Simulation of Transitions between "Pasta" Phases in Dense Matter

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by  Gentaro Watanabe

Simulation of Transitions between “Pasta” Phases in Dense Matter
Gentaro Watanabea,b,c, Toshiki Maruyamad, Katsuhiko Satob,e, Kenji Yasuokaf and Toshikazu Ebisuzakic
aNORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
bDepartment of Physics, University of Tokyo, Tokyo 113-0033, Japan
cThe Institute of Chemical and Physical Research (RIKEN), Saitama 351-0198, Japan
dASRC, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
eResearch Center for the Early Universe, University of Tokyo, Tokyo 113-0033, Japan
fDepartment of Mechanical Engineering, Keio University, Yokohama, 223-8522, Japan
(Dated: February 8, 2008)
Calculations of equilibrium propertiesof densematterpredict thatat subnucleardensitiesnuclei
can be rodlike or slablike. To investigate whether transitions between phases with non-spherical
nucleican occur duringthecollapse of a star, we perform quantummolecular dynamicsimulations
ofthecompressionofdensematter. Wehavesucceededinsimulatingthetransitionsbetweenrodlike
5 andslablikenucleiandbetweenslablikenucleiandcylindricalbubbles. Ourresultsstronglysuggest
0 that non-spherical nuclei can be formed in theinner cores of collapsing stars.
0
2 PACSnumbers: 26.50.+x,21.65.+f,02.70.Ns,97.60.Bw

n
a
J Inordinarymatter,atomicnucleiareroughlyspherical attention of researchers(see, e.g., Ref. [9] and references
because, in the liquid droppicture of the nucleus, effects therein). The mechanism of the collapse-driven super-
8
2 ofthenuclearsurfacetensionaregreaterthanthoseofthe novaexplosionhasbeenacentralmysteryinastrophysics
Coulombforces. When the density of matter approaches for almost half a century. Previous studies suggest that

3 thatofatomicnuclei,calculationspredictthat,inequilib- therevivaloftheshockwavebyneutrinoheatingisacru-
v
riumstate,thenucleiwilladoptdifferentshapes,suchas cialprocess. As hasbeenpointedoutinRefs.[7,10]and
1
cylindersandslabs,etc. Thesephaseswithnon-spherical elaborated in Refs. [11, 12], the existence of the pasta
6
0 nuclei are often referred to as “pasta” phases [1, 2]. phases instead of uniform nuclear matter increases the
8 In the initial stage of the supernova explosions, mat- neutrino opacity of matter in the inner core significantly
0 ter in the collapsing iron cores experiences an adiabatic [13] due to the neutrino coherent scattering by nuclei
4
compression,which leads to anincreaseof the density in [14, 15]; this affects the total energy transferred to the
0
the centralregionfrom∼109 g cm−3 to aroundthe nor- shockedmatter. Thusthepastaphasescouldplayanim-
/
h mal nuclear density ρ0 =0.165 fm−3 just before the star portantrole in the future study of supernova explosions.
t
- rebounds; the temperature there reaches the order of 1 In the presentstudy, we use a nuclear force givenby a
l
c MeV. The pasta phases are thus expected to be formed QMD Hamiltonian with medium-EOS parameter set in
u in the inner cores during the collapse of stars. How- Ref. [16]. This Hamiltonian contains the momentum de-
n
ever, such a speculation is based on phase diagrams of pendent “Pauli potential”, which reproduces the effects
:
v the equilibrium state (e.g., Refs. [3, 4] for finite temper- of the Pauli principle phenomenologically. Parameters
i atures) or static and perturbative calculations [5, 6]. It in the other terms of the Hamiltonian are determined to
X
is still unclear whether or not the pasta phases can be reproduce the saturation properties and the properties
r
a formedandthetransitionsbetweenthemcanberealized of finite nuclei in the ground state, especially of heav-
during the collapse, which lasts less than a second. Be- ier ones relevant to the present study [16]. It is also
cause of the drastic changes of nuclear shape that occur confirmed that a QMD Hamiltonian close to the present
under non-equilibrium conditions, this problem is more modelprovidesagooddescriptionofnuclearreactionsin-
difficult thanthe realizationof the pasta phases by cool- cluding the low energy region(severalMeV per nucleon)
ing at constant density, as demonstrated in Ref. [7], and [17], which would be important for the present case.
