Nuclear caloric curves have been analyzed using an expanding Fermi gas
hypothesis to extract average nuclear densities. In this approach the observed
flattening of the caloric curves reflects progressively increasing expansion
with increasing excitation energy. This expansion results in a corresponding
decrease in the density and Fermi energy of the excited system. For nuclei of
medium to heavy mass apparent densities ~ 0.4 rho_0 are reached at the higher
excitation energies.Comment: 4 pages, 3 figure

B.K. Srivastava

C. Schwarz

D. Gross

E. Suraud

E.J. Moniz

F. Gulminelli

H. Feldmeier

J. B. Natowitz

J. Besprovany

J. Bondorf

J. Cibor

J. Richert

J. Wang

J.A. Hauger

J.B. Elliott

J.B. Natowitz

K. Hagel

L. Qin

M. Kleban

M. Kleine Berkenbusch

M. Murray

M.A. Hasan

P. Bonche

P. Desesquelles

P.F. Bortignon

R. Hasse

R. Wada

S. Levit

S. Shlomo

W.A. Friedman

W.D. Myers

Y. Ma

Y. Sugawa

English

arXiv

Natowitz, J. B.

Hagel, K.

Ma, Y.

Murray, M.

Qin, L.

Shlomo, S.

Wada, R.

Wang, J.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:nucl-ex/0205005v2  8 Aug 2002Caloric Curves and Nuclear ExpansionJ. B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin, S. Shlomo, R. Wada, and J. WangCyclotron Institute, Texas A&M University,College Station, Texas, 77845(Dated: June 18, 2018)Nuclear caloric curves have been analyzed using an expanding Fermi gas hypothesis to extractaverage nuclear densities. In this approach the observed flattening of the caloric curves reflectsprogressively increasing expansion with increasing excitation energy. This expansion results in acorresponding decrease in the density and Fermi energy of the excited system. For nuclei of mediumto heavy mass apparent densities ∼ 0.4ρ0 are reached at the higher excitation energies.PACS numbers: 24.10.i,25.70.GhIn a recent paper measurements of nuclear specificheats from a large number of experiments were employedto construct caloric curves for five different regions of nu-clear mass[1]. Within experimental uncertainties each ofthese caloric curves exhibits a plateau region at higherexcitation energy. In reference 1 it was pointed out thatapplying the simple Fermi gas relationship between exci-tation energy, E∗, and temperature, T , for a nucleus ofmass ,A,E∗ =AKT 2 (1)whereK is the inverse nuclear level density parameter, toa large body of caloric curve data leads, at low excitationenergy, to an increase in the apparent value ofK, followedat higher excitation energies by a rather rapid decrease inthe apparent value of K[1, 2, 3, 4, 5, 6]. These results arepresented in Figure 1. For this figure the data presentedin Figure 5 of reference 1 have been smoothed by makinga running average of three points adjacent in excitationenergy. This reduces the scatter in the points and makesthe trends in the data clearer. Together these data spana range of fragmenting system masses 30 < A < 240.Previous work has shown that the initial heating ofthe system leads to a small expansion of the nucleus,the washing out of shell effects and an increase in thenucleon effective mass[2, 3, 4, 5, 6]. This results in a de-crease of the nuclear level density, reflected as an increasein the inverse level density parameter, K[2, 3, 4, 5, 6].The magnitudes of these effects are mass dependent[4, 5].These trends are seen very clearly in Figure 1 at excita-tion energies < 3 MeV/nucleon. However, at excitationenergies ranging from ∼ 3 MeV/nucleon for the heaviestsystems to ∼ 8 MeV/nucleon for the lighter systems, theinverse level density parameters start to decrease withincreasing excitation energy, reaching low values, near 6MeV, at the highest excitation. (Exceptions are the threehighest points in the A= 100-140 window. As discussedin reference 1, these data points may indicate onset of aqualitatively different feature but this is not verified bythe other data available.)As can be seen in Figure 1, the general trend of thedecrease of K is similar to that which results from atemperature which remains constant with increasing ex-citation energy. This onset of the decrease, reflecting theonset of a plateau in the caloric curves has been takenas the signal of a new behavior. Based on comparisonswith calculations[2, 3, 4, 5, 7] we and others have previ-ously described it as a departure from normal Fermi gasbehavior.To explore the nature of that departure in greater de-tail we note that in the simple, non