Symmetry energy in nuclei and neutron stars

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Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics A ••• (••••) •••–••• www.elsevier.com/locate/nuclphysa

Symmetry energy in nuclei and neutron stars James M. Lattimer a,b,∗ a Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA b Yukawa Institute of Theoretical Physics, Kyoto 606-8317, Japan

Received 5 April 2014; accepted 8 April 2014

Abstract The symmetry energy plays an important role in nuclear astrophysics, ranging from the structure of nuclei to gravitational collapse to neutron stars. The BBAL paper by Bethe, Brown, et al., was among the first to recognize its importance. The role of the symmetry energy has evolved since that time when it represented one of the major uncertainties in modeling the equation of state of dense nuclear matter. At present, the parameters of the nuclear symmetry energy are tightly constrained by a concordance achieved from nuclear experiment, astrophysical observations, and ab initio theoretical calculations of neutron matter. © 2014 Published by Elsevier B.V. Keywords: Nuclear matter; Neutron matter; Equation of state; Neutron stars; Supernovae

1. Introduction The symmetry energy of nuclear matter can be defined as the difference between the energies of pure neutron and symmetric nuclear matter as a function of density. However, there are alternative definitions in the literature, most usually as the quadratic coefficient of an expansion of the energy in neutron excess. To be definite, we shall write S(n) = E(n, 1/2) − E(n, 0);

S2 (n) =

  1 ∂ 2 E(n, x) 8 ∂x 2 x=1/2

(1)

* Address for correspondence: Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA.

http://dx.doi.org/10.1016/j.nuclphysa.2014.04.008 0375-9474/© 2014 Published by Elsevier B.V.

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where E(n, x) is the energy per baryon of uniform nuclear matter with baryon density n and proton fraction x. In Eq. (1), S(n) is the nuclear symmetry energy and S2 (n) is the symmetry energy to second order in a symmetry expansion. Evaluated at the usual nuclear saturation density, ns  0.16 fm−3 or ρs  2.7 × 1014 g cm−3 , we have S2 (ns ) ≡ Sv where Sv is the volume symmetry energy in the liquid drop model. Because of symmetries in the nuclear force, only even powers of neutron richness, 1 − 2x, appear in the symmetry energy expansion. If higher order terms (i.e., quartic) in a symmetry expansion of E are negligible, then S2 (n)  S(n) and the energy of pure neutron matter is given simply by E(n, 0) = S2 (n) − E(n, 1/2). However, this need not be the case, and recent theoretical investigations of neutron and neutron-rich matter [1] provide evidence to suggest that higher order terms in the symmetry energy expansion are not completely negligible. We shall return to this point in Sections 4 and 5. It is important that the symmetry energy has a density dependence. It is useful to further expand S2 in powers of density near the saturation density:     Ksym n − ns 2 L n − ns S(n)  Sv + + ··· (2) + 3 ns 18 ns Then, assuming higher-than-quadratic terms are ignored, the neutron matter energy at the saturation density is determined by standard nuclear parameters, E(ns , 0) = B + Sv . The L parameter is especially important. For example, if S  S2 , then the pressure of pure neutron matter at the saturation density is p(ns , 0) = Lns /3. The parameters Sv and L can be extracted from nuclear experiments, as will be discussed in Section 3.2. The parameter Ksym is more difficult to determine from experiments and is not well known. Fortunately, it does not seem to play a major role in the determination of the other parameters from experiment and observations, but this remains to be fully checked. The symmetry energy plays an important role in many aspects of nuclear physics and astrophysics. Experimental evidence for it was first suggested by the semi-empirical mass formula of Bethe and von Weizsäcker and Bethe [2], which resembles a liquid drop model, contained a term with a coefficient Sv ∼ 25 MeV, and proportional to AI 2 , that represented the volume symmetry energy decrease in binding with increasing asymmetry I = (N − Z)/(N + Z). Later, Myers and Swiatecki [3] refined the drop model by introducing a surface symmetry term, −Ss A2/3 I 2 , representing the decrease in the total symmetry energy due to the fact that some nucleons are in the nuclear surface at lower densities than the interior. When they further refined their model, formulating the liquid droplet model [4] in order to account for the different distributions of neutrons and protons in the nucleus, and to explain the existence of neutron skins on nuclei, the total symmetry energy became written as   Ss A−1/3 −1 Sv AI 2 1 + . (3) Sv Fits to binding energies suggest that Sv ∼ 30 MeV and Ss ∼ 45 MeV, the decrease in Sv being due to the surface energy. However, the precise values of these coefficients cannot be determined from binding energies alone because the range of A1/3 of laboratory nuclei is too small. At best, a strong correlation between Sv and Ss can be obtained. In Section 3.1 we will formulate a simple description of nuclei beginning from basic principles that is able to explain a wide variety of nuclear phenomena, including binding energies, dipole polarizabilities, and neutron skin thicknesses. Importantly, we can show that these phenomena predict different trends for the correlations between Ss and Sv . Comparing these to data allows a more precise determination of Sv and Ss . Furthermore, this model also allows the determination of a relatively accurate prediction

