Exploring laser-driven neutron sources for neutron capture cascades and the production of neutron-rich isotopes

The production of neutron-rich isotopes and the occurrence of neutron capture cascades via laser-driven (pulsed) neutron sources are investigated theoretically. The considered scenario involves the interaction of a laser-driven neutron beam with a target made of a single type of seed nuclide. We present a comprehensive study over $95$ seed nuclides in the range $3\le Z \le 100$ from $^7_3$Li to $^{255}_{100}$Fm. For each element, the heaviest sufficiently-long-lived (half life $>1$ h) isotope whose data is available in the recent ENDF-B-VIII.0 neutron sublibrary is considered. We identify interesting seed nuclides with good performance in the production of neutron-rich isotopes where neutron capture cascades may occur. The effects of the neutron number per pulse, the neutron-target interaction size and the number of neutron pulses are also analyzed. Our results show the possibility of observing up to $4$ successive neutron capture events leading to neutron-rich isotopes with $4$ more neutrons than the original seed nuclide. This hints at new experimental possibilities to produce neutron-rich isotopes and simulate neutron capture nucleosynthesis in the laboratory. With several selected interesting seed nuclides in the region of the branching point of the $s$-process ($^{126}_{51}$Sb, $^{176}_{71}$Lu and $^{187}_{75}$Re) or the waiting point of the $r$-process (Lu, Re, Os, Tm, Ir and Au), we expect that laser-driven experiments can shed light on our understanding of nucleosynthesis.


INTRODUCTION
Neutron-rich isotopes are of particular interest in both fundamental nuclear physics (Thoennessen 2013;Ahn et al. 2019;Gorges et al. 2019;Zhang et al. 2019;Crawford et al. 2019;Gates et al. 2018;Tarasov et al. 2018) and the neutron capture processes of nucleosynthesis in astrophysics (Burbidge et al. 1957;Käppeler et al. 2011;Arnould et al. 2007). On the fundamental nuclear physics side, neutron-rich isotopes could provide key information to test nuclear models and to understand the nuclear interaction (Thoennessen 2013;Ahn et al. 2019;Gorges et al. 2019;Zhang et al. 2019;Crawford et al. 2019;Gates et al. 2018;Tarasov et al. 2018). On the astrophysics side, the slow (s-process) and the rapid neutron capture process (r-process) of neutron capture nucleosynthesis contribute in roughly equal amount to the total elemental abundances beyond iron (Käppeler et al. 2011;Arnould et al. 2007;Meyer 1994;Cowan & Thielemann 2004;Sneden & Cowan 2003). Although the neutron capture nucleosynthesis has been studied extensively, some issues still remain open, such as the s-process branching (Käppeler et al. 2011;Wallerstein et al. 1997) and the astrophysical sites of the r-process (Arnould et al. 2007;Wallerstein et al. 1997;Panov & Janka 2009;Wanajo 2018;Thielemann et al. 2017;Freiburghaus et al. 1999;Surman et al. 2008). Measurements of the properties of neutron-rich nuclei on or near the s-process and r-process paths will improve our understanding of the neutron capture nucleosynthesis (Käppeler et al. 2011;Arnould et al. 2007;Dillmann & Litvinov 2011;Negoita et al. 2016;Habs et al. 2011). Furthermore, radioisotopes also have extensive applications in industry (Charlton 1986) and medicine (de Lima 1998), as well as in the study of nuclear batteries con-sidered as potential long-lived small-scale power sources (Duggirala et al. 2010;Prelas et al. 2016).
So far, the production of neutron-rich isotopes mainly relies on accelerator-and reactor-based facilities, via neutron capture, projectile fragmentation, projectile fission or nuclear fusion reactions (Thoennessen 2013;Ahn et al. 2019;Gorges et al. 2019;Zhang et al. 2019;Crawford et al. 2019;Gates et al. 2018;Tarasov et al. 2018;Thoennessen 2015a;Dillmann & Litvinov 2011;IAEA 2003;Thoennessen 2016a). However, the development of laser technology in the past decades provides a powerful tool for the study of nuclear physics and nuclear astrophysics in laser-generated plasmas (Negoita et al. 2016;Wu & Pálffy 2017;Casey et al. 2012;Zylstra et al. 2016;Bleuel et al. 2016;Cerjan et al. 2018). High-power lasers offer the opportunity of generating intense neutron beams at comparatively small-scale laser facilities, allowing for the production of neutron-rich isotopes via neutron capture. With such lasers, the neutron beams can be produced via laser-induced particle acceleration leading to high-energy particle beams that subsequently interact with a secondary target (beam-target interaction) (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015), or via thermonuclear reactions (Wu 2020;Ren et al. 2017;Ma et al. 2015;Döppner et al. 2015;Olson et al. 2016). While they provide a comparatively low number of neutrons per pulse, the achievable neutron fluxes can be a few orders of magnitude higher than the ones in the conventional neutron sources at largescale rector and accelerator-based facilities (Pomerantz et al. 2014;Pomerantz September 21-25, 2015;Vogel & Carpenter 2012;Carlile et al. 2016). With the highpower lasers (short pulse duration Petawatt-class lasers, or nanosecond duration kilo-Joule or Mega-Joule lasers) available nowadays or in the near future, intense pulses of 10 10 neutrons or higher within durations on the order of picoseconds or nanoseconds can be obtained (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Kar et al. 2016;Mirfayzi et al. 2017;Kleinschmidt et al. 2018;Wu 2020;Ren et al. 2017;Ma et al. 2015;Döppner et al. 2015;Olson et al. 2016), and consequently very high neutron fluxes on the level of 10 20 n/cm 2 /s are achievable. With the intense laser-driven neutron beams, neutron capture experiments and the production of neutron-rich isotopes via neutron capture become possible (Pomerantz et al. 2014;Pomerantz September 21-25, 2015;Cerjan et al. 2018) at such small-scale laser facilities. As achievable neutron fluxes are that high, another advantage of studying neutron capture processes with laser-driven neutron sources is that they provide an opportunity to analyze neutron capture cascades similar to the ones occurring during neutron capture nucleosyn-thesis in astrophysics. This would give us the chance to simulate the neutron capture nucleosynthesis in the laboratory for the first time (Pomerantz et al. 2014;Pomerantz September 21-25, 2015;Cerjan et al. 2018), leading to an improved understanding of the ongoing processes.
