Dynamic evolution and the caloric curve for medium mass nuclei

27 January 2000

Physics Letters B 473 Ž2000. 29–34

Dynamic evolution and the caloric curve for medium mass nuclei J. Cibor a,1, R. Wada a , K. Hagel a , M. Lunardon a , N. Marie a , R. Alfaro a , W. Shen a , B. Xiao a , Y. Zhao a , J. Li a , B.A. Li a , M. Murray a , J.B. Natowitz a , Z. Majka b, P. Staszel b a b

Cyclotron Institute, Texas A & M UniÕersity, College Station, TX 77843-3366, USA Institute of Physics, Jagellonian UniÕersity, ul. Reymonta 4, 30-059 Krakow, ´ Poland

Received 9 September 1999; received in revised form 9 December 1999; accepted 10 December 1999 Editor: V. Metag

Abstract Self-consistent coalescence model analyses of light particle emission have been used to follow the evolution of the temperatures and densities of A f 110 nuclei produced in violent collisions induced by four different 47 AMeV projectiles. The degree of expansion of the emitting system increases with increasing projectile mass. The caloric curve derived for these expanding A f 110 nuclei plateaus near T s 7 MeV. The plateau extends from 3.5 to 6.9 MeVru excitation energy. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 25.70.Pq Keywords: Heavy ions collisions; Coalescence model; Caloric curve

Reactions leading to multifragment final states are expected to yield information on the nuclear equation of state at non-normal densities w1,2x. Here we present results of a study of the reactions 12 C q116 Sn, 108 40 100 22 Ne q Ag, Ar q Mo and 64 Zn q89 Y, all at 47 AMeV projectile energy. Self-consistent coalescence model analyses of the light particle emission in these collisions provide quantitative evidence for increasing expansion as the projectile mass increases. Densities as low as 0.35 r 0 are reached. The caloric curve for these expanded nuclei exhibits a temperature plateau near 7 MeV at excitation energies from 3.5 to 6.9 MeVru.

1

E-mail: [email protected]

The choice to study this series of collisions was guided by QMD transport model calculations carried out with the code CHIMERA w3x using a ‘‘soft’’ equation of state, K s 200 Žsee Fig. 1.. The calculations indicate that the degree of compression and the excitation energy increase monotonically with increasing projectile mass. Both excitation energy and density are those of the heaviest fragment at a given time. The combined thermal and compressional energies which are deposited in the composite nucleus lead to its increasing expansion. The first light particles are emitted at f 50 fmrc after contact, prior to the time that global thermal equilibrium is achieved. These and subsequent pre-equilibrium particles remove both mass and energy from the expanding composite nuclei. The nuclei reach their minimum

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 4 5 7 - 4

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J. Cibor et al.r Physics Letters B 473 (2000) 29–34

Fig. 1. QMD calculated trajectories in the excitation energy per nucleon – normalized density plane for the four different systems studied Žcentral collisions, bF 3 fm, were selected.. The trajectories start at the time of maximum density. The small dots mark time increments of 10 fmrc. Arrows indicate the time of first emission of particles Žnear 50 fmrc after contact.. Both calculated mass and Q z z values are indicated at the minimum calculated density. To the left of the dashed line is the spinodal region. The points with error bars represent the experimental excitation energy – density values derived with coalescence model analyses Žsquares: Mekjian, circles: Sato-Yazaki, see text.. Estimated errors on density are "20%.

densities near f 100 fmrc, at which point thermal equilibrium appears to be essentially established as indicated by Q z z , the second moment of the nucleon momentum distribution. At these minimum densities the expanded nuclei are predicted to have mass numbers essentially the same but, as seen in Fig. 1, excitation energies which range from 2 to 8 MeVru. For the heavier projectiles entrance into the spinodal region is indicated w1,2x and multifragment production is observed in the calculation. The calculated energy spectra of the emitted particles exhibit a monotonic decrease of average kinetic energy with increasing time. If light particle energies are well correlated with emission times, and evaporative or secondary emission components contribute to the spectra primarily at the lower kinetic energies, analyses of the higher energy parts of the particle energy spectra can provide information on the early stage evolution relatively uncontaminated by later

