- Email: [email protected]
Heating nuclear matter with GeV 3He beams1
19 March 1998
Physics Letters B 423 Ž1998. 21–26
Heating nuclear matter with GeV 3 He beams
1
K. Kwiatkowski a , A.S. Botvina b, D.S. Bracken a,2 , E. Renshaw Foxford a,3, W.A. Friedman c , R.G. Korteling d , K.B. Morley a,2 , E.C. Pollacco e, V.E. Viola a , C. Volant e a
Department of Chemistry and Physics and IUCF, Indiana UniÕersity, Bloomington, IN 47405, USA b Institute for Nuclear Research, Moscow, Russia c Department of Physics, UniÕersity of Winconsin, Madison, WI 53706, USA d Department of Chemistry, Simon Fraser UniÕersity, Burnaby, BC V5A S16, Canada e CEA DAPNIAr SPhN, C.E. Saclay, F-91191 Gir-sur-YÕette Cedex, France Received 15 August 1997; revised 3 December 1997 Editor: J.P. Schiffer
Abstract The heating curve for hot nuclei formed in the 4.8 GeV 3 He qnatAg, 197Au reactions has been derived from reconstructed excitation-energy distributions and temperatures based on 2,3 Hr 3,4 He isotope ratios. Intranuclear-cascade predictions over-estimate the excitation-energy distributions for hot thermal-like residues, but are in general agreement with the data when contributions from preequilibrium emission are included. Both targets exhibit nearly an identical temperature vs. excitation energy dependence. The heating curve initially increases rapidly, undergoes a slope change near E )rA s 2–3 MeV, and then follows a gradual monotonic increase up to E )rA s 10 MeV. The results are in qualitative agreement with predictions of EES and SMM multifragmentation models. q 1998 Published by Elsevier Science B.V. PACS: 25.70 pq; 25.55-e
The heating curve for finite nuclear matter, and its implications for the nuclear equation of state, has been investigated in several recent experiments w1–4x. Light-ion-induced multifragmentation of heavy nuclei provides a unique environment for such studies, since the disassembly of these hot systems appears to be driven primarily by thermal forces and the effects 1
Research supported by the U.S. Department of Energy, the National Science Foundation, CEA Saclay and NSERC of Canada. 2 Present address: Los Alamos National Laboratory, Los Alamos, NM 87545. 3 Present address: Microsoft Corp., Seattle, WA 98195.
of angular momentum and the compressionrdecompression cycle are minimal. The fast cascade stage of GeV light-ion-induced reactions leaves the target residue in a highly excited and density-depleted state, in the vicinity of the spinodal Ždecomposition. region w5,6x. During the ensuing final stages of equilibration, the system undergoes further cooling due to thermal expansion, accompanied by late pre-equilibrium emission of light-particles w7x and IMFs w8x ŽIMF: 3 F Z Q 15.. The residue pre- and finalfreezeout density and temperature should be reflected in the observed ejectile energy spectra, particle-particle correlations and their isotopic composi-
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 5 4 - 2
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
22
tion w9x. In this letter we examine such data to derive the heating curve for two systems formed with 4.8 GeV 3 He ions; these results are then compared with theoretical predictions w8,10x. The experiment was carried out at the Laboratoire National Saturne, using the ISiS detector array w11x to investigate the transport and thermal properties of the 3 He qnatAg, 197Au systems at 1.8, 3.6 and 4.8 GeV. In prior publications we have examined the properties of the fragmenting source as a function of various gauges of collision violence w6,12–14x. These studies have shown that IMFs ŽIMF: 3 F Z Q 15. produced in the most violent events originate primarily from a rather slowly moving source Ž Õ I rc ; 0.01. with a nearly isotropic emission pattern. This behavior suggests that the final breakup stage of multifragmentation in these very asymmetric collisions occurs from systems that are at least in ‘‘kinetic equilibrium’’. The most striking feature of the spectra is the large yield of sub-Coulomb IMFs Ž ErA - 3 MeVrnucleon. for the highest multiplicity events – which gives evidence for thermal expansion to a breakup density Q 1r3 normal density. Due to the extended nature of the Coulomb source and the lack of any substantial radial flow, the resulting ejectile energy spectra are soft, with the majority of particles having ErA Q 3 MeV. This fact imposes the need for low detection and Z-identification thresholds, which ISiS provides. Small-angle IMF-IMF correlation studies have further shown that the breakup time scale for these events is quite short, of the order of 50 fmrc w14x. In order to test transport-code predictions of deposited excitation energy and to determine the ‘‘caloric curve’’ w1x for these hot systems formed with light-ion beams, excitation-energy distributions have been constructed from the experimental spectra on an event-by-event basis. Excitation energies E ) were assigned to each event according to the prescription, Mc
E ) s Ý K iC P q Mn² K n : q Q q Eg .
