-scattering and -induced reaction cross sections of Zn at low energies
-nucleus potentials play an essential role for the calculation of -induced reaction cross sections at low energies in the statistical model. Uncertainties of these calculations are related to ambiguities in the adjustment of the potential parameters to experimental elastic scattering angular distributions and to the energy dependence of the effective -nucleus potentials.
The present work studies the total reaction cross section of -induced reactions at low energies which can be determined from the elastic scattering angular distribution or from the sum over the cross sections of all open non-elastic channels.
Elastic and inelastic Zn(,)Zn angular distributions were measured at two energies around the Coulomb barrier at 12.1 MeV and 16.1 MeV. Reaction cross sections of the (,), (,), and (,) reactions were measured at the same energies using the activation technique. The contributions of missing non-elastic channels were estimated from statistical model calculations.
The total reaction cross sections from elastic scattering and from the sum of the cross sections over all open non-elastic channels agree well within the uncertainties. This finding confirms the consistency of the experimental data. At the higher energy of 16.1 MeV, the predicted significant contribution of compound-inelastic scattering to the total reaction cross section is confirmed experimentally. As a by-product it is found that most recent global -nucleus potentials are able to describe the reaction cross sections for Zn around the Coulomb barrier.
Total reaction cross sections of -induced reactions can be well determined from elastic scattering angular distributions. The present study proves experimentally that the total cross section from elastic scattering is identical to the sum of non-elastic reaction cross sections. Thus, the statistical model can reliably be used to distribute the total reaction cross section among the different open channels.
Present address:] RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Present address:] HZDR Dresden-Rossendorf, D-01314 Dresden, Germany Present address:] LIP Lisboa, 1000-149 Lisbon, Portugal
The nucleosynthesis of so-called -nuclei on the neutron-deficient side of the chart of nuclides requires a huge reaction network including more than 1000 nuclei and more than 10000 nuclear reactions (e.g. Woo78 (); Arn03 (); Rau06 (); Rap06 ()). The resulting production factors depend sensitively on branchings between (,n) and (,) reactions which are located several mass units “north-west” of stability on the chart of nuclides for heavy nuclei and close to stability for nuclei in the mass range around . It is impossible to measure all the required (,) reaction rates in the laboratory. Instead, theoretical predictions have been used which are based on statistical model (StM) calculations. In most cases it turns out that the (,) reaction rate is better constrained by experimental (,) data than by (,) data because of the strong influence of thermally excited states in the target nucleus Mohr07 (); Rau11 (); Rau11b (); Rau13 ().
Calculations of (,) and (,) cross sections and reaction rates depend on the transmission which in turn depends on the chosen -nucleus potential. Angular distributions of (,) elastic scattering have been measured with high precision in the last decade Mohr13 (); Pal12 () to determine the -nucleus potential at low energies. However, elastic scattering at low energies is dominated by the Coulomb interaction, and the angular distributions approach the Rutherford cross section for point-like charges at astrophysically relevant energies. Consequently, -nucleus potentials from elastic scattering are determined at somewhat higher energies and have to be extrapolated down to the astrophysically relevant energies.
It has turned out over the years that standard -nucleus potentials like the extremely simple McFadden-Satchler potential McF66 () are able to reproduce (,n) and (,) cross sections around and above the Coulomb barrier, whereas at very low energies below the Coulomb barrier (i.e., in the astrophysically relevant energy range) an increasing overestimation of the experimental reaction cross sections has been found in many cases (e.g., Som98 (); Gyu06 (); Ozk07 (); Cat08 (); Yal09 (); Gyu10 (); Kis11 (); Avr10 (); Sau11 (); Mohr11 (); Pal12b (); Qui14 (); Qui15 (); Sch16 ()). Interestingly, in the mass range, experimental (,p) and (,n) data are well described using the McFadden-Satchler potential Mohr15 (). Several alternative suggestions for low-energy -nucleus potentials have been made in the last years in Su15 (); Avr15 (); Avr14 (); Mohr13 (); Sau11 (); Avr10 (); Dem02 (); Dem09 (), and the related uncertainties of -induced reaction cross sections at low energies were studied very recently by Per16 (); Avr16a (); Mohr16a ().
The motivation of the present work is manyfold. Firstly, we attempt to extend the high-precision elastic scattering measurements of the last decade in the mass range Mohr13 () towards lower masses. Secondly, we include inelastic (,) scattering into our analysis which may play a significant role for the total (non-elastic) reaction cross section at very low energies Rau13b (); Avr16 (). Thirdly, we have measured reaction cross sections of the (,n), (,p), and (,) reactions at exactly the same energy as (,) elastic and inelastic scattering. This avoids any complications from the extrapolation of the energy-dependent -nucleus potential. In our previous study Gyu12 () such experimental data have been used for a sensitive test of the basic quantum-mechanical equation which relates the total reaction cross section to the reflexion coefficients of elastic scattering. In the present work the reduced experimental uncertainties allow to use the quantum-mechanical relation for to constrain the cross sections of unobserved non-elastic channels. Note that the chosen target nucleus Zn is well-suited for such a study because most of the reaction products of -induced reactions are unstable. This allows a simple and robust determination of the total cross section for each reaction channel by activation measurements.
The paper is organized as follows. In Sect. II we describe our experimental procedures for the measurement of -induced reaction cross sections on Zn and Zn(,)Zn scattering. Sect. III presents the analysis of our new scattering data and further scattering data from literature. A comparison between the total reaction cross sections from scattering and from the sum of the individual reaction cross sections will be given in Sect. IV. The predictions of recent global -nucleus potentials will be compared to our experimental reaction data in Sec. V. A final discussion and conclusions will be provided in Sect. VI.