dissipative, uniformdensity Fermi gas model[8] the relationship between thethermal excitation energy per nucleon, ǫth , the temper-ature, T , and the Fermi energy , ǫF , is justǫth =π2T 24ǫF (ρ)(2)Using the usual expression for a, the Fermi gas level den-sity parameter,a =AK(ρ)=π24ǫF (ρ)(3)K(ρ), the inverse level density parameter for an expandednucleus of equilibrium density, ρeq, may be written[9]K(ρeq) =T 2ǫth(ρeqρ0) 23(m∗(ρ0)m∗(ρeq))(4)where ρ0 is the normal nuclear density and m∗ is theratio of the effective mass of the nucleon to the mass ofthe free nucleon. At the temperatures at which we areinterested in using this expression m∗ should be close to1, indicating that the shell effects and collective effectsleading to increased level density above that predicted bythe Fermi gas model are no longer important[2, 3, 4, 5].Above the excitation energy wherem∗ goes to 1, the rela-tive density of a heated uniform Fermi gas at equilibriumcan be extracted from the ratio of the apparent inverselevel density parameter, K(ρeq), to K0, the inverse level2density parameter for a uniform Fermi gas, with m∗ as-sumed to be 1, at T = 0.ρeqρ0=(K(ρeq)K0) 32(5)We have used this relationship of equation (5) to derivethe excitation energy dependence of the relative nucleardensity, ρeq/ρ0. We applied this method to the deter-mination of ρeq/ρ0 points for each mass region only atexcitation energies at and above the entry points intothe caloric curve plateaus. These points have been de-rived from the caloric curves in reference 1 and shown tohave a mass dependence consistent with that predictedby calculations which indicate the onset of Coulomb in-stabilities in the heated nucleus[7, 10, 11]. In evaluatingthis expression we have used the Fermi energy determi-nations of Moniz et al.[12] in equation (3) to determinethe appropriate reference value of K0 for the mass regionbeing investigated.To evaluate K(ρeq) and the density it is required thatǫth be determined. We used an iterative technique to de-rive of ρeq/ρ0 and ǫth. This technique consisted of firstestimating the relative density as (Kapp/K0)3/2, employ-ing this density estimate to evaluate the binding energydifference associated with expansion and subtracting thatfrom the total excitation energy E∗ to obtain ǫth. Usingthis estimate of ǫth allowed a new estimate of K(ρeq) anddensity to be made. The process was repeated until selfconsistent values of ǫth and ρeq/ρ0 were obtained. Thebinding energies at different densities were calculated us-ing a liquid drop model mass formula[13] with the bulkterm extracted from the density and relative neutron ex-cess dependent nuclear matter equation of state of My-ers and Swiatecki[14]. The surface term was scaled asthe bulk term and the Coulomb term corrected for theexpansion. The binding energy changes evaluated in thisway show a near parabolic dependence on density simi-lar to that employed in the Expanding Emitting SourceModel of Friedman[15]. However the absolute values areslightly lower.The results, derived from the inverse level density dataof Figure 1, are presented in Figure 2 as a plot of rela-tive density, ρeq/ρ0, versus excitation energy per nucleon.While the mass dependence seen in Figure 1 continues tobe apparent in Figure 2, the general trend of decreasingnuclear density is similar in each mass region. In thepresent ansatz, the decreasing values of K with excita-tion energy, seen in figure 1 lead to decreasing nucleardensities. We see that at the entry into the caloric curveplateaus, densities near 0.7ρ0 are indicated. Over a spanof a few MeV/nucleon in excitation energy the densitiesdrop into the range of ∼ 0.4ρ0, a value indicated by thethin horizontal line on the figure. At higher excitationthe densities extracted in this manner remain near 0.4ρ0.Modest expansion is a general feature of calculationsfocusing on the thermal properties of heated nuclei[5, 7,10, 16]. For comparison to the present results we haveplotted in Figure 2 calculated average densities derivedfrom the recent finite temperature Hartree-Fock (FTHF)calculations of Hasan and Vary for 90Zr[16] and densi-ties for A = 120 estimated from a model which assumesa trapezoidal density distribution and phenomenologi-cal dependencies on temperature derived from previousworks[5, 17]. As seen the two results are quite similar.The result of the FTHF calculations are limited to en-ergies below ∼ 4 MeV because the model space takes