Assuming a schematic form of the nucleon effective mass as a function of
nuclear excitation energy and mass, we provide a simple explanation for
understanding the experimentally observed mass dependence of the nuclear
caloric curve. It is observed that the excitation energy at which the caloric
curve enters into a plateau region, could be sensitive to the nuclear mass
evolution of the effective nucleon mass.Comment: 5 pages, 5 figures, Accepted for publication in Phys. Rev. C. Minor
  changes mad

Galanopoulos, S.

Shetty, D. V.

Souliotis, G. A.

Yennello, S. J.

English

arXiv

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:0902.2978v2  [nucl-ex]  20 Feb 2009Effective nucleon mass and the nuclear caloric curveD.V. ShettyCyclotron Institute, Texas A&M University, College Station, TX 77843, USA andPhysics Department, Western Michigan University, Kalamazoo, MI 49008, USAG.A. Souliotis, S. Galanopoulos, S.J. YennelloCyclotron Institute, Texas A&M University, College Station, TX 77843, USA(Dated: May 31, 2018)Assuming a schematic form of the nucleon effective mass as a function of nuclear excitationenergy and mass, we provide a simple explanation for understanding the experimentally observedmass dependence of the nuclear caloric curve. It is observed that the excitation energy at which thecaloric curve enters into a plateau region, could be sensitive to the nuclear mass evolution of theeffective nucleon mass.PACS numbers: 25.70.Pq, 25.70.MnThe experimentally observed “plateau” in the nuclear“caloric curve” (temperature versus excitation energy)has long been seen as a signature of liquid-gas phase tran-sition, similar to that in real fluids. Recently, Sobotka etal. [1, 2], have shown that the plateau in the nuclearcaloric curve is a consequence of a combined effect of de-creasing density due to thermal expansion, and the evo-lution of in-medium nucleon effective mass, rather thanan indication of liquid-gas phase coexistence. This modelfor describing the caloric curve is based on the relaxationof density profile of the mononucleus that results in maxi-mum entropy under a local density approximation for thelevel density parameter. A parametric form of the basedensity was assumed and the entropy was calculated inthe Fermi-gas model. The evolution of effective masswith density and excitation was included in a schematicfashion as it is currently unknown.The plateau in the caloric curve from this model is es-tablished at a rather modest excitation energy (about 2MeV/nucleon), well below the excitation energy wherethe experimentally observed caloric curve enters into aplateau region. Also, the value of the plateau tempera-ture remains the same for a mass, A = 90 system as itdoes for a mass, A = 197 system. However, experimen-tally determined caloric curve [3] shows plateau temper-ature that decreases with increasing mass, A.In this work, we investigate how the mass dependenceof the caloric curve, can be understood using the aboveconcept of effective nucleon mass and thermal expansionin a simple phenomenological approach. In particular,we present a schematic expression to understand the ex-perimentally observed plateau in the caloric curve, andshow that the observed mass dependence of the caloriccurve can be reproduced using effective nucleon mass asa function of excitation energy, that is mass dependent.We begin with the assumption that the relation be-tween the temperature, T , and the experimentally mea-sured total excitation energy, E∗, for an expanding nu-cleus of mass A, can be expressed by a form analogousto that for the Fermi gas,E∗ =T 2Keff (ρ,A)(1)whereKeff is the inverse nuclear level density parame-ter of the hot and dilute nucleus at density ρ, and writtenas,Keff (ρ,A) =4ǫF (ρ,A)π2(2)where, ǫF (ρ,A), is the Fermi energy of the finite andexpanding nucleus and given as,ǫF (ρ,A) = ǫoF[m∗(ρo, A)m∗(ρ,A)](ρρo)2/3(3)The quantity m∗(ρo, A), is the ratio of the effectivemass of the nucleon to the mass of the free nucleon, as-suming the nucleus to be a Fermi gas and at ground state(T = 0). The quantity m∗(ρ,A) is the ratio of the effec-tive mass of the nucleon to the mass of the free nucleon,in hot (T > 0) and expanding finite nucleus density ρ.From the above equations, one