Research Methodology
According to p-process models, p-nuclei are produced at stellar temperatures between 1.8 to 3.3 billion degrees Kelvin. Under these temperature conditions, the stellar matter is highly ionized and can be treated as a nondegenerate Fermi Gas in thermodynamic equilibrium consisting of nuclei moving non-relativistically. Their velocities, and hence their energies Ε, can be described by a Maxwell-Boltzmann distribution φ(Ε). Obviously, the probability of two interacting charged particles, is determined by the probability of penetration through the corresponding repulsive Coulomb barrier, which is known as tunneling probability or penetrability P(E). It is then the product φ(E)×P(E) that yields the relative probability of a nuclear reaction between two charged particles as a function of kinetic energy. This product is the so-called Gamow distribution G(E) that is usually approximated by a Gauss-like function, with a maximum value at an energy E0 and a width ΔEG that are given, respectively, by
In these equations, stands for the stellar temperature in billion degrees, is the charge of the target-nucleus, the charge of the projectile-nucleus resulting from these and equations μ is the are reduced in MeV mass units. of the two interacting particles t and p. Quantities E0 and ΔEG resulting from these equations are in MeV units.
The Gamow window ΔEG is a very important quantity for nuclear astrophysics studies: All nuclear burning reactions occurring in a stellar gas take place predominantly within the energy window δE = E0 ± ΔEG/2. Therefore, in order to investigate nuclear reactions of relevance to an astrophysics process taking place in a stellar environment of a given temperature T, one has to determine cross-sections within the corresponding energy range δEG. By applying the p-process relevant temperatures, i.e. 1.8 ≤ T9 ≤ 3.3, in the last two equations, one obtains the beam energy regions, where cross-sections for proton and α-particle capture reactions on nuclei heavier than iron have to be measured. Hence, for the (p,γ) and (α,γ) reactions, these energies range from ≈1 to ≈5 MeV and from ≈4 to ≈12 MeV, respectively. These p-process relevant energies can easily be achieved by low-energy accelerators, mainly Tandems, like those that will be used in the proposed project.
Capture reactions, which are the focus point of this project, lead to the formation of a stable or an unstable compound nucleus. In general, the cross-section is determined from the absolute number of the produced compound nuclei. This can be done by either determining the absolute number of photons emitted by the reaction or by measuring the absolute activity from the subsequent decay of the produced nuclei. The latter method obviously presupposes that the produced nuclei are unstable and is referred as to the “activation” technique. In the former case, the photons are detected during bombardment of the target and the methods applied are all referred as to the “in-beam” methods for cross section measurements. It is worth noting that all cross-section measurements are “absolute” measurements, and hence associated with specific and very challenging experimental requirements.
The activation technique is the oldest one and can provide reliable results within short beam times. However, its applicability is limited because of the nucleus to be investigated has to be unstable with a proper half-live and will, therefore, not be applied in the present work. Instead, we will use three different in-beam methods, namely the γ-angular distributions, the 4π γ-summing technique and the differential cross-section integration (DCI) method, which all by-pass this condition.
Our group has been extensively using both methods form more than 20 years. In fact, among the leading activities of the group is the development of the 4π γ-summing technique with many novel features, coined (see e.g. Refs [8, 14-18]), as well as the extensive exploitation (see, e.g., in Refs, [8, 17, 19-21]) of highly efficient Ge detector technologies that enable measurements of very low cross sections of weak γ transitions by measuring γ-angular distributions, as described below.
Before presenting the relevant methodology, it is useful to discuss the decay scheme of a compound nucleus, that is formed by a (p,γ) reaction, which is considered as a more general case compared to that of an (α,γ). For this purpose, we will refer in the following to figure 3, where a proton of energy Ep impinges on a target nucleus with atomic number Z and a mass number A and, thus, a compound nucleus is formed at a highly excited state, in the following called the “entry” state, with an excitation energy EX = Q + Ep, where Q is the Q-value of the reaction and Ep is the energy of the projectile-target system in the center of mass system.
As shown in figure 3, the “entry” state is de-excited via γ transitions populating either the ground state (g.s.) or other excited discrete levels (L1, L2, L3, . . .), which decay further to other lower-lying discrete levels or to the ground state. The γ transitions de-exciting the entry state are shown in Fig. 3 with black arrows. They are often referred to as primary γ-rays. The primary γ-ray from the “entry” to the ground state is often called the γ0 transition. The γ transitions not de-exciting the entry state are known as secondary γ transitions. These are indicated in figure 3 with grey arrows.