Ii Experimental set-up and procedure
In order to give a comprehensive experimental description of the Zn + system, the cross sections of the following reaction channels were measured in the present work: The elastic -scattering cross section at E = 12.1 and 16.1 MeV was measured in a wide angular range. Inelastic scattering leading to the first four exited states of Zn was also measured in an angular range limited to a more backward region. Using the activation technique, the cross sections of the Zn(,)Ge, Zn(,n)Ge, and Zn(,p)Ga reactions were also measured at the same energies. Since the experimental techniques of both the scattering and activation experiments were already described in detail elsewhere Gyu12 (); Mohr13 (), here only the most important features of the measurements and the results are presented.
ii.1 Scattering experiments
The scattering experiments were carried out at the cyclotron accelerator of Atomki which provided beams of 12.05 and 16.12 MeV energy with typical intensity of 150 nA. Targets were produced by evaporating highly enriched (99.71 %) metallic Zn onto thin (40 g/cm) carbon foils. The thickness of the Zn layer was about 150 g/cm determined by energy loss measurements. The energy loss at the energies of the scattering experiment is negligible compared to the energy width of the beam.
The angular distributions were measured in a scattering chamber equipped with seven ion implanted Si particle detectors. The detectors were fixed on turntables enabling the measurement of scattered particle spectra in an angular range between 20 and 175. In addition, two detectors were fixed at 15 with respect to the beam. These monitor detectors were used for normalization purposes. Two typical scattering spectra are shown in Fig.1 recorded in the forward and backward regions, respectively. Peaks corresponding to elastic and inelastic scattering events on Zn and on carbon and oxygen target components are indicated.
The elastic scattering cross sections were measured at both energies in the complete angular range between 20 and 175. The excitation energies of the first four excited states of Zn are 991.6 keV (), 1799.4 keV (), 1910.3 keV (), and 2306.8 keV () ENSDF (); NDS (). The energy of the first excited state is far away enough from both the ground state and the higher energy excited states so that the peak corresponding to the inelastic scattering leading to this excited state is well separated in the particle spectra. Therefore, the inelastic scattering cross section for this excited state could be determined over a wider angular range. The only limitation is caused by the elastic scattering events on carbon and oxygen in the target which start to overlap with the inelastic peak in the forward angle region. Therefore, the inelastic scattering cross section to the first excited state could be only determined in the angular range between 40 and 175.
The second and third excited states could not be fully resolved because of their energy difference of about 100 keV. Therefore, only the sum of the cross sections leading to these states could be measured in the angular range between 60 and 175. Owing to the sufficient separation, the inelastic cross section leading to the fourth excited state could be determined from 60 to 175. Above the fourth exited state the level density becomes too high, and hence no further inelastic cross sections could be measured to higher lying levels.
The experimental data will be provided to the community in tabular form through the EXFOR database EXFOR ().
ii.2 Activation experiments
The activation cross section measurements had been already carried out and published in full details Gyu12 (). In the present work only one additional data point was determined since in Gyu12 () no measurement was carried out at 16.12 MeV -energy, which is one of the energies where the scattering cross section was measured in the present work and the total cross section is determined.
The new measurement at 16.12 MeV -energy was carried out with exactly the same conditions as the experiments of Gyu12 (). The cyclotron accelerator provided the -beam which bombarded a thin natural isotopic composition Zn target on Al foil backing. The number of the Ge, Ge, and Ga isotopes produced by the Zn(,)Ge, Zn(,n)Ge, and Zn(,p)Ga reactions, respectively, was determined off-line by measuring the -radiation following the decay of the reaction products with a shielded 100 % relative efficiency HPGe detector.
Table 1 summarizes the results of the activation cross section measurements. The first row at 12.05 MeV -energy is taken from Gyu12 () while the second one at 16.12 MeV is the result of the present work. The last column shows the sum of the cross sections of the three reactions. These values will be used in the next sections for the determination of the total cross sections.
|(MeV)||(MeV)||cross section (mbarn)|
The energies in the scattering experiments and in the activation experiments were identical because exactly the same settings for the cyclotron have been used. At the higher energy of 16.1 MeV the effective energies of the scattering and activation experiments are the same ( MeV) because relatively thin targets were used here. However, at the lower energy of 12.1 MeV a thicker target had to be used for the activation experiment leading to a slightly lower effective energy in the activation experiment ( MeV compared to MeV). The beamline for the scattering experiments allows a more precise measurement of the beam energy using the well-calibrated analyzing magnet in this beamline. Compared to the data in Gyu12 () ( MeV), this leads to a minor change of 50 keV in the effective energy for the 12.1 MeV activation experiment. This minor change remains within the given uncertainties of Gyu12 ().
Iii Analysis of scattering data
In a first step, the new elastic scattering data at low energies together with data from literature are analyzed in the optical model (OM). Fortunately, several angular distributions of Zn(,)Zn elastic scattering at higher energies are also available in literature. Therefore, we can study the energy dependence of the angular distributions, the total reaction cross section , and the resulting optical model potentials (OMP). We restrict ourselves to data below about 50 MeV.
The complex OMP is given by
with the real part and the imaginary part of the nuclear potential and the Coulomb potential . The Coulomb potential is calculated as usual from the model of a homogeneously charged sphere with the Coulomb radius . Various parametrizations have been used for the nuclear potentials and .