can write the inversenuclear level density parameter of the hot and dilute nu-cleus as [4],Keff (ρ,A) = Ko(ρρo)2/3[m∗(ρo, A)m∗(ρ,A)](4)where, Ko = 4ǫoF/π2≈ 15, is the inverse level densityparameter of uniform, non−dissipative Fermi gas. ǫoF ,and ρo are the Fermi energy and the nuclear saturationdensity at T = 0 MeV.From the above two expressions, Eqs. 1 and 4, one canwrite the temperature versus the excitation energy for anucleus expanding to an equilibrium density ρ,T 2 = Ko[m∗(ρo, A)m∗(ρ,A)](ρρo)2/3E∗ (5)2Alternatively, one can also start with the assumptionthat the total excitation energy E∗, of an expanding nu-cleus can be written as,E∗ = E∗ther + Eexp (6)where, E∗ther = T2/Ko[m∗(ρo, A)/m∗(ρ,A)] is thethermal part of the excitation energy, and Eexp = ǫb(1−ρ/ρo)2 is the expansion energy of the finite nucleus. Theexpansion energy assumed in the above expression is asimple upside down bell shaped, suggested by Friedman[5], with the ground state binding energy, ǫb = 8 MeV.The temperature versus the total excitation energy rela-tion for a nucleus expanding to an equilibrium density,ρ, can then be written as,T 2 = Kom∗(ρo, A)m∗(ρ,A)[E∗ − ǫb(1− ρ/ρo)2] (7)Using equations, 5 and 7, we can now study the exper-imental caloric curve and its mass dependence.Fig. 1 shows the experimentally measured caloriccurve for the mass range of A = 100 - 140, from variousmeasurements compiled by Natowitz et al. [3]. The datafrom all different measurements are shown collectively inthe figure by inverted triangle symbols and no distinctionis made between them. The dotted (black) curve in thefigure is the simple Fermi gas relation, E∗ther = T2/Ko,with the inverse level density parameter, Ko = 15. Theresults of the equations 5 and 7 are shown by the solid(blue) and dashed (red) curves, respectively. In bothequations, the density ρ/ρo, for a given excitation energyE∗, was taken to be that adopted by Bondorf et al. [6, 7](shown by the solid black curve in Fig. 2(b)). The ef-fective mass m∗(ρ,A), as a function of excitation energywas assumed to have an empirical dependence of the formshown by the dashed (blue) curve in Fig. 2(a). From Fig.1, one observes that Eq. 5 (solid, blue curve) and Eq.7 (dashed, red curve) with the same effective mass anddensity dependence of the excitation energy gives similarresults up to excitation energy of 6 MeV/nucleon, with asmall difference at higher excitation energies. The differ-ence at higher energies is due to the different form of theexpansion energy assumed in the two expressions. Bothcalculations, show a plateau at excitation energy above 3MeV/nucleon, in good agreement with the experimentaldata.The caloric curve obtained from Eq. 5 and 7 is inmuch better agreement with the data compared to thoseobtained by Sobotka et al. [2]. The result of Sobotkaet al., is shown by the dot-dashed (orange) curve. Alsoshown in the figure is the result of De et al. [8], for A =150 (dot-dot-dashed, black curve). The calculation of Deet al. uses a realistic effective Hamiltonian to calculatethe base density profile in a Thomas-Fermi frameworkwith the entropy calculated microscopically. The abovecomparison shows that the phenomenological expressions0 2 4 6 8 10024681012 Temperature (MeV)E* (MeV/nucleon)A = 100 - 140FIG. 1: (Color online) Temperature as a function of excitationenergy for mass A = 100 - 140. The data points (invertedtriangles) are from Ref. [3]. The dotted curve is the Fermigas relation. The dot-dot-dashed curve is from Ref. [8]. Thedot-dashed curve is from Ref. [2]. The solid and the dashedcurve are from Eq. 5 and 7, respectively.TABLE I: The ground state effective mass, m∗(ρo, A), usedin Eq. 3 and 5, for various nuclear mass range.Mass (A) m∗(ρo, A)30 - 60 0.8760 - 100 0.73100 - 140 0.67180 - 240 0.535 and 7 can be used to understand the characteristic fea-tures of the caloric curve.In the following we use Eq. 5 and 7, to study the massdependence of the caloric curve. Fig. 3 and 4 shows theexperimental caloric curve data (inverted triangles) forthe different mass ranges obtained from the work of Na-towitz et al. [3]. The results of Eq. 5 and 7 are shown bythe solid curves in Fig. 3 and 4, respectively. The empir-ical form of the effective nucleon mass and the density asa function of excitation energy used in Eq. 5 and 7 fordifferent mass regions are as shown in Fig. 2(a) and 2(b).These were obtained by tuning the data with a fixed Koparameter. The effective mass for the ground state nu-cleus m∗(ρo, A), is as shown in Table I. The choice of them∗(ρo, A) value was dictated by its sensitivity to the tem-perature at which the caloric curve reaches the plateauregion. This is discussed in the following paragraph. TheFermi gas caloric curve is shown by the dotted curves inFig. 