Unified description of the coupled-channels and statistical Hauser-Feshbach nuclear reaction theories for low energy neutron incident reactions
EEPJ manuscript No. (will be inserted by the editor)
Unified description of the coupled-channels and statisticalHauser-Feshbach nuclear reaction theories for low energyneutron incident reactions
Toshihiko Kawano Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, e-mail: [email protected]
Received: date / Revised version: date
Abstract.
We incorporate the coupled-channels optical model into the statistical Hauser-Feshbach nuclearreaction theory, where the scattering matrix is diagonalized by performing the Engelbrecht-Weidenm¨ullertransformation. This technique has been implemented in the coupled-channels optical model code ECISby J. Raynal, and we extend this method so that all the open channels in a nucleon-induced reaction ona deformed nucleus can be calculated consistently.
PACS.
It is well known that the nucleon-induced scattering pro-cess from a nucleus can be described by the optical model,where the imaginary part of the potential represents adeficit of incoming flux, hence the scattering S -matrix ofthe optical model is no longer unitary. At relatively lowincident energies, where only a few open channels are in-volved, an absorbed particle once forms a compound statein the target nucleus, and it comes back to the entrancechannel as the compound elastic scattering process. Thetotal wave-function is the coherent sum of the incom-ing and out-going waves [1,2]. Although the direct (or“shape”) elastic scattering ( dσ/dΩ ) SE and the compoundelastic scattering ( dσ/dΩ ) CE cannot be distinguished ex-perimentally, theoretical interpretation divides these twoscattering processes into different time-scale domains —the fast and slow parts. The optical model gives the fasterpart, while the statistical model for nuclear reactions [3,4,5,6,7] accounts for the slower compound elastic scat-tering process. We understand the experimental elasticscattering data are the incoherent sum of both processes.The optical model codes, which have been utilized to an-alyze the nucleon scattering data, for example ECIS [8],ELIESE-3 [9], CASTHY [10], ABAREX [11], and so forth,are capable for calculating the compound nucleus (CN)process, yet they are limited to a binary reaction with rel-atively small channel space. Note that the aforementionedcodes are just examples, and there exist more computerprograms that calculate the optical model in the nuclearscience field.As a general descriptions of nuclear reaction process,the Hauser-Feshbach (HF) statistical theory [12] is ex-tended to a multi-stage reaction, where a residual nucleus formed after particle emission is allowed to further decayas another CN process. In this case, the main part of thecalculation is the compound nuclear reaction, and the op-tical model is somewhat hidden behind it. The so-calledHF codes, GNASH [13,14], TNG [15], STAPRE [16], EM-PIRE [17], TALYS [18,19] and so on, invoke an opticalmodel code to generate transmission coefficients T lj as amodel input, where the quantum numbers lj are the or-bital angular momentum and the spin. J. Raynal’s ECIScode has been widely used for this purpose. Exceptions arethe CCONE [20,21] and CoH [22,23] codes, those includea private T lj generator internally.These HF codes, despite they are capable for calculat-ing nuclear reaction cross sections for all the open chan-nels, do not furnish a strict connection between the op-tical and statistical models. This issue becomes more se-rious when the target nucleus is strongly deformed, andthe single channel optical model has to be extended tothe coupled-channels (CC) formalism [24]. There are twovague approximations made by these codes to deal withthe nuclear deformation effect; (a) T lj for the excited statesare replaced by the one for the ground state by correct-ing the channel energy, and (b) the direct reaction effectin the statistical theory [4,25,26] is ignored. Raynal care-fully dealt with these issues in ECIS, and established aunique integration of the optical and HF statistical mod-els, nevertheless the formalism was limited to the binaryreactions only.Like ECIS or ELIESE-3, CoH was originally developedas a nucleon scattering data analysis code [27], which in-cludes the statistical HF theory with the width fluctuationcorrection