Systematic Extraction of Spectroscopic Factors from the 12 C(d,p) 13 C and 13 C(p,d) 12 C Reactions

Transcription

1 Abstract Systematic Extraction of Spectroscopic Factors from the 12 C(d,p) 13 C and 13 C(p,d) 12 C Reactions X. Liu, M.A. Famiano, W.G. Lynch, M.B. Tsang, National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI J.A. Tostevin Department of Physics, School of Physics and Chemistry, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom Past measurements of the angular distributions for the 12 C(d,p) 13 C and 13 C(p,d) 12 C reactions leading to the ground state have been analyzed systematically using the theory of distorted-wave Born-approximation (DWBA). By fitting the first peak of the angular distributions and developing a consistent set of input parameters to the DWBA calculations including the global parameterizations of optical model potentials, spectroscopic factors which scattered around the mean values by about 20% at deuteron energy between 6 60 MeV are extracted. In contrast, the published spectroscopic factors fluctuate by about a factor of three in the same energy range. Such analyses provide useful reference points for future extraction of spectroscopic factors in other systems including spectroscopic factors obtained in reverse kinematics.

2 Single nucleon transfer reactions such as (d,p) or (p,d) reactions have been used extensively to study the spectroscopy of nuclei [38,39,52,53,56]. In particular, these measurements allow the extraction of the spectroscopic factors, which are measured by taking the ratios of the experimental cross-sections to the predicted cross-sections from Distorted Born Wave Approximation (DWBA) models [52,53]. The spectroscopic factors describe the overlap between the initial and final state in the reaction channels and yield important information about single-particle orbitals in many nuclei [39,52,53]. Thus the single nucleon transfer reaction is ideally suited to extract the spectroscopic information in extremely neutron rich or proton rich nuclei [52,53, 56]. As most rare nuclei are not stable and cannot be made into targets, inverse kinematics using these nuclei as projectile is an optimum way to extract spectroscopic information of these nuclei [55]. With the availability of rare isotopes, transfer reactions using inverse kinematics become a valuable tool to explore the single particle orbitals of these rare nuclei [36,37,52,53]. There are other advantages of using inverse kinematics. As shown below, the forward angle region where the first peak of the angular distributions usually occurs, is critical to the accurate extraction of spectroscopic factors [35]. Since the reaction products peak mostly forward, the peak cross-sections can be measured much easier in inverse kinematics than normal kinematics reactions especially at high energy [56]. However, beam intensities from these rare isotopes are much less than stable beams and the history of using rare isotopes to study nuclei spectroscopy is much shorter, it becomes important that one should understand the limitation of the current method by first examining the published measurements with the normal kinematics reactions. It is well known that large uncertainties are associated with the extraction of the spectroscopic factors. For example, in the systematic compilations of the spectroscopic factors in the sd-shell nuclei, Endt found that different values of the spectroscopic factors arise from different experiments and different analyses [27]. By examining large amount of data and using consistency checks when available, Endt compiled a list of the best spectroscopic factor values for the sd-shell nuclei. However, the above analysis performed in 1977 with limited data sets, is limited to heavy nuclei and does not provide