For the imaginary part Woods-Saxon (WS) potentials of volume and surface type were applied:
with the Woods-Saxon function
where and . and are the strengths of the volume and surface imaginary potentials, are the radius parameters, the diffuseness parameters, and for the target Zn. Note that the maximum depth of a surface WS potential in the chosen convention in Eq. (3) is . Following Mohr13 (), at low energies below 25 MeV only a surface imaginary potential was used in combination with a real folding potential. At the higher energies a combination of volume and surface Woods-Saxon potentials is necessary to obtain excellent fits to the experimental angular distributions.
The real part of the nuclear potential was either taken as a volume WS potential (similarly defined as above for the imaginary part) or calculated from a folding procedure. Two parameters for the strength and for the width were used to adjust the folding potential to the experimental data:
Obviously, the width parameter should remain close to unity; otherwise, the folding procedure would become questionable. The strength parameter is typically around leading to volume integrals per interacting nucleon pair MeV fm for heavy nuclei with a closed proton or neutron shell whereas slightly higher values have been found for lighter nuclei and non-magic nuclei. Further details on the folding procedure and the chosen interaction can be found in Mohr13 ().
The total (non-elastic) reaction cross section is related to the elastic scattering angular distribution by
Here is the wave number, is the energy in the center-of-mass (c.m.) system, and are the real reflexion coefficients. These and the scattering phase shifts define the angular distribution of elastic scattering, whereas depends only on the , but is independent of the . The are the contributions of the -th partial wave to the total reaction cross section which show a characteristic behavior as discussed e.g. in Mohr11 (); Mohr13b ().
iii.1 Elastic scattering data from literature
Elastic Zn(,)Zn scattering has been studied in many experiments over a broad energy range. Here we focus on the data up to energies of about 50 MeV. The determination of an optical potential from angular distributions requires high-quality scattering data over the full angular range. A careful determination of the uncertainties is also mandatory because these uncertainties have dramatic impact on the minimization procedure. Therefore, we briefly review the status of the data from literature as well as the availability and reliability of the data in the EXFOR data base EXFOR (). The data are compared to the theoretical angular distributions in Fig. 2. The obtained OMP parameters are listed in Table 2.
DiPietro et al. DiP04 () have primarily studied Zn(He,He)Zn elastic scattering, and for comparison also the Zn(,)Zn reaction was studied at MeV. The data cover a broad angular range but there are also two larger gaps around and . The data (including uncertainties) are available from the EXFOR data base EXFOR ().
Robinson and Edwards Rob78 () have measured angular distributions at , 17.94, and 18.99 MeV. The data cover a broad angular range from very forward () to backward angles around . The data are not yet available at EXFOR, and therefore the data had to be read from Fig. 2 in Rob78 (). The digitization has been performed using a high-resolution medical scanner. However, the quality of the data is essentially limited by the presentation of Fig. 2 in Rob78 (). Error bars are typically smaller than the symbols in Fig. 2 of Rob78 (); therefore, a fixed uncertainty of 5 % has been assigned to all data points. Note that the chosen absolute value of 5 % does affect the resulting but does not affect the minimization procedure and the resulting OMP.
The data of Fulmer et al. Ful68 () at MeV are available in tabular form (Table V of Ful68 ()). The data reach the backward angular range up to . However, the data unfortunately do not cover the forward region (), and thus the absolute normalization cannot be fixed in the usual way by Rutherford scattering. Here it is interesting to note that a much better description of the experimental data can be obtained as soon as the absolute normalization of the data is considered as a free parameter. The best fit is obtained using a normalization of 0.787 for the data in Table V of Ful68 (). Nevertheless, the resulting parameters of the fits do not change dramatically; i.e., already the shape of the angular distribution is able to constrain the fit reasonably well. For completeness it should also be mentioned that the data shown in Fig. 4 of Ful68 () are also about 10 % lower than the values given in Table V of Ful68 ().
There are two data sets at MeV in EXFOR which reference England et al. Eng82 () and Ballester et al. Bal88 (). Both data sets have been digitized from the given Figures in Eng82 (); Bal88 () (Fig. 2 of Eng82 () and Fig. 3 of Bal88 ()). The focus of Bal88 () is inelastic scattering, and it is explicitly stated that “The elastic scattering data have already been published.” where the syperscript “” is a reference to Eng82 (). Thus, both data sets should be identical. However, due to uncertainties of the digitization process, in fact both data sets show minor discrepancies (and not even the number of data points agree). We have decided to analyze both data sets; such an analysis can provide some insight into the uncertainties of the resulting OMP parameters which result from the re-digitization procedure. The angular distribution published in Eng82 (); Bal88 () covers the full angular range from forward angles () to backward angles () with small uncertainties (typically smaller than the shown point size). Unfortunately, these small uncertainties are not available anymore, and a fixed uncertainty of 5 % has been used in the fitting procedure. It turns out that the resulting OMP parameters from the two EXFOR data sets are close to each other. As with the analysis of the 21.3 MeV data (see discussion below), small discrepancies in the shape of the imaginary potential at large radii ( fm) lead to noticable effects in the total reaction cross section which changes by about 10 % from 1317 mb from the data of Bal88 () to 1481 mb from the data of Eng82 (). Therefore, we recommend the average = mb at 25 MeV, and we assign the same 6 % uncertainty to all data which are derived from digitized angular distributions.
The angular distributions at , 38, and 50.5 MeV measured by Baktybaev et al. Bak75 () cover only a limited angular range up to about . The data are available at EXFOR, but have larger uncertainties of about % which are further increased by the re-digitization. Consequently, potentials derived from these data are not as reliable as in the other cases of this work. Further data from the same institute Ais89 () are available at EXFOR. But these data Ais89 () contain only very few points and are thus not included in the present analysis.
