Mechanical and chemical spinodal instabilities in finite quantum systems

4 Pages·0.42 MB·English

by  M. Colonna

Mechanical and chemical spinodal instabilities in finite quantum systems
M. Colonna1, Ph. Chomaz2 and S. Ayik3
1 Laboratori Nazionali del Sud, Via S. Sofia 44, I-95123 Catania, Italy
2 GANIL (DSM-CEA/IN2P3-CNRS),
B.P.5027, F-14021 Caen cedex, France
3 Tennessee Technological University,
Cookeville TN38505, USA
sity matrix ρ˜(t) is determined by, i¯h∂ ρ˜(t) = [h[ρ˜],ρ˜(t)],
t
Self consistent quantumapproachesare usedto studythe whereh[ρ˜]=p2/2m+U[ρ˜]isthemean-fieldHamiltonian
instabilitiesoffinitenuclearsystems. Thefrequenciesofmul-
with U[ρ˜] as the self-consistent potential. To investigate
tipole density fluctuations are determined as a function of
instabilities encountered during the evolution of an ex-
2 dilution and temperature, for several isotopes. The spinodal
panding system, one should study the dynamics of the
0 region of the phase diagrams is determined and it appears
small deviations δρ˜(t) around the TDHF trajectory ρ˜(t)
0 thatinstabilitiesarereducedbyfinitesizeeffects. Theroleof
2 surface and volume instabilities is discussed. It is indicated [10]. It is more convenient to carry out such an investi-
thattheimportantchemical effectsassociated with mechani- gation in the ”co-moving frame” of the system, which is
n
a cal disruption may lead toisospin fractionation. described by the density matrix, ρ(t)=U†(t) ρ˜(t) U(t)
J PACS numbers: 21.65.+f, 25.70.Pq,21.60.Ev where U(t) = exp[ iλtQ] with Q as a one-body opera-
h¯
5 tor. For example, in our case Q could be a suitable con-
2 straining operator for preparing the system at low den-
The dynamics of first order phase transitions is often

1 induced by instabilities against fluctuations of the order sities andλ is the associatedLagrangemultiplier. Then,
theTDHFequationinthemovingframetransformsinto,
v parameter. For instance, mechanical instabilities lead to
0 liquid-gasphase transitionsandchemicalinstabilitiesin-
∂
7 duce spinodal decomposition of binary alloys. In violent i¯h ρ(t)=[h(t) λQ, ρ(t)], (1)
0 ∂t
heavy ion collisions, nuclear matter may be quenched in
1
0 thecoexistenceregionofthenuclearliquid-gasphasedia- where the mean-field Hamiltonian in the moving frame
2 gram. Then, the observedabundant fragment formation is given by, h(t) = U†(t) h[ρ˜] U(t). The small density
0 maytakeplacethrougharapidamplificationofspinodal fluctuationsδρ(t)inthemovingframearedeterminedby
h/ instabilities. New experimental results pleading in fa- the time-dependent RPA equations,
t vor of such a spinodal decomposition have recently been
- ∂
l reported [1,2]. From the theoretical point of view, the i¯h δρ=[h(t)−λQ,δρ]+[δU,ρ]. (2)
c spinodal instabilities in finite systems have been mainly ∂t
u
n studied within semi-classical or hydrodynamical frame- Here, we consider the early evolution of instabilities
: work[3–7]. However,sincetherelevanttemperaturesare
v in the vicinity of an initial state ρ0 determined by a
comparable to the shell spacing and the wave numbers
Xi constrained Hartree-Fock solution [h(0)−λQ, ρ0] = 0,
of the unstable modes are of the order of Fermi momen-
where h(0) = h[ρ0] ≡ h0 is the mean-field Hamiltonian
r tum, quantum effects are expected to be important, as
a at the initial state. Then, small density fluctuations are
stressed in [8,9].
