EExit-Channel Suppression in Statistical Reaction Theory
G.F. Bertsch and T. Kawano Department of Physics and Institute of Nuclear Theory,University of Washington, Seattle, Washington 98915, USA Theoretical Division, Los Alamos National Laboratory,Los Alamos, New Mexico 87545, USA
Abstract
Statistical reaction theories such as Hauser-Feshbach assume that branching ratios follow Bohr’scompound nucleus hypothesis by factorizing into independent probabilities for different channels.Corrections to the factorization hypothesis are known in both nuclear theory and quantumtransport theory, particularly an enhanced memory of the entrance channel. We apply theGaussian orthogonal ensemble to study a complementary suppression of exit channel branchingratios. The combined effect of the width fluctuation and the limitation on the transmissioncoefficient can provide a lower bound on the number of exit channels. The bound is demonstratedfor the branching ratio in neutron-induced reactions on a
U target. a r X i v : . [ nu c l - t h ] N ov ntroduction. Statistical approximations are extremely useful in nuclear physics, particu-larly in reaction theory. Examples are the Hauser-Feshbach and Weisskopf-Ewing formulasfor reaction cross sections [1–3]. The underlying assumption of both is the factorizabilityof the cross section σ ab from one channel to another as σ ab ∼ Γ a Γ b , where Γ i is the averagedecay width through a channel. The factorization follows from Bohr’s compound nucleushypothesis [4], that the decay of a heavy nucleus has no memory of how it was formed. How-ever, factorization is only justified when the reaction takes place through discrete resonancesand there are many channels contributing to each decay mode. Otherwise, the fluctuationsin the widths of the resonance gives rise to the well-known “width fluctuation correction”(WFC) to the statistical models [5], most prominently as the “elastic enhancement factor.”There is now an extensive literature on the subject cited in Refs. [6] and [7]. Similar effectsin electron propagation through mesoscopic conductors are known as the “weak localizationcorrection” and the “dephasing” effect [8, Sect. IV.C and IV.E].While the nuclear correction is best known as an entrance channel effect, it can also bepresent in exit channels if the reaction branching ratios highly favor a decay mode withlarge fluctuations [9]. In this work we show that such situations can lead to effects largeenough to provide bounds on the number of channels in the decay mode, even though themeasurements are on averaged quantities and not on their fluctuations. This study wasmotivated in part by the quest for a theory of fission dynamics based on nucleon-nucleoninteractions. That requires an understanding not only of the distribution of the fissionchannels but their coupling matrix elements to the other states. The GOE statistical model.
The factorization hypothesis and other statistical aspects ofreaction theory can be tested theoretically by models that consider ensembles of Hamilto-nians that mix the constituent configurations. The Gaussian orthogonal ensemble (GOE)has been especially successful in this regard [6, 7]. The reaction theory is expressed in thematrix equations K = πγ T E − H γ (1) S = 1 − iK iK (2)giving for the non-elastic cross sections [10] σ nf = πk n (cid:88) c ∈ f | S nc | . (3)2n Eq. (1) K is a matrix of dimension N ch × N ch , where N ch is the number of reactionchannels in the model. H is the N µ × N µ Hamiltonian matrix for the N µ internal states inthe model. The internal states are connected to the channels by the N ch × N µ reduced-widthmatrix γ . Eq. (2) relates the K -matrix to the familiar S -matrix of scattering theory. Thereis an additional overall phase factor in Eq. (2) which plays no role in the reaction crosssections. In Eq. (3) n is the entrance channel, f is a set of exit channels that are groupedtogether in an experimental cross section, and c are the individual channels. The crosssection depends explicitly on the entrance channel energy E n via the neutron wave-number k n = √ E n M n , with M n the reduced mass.In the GOE statistical model, the Hamiltonian H is sampled from the distribution [11] H µ,µ (cid:48) = H µ (cid:48) ,µ = v µ,µ (cid:48) (1 + δ µ,µ (cid:48) ) / , (4)where µ ≥ µ (cid:48) and v µ,µ (cid:48) is a Gaussian-distributed random variable. The ensemble is com-pletely specified by N µ and the r.m.s. Hamiltonian matrix element (cid:104) v (cid:105) / . Here we shallcharacterize the