3 the systematic uncertainties associated with the method nor did it take advantage of recent improvement with model calculations. One of the problems in the past regarding the extraction of spectroscopic factors lies in the ambiguity of choosing the optical model potentials to describe the nuclei [56]. Since spectroscopic factors are potential dependent, ideally, elastic scattering data on the appropriate nuclei should be measured and fitted to extract the optical model parameters. However, due to the difficulties associated with measuring elastic crosssections at forward angles, such data are not available for most of the selected reactions at the desired energy resulting in experiments where multiple values of the spectroscopic factors are extracted depending on the optical model potential used [16,22,45,47,56]. A lot of data on transfer reactions with normal kinematics has been accumulated in the past 50 years. In the mean time, advances have been made to provide global optical potentials with averaged parameters in the vicinity of the nuclei [23,24,31,32,33,34] or calculating these potentials using a microscopic method such as, the Jeukenne-Lejeune- Mahaux (JLM) potentials [28,29]. To assess the systematic uncertainties associated with the extraction of the spectroscopic factors, we need a reaction with measurements at many incident energies using different detection systems. In the present article, we analyze the angular distributions of the differential cross-sections of the reaction ¹²C(d, p)¹³c and its inverse reaction 13 C(p,d) 12 C using the distorted-wave Born-approximation (DWBA) calculations with the code TWOFNR[50]. A list of the published angular distributions from the ¹²C(d, p)¹³c reaction at incident deuteron energy, range from 0.4 MeV upward to 56 MeV, is given in Table 1. Those of the inverse reaction, 13 C(p,d) 12 C are listed in Table 2. The proton incident energy for the inverse reactions is higher, from 35 to 70 MeV. Until now, spectroscopic factors have been extracted only in a subset of these experiments. The analysis mainly relied on DWBA calculations but the codes used may be different and there is no consistent method of choosing the input parameters needed in the calculations. The published spectroscopic factors for the ¹²C(d, p)¹³c (closed points) are plotted in Figure 1 as a function of deuteron energy and listed in Table 1. The corresponding deuteron incident energy corrected for Q value is plotted for the inverse reactions 13 C(p,d) 12 C (open points) and listed in Table 2. The values fluctuate from 0.3 to 1.4 with no

4 correlation to the incident energy. In some experiments [16,22,45,47], multiple values were published from the use of different parameter sets of optical model potentials. For these experiments, the higher spectroscopic factor values are plotted as squares in Figure 1. The theoretical predictions provided by Cohen and Kurath in shell model calculations is 0.62, shown as a dashed line in Figure 1. With such large scatter of the extracted values, it is difficult to determine the true value of the spectroscopic factor. If the uncertainty arising from the scattered values is an inherent feature in the method to extract spectroscopic factors, an alternative way should be developed. In the present analysis, we try to incorporate as many as possible the angular distribution measurements for the ¹²C(d, p)¹³c and the inverse reactions we can find in the literature. At low energy, the data could be dominated by compound nucleus emissions. In addition, the angular distributions measured at 4.5 MeV in ref. [7,8,9] do not agree with each other. Thus we decide to exclude the data below 6 MeV. To reduce the uncertainty associated with the analysis of using different versions of DWBA as the authors who published the spectroscopic factors undoubtedly must have done, we used the Surrey version of the program TWOFNR [50] which was first written in 1972 ( [57]) to analyze all the data in a consistent way. TWOSTP [58] calculates the scattering differential cross section for general form of the distorted wave Born approximation up to the second order. It is the extended version of the program TWOSTP which calculates the finite-range form factor[58]. The program was chosen for its ease of use and the many desirable features such as providing reasonable default input parameters and the ability to choose a global parametrization of the optical potential as well as the JLM potentials. We have also performed selected calculations with another DWBA code DWUCK5 [30] to ensure that the outputs are consistent when the same parameters are used as input in both programs [59]. We first discuss the analysis of the ¹²C(d, p)¹³c reactions and the comparison of the angular distributions to DWBA calculations. Figure 2 shows the measured angular distributions for data with incident deuteron energy from 7 MeV to 56 MeV. They are displaced by factors of 10 to avoid data from different angular distributions to overlap each other. The displacement factor is 1 for the angular distributions with incident energy of 19.6 MeV. The solid and dash curves in the figures correspond to the calculations

