Density determinations in heavy ion collisions
G. Röpke, S. Shlomo, A. Bonasera, J. B. Natowitz, S. J. Yennello, A. B. McIntosh, J. Mabiala, L. Qin, S. Kowalski, K. Hagel, M. Barbui, K. Schmidt, G. Giulani, H. Zheng, S. Wuenschel
DDensity determinations in heavy ion collisions
G. R¨opke, ∗ S. Shlomo, A. Bonasera,
2, 3
J. B. Natowitz, S. J. Yennello, A. B. McIntosh, J. Mabiala, L. Qin, S. Kowalski, K. Hagel, M. Barbui, K. Schmidt, G. Giulani, H. Zheng, and S. Wuenschel Institut f¨ur Physik, Universit¨at Rostock, Universit¨atsplatz 3, D-18055 Rostock, Germany Cyclotron Institute, Texas A & M University, College Station, Texas, 77843, USA Laboratorio Nazionale del Sud-INFN, v. S. Sofia 64, 95123 Catania, Italy (Dated: November 10, 2018)The experimental determination of freeze-out temperatures and densities from the yields of lightelements emitted in heavy ion collisions is discussed. Results from different experimental approachesare compared with those of model calculations carried out with and without the inclusion of mediumeffects. Medium effects become of relevance for baryon densities above ≈ × − fm − . A quantumstatistical (QS) model incorporating medium effects is in good agreement with the experimentallyderived results at higher densities. A densitometer based on calculated chemical equilibrium con-stants is proposed. PACS numbers: 24.10.Pa,24.60.-k,25.75.-q
Heavy ion collisions (HIC) are often used as a toolto investigate the properties of excited nuclear matter.Measured yields of different ejectiles as well as theirtheir energy spectra and their correlations in momentumspace can be used to infer the properties of the emit-ting source. Despite the fact that a great deal of experi-mental data has been accumulated from HIC during thelast few decades, reconstruction of the properties of thehot expanding nuclear system remains a difficult task.Two major problems are the complications inherent inincorporating non-equilibrium effects and in the treat-ment of strong correlations that are already present inequilibrated nuclear matter.An often employed simple approach to handling theseeffects is the freeze-out approximation. Starting from hotdense matter produced in HIC, this approach assumesthe attainment of local thermodynamic equilibrium aftera short relaxation time. Chemical equilibrium may alsobe established in the expanding fireball if the collisionrates in the expanding hot and dense nuclear system areabove a critical value.While more microscopic approaches employing trans-port models that describe the dynamical evolution of themany particle system are being pursued, a freeze-out ap-proach provides a very efficient means to get a generaloverview of the reaction. Such approaches have beenapplied in heavy-ion reactions, to analyze the equationof state of nuclear matter, see [1], but also recently inhigh-energy experiments (RHIC, LHC) to describe theabundances of emitted elementary particles [2]. Muchinformation on the symmetry energy, on phase instabil-ity, etc., has been obtained using this concept.Within the freeze-out approximation, the abundancesof emitted particles and clusters at freeze-out are deter-mined by the temperature T , the baryon density n B , andthe isospin asymmetry δ = ( n n − n p ) /n B , which is related ∗ Electronic address: [email protected] to the total proton fraction Y e = (1 − δ ) /