Ž 1.
i
Here K iC P is the kinetic energy of each thermal-like charged-particle Ždefined below. detected in an event of thermal multiplicity Mc , transformed event-byevent into the source frame. The second term in-
volves the neutron multiplicity, Mn , and the average neutron kinetic energy ² K n :. Measured charged-particle vs. neutron correlations were used to determine Mn w15x and ² K n : was initially estimated from Coulomb-corrected proton spectra and then iterated to obtain a self-consistent value ² K n : s 2Tth , where Tth s Ž E ) ra.1r2 and ² a: Ar11 MeVy1 w3x. The reconstructed event serves to define the binding-energy difference Q, and the energy released in photons is assumed to be Eg s 2Ž Mc q Mn . MeV. Corrections are also included to account for ISiS geometry. Based upon a detailed analysis of the spectra as a function of total observed charge and deposition energy w12,16x, two assumptions have been made regarding the definition of thermal particles. The first includes only thermal-like ejectiles, defined as protons with K F 30 MeV and complex particles with K iC P - Ž 9.0Zi q 40 . MeV for 197Au , and K iC P - Ž 5.8 Zi q 46 . MeV for natAg .
Ž 2.
The second approach expands the energy acceptance to include preequilibriumrcoalescence-like ejectiles up to ErA - 30 MeVrnucleon w3x. Because of the experimental trigger and geometrical inefficiencies, the reconstruction procedure is uncertain below E ) - 200 MeV. The average residue charge for each event was calculated by subtracting from the target charge the sum of all fast-charged-particles charges; i.e. those with K iC P ) the cutoff energies in Eq. Ž2.. The average mass was obtained by scaling the total fast-charged-particle charge, as predicted by transport calculations w5,17x, and subtracting from the target mass. In Fig. 1 the top panel shows the reconstructed distribution of deposited excitation energy for the 197 Au target with both assumptions for the fragment kinetic energy acceptance. At the 10y4 probability level, the values of E ) extend up to ; 1500 MeV for the thermal-like ejectiles and up to ; 1700 MeV for the cut that includes preequilibrium emission. For the Ag target Žnot shown., these values are 900 and 1100 MeV, respectively. Also shown in Fig. 1 is a comparison with predictions of the ISABEL intranuclear-cascade ŽINC. code w17x. Relative to the reconstructed thermal-like events, the INC distribution extends to excitation energies about 200 MeV
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
higher. However, the ISABEL INC calculates the energy dissipated into the residue at the end of the fast cascade – which does not necessarily correspond to equipartition. Some of this trapped energy appears as preequilibrium H, He and IMF emission w13x, which is not in the model. Thus, the actual excitation energy should be lower than predicted by INC. This appears to be consistent with the observation in Fig. 1 that when the fragment energy acceptance is broadened to include all H, He and IMFs up to energies of ErA s 30 MeV, better agreement with the INC is obtained. In the bottom frame of Fig. 1, the mass distribution, obtained by folding the experimentally-estimated excitation-energy distribution against the average residue mass for each excitation energy, is plotted against the INC predictions. Here the agreement is less satisfactory, especially in predicting the masses of the lightest residues, but to first order, the INC code appears to provide a qualitative account of mass loss. Using the average thermal-like excitation energies and residue masses from the reconstructed data, we have determined the heating curve for the natAg and 197 Au reactions, shown in Fig. 2. Here the temperatures T corresponding to a given excitation energy per nucleon have been derived from the double-isotope-ratio method w9x using the ratio R s Ž 2 Hr 3 H.rŽ 3 Her 4 He. isotope ratios, measured at backward angles Ž1378. to minimize preequilibrium effects. For this set of ratios, the apparent isotopic temperature is given by Tapp s 144.29rln Ž1.59 R.. The energy acceptance for D E for 3,4 He ranged from the ISiS isotope-identification threshold of 32 MeV up to the cutoff for thermal particles defined by Eq. Ž2.. For 2,3 H isotopes, D E was identical, with a shift imposed to match the Coulomb peaks of the elemental spectra. The correction factor proposed by Tsang w18x is negligible for this isotope set Ž DT 0.3 MeV.. Also included in this plot is a low-energy inclusive point from the 180 MeV 4 He q116,124 Sn systems, derived from a large set of separate isotope ratios for carbon and heavier fragments w19x. The heating curve increases rapidly at low E )rA and then exhibits a distinct slope change near 2–3 MeVrnucleon as the system evolves from the liquid to cluster regimes. Thereafter, there is a gradual increase – but no plateau w1x – up to 10 MeVrnucleon. Our result for these very asymmetric
23
Fig. 1. Upper panel: Probability distribution for excitation energies with E ) ) 200 MeV for reconstructed events from 4.8 GeV 3 HeqnatAg, 197Au reactions. Squares correspond to upper energy cuts defined in Eq. Ž2. and circles are for ejectiles with Er A 30 MeV. Lines show INC predictions w17x. Lower panel: Average residue mass distribution for 197Au nuclei bombarded by 4.8 GeV 3 He.