emission processes. Techniques based on analysis of higher energy particles have, in fact, been incorporated into evaluations of slope temperatures w4x and, more recently, double isotope ratio temperatures w5– 7x. The combined TAMU CsI Ball-Neutron Ball detection system was used for the experiments. The CsI Ball consists of 96 ionization chamber-CsI telescopes In seven rings and spans an angular range 208 - u L - 1708. One telescope in each CsI ring also had a 150 mm transmission-mount Si detector between the IC and CsI detectors. Forward hodoscopes combining plastic-CsI and Si-CsI detectors were also employed at smaller angles. Energy spectra and angular distributions of identified p, d, t, 3 He, 4 He and of elements of 3 - Z - 14 were obtained. Violent events corresponding to the 10% of the reaction cross section having the highest charged particle and neutron multiplicities were selected for detailed analysis. For these events multifragment emission is experimentally observed to become increasingly more probable as the projectile mass increases. Invariant velocity plots of the light particle data Žnot shown. reveal very strong similarities in the pattern of light particle emission for the four systems. A common technique to characterize light particle emission in this energy range has been to fit the observed spectra assuming contributions from three sources, a projectile-like source, an intermediate velocity source and a target-like source w5,8–10x. However, given the continuous dynamic evolution of the system, source fits should be considered as providing only a schematic picture of the emission process w10x. We have employed them to estimate the multiplicities and energy emission at each stage of the reaction. To follow the time evolution of the system in more detail a more sophisticated analysis of the particle emission is necessary. The very similar formal structures of coalescence models applied in both non-equilibrium and equilibrium conditions w8,11–13x suggest that they can provide a natural vehicle for following the time evolution of light composite particle emission from the first emission of such particles through freeze-out. In coalescence models the yields, energy spectra and angular distributions of ejected light composite particles are directly related to those of the ejected nucleons of the same velocity w8,11–13x. The phase space

J. Cibor et al.r Physics Letters B 473 (2000) 29–34

correlations which lead to these relationships are parameterized in terms of P0 , the radius of the momentum space volume within which the correlations exist. To determine the coalescence parameter, P0 , we have adopted the Coulomb corrected coalescence model formalism of Awes et al. w8x. In the laboratory frame the relationship between the observed cluster and proton differential cross sections is: d 2 N Ž Z, N, EA . dEA d V

s

ž

Nt q Np Zt q Z p

N

/

N !Z! 4

=

=

 ž

Ay1

3

Ay 1

p P03 1r2

3

2 m Ž E y Ec . d 2 N Ž 1,0, E . dEd V

0

A

/

Ž 1.

Here the double differential multiplicity for a cluster of mass number A containing Z protons and N neutrons and having a Coulomb corrected energy EA is related to the proton double differential multiplicity at the same energy per nucleon, E. Np , Nt and Z p , Zt are respectively the numbers of neutrons and protons in the projectile and target. m is the nucleon mass and Ec is the Coulomb repulsion. A strict quantitative application of the coalescence model requires knowledge of cluster, neutron and proton differential cross-sections with proper absolute normalizations. Many past applications have used only average normalizations, relating the observed inclusive yields to the total reaction cross section and assuming that the effective neutron to proton yield ratios reflected in the coalescence are the same as the neutron to proton ratios in the composite systems considered. As a result, many previous size measurements using the coalescence model provide only qualitative or semi-quantitative support for its application w11x. In this work absolute measured multiplicities for the selected violent events are employed. Further, since in the framework of the coalescence model the triton to 3 He yield ratio defines the appropriate neutron to proton ratio, we have used values derived directly from those ratios in this analysis. For this purpose we have assumed identical P0 for t and 3 He. Our measured values of the tr 3 He

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are plotted in the top of Fig. 2 as a function of Õsurf , the particle emission velocity at the nuclear surface prior to Coulomb acceleration w8x. These ratios of the ‘‘free nucleon densities’’ w7x are significantly higher than the NrZ ratios of the composite systems. Using integrated yields Albergo et al. w7x report a similar result for many of the systems they considered. Bonasera and Bertsch w29x have argued that the change in chemical potential which occurs when two separated nuclei merge is dominated by the Coulomb energy contributions and therefore the yield ratio of like mass species, which become unbound in the collision of two heavy ions, is derived from this change. Their model would strongly favor production of free neutrons over free protons in the collisions considered here. Using the observed experimental yields and the neutron to proton ratios derived from the observed triton to 3 He yield ratios, we have determined P0 as a function of the Coulomb corrected emission velocity at the nuclear surface, Õsurf , for d, t, 3 He and 4 He particles. For each individual light species all four reaction systems show similar values of P0 at high Õsurf , supporting the idea of a common emission

Fig. 2. Experimental ratios of t to 3 He emission Ža. and double isotope yield ratio temperatures Žb. as a function of Coulomb corrected velocity. Emission from the target-like fragment has been removed. The horizontal lines in Ža. indicate the range of composite nucleus NrZ values for the systems studied.

