Relation between total cross sections from elastic scattering and \alpha-induced reactions: the example of {}^{64}Zn

Relation between total cross sections from elastic scattering and -induced reactions: the example of Zn

Gy. Gyürky Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary    P. Mohr Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary Diakonie-Klinikum, D-74523 Schwäbisch Hall, Germany    Zs. Fülöp Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary    Z. Halász Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary    G.G. Kiss Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary    T. Szücs Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary    E. Somorjai Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, Hungary
July 16, 2019

The total reaction cross section is related to the elastic scattering angular distribution by a basic quantum-mechanical relation. We present new experimental data for -induced reaction cross sections on Zn which allow for the first time the experimental verification of this simple relation at low energies by comparison of the new experimental reaction data to the result obtained from Zn(,)Zn elastic scattering.


A main application of quantum mechanics is nuclear physics. Here basic theoretical relations can be tested experimentally with high precision. An interesting example is the simple relation between the total (non-elastic) reaction cross section  and the elastic scattering cross section:


Here is the wave number, is the energy in the center-of-mass (c.m.) system, and and are the real reflexion coefficients and scattering phase shifts which define the angular distribution of elastic scattering. Eq. (1) is derived from a partial wave analysis using the standard two-body Schrödinger equation. This Eq. (1) is widely used, in particular in the calculation of reaction cross sections using the statistical model (StM; to avoid confusion with the widely used abb. “SM” for “shell model”). In the following discussion we will focus on -induced reactions at low energies.

In the StM the reaction cross section of an -induced (,) reaction is calculated in two steps. In the first step the total reaction cross section  is calculated using Eq. (1); the reflexion coefficients are determined by solving the Schrödinger equation using a global -nucleus potential, e.g. the widely used potential by McFadden and Satchler McF66 (). Compound formation is the dominating absorption mechanism at energies from a few MeV up to about several tens of MeV; thus, it is assumed that the compound formation cross section is approximately given by the total reaction cross section: . In the second step is distributed among all open channels. The decay branching is obtained from the transmission factors into the various open channels which are again calculated using global potentials for each (outgoing particle + residual nucleus) channel or from the photon strength function in the case of the (,) channel. Further details of the StM can be found in the recent review Rau11 ().

To our knowledge, the underlying basic relation in Eq. (1) has never been verified experimentally for -induced reactions at low energies around or below the Coulomb barrier. Although there is no special reason to suspect to Eq. (1) for the particular case of -induced reactions, an experimental verification assures its application for the calculation of reaction cross sections which has turned out to be difficult especially at low energies (see discussion below). Alternatively, if the validity of Eq. (1) is assumed a priori, it can be used as a stringent test for consistency between different methods for the determination of .

 has been measured at higher energies up to 200 MeV using transmission experiments Igo63 (); Lab73 (); Auce94 (); Ing00 (). The experimental results have significant uncertainties, and earlier results Auce94 () have been questioned later by the same group Ing00 (). For the early experiments Auce94 () it was found that  from the transmission data is significantly smaller than  derived from (,) elastic scattering using Eq. (1) Abe94 (); Auce94 (); Ait95 () whereas agreement was found using the latest transmission data Ing00 () where  is about  % higher compared to earlier results Auce94 ().

At lower energies -induced reactions have been studied intensively in the last decades to determine a global -nucleus potential which is then used for the prediction of -induced reaction cross sections and their inverse, mainly (,), reaction cross sections. Often the motivation came from astrophysics where (,) reaction rates under stellar conditions play an important role for the nucleosynthesis of heavy neutron-deficient nuclei (the so-called -nuclei) Woo78 (); Arn03 (); Rau06 (); Rap06 (). It turned out over the years that it is very difficult or even impossible to obtain a consistent description of elastic (,) scattering and (,) cross sections. Especially the reproduction of (,) capture cross sections for heavy targets (above ) at the lowest experimentally accessible energies (i.e. the most relevant energy range for the calculation of stellar reaction rates) was poor Som98 (); Gyu06 (); Ozk07 (); Cat08 (); Yal09 (); Gyu10 (); Kis11 (); with significant efforts better results have been obtained mainly for (,n) reactions at slightly higher energies very recently Avr10 (); Sau11 (); Mohr11 (); Pal12 ().