The ground state as well as the higher-lying excited levels can be “fed” or “de-populated” by a sequence of more than one γ transitions, i.e. via a γ cascade. In this case, the term cascade feeding is used. In some cases, an excited state, like the L4 may decay by emitting particles instead of γ-rays. Independently of the method used, the basic equation to obtain the cross section σ is:
where Y0 is the absolute number of photons emitted by a capture reaction in 4π, A is atomic weight of the target in atomic mass units (amu), NA the Avogadro number, ξ the areal density of the target in g/cm2 and Np the number of incoming beam particles (projectiles) that is equal to Q/(Zqe ), with Q the charge collected by a beam-current integrator (CI), Z the atomic number of the target nucleus and qe the electron charge.
Eq. 4 shows that accurate knowledge of the radial density ξ (“target thickness”) is mandatory to obtain cross-sections with small uncertainties. The various in-beam methods used to determine the cross-section of a charge-particle–induced capture reaction differ, in fact, only in the way Y0 in Eq. 4, is determined.
In the γ-angular distributions method [8, 17, 19-21] the absolute number of the emitted photons Y0 is obtained by measuring the angular distributions of all γ transitions feeding the ground state of the produced compound nucleus. Y0 is, thus, the sum of the yields Yi of all transitions feeding the ground state and is given by:
in which, N is the number of transitions feeding the ground state and A0 are the absolute A0 coefficients of the corresponding γ-angular distributions. The A0 coefficients are determined from the fitting of a sum of Legendre polynomials Pk(θ) to the experimental angular distributions. The latter consist of data points I(θ,ε,Np) measured at a number of angles θ with respect to the beam axis, corrected for the absolute efficiency ε of the detector placed at angle θ, the corresponding number of the incoming beam particles Np and dead time. Hence,
where td is the dead-time correction factor, Fγ the area under the peak of the γ transition with energy Eγ detected at angle θ and Q the accumulated charge.
Determining cross sections from γ-angular distributions requires, first, a detailed knowledge of the level scheme of the produced nucleus. The determination of the angular distribution of the γ0 transition sets an additional requirement: the energy Eγ0 of this transition can be very high, even above 10 MeV, as it equals to the energy EX of the “entry” state. Therefore, high-efficiency γ-detectors are necessary.
To date, γ-angular distributions are measured by means of an array of high-purity Ge detectors (HPGe) placed on a motor-driven table that could rotate. This way, γ-singles spectra are measured at five, at least, angles with respect to the beam direction. Such an array is described in detail in Refs. [8, 19]. In the left panel of Fig. 4, a typical γ-singles spectrum measured at 3 MeV for the 89Y(p,γ)90Zr reaction is displayed, whereas a few typical γ-angular distributions and the corresponding fittings (curves) of are shown in the right panel.
The 4π γ-summing technique developed by our group: The use of high-efficiency Ge detectors equipped with BGO shields in in-beam experiments enables us to measure very low capture cross-sections, often smaller than 1 micro-barn. However, the data analysis itself is still a time consuming task, as a high number of γ transitions have to be taken into account and numerous γ-ray spectra have very often to be analyzed (see Eq. 5; index i often runs up to 20 or even higher!). For this reason, a new technique that would enable to measure angle- integrated γ spectra instead of numerous γ-angular distributions was developed by our group and coined 4π γ-summing technique [8, 14-18].
The 4π γ-summing technique is based on the use of a large-volume NaI(Tl) crystal detector that, ideally, sums all γ-rays, which de-excite the entry state of a compound nucleus and form γ cascades. The working principle of such a 4π γ-summing spectrometer relies on the long time response of the NaI(Tl) detector and its large volume. The latter enables to fully absorb a photon, whereas the former renders the photomultipliers unable to distinguish between different photons emitted within a time interval smaller than the decay time of the crystal, which is typically ≥300 ns. This way, one obtains, ideally, one single peak, coined “sum peak” at an energy EΣ equal to the excitation energy EX of the entry state. These are depicted in Fig. 4.
According to panels (a), (b) and (c) of this figure, if the five different γ transitions of the simple level scheme shown in part (a) are detected with a small detector, like, e.g., a 3 inch × 3 inch NaI(Tl) crystal or a small size HPGe detector, one obtains basically the γ spectrum shown in part (b). Each of these different γ transitions can be found in the spectrum as a peak joined by its Compton continuum that is produced also because of the small detector size. The Compton background can be eliminated or at least decreased by increasing, as technically possible, the size of the crystal, i.e. in a sufficiently large NaI crystal almost all photons can be fully absorbed.
