CConfiguration-interaction approach to nuclear fission
G.F. Bertsch and K. Hagino Department of Physics and Institute of Nuclear Theory, Box 351560,University of Washington, Seattle, Washington 98915, USA Department of Physics, Kyoto University, Kyoto 606-8502, Japan
We propose a configuration-interaction (CI) representation to calculate induced nuclear fissionwith explicit inclusion of nucleon-nucleon interactions in the Hamiltonian. The framework is de-signed for easy modeling of schematic interactions but still permits a straightforward extension torealistic ones. As a first application, the model is applied to branching ratios between fission andcapture in the decay modes of excited fissile nuclei. The ratios are compared with the Bohr-Wheelertransition-state theory to explore its domain of validity. The Bohr-Wheeler theory assumes that therates are insensitive to the final-state scission dynamics; the insensitivity is rather easily achievedin the CI parameterizations. The CI modeling is also capable of reproducing the branching ratiosof the transition-state hypothesis which is one of the key ingredients in the present-day theory ofinduced fission.
Introduction.
The theory of induced fission is one ofthe most challenging subjects in many-fermion quantumdynamics. In a recent review [1] of future directions infission theory, the authors omitted the topic “becausethere has been virtually no coherent microscopic theoryaddressing this question up to now.”In this Letter we propose a microscopic approachbased on many-body Hamiltonians in the configuration-interaction (CI) framework. The idea is not new [2],but the methodology has yet to be applied in practice .Before realistic calculations can be contemplated, it isuseful to consider simplified models in the many-particleframework that are extensible to the realistic domain [5–7]. Such models may validate the phenomenological ap-proaches that have been with us since the beginnings offission theory, or it may suggest modifications to them.The focus here is on how the system crosses the fissionbarrier; key observables are the excitation function forfission cross sections and the branching ratio betweenfission and other decay channels. The model proposedbelow incorporates microscopic mechanisms to propagatethe systems from the initial ground-state shape to a re-gion beyond the fission barriers(s). Hamiltonian.
We build the Hamiltonian on a set ofreference states Q , each such state generating a spectrumof quasiparticle excitations which we call a Q -block. The Q -blocks are ordered by deformation Q . The Hamilto-nian is constructed from these elements asˆ H = X Q E gs ( Q ) + ˆ H qp ( Q ) + ˆ H v ( Q ) + f od X Q = Q ± ˆ H od v . (1)Here E gs is the energy of the reference state, calculatedby constrained Hartree-Fock or density functional the- There has been earlier work calculating the dynamics by a dif-fusion equation with a microscopic treatment of the diffusioncoefficient [3, 4]. ory (DFT). In the model below we choose appropriatesets of energies E gs ( Q ) to explore various limits of thetheory. The circumflexes denote terms containing Fock-space operators acting within a Q -block or between or-bitals in adjacent Q -blocks. We detail them below. Constructing the configurations.