systems is consistent with those observed in heavyion studies w2–4x. Both natAg and 197Au residues yield essentially identical results. Also shown in Fig. 2 is the behavior expected for a simple Fermi gas with temperature Tth , and predictions based on the expanding emitting source ŽEES. model of Friedman w8x and the Copenhagen statistical multifragmentation ŽSMM. model w10x, using initial residue excitation-energy, mass and charge distributions taken from INC calculations w17x. In each calculation the dashed line gives the full model prediction and the solid line shows the effect when energy cuts identical to those imposed on the experimental H and He spectra are applied to the model calculations. The ; 1 MeV
24
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
temperature difference between the full model curve and that restricted to the energy thresholds imposed on the data is a consequence of the strong dependence of the 3 Her 4 He ratio on He kinetic energy, discussed below. In comparison with the Fermi-gas estimate for the initial temperature, the data exhibit a flatter slope. Overall, the agreement of the EES and SMM models – both of which assume a phase transition – is fairly good, with the data falling ; 0.5–1 MeV above the model predictions. Qualitatively, the slope of the SMM prediction is in somewhat better accord with the data, although the EES model describes the shoulder near E )rA ; 2–3 MeV somewhat better. Both models reproduce the gradual increase of temperature with increasing E )rA res , which arises in part from the decreasing size of the residues with E ) . Also shown for the SMM model is the intrinsic thermodynamical temperature of the model, which exceeds the model isotope-ratio temperatures below E )rA - 7 MeV. This difference has been previously pointed out in Ref. w20x and more recently discussed in w21x. Two caveats should be added concerning double isotope temperatures derived from 3 Her 4 He ratios.
First, the sequential decay of hot primary fragments has a strong influence on the H and He isotope yields w22–24x. The EES and SMM models treat this process schematically and there may be significant uncertainties in their predictions for secondary decay corrections. SMM describes the initial properties of the fragments in the liquid-drop approximation, and assumes Fermi breakup for fragments with A F 16 and evaporation for heavier fragments. EES populates a limited number of excited states and unbound resonances according to a Boltzmann weight factor, followed by statistical decay. Both models predict that the major contributions to the spectra are at low energies – below 35 MeV for He. Since our experimental He isotope-identification threshold is ; 32 MeV, the net effect on the 3 Her 4 He ratios should be minor for the present analysis. In Fig. 2, it is observed that up to E )rA ; 7–8 MeV, both models predict similar results. Above this excitation energy, SMM increases much more rapidly than EES. Second, the temperatures calculated from the experimental ratios are also sensitive to the range of He kinetic energies that are included in the ratios. Whereas the 2 Hr 3 H ratios depend only weakly on fragment kinetic energy, 3 Her 4 He ratios vary
Fig. 2. Isotope-ratio temperature versus reconstructed E )rA for 4.8 GeV 3 He qnatAg, 197Au reactions. The symbols are identified on the figure; the square is from the 4 He q Sn data of Ref. w19x. Left frame compares data with the INC-EES model and right frame compares with the INC-SMM model. Solid curves are model predictions with experimental cuts imposed on H and He energy spectra. Dashed curves show the effect of removing the experimental cuts. Dotted curves show Fermi gas behavior with a s 11 MeVy1 .
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
Fig. 3. Isotope-ratio temperatures Žobserved at 137 deg. as a function of He kinetic energy for three cuts on excitation energy, as described on figure. Fragment energy acceptance is Er A 30 MeV. Dashed lines are EES model predictions for E ) r A values indicated. Error bars are statistical only.