by Moldauer [5]. Later, CoH was extended tothe full multi-stage HF code, so that the optical model andthe statistical model were naturally unified just like ECIS, a r X i v : . [ nu c l - t h ] S e p Toshihiko Kawano: Title Suppressed Due to Excessive Length but in more general sense. In this paper we present theunified description of CC optical and HF statistical mod-els implemented in the CoH code, where Raynal’s ideasin ECIS were important resources and clues. We demon-strate how the approximations made by the existing HFcodes bring systematic uncertainties in the predicted crosssections. We limit ourselves to the low energy neutron in-duced reactions only, where the compound elastic scatter-ing plays more important role than the charged-particlecases. The energy-averaged cross section for a reaction from chan-nel a to channel b is written by the partial decay width Γ c as σ CN ab = πk a g a πD (cid:28) Γ a Γ b (cid:80) c Γ c (cid:29) , (1)where D is the average resonance spacing, k a is the wavenumber of incoming particle, and g a is the spin factorgiven later. By applying a relation between the single-channel transmission coefficient T a and the decay width Γ a , T a (cid:39) π (cid:104) Γ a (cid:105) D , (2)the width fluctuation corrected HF cross section [1,12]reads σ CN ab = πk a g a T a T b (cid:80) c T c W ab = σ HF ab W ab , (3)where σ HF ab is the original HF cross section. The widthfluctuation correction factor W ab is also a function of T a .When the resonance decay width Γ a forms the χ distri-bution with the channel degree-of-freedom ν a , the widthfluctuation correction factor can be evaluated numericallyas [26,28,29] W ab = (cid:18) δ ab ν a (cid:19) (cid:90) ∞ dtF a ( t ) F b ( t ) (cid:81) k F k ( t ) ν k / , (4) F k ( t ) = 1 + 2 ν k T k (cid:80) c T c t . (5)The Gaussian Orthogonal Ensemble (GOE) model [6] pro-vides accurate estimates of ν a [5,7] in terms of T a , and allof the reaction cross sections σ ab are determined by T a and T b only.When the ground state of nucleus does not coupleso strongly with other states by collective excitation, thescattering matrix element S aa is given by solving the single-channel Schr¨odinger equation for a spherical optical po-tential. Because the S -matrix is diagonal, the transmissioncoefficient T a is defined as a unitarity deficit T lj = 1 − | S lj,lj | , (6) CNGround StateExcited State T lj(1) (E) T lj(0) (E)T lj(1) (E) = T lj(0) (E-E x ) CN' E x Fig. 1.
The ground and excited states in a target nucleus andthe compound nucleus (CN) connected by the particle trans-mission coefficients, T (0) lj and T (1) lj . T (1) lj is often approximatedby shifting to the ground state, as shown by the dotted arrow. where we denote the channel quantum numbers explicitlyby the orbital angular momentum l , and spin j . We alsoadd a discrete level index n as T ( n ) lj . T (0) lj stands for aprobability to form a CN state from the ground state.The detailed-balance equation in Eq. (3) is schematicallyshown in Fig. 1 for the two level case. T (1) lj stands for aprobability to form the same CN from the first excitedstate. However, since optical potentials for excited nucleiare usually unknown, they are replaced by T (0) lj and shiftthe energy by the excitation energy E ( n ) x to take accountof the energy difference, T ( n ) lj ( E ) (cid:39) T (0) lj ( E − E ( n ) x ) . (7)There exist many phenomenological global optical poten-tials that are energy-dependent, and the assumption ofEq. (7) enables all of the HF codes to perform cross sec-tion calculations in wide energy and target-mass ranges.In fact, almost all of the HF codes generate T (0) lj on a fixedenergy grid before performing a CN calculation, and in-terpolate T (0) lj to obtain a required value at each energypoint. When strongly coupled collective levels are involved in thetarget system, Eq. (3) must be calculated by the transmis-sion coefficients in the CC formalism, and the S -matrix isno longer diagonal. ECIS and CoH calculate generalizedtransmission coefficients for all the included states from oshihiko Kawano: Title Suppressed Due to Excessive Length 3 the CC S -matrix [30]. The time-reversal symmetry of S -matrix yields the transmission coefficient for all of the n -thexcited state simultaneously as T ( n ) lj = (cid:88) JΠ (cid:88) a s + 12 j a + 1 g J (cid:32) − (cid:88) b | S JΠab | (cid:33) × δ n a ,n δ l a ,l δ j a ,j , (8)where JΠ is the total spin and parity. This is shown inFig. 