Q s H π (A-) where Q H = (3348 ± 6) kev is the Q value of the EC decay of the H like 40 Pr 58+ ion. 7

8 β + decay of H like 40 Pr 58+ ion The β + decay 40 Pr 58+ F= ion 40 Pr 58+ F= 40 Ce 57+ +e + +ν F = e describes a transition of the H like mother from the hyperfine ground state F M F with F = / and M F = ±/ in the ground state F M F with F = / and into the H like daughter ion 40 Ce 57+ F = M F = ±/. The β+ decay constant of the H like 40 Pr 58+ ion is defined by β + = M m F + M F M F F(Z E FM F F M F + ) (π) 4 δ (4) d 3 p d d 3 k d 3 p + (p d + k + p + p m ) (π) 3 E d (π) 3 E ν (π) 3 E + (A-3) where p m p d k and p + are 4 momenta of the mother ion the daughter ion the neutrino and the positron respectively F(Z E + ) is the Fermi function [7]. The amplitudes of the β + decay are equal to FM F F M F + M F δ M F M F δ M F + = M m E d { 6 [ϕ σ ϕ n+ p+ ϕ n = M m E d 6 [ϕ σ ϕ n+ p+ ϕ n 3 [ϕ n σ ϕ p+ ] [ū ν ( k ) γ ( γ5 ) v e +( p + σ + )] ] [ū ν ( k ) γ ( γ5 ) v e +( p + σ + )]δ M F + { 3 [ū ν( k )(γ + iγ ) ( γ 5 ) v e +( p + σ + )]δ M F + ] [ū ν ( k ) γ ( γ5 ) v e +( p + σ + )]δ M F (A-4) where M F = ±/ defines a polarisation of the electron in the final state. The matrix element of the Gamow Teller transition is defined by = g G F A V ud ψ (Z ) s ψ (Z) s d 3 xρ(r) = M ψ (Z ) s ψ (Z) s (A-5) where ψ (Z ) s ψ (Z) s is the matrix element of the transition of the bound electron from the initial state with the wave function ψ (Z) s into the final state with the wave function ψ (Z ) s. Since for Z = 59 this matrix element is equal to unity with a good accuracy we set ψ (Z ) s ψ (Z) β+ s = and M = M / where M is defined by Eq.(A-6). Having calculated the proton neutron matrix elements [ϕ n σ j ϕ p+ ] = δ j i δ j [ϕ n+ σ j ϕ p ] = δ j + i δ j [ϕ n+ σ j ϕ p+ ϕ σ n j ϕ p ] = δ 3j (A-6) 8

9 we get + M F = M m E d M [ū ν ( k )γ 3 ( γ 5 ) v e +( p + σ + )] δ M F + M F = M m E d M [ū ν ( k )γ 3 ( γ 5 ) v e +( p + σ + )] δ M F { [ū ν ( 3 k )(γ iγ ) ( γ 5 ) v e +( p + σ + )] δ M F { [ū ν ( 3 k )(γ + iγ ) ( γ 5 ) v e +( p + σ + )]δ M F + (A-7) The squared absolute values of the amplitudes summed over the polarisations of the final state are equal to M F =±/ + M F M F =±/ M F { tr{ˆkγ 3 ( γ 5 )(ˆp + m e )γ 3 ( γ 5 ) tr{ˆk(γ + iγ )( γ 5 )(ˆp + m e )(γ iγ )( γ 5 ) = M m E d M M { = M m E d tr{ˆkγ 3 ( γ 5 )(ˆp + m e )γ 3 ( γ 5 ) tr{ˆk(γ iγ )( γ 5 )(ˆp + m e )(γ + iγ )( γ 5 ). (A-8) The results of the calculation of the traces over Dirac matrices are given by tr{ˆk(γ iγ )( γ 5 )(ˆp + m e )(γ + iγ )( γ 5 ) = tr{ˆk(γ iγ )ˆp + (γ + iγ ) = = tr{ˆkγ ˆp + γ + tr{ˆkγ ˆp + γ + itr{ˆkγ ˆp + γ itr{ˆkγ ˆp + γ = = 8 (k x p +x + k p + ) + 8 (k y p +y + k p + ) + 8i (k x p +y + k y p +x ) 8i (k y p +x + k x p +y ) = = 6 (E ν E + k z p +z ) tr{ˆk(γ + iγ )( γ 5 )(ˆp + m e )(γ iγ )( γ 5 ) = = tr{ˆk(γ iγ )( γ 5 )(ˆp + m e )(γ + iγ )( γ 5 ) = 6 (E ν E + k z p +z ) tr{ˆkγ 3 ( γ 5 )(ˆp + m e )γ 3 ( γ 5 ) = 8 (E ν E + + k z p +z ). (A-9) Using Eq.(A-9) for the sum of the squared absolute values of the amplitudes Eq.(A-8) we obtain the following expression M F M F =±/ FM F F M F = M m E d M 4 (E ν E + 3 k zp +z ). (A-0) Substituting Eq.(A-0) into Eq.(A-3) and integrating over the phase volume of the final state we get β = + F + M F(Z E + ) δ (4) (p d + k + p + p m ) d3 p d d 3 kd 3 p + = 3π 5 M = f(q F + 4π 3 β + Z ) (A-) 9