3 and 4.An important point to note in Fig. 3 and 4 is theevolution of the excitation energy of the entry point intothe caloric curve plateau with nuclear mass. To illustrate30 2 4 6 8 10051015  Keff(, A)E* (MeV/nucleon)c)0.00.30.60.9  b)012  (A =     30 - 60)  (A =   60 - 100)  (A = 100 - 140)  (A = 180 - 240)  1/m*(, A)a)FIG. 2: (Color online) a) Effective nucleon mass ratio, b)Density and, c) Keff , as a function of excitation energy. Thedifferent curves in a) and c) correspond to the different massranges.this point more clearly, we show in Fig. 5, a comparisonbetween the caloric curves for A = 30 - 60, A = 60 -100, and A = 180 - 240 mass regions, obtained using Eq.5. The arrows in the figure correspond to the approxi-mate values of the excitation energy at which the caloriccurve for each of the three masses enter into a plateauregion. The figure shows that the temperature at whichthe plateau is reached is sensitive to the value of them∗(ρo, A). A higher m∗(ρo, A) results in higher plateautemperature for lighter mass. The excitation energy atwhich the plateau is reached is, on the other hand, sensi-tive to the mass dependence of the effective nucleon mass,m∗(ρ,A). This excitation energy corresponds to the en-ergy at which the 1/m∗(ρ,A), shown in Fig. 2(a), peaks.The shift in the peak to higher excitation energy for de-creasing mass of the system results in a plateau beingreached at higher excitation energy for lighter mass. Toexplain the experimentally observed caloric curve, effec-tive nucleon mass ratio that is dependent on the excita-tion energy/density and the nuclear mass therefore seemsimperative. The effective nucleon mass as a function ofexcitation energy for different masses, shown in Fig. 2(a),0 2 4 6 8 10 12 140510A = 180 - 240  T (MeV)E* (MeV/nucleon)0510A = 100 - 140  T (MeV)0510A = 60 - 100  T (MeV)0510  T (MeV)A = 30 - 60FIG. 3: (Color online) Temperature as a function of excitationenergy for various mass ranges. The data points (invertedtriangles) are from Ref. [3]. The solid curves are obtainedfrom Eq. 5. The dashed curve is the Fermi gas relation.and used in the above analysis, is an empirical deduction.At present, we do not know of any formal approach todeduce such a dependence. Qualitatively such a depen-dence is expected in the interior of the nucleus, wherethe effective mass is reduced in the bulk, peaks at thesurface and reduces to one with decreasing density andincreasing excitation energy [9]. In such a prescription,the effective nucelon mass is often given by phenomeno-logical expressions that includes a momentum dependentand a frequency dependent term. The detailed densityand excitation energy dependence of these terms are un-known. Theoretical investigation in this direction would40 2 4 6 8 10 12 140510A = 180 - 240  T (MeV)E* (MeV/nucleon)0510A = 100 - 140  T (MeV)0510A = 60 - 100  T (MeV)0510  T (MeV)A = 30 - 60FIG. 4: (Color online) Temperature as a function of excitationenergy for various mass ranges. The data points (invertedtriangles) are from Ref. [3]. The solid curves are obtainedfrom Eq. 7. The dashed curve is the Fermi gas relation.therefore be interesting. While a formal understandingof the effective nucleon mass as a function of excitationenergy and density over a range of nuclear mass would bevery fruitful, the emperical approach utilizing the caloriccurve in this work can aid in studying the isospin (N/Z)dependence of the effective mass in asymmetric nuclei.In the following we show that the mass dependenceof the inverse level density parameter Keff , obtainedusing the above empiricaly deduced effective nucleonmass, m∗(ρ,A), is consistent with the experimentallyand theoretically deduced inverse level