2. The spin-factor g J is given by g J = 2 J + 1(2 s + 1)(2 I n + 1) , (9)where s is the intrinsic spin of the projectile (= 1/2 forneutron), and I n is the target spin of n -th level. The sum-mation runs over the parity conserved channels, albeit weomit a trivial parity conservation. In this expression, crosssections to the directly coupled channels are eliminated toensure the sum of T ( n ) ij gives a correct CN formation crosssection from the n -th level σ ( n ) R = πk n (cid:88) JΠ (cid:88) a δ n a ,n g J (cid:32) − (cid:88) b | S JΠab | (cid:33) = πk n (cid:88) lj j + 12 s + 1 T ( n ) lj . (10)Now the off-diagonal elements in the S -matrix are effec-tively eliminated. By substituting T ( n ) lj into Eq. (3), theHF cross section is determined in terms of the detailedbalance. This formulation is, however, valid only when thewidth fluctuation correction is not so significant, becausethe off-diagonal elements in S to calculate W ab are ig-nored. We will discuss this later.The HF codes, exept for ECIS and CoH to our knowl-edge, simplify Eq. (8) by applying the approximation ofEq. (7), namely T ( n ) lj ( E ) (cid:39) T (0) lj ( E − E ( n ) x ), hence all theCC S -matrix properties are lost. To examine this approx-imation, we calculate transmission coefficients for the ex-cited states T ( n ) lj ( E ) and compare with those for the energy-shifted ground state T (0) lj ( E − E ( n ) x ). The comparison in-cludes two cases of rotational band head; the target groundstate spin is zero, and it is half-integer. The first exampleis for the fast-neutron induced reaction on U, in whichwe couple 5 levels (0, 45, 148, 307, and 518 keV, from 0 + to 8 + ) in the ground state rotational band. The opticalpotential of Soukhovitskii et al. [31] is used. A naturalchoice for the second case, the half-integer ground statespin, is Pu. However, its large fission cross sections atlow energies blur the difference coming from the definitionof T lj . Instead, we adopt Tm that has a similar levelstructure to
Pu [30]. The coupled levels are 0, 8.4, 118,139, and 332 keV from (1 / + to (9 / + , and the opticalpotential of Kunieda et al. [32] is employed.Figure 3 shows the difference in the l = 0 and 1 trans-mission coefficients for U. The approximation by T (0) lj CNGround StateExcited State Coupled States T lj(1) T lj(0) E x Fig. 2.
In the coupled-channels method, the transmission co-efficients T (0) lj and T (1) lj can be derived simultaneously from the S -matrix. seems to be reasonable for the s -wave transmission co-efficient, while the difference reaches about 10% for the p -wave case. The calculations for Tm, shown in Fig. 4,show the opposite tendency; a notable difference appearsin the s -wave. It is difficult to draw a general conclusionby these limited examples. However, it is obvious thatthe calculated cross sections by feeding these transmissioncoefficients into the statistical HF theory are no longerequivalent, and the energy-shifted transmission coefficientinflates uncertainty in the calculated results. To see the actual impact of the generalized transmissioncoefficients on the cross section calculation, we have to cal-culate the HF equation with these actual/approximatedtransmission coefficients. Unfortunately, this is not so easyin general, because it requires extensive modification tothe computer programs. Instead, we made an ad-hoc mod-ification to CoH to test this. For a 100-keV neutron in-duced reaction on U, CoH with the generalized trans-mission coefficient gives the inelastic scattering cross sec-tion of 553 mb to the 45 keV 2 + level. At this energy,an equivalent center-of-mass (CMS) energy to the first45 keV level is 100 A/ ( A + 1) − . . T (0) lj at E CMS = 54 . T (1) lj by thesevalues. The calculated inelastic scattering cross section is488 mb, which is 9% smaller than the generalized trans-mission case, and closer to the evaluated value in JENDL-4 [20,33] of 461 mb. This observation is consistent with the Toshihiko Kawano: Title Suppressed Due to Excessive Length T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV]actualshifted GS T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV]actualshifted GS T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV]actualshifted GS
Fig. 3.