The configurationspace is built in the usual way, defining configurations asSlater determinants of nucleon orbitals. The orbitals areenvisioned as eigenstates of an axially deformed single-particle potential. Ultimately their properties would bedetermined by the density functional theory, but for mod-eling purposes we found it convenient to assume a uni-form spectrum of orbital energies with the same spacing d for protons and neutrons. The ladder of orbital statesextends infinitely in both directions above and below theFermi surface. The operator for the quasiparticle excita-tion energy E qp is given byˆ H qp = d X α : n α > n α ˆ a † α ˆ a α + d X α : n α ≤ n α ˆ a α ˆ a † α . (2)The label α includes all quantum numbers associatedwith the orbital, α = ( Q, n, K, t ). Here n indexes theorbital position in the ladder, with n = 0 correspondingto the Fermi level, and K is its angular momentum aboutthe symmetry axis. To keep the model as transparent aspossible, we restrict K to ± /
2. The label t distinguishesneutrons (n) and protons (p).The orbital excitation energies of many-particle config-urations are integral multiples of d , E qp = E k = k d . Asa function of k , the multiplicity of configurations hav-ing P K = 0 is N k = (1 , , , , , , , ... ) for k = (0 , , , , , , , · · · ). The spectrum up to k = 11 isshown in Fig. 1. Its functional form agrees well with theleading behavior of the Fermi-gas level density formula [9]as described in the figure caption. The histogram shows We note that theory based on Hamiltonian interactions togetherwith orbitals from DFT has been successfully applied elsewhere[8]. a r X i v : . [ nu c l - t h ] F e b k = E k / d N k FIG. 1: Spectrum of many-body configurations in the uni-form model. Here, the total K quantum number of the systemis restricted to K = 0. N k denotes the number of configura-tions at the excitation energy E ∗ = kd . The filled circles andthe histogram show the non-interacting and the interactingspectra, respectively. The dotted red curve shows a fit to thefunctional form log( N k ) ∼ √ k . …… …… …… …… …… FIG. 2: The vertical towers of levels are the Q -blocks in thebasis of configurations. They are ordered by deformation Q . E is the incident energy for a fission reaction. the level density after including ˆ H v in the Hamiltonianof the Q -block.The parameter d will be left unspecified below. It canvary greatly due to shell effects, but in the actinide nu-clei it is in the range 0 . − . k ≈ −
16 in excitation energy and somewhat higher k ≈
17 in level density. What we have described here is asingle-reference basis of configurations. Fission requireslarge-amplitude shape changes, which cannot be reason-ably treated in a single-reference basis. As a minimum,one needs to extend the space by including as referencestates the local DFT minima across the saddle point ofthe barrier [10]. In fact there are many such minimaalong typical fission paths [11]. In our model we organizethe reference configurations as a chain along a path of increasing deformations, as depicted in Fig. 2. One com-plication at this point is that the resulting basis may notbe orthogonal. We shall come back to this point later.
Model nucleon-nucleon Hamiltonians.
In the quasipar-ticle representation, there are three kinds of two-particleinteraction. The interactions that are diagonal in quasi-particle occupations factors are taken into account in E gs ,the ground state energy of the reference configuration.The interactions changing the orbital of one of the nucle-ons do not contribute in the reference configuration if itis a stationary state of the DFT; otherwise it induces adiabatic transformation of the configuration. Such trans-formations are generally unfavored on energetic grounds[12] and they are omitted in the present model. We areleft with the interactions that change the orbitals of bothparticles. In this Letter we deal only with the neutron-proton interaction. The pairing interaction between iden-tical particles is certainly important as well; in fact, it islikely to be more important in non-diabatic collective dy-namics [7, 14]. However, it is also important to assess theeffects of the residual neutron-proton interaction [3], andthis has not be done until now.We write the interaction Hamiltonian asˆ H v = v np X r ˆ a † α ˆ a † α ˆ a α ˆ a α . (3)where the parameter v np is the strength of the interactionand r is a random variable from a Gaussian ensemble ofunit variance. The summation is over the four α indicesrestricted to a fixed Q in ˆ H v and to neighboring Q -blocks[3–7, 10] in ˆ H od v . Also, the sum is restricted to α sets sat-isfying K + K = K + K . The assumption that theneutron-proton interaction is Gaussian distributed is cer-tainly not justified for the low-energy states in a Q -blockwhere collective excitations can be built up. However,high in the spectrum where only the overall interactionstrength is important the mixing approaches the randommatrix limit. The strength can be determined by sam-pling with more realistic interactions that could range insophistication from simple contact interactions or separa-ble interactions to those used in present-day shell modelHamiltonians. The strength of neutron-proton contactinteractions for shell model Hamiltonians is typically inthe range 250 −