alimited number of oscillator shells in to account. Com-parison of the data with these calculations indicates thatabove the Coulomb instability point the density decreasesmore rapidly with increasing excitation energy.While direct measurement of the size or density ofthe fragmenting system are difficult to obtain, manyprevious theoretical and experimental results have sug-gested that low densities such as those indicated bythe present analysis are reached in multi-fragmentingsystems[15, 18, 19, 20, 21, 22, 23, 24, 25, 26]. We havepreviously reported average densities deduced from co-alescence measurements for fragmenting medium massnuclei[26]. In Figure 3, the average densities obtainedwith that coalescence approach are compared with thepresent results for the 100 < A < 140 mass region.The average densities indicated by these two independenttechniques are seen to be in good agreement. Bracken etal.[27] have extracted sizes and average densities versusexcitation energy by analyzing barriers for ejectiles inhigh energy 3He + 197Au collisions. These, also plottedin figure 3, result in slightly lower densities.In conclusion, it appears that this simple hypothesisof expansion of an equilibrated non-dissipative Fermi gasis sufficiently accurate to allow density information to bederived from reported caloric curve measurements. Thepresent experimental results indicate that, as nuclei areheated, a more rapid rate of expansion with increasingexcitation energy commences at the Coulomb instabilitytemperature. This was suggested already in the earlywork of Bonche, Vautherin and Levit[7, 10]. The massdependence of the Coulomb instability temperature issuch that the excitation energy and temperature at whichthis occurs increases with decreasing mass. At the higherexcitation energies densities near 0.4ρ0 are reached. Formedium to heavy mass systems, some recent studies havepresented evidence for observation of critical behavior inthe excitation energy range where the present work indi-cates a rapid decrease in density[28, 29, 30]. The criticaltemperatures reported in those studies are close to theCoulomb instability temperatures. The influence of theCoulomb energy on observables used to determine thecritical point is not completely clarified at present[31, 32].3ACKNOWLEDGEMENTSThis work was supported by the U S Department ofenergy under Grant DE-FG03- 93ER40773 and by theRobert A. Welch Foundation.[1] J.B. Natowitz, R. Wada, K. Hagel, T Keutgen, M. Mur-ray, Y. G. Ma*, A. Makeev, L. Qin , P. Smith and C.Hamilton, Phys. Rev. C 65, 034618 (2002).[2] R. Hasse and P. Schuck, Phys. Lett. B 179, 313 (1986).[3] P.F. Bortignon and C.H. Dasso, Phys. Lett. B189, 381(1987).[4] S. Shlomo and J.B. Natowitz, Phys. Lett. B252, 187(1990).[5] S. Shlomo and J.B. Natowitz, Phys. Rev. C 44, 2878(1991).[6] J.B. Natowitz, D. Fabris, F. Haddad, K. Hagel, J. Li, Y.Lou, N. Mdeiwayeh, G. Nebbia, G. Prete, R. Tezkratt,D. Utley, G. Viesti, R. Wada and B. Xiao. Proceedings ofthe Nuclear Chemistry Award Symposium, 209th Meet-ing of the American Chemical Society, Anaheim, WorldScientific Press, Singapore, (1995) p 1.[7] P. Bonche, S. Levit and H. Vautherin, Nucl. Phys.A427,278 (1984); Ibid, A436, 265 (1986).[8] M.A. Preston, Physics of the Nucleus, Addison-WesleyPublishing Company, Massachusetts (1962) p 125.[9] W. Norenberg, G. Papp and P. Rozmej GSI Preprint2002-3, January, 2002 B.[10] S.Levit and P.Bonche, Nucl. Phys. A437, 426 (1985).[11] J. Besprovany and S. Levit Phys. Lett. B217, 1 (1989).[12] E. J. Moniz, I. Sick, R.R. Whitney, JR Ficenec, R.P.Kephart, W. Trower, Phys. Rev. Lett. 26, 445 (1971).[13] M. Kleban, B. NerloPomorska, J. F. Berger, J. Decharge´,M. Girod, and S. Hilaire, Phys. Rev. C 65, 024309 (2002).[14] W. D. Myers and W. J. Swiatecki, Phys. Rev. C 57, 3020(1998).[15] W. A. Friedman, Phys. Rev. Lett. 60, 2125 (1988).[16] M. A. Hasan and J. P. Vary, Phys. Rev. C 58, 2754(1998).[17] S. Shlomo, private communication.[18] E. Suraud, C. Gregoire and B. Tamain, Prog Part andNucl. Phys. 23, 357 (1989) and references therein.[19] D. Gross et al., Prog. Part.and Nucl. Phys. 30, 155 (1993)and references therein.[20] J. Bondorf et al., Nucl. Phys. A433, 321 (1985).[21] Y. Sugawa and H. Horiuchi, Phys. Rev. C 60, 607 (1999).[22] H. Feldmeier and J. Schnack, Rev. Mod. Phys. 72, 655(2000).[23] F. Gulminelli and D. Durand, Nucl. Phys. A615, 117(1997).