Cross sections for neutron-induced reactions from surrogate data: revisiting the Weisskopf-Ewing approximation for (n,n') and (n,2n) reactions
CCross sections for neutron-induced reactions from surrogate data:revisiting the Weisskopf-Ewing approximation for (n,n (cid:48) ) and (n,2n) reactions
Oliver C. Gorton ∗ San Diego State University, San Diego, California 92182, USA † Jutta E. Escher ‡ Lawrence Livermore National Laboratory, Livermore, CA 94550 (Dated: February 9, 2021)
Background:
Modeling nuclear reaction networks for nuclear science applications and for simulations of astro-physical environments relies on cross section data for a vast number of reactions, many of which have never beenmeasured. Cross sections for neutron-induced reactions on unstable nuclei are particularly scarce, since they arethe most difficult to measure. Consequently, we must rely on theoretical predictions or indirect measurements toobtain the requisite reaction data. For compound nuclear reactions, the surrogate reaction method can be usedto determine many cross sections of interest.
Purpose:
Earlier work has demonstrated that cross sections for neutron-induced fission and radiative neutroncapture can be determined from a combination of surrogate reaction data and theory. For the fission case, it wasshown that Weisskopf-Ewing approximation, which significantly simplifies the implementation of the surrogatemethod, can be employed. Capture cross sections cannot be obtained, and require a detailed description of thesurrogate reaction process. In this paper we examine the validity of the Weisskopf-Ewing approximation fordetermining unknown ( n, n (cid:48) ) and ( n, n ) cross sections from surrogate data. Methods:
Using statistical reaction calculations with realistic parametrizations, we investigate first whetherthe assumptions underlying the Weisskopf-Ewing approximation are valid for ( n, n (cid:48) ) and ( n, n ) reactions onrepresentative target nuclei. We then produce simulated surrogate reaction data and assess the impact of applyingthe Weisskopf-Ewing approximation when extracting ( n, n (cid:48) ) and ( n, n ) cross sections in situations where theapproximation is not strictly justified. Results:
We find that peak cross sections can be estimated using the Weisskopf-Ewing approximation, but theshape of the ( n, n (cid:48) ) and ( n, n ) cross sections, especially for low neutron energies, cannot be reliably determinedwithout accounting for the angular-momentum differences between the neutron-induced and surrogate reaction. Conclusions:
To obtain reliable ( n, n (cid:48) ) and ( n, n ) cross sections from surrogate reaction data, a detailed de-scription of the surrogate reaction mechanisms is required. To do so for the compound-nucleus energies and decaychannels relevant to these reactions, it becomes necessary to extend current modeling capabilities. I. BACKGROUND AND NEED
Nuclear reaction data is required for many applica-tions in both basic and applied science, whether it be formodeling the origin of elements in the universe, the safeoperation of a next-generation reactors, or for national-security applications [1, 2]. Nuclear reaction librariesprovide evaluated reaction data for many such applica-tions [3]. These evaluations are based on nuclear reactioncalculations anchored to experimental data and state-of-the-art nuclear theory. As many reaction cross sectionsof interest cannot be measured directly, due to short life-times or high radioactivity of the target nuclei involved,indirect methods are being developed [4–7] to address thegaps and shortcomings in present databases.In this paper we focus on the “surrogate reac-tion method” [6, 8], an indirect approach for deter-mining cross sections for compound-nuclear reactions.Compound-nuclear, or “statistical” reactions, proceed ∗ [email protected] † University of California, Irvine, California 92679, USA ‡ [email protected] through the formation of an intermediate “compound”nucleus n + A → B ∗ , followed by a decay into reac-tion products B ∗ → c + C . The appropriate formal-ism for calculating cross sections for these reactions isthe Hauser-Feshbach formalism [9, 10]. Hauser-Feshbachcalculations are often quite limited in accuracy due to un-certainties in the nuclear physics inputs needed, in par-ticular the nuclear structure inputs associated with thedecay of the compound nucleus (CN).In a surrogate reaction experiment, the CN of interestis produced via an alternative, experimentally accessi-ble reaction, and the probability for decay into the re-action channel of interest is measured. From this data,constraints for the Hauser-Feshbach calculations can beobtained.The surrogate method has some significant advantagesover alternative indirect approaches: 1) The method doesnot make use of auxiliary nuclear properties that are notavailable for unstable nuclei and for which interpolationor extrapolation procedures are associated with uncon-trolled uncertainties [14, 15], and 2) The method can beused for reactions that populate energies well above par-ticle separation thresholds in the CN, i.e. it is applicablenot only to ( n, γ ), but also to ( n, n (cid:48) ), ( n, n ), ( n, p ), ( n, f ) a r X i v : . [ nu c l - t h ] F e b FIG. 1. (Color online.)
Surrogate reactions approach for the simultaneous measurement of Zr( n, γ ), Zr( n, n (cid:48) ), and Zr( n, n ) cross sections. A recent inelastic scattering experiment has produced the CN up to about 30 MeV, i.e. abovethe two-neutron threshold [11]. Subsequent decay via emission of γ s, one neutron, and two neutrons, produces final Zr, Zr,and Zr nuclei, respectively. The example here displays a situation in which discrete γ transitions between low-lying states inthree nuclei are used to determine the decay channel probabilities. A complementary decay measurement that focuses on thedetection of neutrons is under development as well [12]. The Zr experiment serves as a benchmark, since the neutron-inducedreactions for the stable Zr nucleus are known from direct measurements [13]. reactions (and similarly to proton-induced reactions).Applications of the surrogate method to ( n, f ) reac-tions have a long history [6] and in recent years scientistssuccessfully used the approach to obtain neutron capturecross sections [14, 15]. In this paper, we focus on possibleapplications to ( n, n (cid:48) ) and ( n, n ) reactions.Figure 1 illustrates how the surrogate approach can beused to determine Zr( n, γ ), Zr( n, n (cid:48) ), and Zr( n, n )cross sections from a surrogate inelastic scattering exper-iment. For incident neutron energies below a few MeV,neutron capture and inelastic neutron scattering com-pete with each other, above E n ≈
10 MeV, one- andtwo-neutron emission compete with each other. Protonand α emission compete only weakly and have to be ac-counted for, but are not shown here. In actinides, fissionmay compete at all energies. If the surrogate reactionmeasurement is designed to cover a broad energy range,it becomes possible to determine cross sections for allthree neutron-induced reactions in one experiment. Thedecay channel of interest is determined by either mea-suring γ transitions specific to one of the three decay products, or by detecting outgoing neutrons, in coinci-dence with the scattered α (cid:48) particle. Experimentalistsconducting these measurements have utilized discrete γ rays and are currently developing the capability to useneutron measurements.In principle, a careful description of the surrogate re-action mechanism is required to obtain the cross sectionof the desired reaction. This is because one must ac-count for the differences in the decay of the CN due tothe angular-momentum and parity differences in the sur-rogate and desired reactions (the spin-parity mistmatch).Indeed, ( n, γ ) reactions are very sensitive to spin effects,particularly in nuclei with low level density [16–18]. Onthe other hand, sensitivity studies for surrogate ( n, f )applications have shown that neglecting the spin-paritymismatch yields reasonable results, except at low neutronenergies [19–21]. Neglecting the spin-parity mismatch be-tween the surrogate and desired reactions is known as theWeisskopf-Ewing approximation, and it greatly simplifiesthe extraction of the cross sections from surrogate data,as only a simple theoretical treatment is required.It is the purpose of this paper to investigate what isrequired to determine reliable cross sections for ( n, n (cid:48) )and ( n, n ) reactions from surrogate data. Specifically,we carry out sensitivity studies that examine the valid-ity of the Weisskopf-Ewing approximation for these tworeactions for several regions of the nuclear chart.In the next section, we review the surrogate reactionformalism and provide details on the Weisskopf-Ewingapproximation. In Section III, we describe our proce-dure for testing the assumption of the approximation,and for investigating the consequences of applying the ap-proximation in situations where its assumptions are notstrictly valid. In Section IV, we present results for zirco-nium, gadolinium and uranium nuclei, which are repre-sentative of spherical and deformed nuclei, respectively.We summarize our findings and make recommendationsin Section V. II. REACTION FORMALISM
Here we summarize the Hauser-Feshbach formalism forcalculating the cross section of a compound-nuclear re-action and its relationship to the description of a surro-gate reaction. This clarifies how surrogate reaction datacan be used to constrain calculations for unknown crosssections. We outline the circumstances under which theWeisskopf-Ewing approximation can be used to simplifythe analysis used to obtain the desired compound crosssection.
