Surface tension of compressed, superheavy atoms

0.37 MB·

by  Jorge A. Rueda

Surface tension of compressed, superheavy atoms
JorgeA.Rueda,Yuan-BinWu‡§,andShe-ShengXue
DipartimentodiFisicaandICRA,SapienzaUniversita`diRoma,PiazzaleAldoMoro5,
I-00185Rome,Italy
7 ICRANet,PiazzadellaRepubblica10,I-65122Pescara,Italy
1
0 Abstract. BasedontherelativisticmeanfieldtheoryandtheThomas-Fermiapproximation,
2 westudythesurfacepropertiesofcompressed,superheavyatoms.Bycompressed,superheavy
n atomwemeananatomcomposedbyasuperheavynuclearcore(superheavynucleus)with
mass number of the order of 104, and degenerate electrons that neutralize the system.
a
J Someelectronspenetrateintothesuperheavynuclearcoreandtherestsurroundituptoa
distancethatdependsuponthecompressionlevel. Takingintoaccountthestrong,weak,and
6
electromagneticinteractions,wenumericallystudythestructureofcompressed,superheavy
2 atomsandcalculatethenuclearsurfacetensionandCoulombenergy.Weanalyzetheinfluence
of the electron component and the background matter on the nuclear surface tension and
] Coulombenergyofcompressed,superheavyatoms.Wealsocompareandcontrasttheseresults
h
inthecaseofcompressed,superheavyatomswithphenomenologicalresultsinnuclearphysics
t
- andtheresultsofthecore-crustinterfaceofneutronstarswithglobalchargeneutrality.Based
l onthenumericalresultswestudytheinstabilityagainstBohr-Wheelersurfacedeformationsin
c
thecaseofcompressed,superheavyatoms.Theresultsinthisarticleshowthepossibilityofthe
u
existenceofsuchcompressed,superheavyatoms,andprovidetheevidenceofstrongeffects
n
oftheelectromagneticinteractionandelectronsonthestructureofcompressed,superheavy
[
atoms.

1
v
6
PACSnumbers:21.10.-k,05.30.Fk,26.60.-c
4
1
8 1. Introduction
0
.
1 It has been shown recently that the Einstein-Maxwell-Thomas-Fermi (EMTF) equations [1]
0 supersede the traditional Tolman-Oppenheimer-Volkoff (TOV) [2,3] equations used for the
7 constructionofneutronstarequilibriumconfigurations,whentakingintoaccountthestrong,
1
weak, electromagnetic, and gravitational interactions. In contrast to the imposing of the
:
v conditionoflocalchargeneutralityinthetraditionalTOVapproach, theconditionofglobal
i
X charge neutrality is applied in the EMTF approach, owing to the fact that the traditional
treatment imposing the condition of local charge neutrality is not consistent with the field
r
a equations and microphysical conditions of equilibrium for the system of neutrons, protons,
andelectronsinβ equilibriumandobeyingrelativisticquantumstatistics[4].
Inordertodescribethestronginteractionsbetweennucleons,theσ-ω-ρ nuclearmodel
of relativistic mean field theory (RMFT) [5–12] is adopted in the EMTF approach. This
modelcontainsDiracnucleonstogetherwithascalarmesonσ andavectormesonω aswell
as an isovector meson ρ. The RMFT has gained great successes in giving a quantitative
descriptionofnuclearproperties[13–15]andunderstandingtheinhomogeneousstructuresof
‡ [email protected]
§ Presentaddress:Max-Planck-Institutfu¨rKernphysik,Saupfercheckweg1,D-69117Heidelberg,Germany
2
low-densitynuclearmatterwhichcanberealizedinsupernovaecoresorinneutronstarcrusts
(seee.g.Refs.[16–26]aboutthenuclearpastastructures).