In this paper, we conduct a comprehensive theoretical study of the production of neutron-rich isotopes and neutron capture cascades taking place in singlecomponent targets being irradiated by a laser-driven (pulsed) neutron source. A comprehensive study over 95 potential seed nuclides in the range 3 ≤ Z ≤ 100 from 7 3 Li to 255 100 Fm is conducted. For each element, the heaviest sufficiently-long-lived (half life > 1 h) isotope whose data is available in the recent ENDF-B-VIII.0 neutron sublibrary ) is considered as a potential seed nuclide. We are interested in the seed nuclides which are in the region of the branching point of the s-process or the waiting point of the r-process. We are also interested in the production of neutron-rich isotopes in a regime which has never been accessed by other means in the laboratory.
We develop a theoretical approach describing the production of neutron-rich isotopes, accounting for the successive radiative neutron capture process, the damping of the incident neutron beam, and the loss of nuclei by transmutation and radioactive decay. Both single and multi neutron pulse scenarios are analysed. Laser-driven (pulsed) neutron beams with average energy in the range between 50 keV and 10 MeV are considered. According to our numerical results, interesting seed nuclides are identified from the large list of potential seed nuclides, with good performance in the production of neutron-rich isotopes where successive neutron capture process may occur. For a scenario of 10 4 shots, from a laser-driven neutron source generating 10 12 neutrons per pulse (10 12 n/pl) at a repetition rate of 1 Hz, our results show the possibility of observing up to 4 successive neutron capture events leading to the production of neutron-rich isotopes with 4 more neutrons than the original seed nuclide. This provides the chance of simulating the astrophysical neutron capture nucleosynthesis in the laboratory. Among the identified interesting seed nuclides, some are in the region of the branching point of the sprocess ( 126 51 Sb, 176 71 Lu and 187 75 Re) or the waiting point of the r-process (Lu, Re, Os, Tm, Ir and Au). It is also possible to produce neutron-rich isotopes ( 248 95 Am, 258 99 Es and 259 99 Es) in a regime that has not been accessed in the laboratory. We note that such intense laser-driven neutron beams with a high repetition rate are expected to be achievable in the Petawatt-class laser facilities available in the near future (Negoita et al. 2016;Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Wu 2020;Pomerantz September 21-25, 2015), such as the Extreme Light Infrastructure (ELI) facilities which are under construction (Negoita et al. 2016;ELI 2020).
The paper is organized as follows. In Sec. 2, we present the theoretical approach used to compute neutron captures in a target being irradiated by a laser-driven neutron beam. In Sec. 3, we present the target configuration and the potential seed nuclides as well as the data sources for the reaction cross sections. Our numerical results and discussions are presented in Sec. 4 for the case of one neutron pulse and in Sec. 5 for the case of multiple neutron pulses. We finally summarize and conclude in Sec. 6.

THEORETICAL APPROACH
We investigate a setup in which a rectangular target made of a pure seed material is irradiated by a laserdriven neutron beam. The number of neutrons per pulse of the neutron beam is denoted by N p . Average neutron energies in the range between 50 keV and 10 MeV are considered, which cover the range of the neutron energies of the laser-driven neutron sources (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Kar et al. 2016;Mirfayzi et al. 2017;Kleinschmidt et al. 2018;Wu 2020;Ren et al. 2017;Ma et al. 2015;Döppner et al. 2015;Olson et al. 2016) for the purpose of the present study. The target has an interacting surface area A hit by the neutron beam and a thickness L. The successive radiative neutron capture process producing neutronenriched nuclei, the damping of the incident neutron beam, and the loss of nuclei by transmutation and radioactive decay are of interest and have been taken into account. We note that it is a one-dimensional model, i.e., the neutron-target interaction happens only in the volume A×L and is homogeneous in the interacting area A.
