Symmetry energy, unstable nuclei, and neutron star crusts
aa r X i v : . [ nu c l - t h ] S e p EPJ manuscript No. (will be inserted by the editor)
Symmetry energy, unstable nuclei, and neutron star crusts
Kei Iida and Kazuhiro Oyamatsu Department of Natural Science, Kochi University, Akebono-cho, Kochi 780-8520, Japan RIKEN Nishina Center, Wako-shi, Saitama 351-0198, Japan Department of Human Informatics, Aichi Shukutoku University, 9 Katahira, Nagakute, Aichi 480-1197, JapanReceived: date / Revised version: date
Abstract.
Phenomenological approach to inhomogeneous nuclear matter is useful to describe fundamentalproperties of atomic nuclei and neutron star crusts in terms of the equation of state of uniform nuclearmatter. We review a series of researches that we have developed by following this approach. We start withmore than 200 equations of state that are consistent with empirical masses and charge radii of stable nucleiand then apply them to describe matter radii and masses of unstable nuclei, proton elastic scattering andtotal reaction cross sections off unstable nuclei, and nuclei in neutron star crusts including nuclear pasta.We finally discuss the possibility of constraining the density dependence of the symmetry energy fromexperiments on unstable nuclei and even observations of quasi-periodic oscillations in giant flares of softgamma-ray repeaters.
PACS.
Determining the equation of state (EOS) of uniform nu-clear matter is an old and fundamental issue in nuclearphysics, but it is rather hard to solve [1]. Thus, it is stillimportant to keep studying the EOS of nuclear matterphenomenologically and microscopically. Thanks to devel-opments of neutron star observations and nuclear experi-ments, our interest in the EOS of nuclear matter has to ex-tend for a very large region of density and neutron excess.In fact, nuclear matter associated with various systems,e.g., stable nuclei, unstable nuclei, nuclear pasta, neutronstars, supernova cores, heavy-ion collisions at intermediateenergies, etc., has different density and neutron excess. Onthe other hand, our understanding is far from sufficient.Relatively well-known are pure neutron matter, which hasbeen recently investigated from chiral effective theory in-teractions [2], and symmetric nuclear matter near normalnuclear density, which reflects the saturation of the nu-clear binding energy and density. From there, theoreticalextrapolations are more or less required. Moreover, onemay still ask how large the saturation density of symmet-ric nuclear matter is within five percent errors. We remarkthat thermal effects are also important for supernova coresand heavy-ion collisions.In describing the energy of uniform nuclear matter asfunction of density n and neutron excess α = 1 − x withthe proton fraction x , it is convenient to use an expansionof the energy per nucleon w around the saturation point of symmetric nuclear matter [3], w = w + K n ( n − n ) + (cid:20) S + L n ( n − n ) (cid:21) α . (1)The parameters characterizing this expansion include thesaturation density n and energy w of symmetric nuclearmatter, the symmetry energy coefficient S , the incom-pressibility of symmetric nuclear matter K , and the den-sity symmetry coefficient L . The parameters L and S areassociated with the density dependent symmetry energycoefficient S ( n ) as S = S ( n ) and L = 3 n ( dS/dn ) n = n .Basically, the parameter L corresponds to the pressure ofpure neutron matter at n = n . Generally, higher ordercoefficients with respect to density such as K and L aremore difficult to determine. We remark that expression (1)does not contain even higher order terms, one of which isassociated with the isospin dependence of the incompress-ibility.From the viewpoint of microscopic calculations, evenpure neutron matter at low densities is not simple . Thisis because of strong coupling effects and uncertainties inthe nuclear force. In fact, the Lee-Yang low density ex-pansion only works at very low densities, while we can seea behavior close to the unitarity limit at densities wherethe scattering length is very large compared with inter-particle spacing, which is in turn far larger than the rangeof the interaction [4]. Fortunately, in addition to varia-tional calculations [5], elaborate Green’s function MonteCarlo calculations [4] are available for pure neutron mat- Kei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts ter below normal nuclear density, and they are consistentwith each other. More recently, systematic many-body cal-culations based on chiral effective field theory have beenperformed [2]. In these calculations, effects of three-bodyinteractions play a role in determining the high density be-havior of neutron matter EOS. On the other hand, sym-metric nuclear matter is still more elusive. In fact, thesaturation properties cannot be reproduced by variationalcalculations without a phenomenological three-body force[5]. This is partly due to complexity involving a strongtensor force.In this article, we focus on a phenomenological ap-proach to the EOS of nuclear matter. In sect. 2, a macro-scopic nuclear model, which is constructed in such a wayas to depend on the EOS of uniform nuclear matter, isreviewed. We show that L and K remain uncertain whileempirical masses and charge radii of stable nuclei are equallywell reproduced. The nuclear model is then used to de-scribe matter radii and masses of unstable nuclei, pro-ton elastic scattering and total reaction cross sections offunstable nuclei, and nuclei in neutron star crusts includ-ing nuclear pasta, which are given in sect. 3–6. In sect.7, possible constraints on the density dependence of thesymmetry energy from experiments on unstable nuclei andobservations of quasi-periodic oscillations (QPOs) in giantflares of soft gamma-ray repeaters (SGRs) are discussed.Concluding remarks are finally given in sect. 8. From now on, we will focus on a phenomenological ap-proach to the EOS of nuclear matter. First we discuss thestatic properties of atomic nuclei and their relation withthe EOS of nuclear matter on the basis of ref. [6]. To thisend, we describe a macroscopic nuclear model in a mannerthat depends on the EOS of nuclear matter. Various appli-cations can arise therefrom. For example, we will addressin the next section how one can extract the saturationproperties of asymmetric nuclear matter from the size ofunstable nuclei. The main conclusion will be that it mightbe possible that we determine the density dependence ofthe symmetry energy from future systematic data for radiiof unstable nuclei.Application to neutron stars is of great significance be-cause the EOS of nuclear matter at large neutron excess isrelevant to the structure and evolution of neutron stars [7],which are expected to be further clarified by future spaceand ground-based observations. In the outer part (crust)of a star, nuclei present are considered to be very neutronrich or even drip neutrons in the presence of a neutralizingbackground of electrons. Near normal nuclear density, thesystem is considered to melt into uniform nuclear matter.This nuclear matter mainly constitutes the star’s core andthus controls the structure of a neutron star.The symmetry energy is related to the structure andevolution of neutron stars in many respects. For exam-ple, the mass and radius of a neutron star are mainlydetermined by the EOS of uniform nuclear matter. Thesymmetry energy acts to stiffen the EOS in a neutron-rich situation as encountered in a star. Also, neutron star cool-ing is related to the symmetry energy since it controls theproton fraction in the core region. Fast neutrino emissionprocess, i.e., direct URCA, can only occur at relativelyhigh proton fraction.Let us now start with the phenomenological expressionfor the energy of nuclear matter having the neutron andproton number densities n n and n p , which is divided intothe kinetic energy part and the potential energy part as[8] w = 3¯ h (3 π ) / m n n ( n / n + n / p ) + (1 − α ) v s ( n ) n + α v n ( n ) n , (2)where v s = a n + a n a n (3)and v n = b n + b n b n (4)are the potential energy densities for symmetric nuclearmatter and pure neutron matter, and m n is the neutronmass (for simplicity we here identify the proton mass m p with m n ). The parameter b , which controls the EOS ofmatter for large neutron excess and high density and thushas little effect on the saturation properties of nearly sym-metric nuclear matter, is set to 1.58632 fm , which wasobtained by one of the authors [8] in such a way as toreproduce the neutron matter energy of Friedman andPandharipande [5]. In the present energy expression, thepotential energy part is a parabolic function of α , whilethe kinetic energy part includes higher order terms in α .Such α dependence of the potential energy part is par-tially justified by variational calculations of Lagaris andPandparihande [9].Expression (2) is one of the simplest that reduces tothe usual expression (1) near the saturation point of sym-metric nuclear matter. From empirical masses and radiiof stable nuclei, as we shall see, one can well determinethe saturation density n and energy w of symmetric nu-clear matter and the symmetry energy coefficient S [6].The incompressibility K and the density symmetry co-efficient L are relatively uncertain, but they control thesaturation points at finite neutron excess. In fact, as theneutron excess increases from zero, the saturation pointmoves in the density versus energy plane. Up to secondorder in α , the saturation energy w s and density n s aregiven by w s = w + S α (5)and n s = n − n LK α . (6)The slope, y , of the saturation line near α = 0 ( x = 1 / y = − K S n L . (7) ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 3
D E FA B CG H I
Fig. 1.