As shown in Ref. [27], the self-consistent solution of the EMTF equations leads to a
newstructureofneutronstars,whichissignificantlydifferentfromtheneutronstarstructure
obtainedfromtheTOVequationsimposinglocalchargeneutrality[28]. Inthisnewstructure
of neutron stars, a transition layer (interface) appears between the core and the crust of the
star, near the nuclear saturation density. There is a discontinuity in the density at the core-
crust transition in this new structure of neutron stars. The core (bulk region) inside this
transitionlayerisahadronicphaseandthecrustoutsidethistransitionlayeriscomposedof
anucleilatticeandrelativisticdegenerateelectronsandpossiblyneutronsatdensitiesbelow
thenuclearsaturationdensityandhigherthantheestimatedneutron-dripvalue∼4.3×1011
g cm−3 [29,30]. Inside the transition region, a very strong electric field overwhelming the
critical field Ec =m2ec3/(eh¯) for vacuum breakdown appears [27], where me is the electron
mass.
The surface properties of nuclear matter such as the surface tension and the curvature
energy play an important role in many situations and phenomena such as the stability of
nuclei,fragmentdistributionsinheavy-ioncollisions,andphasetransitionsbetweendifferent
phasesofnuclearmatter. Thesurfacepropertiesofnuclearmatterhavebeenanalyzedalotin
thepastfewdecaysforthematteratthenuclearsaturationdensity[9,31–39],aswellasthe
matteratthesupranuclearregimerealizedintheinteriorofneutronstars[40,41]forthephase
transitionregionandthepastastructuresofthelow-densitynuclearmatter[16–18,20].
The surface properties of the core-crust interface of the new neutron star structure
obtained from the solution of the EMTF equations has been studied in Ref. [42] (see also
Ref.[43]forabriefdescription). WecalculatedinRef.[42]thesurfacetensionaswellasthe
electrostaticenergystoredinthiscore-crusttransitionlayer.Weanalyzedthestabilityofthese
systems through the Bohr-Wheeler fission mechanism [44]. It was shown in Ref. [42] that
theelectromagneticinteractionandthepresenceofdegenerateelectronshaveevidenteffects
on the surface properties of the core-crust interface. In the analyses of Refs. [27,42], we
employedtheconditionthattheelectrondensityisapproximatelyequaltotheprotondensity
in the core bulk region. Here we consider a more general case that the electron density is
smallerthantheprotondensityincorebulkregion. Actually,thisisthecaseofcompressed,
superheavy atoms in which some of the electrons have penetrated into superheavy nuclear
cores (superheavy nuclei) [45,46] (we call them compressed, superheavy atoms according
to Ref. [46] in which a similar object was studied). A compressed, superheavy atom is an
atomcomposedbyasuperheavynuclearcore(superheavynucleus),anddegenerateelectrons
that neutralize the system. Some electrons penetrate into the superheavy nuclear core and
the rest surround it up to a distance that depends upon the compression level. Such kind of
compressed,superheavyatomsarehypotheticalobjectsandcouldbepossibletoappearinthe
highdensityregionoftheneutronstarcrustorinothersystemsforexampleinther-processes
in gamma-ray bursts; studies of such kind of objects could provide a better understanding
inthefieldofnuclearphysicsandnuclearastrophysics. Inthisarticle, westudythesurface
propertiesofthesecompressed,superheavyatoms.
The article is organized as follows. In Sec. 2, we formulate the relativistic equations
of motion for the system of neutrons, protons and electrons fulfilling the strong and
electromagneticinteractionsandβ equilibrium, andtheequationsforgoverningthenuclear
surfacetensionandCoulombenergyofcompressed,superheavyatoms. InSec.3,wepresent
our discussions on the basis of the numerical analysis of the structure, the nuclear surface
tension,andtheCoulombenergyofcompressed,superheavyatoms. WealsoapplytheBohr-
Wheelerfissionmechanism[44]toanalyzethestabilityofcompressed,superheavyatoms,in
3
Sec. 3. We finally give a summary in Sec. 4. We use units with h¯ =c=1 throughout the
article.