A. Theory for the desired reaction
The Hauser-Feshbach (HF) statistical reaction formal-ism properly accounts for conservation of angular mo-mentum and parity in compound-nuclear reactions. Fora reaction with entrance channel α = a + A that forms theCN B ∗ , which subsequently decays into the exit channel χ = c + C , a + A → B ∗ → c + C, the HF cross section can be written as σ αχ ( E a ) = (cid:88) J,π σ CNα ( E ex , J π ) G CNχ ( E ex , J π ) . (1)Here E a and E ex are the kinetic energy of the projec-tile a and the excitation energy of the compound nu-cleus B ∗ , respectively. They are related to each other via E a = m A m a + m A ( E ex − S a ), where S a is the energy needed toseparate the particle a from the nucleus B ∗ . m a and m A are the masses of the projectile and target, respectively. J and π are the spin and parity of the compound nucleusand σ CNα ( E ex , J π ) is the cross section for the forming thecompound nucleus B ∗ with spin and parity J π at energy E ex . The σ CNα ( E ex , J π ) and their sum, the compound-formation cross section σ CNα ( E ex ) = (cid:80) J,π σ CNα ( E ex , J π ), can be determined using an appropriate optical model forthe a -nucleus interaction. Width fluctuation correctionshave been omitted to simplify the notation in Equation 1,but are included in the calculations. G CNχ ( E ex , J π ) is the probability that the CN decaysvia the exit channel χ . It depends on the convolutionof the transmission coefficient T Jχl c j χ with the level den-sity ρ j C ( U ) for the residual nucleus, divided by analogousterms for all competing decay modes χ (cid:48) : G CNχ ( E ex , J π ) = (cid:80) l c j χ j C (cid:82) T Jχl c j χ ρ j C ( U ) dE χ (cid:80) (cid:48) χ (cid:48) l (cid:48) c j (cid:48) χ T Jχ (cid:48) l c j (cid:48) χ + (cid:80) χ (cid:48) l (cid:48) c j (cid:48) χ j (cid:48) C (cid:82) T Jχ (cid:48) l (cid:48) c j (cid:48) χ ( E χ (cid:48) ) ρ j (cid:48) C ( U (cid:48) ) dE χ (cid:48) . (2)The quantities l c and l (cid:48) c are the relative orbital angularmomenta in the exit channels. (cid:126)j χ = (cid:126)j c + (cid:126)j C is the exitchannel spin, related to the total spin (cid:126)J = (cid:126)l a + (cid:126)j α = (cid:126)l c + (cid:126)j χ by conservation of momentum with the entrance channelspin, (cid:126)j α = (cid:126)j a + (cid:126)j A . ρ C ( U, j C ) is the density of levelsof spin j C at energy U in the residual nucleus. Con-tributions from decays to discrete levels and to regionsdescribed by a level density have to be accounted for, asshown in the denominator of (2). All sums over quantumnumbers must respect parity conservation, although thisis not explicitly expressed here.In this paper, we focus on neutron-induced reactions, i.e. α = n + A . For such reactions, the optical modelpotential, used to calculate the first factor in (1), is wellapproximated by a one-body potential [22]. By far thegreatest source of uncertainty comes from the decay prob-abilities, a fact that can be attributed to uncertaintiesin the nuclear structure inputs. ab initio shell-modelcalculations can provide nuclear structure informationfor nuclei with only a dozen or so nucleons, and tradi-tional shell-model calculations cover a limited numberof nuclei, primarily near closed shells, containing up toaround 100 nucleons. Mean-field and beyond-mean fieldapproaches cover a wider range of nuclei, but calculat-ing the relevant structure quantities (level densities andgamma-ray strength functions) is nontrivial. While muchprogress has been made toward achieving microscopicnuclear structure inputs for HF calculations of medium-mass and heavy nuclei, many isotopes needed for appli-cations and for simulating stellar environments are cur-rently out of reach.In the absence of microscopic predictions of structuralproperties, phenomenological models are used for nuclearlevel densities and electromagnetic transition strengths,with parameters that are fit to available data. Mucheffort has been devoted to generate global or regionalparameter systematics [3] that can be utilized as to per-form HF calculations and build nuclear reaction evalua-tions [23–26]. Alternatively, it is possible to use surrogatereaction data to obtain experimental constraints on thedecay probabilities. B. Full modeling of the surrogate reaction
In a surrogate experiment, such as the one schemat-ically shown in Figure 1, the compound nucleus B ∗ isproduced by an inelastic scattering or transfer reaction d + D → b + B ∗ , and the desired decay channel is ob-served in coincidence with the outgoing particle b at angle θ b .The probability for forming B ∗ in the surrogatereaction (with specific values for E ex , J , π ) is F CNδ ( E ex , J, π, θ b ), where δ refers to the surrogate re-action d + D → b + B ∗ . The quantity P δχ ( E ex , θ b ) = (cid:88) J,π F CNδ ( E ex , J π , θ b ) G CNχ ( E ex , J π ) , (3)which gives the probability that the CN B ∗ was formedwith energy E ex and decayed into channel χ , can be ob-tained experimentally by detecting a discrete γ -ray tran-sition characteristic of the residual nucleus (or some othersuitable observable).The distribution F CNδ ( E ex , J, π, θ b ), which may bevery different from the CN spin-parity populations fol-lowing the absorption of a neutron in the desired reaction,has to be determined theoretically, so that the branchingratios G CNχ ( E ex , J π ) can be extracted from the measure-ments.In practice, the decay of the CN is modeled us-ing a Hauser-Feshbach-type decay model and the G CNχ ( E ex , J π ) are obtained by adjusting parametersin the model to reproduce the measured probabilities P δχ ( E ex , θ b ). Subsequently, the sought-after cross sec-tion for the desired (neutron-induced) reaction can beobtained by combining the calculated cross sections σ CNn + A ( E ex , J π ) for the formation of B ∗ (from n + A )with the extracted decay probabilities G CNχ ( E ex , J π ), seeEq. 1. Modeling the CN decay begins with an initial(“prior”) description of structural properties of the reac-tion products (level densities, branching ratios, internalconversion rates), plus a fission model for cases which in-volve that decay mode. Finally, a procedure for fittingthe parameters of the decay models, e.g. via a Bayesianapproach as introduced in Ref. [14], needs to be imple-mented to determine the desired cross section, along withuncertainties.This procedure was recently employed to determinecross sections for neutron capture on the stable Zr and Mo isotopes (for benchmark purposes), as well as forneutron capture on the unstable Y nucleus [14, 15].Such a full treatment of a surrogate experiment is chal-lenging: It involves taking into account differences in theangular momentum J and parity π distributions betweenthe compound nuclei produced in the desired and Sur-rogate reactions, as well as their effect on the decay ofthe compound nucleus. Predicting the spin-parity dis-tribution F CNδ ( E ex , J, π, θ b ) resulting from a Surrogatereaction is a nontrivial task since a proper treatment ofdirect reactions leading to highly excited states in the in- termediate nucleus B ∗ involves a description of particletransfers, and inelastic scattering, to unbound states.For capture cross sections, it was shown that this typeof approach is needed to account for the spin-parity mis-match in the surrogate experiment [16, 17], while for fis-sion applications it often suffices to employ the muchsimpler Weisskopf-Ewing or ratio approximations [21]. C. Weisskopf-Ewing Approximation forneutron-nucleus reactions and surrogate coincidenceprobabilities