2. In this paperwe discuss the extraction of densities and temperaturesfrom the measured yields of ejectiles in HIC. We focuson the information content of neutrons ( n ), protons ( p ),deuterons ( d ), tritons ( t ), He ( h ), and He ( α ) parti-cles, emitted in near Fermi energy reactions. To extractthe relevant information we optimize the freeze-out ap-proach by including correlations and density effects usingsystematic, consistent quantum statistical approaches.We are considering only the yields Y i of these particles,the energy spectra are established by long-range interac-tions and will not be discussed here. It is possible toextend the approach also to other situations where notonly particles with A ≤ Experiments and Data Analysis Using the Nuclear Sta-tistical Equilibrium (NSE) . The NIMROD collaborationhas recently measured yields of light particles in three dif-ferent experiments performed at energies near the Fermienergy. Collisions of Zn projectiles with Mo and
Au target nuclei [3] and the collisions Zn+ Zn, Zn+ Zn, and Ni+ Ni were studied at
E/A = 35MeV/nucleon [4]. Collisions of Ar +
Sn,
Sn and Zn +
Sn,
Sn [5] were studied at 47 MeV/ nu-cleon . These experiments have been described in severalpapers [3, 5–8] where the details are given.Our goal is to derive
T, n B , and δ from the five ex-perimental yields, Y p ; Y d ; Y t ; Y h ; Y α , of the light charged Z = 1 , t/h ra-tios as indicated below.) This problem is easily solvedin the low-density limit where the NSE can be applied, a r X i v : . [ nu c l - t h ] M a y i.e. below n B ≈ − fm − and at moderate tempera-tures where medium effects can be neglected. Using thesimple relations for the non- degenerate ideal mixture ofreacting components n i = 2 s i + 1Λ i e ( E i + Z i µ p + N i µ n ) /T , (1)where Λ i = 2 π ¯ h / ( m i T ) denotes the thermal wavelength, m i the mass, s i the spin, and E i the bindingenergy of the different components, one can construct ex-pressions that are almost directly related to the differentthermodynamic parameters.In particular the ratio Y h /Y t can be used to determinethe asymmetry of the nuclear system. It can also be usedto give an estimate of the neutron yield Y n = Y p Y t Y h f δ ( T ) (2)where f δ ( T ) = exp[( E h − E t ) /T ][( m n m h ) / ( m p m t )] / isa correction that accounts for the difference in the bind-ing energies of H and He. For the sake of simplicity weuse in the following the approximation m A = Am withthe average baryon mass, m .The temperature can be determined by a double ra-tio of yields chosen so that the chemical potentials arecompensated in the NSE. According to Albergo [9] thetemperature can be obtained from T HHe = 14 . (cid:104) . Y α Y d Y t Y h (cid:105) . (3)Within the NSE framework, knowledge of the temper-ature allows the extraction of the baryon density. In [3],the yield ratio of He to H was used to determine thefree proton density according to n p = 0 . × T / exp[ − . /T ] Y α Y t , (4)similarly n n = 0 . × T / exp[ − . /T ] Y α Y h . (5)Here T is the temperature in MeV, and n i has units ofnucleons/cm .The total baryon density follows as n B =( n p /Y p ) (cid:80) i A i Y i . Consistency Test for the NSE.
Note that only ratios ofyields of bound states were used to infer the temperature,Eq. (3) and the chemical potentials, Eqs. (4), (5). Toinfer the thermodynamic parameter, also other ratios canbe considered that contain the free nucleon ( p, n ) yields.If we focus on five measured yields Y p , Y d , Y t , Y h , Y α , wehave four ratios that are of relevance to infer the threeparameters T, n B , δ that characterize the thermodynamicstate of the nuclear system. There is one additional de-gree of freedom that can be used for a consistency check.In particular, we can consider the ratio R test = 4 (cid:15) (cid:18) (cid:19) (cid:15)/ (cid:18) (cid:19) / Y (cid:15) − α Y (cid:15)p Y (cid:15) − h Y (cid:15) − t Y d (6) with (cid:15) = ( E α + E d − E t − E h ) / (2 E α − E t − E h ) = 0 . R NSEtest = 1. This quantity is alsoeasily determined from measured yields. In particular,the data obtained in the experiment Ref. [3] give in total(summed over v s ) R test = 1 .
22, the data of [5] lead to R test = 1 .