Calculated transmission coefficients for the first andsecond coupled levels of
U; (a) for ( l, j ) = (0 , / , / , / S -matrix. The dashedcurves are approximation by the ground state transmission co-efficient shifted by the level excitation energies. The black, red,and blue curves are for the first, second, and third levels. Thecorresponding level should also be distinguished by the shiftedthreshold energies. T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV] actualshifted GS T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV] actualshifted GS T r an s m i ss i on C oe ff i c i en t, L = , J = / C.M. Energy [MeV] actualshifted GS
Fig. 4.
Same as Fig. 3 but for
Tm. larger p -wave transmission coefficient as shown in Fig. 3(b).In the past, a code comparison was carried out [34]by including EMPIRE [17], TALYS [19,35], CCONE [36],and CoH . The result revealed that the inelastic scatteringcross section by CoH tends to be higher than those bythe other codes for the U case at low energies. Thedifference is about 10% at 100 keV, and this confirmsour numerical exercise here; the approximation by the oshihiko Kawano: Title Suppressed Due to Excessive Length 5 ground state transmission coefficient systematically un-derestimates the inelastic scattering cross section for the
U case.
There might be many uncoupled levels involved in actualCN calculations, as schematically shown in Fig. 5. In thetarget nucleus, there are uncoupled discrete levels up tosome critical energy, then a level density model is used todiscretize the continuum above there. The transmissioncoefficients to these states are calculated by the single-channel case.The formed CN state can decay by emitting a charged-particle, γ -ray or fission. The charged-particle transmis-sion coefficients are basically the same as the uncoupledneutron channel case, except for the Coulomb interaction.The transmission coefficients for the γ decay are calculatedby applying the giant dipole resonance (GDR) model [37],where GDR parameters derived from experimental dataor theoretically predicted are often tabulated [37,38]. Al-though there are a large number of final states availableafter a γ -ray emission, these probabilities are very smallcompare to the neutron transmission coefficients. Often itis good enough to lump the γ -ray channels into a single γ -ray transmission coefficient as T γ = (cid:88) XL (cid:90) E n + S n T XL ( E γ ) ρ ( E x ) dE x , (11)where S n is the neutron separation energy, X stands forthe type of radiation (E: electric, M: magnetic), L is themultipolarity, and ρ ( E x ) is the level density at excitationenergy E x = E n + S n − E γ . When the final state is in adiscrete level, the integration in Eq. (11) is replaced by anappropriate summation.When the CN fissions, the simplest expression of thefission transmission coefficients is the WKB approxima-tion to the inverted parabola shape of fission barriers pro-posed by Hill and Wheeler [39]. Albeit it is know that thisform has an issue to reproduce experimental fission crosssections, this is beyond the scope of current paper, and wedo not discuss it further. The fission takes place thoughmany states on top of the fission barrier, so that it is con-venient to lump these partial fission probabilities into thefission transmission coefficient T f . The width fluctuation correction factor W ab consists ofthe elastic enhancement factor and the actual width fluc-tuation correction factor [28]. A convenient definition issuggested by Hilaire, Lagrange, and Koning [40], whichis to define as a ratio to the pure HF cross section as inEq. (3). When we ignore the channel coupling effect, W ab in Eq. (4) can be calculated by the generalized transmis-sion coefficients in Eq. (8). However, we cannot employ CNGround State C oup l ed S t a t e s T (0) T (1) T (2) Fig. 5.