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of the dependence of Ss on the symmetry parameters Sv and L. Thus, nuclear experiments can be directly connected to the properties of the nuclear force itself, as will be discussed below. The liquid drop model has long been applied to nuclear astrophysics problems such as the equation of state in the crust of neutron stars and in matter encountered during the gravitational collapse of the cores of massive stars leading to supernovae. The state-of-the-art model when the BBAL [5] paper appeared was due to Baym, Bethe and Pethick (BBP) [6]. BBP recognized that nuclei in the dense matter conditions of a neutron star crust would be subject to interactions with both a neutron gas and other nuclei. In particular, the bulk, volume and surface energies would be altered from that of laboratory nuclei. They showed that a liquid drop approach could take these important effects into account as long as the matter inside and outside of nuclei was described with the same energy density functional. BBAL was interested in a somewhat different problem: the changes of nuclear energy resulting from the higher densities of stellar collapse rather than of moving deeper inside a neutron star. Neutrino trapping during stellar collapse prevents the proton fraction of matter from falling as rapidly as in a neutron star interior at the same density. It seems appropriate in Gerry’s memorial volume to reflect on the main arguments of BBAL and to see how well they stand up today. Hans Bethe and Gerry suggested that I apply the liquid drop model used by BBP to the study of gravitational collapse. They already knew what the answer would be, but they wanted a platform to explain it. I was not convinced of their arguments at first, and with some reluctance joined the collaboration. For me it was a fortunate decision that I set aside such youthful arrogance. It is a tribute to Gerry that we will conclude that his nuclear physics intuition, coupled with Hans’ knowledge of thermodynamics, has survived the test of time. 2. BBAL and the EOS BBAL introduced two important rules concerning the state of matter during gravitational collapse. First, the specific entropy of matter stays nearly constant, near the value of 1 in units of Boltzmann’s constant. Second, although the natural tendency is for matter to become more neutron rich as the density increases due to the condition of beta equilibrium, the trapping of neutrinos in collapsing matter counteracts this tendency. In explanation of the first rule, before neutrinos become trapped, beta equilibrium cannot be maintained during collapse, and weak interactions occurring in the matter produce entropy. The deleptonization, or loss of proton fraction due to electron captures on nuclei or free protons [7–9] is thus relatively slow. At the same time, the neutrinos escaping from the collapsing matter reduce entropy, with the result that the entropy remains nearly constant. The second rule stems from the fact that the more the matter goes out of beta equilibrium (the higher its “out-of-whackness”), the greater is the average neutrino energy produced by weak interactions. The entropy lost from escaping neutrinos is thus able to keep up with the entropy gained from being out of beta equilibrium. And the higher neutrino energies ensure that neutrino trapping occurs earlier in the collapse (i.e., at lower densities), since neutrino cross sections scale as neutrino energy squared. Once neutrino trapping sets in, the entropy is frozen. BBAL claimed that their scenario is not sensitive to the details of the liquid drop model. Nevertheless, it is interesting to compare the specific model that was employed with models thought to be realistic today. The fundamental quantities of interest are μ(n, ˆ x) ≡ μn − μp , the difference between the neutron and proton chemical potentials, A(x, n), the nuclear mass number, and the net electron capture rate, all as functions of the proton fraction x and the baryon density n. They assumed during gravitational collapse and before bounce that the baryons are

