Additive empirical parametrization and microscopic study of the deuteron breakup

TL;DR

This study compares empirical and microscopic models of deuteron breakup cross sections across various energies and targets, highlighting their similarities and discrepancies to guide future experimental and theoretical work.

Contribution

It introduces an additive empirical parametrization and assesses microscopic calculations against experimental data for deuteron breakup cross sections.

Findings

Comparable mass dependences of EB cross sections from both models.

Variance in absolute values up to a factor of two due to limited high-energy data.

Validation of microscopic calculations through observed similarities.

Abstract

Comparative assessment of the total breakup proton-emission cross sections measured for 56 MeV deuteron interaction with target nuclei from $^{12}$C to $^{209}$Bi, with an empirical parametrization and recently calculated microscopic neutron-removal cross sections has been done at the same time with similar data measured at 15, 25.5, 70, and 80 MeV. Comparable mass dependances of the elastic-breakup (EB) cross sections provided by the empirical parametrization and the microscopic results have been also found at the deuteron energy of 56 MeV, while the assessment of absolute-values variance up to a factor of two has been not possible due to the lack of EB measurements at energies higher than 25.5 MeV. While the similarities represent an additional validation of the microscopic calculations, the cross-section difference should be considered within the objectives of further measurements.

Equations

$$f_{BU}^{n/p}=0.087-0.0066Z+0.00163Z{A}^{1/3}+$$

$$0.0017A^{1/3}E-0.000002Z{E}^2,$$

$$f_{EB}=0.031-0.0028Z+0.00051Z{A}^{1/3}+$$

$$0.0005A^{1/3}E-0.000001Z{E}^2.$$

$$f_{EB}^{norm}(E)=f_{BU}^{n/p}(E)\frac{f_{EB}(E_{max})}{f_{BU}^{n/p}(E_{max})},E>E_{max}.$$

$$f_{BF}^{n/p}=f_{BU}^{n/p}-f_{EB}^{norm},$$

$$f_{BU}=2f_{BU}^{n/p}-f_{EB}^{norm},$$

Full text

I Introduction

An update of the theoretical analysis of deuteron-nuclei interaction within an unitary and consistent account of the related reaction mechanisms is highly requested by on-going strategic research programs (ITER, IFMIF, SPIRAL2-NFS) iter and medical investigations using accelerated deuterons, on the basis of improved nuclear databases FENDL . The need of this update comes essentially from specific noncompound processes that should be considered in the case of the incident deuterons, making them substantially different from other incident particles. Thus, the deuteron breakup (BU) is particularly quite important due to the large variety of reactions initiated by the breakup nucleons along the whole incident energy range BU1 ; BU2 ; BU3 . Otherwise, the deuteron interaction with low and medium mass target nuclei and incident energies below and around the Coulomb barrier proceeds largely through stripping and pick-up direct reaction (DR) mechanisms, while pre-equilibrium emission (PE) and evaporation from fully equilibrated compound nucleus (CN) become important at higher energies RC2015 . On the other hand, the scarce deuteron-breakup experimental data systematics pamp78 ; wu79 ; klein81 ; mats80 ; must87 related to the high complexity of the breakup mechanism has constrained so far a comprehensive analysis of the deuteron interactions within wide ranges of target nuclei and incident energies.

Moreover, unlike the DR, PE, and CN theoretical models, various current studies concern the theoretical description of the breakup mechanism and its components, namely the elastic breakup (EB), in which the target nucleus stays in its ground state and both deuteron constituents fly apart, and the inelastic breakup or breakup fusion (BF), where one of these constituents interacts non-elastically with the target nucleus. Microscopic EB calculations have been performed using the continuum-discretized coupled-channels (CDCC) method (kamimura86 ; austern ; deltuva ; ogata and Refs. therein), treating the deuteron scattering on a target nucleus $A$ by a three-body reaction model. The EB component is treated as an inelastic excitation of the projectile, due to the nuclear and Coulomb interactions with the target, through the coupled channels approach. In order to deal with a finite set of coupled equations, an essential feature of the CDCC method is the introduction of a discretization procedure, in which the continuum spectrum is represented by a finite and discrete set of square-integrable functions. However, since the deuteron elastic breakup component is almost one order of magnitude weaker than the total EB+BF process klein81 ; must87 , a model of either the total breakup or the inelastic breakup formalism is highly requested.