Energy per nucleon of nuclear matter for nine extreme cases. In each panel, the solid lines are the energy at neutronexcess α = 0 , . , . , . ,
1, and the dotted line is the saturation line. From ref. [10].
Figure 1 illustrates nine of the present EOS models,which can be obtained for various sets of the incompress-ibility K and the density symmetry coefficient L as will beshown below. In each panel we plot the energy as functionof nucleon density ranging from symmetric nuclear mat-ter to pure neutron matter. The saturation line is writtenin dashed line. As the incompressibility K increases, thecurvature at the saturation point becomes larger. While,as the density symmetry coefficient L increases, the slopeof the saturation line becomes gentler. The question of in-terest here is what kind of EOS is favored by empiricaldata on nuclear masses and radii. For this purpose, we describe macroscopic nuclear prop-erties in a way that is dependent on the EOS of nuclearmatter by using a simplified version of the Thomas-Fermimodel [6]. A similar approach was independently utilizedby Bodmer and Usmani [11]. The essential point of thepresent model is to write down the binding energy of anucleus of mass number A and charge number Z in a den-sity functional form: E = E b + E g + E C + N m n c + Zm p c , (8)where E b = Z d rn ( r ) w [ n n ( r ) , n p ( r )] (9) Kei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts is the bulk energy, E g = F Z d r |∇ n ( r ) | (10)is the gradient energy with adjustable constant F , E C = e Z d r Z d r ′ n p ( r ) n p ( r ′ ) | r − r ′ | (11)is the Coulomb energy, and N = A − Z is the neutronnumber. This form allows us to connect the EOS and thebinding energy through the bulk energy part. For simplic-ity we use the following parametrization for the nucleondistributions ( i = n, p ): n i ( r ) = n in i " − (cid:18) rR i (cid:19) t i , r < R i , , r ≥ R i . (12)Here, R i roughly represents the nucleon radius, t i the rel-ative surface diffuseness, and n in i the central number den-sity. The proton distribution of the form (12) can fairlyaccurately reproduce the empirical behavior from electronelastic scattering off stable nuclei.In order to construct the nuclear model in such a wayas to reproduce empirical masses and radii of stable nu-clei, we first extremized the binding energy with respectto the particle distributions for fixed mass number A , fiveEOS parameters ( n , w , K , S , and L ), and gradient co-efficient F . Next, for various sets of the incompressibility K and the density symmetry coefficient L , we obtainedthe remaining three EOS parameters as well as F by fit-ting the calculated optimal values of the nuclear chargenumber, mass, and root-mean-square (rms) charge radiusto empirical data for stable nuclei on the smoothed betastability line. Here we have defined the rms charge radiusas R c = (cid:20) Z − Z d rr ρ c ( r ) (cid:21) / , (13)where ρ c ( r ) = ( π / a p ) − Z d r ′ exp (cid:0) −| r − r ′ | /a p (cid:1) n p ( r ′ )(14)with a p = 0 .
65 fm represents the charge distribution foldedwith the proton form factor [12]. The rms deviations ofthe calculated masses from the measured values [13] areof order 3 MeV, which are comparable with the deviationsobtained from the Weizs¨acker-Bethe mass formula, whilethe rms deviations of the calculated charge radii from themeasured values [14] are about 0.06 fm, which are compa-rable with the deviations obtained from the A / law.In fig. 2 we exhibit the EOS parameter region that canbe constrained from the fitting to empirical masses andradii of stable nuclei, together with various mean-field-model predictions. The saturation density n and energy w and the symmetry energy coefficient S are fairly well Fig. 2.
Various optimal relations among the parameters S , n , w , L , and K characterizing the EOS of nearly symmetricnuclear matter. In addition to the present results (crosses),the Skyrme-Hartree-Fock predictions [dots except for SIII(square)] and the TM1 prediction (triangle) are plotted. In(c), the thin lines are lines of constant y . From ref. [6]. constrained, while about 200 sets of the incompressibility K and the density symmetry coefficient L can providereasonable fitting. In fact, the data fitting based on theleast squares method depends on how to assign weights,which makes it impractical to find the optimal values of K and L among the sets, while for y higher than ∼ − , the fitting becomes no longer effective. This K - L region is a starting point of our study. We want tonarrow this region by using future experiments for unsta-ble nuclei. In the present analysis we rule out the pos-sibility that the slope y is positive. A positive y is in-consistent with the fact that the empirical matter radiiof A = 17 , ,
31 isobars [15,16,17] tend to increase withneutron/proton excess. This is because a positive y playsa role in increasing the saturation density n s with neu-tron/proton excess, as can be seen from eqs. (6) and (7).It is also interesting to see the roughly linear relationbetween L and S in fig. 2. This tendency is not clearin the Skyrme-model calculations with zero-range force,but is known to be seen in the Gogny-model calculationswith finite-range force [18]. How this tendency is related tothe range of the three-nucleon force has been recently dis-cussed [19]. We remark, however, that the roughly linearrelation is consistent with a recent 1 σ fit to experimen-tal masses and radii using a Skyrme-parametrized energydensity functional [20].The present L - S relation can be compared with otherconstraints on L and S . It turns out that various con-straints from heavy-ion data associated with isospin dif- ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 5 Fig. 3.