density param-eter at low exciation energy. In the past, temperature0 2 4 6 8 100246810  (A = 30 - 60)  (A = 60 - 100)  (A = 180 - 240) Temperature (MeV)E* (MeV/nucleon)FIG. 5: (Color online) Comparison between the caloric curvesobtained from Eq. 5 for three different mass ranges.dependence of the nuclear level density parameter hasbeen investigated extensively by studying the spectra oflight particles emitted in hot nuclei populated at E∗ >1 MeV/nucleon (T > 2 MeV). It has been shown fromthese studies [10, 11] that the inverse level density param-eter K, increases from 8 to 13 for temperature increasingfrom T = 2 MeV to T = 5.5 MeV in nuclei of mass A∼ 160. However, similar studies [12, 13, 14] carried outfor light nuclei, such as A ∼ 40, failed to show an in-crease. The K remaining nearly constant at 9 - 10 in theexcitation energy range of 2.5 to 5.0 MeV/nucleon. InFig. 2(c), we show the effective inverse level density pa-rameter, Keff , of the hot and dilute nucleus for variousmass regions obtained from the present analysis. For T= 0 MeV, one observes that the inverse level density pa-rameter, Keff ∼ 8, in agreement with the low excitationenergy studies. For light nuclei, A = 30 - 60, the Keffremains essentially constant up to excitation energy of 5MeV/nucleon. While for heavier nuclei it varies between8 and 16 for excitation energies below 2 - 3 MeV. Sim-ilar dependence was also obtained from the theoreticalcalculations of Ref. [15, 16] that investigated the massdependence of the level density parameter at low exci-ation energy. For higher excitation energies, Fig. 2(c)shows a steady decline in the inverse level density pa-rameter, which results in a plateau like behavior in thecaloric curve (temperature versus excitation energy plot)for the hot and expanding nucleus.In conclusion, it is shown that the mass dependenceof the nuclear caloric curve can be modelled using Equa-tions 5 and 7, with the effective nucleon mass of the formshown in Fig. 2(a), or alternatively, using Eq. 1 withthe inverse level density parameter of the form shown inFig. 2(c). The mass dependence of the inverse level den-sity parameter thereby obtained is consistent with theexperimentally and theoretically deduced level density5parameter for low excitation energies. Furthermore, thepresent study demonstrates that the caloric curve canbe used as a tool to fine tune the effective nucleon massand study the nuclear interaction. A natural extensionof the present study would be to apply the present ap-proach to fine tune the effective mass further by includingthe isospin (N/Z) dependence. Such a study would re-quire caloric curve measurements of asymmetric (N/Z >1) nuclei using beams of radioactive nuclei. This wouldprovide a complete understanding of the interaction un-der extreme conditions of excitation energy, density andisospin (N/Z).This work was supported in part by the Robert A.Welch Foundation through grant No. A-1266, and theDepartment of Energy through grant No. DE-FG03-93ER40773.[1] L.G. Sobotka, R.J. Charity, J. Toke, and W.U. Schroder,Phys. Rev. Lett. 93, 132702 (2004).[2] L.G. Sobotka and R.J. Charity, Phys. Rev. C 73, 014609(2006).[3] J.B. Natowitz, R. Wada, K. Hagel, T. Keutgen, M. Mur-ray, A. Makeev, L. Qin, P. Smith, and C. Hamilton, Phys.Rev. C 65, 034618 (2002).[4] W. Norenberg, G. Papp, and P. Rozmej, Eur. Phys. J.A14, 43 (2002).[5] W.A. Friedman, Phys. Rev. Lett. 60, 2125 (1988).[6] J.P. Bondorf, R. Donangelo, I.N. Mishustin, and H.Schulz , Nucl. Phys. A444, 460 (1985).[7] J.P. Bondorf, A.S. Botvina, and I.N. Mishustin, Phys.Rev. C 58, 27 (1998).[8] J.N. De, S.K. Samaddar, X. Vinas, and M. Centelles,Phys. Lett. B638, 160 (2006).[9] R.W. Hasse and P. Schuck, Phys. Lett. B179, 313 (1986).[10] G. Nebbia et al., Phys. Lett. B 176, 20 (1986).[11] K. Hagel et al., Nucl. Phys. A486, 429 (1988).[12] A. Chbihi, L.G. Sobotka, N.G. Nicolis, D.G. Sarantities,D.W. Stracener, Z. Majka, D.C. Hensley, J.R. Beene, andM.L. Halbert, Phys. Rev. C 43, 666 (1991).[13] B. Fornal et al., Phys. Rev. C 44, 2588 (1991).[14] K. Yoshida et al., Phys. Rev. C 46, 961 (1992).[15] S. Shlomo and J.B. Natowitz, Phys. Rev. C 44, 2878(1991).[16] J.N. De, S. Shlomo, and S.K. Samaddar, Phys. Rev. C57, 1398 (1998).