[24] P. Desesquelles, M. D’Agostino, A. S. Botvina, M. Bruno,N. Colonna, A. Ferrero, M. L. Fiandri, E. Fuschini, F.Gramegna, I. Iori, G. V. Margagliotti, P. F. Mastinu, P.M. Milazzo, A. Moroni, F. Petruzzelli, R. Rui, G. Van-nini, J. D. Dinius, C. K. Gelbke, T. Glasmacher, D.O.Handzy, W. C. Hsi, M. Huang, M. A. Lisa, W. G. Lynch,C. P. Montoya, G. F. Peaslee, L. Phair, C. Schwarz, M.B.Tsang and C. Williams, Nucl. Phys. A633, 547 (1998).[25] C. Schwarz, S. Fritz, R. Bassini, M. Begemann-Blaich,S.J. Gaff-Ejakov, D. Gourio, G. C ross, G. Imme, I.Iori, U. Kleinevoss, G. J. Kunde, W.D. Kunze, U. Ly-nen, V. Maddalena, M. Mahi, T. Mohlenkamp, A. Mo-roni, W.F.J. Muller, C. Nociforo, B. Ocker, T. Odeh, F.Petruzzelli, J. Pochodzalla, G. Raciti, G. Riccobene, F.P.Romano, A. Saija, M. Schnittker, A. Schuttauf, W. Sei-del, V. Serfling, C. Sfienti, W. Trautmann, A. Trzcinski,G. Verde, A. Worner, H.F. Xi and B. Zwieglinski, Nucl.Phys. A681, 279 (2001).[26] J. Cibor, R. Wada, K. Hagel, M. Lunardon, N. Marie, R.Alfaro, W. Q. Shen, B. Xiao, Y. Zhao, J. Li, B. A. Li, M.Murray, J. B. Natowitz, Z. Majka and P. Staszel, Phys.Lett. B473, 29 (2000).[27] D. Bracken, Thesis, Indiana University (1996), To bepublished.[28] J. A. Hauger, B. K. Srivastava, S. Albergo, F. Bieser,F. P. Brady, Z. Caccia, D. A. Cebra, A. D. Chacon,J. L. Chance, Y. Choi, S. Costa, J. B. Elliott,1, M. L.Gilkes, A. S. Hirsch, E. L. Hjort, A. Insolia, M. Justice,D. Keane, J. C. Kintner, V. Lindenstruth, M. A. Lisa, H.S. Matis, M. McMahan, C. McParland, W. F. J. Mueller,D. L. Olson, M. D. Partlan, N. T. Porile, R. Potenza,G. Rai, J. Rasmussen, H. G. Ritter, J. Romanski, J. L.Romero, G. V. Russo, H. Sann, R. P. Scharenberg, A.Scott, Y. Shao, T. J. M. Symons, M. Tincknell, C. Tuve,S. Wang, P. Warren, H. H. Wieman, T. Wienold, and K.Wolf, Phys. Rev. C 62, 024616 (2000).[29] J. B. Elliott, L. G. Moretto, L. Phair, G. J. Wozniak,T. Lefort, L. Beaulieu, K. Kwiatkowski, W.-C. Hsi, L.Pienkowski, H. Breuer, R. G. Korteling, R. Laforest, E.Martin, E. Ramakrishnan, D. Rowland, A. Ruangma, V.E. Viola, E. Winchester, S. J. Yennello Preprint LBNL-47711, Phys. Rev. Lett. 88, 042701 (2002)[30] M. Kleine Berkenbusch, W. Bauer, K. Dillman, and S.Pratt, L. Beaulieu, K. Kwiatkowski, T. Lefort, W.-c. Hsi,and V. E. Viola, S. J. Yennello, R. G. Korteling and H.Breuer, Phys. Rev. Lett. 88, 022701 (2002).[31] B. K. Srivastava, S. Albergo, F. Bieser, F. P. Brady, Z.Caccia, D. A. Cebra, A. D. Chacon, J. L. Chance, Y.Choi, S.Costa, J. B. Elliott, M. L. Gilkes, J. A. Hauger,A. S. Hirsch, E. L. Hjort, A. Insolia, M. Justice, D.Keane, J. C. Kintner, V. Lindenstruth, M. A. Lisa, H.S. Matis, M. McMahan, C. McParland, W. F. J. Mu¨ller,D. L. Olson, M. D. Partlan, N. T. Porile, R. Potenza,G. Rai, J. Rasmussen, H. G. Ritter, J. Romanski, J. L.Romero, G. V. Russo, H. Sann, R. P. Scharenberg, A.Scott, Y. Shao, T. J. M. Symons, M. Tincknell, C. Tuv,S. Wang, P. Warren,H. H. Wieman, T. Wienold, and K.Wolf, Phys. Rev. C 64, 041605(R) (2001).[32] J. Richert, and P. Wagner, Phys. Rep. 350, 1 (2001).430 < A < 6060 < A < 100100 < A < 140140 < A < 180180 < A < 240 E*/A, MeV/u0 5 10 15 K, MeV05101520FIG. 1: Apparent inverse level density parameter K versusexcitation energy. Symbols show results for five different massregions; The meaning of the various symbols is indicated inthe figure. The thicker lines represent K values calculatedassuming constant temperatures of 6 MeV(solid), 7 MeV(dot-dashed) and 8 MeV (dashed).30 < A < 6060 < A < 100100 < A < 140140 < A < 180180 < A < 240A=120 CalcZr Calc90  E*/A (MeV/nucleon)0 5 10 150ρ/eqρ 00.51FIG. 2: Derived average nuclear density versus excitationenergy. Symbols show results for five different mass regions;The meaning of the various symbols is indicated in the figure.The thin horizontal line marks ρ = 0.4. The two lines startingat ρ = 1 represent calculated average densities for 90Zr[16](solid) and A=120[5, 17] (dot-dashed)(See text)5100 < A < 140Coalescence AnalysisAu197He + 3 A=120 CalcZr Calc90  E*/A (MeV/nucleon)0 5 10 150ρ/eqρ 00.51FIG. 3: Average nuclear density versus excitation energy.Open diamonds show results for 100 < A < 140, derivedin the present work. Solid squares with error bars repre-sent results of a coalescence model analysis of light particleemission[26]. Solid circles represent results from reference 27.The thin horizontal line marks ρ = 0.4. The two lines start-ing at ρ = 1 represent calculated average densities for 90Zr[16](dot-dashed) and A