Optimal relation of the gradient coefficient F with thesaturation energy w of symmetric nuclear matter. fusion and neutron-proton ratio, pigmy dipole resonances,excitation energies of isobaric analog states are not al-ways consistent with each other [21]. Something has to bewrong, but we do not know which one. This may implythat data from stable nuclei are not enough to reasonablyconstrain L .For completeness, in fig. 3 we exhibit the optimal val-ues of F , which ranges ∼ , as a function ofthe optimal w . We note that there is a clear correlationbetween F and w because fitting to empirical masses re-quires a larger gradient energy for a larger bulk bindingenergy.We conclude this section by summarizing salient fea-tures of the macroscopic nuclear model used here. Thismodel can describe global nuclear properties such as massesand rms radii in a manner that is dependent on the EOSof nuclear matter, although it is not good at describingtails of the density distribution and does not allow forshell or pairing effects. As will be shown in the next sec-tion, the present macroscopic approach predicts that thematter radii depend appreciably on the density symme-try coefficient L , while being almost independent of theincompressibility K . We now address how the EOS-dependent macroscopic nu-clear model predicts the rms matter radii of unstable nu-clei. As clarified in the previous section, the saturationdensity n and energy w and the symmetry energy coef-ficient S can be well determined from systematic data on masses and radii of stable nuclei, while the incompressibil-ity K and the density symmetry coefficient L are moredifficult to determine. There are many trials of extractingthe incompressibility from empirical data such as giantmonopole resonances in stable nuclei [22] and even caloriccurves in nuclear collisions [23]. Unfortunately this kindof extraction depends on models for the effective nucleon-nucleon interaction [24]. A decade ago, we proposed amethod for extracting the density derivative of the sym-metry energy from future systematic data on the matterradii of unstable nuclei on the basis of the macroscopicnuclear model [6]. In RIKEN and GSI, it is expected thatRI beams of heavy nuclides incident on proton targetswill provide elastic scattering data with reasonable accu-racy, from which one may be able to deduce the matterradii, e.g., through an empirical relation between the firstdiffraction peak angles and the matter radii [25]. In GSI,this type of experiment named S272 was performed for Ni several years ago, but the data remain unpublished.Figure 4 shows the rms matter and charge radii for Niand Sn isotopes calculated for various sets of K and L .Here we have defined the rms matter radius as R m = (cid:20) A − Z d rr ρ m ( r ) (cid:21) / , (15)where ρ m ( r ) = ( π / a p ) − Z d r ′ exp (cid:0) −| r − r ′ | /a p (cid:1) n ( r ′ )(16)is the matter distribution folded with the proton chargeform factor equally for neutrons and protons. At fixed K ,differences of order 0.1 fm occur in the prediction of thematter radii of very neutron-rich nuclei due to uncertain-ties in the density symmetry coefficient L . This tendencyarises because the saturation density at nonzero neutronexcess decreases with increasing L as in eq. (6). On theother hand, the matter radii are almost independent of theincompressibility K . Note that as K increases, the sur-face diffuseness is reduced, while the saturation density n is also reduced as shown in fig. 2(d). We can thus concludethat these effects counteract with each other. Such K in-dependence is promising for the purpose of deriving thevalue of L from the experimentally deduced matter radii.These are just plotted for stable nuclei by crosses, whichare deduced from proton elastic scattering data by usingoptical potential models, but are not useful for derivationof L . Data for unstable nuclei are thus strongly desired.It is often claimed that the neutron skin thickness ofneutron-rich stable nuclei such as Sn and
Pb canseverely constrain L (e.g., ref. [27]). However, theoreticalpredictions of the neutron skin thickness depend not onlyon the nuclear bulk properties but also on the nuclear sur-face properties. In fact, within a compressible liquid-dropmodel [28,29], one can show that the predicted neutronskin thickness, which is determined by a balance betweenthe bulk and surface symmetry energies, has a linear de-pendence on L , but the poorly known density dependenceof the surface tension prevents a model-independent con-straint on L . Although the macroscopic nuclear model Kei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts
Fig. 4.
The rms charge and matter radii, R c and R m , of Ni and Sn isotopes for combinations of L = 0 , ,
80 MeV and K = 180 , ,
360 MeV. The experimental data for the rms charge radii (dots) and matter radii (crosses) are taken from refs.[14] and [26], respectively. From ref. [6]. used here and various mean-field models give a roughlylinear correlation between L and the neutron skin thick-ness, respectively, such a correlation is significantly dif-ferent between the two types of models. This kind of dif-ference would affect a possible constraint on L . Note thatthe present macroscopic model tends to underestimate thesurface diffuseness, which is not favorable for the predic-tion of the neutron skin thickness.Several summarizing remarks on the contents in thisand previous sections are in order. By using the macro-scopic nuclear model, we derived the relations between the EOS parameters from experimental data on the massesand charge radii of stable nuclei, and we found that L and K are still uncertain. The important prediction is thatthe density symmetry coefficient L may be determined ifa global behavior of the matter radii at large neutron ex-cess is obtained from future systematic measurements ofthe matter radii of unstable nuclei. Lastly, we remark thatthe parameter L , which characterizes the dependence ofthe EOS on neutron excess, is relevant to the structureand evolution of neutron stars through mass-radius rela-tion, crust-core boundary, cooling, etc. ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 7 It is not straightforward to deduce the matter radii ofunstable nuclei from experimental data such as proton-nucleus elastic differential cross sections and total reactioncross sections because it requires the approximate scatter-ing theory [26,30]. It is thus instructive to examine howthe cross sections themselves are related to the parame-ter L in a proper theoretical framework, namely, the opti-cal limit approximation of the Glauber multiple scatteringmodel [31] that incorporates the nucleon distributions asobtained in sect. 2.We first consider proton elastic scattering on the basisof ref. [32]. For sufficiently high proton incident energiesand small momentum transfers to validate the Glaubermodel, we find that the angle of the scattering peak de-creases with L more remarkably for larger neutron excess,while the peak height increases with K almost indepen-dently of neutron excess. We suggest the possibility thatcomparison of the calculations with experimental data forthe peak angle may be useful for determination of L .The elastic differential cross section at given momen-tum transfer q and incident proton energy T p can be writ-ten as (e.g., ref. [33]) dσdΩ = | F ( q ) | , (17)with the elastic scattering amplitude, | F ( q ) | = | F C ( q )+ ik π Z d b e − i q · b +2 iη ln( k | b | ) h − e iχ N ( b ) i | . (18)Here, ¯ hk = p ( T p /c + m p c ) − ( m p c ) is the incident pro-ton momentum, b is the impact parameter, η = Ze / ¯ hv with the incident proton velocity v = ¯ hkc/ ( T p /c + m p c ) isthe Sommerfeld parameter, F C ( q ) = − ηk q exp (cid:20) − iη ln (cid:18) | q | k (cid:19) + 2 i arg Γ (1 + iη ) (cid:21) (19)is the amplitude of the Coulomb elastic scattering, whichwe approximate as a usual Rutherford scattering off apoint charge, and iχ N ( b ) = − Z d r [ n p ( r ) Γ pp ( b − s )+ n n ( r ) Γ pn ( b − s )] (20)is the phase shift function with the projection s of the co-ordinate r on a plane perpendicular to the incident protonmomentum and with the profile function Γ pN of the freeproton-nucleon ( pN ) scattering amplitude, for which weuse a simple parametrization, Γ pN ( b ) = 1 − iα pN πβ pN σ pN exp( − b / β pN ) , (21)where α pN = − Im Γ pN (0) / Re Γ pN (0), σ pN is the pN to-tal cross section, and β pN is the slope parameter. Here Fig. 5.
