Neutron width statistics in a realistic resonance-reaction model
NNeutron width statistics in a realistic resonance-reaction model
P. Fanto , G. F. Bertsch , and Y. Alhassid Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, CT 06520 Department of Physics and Institute for Nuclear Theory, Box 351560University of Washington, Seattle, WA 98195 (Dated: July 30, 2018)A recent experiment on s -wave neutron scattering from , , Pt found that the reduced neu-tron width distributions deviate significantly from the expected Porter-Thomas distribution (PTD),and several explanations have been proposed within the statistical model of compound nucleus re-actions. Here, we study the statistics of reduced neutron widths in the reaction n + Pt within amodel that combines the standard statistical model with a realistic treatment of the neutron chan-nel. We find that, if the correct secular energy dependence of the average neutron widths is used,then the reduced neutron width distribution is in excellent agreement with the PTD for a reasonablerange of the neutron-nucleus coupling strength and depth of the neutron channel potential. Withinour parameter range, there can be a near-threshold bound or virtual state of the neutron channelpotential that modifies the energy dependence of the average width from the √ E dependence, com-monly assumed in experimental analysis, in agreement with the proposal of H. A. Weidenm¨uller [1].In these cases, the reduced neutron width distributions extracted using the √ E dependence aresignificantly broader than the PTD. We identify a relatively narrow range of parameters where thiseffect is significant. PACS numbers: 24.60.Dr, 24.60.Ky, 24.30Gd, 24.60.Lz
Introduction .— The statistical model of compoundnucleus (CN) reactions predicts that reduced widthsfor any channel follow the Porter-Thomas distribution(PTD) [2, 3], a χ distribution in ν = 1 degrees of free-dom. Recently, an experiment on s -wave neutron scat-tering from , , Pt found a much broader distribu-tion of the reduced neutron widths [4]. Several explana-tions have been proposed for this deviation from the PTDwithin the statistical model, but none has fully resolvedthe issue.In Ref. [1], it was argued that the secular energy de-pendence of the average neutron widths can deviate fromthe usually assumed √ E form for Pt isotopes because ofa near-threshold bound or virtual state of the neutronchannel potential. The authors of Ref. [4] showed thatusing the modified normalization proposed in Ref. [1][see (4) below] to extract the reduced widths did not im-prove the agreement between their data and the PTD [5].However, their procedure for determining the resonancesmight not hold in the presence of a state very close tothreshold [1, 5], so the possible existence of this state isstill an open question.Other work has attempted to explain the experimentalresults through the non-statistical interactions betweenthe CN states due to coupling to the neutron channel. Ithas been shown that the imaginary non-statistical inter-action can cause deviation from the PTD even for fairlyweak coupling [6, 7]. However, it is not clear how strongthis effect would be in Pt isotopes. In Ref. [8], it was pro-posed that the real shift due to off-shell coupling to theneutron channel perturbs the GOE near threshold. How- ever, it was subsequently proven [9] that in the modelof Ref. [8] the PTD would hold locally in the resonancespectrum. Many-body correlations beyond the statisticalmodel have also been studied [10].However, no study has incorporated all the relevantphysics of the statistical model. Importantly, nearthreshold, the real and imaginary terms have a strong en-ergy dependence that has been neglected in all prior nu-merical and analytical work [6–10]. Moreover, no studyhas used realistic parameters for neutron scattering fromPt isotopes. For these reasons, prior work has not fullysettled the question of whether PTD violation within thestatistical model could occur for this reaction. This prob-lem is of considerable importance because the statisticalmodel is widely used in reaction calculations.Here, we study neutron scattering off Pt within areaction model that combines a realistic treatment of theneutron channel with the usual description of the internalCN states by the Gaussian orthogonal ensemble (GOE) ofrandom-matrix theory [3]. Our model enables us to studyaverage neutron widths, the reduced width distribution,and the elastic and capture cross sections within the sameframework. We start with a baseline physical parameterset for the model taken from the literature. We thenvary the parameter set to produce the conditions underwhich the proposed mechanisms for PTD violation couldbe operative. Finally, we discuss the compatibility ofthese varied parameter sets with the scattering data.Our main conclusion is that, within the reason-ably large parameter range studied, the reduced neu-tron width distribution is in excellent agreement with a r X i v : . [ nu c l - t h ] J un the PTD. Thus, when described realistically, the non-statistical interactions cannot explain the observed devi-ation from the PTD within the parameter range used.Evidence of PTD violation may be observed only if thesecular energy dependence of the average neutron widthis not described correctly. Within our parameter range,there can be a near-threshold bound or virtual state ofthe neutron channel potential. In the presence of sucha state, the energy dependence of the average neutronwidth differs significantly from the √ E dependence [1],and reduced width distributions extracted with the √ E assumption are significantly broader than the PTD. Weidentify measurable signatures of this state’s existence. Hamiltonian and resonance determination .— Ourmodel Hamiltonian matrix H combines a mesh represen-tation of the neutron channel with the GOE descriptionof the internal states. The neutron channel mesh hasspacing ∆ r and radial sites r i = i ∆ r , ( i = 1 , ..., N n ). Thechannel Hamiltonian matrix is H n ,ij = [2 t + V ( r i )] δ ij − tδ i,j +1 − tδ i,j − , where t = (cid:126) / m (∆ r ) and V ( r ) is thechannel potential. The energies of the N c internal statesfollow the middle third of a GOE spectrum with averagespacing D . To each internal energy we add the imagi-nary constant ( − i/ γ to account for resonance decayby gamma-ray emission. The neutron channel couplesto each internal state µ at a single site r e = i e ∆ r withstrength v µ = v (∆ r ) − / s µ , where v is a coupling con-stant and s µ are drawn from a normal distribution withzero average and unit variance. The explicit ∆ r depen-dence of v µ is required to achieve a fixed v in the con-tinuum limit ∆ r →
0. All results shown below werecalculated using (∆ r, N n , N c ) = (0 .
01 fm , , k r that correspondto the neutron resonances by solving the Schr¨odingerequation H (cid:126)u = E(cid:126)u ( (cid:126)u is a column vector with N n + N c components) with the appropriate boundary conditionsfor the neutron wavefunction u ( r ). We impose u (0) = 0for the wavefunction to be regular at the origin. A reso-nance is a pole of the S matrix corresponding asymptot-ically to a purely outgoing wave, i.e. u ( r ) → B ( k ) e ikr forlarge r . For sufficiently large N n , this condition yields u ( N n + 1) = u ( N n ) e ik ∆ r . We obtain the nonlinear eigen-value problem M ( k ) (cid:126)u = (cid:2) H − te ik ∆ r C − E (cid:3) (cid:126)u = 0 (1)where C ij = δ i,j δ i,N n . We solve (1) iteratively to findthe resonances k r , adapting a method from Ref. [11].The resonance energies E r and total widths Γ r are de-termined from (cid:126) k r / m = E r − ( i/ r . The partialneutron widths Γ n,r are then given by Γ n,r = Γ r − Γ γ .Elastic and capture cross sections are calculated from theelastic scattering amplitude, which is determined usingthe boundary conditions of a scattering wave. Furtherdetails and the relevant computer codes are provided inthe Supplementary Material [12]. σ ( b ) σ ( n , γ ) ( b ) E (keV)
FIG. 1. Elastic scattering (top panel) and capture (bot-tom panel) cross sections. Our baseline calculations, aver-aged over 1 keV bins (black circles joined by dashed line), arecompared with cross sections from the JEFF-3.2 library [15],averaged over the same bins (blue squares joined by dashed-dotted line). Error bars indicate standard deviations from 10realizations of the GOE. The red histogram shows experimen-tal average capture cross sections [17].
Application to n + Pt.— We determine a baseline pa-rameter set as follows. We take a Woods-Saxon poten-tial in the neutron channel with parameters V = − . r , a ) = (1 . , .
