A uniﬁed understanding of ( γ ,n) and (n, γ ) reactions and direct neutron-multiplicity sorting

. We discuss the γ -ray strength function toward a uniﬁed understanding of ( γ ,n) and (n, γ ) reactions and propose a novel technique of direct neutron-multiplicity sorting to resolve the long-standing discrepancy between the Livermore and Scalya data of partial photoneutron cross sections.


Introduction
The γ -ray strength function (γ SF) is a nuclear statistical quantity that governs electromagnetic processes in radiative neutron capture and photoneutron emission. In the statistical model, the (γ ,n) and (n,γ ) reaction cross sections are interconnected through the γ SF and the Brink hypothesis linking the downward γ SF in photo-deexcitation to the upward γ SF in photoabsorption in the approximate equality [1]. An indirect method called the γ -ray strength function method [2] was devised for constraining radiative neutron capture cross sections for unstable nuclei of direct relevance to nuclear astrophysics and nuclear engineering. This method requires a systematic measurement of (γ ,n) cross sections over an isotopic chain and uses known (n,γ ) cross sections for qualification of the model γ SF. Thus, this method leads to a unified understanding of (γ ,n) and (n,γ ) reaction cross sections over an isotopic chain. The method was applied to such isotopic chains as Zr [3], Sn [4], Mo [5] and more recently, Sm [6] and Nd [7].
The γ SF can also be constructed by putting experimental data of (γ ,n), (γ ,γ ) and particle-gamma coincidences together.
The IAEA-CRP F41032 was launched in 2016 as a 4-year project to generate a reference database for photon strength functions and update the photonuclear library, IAEA-TECDOC-1178 published in 2000 [8], the compilation and evaluation of photonuclear data for 164 isotopes of 48 elements from 2 H to 241 Pu. In this a e-mail: hiro@konan-u.ac.jp project, the main purpose of the update is to resolve the discrepancy between the Livermore and Saclay data of partial photoneutron cross sections compiled in the library by providing new reliable data.
The emergence of quasi-monochromatic γ -ray beams produced by Compton backscattering of laser photons from relativisitiv electrons has made these research activities feasible at the turn of the 21st century.

Laser compton scattering γ -ray beam
The laser-Compton scattering (LCS) from relativistic electrons converts a laser beam to a quasi-monochromatic γ -ray beam with an energy-amplification factor, ∼ 4γ 2 , where γ is the Lorentz factor, γ = E e /mc 2 . The Lorentz factor is about 2000 at the electron energy E e = 1 GeV. Thus, the amplification factor is ∼ 16 million, producing a ∼ 16 MeV γ -ray beam from a 1 eV (λ = 1064 nm) photon beam of a Nd:YVO 4 laser. The LCS γ -ray beam, whose energy varies with the square of the electron energy, is energy-tunable from 4 to 38 MeV (8 to 76 MeV) with the use of the λ = 1064 nm (532 nm) photons at the NewSUBARU synchrotron radiation facility where the electron beam energy can be changed from 0.5 to 1.5 GeV. The LCS γ -ray beam is quasi-monochromatic with energy spreads essentially determined by the electron beam emittance and size of the collimator. A typical energy spread is 1-2% in FWHM in photoneutron cross section measurements around neutron threshold [6,7]. The LCS γ -ray beam is background-free in the quasi-monochromatic region, which is in sharp contrast to the fact that γ -ray beams produced by the positron annihilation in flight are accompanied by considerable background γ -rays generated by the positron bremsstrahlung [9].

A unified understanding of (γ , n) and (n, γ ) reactions
Photoneutron cross sections measured over the Nd and Sm isotopic chain were compared with theoretical calculations obtained by the TALYS nuclear reaction code with two models of γ SF, the Skyrme Hartree-Fock-Bogoliubov (HFB) plus quasi-particle random phase approximation (QRPA) model [10] based on the BSk7 interaction and the axially symmetric deformed Gogny HFB plus QRPA model based on the D1M interaction [11]. The experimental data are best reproduced by the latter model of γ SF. Figures 1 and 2 show (γ ,n) cross sections for Nd and Sm isotopes, reapectively. Figures 3 and 4 show comparisons between the existing (n,γ ) cross section data for Nd and Sm isotopes and the TALYS calculations with the D1M+QRPA model of the E1 strength. The shaded area represents the sensitivity to the nuclear level density.

Constructing experimental γ -ray strength function
Experimentally photoneutron cross section measurements provide the upward γ SF above the neutron threshold. The nuclear resonance fluorescence measurement provides the downward γ SF through the ground-state decay of the populated both resolved and unresolved (quasicontinuum) states below the neutron threshold The Oslo method [12] of deducing the primary γ -transitions from multiple gamma-decay data obtained in inelastic scattering and transfer/pickup reactions can also provide the downward γ SF below the neutron threshold. Based on the Brink hypothesis of the approximate equality of the upward and downward γ SF, one can construct the γ SF by combining (γ ,n), (γ ,γ ) and the Oslo data with the (γ ,n) cross section as absolute normalization. Figures 6 and 7 show thus constructed γ SF for 117 Sn [13] and 74 Ge [15].

