Neutrino-“pasta” scattering: The opacity of nonuniform neutron-rich matter

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Neutrino-pasta scattering: the opacity of nonuniform neutron-rich matter

C.J. Horowitz∗ and M.A. Perez-Garcıa†

Nuclear Theory Center and Department of Physics, Indiana University, Bloomington, Indiana 47405

J. Piekarewicz‡

Department of Physics, Florida State University, Tallahassee, FL 32306

(Dated: February 5, 2008)

Neutron-rich matter at subnuclear densities may involve complex structures displaying a varietyof shapes, such as spherical, slablike, and/or rodlike shapes. These phases of the nuclear pasta areexpected to exist in the crust of neutron stars and in core-collapse supernovae. The dynamics ofcore-collapse supernovae is very sensitive to the interactions between neutrinos and nucleons/nuclei.Indeed, neutrino excitation of the low-energy modes of the pasta may allow for a significant energytransfer to the nuclear medium, thereby reviving the stalled supernovae shock. The linear responseof the nuclear pasta to neutrinos is modeled via a simple semi-classical simulation. The transportmean-free path for µ and τ neutrinos (and antineutrinos) is expressed in terms of the static structurefactor of the pasta, which is evaluated using Metropolis Monte Carlo simulations.

PACS numbers:

I. INTRODUCTION

Neutron-rich matter may have a complex structure atdensities just below that of normal nuclei. This is becauseall conventional matter is frustrated. While nucleons arecorrelated at short distances by attractive strong interac-tions, they are anti-correlated at large distances becauseof the Coulomb repulsion. Often these short and largedistance scales are well separated, so nucleons bind intonuclei that are segregated in a crystal lattice. However,at densities of the order of 1013−1014 g/cm3 these lengthscales are comparable [1]. Competition among these in-teractions (i.e., frustration) becomes responsible for thedevelopment of complex structures with many possiblenuclear shapes, such as spheres, cylinders, flat plates, aswell as spherical and cylindrical voids [2]. The term pasta

phases has been coined to describe these complex struc-tures [1], and many calculations of their ground-statestructure have already been reported [1, 2, 3, 4, 5, 6].While the study of these pasta phases is interesting in itsown right, it becomes even more so due to its relevanceto the structure of the inner crust of neutron stars andto the dynamics of core-collapse supernovae.

Frustration, a phenomenon characterized by the exis-tence of a very large number of low-energy configurations,emerges from the impossibility to simultaneously mini-mize all “elementary” interactions. Should a proton inthe pasta join a nuclear cluster to benefit from the nuclearattraction or should it remain well separated to minimizethe Coulomb repulsion? Frustration, a term that appearsto have been coined in the late seventies [7, 8], is preva-lent in complex systems ranging from magnetism [9, 10]

∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

to protein folding [11]. In condensed-matter systems,frustration dates back to the 1950 study of Ising an-tiferromagnets on triangular lattices by Wannier [12].Three antiferromagnetically coupled spins fixed to thesites of an equilateral triangle cannot minimize all inter-actions simultaneously: once two spins are anti-aligned,the third one cannot be antiparallel to both of them. Fur-ther, in Ref. [13] it has been shown that finding the trueground state—among the many metastable states—of aspin glass shares features in common with NP-complete

problems, such as the traveling salesman problem of famein the theory of combinatorial optimization [14]. Finally,because of the preponderance of low-energy states, frus-trated systems display unusual low-energy dynamics.

Because of the complexity of the system, almost nowork has been done on the low-energy dynamics of thepasta or on its response to weakly-interacting probes. Inthis paper we study the excitations of the pasta via asimple semi-classical simulation similar to those used todescribe heavy-ion collisions. Heavy-ion collisions canproduce hot, dense matter. However, by carefully heat-ing the system and allowing it to expand, heavy-ion col-lisions can also study matter at low densities. Multi-fragmentation, the breakup of a heavy ion into severallarge fragments, shares many features with pasta for-mation, as they are both driven by the same volume,surface, and Coulomb energies. There have been sev-eral classical [15, 16] and quantum-molecular-dynamics(QMD) [17] simulations of heavy-ion collisions. Thesesame approaches may be applied to the nuclear pasta byemploying a simulation volume and periodic boundaryconditions. One great advantage of such simulations isthat one can study pasta formation in an unbiased waywithout having to assume particular shapes or configura-tions from the outset. While QMD has been used beforeto study the structure of the pasta [18, 19, 20], no calcu-lations of its linear response to weakly-interacting probes(e.g., neutrinos) have been reported.

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Neutrino interactions are crucial in the dynamics ofcore-collapse supernovae because neutrinos carry 99% ofthe energy. Neutrinos are initially trapped due to thelarge coherent neutrino-nucleus elastic scattering crosssection. This trapping is important for the electron-per-baryon fraction Ye of the supernova core, as it hindersany further conversion of electrons into neutrinos. How-ever, with increasing density Coulomb interactions be-tween ions lead to significant ion screening of neutrino-nucleus elastic scattering [21]. As the density is increasedstill further, the ions react to form nuclear pasta and oneneeds to calculate neutrino-pasta interactions.