strongly with energy w16,20x. Inclusion of the He spectrum below the ISiS identification threshold would tend to lower the calculated temperature somewhat, as shown for both the EES and SMM models in Fig. 2. The strong variations in the He isotope ratios are shown in Fig. 3, where the differential isotope-ratio
25
temperature is plotted versus He kinetic energy for three different cuts in E )rA. Here we have included all ejectiles with ErA - 30 MeV to include effects of preequilibrium emission. The most highly excited residues give the highest temperatures and these increase systematically with increasing He energy. Also shown are the corresponding EES predictions for several E )rA values. This behavior can be thought of as due to a ‘‘cooling effect’’ w20,25,26x; i.e. the hotter sources produce more energetic fragments that evolve toward the lower thermodynamic temperature at low fragment energies. Since the He spectra decrease exponentially with increasing energy, the effect on the integral He yield translates into a temperature increase of only about 1–2 MeV. Consistent with the gates set on the He spectra used to calculate T app , this may partially rationalize the differences between the data and the models in Fig. 2. In Fig. 4, the 3 Her 4 He ratio is plotted versus He kinetic energy for both Ag and Au targets at 4.8 GeV bombarding energy. The variation of these ratios with energy and angle is apparent. When gated on excitation energy, the ratio increases as a function of excitation energy, but the slope remains essentially unchanged Žas is apparent in Fig. 3.. In fact, the slope is larger at backward angles than in the for-
Fig. 4. He isotope ratios as a function of He energy observed at 43 and 137 deg for the 4.8 GeV 3 He reaction on Ag Žleft panel. and Au Žright panel.. Lines are INC-EES model predictions for 1378 Žsolid. and 438 Ždashed.. Error bars are statistical only.
26
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
ward hemisphere, suggesting that there is no kinematic regime where this effect can be avoided. Also shown in Fig. 4 are INCrEES model predictions for the He isotope ratios. The model values are consistent with the data below about 50 MeV, but deviate above this value. The INC-SMM calculations predict even less sensitivity to He kinetic energy for K i R 40 MeV. Improved agreement can be obtained up to about 90 MeV Žwhere 3 Her 4 He ; 1. by implementing a coalescence option in the code. This behavior suggests that preequilibriumrcoalescence processes play an important role in producing the observed ratios, especially at high spectral energies. In summary, the heating curve for hot residues formed in the bombardment of Ag and Au with 4.8 GeV 3 He ions has been determined from the reconstructed excitation-energy distributions and temperatures based on Ž 2 Hr 3 H.rŽ 3 Her 4 He. isotope ratios. The results show that residue excitation energies up to E )rA ; 10 MeV can be achieved in light-ion-induced reactions. The ISABEL INC code overestimates the excitation-energy distribution for thermallike events, but provides a reasonable description of the data when preequilibrium particles are included in the energy sum. The heating curve exhibits a slope discontinuity near E )rA ; 2–3 MeV after which the temperature increases more gradually up to E )rA ; 10 MeV. This is a consequence of the continuously varying system size with E )rA and the secondary de-excitation of fragments in the phase transition region. The results are in approximate agreement with both EES and SMM models; both of which assume a phase transition, although the data are about 0.5–1 MeV higher than the model predictions. The strong sensitivity of He isotope ratios on fragment kinetic energy is pointed out. This may be evidence for a ‘‘cooling’’ effect, but the failure of statistical models to reproduce these ratios at higher He kinetic energies also implies contributions to the 3 He yield from preequilibriumrcoalescence processes.