Physical Review C

Effective nucleon mass and the nuclear caloric curve

Texas A&M University

ar
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:0
90
2.
29
78
v2
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nu
cl-
ex
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00
9
Effective nucleon mass and the nuclear caloric curve
D.V. Shetty
Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA and
Physics Department, Western Michigan University, Kalamazoo, MI 49008, USA
G.A. Souliotis, S. Galanopoulos, S.J. Yennello
Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA
(Dated: May 31, 2018)
Assuming a schematic form of the nucleon effective mass as a function of nuclear excitation
energy and mass, we provide a simple explanation for understanding the experimentally observed
mass dependence of the nuclear caloric curve. It is observed that the excitation energy at which the
caloric curve enters into a plateau region, could be sensitive to the nuclear mass evolution of the
effective nucleon mass.
PACS numbers: 25.70.Pq, 25.70.Mn
The experimentally observed “plateau” in the nuclear
“caloric curve” (temperature versus excitation energy)
has long been seen as a signature of liquid-gas phase tran-
sition, similar to that in real fluids. Recently, Sobotka et
al. [1, 2], have shown that the plateau in the nuclear
caloric curve is a consequence of a combined effect of de-
creasing density due to thermal expansion, and the evo-
lution of in-medium nucleon effective mass, rather than
an indication of liquid-gas phase coexistence. This model
for describing the caloric curve is based on the relaxation
of density profile of the mononucleus that results in maxi-
mum entropy under a local density approximation for the
level density parameter. A parametric form of the base
density was assumed and the entropy was calculated in
the Fermi-gas model. The evolution of effective mass
with density and excitation was included in a schematic
fashion as it is currently unknown.
The plateau in the caloric curve from this model is es-
tablished at a rather modest excitation energy (about 2
MeV/nucleon), well below the excitation energy where
the experimentally observed caloric curve enters into a
plateau region. Also, the value of the plateau tempera-
ture remains the same for a mass, A = 90 system as it
does for a mass, A = 197 system. However, experimen-
tally determined caloric curve [3] shows plateau temper-
ature that decreases with increasing mass, A.
In this work, we investigate how the mass dependence
of the caloric curve, can be understood using the above
concept of effective nucleon mass and thermal expansion
in a simple phenomenological approach. In particular,
we present a schematic expression to understand the ex-
perimentally observed plateau in the caloric curve, and
show that the observed mass dependence of the caloric
curve can be reproduced using effective nucleon mass as
a function of excitation energy, that is mass dependent.
We begin with the assumption that the relation be-
tween the temperature, T , and the experimentally mea-
sured total excitation energy, E∗, for an expanding nu-
cleus of mass A, can be expressed by a form analogous
to that for the Fermi gas,
E∗ =
T 2
Keff (ρ,A)
(1)
whereKeff is the inverse nuclear level density parame-
ter of the hot and dilute nucleus at density ρ, and written
as,
Keff (ρ,A) =
4ǫF (ρ,A)
π2
(2)
where, ǫF (ρ,A), is the Fermi energy of the finite and
expanding nucleus and given as,
ǫF (ρ,A) = ǫ
o
F
[
m∗(ρo, A)
m∗(ρ,A)
](
ρ
ρo
)2/3
(3)
The quantity m∗(ρo, A), is the ratio of the effective