Physical Review C

Caloric curves and nuclear expansion

Texas A&M University

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Caloric Curves and Nuclear Expansion
J. B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin, S. Shlomo, R. Wada, and J. Wang
Cyclotron Institute, Texas A&M University,
College Station, Texas, 77845
(Dated: June 18, 2018)
Nuclear caloric curves have been analyzed using an expanding Fermi gas hypothesis to extract
average nuclear densities. In this approach the observed flattening of the caloric curves reflects
progressively increasing expansion with increasing excitation energy. This expansion results in a
corresponding decrease in the density and Fermi energy of the excited system. For nuclei of medium
to heavy mass apparent densities ∼ 0.4ρ0 are reached at the higher excitation energies.
PACS numbers: 24.10.i,25.70.Gh
In a recent paper measurements of nuclear specific
heats from a large number of experiments were employed
to construct caloric curves for five different regions of nu-
clear mass[1]. Within experimental uncertainties each of
these caloric curves exhibits a plateau region at higher
excitation energy. In reference 1 it was pointed out that
applying the simple Fermi gas relationship between exci-
tation energy, E∗, and temperature, T , for a nucleus of
mass ,A,
E∗ =
A
K
T 2 (1)
whereK is the inverse nuclear level density parameter, to
a large body of caloric curve data leads, at low excitation
energy, to an increase in the apparent value ofK, followed
at higher excitation energies by a rather rapid decrease in
the apparent value of K[1, 2, 3, 4, 5, 6]. These results are
presented in Figure 1. For this figure the data presented
in Figure 5 of reference 1 have been smoothed by making
a running average of three points adjacent in excitation
energy. This reduces the scatter in the points and makes
the trends in the data clearer. Together these data span
a range of fragmenting system masses 30 < A < 240.
Previous work has shown that the initial heating of
the system leads to a small expansion of the nucleus,
the washing out of shell effects and an increase in the
nucleon effective mass[2, 3, 4, 5, 6]. This results in a de-
crease of the nuclear level density, reflected as an increase
in the inverse level density parameter, K[2, 3, 4, 5, 6].
The magnitudes of these effects are mass dependent[4, 5].
These trends are seen very clearly in Figure 1 at excita-
tion energies < 3 MeV/nucleon. However, at excitation
energies ranging from ∼ 3 MeV/nucleon for the heaviest
systems to ∼ 8 MeV/nucleon for the lighter systems, the
inverse level density parameters start to decrease with
increasing excitation energy, reaching low values, near 6
MeV, at the highest excitation. (Exceptions are the three
highest points in the A= 100-140 window. As discussed
in reference 1, these data points may indicate onset of a
qualitatively different feature but this is not verified by
the other data available.)
As can be seen in Figure 1, the general trend of the
decrease of K is similar to that which results from a
temperature which remains constant with increasing ex-
citation energy. This onset of the decrease, reflecting the
onset of a plateau in the caloric curves has been taken
as the signal of a new behavior. Based on comparisons
with calculations[2, 3, 4, 5, 7] we and others have previ-
ously described it as a departure from normal Fermi gas
behavior.
To explore the nature of that departure in greater de-
tail we note that in the simple, non dissipative, uniform
density Fermi gas model[8] the relationship between the
thermal excitation energy per nucleon, ǫth , the temper-
ature, T , and the Fermi energy , ǫF , is just
ǫth =
π2T 2
4ǫF (ρ)
(2)
Using the usual expression for a, the Fermi gas level den-
sity parameter,
a =
A
K(ρ)
=
π2
4ǫF (ρ)
(3)
K(ρ), the inverse level density parameter for an expanded
nucleus of equilibrium density, ρeq, may be written[9]
K(ρeq) =
T 2
ǫth
(
ρeq
ρ0
) 2
3
(
m∗(ρ0)
m∗(ρeq)
)
(4)
where ρ0 is the normal nuclear density and m
∗ is the
ratio of the effective mass of the nucleon to the mass of
the free nucleon. At the temperatures at which we are
interested in using this expression m∗ should be close to
1, indicating that the shell effects and collective effects
leading to increased level density above that predicted by
the Fermi gas model are no longer important[2, 3, 4, 5].
Above the excitation energy wherem∗ goes to 1, the rela-
tive density of a heated uniform Fermi gas at equilibrium
can be extracted from the ratio of the apparent inverse
level density parameter, K(ρeq), to K0, the inverse level
2density parameter for a uniform Fermi gas, with m∗ as-
sumed to be 1, at T = 0.
ρeq
ρ0
=
(
K(ρeq)
K0
) 3
2
(5)
We have used this relationship of equation (5) to derive
the excitation energy dependence of the relative nuclear
density, ρeq/ρ0. We applied this method to the deter-
mination of ρeq/ρ0 points for each mass region only at
excitation energies at and above the entry points into
the caloric curve plateaus. These points have been de-
rived from the caloric curves in reference 1 and shown to
have a mass dependence consistent with that predicted
by calculations which indicate the onset of Coulomb in-