The angles and heights of the scattering peak in thesmall angle regime, calculated as functions of L and K for p - Sn and p - Sn elastic scattering at T p = 800 MeV. The ex-perimental angles and heights including errors (from ref. [35])are denoted by the horizontal lines (thick lines: central values,thin lines: upper and lower bounds). From ref. [32]. the values of α pN , β pN , and σ pN at given incident protonenergy T p are taken from ref. [34].Generally, the peak angles are related to the nuclearsize, while the peak heights are related to the surface dif-fuseness. In our macroscopic calculations, the radius anddiffuseness are in turn related to the EOS parameters. Forlarger density symmetry coefficient L we obtain a largerradius, while for larger incompressibility K we obtain asmaller surface diffuseness. So it is interesting to investi-gate the detailed peak structure in the small angle regimeand its relation with the EOS parameters.In fig. 5 we illustrate the scattering angles and heightsin the first peak, calculated for about 200 sets of the EOSparameters in the case of stable Sn isotopes at incident en-ergy of 800 MeV. The peak angle decreases with L . Thisis an important property which might enable us to ex-tract L from comparison with the experimental peak an-gle. However, such extract is difficult in this case becausethe experimental uncertainty due to the absolute anglecalibration, which is taken to be 0.05 deg, is too large todistinguish between different L ’s for nuclei having neutronexcess of order or smaller than 0.2.On the other hand, the peak height increases with theincompressibility K in a way almost independent of neu-tron excess. However, it is also difficult to extract K fromcomparison with the experimental peak height mainly be-cause our semi-classical nuclear model tends to underesti-mate the surface diffuseness. Kei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts
Fig. 6.
Same as fig. 5 for p - Ni and p - Ni elastic scatteringat T p = 800 MeV. From ref. [32]. We move on to scattering off unstable nuclei, whosebeams incident on proton targets can provide elastic scat-tering data. Here we consider a very neutron-rich nucleus Ni. In fact, the neutron excess for Ni amounts to 0.3.We perform the calculations for Ni and Ni at incidentenergy of 800 MeV. We find from fig. 6 that the L depen-dence of the difference in the peak angle between the Niand Ni cases looks large enough to enable us to extract L . It is now useful to take the difference because our cal-culations based on the macroscopic nuclear model containsystematic errors by ignoring pairing and shell effects andtails of the nucleon distributions. In order to take full careof such systematic errors, systematic measurements of thepeak angles in the small angle region for various nuclidesare desired for as large neutron excess as possible withaccuracy of order 0.01 deg. According to the usual Fraun-hofer diffraction, the difference of order 0.01 deg in thepeak angle corresponds to the difference of order 0.01 fmin the rms matter radius. We remark in passing that thepresent analysis eventually developed into a black spheremodel [25] that gives a nuclear length scale that charac-terizes measured total reaction cross sections and elasticdiffraction peak angles simultaneously.We finally consider proton-nucleus total reaction crosssections, which can be calculated from the same Glaubermodel as used for the elastic scattering calculations [36].The total reaction cross section can be written as σ R = Z d b (cid:18) − (cid:12)(cid:12)(cid:12) e iχ N ( b ) (cid:12)(cid:12)(cid:12) (cid:19) , (22)where χ N is given by eq. (20).Figure 7 shows the results for the selected isotopes at T p = 800 MeV. The dependence of the total reaction cross σ R ( m b ) Sn σ R ( m b ) Sn σ R ( m b ) Ni σ R ( m b ) Ni σ R ( m b ) K (MeV) Sn σ R ( m b ) L (MeV) Sn Fig. 7. (Color online) The total reaction cross sections calcu-lated as a function of K and L for p - , Sn and p - Ni at T p = 800 MeV. From ref. [36]. section σ R on the EOS parameters is weak even for veryneutron-rich nuclei such as Ni, in contrast to the case ofelastic scattering in which a strong L (or size) dependenceof the calculated diffraction peak angle appears for Ni asshown in fig. 6. This was not expected from a standard pic-ture that the larger size, the larger σ R , but in fact reflectsa feature of the optical limit Glauber theory in which anunphysical exponential dependence of the reaction crosssection on the neutron skin thickness remains when thetotal proton-neutron cross section is small enough [36].For duly describing the size dependence of the total reac-tion cross sections, therefore, alternative approaches basedon empirical data for the total reaction cross sections suchas those in refs. [37] and [38] might be useful even for highincident energies where the Glauber theory is usually as-sumed to be valid.We remark that differences between interaction crosssections and total reaction cross sections are often ignored,which causes interaction cross sections, whose data are fareasier to obtain experimentally, to be identified with to-tal reaction cross sections. However, this is not alwaysthe case even for high energy data, as suggested by us-ing pseudodata for total reaction cross sections that canbe obtained from the measured elastic diffraction peakangles for stable nuclei via the black sphere model [39].Within the framework of the full Glauber scattering the-ory, Novikov and Shabelski [40] also confirmed that thedifferences are appreciable. ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 9 Let us move on to nuclear masses. The mass has an ad-vantage over the size because empirical mass data havebeen already accumulated for unstable nuclei. Then, it isnatural to ask if the existing data for masses of unstablenuclei is useful for determination of L . We shall give atentative answer on the basis of refs. [41,42].We want to know the global neutron excess dependenceof nuclear masses. To this end, some kind of differentialsare useful. Here we focus on the two proton separationenergy S p ( Z, N ) = E B ( Z, N ) − E B ( Z − , N ) with thenuclear binding energy E B . As illustrated in ref. [43], forfixed proton number Z , the empirical values of S p andalso the values from the Koura-Tachibana-Uno-Yamada(KTUY) mass formula show a very smooth isospin depen-dence except for proton shell gaps. Moreover, the even-oddstaggering essentially disappears in the two proton sepa-ration energy S p . On the other hand, the two neutronseparation energy S n = E B ( Z, N ) − E B ( Z, N −