The Hauser-Feshbach expression for the cross sectionof the desired neutron-induced reaction, Eq. (1), con-serves total angular momentum J and parity π . Un-der certain conditions the branching ratios G CNχ ( E ex , J π )can be treated as independent of J and π and the crosssection for the desired reaction simplifies to σ W En + A,χ ( E a ) = σ CNn + A ( E ex ) G CNχ ( E ex ) (4)where σ CNn + A ( E ex ) = (cid:80) Jπ σ CNn + A ( E ex , J π ) is the crosssection describing the formation of the compound nu-cleus at energy E ex and G CNχ ( E ex ) denotes the Jπ -independent branching ratio for the exit channel χ . Thisis the Weisskopf-Ewing limit of the Hauser-Feshbach the-ory [22].The Weisskopf-Ewing limit provides a simple and pow-erful approximate way of calculating cross sections forcompound-nucleus reactions. In the context of surro-gate reactions, it greatly simplifies the application of themethod. In section II B we described the process requiredto obtain the Jπ -dependent branching ratios G CNχ frommeasurements of P δχ ( E ex ). Under the Weisskopf-Ewinglimit, and because (cid:80) Jπ F CNδ ( E ex , J π ) = 1, P δχ ( E ex ) = G CNχ ( E ex ) . (5)Calculating the direct-reaction probabilities F CNδ ( E ex , J, π, θ b ) and modeling the decay of thecompound nucleus are no longer required in this ap-proximation. (In actual applications, experimentalefficiencies have to be included when determining P δχ ( E ex ); these are omitted for simplicity here, but areaccounted for in the analysis of surrogate experiments.)The conditions under which the approximate expres-sions (4) and (5) are obtained from equations (1) and (3)are discussed in the appendix.In addition, the Weisskopf-Ewing approximation canbe used in situations in which the surrogate reaction pro-duces a spin distribution that is very similar to that ofthe desired reaction, i.e. F CNδ ( E ex , J π ) ≈ F CNn + A ( E ex , J π ) , (6)where, F CNn + A ( E ex , J π ) ≡ σ CNn + A ( E ex , J π ) (cid:80) Jπ (cid:48) σ CNn + A ( E ex , J π (cid:48) ) , (7)since the weighting of the J π -dependent decay probabili-ties in the measured P δχ ( E ex ) is the same as the weight-ing relevant to the desired reaction. While some intuitivearguments have been forwarded in favor of specific sur-rogate reaction mechanisms that might satisfy the con-dition (6), not much is actually known about what spin-parity distributions F CNδ are obtained when producinga CN at high excitation energies ( E ex > G CNχ ( E ex , J π ) on spin and parity (Section III A)and the impact of using the Weisskopf-Ewing approxi-mation in situations in which G CNχ ( E ex , J π ) depends onspin and parity (Section III B). III. ASSESSING THE VALIDITY OF THEWEISSKOPF-EWING APPROXIMATION
As discussed in the previous section, there are two sce-narios in which it is clearly valid to employ the Weisskopf-Ewing approximation in the analysis of a surrogate ex-periment: (a) The decay probabilities G CNχ ( E ex , J π ) areindependent of Jπ for the decay channel χ of interest;or (b) The surrogate and desired reactions produce iden-tical spin distributions (“serendipitous” or “matching”approach [6]). In addition, there are some intermedi-ate situations in which a Weisskopf-Ewing analysis cangive a good approximation to the true cross section.For instance, it is possible that the decay probabilities G CNχ ( E ex , J π ) are only moderately sensitive to Jπ , andthat the surrogate and desired reactions populate some-what similar compound nucleus spins and parities, sothat violations of the Weisskopf-Ewing limit may havelittle impact on the extracted cross section. Investiga-tions into the possibility of using the Weisskopf-Ewingapproximation must therefore consider both the behaviorof the decay probabilities G CNχ ( E ex , J π ) for the decaychannel χ of interest and their influence in typical surro-gate reaction analyses.Earlier studies, which have done that, demonstratedthat it is not a priori clear whether the Weisskopf-Ewinglimit applies to a particular reaction in a given energyregime [16–18, 21]. For fission applications, it was foundthat using the Weisskopf-Ewing approximation gives rea-sonable cross sections, with violations of the Weisskopf-Ewing limit occurring primarily at low energies ( E n be-low 1-2 MeV) and at the onset of first and second-chancefission [21]. For neutron capture reactions, however, the G CNγ ( E ex , J π ) were found to be very sensitive to the Jπ and no circumstances have been identified so far in whichthe Weisskopf-Ewing limit can be used to obtain capturecross sections [17].In the present study we focus on the proposed use ofthe surrogate method to determine ( n, n (cid:48) ) and ( n, n )cross sections. To study the validity of the Weisskopf-Ewing approximation, we proceed in two steps:1. Investigation of the Jπ dependence of the decay probabilities G CNχ ( E ex , J π ) for χ = 1 n and 2 n , i.e. for one- and two-neutron emission.2. Assessment of the impact of the Jπ dependence ofthe G CNχ ( E ex , J π ) on cross sections extracted byusing the Weisskopf-Ewing approximation. A. Method for determining spin-parity dependence
In the first step, we obtain the G CNχ ( E ex , J π ) fromwell-calibrated Hauser-Feshbach calculations that in-volve the relevant decay channels. We selected n + Zr, n + Gd, and n + U as representative cases for neu-tron reactions on spherical and deformed nuclei, withthe uranium case representing a nucleus for which fissioncompetes with particle evaporation and γ emission.For each nucleus, we carried out a full Hauser-Feshbachcalculation of the neutron-induced reaction and cali-brated the model parameters to give an overall good fitof the known neutron cross sections. This local opti-mization of model parameters allows us to isolate thespin-parity effects from model uncertainties. Our opti-mization procedure accounted for pre-equilibrium effects.This is necessary to accurately and realistically reproducethe data without biasing the model-space parameters. Incontrast, the calculations described in this and the fol-lowing section include only contributions from compoundnucleus decay. This is consistent with the goal of investi-gating the ability to determine the compound cross sec-tion from a Weisskopf-Ewing analysis of surrogate data.The calculations were carried out with Hauser-Feshbach codes Stapre [27] and
YAHFC-MC [25]. Theresults discussed here are obtained using the latter. Weextracted the branching ratios G CNxn ( E, J π ) for one- andtwo-neutron emission ( x = 1 and 2, respectively) for arange of spin and parity values of the initially formedcompound nuclei Zr ∗ , Gd ∗ , and U ∗ , and investi-gated their behavior as a function of the excitation energy E ex of the CN. Our findings are discussed in Section IV A. B. Method for demonstrating impact of spin-paritydependence
In the second step, we employ the decay probabilities G CNxn ( E ex , J π ) extracted above to simulate the results ofpossible surrogate measurements. This is done by calcu-lating the coincidence probabilities given by equation (3),which are ordinarily measured in a surrogate experiment,by multiplying the G CNxn ( E ex , J π ) with several schematicspin-parity distributions F CNδ ( E ex , J π ), summed over allrelevant spins and parities: P simxn ( E ex ) = (cid:88) Jπ F CNδ ( E ex , J π ) G CNxn ( E ex , J π ) . (8)We normalized the distributions (cid:80) Jπ F CNδ ( E ex , J π ) = 1and did not consider angle dependencies. Multiplicationof these simulated coincidence probabilities P simxn ( E ex ) bythe CN-formation cross section σ CNn + A ( E ex ) then yieldscross sections σ W E ( n,n (cid:48) ) ( E n ) and σ W E ( n, n ) ( E n ) that corre-spond to a Weisskopf-Ewing analysis of the simulatedsurrogate measurement: σ W E ( n,xn ) ( E n ) = σ CNn + A ( E ex ) P simxn ( E ex ) (9)for x = 1 ,
2. In Section IV B, we compare the so extractedcross sections for various spin-parity distributions F CNδ to each other and to the known desired cross sections.To select relevant Jπ distributions for our study, webriefly summarize what is known about Jπ distributionsthat typically occur in neutron-induced as well as surro-gate reactions.
1. Spin-parity distributions in neutron-induced reactions.
Figure 2 shows spin-parity distributions relevant toneutron-induced reactions, as predicted by the Koning-Delaroche optical model [28]. For the ( n, n (cid:48) ) and ( n, n )applications considered here, neutron energies betweenabout 5 and 20 MeV are relevant. The examples selectedhere involve target nuclei with low spins (3 / − for Gdand 0 + for the even-even Zr and
U nuclei), so thespin-distributions are closely connected to the angular-momentum transferred in the reaction.Panel (a) shows the population of positive and neg-ative parity states for the n+ Zr example, for severalneutron energies E n . At E n ≈ Gd, and (c), for n+
U, are rep-resentative of the situations one encounters for deformedrare-earth and actinide nuclei, respectively. Overall, thedistributions are smoother for the deformed nuclei thanfor the Zr case and involve larger values of angular mo-mentum. With increasing E n , the positive and negativeparity distributions become similar, while at low energies, E n <
2. Spin-parity distributions in surrogate reactions.
The findings of the following illustrate that it is not correct to assume that the spin-parity distribution of acompound nucleus produced in a surrogate reaction isgiven by the spin and parity behavior of the level densityfor that nucleus. The reaction mechanism plays a crit-ical role in selecting which states act as doorways intothe compound nucleus. The population of these doorwaystates determines the Jπ distribution for the surrogatereaction. E n = 4.94 MeV0.050.150.25 E n = 9.98 MeV0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.50.050.150.25 E n = 19.86 MeV P r o b a b ili t y (a) n + Zr E n = 5.90 MeV0.050.150.25 E n = 10.10 MeV0 1 2 3 4 5 6 7 8 9 10 110.050.150.25 E n = 15.90 MeV P r o b a b ili t y (b) n + Gd E n = 5.95 MeV0.050.150.25 E n = 10.15 MeV0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.50.050.150.25 E n = 15.15 MeV Spin J P r o b a b ili t y (c) n + U FIG. 2. (Color online.)