It is the aim of the present work to provide an experimental confirmation of Eq. (1) at relatively low energies around the Coulomb barrier because it is implicitly used in all the above mentioned studies of (,) and (,n) reactions Som98 (); Gyu06 (); Ozk07 (); Cat08 (); Yal09 (); Gyu10 (); Kis11 (); Avr10 (); Sau11 (); Mohr11 (); Pal12 (). The target nucleus Zn is an almost perfect candidate for such a study because () a series of experiments have measured angular distributions of elastic (,) scattering in the energy range from 12 to 50 MeV DiP04 (); OLDSCAT () which enables to study the Zn- potential in a wide energy range, and () almost all relevant reaction channels at low energies lead to unstable residual nuclei which can be measured using the activation technique. Thus, this experiment can use a completely different experimental approach for the determination of  where the systematic uncertainties of transmission experiments can be avoided (see discussion in Ing00 ()). In addition we point out that the activation technique is the most appropriate tool for the determination of total (,) reaction cross sections whereas e.g. in-beam (,) experiments might miss weak -ray branches.

Although several data sets for (,) cross sections on Zn are already available (e.g. Por59 (); Rud69 (); Ste64 (); Cog65 (); Scu09 ()), the data quality is relatively poor. So we have remeasured the cross sections of the Zn(,)Ge, Zn(,n)Ge, and Zn(,p)Ga reactions at low energies. Relatively low energies were chosen not only because of the underlying astrophysical request for a low-energy -nucleus potential, but also because of the relatively small numbers of open channels. In the energy range under study, the total reaction cross section is given by the following sum:


In the following we present first our new experimental data for the Zn(,)Ge, Zn(,n)Ge, and Zn(,p)Ga reactions. Then we estimate the cross sections of the remaining open channels in Eq. (2) which are much smaller than the dominant (,p) and (,n) cross sections. Next we compare the sum of the -induced cross sections to the total reaction cross section  from the analysis of an angular distribution of Zn(,)Zn elastic scattering and find agreement within the uncertainties, i.e. we confirm the basic quantum-mechanical relation in Eq. (1). Finally, we suggest potential improvements to reduce the uncertainties.

The cross sections of the three studied -induced reactions have been measured with the activation method. The experimental technique was similar to the one described in one of our recent works hal12 (). Some important aspects are briefly described here. For further details see also gyu12 ().

The targets were prepared by evaporating natural isotopic composition metallic Zn onto 2 m thick Al foils. The target thicknesses (typically between 100 and 500 g/cm) have been measured by weighing and Rutherford backscattering spectrometry. The -irradiations have been carried out at the cyclotron accelerator of ATOMKI which provided an -beam with up to 1 A intensity. The durations of the irradiations varied between half an hour and one day. Changes in the beam intensity were taken into account by recording the current integrator counts in multichannel scaling mode with 1 minute time constant. The stability of the targets was continuously monitored by detecting the elastically scattered alpha particles from the target with an ion implanted Si detector built into the chamber at 150 degrees. With a beam intensity not higher than 1 A no target deterioration has been observed.

The cross sections of the Zn(,)Ge, Zn(,p)Ga, and Zn(,n)Ge reactions have been determined from the measurement of the -radiation following the -decay of the reaction products having half-lives of 270.93 d, 3.26 d, and 18.9 min, respectively. The decay parameters of the reaction products are summarized in Table 1. Owing to the largely different half-lives of the reaction products, the gamma-counting of one target has been done two or three times.

Reaction Product half-life E relative
isotope [keV] intensity [%]
Zn(,) Ge 270.93 d
Ge decay Ga 67.71 min 1077.3 3.22 0.03
Zn(,p) Ga 3.26 d 184.6 21.41 0.01
209.0 2.46 0.01
300.2 16.64 0.12
393.5 4.56 0.24
Zn(,n) Ge 18.9 min 167.0 84.28 4.52
828.3 2.99 0.27
1472.8 4.9 0.2
Table 1: Decay parameters of the reaction products. Only the strongest gamma transitions used for the analysis are listed. Data are taken from NDS68 () and NDS67 (). Since the decay of Ge is not followed by gamma radiation, the decay of its short-lived daughter, Ga has been measured.