Inside the target, beam neutrons interact with the target atoms generating secondary particles, thereby also transmuting the target nuclides. Successive effects of these secondary particles are neglected for the purpose of the present study. Furthermore, as we are not interested in the production of nuclides other than the neutronenriched isotopes of the seed nuclide, we do not calculate abundances or successive transmutations of such nuclides, i.e., we only keep track of the seed nuclide and the neutron-enriched isotopes of the seed nuclide. By doing so, we neglect the contribution of any loops in a transmutation path (e.g., 28 Si , whose effect is assumed to be small. In order to close a loop, processes that kick out a nucleon from the nucleus have to take place. However, such processes like (n, p), (n, 2n), (n, d), etc., are in general highly suppressed below several MeV and could therefore only skew our results towards the high energy end. We note that more sophisticated approaches modelling a variety of different processes, including such loops, can be found in, e.g., Refs. (Wilson 1999;Kum 2018;Sublet et al. 2017).
In the following of Sec. 2, we begin with the simplest case that the neutron capture in a thin target during one neutron pulse in Sec. 2.1. Then we generalise the model to the case of a thick target in Sec. 2.2 to include the effect of the damping of the incident neutron beam. In Sec. 2.3, we further generalise the model to the case of multiple neutron pulses to include the effects of the multiple neutron pulses and the decay of the nuclei. In Sec. 2.4, we calculate the total amount of neutronenriched nuclei produced during the interaction, which in addition to the number of nuclei remaining after the interaction also accounts for the number of nuclei lost due to transmutation and radioactive decay.

Neutron capture in a thin target during one neutron pulse
We begin with the case of a thin target with thickness L λ and further assume that the interaction time T p (the neutron pulse duration) is much shorter than all half lives of the involved particles. Here, λ is the neutron penetration depth of the target material, which marks the scale at which the damping of the incident neutron beam becomes relevant. Furthermore only one neutron pulse is considered for the moment. Based on these assumptions, we can neglect the damping of the incident neutron beam and the decay of the nuclei. We denote the isotope having i more neutrons than the seed isotope (0-species) as the i-species isotope. The populations of these isotopes are coupled via the following set of equationṡ . . .
where N i stands for the number of nuclei of the i-species isotope involved in the interaction, which initially takes on the value N i = N 0 i . Here, the l-species is the cut off species that N l stays sufficiently small such that N i for i > l is negligible. R b is the rate of neutrons irradiating the target, which is related to the neutron current density j b via R b = j b A. Moreover, σ c,i denotes the neutron capture cross section of the i-species isotope, and σ tr,i is its transmutation cross section including all processes changing the neutron or proton number.
The beam rate R b in Eqs.
(1)-(4) is in general time dependent. However, this time dependence can be eliminated by changing t → τ (t) = t 0 R b (t )dt /N p , which effectively replaces R b (t) with N p in the above equations. Eventually, we are only interested in the population at the end of the pulse at T p [τ (T p ) = 1]. For simplicity we just rename τ → t again and evaluate at t = 1. We further define the capture and loss parameters for the i-species isotope, whose values can be estimated for the purpose of this study as µ i , η i 10 −6 − 10 −4 . The above equations (1)-(4) can be rewritten in a more convenient way asṄ where the matrix has been introduced and N = N 0 N 1 · · · N l T . The solution can be directly obtained by integration, yielding N (t) = e Bt N 0 (Jeffrey 2010). Hence we find the populations after one pulse (Jeffrey 2010), 2.2. Neutron capture in a thick target during one neutron pulse Let us now relax the assumption of a thin target and account for the damping of the incident neutron beam. Since the initial seed nuclides will stay the most abundant species, the incident neutron beam is damped in good approximation according to where R b,0 is the initial neutron rate, n t is the initial number density of the nuclei in the target and σ tot,0 is the total neutron interaction cross section of the seed nuclide. The penetration depth λ is then given by We can generalise the result in Sec. 2.1 to this case by The e −x/λ factor can be pulled out of B, such that one effectively has B → e −x/λ B.
Integrating over x can be performed in the sum representation of Eq. (8) and we obtain where in practice we evaluate M up to l-th order, and define γ k = λ 1 − e −kL/λ /(kL).

Neutron capture during multiple neutron pulses
So far we have only considered a single pulse. Let us now turn to the case that the neutron source repeatedly generates neutron pulses with repetition rate f rep , i.e., each T del = 1/f rep there is a neutron pulse. To describe the populations after several neutron pulses, radioactive decay of the nuclei also needs to be taken into account. Since we have assumed that all mean lifetimes τ i are much longer than the neutron pulse duration T p (T p T del ), the model can be easily generalized. The populations after s neutron pulses [plus (T del −T p ) more precisely, i.e., at the time of sT del ] N (s) can be related to the populations after the previous pulse N (s − 1) via where the prefactor accounts for the radioactive decay, and the term M N (s − 1) gives the one-pulse result for the initial conditions N (s − 1) (with the subscript i on a vector, we denote the i-th component of the corresponding vector). With Eq. (12) the populations can be computed recursively. In order to have some qualitative insights into the effect of multiple neutron pulses and Eq. (12), we assume τ i = ∞ and γ k = 1 (i.e., thin target assumption). Then Eq. (12) becomes If sµ i 1 and sη i 1 we may keep terms only to leading order in k for each component of N (s). With the initial condition N 0 = N t 0 0 · · · T , where N t is the initial number of seed nuclei in the neutron-target interaction region, the leading order for the i-species is the B i term and we find that the populations scale polynomially with s (the number of neutron pulses N pl , respectively) as Furthermore the number of nuclei of the isotope that captured one more neutron is suppressed by a factor which is linear in s.

