Observation of large enhancements of charge exchange cross sections with neutron-rich carbon isotopes

Prog. Theor. Exp. Phys. 2016, 043D05 (16 pages) DOI: 10.1093/ptep/ptw034 Observation of large enhancements of charge exchange cross sections with neutron-rich carbon isotopes I. Tanihata1,2,∗, S. Terashima1,∗, R. Kanungo3,F.Ameil 4,J.Atkinson 2,Y.Ayyad 2, D. Cortina-Gil5, I. Dillmann4,6,A.Estrad ´ e3,4, A. Evdokimov4,F.Farinon 4, H. Geissel4,6, G. Guastalla4, R. Janik7, R. Knoebel4, J. Kurcewicz4, Yu. A. Litvinov4,M.Marta 4, M. Mostazo5, I. Mukha4, C. Nociforo4,H.J.Ong 2,S.Pietri 4, A. Prochazka4, C. Scheidenberger4,6, B. Sitar7,P.Strmen 7, M. Takechi4, J. Tanaka2, H. Toki2,J.Vargas 5, J. S. Winfield4, and H. Weick4 1School of Physics and Nuclear Energy Engineering and IRCNPC, Beihang University, Beijing 100191, China 2RCNP, Osaka University, Ibaraki 567-0047, Japan 3Saint Mary’s University, Halifax, NS B3H 3C3, Canada 4GSI Helmholtz Center, 64291 Darmstadt, Germany 5Universidad de Santiago de Compostela, Santiago de Compostela, Spain 6Justus Liebig-Universität Giessen, II. Physikalisches Institut, 35390 Giessen, Germany 7Comenius University, Bratislava, Slovakia ∗E-mail: [email protected] Received November 25, 2015; Revised March 8, 2016; Accepted March 8, 2016; Published April 28, 2016 ............................................................................... Production cross sections of nitrogen isotopes from high-energy (∼950 MeV per nucleon) carbon isotopes on hydrogen and carbon targets have been measured for the first time for a wide range of isotopes (A=12 to 19). The fragment separator FRS at GSI was used to deliver C-isotope beams. The cross sections of the production of N-isotopes were determined by charge measurements of forward-going fragments. The cross sections show a rapid increase with the number of neutrons in the projectile. Since the production of nitrogen is mostly due to charge-exchange (Cex) reactions below the proton separation energies, the present data suggests a concentration of Gamow–Teller and/or Fermi transition strength at low excitation energies for neutron-rich carbon isotopes. It was also observed that the Cex cross sections were enhanced much more strongly for neutron-rich isotopes in the C-target data. ............................................................................... Subject Index D12, D23, D27, D29, D40 Charge exchange (Cex) reactions such as (p,n) and (3He,t) at intermediate and high energies bring about similar transitions to Fermi (F) and Gamow–Teller (GT) βdecays, including transitions to higher excited states that could not be populated by βdecays. Such transitions in nuclei near the stability lines have been studied and the building up of giant GT resonances has been discussed for mid-shell nuclei by Fujita et al. [1]. In that study, a change of transition strength was investigated for (3He,t) reactions at 140 MeV/nucleon using even–even TZ=1target nuclei in the f7/2shell. It was found that most of the transition strength was concentrated in the low-energy states of the nuclei produced at the bottom of the shell, such as for the case of 42Sc. The transition to the isobaric analog state (IAS) and the first GT state carry most of the transition strength below 12 MeV. In contrast, the transition strength in high-energy excitations increases when the number of nucleons © The Author(s) 2016. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. in the shell increases. For example, most of the strength is located from 6 to 12 MeV, i.e. the giant resonance region. The low-energy excited states including IAS contribute a very small amount to the total strength. This is due to the development of giant GT resonances at higher excitation energies. Fujita’s experiment shows a gradual change of the strength distribution when the number of nucleons in a valence shell increases along the stability line. Interest remains in how the strength changes when only the number of neutrons increased to unstable nuclei. Many studies have been reported [2,3] on the relationship between the β-decay strength and the cross section. These two observed values are related to each other and commonly parameterized as σ=ˆσiFi(q,ω)B(i), (1) where i=F or GT distinguishes the Fermi and GT transitions. The