characterized by the RPA modes ρ and the associated
ν
In this work, we present a fully quantal investigation
frequencies ω . Incorporating the representation |i >,
ν
of the spinodal instabilities and related phase diagrams
which diagonalizes h0 −λQ and ρ0 with eigenvalues ǫi
of finite nuclear systems. We determine frequencies and
and occupation numbers n , the equations for the RPA
i
form factors of the unstable collective modes of an ex- functions <i|ρ |j >=ρij become,
ν ν
citedexpandedsystembylinearizingthetimedependent
Hartree Fock (TDHF) equation. We carry out applica- ¯hωυρiνj =(ǫi ǫj) ρiνj +X(nj ni)Vil,kj ρkνl , (3)
tionsforCaandSnisotopes. Thequantumnatureofthe
kl
system is responsible for many dominant features such
as the fact that the first mode to become unstable is the where Vil,kj =< i|∂U/∂ρlk|j > denotes the residual in-
lowlyingoctupolevibration. Slightlydilutedsystemsare teraction [10,11]. When the frequency of a mode drops
unstable against low multipole deformations of the sur- to zero and then becomes imaginary, the system enters
face, which may be associated with asymmetric binary an instability region.
or ternary fission processes. We extend our analysis to Toperformeanextensivestudyofinstabilitieswemay
charge asymmetric systems and show the importance of parametrize the possible densities ρ0 either by a static
chemical effects in the spinodal fragmentation process. Hartree-Fock (HF) calculations constrained by a set of
In TDHF theory, the evolution of the one-body den-
1
collective operators [12], or using a direct parameteriza- resentation using the Skyrme force SLy4 [13]. We note
tion of the density matrix. We follow the second ap- that the isospin symmetry is already broken at the HF
proach by introducing a self-similar scaling of the HF level. The particle and hole states are obtained by diag-
density as suggested by dynamical simulations. onalizing the HF Hamiltonian in a large harmonic oscil-
We solve the HF equation for the ground state lator representation [14], which includes 12 major shells
[h ,ρ ]=0, leading to the single-particle wavefunc- for Ca isotopes and 15 for Sn. We apply the scaling
HF HF
tions |ϕ > and the associated energies ε . We intro- and heating procedures described above to the density
i i
duce the density matrix at a finite temperature T as matrix, and calculate the residual interaction in a self
ρ [T]= 1/ 1+exp h ε [T] /T , where ε [T] consistent manner. We solve the RPA eq.(3) by a direct
HF (cid:0) (cid:0)(cid:0) HF F (cid:1) (cid:1)(cid:1) F
is the corresponding Fermi level that is tuned in or- diagonalization using a discrete two quasi-particle exci-
der to get the correct particle number. We perform a tation representation [15].
scaling transformation, R[α], which inflates the wave The top part of figure 1 shows calculations performed
functions in radial direction by a factor α according for40Ca. Toppanelsshowscontourplotsoftheisoscalar
to < r|R[α]|ϕ >= α−1/3 < r/α|ϕ > . We define strength function associated with the isoscalar operator
the density matrix for a hot and diluted system by As = A rLY , with multipolarity L = 2−5, as
LM Pi=1 i LM
ρ0[α,T]=R[α] ρHF (cid:2)α2T(cid:3) R†[α]. The associated con- a function of the dilution parameterα. In the stable do-
strainedHamiltonianisthush¯0[α]=α2R[α] hHF R†[α] main, the energy associated with the dominant isoscalar
so that the constraint can be identified as −λQ = strength decreases as dilution becomes larger, and at
α
h¯0[α]−h[ρ0[α,T]].Byconstruction(cid:2)h¯0[α], ρ0[α,T](cid:3)= a critical dilution it drops to zero. At larger dilution,
0, so that they can be diagonalized simultaneously. The thesystembecomesunstable,andforeachmultipolarity,
eigenstates of the constrained Hamiltonian are given one or two unstable modes appear. This is illustrated
by |i >= R[α]|ϕ >, and the corresponding energies in the bottom panel, where the ”energy” of the mode
i
and occupation numbers are ǫ = ε /α2 and n = E =−i¯hω is plotted as a function of the dilution.