The recently deuteron-breakup detailed analyzes of both EB and BF components by the distorted wave Born approximation (DWBA) method carlson ; potel ; moro , with prior/post form amplitudes, performed a successful description of proton spectra and angular distributions for the $(d,p)$ reaction on 27Al, 58Ni, 93Nb, and 118Sn at incident energies from 15 MeV to 100 MeV. The corresponding calculated EB and BF cross sections would be, however, also quite useful for the comparison with experimental data pamp78 ; wu79 ; klein81 ; mats80 . More recently, Neoh et al. neoh applied the CDCC extension of the eikonal reaction theory (ERT), using also microscopic optical potentials, to the analysis of the EB and neutron removal cross sections at 28 MeV/nucleon on various target nuclei from 12C to 209Bi, of interest for further studies of unstable nuclei structure.

A comparative assessment of the measured data and the results of microscopic description of the BU process as well as current parametrization already involved within recent systematic studies of deuteron-induced reactions BU1 ; BU2 ; BU3 , aimed by this work, could be equally useful to basic objectives neoh and improved nuclear data calculations within a considerable range of target nuclei and incident energies up to 60 MeV. The former parametrization FED is addressed in Sec II, including a further normalization of the EB that has been proved necessary at energies beyond the restricted range of the available measured data, as well as an additional constraint of the total BU cross section for the target nuclei above $A$=200. The comparison of the total BU proton-emission cross sections measured by Matsuoka et al. mats80 , for 56 MeV deuteron interaction with target nuclei from 12C to 209Bi, with the empirical parametrization and the microscopic neutron-removal cross sections neoh , done at the same time with similar data measured at 15, 25.5, 70, and 80 MeV is discussed in Sec. III.A, while a similar analysis is given for the EB cross sections in Sec. III.B, followed by conclusions in Sec. IV.

II Deuteron breakup parametrization

An empirical parametrization FED of both the total breakup (EB+BF) and EB data has involved the assumption that the inelastic-breakup cross section for neutron emission $\sigma_{BF}^{n}$ is the same as that for the proton emission $\sigma_{BF}^{p}$ (e.g., Ref. must87 ), so that the total breakup cross sections $\sigma_{BU}$ is given by the sum $\sigma_{EB}$+2$\sigma_{BF}^{n/p}$. The parametrization has concerned the total BU nucleon-emission and EB fractions, i.e. $f_{BU}^{n/p}$ = $\sigma^{n/p}_{BU}/\sigma_{R}$ and $f_{EB}$=$\sigma_{EB}/\sigma_{R}$, respectively, where $\sigma_{R}$ is the deuteron total-reaction cross section. Thus, the dependence of these fractions on the deuteron incident energy $E$ and the target-nucleus atomic $Z$ and mass $A$ numbers was obtained FED through analysis of the experimental systematics of deuteron-induced reactions on target nuclei from 27Al to 232Th and incident energies up to 80 MeV for the former pamp78 ; wu79 ; klein81 ; mats80 ,

[TABLE]

but within a more restricted energy range up to 30 MeV klein81 ; must87 for the later:

[TABLE]

The comparison of the experimental data and parametrization results shown in Fig. 1 for deuterons incident on nuclei from 27Al to 232Th, at energies up to 80 MeV, has proven a suitable agreement.

II.1 Deuteron EB normalization

On the other hand, it was found an apparent decrease of the fraction $f_{EB}$ at energies beyond the range E$$<30 MeV of the EB data klein81 ; must87 , unlike the trend of both the fraction $f_{BU}^{p}$ fraction (Fig. 1) and the total-reaction cross section. Consequently, the correctness of this EB extended parametrization has been checked through the comparison of its predictions and results of the CDCC method for the 63Cu and 93Nb target nuclei maCDCC . Thus, while a good agreement was obtained between the EB data klein81 and both the CDCC results and the empirical parametrization FED at the energies of the available data, at higher energies the energy dependence of $\sigma_{R}$ and the $f_{BU}^{p}$ parametrization is common only to the CDCC results maCDCC . Therefore, the necessary caution in extrapolating the $f_{EB}$ empirical parametrization beyond the energies of the corresponding data have had to be considered at the same time with the challenging CDCC calculations for each target/energy of interest. Under these conditions it has been opportune to adopt a normalized EB fraction for the energies beyond the maximum of the former parametrization FED by taking into account the energy dependence of the $f_{BU}^{p}$ fraction ND2016 . Hence, we have chosen to keep unchanged the ratio of the EB and BU fractions at the incident energies above the energy $E_{max}$ corresponding to the maximum of the $f_{EB}$ fraction FED , by means of the relation:

[TABLE]

Thus, the normalized EB fraction follows the behavior of the total BU nucleon-emission fraction shown in Fig. 1, in agreement with the CDCC calculation results maCDCC . Despite the EB component is less than 10% of total BU cross section, this $f_{EB}$ normalization is of particular interest at deuteron energies above $\sim$50 MeV and especially for heavier target nuclei, for the inelastic breakup fraction

[TABLE]

as well as the total breakup fraction

[TABLE]

under the above-mentioned assumption of equal neutron- and proton-emission BU cross sections FED .