proportionality factor ˆσis called the unit cross section and may depend on the beam energy. The factor Fi(q,ω)describes the shape of the cross section distribution and goes to unity in the limit of zero momentum and energy transfer. We assume Fi(q,ω)∼1in the following discussion. The last factor, the beta-decay transition strength B(i), is obtained from the beta-decay ft value. If the relationship between the beta-decay and (p,n) reactions is direct, the unit cross section is expected to be a slowly changing function of the mass of the nuclei, A. Taddeucci et al. [4] studied the unit cross section for many nuclei in a wide range of masses. Also, Sasano et al. [5] studied the unit cross section systematically for medium-heavy nuclei. Those papers reported that the unit cross section changes smoothly with mass number except for light nuclei. Taddeucci et al. observed a peculiar behavior in the strength for C isotopes. The unit cross sections for 12C, 13C, and 14C do not exhibit a smooth dependence on A, but vary greatly. This contrasts strongly with the fact that the same value of unit cross section can be used for transitions to states of different excitation energies within one nuclide. This peculiar behavior could be due to an uncertainty in the distorted wave impulse approximation used, though other possibilities cannot be rejected [4]. Therefore, studies with the longer chains of C-isotopes may shed light on the isotope dependence. The beam energy dependence of such relationships was studied by Fujiwara et al. by (p,n) and (3He,t) reactions [6]. As an example, the ratio of the cross sections σ12C−>12N(gs)/σ13C−>13N(3.51)was found to be constant for beam energies from 150 to 700 MeV/nucleon. It is generally observed that the unit cross section of a Fermi transition is much smaller (1/5 to 1/10) than that of GT transitions [4]. To understand the r-process, i.e., the nucleosynthesis of the heaviest nuclei, the β-decay strengths of very neutron-rich nuclei are essential information. The total β-decay strength of very-neutronrich nuclei is the sum of all transitions and is directly related to the transition strength that may be measured by Cex reactions using high-energy beams of neutron-rich nuclei. To date, no systematic measurements of Cex reactions have been made for neutron-rich unstable nuclei. The known smoothness of the unit cross section for heavy nuclei is advantageous for such a study. Searches for IASs of very-neutron-rich nuclei have been reported for 11Li [7,8]and14Be [9]using (p,n) reactions and strong transitions to IASs have been observed for both these nuclei. However, studies have only been made of selected neutron halo nuclei and no systematic measurements have yet been reported. Charge exchange reactions with heavy ions and their relation to GT transitions have been discussed by Osterfeld et al. [10]. They discussed Cex reactions with particular reference to the (12C, 12B) and (12C, 12N) reactions under a strong absorption model. They found that the L=0 transitions clearly reflect the strength of the GT transition. In contrast, a later theoretical study by Bertulani and Lotti [11] concluded that the determination of GT and Fermi strength from heavy-ion 2/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. Cex is not necessarily straightforward. Consequently, systematic studies of Cex reactions with heavy ions in addition to the (p,n) reaction are necessary, particularly for neutron-rich nuclei. The present paper reports the first measurement of the isotope dependence of the production of nitrogen isotopes from high-energy incident carbon isotopes, 12-19C, on proton and carbon targets. Outgoing Nisotopes were measured at near zero degrees, covering most of the scattering angles of projectile fragmentation. We call this cross section the charge-exchange reaction cross section (σex) because the proton-transfer reaction cross section is expected to be much smaller than the cross section due to a charge exchange between a projectile and a target. This charge exchange at high energies in this study is expected to occur mainly though charged meson exchanges and proton– neutron exchange reactions [12]. A theoretical study was made to estimate the contribution from direct charge exchange due to central