The (t,3He) and (3He,t) reactions as complementary probes of the spin-isospin response of nuclei

Nuclear Physics A 788 (2007) 61c–69c

The (t,3 He) and (3 He,t) reactions as complementary probes of the spin-isospin response of nuclei.∗ R.G.T. Zegersa a

National Superconducting Cyclotron Laboratory, Department of Physics and Astronomy and the Joint Institute for Nuclear Astrophysics, Michigan State University, East Lansing, MI 48824-1321, USA. The (t,3 He) reaction at 115 MeV/nucleon has been developed as a tool to extract Gamow-Teller strengths. A secondary triton beam was used, initially produced from a primary α-beam, but since recently from a primary 16 O beam. An important advantage of the (t,3 He) probe is that the data can be combined with results from the (3 He,t) reaction, so that studies in the ΔTz = −1 and ΔTz = +1 direction can be performed with similar probes and good resolutions. This is important for improving the understanding of the reaction mechanism and associated errors in the extraction of transition strengths. 1. Introduction Charge-exchange reactions have long been recognized as an important tool to study the spin-isospin response of nuclei [1]. These ΔT = 1 reactions can occur either with (ΔS = 1) or without (ΔS = 0) transfer of spin. Gamow-Teller (GT) transitions in particular (ΔS = 1, ΔL = 0) haven been the subject of intensive studies, using a variety of probes. These transitions are mediated through the στ± operator and connect the same initial and final states as β± decays. Since β-decay has access to states only in a limited energy window, charge-exchange reactions with hadronic probes are the preferred tool for mapping the complete GT response and thus provide valuable information to test nuclear structure calculations using shell or mean-field models. It is important to test the reliability of such models, since they are, for example, used in the estimation of weak-transition rates of importance for late-stellar evolution [2] and matrix elements for (neutrinoless) double−β decay [3]. Charge-exchange reactions are also important to study collective isovector transitions. The strength distributions can be linked to macroscopic properties of nuclear matter [4] and provide information on the microscopic structure of nuclei at high excitation energies [5]. Charge-exchange experiments have been performed using a variety of probes. An important boundary condition is that such studies have to carried out with beam energies  100 MeV/nucleon to ensure that the reactions are predominantly of single-step nature. Although the elementary (p,n) and (n,p) charge-exchange reactions have been used intensively [7,8], recent efforts have focused on probes with which high resolutions can be ∗

This work was supported by the US NSF (PHY02-16783 (JINA), PHY-0110253)

0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2007.01.049

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Figure 1. Schematic layout of the NSCL-CCF and the A1900 fragment separator as operated for producing the secondary triton beam of 115 MeV/nucleon from a 150MeV/nucleon 16 O beam.

achieved, in particular (3 He,t) [9] and (d,2 He) [10]. At the NSCL, a secondary triton beam has been developed with an energy of 115 MeV/nucleon, with the aim to perform (t,3 He) charge-exchange studies. Besides the fact that good resolutions can be achieved, the mirror (3 He,t) reaction is available for performing detailed reaction studies. At first, the triton beam was produced from a primary 140-MeV/nucleon α-beam [11,12]. Triton-beam intensities achieved were ∼ 1 × 106 pps. A higher intensity is desirable and, in addition, a significant amount of overhead time is now involved in producing a primary α beam since the  NSCL K1200 cyclotron has to be operated in stand-alone mode, instead of the usual K500 K1200 coupled operation [13]. Therefore, a new method to produce a beam of secondary tritons has been developed [14], employing a primary beam of 150-MeV/nucleon 16 O, as discussed below. Besides the new method for producing a secondary triton beam, we describe the result of a 26 Mg(t,3 He) experiment. The data is combined with data for the 26 Mg(3 He,t) reaction taken at RCNP, Osaka to check the details of the reaction mechanism [15]. The development of better reaction calculations leads to an improved understanding of systematic errors associated with extracting GT strength distributions from experimental data. 2. Production of secondary tritons from a primary

16

O beam.