GOE ensemble by D , the average level spacing in the middle of the distri-bution. The spacing is related to the matrix elements by D = π (cid:104) v (cid:105) / N − / µ . The γ matrixassociated with a GOE Hamiltonian can be assumed to have a diagonal structure of theform γ | µ,c = γ c δ µ,c . In this work, we also make the simplifying assumption that the γ c areequal for all channels within a given decay mode f . When the matrix is transformed to thebasis diagonalizing the GOE Hamiltonian, the amplitudes will be distributed over eigen-states according the Porter-Thomas distribution [12] with a number of degrees of freedomequal to the number of channels N f . It will be convenient to define an effective K -matrixdecay rate for the different modes Γ Kf asΓ Kf = 2 πN µ (cid:88) c ∈ f γ c . (5)If all the Γ Kf are small compared to D , the average S -matrix decay widths satisfyΓ f ≈ Γ Kf . (6)Note also that N chn = 1 for the entrance channel; its reduced width controls the total reactioncross section [13].The calculations reported below were carried out with codes that constructed the GOEdistribution by Monte Carlo sampling of H and applying Eqs. (1-3). The codes and inputdata are provided in the Supplementary Material [14].3 pplication to the branching ratio U(n,f )/
U(n, γ ). Here we show by a physical ex-ample that cross-section branching ratios that heavily favor some particular exit channel canbe severely suppressed. The behavior follows from the GOE statistical model as formulatedin the last section and is thus universal. Our example is the neutron-induced reactions on
U. For neutron energies below ∼
10 keV the predominant reactions are in the s -waveleading to capture by gamma emission or fission. An important quantity is α − , the ratioof the fission cross section σ F to capture cross section σ cap , α − = σ F /σ cap . It varies in therange α − ∼ ∼ . D and Γ cap which will be used to determine the parametersof the K -matrix [16] We will examine the cross section at E n = 10 keV; the experimental TABLE I: Experimental observables for neutron-induced reactions on
U. The cross section datais at a neutron bombarding energy E n = 10 keV. The last two entry are the ratio of cross sections,show the range of the ratios as well as the value at 10 keV.Observable Value Source D . ± .
05 eV [17, 23]Γ cap ± σ cap . ± .
07 b [18, 19] σ F . ± .
06 b [18, 19] α − (10) 2 . ± . α − (1 −
15) 2-3 . values averaged over a 1 keV are also given in Table I.The K -matrix reduced-width parameters are determined as follows. The capture widthis small compared to D and many channels contribute so we can safely apply Eq. (6); theequivalent Γ cap is shown in Table II. The coupling to the entrance channel depends on E n and is usually parameterized by the strength function S as (cid:104) Γ n (cid:105) D = S E / . (7)4rom total cross sections one finds S ≈ ± . × − eV − / [22, 23],[24, Fig. 47] and weuse that value to determine the entry in Table II. We note that this value is consistent withthe coupled-channel analysis of Ref. [25]. For the fission reduced width, we first make thefactorization (Hauser-Feshbach) approximation and assume that the nominal decay widthsscale with the cross sections, i.e. Γ KF / Γ Kcap = σ F /σ cap . TABLE II: K -matrix parameters (eV) describing observed cross sections at E n = 10 keV inHauser-Feshbach theory and assuming Γ << D . We have also include the parameters from theENDF/B-VII.1 evaluation. Γ Kcap Γ KF Γ Kn this work 0 . ± . ± .
01 0 . ± . .
039 0 .
289 0 . The number of channels and states in the K -matrix are still to be specified. As presented,the model is independent of the number of states as long as that number is large. We shalltake N µ = 50 − N chcap >>
1; we take N chcap = 10 in our modeling. The number of fissionchannels is not well known [26] and we consider two possibilities: model A with one fissionchannel and model B with five fission channels.With all parameters now specified in the GOE K -matrix, we can compute the averagecross sections and branching ratios. These are shown in Table III. In model A, one sees TABLE III: Average reaction cross sections at E n = 10 keV, comparing models A and B withexperiment. The uncertainties on the calculated values are the r.m.s. sample-to-sample fluctuationsassociated with the random matrix ensemble of the internal states, taking a 1 keV averaginginterval. We have also included in the table the impact on the elastic scattering S -matrix. N chF σ F (b) σ cap (b) α − | S nn | Exp. 2 . ± .