2) doesnot show an ideal isospin dependence at fixed Z , becauseit has a very large discontinuity around neutron magicnumbers.We now try to compare the empirical S p and the pre-dictions from the EOS models C and G shown in fig. 1 byusing the macroscopic nuclear model. As exhibited in fig.8, the empirical S p shows a smooth dependence on neu-tron excess except for symmetric nuclei, nuclei with neu-tron magic numbers, and deformed nuclei. On the otherhand, the calculated S p shows a larger neutron excessdependence for larger L . Comparison between the empir-ical and calculated S p seems easier for smaller mass. Asfar as the slope of S p with respect to neutron excess isconcerned, a larger L value is more consistent with theempirical data. Note that there are roughly uniform off-sets between the data and the calculations with the EOSmodel C at N > Z , which are presumably due to protonshell gaps ignored in the present calculations. That is whythe fact that the calculated S p from the EOS model G isapparently closer to the empirical S p has to be seen withcaution.In order to understand the L dependence of S p , wego back to the L dependence of the calculated nuclearmasses. For very neutron rich nuclei, as shown in fig. 9,the calculated mass decreases with L , while having a rela-tively weak dependence on K . This suggests that nuclearmasses are not always dominated by the bulk propertiesof nuclear matter. In fact, the L dependence of the cal-culated mass cannot be explained by the bulk asymmetryterm because for a larger L we obtain a larger S (see fig.2), leading to a larger mass according to eq. (5). There-fore, we can conclude that the surface asymmetry termis responsible for the present L dependence of the cal-culated mass. In fact, within a compressible liquid-dropmodel [28,29], one can show that the surface tension forneutron rich nuclei is effectively smaller for larger L , lead-ing to a smaller mass and S p .Note that the macroscopic nuclear model used hereeffectively has a nonvanishing density dependence of thesurface tension. This is a contrast to the cases of many -30-20-10010 S - S W B ( M e V ) exp03 KTUY05 EOS C (L=146 MeV) EOS G (L=5.7 MeV) O Z=8-20-15-10-50510 S - S W B ( M e V ) exp03 KTUY05 EOS C (L=146 MeV) EOS G (L=5.7 MeV) Mg Z=12-15-10-50510 S - S W B ( M e V ) exp03 KTUY05 EOS C (L=146 MeV) EOS G (L=5.7 MeV) Ca Z=20-8-404 S - S W B ( M e V ) exp03 KTUY05 EOS C (L=146 MeV) EOS G (L=5.7 MeV) Ni Z=28
Fig. 8. (Color online) The two-proton separation energy, hav-ing the one calculated from a Weizs¨acker-Bethe mass formula[28] subtracted out, for O, Mg, Ca, and Ni isotopes. The em-pirical values [13], the calculated values from the EOS modelsC and G, and the values obtained from a contemporary massformula [43] are plotted in each panel. From ref. [41].0 Kei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts -36-34-32-30-28-26-24-22 m a ss e xc e ss ( M e V ) Ni m a ss e xc e ss ( M e V ) O Fig. 9. (Color online) The mass excess calculated for Ni and O as a function of L . From ref. [41]. P r o t on nu m be r Exp. (CN2004) mass-measured (AWT03) stable drip line EOS C EOS G KTUY05
Fig. 10. (Color online) The neutron and proton drip linesobtained from the EOS models C and G by using the macro-scopic nuclear model and from a contemporary mass formula[43]. The regions filled with squares correspond to empiricallyknown nuclides [44,45]. From ref. [42]. compressible liquid-drop models that assume vanishingdensity dependence of the surface tension. However, thisassumption applies only for the planar interface betweenthe bulk liquid and gas (vacuum) phases where under me-chanical stability, the nucleon density is fixed at the satu-ration density and zero, respectively. A possible constrainton L from empirical nuclear masses would thus depend onthe adopted density dependence of the surface tension.This situation is similar to the case of the neutron skinthickness. As an important application, we proceed to exhibitthe neutron drip line, which was calculated from the EOSmodels C and G within the macroscopic nuclear modeland from the KTUY mass formula [42]. In fig. 10, theneutron drip line was drawn by identifying nuclides atneutron drip with those neighboring to nuclides for which S n = E B ( Z, N ) − E B ( Z, N −
1) and S n are positive andbeyond which at least one of them is negative. As L in-creases, the calculated drip line shifts to the neutron richside on the chart of nuclides, because for larger L weobtain more binding through the surface properties dis-cussed above. Consequently, this shift is appreciable forsmall masses where some of the drip nuclei are empiri-cally identified [46,47]. It is an interesting open questionto constrain L from the empirical drip line to be expandedin the near future. Note, however, that the neutron dripline is mainly determined by a competition between theCoulomb energy and the symmetry energy coefficient S ,while the present L effect, induced by the surface proper-ties, is just secondary. Let us turn to a different topic of research, namely, neu-tron star crusts. On the basis of ref. [10], we will show thatthe presence of nuclear pasta in neutron stars is sensitiveto the density symmetry coefficient L .Nuclear pasta represents exotic shapes of nuclei, whichmay occur in the deepest region of the crust [48,49]. Inthis region, nuclei are considered to be closely packed ina bcc Coulomb lattice. Then, the total surface area be-comes very large. In order to lower the system energy, itis convenient that the spherical nuclei are elongated andfuse into a nuclear rod. In the presence of Coulomb energy,the nuclei cannot have arbitrary shape. With the densityincreased further, possible changes in nuclear shape areconsidered to be rods, slabs, tubes, bubbles, and uniform.In terms of liquid crystals, the rod and tube phases arecolumnar, while the slab phase is smectic A. Also thesepasta phases can be regarded as liquid-gas mixed phases.As we will see, the symmetry energy at subnuclear densi-ties controls the crust-core boundary and the presence ofnuclear pasta.In describing zero-temperature matter in neutron starcrusts, we again use the macroscopic nuclear model. Thistime, not for a nucleus in vacuum, but for a nucleus orbubble in a Wigner-Seitz cell. New additions are drippedneutrons, a neutralizing uniform background of electrons,and the lattice energy.For each unit cell, we write the total energy as W = W N + W e + W C , (23)where W N , W e , and W C are the nuclear energy, the elec-tron energy, and the Coulomb energy. ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 11 As in eq. (8), the nuclear energy is again expressed inthe density functional form: W N = Z cell d r { n ( r ) w [ n n ( r ) , n p ( r )]+ m n c n n ( r ) + m p c n p ( r ) + F |∇ n ( r ) | (cid:9) . (24)For a spherical nucleus in vacuum, this expression reducesto E − E C [see eq. (8)].The electron energy can be approximated as the energyof an ideal uniform Fermi gas, W e a = m e c π ¯ h { x e (2 x e + 1)( x e + 1) / − ln[ x e + ( x e + 1) / ] } (25)with x e = ¯ h (3 π n e ) / m e c , (26)where m e is the electron mass, and n e is the electron num-ber density that satisfies the charge neutrality condition, a n e = Z cell d rn p ( r ) . (27)We remark that n e is so high that we can safely ignoreinhomogeneity of the electron density induced by the elec-tron screening of nuclei or bubbles and the Hartree-Fockcorrections to the electron energy.The Coulomb energy, which is composed of the protonself-Coulomb energy and the lattice energy, can be writtenas W C = 12 Z cell d re [ n p ( r ) − n e ] φ ( r ) + ∆W , (28)where φ ( r ) is the electrostatic potential in a Wigner-Seitzcell, and ∆W is the difference of the rigorous calculation[50] for a cell in the bcc (triangular) lattice of spherical(cylindrical) nuclei or bubbles having sharp surfaces fromthe Wigner-Seitz value, as parametrized in ref. [8]. We takeinto account ∆W , which is a less than 1 % correction,because ∆W depends sensitively on the dimensionalityof the lattice.For nucleon distributions in the Wigner-Seitz cell, wesimply generalize the parametrization (12) for a nucleusin vacuum into n i ( r ) = ( n in i − n out i ) " − (cid:18) rR i (cid:19) t i + n out i , r < R i ,n out i , R i ≤ r. (29)Here r is the distance from the central point, axis, or planeof the unit cell. In the case of nuclei, n out p = 0, while inthe case of bubbles, n in p = 0.We finally determine the equilibrium configuration ofthe system at given baryon density, n b = a − Z cell d rn ( r ) . (30) nu c l eon nu m be r den s i t y ( f m - ) pasta nuclei fission instability proton clustering GHI AD E B F C