Spin-parity distributions for com-pound nuclei produced in neutron-induced reactions, for sev-eral neutron energies E n . Solid bars are positive- and hatchedbars are negative-parity probabilities. Panels (b) n+ Gdand (c) n+
U are representative of deformed rare-earth andactinide nuclei, respectively, while panel (a) presents the caseof a near-closed shell nucleus, n+ Zr. Neutron energies below1 MeV are important for neutron capture reactions [17]. Forthe ( n, n (cid:48) ) and ( n, n ) applications considered in this paper,neutron energies between about 5 and 20 MeV are relevant. Figure 1 illustrates schematically the excitation ener-gies that a surrogate reaction has to populate in order toproduce decay information relevant to ( n, γ ), ( n, n (cid:48) ) and( n, n ) reactions. For neutron capture, E ex values be-tween about 5 and 10 MeV have to be reached, for inelas-tic scattering, energies between approximately 10 and 20MeV are relevant, and for ( n, n ) reactions, E ex = 20-30MeV are important. These energy regimes exhibit highlevel densities, and transfer reactions aiming to populatethese energy ranges are very different from those usedfor traditional nuclear structure studies. It should there-fore not surprise that standard DWBA or even coupled-channels calculations cannot be used to reliably calculatethe direct (surrogate) reactions that produce such states.Predicting the spin-parity distributions for thesehigher excitation energies requires taking into accountboth the surrogate reaction mechanism and the nuclearstructure at these higher energies. For instance, to cal-culate the Jπ population in the compound nucleus Zr ∗ that was produced via the Zr(p,d) pickup reaction ina recent surrogate experiment with E p = 28.5 MeV [14],it was necessary to consider the structure of deep neu-tron hole states, which exhibit considerable spreading.Furthermore, two-step mechanisms involving ( p, d (cid:48) )( d (cid:48) , d )and ( p, p (cid:48) )( p (cid:48) , d ) combinations of inelastic scattering andpickup contribute significantly to the reaction. Thesehave a strong influence on the final spin-parity distribu-tion in Zr ∗ [14], which is shown for E ex = 7.25 MeVin Figure 3(a). The influence of the reaction mechanismis reflected in the differences between the predicted spin-parity population (bars) and the spin distribution in arepresentative level density model at the same excitationenergy (green curve).Around the neutron separation energy, i.e. in the en-ergy region of interest to neutron capture, the angu-lar behavior of the (p,d) cross section was found to befairly structureless, and the Jπ distribution was seen tovary little over several MeV around E ex = S n ( Zr) =7.195 MeV [29]. These observations reflect the fact thatthe surrogate reaction does not produce a simple single-particle excitation, but populates specific doorway stateswhich mix with neighboring complex many-body statesto form the compound nucleus.The ( d, p ) transfer reaction, which – at first glance –seems to be a well-matched surrogate for neutron-inducedreactions, turns out to involve non-trivial reaction mech-anisms as well. The case of interest is that in whichthe deuteron breaks up in the combined Coulomb-plus-nuclear field, and the neutron is absorbed while the pro-ton escapes and is observed in a charged-particle detec-tor. Calculating the resulting compound nucleus Jπ dis-tribution requires a theoretical description that separateselastic from nonelastic breakup and, in principle, alsoneeds to separate out inelastic breakup, rearrangement,and absorption. This challenge has generated strong in-terest in developing a more detailed formalism for inclu-sive ( d, p ) reactions [30–34]. This formalism was used tocalculate the Jπ distribution relevant to the Mo( d, p )surrogate reaction described in Ref. [15]. The calculated Jπ distribution, for excitation energies near the neutronseparation energy in Mo is shown in Figure 3(b). Here,again, the predicted spin-parity distribution (bars) doesnot follow the distribution of spins that are expected tobe available at this energy, based on a representative leveldensity model (green curve).Inelastic scattering with charged light ions is a thirdtype of reaction that has been employed in surrogate re-action measurements [35–39]. From these experiments,as well as from traditional studies of giant resonances [40–42], it is known that inelastic scattering can produce acompound nucleus at a wide range of excitation energies. To our knowledge, no detailed studies of the associatedspin distributions have been published to date, but thereare indications that this type of reaction is also likelyto produce Jπ distributions that are broad and may becentered at angular momentum values of 5-10 (cid:126) [35, 39].Furthermore, for inelastic α scattering, a staggering ofeven and odd parity populations is expected, since thereaction populates predominantly natural-parity states. Spin J P r o b a b ili t y (a) Zr ( p , d )(b) Mo ( d , p ) FIG. 3. (
Color online. ) Spin-parity distributions (bars) nearthe neutron separation energy, as predicted for use with spe-cific surrogate experiments. Solid bars are positive-parityand hatched bars are negative-parity probabilities. Panel (a)shows the half-integer J distribution in the compound nucleus Zr ∗ resulting from a Zr(p,d) reaction with E p = 28.5 MeVat E ex = 7.25 MeV [14]. Panel (b) shows the integer valuedresult for Mo( d, p ) surrogate reaction with E d = 12.4 MeVat E ex = 9.18 MeV [15]. In both cases, the predicted spin-parity distributions were used in combination with modelsfor the decay of the respective compound nuclei, leading tothe successful determination of (benchmark) neutron capturecross sections. For comparison, the spin distribution calcu-lated from an energy-dependent level density model, whichassumes equal parity distribution, is given by the green solidcurve [43].
3. Schematic spin-parity distributions
In order to investigate the impact of a spin-parity mis-match between the desired and surrogate reaction on thecross section obtained from a Weisskopf-Ewing analysis,we employ the schematic distributions F CNδ ( J π ) shownin Figure 4. We include distributions that are centered atboth low and high angular-momentum values and allowfor more spread-out distributions in the latter case. Thedistributions centered at low J values allow us to investi-gate situations in which the surrogate reaction populateslower spins than the desired reaction.The distributions shown will be combined with the de-cay probabilities G CNχ ( E ex , J π ) extracted from our cal-ibrated Hauser-Feshbach calculations (see Section IV A)to simulate a range of possible surrogate data P δχ ( E ex , θ )using Eq. 3. For simplicity, we will neglect the energydependence of the Jπ distributions. This should be areasonable approach for our sensitivity studies, as recentresults indicate that these distributions vary slowly withenergy. F ( J , = 1.0)0.000.050.100.15 F ( J , = 3.0)0.000.050.100.15 F ( J , = 5.0)0.000.050.100.15 F ( J , = 7.0)0 2 4 6 8 10 12 140.000.050.100.15 F ( J , = 9.0) Spin J P r o b a b ili t y FIG. 4. (
Color online. ) Schematic spin distributions em-ployed in the current study. Each is of the form F ( J, µ ) ∝N ( m = µ, sd = √ µ ), where N is a normal distribution andmean spin µ is indicated in the legend. The spin values J areeither integer or half-integer, for even- A and odd- A nuclei,respectively, and equal probability is assigned to positive andnegative parity states. IV. RESULTS
We first demonstrate that the one- and two-neutrondecay probabilities depend on the spin, and to a lesserextent, the parity of the compound nucleus. The de-pendence is strongest at low energies and for sphericalnuclei, and lesser at higher energies and for deformed nu-clei. Then, we show the impact of the Weisskopf-Ewingapproximation on the outcome of simulated surrogate ex-periments, giving insight into the effect that the spin de-pendence has on predicted cross sections.
A. Decay probabilities for representative nuclei G CNxn ( E ex , J π ) for one- and two-neutron emission fromthe compound nucleus Zr ∗ are shown in Figure 5,for both positive and negative parities and a variety ofspins. The behavior of G CNxn ( E ex , J π ) just above theCN separation energy, corresponding to E ex = S n ( Zr)= 7 .
194 MeV, is governed by the interplay of the neutron-transmission coefficients and the low-energy structure ofthe residual nucleus Zr which is reached by one-neutronemission. The situation is schematically illustrated inFigure 1. Due to the shell structure of the nucleus, thelow-energy spectrum of Zr is very sparse, with the firstexcited state occurring at 1.76 MeV. Since both stateshave J π = 0 + and s- and p-wave neutron emission dom-inates at low energies, the residual nucleus can only be reached from low-spin states in the compound nucleus Zr ∗ . This suppression of neutron emission from all butthe lowest spin states in Zr ∗ is well known from earlierstudies of neutron capture reactions [6, 18, 44].As the excitation energy in Zr ∗ increases, addi-tional states in the residual nucleus become accessibleand the decay probabilities G CNxn ( E ex , J π ) for higher J values take on non-zero values. In the region between E ex = 15 −
20 MeV, the one-neutron emission probabilityis essentially unity, because of the weakness of competingdecay channels.In the energy region between 20 and 27 MeV, weobserve the transition from predominantly one-neutronemission to two-neutron emission. We see significant de-pendence of the branching ratio on the spins of the com-pound nucleus for J ≥ .