stabilities in the heated nucleus[7, 10, 11]. In evaluating
this expression we have used the Fermi energy determi-
nations of Moniz et al.[12] in equation (3) to determine
the appropriate reference value of K0 for the mass region
being investigated.
To evaluate K(ρeq) and the density it is required that
ǫth be determined. We used an iterative technique to de-
rive of ρeq/ρ0 and ǫth. This technique consisted of first
estimating the relative density as (Kapp/K0)
3/2, employ-
ing this density estimate to evaluate the binding energy
difference associated with expansion and subtracting that
from the total excitation energy E∗ to obtain ǫth. Using
this estimate of ǫth allowed a new estimate of K(ρeq) and
density to be made. The process was repeated until self
consistent values of ǫth and ρeq/ρ0 were obtained. The
binding energies at different densities were calculated us-
ing a liquid drop model mass formula[13] with the bulk
term extracted from the density and relative neutron ex-
cess dependent nuclear matter equation of state of My-
ers and Swiatecki[14]. The surface term was scaled as
the bulk term and the Coulomb term corrected for the
expansion. The binding energy changes evaluated in this
way show a near parabolic dependence on density simi-
lar to that employed in the Expanding Emitting Source
Model of Friedman[15]. However the absolute values are
slightly lower.
The results, derived from the inverse level density data
of Figure 1, are presented in Figure 2 as a plot of rela-
tive density, ρeq/ρ0, versus excitation energy per nucleon.
While the mass dependence seen in Figure 1 continues to
be apparent in Figure 2, the general trend of decreasing
nuclear density is similar in each mass region. In the
present ansatz, the decreasing values of K with excita-
tion energy, seen in figure 1 lead to decreasing nuclear
densities. We see that at the entry into the caloric curve
plateaus, densities near 0.7ρ0 are indicated. Over a span
of a few MeV/nucleon in excitation energy the densities
drop into the range of ∼ 0.4ρ0, a value indicated by the
thin horizontal line on the figure. At higher excitation
the densities extracted in this manner remain near 0.4ρ0.
Modest expansion is a general feature of calculations
focusing on the thermal properties of heated nuclei[5, 7,
10, 16]. For comparison to the present results we have
plotted in Figure 2 calculated average densities derived
from the recent finite temperature Hartree-Fock (FTHF)
calculations of Hasan and Vary for 90Zr[16] and densi-
ties for A = 120 estimated from a model which assumes
a trapezoidal density distribution and phenomenologi-
cal dependencies on temperature derived from previous
works[5, 17]. As seen the two results are quite similar.
The result of the FTHF calculations are limited to en-
ergies below ∼ 4 MeV because the model space takes a
limited number of oscillator shells in to account. Com-
parison of the data with these calculations indicates that
above the Coulomb instability point the density decreases
more rapidly with increasing excitation energy.
While direct measurement of the size or density of
the fragmenting system are difficult to obtain, many
previous theoretical and experimental results have sug-
gested that low densities such as those indicated by
the present analysis are reached in multi-fragmenting
systems[15, 18, 19, 20, 21, 22, 23, 24, 25, 26]. We have
previously reported average densities deduced from co-
alescence measurements for fragmenting medium mass
nuclei[26]. In Figure 3, the average densities obtained
with that coalescence approach are compared with the
present results for the 100 < A < 140 mass region.
The average densities indicated by these two independent
techniques are seen to be in good agreement. Bracken et
al.[27] have extracted sizes and average densities versus
excitation energy by analyzing barriers for ejectiles in
high energy 3He + 197Au collisions. These, also plotted
in figure 3, result in slightly lower densities.
In conclusion, it appears that this simple hypothesis
of expansion of an equilibrated non-dissipative Fermi gas
is sufficiently accurate to allow density information to be
derived from reported caloric curve measurements. The
present experimental results indicate that, as nuclei are
heated, a more rapid rate of expansion with increasing
excitation energy commences at the Coulomb instability
temperature. This was suggested already in the early
work of Bonche, Vautherin and Levit[7, 10]. The mass
dependence of the Coulomb instability temperature is
such that the excitation energy and temperature at which
this occurs increases with decreasing mass. At the higher
excitation energies densities near 0.4ρ0 are reached. For
medium to heavy mass systems, some recent studies have
presented evidence for observation of critical behavior in
the excitation energy range where the present work indi-
cates a rapid decrease in density[28, 29, 30]. The critical
temperatures reported in those studies are close to the
Coulomb instability temperatures. The influence of the
Coulomb energy on observables used to determine the
critical point is not completely clarified at present[31, 32].
3ACKNOWLEDGEMENTS
This work was supported by the U S Department of
energy under Grant DE-FG03- 93ER40773 and by the