The authors wish to thank Z. Fraenkel and Y. Yariv for providing the ISABEL INC code. This work was supported by the U.S. Department of Energy, CEA Saclay, the U.S. National Science Foundation and the National Research Council of Canada.
References w1x J. Pochodzalla et al., Phys. Rev. Lett. 75 Ž1995. 1040. w2x J.B. Natowitz et al., Phys. Rev. C 52 Ž1995. R2322; R. Wada et al., Phys. Rev. C 55 Ž1997. 227. w3x J.A. Hauger et al., Phys. Rev. Lett. 77 Ž1996. 235. w4x Y.G. Ma et al., Phys. Lett. B 390 Ž1997. 41. w5x G. Wang et al., Phys. Rev. C 53 Ž1996. 1811. w6x K. Kwiatkowski et al., Phys. Rev. Lett. 74 Ž1995. 3756; E. Renshaw Foxford et al., Phys. Rev. C 54 Ž1996. 749. w7x G. Papp, W. Norenberg, Heavy-Ion Physics 1 Ž1995. 241; Z. He, J. Wu, W. Norenberg, Nucl. Phys. A 489 Ž1988. 421. w8x W.A. Friedman, Phys. Rev. C 42 Ž1990. 667. w9x S. Albergo et al., Nuovo Cimento A 89 Ž1985. 1. w10x A. Botvina et al., Nucl. Phys. A 507 Ž1990. 649. w11x K. Kwiatkowski et al., Nucl. Instrum. Meth. A 360 Ž1995. 571. w12x K.B. Morley et al., Phys. Rev. C 54 Ž1996. 737. w13x G. Wang et al., Phys. Lett. B 393 Ž1997. 290. w14x G. Wang et al., INC Report INC-119, submitted for publication. w15x F. Goldenbaum et al., Phys. Rev. Lett. 77 Ž1996. 1230; L. Pienkowski et al., Phys. Lett. B 336 Ž1994. 147. w16x D.S. Bracken, Ph.D. thesis, Indiana University, 1996. w17x Y. Yariv, Z. Fraenkel, Phys. Rev. C 26 Ž1982. 2138. w18x M.B. Tsang et al., Phys. Rev. Lett. 78 Ž1997. 3836. w19x J. Brzychczyk et al., Phys. Rev. C 47 Ž1993. 1553. w20x K. Kwiatkowski et al., in: S. Costa et al. ŽEds.., Proc. CRIS’96, World Scientific, 1996, p. 357. w21x J.P. Bondorf, A.S. Botvina, I.M. Mishustin, submitted to Phys. Rev. C. w22x M.B. Tsang et al., Phys. Rev. C 53 Ž1991. R1057. w23x F. Gulminelli, D. Durand, Nucl. Phys. A 615 Ž1997. 117. w24x Z. Majka et al., Phys. Rev. C 55 Ž1997. 2991. w25x H. Gutbrod et al., in: D.M. Sheppard ŽEd.., Rel. Heavy-Ion Winter School, Banff, Canada, 1982, p. 1. w26x H. Xi et al., Report MSUCL-1073, May 1997.
Physics Letters B 423 Ž1998. 21–26
Heating nuclear matter with GeV 3 He beams
1
K. Kwiatkowski a , A.S. Botvina b, D.S. Bracken a,2 , E. Renshaw Foxford a,3, W.A. Friedman c , R.G. Korteling d , K.B. Morley a,2 , E.C. Pollacco e, V.E. Viola a , C. Volant e a
Department of Chemistry and Physics and IUCF, Indiana UniÕersity, Bloomington, IN 47405, USA b Institute for Nuclear Research, Moscow, Russia c Department of Physics, UniÕersity of Winconsin, Madison, WI 53706, USA d Department of Chemistry, Simon Fraser UniÕersity, Burnaby, BC V5A S16, Canada e CEA DAPNIAr SPhN, C.E. Saclay, F-91191 Gir-sur-YÕette Cedex, France Received 15 August 1997; revised 3 December 1997 Editor: J.P. Schiffer
Abstract The heating curve for hot nuclei formed in the 4.8 GeV 3 He qnatAg, 197Au reactions has been derived from reconstructed excitation-energy distributions and temperatures based on 2,3 Hr 3,4 He isotope ratios. Intranuclear-cascade predictions over-estimate the excitation-energy distributions for hot thermal-like residues, but are in general agreement with the data when contributions from preequilibrium emission are included. Both targets exhibit nearly an identical temperature vs. excitation energy dependence. The heating curve initially increases rapidly, undergoes a slope change near E )rA s 2–3 MeV, and then follows a gradual monotonic increase up to E )rA s 10 MeV. The results are in qualitative agreement with predictions of EES and SMM multifragmentation models. q 1998 Published by Elsevier Science B.V. PACS: 25.70 pq; 25.55-e
The heating curve for finite nuclear matter, and its implications for the nuclear equation of state, has been investigated in several recent experiments w1–4x. Light-ion-induced multifragmentation of heavy nuclei provides a unique environment for such studies, since the disassembly of these hot systems appears to be driven primarily by thermal forces and the effects 1
Research supported by the U.S. Department of Energy, the National Science Foundation, CEA Saclay and NSERC of Canada. 2 Present address: Los Alamos National Laboratory, Los Alamos, NM 87545. 3 Present address: Microsoft Corp., Seattle, WA 98195.