i i i ν ν
1/ 1+exp ε ε α2T /α2T , respectively. It is seen that at low dilutions, around α = 1.2, only
(cid:0) (cid:0)(cid:0) i F (cid:2) (cid:3)(cid:1) (cid:1)(cid:1)
theoctupolemodebecomesunstable. Ingeneral,density
fluctuations with odd multipolarity become unstable at
relatively smaller values of dilution than those for even
multipolefluctuations. Thisis agenuinequantumeffect,
due to the fact that the majority of the particles have
to jumponly onemajorshelltoproduce anoddnatural-
parity particle-hole excitation while twice this energy is
requiredforanevenone. Innucleiatnormaldensitythis
makes the first 3− a strongly collective low lying state;
in diluted system this state is the first to turn unstable.
The second feature is that large multipole deformations
are hardly becoming unstable. This is due to surface
and quantum effects that prevent the break-upof such a
small system into several fragments. As a consequence,
the fastest growth time, τ = h¯/|E |≈ 28 fm/c, occurs
ν ν
for L = 2 at the dilution α=1.8. However, deep inside
the instability region, the octupole mode is almost as
unstable as the quadrupole mode.
Results of similar calculations performedfor 120Sn are
shown in the bottom part of figure 1. Also in this case,
the octupole mode becomes unstable first, at α ≈ 1.1.
Largemultipolesaremoreunstablein120Snthanin40Ca,
sincethe systemislargerandcanaffordlargermultipole
FIG. 1. Contour plots of the isoscalar strength functions deformations. Thisisthereasonwhythesmallestgrowth
associated with the multipolarity L = 2−5 as a function of time in 120Sn occurs for L=3.
thedilutionparameterαandthecollectiveenergyofthemode It is useful to study the behavior of the RPA solu-
Eν = c¯hων (c = 1 for stable modes, −i for unstable modes) tion in the coordinate space, ρ+p,n(r) ≡ ρ+p,n(r,r), for
for 40Ca (top) and 120Sn (bottom). The strength values are protons and for neutrons separately. Since the unstable
giveninpercentageoftheEWSR:2(yellow),15(orange)and modes have large isoscalar strength, protons and neu-
30 (red).
trons mostly move in phase. However, because of the
isospinsymmetrybreakinginducedbytheCoulombforce
WeperformtheHFcalculationsinthecoordinaterep-
and by the initial asymmetry of the system considered,
2
neutrons and protons oscillations have different ampli- amplitude ofprotonfluctuationsρ+(r) andneutronfluc-
p
tudeandshape. Figure2(a)showstheradialdependence tuations ρ+(r), obtained at a radial distance r ranging
n
of the form factor associated with the unstable octupole from 0 to the system radius R, for the octupole mode.
mode, at the dilution α =1.5, for protons (dotted line), The different panels correspond to the three Sn isotopes
neutrons (full line) and the sum (dashed line), in Sn iso- consideredandto three dilutions,α, indicatedin the fig-
topes. Contourplots ofneutron(proton)perturbedden- ure. The correlationsdescribeclosedpaths,since ρ+ (r)
p,n
sities, ρ0p,n(r)+ρ+p,n(r), are also shown in parts b (c) of starts from zero at r =0 and goes to zero again at large
figure 2. distances.
FIG.3. Correlations between theamplitude of proton and
FIG.2. (a)Radialdependenceoftheformfactorassociated neutron fluctuations, for several dilutions. The full circles
withtheunstablemodewithL=3,atthedilutionα=1.5,for correspond to a radial distance 0 < r < 2.4 α fm, the open
protons(dottedline),neutrons(fullline)andthesum(dashed circles correspond to 2.4 α fm < r < R. The dashed line
line). Theradial distance isscaled bythedilution parameter indicatesρ+p =ρ+n,thefulllinecorrespondstoρ+p =ρ+n Z/N.