In the pasta phase one can have coherent neutrino scat-tering from density contrasts, such as Swiss cheese likevoids. Critical fluctuations could significantly increasethe cross section, thereby greatly reducing the neutrinomean-free path. As the existence of many low-energyconfigurations is the benchmark of frustrated systems,one expects large configuration mixing among the vari-ous different pasta shapes. This mixing often leads tointeresting low-energy collective excitations. As it willbe shown later (see Fig. 2) at subnuclear densities ofthe order of 1013 g/cm3, the pasta resembles a collec-tion of spherical neutron-rich nuclei embedded in a di-lute neutron gas. Neutron-rich nuclei with large neu-tron skins have Pygmy giant resonances, involving col-lective oscillations of the neutron skin against the sym-metric core [22, 23]. We expect that the soft neutron-rich pasta will have many low-energy collective oscil-lations. This could provide important physics that ispresently missing from core-collapse supernovae simula-tions. Neutrino excitation of the low-energy pasta modesmay allow for a significant energy transfer to the nu-clear medium, potentially reviving the stalled supernovaeshock. To our knowledge, there have been no calculationsof these effects. Note, however, that Reddy, Bertsch, andPrakash [24] have found that coherent neutrino scatter-ing from a nonuniform kaon condensed phase greatly de-creases the neutrino mean free path.

Present models of the equation of state for supernovaesimulations, such as that of Lattimer and Swesty [25],describe the system as a liquid drop for a single repre-sentative heavy nucleus surrounded by free alpha parti-cles, protons, and neutrons. One then calculates neu-trino scattering from these constituents—by arbitrarilymatching to a high-density uniform phase [26]. Unfortu-nately, this approximation is uncontrolled as it neglectsmany important interactions between nuclei. By simulat-ing the pasta phase directly in the nucleon coordinates,one hopes to improve on this matching and to understandits limitations.

There is a duality between microscopic descriptions ofthe system in terms of nucleon coordinates and “macro-scopic” descriptions in terms of effective nuclear degreesof freedom. Thus, a relevant question to pose is as fol-lows: when does a neutrino scatter from a nucleus andwhen does it scatter from an individual nucleon? At theJefferson Laboratory a similar question is studied; when,

i.e., at what momentum transfer, does a photon coupleto a full hadron and when to an individual quark? Mod-els of the quark/hadron duality have provided insight onhow descriptions in terms of hadron degrees of freedomcan be equivalent to descriptions in terms of quark coor-dinates [27]. Here we are interested in nucleon/nuclearduality, that is, how can nuclear models incorporate themain features of microscopic nucleon descriptions?

The manuscript has been organized as follows. InSec. II the semi-classical formalism is introduced. A verysimple (perhaps minimal) model is employed that con-tains the essential physics of frustration. The linear re-sponse of the pasta to neutrino scattering, in the form ofa static structure factor, is discussed in Sec. III. Resultsare presented in Sec. IV, while conclusions and futuredirections are reserved to Sec. V.

II. FORMALISM

In this section we introduce a classical model that whilesimple, contains the essential physics of frustration. Thatis, competing interactions consisting of a short-range nu-clear attraction and a long-range Coulomb repulsion. Wemodel a charge-neutral system of electrons, protons, andneutrons. The electrons are assumed to be a degeneratefree Fermi gas of density ρe =ρp and the nucleons inter-act via a classical potential. The only quantum aspects ofthe calculation are the use of an effective temperature andeffective interactions to simulate effects associated withquantum zero-point motion. Of course more elaboratemodels are possible and these will be presented in futurecontributions. For these first simulations we adopted avery simple version that displays the essential physicsof nucleons clustering into pasta in a transparent form.Moreover, this simple model facilitates simulations witha relatively large numbers of particles, a feature that isessential to estimate and control finite-size effects.

The total potential Vtot energy is assumed to be a sumover two-body interactions Vij of the following form:

Vtot =∑

i<j

V (i, j) , (1)

where the “elementary” two-body interaction is given by

V (i, j) = ae−r2

ij/Λ +[

b + cτz(i)τz(j)]

e−r2

ij/2Λ + Vc(i, j) .

(2)Here the distance between the particles is denoted byrij = |ri−rj | and the isospin of the jth particle is τz(j)=1for a proton and τz(j) = −1 for a neutron. The modelparameters a, b, c, and Λ will be discussed below. Suf-fices to say that the above interaction includes the char-acteristic intermediate-range attraction and short-rangerepulsion of the nucleon-nucleon (NN) force. Further,the isospin dependence of the potential insures that whilepure neutron matter is unbound, symmetric nuclear mat-ter is bound appropriately. Finally, a screened Coulomb

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interaction of the following form is included:

Vc(i, j) =e2

rije−rij/λτp(i)τp(j) , (3)

where τp(j)= (1+τz(j))/2 and λ is the screening lengththat results from the slight polarization of the electrongas. That is, the relativistic Thomas-Fermi screeninglength is given by

λ =π

e

(

kF

√

k2F

+ m2e

)−1/2

, (4)

Note that the electron Fermi momentum has been definedby kF =(3π2ρe)

1/3 and me is the electron mass [28, 29].Unfortunately, while the screening length λ defined aboveis smaller than the length L of our simulation box, itis not significantly smaller (unless a prohibitively largenumber of particles is used). Therefore, to control finite-size effects we were forced to arbitrarily decrease thevalue of λ (see Sec. IV).