The -measurements were carried out with a 100 % relative efficiency HPGe detector equipped with a 4 low background shielding. The absolute efficiency of the detector at a large source-to-detector distance of 27 cm was measured with several calibrated sources hal12 (). In this geometry the true coincidence summing effect is completely negligible. The strong activity of the short lived Ge reaction product was measured in this far geometry.

At higher alpha energies the (,p) cross section is large enough so that the target activity could be measured in far geometry. At low energies, on the other hand, a close source-to-detector distance of 1 cm was used where the true coincidence summing effect is strong. In order to take this into account, spectra of strong Ga sources were measured both in far and close geometries and for all studied transitions a conversion factor between the two geometries was calculated. This factor accounts for the ratio of the efficiencies as well as the effect of summing.

The weak source activities from the Zn(,)Ge reaction could only be measured in the close counting geometry. Owing to the simple decay scheme of Ga, however, the summing effect is negligible here. Therefore, for the Ga activity measurement the absolute efficiency of the detector has been measured directly in close geometry using summing-free, single line calibration sources (Be, Mn, Cu and Cs).

A typical -spectrum of the Zn(,p)Ga reaction channel was shown in Ref. gyu12 (). Here in Fig. 1 two spectra are shown which were used to determine the Zn(,n)Ge (upper panel) and Zn(,)Ge (lower panel) cross sections at E = 12.37 MeV. The waiting time between the end of the irradiation and the start of counting as well as the durations of the countings are indicated. The peaks used for the analysis are marked by arrows.

Figure 1: Gamma-spectra measured on a target irradiated with a 13.2 MeV -beam. Upper panel: spectrum measured directly after the irradiation where the decay of the Zn(,n)Ge reaction product is dominant. Lower panel: spectrum taken after 22 days of cooling time of the target in order to detect the low activity of the (,) reaction product. The peaks used for the analysis are indicated.

Table 2 lists the measured cross sections for the three reactions. The cross section of the Zn(,p)Ga reaction was measured in a wide energy range between E = 5.8 and 12.4 MeV. The Zn(,n)Ge reaction was studied from the threshold up to 12.4 MeV at 9 energies. Owing to the long half-life of the (,) reaction product and the weak -branching of the decay, the activation method is not optimal for the measurements of the very low (,) cross sections. Therefore, the (,) cross section was measured only at a few energies. Further (,) measurements are planned using e.g. the AMS method VERA (). Data for the three channels (,p), (,n), and (,) are available at 12.37 MeV (shown in bold face in Table 2); these data are used for the determination of the total reaction cross section .

The effective energies were calculated by taking into account the energy loss of the beam in the target. The uncertainty of the energy is the quadratic sum of the uncertainty of the beam energy (0.5 %) and half of the energy loss in the target. The uncertainty of the cross section values is the quadratic sum of the uncertainties from the following components: target thickness (8 %), detector efficiency (5 %), current integration (3 %), decay parameters ( 4 %), counting statistics (typically about  %, in the worst case  23 %).

Figure 2 shows the results of the three reactions in the form of the astrophysical S-factor. Along with the experimental data the predictions of two different StM codes, the NON-SMOKER NS () and TALYS TALYS () codes, are plotted using their standard parameters. Both codes give a good qualitative description of the measured data. At  = 12.37 MeV, where the total cross section is determined, both codes reproduce well the (,p) and (,n) cross sections. The (,) value is well predicted by TALYS and somewhat underestimated by NON-SMOKER which code is not optimized for high energies. The discussion of the low energy cross sections and the astrophysical consequences will be the subject of a forthcoming publication.