36, and the data of [8] to 1.147 (summed overall excitation energies). Apparently, in comparison withthe yields of bound nucleons, the yield Y p is higher thanexpected within NSE.Different reasons can be given for this deviation:i) The assumption of thermodynamic equilibrium is notrealized. One has to investigate the dynamical non-equilibrium expansion of the fireball produced in HIC.ii) The source is more complex.iii) The assumption of an ideal mixture of colliding, butotherwise non-interacting components (free nucleons andclusters) must be improved.We will not discuss how the freeze-out concept has tobe modified when non-equilibrium and finite size effectsare taken into account. Rather here we focus on the lastpoint - improving the approximation of an ideal mixtureby considering effects of correlations in the medium. Thiscan be done within a systematic quantum statistical ap-proach Quantum statistical (QS) approach . Within a quantumstatistical approach to nuclear matter, correlations andbound state formation are treated using Green’s func-tions to derive in-medium few-body wave equations, see[11]. Comparing to the zeroth order NSE, improvementsare obtained, in particular:i) The classical Boltzmann distribution is replaced by theFermi or Bose distributions if degenerate effects are to beaccounted for. This follows immediately from a quantumstatistical approach. In a similar spirit, the momentumquadrupole and normalized number fluctuations for lightparticle emission in HIC have been analyzed in Ref. [8].In that work it has been proposed to use the the reduc-tion of fluctuations for Fermi systems or enhancement offluctuations for Bose systems to estimate the thermody-namic parameters.ii) With increasing density, medium effects have to beincluded. Within a quasiparticle picture, the binding en-ergies of the bound states are decreasing with increasingdensity due to Pauli blocking. Depending on tempera-ture and center of mass momentum, the bound statesmerge in the continuum at the so-called Mott density.Since the composition is determined by the quasiparti-cle energies, the cluster abundances are suppressed. Asa consequence, the mass fraction of free nucleons is en-hanced compared with the NSE, see Fig. 1. The mediumeffects become of relevance when the baryon density n B exceeds a value of about 5 × − fm − . The expressions(3) - (5) used to derive the thermodynamic parametersbased on the NSE have to be correspondingly corrected,as will be shown in the following. (See also Ref. [10].)iii) In a QS approach, contributions of the continuumto the density also arise (scattering states). Within a baryon density n B [fm -3 ] fr ee p r o t on fr ac ti on X p T =11 MeVT =10 MeVT = 9 MeVT = 8 MeVT = 7 MeVT = 6 MeVT = 5 MeVT = 4 MeV
FIG. 1: (Color online) Free proton fraction as function of den-sity and temperature in symmetric matter. Restricting com-ponents to light elements A ≤
4, the QS calculations (solidlines) are compared with the NSE results (dotted lines). Nocontinuum contributions are included. The Mott effect and itstemperature dependence is clearly seen near 0.01 fm − wherethe bound state fraction disappears and the free proton frac-tion rises. virial expansion, for each channel where a bound state isformed, scattering states will also contribute to the equa-tion of state. An upper limit for the contributions of thecontinuum can be given subtracting for each bound statethe same term with zero binding energy. These contin-uum contributions are small in the region considered hereand are neglected in the present work. Future investiga-tions are needed to account the continuum correlations.Because the ratio of free nucleons to bound clustersis strongly influenced by medium effects the use of theNSE is limited to very small densities. Compared to theNSE, in the QS approach the concentration of boundstates is going down, whereas the fraction of free nucleonsincreases. This modifies the yield fractions that containthe free nucleon yields. Temperature Determinations in Low Density NuclearMatter.
At densities below the Mott point the effect ofmedium modifications on the double isotope ratios is notstrong [10]. Thus, to a good approximation the deter-mination of the temperature can be performed employ-ing the double ratios, Eq. (3). In Refs. [3, 5–7] thistechnique is employed to characterize the temperatureevolution of the expanding nascent fireball (the interme-diate velocity or nucleon-nucleon source) by associatingparticle velocity with emission time. (The Albergo ex-pression, Eq. (3), is modified by a factor (9/8) / infront of the double ratio when applied to particles withthe same surface velocity, see [3].) In Ref. [8] whichfocuses on quasiprojectile sources of different excitationenergy, temperatures have been calculated employing themomentum quadrupole fluctuation method. In the com-parisons which follow, the temperatures are those derived in the quoted references. Density Determinations . The main problem is thedetermination of the density because the influence ofmedium effects can be strong. In the following we com-pare results from four different approaches to determina-tion of the density, i) the Albergo NSE based relations[3], ii) the Mekjian coalescence model which takes intoaccount three body terms which might mimic either ahigher density (three body collisions) or Pauli blocking.