The simulations are carried out in a canonical ensem-ble with a fixed number of particles A at a temperatureT . The volume V at a fixed baryon density ρ is simplyV =A/ρ. To minimize finite-size effects we use periodicboundary conditions, so that the distance rij is calcu-lated from the x, y, and z coordinates of the ith and jthparticles as follows:

rij =√

[xi − xj ]2 + [yi − yj ]2 + [zi − zj ]2 , (5)

where the periodic distance, for a cubic box of side L =V 1/3, is given by

[l] = Min(|l|, L − |l|) . (6)

The potential energy defined in Eq. (1) is indepen-dent of momentum. Therefore, the partition function forthe system factors into a product of a partition functionin momentum space—that plays no role in the compu-tation of momentum-independent observables—times acoordinate-space partition function of the form

Z(A, T, V ) =

∫

d3r1 · · · d3rA exp (−Vtot/T ) . (7)

Note that the 3-dimensional integrals are performed overthe simulation volume V .

The average energy of the system 〈E〉=〈K〉+〈Vtot〉 ismade of kinetic (K) and potential (Vtot) energy contri-butions. As the (momentum-independent) interactionshave no impact on momentum-dependent quantities, theexpectation value of the kinetic energy reduces to its clas-sical value, that is,

〈K〉 =3

2AT . (8)

In turn, the expectation value of the potential energymay be computed from the coordinate-space partition

function as follows:

〈Vtot〉 =1

Z(A, T, V )

∫

d3r1 · · · d3rAVtot exp (−Vtot/T ) .

(9)In summary, a classical system has been constructedwith a total potential energy given as a sum of two-body momentum-independent interactions [see Eq. (2)].Any momentum-independent observable of interest canbe calculated from the partition function [Eq. (7)], whichwe evaluate via Metropolis Monte-Carlo integration [30].

FIG. 1: Energy per particle for symmetric (dashed line) andfor pure-neutron matter (solid) versus baryon density nb at atemperature of T =1 MeV.

We now return to discuss the choice of model param-eters. The constants a, b, c, and Λ in the two-body in-teraction Eq. (2) were adjusted—approximately—to re-produce the following bulk properties: a) the saturationdensity and binding energy per nucleon of symmetric nu-clear matter, b) (a reasonable value for) the binding en-ergy per nucleon of neutron matter at saturation density,and c) (approximate values for the) binding energy of afew selected finite nuclei. The temperature was arbitrar-ily fixed at 1 MeV for all the calculations. Note thatthe parameter set employed in these calculations (anddisplayed in Table I) has yet to be carefully optimized.We reiterate that for these first set of simulations, theinteraction is sufficiently accurate to describe the essen-tial physics of the pasta. Indeed, this is illustrated inFig. 1 and Table II. In Fig. 1 the average potential en-ergy versus density at a temperature of T = 1 MeV isdisplayed for a simulation of symmetric nuclear mattercontaining A=400 particles and, as is customary, assum-ing no Coulomb interactions. Also shown in the figure isthe potential energy for pure neutron matter calculatedwith N = 200 particles. In the case of finite nuclei (alsocalculated at T =1 MeV) the full Coulomb interaction isincluded using a screening length λ much larger than theresulting root-mean-square radius of the nucleus. Simula-tions based on a Metropolis Monte-Carlo algorithm were

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TABLE I: Model parameters used in the calculations.

a b c Λ

110 MeV -26 MeV 24 MeV 1.25 fm2

TABLE II: Binding energy per nucleon for various closed shellnuclei. Note that all energies are in MeV and that the Monte-Carlo results include only 〈Vtot〉.

Nucleus Monte Carlo 〈Vtot〉 Experiment16O −7.56 ± 0.01 −7.9840Ca −8.75 ± 0.03 −8.4590Zr −9.13 ± 0.03 −8.66

208Pb −8.2 ± 0.1 −8.45

used to compute the average potential energy, startingwith nucleons distributed uniformly in a sphere with aradius comparable to the expected size of the nucleus;this sphere was placed in the center of a very large box.Results of the simulations and comparison with experi-mental values have been collected in Table II. Note thatthe simulation results are for the potential energy only. Ifthe kinetic energy per nucleon (3T/2) is added to thesevalues, the nuclei in Table II would be slightly under-bound. Furthermore, finite nuclei are only metastablein this semiclassical approximation. Nucleons can evap-orate over a very long time scale. However, this is notexpected to significantly impact the pasta phases sincethese already have free nucleons.

III. NEUTRINO SCATTERING

The model is used to describe neutrino scattering fromnonuniform neutron-rich matter. As neutrino interac-tions play a predominant role in core-collapse supernovae,one is interested in understanding how the neutrinos dif-fuse and how do they exchange energy. In this first paperwe focus on the transport mean free path for νµ and ντ ,which lack charged-current interactions at low energies.Their mean-free path is dominated by neutral currentneutrino-nucleon scattering.

The free-space cross section for neutrino-nucleon elas-tic scattering is given by

dσ

dΩ=

G2F E2

ν

4π2

[

c2

a(3 − cos θ) + c2

v(1 + cos θ)]

, (10)

where GF is the Fermi coupling constant, Eν the neu-trino energy and θ the scattering angle. Note that thisequation neglects weak magnetism and other correctionsof order Eν/M , with M the nucleon mass [31].