500 MeV-fm [3, 13]. The correspond-ing strength in actinides [33] for our parameterization is0 . d − . d ; we take v np = 0 . d in most of the examplesbelow.The configurations may be characterized by the num-ber of quasiparticles as well as by the energy index k .Each k = 0 subblock contains configurations going fromtwo quasiparticles to the maximum energetically allowed.The subblocks are all connected by the residual interac-tion, although the matrices connecting them are sparse.For example, the N × N matrix has an off-diagonal fill-ing of 5%, while the N × k = 6subblock to the ground state is 27% filled.In the presence of the residual interaction, the eigen-states of a shell-model configuration space are foundto approach the random matrix limit of the GaussianOrthogonal Ensemble (GOE) when the rms interactionstrength is larger than the level spacing between configu-rations [15, 16]. For reaction theory, the most importantGOE characteristic is the Porter-Thomas distribution ofdecay widths, requiring a nearly Gaussian distribution ofconfiguration amplitudes in the eigenstates. The eigen-states of large-dimension k -subblocks do in fact acquirethe properties of the GOE, even though the sparseness ofthe interaction matrix works against a complete mixingof the configurations [33]. More realistic model that donot permit the k grouping will still approach the GOE athigh excitation energy. However, it should be mentionedthat a numerical study [17, 18] of a light-nucleus spec-trum did not confirm the above stated criterion for GOEbehavior.The interaction between Q -blocks is responsible forshape changes [3] and is thus crucial to the modeling.It is clear that the interaction is somewhat suppresseddue to the imperfect overlap of orbitals built on differ-ent mean-field reference states. Another complication isthat the configurations in different Q -blocks will not beorthogonal unless special measures are taken, e.g., re-striction by K -partitioning [19]. These problems havelong been dealt with in other areas of physics [20–22]and can be treated in nuclear physics in the same way.For our model, we simply parameterize the effects by theattenuation factor f od in Eq. (1). Reaction theory.
Induced fission is in the domain ofreaction theory: an external probe, typically a neutron,excites the nucleus leading to its decay by fission. Anumber of reaction-theoretic formalisms are available fortreating CI Hamiltonians. We mention in particular the K -matrix formalism [5, 25–28] and the S -matrix formal-ism [24, 25]. The key quantity is the transmission coeffi-cient from an incoming channel to the decay channels ofinterest, T in ,C = X j ∈ C | S in ,j | . (4)Here C is the set of quantum mechanical channels as-sociated with the type of reaction. For neutron-inducedreactions on heavy nuclei, it could be inelastic scattering,capture, or fission. The relevant S -matrix quantities maybe calculated as [28, 29] | S j,j | = X µ,µ Γ j,µ | ( ˜ H − E ) − | µ,µ Γ j ,µ . (5)Here Γ i,µ is the decay width of the state µ into the chan-nel i . Note that the CI Hamiltonian H is modified byincluding of the coupling to the channels. We assume Early studies also made use of the R -matrix theory [23–25]. How-ever, it requires unphysical boundary conditions that are difficultto implement. od B CN limit
FIG. 3: The branching ratio as a function of the attenuationfactor f od in the interaction between Q -blocks. The modelspace is ( k = 0 − × E gs = (0 . , . , . v np = 0 . d ,Γ cap = 0 . d , and Γ fis = 0 . d . The energy average in Eq.(7) is taken in the range of 3 . d ≤ E ≤ . d . The dashed linedenotes the branching ratio in the compound nucleus limit, B cap , fis = Γ fis / Γ cap = 3 . in the model that each channel couples to a single inter-nal configuration, and we neglect dispersive effects. Themodified Hamiltonian ˜ H then reads,˜ H µ,µ = H µ,µ − iδ µ,µ X j Γ j,µ / . (6)The main observable we are interested in is the branchingratio between fission and capture. We define it as B cap , fis = R dE T in , fis R dE T in , cap . (7)The range of integration is the same for numerator anddenominator and in practice would be determined byexperimental considerations. For simplicity, we assumethat the entrance channel width is small compared tothe decay widths, in which case it cancels out of Eq. (7).For a typical example the experimental quantities areΓ cap ≈ . B cap , fis ≈ Results.