and tensor interactions and from sequential proton and neutron transfer [13]. Their calculations indicated that direct charge exchange is safely dominant for incident energies above 100AMeV. In the final states of the present experiment, only N isotopes were identified and thus the reaction is charge exchange restricted to produce N isotopes below the proton emission threshold. In contrast, neutron emissions from excited states do not change the Z value of the final state and thus those channels are included in the measured cross sections. The relationship between the separation energies and the β-decay Q value (Qβ) in a related pair of nuclei is shown in Fig. 1. The observation window of the present AC(p,n)AN reaction is shown by the shaded area between the two arrows pointing from the ground state of the AN nucleus to its excited states below the proton separation energy. The neutron separation energy Sn(AN)for a neutron-rich N isotope is always smaller than the proton separation energy Sp(AN), so neutron evaporation may occur within this window though the final nucleus remains an N isotope. The separation energies and Qβare related as Sp(N)+0.782 =Qβ+Sn(C), (2) where 0.782 MeV is the mass difference between a neutron and hydrogen (Mn–MH). For a neutronrich nucleus the neutron separation energy is small. In particular, the neutron separation energy is about 1 MeV for nuclei along the R-process. Therefore, in such nuclei the (p,n) transition window is very close to the β-decay window determined by the Qβ-value. Therefore, σex may be closely related to the total β-transition strength for nuclei near the R-process path. In an effort to study the above, the isotope dependences of σex for C-isotopes for A=12–19 were measured using hydrogen and carbon targets at the SIS-18/FRS facility at GSI. Incident beams Fig. 1. Related nuclei and the relation between the separation energies (Sn,Sp) and the β-decay Qvalue. 3/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. Fig. 2. Experimental setup. Sci: plastic scintillation detector, TPC: time-projection chamber, MUSIC: multi-sampling ion chamber. of 1 GeV/nucleon 40Ar and 22Ne were used to produce secondary beams of C isotopes at around 950 MeV/nucleon. The production target was a 5 g/cm2thick Be plate. The measurements were made at the final achromatic focus of FRS [14] after selection of C isotopes. The intensity of total secondary nuclei at the secondary target was kept below a few thousand per second at maximum. The primary beam intensity was accordingly changed for different settings of isotopes but ranged from 108to 109per synchrotron spill, which was 4 seconds. Details of the principle and the method of nuclear separation are described in Ref. [14]. A schematic diagram of the present detector setup is shown in Fig. 2. In the figure, MUSIC indicates a multi-sampling ion chamber, TPC indicates a time-projection chamber, Sci indicates a plastic scintillation detector, and VETO is a plastic scintillation detector with a hole in the middle. Each MUSIC is segmented into eight cells and the signal of each cell was read by an anode pad. Incident particles are identified by Esignals from MUSIC1 and the time of flight (TOF) is determined by the time difference between the signal from the first plastic scintillator (Sci1) and the signal from the plastic scintillator placed at S2, which is the dispersive focus of the FRS located 36 m upstream from Sci1. The track of an incident particle is determined by TPC1 and TPC2, which are placed before and after MUSIC1. From these measurements, the incident positions and incident angles of a particle at the target can be determined. This information is used to select incident particles that satisfy the condition determined by the following detectors. The VETO counter is used to reject events for which the incident carbons are associated with other charged particles. The Zresolution (σZ) of MUSIC1 is 0.12 when all eight cells of the signals are added. The number of incident nuclei is determined by selecting good incident nuclei using TOF, Eby MUSIC1, and the incident position and angle. The mixing of other nuclides in the incident C isotopes is always less than 10−4and thus no effect of such contamination is expected in the measured cross sections. The