10 where f(q β + Z ) is defined by [7] f(q β + Z ) = Qβ + m e (Q β + m e E + ) E+ m e F(Z E +) E + de +. (A-) m e Thus the β + decay constant of the H like 40 Pr 58+ ion is equal to β + = M f(q F + 4π 3 β + Z ) (A-3) and Q β + = (3396 ± 6) kev is the Q value of the β + decay of the H like 40 Pr 58+ ion. EC decay of He like 40 Pr 57+ ion The K shell electron capture decay (the EC decay) 40 Pr 57+ I= 40 Ce ν F= e describes a transition of the He like mother ion 40 Pr 57+ from the ground (s) state I I z with I = and I z = 0 ± into the H like daughter ion 40 Ce 57+ in the ground state F M F with F = / and M F = ±/. The EC decay constant is defined by λ (He) EC = M m I + I zm F IIz FM F (π) 4 δ (4) (p d + k p m ) The amplitudes of the EC decay II z FM F are equal to d 3 p d (π) 3 E d d 3 k (π) 3 E ν. (A-4) = M m E d E ν [ϕ ϕ n σ p+ ] [ϕ n σ ) σ ϕ ν ( e ] = M m E d E ν [ϕ ϕ n σ p+ = M m E d E ν [ϕ n σ ) σ ϕ ν ( e ] = M m E d E ν [ϕ n σ ) σ ϕ ν ( e+ ] [ϕ n+ σ ϕ p+ ϕ n [ϕ n+ σ ϕ p+ ϕ n ] [ϕ n σ ) σ ϕ ν ( e+ ] ] ] = M m E d E ν [ϕ ϕ n+ σ p ] [ϕ n σ ) σ ϕ ν ( e ] = M m E d E ν [ϕ ϕ n+ σ p ] [ϕ n σ ) σ ϕ ν ( e+ ] (A-5) where n = k/e ν is a unit vector alone the 3 momentum of the neutrino and we have denoted ψ (Z ) s ψ (Z) (s) = = M ψ(z ) s ψ (Z) (s) d 3 x d 3 x ψ (Z ) s ( r )ψ (Z) (s) ( r r )ρ(r ) (A-6) d3 x ρ(r ) 0