We can now set the parameters to simulatethe branching between capture and fission processes. Tothis end, we consider chains of three or more Q -blocks;the first represents the spectrum built on the ground stateand the last has the doorway state to fission channels.Imaginary energies − i Γ cap / − i Γ fis / H to account for the decay widths [5].As a warm-up, we find conditions on f od that justifythe compound-nucleus (CN) hypothesis that the relativedecay rates are proportional to the decay widths in ˜ H .The model has three identical Q -blocks composed of k ≤ f od &
1, thebranching ratio is consistent with the formula B cap , fis = Γ fis B FIG. 4: Sensitivity of the branching ratio to the exitchannel widths. The blue open circles are for the modelspace of ( k = 0 − × E gs = (0 , d, k = 0 − × E gs = (0 . , . , . , . , . v np = 0 . d , f od = 1 .
0, and Γ cap = 0 . d . The incomingchannel is assumed to be one of the configurations at E ∗ = 6 d in the first block, and the energy average in Eq. (7) is takenin the range of 5 . d ≤ E ≤ . d . Γ fis / Γ cap , confirming the CN limit. Even with f od = 0 . BW = 12 πρ X i T i . (8)Here i are states on the barrier top, T i are transmissioncoefficients across the barrier, and ρ is the level density ofthe compound nucleus (i.e., the first Q -block) at the givenexcitation energy. Notice that the BW formula does notdepend on the fission widths Γ fis , unlike Eq. (5).First consider the model space of 3 identical Q -blockscomposed of k ≤ E gs = (0 , d,
0) tomake a barrier at the middle block. As may be seen inFig. 4, this model fails the first assumption: the derivedbranching ratio is sensitive to the fission decay width,Γ fis . The reason is that there are many virtual transitionspossible through the higher levels in the barrier-top Q -block. Because the effective number of partially openchannels is large, the communication between the end Q -blocks remains strong.We found two ways to greatly diminish the dependenceon Γ fis in our model. The first way is to increase thechain of Q -blocks on the barrier. Then the path acrossthe barrier requires multiple virtual transitions, resultingin a much stronger suppression factor. This may be seenin Fig. 4 for the 5-block case.The other way is to eliminate the virtual transitionsat the barrier by cutting off the spectrum of the middleblock. Fig. 5 shows the results with 3 Q-blocks. Thefirst and the last blocks are defined as usual in the N k
10 100 1000 N k B uniform model ( Γ fis = 0.3)uniform model ( Γ fis = 0.1)Bohr-Wheeler FIG. 5: Branching ratios in ( k ) / (0) / ( k ) configuration spacesin which the barrier Q -block is a single state degenerate withthe other Q -blocks. Other parameters are v np = 0 . d, f od =1 and Γ cap = 0 . d . The filled circles and the open circles areobtained with Γ fis = 0 . d and 0 . d , respectively. The uncer-tainties are estimated with 100 different random number setsin the Hamiltonian matrix. The filled squares shows the pre-dicted values from a schematic transition-state formula (seetext for details). space. The middle block has only a k = 0 configuration,shifted in energy to E qp = k d . The filled circles and theopen circles show the branching ratios with Γ fis = 0 . d and 0 . d , respectively. There is hardly any differencebetween the two curves.One can make a crude approximation to the Bohr-Wheeler transition state formula Eq. (8) within theframework of the model. The branching ratio for a sin-gle barrier state with a transmission factor T t = 1 reads B cap , fis ∼ / (2 πρ Γ cap ). The spread of the k subblockwith the given parameters is about d , resulting in a leveldensity ρ = N k /d . This estimate gives the reasonableagreement shown by the filled squares in the Figure; itappears that the internal transmission factor for largespaces approaches T = 1. However, the comparisonshould not be considered quantitative because the leveldensity of the first Q -block is not constant over the en-ergy window accessed by the state in the middle. Summary and Outlook.