5, while there is much weakerdependence for J ≤ .
5. The decay probabilities are notvery sensitive to parity. Figure 6 shows the analogousone- and two-neutron emission probabilities for the decayof the rare-earth nucleus
Gd. Here, the dependenceon spin is weaker than in the Zr case, especially near theone-neutron separation energy of the compound nucleus.This is primarily due to the significantly higher level den-sity in the gadolinium nuclei: While the first excited statein Zr is at 1.76 MeV, there are 15 levels below 0.5 MeVin
Gd. In general, the level densities in deformed nu-clei are much higher, and the sensitivity of the compoundnucleus decays to spin and parity is reduced. This is alsotrue at higher energies: The competition between one-and two-neutron emission shows significant dependenceon the compound-nuclear spins, although the sensitivityis not as strong as in the zirconium case. Figure 7 showsthe one- and two-neutron emission probabilities for the
U nucleus. Like the gadolinium case discussed, theuranium nuclei are deformed and have a much higherlevel density than the zirconium nuclei:
U has 16 lev-els below 1 MeV. The transition from one-neutron totwo-neutron emission, which lies near the threshold forsecond-chance fission, is also sensitive to the angular mo-mentum population of the compound nucleus. Multiplechannels compete at all energies considered and no clearplateaus for the probabilities emerge, unlike in the othercases considered.For all three cases discussed, we have observed thatthere is enhanced sensitivity of the neutron emissionprobabilities near the thresholds. It can therefore be ex-pected that a failure to account for the spin-parity mis-match in the analysis of surrogate reaction will result inextracted ( n, n (cid:48) ) and ( n, n ) cross sections that do notreflect the true threshold behavior. This will be investi-gated in more detail in the next subsection. B. Impact of spin dependence of 1n and 2n decayprobabilities
In the previous section, we observed that the one- andtwo-neutron decay probabilities show a significant de- P r o b a b ili t y (a) G n ( E , J ) J = 0.5 J = 3.5 J = 6.5 J = 9.5 J = 12.50.00.20.40.60.81.0 P r o b a b ili t y (b) G n ( E , J + )0.00.20.40.60.81.0 P r o b a b ili t y (c) G n ( E , J )5 10 15 20 25 30Excitation energy of the compound nucleus (MeV)0.00.20.40.60.81.0 P r o b a b ili t y (d) G n ( E , J + ) FIG. 5. (Color online.)
Probabilities for neutron emissionfrom the Zr ∗ nucleus, as function of excitation energy, forvarious Jπ values of the compound nucleus. Both decay chan-nels exhibit a strong dependence on the spin of the compoundnucleus. The variance is seen to be greatest at the onset ofone-neutron emission, near E ex = S n ( Zr) = 7 .
194 MeV. pendence on the spin of the compound nucleus and alesser dependence on parity. Here we study the impactof this dependence on cross sections obtained under theassumption of the validity of the Weisskopf-Ewing ap-proximation. We use the schematic spin distributions F CNδ ( E ex , J π ) discussed in Section III B 3. They are con-veniently parameterized as discretized normal distribu-tions with mean µ and variance σ = µ : F CNδ ( E ex , J π ) ∝ N ( m = µ, sd = √ µ ) . (10)The distributions are cutoff above J = 50 and normalizedto unity. For the even-even compound nucleus Gd ∗ ,we consider the five distributions, µ = 1 , , , ,
9, shownin Figure 4; for the odd nuclei Zr ∗ and U ∗ we use P r o b a b ili t y (a) G n ( E , J ) J = 0.0 J = 3.0 J = 6.0 J = 9.0 J = 12.00.00.20.40.60.81.0 P r o b a b ili t y (b) G n ( E , J + )0.00.20.40.60.81.0 P r o b a b ili t y (c) G n ( E , J )5 10 15 20 25 30Excitation energy of the compound nucleus (MeV)0.00.20.40.60.81.0 P r o b a b ili t y (d) G n ( E , J + ) FIG. 6. (Color online.)
Probabilities for one-and two-neutron emission from the Gd ∗ nucleus, as function of ex-citation energy, for various Jπ values of the compound nu-cleus. The decay probabilities for both channels are seen todepend on the angular-momentum states populated in thecompound nucleus, at the onset of one-neutron emission near E ex = S n ( Gd) = 7 .
937 MeV and in the transition regionwhere the two-neutron channel opens. µ = 1 . , . , . , .
5, and 9 . Zr( n, n (cid:48) ) and Zr( n, n ) cross sectionsobtained from a Weisskopf-Ewing analysis of the simu-lated surrogate data are shown in Figure 8. As expected,the threshold regions for both reactions are particularlysensitive to spin effects. At the onset of inelastic scat-tering, it is not possible to obtain a reliable ( n, n (cid:48) ) crosssection; both shape and magnitude show a very largevariance. Different spin distributions give the same mag-nitude of this cross section in the region of the plateau,but there is again significant uncertainty in the regionwhere the two-neutron channel opens up.0 P r o b a b ili t y (a) G n ( E , J ) J = 0.5 J = 3.5 J = 6.5 J = 9.5 J = 12.50.00.20.40.60.8 P r o b a b ili t y (b) G n ( E , J + )0.00.20.40.60.8 P r o b a b ili t y (c) G n ( E , J )5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0Excitation energy of the compound nucleus (MeV)0.00.20.40.60.8 P r o b a b ili t y (d) G n ( E , J + ) FIG. 7. (Color online.)
Probabilities for one-and two-neutronemission from the U ∗ nucleus, as function of excitation en-ergy, for various Jπ values of the compound nucleus. We ob-serve a strong spin- and parity-dependence of G CN n ( E ex , J π )near E ex = S n ( U) = 4.806 MeV, which lies just below thethreshold for fission.
Given the findings in the previous section, we expectthe situation to be better for the gadolinium case, shownin Figure 9. While the
Gd( n, n (cid:48) ) cross section nearthe onset of inelastic scattering varies less than the anal-ogous zirconium cross section, it is still quite unreliable.The value of the
Gd( n, n (cid:48) ) cross section shows no de-pendence on the simulated spin-parity distribution ina region around E n = 15 MeV. Not surprisingly, theWeisskopf-Ewing approximation for different sets of sim-ulated surrogate data yields results that are consistentwith each other in an energy regime where there is lit-tle to no competition from other decay channels. Themaximum for the Gd( n, n ) cross section occurs near C r o ss s e c t i o n ( b ) = 1.5= 3.5= 5.5= 7.5= 9.50 5 10 15 20 25 30 35 E n (MeV)0.00.20.50.81.01.2 C r o ss s e c t i o n ( b ) (a) Zr ( n , n )(b) Zr ( n , 2 n ) EXFOR FIG. 8. (Color online.)
Cross sections for (a) Zr( n, n (cid:48) ) and(b) Zr( n, n ), obtained from simulated surrogate data, usingthe Weisskopf-Ewing assumption. The underlying schematicspin-parity distributions used are indicated in the legend. Theshape of the transition depends clearly on which simulatedsurrogate data is used, with the cross sections varying by ±
30% at about E n = 15 MeV. The Zr( n, n ) cross sec-tion varies by ±
4% near its maximum, which is located atabout E n = 20 MeV. For comparison, experimental data [45]for Zr( n, n ) is shown in panel (b). The only data for theinelastic scattering case is for scattering to an isomeric state. E n = 15 MeV, where the different sets of surrogatedata differ from each other by about 4%, which is anuncertainty that is similar to the error bands obtainedfrom direct measurements. Overall, it appears that theWeisskopf-Ewing approximation might provide a veryrough estimate of the ( n, n ) cross section of a rare earthnucleus.For the uranium case, shown in Figure 10, we observe afurther decrease in sensitivity to differences in spin. Evenso, the shape of the U( n, n (cid:48) ) cross section cannot bereliably extracted at low energies. With increasing en-ergy, the Weisskopf-Ewing approximation becomes morereliable. In fact, the U( n, n ) cross section obtainedfrom the simulated data is in good agreement with avail-able directly-measured data. Overall, it appears that theWeisskopf-Ewing approximation might provide a reason-able estimate of the ( n, n ) cross section of an actinidenucleus.Overall, we find that the Weisskopf-Ewing approxima-tion can provide rough first estimates for the ( n, n ) crosssections of nuclei with large level densities, such as rareearth and actinide nuclei, while the low-energy behavioris much less reliable. Specifically, near thresholds there1 C r o ss s e c t i o n ( b ) (a) Gd ( n , n )= 1= 3= 5= 7= 90 5 10 15 20 25 E n (MeV)0.00.51.01.52.0 C r o ss s e c t i o n ( b ) (b) Gd ( n , 2 n )ENDF/B.VIII.0EXFOR FIG. 9. (Color online.)