67) fm [see Eqs. (2-181) and(2-182) of Ref. [13]]. The mean resonance spacing D = 82eV and the total gamma decay width Γ γ = 72 meV aretaken from the RIPL-3 database [14]. We choose a cou-pling strength of v = 11 keV-fm / to reproduce roughlythe RIPL-3 neutron strength function S √ E n = ¯Γ n /D at neutron energy of E n = 8 keV (see Table I).Fig. 1 shows the elastic and capture cross sections forthe baseline model averaged over neutron energy in binsof 1 keV width. We also show elastic and capture crosssections from the JEFF-3.2 library [15], which are basedon the reaction code TALYS [16], averaged over the sameenergy bins. The histogram in the bottom panel of Fig. 1shows experimental energy-averaged capture cross sec-tions [17]. Overall, the agreement with other calculationsand experiment is sufficiently close to take the baselineparameter set as our starting point. Reduced neutron width statistics .— The reduced neu-tron width γ n,r is defined by γ n,r = Γ n,r / ¯Γ n ( E r ) , (2)where ¯Γ n ( E ) is the average width that varies smoothlywith the neutron energy E . Fig. 2 shows the averagewidths calculated for various parameter sets. In eachcase, the data was computed for 100 GOE realizations,from each of which we take as data 160 resonances fromthe middle of our model resonance spectrum. The realparts of these resonance energies fall mostly in the in-terval E = 1 −
14 keV, which covers the bulk of the ex-perimental range of Ref. [4]. For the baseline model, thehistogram compares well with the √ E dependence. Theprobability density of the neutron scattering wavefunc-tion [18] at the interaction point, u E ( r e ), is also shown inFig. 2 and, for the baseline model, is hardly distinguish-able from the √ E curve, in agreement with the statisticalmodel prediction [1, 19]. Γ n ( e V ) FIG. 2. Comparison of ¯Γ n ( E ) calculated for the differentmodels of Table I (histograms) with √ E (solid blue lines), theneutron probability density u E ( r e ) (red dashed lines), and theformula in Eq. (4) [1] (green dashed-dotted lines). Functionsare normalized to match the model calculations at E = 8 keV. P ( y ) y -10 -6 -2 2B y FIG. 3. The histograms describe the distributions of thelogarithm of the normalized reduced width for the baselinemodel. The reduced widths are calculated from Eq. (2). Re-duction A (left panel) uses ¯Γ n ( E ) from the model, while re-duction B (right panel) uses ¯Γ n ( E ) ∝ √ E . The solid lines arethe PTD. Next, we determine the reduced widths and comparetheir distribution with the PTD. We consider the dis-tributions extracted using the average widths calculatedfrom the model, which we call reduction A, as well asthose extracted using the ¯Γ n ( E ) ∝ √ E ansatz, which we call reduction B. Fig. 3 shows as histograms the calcu-lated probability distributions of the logarithm y = ln x of the normalized reduced widths x = γ n / (cid:104) γ n (cid:105) for thebaseline model. For both reductions A and B, we findexcellent agreement with the PTD for y P ( y ) = x P PT ( x ) = (cid:114) x π e − x/ . (3)For a quantitative comparison, we compute the reducedchi-squared value χ r , using χ r ≈ χ r ≈ Model baseline M2 M3 M4 M5 M6 V (MeV) -44.54 -41.15 v (keV-fm / ) 11.0 5.5 22.0 1.6 0.8 3.2 S · (eV − / ) 2.0 0.5 5.4 2.0 0.5 8.2¯ σ el (b) 30. 19.0 23. 279. 288. 249.¯ σ γ (b) 0.44 0.32 0.50 0.47 0.39 0.53 χ r PTD A 0.9 1.0 1.1 0.9 1.0 1.4 χ r PTD B 1.0 1.0 1.3 5.8 6.0 6.1 ν fit A 1.0 1.0 0.98 1.0 1.0 0.98 χ r fit A 0.9 1.0 1.0 0.9 1.0 1.3 ν fit B 1.0 1.0 0.97 0.92 0.92 0.92 χ r fit B 1.0 1.1 1.1 3.4 3.8 3.7TABLE I. Calculated resonance properties of the n + Pt re-action for various parameter sets. The neutron strength func-tion parameter S = (¯Γ n /D ) / √ E and average elastic scatter-ing cross section ¯ σ el are evaluated at E = 8 keV. The RIPL-3strength function parameter is 2 · − eV − / [14]. The cap-ture cross section ¯ σ γ is the average over the interval 5-7.5 keVcorresponding to the measured value of 0.6 b [17]. ReductionsA and B are as described in the caption to Fig. 3. The rowlabeled χ r PTD contains the chi-squared results comparingthe reduced width distributions to the PTD. The values ν fit and χ r fit refer to the maximum-likelihood fit to Eq. (5). Parameter variation .