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largely confined in nuclei, which is the case if x  0.3 and T = 0. In this case, we have the net number of electrons per baryon for charge neutrality to be Ye = x, and the nuclear filling factor u = n/ns where the average baryon density in nuclei is assumed to be ns . In practice, during the latter stages of collapse, x  0.3 and kB T is of order a few MeV, so that the abundances of free nucleons is not zero, but still small. One finds under these conditions that Xn  0.1 and ˆ BT  X . Xp  Xn e−μ/k n 2.1. BBAL’s formulation With baryons in nuclei assumed to dominate the abundances, one has     ∂E(n, x) 1 ∂W (A, x = Z/A, u) μˆ = − − ∂x A ∂x Ye ,T A,T ,u

(4)

where the total energy of a nucleus is determined by the liquid drop semi-empirical mass formula [6] W (A, x, u) = −BA + Sv AI 2 + ωs (x)A2/3 + ωC0 x 2 A5/3 D(u),

(5)

where D(u) = 1 − 3u1/3 /2 + u/2 corrects for the Coulomb interactions among nuclei; it disappears for filling factor u = 1. In this expression, B  16 MeV, Sv  31.25 MeV, and ωC0  0.75 MeV. The surface energy function was assumed by BBAL to be ωs (x) = 16ωs0 x 2 (1 − x)2 , where ωs0 = 290/16  18.1 MeV. It is clear that the assumed surface symmetry coefficient Ss = −(∂ 2 ωs /∂x 2 )x=1/2 /8 = 2ωs0  1.2Sv . The Nuclear Virial Theorem found by BBP ensures that the nuclear surface energy is twice the Coulomb energy; it is the result of optimizing W with respect to A for fixed x and u. Therefore, A=

8ωs0 194 (1 − x)2  (1 − x)2 . ωC0 D(u) D(u)

(6)

It now follows that

  2 1/3 μˆ  4Sv I − 8 ωC0 D(u)ωs0 x(1 − x)1/3 (3 − 5x)  125I − 50x(1 − x)1/3 (3 − 5x)D 1/3 MeV,   14x 35x 2 d μˆ  −250 − 150 1 − + D 1/3 (1 − x)−2/3 MeV. dx 3 9

(7)

Eq. (7) shows that d μ/dx ˆ is relatively insensitive to the surface symmetry energy, since the second term is relatively small (about 1/3 of the first term) during gravitational collapse for ˆ which x  0.4. The mean nuclear mass is more sensitive to Ss than is μ. The electron capture rate in BBAL was based on the Fermi gas model, in which it is assumed that only protons and electrons near their respective Fermi energies will participate. Then it is found that the energy-averaged fraction of electrons and protons that can participate is X=

1 3mb Δ4 , 4 m2e h¯ 2 kF2 μe

(8)

where mb (me ) is the baryon (electron) mass, kF = (3π 2 ns x)1/3 /h¯ is the proton Fermi wavenumber, and Δ = μe − μˆ − Δn is the net energy available for the reaction. Here, Δn ≈ 3 MeV was derived by BBAL to be the mean excitation energy in the daughter nucleus which is needed to find the maximum shell-model strength for the weak interaction. This equation shows the importance

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and relevance of μˆ for the description of matter in gravitational collapse. In beta equilibrium, when neutrinos can freely escape, one has the energy minimization condition μˆ = μe and there would be no energy available for electron captures. During collapse, however, beta equilibrium cannot be maintained on the rapid collapse timescales and μˆ becomes smaller than μe . Eventually, the “out-of-whackness” parameter Δ becomes positive and net electron captures proceed, reducing Ye . Since the electron capture rate is sensitive to Δ, Δ cannot become too large, meaning that μˆ tracks μe , although it remains less than μe . BBAL has an extensive discussion illustrating how d μ/dμ ˆ e , which is initially small, increases and approaches 1 as the collapse proceeds. At densities beyond about 1012 g cm−3 , however, neutrinos cease to freely escape. In this case, they build a degenerate sea, and BBAL showed that the Δ4 factor is replaced by   (9) Δ4 −→ (Δ − μν )2 Δ2 + 2Δμν + 3μ2ν , which rapidly decreases as trapped neutrinos increase in abundance. As net electron captures slow, the “out-of-whackness” parameter, which is now Δ − μν , tends to 0 and beta equilibrium is approached. With neutrino trapping, the lepton fraction YL = Ye + Yν freezes, although Ye tends to continue to slowly fall as Yν increases. At high densities in beta equilibrium, the spin factor of the neutrinos, being half that of electrons, ensures that the ratio Yν /Ye increases with density. 2.2. LPRL’s formulation In contrast to BBAL, LPRL [10] extended the BBP-like liquid drop model to finite temperatures and also took into account the different neutron and proton density distributions in nuclei, as Myers and Swiatecki [4] had done earlier. Notably, this allows the modeling of neutron skins of neutron-rich nuclei. In LPRL’s formulation, the surface tension (which is a thermodynamic potential rather than an energy) is properly treated as being a function of neutron chemical potential. The neutrons in nuclei are separated into bulk neutrons (A − Ns ) and the neutrons in the surface Ns , with the result that the proton fraction in the bulk is no longer x = Z/A but xi = Z/(A − Ns ). Even though the surface tension can be written as a function of xi by ωs (xi ) = ωs0 − Ss (1 − 2xi )2 , it is technically a function of the surface chemical potential μs . The total nuclear energy must take into account that the surface free energy is the surface tension (aka the surface thermodynamic potential) plus a μNs energy contribution. The energy of a nucleus in this model is   W (A, x, u) = −B + Sv (1 − 2xi )2 (A − Ns ) + ωs (μs )A2/3 + μs Ns + ωC0 x 2 A5/3 D(u).

