Constraining the in-medium nucleon-nucleon cross section from the width of nuclear giant dipole resonance
aa r X i v : . [ nu c l - t h ] J u l Constraining the in-medium nucleon-nucleon cross section from the width of nucleargiant dipole resonance
Rui Wang a,b , Zhen Zhang c , Lie-Wen Chen d , Che Ming Ko e , Yu-Gang Ma a,b a Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai ,China b Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai , China c Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai , China d School of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University,Shanghai , China e Cyclotron Institute and Department of Physics and Astronomy, Texas A & M University, College Station, Texas , USA
Abstract
We develop a new lattice Hamiltonian method for solving the Boltzmann-Uehling-Uhlenbeck (BUU) equation. Adoptingthe stochastic approach to treat the collision term and using the GPU parallel computing to carry out the calculationsallows for a rather high accuracy in evaluating the collision term, especially its Pauli blocking, leading thus to a newlevel of precision in solving the BUU equation. Applying this lattice BUU method to study the width of giant dipoleresonance (GDR) in nuclei, where the accurate treatment of the collision term is crucial, we find that the obtained GDRwidth of
Pb shows a strong dependence on the in-medium nucleon-nucleon cross section σ ∗ NN . A very large mediumreduction of σ ∗ NN is needed to reproduce the measured value of the GDR width of Pb at the Research Center forNuclear Physics in Osaka, Japan.
Keywords:
Heavy-ion collisions, Transport models, Nuclear giant dipole resonance width, In-medium nucleon-nucleoncross section
Introduction.
The in-medium nucleon-nucleon (NN)cross section σ ∗ NN has significant effects on the dynamicsof heavy-ion collisions (HICs), and it thus plays a crucialrole in understanding the reaction mechanisms as well asvarious phenomena and observables in these collisions [1–6]. The importance of σ ∗ NN also lies in its intimate relationto the transport properties of nuclear matter [7, 8] andthe nucleon effective interactions [9]. Since a major goalof studying HICs is to extract the equation of state (EOS)of nuclear matter from experimental data [10–15], a thor-ough understanding of σ ∗ NN helps reduce the uncertaintiesin transport models [16, 17] that are used for describingthese reactions. While the NN cross section in free space σ freeNN can be directly measured in experiments, the determi-nation of the value of σ ∗ NN in nuclear medium usually relieson theoretical investigations. These include calculationsbased on microscopic theories, such as the nonrelativis-tic and relativistic Brueckner theories [9, 18–23] and theclosed time path Green’s function approach [24, 25]. Also,there have been attempts to extract σ ∗ NN from experimentsby comparing results from transport model calculations,where σ ∗ NN is a crucial input, with observables measured Email addresses: [email protected] (Rui Wang), [email protected] (Zhen Zhang), [email protected] (Lie-Wen Chen), [email protected] (Che Ming Ko), [email protected] (Yu-Gang Ma) in HICs that are sensitive to σ ∗ NN , e.g., the collective flowand nuclear stopping [1, 8, 26–30]. Although these studieshave reached the consensus that the NN cross section issuppressed in nuclear medium, the reduction factor is stillfar from certainty.The transport model used in describing HICs is a straight-forward tool for studying σ ∗ NN because one of its main in-gredients, the NN collision term, embodies the informationof σ ∗ NN . Since the mean field or the EOS of nuclear matteris another major ingredient of one-body transport models,finding the proper observables that depend on σ ∗ NN ratherthan the nuclear EOS is essential for studying σ ∗ NN . Onesuch observable is the width of nuclear giant dipole res-onance (GDR) as it is naturally related to σ ∗ NN throughthe NN collision term in transport models. In general, thedamping width of nuclear collective motion originates fromthree sources: 1) the escape width associated with parti-cle emissions; 2) the fragmentation or the Landau damp-ing width due to couplings between single particle statesand the mean field; 3) the spreading or collisional damp-ing width caused by the coupling to more complex stateslike the two-particle-two-hole (2 p -2 h ), 3 p -3 h , etc. For aheavy nucleus at zero temperature, the width of its GDRis mainly exhausted by collisional damping [31–33] beforethe contribution from deformation fluctuations appears asa result of the finite temperature effect [34]. In the trans-port model, the collisional damping is incorporated in the Preprint submitted to Elsevier July 24, 2020 inary collisions of nucleons and thus depends directly on σ ∗ NN . It is therefore expected that the GDR width of aheavy nucleus in studies based on the transport model de-pends strongly on σ ∗ NN and weakly on the nuclear EOS.The major obstacle that has so far prevented the useof transport models to accurately calculate the spreadingwidth of GDR is due to the fermionic nature of nucle-ons. Specifically, the accurate treatment of Pauli blockingin transport models is challenging [16, 17], especially forsmall amplitude nuclear collective motions. Both subtleimplementations and advanced computing techniques arerequired for overcoming this difficulty. In the present work,we extend the previous study using the lattice Hamilto-nian Vlasov method based on the next-to-next-to-next-toleading order (N3LO) Skyrme pseudopotential [35] to in-clude a stochastic elastic NN collision term. Solving theresulting Boltzmann-Uehling-Uhlenbeck (BUU)-type one-body transport model with the high computation efficiencyprovided by GPU parallel computing [36], which enablesthe accurate treatment of Pauli blocking in the collisionterm of the BUU equation, allows us to calculate preciselythe spreading width of the GDR in nuclei. We then ob-tain a stringent constraint on the in-medium NN cross sec-tion σ ∗ NN by comparing the GDR width of Pb from thepresent lattice BUU (LBUU) method with that measuredfrom
Pb( ~p, ~p ′ ) reaction with polarized protons at theResearch Center for Nuclear Physics (RCNP) in Osaka,Japan [37]. Model description.