A series of test experiments were performed to find the optimal method for producing a secondary triton beam from Oxygen via fast fragmentation [14]. Primary beams of 120MeV/nucleon 18 O and 150-MeV/nucleon 16 O were employed. Although the best yields achieved were very similar for the two different primary beams, the maximum yield with the 16 O beam was obtained at triton energies above 100-MeV/nucleon and with the 18 O beam at 85-MeV/nucleon. Since the higher triton beam-energy is preferred (see above), further studies were only performed with the 16 O primary beam. Fig. 1 shows a schematic layout of the coupled-cyclotron facility [13] and the A1900 fragment separator [17]. With the K500 cyclotron, 16 O3+ ions produced in an ion source

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were pre-accelerated to 13 MeV/nucleon and then further accelerated to 150-MeV/nucleon with the K1200 cyclotron. The 16 O beam was then impinged on a Be production-target. The secondary particles were detected at the focal-plane of the A1900 and identified by measuring the time-of-flight (TOF), relative to the radio-frequency signal (RF) of the cyclotron, and the energy loss in a 0.5-mm thick silicon PIN detector. In order to obtain high-resolution (t,3 He) data, the beam lines and S800 spectrometer [18] must be operated in dispersion-matching mode. For this purpose, slits at the intermediate image of the A1900 are used to restrict the momentum spread of the triton beam to dp/p0 = 5 × 10−3 , where p0 is the central beam momentum and dp the full momentum spread. Tritons were cleanly separated at the focal plane of the A1900 from other particles produced in the Be target, as shown in Fig. 2a. For tritons with Et > 110 MeV/nucleon, the contamination from other particles in the beam was less than 15%. In (t,3 He) experiments, the background due to contaminants is small and can easily be removed by using the TOF information. Therefore, to avoid reduction of the triton-beam intensity, no additional measures to reduce the fraction of contaminants were taken. In Fig. 2b, the triton-production rate in the A1900 is shown as a function of Be-target thickness for Et = 115 MeV/nucleon. Rates at the A1900 focal plane for tritons of higher energy were comparable to these results, such tritons cannot be transported to the S800 target position because of their high magnetic rigidity. A Be-target thickness of 3.5 × 104 mg/cm2 was chosen to produce the triton beam used in (t,3 He) experiments. The triton rate at the A1900 focal plane with this target was ∼ 105 /pnAs (per pnA of 16 O, per second). The transmission from the A1900 to the S800 target was about 50%, so that at primary beam intensities of 100 pnA, triton intensities of ∼ 5 × 106 pps can be achieved. As a first test for (t,3 He) experiments with the triton beam produced from 16 O, the reaction on a 9.86-mg/cm2 thick, 99.92% isotopically-enriched 24 Mg was studied. From the measurement of the momentum and angle of the scattered 3 He particles at the focal plane of the S800, the excitation energies in 24 Na and the scattering angles were determined. For details on the reconstruction, see Refs. [12,15]. The spectrum at forward scattering angles is shown in Fig. 2c. The excitation-energy in 24 Na is measured with a resolution of 190 keV full-width at half-maximum (FWHM) using the transition to the strong 1+ state at Ex (24 Na)=1.35 MeV. This corresponds to an intrinsic energy resolution (i.e. not due to energy losses and straggling in the target) of 170 keV. 3. Study of the

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Mg(3 He,t) reactions.

A detailed study of the method to extract GT strengths (B(GT)) from (3 He,t) and (t,3 He) was performed by combining results from both reactions on the same target (26 Mg) [15]. The B(GT) for the transitions to the first four 1+ states in 26 Al excited via the 26 Mg(3 He,t) reaction are known from β-decay measurements in the mirror system (26 Si(β + )26 Al) [19]. Using these transitions, the proportionality between B(GT) and cross section at zero momentum transfer can derived. This proportionality, deduced in eikonal dσ approximation [7] states that dΩ =σ ˆ B(GT ), where σ ˆ is the unit cross section and q (q=0) the momentum transfer. In both experiments, absolute differential cross sections for GT transitions were extracted from the data. Under the assumption of isospin symmetry, the consistency of the (t,3 He) and (3 He,t) data sets was checked by comparing cross sections

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Figure 2. a) Energy loss in the A1900 focal-plane PIN detector versus time-of-flight relative to the cyclotron-RF signal. Tritons make up 85% or more of the particles produced by fast fragmentation of 150-MeV/nucleon 16 O if Et > 110 MeV/nucleon. b) Triton production rate (Et = 115 MeV/nucleon) as a function of Be production-target thickness. The maximum is achieved at thickness of 3 − 4 × 103 mg/cm2 . c) Excitation energy spectrum of 24 Na measured via 24 Mg(t,3 He) at forward scattering angles. The strongest peak is the transition to the 1.35-MeV 1+ state, measured with a resolution of 190 keV (FWHM).