Esignals from MUSIC2 are used to determine the Z of the particles after the reaction target located just upstream of MUSIC2. MUSIC2 measures eight layers of Efor a particle. The active area of MUSIC2 is 200 ×80 mm2and the active length is 400 mm. The smallest covering polar angle of MUSIC2 from the target was 106 mrad, which is large enough to cover almost all the projectile fragments. The position distribution of Z=7particles was also measured by the positionsensitive detector TPC3 after MUSIC2, which confirmed that all the N production events are well contained within the MUSIC2 active area. Two types of reaction targets were used: a graphite plate of 4.010 g/cm2and a polyethylene plate of 3.625 g/cm2in thickness. Data were also accumulated without a target (empty target) to estimate the number of reactions that occur at places other than at the target. The experimental setup is the same as that presented in previous papers [15,16]. Figure 3shows the Espectra of the MUSIC2 detector after the C target for a measurement with an incident 18C beam. The upper panel shows the E spectrum obtained by summing all the signals from the eight layers of MUSIC2. The highest peak in the histogram is from the non-interacting 18C and a small peak at the right-hand side of the 18C is the peak for Z=7nuclei. The Zresolution (σZ) of MUSIC2 is 0.12. The number of produced Z=7nuclei is determined from the total count 4/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. 1 10 100 1000 10 000 0 500 1000 1500 2000 2500 3000 Counts ΔE of M2 [arbitrary] σz= 0.12 Z = 6 Z = 7 Z = 5 (a) (b) Fig. 3. Espectra in MUSIC2 for 18C incident on a C target. Upper panel: pulse height spectrum of sum of all eight layers (M2) in MUSIC2. Lower panel: scatter plot of the sum of the front four layers (M2F) and the sum of the back four layers (M2B). of events that have an energy loss larger than the energy determined by the minimum counts of the spectrum between Z=6and Z=7.TheEsignals in MUSIC2 can also be divided into the front four layers (M2F) and the back four layers (M2B). The lower panel of Fig. 3shows the scatter plot of Efor M2F and M2B. Almost all of the Z=7events show a consistent E between M2F and M2B. This indicates that the loss of Z=7particles, by scattering, reactions, or anything else in the detector, is small. The amount of loss is estimated to be less than 5% of the events and thus is not corrected for in the estimates of the cross sections. The cross sections σN=7for C and polyethylene targets were determined after subtracting the empty target background. The background here mainly comes from the reactions which occurred in the detectors after the incident identifications and the admixture of Z=7nuclides in the incident beam, if any. The typical rate of the background was 6×10−5of the incident beam. The cross section for a proton target was obtained by subtracting the C target cross section from that of the polyethylene target. 5/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. Table 1. Observed production cross section of N from C isotopes. Aσz=7(H) [mb] σz=7(C) [mb] Sp(N) [MeV] Sn(C) [MeV] Qβ-(C) [MeV] IAS [MeV] Ntr 12 0.11 ±0.07 0 ±0.08 0.6 18.72 −17.34 — 4 13 0±0.20.33 ±0.33 1.94 4.95 −2.22 0 1 14 0.30 ±0.08 0.36 ±0.08 7.55 8.18 0.16 2.313 2 15 0.29 ±0.09 0.61 ±0.11 10.21 1.22 9.77 11.615 3 16 0.76 ±0.15 1.91 ±0.20 11.48 4.25 8.01 9.93 14 17 1.45 ±0.29 3.14 ±0.36 13.13 0.73 13.16 unknown 26 18 1.27 ±0.21 5.87 ±0.27 15.21 4.19 11.81 unknown 38 19 1.88 ±0.65 7.66 ±0.80 16.97 0.16 16.55 unknown 50 A: mass number of incident carbon, σz=7(H): cross section with H target, σz=7(C): cross section with C target, Sp: proton separation energy, IAS: excitation energy of isobaric analog state of ACinAN, Ntr: number of possible L=0transitions. Fig. 4. Observed charge exchange cross sections of C isotopes on H and C targets. The arrows for A=12 and 13 indicate that the error bars extend below the bottom of the figure. Two simple model expectations of neutron number dependence of the cross sections are shown by the dashed line (see text for an explanation). The cross sections determined are listed in Table 1and are presented in Fig. 4. The cross section increases rapidly as the number of neutrons increases in the C isotopes. The rate of increase