an ab-initio approach is called for. Using the aboveQMDHamiltonian,we performsimu-
In the present Letter, we solve the problem about lationsof symmetricnuclearmatter with 16384nucleons
the transitions between pasta phases using a dynami- inacubicboxwithperiodicboundarycondition(seeRef.
calframeworkfor nucleonmany-bodysystems calledthe [12]forothercases). Thesystemcontainsequalnumbers
quantummoleculardynamics(QMD)[8]. QMDisasuit- of protons (and neutrons) with spin up and spin down.
able approachto describe thermal fluctuations and is ef- Therelativisticdegenerateelectronswhichensurecharge
ficient enough to treat large systems consisting of sev- neutrality are treated as a uniform background because,
eral nuclei. Furthermore, at the relevant temperatures at subnuclear densities, the effect of the electron screen-
of several MeV, shell effects, which cannot be described ing is small[5,18]. The Coulombenergy,taking account
byQMD,arelessimportantbecausethey washedoutby of the Gaussian charge distribution of the proton wave
thermal fluctuations. packets,iscalculatedbytheEwaldmethod. Thetemper-
The pasta phases have recently begun to attract the ature T is measured by the effective kinetic temperature
2
for momentum-dependent potentials, which is consistent boringnucleiinthesamelatticeplane[seeFigs.1-(5)and
withthetemperatureintheBoltzmannstatistics[7]. The 1-(6)]. Eachfusionprocessinthechainreactionproceeds
QMD equations of motion are integrated by the fourth- on a time scale of order 102 fm/c, which is much shorter
order Gear predictor-corrector method with a multiple than the time scale of the density change [19].
timestepalgorithm. Integrationtimesteps∆tareadap- The transition from the phase with slablike nuclei
tive in the range of ∆t<0.1 – 0.2 fm/c. to the phase with cylindrical holes is shown in Fig. 2
As the initial condition, we use samples of the colum- (u = 0.42 – 0.45 and 0.55 – 0.59 at t = 0 and 27370
nar phase and of the laminar phase of 16384-nucleon fm/c). When the internuclear spacing decreases enough,
system at T ≃ 1 MeV obtained in Ref. [4]. In prepar- neighboringslablikenucleitouchduetothethermalfluc-
ing them, we first combine eight replicated 2048-nucleon tuation as in the above case. Once nuclei begin to touch
samplesinthegroundstate(T ≃0MeV;nucleonnumber (u =0.49 – 0.52 at t = 8460 fm/c), bridges between the
densityρ=0.225ρ0 forthe phasewithrodlikenucleiand slabs are formed at many places on a time scale (of or-
0.4ρ0 for the caseofslablikenuclei)intoa 16384-nucleon der100fm/c)muchshorterthanthatofthecompression
sample, and then put randomnoise on the positions and [cf. Figs. 2-(3) and 2-(4)]. After that the bridges cross
the momenta of nucleons up to 0.1 fm and 1 MeV/c, re- the slabs nearly orthogonally for a while, which makes
spectively. We equilibrate the sample at T = 1 MeV for hollow regions on a square lattice rather than the final
∼ 4000 – 5000 fm/c using the Nos´e-Hoover thermostat triangular one. Nucleons in the slabs continuously flow
for momentum-dependent potentials [4]. We further re- intothebridges,whichbecomewiderandmergetogether
lax the sample for ∼ 5000 fm/c without the thermostat. to form cylindrical holes. Afterwards, the connecting re-
Starting from the above sample, we simulate the adi- gions consisting of the merged bridges move gradually,
abatic compression. In the case starting from the phase and the cylindrical holes relax to form a triangular lat-
with rodlike nuclei [slablike nuclei], the density is in- tice. The final temperature in this case is ≃1.3 MeV.
creased by 2×10−4ρ0 [1×10−4ρ0] every 100 steps by Let us now investigate the detailed time evolution of
changingthe box size (the particle positions are rescaled the nuclear structure. The integral mean curvature and
at the same time). The average rate of change for the the Euler characteristic(see, e.g., Ref. [21]) are powerful
density is ≃1.3×10−5ρ0/(fm/c) [≃7.1×10−6ρ0/(fm/c)]; toolsforthispurpose. SupposethereisasetofregionsR,
this rate ensures the adiabaticity of the simulated com- where the density is higher than a threshold density ρth.