In the absence of weak magnetism, the weak neutralcurrent Jµ of a nucleon has axial-vector (γ5γµ) and vectorγµ contributions. That is,

Jµ = caγ5γµ + cvγµ . (11)

The axial coupling constant is,

ca = ±ga

2(ga = 1.26) , (12)

with the + sign for neutrino-proton and the − sign forneutrino-neutron scattering. The weak charge of a protoncv is suppressed by the weak-mixing (or Weinberg) anglesin2 θW =0.231,

cv =1

2− 2 sin2 θW = 0.038 ≈ 0 . (13)

In contrast, the weak charge of a neutron is both largeand insensitive to the weak-mixing angle: cv =−1/2. Thetransport mean-free path λt is inversely proportional tothe transport cross section σt and is given by the follow-ing expression:

σt =

∫

dΩdσ

dΩ(1 − cos θ) =

2G2F E2

ν

3π

(

5c2

a + c2

v

)

. (14)

The weighting factor (1−cosθ) included in the definitionof the transport cross section favors large-angle scatter-ing, as momentum is transferred more efficiently into themedium. As a result, the axial-vector contribution c2

a

dominates the cross section. Assuming that the scat-tering in the medium is the same as in free space, thetransport mean-free path becomes

λt = (ρpσpt + ρnσn

t )−1

. (15)

Here ρp(ρn) is the proton(neutron) density and σpt (σn

t )is the transport cross section for scattering from a pro-ton(neutron).

If nucleons cluster tightly into nuclei or into pasta,then the scattering from different nucleons could be co-herent. This will significantly enhance the cross sectionas it would be proportional to the square of the number ofnucleons [32]. The contribution from the vector currentis expected to be coherent. Instead, the strong spin andisospin dependence of the axial current should reduce itscoherence. This is because in nuclei—and presumably inthe pasta—most nucleons pair off into spin singlet states(note that in the nonrelativistic limit the nucleon axial-vector current becomes γ5γτz → −στz). Therefore, inthis paper we focus exclusively on coherence effects forthe vector current.

Coherence is important in x-ray scattering from crys-tals. Because the x-ray wavelength is comparable to theinter-particle spacing, one needs to calculate the rela-tive phase for scattering from different atomic planes andthen sum over all planes. Neutrino-pasta scattering in-volves a similar sum because the neutrino wavelength iscomparable to the inter-particle spacing and even to theinter-cluster spacing. Therefore, one must calculate therelative phase for neutrino scattering from different nu-cleons and then add their contribution coherently. Thisprocedure is embodied in the static structure factor S(q).

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The dynamic response of the system to a probe of mo-mentum transfer q and energy transfer ω>0 that couplesto the weak charge density ρ(q) is given by [28]

S(q, ω) =∑

n6=0

∣

∣

∣〈Ψn|ρ(q)|Ψ0〉∣

∣

∣

2

δ(ω − ωn) , (16)

where ωn is the energy difference between the excitedstate |Ψn〉 and the ground state |Ψ0〉. In linear responsetheory, namely, assuming that the process can be treatedin lowest order (an excellent approximation for neutrinoscattering) the cross section can be directly related tothe dynamic response of the system. In the case thatthe individual excited states may not be resolved, thenone integrates over the energy transfer ω to obtain thestatic structure factor. Here we define the static structurefactor per neutron as follows:

S(q) =1

N

∫ ∞

0

S(q, ω)dω =1

N

∑

n6=0

∣

∣

∣〈Ψn|ρ(q)|Ψ0〉∣

∣

∣

2

,

(17)with the weak vector-charge density given by

ρ(q) =

N∑

i=1

exp(iq · ri) , (18)

where the sum in Eq. (18) is only over neutrons.The cross section per neutron for neutrino scattering

from the whole system is now given by

1

N

dσ

dΩ= S(q)

G2F E2

ν

4π2

1

4(1 + cos θ) . (19)

Note that the weak charge of the nucleon (cv ≈0 for pro-tons and cv =−1/2 for neutrons) has been incorporatedinto the above cross section, so that the normalizationof the weak vector-charge density is ρ(q = 0) = N . Fur-ther, Eq. (19) is the cross section per neutron obtainedfrom Eq. (10) (with ca = 0) multiplied by S(q). Thisindicates that S(q) embodies the effects from coherence.Finally, note that the momentum transfer is related tothe scattering angle through the following equation:

q2 = 2E2

ν(1 − cos θ) . (20)

Two assumptions have been made in the derivation ofEq. (19). First, no contribution from the axial currentto the cross section has been included, because nucleonspair into spin-zero states. Second, the excitation energytransferred to the nucleons is small and we have summedover all possible excitation energies.

The static structure factor has important limits. Aparticularly useful form in which to discuss them invokescompleteness on Eq. (17). That is,

S(q) =1

N

(

〈Ψ0|ρ†(q)ρ(q)|Ψ0〉 −∣

∣

∣〈Ψ0|ρ(q)|Ψ0〉∣

∣

∣

2)

.

(21)

The last term in the above expression represents the elas-tic form factor of the system, which only contributes atq = 0. In the limit of q → 0, the weak charge den-sity [Eq. (18)] becomes the number operator for neutrons

ρ(q = 0) = N , so that the static structure factor reducesto,

S(q = 0) =1

N

(

〈N2〉 − 〈N〉2)

. (22)

Thus, the q→0 limit of the static structure factor is re-lated to the fluctuations in the number of particles, orequivalently, to the density fluctuations. These fluctua-tions are, themselves, related to the compressibility anddiverge at the critical point [33]. To discuss the large qlimit, Eq. (18) is substituted into Eq. (21) to yield

S(q)=1

N

N∑

i,j

〈Ψ0| exp(iq · rij)|Ψ0〉−∣

∣

∣〈Ψ0|ρ(q)|Ψ0〉∣

∣

∣

2

,

(23)In the q → ∞ limit, all the terms in the sum with i 6=j, aswell as the second term in the above expression, oscillateto zero. This only leaves the i=j terms, which there areN of them so that

S(q → ∞) = 1 . (24)

This result indicates that if the neutrino wavelength ismuch shorter than the inter-particle separation, the neu-trino only resolves one nucleon at a time. This corre-sponds to quasielastic scattering where the cross sectionper nucleon in the medium is the same as in free space.