The data at 31 MeV by Alpert et al. Alp70 () also cover only a limited angular range. The above statements on the Baktybaev et al. data also hold here. As the data are not available at EXFOR, we have re-digitized the angular distribution from Fig. 3 of Alp70 (). Error bars are not visible in this Fig. 3. We have used a fixed 5 % uncertainty in the fitting procedure.
A limited angular distribution is available by McDaniels et al. McD60 () at 41 MeV. Re-digitized data without uncertainties are available at EXFOR. Again we have used a fixed 5 % uncertainty for the fitting procedure of these data.
The data at 43 MeV by Broek et al. Bro62 () cover a very limited angular range from about . In addition, the scanned pages of this article (as provided at the ScienceDirect web page) are distorted. We decided to exclude these data from our analysis.
An excellent angular distribution at 48 MeV from about to is available in Pirart et al. Pir78 (). Unfortunately, again the data in EXFOR had to be re-digitized from Fig. 1 of Pir78 () where uncertainties are not visible. Similar to the previous cases, we have used a fixed 5 % uncertainty in our analysis.
For all angular distributions fits were performed in the following way. First, a folding potential was calculated at the average energy of 21.3 MeV using the energy-dependent parameters of the nucleon-nucleon interaction listed in Mohr13 (). The energy dependence of these parameters is relatively weak and has practically no impact on the final results. For a detailed discussion, see Mohr13 (). Next, the strength parameter and the width parameter of the folding potential and the parameters , , and of the imaginary part were fitted simultaneously to the experimental angular distributions. In addition, the absolute normalization of the angular distributions was allowed to vary because this absolute normalization often has much larger uncertainties. The normalization factors deviate from unity by not more than about 30 %. Fits with fixed lead in many cases to much poorer of the fit. E.g., in the case of the 21.3 MeV data of Fulmer et al. Ful68 () reduces by about a factor of 4 from fixed to fitted . The resulting OMP parameters remain relatively stable with variations of about 2 % for the volume intgrals and . However, relatively small changes in the shape of the imaginary potential at large radii ( fm) result in a change of the total reaction cross section of about 10 % from 1200 mb for fixed to 1327 mb for fitted . In cases where the additional free parameter did not improve the reduced , a fixed normalization was used. The results of these fits are shown in Fig. 2 and listed in Table 2. In general, an excellent reproduction of the experimental angular distributions could be achieved in the full energy range under study. Further information on the uncertainties of the total reaction cross section from elastic scattering angular distributions is given in Mohr10 ().
|111from the local potential fit using Eq. (5); uncertainties estimated as discussed in the text||Ref.|
|(MeV)||(MeV)||(–)||(–)||(MeV fm)||(fm)||(MeV)||(fm)||(fm)||(MeV)||(fm)||(fm)||(MeV fm)||(fm)||(–)||(mb)||Exp.|
|13.4||12.40||1.314||1.019||377.7||4.757||–||139.0||1.603||0.380||107.9||6.593||0.946||610||80222uncertainty from Gyu12 ()||DiP04 ()|
|15.0||14.11||1.292||1.029||378.3||4.808||–||105.3||1.607||0.410||88.7||6.638||1.0333fixed normalization||858||52||Rob78 ()|
|17.9||16.88||1.336||1.012||371.3||4.726||–||102.5||1.530||0.468||89.8||6.404||1.033footnotemark: 3||1083||65||Rob78 ()|
|19.0||17.87||1.344||1.015||377.9||4.744||–||111.0||1.525||0.456||94.1||6.371||1.033footnotemark: 3||1149||69||Rob78 ()|
|25.0||23.53||1.435||0.980||367.6||4.578||1.080||0.985||1.544||0.255||54.9||4.659||1.033footnotemark: 3||1481||89||Eng82 ()|
|25.0||23.53||1.457||0.986||379.9||4.605||1.591||0.437||1.478||0.302||59.6||4.565||1.033footnotemark: 3||1317||79||Bal88 ()|
|12.1||11.34||1.288||1.038||394.5||4.845||–||186.5||1.697||0.275||116.6||6.876||1.033footnotemark: 3||428||7444discussion of uncertainty: see text||555this work|
|16.1||15.17||1.330||1.006||371.2||4.696||–||93.7||1.542||0.457||81.3||6.439||1.033footnotemark: 3||905||1844footnotemark: 4||55footnotemark: 5|
iii.2 New elastic scattering data at 12 and 16 MeV
After the successful description of the elastic Zn(,)Zn scattering data from literature DiP04 (); Rob78 (); Ful68 (); Eng82 (); Bal88 (); Bak75 (); Alp70 (); McD60 (); Pir78 () we expected a similar behavior for the analysis of our new data at 12.1 and 16.1 MeV. However, the new data cover backward angles up to about which exceeds the angular range of the literature data at low energies DiP04 (); Rob78 (). We found an unexpected increase of the Rutherford-normalized cross section which is more pronounced at the lower energy, and as it turns out it is practically impossible to describe the full angular distributions using an OMP composed of a real folding potential and an imaginary surface WS potential (similar to the fits to literature data at low energies). The reduced of the OM fits does not reach values around 1.0, but remains at about 2.0 (2.4) for the 12 MeV (16 MeV) data.
Therefore, we have performed a phase shift fit (PSF) using the method of Chi96 (). The PSFs are able to reproduce the full angular distributions at both energies with (see Figs. 3 and 4). Values of in a PSF would have been an indication for experimental problems.