pression process with respect to the change of nuclear TheintegralmeancurvatureandtheEulercharacteristic
structure [19]. Finally, we relax the compressed sample forthesurfaceofthisregion∂Raredefinedassurfacein-
at ρ=0.405ρ0 [0.490 ρ0]. These final densities are those tegralsofthemeancurvatureH andtheGaussiancurva-
of the phase with slablike nuclei [cylindrical bubbles] in ture G, respectively; i.e., R HdA andχ≡ 1 R GdA,
∂R 2π ∂R
the equilibrium phase diagram at T ≃1 MeV [4]. where dA is the area element of the surface of R. The
The resulting time evolution of the nucleon distribu- topological quantity χ = (number of isolated regions)−
tion is shown in Figs. 1 and 2. Starting from the phase (number of tunnels)+(number of cavities) [22]. Wecal-
with rodlike nuclei [Fig. 1-(1); ρ=0.225ρ0 (volume frac- culatethesequantitiesusingthe densityfieldofnucleons
tionofnuclearmatterregionu=0.20–0.22[20])attime on1283 gridpoints (seeRef. [7]fordetailedprocedures).
t =0], the phase with one-dimensional layered lattice of In Figs. 3-(A) and 3-(B), the quantities R HdA and
∂R
slablikenucleiisformed[Fig.1-(9);ρ=0.405ρ0(u=0.41 χ calculated for each nucleon distribution in Fig. 1 are
– 0.45) at t = 17720 fm/c]. During the compression, shown as functions of ρth. In this case, the initial con-
the temperature increases gradually up to ≃ 1.35 MeV dition is the phase with rodlike nuclei, whose structure
in the final state. We note that, in the process of the is well characterized by the plateau of R HdA ≃ 4000
∂R
compression, the phase with rodlike nuclei persists as fm [see the red curve in Fig. 3-(A)]. The slablike nuclei
a metastable state, and moreover, until nuclei begin to in the final state, on the other hand, are characterized
touch and fuse they are not elongatedalong the plane of by R HdA≃0 fm, correspondingto the plateau of the
∂R
the final slabs [see Fig. 1-(2)]. This shows that the tran- gray curve in Fig. 3-(A).
sition from the phase with rodlike nuclei to the slablike Thebehaviorofχclearlyshowsthattherodlikenuclei
nuclei is not triggered by the fission instability. This re- begintotouchbetweent=5290and6050fm/c,whenthe
sult is consistent with a previous study, which shows the plateauvalue ofχ starts to deviate fromzero,character-
stability of the rodlike nuclei against a small quadrupo- izingtherodlikenuclei,tonegativevalues,characterizing
lar deformation of the cross section [6]. When the in- the multiply connected structures such as sponges. We
ternuclear spacing becomes small enough and once some notethatbeforethenucleitouch,thechangeinR HdA
∂R
pair of neighboring rodlike nuclei touch due to thermal is small except for lower values of ρth <∼ 0.1ρ0. This re-
fluctuations, they fuse [see the lower two nuclei in the flectsthefactsthatthephasewithrodlikenucleipersists
middle column in Figs. 1-(3) and 1-(4); u = 0.27 – 0.30 as a metastable state and that the transition is not in-
at t=6050 fm/c]. Like a chain reaction, such connected duced by the fission instability [23]. Also the behavior
pairs ofrodlike nuclei further touchand fuse with neigh- of χ should be noted; as can be seen from Fig. 4-(A), it
3
8000 3000 0 fm/c
(A) (C) 8460 fm/c
6000 2000 99186700 ffmm//cc
10590 fm/c
4000 12000 fm/c
H (fm) 20 000 10 000 112397563117000 fffmmm///ccc
dA -2000 562095000 fffmmm///ccc -1000
∫ -4000 78536100 ffmm//cc -2000
9840 fm/c
-6000 1114308500 ffmm//cc -3000
17720 fm/c
-8000 -4000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
10 10
0
0
-10
-20 -10
-30
χ -20
----76540000 (B)111567891472053830795614852000000000 fffffffffmmmmmmmmm/////////ccccccccc --4300 (D)11112899023974185056366790117000000000 fffffffffmmmmmmmmm/////////ccccccccc
-80 -50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
ρth/ρ0 ρth/ρ0
FIG. 3: (Color) The quantities R HdA and χ as functions
∂R
ofthethresholddensityρth calculatedforthenucleondensity
fieldsof Fig. 1 [(A) and (B)] and of Fig. 2 [(C) and (D)].