Total amount of produced nuclides
A further quantity of interest is the total amount of nuclei of the different species N tot i produced during the interaction, which is governed by the differential equation for the total amount of nuclei of the (i + 1)-species isotope produced during one neutron pulse. Summing up the contributions for s neutron pulses gives 3. SEED NUCLIDES AND DATA SOURCES

Seed nuclides
Laser-driven neutron sources generate neutron pulses on the scale of micrometers with pulse durations on the order of picoseconds or nanoseconds (T p ∼ picoseconds to nanoseconds) (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Kar et al. 2016;Mirfayzi et al. 2017;Kleinschmidt et al. 2018;Wu 2020;Ren et al. 2017;Ma et al. 2015;Döppner et al. 2015;Olson et al. 2016). Thus, for the purpose of the present work, we assume a rectangular target with an interacting surface area A = 25 µm (perpendicular to the incident beam) and thickness L = 100 µm. Neutron beams with average energy E inc in the range between 50 keV and 10 MeV are considered, and a Gaussian profile energy spectrum with relative width of w/E inc = 10% is assumed. We note that, according to Eqs. (1)-(4) and the discussion in Sec. 2.1, the production of neutron-rich isotopes depends on the incident neutron beam via the parameter N p /A. Thus, assuming a fixed L and a given neutron beam energy, , is a function of N p /A. This relation only holds for N i /A, since the initial conditions of N i depend linearly on A and N i itself depends linearly on these initial conditions.
The main cross section data employed in the present work is the recent Evaluated Nuclear Data File (ENDF) release ENDF/B-VIII.0 by the National Nuclear Data Center (NNDC) from 2018 ). We consider the heaviest sufficiently-long-lived (half life > 1 h) isotope per element whose data is available in ENDF/B-VIII.0 neutron sublibrary to serve as a potential seed target material. In addition, the NON-SMOKER library (Rauscher & Thielemann 2001) which provides theoretical neutron-capture cross sections is used for neutronenriched isotopes.
A list of the seed nuclides considered in the present study is shown in Table 1, together with the half lives T 1/2 and T +i 1/2 (i: 1, 2, 3, and 4) of each seed nuclide and its i-species neutron-enriched isotope, respectively. Furthermore, we note that the number density of the seed nuclei in the target is estimated by their atomic weight and the corresponding elemental mass density (Haynes 2016;RSC 2020;Morss et al. 2006).

ENDF/B-VIII.0 data and NON-SMOKER data
The ENDF/B-VIII.0 neutron sublibrary (denoted as ENDF-B) data comes in one data file [in the ENDF6 format (Trkov et al. 2018)] per nuclide containing a variety of different cross sections and other information, such as energies of excited states. Before usage the data needs to be preprocessed using the PREPRO (PREPRO 2018) code to interpolate between the experimental data and add the contribution of resonances to the cross sections.
For the purpose of the present study the ENDF-B library provides us with cross sections for the considered seed nuclides. From the ENDF-B library, we obtain the total neutron interaction cross section σ tot,0 that includes all possible processes of elastic scattering (n, n), inelastic scatting with excitation of the nucleus (n, n ) and non-elastic scattering. Furthermore we employ Note-The half life of the seed nuclides T 1/2 and of the corresponding i-species neutron-enriched isotopes T +i 1/2 presented are taken from the ENDF-B-VIII.0 decay sublibrary . For the cases in which only a bound for the half life is available, this bound is used during all calculations, and similarly unknown half lives are treated as zero. The neutron penetration depth λ is the minimal penetration depth in the energy (average neutron energy) range between 50 keV and 10 MeV for the seed material. Half lives fulfilling T +i 1/2 > 1 h are marked bold and correspondingly seed nuclides that have at least one such sufficiently-stable neutron-enriched isotope are also marked bold. In this table the notation aEb stands for a × 10 b . the neutron-capture cross section σ c,0 , the elastic cross section σ el,0 , the inelastic cross section σ inel,0 and the non-elastic cross section σ non−el,0 . This allows us to calculate the transmutation cross section σ tr,0 , describing all neutron interaction processes changing the proton or neutron number of the target seed nuclei σ tr,0 = σ non−el,0 − σ inel,0 .
As the non-elastic cross section σ non−el,0 is not available for most of the seed nuclides we study, we calculate it for the missing cases by means of the total and elastic cross section via In order to check the precision of σ non−el,0 obtained by this procedure we compare it with the values directly taken from the ENDF-B library for the cases where this data exist. The comparison shows that the mean relative error of σ non−el,0 calculated by Eq. (20) is around 1% for most cases. For a few cases, however, one finds a mean relative error of about 50%. The error in the calculated non-elastic cross section is possibly due to miss alignments of the energy grids on which the different cross sections are given. Thus, we conclude that σ non−el,0 provided by Eq. (20) is a good approximation. Furthermore, we note that in the ENDF-B library, inelastic cross sections σ inel,0 are missing for 18 O, 13 C, 233 Pa and 9 Be. However, for 18 O, 13 C and 233 Pa inelastic cross section data is added during preprocessing by PREPRO.