(10)

The two parameters, μs (the surface neutron chemical potential) and the number of surface neutrons Ns , are determined by optimizing the total energy with respect to them:    ∂W ∂ωs 2/3 ∂ωs ∂μs −1 2/3 = A + Ns = A + Ns = 0, ∂μs ∂μs ∂xi ∂xi ∂W = B − Sv (1 − 2xi )2 − 4Sv (1 − 2xi )xi + μs = 0. (11) ∂Ns The second equation tells us that μs = −B + Sv (1 − 4xi2 ). Then the first equation can determine that

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Ns =

Ss 2/3 1 − 2xi A . Sv 2xi

(12)

With the definition of xi = Z/(A − Ns ) one can then show that 1 − 2xi =

1 − 2x I = 1 + Ss A−1/3 /Sv 1 + Ss A−1/3 /Sv

and by substituting our expressions into Eq. (10) that  Sv I 2 W (A, x, u) = −B + A + ωs0 A2/3 + ωC0 x 2 A5/3 D(u). 1 + Ss A−1/3 /Sv

(13)

(14)

Now one sees that the expression for μˆ is different than for BBAL, 4Sv I − 2ωC0 xA2/3 D(u) 1 + Ss A−1/3 /Sv  xi4 1/3 4Sv I − 8 ω ω D(u) (1 − xi )4/3 = C0 s0 x 1 + Ss A−1/3 /Sv

μˆ =

(15)

where we used the fact that the prescription for A remains the same as in BBAL and is given by Eq. (6), and we used the BBAL prescription for the xi dependence of ωs . For Ss A−1/3 /Sv ≈ 1/4, one finds that μˆ  100I

4/3 1/3 xi − 50D (1 − xi )4/3 x 1/3



MeV,



d μˆ 7  −200 − 50D 1/3 1 − x (1 − x)1/3 ; dx 3

(16)

in the second equation we simplified by setting xi = x. Despite the different formulation and specific results, the general relation between μˆ and x follows that of the simpler formulation of BBAL; at x = 1/2 the values of μˆ differ by 15%; at x = 1/3 they differ by 7%. As a result, we conclude that the scenario outlined in BBAL is relatively independent of the nuclear model and the nuclear force parameters. 3. Experimental constraints on the nuclear symmetry energy The symmetry energy changes nucleon distributions in the nucleus, which affects its energy and oscillation frequencies. A transparent way of establishing the liquid droplet model and also visualizing how nuclear observables such as binding energies, dipole resonance energies and neutron skin thicknesses depend on the nuclear force, including its symmetry parameters Sv and L, is through an analytic nuclear model developed by Lipparini and Stringari [11] and generalized by Lattimer and Steiner [12]. 3.1. The nuclear model We assume a simplified Hamiltonian energy density consisting of uniform matter, Coulomb and gradient contributions. Using n = nn + np for the isoscalar density and α = nn − np for the isovector density, we have

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 2 VC dn H = HB (n, α) + np , + Q(n) 2 dr   2   K α n 2 + S2 (n) 1− HB (n, α) = n −B + 18 ns n