The BUU equation is a semi-classicalapproximation to the quantum transport equation [38, 39].For a momentum-dependent mean-field potential U ( ~r, ~p ),it reads as ∂f∂t + ~pE ·∇ ~r f + ∇ ~p U ( ~r, ~p ) ·∇ ~r f −∇ ~r U ( ~r, ~p ) ·∇ ~p f = I c , (1)where f = f ( ~r, ~p ) is the one-body phase-space distributionfunction of nucleons or their Wigner function. The r.h.sof Eq. (1) is the NN collision term including the Pauliblocking effect due to the Fermi statistics of nucleons, i.e., I c = − g Z d ~p (2 π ¯ h ) d ~p (2 π ¯ h ) d ~p (2 π ¯ h ) |M → | × (2 π ) δ ( p + p − p − p ) × [ f f (1 − f )(1 − f ) − f f (1 − f )(1 − f )] , (2)where g is the degeneracy, M → is the in-medium tran-sition matrix element, and (1 − f i ) is the Pauli suppressionfactor. It is worth mentioning that higher-order quantumcorrections to Eq. (1) can be added perturbatively [40].In the present work, we solve the BUU equation bythe lattice Hamiltonian (LH) method [41–43], which is avariant of the usual test particle method [44]. In the LHmethod, the total Hamiltonian H of the system is approx-imated by the lattice Hamiltonian H L , i.e., H = Z H ( ~r )d ~r ≈ l X α H ( ~r α ) ≡ H L , (3) where H is the Hamiltonian density, ~r α represents the co-ordinate of certain lattice site α , and l is the lattice spac-ing. For the nucleon one-body phase-space distributionfunction f τ ( ~r α , ~p ), it is expressed as f τ ( ~r α , ~p, t ) = (2 π ¯ h ) gN E α,τ X i S (cid:2) ~r i ( t ) − ~r α (cid:3) δ (cid:2) ~p i ( t ) − ~p (cid:3) , (4)where S is the form factor and N E is the number of en-sembles or test particles used in the calculation. The sumin Eq.(4) runs over all test nucleons of isospin state τ thatcontribute to the lattice site α . In the present work, weadopt a triangular form factor S with the size of 4 l , andits detail can be found in Ref. [35]. The Hamiltonianin Eq. (3) contains both the Coulomb and the nuclearpart [35] with the latter obtained from the N3LO Skyrmepseudopotential [45] SP6h, whose details can be found inRef. [46].In the present LBUU method, the ground state of aspherical nucleus at zero-temperature is obtained from theThomas-Fermi approach [41, 47–49] via the variation ofthe Hamiltonian with respect to the radial nucleon density ρ τ ( r ). The obtained ρ τ ( r ) is then used to determine theinitial coordinates of test nucleons, while their initial mo-menta are generated according to zero-temperature Fermidistribution with local Fermi momentum given by p Fτ ( r )= ¯ h (cid:2) π ρ τ ( r ) (cid:3) / . This method for initialization ensuresthe stability of ground-state nuclei in BUU-like transportmodels [35, 49].For the collision term in the BUU equation, we imple-ment it using the stochastic approach [50], which is morereliable than the commonly used geometric method whenthe mean free path λ MFP of a test nucleon is not muchlarger than the interaction length between two test nucle-ons [51] or when the NN scattering cross section is verylarge. The collision probability P ij of two test nucleons inthe stochastic approach is determined from the NN colli-sion term in Eq. (2), which is P ij = v rel σ ∗ NN S ( ~r i − ~r α ) S ( ~r j − ~r α ) l ∆ t. (5)To reduce the statistical fluctuations of collision eventsand better reflect the nature of the BUU equation, weinclude collisions of test nucleons from different ensembles.In this case, the collision probability is reduced to P ij /N E ,because of the scaling σ ∗ NN → σ ∗ NN /N E of the in-mediumNN cross section between test nucleons. Under such ascaling, the diluteness of the system, which is characterizedby p σ ∗ NN /λ MFP , is reduced by the factor √ N E , and thismakes it possible to solve the BUU equation almost exactlywith a sufficiently large N E achieved by adopting the GPUparallel computing.For the i -th and j -th test nucleons colliding at the lat-tice site ~r α , the direction of their final momenta ~p and ~p are sampled according to the differential cross-sectiongiven in Ref. [52]. However, this collision can only hap-pen if it is allowed by the Pauli principle via the factor21 − f ( ~r α , ~p )] × [1 − f ( ~r α , ~p )]. In the present LBUUmethod, the distribution function f τ ( ~r α , ~p ) is calculatedfrom averaging its value in Eq. (4) over a given momentum-space sphere centered at ~p with radius R pτ ( ~r α , ~p ). In typ-ical transport model calculations, R pτ ( ~r α , ~p ) is taken tohave a constant value of about one hundred MeV. In thepresent work, we use an improved form for R pτ ( ~r α , ~p ) thatis specifically proposed for small-amplitude nuclear col-lective dynamics near ground state [48], i.e., R pτ ( ~r α , ~p ) =max[∆ p, p Fτ ( ~r α ) − | ~p | ], where p Fτ = ¯ h (3 π ρ τ ) / is the lo-cal nucleon Fermi momentum and ∆ p is a constant withthe dimension of momentum that needs to be taken to besufficiently small.The treatment of Pauli blocking in transport modelsis crucial in calculating the width of nuclear collectiveexcitations. At low incident energy or temperature, thePauli blocking is notoriously difficult to handle in trans-port models [16, 17]. This is mainly caused by the in-accuracy in calculating the local momentum distribution f τ ( ~r α , ~p ), which then leads to numerically spurious colli-sions and thus an overestimated GDR width as a result ofthe enhanced collisional damping. There are three mainorigins for the numerically spurious collisions in trans-port models: 1) fluctuations in calculating f τ ( ~r α , ~p ) causedby insufficiently large N E ; 2) spurious thermal excitationcaused by finite ∆ p in calculating f τ ( ~r α , ~p ) (also see Ref. [48]);and 3) diffusion in local momentum caused by finite latticespacing l when averaging over different local densities onthe nuclear surface.In choosing the parameter values in the LBUU calcula-tions, we use the following criteria. For a given l and ∆ p , N E should be large enough to eliminate the overwhelmingmajority of the spurious collisions caused by the first originmentioned above, and at the same time l and ∆ p shouldbe chosen to be sufficiently small to suppress the effectsdue to the second and third origins on the GDR width.After careful tests based on considerations of numericalaccuracy and computation efficiency, we find the optionalvalues of l = 0 . p = 0 .
05 GeV and N E = 30000. Itis worth to mention that with the adoption of GPU par-allel computing, it is possible to use a value for N E thatfar exceeds those used in all previous calculations basedon the BUU transport equation. Further reducing ∆ p and l and increasing N E only leads to a negligible variation inthe calculated GDR width.We note that for the case of free NN cross section,an average of 97 .
93% of the attempted collisions in theground state of
Pb are blocked by the Pauli princi-ple, resulting in an average of 1 .
30 successful collisionsof physical nucleons per fm /c during the time evolution of0 −
500 fm /c . Also, the root mean square (rms) radius andthe ground-state energy of Pb vary by less than 3 . . .
2% (50 MeV), respectively, during this timeevolution. With a reduced in-medium NN cross section,both the number of successful collisions and the change inthe radius and binding energy are even smaller. Since thebinding energy decreases monotonically with time without oscillations, it is not expected to have much effect on thecalculated excitation energy of GDR. The energy violation,which is caused by our use of in-vacuum energy conserva-tion in NN scatterings, instead of the in-medium energyconservation in the presence of the momentum-dependentpotential, is not expected to affect the calculated widthof GDR either. This is because the latter is controlled bythe NN scattering rate, which depends on the NN scatter-ing cross section and the Pauli blocking factor. We alsonote that although the radius and binding energy varia-tions in LBUU are larger than those in the lattice Hamil-tonian Vlasov approach of Ref. [35], where the rms radiusand the binding energy almost do not change, and the ra-dial density profile only changes slightly during the timeevolution of 0-1000 fm/c, the LBUU method used in thepresent study is sufficiently accurate for investigating theGDR width.
Results and discussions.