C
FIG. 1: (Color) Snapshots of the transition process from the (A) (B)
phase with rodlike nuclei to the phase with slablike nuclei S CH S
(the whole simulation box is shown). The red particles show (S,CH)
(C,S)
protonsandthegreenonesneutrons. Afterneighboringnuclei
touch as shown by the circle in Fig. 1-(3), the “compound
nucleus” elongates along the arrow in Fig. 1-(4). The box
size is rescaled tobe equal in this figure.
FIG. 4: (Color) Time evolution of R HdA and χ during
∂R
the simulations. The data points and the error bars show,
respectively, the mean values and the standard deviations in
therange of ρth =0.3 – 0.5ρ0, which includes thevalues cor-
responding to the nuclear surface assumed in [20]. We set
the averaging range relatively wide because the nuclear sur-
face cannot always be characterized by a single value of ρth
asmentionedin[20]. Thepanel(A)isforthetransitionfrom
cylindrical [C] to slablike nuclei [S]and thepanel (B) for the
transitionfromslablikenucleitocylindricalholes[CH].Tran-
sientstatesareshownas(C,S)and(S,CH)foreachtransition.
becomes negative between the phase with rodlike nuclei
(χ ≃ 0 and R HdA > 0) and the phase with slablike
∂R
nuclei (χ ≃ 0 and R HdA ≃ 0). This implies that the
∂R
transitionproceedsthroughatransientstatewith“spon-
gelike”structure. Thestatewhichgivesthesmallestχat
t≃9840fm/ccorrespondstothemomentwhenallofthe
rodlike nuclei are connected to others by small bridges;
after that, the connected nuclear rods relax into slablike
nuclei, i.e., the bridges in the slablike structures merge
FIG.2: (Color)ThesameasFig.1forthetransitionfromthe
to form the nuclear slabs and those across the slabs dis-
phase with slablike nuclei to the phase with cylindrical holes
appear. Thewholetransitionprocesscanbedividedinto
(theboxsize isnot rescaled in thisfigure). Aftertheslablike
the“connectingstage”andthe“relaxationstage”before
nucleibegin totouch[seethecircle inFig.2-(3)],thebridges
firstcrossesthemalmostorthogonallyasshownbythearrows andafterthis moment;theformerstartswhenthenuclei
in Fig. 2-(5). Then thecylindrical holes are formed and they begin to touch and it takes ≃3000 – 4000 fm/c and the
relax intoatriangularlattice, asshownbythearrows inFig. latter takes more than 8000 fm/c.
2-(9).
4
The same quantities are shown for the transition from [2] M. Hashimoto, H. Seki and M. Yamada, Prog. Theor.
the phasewith slablikenuclei to the phase with cylindri- Phys. 71, 320 (1984).
cal bubbles in Figs. 3-(C), 3-(D), and 4-(B). The initial [3] M. Lassaut et al., Astron. Astrophys.183, L3 (1987).
[4] G. Watanabeet al.,Phys.Rev. C 69, 055805 (2004).
and the final structures are characterizedby the plateau
values of R HdA ≃ 0 and ≃ −2000 fm, respectively. [5] C. J. Pethick and D. G. Ravenhall, Annu. Rev. Nucl.
∂R Part. Sci. 45, 429 (1995).
FromFigs.3-(D),and4-(B),we see that the slablike nu-
[6] K. Iida, G. Watanabe and K. Sato, Prog. Theor. Phys.
clei begin to touch at t <∼ 8460 fm/c and the connection 106, 551 (2001).
of the slablike nuclei by the small bridges are completed [7] G.Watanabeet al.,Phys.Rev.C66, 012801(R) (2002);
at t <∼ 12000 fm/c corresponding to the state with the ibid.,68, 035806 (2003).
lowest χ. In this case, the connecting stage lasts for [8] J. Aichelin, Phys.Rep. 202, 233 (1991).
≃ 3000 – 4000 fm/c and the relaxation stage for more [9] A.Burrows,S.ReddyandT.A.Thompson,Nucl.Phys.