for excitation of analog, T=2, 1+ states. The 26 Mg(3 He,t) experiment was performed at RCNP, Osaka using a 140-MeV/nucleon 3 He beam. Tritons were detected in the Grand Raiden Spectrometer [20] set at 0◦ . The experiment was run in achromatic mode and the resolution of the 26 Al excitation-energy measurement was 100 keV (FWHM). GT transitions are easily identified through their unique forward-peaking angular distribution2 . Besides the angular distributions of transitions to states with J π = 0+ , 1+ (ΔL = 0), the angular distributions of known transitions involving ΔL = 1, 2 were also well reproduced in distorted-wave Born approximation ◦ (DWBA) using the code fold [21]. The 0◦ cross section for each GT transition ( dσ(0dΩ,Ex ) , where Ex is the excitation energy for the state) was extracted by fitting the DWBA angular distribution to the data. An extrapolation to q = 0 was then made based on the ◦ ratio dσ(0dΩ,Ex ) / dσ(q=0) calculated in DWBA. dΩ In the DWBA calculation, the Love-Franey effective NN-interaction [22] is doublefolded over the transition densities of the target-residual and projectile-ejectile systems. Central and tensor components of the interaction are included and exchange is treated in a short-range approximation [22]. The latter approximation is known to result in an overprediction of the absolute cross sections [23] and, indeed, the calculations had to be scaled down by a factor ∼ 1.7 to reproduce the data for GT transitions with known B(GT) and the IAS. One-body transition densities (OBTDs) for the 26 Mg-26 Al system were calculated using the USB-05B interaction [24] using the code OXBASH [25]. Densities for the 3 He-t system were taken from Variational Monte-Carlo results [26]. 2

With the exception of the isobaric analog state. Its location in the spectrum is well-known, however.

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The 26 Mg(t,3 He) experiment, populating states in 26 Na, was performed at the NSCL using a 115-MeV/nucleon triton beam with an intensity of 1.3 × 106 pps, produced from a primary α-beam [12]. Due to relatively high level-densities in 26 Na, even at low excitation energies, GT transitions to 1+ states were not fully separable. Therefore, a multipoledecomposition analysis, based on angular distributions calculated in DWBA, was performed to extract the GT contributions, their 0◦ cross sections and, by extrapolation, their cross section at q=0. Knowing from the (3 He,t) analysis that experimental angular distributions are well-reproduced by the DWBA gave confidence that the procedure could be carried out with relatively small systematic errors. The unit cross section for extracting GT strength in the (3 He,t) experiment was calibrated using the transitions to the two lowest-lying 1+ states in 26 Al for which the strength is known from β-decay [WIL80] (at Ex = 1.06 MeV with B(GT)= 1.098 ± 0.022 and Ex = 1.85 MeV with B(GT)= 0.536 ± 0.014) and then used for extracting the GT transition strengths to all 1+ states in 26 Al. After correction for a kinematical factor (k) due to the small difference in beam energies of the (3 He,t) and (t,3 He) experiments, the ratio of cross sections for transitions to T=2 analog states in 26 Al and 26 Na should, under assumption of isospin symmetry, be different by a factor of 6 due to a difference in isospin kσ 3 He) Clebsch-Gordan coefficients ( σ (t, (q = 0) = 6, with k = 1.2). A factor of 6.1 ± 0.5, 3 ( He,t)