2. Equationsofmotionandsurfacetension
Thesystemofcompressed,superheavyatomsunderconsiderationiscomposedofdegenerate
neutrons,protons,andelectronsincludingthestrong,electromagnetic,andweakinteractions
and fulfilling global charge neutrality. In this system, the electron density in the inside
bulk region (neb) smaller than the proton one (nbp), i.e., neb <nbp. We adopt the σ-ω-ρ
phenomenologicalnuclearmodelofBogutaandBodmer[9]todescribethestronginteractions
betweenthenucleons. TheLagrangiandensityofthemodelweconsideredhereisgivenby
L =Lf+Lσ+Lω+Lρ+Lγ+Lint, (1)
including the free-field Lagrangian densities L , L , L , and L , respectively for the
γ σ ω ρ
electromagnetic and the three mesonic fields, the three fermion species (electrons, protons
andneutrons)LagrangiandensityLf andtheinteractingpartLint. Adetaileddescriptionof
thismodelcanbefoundinRef.[27].
We adopt the compressed, superheavy atom as a spherical droplet, so we have
sphericalsymmetryinthissystem. Withinthemean-fieldapproximationandThomas-Fermi
approximation,theequationsofmotionforthissystemaregivenby
d2V 2dV
+ =−4πe(np−ne), (2)
dr2 r dr
d2σ 2dσ
dr2 + r dr =[∂σU(σ)+gsns], (3)
d2ω 2dω
+ =−(g Jω−m2ω), (4)
dr2 r dr ω 0 ω
d2ρ 2dρ
+ =−(g Jρ−m2ρ), (5)
dr2 r dr ρ 0 ρ
EeF =µe−eV =constant, (6)
EpF =µp+gωω+gρρ+eV =constant, (7)
EnF =µn+gωω−gρρ =constant. (8)
This is a special case of the EMTF system of equations [1,27] without the presence of
the gravitational interaction. Here we have introduced the notation ω ≡ω, ρ ≡ρ, and
0 0
A ≡V for the time components of the meson fields, where A is the electromagnetic field.
0
(cid:113)
µi = (PiF)2+m˜2i and ni =(PiF)3/(3π2) are the free chemical potential and the number
densityofthei-fermionspecies(i=n,p,e)withFermimomentumPF. Theparticleeffective
i
massesarem˜N =mN+gsσ andm˜e=me,wheremiistherestmassofeachi-fermionspecies.
gs,gω,andgρ arethecouplingconstantsoftheσ,ω andρ fields,andeisthefundamental
electric charge. m , m , and m are the masses of σ, ω, and ρ. U(σ) is the scalar self-
σ ω ρ
interactionpotentialwhichcanbefoundine.g. Refs.[27,42].
The generalized Fermi energies of electrons, protons, and neutrons, EF, EF, and EF,
e p n
derived from the thermodynamic equilibrium conditions given by the statistical physics of
multicomponent systems, are linked by the β-equilibrium [47] of protons, neutrons, and
electrons,
EF =EF+EF. (9)
n p e
4
Thescalardensitynsisgivenbytheexpectationvalue
ns= (2π2)3i=∑n,p(cid:90)0PiFd3kεmik˜(Nk), (10)
(cid:113)
where εk(k)= k2+m˜2 is the single particle energy. In the static case, the nonvanishing
i i
componentsofthecurrentsare
J0ch=(np−ne), (11)
J0ω =nb=(nn+np), (12)
J0ρ =(np−nn), (13)
herenb=np+nnisthebaryonnumberdensity.
WewouldliketomentionherethattheThomas-FermiapproximationandtheThomas-
Fermi approximation combined with the RMFT applied to nuclei are well-known and have
gained great successes in understanding nuclear structures (see, e.g., Refs. [48–50]). In the
present study, we apply this approach of the Thomas-Fermi approximation combined with
RMFTtocompressed,superheavyatoms,inspiredbyournewneutronstarmodelstudiedin
Refs.[27,42]. Oneofourmajorpurposeshereistoanalyzethepossibilityoftheexistenceof
such“exotic”neutronrichnucleiwhosemassnumbersaremuchlargerthanthatofordinary
nuclei.Anothermajorpurposehereistostudytheeffectsoftheelectronsandelectromagnetic
interaction on the surface properties of such a system. The study presented in this article
would give us a further understanding of the influence of the electromagnetic interaction
and electrons on the surface properties of the core-crust interface of our new structure of
neutron stars [27,42], hence give us a further understanding of global charge neutrality and
thestructureofneutronstars.