11 where ψ (Z) (s) ( r r ) is the wave function of the ground (s) state of the He like 40 Pr 57+ ion and M is the matrix element of the Gamow Teller transition defined by Eq.(A-6). For the amplitudes of the EC decay of the He like 40 Pr 57+ ion we get = + = 0 = M m E d E ν [ϕ n σ ) σ ϕ ν ( e ] = + = 0 [ϕ n+ σ ϕ p+ ϕ n ] = M m E d E ν [ϕ ϕ n+ σ p ] [ϕ n σ ) σ ϕ ν ( e+ ]. (A-7) Having calculated the spinorial matrix elements we obtain = = = = = M m E d E ν = M m E d E ν M ψ (Z ) s ψ (Z) (s) = 4 M m E d E ν = M m E d E ν M ψ (Z ) s ψ (Z) (s) (A-8) where we have used Eq.(A-6). As a result the EC decay constant of the He like 40 Pr 57+ I= ion is defined by λ (He) EC = M m I + ( + )(π) 4 δ (4) (p 0 + d + k p m ) d3 p d d 3 k = (π) 3 E d (π) 3 E ν I + 3 M ψ (Z ) s Thus the EC decay constant of the He like 40 Pr 57+ ion is equal to λ (He) EC = 3 I + M ψ (Z ) s ψ (Z) (s) Q He π ψ (Z) (s) Q He π. (A-9) (A-30) where Q He = (335 ± 6) kev is the Q value of the EC decay of the He like 40 Pr 57+ ion. β + decay of the He like 40 Pr 57+ ion The β + decay 40 Pr 57+ I= 40 Ce 56+ I =0 +e+ +ν e describes a transition of the He like mother ion 40 Pr 57+ I= from the ground (s) state I I z with I = and I z = 0 ± into the He like daughter ion 40 Ce 56+ I =0 in the ground (s) state I I z with I = 0 and I z = 0. The β + decay constant of the He like 40 Pr 57+ ion is defined by λ (He) β + = M m I + I z=0± d3 p d d 3 k d 3 p + (π) 3 E d (π) 3 E ν (π) 3 E + IIz 00 (π) 4 δ (4) (p d + k + p + p m ) F(Z E + ) Using the results obtained above and following [5 7 7] for the amplitudes the β + decay 40 Pr 57+ I= 40 Ce 56+ I =0 + e+ + ν e we obtain the following expressions (A-3) IIz 00 of + 00 = M M m E d [ϕ σ ϕ n p+ ] [ū ν ( k ) γ ( γ5 )v e +( p + σ + )]

12 0 00 = M m E d M [ϕ n+ σ ϕ p+ ϕ n [ū ν ( k ) γ ( γ5 )v e +( p + σ + )] ] 00 = M M m E d [ϕ σ ϕ n+ p ] [ū ν ( k ) γ ( γ5 )v e +( p + σ + )]. (A-3) Using Eq.(A-6) for the calculation of the proton neutron matrix elements we get + 00 = M M m E d [ū ν( k )(γ iγ ) ( γ 5 )v e +( p + σ + )] [ūν ( k )γ 3 ( γ 5 )v e +( p + σ + )] 0 00 = M m E d M 00 = M m E d M [ū ν( k )(γ + iγ ) ( γ 5 )v e +( p + σ + )]. (A-33) The squared absolute values of the amplitudes and 00 are M + 00 = M m E d tr{ˆk(γ + iγ )( γ 5 )(ˆp + m e )(γ iγ )( γ 5 ) = 8 = M m E d M (E ν E + k z p +z ) M 0 00 = M m E d tr{ˆkγ 3 ( γ 5 )(ˆp + m e )γ 3 ( γ 5 ) = 8 = M m E d M (E ν E + + k z p +z ) M 00 = M m E d tr{ˆk(γ iγ )( γ 5 )(ˆp + m e )(γ + iγ )( γ 5 ) = 8 = M m E d M (E ν E + k z p +z ) (A-34) where we have used Eq.(A-9). The sum of the squared absolute values of the amplitudes Eq.(A-34) is I z II z 00 = M m E d M 6 (E ν E + 3 k zp +z ). (A-35) Substituting Eq.(A-35) into Eq.(A-3) and integrating over the phase volume of the final state we get λ (He) 3 β = + I + M δ (4) (p d + k + p + p m ) F(Z E + ) d3 p d d 3 kd 3 p + = 64π 5 3 M = f(q I + 4π 3 β + Z ) (A-36) where f(q β + Z ) is defined by Eq.(A-). Thus the β + decay constant of the He like 40 Pr 57+ ion is equal to λ (He) β = 3 M f(q + I + 4π 3 β + Z ) (A-37) where Q β + = (3396 ± 6) kev is the Q value of the β + decay of the He like 40 Pr 57+ ion.