As we select the potential seed nuclides by the heaviest sufficiently-long-lived isotope per element whose data is available in the ENDF-B library, the cross sections for almost all of the neutron-enriched isotopes are not available. We therefore also make use of theoretically predicted neutron-capture cross sections by the NON-SMOKER code (Rauscher & Thielemann 2001) for the neutron-capture cross sections σ c,i (i > 0) of the neutron-enriched isotopes. While neutron cross sections are only provided for 10 ≤ Z ≤ 83, the range of covered neutron numbers in the NON-SMOKER data exceeds that of the ENDF-B data. The NON-SMOKER code predicts two sets of cross sections based on the Extended Thomas-Fermi Approach with Strutinski Integral model (ETFSI-Q) and the Finite Range Droplet Model (FRDM) respectively, where the first one only covers 26 ≤ Z ≤ 83.
Comparisons of the neutron capture cross sections from the ENDF-B library and the NON-SMOKER data show that the latter mostly fit the data from ENDF-B within one order of magnitude, except at resonances, where they just follow the value averaged over the resonance. For a quantitative comparison we look at the N 1 abundances obtained from the neutron-capture cross sections taken from the NON-SMOKER data and the ENDF-B data, respectively. The relative deviations of the two results for N 1 after one pulse are computed for 300 log-spaced energies (average incident neutron energy) between 50 keV to 10 MeV and for different seed nuclides. Both models of the NON-SMOKER data perform equally well and have a mean relative deviation to the ENDF-B result of around 50%. The details of the comparison are shown in the Appendix A. In the present work, the FRDM data is employed for the neutron-capture cross sections σ c,i with i > 0, as it covers a larger range of elements than the ETFSI-Q data. We note that if no neutron-capture cross section is available in the NON-SMOKER data, the last available cross section (less neutrons) is used as an approximation, µ i = µ i−1 . For the elements with Z < 10 or Z > 83 this leads to the approximation µ 4 ≈ µ 3 ≈ µ 2 ≈ µ 1 ≈ µ 0 .
The non-elastic and inelastic cross sections of neutronenriched isotopes are estimated by the one of the seed nuclide as they are not available for almost all neutronenriched isotopes in the ENDF-B or NON-SMOKER data (only for the seed nuclides 113 47 Ag, 193 77 Ir and 240 92 U, non-elastic and inelastic cross section data is available for some of their neutron-enriched isotopes in the ENDF-B). Analysis of the non-elastic cross sections for the seed nuclide shows that, up to a few 100 keV the non-elastic cross section is dominated by neutron capture for most (non fissile) nuclides. Above this energy scale, excitations of the nucleus and then high energy processes such as proton production, etc. become dominating. We therefore estimate for i ≥ 1 For the situation that the neutron capture might no be the dominating process of the non-elastic cross section, the above approximation (21) Figure 1. N1 and N2 after one neutron pulse, without taking the radioactive decay of the nuclei into account since τi Tp, for the seed nuclide listed in Table 1 and selected average neutron energies: (a) Einc = 100 keV, (b) Einc = 500 keV, (c) Einc = 1 MeV and (d) Einc = 5 MeV. A neutron beam with Np = 10 12 n/pl is assumed. N1 is represented by the blue bars, where their values need to be read from the left axis. N2 is represented by the orange bars, where their values need to be read from the right axis. For some seed nuclides, N2(1pl) is not shown, because it is too small to be visible in the range shown in the plots.

Penetration depths
In order to have an idea of the damping of the incident neutron beam inside the target, we present the minimal  Table 1. A neutron beam with Np = 10 12 n/pl is assumed and results for an average neutron energies Einc = 100 keV (f1: blue curve with filled circles; f2: black curve with filled up-pointing triangles) and Einc = 1 MeV (f1: orange curve with crosses; f2: purple curve with filled down-pointing triangles) are shown. penetration depth for all potential seed materials in Table (1). By means of Eq. (10) the penetration depth is calculated over an average incident neutron energy range of 50 keV to 10 MeV for each seed nuclide. Afterwards the minimal value per seed nuclide is selected for presentation, since the smallest penetration depths have the strongest effect on the actual quantities (N i and N tot i ) we are interested in. As the minimal penetration depth found is about 5 mm, the effect of the thickness of the target on the abundances is small (∼ 1% or less).

NUMERICAL RESULTS FOR ONE NEUTRON PULSE
In order to acquire an overall picture, we first consider the case of a single neutron pulse for each of the seed nuclides listed in Table 1. The results for N 1 (1pl) and N 2 (1pl) without accounting for any decay of the nuclei (τ i T p ), calculated by Eq. (11), are presented in Fig. 1 for the average neutron energies 100 keV, 500 keV, 1 MeV, and 5 MeV. We assume a neutron beam with N p = 10 12 n/pl. The results show that the number of neutron capture events decreases in general as the incident neutron energy increases. This is due to the decreasing neutron-capture cross section. At an average neutron energy of E inc = 100 keV, N 1 (1pl) reaches an order of 10 9 and N 2 (1pl) reaches an order of 10 3 , for some of the seed nuclides. For an average neutron energy of E inc = 1 MeV, N 1 (1pl) reaches an order of 10 8 , while N 2 (1pl) reaches an order of 10 2 . This indicates that even with one neutron pulse, it is possible to observe 2 successive neutron capture events.