By adding the energy signals from the photomultipliers, one obtains the spectrum of part (c) consisting, ideally, of the sum peak only that is labeled as γΣ. The main advantage in using the 4π γ-summing method is that instead of measuring and analyzing numerous peaks in the γ spectra, one needs to derive only the area Fγ under the sum peak, correct it for the corresponding absolute efficiency εΣ and then the yield Y0 in Eq. 5, is obtained by:
A crucial point in cross-section measurements with the 4π γ-summing technique is the application of Eq. 7, as it requires accurate knowledge of the sum-peak efficiency eΣ that depends strongly on the γ-multiplicities of the γ cascades de-exciting a compound nucleus, which is “summed” by our detector. These multiplicities are, in almost all cases, unknown. For this reason, we developed a new approach to determine first “average” multiplicities and then the sum-peak efficiency εΣ. This approach is presented in detail in Ref. [15].
The 4π γ-summing technique was first applied by our group by employing a large-volume eightfold segmented NaI(Tl) crystal as we already described in Refs. [8, 14]. Meanwhile, the method has been improved by utilizing a cylindrical (12inch ×12inch) single NaI(Tl)-crystal installed at the Dynamitron Tandem lab. of the Ruhr-Universität Bochum, Germany. A typical γ-singles spectrum measured at Ep = 3 MeV for the 89Y(p,γ)90Zr reaction with this detector is shown in panel (d) of Fig. 5. As can be seen, the “ideal” picture depicted in panel (c) is not observed in “real” experiments. The spectrum does not consist of just a single peak, as expected, but of many other peaks, as well as a Compton continuum arising because some photons are not fully absorbed.
In addition, the spectrum includes strong peaks from compound reactions or inelastic scattering of the proton beam with isotopes contained either in the target (19F, 16O, 12C) or with the material surrounding the target (27Al+p→28Si), as well as the known natural background lines at 1.461, 2.2 and 2.614 MeV. In particular, the spectrum shown in panel (d) includes three sum peaks, labeled as γΣ0, γΣ1, and γΣ3. The first one is at 11.317 MeV, whereas the other appear at lower energies differing by ≈1.761 and 2.319 MeV from the first one. All three sum peaks belong to 90Zr. The first (γΣ0) results from the sum of the γ0 transition from the entry state and all the γ cascades that by-pass the first excited level and feed the ground state, as shown in the inset of (d).
The γΣ1 transition is the sum of all cascades starting from the entry state and feeding the first excited state at 1.761MeV, a J π= 0+ state decaying to the ground state via an E0 transition. The γΣ3 transition is the summing all γ cascades arriving at the 3rd excited state at 2.319 MeV (Jπ = 5− ). This state has a half-life of 809 ms that is much longer than the decay time of the NaI(Tl) crystal. As a result, the associated cascading transitions form a 3rd sum peak and the “delayed” 5−→ 0+ transition arises in the spectrum as an individual peak. In addition to these sum peaks, a fourth one at ≈9.13 MeV, labeled as IΣ2, is due to the poor resolution of the NaI(Tl) detector and cannot be resolved from γΣ3. IΣ2 is simply the result of the incomplete summing; i.e. some portion of the 2186 keV photons emitted by 90Zr are not summed by the detector with the γ cascades feeding into the 2nd excited level that they de-excite (see in [17] for details).
Experiments based on γ-angular distributions may, comparatively, be the most time-consuming ones. However, the amount and quality of information gained through these experiments is superior compared to the other methods. On the other hand, the γ-summing method is powerful in terms of detection efficiency. The spectrum shown in panel (d) of Fig. 4 for 89Y(p,γ)90Zr was accumulated within less than hour, whereas that in the left panel of Fig. 4 was taken within ≈6 hours. Determining the cross section of this reaction from γ-angular distributions would require measuring at least four more spectra at different angles, i.e. at least 30 hours, in contrast to the spectrum shown in panel (d) of Fig. 5 which is angle integrated.
The differential cross-section integration (DCI) method: This is also an in-beam technique first formulated and applied by Mihailescu et al. [22,23], however, for neutron-induced capture reactions and not for charged-particle ones. For the first time, our group has tested successfully its applicability to proton-induced capture reactions using spectra measured previously for the 88Sr(p,γ)89Y reaction. The DCI method looks very promising in terms of shorter measurement times as, under certain conditions, it does not require measuring γ-angular distributions, at least at five angles. As our test of the DCI-method’s applicability in charged-particle induced capture reactions was successful, we plan to employ it also in the ARENA project.