J. Cibor et al.r Physics Letters B 473 (2000) 29–34

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mechanism for the higher energy particles. The derived values of P0 decrease with decreasing surface velocity reflecting changes in the emitting system, a variation in the contributions from different emission sources or changes in the emission mechanism. A comparison of the data from the different reactions indicates that much of the decrease in P0 observed at the lowest velocities results from the latter two causes, i.e., increasing contributions from late stage evaporative decay of the target-like source or secondary decay from light fragments w14x. We have attempted to remove that contribution by extracting P0 values for isotopes of Z s 1 and 2 from observed experimental yields from which the target-like-source yields, obtained in source fits, have been subtracted. The TLF yield contributions to the total are negligible for C q Sn for the whole velocity range considered here and amount to 30% for protons in Zn q Y reaction being significantly smaller for other species. To determine the size of the emitting system we have used the thermal coalescence model of Mekjian w12x. Many experimental observations indicate that, rather than being related to ‘‘primary’’ nucleon yields, as expected in non-equilibrium coalescence models, the cluster yields are related to observed nucleon yields as predicted by equilibrium coalescence models w11,12x. This is consistent with results of transport model calculations which indicate a rapid equilibration of the emitting system w3,15–17x as well as with the apparent successes of statistical models which assume the existence of equilibrium or near-equilibrium conditions in hot expanding nuclei w18–20x. Further, the assumed validity of this model is implicit in all recent works which use double isotope yield ratios to determine temperatures w5– 7,21–26x. In the Mekjian model there is a direct relationship between the coalescence parameter and the volume of the emitting system. In terms of the P0 derived from Eq. Ž1. the relationship is: Vs

žž

Z!N ! A3 2A

1r Ž Ay1 .

/

Ž 2 s q 1. e

Ž E 0 rT .

/

3h 3 4p P03

Ž 2. Here Z, N and A are the same as in Eq. Ž1., E0 is the binding energy and s the spin of the emitted cluster and T is the temperature.

Thus, if the temperature is known the radius can be derived from the observed P0 value. We present in Fig. 2Žb., also as a function of Õsurf , double isotope yield ratio temperatures, TAlb w7x, defined from the double isotope yield ratio Ž YdrYt .r Ž Y3 rY4 .. We note that if these particles are emitHe He ted from a source of the temperature, T, having a volume maxwellian spectrum, the double isotope yield ratio evaluated at equal Õsurf is Ž 8r9. times the ratio derived from either the integrated particle yields ŽAlbergo. or the yields at a given energy above the barrier. These temperatures increase with projectile mass, decrease with decreasing Õsurf and vary only slowly in the 4–6 cmrns range. At even lower Õsurf values of TAlb f 4–5 MeV, similar to those integral values seen in other experiments w5– 7,21–27x are observed. We take this latter observation, coupled with the sharp changes in the NrZ below 3 cmrns as an indication that the spectra at these lower velocities still contain a contribution from late stage evaporation. Indeed the NrZ values and TAlb values observed in this low velocity region are very similar to those calculated using the sequential evaporation code GEMINI w28x. For this reason we do not attempt to extract emission system sizes for Õsurf - 4 cmrns. The QMD calculated relationship between Õsurf and emission time leads to an approximate time scale presented at the top of Fig. 2 and indicates that emission times near 100 fmrc, where equilibration is predicted by the calculation, are sampled at Õsurf s 4 cmrns. For the four systems studied we present, in Fig. 3Ža,b., the values of P0 and the corresponding equivalent sharp cut-off radii Žassuming a spherical volume., derived at Õsurf s 7 and 4 cmrns for triton emission observed in the angular range of 388 to 528 in the laboratory. This selection of angular range minimizes the relative contributions from secondary evaporative decay of projectile-like or target-like sources. The calculations indicate that global thermal equilibrium is not completely established when the first particles are emitted. Thus the temperatures at high Õsurf should be considered only as approximations. Inspection of Eq. Ž2. shows, however, that for T ) 5 MeV the radii derived from the measured P0 values are relatively insensitive to T. We have also derived sizes from P0 following the density matrix formalism of Sato and Yazaki w13x. In

(

J. Cibor et al.r Physics Letters B 473 (2000) 29–34

Fig. 3. Radii and relative radii derived from the coalescence model analyses for all four systems at Õsurf s 4 and 7 cmrns. Top: P0 values derived from triton data. Middle: Radii derived from triton data Žsquares: Mekjian, circles: Sato-Yazaki, see text.. Bottom: Relative radii averaged over all four particles. Error bars represent statistical errors.

that formulation of the coalescence model the size of the emitted fragment is explicitly taken into account in an attempt to refine the source size measurement. Neither chemical or thermal equilibrium is assumed. A term which appears related to temperature, in fact characterizes the nucleon momentum distribution and is often defined from the slope of the spectrum. For this evaluation as a function of time, we set the instantaneous temperature parameter equal to the Albergo temperature at the corresponding surface velocity. At 7 cmrns the extracted radii are observed to be 7 to 7.4 fm, for all four systems. At 4 cmrns the radii increase with projectile mass, indicating progressively larger sizes at later emission times. Results for other Z s 1 and 2 particles show very similar trends to those seen for the tritons, but the absolute values of the radii are larger for deuterons and smaller for alpha particles, apparently reflecting the very different binding and spatial extent of these clusters w11,13x. In relative terms, i.e., RŽ Õsurf s 4 cmrns.rRŽ Õsurf s 7 cmrns. the size increase is found to be essentially the same for t, 3 He and 4 He but appears larger for deuteron emission ŽFig. 3Žc...