of angular momentum and the compressionrdecompression cycle are minimal. The fast cascade stage of GeV light-ion-induced reactions leaves the target residue in a highly excited and density-depleted state, in the vicinity of the spinodal Ždecomposition. region w5,6x. During the ensuing final stages of equilibration, the system undergoes further cooling due to thermal expansion, accompanied by late pre-equilibrium emission of light-particles w7x and IMFs w8x ŽIMF: 3 F Z Q 15.. The residue pre- and finalfreezeout density and temperature should be reflected in the observed ejectile energy spectra, particle-particle correlations and their isotopic composi-
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 5 4 - 2
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
22
tion w9x. In this letter we examine such data to derive the heating curve for two systems formed with 4.8 GeV 3 He ions; these results are then compared with theoretical predictions w8,10x. The experiment was carried out at the Laboratoire National Saturne, using the ISiS detector array w11x to investigate the transport and thermal properties of the 3 He qnatAg, 197Au systems at 1.8, 3.6 and 4.8 GeV. In prior publications we have examined the properties of the fragmenting source as a function of various gauges of collision violence w6,12–14x. These studies have shown that IMFs ŽIMF: 3 F Z Q 15. produced in the most violent events originate primarily from a rather slowly moving source Ž Õ I rc ; 0.01. with a nearly isotropic emission pattern. This behavior suggests that the final breakup stage of multifragmentation in these very asymmetric collisions occurs from systems that are at least in ‘‘kinetic equilibrium’’. The most striking feature of the spectra is the large yield of sub-Coulomb IMFs Ž ErA - 3 MeVrnucleon. for the highest multiplicity events – which gives evidence for thermal expansion to a breakup density Q 1r3 normal density. Due to the extended nature of the Coulomb source and the lack of any substantial radial flow, the resulting ejectile energy spectra are soft, with the majority of particles having ErA Q 3 MeV. This fact imposes the need for low detection and Z-identification thresholds, which ISiS provides. Small-angle IMF-IMF correlation studies have further shown that the breakup time scale for these events is quite short, of the order of 50 fmrc w14x. In order to test transport-code predictions of deposited excitation energy and to determine the ‘‘caloric curve’’ w1x for these hot systems formed with light-ion beams, excitation-energy distributions have been constructed from the experimental spectra on an event-by-event basis. Excitation energies E ) were assigned to each event according to the prescription, Mc
E ) s Ý K iC P q Mn² K n : q Q q Eg .
Ž 1.
i
Here K iC P is the kinetic energy of each thermal-like charged-particle Ždefined below. detected in an event of thermal multiplicity Mc , transformed event-byevent into the source frame. The second term in-
volves the neutron multiplicity, Mn , and the average neutron kinetic energy ² K n :. Measured charged-particle vs. neutron correlations were used to determine Mn w15x and ² K n : was initially estimated from Coulomb-corrected proton spectra and then iterated to obtain a self-consistent value ² K n : s 2Tth , where Tth s Ž E ) ra.1r2 and ² a: Ar11 MeVy1 w3x. The reconstructed event serves to define the binding-energy difference Q, and the energy released in photons is assumed to be Eg s 2Ž Mc q Mn . MeV. Corrections are also included to account for ISiS geometry. Based upon a detailed analysis of the spectra as a function of total observed charge and deposition energy w12,16x, two assumptions have been made regarding the definition of thermal particles. The first includes only thermal-like ejectiles, defined as protons with K F 30 MeV and complex particles with K iC P - Ž 9.0Zi q 40 . MeV for 197Au , and K iC P - Ž 5.8 Zi q 46 . MeV for natAg .
Ž 2.
The second approach expands the energy acceptance to include preequilibriumrcoalescence-like ejectiles up to ErA - 30 MeVrnucleon w3x. Because of the experimental trigger and geometrical inefficiencies, the reconstruction procedure is uncertain below E ) - 200 MeV. The average residue charge for each event was calculated by subtracting from the target charge the sum of all fast-charged-particles charges; i.e. those with K iC P ) the cutoff energies in Eq. Ž2.. The average mass was obtained by scaling the total fast-charged-particle charge, as predicted by transport calculations w5,17x, and subtracting from the target mass. In Fig. 1 the top panel shows the reconstructed distribution of deposited excitation energy for the 197 Au target with both assumptions for the fragment kinetic energy acceptance. At the 10y4 probability level, the values of E ) extend up to ; 1500 MeV for the thermal-like ejectiles and up to ; 1700 MeV for the cut that includes preequilibrium emission. For the Ag target Žnot shown., these values are 900 and 1100 MeV, respectively. Also shown in Fig. 1 is a comparison with predictions of the ISABEL intranuclear-cascade ŽINC. code w17x. Relative to the reconstructed thermal-like events, the INC distribution extends to excitation energies about 200 MeV
K. Kwiatkowski et al.r Physics Letters B 423 (1998) 21–26