One can calculate the static structure factor from theneutron-neutron correlation function which is defined asfollows:

g(r) =1

Nρn

N∑

i6=j

〈Ψ0|δ(r − rij)|Ψ0〉 . (25)

The two-neutron correlation function “asks” (and “an-swers”) the following question: if one “sits” on a neutron,what is the probability of finding another one a distance|r| away. The correlation function is normalized to oneat large distances g(r → ∞) = 1; this corresponds tothe average density of the medium. The static structurefactor is obtained from the Fourier transform of the two-neutron correlation function. Comparing with Eq. (23)this yields,

S(q) = 1 + ρn

∫

d3r(

g(r) − 1)

exp(iq · r) . (26)

The i=j terms in Eq. (23) gives the leading 1 in the aboveexpression, while the elastic form factor |〈Ψ0|ρ(q)|Ψ0〉|2yields the −1 in the integrand of Eq. (26).

To obtain the transport cross section we proceed, as inEq. (10), to integrate the angular-weighted cross sectiondσ/dΩ(1 − cos θ) over all angles. That is,

σt =1

N

∫

dΩdσ

dΩ(1 − cos θ) = 〈S(Eν)〉σ0

t . (27)

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Note that the free neutron cross σ0t follows directly from

Eq. (14) in the limit of ca≡0,

σ0

t =G2

F E2ν

6π. (28)

Further, the angle averaged static structure factor hasbeen defined as follows:

〈S(Eν)〉 ≡ 3

4

∫ 1

−1

dx(1 − x2)S(

q(x, Eν))

, (29)

where the static structure factor S(q) depends on neu-trino energy and angle through Eq. (20), that is, q2 =2E2

ν(1 − x) with x=cos θ. Using Eq. (26) and switchingthe orders of integration, this can be written as

〈S(Eν)〉 = 1 +4πρn

E2ν

∫ ∞

0

drf(2Eνr)(

g(r) − 1)

. (30)

Note that the following function has been introduced

f(t) = 72(cos t+ t sin t−1)/t4−6(5 cos t+ t sin t+1)/t2 .(31)

Finally, the transport mean free path (λt = 1/σtρn) isgiven by

λ−1

t = σ0

t ρn〈S(Eν)〉 . (32)

In this way 〈S(Eν)〉 describes how coherence modifies themean-free path. In the next section, simulation results forthe two-neutron correlation function, the static structurefactor, and the angle averaged static structure factor willbe presented.

IV. RESULTS

In this section we present our simulation results. As anexample of a typical low-density condition we consider asubnuclear density of ρ = 0.01 fm−3 (about 1/16 of nor-mal nuclear density), a temperature T =1 MeV, and anelectron fraction of Ye = 0.2. Proto-neutron stars haveelectron fractions that start out near 1/2 and drop withtime, so Ye =0.2 represents a typical neutron-rich condi-tion. Monte-Carlo simulations for a total of A=4000 par-ticles (N =3200, Z =800) have been performed. Becauseof the many competing minima, a significantly larger sys-tem would take an unreasonably long time to thermalizeon a modest work station (see details below).

The simulation volume for the above conditions con-sists of a cube of length L = 73.7 fm. While this value islarger than the electron screening length λ = 26.6 fm [seeEq. (4)], it is not sufficiently larger. Indeed, to minimizefinite-size effects in a simulation with periodic boundaryconditions one would like exp

(

−L/(2λ))

≪ 1. Clearly,this condition is not adequately satisfied. Therefore,in an effort to minimize the contamination from finite-size effects, we reduce the electron screening length—arbitrarily—to the following value:

λ ≡ 10 fm . (33)

This value for λ is adopted hereafter for all of our sim-ulations (see also [18]). This smaller screening lengthdecreases slightly the Coulomb interaction at large dis-tances, which could promote the growth of slightly largerclusters. However, we do not expect this decrease in λ toqualitatively change our results. We note that λ is stilllarger than the typical size of a heavy nucleus.

FIG. 2: (Color online) Monte-Carlo snapshot of a configura-tion of N =3200 neutrons (light gray circles) and Z =800 pro-tons (dark red circles) at a baryon density of ρ = 0.01 fm−3,a temperature of T = 1 MeV, and an electron fraction ofYe =0.2. 3D imaging courtesy of the FSU Visualization Lab-oratory.

By far, the most time consuming part of the simula-tion is producing suitable initial conditions. The simu-lations are started with the A=4000 nucleons randomlydistributed throughout the simulation volume. Next, weperform a total of about 325,000 Metropolis sweeps start-ing at the higher temperature of T = 2 MeV and reduc-ing the temperature until eventually reaching the targettemperature of T = 1 MeV, in a “poor’s-man” attemptat simulated annealing. Note that a Metropolis sweepconsists of a single trial move for each of the A = 4000particles in this system. We call this procedure cookingthe pasta.