The comparison between the OM fits and the PSFs shows clearly that the poor description of the angular distributions in the OM fits is related to the rise of the elastic scattering cross section at extreme backward angles. Therefore, we have made additional fits which have been restricted to data at (, ). Already the restriction to leads to a dramatic improvement of at both energies by about a factor of two whereas the do not improve further for the fits restricted to or . Therefore, we list the results for the OM fits truncated at in Table 2.
Although the of the various fits differ significantly, it is difficult to visualize the differences. In the standard presentation (upper parts of Figs. 3 and 4) the various fits are mostly hidden behind the experimental data. Therefore, we consider the PSFs (with ) as quasi-experimental data and show the ratio between the OMP fits and the PSFs in the middle part of Figs. 3 and 4. Here it is nicely visible that all truncated fits underestimate the most backward cross sections by about %. As soon as the full data set is used for fitting, the underestimation at most backward angles becomes smaller (about %). However, at the same time the data between and are overestimated by about %; this leads to the overall significantly worse in this fit.
A relatively poor fit with may also result from an inappropriate OMP. Although it is very unlikely that the otherwise successful folding potential Mohr13 () fails in the particular case of Zn at low energies, we have repeated the above procedure of fitting the full angular distribution and truncated angular distributions using WS potentials of volume type in the real and imaginary part of OMP. Almost exactly the same behavior was found in this case (see lower parts of Figs. 3 and 4).
From all the above calculations the total reaction cross sections have been determined using Eq. (5). Fortunately, the results for turn out to be very stable. At the lower energy we find an average value of = mb. The highest (lowest) value of = 440 mb (420 mb) is found for the folding potential truncated at (). At the higher energy we find an average value of = mb. The highest (lowest) value of = 928 mb (895 mb) is found for the folding potential truncated at (WS potential truncated at ). The result of the PSF ( = 898 mb) is relatively close to lowest result, and there seems to be a small systematic deviation between the folding potential fits (average = mb) and the WS potential fits (average = mb). Therefore, we recommend = mb (with a slightly increased 2 % uncertainty) at the higher energy.
At the lower energy of 12.1 MeV, there is an obvious discrepancy between the PSF and the OMP fits. The OMP fits lead to a relatively smooth variation of the phase shifts with the angular momentum number . Contrary to the OMP fits, the PSF shows stronger variations in and which cannot be reproduced by a typical -nucleus potential. It is interesting to note that although the underlying reflexion coefficients are not identical, the resulting total reaction cross section is almost the same for all fits (see above).
At the higher energy of 16.1 MeV the discrepancies between the and from the PSF and from the OMP fits are smaller. This is not surprising as the backward rise at the higher energy is not as pronounced as at 12.1 MeV. The small systematic discrepancy between the OMP fits using either a folding potential or a WS potential is mainly related to tiny differences in for .
iii.3 Reduced reaction cross section
Total reaction cross sections of -induced reactions for many target nuclei and in a broad energy range follow a systematic behavior which becomes visible in a plot of so-called reduced cross sections versus reduced energy as suggested in Gom05 ().
The result is shown in Fig. 9. Contrary to the common trend for all nuclei with masses above , the data for Zn are slightly higher than the general trend at all energies under study. Very recently, the analysis of reaction data for lighter targets (Na Alm14 () and S Bow13 ()) has shown that for these light nuclei is dramatically higher than the general trend for heavy nuclei Mohr14 (). However, the dramatically increased reduced cross sections for Na from the Na(,p)Mg data of Alm14 () were not confirmed by later experiments How15PRL (); Tom15PRL () and turned out to be an experimental error Alm15 (). Fig. 9 shows also the predictions from four -nucleus potentials McF66 (); Avr10 (); Mohr13 (); Su15 (). These results will be discussed later.
For completeness it should be noted that there is an approximate relation between reduced energies and the Gamow window Mohr16 (): . Consequently, the astrophysically relevant range for the reduced energy is located below the shown range of Fig. 9 which was chosen from the availability of experimental scattering data.
iii.4 ALAS for Zn?
Anomalous large-angle elastic scattering (ALAS) has been discussed in literature already many years ago (e.g., Lan82 (); Mic95 ()). However, there is no strict definition for ALAS. The phenomenon was first discussed in connection with O(,)O and Ca(,)Ca elastic scattering. For these reactions it was noticed later that the so-called anomalous cross sections are related to weak absorption of the doubly-magic target nuclei, and it is possible to describe the angular distributions within the OM Lan82 (); Mic95 (); Abe93 (); Atz96 (); Hir13 (); Abd03 (). Contrary to these findings, it turned out that the reproduction of the backward angular range in e.g. Li(,)Li or Ne(,)Ne remains extremely difficult, and various explanations for the backward rise have been suggested: inelastic coupling to low-lying excited states, compound-elastic contributions, elastic transfer, angular-momentum-dependent absorption (e.g., Bac72 (); Sam92 (); Yan11 (); Fri71 ()).
As we have seen above, fortunately the influence of the backward rise on the derived total cross section remains small. Therefore, a complete theoretical analysis of the backward rise remains beyond the scope of the present paper. Nevertheless, two of the above effects will be analyzed in more detail. Firstly, inelastic scattering may contribute significantly to the total reaction cross section . For a quantitative analysis we have measured several angular distributions (See Sect. III.5). Secondly, compound-elastic scattering may contribute to the elastic scattering angular distribution but by definition it is not included in the OM analysis (see Sect. III.6).
iii.5 Analysis of inelastic scattering
One focus of the present study is the comparison between the total reaction cross section from elastic scattering in Eq. (5) and the sum over all open non-elastic channels. This will be discussed in further detail in Sec. IV. Besides the real reaction channels like (,n), (,p), and (,), inelastic (,) may contribute to this sum. In our previous study Gyu12 () we have estimated the (,) cross section from coupled-channel calculations and from Coulomb excitation. Now we are able to provide experimental constraints for the (,) cross section for the low-lying states.