Theparametersofthenuclearmodel,namelythecouplingconstantsgs,gω,andgρ,the
meson masses m , m , and m , and the third- and fourth- order constants of the self-scalar
σ ω ρ
interactionsg andg arefixedbyfittingnuclearexperimentaldata,suchassaturationdensity,
2 3
bindingenergypernucleon,symmetryenergy,surfaceenergy,andnuclearincompressibility.
WehereusetheparametersoftheNL3parametrization[51]astheoneusedinRefs[27,42],
showninTable1.
m (MeV) 508.194 g 12.868
σ ω
m (MeV) 782.501 g 4.474
ω ρ
m (MeV) 763.000 g (fm−1) −10.431
ρ 2
gs 10.217 g3 −28.885
Table1.TheparametersofthenuclearmodelfromNL3[51].
Now we turn to the analyze of the surface tension of this system. We construct the
surface tension following a similar method in Ref. [42]. Since we treat the compressed,
superheavyatomasasphericaldroplet,weassumeasphericalsurface(thesizeofthesystem
underconsiderationislargerthantheoneofordinarynuclei,sothenuclearcurvatureenergy
hereissmallcomparedtothenuclearsurfaceenergy)withasmallthicknessseparatingone
finiteregion(insidethenuclearcoreregion)andonesemi-infiniteregion(outsidebackground
region,similartotheoutsidecrustregioninthediscussionofRef.[42]). Thenumberdensity
ofthei-fermionspeciesni(r)approachesthedensityofthei-fermionspeciesnibintheorigin
5
(theinsideregion)asthepositionr→0,andapproachesthedensityintheoutsideregionof
the i-fermion species nio as the r→+∞. To construct the surface tension, as in the case of
the semi-infinite matter model, we imagine a reference system with sharp surfaces at radii
ri (i=n,p,e,σ,ω,ρ)atwhichfermiondensitiesandmesonfieldsfalldiscontinuouslyfrom
the bulk region to the outside region. Following a similar method of Baym-Bethe-Pethick
(BBP) [29], the location of the reference surface for the i-fermion species is defined by the
conditionthatthereferencesystemhasthesamenumberofi-fermionspeciesastheoriginal
system,
(cid:90) ri (cid:90) ∞
4π r2dr[ni(r)−nib]+4π r2dr[ni(r)−nio]=0, i=n,p,e. (14)
0 ri
Similar to the definition of reference surfaces for fermions, the location of the reference
surfacesformesonfieldsaredefinedby
(cid:90) ri (cid:90) ∞
4π r2dr[Fi(r)−Fib]+4π r2dr[Fi(r)−Fio]=0, i=σ,ω,ρ, (15)
0 ri
where Fi(r) is the time component of the i-meson field, Fib is the time component of the i-
meson field in the inside region, and Fio is the time component of the i-meson field in the
outsideregion.
SimilartothewayofBBP[29],thenuclearsurfaceenergycanbecomputedasthetotal
energysubtractingoffthebulkenergy,
(cid:26)(cid:90) ri (cid:90) ∞ (cid:27)
Esur= ∑ 4π r2[εi(r)−εib]dr+ r2[εi(r)−εio]dr , (16)
i=n,p,σ,ω,ρ 0 ri
andtheCoulombenergyis
(cid:90) ∞
Ecoul=4π r2εE(r)dr, (17)
0
whereεi(r)istheenergydensityoftheispeciesoffermionormesonfields,εib istheenergy
density of the i species of fermion or meson fields in the center of the system (the inside
region), εio is the energy density of the i species of fermion or meson field in the outside
region, and εE(r) is the energy density of the electric field. Similar to the energy densities
giveninRef.[42],theenergydensityofthei-fermionspeciesεi(r)is
εi(r)= 8π12(cid:26)PiF(cid:113)(PiF)2+m˜2i (cid:2)2(PiF)2+m˜2i(cid:3)
(cid:113)
PF+ (PF)2+m˜2(cid:27)
i i i
−m˜4ln , (18)
m˜i
andtheenergydensitiesofthemesonfieldsinthissphericalsystemare
1(cid:18)dσ(cid:19)2
ε (r)= +U(σ), (19)
σ 2 dr
1(cid:18)dω(cid:19)2 1
ε (r)= + m2ω2, (20)
ω 2 dr 2 ω
1(cid:18)dρ(cid:19)2 1
ε (r) = + m2ρ2, (21)
ρ 2 dr 2 ρ
1 (cid:18)dV(cid:19)2
εE(r)= . (22)
8π dr
6
Thenuclearsurfacetensionisgivenasthenuclearsurfaceenergyperunitarea,
E
σ = sur , (23)
Ns 4πr2
n
andsimilarlyweobtaintheCoulombenergyperunitarea(thesurfacetensionfortheelectric
field)
E
σ = coul, (24)
Cs 4πr2
n
where rn is the reference radius of neutrons defined by Eq. (14). Since the neutron number
is much larger than the proton number in the system, so it is reasonable to set the radius of
neutronstobetheradiusofthenucleustoestimatethesurfacetensions;thisisconsistentwith
theexistenceoftheneutronshaloorneutronskineffect[52].