mass of the nucleon to the mass of the free nucleon, as-
suming the nucleus to be a Fermi gas and at ground state
(T = 0). The quantity m∗(ρ,A) is the ratio of the effec-
tive mass of the nucleon to the mass of the free nucleon,
in hot (T > 0) and expanding finite nucleus density ρ.
From the above equations, one can write the inverse
nuclear level density parameter of the hot and dilute nu-
cleus as [4],
Keff (ρ,A) = Ko
(
ρ
ρo
)2/3[
m∗(ρo, A)
m∗(ρ,A)
]
(4)
where, Ko = 4ǫ
o
F/π
2
≈ 15, is the inverse level density
parameter of uniform, non−dissipative Fermi gas. ǫoF ,
and ρo are the Fermi energy and the nuclear saturation
density at T = 0 MeV.
From the above two expressions, Eqs. 1 and 4, one can
write the temperature versus the excitation energy for a
nucleus expanding to an equilibrium density ρ,
T 2 = Ko
[
m∗(ρo, A)
m∗(ρ,A)
](
ρ
ρo
)2/3
E∗ (5)
2Alternatively, one can also start with the assumption
that the total excitation energy E∗, of an expanding nu-
cleus can be written as,
E∗ = E∗ther + Eexp (6)
where, E∗ther = T
2/Ko[m
∗(ρo, A)/m
∗(ρ,A)] is the
thermal part of the excitation energy, and Eexp = ǫb(1−
ρ/ρo)
2 is the expansion energy of the finite nucleus. The
expansion energy assumed in the above expression is a
simple upside down bell shaped, suggested by Friedman
[5], with the ground state binding energy, ǫb = 8 MeV.
The temperature versus the total excitation energy rela-
tion for a nucleus expanding to an equilibrium density,
ρ, can then be written as,
T 2 = Ko
m∗(ρo, A)
m∗(ρ,A)
[E∗ − ǫb(1− ρ/ρo)
2] (7)
Using equations, 5 and 7, we can now study the exper-
imental caloric curve and its mass dependence.
Fig. 1 shows the experimentally measured caloric
curve for the mass range of A = 100 - 140, from various
measurements compiled by Natowitz et al. [3]. The data
from all different measurements are shown collectively in
the figure by inverted triangle symbols and no distinction
is made between them. The dotted (black) curve in the
figure is the simple Fermi gas relation, E∗ther = T
2/Ko,
with the inverse level density parameter, Ko = 15. The
results of the equations 5 and 7 are shown by the solid
(blue) and dashed (red) curves, respectively. In both
equations, the density ρ/ρo, for a given excitation energy
E∗, was taken to be that adopted by Bondorf et al. [6, 7]
(shown by the solid black curve in Fig. 2(b)). The ef-
fective mass m∗(ρ,A), as a function of excitation energy
was assumed to have an empirical dependence of the form
shown by the dashed (blue) curve in Fig. 2(a). From Fig.
1, one observes that Eq. 5 (solid, blue curve) and Eq.
7 (dashed, red curve) with the same effective mass and
density dependence of the excitation energy gives similar
results up to excitation energy of 6 MeV/nucleon, with a
small difference at higher excitation energies. The differ-
ence at higher energies is due to the different form of the
expansion energy assumed in the two expressions. Both
calculations, show a plateau at excitation energy above 3
MeV/nucleon, in good agreement with the experimental
data.
The caloric curve obtained from Eq. 5 and 7 is in
much better agreement with the data compared to those
obtained by Sobotka et al. [2]. The result of Sobotka
et al., is shown by the dot-dashed (orange) curve. Also
shown in the figure is the result of De et al. [8], for A =
150 (dot-dot-dashed, black curve). The calculation of De
et al. uses a realistic effective Hamiltonian to calculate
the base density profile in a Thomas-Fermi framework
with the entropy calculated microscopically. The above
comparison shows that the phenomenological expressions
0 2 4 6 8 10
0
2
4
6
8
10
12
 