A precise experimental determination of the total (,) cross section is very difficult for at least two reasons. The total inelastic (,) cross section is composed of contributions to all excited states in the target nucleus Zn with excitation energies below of the scattering experiment. In practice, the spectra in Fig. 1 allow a determination of the (,) cross section only for the lowest excited states in Zn, and in particular at backward angles a significant yield appears about 5 MeV below the elastic peak. Besides inelastic scattering from Zn, this yield may also come from reaction products of -induced reaction on the Zn target and all target contaminations because the detectors do not allow the identification of the ejectiles. Furthermore, the measurement of each angular distribution is complicated because inelastic peaks may overlap with elastic scattering from lighter nuclei in the target (e.g., in the carbon backing). In addition, at forward angles the (,) cross section is much smaller than the elastic cross section which approaches the Rutherford cross section and thus increases dramatically to small scattering angles with with .
iii.5.1 Inelastic scattering to low-lying excited states
The first excited states in the level scheme of Zn consist of a state at 992 keV and a triplet of states (, , ) with almost twice the excitation energy of the first state, i.e. a typical vibrational behavior. Thus, these inelastic angular distributions were analyzed within the anharmonic vibrator model which is implemented in the widely used coupled-channels ECIS code ECIS ().
Experimental angular distributions were measured for the first state at 992 keV and the state at 2307 keV. The experimental resolution was not sufficient to separate the second state at 1799 keV and the state at 1910 keV; only the sum of both states could be determined. For experimental details, see Sec. II.1.
It has been difficult to fit the elastic scattering angular distributions, see Sec. III.2. Obviously, these problems appear also when simultaneous fits are made to elastic and inelastic angular distributions. Several fits with different potentials and a varying number of adjustable parameters have been made. These fits show significant differences in the reproduction of the angular distributions, but fortunately the angle-integrated inelastic cross sections are quite stable. The results are shown in Fig. 10 for the 12.1 MeV data and in Fig. 11 for the 16.1 MeV data.
As a first approximation, the widely used potential of McFadden and Satchler McF66 () was applied in combination with an adjustment of the couplings to the inelastic states (dotted lines in Figs. 10 and 11). The reproduction of the elastic angular distributions is reasonable but not perfect. Such a behavior is expected because the potential parameters were not re-adjusted. The angular distribution of the state is reasonably well described, but the two-phonon states and in particular the state cannot be reproduced.
In a next step, the Woods-Saxon potential from the optical model fits (restricted to scattering angles below ) was used (dashed lines, label “WS close to OM”); see also Sec. III.2. Again, the coupling to the inelastic states was fitted, and a minor readjustment was allowed for the depths of the real and imaginary parts of the Woods-Saxon potential. Of course, from the fitting procedure the agreement for the elastic angular distribution improves. But simultaneously also the description of the inelastic angular distributions improves.
In a third calculation, additionally all parameters of the Woods-Saxon potentials (real and imaginary depths, radii, and diffusenesses) were adjusted simultaneously (full lines, label “fit ()”). Although a smaller is obtained, the visible changes in the angular distributions remain relatively small.
The final calculation repeats the third calculation, but includes all experimental data, i.e. including the backward rise of the elastic angular distributions beyond . The results (dash-dotted lines, label “fit (all data)”) show significantly worse agreement in the backward angular region of the inelastic angular distributions. This is related to a wide variation of the WS parameters, similar to the problems found in the OM study in Sec. III.2. Hence, the backward rise of the elastic cross sections cannot be explained by the inelastic coupling to low-lying excited states.
Interestingly, despite the relatively wide changes of the inelastic angular distributions, the angle-integrated inelastic cross sections remain relatively stable. E.g., at 12.1 MeV for the dominating first state a cross section of 27.3 mb is obtained from the McFadden/Satchler potential, the Woods-Saxon potential from optical model fit gives 32.9 mb, and the fits to data up to (all data) result in 34.3 mb (30.1 mb). Excluding the McFadden/Satchler result (without any adjustment of the potential to the experimental data), we adopt a semi-experimental angle-integrated cross section of mb in this case. Similar results are found for all angle-integrated inelastic cross sections at both experimental energies. The results are listed in Table 3. For each level, the given uncertainties are estimated from the variations of the different fits. However, it should be kept in mind that the experimental inelastic angular distributions do not cover the full angular range and thus cannot fully constrain the fits; this holds in particular for the significant contribution of the state at forward angles. Therefore, a somewhat increased uncertainty of about 15 % is carefully estimated for the sum over the experimentally determined inelastic cross sections to low-lying excited states in Table 3.
|E||E||(992 keV)||(1799 keV)||(1910 keV)||(2307 keV)|
|12.05||11.29||0.09||33||3||3.1||0.5||10.2||0.5||3.2||0.5||49.5||7.5666discussion of uncertainty: see text||31.5||6.3777estimated uncertainty of 20 % (see text)|
|16.12||15.17||0.08||28||1||0.6||0.2||13.6||0.5||0.9||0.1||43.1||6.511footnotemark: 1||157.9||30.622footnotemark: 2|
iii.5.2 Inelastic scattering to higher-lying excited states
The inelastic (,) cross sections to higher-lying states above the (, , )-triplet were estimated using the combination of direct and compound contributions as implemented in the widely used nuclear reaction code TALYS (version 1.8). Contrary to the first excited states with their dominating direct contributions, inelastic scattering to higher-lying states is dominated by compound contributions. In these TALYS calculations 30 low-lying levels below MeV in Zn were taken into account explicitly; for MeV a continuum contribution is estimated using a theoretical level density.