Direct neutron-multiplicity sorting
After the elaborate efforts of evaluating the Livermore and Saclay data of partial photonuetron cross sections, especially those based on the transitional neutron-multiplicity  function [15] as well as (γ , n) cross sections below twoneutron separation energies compiled in the photonuclear data library [8], it turned out that the discrepancy between the Livermore and Saclay data cannot be resolved in any systematic way. In many cases of the partial cross sections, Saclay provided larger cross sections for (γ , n) reactions than Livermore and vice versa for (γ , 2n) reactions. An important mission of the IAEA-CRP F41032 is to provide new data of total and partial photoneutron cross sections that help resolve the long-standing discrepancy. We propose here direct neutron-multiplicity sorting with a flat-efficiency neutron detector.

Multi-neutron coincidences vs partial photoneutron reactions
For simplicity, we treat the neutron-multiplicity up to 3. However, the discussion given below holds for any neutron-multiplicity. The number of neutrons, N x , emitted in the (γ , xn) reaction with the neutron-multiplicity x is expressed by where N γ is the number of γ -rays incident on a target, N T is the number of target nuclei per unit area, and σ (γ, xn) is the partial photoneutron cross section. Let us consider a multi-neutron coincidence measurement for (γ, xn) cross sections using a neutron detector with an efficiency ε. We experimentally separate single, double, and triple neutroncoincidence events that are expressed with N 1 , N 2 , and N 3 , respectively, as follows.
One can see that there are contributions from all (γ, xn) reactions with x = 1, 2, and 3 to the single neutron events. The second term, for example, comes from the fact that we can detect one neutron out of two neutrons emitted in the (γ, 2n) reaction and undetect the other neutron.
The Livermore detector had 45-30% efficiencies over neutron energies up to 5 MeV. The neutron energy dependence of the detection efficiency was still large so that the ring-ratio (RR) technique [16] was used to determine the average neutron energy. The RR technique, which can be applied to multi-neutron coincidence events (N s , N d , and N t ), however, does not determine the average neutron energies for individual (σ, xn) reactions.

Flat-efficiency neutron detector
It is essential to develop a flat-efficiency neutron detector to measure partial photoneutron cross sections by direct neutron-multiplicity sorting.
We previously developed a high-efficiency neutron detector (HED) that has efficiencies more than 60% for neutron energies below 1 MeV. This detector has long been used for (γ , n) cross section measurements below twoneutron separation energies. We have newly developed a flat-efficiency neutron detector (FED) by modifying the triple-ring configuration of the HED. The flat response was achieved by weakening the strong-energy dependence of the inner ring, which shows a rapid decrease with increasing neutron energy, at the cost of the large detection efficiency and compensating the energy dependence with increasing efficiencies of the middle and outer rings. The number of 3 He counters and the distance of the three rings were optimized by Monte Carlo simulations.
The FED consists of three concentric rings of four, nine, and eighteen 3 He counters embedded in a polyethylene moderator at the distances of 5.5, 13.0 and 18.0 cm from the γ -ray beam axis, respectively. The HED and FED share the same 3 He counters of 45 cm long active volume filled with a 10 atm gas mixture of 3 He and CO 2 . Figure 8 shows a picture of the FED.
The efficiency of the FED was calibrated with a 252 Cf source whose absolute emission rate was determined to be (1.62 ± 0.04) × 10 4 neutrons per second at the National Metrology Institute of Japan. The results of the calibration are shown in Fig. 9 in comparison with Monte Carlo simulations. The experimental results are well reproduced by the MCNP [17] simulation (dark lines). The GEANT4 [18] simulations slightly overestimates the efficiency of the inner ring and thus the total efficiency. The total efficiency summed over the three rings is 37.27 ± 0.82% at 2.13 MeV, the average energy of the 252 Cf neutron spectrum and 38.0-32.9% over 0-5 MeV.
Besides the flatness, a reasonably-high efficiency is required in multi-neutron coincidence measurements because the efficiency of detecting i neutrons in  coincidences is given by ε i . The present efficiency assures safely 3-and possibly 4-fold coincidences.

First measurements by direct neutron multiplicity sorting
First measurements by direct neutron-multiplicity sorting were carried out for 209 Bi [19] and 9 Be at the NewSUBARU facility in 2015. Figure 10 shows arrival-time distributions of triplecoincidence neutrons detected with the 3 He counters after moderation in the polyethylene for 209 Bi at E γ = 33.3 MeV. A detailed analysis of the multiple-neutron events is in progress.

The PHOENIX collaboration for the IAEA-CRP
The PHOENIX, which stands for "photoexcitation and neutron emission cross (x) sections" collaboration for the IAEA-CRP has begun at the NewSUBARU facility in Japan. Four teams of the Konan University, the ELI-NP [20], the University of Oslo, and the Lomonosov Moscow State University are formally involved in this collaboration. We measure (γ , n) cross sections for the IAEA-CRP of generating a reference database for photon strength functions and (γ , xn) cross sections with x = 1-3 for that of updating the photonuclear data library.

ND2016
The Konan, ELI-NP, MSU, and Oslo teams are officially assigned their own (γ , xn) and (γ , n) data within the IAEA-CRP F41032 as follows. In addition to the official assignment, the PHOENIX collaboration may go beyond the IAEA-CRP, depending on the availability of enriched samples. Additional data of (γ ,n) cross sections were obtained for four nuclei, 138 Ba, 137 Ba, 61 Ni, and 13 C in 2016.