[5, 12], iii) the quantum fluctuation analysis method [8],and iv) an approach based on use of the Chemical equi-librium constant employed in Refs. [5–7]. The use of thefirst three of these techniques to extract temperaturesand densities have been well described in the referencescited. The use of the chemical equilibrium constant, in-troduced in [5], to characterize the relative yields K c ( A, Z ) = ρ ( A, Z ) n Zp n ( A − Z ) n , (7)has some particular advantages. In contrast to the freeproton fraction, these chemical equilibrium constants,while sensitive to the effects of the medium, are not de-pendent on the asymmetry parameter or the choice ofcompeting species present in a model in the low-densitylimit where the NSE can be applied. Specifically, to inferthe values for the thermodynamic parameters of nuclearmatter in HIC at freeze-out from experimental data wedefine the quantity ˜ K α that is related to the chemicalequilibrium constant for α particle formation and canbe directly determined from the observed experimentalyields, ˜ K α = Y α Y p Y h Y t (cid:32)(cid:88) i A i Y i (cid:33) = n α n p n h n t n B . (8)The second relation is found by dividing the particlenumbers by their common volume. This modified chem-ical constant ˜ K α does not depend on the volume of thesystem. Note that the baryon density equals n B = n p + n n + 2 n d + 3 n t + 3 n h + 4 n α , if the ejectiles arerestricted to A ≤
4. In general, clusters with higher A must included if they are formed from the source underconsideration.Within NSE we can show thatln ˜ K NSE α = 3 ln n B + f α ( T ) . (9)Is applicable for the low-density region, f α ( T ) = ( E α +2 E h − E t ) /T + (9 /
2) ln[2 π ¯ h / ( mT )] − ln 2. The quasi-particle shifts we have previously calculated for the singlenucleons as well as for the light clusters [11], indicate thatmedium effects are relevant above the density of about n B = 5 × − fm − . In Figure 2 we present theoret-ical values of ˜ K α which have been calculated assumingincluding QS corrections (symmetric matter). The de-crease of ˜ K α for densities above 10 − fm − is due to theMott effect that bound states disappear because of Pauliblocking, see [6]. n B [fm ] Y a / Y p4 * Y h2 / Y t * ( Y p + Y n + Y d + Y t + Y h + Y a ) Data [4]T = 4 MeVT = 5 MeVT = 6 MeVT = 7 MeVT = 8 MeVT = 9 MeVT =10 MeVT =11 MeV
FIG. 2: (Color online) Chemical constant ˜ K α as func-tion of density and temperature. Data (stars) for T =5 , , , , , ,
11 MeV [5] in comparison with the NSE values(thin dotted lines) and QS calculations (bold straight lines).
In our calculations we find essentially no dependenceon the asymmetry parameter as should be expected forthe chemical equilibrium expression. In principle thisplot constitutes a densitometer which may be employedto estimate the density from experimental yields if thetemperature has been determined. However, in generalthere are two solutions so that one has to select out thecorrect one. For comparison to the theoretical valuespresented in Figure 2 we present also in that figure ex-perimental values for T = 5 to 11 MeV, derived from themeasured data discussed in Refs. [5–7]. These data rein-force the interpretation that the natural evolution of thesystems under investigation in those works encompassesdensities approaching the Mott point as was previouslyconcluded.To compare the results of using this densitometer(number iv) in our list of possible techniques), with re-sults from the other three techniques in the list we nowuse comparisons to the theoretical curves to derive den-sities from the observed experimental values of reference[5]. These derived values are slightly different than thoseextracted using a coalescence model. The comparison ofresults from different techniques of extracting T and n B from experimental data are presented in Figure 3. Theuse of the basic NSE gives unrealistically low densitiesreflecting the limitations of that model and its region ofapplicability [5, 12, 13] . This point was already apparentin the results for laboratory tests of the the astrophysicalEoS [5] that also demonstrate the relevance of medium ef-fects above n B ≈ − nuc/fm . Interestingly the resultsof the coalescence model analysis, the quantum fluctua-tion analysis presented in Figure 2 lead to very similarresults even though different systems and sources havebeen explored. Both are quite similar to the densitome-ter analysis based on QS model results. We return tothis point below. Discussion.