All the possible decay channels from a compound stateare schematically shown by the arrows. The solid arrows arethe transmission coefficients by the CC model. All the otherchannels shown by the dotted arrows are uncoupled levels. this prescription when a strong channel-coupling resultsin non-negligible off-diagonal elements in S . Instead, weperform the Engelbrecht-Weidenm¨uller (EW) transforma-tion [25] to correctly eliminate the off-diagonal elements.Satchler’s transmission matrix [41] is defined by theCC S -matrix as P ab = δ ab − (cid:88) c S ac S ∗ bc . (12)Since P is Hermitian, we can diagonalize this by a unitarytransformation [25]( U P U † ) αβ = δ αβ p α , ≤ p α ≤ , (13)where α and β are the channel indices in the diagonalizedspace. The diagonal element p α is the new transmissioncoefficient, because the S -matrix is also diagonalized as˜ S = U SU T , (14)which defines the single-channel transmission coefficient T α = 1 − (cid:12)(cid:12)(cid:12) ˜ S αα (cid:12)(cid:12)(cid:12) = p α . (15)We now calculate the width fluctuation in the diagonalchannel space, then transform back to the cross-sectionspace by [4] σ ab = (cid:88) α | U αa | | U αb | σ αα + (cid:88) α (cid:54) = β U ∗ αa U ∗ βb ( U αa U βb + U βa U αb ) σ αβ + (cid:88) α (cid:54) = β U ∗ αa U ∗ αb U βa U βb (cid:68) ˜ S αα ˜ S ∗ ββ (cid:69) . (16) Toshihiko Kawano: Title Suppressed Due to Excessive Length σ αα and σ αβ are the width fluctuation corrected cross sec-tion with the transmission coefficient of p α . The last term (cid:68) ˜ S αα ˜ S ∗ ββ (cid:69) was evaluated by applying the Monte Carlotechnique to GOE [42],˜ S αα ˜ S ∗ ββ (cid:39) e i ( φ α − φ β ) (cid:18) ν α − (cid:19) / (cid:18) ν β − (cid:19) / σ αβ , (17)where φ α = tan − ˜ S αα . Here we replaced the energy av-erage (cid:104)∗(cid:105) by the ensemble average ∗ . Applying the GOEmodel to the channel degree-of-freedom ν α [7], the HFcross section with the width fluctuation correction is fullycharacterized in the CC framework.When uncoupeld-channels, such as the inelastic scat-tering to the higher levels, γ -decay and fission channels,exist, the transmission matrix has these sub-space P = P C T n T γ T f , (18)where P C is the coupled channels P matrix in Eq. (12).Because T n , T γ , and T f are still diagonal, the unitarytransformation is only applied to P C . The uncoupled crosssection is calculated by [42] σ ab = (cid:88) α | U αa | σ αβ δ βb . (19)Here we take U and
Tm as examples again. Thewidth fluctuation correction is defined as a ratio to theHF cross section, and we calculate two cases; (a) the widthfluctuation factor W ab in Eq. (4) is calculated by using thegeneralized transmission coefficient of Eq. (8), and the off-diagonal elements in the S -matrix are ignored, and (b) thefull EW transformation is performed.It is known that an asymptotic value of W aa (elas-tic enhancement factor) is 2 when all the channels areequivalent. Figure 6 (a), which is the calculated W aa for U, shows this behavior, but the EW transformationslightly deters W aa from approaching the asymptote. Theweaker elastic enhancement results in increase in the in-elastic scattering channels. This is also demonstrated bythe Monte Carlo simulation for the GOE scattering matrixwhen direct reaction components are involved [7]. In otherwords, the directly coupled channels squeeze the elasticscattering channel due to constraint by the S -matrix uni-tarity, hence the enhancement in the elastic channel willhave less influence on the other channels.In the case of Tm, shown in Fig. 7, the asymptoticvalue of W aa does not reach 2 but stays about 1.6. Thismight be because the s -wave transmission coefficient forthe second excited state is very different from the otherchannels. Because the number of channels is larger thanthe U case, the EW transformation less impacts theCN calculations. In addition, other uncoupled channels,e.g. radiative capture and fission channels if exist, furthermitigate the elastic enhancement effect. Therefore the EW W i d t h F l u c t ua t i on C o rr e c t i on Energy [MeV]without EW transwith EW trans W i d t h F l u c t ua t i on C o rr e c t i on Energy [MeV]without EW transwith EW trans W i d t h F l u c t ua t i on C o rr e c t i on Energy [MeV]without EW transwith EW trans
Fig. 6.