Cross sections for
Gd( n, n (cid:48) ) and
Gd( n, n ), obtained from simulated surrogate data, usingthe Weisskopf-Ewing assumption and several schematic spin-parity distributions. In the energy region where the transitionfrom one- to two-neutron emission occurs, the cross sectionsexhibit greater uncertainty, varying by ±
33% for ( n, n (cid:48) ) and ±
22% for ( n, n ) at E n = 10 MeV. The maximum for ( n, n )near E n = 15 MeV, exhibits smaller uncertainties, on theorder of 4%. For comparison, directly measured data is [45]shown for the Gd( n, n ) cross section; no data is availablefor the inelastic cross section. is clearly increased sensitivity of the decay to the under-lying spin-parity distribution in the compound nucleus.As a result, the shape of the extracted cross sections donot reproduce the true cross sections very well. Notably,the Weisskopf-Ewing approximation fails at the onset ofone-neutron emission. This is in line with earlier findingsabout the limitations of this approximation for neutroncapture cross sections.In addition, it should be stressed that we have focusedon the compound contributions to the ( n, n (cid:48) ) and ( n, n )cross sections here. For inelastic scattering it is wellknown that direct (pre-equilibrium) mechanisms providesignificant additional contributions, which are not con-sidered here. These have to be calculated separately andadded to the cross section, similar to what is done forthe direct-reaction component in an evaluation. Unfor-tunately, for many nuclei there is little data availablefor neutron inelastic scattering, and the calculations arechallenging, so this reaction channel requires additionalstudies, both experimentally and theoretically. C r o ss s e c t i o n ( b ) (a) U ( n , n ) = 1= 3= 5= 7= 9EXFOR0 5 10 15 20 250.000.250.500.751.001.251.50 C r o ss s e c t i o n ( b ) (b) U ( n , 2 n ) ENDF/V.VIII0EXFOR E n (MeV) FIG. 10. (Color online.)
Cross sections for (a) U( n, n (cid:48) )and (b) U( n, n ), obtained from simulated surrogate data,using the Weisskopf-Ewing assumption and several schematicspin-parity distributions. The U( n, n ) results agree rea-sonably well with the existing data [45]. For the inelasticcase, data is only available data for low energies, where directreaction mechanisms are known to contribute. V. OUTLOOK
We have investigated the potential use of theWeisskopf-Ewing approximation for determining ( n, n (cid:48) )and ( n, n ) cross sections from surrogate reaction data.Earlier work for neutron-induced fission and radiativeneutron capture demonstrated that this approximationyields reasonable approximations for the fission cross sec-tions, but fails for capture, making it necessary to employmore detailed theoretical modeling in the latter case.We modeled the nuclear structure properties that de-termine the decay of a compound nucleus via 1 n and 2 n emission, as well as the combined effect of the nuclearstructure and the surrogate reaction mechanisms on thecross-section results that one obtains from a Weisskopf-Ewing analysis of the indirect data. We found thatthe Weisskopf-Ewing approximation fails to give consis-tent cross section shapes in the presence of a spin-paritymismatch between the desired and surrogate reactions.The outcomes are worse for nuclei with low level den-sity, i.e. for lighter nuclei and for those in regions nearclosed shells. While rough estimates for the cross sectionsmight be obtained for ( n, n ) reactions on well-deformedrare-earth and actinide nuclei, we find that nuclei in themass-90 region are more sensitive to the effects of spinand parity. Furthermore, inelastic neutron scatteringcross sections are found to be quite sensitive to angular-2momentum effects and thus require a detailed treatmentof the reaction mechanism, similar to that recently usedfor extracting capture cross sections from surrogate data.Suggestions to find a surrogate reaction that approxi-mates the spin-parity distribution relevant to the desiredreaction are well-motivated, as the use of the Weisskopf-Ewing approximation greatly simplifies surrogate appli-cations. However, not enough is known about the angularmomentum and parity of the compound states that arepopulated in a surrogate reaction to plan an appropri-ate experiment. Recent work has demonstrated that thesurrogate reactions that produce a compound nucleus atthe high energies of interest involve higher-order reactionmechanisms, which render the type of simple angular-momentum estimates that are often used in traditionalnuclear structure studies inadequate. It is also not neces-sarily true that a surrogate reaction produces spins in acompound nucleus that are higher than those relevant toneutron-induced reactions. This means that in order toachieve cross section results with appropriate shapes anderrors less than about 30%, surrogate reaction data willneed to be combined with full modeling of the reactionmechanism, as described in section II B.In light of our findings that the Weisskopf-Ewing ap-proximation is insufficient for determining ( n, n (cid:48) ) and( n, n ) cross sections, we believe that further develop-ment of surrogate reaction theory is important for ad-dressing existing nuclear data needs. Inelastic scatter-ing ( n, n (cid:48) ) reactions in particular are poorly constrainedby direct measurement techniques. Alternative indirectmethods [46] do not address ( n, n (cid:48) ) and ( n, n ) reac-tions. Recent surrogate reaction applications to neu-tron capture have demonstrated how to proceed to accu-rately extract cross sections from surrogate data in sit-uations where the Weisskopf-Ewing approximation fails[14, 15, 39]. Given the limited utility of the Weisskopf-Ewing approximation for neutron induced one- and two-neutron emission reactions, we conclude that additionaldevelopments are needed in order to describe the rele-vant reaction mechanisms, such as those involved in the( He, He (cid:48) ) scattering experiment described in Figure 1. VI. ACKNOWLEDGEMENTS
This work was performed under the auspices of theU.S. Department of Energy by Lawrence LivermoreNational Laboratory under Contract No. DE-AC52-07NA27344 with support from LDRD Project No. 19-ERD-017, and the HEDP summer student program. Apart of this work was supported by DOE grant DE-FG02-03ER41272.
VII. APPENDIX
As discussed in Section II, if the decay probabilities G CNχ ( E ex , J π ) are independent of spin and parity, or thesurrogate reaction produces a compound nucleus spin dis-tribution which is very similar to that produced by theneutron-induced reaction, the cross section for the de-sired reaction can be obtained very simply as: σ n + A,χ ( E n ) = σ CNn + A ( E ex ) P CNδχ ( E ex ) , (11)where P CNδχ ( E ex ) is the coincidence probability deter-mined from the surrogate measurement.The latter of these options, the ‘serendipitous’or ‘matching’ condition requires that F CNδ ( J π ) ≈ F CNn + A ( E ex , J π ) holds. A comparison of F CNn + A ( E ex , J π )for representative nuclei and energies E ex , shown in Fig-ure 2 of this paper and in Figure 3 of Ref. [17], with re-alistic surrogate spin-parity distributions, such as thoseshown in Figure 3, indicates that it is difficult to identifyand carry out a surrogate reaction experiment that canachieve this condition.Here, we briefly review the conditions in which thedecay probabilities become approximately independentof J π , i.e. G CNχ ( E ex , J π ) → G CNχ ( E ex ) (see also Refs. [21,22]):First, the energy of the compound nucleus has to besufficiently high, so that almost all channels into whichthe nucleus can decay are dominated by integrals over thelevel density. In that case, the denominator in Equation2 reduces to the second term only.Second, correlations between the incident and outgoingreaction channels, which can be formally accounted forby including width fluctuation corrections [47], have to benegligible. These correlations enhance elastic scattering,at the expense of the inelastic and reaction cross sections,and are most prominent at the low energies relevant tocapture reactions. Width fluctuations are negligible ifthe first condition (above) is satisfied.Third, the transmission coefficients T Jχ (cid:48) l (cid:48) c j (cid:48) χ associatedwith the available exit channels have to be independent ofthe spin of the states reached in these channels. This con-dition is sufficiently well satisfied since the dependence oftransmission coefficients on target spin is very weak and,in fact, is ignored in many Hauser-Feshbach codes.Fourth, the level densities ρ j C in the available channelshave to be independent of parity and their dependenceon the spin of the relevant nuclei has to be of the form ρ j C ∝ (2 j C + 1). While level densities are known todepend on parity, that