Results in this section are based on a statistical av-erage of the final 50,000 sweeps. This yields a potentialenergy of −5.385±0.003 MeV/A. A sample configurationof the 4000 particles is shown in Fig. 2. The protons arestrongly correlated into clusters (“nuclei”) as are a largenumber of neutrons. In addition, there is a low densityneutron gas between the clusters. At this density it maybe reasonable to think of the system as a high-densityliquid of “conventional” nuclei immersed in a dilute neu-tron gas. A great virtue of the simulation is that one doesnot have to arbitrarily decide which nucleons cluster innuclei and which ones remain in the gas. These “deci-sions” are being answered dynamically. Further, one can

7

calculate modifications to nuclear properties due to theinteractions. In a future work we plan to compare oursimulation results to some conventional nuclear models.

Protons moving between clusters face a Coulomb bar-rier. This may inhibit the thermalization process andwith it the formation of larger clusters. This could in-crease our results for S(q). To test the thermalization ofthe pasta, our Metropolis Monte Carlo configuration wasevolved further via molecular dynamics for a total timeof 46,500 fm/c. The molecular dynamics calculations willbe described in future work. This led to an increase in thepeak of S(q) by only about 10 percent. Although thesemolecular dynamics results did not reveal a large secu-lar change in the system with time, we caution that ourcooking procedure may not have converged to the truethermal-equilibrium state. The static structure factorS(q) may still change with additional Metropolis MonteCarlo or molecular dynamics evolution.

0 10 20 30 40 50r (fm)

0

1

2

3

4

5

6

g(r)

FIG. 3: Neutron-neutron correlation function at a tempera-ture of T = 1 MeV, an electron fraction of Ye = 0.2, and abaryon density of ρ = 0.01 fm−3. These results show largefinite-size effects beyond r=L/2 = 36.9 fm.

The neutron-neutron correlation function g(r) is dis-played in Fig. 3. The two-neutron correlation function iscalculated by histograming the relative distances betweenneutrons. The correlation function is very small at shortdistances because of the hard core in our NN interac-tion. At intermediate distances g(r) shows a large broadpeak between r =2 fm and r≃ 10 fm. This correspondsto the other neutrons bound into a cluster. Superim-posed on this broad peak one observes three (or four)sharper peaks corresponding to nearest, next-to-nearest,and next-to-next-to-nearest neighbors. These structuresdescribe two-neutron correlations within the same clus-ter. At larger distances, between 10 and 20 fm, the cor-relation function shows a modest dip below one, suggest-ing that the attractive NN interaction has shifted someneutrons from larger to smaller distances in order to form

the clusters. Alternatively, Coulomb repulsion makes itless likely to find two clusters separated by these radii.Finally, there is an abrupt drop in g(r) at a distancecorresponding to half the value of the simulation length(r=L/2=36.9 fm) caused by finite-size effects. We notethat under our assumptions of periodic boundary condi-tions, g(r) is identically zero for r>

√3L/2.

FIG. 4: (Color online) Monte-Carlo snapshot of a configura-tion of N =3200 neutrons (light gray circles) and Z =800 pro-tons (dark red circles) at a baryon density of ρ=0.025 fm−3,a temperature of T = 1 MeV, and an electron fraction ofYe =0.2. 3D imaging courtesy of the FSU Visualization Lab-oratory.

Increasing the density can change the nature of thepasta. Figure 4, shows a sample configuration of 4000particles at a density of 0.025 fm−3. Note that althoughthe density has increased, both the temperature and theelectron fraction have remained fixed at T =1 MeV andYe = 0.2, respectively. At a density of ρ = 0.025 fm−3

(about 1/6 of normal nuclear-matter saturation density)the spherical clusters of Fig. 2 start to coalesce intocylindrical-like structures. The system appears to besegregated into high-density regions of cylindrical nucleiimmersed in a dilute neutron gas. As we continue toperform additional simulations, high-quality renderingsof nucleon configurations for a variety of densities, tem-peratures, and electron fractions will be developed.

To compute the static structure factor S(q), one nu-merically transforms the two-neutron correlation func-tion, as indicated in Eq. (26). However, because of finite-size effects we truncate the integral at r=L/2 and assumeg(r)≡1 for r>L/2. This procedure yields the results dis-played in Fig. 5. There is a modest peak in S(q=0) dueto density fluctuations. Of course, the number of neu-trons in our simulation remains fixed, yet fluctuationscan take neutrons across the r = L/2 cutoff and thesefluctuations will contribute to the value of S(q) at q=0.

The error bars in Fig. 5 are statistical only, based onthe last 50,000 sweeps. We caution that there may befinite-size effects at small momentum transfers. The box

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7q (fm

-1)

0

5

10

15

20

25

S(q)

FIG. 5: Static structure factor S(q) versus momentum trans-fer q at a temperature of T =1 MeV, an electron fraction ofYe = 0.2, and a baryon density of ρ = 0.01 fm−3. The errorbars are statistical only. Finite size effects may be importantat small q as indicated by the dotted error bars.

size for our simulation at ρ = 0.01 fm−3 is L= 73.7 fm.This corresponds to a minimum momentum transfer of

qmin ≈ 2π

L= 0.085 fm−1 . (34)

Momentum transfers smaller than qmin correspond towavelengths larger than the simulation volume, so ourresults for q. 2qmin may be sensitive to finite-size effects.Indeed, for q . 2qmin the static structure factor was ob-served to change significantly from one Metropolis runto the next. To indicate the sensitivity of our results tofinite-size effects, Fig. 5 displays the static structure fac-tor in the q≤ 2qmin region with dotted error bars. Notethat the point q = qmin has been signaled out in Fig. 5to indicate that it is more stable from one Metropolisrun to the next, because the weak vector-charge density[Eq. (18)] evaluated at q = qminq is invariant under atranslation of the system by a distance L along q.