α. (b)Contourplotsoftheneutron perturbeddensity. From Results are shown for 100Sn (a), 120Sn (b) and 132Sn (c).
thesurface to thecenter,thecontourlines correspond to the
density values: 0.0037, 0.0075, 0.015, and 0.023 fm−3. For Whenprotonsandneutronsfluctuatewhithoutchang-
larger density,colors show small density variations (3%). (c) ing the local chemical ratio one should have ρ+(r) =
p
Same for protons ρ+(r) Z/N (full line). An indentical motion of the two
n
fluids should follow the diagonal ρ+(r) = ρ+(r) (dashed
p n
We observethatprotonoscillationsaremostlylocated line). InN =Z nuclei(parta)themotionofprotonsand
at the surface of the system, which is a way to mini- neutronsshouldbethesame(isoscalarexcitations),how-
mize the Coulomb repulsion energy. At the same time ever the Coulomb force introduces small differences. As
neutrons try to follow protons, however this motion is seen in figure 2, in diluted systems (α > 1), we observe
strongly affected by the isospin initial asymmetry and that,inthevicinityofthenuclearsurface,densityfluctu-
the difference between the neutron and proton orbitals ationsarelargerforprotonsthanforneutrons,expecially
at the Fermi energy. In fact, in neutron rich systems, in the case of neutron-rich unstable isotopes, 120Sn and
much larger proton oscillations are observed, that is a 132Sn (see panels (b) and (c) in figure 3), leading to a
way to form more symmetric fragments and hence to re- reduction of the asymmetry of the fragments produced
ducethe symmetryenergy. Thelasteffectisparticularly at the surface. In the interior we observe, for these iso-
evident in the neutron rich 132Sn for which the neutrons topes, an interesting evolution from ρ+(r)<ρ+(r) Z/N
p n
are more difficult to put in motion. We observe a quite for α = 1 (stable modes) to ρ+(r) > ρ+(r) Z/N for the
p n
complex structure of the unstable modes: Volume and dilute systems. Hence we observe a change of the local
surface instabilities are generally coupled and cannot be chemical ratio, leading to a reduction of the symmetry
easily disantangled, as well as isoscalarand isovectorex- energy, also in the interior of the system. This effect
citations, since protonsand neutrons do not move in the may be related to the isospin fractionation that occurs
same way. Figure 3 shows the correlations between the in unstable asymmetric nuclear matter [16]. The pro-
3
tonmigrationtowardsthe largedensity domains is more isospin effects on instabilities by carrying out calcula-
frequent than the neutron migration, which may lead to tions for Ca and Sn isotopes. We find that instabilities
formation of more symmetric fragments. aremostlyofisoscalarnature,butcontainalsoanimpor-
We, also, carry out calculations at finite temperature tant isovector component in asymmetric matter. Hence
and determine the dilutions at which different unstable the liquid-gas separation of asymmetric systems is al-
modesbegintoappear. Thisallowsustospecifythebor- wayslinkedtoachemicalseparationinducingafractional
der of the instability region in the density-temperature distillation of the system. The degree of instability in
planefordifferentunstablemodes. Figure4showsphase neutron-richandalsoproton-richnucleiappearsreduced
diagramsforoctupoleinstabilitiesinSnandCaisotopes. as compared to that of symmetric nuclei of comparable
size. Theinstabilitiesarealsoreducedinsmallnuclei. Fi-
nally we have stressed important quantum effects, such
astheoctupolarnatureofthemostimportantinstability.
Authors acknowledge the Yukawa Institute for Theo-
reticalPhysics,S.A.andM.C.acknowledgeGANIL,and
Ph.C. acknowledges LNS, where parts of this work have
beencarriedout,forsupportandwarmhospitality. Stim-
ulating discussions with M.Di Toro and D.Lacroix are
also acknowledged. This work is supported in part by
the US DOE grant DE-FG05-89ER40530.