The relation between the nuclear surface energy and the Coulomb energy is very
importantforanucleus. AsshownbyBohrandWheeler[44]whenthecondition
E >2E (25)
coul sur
is satisfied, the nucleus becomes unstable against nuclear fission. A careful analysis on the
derivationofthisconditionshowsthattheBohr-WheelerconditiongivenbyEq.(25)applies
alsotooursystem[42].
3. Numericalanalysis
Following a similar procedure in Refs. [27,42], we can solve the equations (2)-(8) together
with the β-equilibrium (9) to obtain the fermion-density and meson-field profiles. This
system of equations can be numerically solved with appropriate boundary conditions and
approximations,asshowninRefs.[27,42].
In order to obtain a solution of these equations, we set a value for the baryon number
density nbb =nnb+npb in the region near the center, and we set a small electron density
neb = yenpb in the region near the center with electron fraction ye < 1. As described in
Refs. [27,42], the fermion densities nio in the outside region depend on the density at
the base of the background under consideration (similar to the crust in the discussion of
Ref. [42]). The background matter is composed of a nuclei lattice in a background of
degenerateelectrons,whosedensityisdenotedhereasnbeg.Inaddition,therearefreeneutrons
in the background when the density ρ of the background is higher than the neutron-drip
bg
densityρ ≈4.3×1011gcm−3[29]. Sowhenthedensityρ ofthebackgroundissmaller
drip bg
thantheneutron-dripdensityρ ,i.e.,ρ <ρ ,wesettheprotondensityandtheneutron
drip bg drip
density to zero in the outside region while the electron density matches the value nbeg of the
densityofbackgroundelectrons,i.e.,neo=nbeg.Whenρbg>ρdripboththeneutrondensityand
the electron density have to match their corresponding background densities, i.e., neo =nbeg
andnno=nbng,wherenbngistheneutrondensityinthebackground.AsshowninRef.[29]there
is no proton drip in the systems under consideration, so we keep the outside proton density
valueaszero.Inordertosetthematchingdensityvaluesforelectronsandneutronsweusethe
relationbetweenthefreeneutrondensityandtheelectrondensityinSection6ofRef.[29].
As shown in Refs. [27,42], the transition interface that we are interesting in appears
near the nuclear saturation density n =0.16 fm−3. In order to study the compressed,
nucl
superheavy atoms and the influence of the electrons and electromagnetic interaction on the
surfacepropertiesofthesystem,weassumeatfirstthebaryonnumberdensityintheregion
near the center to be the nuclear saturation density (results presented in Figs. 1-5), i.e.,
7
n =n =0.16fm−3. Attheendofthissection,wewillalsostudytheinfluenceofbaryon
bb nucl
numberdensity(resultspresentedinFig.6andTable2).