21 1 . ± .
07 2.8A 1 1 . ± .
05 1 . ± .
05 1 . ± .
07 0.954B 5 2 . ± .
10 1 . ± .
06 2 . ± .
11 0.950 ABLE IV: Average fission widths Γ KF required to reproduce the observed cross-section ratio α − = 2 . ± . α − is propagated through the models to give the uncertainty bars in the table. Units are eV.Model N chcap N chF HW HW/WFC KtoSA 10 1 0 . ± .
008 0 . ± .
06 noneB 10 5 0 . ± .
008 0 . ± .
01 0 . ± .
100 200 300 400 500 Γ KF (meV) α − FIG. 1: Cross section ratio α − = (cid:104) σ F (cid:105) / (cid:104) σ cap (cid:105) as a function of average fission width Γ KF assuminga single fission channel. Solid line: Eq. (1-3). Dotted line: Hauser-Feshbach approximation, i.e. α − = Γ KF / Γ Kcap . Dashed line: Hauser-Feshbach including the WFC correction, Eq. (8). Blue band:experimental range, taking uncertainty from Table I. Widths are the statistical errors associatedwith the 1 keV cross-section averaging interval. an enhancement of the capture cross section and a corresponding suppression of the fissioncross section. Clearly factorization is violated.Let us see if we can reproduce the experimental branching ratio simply by increasing thefission width, but keeping only a single channel. Taking the width as a free parameter, weobtain the branching ratios shown as the solid line in Fig. 1. Also, we show in Table IV thevalues of Γ KF required to fit the observed branching ratio. Model A saturates at α − ∼ . KF that can reproduce experiment. Thus, we can exclude6ssion models having only a single channel, based solely on average cross section data. Ofcourse, the fluctuation in cross-section ratios also carries information on the number ofchannels and is the basis of previous estimates that the effect channel count is of the orderof a few. Finally, one can see from the second line of Table IV that model B can fit the datataking the decay with close to the Hauser-Feshbach value. Discussion.
The exit channel suppression comes about by two mechanisms that can beunderstood as follows. The part coming from Porter-Thomas fluctuations can be analyzedat the level of the K -matrix: assuming isolated resonances, the branching ratio can becalculated as in Ref. [22], α − = (cid:42) Γ KF (cid:80) f Γ Kf (cid:43) (cid:44)(cid:42) Γ Kcap (cid:80) f Γ Kf (cid:43) . (8)The results are shown as the dashed line in Fig. 1. For Γ KF = 0 .
105 eV, Eq. (8) gives a WFCfactor of 0.43, close to that of the full S -matrix treatment. However, to explain the observed α − , we have to go to much larger fission width, Γ KF ≈ .
46 eV, as may be seen in TableIV. At that width the WFC factor is 0.23 in the HF/WFC treatment and 0.16 in the full S -matrix treatment. Increasing Γ KF further does not raise the S -matrix value significantly.We attribute the additional suppression in the S -matrix treatment to the constraint onstatistical decay rates W f imposed by the Bohr-Wheeler formula [27] W f = Γ f = 12 π DT, (9)where T is the transmission coefficient of the channel. General considerations of detailedbalance require T ≤
1. The nominal fission width in the K -matrix reduced width is close tothe bound, so it is not unexpected that there is a further suppression in the S -matrix. Conclusion.
We have demonstrated that branching ratios can be a useful observable inthe study of fission dynamics near threshold. Namely, effects not included in the Hauser-Theory can severely constrain the number of exit channels. In the example presented here,the energy of the fissioning nucleus is above the fission barrier. It might be of interest toapply the analysis to below-barrier fission as well [28]. There one sees sharp peaks in thefission cross section, ascribed to individual states along the fission path. These states act asfission channels with N chF = 1 in the K -matrix modeling.We confirmed the generality of our conclusion by exploring a variety of channel numberand transmission combinations. Several examples are provided in Supplemental Material.7he repository also contains the main code implementing Eq. (1-4) and the script to computethe branching ratio and its uncertainty. Acknowledgments.
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