Te
m
pe
ra
tu
re
 (M
eV
)
E* (MeV/nucleon)
A = 100 - 140
FIG. 1: (Color online) Temperature as a function of excitation
energy for mass A = 100 - 140. The data points (inverted
triangles) are from Ref. [3]. The dotted curve is the Fermi
gas relation. The dot-dot-dashed curve is from Ref. [8]. The
dot-dashed curve is from Ref. [2]. The solid and the dashed
curve are from Eq. 5 and 7, respectively.
TABLE I: The ground state effective mass, m∗(ρo, A), used
in Eq. 3 and 5, for various nuclear mass range.
Mass (A) m∗(ρo, A)
30 - 60 0.87
60 - 100 0.73
100 - 140 0.67
180 - 240 0.53
5 and 7 can be used to understand the characteristic fea-
tures of the caloric curve.
In the following we use Eq. 5 and 7, to study the mass
dependence of the caloric curve. Fig. 3 and 4 shows the
experimental caloric curve data (inverted triangles) for
the different mass ranges obtained from the work of Na-
towitz et al. [3]. The results of Eq. 5 and 7 are shown by
the solid curves in Fig. 3 and 4, respectively. The empir-
ical form of the effective nucleon mass and the density as
a function of excitation energy used in Eq. 5 and 7 for
different mass regions are as shown in Fig. 2(a) and 2(b).
These were obtained by tuning the data with a fixed Ko
parameter. The effective mass for the ground state nu-
cleus m∗(ρo, A), is as shown in Table I. The choice of the
m∗(ρo, A) value was dictated by its sensitivity to the tem-
perature at which the caloric curve reaches the plateau
region. This is discussed in the following paragraph. The
Fermi gas caloric curve is shown by the dotted curves in
Fig. 3 and 4.
An important point to note in Fig. 3 and 4 is the
evolution of the excitation energy of the entry point into
the caloric curve plateau with nuclear mass. To illustrate
30 2 4 6 8 10
0
5
10
15
 
 
K
ef
f(
, A
)
E* (MeV/nucleon)
c)
0.0
0.3
0.6
0.9
 
 
b)
0
1
2
  (A =     30 - 60)
  (A =   60 - 100)
  (A = 100 - 140)
  (A = 180 - 240)
 
 
1/
m
*(
, A
)
a)
FIG. 2: (Color online) a) Effective nucleon mass ratio, b)
Density and, c) Keff , as a function of excitation energy. The
different curves in a) and c) correspond to the different mass
ranges.
this point more clearly, we show in Fig. 5, a comparison
between the caloric curves for A = 30 - 60, A = 60 -
100, and A = 180 - 240 mass regions, obtained using Eq.
5. The arrows in the figure correspond to the approxi-
mate values of the excitation energy at which the caloric
curve for each of the three masses enter into a plateau
region. The figure shows that the temperature at which
the plateau is reached is sensitive to the value of the
m∗(ρo, A). A higher m
∗(ρo, A) results in higher plateau
temperature for lighter mass. The excitation energy at
which the plateau is reached is, on the other hand, sensi-
tive to the mass dependence of the effective nucleon mass,
m∗(ρ,A). This excitation energy corresponds to the en-
ergy at which the 1/m∗(ρ,A), shown in Fig. 2(a), peaks.
The shift in the peak to higher excitation energy for de-
creasing mass of the system results in a plateau being
reached at higher excitation energy for lighter mass. To
explain the experimentally observed caloric curve, effec-
tive nucleon mass ratio that is dependent on the excita-
tion energy/density and the nuclear mass therefore seems
imperative. The effective nucleon mass as a function of
excitation energy for different masses, shown in Fig. 2(a),
0 2 4 6 8 10 12 14
0
5
10
A = 180 - 240
 
 
T
 (
M
e
V
)
E* (MeV/nucleon)
0
5
10
A = 100 - 140
 
 
T
 (
M
e
V
)
0
5
10
A = 60 - 100
 
 
T
 (
M
e
V
)
0
5
10
 
 
T
 (
M
e
V
)
A = 30 - 60
FIG. 3: (Color online) Temperature as a function of excitation
energy for various mass ranges. The data points (inverted
triangles) are from Ref. [3]. The solid curves are obtained
from Eq. 5. The dashed curve is the Fermi gas relation.
and used in the above analysis, is an empirical deduction.
At present, we do not know of any formal approach to
deduce such a dependence. Qualitatively such a depen-
dence is expected in the interior of the nucleus, where
the effective mass is reduced in the bulk, peaks at the
surface and reduces to one with decreasing density and
increasing excitation energy [9]. In such a prescription,
the effective nucelon mass is often given by phenomeno-
logical expressions that includes a momentum dependent
and a frequency dependent term. The detailed density
and excitation energy dependence of these terms are un-
known. Theoretical investigation in this direction would
40 2 4 6 8 10 12 14
0
5
10
A = 180 - 240
 
