LA-UR-16-25845 Neutron-gamma competition for β-delayed neutron emission M. R. Mumpower,∗ T. Kawano, and P. M¨oller Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: August 8, 2016) We present a coupled Quasi-particle Random Phase Approximation and Hauser-Feshbach (QRPA+HF) model for calculating delayed particle emission. This approach uses microscopic nu- clear structure information which starts with Gamow-Teller strength distributions in the daughter nucleus, and then follows the statistical decay until the initial available excitation energy is ex- hausted. Explicitly included at each particle emission stage is γ-ray competition. We explore this model in the context of neutron emission of neutron-rich nuclei and find that neutron-gamma com- petition can lead to both increases and decreases in neutron emission probabilities, depending on the system considered. A second consequence of this formalism is a prediction of more neutrons on average being emitted after β-decay for nuclei near the neutron dripline compared to models that do not consider the statistical decay. PACS numbers: 23.40.-s, 24.60.Dr I. INTRODUCTION ∼ 250 measurements exist on neutron emitters in the literature [4]. Delayed neutron emission is a common decay channel Very few measurements exist for the branching ra- found in neutron-rich nuclei. It was first discovered in tios or Pn values of multi-neutron emission. The largest 1939 during an investigation of the splitting of uranium known two-neutron emitter (86Ga) was observed at the and thorium when these elements were bombarded with Holifield Radioactive Ion Beam Facility with a P2n = neutrons [1]. This phenomenon is energetically possi- 20(10)% [7]. Future studies at radioactive beam facilities ble when a precursor nucleus with β−-decay energy win- will extend the reach of measureable nuclei well into the dow (Qβ) populates excited states in the daughter nu- regime of multi-neutron emission predicted by theory [8{ cleus above the one neutron separation energy (Sn). The 10]. However, it will still be many years before there is excited states may decay by particle or γ-ray emission. sufficient data to fully benchmark theoretical models. Accurately accounting for β-delayed neutron emission is Delayed neutron emission is challenging to describe necessary in, for example, nuclear reactor operation, to theoretically, relying on the complex description of the the rapid neutron capture or r-process of nucleosynthesis. nucleus, its excited states, and the likelihood of particle For r-process nucleosynthesis, the predicted produc- emission. In particular it is difficult in any model to cal- tion pathway ventures far from the currently known nu- culate the energies of the high-lying states near the neu- clides towards the neutron dripline where Sn is small tron separation energy very accurately. Two approaches and Qβ is large. In this case, multiple-neutron emis- to the description of delayed neutron emission can be sion is possible and may even be the dominant decay found in the literature. The first approach involves mod- pathway for many nuclei near the neutron dripline. This els using systematics, such as the Kratz-Herman formula provides a secondary source of free neutrons during late- [11] or more recent approaches, see e.g. [12]. Models us- times when the nuclei decay back to stability [2]. See ing systematics produce a good global fit to known data Ref. [3] for a recent review and further discussions of the and improvements have recently been made by Miernik impact of β-delayed neutron emission on r-matter flow. et al. where they provide an approximation to neutron- arXiv:1608.01956v1 [nucl-th] 5 Aug 2016 Experimental studies of β-delayed neutron emission gamma competition [13]. The applicability of system- are difficult due to the challenge of producing short-lived atics to multi-neutron emission towards the dripline is neutron-rich nuclei. Studies of these nuclei are further unknown, as they are based on simple relations which compounded by the requirement of high efficiency neu- factor in Qβ and Sn. tron detectors [4]. Alternative techniques seek to sidestep A second approach is to use a microscopic descrip- the problem of developing high efficiency neutron detec- tion, usually starting with some form of Quasi-particle tors by using recoil-ion spectroscopy coupled with a β- Random Phase Approximation (QRPA) that provides Paul trap [5]. Only recently has it been possible to mea- the initial population of the daughter nucleus [8{10, 14{ sure delayed neutron branching ratios for heavy nuclei 16]. Rapid progress has been made for the description near the N = 126 shell closure [6]. In total, roughly of half-lives using the Finite Amplitude Method which is equivalent to QRPA, but this method has not yet been applied to Pn predictions [17]. To calculate Pn values from the QRPA approaches, a schematic formula is used ∗Electronic address: [email protected] based off the available energy window [8{10]. This ap- 2 proach to calculating Pn values is sometimes referred to calculation we smooth the β-strength distribution by a the ‘cutoff' method in the literature. For ease of cita- Gaussian, tion later in the paper, we define three reference mod- (k) 2 els: QRPA-1 (Gamow-Teller (GT) only) [8] and QRPA-2 X (k) 1 [E − Ex] !