Another important quantity is describing the fraction of the seed nuclei transmuting to the i-species neutron-enriched nuclei to the number of the seed nuclei N t . The results for f 1 (1pl) and f 2 (1pl), after one neutron pulse and without taking the decay of nuclei into account (τ i T p ), are shown in Fig. 2 as functions of the atomic number Z of the seed nuclide. As in Fig. 1, we assume again a neutron beam with N p = 10 12 n/pl. One observes in Fig. 2 that f 1 (1pl) and f 2 (1pl) have 3 terrace regions in Z where f i (1pl) have similar and quite good values: 41 − 51, 60 − 79, and 88 − 100.
Among the good seed-candidates showing large N 1 (1pl) and N 2 (1pl) in Fig. 1 and Fig. 2, seed nuclides which have at least one sufficiently-long-lived neutronenriched isotope (T +i 1/2 > 1 h) are of particular interest. On the one hand, they provide the possibility of conducting further experimental studies of the nuclear properties of their neutron-enriched isotopes. On the other hand, they have advantages for the case of multiple neu-  tron pulses, as the abundance of long-lived isotopes can accumulate in the target, thus enhancing the successive neutron capture, simultaneously allowing for the production of further neutron-enriched isotopes. In Fig. 3, we present the results of N 1 (1pl), N 2 (1pl), N 3 (1pl) and N 4 (1pl), again calculated by Eq. (11), as functions of the average neutron energy for a selection of such seed nuclides (  95 Am). We once more assume a neutron beam with N p = 10 12 n/pl.
As shown in Fig. 3, N 1 (1pl) and N 2 (1pl) have values larger than 1 for average incident neutron energies up to a few MeV. This, in principle, suggests the possibility of observing isotopes with 1 and 2 more neutrons than the seed nuclide after only a single neutron pulse. For all seeds and considered neutron energies, N 3 (1pl) and N 4 (1pl) are smaller than 1, indicating that the successive neutron capture process of capturing more than 2 neutrons is negligible in the single neutron pulse case. Figure 3 also shows that the seed nuclides 233 91 Pa, 187 75 Re and 176 71 Lu show in general a good performance in the considered neutron energy range (50 keV to 10 MeV). Furthermore, 244 95 Am performs well around a neutron energy of 1 MeV. The seed nuclides 192 76 Os and 226 88 Ra exhibit a good performance for neutron energies of a few MeV, while they show a not as good performance at other neutron energies.

NUMERICAL RESULTS FOR MULTIPLE NEUTRON PULSES
We now turn to the scenario consisting of multiple neutron pulses, assuming a repetition rate of f rep = 1  Hz for the neutron source, corresponding to T del = 1 s.
The target is exposed to 10 4 neutron pulses (N pl = 10 4 ), equivalent to roughly three hours of interaction time.
As discussed previously, seed nuclides with at least one sufficiently-long-lived neutron-enriched isotope (T +i 1/2 > 1 h) are of primary interest and hence we restrict our study to such seed nuclides, marked bold in Table 1. We first consider a neutron beam with N p = 10 12 n/pl. The abundance N i and the total produced amount N tot i after 10 4 neutron pulses (at time 10 4 T del ) is calculated by Eq. (12) and Eq. (18), respectively. After analyses of the results we select a set of interesting seed nuclides ( 75 33 As, Lu exhibit a good performance for the entire neutron energy range we consider, 244 95 Am performs well at neutron energies around 1 MeV and 192 76 Os and 226 88 Ra show a good performance for neutron energies of a few MeV. We note that only the seed nuclides 75 33 As, 126 51 Sb and 176 71 Lu have sufficiently-stable 3-species isotopes (T +3 1/2 > 1 h), while 226 88 Ra has the 4-species isotope with a half life of T +4 1/2 > 1 h. After 10 4 neutron pulses and at an average neutron energy of ∼ 100 keV, approximatively 1% to 10% of the seed nuclei can be neutron-enriched, as shown in Fig. 4. N 3 and N 4 remain observable (> 1) up to a few MeV, indicating the possibility of producing isotopes with 4 more neutrons than the seed nuclide. A comparison between Fig. 4 and Fig. 5 shows the effect of the decay and transmutation of nuclei, indicating that more neutron- N tot 4 is still observable (> 1) up to an average neutron energy of a few MeV. This, in principle, implies that it is possible to observe 4 successive neutron capture events.
Comparisons of the results in Fig. 5 with the results in Fig. 3 show that, the production of the nuclei of the ispecies isotope, for i ≥ 2, depends on the number of neutron pulses N pl in a non-linear manner. The dependance of N tot i is approximatively given by N tot compare also to Eq. (14)]. This interesting feature shows the advantage of multiple neutron pulses, where such nonlinear increasing of N tot i for i > 1 is due to the accumulation of neutron-enriched nuclei during the multiple neutron pulses.
Furthermore, for all of these seed nuclides of the selected set (   in these cases is slightly weaker than (N pl ) i . This is due to the quite short lifetime of the isotopes of the 3-species of these seed nuclides.