II.2 Deuteron BU additional constraint

The so scarce total BU proton-emission systematics for heaviest nuclei (A$$>200) at incident energies around the Coulomb barrier, of great interest for deuteron interaction with actinide nuclei Pad ; wilson ; ducasse , includes only one single datum for 232Th at $E$=15 MeV klein81 . It is properly described by the present parametrization as well as that at 70 MeV reported by Wu et al. wu79 (Fig 1). However, following the EB fraction normalization, i.e. Eqs. ( 3) and (5), the total BU fraction corresponding to this target nucleus exceeds unity (e.g., $f_{BU}$=1.0215 at $E$=32 MeV).

This unphysical overrun of the total-reaction cross sections should be firstly considered with respect to the systematics accuracy. Since the corresponding data errors amount to 10-15% FED , we have adopted an additional constraint for A$$>200, namely that the $f_{BU}$ fraction should not exceed 0.9. This figure is only presently a simple hard limit which has been also included in the latest version of the computer code TALYS$-$1.8 talys as well as the above-mentioned EB normalization. Nevertheless, it should be confirmed by further data measurements and also theoretical modeling progress.

III Microscopic and parametrization comparison

III.1 The total BU proton emission

The comparison of the total BU proton-emission cross sections $\sigma^{p}_{BU}$ measured by Matsuoka et al. mats80 , for 56 MeV deuteron interaction with target nuclei from 12C to 209Bi, with the above-described parametrization and the microscopic neutron-removal cross sections neoh is shown in Fig. 2 (a-d) at the same time with similar data measured at 15 klein81 , 25.5 pamp78 ; klein81 , and 70 and 80 MeV wu79 . Since the absolute cross sections may depend on the model ingredients of reaction mechanisms involved within the experimental data analysis, e.g., optical and PE model parameters, a similar comparative analysis concerns at the same time in Fig. 2 (e-h) the corresponding total BU proton-emission fractions $f_{BU}^{p}$. On the other hand, the $f_{BU}^{p}$ values may illustrate the importance of the breakup process among the other reaction mechanisms related to the deuteron interaction. Moreover, the same scale is used for the $\sigma^{p}_{BU}$ as well as $f_{BU}^{p}$ values at all incident energies of the available experimental data, in order to make possible also an assessment of their energy dependence.

There are several features which are pointed out by this comparative analysis. First, the increase of $\sigma^{p}_{BU}$ with the mass of the target nucleus is well described by the empirical parametrization for all deuteron energies from 15 to 80 MeV. There is a similar trend of the microscopic results for medium-mass nuclei with 40<$$A<120, while it is apparent an overestimation of the measured data for light nuclei (A$$<40) as well as an underestimation for heavier ones (A$$>120).

Second, the $f_{BU}^{p}$ values show that the importance of the BU mechanism is increasing with the target-nucleus mass, from 27Al up to 232Th, at the lower incident energies of 15 and 25.5 MeV. This increase is less significant at the energy of 56 MeV, and even reversed at 70-80 MeV. Actually it seems that the fraction $f_{BU}^{p}$ has reached its maximum at 56 MeV, for the target nuclei with A$$>120, while for 40<$$A$$<120 this maximum moves at energies over 56 MeV but lower than 70-80 MeV. Moreover, the $f_{BU}^{p}$ values are still increasing with the incident energy even at 80 MeV for the deuteron interaction with light target nuclei (A$$<40). These energy dependences of the measured $f_{BU}^{p}$, which are obvious also in Fig. 1 for the target nuclei from 27Al up to 232Th, are satisfactorily described by the empirical parametrization. The microscopic results at the energy of 56 MeV neoh show a steep decrease for target nuclei from $A$=12 up to A$$\sim120, apart from the data, while for A$$>120 their underestimated values describe however the target-nucleus mass dependence. Thus, one may note that the microscopic theory provides at the incident energy of 56 MeV a mass dependence which becomes real at the higher energies of 70-80 MeV.