7

(17)

where HB is the uniform matter energy density, K is the incompressibility parameter, VC is the Coulomb potential, and Q(n) controls the gradient contributions. If the protons are uniformly distributed for r < R,   Ze2 3 r2 VC = − (18) R 2 2R 2 for r < R and VC = Ze2 /r for r > R. However, a reasonable approximation is to employ Eq. (18) for all r, which also serves to keep the treatment analytic. Where this approximation is inaccurate, the proton density is small or zero, and the influence on the total energy is negligible (cf., Eq. (34)). We will optimize the total energy of a nucleus subject to the constraints



A = nd 3 r, N − Z = αd 3 r, (19) producing the chemical potentials μ and μ: ¯ δ δ [H − μn] = 0, [H − μα] ¯ = 0. δn δα Eq. (20) leads to    d dn ∂Q dn 2 ∂HB 2 = Q − − μ, dr dr ∂n dr ∂n α VC 1 ∂(np VC ) ∂HB − μ¯ = 2S2 − + − μ, ¯ 0= ∂α 4 ∂np n 2

(20)

(21)

implicitly assuming VC ∝ np = (n − α)/2 and also that VC is not a function of n. First, examine the second equation, which tells us that α=

n(2μ¯ + VC ) 4S2

and, together with the second constraint in Eq. (19), that   Sv I 3 Ze2 H2 μ¯ = 2 + 1− , H0 4 R 3H0 where I = (N − Z)/A and we define the dimensionless integrals  

n Sv r i 3 Hi = d r. A S2 R Solving for α, we find    n Sv I Ze2 H2 r2 α= + − 2 . S2 H0 8R H0 R

(22)

(23)

(24)

(25)

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Inclusion of the Coulomb potential results in an increase in asymmetry at the center of the nucleus: protons are pushed outwards by Coulomb repulsion. To make further progress, we assume Q = Q/n. This is not mandatory, but this choice can yield an analytic solution for the density profile. One finds, after multiplying both sides of the first of Eq. (21) by dn/dr and dropping terms proportional to α 2 (justified when I 2  1), that it becomes a perfect differential:       d Q dn 2 Kn n 2 d − μn . (26) −Bn + 1− = dr n dr dr 18 ns Symmetry demands that dn/dr = 0 for r = 0, which gives μ = −B√if n(0) = ns . Integrating Eq. (26) again, one finds n = ns (1 + ex−y )−1 where r = ax and a = 18Q/K. The value of y is determined from the first constraint in Eq. (19):

∞ A=

nd 3 r = 4πns a 3 F2 (y),

(27)

0

where the standard Fermi integral is

∞ Fi (y) = 0

   x i dx y 1+i i(i + 1) π 2  + ··· . 1+ 1 + ex−y 1+i 6 y

(28)

This converges rapidly since y  ro A1/3 /a  13 for 208 Pb, where 4πns ro3 = 3. In reality, we expect the gradient contributions to H to have isospin dependence; this plus the assumptions made about the Coulomb potential and dropping terms proportional to α 2 alters the density profile, the density at r = 0, and the condition μ = −B. However, in the limit I → 0 these modifications are small; for 56 Ni near the center, one has α(0)/ns  0.015 using Eq. (25) so that corrections to n(0) and μ are small. The parameter K  240 MeV from experiment, and Q follows from the observed 90–10 surface thickness value:

0.9 t90–10 = a

dn = 4a ln 3  2.3 fm, dn/dr

(29)

0.1

yielding a = 0.523 fm and   K t90–10 2 Q=  3.65 MeV fm2 . 18 4 ln 3

(30)

As a check, the surface energy parameter is the semi-infinite, symmetric matter, surface thermodynamic potential:

∞ 

∞ ωs0 = lim

α→0

4πro2 −∞

[H − μn − μα]dz ¯ = 8πQro2 −∞

dn dr

2

dz 3Q  18.3 MeV. (31) = n aro

This is close to the accepted value [4] and exactly the value used by BBAL. The adopted energy density functional therefore fits the most important observed properties of the symmetric matter nuclear interface, its tension and thickness, as well as the observed nuclear incompressibility.