higher. However, the ISABEL INC calculates the energy dissipated into the residue at the end of the fast cascade – which does not necessarily correspond to equipartition. Some of this trapped energy appears as preequilibrium H, He and IMF emission w13x, which is not in the model. Thus, the actual excitation energy should be lower than predicted by INC. This appears to be consistent with the observation in Fig. 1 that when the fragment energy acceptance is broadened to include all H, He and IMFs up to energies of ErA s 30 MeV, better agreement with the INC is obtained. In the bottom frame of Fig. 1, the mass distribution, obtained by folding the experimentally-estimated excitation-energy distribution against the average residue mass for each excitation energy, is plotted against the INC predictions. Here the agreement is less satisfactory, especially in predicting the masses of the lightest residues, but to first order, the INC code appears to provide a qualitative account of mass loss. Using the average thermal-like excitation energies and residue masses from the reconstructed data, we have determined the heating curve for the natAg and 197 Au reactions, shown in Fig. 2. Here the temperatures T corresponding to a given excitation energy per nucleon have been derived from the double-isotope-ratio method w9x using the ratio R s Ž 2 Hr 3 H.rŽ 3 Her 4 He. isotope ratios, measured at backward angles Ž1378. to minimize preequilibrium effects. For this set of ratios, the apparent isotopic temperature is given by Tapp s 144.29rln Ž1.59 R.. The energy acceptance for D E for 3,4 He ranged from the ISiS isotope-identification threshold of 32 MeV up to the cutoff for thermal particles defined by Eq. Ž2.. For 2,3 H isotopes, D E was identical, with a shift imposed to match the Coulomb peaks of the elemental spectra. The correction factor proposed by Tsang w18x is negligible for this isotope set Ž DT 0.3 MeV.. Also included in this plot is a low-energy inclusive point from the 180 MeV 4 He q116,124 Sn systems, derived from a large set of separate isotope ratios for carbon and heavier fragments w19x. The heating curve increases rapidly at low E )rA and then exhibits a distinct slope change near 2–3 MeVrnucleon as the system evolves from the liquid to cluster regimes. Thereafter, there is a gradual increase – but no plateau w1x – up to 10 MeVrnucleon. Our result for these very asymmetric
23
Fig. 1. Upper panel: Probability distribution for excitation energies with E ) ) 200 MeV for reconstructed events from 4.8 GeV 3 HeqnatAg, 197Au reactions. Squares correspond to upper energy cuts defined in Eq. Ž2. and circles are for ejectiles with Er A 30 MeV. Lines show INC predictions w17x. Lower panel: Average residue mass distribution for 197Au nuclei bombarded by 4.8 GeV 3 He.
systems is consistent with those observed in heavyion studies w2–4x. Both natAg and 197Au residues yield essentially identical results. Also shown in Fig. 2 is the behavior expected for a simple Fermi gas with temperature Tth , and predictions based on the expanding emitting source ŽEES. model of Friedman w8x and the Copenhagen statistical multifragmentation ŽSMM. model w10x, using initial residue excitation-energy, mass and charge distributions taken from INC calculations w17x. In each calculation the dashed line gives the full model prediction and the solid line shows the effect when energy cuts identical to those imposed on the experimental H and He spectra are applied to the model calculations. The ; 1 MeV
24
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temperature difference between the full model curve and that restricted to the energy thresholds imposed on the data is a consequence of the strong dependence of the 3 Her 4 He ratio on He kinetic energy, discussed below. In comparison with the Fermi-gas estimate for the initial temperature, the data exhibit a flatter slope. Overall, the agreement of the EES and SMM models – both of which assume a phase transition – is fairly good, with the data falling ; 0.5–1 MeV above the model predictions. Qualitatively, the slope of the SMM prediction is in somewhat better accord with the data, although the EES model describes the shoulder near E )rA ; 2–3 MeV somewhat better. Both models reproduce the gradual increase of temperature with increasing E )rA res , which arises in part from the decreasing size of the residues with E ) . Also shown for the SMM model is the intrinsic thermodynamical temperature of the model, which exceeds the model isotope-ratio temperatures below E )rA - 7 MeV. This difference has been previously pointed out in Ref. w20x and more recently discussed in w21x. Two caveats should be added concerning double isotope temperatures derived from 3 Her 4 He ratios.
First, the sequential decay of hot primary fragments has a strong influence on the H and He isotope yields w22–24x. The EES and SMM models treat this process schematically and there may be significant uncertainties in their predictions for secondary decay corrections. SMM describes the initial properties of the fragments in the liquid-drop approximation, and assumes Fermi breakup for fragments with A F 16 and evaporation for heavier fragments. EES populates a limited number of excited states and unbound resonances according to a Boltzmann weight factor, followed by statistical decay. Both models predict that the major contributions to the spectra are at low energies – below 35 MeV for He. Since our experimental He isotope-identification threshold is ; 32 MeV, the net effect on the 3 Her 4 He ratios should be minor for the present analysis. In Fig. 2, it is observed that up to E )rA ; 7–8 MeV, both models predict similar results. Above this excitation energy, SMM increases much more rapidly than EES. Second, the temperatures calculated from the experimental ratios are also sensitive to the range of He kinetic energies that are included in the ratios. Whereas the 2 Hr 3 H ratios depend only weakly on fragment kinetic energy, 3 Her 4 He ratios vary
Fig. 2. Isotope-ratio temperature versus reconstructed E )rA for 4.8 GeV 3 He qnatAg, 197Au reactions. The symbols are identified on the figure; the square is from the 4 He q Sn data of Ref. w19x. Left frame compares data with the INC-EES model and right frame compares with the INC-SMM model. Solid curves are model predictions with experimental cuts imposed on H and He energy spectra. Dashed curves show the effect of removing the experimental cuts. Dotted curves show Fermi gas behavior with a s 11 MeVy1 .