In order to understand the effect of the number of neutrons per pulse N p and the parameter N p /A, respectively, we also perform calculations for the case of N p = 10 10 n/pl. The results for N i and N tot i (i up to 3) after 10 4 neutron pulses (at time 10 4 T del ) as functions of the average neutron energy are shown in Fig. 6, again for the same set of seed nuclides (  only up to a few 100 keV of the average neutron energy. One finds that N i and N tot i scale as N i p , since the parameter N p /A and the number of the neutron pulses N pl are too small to lead to saturation. This scaling can also be anticipated from the qualitative discussion leading to Eq. (14). As discussed in Sec. 3.1, N i /A is a function of N p /A. Thus, N i and N tot i scale as (N p ) i /A i−1 .
As mentioned above, we analyze all seed nuclides marked bold in Table 1 having at least one sufficientlylong-lived neutron-enriched isotope (T +i 1/2 > 1 h). Beside the set of seed nuclides discussed so far and presented in Figs. 4-6, there are also some other seed nuclides exhibiting good results. We therefore also present results for the seed nuclides 171 69 Tm,193 77 Ir,197 79 Au, 227 89 Ac and 255 99 Es. The corresponding values for N i (i up to 4) after 10 4 neutron pulses (at time 10 4 T del ) are shown in Fig. 7 as functions of the average incident neutron energy. Here, we again assume a neutron beam with N p = 10 12 n/pl. While these nuclides perform worse than the best ones shown in Figs. 4-5, they still achieve fairly good results. We note that, since we assume the lifetime of 259 99 Es to be zero, as described in Table 1, the N 4 result for the seed nuclide 255 99 Es is not shown in Fig. 7. However, 259 99 Es nuclei are still produced during the interaction. In total a number N tot 4 of about 10 3 of such nuclei is produced at an average neutron energy of 100 keV, while it decreases to ∼ 1 as the incident neutron energy goes up to ∼ 1 MeV.
Furthermore, the maximum values of the abundances N i (i up to 4) shown in Fig. 7 are found at the end of the 10 4 neutron pulses, with the only exception being the 256 99 Es isotope of 255 99 Es. For this case saturation occurs at low and high neutron energy, where N 1 already peaks at around 9000 neutron pulses. This is due to the quite short lifetime of 256 99 Es. We note that 126 51 Sb, 176 71 Lu and 187 75 Re are at or are near the branching point isotopes of the s-process (Klay et al. 1991;Doll et al. 1999;Käppeler et al. 1993Käppeler et al. , 1991Käppeler 1999;Battaglia et al. 2016). The elements Lu, Re, Os, Tm, Ir and Au are close to the region of the the waiting point N = 126 of the r-process (Panov & Janka 2009;Negoita et al. 2016), where N is the neutron number of isotopes. Measurements of the properties of the neutron-rich nuclei produced from these seed nuclides, as well as the neutron capture cascade itself, would improve our understanding of the neutron-capture nucleosynthesis in astrophysics. In addition, Es capturing 3 and 4 neutrons, respectively, are beyond the heaviest isotopes of Am and Es that have been accessed so far in the laboratory (Thoennessen 2013(Thoennessen , 2016a(Thoennessen , 2014(Thoennessen , 2015b(Thoennessen , 2016b(Thoennessen , 2017(Thoennessen , 2018(Thoennessen , 2019. This indicates that we could get access to such neutron-rich isotopes in a regime that has never been access to by other means in the laboratory. Furthermore, 129 51 Sb is also an interesting case for fundamental nuclear physics as it has recently been found to show the signal of emerging nuclear collectivity (Gray et al. 2020).
In the Petawatt-class laser facilities such as the ELI facilities (Negoita et al. 2016;ELI 2020) available in the near future, lasers with a power on the Petawatt level and a repetition rate of around 1 Hz will be in operation. The intense laser-driven neutron beams with 10 12 neutrons per pulse and a high repetition rate that we have mainly focused on in the present work are expected to be achievable in such laser facilities. In highpower laser facilities currently available, with less power or smaller repetition rate of the lasers than the upcoming ones, the achievable neutron beams are less intense or have a smaller repetition rate compared to the one (10 12 neutrons per pulse and a repetition of 1 Hz) that we have mainly focused on (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Wu 2020;Pomerantz September 21-25, 2015). In this case, as shown in Fig. 6, up to 3 successive neutron capture events leading to neutron-rich isotopes with 3 more neutrons than the original seed nuclide are expected.