10-1 (a) neutrons 3000 (b)
2500
-3Lm10-2 eplreocttornosns E 2000
nHfi10-3 Ec 11050000
500
10-4
0
0 100 200 300 400 0 100 200 300 400
r(cid:144)ΛΣ r(cid:144)ΛΣ
20
(c)
r(cid:144)Λ
20 40 60 80 Σ
-20 Σ
Ω
Ρ
-40
Figure1. (Coloronline)(a): fermiondensityprofilesinunitsoffm−3. (b): electricfieldin
unitsofthecriticalfieldEc. (c): mesonfieldsσ,ω,andρ inunitsofMeV. Hereweset
PF =0.95PF,thebaryonnumberdensityintheregionnearthecenteristhenuclearsaturation
eb pb
densitynnucl,andthedensityintheoutside(background)regionistheneutron-dripdensity
ρbg=ρdrip≈4.3×1011gcm−3. λσ =h¯/(mσc)∼0.4fmistheComptonwavelengthofthe
σmeson.
10-1 (a) neutrons 3000 (b)
-3Lm10-2 eplreocttornosns E 22050000
nHfi10-3 Ec 11050000
500
10-4
0
0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350
r(cid:144)ΛΣ r(cid:144)ΛΣ
20
(c)
r(cid:144)Λ
10 20 30 40 50 60 Σ
-20 Σ
Ω
Ρ
-40
Figure2. (Coloronline)(a): fermiondensityprofilesinunitsoffm−3. (b): electricfieldin
unitsofthecriticalfieldEc. (c): mesonfieldsσ,ω,andρ inunitsofMeV. Hereweset
PF =0.8PF,thebaryonnumberdensityintheregionnearthecenteristhenuclearsaturation
eb pb
densitynnucl,andthedensityintheoutside(background)regionistheneutron-dripdensity
ρbg=ρdrip≈4.3×1011gcm−3.
8
TheresultsofthesolutionsoftwoexamplesareshowninFig.1forthecasePF =0.95PF
eb pb
andinFig.2forthecasePF =0.8PF,whenthedensityintheoutside(background)regionis
eb pb
theneutron-dripdensityρ =ρ ≈4.3×1011 gcm−3. Wehaveintroducedthenotations
bg drip
PF fortheFermimomentumofelectronsintheregionnearthecenterofthesystem,andPF
eb pb
for the Fermi momentum of protons in the region near the center of the system. It is also
worthmentioningherethatthetypicalmassnumberofthesecompressed,superheavyatoms
is ∼104; e.g., A∼35000 and Z/A∼0.154 for the case shown in Fig. 1, and A∼12000
and Z/A ∼ 0.189 for the case shown in Fig. 2, where A is the total number of nucleons
(massnumber)andZisthetotalnumberofprotons. Themassnumbersofthesecompressed,
superheavyatomsaremuchlargerthanthatofordinarynuclei.
As shown in Fig. 1, when the difference between the electron density and the proton
density in the region near the center of the system (npb−neb) is small, the fermion-density
and meson-field profiles are similar to their counterparts in the case of semi-infinite matter
(electron density nearly equal to the proton density in the inside bulk region neb (cid:39) npb).
Comparing to the results in the case of the electron density being approximately equal to
theprotondensityinthecorebulkregionshowninRef.[42],thebumpoftheprotonprofile
islargerinthiscase,asexpectedfromthefactthattheinternalelectricfieldislessscreened
thanthecaseofneb(cid:39)npb. WecanalsoseefromFigs.1-2,howthefermionandmeson-field
profileschangeforincreasingchargeseparations(npb−neb).
ΣNsHMeV(cid:144)fm2L (a) ΣC1sH1M.0eV(cid:144)fm2L (b)
8.5 10.5
10.0
8.0
9.5
0.6 0.7 0.8 0.9 PFeb(cid:144)PFpb 0.6 0.7 0.8 0.9 PFeb(cid:144)PFpb
Figure3. (Coloronline)ThedependenceofthesurfacetensionontheratioPF/PF. The
eb pb
baryonnumberdensityintheregionnearthecenteristhenuclearsaturationdensitynnucl,
andthefermiondensitiesandmesonfieldstendtobezerointheoutsideregion. (a): nuclear
surfacetensionσNs.(b):CoulombenergyperunitareaσCs.