Fig. 11. (Color online) The density region containing bubblesand nonspherical nuclei as a function of L , calculated for theEOS models A–I in fig. 1. For comparison, the density corre-sponding to u = 1 / First, for each of the five inhomogeneous phases, we min-imize the total energy density
W/a with respect to theeight parameters a , n in n , n out n , n in p (for nuclei) or n out p (forbubbles), R n , R p , t n , and t p . This minimization implicitlyallows for the stability of the nuclear matter region (theregion containing protons) with respect to change in thesize, neutron drip, β -decay, and pressurization. In additionto the five inhomogeneous phases, we consider a uniformphase of β -equilibrated, neutral nuclear matter. The en-ergy density of this phase is the sum of the nucleon part nw + m n c n n + m p c n p [see eq. (2)] and the electron part(25). By comparing the resultant six energy densities, wecan determine the equilibrium phase of energy density ρ .In fig. 11, we show the resultant density region wherepasta nuclei are predicted to appear. We find that thelarger L , the narrower pasta region. This tendency sug-gests that for smaller symmetry energy at subnuclear den-sities, protons become more difficult to cluster in uniformmatter.The lower end of the pasta region can be naively under-stood from fissionlike instability of spherical nuclei. In aliquid-drop model, it is predicted that nuclei tend towardquadrupolar deformations when the Coulomb energy istwice as large as the surface energy. In neutron stars, dueto the lattice energy, this condition can be essentially meteven in equilibrium when the volume fraction of nucleireaches 1/8. At this volume fraction, the baryon densityis of order 0.06 fm − and almost independent of the EOSmodels, as shown in fig. 12.On the other hand, the upper end of the pasta regioncan be naively understood from proton clustering instabil-ity of uniform matter. The tendency to proton clusteringcan be measured by the sign change of the effective po-tential between proton density fluctuations. The drivingforce of proton clustering is the symmetry energy at sub- a v e r age nu c l eon den s i t y ( f m - ) proton clustering fission instability Fig. 12. (Color online) The onset density of proton clusteringin uniform nuclear matter calculated from the present EOSmodels. For comparison, we plot the density corresponding to u = 1 / nuclear densities. In fact, for larger L , the system keepshomogeneous down to lower density, as shown in fig. 12.Comparing the onset densities of proton clustering andfission instability with the equilibrium calculations of thepasta region, we find that the onset densities are a goodmeasure of the pasta region (see fig. 11). Judging from fig.12, the critical value of L for the presence of pasta nucleiin neutron stars is about 100 MeV.Recently, more complicated structures with intersect-ing rods have been predicted by calculations beyond theWigner-Seitz approximation [51,52,53,54,55,56]. On theother hand, a liquid-drop approach [57,58] can be usedfor the purpose of examining the possible occurrence ofa periodic bicontinuous structure, namely, gyroid, whichis known to occur in polymer systems [59]. Energetically,this structure is to be reckoned with seriously judging fromthe evaluated energy difference from the ground state atsubnuclear densities, but even with shape-dependent cur-vature effects included, no density region where the gyroidis most stable was found.At finite temperatures, pasta nuclei do not always haveperfect structure. In fact, they thermally fluctuate justlike molecular liquid crystals. At typical temperatures ofneutron star interiors, however, the amplitude of the dis-placements involved is smaller than internuclear spacing[60].We conclude this section by showing the size of spher-ical nuclei in the inner crust estimated for various EOSmodels within the macroscopic nuclear model. We findfrom fig. 13 that the larger L , the smaller size. This ten-dency suggests that for larger L or, equivalently, smallersymmetry energy at subnuclear density, the density ofdripped neutrons becomes larger and hence the surfacetension becomes smaller. This is essential to possible con-straints on L from neutron star asteroseismology as willbe discussed in the next section. p r o t on nu m be r b (fm -3 ) inner crustproton number Z IHG FEDC AB
Fig. 13. (Color online) The charge number of spherical nucleias a function of n b , calculated for the EOS models A–I. Fromref. [10]. L In this section we consider possible constraints on L fromempirical data via the macroscopic nuclear model. As dis-cussed in sect. 5, experimental data on masses of unstablenuclei could give a stringent constraint on L if the modelwere free from systematic errors associated with the de-scription of the isospin-dependent surface properties. Infact, constraints on L that were obtained from the massdata in earlier publications scatter [61]. All we can con-clude at this stage is that a very small L cannot reproducethe empirical isotope dependence of the two-proton sepa-ration energy S p depicted in fig. 8. On the other hand,future systematic data associated with the size of unsta-ble nuclei are expected to help constrain L as discussed inSecs. 3 and 4.We finally turn to QPOs in giant flares from SGRs andtheir possible relation with crustal torsional oscillations.Usually, SGRs are considered to be magnetars, i.e., neu-tron stars with surface magnetic fields of order 10 G.About a decade ago, one of them exhibited a giant flareand fortunately, its X-ray afterglow was detected by theRossi X-ray Timing Explorer [62]. It turns out that theafterglow oscillates quasi-periodically.Steiner and Watts [63] tried to explain these QPOs interms of crustal shear modes. They succeeded in repro-ducing some of the measured frequencies, but the analysisis model dependent. This is because the shear modulusis controlled by the charge number of neutron-rich nucleithat constitute the crust, and the charge number is pre-dicted to be dependent on L as shown in fig. 13.With this L dependence of the charge number in mind,we evaluated the frequency of the fundamental mode ofcrustal torsional oscillation by ignoring and allowing forthe effects of neutron superfluidity [64,65]. For such evalu-ations, we first consider the equilibrium neutron star con-figurations. Since the magnetic energy is much smallerthan the gravitational binding energy even for magne-tars, we can neglect the deformation due to the mag-netic pressure. Additionally, since the magnetars are rel-atively slowly rotating, we can also neglect the rotational ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 13 effect. Hereafter, therefore, we consider spherically sym-metric neutron stars, whose structure is described by thesolutions of the well-known Tolman-Oppenheimer-Volkoff(TOV) equations. In this case, the metric can be expressedin terms of the spherical polar coordinates r , θ , and φ as ds = − e Φ dt + e Λ dr + r dθ + r sin θ dφ , (31)where Φ and Λ are functions of r . (Hereafter, we use unitsin which G = c = 1.) We remark that Λ ( r ) is associatedwith the mass function m ( r ) = Z r dr ′ πr ′ ρ ( r ′ ) (32)as e Λ = [1 − m ( r ) /r ] − .To solve the TOV equations, one generally uses