— Here we vary the parameters v and V to investigate proposed explanations for PTDviolation. First, we vary the coupling strength v by afactor of two smaller or larger than the baseline value,keeping V fixed at its baseline value. These sets are la-beled, respectively, by M2 and M3 in Table I. As shownin Table I, the average elastic scattering cross section at E = 8 keV varies only in the range 19–30 b, and the aver-age capture cross section in the interval 5–7.5 keV variesby a similar fractional amount. The reduced width dis-tributions from reductions A and B are nearly identicalto the corresponding baseline distributions in Fig. 3. The χ r values for the PTD are all close to 1, indicating goodagreement with the PTD. In the strong coupling case M3,the average width shown in Fig. 2 deviates somewhatfrom the expected √ E dependence. This is a numericaleffect due to the finite bandwidth of internal states [12].Next, we vary V to investigate the effect of a near-threshold bound or virtual state in the neutron channel.With our baseline potential, there is a bound 4 s neu-tron level at energy ≈ − . V to − .
15 MeV results in a weakly bound state with energy E ≈ − V is sufficiently moderateto justify its inclusion in our parameter set [22]. We ad-just v in model M4 to reproduce the RIPL-3 strengthfunction parameter S and vary v by a factor of twosmaller or larger for models M5 and M6, respectively.The average capture cross sections for models M4–M6,shown in Table I, are only slightly larger than those ofthe baseline model. However, the elastic cross sectionsare much larger than the baseline values. Thus, experi-mental elastic cross sections could be used to narrow theparameter values of our model. Unfortunately, we knowof no published experimental elastic cross sections for thisreaction.As shown in Fig. 2, the average neutron widths formodels M4–M6 have an energy dependence that differssignificantly from √ E . However, the quantity u E ( r e ) re-mains an excellent estimator of the correct energy depen-dence of the average widths. An analytic expression wasderived in Ref. [1] for a near-threshold bound or virtualstate with energy E ( E < u E ( r e ) ∝ √ EE + | E | . (4)Using E ≈ − u E ( r e ) and the averagewidths (see Fig. 2).The reduced width distributions for model M4 areshown in the upper panels of Fig. 4 (similar results areobtained for models M5 and M6). The distributions ex-tracted with the calculated ¯Γ n ( E ) (reduction A) are welldescribed by the PTD, as is confirmed by the χ r valuesin Table I. In contrast, the distributions obtained usingthe √ E dependence (reduction B) are noticeably broaderthan the PTD, and the χ r values for this reduction aresignificantly larger than 1.As we make the neutron potential slightly less attrac-tive, the weakly bound state becomes a virtual statewhose energy E is also negative but on the second Rie-mann sheet [23]. For example, when V = − .
85 MeV,we have a virtual state with E ≈ − √ E occurs for E = 0. We then expect to see the maximaldeviation from a PTD in reduction B. In our model, thisoccurs for V = −
41 MeV. The reduced width distribu-tions for this case are shown in the lower panels of Fig. 4.For reduction B, we observe an even stronger deviationfrom the PTD, as expected.Finally, for all the parameter sets considered, we madea maximum-likelihood fit of the calculated distributionsto a χ distribution in ν degrees of freedom P ( x | ν ) = ν ( νx ) ν/ − ν/ Γ( ν/ e − νx/ . (5) P ( y ) B 0 0.1 0.2-10 -6 -2 2A P ( y ) y -10 -6 -2 2B y FIG. 4. As in Fig. 3 but for model M4 (top panels), andfor the model with E ≈ χ distributionsin ν = ν fit degrees of freedom are shown by the dashed graylines. See text for details. More specifically, we find the value ν fit that maximizesthe likelihood function L ( ν ) = (cid:81) i P ( x i | ν ), where x i arethe reduced width data values. The PTD is recovered for ν = 1. As shown in Table I, for reduction A, all modelsreproduce the PTD. Moreover, for reduction B, modelsM1-M3 also reproduced the PTD. However, for modelsM4-M6 and for reduction B, we obtain ν fit = 0 .