proving good consistency, was found for the first 1+ state in 26 Na at Ex = 0.08 MeV and its analog in 26 Al at Ex = 13.57 MeV. In Fig. 3a, the GT strength distributions extracted from the 26 Mg(3 He,t) experiment are compared with the results from the shell-model calculation. The results from the 26 Mg(t,3 He) experiment are also added, by taking into account the difference in Coulomb-energy and divided by the Clebsch-Gordan factor of 6. Except for one state, at Ex (26 Al)=14.53 MeV (which is likely a T=1 state) a good consistency between the T=2 states in 26 Al and 26 Mg is found. In Fig. 3b the cumulative sums of the GT strengths for both data and shell-model are shown. Included are experimental results from 26 Mg(p,n) [27] and 26 Mg(d,2 He) [28], which also agree well with the (3 He,t) and (t,3 He) data, respectively, showing that the extraction of GT strength in this case is probe independent. From the (3 He,t) and (t,3 He) results, the exhaustion of the model-independent sum-rule for GT strength in nuclei Sβ − (GT ) − Sβ + (GT ) = 3(N − Z) [29] can be calculated. A value of 3.61 ± 0.08 (where the error is due to statistical and systematical uncertainties in background subtraction only) is found, corresponding to 60% ± 1% of 3(N − Z) = 6. This factor agrees well to the established GT-strength quenching factor for sd-shell nuclei of 59% ± 3% [30]. In Fig. 3, the shell-model calculations were, therefore, multiplied by the quenching factor of 0.6 . 4. Systematic errors in the extraction of Gamow-Teller strength.

It is well-known that the proportionality between B(GT) and cross section at q = 0 is not perfect [7] and the breaking of proportionality is stronger for transitions with small B(GT). To quantify this breaking for the reactions under consideration here, a theoretical study was performed, using the DWBA formalism and shell-model inputs discussed above. OBTDs for the first one hundred T=0,1 and 2, 1+ in in 26 Al via 26 Mg(3 He,t) at 140 MeV/nucleon were generated in the shell-model. The Gamow-Teller strength and subse-

B(GT)

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Figure 3. a) Measured Gamow-Teller strengths from 26 Mg(3 He,t) and comparison with the shell-model predictions using the USD-05B interaction, multiplied by the extracted quenching factor of 0.6. For Ex > 13 MeV, Gamow-Teller strengths extracted from 26 Mg(t,3 He) are included. The 26 Mg(t,3 He) spectrum has been shifted by the Coulombenergy difference so that the first Gamow-Teller state at Ex = 0.08 keV is aligned with the first T=2 Gamow-Teller state at Ex =13.6 MeV in the 26 Mg(3 He,t) spectrum. The strengths have been divided by 6 to account for the difference in isospin Clebsch-Gordan coefficients. Regions where T = 0, 1 or 2 are dominant are indicated. b) Cumulative sums of Gamow-Teller strengths; below Ex = 13 MeV, results from 26 Mg(3 He,t), 26 Mg(p,n) [27] and the shell-model using the USD-05B interaction [24] are compared. Above Ex = 13 MeV, results from 26 Mg(t,3 He), 26 Mg(d,2 He) [28] and the shell-model are compared (T=2 states only). Note the changes in vertical scale for Ex > 13 MeV.

quently the DWBA differential cross section were calculated for each of the transitions. From this point onward, we treated these calculated cross sections as if they were data and extracted the Gamow-Teller strengths for each state in exactly the same manner as was done in the analysis of the experimental data. A relative systematic error was then −B(GT )SM defined by B(GT )DWBA , where B(GT )DW BA is the GT strength extracted from the B(GT )SM cross section calculation, treating it as if it were data, and B(GT )SM the GT strength as calculated in the shell-model. The result of this study is shown in Fig. 4. The increase in systematic error with decreasing B(GT) can clearly be seen. By changing the input for the theoretical calculations, the main cause for proportionality breaking was investigated. It was found that the dominant contribution is due to interference between the ΔL = 0, ΔS = 1 and ΔL = 2, ΔS = 1 amplitudes. The former amplitude is mainly mediated through the στ component of the effective NN interaction, whereas the latter amplitude near q = 0 is largely due to the tensor-τ component. It was found that, by setting the tensor-τ term of the interaction equal to zero, the proportionality between cross section at q = 0 and GT strength was valid on the level of ∼ 2%. Since the interference can be destructive or constructive, the systematic error in B(GT) when summing over many transitions, becomes small ( 1% for the case studied here).

Relative systematic error in B(GT)

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Figure 4. Results of a theoretical study into the breaking of the proportionality between GT strength and cross section at q = 0. Only transitions with B(GT)> 0.001 are shown and the approximate experimental detection limit is indicated. For details, see text.