thezero-temperature EOS, i.e., the pressure p as a functionof the mass density ρ . For matter in the crust, we use thesame EOS models as described above. Unfortunately, thecore EOS is still uncertain in the absence of clear under-standing of the constituents and their interactions bothin vacuum and in medium. Since we will focus on sheartorsional oscillations that occur in the crust, we can ef-fectively describe such uncertainties in the core EOS bysolely setting the star’s mass M and radius R as free pa-rameters, without using specific models for the core EOS.In fact, for various sets of M and R , we systematicallyconstruct the equilibrium configuration of the crust by in-tegrating the TOV equations with the crust EOS from thestar’s surface all the way down to the crust-core boundary.This is a contrast to the usual way of constructing a starby initially giving a value of the central mass density andthen integrating the TOV equations with a specific modelfor the EOS from the star’s center to surface. Hereafter,we will consider 1 . ≤ M/M ⊙ ≤ . ≤ R ≤ M and R . Such choice of M and R can duly encapsulate uncertainties of the core EOS.Generally, a restoring force for shear torsional oscilla-tions is provided by shear stress, which comes from theelasticity of the oscillating body and is characterized bythe shear modulus µ . In the case of torsional oscillationsin the crust of a neutron star, the shear modulus is deter-mined by the lattice energy of the Coulomb crystal thatconstitutes the crust. Since the crystal is generally con-sidered to be of bcc type one can use the correspondingshear modulus, which is calculated for Ze point charges ofnumber density n i in the uniform neutralizing backgroundas µ = 0 . n i ( Ze ) a , (33)where a = (3 Ze/ πn e ) / is the radius of a Wigner-Seitzcell [66]. Note that this formula is derived in the limit ofzero temperature from Monte Carlo calculations for theshear modulus averaged over all directions [67].The shear modulus depends strongly on the value of L ,which comes mainly from the L dependence of the calcu-lated Z [10]. It is natural that one should take into accountthe shear modulus in pasta phases, if present, but here-after we simply assume µ = 0 for the pasta phases. This is because the shear modulus in the pasta phases excepta phase of spherical bubbles has at least one direction inwhich the system is invariant with respect to translationand hence is expected to be significantly smaller than thatin a phase of spherical nuclei [68]. Under this assumption,we have only to consider the shear torsional oscillationsthat are excited within a crustal region of spherical nu-clei. Anyway, the constraint on L that will be given belowcan be considered to be robust, because the pasta regionas shown in fig. 11 is highly limited given the resultingconstraint on L .We now consider the shear torsional oscillations on theequilibrium configuration of the crust of a neutron stardescribed above. In order to determine the frequencies,we adopt the relativistic Cowling approximation, i.e., weneglect the metric perturbations on eq. (31) by setting δg µν = 0. In fact, one can consider the shear torsionaloscillations with satisfactory accuracy even with the rel-ativistic Cowling approximation, because the shear tor-sional oscillations on a spherically symmetric star are in-compressible and thus independent of the density vari-ation during such oscillations. Additionally, due to thespherically symmetric nature of the background, we haveonly to consider the axisymmetric oscillations. Then, theonly non-zero perturbed matter quantity is the φ com-ponent of the perturbed four-velocity, δu φ , which can bewritten as δu φ = e − Φ ∂ t Y ( t, r ) 1sin θ ∂ θ P ℓ (cos θ ) , (34)where ∂ t and ∂ θ denote the partial derivatives with respectto t and θ , respectively, while P ℓ (cos θ ) is the ℓ -th orderLegendre polynomial. We remark that Y ( t, r ) character-izes the radial dependence of the angular displacementof a matter element. By assuming that the perturbationvariable Y ( t, r ) has such a harmonic time dependence as Y ( t, r ) = e i ωt Y ( r ), the perturbation equation that gov-erns the shear torsional oscillations can be derived fromthe linearized equation of motion as [69] Y ′′ + (cid:20)(cid:18) r + Φ ′ − Λ ′ (cid:19) + µ ′ µ (cid:21) Y ′ + (cid:20) Hµ ω e − Φ − ( ℓ + 2)( ℓ − r (cid:21) e Λ Y = 0 , (35)where H is the enthalpy density defined as H ≡ ρ + p with the mass density ρ and pressure p as described inthe previous section, and the prime denotes the derivativewith respect to r . Note that under the present definition of H , effects of neutron superfluidity are essentially ignoredas in ref. [64], while such effects will be included below asin ref. [65].Once appropriate boundary conditions are imposed,the problem to solve reduces to an eigenvalue problemwith respect to ω . Since there is no matter outside the star,we adopt the zero-torque condition at the star’s surface.Meanwhile, since there is no traction force in the regionwith µ = 0, we adopt the zero-traction condition at theposition where spherical nuclei disappear in the deepest L [MeV] t [ H z ]
18 Hz
Fig. 14. (Color online) t as a function of L for 10 km ≤ R ≤
14 km and 1 . M ⊙ ≤ M ≤ . M ⊙ . The horizontal dot-dashed line denotes the lowest QPO frequency observed fromSGR 1806-20 [62]. From ref. [64]. region of the crust. In practice, one can show that bothconditions reduce to Y ′ = 0.For the EOS models A–I, we then calculate t , namely,the frequency of the mode with zero radial node and spher-ical harmonics ℓ = 2, which is theoretically the lowestamong various eigenfrequencies of the torsional oscilla-tions. By interpolating the results for t and assuming1 . ≤ M/M ⊙ ≤ . ≤ R ≤
14 km, we obtainan allowed region as in fig. 14. We can clearly see the L dependence of the frequency, while the width comes fromvarious sets of the star’s radius and mass. These featuresoccur because t is basically determined by the ratio ofthe shear velocity p µ/H over the oscillation path lengththrough the crust. Note that the lower (upper) boundaryof the allowed region in fig. 14 corresponds to the largest(smallest) and heaviest (lightest) case. In these calcula-tions, we ignore the effects of neutron superfluidity andpasta elasticity, which act to enhance the frequency. As aresult, these calculations are expected to provide a lowerlimit of L , hereafter referred to as L min , under the as-sumption that the observed QPOs in SGR giant flaresarise from the torsional oscillations in neutron star crusts.In fact, since, under this assumption, t would be equalto or even lower than the lowest QPO frequency observed,we obtain L min of order 50 MeV, which is fairly stringentgiven that L is still uncertain [61]. Note that we ignoreelectron screening and finite-size effects on charge [70,71],which act to reduce the shear modulus and hence t , aswell as nuclear shell and pairing effects [72], which actto shift the nuclear charge number by keeping the localaverage almost unchanged. Such ignorance could modify L min .Now, we take into account the effect of neutron super-fluidity on the shear torsional oscillations [65]. In general,it is considered that neutrons confined in the nuclei startto drip therefrom when the mass density becomes higherthan ∼ × g cm − . Then, some of the dripped neu- trons can behave as a superfluid. A significant fractionof the dripped neutrons may move non-dissipatively withprotons in the nuclei as a result of Bragg scattering off thebcc