Width fluctuation corrections for
U defined as aratio to the Hauser-Feshbach cross section. The panels (a),(b), and (c) are the compound elastic, inelastic to the first,and second levels for the neutron-induced reaction on
U.The solid curves are calculated by performing the Engelbrecht-Weidenm¨uller (EW) transformation, and the dashed curves arewithout the EW transformation. transformation is mostly important for rotating even-evennuclei with large deformation. Having said that, the dif-ference seen in Fig. 7 implies an inherent deficiency in thesimplified HF calculations widely adopted nowadays.
In reality, the EW transformation does not modifies thecalculated cross sections so largely. As seen in Fig. 6, the oshihiko Kawano: Title Suppressed Due to Excessive Length 7 W i d t h F l u c t ua t i on C o rr e c t i on Neutron Incident Energy [MeV]without EW transwith EW trans W i d t h F l u c t ua t i on C o rr e c t i on Neutron Incident Energy [MeV]without EW transwith EW trans W i d t h F l u c t ua t i on C o rr e c t i on Neutron Incident Energy [MeV]without EW transwith EW trans
Fig. 7.
Same as Fig. 6 but for
Tm. difference between the EW and single-channel W ab cases isat most 15%. Such a difference may occur due to other un-certain inputs to the calculation. One of the most crucialmodel parameters is the optical potential. The optical po-tential parameters are often obtained phenomenologicallyby fitting to experimental elastic scattering and total crosssections. In general, similar quality of data fitting can beachieved by different potential parameters, while they mayhave slightly different partial wave contributions. Fluctu-ation in the partial wave contribution is sometimes visi-ble in the inelastic scattering cross sections, where limitednumbers of partial waves are involved.Figure 8 (a) shows a comparison of the calculated in-elastic scattering cross section to the first 45 keV level of U with available experimental data of Miura et al. [43],Kornilov and Kagalenko [44], Moxon et al. [45], Litvin- skii et al. [46], Winters et al. [47], Guenther et al. [48],Haouat et al. [49], and Tsang and Brugger [50]. We per-formed this calculation with the CC optical model poten-tial of Soukhovitskii et al. [31]. While the measurementsare largely scattered in the hundreds keV region, the EWtransformation moves the calculation into a preferable di-rection. However, the enhancement due to the EW trans-formation is rather modest, which is also seen in Fig. 6(b).When we switch the optical potential into the updatedSoukhovitskii potential [51], the EW transformation be-comes noticeable as shown in Fig. 8 (b). In this case the en-hancement is visible in the wider energy range. Of courseit is not rational to verify an optical potential by apply-ing it to the statistical model, as ambiguity caused byother model inputs persists. In the CC formalism, calcu-lated cross sections are also influenced by the couplingscheme [52]. Despite other available optical potentials for
U may provide different excitation functions of 45-keVlevel, we may say generally that the EW transformationincreases the 45-keV level cross section due to the hinderedelastic enhancement.The increase in the inelastic scattering of the 148-keVlevel, as well as the total inelastic scattering cross section,is shown in Fig. 9. The 2 + cross sections are the same asthose in Fig. 8. Since the relative magnitude of the 4 + levelcross section is smaller than the 2 + level, this has a minorimpact on the total inelastic scattering cross section. Thisis also true for the higher spin states (6 + , + . . . )It might be worth reminding that these “without EW”cases employ the generalized transmission coefficient tocalculate both the HF cross section and the W ab factor.When one adopts a conventional prescription of T ( n ) ( E ) (cid:39) T (0) ( E − E ( n ) x ), the calculated inelastic scattering crosssection would be further lower than the no-EW case. Ev-idently this approximation cannot be justified anymorewhen the nuclear deformation plays an important role.Use of the generalized transmission calculation is still ap-proximated, albeit it mitigates this deficiency to some ex-tent, and afford us not so heavy computation. However, aswe