(Ex) = C b p exp − ; (2) (additional inclusion of First-Forbidden (FF) transitions) 2πΓ 2Γ2 [9], and D3C∗ [10]. The latter is a microscopic approach k based off covariant density functional theory and proton- where b(k) are the branching ratios from the parent state neutron relativistic QRPA. (k) to the daughter states E , Ex is the excitation energy In this paper we present a further upgrade to the of the daughter nucleus, and the Gaussian width is taken microscopic framework of computing delayed neutron- to Γ = 100 keV, as in [18]. The normalization constant, emission probabilities. Our approach combines both the C, is the given by the condition QRPA and statistical methods into a single theoretical framework. This is achieved by first calculating the β- Z Qβ decay intensities to accessible states in the daughter nu- !(Ex)dEx = 1; (3) cleus using QRPA. The subsequent decay of these states 0 by neutron or γ-ray emission is then treated in the sta- where Q is the β−-decay energy window. tistical Hauser-Feshbach (HF) theory. This work extends β Our QRPA solution may be fully replaced or supple- that of Ref. [18], which was limited to near stable iso- mented with experimental data when available. In the topes, to the entire chart of nuclides. case where the level structure and β-decay scheme are known completely, data are taken from the RIPL-3 / ENSDF and no QRPA strength is used. If the database is II. MODEL incomplete, we combine the QRPA solutions with renor- malized ENSDF data. In some cases, energy levels are The calculation of β-delayed neutron emission in known but the spin and parity assignments are uncertain. the Quasi-particle Random Phase Approximation plus In this case, or in the case of that no level information is Hauser-Feshbach (QRPA+HF) approach is a two step found in ENSDF, the QRPA solution is relied upon. process. It involves both the β-decay of the precursor When applicable, ground state masses are taken from nucleus (Z,A) where Z represents the atomic number, the latest compilation of the Atomic Mass Evaluation and A is the total number of nucleons, and the statisti- (AME2012) [22]. Ground state masses and deformations cal decay of the subsequent generations, (Z + 1,A − j) from the 1995 version of the Finite-Range Droplet Model where j is the number of neutrons emitted, ranging from (FRDM1995) are used unless otherwise noted [23]. 0 to many. The statistical decay continues until all the In the second stage of the calculation we assume the available excitation energy is exhausted. This is shown daughter nucleus is a compound nucleus initially pop- schematically in Fig 1. ulated by the β-decay strength from the QRPA calcu- Beginning with the precursor nucleus (Z,A), the de- lations. This assumption is based off the independence cay to the daughter (Z + 1, A) is calculated using the hypothesis from N. Bohr, from which it follows that the QRPA framework of M¨olleret al. [8, 9, 19, 20]. In this compound nucleus depends only on its overall properties framework, the Schr¨odingerequation is solved for the nu- rather than the details of formation [24]. We follow the clear wave functions and single-particle energies using a statistical decay of this nucleus and subsequent genera- folded-Yukawa Hamiltonian. Residual interactions (pair- tions until all the available excitation energy is exhausted ing and Gamow-Teller) are added to obtain the decay using the Los Alamos CoH statistical Hauser-Feshbach matrix element from initial to final state, hfjβGT jii.A code [18, 25, 26]. β-decay strength function is used to describe the behav- For calculation of the neutron transmission coefficients ior of the squares of overlap integrals of nuclear wave we employ the Koning-Delaroche global optical potential functions. The β-decay strength function is given by [21] that has an iso-spin dependent term [27]. This model has been shown to provide a very good description of I(E) neutron-rich nuclei, see e.g. Ref. [28, 29]. We have also Sβ(Ex) = (1) f(Z; Qβ − Ex)T1=2 used the Kunieda deformed optical potential [30], how- ever we find only minor changes to our results and so do where I(Ex) β-intensity per MeV of levels at energy Ex not report on this model here. in the daughter nucleus, f is the usual Fermi function As shown in Fig.