dependence becomes weaker withincreasing excitation energy and is often ignored in statis-tical reaction calculations. In addition, many successfulapplications use level densities that are parametrized ina form that is factorized (for each parity) as: ρ j C ( U C ) = w ( U C )(2 j C + 1) exp (cid:18) − j C ( j C + 1)2 σ C (cid:19) , (12)3where w ( U C ) contains the energy dependence of the leveldensity and σ C is the spin cut-off factor. At low energies( E ex ≤ σ C are 7-10 in the Zrregion and 12-16 in the Gd region [48]. As E ex increasesfrom a few MeV to about 20 MeV, σ C can increase by afactor 4 or more for these mass regions [43]. If we thenassume that the spins populated in the residual nucleusare small compared to the σ C , the level density can bewritten as ρ j C ( U C ) ≈ w C ( U C )(2 j C + 1) . (13)When the above conditions are satisfied, the decayprobabilities from Equation 2 take the form: G CNχ ( E ex , J π ) = (cid:80) l c j χ j C (cid:82) T Jχl c j χ w C ( U C )(2 j C + 1) dE χ (cid:80) χ (cid:48) l (cid:48) c j (cid:48) χ j (cid:48) C (cid:82) T Jχ (cid:48) l (cid:48) c j (cid:48) χ ( E χ (cid:48) ) w C (cid:48) ( U (cid:48) C )(2 j (cid:48) C + 1) dE χ (cid:48) . (14)We can carry out the sum over j C if we use the trianglerule | j χ − j c | < j C < | j χ + j c | to obtain the identity (cid:88) j C (2 j C + 1) = (2 j χ + 1)(2 j c + 1) . and analogously for the j χ : (cid:88) j χ (2 j χ + 1) = (2 J + 1)(2 l c + 1) , to obtain the spin-independent decay probabilities: G CNχ ( E ex ) = (15) (cid:0)(cid:80) l c (2 l c + 1) T χl c (cid:1) (cid:82) (2 j c + 1) w C ( U C ) dE χ (cid:16)(cid:80) χ (cid:48) l (cid:48) c (2 l c + 1) T χ (cid:48) l (cid:48) c ( E χ (cid:48) ) (cid:17) (cid:82) (2 j (cid:48) c + 1) w C (cid:48) ( U (cid:48) C ) dE χ (cid:48) . (16)In summary, in order for the G CNχ ( E ex , J π ) to becomeindependent of spin and parity, the energy E ex of thecompound nucleus must be high enough so that decaysto the continuum of residual nuclei dominate, and the re-action must populate spins that are small relative to thespin cutoff parameter. Since neutron-induced reactionsand surrogate reactions can produce different spin distri-butions, it is possible that the conditions for the validityof the Weisskopf-Ewing approximation are satisfied forone type of reaction, but not the other. [1] A. Arcones, D. W. Bardayan, T. C. Beers, L. A. Bern-stein, J. C. Blackmon, B. Messer, B. A. Brown, E. F.Brown, C. R. Brune, A. E. Champagne, A. Chieffi, A. J.Couture, P. Danielewicz, R. Diehl, M. El-Eid, J. E. Es-cher, B. D. Fields, C. Fr¨ohlich, F. Herwig, W. R. Hix,C. Iliadis, W. G. Lynch, G. C. McLaughlin, B. S. Meyer,A. Mezzacappa, F. Nunes, B. W. O’Shea, M. Prakash,B. Pritychenko, S. Reddy, E. Rehm, G. Rogachev, R. E.Rutledge, H. Schatz, M. S. Smith, I. H. Stairs, A. W.Steiner, T. E. Strohmayer, F. Timmes, D. M. Townsley,M. Wiescher, R. G. Zegers, and M. Zingale, White paperon nuclear astrophysics and low energy nuclear physicspart 1: Nuclear astrophysics, Progress in Particle andNuclear Physics , 1 (2017).[2] A. C. Hayes, Applications of nuclear physics, Reports onProgress in Physics , 026301 (2017).[3] R. Capote, M. Herman, P. Oblozinsk´y, P. Young,S. Goriely, T. Belgya, A. Ignatyuk, A. Koning, S. Hi-laire, V. Plujko, M. Avrigeanu, O. Bersillon, M. Chad-wick, T. Fukahori, Z. Ge, Y. Han, S. Kailas, J. Kopecky,V. Maslov, G. Reffo, M. Sin, E. Soukhovitskii, andP. Talou, Ripl - reference input parameter library forcalculation of nuclear reactions and nuclear data eval-uations, Nuclear Data Sheets , 3107 (2009).[4] G. Baur and H. Rebel, Coulomb breakup of nuclei - ap-plications to astrophysics, Annual Review of Nuclear andParticle Science , 321 (1996).[5] S. Typel and G. Baur, Theory of the Trojan–Horsemethod, Ann. Phys. , 228 (2003).[6] J. E. Escher, J. T. Burke, F. S. Dietrich, N. D. Scielzo,I. J. Thompson, and W. Younes, Compound-nuclear re-action cross sections from surrogate measurements, Rev. Mod. Phys. , 353 (2012).[7] A. Larsen, A. Spyrou, S. Liddick, and M. Guttorm-sen, Novel techniques for constraining neutron-capturerates relevant for r-process heavy-element nucleosynthe-sis, Progress in Particle and Nuclear Physics , 69(2019).[8] Escher, J. E., Tonchev, A. P., Burke, J. T., Bedrossian,P., Casperson, R. J., Cooper, N., Hughes, R. O., Humby,P., Ilieva, R. S., Ota, S., Pietralla, N., Scielzo, N. D., andWerner, V., Compound-nuclear reactions with unstablenuclei: Constraining theory through innovative experi-mental approaches, EPJ Web of Conferences , 12001(2016).[9] W. Hauser and H. Feshbach, The inelastic scattering ofneutrons, Phys. Rev. , 366 (1952).[10] P. Fr¨obrich and R. Lipperheide, Theory of Nuclear Reac-tions (Clarendon Press, Oxford, 1996).[11] N. Scielzo, Private communication.[12] R. O. Hughes, J. T. Burke, and J. E. Escher,
Toward(n,n’) and (n,2n) cross sections for 155Gd using the sur-rogate reaction method and the NeutronSTARS detector ,Tech. Rep. (Lawrence Livermore National Laboratory,2020).[13] V. Semkova, E. Bauge, A. Plompen, and D. Smith, Neu-tron activation cross sections for zirconium isotopes, Nu-clear Physics A , 149 (2010).[14] J. E. Escher, J. T. Burke, R. O. Hughes, N. D. Scielzo,R. J. Casperson, S. Ota, H. I. Park, A. Saastamoinen,and T. J. Ross, Constraining neutron capture cross sec-tions for unstable nuclei with surrogate reaction data andtheory, Phys. Rev. Lett. , 052501 (2018).[15] A. Ratkiewicz, J. A. Cizewski, J. E. Escher, G. Potel, J. T. Burke, R. J. Casperson, M. McCleskey, R. A. E.Austin, S. Burcher, R. O. Hughes, B. Manning, S. D.Pain, W. A. Peters, S. Rice, T. J. Ross, N. D. Scielzo,C. Shand, and K. Smith, Towards neutron capture on ex-otic nuclei: Demonstrating ( d, pγ ) as a surrogate reactionfor ( n, γ ), Phys. Rev. Lett. , 052502 (2019).[16] S. Chiba and O. Iwamoto, Verification of the surrogateratio method, Phys. Rev. C , 044604 (2010).[17] J. E. Escher and F. S. Dietrich, Cross sections for neu-tron capture from surrogate measurements: An examina-tion of Weisskopf-Ewing and ratio approximations, Phys.Rev. C , 024612 (2010).[18] C. Forss´en, F. Dietrich, J. Escher, R. Hoffman, andK. Kelley, Determining neutron capture cross sectionsvia the surrogate reaction technique, Phys. Rev. C ,055807 (2007).[19] W. Younes and H. C. Britt, Neutron-induced fission crosssections simulated from ( t, pf ) results, Phys. Rev. C ,024610 (2003).[20] W. Younes and H. C. Britt, Simulated neutron-inducedfission cross sections for various Pu, U, and Th isotopes,Phys. Rev. C , 034610 (2003).[21] J. E. Escher and F. S. Dietrich, Determining ( n, f ) crosssections for actinide nuclei indirectly: Examination of thesurrogate ratio method, Phys. Rev. C , 054601 (2006).[22] E. Gadioli and P. E. Hodgson, Pre-Equilibrium NuclearReactions (Clarendon Press, Oxford, 1992).[23] A. Koning, S. Hilaire, and M. Duijvestijn, TALYS: Nu-clear Reaction Simulator (2012), ascl:1202.004.[24] A. Koning, D. Rochman, J. Kopecky, J. C. Sub-let, E. Bauge, S. Hilaire, P. Romain, B. Morillon,H. Duarte, S. van der Marck, S. Pomp, H. Sjostrand,R. Forrest, H. Henriksson, O. Cabellos, S. Goriely,J. Leppanen, H. Leeb, A. Plompen, and R. Mills,Tendl-2015: Talys-based evaluated nuclear data library,https://tendl.web.psi.ch/tendl 2015/tendl2015.html(2015).[25] W. E. Ormand, YAHFC-MC: A Monte Carlo basedHauser-Feshbach reaction code system (2018).[26] D. Brown, M. Chadwick, R. Capote, A. Kahler, A. Trkov,M. Herman, A. Sonzogni, Y. Danon, A. Carlson,M. Dunn, D. Smith, G. Hale, G. Arbanas, R. Arcilla,C. Bates, B. Beck, B. Becker, F. Brown, R. Casper-son, J. Conlin, D. Cullen, M.-A. Descalle, R. Firestone,T. Gaines, K. Guber, A. Hawari, J. Holmes, T. John-son, T. Kawano, B. Kiedrowski, A. Koning, S. Kopecky,L. Leal, J. Lestone, C. Lubitz, J. Marquez Damian,C. Mattoon, E. McCutchan, S. Mughabghab, P. Navratil,D. Neudecker, G. Nobre, G. Noguere, M. Paris, M. Pigni,A. Plompen, B. Pritychenko, V. Pronyaev, D. Roubtsov,D. Rochman, P. Romano, P. Schillebeeckx, S. Simakov,M. Sin, I. Sirakov, B. Sleaford, V. Sobes, E. Soukhovit-skii, I. Stetcu, P. Talou, I. Thompson, S. van der Marck,L. Welser-Sherrill, D. Wiarda, M. White, J. Wormald,R. Wright, M. Zerkle, G. ˇZerovnik, and Y. Zhu, Endf/b-viii.0: The 8th major release of the nuclear reaction datalibrary with cielo-project cross sections, new standardsand thermal scattering data, Nuclear Data Sheets ,1 (2018), special Issue on Nuclear Reaction Data.[27] M. Uhl and B. Strohmaier,
STAPRE, A Computer Codefor Particle Induced Activation Cross Sections and Re-lated Quantities , Tech. Rep. IRK 76/01, rev. 1978 (Insti-tut f¨ur Radiumforschung und Kernphysik, Vienna, Aus-tria, 1976). [28] A. J. Koning and J.-P. Delaroche, Local and global nu-cleon optical models from 1 keV to 200 MeV, Nucl. Phys.