The first 4 excited states were already taken into account in the previous section III.5.1. Thus, the summed inelastic cross section to higher-lying states is estimated from the calculated total inelastic cross section in TALYS minus the calculcated inelastic cross sections to the first 4 excited states. Fortunately, even a broad variation of the TALYS parameters (mainly a variation of the -nucleus potential) leads to relatively stable values for the inelastic cross sections. At the lower energy of 12.1 MeV, varies between 26.6 mb and 34.3 mb with an average value of mb. Because of the missing experimental constraint, we finally assign a larger 20 % uncertainty to this value. At the higher energy of 16.1 MeV shows significantly larger values between 139 mb and 168 mb with an average of mb. Again, we finally assign a 20 % uncertainty (see Table 3). Indeed, this choice of the uncertainty for is somewhat arbitrary. But we think that increasing the TALYS uncertainties by a factor of about two should provide a careful estimate of the real uncertainty of .
iii.6 Compound-elastic contributions to low-lying states
In general, the compound mechanism may also contribute to the elastic angular distribution and to the inelastic angular distributions of the low-lying states. The angle-integrated compound-elastic contribution is small with about mb at 12.1 MeV and around 0.5 mb at 16.1 MeV. Angle-integrated compound-inelastic cross sections to the low-lying states in Sec. III.5.1 are of the order of a few mb at 12.1 MeV and below 1 mb at 16.1 MeV (calculated by TALYS, again mainly varying the -nucleus potential). Thus, the compound contributions may slightly affect the elastic angular distribution in particular at backward angles where the direct cross section is small, and may also contribute to the unexpected elastic cross sections at backward angles (see Sec. III.4). The inelastic angular distributions will also be somewhat affected by the compound contributions; however, as long as a reasonable description of the experimental angular distributions is achieved within the direct coupled-channels model, the angle-integrated inelastic cross section should be well-defined by these calculations in the coupled-channels approach (as done in Sec. III.5.1).
Iv Comparison of total cross sections from elastic scattering to reaction cross sections
The total non-elastic cross section is given by the sum over all open channels. For Zn at the energies under study this means:
in the above Eq. (8) can be derived from Eq. (5), i.e. from the angular distribution of elastic scattering (see Sec. III.2 and Table 2). The determination of the sum on the right-hand-side of Eq. (8) will be discussed in detail below.
The identity of from elastic scattering in Eq. (5) and from the sum over non-elastic channels in Eq. (8) follows directly from basic quantum-mechanics. In our previous work Gyu12 () we have calculated the ratio between the result from elastic scattering and the sum over the non-elastic channels, and was found within the experimental uncertainties. As there is no reason to question this theoretically expected ratio of , we can also reverse the arguments: fixing from Eq. (5) allows to obtain experimental constraints for the cross sections of unobserved non-elastic channels in the sum on the right-hand-side of Eq. (8). This will become important in particular for the compound-inelastic (,) cross section at the higher energy of 16.1 MeV.
At the lower energy of 12.1 MeV, all channels with two outgoing particles in the third line of Eq. (8) were neglected because these channels are either closed or have negligible cross sections. The same holds for the last line in Eq. (8). The remaining contributions can be taken almost completely from experiment. The cross sections of the (,), (,n), and (,p) reactions in the second line of Eq. (8) are taken from Table 1; their sum amounts to mb. The (,) inelastic scattering cross section is taken from experiment for the low-lying excited states with mb and from theoretical estimates for the higher-lying states with mb (see Table 3). Summing up all these values leads to a total non-elastic cross section of mb. Within the uncertainties, this result is in excellent agreement with the value of mb derived from elastic scattering (see Table 2). As expected, the ratio between the result from elastic scattering and the sum over the contributing channels results in , i.e. it is identical to unity within the uncertainties. Compared to our previous study Gyu12 (), the uncertainty of the ratio could be reduced by a factor of two. This reduction is based on improved scattering data at the same energy of the reaction cross sections. Note that the sum in Eq. (8) is based on experimental data with the only exception of inelastic scattering to higher-lying states in Zn which contributes only with 31.5 mb (or 7 %) to the sum of 447 mb.
At the higher energy of 16.1 MeV the situation is somewhat more complicated because more channels are open. Nevertheless, because of strongly negative -values, the cross sections of the (,) and (,He) reactions remain negligibly small, and the (,) channel is still closed; thus, the forth line in Eq. (8) still can be neglected. The same holds for the reactions with two outgoing particles in the third line of Eq. (8) with the exception of the (,) reaction. According to TALYS calculations, this reaction contributes with mb. Because there is no experimental constraint, we estimate a contribution of 18 mb with an uncertainty of 25 %. The other contributions are determined in the same way as for the lower energy of 12.1 MeV. The sum of the (,), (,n), and (,p) cross sections amounts to mb. The (,) inelastic scattering cross section is composed of the experimental result for the low-lying excited states ( mb) and of the theoretical estimates for the higher-lying states ( mb). The sum of mb again agrees well with the result of mb from elastic scattering, leading to a ratio of between the result from elastic scattering and the sum over the reaction channels.