33

To determine the average density associated with the system at 4 cmrns we first assumed that the highest velocity particles are emitted from an object of mass equal to the sum of the masses of the target and projectile nuclei and of density 0.90 r 0 Žsee Fig. 1.. We than used the relative values of the radii derived from the data ŽFig. 3., and the mass remaining in the system at Õsurf s 4 cmrns, i.e. the mass remaining in the target–like source, to find for the thermal model w12x the average densities sampled at Õsurf s 4 cmrns. We find these values to be 0.81 r 0 , 0.54 r 0 , 0.45 r 0 , 0.35 r 0 for the 12 C, 22 Ne, 40Ar, 64 Zn induced reactions, respectively, with uncertainties of "20%. The values obtained with Sato–Yazaki w13x model are slightly different Žsee Fig. 1.. Excitation energies remaining in the TLF source were determined using calorimetric techniques. For this purpose we combined the information on the multiplicities and kinetic energies of light charged particles and intermediate mass fragments associated with the TLF source and also neutron multiplicity information from the neutron ball. Since the neutron energies were not measured we assume an average neutron emission energy over the TLF deexcitation cascade of 2T where T s 3 MeV for the C q Sn reaction and 3.5 MeV for others. Combining this information we calculated the primary TLF excitation energy as: E w s Ý i Ž Mc p Ž i . Ec p Ž i .. q Mn En q Q q Eg , where the sum extends over particle type i, Mc p Ž i ., Mn are the average multiplicities of charged particles and neutrons, Ec p Ž i ., En are their average kinetic energies, Eg is the energy in gamma rays Ž10 MeV. and Q is the Q value for the observed de-excitation starting from the primary TLF and assuming A s 2 Z for IMF’s. We take these excitation energies to be those of the equilibrated expanded system after intermediate source emission and thus appropriate to the temperatures determined at that point. The excitation energies for the reactions under investigation range from 2.6 MeVru to 6.9 MeVru. Our results are shown by the large solid points in Fig. 1. Estimated errors on density are "20% of the plotted values. We present in Fig. 4 the double isotope yield ratio temperatures at 4.0 cmrns plotted against excitation energy. The results indicate a nearly flat caloric curve with T f 7 MeV at excitation energies from 3.5 to 6.9 MeVru. These temperatures near 7 MeV

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J. Cibor et al.r Physics Letters B 473 (2000) 29–34

out of the expanding system creating a natural limit to the excitation energy of the remaining nucleus w30,31x. We have previously suggested a similar explanation for the observed limiting temperature w9x. Acknowledgements The authors appreciate very useful conversations with A. Bonasera, C.-M. Ko, E. Gadioli and E. Fabrici. This work was supported by The Robert A. Welch Foundation and the United States Department of Energy ŽGrant No. DE-FE05-86ER40256. and the Polish Scientific Research Committee ŽGrant No. 2P03B 103 12.. References Fig. 4. Caloric curve for medium mass nuclei. Double isotope yield ratio temperatures derived in the present work are combined with results reported previously by Wada et al. w9x. Dashed lines indicate trends of a Fermi gas model calculation with two different choices of level density parameter.

which result for the expanded low density systems are significantly higher than those derived from integrated isotope yields w5–7,21–27x. They are also very close to those derived in our earlier work on the caloric curve for A f 125 nuclei in which we found a temperature of 6.8 " 0.5 MeV at 4.3 MeVru excitation energy w9x. The overlap between velocity selected double isotope ratio temperatures and temperatures deduced from the excitation dependence of our earlier slope measurements, seen in Fig. 4, suggest the reasonableness of using isotope ratio temperatures above 3 MeV excitation energy. This is consistent with the results of Ref. w23x which, for primary emissions indicate that errors should be small at density ratios, rrr 0 , lower than 0.9 . This observation, coupled with the more detailed understanding of the particle emission dynamics and system evolution obtained from the coalescence model analysis, may indicate that the temperature limit of thermally equilibrated nuclei produced in these collisions results from binding energy limitations, i.e., the unbound nucleons with high kinetic energies stream

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