FIG.4. Border of the instability region (full fine) associ-
ated with L = 3, for Ca and Sn isotopes. The dashed line [1] L. Beaulieau et al., Phys. Rev.Lett. 84, 5971 (2000).
connects the points having the instability growth time τ = [2] B. Borderie et al., Phys. Rev.Lett. 86, 3252 (2001).
100 fm/c. The dots are associated with τ = 50 fm/c. [3] G. F. Bertsch and P. J. Siemens, Phys. Lett. B126, 9
(1983).
Here,forsimplicity,wedefinethedensityasρ=ρ0/α3. [4] H.Heiselberg,C.J.Pethick,andD.G.Ravenhall,Phys.
Thefulllineindicatestheborderoftheinstabilityregion. Rev. Lett.61, 818 (1988).
The dashedline connectspointsthatareassociatedwith [5] W. N¨orenberg, G. Papp and P. Rozmej, Eur. Phys. J.
the instability growth time τ = 100 fm/c, and the dots A9, 327 (2000).
corresponds to situations (if any) with a shorter growth [6] M. Colonna, Ph. Chomaz, and J. Randrup, Nucl.
time τ = 50 fm/c. The instability region appears quite Phys. A567, 637 (1994); A.Guarnera, M.Colonna and
Ph.Chomaz, Phys.Lett. B373,267 (1996).
reducedascomparedtothatofnuclearmatter. Thelim-
[7] B.Jacquot,S.Ayik,Ph.ChomazandM.Colonna,Phys.
itingtemperatureforinstabilitytooccurisaround6MeV
Lett. B383, 247 (1996).
for Sn and 4.5 MeV for Ca while it is about 16 MeV in
[8] S.Ayik,M.ColonnaandPh.Chomaz,Phys.Lett.B353,
symmetric nuclear matter. Heavier systems have larger
417 (1995).
instability region than the lighter ones. Moreover, more
[9] B.Jacquot,M.Colonna,S.AyikandPh.Chomaz,Nucl.
asymmetricsystems arelessunstable. Infact, inspite of Phys. A617, 356 (1997).
thelargermass,132Snislessunstablethan120Sn. Larger [10] D.VautherinandM.Veneroni,FirstInt.SpringSeminar
instability growth times are obtained for 132Sn. As seen on Nuclear Physics, Sorrento, Italy (1986).
from phase diagrams for 36Ca and 40Ca, the region of [11] P.RingandP.Shuck,TheNuclearMany-Bodyproblem,
instability is reduced also in proton-rich systems. This Springer-Verlag, New York (1980).
behavior is in agreement with nuclear matter calcula- [12] H. Sagawa and G.F. Bertsch, Phys. Lett. B146, 138
(1984); B155, 11 (1985).
tions, which indicates that the instability region shrinks
[13] E.Chabanatetal.,Nucl.Phys.A627,710(1997); Nucl.
in asymmetric nuclear matter [16].
Phys. A635, 231 (1997); Erratum: Nucl. Phys. A643,
In conclusion, we have presented a study of the early
441 (1998).
developmentofspinodalinstabilitiesinfinitenuclearsys- [14] N.VanGiaiandH.Sagawa, Nucl.Phys.A371,1(1981).
temsbyemployingaquantalRPAapproach. Resultsare [15] D. Vautherin and N. Vinh Mau, Nucl. Phys. A422, 140
relevantformultifragmentationstudies,infactdominant (1984).
unstable modes determine the onset of the subsequent [16] Bao-An Li, C.M. Ko, Nucl. Phys. A 618 498 (1997);
fragmentation of the system, that has to be followed us- V.Baran, M.Colonna, M.Di Toro and V.Greco, Phys.
ing a full dynamical calculation. We have investigated Rev. Lett.86, 4492 (2001).
4