The model presented here forinduced fission in a CI representation appears to be suf-ficiently detailed to examine the validity of transition-state theory in a microscopic framework. Depending onthe interaction and the deformation-dependent configu-ration space, one achieves conditions in which branchingratios depend largely on barrier-top dynamics and areinsensitive to properties closer to the scission point. Theinsensitive property is one of the main assumptions inthe well-known Bohr-Wheeler formula for induced fission,but up to now it had no microscopic justification.Whether the transition-state hypothesis is valid underrealistic Hamiltonians remains to be seen and will re-quire a large computational effort to answer. In the nearterm, the model can be applied in a number of differentways. We plan to study the barrier transmission factor T i as a function of barrier height to test another basicassumption in present-day theory, namely treating themby the Hill-Wheeler formula [32, p. 1140]. It also appearsquite straightforward to include a pairing interaction inthe Hamiltonian. This would allow one to explore for the first time the competition between the two kinds ofinteraction in barrier-crossing dynamics.This work was supported in part by JSPS KAKENHIGrant Number JP19K03861. [1] M. Bender et al. , J. Phys. G , 113002 (2020).[2] F. D¨onau,J. Zhang and L. Riedinger, Nucl. Phys. A496 ,333 (1989).[3] B.W. Bush, G.F. Bertsch and B.A. Brown, Phys. Rev.C , 1709 (1992).[4] D. Cha and G.F. Bertsch, Phys. Rev. C , 306 (1992).[5] G.F. Bertsch, Phys. Rev. C , 034617 (2020).[6] K. Hagino and G.F. Bertsch, Phys. Rev. C , 064317(2020).[7] K. Hagino and G.F. Bertsch, Phys. Rev. C . 024316(2020).[8] D. Sangalli et al. , Phys. Rev. B , 195205 (2016).[9] A. Bohr and B.R. Mottelson, Nuclear Structure (W.A.Benjamin, Reading, MA, 1969), Vol. I.[10] F. Barranco, G.F. Bertsch, R.A. Broglia, and E. Vigezzi,Nucl. Phys.
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1. Estimates of the physical parameters2. Compound nucleus limit3. Codes
Estimation of physical parameters
Orbital energy spacing d . The single-particle levelspacing in the uniform model d sets the energy scalefor the model and does not play any explicit role inthe model. However, it is required to determine otherenergy parameters which are expressed in units of d .Several estimates of d for U are given in Table I.The first is based on orbital energies in a deformedWoods-Saxon potential with the parameters given inRef. [1]; see Table II for the calculated orbital energies.In more realistic theory, the momentum dependence d (MeV) Source0.45 Woods-Saxon well0.51 FRLDM [5]0.33 FGM [6]TABLE I: Estimated orbital level spacing in U. The firsttwo are from potential models and the last extracted from theFermi gas formula and measured level densities.protons neutrons2
K π ε Kπ /d K π ε Kπ /d − − − − − − − − − − − − − − − − − − − U at de-formation ( β , β ) = (0 . , . of the potential tends to increase the spacing, but thecoupling to many-particle degrees of freedom decreasesthe spacing of the quasiparticle poles. The combinedeffect seems to be to somewhat decrease the spacing . Level density.
It is important to know the compositionof the levels in the compound nucleus to construct mi- We note that in the energy density functional fit including fissiondata[7] the effective mass in the single-particle Hamiltonian wasvery close to 1. croscopic models that involve those levels. For a concreteexample, consider the levels at the neutron threshold en-ergy S n = 6 . U. The predominating configu-rations at this energy should be k subblocks at k ≈ S n /d in the independent quasiparticle approximation. Anotherapproach that is less sensitive to the residual interactionis to estimate the total number of states below S n andcomparing it to the number obtained by summing the N k degeneracies in the Q -block spectrum. In the U ex-ample, the combined level spacing of J π = 3 − and 4 − isabout 0.45 eV at S n [10]. At that excitation energy thelevel density is the same for even and odd parities, and itvaries with angular momentum as 2 J + 1. The inferredlevel spacing of J π = 0 + levels is thus about 7 eV. Theaccumulative number of levels can be approximated by N = ρT where T is the nuclear temperature, defined as T = d log( ρ ( E )) /dE . A typical estimate for our exampleis T = 0 .
