a r X i v :n u c l -t h /0302068v 2 12 J u n 2003
Light clusters production as a probe to nuclear symmetry energy
Lie-Wen Chen,1,?C.M.Ko,1and Bao-An Li 2
1
Cyclotron Institute and Physics Department,Texas A&M University,College Station,Texas 77843-3366
2
Department of Chemistry and Physics,P.O.Box 419,
Arkansas State University,State University,Arkansas 72467-0419
(Dated:February 8,2008)Using an isospin-dependent transport model for heavy-ion collisions induced by neutron-rich nuclei at intermediate energies,we study the production of light clusters such as deuteron,triton,and 3He via coalescence of nucleons.We ?nd that both the yields and energy spectra of these light clusters are a?ected signi?cantly by the density dependence of nuclear symmetry energy,with a sti?er symmetry energy giving a larger yield.
PACS numbers:25.70.-z,25.70.Pq.,24.10.Lx
The energy per particle of an asymmetric nuclear matter with density ρand an isospin asymmetry δ=(ρn ?ρp )/ρ,where ρn and ρp are,respectively,its neu-tron and proton densities,is usually approximated by a parabolic law [1],i.e.,
E (ρ,δ)=E (ρ,0)+E sym (ρ)δ2,
(1)
where E (ρ,0)is the energy per particle of symmetric nu-clear matter and E sym (ρ)is the nuclear symmetry energy.Although the nuclear symmetry energy at normal nuclear matter density ρ0=0.16fm ?3has been determined to be around 30MeV from the empirical liquid-drop mass formula [2,3],its values at other densities are poorly known.Studies based on various theoretical models also give widely di?erent predictions [4].Lack of this knowl-edge has hampered our understanding of both the struc-ture of radioactive nuclei [5,6,7,8]and many important issues in nuclear astrophysics [9,10,11],such as the nu-cleosynthesis during pre-supernova evolution of massive stars and the properties of neutron stars [9,11].However,recent advance in radioactive nuclear beam facilities pro-vides the opportunity to study the density dependence of the nuclear symmetry energy.Theoretical studies have already shown that in heavy-ion collisions induced by neutron-rich nuclei,the e?ect of nuclear symmetry energy can be studied via the pre-equilibrium neutron/proton ratio [12],the isospin fractionation [13,14,15,16],the isoscaling in multifragmentation [17],the proton di?er-ential elliptic ?ow [18],the neutron-proton di?erential transverse ?ow [19],the π?to π+ratio [20],and two-nucleon correlation functions [21].
In this work,we study the production of deuteron,tri-ton,and 3He in heavy-ion collisions induced by neutron-rich nuclei by means of the coalescence model based on the nucleon phase space distribution functions from an isospin-dependent transport model.It is found that both the multiplicities and energy spectra of these light clus-ters are sensitive to the density dependence of nuclear symmetry energy but not to the isospin-independent part of nuclear equation of state and the in-medium nucleon-nucleon cross sections.Therefore,light clusters produc-
tion in heavy-ion collisions induced by neutron-rich nuclei provides another possible method for extracting useful information about the nuclear symmetry energy.
The coalescence model has been used extensively in describing the production of light clusters in heavy-ion collisions at both intermediate [22,23,24,25]and high energies [26,27].In this model,the probability for pro-ducing a cluster is determined by its Wigner phase-space density and the nucleon phase-space distribution at freeze out.Explicitly,the multiplicity of a M -nucleon cluster in a heavy ion collision is given by [26]
N M =G
d r i 1d q i 1···d r i M ?1d q i M ?1
×
i 1>i 2>...>i M
ρW i (r i 1,q i 1···r i M ?1,q i M ?1) .(2)
In the above,r i 1,···,r i M ?1and q i 1,···,q i M ?1are,re-spectively,the M ?1relative coordinates and momenta
taken at equal time in the M -nucleon rest frame;ρW i is the Wigner phase-space density of the M -nucleon cluster;and ··· denotes event averaging.The spin-isospin sta-tistical factor for the cluster is given by G ,and its value is 3/8for deuteron and 1/3for triton or 3He,with the lat-ter including the possibility of coalescence of a deuteron with another nucleon to form a triton or 3He [28].
For the deuteron Wigner function,it is obtained from the Hulth′e n wave function,i.e.,
φ=
2π(α?β)2
e ?αr ?e ?
βr
2
root-mean-square radii
of triton and 3He,i.e.,1.61fm and 1.74fm,respectively [30].Normal Jacobian coordi-nates for a three-particle system are then introduced to derive the Wigner functions for triton and 3He as in Ref.[26].