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Fig. 3. Isotope-ratio temperatures Žobserved at 137 deg. as a function of He kinetic energy for three cuts on excitation energy, as described on figure. Fragment energy acceptance is Er A 30 MeV. Dashed lines are EES model predictions for E ) r A values indicated. Error bars are statistical only.
strongly with energy w16,20x. Inclusion of the He spectrum below the ISiS identification threshold would tend to lower the calculated temperature somewhat, as shown for both the EES and SMM models in Fig. 2. The strong variations in the He isotope ratios are shown in Fig. 3, where the differential isotope-ratio
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temperature is plotted versus He kinetic energy for three different cuts in E )rA. Here we have included all ejectiles with ErA - 30 MeV to include effects of preequilibrium emission. The most highly excited residues give the highest temperatures and these increase systematically with increasing He energy. Also shown are the corresponding EES predictions for several E )rA values. This behavior can be thought of as due to a ‘‘cooling effect’’ w20,25,26x; i.e. the hotter sources produce more energetic fragments that evolve toward the lower thermodynamic temperature at low fragment energies. Since the He spectra decrease exponentially with increasing energy, the effect on the integral He yield translates into a temperature increase of only about 1–2 MeV. Consistent with the gates set on the He spectra used to calculate T app , this may partially rationalize the differences between the data and the models in Fig. 2. In Fig. 4, the 3 Her 4 He ratio is plotted versus He kinetic energy for both Ag and Au targets at 4.8 GeV bombarding energy. The variation of these ratios with energy and angle is apparent. When gated on excitation energy, the ratio increases as a function of excitation energy, but the slope remains essentially unchanged Žas is apparent in Fig. 3.. In fact, the slope is larger at backward angles than in the for-
Fig. 4. He isotope ratios as a function of He energy observed at 43 and 137 deg for the 4.8 GeV 3 He reaction on Ag Žleft panel. and Au Žright panel.. Lines are INC-EES model predictions for 1378 Žsolid. and 438 Ždashed.. Error bars are statistical only.
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ward hemisphere, suggesting that there is no kinematic regime where this effect can be avoided. Also shown in Fig. 4 are INCrEES model predictions for the He isotope ratios. The model values are consistent with the data below about 50 MeV, but deviate above this value. The INC-SMM calculations predict even less sensitivity to He kinetic energy for K i R 40 MeV. Improved agreement can be obtained up to about 90 MeV Žwhere 3 Her 4 He ; 1. by implementing a coalescence option in the code. This behavior suggests that preequilibriumrcoalescence processes play an important role in producing the observed ratios, especially at high spectral energies. In summary, the heating curve for hot residues formed in the bombardment of Ag and Au with 4.8 GeV 3 He ions has been determined from the reconstructed excitation-energy distributions and temperatures based on Ž 2 Hr 3 H.rŽ 3 Her 4 He. isotope ratios. The results show that residue excitation energies up to E )rA ; 10 MeV can be achieved in light-ion-induced reactions. The ISABEL INC code overestimates the excitation-energy distribution for thermallike events, but provides a reasonable description of the data when preequilibrium particles are included in the energy sum. The heating curve exhibits a slope discontinuity near E )rA ; 2–3 MeV after which the temperature increases more gradually up to E )rA ; 10 MeV. This is a consequence of the continuously varying system size with E )rA and the secondary de-excitation of fragments in the phase transition region. The results are in approximate agreement with both EES and SMM models; both of which assume a phase transition, although the data are about 0.5–1 MeV higher than the model predictions. The strong sensitivity of He isotope ratios on fragment kinetic energy is pointed out. This may be evidence for a ‘‘cooling’’ effect, but the failure of statistical models to reproduce these ratios at higher He kinetic energies also implies contributions to the 3 He yield from preequilibriumrcoalescence processes.
The authors wish to thank Z. Fraenkel and Y. Yariv for providing the ISABEL INC code. This work was supported by the U.S. Department of Energy, CEA Saclay, the U.S. National Science Foundation and the National Research Council of Canada.
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