Robert A. Welch Foundation.
[1] J.B. Natowitz, R. Wada, K. Hagel, T Keutgen, M. Mur-
ray, Y. G. Ma*, A. Makeev, L. Qin , P. Smith and C.
Hamilton, Phys. Rev. C 65, 034618 (2002).
[2] R. Hasse and P. Schuck, Phys. Lett. B 179, 313 (1986).
[3] P.F. Bortignon and C.H. Dasso, Phys. Lett. B189, 381
(1987).
[4] S. Shlomo and J.B. Natowitz, Phys. Lett. B252, 187
(1990).
[5] S. Shlomo and J.B. Natowitz, Phys. Rev. C 44, 2878
(1991).
[6] J.B. Natowitz, D. Fabris, F. Haddad, K. Hagel, J. Li, Y.
Lou, N. Mdeiwayeh, G. Nebbia, G. Prete, R. Tezkratt,
D. Utley, G. Viesti, R. Wada and B. Xiao. Proceedings of
the Nuclear Chemistry Award Symposium, 209th Meet-
ing of the American Chemical Society, Anaheim, World
Scientific Press, Singapore, (1995) p 1.
[7] P. Bonche, S. Levit and H. Vautherin, Nucl. Phys.A427,
278 (1984); Ibid, A436, 265 (1986).
[8] M.A. Preston, Physics of the Nucleus, Addison-Wesley
Publishing Company, Massachusetts (1962) p 125.
[9] W. Norenberg, G. Papp and P. Rozmej GSI Preprint
2002-3, January, 2002 B.
[10] S.Levit and P.Bonche, Nucl. Phys. A437, 426 (1985).
[11] J. Besprovany and S. Levit Phys. Lett. B217, 1 (1989).
[12] E. J. Moniz, I. Sick, R.R. Whitney, JR Ficenec, R.P.
Kephart, W. Trower, Phys. Rev. Lett. 26, 445 (1971).
[13] M. Kleban, B. NerloPomorska, J. F. Berger, J. Decharge´,
M. Girod, and S. Hilaire, Phys. Rev. C 65, 024309 (2002).
[14] W. D. Myers and W. J. Swiatecki, Phys. Rev. C 57, 3020
(1998).
[15] W. A. Friedman, Phys. Rev. Lett. 60, 2125 (1988).
[16] M. A. Hasan and J. P. Vary, Phys. Rev. C 58, 2754
(1998).
[17] S. Shlomo, private communication.
[18] E. Suraud, C. Gregoire and B. Tamain, Prog Part and
Nucl. Phys. 23, 357 (1989) and references therein.
[19] D. Gross et al., Prog. Part.and Nucl. Phys. 30, 155 (1993)
and references therein.
[20] J. Bondorf et al., Nucl. Phys. A433, 321 (1985).
[21] Y. Sugawa and H. Horiuchi, Phys. Rev. C 60, 607 (1999).
[22] H. Feldmeier and J. Schnack, Rev. Mod. Phys. 72, 655
(2000).
[23] F. Gulminelli and D. Durand, Nucl. Phys. A615, 117
(1997).
[24] P. Desesquelles, M. D’Agostino, A. S. Botvina, M. Bruno,
N. Colonna, A. Ferrero, M. L. Fiandri, E. Fuschini, F.
Gramegna, I. Iori, G. V. Margagliotti, P. F. Mastinu, P.
M. Milazzo, A. Moroni, F. Petruzzelli, R. Rui, G. Van-
nini, J. D. Dinius, C. K. Gelbke, T. Glasmacher, D.O.
Handzy, W. C. Hsi, M. Huang, M. A. Lisa, W. G. Lynch,
C. P. Montoya, G. F. Peaslee, L. Phair, C. Schwarz, M.B.
Tsang and C. Williams, Nucl. Phys. A633, 547 (1998).
[25] C. Schwarz, S. Fritz, R. Bassini, M. Begemann-Blaich,
S.J. Gaff-Ejakov, D. Gourio, G. C ross, G. Imme, I.
Iori, U. Kleinevoss, G. J. Kunde, W.D. Kunze, U. Ly-
nen, V. Maddalena, M. Mahi, T. Mohlenkamp, A. Mo-
roni, W.F.J. Muller, C. Nociforo, B. Ocker, T. Odeh, F.
Petruzzelli, J. Pochodzalla, G. Raciti, G. Riccobene, F.P.
Romano, A. Saija, M. Schnittker, A. Schuttauf, W. Sei-
del, V. Serfling, C. Sfienti, W. Trautmann, A. Trzcinski,
G. Verde, A. Worner, H.F. Xi and B. Zwieglinski, Nucl.
Phys. A681, 279 (2001).
[26] J. Cibor, R. Wada, K. Hagel, M. Lunardon, N. Marie, R.
Alfaro, W. Q. Shen, B. Xiao, Y. Zhao, J. Li, B. A. Li, M.
Murray, J. B. Natowitz, Z. Majka and P. Staszel, Phys.
Lett. B473, 29 (2000).
[27] D. Bracken, Thesis, Indiana University (1996), To be
published.
[28] J. A. Hauger, B. K. Srivastava, S. Albergo, F. Bieser,
F. P. Brady, Z. Caccia, D. A. Cebra, A. D. Chacon,
J. L. Chance, Y. Choi, S. Costa, J. B. Elliott,1, M. L.
Gilkes, A. S. Hirsch, E. L. Hjort, A. Insolia, M. Justice,
D. Keane, J. C. Kintner, V. Lindenstruth, M. A. Lisa, H.
S. Matis, M. McMahan, C. McParland, W. F. J. Mueller,
D. L. Olson, M. D. Partlan, N. T. Porile, R. Potenza,
G. Rai, J. Rasmussen, H. G. Ritter, J. Romanski, J. L.
Romero, G. V. Russo, H. Sann, R. P. Scharenberg, A.
Scott, Y. Shao, T. J. M. Symons, M. Tincknell, C. Tuve,
S. Wang, P. Warren, H. H. Wieman, T. Wienold, and K.
Wolf, Phys. Rev. C 62, 024616 (2000).
[29] J. B. Elliott, L. G. Moretto, L. Phair, G. J. Wozniak,
T. Lefort, L. Beaulieu, K. Kwiatkowski, W.-C. Hsi, L.
Pienkowski, H. Breuer, R. G. Korteling, R. Laforest, E.
Martin, E. Ramakrishnan, D. Rowland, A. Ruangma, V.
E. Viola, E. Winchester, S. J. Yennello Preprint LBNL-
47711, Phys. Rev. Lett. 88, 042701 (2002)
[30] M. Kleine Berkenbusch, W. Bauer, K. Dillman, and S.
Pratt, L. Beaulieu, K. Kwiatkowski, T. Lefort, W.-c. Hsi,
and V. E. Viola, S. J. Yennello, R. G. Korteling and H.
Breuer, Phys. Rev. Lett. 88, 022701 (2002).
[31] B. K. Srivastava, S. Albergo, F. Bieser, F. P. Brady, Z.
Caccia, D. A. Cebra, A. D. Chacon, J. L. Chance, Y.
Choi, S.Costa, J. B. Elliott, M. L. Gilkes, J. A. Hauger,
A. S. Hirsch, E. L. Hjort, A. Insolia, M. Justice, D.
Keane, J. C. Kintner, V. Lindenstruth, M. A. Lisa, H.
S. Matis, M. McMahan, C. McParland, W. F. J. Mu¨ller,
D. L. Olson, M. D. Partlan, N. T. Porile, R. Potenza,
G. Rai, J. Rasmussen, H. G. Ritter, J. Romanski, J. L.
Romero, G. V. Russo, H. Sann, R. P. Scharenberg, A.
Scott, Y. Shao, T. J. M. Symons, M. Tincknell, C. Tuv,
S. Wang, P. Warren,H. H. Wieman, T. Wienold, and K.
Wolf, Phys. Rev. C 64, 041605(R) (2001).
[32] J. Richert, and P. Wagner, Phys. Rep. 350, 1 (2001).
430 < A < 60
60 < A < 100
100 < A < 140
140 < A < 180
180 < A < 240
 E*/A, MeV/u
0 5 10 15
 