65 MeV, giving N ≈ . × . To estimate thelevel density in the present model, we start with the set ofquasiparticle configurations including both parities andall K values. The resulting k -blocks have multiplicitiesthat are well fit by the formula N k ≈ exp( − .
23 + 4 . k / ) . (1)Projection on good parity decreases this by a factor oftwo. The projection on angular momentum J = 0 ismore subtle. The J = 0 states are constructed by pro-jection from K = 0 configurations; other configurationsdo not contribute. However, there may be two distinctconfigurations that project to the same J = 0 state. Thisgives another factor of nearly two reduction in the mul-tiplicity. The remaining task is to estimate the fractionof K = 0 configurations in the unprojected quasiparti-cle space. The distribution of K values is approximatelyGaussian with a variance given by h K i = h n qp ih K i sp (2)where h n qp i ≈ k block and h K i sp ≈ K ’s near the Fermi level. Including theseprojection factors, the integrated number of levels up to S n is achieved by including all k -subblocks up to k = 17in the entry Q -block. Neutron-proton interaction v np . To set the scale for ourneutron-proton interaction parameter v np we comparewith phenomenological contact interactions that havebeen used to model nuclear spectra. The matrix elementof the neutron-proton interaction is h n p | v | n p i = − v I (3)where I = Z d rφ ∗ n ( r ) φ ∗ p ( r ) φ n ( r ) φ p ( r ) . (4)The parameter v is the strength of the interaction, typ-ically expressed in units of MeV fm . Some values of v a r X i v : . [ nu c l - t h ] F e b Basis of estimate v (MeV fm ) Citation G -matrix 530 [4] sd -shell spectra 490 [3] β -decay 395,320 [2]TABLE III: Estimates of neutron-proton interaction strength. -0.0002 -0.0001 0 0.0001 0.0002 I (fm -3 )00.020.040.060.08 d i s t r i bu ti on FIG. 1: Integrals I in Eq. (4) of orbitals near the Fermienergy. from the literature are tabulated in Table III. We shalladopt the value v = 500 MeV fm for most of the modelcalculations.If the wave functions of the eigenstates approach thecompound nucleus limit, the only characteristic we needto know is its mean-square average among the activeorbitals. We have calculate the integral Eq. (4) for allthe fully off-diagonal matrices of the orbitals within 2MeV of the Fermi energy. Fig. 1 shows a histogram oftheir distribution . The variance of the distributionis h I i / = 5 . × − fm − . Combining this withour estimate of v we find ( h n p | v | n p i ) / = 0 . v np ∼ . d with our estimatedsingle-particle level density. Decay widths
Experimentally in the nucleus
U at anenergy near S n , the compound-nucleus gamma decaywidths are about 0 .
040 eV[10]. Such widths are smallerthan any of the other energy scales in the reaction. Thebranching ratio between gamma capture and fission fa-vors fission by about a factor of 3, but B cap , fiss has strongfluctuations around that average. There is no direct in-formation about the exit channel decay widths near thescission point. However, for several of the examples inthe text we have taken Γ fis / Γ cap = 3, which would givethe observed branching in the (unphysical) compound If the orbitals are restricted only to those in TABLE II, thehistogram is more structured. nucleus limit.On the theoretical side, the decay widths of thestates in the Hamiltonian enter into the reaction crosssections into two ways, explicitly as a factor in Eq.(5) of the text and implicitly in the Green’s function( ˜ H − E ) − . If the decay width is smaller than any of theother internal energy scales, one can neglect its effecton the Green’s function. It is also the case that thetransmission factors depend strongly on the entrancechannel widths, but the branching ratios are insensitive.Our reported calculations were carried out in the limitof small entrance channel widths, but realistic onesderived from optical model phenomenology can be easilyincorporated, as was done in the MAZAMA code [9]. Compound nucleus limit.