 
T
 (
M
e
V
)
E* (MeV/nucleon)
0
5
10
A = 100 - 140
 
 
T
 (
M
e
V
)
0
5
10
A = 60 - 100
 
 
T
 (
M
e
V
)
0
5
10
 
 
T
 (
M
e
V
)
A = 30 - 60
FIG. 4: (Color online) Temperature as a function of excitation
energy for various mass ranges. The data points (inverted
triangles) are from Ref. [3]. The solid curves are obtained
from Eq. 7. The dashed curve is the Fermi gas relation.
therefore be interesting. While a formal understanding
of the effective nucleon mass as a function of excitation
energy and density over a range of nuclear mass would be
very fruitful, the emperical approach utilizing the caloric
curve in this work can aid in studying the isospin (N/Z)
dependence of the effective mass in asymmetric nuclei.
In the following we show that the mass dependence
of the inverse level density parameter Keff , obtained
using the above empiricaly deduced effective nucleon
mass, m∗(ρ,A), is consistent with the experimentally
and theoretically deduced inverse level density param-
eter at low exciation energy. In the past, temperature
0 2 4 6 8 10
0
2
4
6
8
10
  (A = 30 - 60)
  (A = 60 - 100)
  (A = 180 - 240)
 
T
em
pe
ra
tu
re
 (
M
eV
)
E* (MeV/nucleon)
FIG. 5: (Color online) Comparison between the caloric curves
obtained from Eq. 5 for three different mass ranges.
dependence of the nuclear level density parameter has
been investigated extensively by studying the spectra of
light particles emitted in hot nuclei populated at E∗ >
1 MeV/nucleon (T > 2 MeV). It has been shown from
these studies [10, 11] that the inverse level density param-
eter K, increases from 8 to 13 for temperature increasing
from T = 2 MeV to T = 5.5 MeV in nuclei of mass A
∼ 160. However, similar studies [12, 13, 14] carried out
for light nuclei, such as A ∼ 40, failed to show an in-
crease. The K remaining nearly constant at 9 - 10 in the
excitation energy range of 2.5 to 5.0 MeV/nucleon. In
Fig. 2(c), we show the effective inverse level density pa-
rameter, Keff , of the hot and dilute nucleus for various
mass regions obtained from the present analysis. For T
= 0 MeV, one observes that the inverse level density pa-
rameter, Keff ∼ 8, in agreement with the low excitation
energy studies. For light nuclei, A = 30 - 60, the Keff
remains essentially constant up to excitation energy of 5
MeV/nucleon. While for heavier nuclei it varies between
8 and 16 for excitation energies below 2 - 3 MeV. Sim-
ilar dependence was also obtained from the theoretical
calculations of Ref. [15, 16] that investigated the mass
dependence of the level density parameter at low exci-
ation energy. For higher excitation energies, Fig. 2(c)
shows a steady decline in the inverse level density pa-
rameter, which results in a plateau like behavior in the
caloric curve (temperature versus excitation energy plot)
for the hot and expanding nucleus.
In conclusion, it is shown that the mass dependence
of the nuclear caloric curve can be modelled using Equa-
tions 5 and 7, with the effective nucleon mass of the form
shown in Fig. 2(a), or alternatively, using Eq. 1 with
the inverse level density parameter of the form shown in
Fig. 2(c). The mass dependence of the inverse level den-
sity parameter thereby obtained is consistent with the
experimentally and theoretically deduced level density
5parameter for low excitation energies. Furthermore, the
present study demonstrates that the caloric curve can
be used as a tool to fine tune the effective nucleon mass
and study the nuclear interaction. A natural extension
of the present study would be to apply the present ap-
proach to fine tune the effective mass further by including
the isospin (N/Z) dependence. Such a study would re-
quire caloric curve measurements of asymmetric (N/Z >
1) nuclei using beams of radioactive nuclei. This would
provide a complete understanding of the interaction un-
der extreme conditions of excitation energy, density and
isospin (N/Z).
This work was supported in part by the Robert A.
Welch Foundation through grant No. A-1266, and the
Department of Energy through grant No. DE-FG03-
93ER40773.
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Effective nucleon mass and the nuclear caloric curve

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