The collective excitation of anucleus consisting of A nucleons can be induced by addinga perturbation to its Hamiltonian at the initial time t ,i.e., ˆ H ex ( t ) = λ ˆ Qδ ( t − t ), where ˆ Q is an appropriate ex-citation operator and λ is a small parameter. The widthof a collective excitation is defined by the full width athalf maximum (FWHM) of its strength function S ( E ) asa function of the excitation energy E . In the linear re-sponse theory [53], the S ( E ) is obtained from the Fourierintegral S ( E ) = − πλ Z ∞ d t ∆ h ˆ Q i ( t )sin Et ¯ h , (6)where ∆ h ˆ Q i ( t ) = h ′ | ˆ Q | ′ i − h | ˆ Q | i is the time evolutionof the response function of the nucleus to the excitationoperator ˆ Q with | i and | ′ i denoting the nuclear statesbefore and after the perturbation, respectively. In termsof the Wigner transform q ( ~r, ~p ) of the one-body excita-tion operator ˆ q , which is related to ˆ Q by ˆ Q = P Ai ˆ q , theexpectation values in the above can be evaluated accord-ing to h ˆ Q i ( t ) = R f ( ~r, ~p, t ) q ( ~r, ~p )d ~r d ~p using the nucleonphase-space distribution function f ( ~r, ~p, t ). Details on thesingle-particle operator used in exciting a ground state nu-cleus in transport models can be found in Ref. [35].We first employ the present LBUU method to studythe effect of NN scatterings on the isovector dipole re-sponse of Pb using the excitation operator ˆ Q IVD = NA P Zi ˆ z i − ZA P Ni ˆ z i . In Fig. 1, we show the results ob-tained by using the free NN elastic scattering cross sectiontaken from Ref. [52] with σ freeNN ( p lab ) = σ freeNN (0 . /c )for neutron-neutron ( nn ) or proton-proton ( pp ) collisionsat p lab ≤ . /c and σ freeNN ( p lab ) = σ freeNN (0 .
05 GeV /c )for neutron-proton ( np ) collisions at p lab ≤ .
05 GeV /c ,as experimental data for lower incident momenta ( p lab )are unavailable. For comparison purpose, results from theLBUU calculation without NN scatterings, i.e., the Vlasovcalculation, are also shown in Fig. 1. In both cases, weuse in the initial perturbation the same parameter λ =15 MeV /c , which is also used in all the calculations in the3
100 200 300 400 500-18-12-6061218 5 10 15 20 25051015202530 D Q I V D ( f m ) t (fm/c) Vlasov s freeNN ground state S ( E ) ( f m / M e V ) E (MeV)
Vlasov s freeNN
RCNP
Figure 1: Time evolution of the isovector dipole response function∆ h ˆ Q IVD i (left) and strength function S ( E ) (right) of Pb due tothe perturbation of ˆ H ex = λ ˆ Q IVD δ ( t − t ) with λ = 15 MeV /c from the Vlasov calculation (solid lines) and the LBUU calculation(dashed lines) with σ freeNN . The dotted cyan curve in the left windowrepresents the expectation value of the ˆ Q IVD in the ground state of
Pb from the LBUU calculation with σ freeNN . present study, and we find that varying the value of λ by2 / h ˆ Q IVD i ( t ), the inclusion of NN scatterings sig-nificantly enhances the damping of the oscillations. Thedotted cyan curve in the left window of Fig. 1 representsthe expectation value of the ˆ Q IVD in the ground state of
Pb from the LBUU calculation with σ freeNN , which is neg-ligible compared with that in the excited cases. To il-lustrate more clearly the effect of collisional damping, weshow in the right window of Fig. 1 the GDR strength func-tion S ( E ) from the Fourier transformation of the responsefunction. Note that the Vlasov calculation is carried outfor a long evolution time of 1000 fm /c when the amplitudeof the oscillation of ∆ h ˆ Q i ( t ) almost vanishes so that thefluctuation in the calculated strength function from theFourier transform of ∆ h ˆ Q i ( t ) is negligible. We clearly seethe large increase of GDR width after including NN scat-terings, namely, the GDR width of Pb are 6 . . Pb has been welldetermined to be 4 . Pb( ~p, ~p ′ ) reactioncarried out at RCNP [37]. Our result from the LBUUcalculation with σ freeNN thus significantly overestimates theGDR width of Pb. This is understandable because ofthe absence of medium effect on the NN scattering in thecalculation. Its inclusion is expected to reduce the NNcross section, weaken the collisional damping, and result ina smaller GDR width. The sensitivity of the GDR widthto NN scatterings shown in Fig. 1 makes it possible toconstrain the medium effect on the NN scattering crosssection.For σ ∗ NN , we parameterize it by multiplying the free NNcross section with a medium-dependent correction factor.Specifically, we choose an exponential reduction factor as G ( M e V ) a SP6hRCNP a = 1.8 a = 1.8 shifted S ( E ) ( f m / M e V ) E (MeV)
Expt.