The concept of the compound nucleus is a major ingre-dient of the reaction theory for fission of heavy nuclei.From the side of theory, the compound nucleus is char-acterized by a set of properties derived from Wigner’srandom matrix model (RME) [11–17], or more specifi-cally the Gaussian orthogonal ensemble (GOE). In ourframework, the RME is most closely approached whenthe space is restricted to a single k -block. Since the di-agonal elements of the Hamiltonian are all the same, theonly energy scale relevant to the diagonalization is v np ,the strength of the interaction. However, it is far fromguaranteed that the model will satisfy the expected prop-erties because our Hamiltonian is a sparse matrix, unlikethe RME. Here we examine the properties of the statesin the k = 6 space to show that indeed the model doesapproach the RME limit. The Hamiltonian matrix has adimension of N = 784 and there are 29688 off-diagonalmatrix elements. To be specific, the interaction strengthis taken as v np = 0 . d .Properties of the RME that are independent of thebasis are the semicircular distribution of eigenvalues andthe repulsion between neighboring eigenvalues. The topand the middle panels of Fig. 2 show the eigenvaluedistribution and the distribution of the nearest neighborlevel spacing compared with the semi-circle formula ρ ( E ) = 2 π N E s − (cid:18) EE (cid:19) , (5)and the eigenvalue repulsion formula (i.e., the Wignerdistribution), P ( x ) = π xe − πx / (6)respectively. Here, E in Eq. (5) is a parameter charac-terizing the width of the eigenvalue distribution, whichwe take E = 0 . d , and x in Eq. (6) is defined as x = s/ h s i , where s is a nearest neighbor level spacingand h s i is its average. The figure indicates that bothproperties are reasonably well satisfied.For the problem of fission, the most importantproperty of the compound nucleus is a distribution of d i s t r i bu ti on semi-circle0 1 2 3s / s av P ( s ) Wigner0 0.05 0.1WF amplitudes0100002000030000400005000060000 d i s t r i bu ti on Porter-Thomas (a)(b)(c)
FIG. 2: (a) Eigenvalue spectrum of the neutron-proton inter-action with v np = 0 . d in the E ∗ = 6 d configuration space.The red solid curve shows the semi-circle distribution, Eq.(5), with E = 0 . d . (b) The nearest neighbor level spac-ing distribution compared with the Wigner distribution, Eq.(6). (c) The distribution of wave function amplitudes for allthe components of all the eigenstates within the model space.The red solid line shows the expected Gaussian distribution. decay widths to the individual channels. This shouldobey the Porter-Thomas distribution if the configurationamplitudes are Gaussian distributed. Taking all theeigenfunctions and configurations of the model space,the bottom panel of Fig. 2 shows the distribution of thewave function amplitudes compared with the Gaussiandistribution having the width parameter σ = N − / . Aswe see, the distribution agrees well with the Gaussiandistribution. Codes
Codes to compute selected data points in Figs. 1,3,4,and 5 are provided in the subdirectory codes . TheHamiltonian matrices are constructed by Fortran codes ham*.f . The subsequent analysis is carried out byPython scripts and shell scripts. The Fortran codes havebeen compiled and tested with the gfortran compiler.The Python scripts are listed below. They compile thenecessary Fortran code to generate the Hamiltoniansin the ensembles and then calculate the observablesaccording to the formulas in the text.Fig. 1, histogram: fig1.pyFig. 3, branching ratio at f od = 0 .
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