E cross section E cross section
[MeV] [mbarn] [MeV] [mbarn]
Zn(,p)Ga reaction Zn(,p)Ga reaction (cont.)
5.79 0.08 (2.650.38) 10.38 0.08 177 20
5.86 0.07 (3.030.40) 11.09 0.09 254 28
6.16 0.08 (3.730.42) 11.24 0.09 255 28
6.26 0.05 (3.690.42) 11.60 0.06 265 29
6.39 0.08 (2.940.33) 12.22 0.06 317 36
6.52 0.08 (2.760.31) 12.37 0.07 344 40
6.80 0.08 (4.380.49) Zn(,n)Ge reaction
7.01 0.06 (7.080.80) 9.04 0.08 10.6 1.2
7.04 0.05 (5.880.66) 9.68 0.08 41.7 4.6
7.15 0.08 2.52 0.28 10.30 0.07 70.4 7.7
7.51 0.04 5.63 0.63 10.38 0.08 73.6 8.1
7.51 0.04 5.95 0.67 11.09 0.09 106 12
7.70 0.04 7.78 0.88 11.24 0.09 109 12
7.96 0.06 7.96 0.89 11.60 0.06 120 13
8.07 0.05 12.4 1.4 12.22 0.06 165 18
8.47 0.10 36.6 4.1 12.37 0.07 179 19
8.79 0.10 49.8 5.6 Zn(,)Ge reaction
9.04 0.08 72.3 7.3 7.96 0.06 0.92 0.23
9.68 0.08 123 13 11.24 0.09 2.42 0.21
10.30 0.07 172 19 12.37 0.07 1.81 0.19
Table 2: Measured cross sections of the three studied reactions. See text for details.
Figure 2: (color online) Astrophysical S-factor of the measured reactions and the predictions of statistical model calculations.

Now we focus on the measurement at the energy  MeV and determine the sum on the right-hand side (r.h.s.) of Eq. (2). The first three terms can be taken directly from our experimental results in Table 2. The summed cross section of these three channels is 525  57 mb where for the calculation of the uncertainty the common uncertainties of the three measurements have been taken into account. The remaining five terms have to be estimated using theoretical considerations. Because these terms are relatively small compared to the dominating (,p) and (,n) cross sections, the sum in Eq. (2) is mostly defined by our experimental data.

(,): Inelastic scattering is dominated by Coulomb excitation at energies below the Coulomb barrier. We have performed coupled-channel calculations with the code ECIS ECIS () in the vibrational model using the potential of McF66 () and the deformation parameter which is taken from compilation of Raman et al. Ram01 (). For the first excited state at  keV we find a Coulomb contribution of 34 mb (close to the semi-classical result for Coulomb excitation Ald56 () using the adopted transition strength of  W.u. NDS ()) and a smaller nuclear contribution of 12 mb leading to a total excitation cross section of 46 mb with an estimated 20 % uncertainty of 10 mb. The Coulomb-nuclear interference is practically negligible at the low energies under study. The cross sections of higher-lying excited states are much smaller which is confirmed by the spectrum of inelastically scattered  particles at somewhat higher energies (see e.g. Fig. 2 of Scu09 ()). So we estimate  mb for the higher-lying states, i.e. a lower limit of zero and an upper limit identical to the dominating cross section of the first excited state. By summing the dominating contribution of the first excited state ( mb) and the estimated contribution of higher-lying states ( mb) we find in total (,) mb.

(,), (,p), (,2p), (,n): The -values for all two-particle emission reactions are strongly negative. Thus, the probability to emit one or even two charged particles from the compound nucleus Ge with very low energies is extremely small. Calculations using the code TALYS TALYS () with different -nucleus optical potentials always lead to negligible cross sections below 1 mb for each of the two-particle emission channels. Thus, we adopt  mb for each of the two-particle channels.

Summing up all the above cross sections, we find a total reaction cross section of  mb for the r.h.s. of Eq. (2). Next, this number has to be compared to the total reaction cross section  derived from the elastic scattering angular distribution using Eq. (1).

Di Pietro et al. DiP04 () have measured Zn(,)Zn elastic scattering at the energy  = 12.4 MeV. From their analysis using Woods-Saxon potentials of volume type they determine a total reaction cross section of   mb. A reanalysis of these scattering data using a folding potential in the real part and an imaginary surface Woods-Saxon potential (consistent with a global study of -nucleus potentials at low energies Mohr_ADNDT ()) leads to a slightly lower value of 610 mb with a significantly reduced . So we adopt   mb in the following discussion. This result also fits nicely into the systematics of so-called reduced cross sections  versus reduced energy  Far10 (); Mohr10 () for Zn(,)Zn scattering data at higher energies DiP04 (); OLDSCAT (); however, the obtained  for Zn are slightly higher than  for other -nucleus systems Mohr10 (); Mohr11 ().