demonstrated the quantitative deficiencies in the ap-proximations and simplifications made so far, we shouldconsider implementing the EW transformation in the HFmodel codes for better prediction of nuclear reaction crosssections for deformed nuclei.A remaining complication is the angular distributionsof scattered particles in the CN process. The differentialcross section is expanded by the Legendre polynomials inthe Blatt-Biedenharn formalism [53], (cid:18) dσdΩ (cid:19) ab = (cid:88) L B L P L (cos θ b ) , (20)where the scattering angle θ b is in the center-of-mass sys-tem. A full expression of the B L coefficients is given inthe single-channel width fluctuation case [2,54]. However,a complete formulation of the B L coefficient becomes verydifficult to calculate when the EW transformation is per-formed. Alternatively, we can apply the generalized trans-mission coefficients without the EW transformation for Toshihiko Kawano: Title Suppressed Due to Excessive Length k e V I ne l a s t i c S c a tt e r i ng C r o ss S e c t i on [ m b ] Neutron Incident Energy [MeV]Miura (2000)Kornilov (1996)Moxon (1994)Litvinskii (1985)Winters (1981)Guenther (1975)Haouat (1982)Tsang (1978)without EW transwith EW trans 0 500 1000 1500 2000 0.01 0.1 1(b) k e V I ne l a s t i c S c a tt e r i ng C r o ss S e c t i on [ m b ] Neutron Incident Energy [MeV]Miura (2000)Kornilov (1996)Moxon (1994)Litvinskii (1985)Winters (1981)Guenther (1975)Haouat (1982)Tsang (1978)without EW transwith EW trans
Fig. 8.
Comparison of calculated inelastic scattering cross sec-tion to the first 45 keV level of
U. The top panel (a) is thecase of 2004 Soukhovitskii potential [31], and (b) is the 2005Soukhovitskii potential [51] calculating B L . This is roughly the Legendre coefficientsin the HF case B HF L times the width fluctuation correctionfactor W ab , but more correction terms are involved [54]. We presented a general formulation of the statistical Hauser-Feshbach (HF) theory with width fluctuation correctionfor a deformed nucleus, and applied to the low-energy neu-tron induced reactions on
U and
Tm. The main dif-ference between the conventional HF model is; (a) we cal-culate generalized transmission coefficients from the coupled-channels (CC) S -matrix, and (b) the width fluctuation cal-culation is performed in the diagonalized channel space,which is the so-called Engelbrecht-Weidenm¨uller (EW)transformation. Whereas these ingredients were alreadyimplemented into J. Raynal’s coupled-channels code ECIS,the coupled-channels HF code, CoH , offers more generalfunctionality for calculating nuclear reactions at low ener-gies. We demonstrated that both the generalized transmis-sion coefficients and the EW transformation increase theneutron inelastic scattering cross section when stronglycoupled direct reaction channels exist. This happens dueto the fact that contributions from each partial wave aredifferent, and that constraints by the unitarity of S -matrix I ne l a s t i c S c a tt e r i ng C r o ss S e c t i on [ m b ] Neutron Incident Energy [MeV]without EW transwith EW trans 0 1000 2000 3000 0 0.5 1 1.5 2(b)2+ 4+ Total I ne l a s t i c S c a tt e r i ng C r o ss S e c t i on [ m b ] Neutron Incident Energy [MeV]without EW transwith EW trans
Fig. 9.
The inelastic scattering cross section to the 45 keVand 148 keV levels, and the total inelastic scattering cross sec-tion of
U. The top panel (a) is for the 2004 Soukhovitskiipotential [31], and (b) is the 2005 Soukhovitskii potential [51] is somewhat relaxed. The HF nuclear reaction calculationcodes currently available in the market often simplify thedeformed nucleus calculations by assuming a nuclear de-formation effect is negligible. Our numerical calculationsfor a few examples evidently demonstrated that such thesimplification results in underestimation of the inelasticscattering cross sections.
The author is grateful to E. Bauge, S. Hilaire, and P. Chau ofCEA Bruy`eres-le-Chˆatel and P. Talou of LANL for encourag-ing this work. This work was carried out under the auspices ofthe National Nuclear Security Administration of the U.S. De-partment of Energy at Los Alamos National Laboratory underContract No. 89233218CNA000001.
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