Using the definitions in Eqs. (23) and (24), we obtain the surface tensions for
compressed, superheavy atoms. The dependence of the surface tension on the ratio of the
electron Fermi momentum and the proton Fermi momentum in the region near the center
of the system (PF/PF) is shown in Fig. 3 for the case of the fermion densities and meson
eb pb
fields tending to be zero in the outside region, and Fig. 4 for the case of the density in the
outside (background) region is the neutron-drip density ρ =ρ ≈4.3×1011 g cm−3.
bg drip
Fromtheresults,thesystemisstablewithrespecttotheBohr-Wheelercondition(25)ofthe
stability,inallratiosPF/PF weconsider.Thisistheresultofthepenetrationoftherelativistic
eb pb
electrons into the nucleus (see also Refs. [45,46]). This in principle implies the possibility
of the existence of such kind of compressed, superheavy atoms. As shown in Fig. 3, the
nuclear surface tension σNs first increases and then decreases when the difference between
theelectrondensityandtheprotondensityincreases,andthenuclearsurfacetensiontendsto
thephenomenologicalresult(∼1MeVfm−2)withoutthepresenceofelectronsintheinside
bulk region studied in the nuclear physics [29]. There are two effects which influence on
the nuclear surface tension σNs: (I) for neb <npb the bump of the proton profile around the
nuclearsurfacechangesasshowninFigs.1–2,and(II)thehigherthedifference(npb−neb)is,
thelowerthenuclearasymmetry. Asaconsequence,thetotalenergyofthesystemdecreases.
9
The combination of these two effects leads to the results of the nuclear surface tension σNs
showninFig.3.
ΣNsHMeV(cid:144)fm2L (a) Σ HMeV(cid:144)fm2L (b)
Cs
8.5
8.0
8.0 7.5
7.5 7.0
6.5
7.0
6.0
PF (cid:144)PF PF (cid:144)PF
0.6 0.7 0.8 0.9 eb pb 0.6 0.7 0.8 0.9 eb pb
Figure4. (Coloronline)ThedependenceofthesurfacetensionontheratioPF/PF. The
eb pb
baryonnumberdensityintheregionnearthecenteristhenuclearsaturationdensitynnucl,and
thedensityintheoutsideregionistheneutron-dripdensityρbg=ρdrip≈4.3×1011gcm−3.
(a):nuclearsurfacetensionσNs.(b):CoulombenergyperunitareaσCs.
Comparing the results of Fig. 3 and Fig. 4, we can find that the electrons in the
outsideregionhavestrongeffectsonthesurfacestructureofcompressed,superheavyatoms
consideredhere. Theincreaseoftheelectrondensityintheoutsideregioneffectivelyreduces
theCoulombenergyperunitareaσ ,aswellasthenuclearsurfacetensionσ . Thiseffect
Cs Ns
is enhanced when increasing difference between the electrondensity and the proton density
in the region near the center of the system (npb−neb), as shown in Figs. 3-4. This effect is
mainlyduetothereasonthattheelectronshaveastronginfluenceonthebumpontheprofiles,
leadingtoastrongeffectonthesurfacestructureandthesurfacetensionsσ andσ . These
Ns Cs
resultsprovidetheevidenceofstrongeffectsoftheelectromagneticinteractionandelectrons
onstructureofthesystem. Thisresultoftheeffectduetotheelectronsintheoutsideregion
asshownbythecomparisonofFig.3andFig.4isdifferentfromthecasestudiedinRef.[42]
where the electron density in the inside bulk region (neb) is nearly equal to the proton one
(nbp). InthecaseshowninRef.[42],theeffectoftheelectronsintheoutsideregionissmall
whenthedensityρ intheoutsideregionissmallerthantheneutron-dripdensity,ρ <ρ .
bg bg drip
We now turn to study the effect of the free neutrons in the background (the outside
region) on the surface properties of compressed, superheavy atoms. The dependence of the
surfacetensiononthedensityρ ofthebackgroundforthecaseofPF =0.8PF isshownin
bg eb pb
Fig.5. AsshowninFig.5(c),theBohr-Wheelercondition(25)fortheinstabilityisreached
at a background density ρcrit ∼ 9.7×1013 g cm−3, so the system becomes unstable against
bg
fissionwhenρ >ρcrit. Thisimposesaphysicalupperlimittothedensityofthebackground
bg bg
forcompressed,superheavyatomswithPF =0.8PF. Thiscriticalbackgrounddensityρcritis
eb pb bg
smallerthantheoneforthecaseoftheelectrondensityintheinsidebulkregionbeingnearly
equal to the proton one (neb (cid:39)nbp) discussed in Ref. [42]. This implies that the difference
betweentheelectrondensityandtheprotondensityintheregionnearthecenterofthesystem
(npb−neb)candecreasethestabilityofcompressed,superheavyatoms.