The energy difference between the elastic scattering data at 12.4 MeV and the activation data at 12.37 MeV is very small. The calculated  at 12.37 MeV differs by less than 0.5 % from the result at 12.4 MeV. This tiny difference is neglected, and we take   mb for  in Eq. (1) at 12.37 MeV.

The ratio between the left-hand side and right-hand side in Eq. (2) should be unity if the l.h.s. can be taken from Eq. (1). The experimental result of this work confirms the validity of Eqs. (1) and (2) within about 17 % uncertainty which is the first experimental confirmation of Eqs. (1) and (2) for -induced reactions at low energies around the Coulomb barrier. A more precise confirmation requires a reduction of the uncertainties. Such a reduction is possible for the total reaction cross section from elastic scattering where uncertainties of the order of a few per cent can be achieved if the angular distribution is measured over the full angular range with small uncertainties Mohr10 (). Such an experiment is in preparation at ATOMKI. It is also planned to measure inelastic scattering cross sections in this experiment; this will lead to a further reduction of the uncertainty of because of a more precise determination of the (,) cross section in Eq. (2).

In the previous work Auce94 (); Abe94 (); Ait95 () ratios of about were found for various nuclei at energies above the Coulomb barrier whereas was found using the latest transmission data Ing00 (). However, there is no explicit determination of the ratio in recent literature Auce94 (); Abe94 (); Ing00 () because in most cases the energies of the transmission data did not exactly match the energies of the available elastic scattering angular distributions, and thus also no uncertainty for is given. The experimental uncertainties for the transmission data are about  % in Auce94 () and  % in Ing00 (). As can be seen from Fig. 1 of Abe94 (), different parametrizations of the -nucleus potential differ by about  % in the calculation of ; the differences may result from the limited angular range of the angular distributions at higher energies. Together with an additional uncertainty from the mismatch of the energies between the transmission data and the scattering data we estimate a total uncertainty of at least 20 % for the ratio at higher energies. We recommend to perform a new transmission experiment for a properly chosen target nucleus and a simultaneous study of the elastic and inelastic scattering angular distributions (with small uncertainties over a broad angular range) at exactly the same energy. With these data it should be possible to confirm Eqs. (1) and (2) also at energies significantly above the Coulomb barrier with small uncertainties.

We have also determined the total reaction cross section at the other energies where we have measured three reaction channels. The summed cross sections are 366 39 mb (8.9 1.0 mb) at 11.24 MeV (7.96 MeV). The given numbers do not include inelastic cross sections of 62 22 mb (16 5 mb). From these data it is possible to determine  = 428 44 mb at 11.24 MeV. We do not provide  at 7.96 MeV because  is dominated by inelastic scattering and not by our experimental data. A test of Eq. (1) is not possible at these lower energies because experimental angular distributions are not available. Such scattering experiments will be very difficult in particular at the lower energy because the elastic scattering cross section will deviate from the Rutherford cross section by less than 10 % here.

In conclusion, this work has confirmed for the first time experimentally that the total reaction cross section  of -induced reactions from the sum over all open reaction channels (measured by the activation technique) and  from the analysis of elastic scattering angular distributions is identical. This is an important experimental confirmation of the basic quantum-mechanical relations in Eqs. (1) and (2) which are widely used in the prediction of reaction cross sections e.g. in the statistical model, in particular because some previous studies Auce94 (); Abe94 (); Ait95 () failed to confirm this identity. A close relation between reaction cross sections and backward angle elastic scattering was also found by the analysis of barrier distributions (e.g. Tim95 ()). If the validity of Eqs. (1) and (2) is assumed a priori, our result may also be interpreted as consistency check of different methods for the determination of the total reaction cross section . This result also strengthens the motivation for the study of elastic -scattering at low energies to determine  and to obtain a low-energy -nucleus potential for a better prediction of (,) capture cross sections Mohr_ADNDT (); Pal12a ().

We thank Zs. Dombradi for the ECIS calculations and A. Di Pietro and V. Scuderi for encouraging discussions. This work was supported by ERC St.G. 203175 and OTKA PD104664, K101328 and NN83261 (EuroGENESIS).


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