for the proton target is faster than linear with respect to the neutron number. The cross section increases even faster for the C target. We first consider reactions with the H target. The present measurements do not allow the mass number of N isotopes in the final states to be determined; therefore, neutron emissions followed by (p,n) reactions are not distinguished. In other words, (p,n) reactions below the proton emission threshold are all integrated in the measured cross sections whether or not they emit neutrons. At the present beam energy, the rate of proton capture reactions followed by neutron evaporation is considered to be negligibly small [11,12]. Therefore, almost all the reactions for the production of N isotopes are charge exchange (p,n) reactions. The (p,n) cross section at small scattering angles at the present high energy is expected to be dominated by Fermi and GT transitions. The present measurement, however, covers almost all the scattering angle of the (p,n) section and therefore transitions corresponding to other types of selection rules may be included. In the following, however, we assumed that the 6/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. Table 2. Beta decay strength of C isotopes. β-transition Final state log ft 1/ft ×10−7¯σ∗8 exβAforsum 12N(1+)→12C*1gs (0+)4.12 ±0.003 2276 ±16 1.403 ±0.009 12 13N(1/2−)→13C*2gs (1/2−)3.667 ±0.001 2153 ±50.427 ±0.001 13 14C(0+)→14N*3gs (1+)9.04 ±?0.009 ±? 14O(0+)→14N*3gs (1+)7.266 ±0.009 0.524 ±0.011 (Mirror) 2.31 (0+)3.4892 ±0.0002 3241.9±1.5 3.95 (1+)3.15 ±0.02 7080 ±330 4.56 ±0.14∗714 15C(1/2+)→15N*4gs (1/2−)5.99 ±0.03 10.2±0.7 5.30 (1/+)4.11 ±0.01 776 ±18 7.30 (3/2+)6.89 ±0.05 1.29 ±0.15 8.31 (1/2+)5.18 ±0.05 66 ±8 8.57 (3/2+)5.34 ±0.07 46 ±7 9.05 (1/2+)4.05 ±0.04 891 ±82 1.09 ±0.05∗715 16C(0+)*5gs (2−)— 0.12 (0−)6.7±0.07 2.00 ±0.32 3.35 (1+)3.551 ±0.012 2812 ±78 4.32 (1+)3.83 ±0.05 1480 ±170 2.63 ±0.12∗716 18C(0+)*6gs (1−)— 1.735 (2+)5.2±0.463±58 2.614 (1+)4.08 ±0.08 830 ±150 0.51 ±0.09∗718 *1Ref. [17], *2Ref. [18], *3Ref. [19], *4Ref. [20], *5Ref. [21], *6Ref. [22] *7Value obtained only from the ft values of 14Oβdecay. It is the summed values of all listed states. ∗8See Eqs. (4)–(10) for definitions. ?: The error is not shown in reference [19]. main contributions are mainly from “allowed” transitions that are dominant in beta decays. We have started collaboration with theory to estimate the contributions of such transitions as well as GT and F transitions in the total-charge-changing cross sections (C. A. Bertulani and D. Y. Pang, private communications). Next we compare these cross sections with observed beta-decay transitions. The known log ft values are listed in Table 2. The beta-decay transition strength B(α) in Eq. (1) is related to the ft value by G2 VB(F)+G2 AB(GT)=K ft,(3) where K,GV,andGAare the coupling constants of beta decay and are common for all nuclei. The ratios of the axial-vector coupling constant and the vector coupling constant are R=(GA/GV)2= 1.56 ±0.2and K/(GV)2=6163 ±4s. Because we are discussing the relationship between the summed cross sections of charge exchange and beta decays, it is in general not possible to separate Fermi and GT transitions. Under this assumption: B(F)+RB(GT)=6163 ft .(4) Because the present measurement deals with values of σex below the proton emission threshold (Sp), which is close to the corresponding beta decay Q-value (Qβ), comparisons between the integrated βstrength below Spand σex are meaningful. Because the unit cross sections of Fermi and GT transitions are different, the total charge-exchange cross section evaluated from beta-decay σexβcan be 7/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. written as σexβ= all transitions ˆσFB(F)+ˆσGT RB(GT).(5) For a pure Fermi or GT transition, adding these transition strengths is straightforward. However, for a mixed transition such as 13C→13N, this addition should be treated carefully. Note that the unit cross sections ˆσFand ˆσGT are not the same, but the unit cross section of the Fermi transition is much smaller than that of GT transitions: ˆσF/ˆσGT ∼1/10 [4]. Therefore, σexβ= all transitions ˆσGT [B(F)/10 +RB(GT)].