The results in Fig. 5 clearly show the strong effect of the fermions in the outside
(background) region on the surface structure of compressed, superheavy atoms, as we have
discussed above in the comparison of Fig. 3 and Fig. 4. The Coulomb energy per unit area
σ and the nuclear surface tension σ change significantly as changing the density ρ of
Cs Ns bg
the background (the outside region), in both cases: (I) the density ρ of the background is
bg
higher than the neutron-drip density ρ ; (II) the density ρ of the background is smaller
drip bg
thantheneutron-dripdensityρ .
drip
In the previous discussions, we have assumed the baryon number density in the region
10
ΡbgHg(cid:144)cm3L (a) ΡbgHg(cid:144)cm3L (b)
1013 1013
1011 1011
109 109
Σ Σ
2 3 4 5 6 7 8 Ns 4 5 6 7 8 9 10 11 Cs
ΡbgHg(cid:144)cm3L (c)
1013
1011
109
Σ (cid:144)Σ
0.5 0.6 0.7 0.8 0.9 1.0 1.1 Ns Cs
Figure5. (Coloronline)Thedependenceofthesurfacetensiononthedensityρbg ofthe
background. HerewesetPF =0.8PF andthebaryonnumberdensityintheregionnearthe
eb pb
centeristhenuclearsaturationdensitynnucl.(a):nuclearsurfacetensionσNs,inunitsofMeV
fm−2.(b):CoulombenergyperunitareaσCs,inunitsofMeVfm−2.(c)Ratioofthenuclear
surfacetensionandtheCoulombenergyperunitarea,σNs/σCs.
ΣNsHMeV(cid:144)fm2L (a) ΣCsHMeV(cid:144)fm2L (b)
11
12 10
9
10
8
8 7
6
6
5
0.8 0.9 1.0 1.1 1.2nbb(cid:144)nnucl 0.8 0.9 1.0 1.1 1.2nbb(cid:144)nnucl
Figure6.(Coloronline)Thedependenceofthesurfacetensiononthebaryonnumberdensity
intheregionnearthecenter(nbb). HerePeFb=0.8PpFb,andthedensityintheoutsideregion
istheneutron-dripdensityρbg=ρdrip≈4.3×1011gcm−3.(a):nuclearsurfacetensionσNs.
(b):CoulombenergyperunitareaσCs.
near the center to be the nuclear saturation density n in symmetric matter, to study
nucl
the influence of the electrons on the surface properties of the transition interface [27,42].
However, the saturation density in nuclei can be different while changing the asymmetry
parameter(see,e.g.,Refs.[53,54]). Therefore,itwouldbenecessarytoanalyzetheinfluence
ofthebaryonnumberdensityonthesurfacetensions. Thedependenceofthesurfacetension
onthebaryonnumberdensityintheregionnearthecenter(n )isshowinFig.6,forthecase
bb
ofPF =0.8PF andρ =ρ ≈4.3×1011gcm−3. ComparingwiththeresultsinRef.[42],
eb pb bg drip
the dependence of the surface tension on the baryon number density shown in Fig. 6 for
thecaseofcompressed,superheavyatomshasasimilarbehaviorasinthecasediscussedin
Ref.[42]forthecore-crustinterfaceofneutronstars(neb≈npb). Therefore,wecanconclude
thattheeffectsofthebaryonnumberdensityonthesurfacetensionsσ andσ forthecase
Ns Cs
ofcompressed,superheavyatomsaresimilartotheonesforthecasethecore-crustinterface
ofneutronstars(neb≈npb)[42].