K
, M
eV
0
5
10
15
20
FIG. 1: Apparent inverse level density parameter K versus
excitation energy. Symbols show results for five different mass
regions; The meaning of the various symbols is indicated in
the figure. The thicker lines represent K values calculated
assuming constant temperatures of 6 MeV(solid), 7 MeV(dot-
dashed) and 8 MeV (dashed).
30 < A < 60
60 < A < 100
100 < A < 140
140 < A < 180
180 < A < 240
A=120 Calc
Zr Calc90 
 E*/A (MeV/nucleon)
0 5 10 15
0ρ/
eqρ
 
0
0.5
1
FIG. 2: Derived average nuclear density versus excitation
energy. Symbols show results for five different mass regions;
The meaning of the various symbols is indicated in the figure.
The thin horizontal line marks ρ = 0.4. The two lines starting
at ρ = 1 represent calculated average densities for 90Zr[16]
(solid) and A=120[5, 17] (dot-dashed)(See text)
5100 < A < 140
Coalescence Analysis
Au197He + 3 
A=120 Calc
Zr Calc90 
 E*/A (MeV/nucleon)
0 5 10 15
0ρ/
eqρ
 
0
0.5
1
FIG. 3: Average nuclear density versus excitation energy.
Open diamonds show results for 100 < A < 140, derived
in the present work. Solid squares with error bars repre-
sent results of a coalescence model analysis of light particle
emission[26]. Solid circles represent results from reference 27.
The thin horizontal line marks ρ = 0.4. The two lines start-
ing at ρ = 1 represent calculated average densities for 90Zr[16]
(dot-dashed) and A


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