Figure 2: The GDR width of
Pb from LBUU calculations for dif-ferent values of α in σ ∗ NN . The horizontal line represents the RCNPexperimental value of 4 . α = 1 . suggested by the T -matrix approach in Ref. [20], i.e., σ ∗ NN = σ freeNN exp (cid:20) − α ρ/ρ nuc T c . m . /T ) (cid:21) . (7)In the above, T c . m . is the the total kinetic energy of twoscattering test nucleons at the rest frame of the local mediumor cell, ρ nuc = 0 .
16 fm − is the nuclear normal density, and T = 0 .
015 GeV. For the parameter α , its original value inRef. [20] is 0 .
6, which is called the Rostock cross section.In the present study, we treat it as a free parameter tocontrol the strength of medium effect. Displayed in Fig. 2is the GDR width Γ of
Pb obtained with different val-ues of α . As expected, the GDR width decreases withincreasing α . To reproduce the experimental value of Γ =4 . α to be as large asabout 1 .
8, which indicates a very large medium reductionof the NN scattering cross section. T Fc.m. /8 s *NN / s FreeNN T Fc.m.
FU4FP6 np nn pp a = 1.8 a = 0.6 r (fm -3 ) Fc.m.
Figure 3: Density dependence of the medium correction at differentvalues for the total kinetic energy T c . m . of two scattering nucleonsusing NN cross sections from Eq. (7) with α = 1 .
8, the Rostock crosssection with α = 0 .
6, and the FU4FP6 parameterization.
Although an early study on the balance energy, atwhich the nucleon direct flow in HICs vanishes, favors asmall medium reduction of the NN cross section [1], morerecent studies based on the analysis of the collective flowand nuclear stopping data [6, 29] as well as the nucleon4nduced reaction cross section [30] require a large mediumreduction. For comparisons, we also calculate the GDRwidth from the LBUU method with two different σ ∗ NN ,namely, the Fuchs cross section [23], which is obtainedfrom the in-medium Dirac-Brueckner T matrix, and theFU4FP6 parameterization, which is preferred by the nu-cleon induced reaction cross section [30]. The value ofGDR width of Pb calculated using the FU4FP6 param-eterization is 4 .
32 MeV, which is consistent with the exper-imental data. On the other hand, the values obtained withthe Fuchs cross section and the Rostock cross section with α = 0 . .
39 MeVand 5 .
59 MeV, respectively, which both overestimate theexperimental value. In Fig. 3, we show the density depen-dence of the medium correction σ ∗ NN /σ freeNN at three differ-ent T c . m . values for the NN cross section in Eq. (7) with α = 1 .
8, the Rostock cross section with α = 0 .
6, and theFU4FP6 parameterization with the isospin asymmetry δ set to be 0 .
21 as in
Pb. The T Fc . m . ≈ .
073 GeV in thisfigure represents the T c . m . of two nucleons at the Fermisurface of normal nuclear matter density ρ nuc . It is seenthat both the α = 1 . Pb, show similar medium reductions, which are verylarge compared with that from the Rostock cross section.In the inset of Fig. 2, we further show the strengthfunction of the iso-vector excitation of
Pb from theLBUU calculation uisng the cross section in Eq. (7) with α = 1 . . Conclusions.
We have used the LH method to solvethe BUU transport equation with the binary collisionsin the collision term treated via the stochastic approach.With the use of a sufficiently large number of test par-ticles, the present LBUU method treats the Pauli block-ing in the collision term of BUU equation with very highprecision and thus significantly increases the accuracy insolving the BUU equation. From the accurately calcu-lated GDR width of
Pb, we have found that it dependsstrongly on the magnitude of the in-medium NN cross sec-tion σ ∗ NN , and the experimentally measured GDR width of Pb from the
Pb( ~p, ~p ′ ) reaction at RCNP can only bereproduced with a NN cross section that is significantly re-duced in nuclear medium. The large medium reduction of σ ∗ NN raises challenges to microscopic calculations based onrealistic NN interactions. Also, the effects of such a largemedium reduction of σ ∗ NN on the widths of other modesof giant resonances in nuclei and on the dynamics of HICsneed to be studied as it may significantly affect the ex-tracted information on the properties of nuclear matter atvarious densities. Acknowledgements.
We thank Bao-An Li and Jun Sufor useful discussions, and Meisen Gao, Jie Pu, Chen Zhongand Ying Zhou for the maintenance of the GPU severs.This work was partially supported by the National Nat-ural Science Foundation of China under Contracts No.11905302, No. 11947214, No. 11890714, No. 11625521and No. 11421505, the Key Research Program of Fron-tier Sciences of the CAS under Grant No. QYZDJ-SSW-SLH002, the Strategic Priority Research Program of theCAS under Grants No. XDB16 and No. XDB34000000,the Major State Basic Research Development Program(973 Program) in China under Contract No. 2015CB856904,the US Department of Energy under Contract No. DE-SC0015266, and the Welch Foundation under Grant No.A-1358.