5 based on the global and JLM potentials respectively. These calculations will be discussed in detail below. All the curves are normalized by the spectroscopic factors listed in Table 1 and plotted in the left and right panel of Figure 3. Most calculations predict that a peak should occur in forward angle, θ cm < 30 and that the backward angle data are more sensitive to the effect of coupled channels. Thus we want to restrict our analysis mainly to the first peak. To be consistent, the spectroscopic factors are extracted by minimum χ 2 fitting to the angular distributions bounded by 30% of the predicted maximum yield and θ cm < 30. To minimize the uncertainty in determining the optical model potentials, we first use the global parametrization of the optical potential option in the calculations. The potentials of Perey & Perey [23] are used for protons when the proton energy is less than 26 MeV. For proton with energies larger than 26 MeV, the potentials of Menet et al. are used [24]. However, using the current analysis criterion, the spectroscopic factors are not affected much if only one parametrization (either Menet or Perey and Perey) is used throughout the whole energy range. The two potentials predict different angular distributions mainly at angles larger than 20 deg and at high incident energy. The deuteron channel potentials are constructed by the Johnson-Soper adiabatic potential [51] using the Chapel-Hill 89 nucleon potentials [31]. For the neutron binding energy potential, the customary values of the radius and the diffuseness parameter of 1.25 fm and 0.65 fm respectively are used. Non-locality correction with the range parameters of 0.85 and 0.54 for the proton and deuteron potentials is included [48]. Such correction which affects the transition amplitude in the nuclear interior is found to be important at incident energy higher than 40 MeV as well as single nucleon transfer reactions using heavy nuclei such as 40 Ca [59]. We chose calculations with the finite-range approximation because the resulting angular distributions agree with the shape of the measured angular distributions better than the distributions calculated with the zero-range option, even though the extracted average spectroscopic factor values are similar. The standard parameters with the finite range factor = [25] and D0 2 = MeV 2 fm 3 [25] are used. The predicted angular distributions are plotted as solid curves in figure 2. The spectroscopic factor extracted using global parametrization of optical model potentials

6 are shown in the left panel of Figure 3. The spectroscopic factors based on the same experimental data set corresponding to those used in Figure 1 are plotted as closed circle, while those from additional data sets are plotted as squares. Above 50 MeV, the amount of data is limited. To supplement this region, we analysed the angular distributions from the inverse reaction 13 C(p,d) 12 C listed in Table 2 similarly. The fits to the angular distributions are of similar quality as those shown in Figure 2. The corresponding spectroscopic factors are plotted as open symbols in Figure 3 and listed in Table 2. The spectroscopic factors from three incident energies (E_d=28 and 56 MeV and E_p=67 MeV) are not included in Figure 3 because the angular distributions for these three sets of data only have data at the backward angles without the first peak. Thus the quality of the fits is very poor. Over the range of incident energy from 6 to 60 MeV, the average value of the spectroscopic factor is 0.74±0.07 while the published values vary from 0.5 to 1.5. Thus analyzing the data consistently reduces the fluctuations substantially. The energy range studied here is well within the optimum angular and linear momentum matching conditions of the current transfer reactions. To provide an alternate proton and deuteron potentials to the calculations, we also analyze the data using the JLM potentials [28,29]. To be consistent, the deuteron potentials are constructed using the Johnson-Soper adiabatic potential [51] based on the JLM potentials. The Woods-Saxon density form [28,29] is used with the diffuseness of 0.54fm and the potential scaling of 1.0/0.8 [28,29] obtained from systematic is used. The n-binding potentials, and other options in the calculations such as the non-locality corrections and finite range options used are the same as those adopted in the calculations using the global parametrizations. For the calculations using JLM potentials, the rms radius of ¹²C and ¹³C are both set to 2.468fm, which corresponds to the charge radius of the ¹²C nucleus [49]. (The predicted angular distributions from the JLM potentials are very sensitive to the nuclear rms radii. For example, choosing the matter radius of fm and keeping all the other parameters the same, the shape of the angular distributions no longer resembles the experimental data and the predicted cross-sections are 3 to 4 times higher than the experimental data.)