It is confirmed that calibration of the proportionality is best performed using transitions with large GT strengths, hence the choice made earlier to only use the first two strong GT transitions for this purpose. For the states at Ex = 2.07 MeV (B(GT)=0.091) and at Ex = 2.74 MeV (B(GT)=0.113) for which the strength is also known from β-decay, errors of 10− 20% are expected, based on the results shown in 4. The measured deviations, assuming that the proportionality calibration using the transitions to the states at Ex =1.06 MeV and Ex =1.85 MeV is perfect, were 10.6% and 18.5%, respectively, consistent with the expectation. It can be concluded that some caution is advised when trying to decouple the reaction mechanism from the nuclear structure in the extraction of GT strength from the data, as is done in the eikonal approximation used above. Since the proportionality breaking depends on the interplay between reaction mechanism and structure, the effects are not necessarily identical when studying the same target nucleus with different probes and this feature provides a way to identify such transitions. This is especially important if a particular transition is used as the calibrator for the unit cross section. A good example is the comparison between the 58 Ni(p,n) and 58 Ni(3 He,t) reactions. The ratios of the cross section at 0◦ for populating the 1+ ground state to that for the 1+ excited state at 1.05 MeV in 58 Cu are 0.33 for (p,n) at 120 MeV, 0.41 for (p,n) at 160 MeV [31] and 0.57 for (3 He,t) [32]. The differences between the (p,n) and (3 He,t) data are too large to explain by experimental uncertainties only. In a simple independent-particle picture, the neutrons and protons in 58 Ni fill all orbitals up to f7/2 . The two remaining neutrons populate the p3/2 and f5/2 orbitals. Therefore, relatively strong contributions from νp3/2 -πf5/2 and νf5/2 -πp3/2 particle-hole components are to be expected for excitations of the lowest lying 1+ states in 58 Cu, even though these components are purely of ΔL = 2, ΔS = 1 nature (i.e. not Gamow-Teller). As discussed in Ref. [33], in more realistic models using the

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GXPF1 and KB3G interactions, this is still largely true: transitions to the 1+ ground state and excited state at Ex =1.05 MeV have strong contributions from the p-orbit. OBTDs obtained from the shell-models using GXPF1 and KB3G [33] were used in reaction calculations using the fold code for 58 Ni(3 He,t) and the dw81 [34] code for 58 Ni(p,n) [16]. The ratios of theoretical unit cross sections for the transitions to the 58 Cu σ ˆg.s. ground state and 1.05 MeV excited state (Rσˆ = σˆ1.05 ) were calculated for each reaction. MeV If proportionality were maintained, Rσˆ = 1. It was found, however, that Rσˆ = 1.19 for the (p,n) reaction and Rσˆ = 1.44 for (3 He,t) when using the OBTDs obtained with the GXPF1 interaction. Results with the KB3G interaction were similar [16]). For the (p,n) reaction, the interference between ΔL = 0, ΔS = 1 and ΔL = 2, ΔS = 1 amplitudes was constructive (but slightly different in magnitude) for both transitions. In the case of the (3 He,t) reaction the interference was constructive for the transition to the ground state and destructive for the transition to the first excited state, hence the larger effect on the ratio Rσˆ . This result qualitatively explains the discrepancy found between the (p,n) and (3 He,t) reactions. For heavier nuclei, for which reliable shell-model transition densities are not available, it is at present difficult to perform detailed studies of the effect of the reaction mechanism on the extraction of strength from cross sections. It is important, however, to develop such capabilities so that increasingly sophisticated mean-field calculations can be compared in detail with data. This not only holds for the extraction of Gamow-Teller strengths, but also for other isovector multipole transitions and giant resonances. To bridge the gap between experiment and theory, a project [35] is currently underway to perform cross section calculations in the DWBA code fold using transition densities from self-consistent QRPA [6] calculations. 5. Conclusions & Outlook The (t,3 He) reaction at 115 MeV/nucleon has been developed as a tool to extract Gamow-Teller transition strengths. Besides the detailed study of this reaction on a 26 Mg target, which was combined with results from 26 Mg(3 He,t) and presented here, data on 58 Ni, of importance to test the validity of models used for estimating weak rates in latestellar evolution, has also already been analyzed [16]. Since the development of a new method to produce a secondary triton beam, employing a primary 16 O beam instead of an α-beam, the triton beam intensity has been increased and data on a number of nuclei of importance for astrophysics have been taken. High-statistics (3 He,t) data is used to study in detail the reaction mechanism and improve the reaction calculations, so that a better comparison between structure theory and data can be performed and systematic errors in the extraction of transition strength from data can be estimated. REFERENCES 1. 2. 3. 4.

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