References [1] G. D. Westfall, W. Bauer, D. Craig, M. Cronqvist, E. Gaultieri,S. Hannuschke, D. Klakow, T. Li, T. Reposeur, A. M. Van-der Molen, W. K. Wilson, J. S. Winfield, J. Yee, S. J. Yennello,R. Lacey, A. Elmaani, J. Lauret, A. Nadasen, E. Norbeck, Massdependence of the disappearance of flow in nuclear collisions,Phys. Rev. Lett. 71 (13) (1993) 1986–1989.[2] X. Cai, J. Feng, W. Shen, Y.-G. Ma, J. Wang, W. Ye, In-medium nucleon-nucleon cross section and its effect on totalnuclear reaction cross section, Phys. Rev. C 58 (1) (1998) 572–575.[3] L.-W. Chen, F.-S. Zhang, G.-M. Jin, Z.-Y. Zhu, Isospin de-pendence of radial flow in heavy-ion collisions at intermediateenergies, Phys. Lett. B 459 (1-3) (1999) 21–26.[4] J.-Y. Liu, W.-J. Guo, S.-J. Wang, W. Zuo, Q. Zhao, Y.-F. Yang,Nuclear Stopping as a Probe for In-Medium Nucleon-NucleonCross Sections in Intermediate Energy Heavy Ion Collisions,Phys. Rev. Lett. 86 (6) (2001) 975–978.[5] T.-T. Wang, Y.-G. Ma, C.-J. Zhang, Z.-Q. Zhang, Effect ofin-medium nucleon-nucleon cross section on proton-proton mo-mentum correlation in intermediate-energy heavy-ion collisions,Phys. Rev. C 97 (3) (2018) 034617.[6] P.-C. Li, Y.-J. Wang, Q.-F. Li, H.-F. Zhang, Collective flow andnuclear stopping in heavy ion collisions in Fermi energy domain,Nucl. Sci. Tech. 29 (12) (2018) 177.[7] H. L. Liu, Y. G. Ma, A. Bonasera, X. G. Deng, O. Lopez,M. Veselsk´y, Mean free path and shear viscosity in central
Xe+
Sn collisions below 100 MeV/nucleon, Phys. Rev. C96 (6) (2017) 064604.[8] B. Barker, P. Danielewicz, Shear viscosity from nuclear stop-ping, Phys. Rev. C 99 (3) (2019) 034607.[9] F. Sammarruca, P. Krastev, Effective nucleon-nucleon cross sec-tions in symmetric and asymmetric nuclear matter, Phys. Rev.C 73 (1) (2006) 014001.[10] B.-A. Li, C. M. Ko, Z. Ren, Equation of State of AsymmetricNuclear Matter and Collisions of Neutron-Rich Nuclei, Phys.Rev. Lett. 78 (9) (1997) 1644–1647.[11] P. Danielewicz, R. Lacey, W. G. Lynch, Determination of theEquation of State of Dense Matter, Science 298 (5598) (2002)1592–1596.
12] L.-W. Chen, C. M. Ko, B.-A. Li, Determination of the Stiffnessof the Nuclear Symmetry Energy from Isospin Diffusion, Phys.Rev. Lett. 94 (3) (2005) 032701.[13] M. A. Famiano, T. Liu, W. G. Lynch, M. Mocko, A. M. Rogers,M. B. Tsang, M. S. Wallace, R. J. Charity, S. Komarov, D. G.Sarantites, L. G. Sobotka, G. Verde, Neutron and Proton Trans-verse Emission Ratio Measurements and the Density Depen-dence of the Asymmetry Term of the Nuclear Equation of State,Phys. Rev. Lett. 97 (5) (2006) 052701.[14] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li,W. G. Lynch, A. W. Steiner, Constraints on the Density De-pendence of the Symmetry Energy, Phys. Rev. Lett. 102 (12)(2009) 122701.[15] L. Ou, Z. Xiao, H. Yi, N. Wang, M. Liu, J. Tian, DynamicIsovector Reorientation of Deuteron as a Probe to Nuclear Sym-metry Energy, Phys. Rev. Lett. 115 (21) (2015) 212501.[16] J. Xu, L.-W. Chen, M. B. Tsang, H. Wolter, Y.-X. Zhang,J. Aichelin, M. Colonna, D. Cozma, P. Danielewicz, Z.-Q. Feng,A. Le F`evre, T. Gaitanos, C. Hartnack, K. Kim, Y. Kim, C.-M. Ko, B.-A. Li, Q.-F. Li, Z.-X. Li, P. Napolitani, A. Ono,M. Papa, T. Song, J. Su, J.-L. Tian, N. Wang, Y.-J. Wang,J. Weil, W.-J. Xie, F.-S. Zhang, G.-Q. Zhang, Understandingtransport simulations of heavy-ion collisions at 100 a and 400 a MeV: Comparison of heavy-ion transport codes under controlledconditions, Phys. Rev. C 93 (4) (2016) 044609.[17] Y.-X. Zhang, Y.