7 The predicted distributions are plotted as dashed lines in Figure 2. The angular distributions based on JLM potentials and the distributions based on global potentials disagree mainly at large angle where the JLM potentials seem to agree with the data better. The extracted spectroscopic factors are plotted in the right panel of Figure 3 with the same convention as the left panel. The extracted spectroscopic factors are about 20% lower than those extracted using the global parameters potentials. However, the extracted values using the global parameters and the JLM potentials behave similarly over the entire energy range. For deuteron incident energy ranging from 6-60 MeV, the average spectroscopic factor value is 0.62±0.09. In summary, the angular distributions measured in the reaction of ¹²C(d,p)¹³C gs and the inverse reaction 13 C(p,d) 12 C in the past 50 years have been evaluated. Extensive analysis using many measurements provides an overview of the uncertainties associated with extracting the spectroscopic factors and the region where reliable values maybe obtained in normal kinematics reactions. It is shown that, fitting the first maximum in the angular distributions, spectroscopic factors could be extracted consistently to within 20% using a well-defined analysis procedure and comparison to a consistent set of calculations. Calculations using global parametrization of optical potentials or JLM potentials give similar behavior of the spectroscopic factors even though the absolute values depend on the potentials used. As it may not be important to extract the absolute spectroscopic factors, the global parametrization of the optical model potentials has the obvious advantage of requiring less input parameter choice. The current analysis of 12 C(d,p) 13 C and 13 C(p,d) 12 C reactions provide reference points where relative spectroscopic factors can be measured. The set of input parameters used in this work should be applicable to other systems and they are summarized in Table 3. However, further work is needed to see if similar analysis could be extended to other systems to provide guidance to extract spectroscopic factors in inverse kinematics reactions, which become more widely used with the availability of rare isotope beams. This work is supported by the National Science Foundation under Grant Nos. PHY

8 Table 1. List of references and spectroscopic factors for 12 C(d,p) 13 C gs Author Ref. Energy (MeV) s.f. (published) s.f. (global) s.f. (JLM) code G.D.Putt [42] DWBA, (WUCK) N.E.Davison [41] N/A R.V.Poore [40] DWBA J.W.Leonard [43] DWBA A.Gallmann [7] DWBA A.Gallmann [7] DWBA A.Gallmann [7] DWBA T.W.Bonner [8] N/A H.Guratzsch [9] DWBA N.Zaika [10] N/A D.Robson [11] DWBA U.Schmidt-Rohr [12] PWBA J.Lang [13] DWBA (PTOLEM Y) E.W.Hamburger [14] PWBA E.W.Hamburger [14] PWBA E.W.Hamburger [14] PWBA J.N.McGruer [15] N/A S.E.Darden [16] , DWBA (WUCK) Morita. S. [17] N/A Morita. S. [17] N/A R.Van Dantzig [18] N/A R.J.Slobodrian [19] 28 (0.96) (0.77) N/A H.Ohnuma [20] DWBA

9 (NDWUC K), CCBA W. Fetscher [21] DWBA K.Hatanaka [22] , 0.75, 1.26 (1.07) (1.01) DWBA (TWOFN R) Table 2. List of references and spectroscopic factors for 13 C(p,d) 12 C Author Ref. Energy s.f. s.f. s.f. code (MeV) (published) (global) (JLM) H.Toyokawa [44] DWBA(TWOFNR) CDCC- CCBA(CDC2RT,HICALST) J.R.Campbell [45] , DWBA(PTOLEMY) H.Taketani [46] DWBA K.Hosono [47] , 0.31, 0.43 (1.22) (1.43) DWBA(TWOSTP) Table 3: Summary of the input parameters used in TWOFNR Global JLM Proton potential E_p<26 MeV Perey&Perey [23], E_p>26 Menet, [24] JLM, rms=2.45 fm [29] Deuteron potential Adiabatic [51] from Chapel-Hill Adiabatic [51] from JLM [31] shape Wood-Saxon Wood-Saxon; a =0.54fm n-binding potential Wood-Saxon, r0=1.25, a=0.65, depth adjusted to fit the binding energy, no spin-orbit strength Wood-Saxon, r0=1.25, a=0.65, depth adjusted to fit the binding energy, no spinorbit strength Finite range calculation Yes Yes Hulthen finite range factor [25] Vertex constant D0^2, [25] potential scaling lamda N/A 1.0/0.8 for real/imaginary [29]

10 Non-Locality [48] p=0.85, n=0, d=0.54 p=0.85, n=0, d=0.54

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