SUMMARY
We have studied the neutron capture cascades, and consequently, the production of neutron-rich isotopes taking place in single-component targets being irradiated by a laser-driven (pulsed) neutron source. Specifically, we have considered a rectangular target, and investigated the effects of the neutron irradiation for a variety of different target seed nuclides. These seed nuclides have been taken from the recent ENDF-B-VIII.0 neutron sublibrary , where we have chosen the heaviest sufficiently-long-lived isotope (T 1/2 > 1 h) per element. In this way a total of 95 different seed nuclides from 7 3 Li to 255 100 Fm have been studied. We have presented a theoretical approach describing the production of neutron-enriched isotopes, accounting for the successive radiative neutron capture process, the damping of the incident neutron beam, the loss of target nuclei by transmutation and radioactive decay of the nuclei, and the effect of multiple neutron pulses. Our calcu-lations show that, even with a single neutron pulse of 10 12 neutrons per pulse, observing 2 successive neutron capture events is possible. Furthermore, our results for the scenario of 10 4 neutron pulses, provided by a laserdriven neutron source delivering 10 12 neutrons per pulse at a repetition rate of 1 Hz, show the possibility of observing up to 4 successive neutron capture events leading to the production of neutron-rich isotopes with 4 more neutrons than the original seed nuclide. Such intense laser-driven neutron beams with a high repetition rate are expected to be achievable in the Petawatt-class laser facilities, such as ELI facilities (Negoita et al. 2016;ELI 2020), available in the near future. With the neutron beams with less intensity or smaller repetition rate currently available (Roth et al. 2013;Pomerantz et al. 2014;Higginson et al. 2015;Wu 2020;Pomerantz September 21-25, 2015), up to 3 successive neutron capture events, leading to neutron-rich isotopes with 3 more neutrons than the original seed nuclide, are expected.
The Re) or the waiting point of the r-process (Lu, Re, Os, Tm, Ir and Au).
Moreover it is also possible to produce neutron-rich isotopes ( 248 95 Am, 258 99 Es and 259 99 Es) in a regime that has not been accessed by other means in the laboratory. Measuring the properties of the produced neutron-rich nuclei, as well as observing the neutron capture cascades for these seed nuclides which could allow us to simulate the astrophysical neutron capture nucleosynthesis in the laboratory, could improve our understanding of the astrophysical nucleosynthesis. In addition, 129 51 Sb produced from the seed nuclide 126 51 Sb is an interesting nuclide for fundamental nuclear physics. Our study also shows that the production of neutron-enriched isotopes scales as N tot i ∝ (N pl ) i with the number of neutron pulses N pl , as long as saturation due to competition among neutron capture, radioactive decay of nuclei, and the loss of nuclei due to transmutation does not occur. Furthermore, the abundance N i and the total produced amount N tot i of neutron-enriched nuclei scale as N i p /A i−1 before saturation occurs. Our study would be interesting for the industry of radioisotope production, astrophysics concerned with neutron-capture nucleosynthesis, and fundamental nuclear physics.

ACKNOWLEDGMENTS
The authors gratefully acknowledge fruitful discussions with A. Pálffy and C. H. Keitel.

APPENDIX
A. COMPARISON OF NON-SMOKER DATA AND ENDF-B DATA In order to have an idea of the quality of the NON-SMOKER data we employ in the present work, we make quantitative comparisons between the NON-SMOKER data (Rauscher & Thielemann 2001) and ENDF-B data ). We therefore calculate N 1 by Eq. (8) (one neutron pulse, no radioactive decay and no damping of the incident neutron beam) for different seed nuclides and different average neutron energies. Here, we only keep terms up to the leading order for N 1 . The calculation is performed twice using the two NON-SMOKER datasets and once using the ENDF-B cross section data. We define the relative deviation by where N 1 | NON-SMOKER denotes N 1 calculated using the NON-SMOKER neutron-capture cross sections and N 1 | ENDFB is N 1 calculated using the neutron-capture cross sections from the ENDF-B library. These relative deviations are calculated for 300 log-spaced energies (average incident neutron energy) between 50 keV to 10 MeV. The maximal relative deviation, the minimal relative deviation and the mean relative deviation for each seed nuclide are shown in Fig. 8. Only some of the seed nuclides listed in Table 1 have been selected for this calculation, since not for all nuclides data is available in all of the three datasets. As in the main text of the paper, we have assumed a target with interacting surface area A = 25 µm and thickness L = 100 µm. The neutron beam has a Gaussian profile energy spectrum with relative width of w/E inc = 10%, and the neutron number per pulse assumed here is N p = 10 11 n/pl. Both models used in the NON-SMOKER code yield a mean relative deviation (average over all the seed nuclides presented in Fig. 8) to the ENDF-B data of around 50%, i.e., for the NON-SMOKER ETFSI-Q  FRDM NON-SMOKER data against experimental data in N1 evaluation mean relative deviation for FRDM model max relative deviation for FRDM model min relative deviation for FRDM model Figure 8. The deviations for N1 after one neutron pulse between the NON-SMOKER data and ENDF-B data. The deviations are obtained by calculating N1 for 300 log-spaced energies (average incident neutron energy) between 50 keV to 10 MeV. A neutron beam with Np = 10 11 n/pl is assumed. The damping of the incident neutron beam due to the target thickness, and the radioactive decay of nuclei are not taken into account. (a): Relative deviations between N1 calculated using the NON-SMOKER ETFSI-Q data and N1 calculated using the ENDF-B data. The mean deviation (for the NON-SMOKER ETFSI-Q data) averaged over all the seed nuclides analysed here is 52%. (b): Relative deviations between N1 calculated using the NON-SMOKER FRDM data and N1 using the ENDF data. The mean deviation (for the NON-SMOKER FRDM data) averaged over all the seed nuclides analysed here is 54%. The blue bar shows the maximal relative deviation and the orange bar shows the minimal relative deviation. The filled circle shows the mean relative deviation for each seed nuclide.
data, the mean relative deviation is 52%, while for the NON-SMOKER FRDM data, the mean relative deviation is 54%.