The static structure factor displays a large peak atq≈ 0.3 fm−1, corresponding to coherent scattering frommany neutrons bound into a single cluster. At smallermomentum transfers, q ≈ 0.2 fm−1, S(q) decreases be-cause of ion screening. Here the neutrino wavelength isso long that it probes multiple clusters. These other clus-ters screen the weak charge and reduce the response. Atmomentum transfers larger than q≈0.3 fm−1, the staticstructure factor decreases with increasing q. This is theeffect of the cluster form factor. As the momentum trans-fer increases the neutrino can no longer scatter coherentlyfrom all the neutrons in a cluster because of the clustersextended size. Thus, the observed peak in S(q) developsas a trade off between ion screening, which favors largeq, and the cluster form factor, which favors small q.

In summary, one can divide the response of the pastainto the following regions. At low-momentum transfers(q . 0.2 fm−1) the response is dominated by ion screen-ing and density fluctuations. For momentum transfersin the region q = 0.2−0.4 fm−1 one observes coherentscattering from the pasta. At the larger momenta ofq = 0.4−1 fm−1, the falling response reflects the pastaform factor. Finally, the large momentum transfer regionabove q = 1 fm−1 corresponds to quasielastic scatteringfrom nearly free neutrons, as S(q → ∞)=1 [see Eq. (24)].

0 20 40 60 80 100Eν (MeV)

0

5

10

15

20

<S(

Eν)>

FIG. 6: Angle averaged static structure factor 〈S(Eν)〉 ver-sus neutrino energy Eν at a temperature of T = 1 MeV,an electron fraction of Ye = 0.2, and a baryon density ofρ=0.01 fm−3. The error bars are statistical only, see text.

The angle averaged structure factor 〈S(Eν)〉 [definedin Eq. (29)] is shown in Fig. 6. Note that the integralin Eq. (30) was also truncated at r = L/2 because offinite-size effects. The averaged structure factor showsa broad peak for neutrino energies near 40 MeV. In-deed, the transport cross section is significantly enhancedby coherence effects for neutrino energies from about 20to 80 MeV. The impact of this coherence on the neu-trino mean-free path [Eq. (32)] is displayed in Fig. 7.Also shown in the figure is the mean-free path obtainedby ignoring coherence effects by setting 〈S(Eν)〉 = 1 inEq. (32). Coherence significantly reduces the mean-freepath for neutrino energies in the range Eν =15−120 MeV.Again, finite size effects may be important for low neu-trino energies.

V. CONCLUSIONS

Neutron-rich matter is expected to have a complexstructure at subnuclear densities. Complex pasta phasesmay result from frustration through the competition be-tween an attractive nuclear interaction and the Coulombrepulsion. Neutrino interactions with the pasta may

9

0 20 40 60 80 100Eν (MeV)

10

100

1000

10000

λ t (m

)

FIG. 7: Transport mean-free path λt for νµ or ντ versus neu-trino energy Eν at a baryon density of ρ=0.01 fm−3, a tem-perature of T = 1 MeV, and an electron fraction of Ye = 0.2.The solid line and error bars include full coherence effectswhile the dashed line is obtained by using 〈S(E)〉 = 1 inEq. (32). Error bars are statistical only. Finite size effectsmay be important for low Eν as indicated by the dotted errorbars.

be important for properties of core-collapse supernovae,such as its electron fraction.

In this work we have employed a semi-classical modelto simulate the dynamics of the pasta phase of neutron-rich matter. Although our model is very simple, itnonetheless retains the crucial physics of frustration. Us-ing a Metropolis Monte-Carlo algorithm, the partitionfunction was computed for a system of 4000 nucleons ata given temperature and density. We find that almost allprotons and most of the neutrons cluster into “nuclei”that are surrounded by a dilute neutron gas.

Observables computed in our simulations included theneutron-neutron correlation function. This calculationwas implemented by constructing a histogram of all rel-ative neutron distances. The two-neutron correlationfunction g(r) gives the probability of finding a neutronat a distance r away from a reference neutron. A largepeak in g(r) at intermediate distances (r = 2−10 fm) isfound. This reflects the presence of the other neutronsin the cluster.

The static structure factor S(q), a fundamental ob-servable obtained from the Fourier transform of the two-neutron correlation function, describes the degree of co-herence for neutrino-nucleon elastic scattering. For smallmomentum transfers, S(q) describes density fluctuationsand ion screening. In this region the neutrino wave-length is longer than the average inter-cluster separation,thereby allowing other clusters to screen the weak chargeof a given cluster. At momentum transfers of approx-imately q = 0.2− 0.4 fm−1, the static structure factordevelops a large peak, associated to the coherent scatter-

ing from all the neutrons in the cluster. This coherenceis responsible for a significant reduction in the neutrinomean-free path. To our knowledge, these represent thefirst consistent calculation of the neutrino mean-free pathin nonuniform neutron-rich matter.