The space-time distribution of nucleons at freeze out is obtained from an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU)transport model (e.g.,[12,19,20,31]).For a review of the IBUU model,we refer the reader to Ref.[1].The IBUU model includes explicitly the isospin degree of freedom through di?erent proton and neutron initial distributions as well as their di?erent mean-?eld potentials and two-body collisions in subse-quent dynamic evolutions.For the nucleon-nucleon cross sections,we use as default the experimental values in free space.In order to study the e?ects due to the isospin-dependence of in-medium nucleon-nucleon cross sections σmedium ,we also use a parameterization obtained from the Dirac-Brueckner approach based on the Bonn A po-tential [32].For experimental free-space cross sections,the neutron-proton cross section is about a factor of 3larger than the neutron-neutron or proton-proton cross sections.On the other hand,the in-medium nucleon-nucleon cross sections used here have smaller magnitudes and weaker isospin dependence than σexp but strong den-sity dependence.For the isoscalar potential,we use as default the Skyrme potential with an incompressibility K 0=380MeV.This potential has been shown to re-produce the transverse ?ow data from heavy ion colli-sions equally well as a momentum-dependent soft poten-tial with K 0=210MeV [33,34].
The IBUU model is solved with the test particle method [35].Although the mean-?eld potential is evalu-ated with test particles,only collisions among nucleons in each event are allowed.Light cluster production from co-alescence of nucleons is treated similar as nucleon-nucleon collisions,i.e.,only nucleons in the same event are al-lowed to coalesce to light clusters.Results presented in the following are obtained with 20,000events using 100test particles for a physical nucleon.
For the density dependence of symmetry energy,we adopt the parameterization used in Ref.[36]for studying the properties of neutron stars,i.e.,
E sym (ρ)=E sym (ρ0)·u γ,
(4)
where u ≡ρ/ρ0is the reduced density and E sym (ρ0)=30MeV is the symmetry energy at normal nuclear matter density.In the following,we consider the two cases of γ=0.5(soft)and 2(sti?)to explore the large range of E sym (ρ)predicted by many-body theories [10].
We consider the reaction of 52Ca +48Ca,which has an isospin asymmetry δ=0.2and can be studied at fu-ture Rare Isotope Accelerator Facility.In the present study,nucleons are considered as being frozen out when their local densities are less than ρ0/8and subsequent interactions do not cause their recapture into regions of
050100150200
A v e r a g e m u l t i p l i c i t y
t (fm /c)
FIG.1:Time evolutions of the average multiplicity of (a)
deuteron,(b)triton,and (c)3He from central collisions of 52
Ca+48Ca at E =80MeV/nucleon by using the soft (solid curves)and sti?(dashed curves)symmetry energies with a sti?nuclear compressibility K 0=380MeV and free nucleon-nucleon cross sections.Results using the soft symmetry en-ergy and free nucleon-nucleon cross sections but K 0=201MeV are shown by dotted curves,while those from the soft symmetry energy and K 0=380MeV but in-medium nucleon-nucleon cross sections are given by dash-dotted curves.
higher density.Shown in Figs.1(a),(b)and (c)are time evolutions of the average multiplicity of deuteron,triton,and 3He from central collisions of 52Ca +48Ca at E =80MeV/nucleon by using the soft (solid curve)and sti?(dashed curve)symmetry energies.It is seen that production of these light clusters from a neutron-rich reaction system is very sensitive to the density depen-dence of nuclear symmetry energy.Final multiplicities of deuteron,triton,and 3He for the sti?symmetry energy is larger than those for the soft symmetry energy by 51%,73%,and 100%,respectively.This is due to the fact that the sti?symmetry energy induces a stronger pressure in the reaction system and thus causes an earlier emission of neutrons and protons than in the case of the soft symme-try energy [37],leading to a stronger correlations among nucleons.Furthermore,the soft symmetry energy,which gives a more repulsive symmetry potential for neutrons and more attractive one for protons in low density region (≤ρ0)than those from the sti?symmetry energy,gen-erates a larger phase-space separation between neutrons and protons at freeze out,and thus a weaker correlations among nucleons.The larger sensitivity of the multiplic-ity of 3He to the nuclear symmetry energy than those of triton and deuteron as seen in Fig.1re?ects the fact
3
that the symmetry energy e?ect
is stronger on lower mo-mentum protons than neutrons [37].
Although the density dependence of nuclear symme-try energy a?ects appreciably the yield of light clusters,changing the incompressibility from K 0=380to 201MeV (dotted curves)or using σmedium instead of σexp (dash-dotted curves)only leads to a small change in the yield of these clusters as shown in Fig. 1.This implies that the relative space-time structure of neutrons and protons at freeze out is not sensitive to the equation of state (EOS)of symmetric nuclear matter and in-medium nucleon-nucleon cross sections.
020406080100120
d N /d E
k i n
(1/M e V )
E ki n
(MeV)
FIG.2:Kinetic energy spectra in the center-of-mass system
for (a)deuteron,(b)triton,and (c)3He from central collisions of 52Ca+48Ca at E =80MeV/nucleon by using the soft (solid squares)and sti?(open squares)symmetry energies with a sti?nuclear compressibility K 0=380MeV and free nucleon-nucleon cross sections.