13 Appendix B: Calculation of the EC decay constant of the H-like 40 Pr 58+ ion from the hyperfine state 40 Pr 58+ F= 3 The calculation of the EC decay constant of the H like 40 Pr 58+ ion from the hyperfine state 40 Pr 58+ F= 3 with F = 3 is similar to that for the EC decay from the hyperfine ground state 40 Pr 58+ F= with F =. The EC decay constant is defined by EC = M m F + M F =± ±3 FM F (π) 4 δ (4) (p d + k p m ) d 3 p d (π) 3 E d d 3 k (π) 3 E ν (B-) where p m p d and k are 4 momenta of the mother ion the daughter ion and the neutrino respectively. The amplitudes FM F of the EC decay are given by = M m E d E ν [ϕ ϕ n σ p+ ] [ϕ ν = { M m E d E ν 3 [ϕ n + 3 [ϕ σ ϕ n+ p+ ϕ n { = M m E d E ν + 3 [ϕ σ ϕ n+ p+ ϕ n σ ϕ e+ ] σ ϕ p+ ] [ϕ ν ] [ϕ ν 3 [ϕ n+ σ ϕ e ] σ ϕ e+ ] ] [ϕ ν ] [ϕ ν = M m E d E ν [ϕ ϕ n+ σ p ] [ϕ ν σ ϕ e+ ] σ ϕ e ] σ ϕ e ]. (B-) Since the spinorial matrix elements vanish [ϕ ϕ n+ σ p ] [ϕ ν [ϕ n+ σ ϕ p+ ϕ n σ ϕ e+ ] + [ϕ ϕ n+ σ p+ ϕ n [ϕ ϕ n σ p+ ] [ϕ ν [ϕ ϕ n σ p+ ] [ϕ ν ] [ϕ ν ] [ϕ ν [ϕ ϕ n+ σ p ] [ϕ ν σ ϕ e+ ] = 0 σ ϕ e ] = 0 σ ϕ e+ ] = 0 σ ϕ e ] = 0 σ ϕ e ] = 0 (B-3) the amplitudes Eq.(B-) of the EC decay from the hyperfine state 40 Pr 58+ zero F= 3 are equal to 3 +3 = 3 + = 3 = 3 3 = 0. (B-4) This confirms the suppression of the EC decay of the H like 40 Pr 58+ ion from the hyperfine state 40 Pr 58+ pointed out in [6 9]. F= 3 3

14 References [] R. B. Firestone et al. in Table of Isotopes Eighth Edition Vol.I John Wiley & Sons Inc New York 996. [] E. I. Biryukov and N. S. Shimanskaya Sov. J. Phys. (970) 38. [3] J. L. Evans et al. Phys. Rev. C 6 (97) 37. [4] M. Campbell K. W. D. Ledingham and A. D. Baillie Nucl. Phys. A 83 (977) 43. [5] W. Bambynek et al. Rev. Mod. Phys. 49 (977)77. [6] Yu. A. Litvinov et al. (the GSI Collaboration) Phys. Rev. Lett (008). [7] H. F. Schopper in Weak interactions and nuclear beta decay North Holland Publishing Co. Amsterdam 966. [8] W. M. Yao et al. (Particle Data Group) J. Phys. G 33 (006) [9] L. M. Folan and V. I. Tsifrinovich Phys. Rev. Letters 74 (995) 499. [0] In our approach the matrix element of the Gamow Teller transition is defined by M = g A G F V ud d 3 xψ d (r)ψ m(r) where Ψ d (r) and Ψ m(r) are the wave functions of the daughter and mother nuclei respectively. We set Ψ d (r)ψ m(r) ρ(r) where ρ(r) has a Woods Saxon shape with R =. A /3 fm and slope parameter a = 0.50 fm taken from W. N. Cottingham and D. A. Greenwood in An introduction to nuclear physics Cambridge University Press Second Edition 00. [] C. Itzykson and J. B. Zuber in Quantum field theory McGraw Hill Book Co. New York 980. [] ǫ (He) (s) (Z) = kev and ǫ (He) (s) (Z ) = kev are binding energies of the He like ions 40 Pr 57+ and 40 Ce 56+ with Z = 59. [3] V. M. Shabaev J. Phys. B: At. Mol. Opt. Phys (994); V. M. Shabaev M. B. Shabaeva and I. I. Tupitsyn Phys. Rev. A (995); V. M. Shabaev et al. Phys. Rev. A 56 5 (997). [4] A. Bohr and V. F. Weisskopf Phys. Rev (950); A. Bohr Phys. Rev (95); M. Le Bellac Nucl. Phys (963). [5] D. E. Zwanziger Phys. Rev. 8 (960); S. J. Brodsky and G. W. Erickson Phys. Rev (966); J. R. Sapirstein Phys. Rev. Lett (983). [6] Z. Patyk et al. Phys. Rev. C (008). [7] E. J. Konopinski in The theory of beta radioactivity Oxford at the Clarendon Press