lattice of the nuclei; only neutrons in the conductionband can freely flow with respect to the lattice. In fact, therecent band calculations beyond the Wigner-Seitz approx-imation by Chamel [73] show that the superfluid density,which is defined here as the density of neutrons unlockedto the motion of protons in the nuclei, depends sensitivelyon the baryon density above neutron drip and that a con-siderable portion of the dripped neutrons can be locked tothe motion of protons in the nuclei. On the other hand,since the shear torsional oscillations are transverse, the re-maining superfluid neutrons, whose low-lying excitationsare longitudinal, do not contribute to such oscillations [74].We build the effect of neutron superfluidity into theeffective enthalpy density ˜ H , which can be determinedby subtracting the superfluid mass density from the totalenthalpy density H in eq. (35) that fully contains the con-tributions of the superfluid neutrons as well as the nucleiand companions. Since we assume that the temperature ofneutron star matter is zero, the baryon chemical potential µ b can be expressed as µ b = H/n b . Thus, one can writedown ˜ H = (cid:18) − N s A (cid:19) H, (36)where N s denotes the number of neutrons in a Wigner-Seitz cell that do not comove with protons in the nucleus,while A is the total nucleon number in the Wigner-Seitzcell. Finally, substituting ˜ H for H in eq. (35), one candetermine the frequencies of the shear torsional oscilla-tions, which include the effect of neutron superfluidityin a manner that depends on the value of N s . Hereafter,we will assume that N s comes entirely from the drippedneutron gas. Even so, it is still uncertain how much frac-tion of the dripped neutrons behave as a superfluid. Thus,we introduce a new parameter N s /N d , where N d is thenumber of the dripped neutrons in the Wigner-Seitz cell.For N s /N d = 0, all the dripped neutrons behave as nor-mal matter and contribute to the shear motion, while for N s /N d = 1, all the dripped neutrons behave as a super-fluid. We remark that N d − N s denotes the number ofthe dripped neutrons bound to the nucleus. Typically, thevalue of N s /N d depends on the density inside a neutronstar [73], but the case of N s /N d = 0 in the whole crust iscloser to the typical behavior than the case of N s /N d = 1.We proceed to show how neutron superfluidity affects L min . For constant values of N s /N d , we calculate t byfollowing the same line of argument as in the absence ofneutron superfluidity (i.e., N s /N d = 0) and therefrom ob-tain L min , which is illustrated in fig. 15. One can observethat the value of L min , which is 47.6 MeV in the case of N s /N d = 0, can be as large as 125.9 MeV in its presence(i.e., 0 < N s /N d ≤ L min = 55 . N s /N d in ref. [73]. This L min is expected to give a reli-able constraint on L , although a possible L dependence of N s /N d through the band structure remains to be investi-gated. ei Iida, Kazuhiro Oyamatsu: Symmetry energy, unstable nuclei, and neutron star crusts 15 N s /N d L m i n [ M e V ] Chamel (2012)
Fig. 15. (Color online) L min as a function of N s /N d , the frac-tion of superfluid neutrons. The horizontal broken line corre-sponds to the result from the band calculations of N s /N d byChamel [73]. From ref. [65]. Instead of just considering L min , we can obtain a morestringent constraint on L by fitting the predicted frequen-cies of fundamental torsional oscillations with differentvalues of ℓ to the low-lying QPO frequencies observed inSGRs. To this end, we use the values of N s /N d derivedby Chamel [73]. In the present analysis, we focus on theobserved QPO frequencies lower than 100 Hz, i.e., 18, 26,30, and 92.5 Hz in SGR 1806-20 and 28, 54, and 84 Hzin SGR 1900+14 [62]. In fact, the even higher observedfrequencies would be easier to explain in terms of mul-tipolar fundamental and overtone frequencies. Because ofthe small interval between the observed frequencies 26 and30 Hz in SGR 1806-20, it is more difficult to explain theQPO frequencies observed in SGR 1806-20 than those inSGR 1900+14. If one identifies the lowest frequency inSGR 1806-20 (18 Hz) as the fundamental torsional oscil-lation with ℓ = 3, one can reasonably explain 26, 30, and92.5 Hz in terms of those with ℓ = 4, 5, and 15. In the caseof the typical neutron star model with M = 1 . M ⊙ and R = 12 km, we compare the predicted frequencies withthe observed ones as shown in fig. 16. One can observefrom this figure that the best value of L to reproduce theobserved frequencies is L = 127 . ℓ = 3), 24.9 Hz ( ℓ = 4), 31.0Hz ( ℓ = 5), and 90.3 Hz ( ℓ = 15), are within less than 5%deviations from the observations.Let us now extend the analysis to different stellar mod-els and to SGR 1900+14. We find that the QPOs ob-served in SGR 1806-20 can be explained in terms of theeigenfrequencies of the same ℓ within similar deviationseven for different stellar models except for the case with M = 1 . M ⊙ and R = 10 km, while the low-lying QPOsobserved in SGR 1900+14 can be similarly explained interms of the fundamental torsional oscillations with ℓ = 4,8, and 13. As a result, the allowed region of L where theQPO frequencies observed in SGR 1806-20 and in SGR1900+14 are reproducible simultaneously lies in the range
40 80 120 16010100 L [MeV] fr e qu e n c y [ H z ] t t t t t R =12km M =1.4 M ! Fig. 16. (Color online) Comparison of the calculated frequen-cies of torsional oscillations (solid lines) with the QPO fre-quencies observed in SGR 1806-20 (dot-dashed lines), wherethe stellar model adopted in the calculations is
M/M ⊙ = 1 . R = 12 km. The vertical line corresponds to the value of L that is consistent with the observations. From ref. [65]. ≤ L ≤ . . ≤ M/M ⊙ ≤ . ≤ R ≤
14 km. It is interestingto compare this constraint with various experimental con-straints on L [61], which have yet to converge but seem-ingly favor smaller L . Note that if we extend the massrange to, e.g., 1 . ≤ M/M ⊙ ≤ .
8, this constraint wouldbe broader with the lower bound unchanged, while the val-ues of L min in fig. 15 would be unchanged. This is becauseof larger t for smaller M .In principle, there are many other ways of identifyingthe observed QPOs. For example, magnetic oscillationsand magneto-elastic oscillations [75,76] have been alreadyinvoked as such candidates. Our analyses, however, im-ply that as good a reproduction of the observed low-lyingfrequencies as shown in fig. 16 would be desired. It is anopen issue to ask if there could be a new way of reasonableidentification that predicts a lower L . As we have shown by using the macroscopic nuclear model,the parameter L controlling the density dependence of thesymmetry energy is closely related to the size and massof unstable nuclei in laboratories and to the pasta regionand shear modes in neutron star crusts. Ongoing and fu-ture developments of neutron star observatories and RIbeam facilities are thus expected to help determining L sufficiently well. For such determination, systematic errorsinvolved in connecting L with observables would have tobe duly taken into account, no matter whether the macro-scopic nuclear model is used or not. Acknowledgments
We are indebted to our collaborators of the works under-lying the present contribution, namely, B. Abu-Ibrahim,A. Kohama, H. Koura, K. Nakazato, and H. Sotani. Thiswork was supported in part by Grants-in-Aid for Scien-tific Research on Innovative Areas through No. 24105008provided by MEXT.
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