-J. Wang, M. Colonna, P. Danielewicz, A. Ono,M. B. Tsang, H. Wolter, J. Xu, L.-W. Chen, D. Cozma, Z.-Q.Feng, S. Das Gupta, N. Ikeno, C.-M. Ko, B.-A. Li, Q.-F. Li,Z.-X. Li, S. Mallik, Y. Nara, T. Ogawa, A. Ohnishi, D. Oliiny-chenko, M. Papa, H. Petersen, J. Su, T. Song, J. Weil, N. Wang,F.-S. Zhang, Z. Zhang, Comparison of heavy-ion transport sim-ulations: Collision integral in a box, Phys. Rev. C 97 (3) (2018)034625.[18] G. Q. Li, R. Machleidt, Microscopic calculation of in-mediumnucleon-nucleon cross sections, Phys. Rev. C 48 (4) (1993) 1702–1712.[19] G. Q. Li, R. Machleidt, Microscopic calculation of in-mediumproton-proton cross sections, Phys. Rev. C 49 (1) (1994) 566–569.[20] T. Alm, G. R¨opke, M. Schmidt, Critical enhancement of thein-medium nucleon-nucleon cross section at low temperatures,Phys. Rev. C 50 (1) (1994) 31–37.[21] H.-J. Schulze, A. Schnell, G. R¨opke, U. Lombardo, Nucleon-nucleon cross sections in nuclear matter, Phys. Rev. C 55 (6)(1997) 3006–3014.[22] M. Kohno, M. Higashi, Y. Watanabe, M. Kawai, In-mediumnucleon-nucleon cross sections from nonrelativistic reaction ma-trices in nuclear matter, Phys. Rev. C 57 (6) (1998) 3495–3498.[23] C. Fuchs, A. Faessler, M. El-Shabshiry, Off-shell behavior of thein-medium nucleon-nucleon cross section, Phys. Rev. C 64 (2)(2001) 024003.[24] G. Mao, Z. Li, Y. Zhuo, Y. Han, Z. Yu, Study of in-mediumNN inelastic cross section from relativistic Boltzmann-Uehling-Uhlenbeck approach, Phys. Rev. C 49 (6) (1994) 3137–3146.[25] Q. Li, Z. Li, E. Zhao, Density and temperature dependence ofnucleon-nucleon elastic cross section, Phys. Rev. C 69 (1) (2004)017601.[26] B.-A. Li, P. Danielewicz, W. G. Lynch, Probing the isospin de-pendence of the in-medium nucleon-nucleon cross sections withradioactive beams, Phys. Rev. C 71 (5) (2005) 054603.[27] B.-A. Li, L.-W. Chen, Nucleon-nucleon cross sections inneutron-rich matter and isospin transport in heavy-ion reactionsat intermediate energies, Phys. Rev. C 72 (6) (2005) 064611.[28] Y. Zhang, Z. Li, P. Danielewicz, In-medium NN cross sec-tions determined from the nuclear stopping and collective flowin heavy-ion collisions at intermediate energies, Phys. Rev. C75 (3) (2007) 034615.[29] O. Lopez, D. Durand, G. Lehaut, B. Borderie, J. D. Frankland,M. F. Rivet, R. Bougault, A. Chbihi, E. Galichet, D. Guinet,M. La Commara, N. Le Neindre, I. Lombardo, L. Manduci,P. Marini, P. Napolitani, M. Pˆarlog, E. Rosato, G. Spadaccini, E. Vient, M. Vigilante, In-medium effects for nuclear matter inthe Fermi-energy domain, Phys. Rev. C 90 (6) (2014) 064602.[30] L. Ou, X.-y. He, In-medium nucleon-nucleon elastic cross-sections determined from the nucleon induced reaction cross-section data, Chinese Phys. C 43 (4) (2019) 044103.[31] V. M. Kolomietz, V. A. Plujko, S. Shlomo, Interplay betweenone-body and collisional damping of collective motion in nuclei,Phys. Rev. C 54 (6) (1996) 3014–3024.[32] N. D. Dang, Shear-viscosity to entropy-density ratio from giantdipole resonances in hot nuclei, Phys. Rev. C 84 (3) (2011)034309.[33] U. Garg, G. Col`o, The compression-mode giant resonances andnuclear incompressibility, Prog. Part. Nucl. Phys. 101 (2018)55–95.[34] A. Bracco, F. Camera, M. Mattiuzzi, B. Million, M. Pignanelli,J. J. Gaardhøje, A. Maj, T. Ramsøy, T. Tveter, Z. ´Zelazny,Increase in Width of the Giant Dipole Resonance in Hot Nuclei:Shape Change or Collisional Damping?, Phys. Rev. Lett. 74 (19)(1995) 3748–3751.[35] R. Wang, L.