However, much remains to be done. First, one needs tofocus on the thermalization of our simulations. It is diffi-cult to insure that the system has reached thermal equi-librium because the Coulomb barrier hinders the motionof individual protons. Second, one must further investi-gate the impact of finite-size effects and the simple treat-ment of long-range Coulomb interactions on our simula-tions. This may require simulations with larger numbersof particles, as it is difficult to fit a long-wavelength neu-trino into the present simulation volume. Third, whilewe have focused here on the vector part of the weak-neutral-current response, because it can be greatly en-hanced by coherence, one should extend the study tothe axial-vector (or spin) response, as it dominates thescattering when it is coming from uncorrelated nucleons.Further, one should also calculate charged-current inter-actions in nonuniform matter. Finally, in the presentcontribution no effort was made to optimize the NN in-teraction. While it may be advantageous to do so, anyaccurately calibrated interaction must retain the essen-tial features of frustration. Moreover, more sophisticatedinteractions that include momentum and/or density de-pendence will significantly increase the computational de-mands. At present, we are checking our results againstmore sophisticated simulations using molecular dynam-ics, studying finite size effects in larger simulations, ex-ploring the temperature and density dependence of ourresults, and calculating the dynamical response. Theseresults will be presented in a future contribution [34].

Acknowledgments

We acknowledge useful discussions with Wolf-gang Bauer, Adam Burrows, Vladimir Dobrosavljevic,Thomas Janka, Sanjay Reddy, Pedro Schlottmann, Ro-mualdo de Souza, and Victor Viola. C.J.H. thanks theAspen Center for Physics, the Max Plank Institute forAstrophysics, and the Institute for Nuclear Theory at theUniversity of Washington for their hospitality. M.A.P.G.acknowledges partial support from Indiana Universityand FICYT. J.P. thanks David Banks and the staff atthe FSU Visualization Laboratory for their help. Thiswork was supported in part by DOE grants DE-FG02-87ER40365 and DE-FG05-92ER40750.

10

[1] D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Phys.Rev. Lett. 50, 2066 (1983).

[2] M. Hashimoto, H. Seki, and M. Yamada, Prog. Theor.Phys. 71, 320 (1984).

[3] R. Williams and S. E. Koonin, Nucl. Phys. A435, 844(1985).

[4] K. Oyamatsu, Nucl. Phys A561, 431 (1993).[5] K. Sumiyoshi, K. Oyamatsu, and H. Toki, Nucl. Phys

A595, 327 (1995).[6] C. P. Lorenz, D. G. Ravenhall, and C. J. Pethick, Phys.

Rev. Lett. 70, 379 (1993).[7] G. Toulouse, Commun. Phys. 2, 115 (1977).[8] J. Villain, Jour. Phys. C 10, 1717 (1977).[9] R. Liebmann, Statistical Mechanics of Periodic Frus-

trated Ising Systems, vol. 251 (Springer-Verlag, BerlinHeidelberg, 1986).

[10] H. T. Diep, Magnetic Systems with Competing Interac-

tions (World Scientific, Singapore, 1994).[11] C. J. Camacho, Phys. Rev. Lett. 77, 2324 (1996).[12] G. H. Wannier, Phys. Rev. 79, 357 (1950).[13] S. Kirkpatrick and D. Sherrington, Phys. Rev. B 17, 4384

(1978).[14] C. H. Papadimitriou and K. Kenneth Steiglitz, Combina-

torial Optimization: Algorithms and Complexity (Dover,Mineola, New York, 1998).

[15] A. R. Bodmer, C. N. Panos, and A. D. MacKellar, Phys.Rev. C 22, 1025 (1980).

[16] T. J. Schlagel and V. R. Pandharipande, Phys. Rev. C36, 162 (1987).

[17] G. Peilert, J. Randrup, H. Stocker, and W. Greiner,Phys. Lett. B 260, 271 (1991).

[18] T. Maruyama, K. Niita, K. Oyamatsu, T. Maruyama,

S. Chiba, and A. Iwamoto, Phys. Rev. C 57, 655 (1998).[19] G. Watanabe, K. Sato, K. Yasuoka, and T. Ebisuzaki,

Phys. Rev. C 68, 035806 (2003), nucl-th/0308007.[20] G. Watanabe, K. Sato, K. Yasuoka, and T. Ebisuzaki

(2003), nucl-th/0311083.[21] C. J. Horowitz, Phys. Rev. D 55, 4577 (1997).[22] M. Igashira, H. Kitazawa, H. Komano, and N. Yama-

muro, Nucl. Phys. A457, 301 (1986).[23] D. Vretenar, N. Paar, P. Ring, and G. A. Lalazissis, Phys.

Rev. C 63, 047301 (2001), nucl-th/0009057.[24] S. Reddy, G. Bertsch, and M. P. Prakash, Phys. Lett.

B475, 1 (2000).[25] J. M. Lattimer and F. D. Swesty, Nucl. Phys. A535, 331

(1992).[26] M. Liebendoerfer, O. E. B. Messer, A. Mezzacappa, S. W.

Bruenn, C. Y. Cardall, and F. K. Thielemann (2002),astro-ph/0207036.

[27] S. Jeschonnek and J. W. Van Orden, Phys. Rev. D 65,094038 (2002), and references therein.

[28] A. L. Fetter and J. D. Walecka, Quantum Theory of

Many-Particle Systems (McGraw-Hill, New York, 1971).[29] S. A. Chin, Ann. of Phys. 108, 301 (1977).[30] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller,

and E. Teller, Jour. Chem Phys. 21, 1087 (1953).[31] C. J. Horowitz, Phys. Rev. D 65, 043001 (2002).[32] D. Z. Freedman, D. N. Schramm, and D. L. Tubbs, Annu.

Rev. Nucl. Sci. 27, 167 (1977).[33] P. A. Egelstaff, An Introduction to the Liquid State (Aca-

demic Press, New York, 1967).[34] C. J. Horowitz, M. A. Perez-Garcıa, and J. Piekarewicz,

in preparation.