The basic idea of the DCI method is to measure differential cross sections, dσi/dΩ, at specific angles θ with respect to the beam direction, instead of complete γ-angular distributions. The total reaction cross section σT then obtained by summing all the partial cross-sections i.e. σT = Σσi. In Refs. [22,23], it was shown that, by applying a Gaussian quadrature formalism one can derive the partial cross sections σi of the individual γ transitions populating the ground state of the compound nucleus produced in a capture reaction from the differential cross sections dσi/dΩ using:
The number and value of the weighting factors w1, w2 and w3 entering Eq. 8, depend on the number of the detectors employed and the angles in which they are positioned. Hence, whenone detector is used placed at 54.7o or 125.3o, then w2 = w3 = 0. The angle conditions for applying the DCI method for up to three detectors is given in Table 1 below. Clearly, the DCI method simplifies cross-section measurements significantly. In order to test the applicability of the DCI method to charged-particle capture reactions, we used the γ-angular distributions we measured previously [19] at a proton beam energy of 2.5 MeV for the 88Sr(p,γ)89Y reaction and derived the partial cross-sections σi of the 12 γ-transitions that were taken into account in [19] to derive the total cross-section σT. Fortunately, three of the detectors used in [19], were placed at angles θ differing less than 2ο from those listed in table 1. By applying Eq. 8, we derived the total cross-section of this reaction with DCI, noted here as σTDCI and compared it with σTW(θ) that was reported in [19]. As shown inFig. 6 (last data point (σΤ), indicated by filled rectangle for the corresponding ration ri = σTDCI/σTW equals unity within less than 5%, indicating that the two independently obtained σΤ values are in excellent agreement.
In corresponding figure 6, we plot the corresponding ratios ri = σTDCI/σTW for all 12 analyzed γ transitions. The corresponding γ-ray energies are given in parenthesis. As shown for all transitions ri ~ 1 within 5%. This result confirms the compatibility of DCI with the other in-beam cross-section methods and, due to its advantages in terms of measurement time, it will be employed in the proposed project accordingly. The candidate reactions of interest were selected so as to: a) cover a wide mass range, b) produce even-even, even-odd, and odd-odd nuclei to look for effects due to nuclear level density, c) re-measure if strong deviations between independent measurements exist and e) use same targets for both type of reactions, (p,γ) and (α,γ), since we shall need enriched, and hence costly, material for their preparation. Hence, for the candidate reactions we intend to use following targets: 59Co, 60Ni, 63Cu, 67Zn, 68Zn, 69Ga, 76Ge, 94Zr, 95Mo, 103Rh, 107Ag, 108 Cd, 112Sn, 120Te, 124Te, and 144Sm.
For our theoretical calculations aiming, at first, at credibility tests of existing phenomenological and microscopic models for the nuclear parameters entering Hauser-Feshbach calculations, we will use the most recent version of the TALYS code [24]. TALYS includes several OMP, NLD, and γ SF models as testing options. In addition to the HF calculations and the basic p-process reaction network code LIBNUCN ET [25] will be installed and reaction network calculations using our experimental results will be performed to evaluate the impact of the associated uncertainties in p-process abundance calculations.
Regarding the planned development of a global fully microscopic αOMP for Hauser-Feshbach calculations, it is worth mentioning that, a global semi-macroscopic alpha-nucleus optical potential was developed in 2002 by P. Demetriou et al., [12] and further updated in 2007 [26], to take into account new experimental data. Since then, the global alpha optical potential has been widely used in calculations of alpha-nucleus reactions, especially since it was implemented in the nuclear reaction code TALYS, by an extended user community.
The widespread use of the global semi-microscopic alpha optical potential in calculations of reaction cross sections of heavy mass nuclei that have been measured in the past decade has revealed that the potential has some deficiency in describing the deformed nuclei. This is expected since this potential is spherical and does not take into account deformation effects.
In this project, we aim to extend the global semi-microscopic alpha-nucleus optical potential to include deformation effects. Since our goal is to maintain the global nature of the potential, the deformation effects will be treated in an empirical way. The real part of the potential will be modified to take into account the radial deformation using an empirical formula. The parameters of the formula will be adjusted to reproduce the existing experimental data on alpha elastic scattering and alpha reactions of deformed nuclei.
For this purpose, we shall perform the following tasks: compilation of available experimental data on elastic scattering and alpha reactions for deformed nuclei, development of empirical formula that includes deformation effects via the quadrupole deformation parameter, adjustment of the parameters of the empirical formula to the compiled experimental database, fine-tuning of the global alpha optical parameters on the entire experimental database of alpha elastic scattering and alpha reactions to update all the parameters of the potential on the most recent experimental data and ensure its reliability globally.