92 for allcases, and the χ r values are significantly larger than 1. Conclusion .— We have studied the statistics of neu-tron resonance widths in the n + Pt reaction within amodel that combines a realistic treatment of the neutronchannel with the GOE description of the internal states.Our model is the first to incorporate all aspects of thestatistical model for a single-channel reaction. Our mainconclusion is that the PTD describes well the distribu-tion of reduced neutron widths (2) for a reasonably largeparameter range around baseline values taken from theliterature. Our results indicate that non-statistical inter-actions do not explain the experimentally observed PTDviolation. These interactions may be more important inother systems, where the coupling between the channelsand the internal states is stronger.Apparent PTD violation may be observed only if thesecular energy dependence of the average neutron widthis not described correctly. Within our parameter range,this can happen in the presence of a near-threshold boundor virtual state of the neutron channel potential. In thiscase, the energy dependence of the average width differssignificantly from √ E , and the distributions of reducedwidths extracted with the usual √ E ansatz are broaderthan the PTD. However, significant deviations from the √ E behavior require that the magnitude | E | of the en-ergy of this near-threshold state be no more than a fewkeV for , , Pt. Moreover, as stated above, the au-thors of Ref. [4] showed that using the form (4) did notimprove their data’s agreement with the PTD [5]. How-ever, a state so close to threshold might undermine theexperimental resonance determination procedure [1, 5].We have found that the magnitude and shape of theelastic neutron cross section are strongly affected by anear-threshold state in the neutron channel potential (seeTable I). Therefore, experimental measurements of theelastic cross section would be useful in determining thepossible existence of such a near-threshold state.
Acknowledgments .— This work was supported in partby the U.S. DOE grant Nos. DE-FG02-00ER411132 andDE-FG02-91ER40608, and by the DOE NNSA Stew-ardship Science Graduate Fellowship under cooperativeagreement No. de-na0002135. We would like to thankH. A. Weidenm¨uller for useful discussions. PF and YAacknowledge the hospitality of the Institute for NuclearTheory at the University of Washington, where part ofthis work was completed during the program INT-17-1a, “Toward Predictive Theories of Nuclear ReactionsAcross the Isotopic Chart.” This work was supported bythe HPC facilities operated by, and the staff of, the YaleCenter for Research Computing. [1] H. A. Weidenm¨uller, Phys. Rev. Lett. , 232501(2010).[2] R.G. Thomas and C.E. Porter, Phys. Rev. , 483(1956).[3] G. E. Mitchell, A. Richter, and H. A. Weidenm¨uller, Rev.Mod. Phys. , 2845 (2010).[4] P. E. Koehler, F. Beˇcv´aˇr, M. Krtiˇcka, J. A. Harvey, andK. H. Guber, Phys. Rev. Lett. , 072502 (2010).[5] P. E. Koehler, F. Beˇcv´aˇr, M. Krtiˇcka, J. A. Harvey, andK. H. Guber, arXiv:1101.4533 (2011).[6] G. L. Celardo, N. Auerbach, F. M. Izrailev, and V. G.Zelevinsky, Phys. Rev. Lett. , 042501 (2011).[7] Y. V. Fyodorov and D. V. Savin, Euro. Phys. Lett. ,40006 (2015).[8] A. Volya, H. A. Weidenm¨uller, and V. Zelevinsky, Phys. Rev. Lett. , 052501 (2015).[9] E. Bogomolny, Phys. Rev. Lett. , 022501 (2017).[10] A. Volya, Phys. Rev. C , 044312 (2011).[11] D.A. Bykov and L.L. Doskolovich, J. Lightwave Techno. , 793 (2013).[12] See the Supplementary Material.[13] A. Bohr and B. R. Mottelson, Nuclear Structure
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