(6) For a pure Fermi transition, the partial strength for a transition σk exβ(F)is proportional to σk exβ(F)=ˆσGT B(F)/10 =ˆσGT 616.3 ft , and ¯σk exβ(F)≡σk exβ(F)/ ˆσGT =616.3 ft ,(7) where kindicates an individual transition and ¯σk exβ(F)is the normalized strength of a Fermi transition. The normalized strength for a pure GT transition is then ¯σk exβ(GT)≡σk exβ(GT)/ ˆσGT =6163 ft ,(8) and the normalized total charge exchange cross section is ¯σexβ= k¯σk exβ(F)+¯σk exβ(GT).(9) Note that care should be taken for a mixed transition; the partial Fermi and GT amplitudes in the transition should be calculated individually and then added. In the following we examine such relationships for pairs of C and N nuclei: ◦12C is stable, so no beta transitions to 12N can be observed. The transitions that affect the Cex reaction occur below 0.601 MeV excitation energy in 12N. Only the ground state of 12Nexists within this range, so the βtransition strength can be obtained from the βdecay of 12Ntothe ground state of 12C, which is a pure GT transition. The observed log ft value is shown in Table 2. To obtain B(GT) for 12C→12N, a spin factor (2J12N +1)/(2J12C +1)should be included because B(GT)=f     k σktk     i 2(2Ji+1), (10) where |iand |fare the initial and final states, respectively, kσktkis the GT operator, kis the nucleon index, σkis the Pauli spin operator, and tkis the isospin operator. The spin-factorcorrected ft value is used to obtain values of ¯σexβ,showninTable2. ◦13C is also stable for beta transitions. Only the ground state of 13N is below the proton emission threshold, very similar to the 12C case, and only transitions between the ground states contribute to the Cex reaction, which includes Fermi and GT mixed transitions between mirror states. In this case, B(F)=1and B(GT)can be calculated directly from the beta decay transition. For the present 13Nand13C case, RB(GT)=0.327 from the ft value listed in Table 2. Therefore, the 8/16 Downloaded from https://academic.oup.com/ptep/article-abstract/2016/4/043D05/2461211 by guest on 04 June 2020 PTEP 2016, 043D05 I. Tanihata et al. 0.01 0.1 1 10 11 12 13 14 15 16 17 18 19 20 Mass Number = 7 [mb] decay strength = 7 [mb] and (1/ft) normalized at A = 12 Fig. 5. Comparison between βdecay strengths ¯σexβand σex of C isotopes with proton target. The central value of A=13 cross section is 0. The upper limit of the error of the cross section is shown in the figure. Fermi transition strength is larger than that of the GT transition. The spin factor is 1 because the spins of the initial and final states are both 1/2. The ¯σexβvalue with mixed transitions is shown in Table 2. ◦14C decays to 14N, but only to the ground state, and this transition is known to be very weak. The proton emission threshold for 14NisEx=7.55 MeV and many states exist below this excitation energy. Although only beta transitions between the ground states can be observed for 14C, the mirror nucleus 14O exhibits transitions up to the Ex=3.95 MeV state. The 2.313 MeV state in 14N is the IAS of the 14Cand14O ground states and thus the transition is a super-allowed Fermi transition. The transition to 3.95 MeV is also very strong: log ft =3.15. The spin factor is 1,sothesumof1/ft is very large compared with that of 12Cand13C. In addition, although βdecay is not possible due to the Q-value, a 1+state exists at Ex=6.024 MeV. The transition to this state is expected to contribute to the Cex transition and thus the obtained ¯σexβshown in Table 2may be an underestimation. Although the ft values of the mirror transitions may have some asymmetry of 10%–20% [23,24], this is low enough that it does not affect the following discussions. The calculated strength from the ft values ¯σexβis much larger than that for 12Cand13C. In fact, the observed σex increases suddenly between 13Cand14C, which is consistent with the increase of the calculated strength. Taddeucci [4] found that the unit cross section varies for the 12C, 13C, and 14C isotopes. The difference is about a factor of 1.5 between 12Cand13C and almost 1 between 12Cand14C, and is much smaller than the present increase of the betatransition strength of more than a factor of 5. The value of σex with a proton target and the strength presently obtained from β-decay ¯σexβare presented in Fig. 5. The strengths obtained from β-decays are normalized for 12C. 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