-W. Chen, Z. Zhang, Nuclear collective dynamicsin the lattice Hamiltonian Vlasov method, Phys. Rev. C 99 (4)(2019) 044609.[36] G. Ruetsch, M. Fatica, CUDA Fortran for Scientists and Engi-neers: Best Practices for Efficient CUDA Fortran Programming,Morgan Kaufmann, Waltham, MA, 2013.[37] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita,T. Adachi, C. A. Bertulani, J. Carter, M. Dozono, H. Fujita,K. Fujita, K. Hatanaka, D. Ishikawa, M. Itoh, T. Kawabata,Y. Kalmykov, A. M. Krumbholz, E. Litvinova, H. Matsubara,K. Nakanishi, R. Neveling, H. Okamura, H. J. Ong, B. ¨Ozel-Tashenov, V. Y. Ponomarev, A. Richter, B. Rubio, H. Sak-aguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu,F. D. Smit, T. Suzuki, Y. Tameshige, J. Wambach, R. Yamada,M. Yosoi, J. Zenihiro, Complete Electric Dipole Response andthe Neutron Skin in
Pb, Phys. Rev. Lett. 107 (6) (2011)062502.[38] P. Carruthers, F. Zachariasen, Quantum collision theory withphase-space distributions, Rev. Mod. Phys. 55 (1) (1983) 245–285.[39] G. Bertsch, S. Das Gupta, A guide to microscopic models for in-termediate energy heavy in collisions, Phys. Rep. 160 (4) (1988)189–233.[40] A. Bonasera, V. N. Kondratyev, A. Smerzi, E. A. Remler, Nu-clear dynamics in the Wigner representation, Phys. Rev. Lett.71 (4) (1993) 505–508.[41] R. J. Lenk, V. R. Pandharipande, Nuclear mean field dynamicsin the lattice Hamiltonian Vlasov method, Phys. Rev. C 39 (6)(1989) 2242–2249.[42] H. M. Xu, W. G. Lynch, P. Danielewicz, G. F. Bertsch, Dis-appearance of fusionlike residues and the nuclear equation ofstate, Phys. Rev. Lett. 65 (7) (1990) 843–846.[43] H. M. Xu, Disappearance of flow in intermediate-energynucleus-nucleus collisions, Phys. Rev. Lett. 67 (20) (1991) 2769–2772.[44] C.-Y. Wong, Dynamics of nuclear fluid. VIII. Time-dependentHartree-Fock approximation from a classical point of view,Phys. Rev. C 25 (3) (1982) 1460–1475.[45] F. Raimondi, B. G. Carlsson, J. Dobaczewski, Effective pseu-dopotential for energy density functionals with higher-orderderivatives, Phys. Rev. C 83 (5) (2011) 054311.[46] R. Wang, L.-W. Chen, Y. Zhou, Extended Skyrme interactionsfor transport model simulations of heavy-ion collisions, Phys.Rev. C 98 (5) (2018) 054618.[47] P. Danielewicz, Determination of the mean-field momentum-dependence using elliptic flow, Nucl. Phys. A 673 (1-4) (2000)375–410.[48] T. Gaitanos, A. B. Larionov, H. Lenske, U. Mosel, Breathingmode in an improved transport approach, Phys. Rev. C 81 (5)(2010) 054316.[49] H. Lin, P. Danielewicz, One-body Langevin dynamics in heavy-ion collisions at intermediate energies, Phys. Rev. C 99 (2)2019) 024612.[50] P. Danielewicz, G. F. Bertsch, Production of deuterons and pi-ons in a transport model of energetic heavy-ion reactions, Nucl.Phys. A 533 (1991) 712–748.[51] Z. Xu, C. Greiner, Thermalization of gluons in ultrarelativisticheavy ion collisions by including three-body interactions in aparton cascade, Phys. Rev. C 71 (6) (2005) 064901.[52] J. Cugnon, D. L’Hˆote, J. Vandermeulen, Simple parametriza-tion of cross-sections for nuclear transport studies up to theGeV range, Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At. 111 (3-4) (1996) 215–220.[53] A. Fetter, J. D. Walecka, Quantum Theory of Many-ParticleSystems, McGraw-Hill, New York, 1971.[54] L. Trippa, G. Col`o, E. Vigezzi, Giant dipole resonance as aquantitative constraint on the symmetry energy, Phys. Rev. C77 (6) (2008) 061304(R).[55] V. Kondratyev, A. Smerzi, A. Bonasera, Dynamics of a quantalsystem, Nucl. Phys. A 577 (3-4) (1994) 813–828.