Substantial progress has been made in the n B [fm -3 ] T [ M e V ] [3], NSE[3]. QS[4], coalescence[4], QS[4], NSE[10], fluctuations FIG. 3: (Color online) Baryon density derived from yieldsof light elements. Data according to [3, 5, 8] are comparedwith the results of the analysis of yields using NSE and QScalculations for ˜ K α . effort to explore nuclear matter at subsaturation densi-ties. There is now experimental evidence that proves therelevance of incorporating medium effects such as Pauliblocking and the Mott effect into theoretical treatments.As expected from a quantum statistical approach, theNSE based on non-interacting components is not suffi-cient to explain the data from experiments that inves-tigate nuclear systems at densities around one tenth ofsaturation density and above. Considering the clustersas quasiparticles, a smooth transition from the NSE atlow densities to mean-field approaches at the saturationdensity can be modeled [14]. The Albergo densitometeris restricted to very low densities. The densitometer pro-posed here, based upon chemical equilibrium constantscalculated within the framework of the QS model, canbe applied at significantly higher densities.According to Fig. 2, measured yields of light elementscan be used to infer the baryon density if the temperatureis known. Despite the double valued solution, this dia-gram may serve as an important tool to derive densitiesfrom measured yields. Two other independent methodshave been used to infer densities from the yields of lightclusters:i) The Mekjian coalescence model [12] has been used. Co-alescence parameters P were calculated for the differentclusters, see [7], and used to determine volumes. Thecorresponding volume was used to convert the measuredyields into densities. The results are shown in Figs. 2and 3.ii) An alternative approach to infer the parameter val-ues for density and temperature, proposed in [8], em-ploys quadrupole momentum fluctuations and the fluc-tuations of fermion and boson numbers in the nuclearsystem. Compared with classical systems number fluc-tuations are decreased for fermion systems and increasedfor boson systems if the temperature approaches the crit-ical temperature.The density values derived by both the coalescence andfluctuation methods in rather good agreement with QSresults that include medium effects, but in disagreementwith the values derived from NSE. Only below densitiesof about 5 × − fm − , the NSE is applicable.The discrepancies with NSE are substantially reducedif medium effects such as Pauli blocking [11] or, alterna-tively, excluded volume [12, 15] are taken into account.The fact that the two different experimental results forthe temperature and density regions explored are consis-tent with each other despite the fact that they are ob-tained from quite different emitting sources and analyses,suggests that an underlying unifying feature of the EOSis responsible. Indeed, further analysis by Mabiala et al. [8] indicate that the data are sampling the vapor branchof the liquid gas coexistence curve and within the frame-work of the Guggenheim systematics may be employed todetermine the critical temperatures of mesoscopic nuclearsystems, in a manner analogous to previous treatments[16, 17]. In conclusion we note some open questions with respectto the present determination of thermodynamic parame-ter values:i) The formation of larger clusters is neglected. For theirinclusion see Ref. [15]. The chemical equilibrium con-stants are not sensitive to the formation of other clus-ters.ii) Continuum correlations are not taken into account.Possibly they are less important if the quasiparticle pic-ture and resonances for nearly bound states are included.A non-equilibrium approach is needed to follow contin-uum correlations during the expansion of the fireball.iii) The freeze-out concept is only a simplified approachand can be improved by more dynamical descriptions ofthe non-equilibrium time evolution during HIC. Acknowledgement : The work was supported by theUS Department of Energy under Grant No. DE-FG03-93ER40773 and the Robert A. Welch Foundation underGrant No. A0330. [1] J. P. Bondorf et al. , Phys. Rept. , 133 (1995).[2] A. Andronic, P. Braun-Munzinger, and J. Stachel, Nucl.Phys.
A772 , 167 (2006).[3] S. Kowalski et al. , Phys. Rev. C , 014601 (2007).[4] Z. Kohley et al. , Phys. Rev. C , 044601 (2011).[5] L. Qin et al. , Phys. Rev. Lett. , 172701 (2012).[6] K. Hagel et al. , Phys. Rev. Lett. , 062702 (2012).[7] R. Wada et al. , Phys. Rev. C , 064618 (2012).[8] H. Zheng and A. Bonasera, Phys. Lett. B , 178(2011); H. Zheng, G. Giuliani, and A. Bonasera,Nucl. Phys. A , 43 (2012); J. Mabiala et al. ,arXiv:1208.3480v1 [nucl-ex].[9] S. Albergo et al. , Nuovo Cimento A , 1 (1985). [10] S. Shlomo et al. , Phys. Rev. C , 034604 (2009).[11] G. R¨opke, Nucl. Phys. A , 66 (2011).[12] A. Z. Mekjian, Phys. Rev. C , 1051 (1978); Phys. Rev.Lett. , 640 (1977).[13] C. J. Horowitz and A. Schwenk, Nucl. Phys. A , 55(2006).[14] S. Typel et al. , Phys. Rev. C , 015803 (2010).[15] M. Hempel et al. , Nucl. Phys. A , 210 (2010).[16] J. B. Elliott et al. , Phys. Rev. C , 024609 (2003); J. B.Elliott et al. , arXiv:1203.5132 [subm. Phys. Rev. C].[17] J.B. Natowitz et al. , Int. J. Mod. Phys. E13