The kinetic energy spectra in the center-of-mass sys-tem for deuteron,triton,and 3He are shown in Fig.2for both the soft (solid squares)and sti?(open squares)symmetry energies.It is seen that the symmetry energy has a stronger e?ect on the yield of low energy clusters than that of high energy ones.For example,the sym-metry energy e?ect on the yield of deuteron,triton,and 3
He is about 60%,100%,and 120%,respectively,if their kinetic energies are around 10MeV,but is about 30%,40%,and 85%,respectively,if their kinetic energies are around 100MeV.This follows from the fact that lower energy clusters are emitted later in time when the size of nucleon emission source is relatively independent of nuclear symmetry energy,leading thus to a similar prob-ability for nucleons to form light clusters.Since there are more low energy nucleons for a sti?er symmetry energy,
more light clusters are thus produced.On the other hand,higher energy nucleons are emitted earlier when the size of emission source is more sensitive to the symmetry en-ergy,with a smaller size for a sti?er symmetry energy.The probability for light cluster formation is thus larger for a sti?er symmetry energy.This e?ect is,however,re-duced by the smaller number of high energy nucleons if the symmetry energy is sti?er.As a result,production of high energy light clusters is less sensitive to the sti?ness of symmetry energy.This is di?erent from that seen in the correlation functions between two nucleons with low relative momentum,where the symmetry energy e?ect is larger for nucleon pairs with higher kinetic energies [21],as they are only a?ected by the size of the emission source,not the number of emitted nucleon pairs.
20406080100120
Y (t )/Y (H e )
E (MeV)
FIG.3:The ratio t/3He as a function of the cluster kinetic energy in the center-of-mass system by using the soft and sti?symmetry energies.The lines are drawn to guide the eye.
The isobaric yield ratio t/3He is less model-dependent and also less a?ected by other e?ects,such as the feed-back from heavy fragment evaporation and the feed-down from produced excited triton and 3He states.In Fig.3,we show the t/3He ratio with statistical errors as a func-tion of cluster kinetic energy in the center-of-mass system for the soft (solid squares)and sti?(open squares)sym-metry energies.It is seen that the ratio t/3He obtained with di?erent symmetry energies exhibits very di?erent energy dependence.While the t/3He ratio increases with kinetic energy for the soft symmetry energy,it decreases with kinetic energy for the sti?symmetry energy.The symmetry energy thus a?ects more strongly the ratio of high kinetic energy triton and 3He.For both soft and sti?symmetry energies,the ratio t/3He is larger than the neutron to proton ratio of the whole reaction system,i.e.,N/Z =1.5.This is in agreement with results from both experiments and statistical model simulations for other reaction systems and incident energies [38,39,40,41].It is interesting to note that although the yield of lower
4
energy triton and3He is more sensitive to symmetry en-ergy than higher energy ones,as shown in Fig.2,their ratio at higher energy is a?ected more by the symmetry energy.Moreover,the energy dependence of the ratio t/3He is insensitive to the EOS of symmetric nuclear mat-ter and in-medium nucleon-nucleon cross sections.These features thus imply that the pre-equilibrium triton to3He ratio is also a sensitive probe to the density dependence of nuclear symmetry energy.
Isospin e?ects on cluster production and isotopic ratios in heavy ion collisions have been previously studied using either the lattice gas model[41]or a hybrid of IBUU and statistical fragmentation model[16].These studies are, however,at lower energies than considered here,where e?ects due to multifragmentation as a result of possible gas-liquid phase transition may play an important role. Except deuterons,both tritons and3He are only about one per event in heavy ion collisions at80MeV/nucleon. The number of other clusters such as the alpha particle is not large either[42,43].In this case,the coalescence model is expected to be a reasonable model for determin-ing the production of light clusters from heavy ion colli-sions.Furthermore,the e?ect obtained in present study will be enhanced if other clusters are emitted earlier as the isospin asymmetry of the residue is increased.
In conclusion,using an isospin-dependent transport model together with a coalescence model for light clus-ter production,we have found that the nuclear symme-try energy a?ects signi?cantly the production of light clusters in heavy-ion collisions induced by neutron-rich nuclei.More deuterons,tritons,and3He are produced with the sti?nuclear symmetry energy than the soft nu-clear symmetry energy.This e?ect is particularly large when these clusters have lower kinetic energies.Also, the isobaric ratio t/3He,especially for higher energy tri-tons and3He,shows a strong sensitivity to the density dependence of nuclear symmetry energy.It is further found that light clusters production is not sensitive to the isospin-independent part of nuclear equation of state and the in-medium nucleon-nucleon cross sections.The study of light clusters production in heavy ion collisions induced by neutron-rich nuclei thus allows us to extract useful information about the density-dependence of nu-clear symmetry energy.
We thank Joe Natowitz for discussions on light clusters production in heavy ion collisions.This paper is based on the work supported by the U.S.National Science Founda-tion under Grant Nos.PHY-0098805and PHY-0088934 as well as the Welch Foundation under Grant No.A-1358.LWC is also supported by the National Natural Science Foundation of China under Grant No.10105008.
?On leave from Department of Physics,Shanghai Jiao Tong University,Shanghai200030,China
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