A713 , 231 (2003).[29] E. Escher, J., T. Burke, J., J. Casperson, R., O. Hughes,R., and D. Scielzo, N., One-nucleon pickup reactions andcompound-nuclear decays, EPJ Web Conf. , 03002(2018).[30] J. Lei and A. M. Moro, Reexamining closed-form formu-lae for inclusive breakup: Application to deuteron- and Li-induced reactions, Phys. Rev. C , 044616 (2015).[31] J. Lei and A. M. Moro, Numerical assessment of post-prior equivalence for inclusive breakup reactions, Phys.Rev. C , 061602 (2015).[32] G. Potel, F. M. Nunes, and I. J. Thompson, Establishinga theory for deuteron-induced surrogate reactions, Phys.Rev. C , 034611 (2015).[33] B. V. Carlson, R. Capote, and M. Sin, Inclusive protonemission spectra from deuteron breakup reactions, Few-Body Systems , 307 (2016).[34] G. Potel, G. Perdikakis, B. V. Carlson, M. C. Atkinson,W. Dickhoff, J. E. Escher, M. S. Hussein, J. Lei, W. Li,A. O. Macchiavelli, A. M. Moro, F. Nunes, S. D. Pain,and J. Rotureau, Toward a complete theory for predictinginclu- sive deuteron breakup away from stability, Eur.Phys. J. A , 178 (2017).[35] N. D. Scielzo, J. E. Escher, J. M. Allmond, M. S. Basu-nia, C. W. Beausang, L. A. Bernstein, D. L. Bleuel, J. T.Burke, R. M. Clark, F. S. Dietrich, P. Fallon, J. Gibelin,B. L. Goldblum, S. R. Lesher, M. A. McMahan, E. B.Norman, L. Phair, E. Rodriquez-Vieitez, S. A. Sheets,I. J. Thompson, and M. Wiedeking, Measurement of γ -emission branching ratios for , , gd compound nu-clei: Tests of surrogate nuclear reaction approximationsfor ( n, γ ) cross sections, Phys. Rev. C , 034608 (2010).[36] J. J. Ressler, J. T. Burke, J. E. Escher, C. T. Angell,M. S. Basunia, C. W. Beausang, L. A. Bernstein, D. L.Bleuel, R. J. Casperson, B. L. Goldblum, J. Gostic,R. Hatarik, R. Henderson, R. O. Hughes, J. Munson,L. W. Phair, T. J. Ross, N. D. Scielzo, E. Swanberg, I. J.Thompson, and M. Wiedeking, Surrogate measurementof the P u ( n, f ) cross section, Phys. Rev. C , 054610(2011).[37] R. O. Hughes, C. W. Beausang, T. J. Ross, J. T.Burke, R. J. Casperson, N. Cooper, J. E. Escher,K. Gell, E. Good, P. Humby, M. McCleskey, A. Saasti-moinen, T. D. Tarlow, and I. J. Thompson, Pu( n, f ), Pu( n, f ), and
Pu( n, f ) cross sections deduced from( p, t ), ( p, d ), and ( p, p (cid:48) ) surrogate reactions, Phys. Rev.C , 014304 (2014).[38] S. Ota, J. T. Burke, R. J. Casperson, J. E. Escher, R. O.Hughes, J. J. Ressler, N. D. Scielzo, I. J. Thompson,R. A. E. Austin, B. Abromeit, N. J. Foley, E. McCleskey,M. McCleskey, H. I. Park, A. Saastamoinen, and T. J.Ross, Spin differences in the zr compound nucleus in-duced by ( p, p (cid:48) ) inelastic scattering and ( p, d ) and ( p, t )transfer reactions, Phys. Rev. C , 054603 (2015).[39] R. Perez Sanchez, B. Jurado, V. Meot, O. Roig,M. Dupuis, O. Bouland, D. Denis-Petit, P. Marini,L. Mathieu, I. Tsekhanovich, M. Aiche, L. Audouin,C. Cannes, S. Czajkowski, S. Delpech, A. Gorgen,M. Guttormsen, A. Henriques, G. Kessedjian, K. Nishio,D. Ramos, S. Siem, and F. Zeiser, Neutron-inducedfission and radiative-capture cross sections from decay probabilities obtained with a surrogate reaction (2020),arXiv:2002.05439 [nucl-ex].[40] F. E. Bertrand, G. R. Satchler, D. J. Horen, J. R. Wu,A. D. Bacher, G. T. Emery, W. P. Jones, D. W. Miller,and A. van der Woude, Giant multipole resonances frominelastic scattering of 152-MeV alpha particles, Phys.Rev. C , 1832 (1980).[41] R. Bonetti, L. Colombo, and K.-I. Kubo, Inelastic α -scattering to the continuum: A probe of α -clustering innuclei, Nucl. Phys. A420 , 109 (1984).[42] P. Martin, Y. Gaillard, P. de Saintignon, G. Perrin,J. Chauvin, G. Duhamel, and J. Loiseaux, Excitation oflow-lying levels and giant resonances in zr via 57.5 mevpolarized proton inelastic scattering, Nuclear Physics A , 291 (1979).[43] T. von Egidy and D. Bucurescu, Experimental energy-dependent nuclear spin distributions, Phys. Rev. C ,054310 (2009).[44] G. Boutoux, B. Jurado, V. M´eot, O. Roig, L. Math-ieu, M. A¨ıche, G. Barreau, N. Capellan, I. Compa-nis, S. Czajkowski, K.-H. Schmidt, J. Burke, A. Bail,J. Daugas, T. Faul, P. Morel, N. Pillet, C. Th´eroine,X. Derkx, O. S´erot, I. Mat´ea, and L. Tassan-Got, Study of the surrogate-reaction method applied to neutron-induced capture cross sections, Physics Letters B ,319 (2012).[45] V. Zerkin and B. Pritychenko, The experimental nuclearreaction data (exfor): Extended computer database andweb retrieval system, Nuclear Instruments and Meth-ods in Physics Research Section A: Accelerators, Spec-trometers, Detectors and Associated Equipment , 31(2018).[46] A. C. Larsen, M. Guttormsen, M. Krtiˇcka, E. Bˇet´ak,A. B¨urger, A. G¨orgen, H. T. Nyhus, J. Rekstad,A. Schiller, S. Siem, H. K. Toft, G. M. Tveten, A. V.Voinov, and K. Wikan, Analysis of possible systematic er-rors in the oslo method, Phys. Rev. C , 034315 (2011).[47] S. Hilaire, C. Lagrange, and A. J. Koning, Comparisonsbetween various width fluctuation correction factors forcompound nucleus reactions, Annals of Physics , 209(2003).[48] T. von Egidy and D. Bucurescu, Spin distribution in low-energy nuclear level schemes, Phys. Rev. C78