A comment should concern the inclusion of the $(d,p)$ stripping direct reaction by the microscopic total BU proton-emission cross sections, unlike the experimental data pamp78 ; wu79 ; mats80 ; klein81 which were obtained through a distinct analysis of the BU, DR, PE, and CN reaction mechanisms. However, the results of systematic analysis of the four different mechanisms for the deuteron interactions with 27Al, 54,56-58Fe, 58,60-62,64Ni, 63,65Cu, 93Nb, and 238U BU1 ; BU2 ; BU3 ; ND2016 (e.g., Fig. 3) point out the decreasing importance of the stripping reaction with the deuteron energy increase. Thus, the corrections for $(d,p)$ DR contribution should be less than 1% at the energy of $56$ MeV, with no real effect on the comparison shown in Fig. 2.

III.2 The elastic breakup

A comparison of the mass dependance of the EB cross sections and corresponding fractions $f_{EB}$ provided by the above-described parametrization and the microscopic results neoh is shown in Fig. 4(a,b) for the deuteron energy of 56 MeV. Comparable trends of these results are obvious, the theoretical values being larger up to a factor of two. Moreover, there have been obtained similarly larger fractions $f_{EB}$ for light (A$$<50) and heavier (A$$>160) target nuclei. Unfortunately, the lack of EB measurements at energies higher than 25.5 MeV does make difficult the assessment of the apparent discrepancy among the microscopic and parametrization predictions for EB cross sections corresponding to the energy of 56 MeV. On the other hand, the parametrized predictions have already been involved within systematic analysis of all available data for deuteron interaction with various nuclei BU1 ; BU2 ; BU3 ; ND2016 , with a general good agreement between the measured and calculated data. Thus, the just above-mentioned similarities represent an additional validation of the microscopic calculations, while the absolute-value variance should be considered within the objectives of further measurements.

Furthermore, the comparison in Fig. 4(c) of the microscopic neoh and empirical EB excitation functions corresponding to the deuteron interaction with 58Ni nucleus proves the maximum pointed out within the discussion of the total BU protons-emission component. However, the comparison of the EB measurements of Kleinfeller et al. klein81 for 62Ni at the incident energies of only 15 and 25.5 MeV, and the related empirical excitation function which describes these two data points, do not allow a certain assertion concerning neither the deuteron energy corresponding to this maximum, nor the accuracy of the theoretical or empirical predictions.

IV Conclusions

Comparative assessment of the measured data and the results of microscopic description neoh of the deuteron BU process as well as current parametrization FED ; ND2016 already involved within recent systematic studies of deuteron-induced reactions BU1 ; BU2 ; BU3 has been carried on. A normalization of the EB parametrization FED has been proved necessary at energies beyond the restricted range of the available measured data. On the other hand, an additional constraint has to concern the total BU cross section for the target nuclei above $A$=200, at 0.9 of the total-reaction cross section.

The comparison of the total BU proton-emission cross sections $\sigma^{p}_{BU}$ measured by Matsuoka et al. mats80 , for 56 MeV deuteron interaction with target nuclei from 12C to 209Bi, with the empirical parametrization and the microscopic neutron-removal cross sections neoh has been done at the same time with similar data measured at 15 klein81 , 25.5 pamp78 ; klein81 , and 70 and 80 MeV wu79 . Actually, the total BU proton-emission cross sections measured by Matsuoka et al. mats80 at 56 MeV have been essential for the evidence of the maximum of the deuteron breakup mechanism around this incident energy and medium-mass target nuclei. Moreover, the opportunity of the comparison of microscopic and parametrization results at this incident energy has been most useful for further development of both methods of deuteron-breakup mechanism study. At the same time, the corrections for $(d,p)$ stripping-reaction contribution to the neutron-removal cross sections have been shown to be less than 1% at the energy of $56$ MeV, with no real effect on the above-mentioned comparison.

Comparable mass dependances of the EB cross sections provided by the empirical parametrization and the microscopic results neoh have been found at the deuteron energy of 56 MeV, while the assessment of a variance of the absolute values up to a factor of two has been not possible due to the lack of EB measurements at energies higher than 25.5 MeV. However, since the parametrized predictions have already been involved within successful analysis of all available data for deuteron interaction with various nuclei BU1 ; BU2 ; BU3 ; ND2016 , these similarities represent an additional validation of the microscopic calculations, while the cross-section difference should be considered within the objectives of further measurements.

V Acknowledgments

The authors acknowledge the useful correspondence with Constance Kalbach on need of Sec. II.A. This work has been partly supported by Autoritatea Nationala pentru Cercetare Stiintifica (PN-16420102) and partly carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.