As pointed out above, the expected ratio allows to constrain the cross sections of unobserved non-elastic channels. For the data at the higher energy of 16.1 MeV this leads to the conclusion that the significant contribution of compound-inelastic scattering to higher-lying excites states of mb is indeed confirmed experimentally by the present data. Starting from the experimental result from elastic scattering in Eq. (5) and subtracting the experimentally determined cross sections, i.e. the inelastic scattering to low-lying states and the reaction channels in the second line of Eq. (8), this leads to a remaining cross section of mb which has to be distributed among the remaining open channels which are mainly inelastic scattering to higher-lying states ( mb from TALYS) and to a minor degree the (,) reaction ( mb from TALYS).
V Predictions from global -nucleus potentials
Finally, the experimental results for the total reaction cross sections and for the (,), (,n), and (,p) cross sections are compared to predictions from global -nucleus potentials. Interestingly, it turns out that the widely used -nucleus potentials predict very similar total cross sections. The predictions of Watanabe (TALYS default) Wat58 (), McFadden and Satchler McF66 (), Demetriou et al. Dem02 (), Avrigeanu et al. Avr14 (), Su and Han Su15 (), and from the ATOMKI-V1 potential Mohr13 () are listed in Table 4. The new potential by Su and Han Su15 () overestimates the total reaction cross section in particular at low energies; at 5 MeV the predicted from the potential by Su and Han is about 11 b whereas the predictions from the other potentials Wat58 (); McF66 (); Dem02 (); Avr14 (); Mohr13 () vary between 0.3 b and 1.6 b with a mean value of about 0.8 b. (For completeness it may be noted here that the present TALYS 1.8 version uses still the Watanabe potential as default, although the TALYS manual states that this has changed to Avrigeanu et al. Avr14 ().)
|E MeV||E MeV||Ref.|
|experiment (this work)|
|408||833||Watanabe Wat58 ()|
|376||809||McFadden/Satchler McF66 ()|
|427||868||Demetriou et al., V1 Dem02 ()|
|405||837||Demetriou et al., V2 Dem02 ()|
|455||845||Demetriou et al., V3 Dem02 ()|
|415||857||Avrigeanu et al. Avr14 ()|
|475||915||ATOMKI-V1 Mohr13 ()|
|552||1010||Su and Han Su15 ()|
The total reaction cross section is mainly composed of the dominating (,p) and (,n) channels. The branching between these two channels is sensitive to the chosen nucleon potential whereas the total reaction cross section is sensitive only to the -nucleus potential. The TALYS default option for the nucleon potential Kon03 () works very well here and was not changed in this work. Excellent agreement for the (,n) and (,p) cross sections is found, see Fig. 12.
The cross section of the Zn(,)Ge reaction is sensitive to the -nucleus potential and to the -ray strength function. Here the best result is obtained using the Hartree-Fock BCS -ray strength from Cap09 (). The TALYS default option using generalized Lorentzian -ray strength from Kop90 () significantly underestimates the Zn(,)Ge cross section (see Fig. 12, thin red long-dashed line).
Vi Summary and conclusions
Elastic and inelastic Zn(,)Zn scattering was measured at the energies of 12.1 MeV and 16.1 MeV. At the same energies the cross sections of the (,), (,n), and (,p) reactions were determined using the activation technique. The experimental angular distributions of elastic scattering cover the full angular range and thus allow for a precise determination of the total non-elastic reaction cross section with uncertainties of about %. A perfect description of the elastic angular distributions could only be achieved using phase shift fits. The surprising rise of the elastic cross sections at very backward angles may be considered as so-called ALAS and could not be fully explained. Fortunately, the behavior of the angular distibutions at these very backward angles practically does not affect the determination of the total cross sections .
The total reaction cross sections follow a general smooth trend when presented as so-called reduced cross sections versus reduced energies . The data for Zn lie in between the common behavior for heavy target nuclei with Mohr13 () and slightly increased values for lighter target nuclei with Mohr15 ().
The total non-elastic reaction cross section was also determined from the sum over the cross sections of all non-elastic channels (including inelastic scattering). At the lower energy of 12.1 MeV excellent agreement between from elastic scattering and from the sum over non-elastic channels was found; here almost all open channels (including inelastic scattering to low-lying states in Zn) could be determined experimentally. At the higher energy of 16.1 MeV we find again excellent agreement for from the two approaches. However, now a significant contribution of inelastic scattering to higher-lying states in Zn is required which is indeed predicted in the statistical model. In turn, this may be considered as an experimental verification of these statistical model predictions. Compared to our previous work Gyu12 (), the experimental uncertainties in the comparison of from the two approaches could be reduced significantly by about a factor of two.
Usually, the cross sections of -induced reactions in the statistical model depend sensitively on the chosen -nucleus potential. At 12.1 MeV and 16.1 MeV, the recent global -nucleus potentials predict very similar total reaction cross sections , and with the exception of the latest potential by Su and Han Su15 () this behavior surprisingly persists down to lower energies. Thus, in the case of Zn the total reaction cross section can be described well, and the (,), (,n), and (,p) data can be used to constrain further ingredients of the statistical model. In particular, it is found that the TALYS default nucleon optical potential Kon03 () works very well, whereas the (,) data can be best described using the Hartree-Fock BCS -ray strength Cap09 () but the default generalized Lorentzian Kop90 () significantly underestimates the experimental data.
Acknowledgements.This work was supported by OTKA (K108459 and K120666), NTKH/FCT Bilateral cooperation program 6818, and Technological Research Council of Turkey (TUBITAK-Grant-109T585). A. O. acknowledges support from ERASMUS scholarship, and M. P. T. acknowledges support from TÁMOP 4.2.4. A/2-11-1-2012-0001 National Excellence Program.
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