A, in press (astro-ph/0404432).
than15000fm/c. In the latter period, the bridgesmerge
[10] G. Watanabe, K. Iida and K. Sato, Nucl. Phys. A687,
to form cylindrical holes shown by the increase of χ to-
512 (2001); Erratum, Nucl.Phys. A726, 357 (2003).
wardzero,and,simultaneously,theirpositionsrelaxinto [11] C. J. Horowitz, M. A.P´erez-Garc´ia, and J. Piekarewicz,
a triangular lattice as mentioned before. Phys. Rev.C 69, 045804 (2004).
Inconclusion,wehavesucceededinsimulatingthe dy- [12] G. Watanabeet al.,in preparation.
namicalprocessoftwotypesoftransitionsbetweenpasta [13] The cross section for the neutrino coherent scattering is
phases at subnucleardensities. Our calculations support approximated to be proportional to the static structure
the idea that transitions between pasta phases can oc- factor of neutrons, whose peak height for the columnar
andlaminarphases(x=0.3,T =1MeV)isO(102)[12].
cur duringstellar collapse. The particulartransitionswe
[14] D. Z. Freedman, Phys.Rev.D 9, 1389 (1974).
have examined are triggered by the thermal fluctuation, [15] K. Sato, Prog. Theor. Phys. 53, 595 (1975).
not by the fission instability. They consist of the con- [16] T. Maruyama et al., Phys.Rev.C 57, 655 (1998).
necting stage and the relaxation stage. The total time [17] K. Niita, JAERI-conf.96-009, 22 (1996) (in Japanese).
of the connecting stage is 3000 – 4000 fm/c in our sim- [18] G. Watanabe and K. Iida, Phys. Rev. C 68, 045801
ulations, which could be shortened by the artificial com- (2003).
pression. However, we can conclude that the connecting [19] According to the typical value of the density difference
stagewouldbecompleteinatimescaleoforder103fm/c between each pasta phase, ∼0.1ρ0, weincrease theden-
sity to the value corresponding to the next pasta phase
taking account of the facts that each connecting process taking theorder of 104 fm/c, which is much longer than
observedinthisstageproceedsmuchfasterthanthecom- thetypicaltimescaleofthenuclearfission,∼1000fm/c.
pression and that the time scale of ordinary nuclear fis- Therefore, if a dynamical phenomenon of nuclei whose
sionisabout1000fm/c. Therelaxationstagetakesabout time scale is less than the order of 104 fm/c is observed
10000fm/c or more. A remainingchallenge is to investi- during the compression, its dynamics is determined by
gatethetransitionfromthebcclatticeofsphericalnuclei theintrinsicphysicalpropertiesofthesystemnotbythe
density change applied externally, whose time scale is
tothe triangularlatticeofrodlikenuclei[12]. Ifthis pro-
much shorter than that of the actual stellar collapse.
cess is confirmed, the existence of the pasta phases in
[20] We here assume that the nuclear surface corresponds to
supernova cores will be almost established.
the isodensity surface for a threshold density (see later)
G.W.isgratefultoC.J.PethickandK.Iidaforvalu- ofρth =0.45–0.5ρ0.Weshouldmentionthatthenuclear
able discussions and comments. G. W. also appreciates surface cannot be characterized by a single value of ρth
M. Shimizu, L. M. Jensen, P. Urkedal, T. Iitaka and T. exceptforthestatebeforethefusionandafterthesystem
Shimobaba. T. M. thanks S. Chiba for discussions and isfullyrelaxedbecausethedensityinthenuclearmatter
region in thetransient states is quiteinhomogeneous.
kindsuggestions. Thisworkwassupportedinpartbythe
[21] K. Michielsen and H. De Raedt, Phys. Rep. 347, 461
Nishina Memorial Foundation, by the Ministry of Edu-
(2001).
cation,Culture,Sports,Science andTechnologythrough
[22] The value of χ is positive for the phases with spherical
Research Grant No. S14102004, No. 14079202 and No. nuclei and spherical bubbles, and is zero for the other
14-7939, and by RIKEN through Research Grant No. idealpastaphases,i.e.,thephasesofrodlikenuclei,slab-
J130026. like nuclei, and cylindrical bubbles.
[23] R HdA should decrease before χ deviates from zero if
∂R
the transition is triggered by the fission instability, i.e.,
nucleideformlargelybeforetheytouch.Wesee,however,
only a minor change of R HdAbefore χ decreases.
∂R
[1] D. G. Ravenhall, C. J. Pethick and J. R. Wilson, Phys.
Rev.Lett. 50, 2066 (1983).