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Although Lipparini and Stringari assumed S2 (n) = Sv + L(n − ns )/(3ns ), this simplification is inaccurate, and, furthermore, is not necessary for the model’s analyticity. If S2 (n) can be  i −1 in the domain 0 < (n) = S ( b (n/n represented by the convergent series expansion S 2 v s) ) i i  n < ns , with i bi = 1, the integrals Hi can be expanded in Fermi integrals and also, therefore, in powers of 1/y: Hi =

 4πns a 3+i  3 3T F2+i (y) − (2 + i)T F1+i (y) + · · ·  − + ···, i A R 3+i y

(32)

keeping only the leading terms in y ∝ A1/3 . In turn, T is given for all integers i ≥ 0 by the expansion [12] T = b1 + 3b2 /2 + 11b3 /6 + 25b4 /12 + 137b5 /60 + · · ·

(33)

This model can be directly compared to the liquid droplet model. The total symmetry and Coulomb energies are



α2 1 np VC d 3 r Esym + EC = S2 d 3 r + n 2    Sv 3Z 2 e2 AI 5H2 + 1+ 1− = AI 2 . (34) H0 5R 8Z 3H0 The first term, representing the symmetry energy in the absence of the Coulomb potential, is identical to that in the liquid droplet model (Eq. (14)) provided we make the association Ss 3A1/3 3a =− T =− T. Sv y ro

(35)

We now have the additional important result that Eqs. (33) and (35) allow one to compute the surface symmetry parameter directly in terms of the nuclear force parameters. If the Lipparini and Stringari ansatz for S2 is used, the values b0 = 1 + p and b1 = −p, where p = L/(3Sv ), are found by expanding around n = ns . In this case, T = −p and Ss = aL/ro . In practice, this provides a poor fit to realistic calculations. However, keeping an additional term in the expansion S2 (n)  Sv + L(n − ns )/(3ns ) + Ksym (n/ns − 1)2 /18 + · · · ,

(36)

and setting Ksym = 6L − 18Sv , so that S2 (0) = 0, provides an improved result. In this case, we find b0 = 2 + p 2 , b1 = −2 + p − 2p 2 , b2 = 1 − p + p 2 ,     Ss /Sv = (3a/2ro ) 1 + p + p 2 . T = − 1 + p + p2 /2,

(37)

This provides a reasonable approximation to better calculations (see Ref. [13]). BBAL estimated Ss /Sv  1.17, which implies p  0.48 or L = 45 MeV. Interestingly, this is centered in the current experimentally-determined range. Irrespective of the form of S2 , the symmetry and Coulomb energies become  AI 2 Sv2 3Z 2 e2 AI Ss Esym + EC = + 1 − , (38) 5R 12Z Sv A1/3 + Ss Sv + Ss A−1/3 showing explicitly the reduction of Coulomb energy due to polarization effects.

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The dipole static polarizability, αD , hereafter referred to simply as the dipole polarizability, is found by performing the constrained variation [11]  

δ 3 3 Hd r − zαd r = 0, (39) δα with a small parameter. Defining αd as the function α(r) which solves Eq. (39), the dipole polarizability is

1 zαd d 3 r. αD = (40) 2

The solutions for αd and the dipole polarizability are  

z + VC H2 AR 2 AR 2 5 Ss αd = n , αD =  1+ , 2S2 12Sv 20Sv 3 Sv A1/3

(41)

where z2 = r 2 /3 within the integral (40). Note that the contribution from VC vanishes because of symmetry. αD is also related to the m−1 sum rule by αD = 4m−1 ; for 208 Pb, m−1  6.9 fm2 MeV−1 . The mean-square radii are   2 

H2 1 R2 3 Ze2 H2 2 2 3 (n ± α)r d r = rn,p = ± − H4 , ±I (42) 2(N, Z) 1±I 5 H0 8ARSv H0 where I = (N − Z)/A. To lowest order, the root mean square radius difference, i.e., the neutron skin thickness, is then    3 Ss 2ro (1 − I 2 )−1/2 3Ze2 10 Ss rnp  − I 1 + . (43) 3 Sv A1/3 3(1 + Ss A−1/3 /Sv ) 5 Sv 140ro Sv The Coulomb potential clearly reduces the neutron skin thickness by pushing protons towards the surface. There is significant competition between volume and surface contributions in the symmetry energy, dipole polarizability and neutron skin thickness of nuclei. As the volume-to-surface ratio scales as A1/3 , which only has an effective range between 3 and 6 in heavy nuclei, no single experiment can hope to sort out the individual contributions and thereby determine Sv and Ss , or equivalently, Sv and L, uniquely. Each case will, however, result in a correlation between these parameters with a different slope dL/dSv . It is straightforward to estimate these slopes from the model:     L 2R Ss dL I (1 + 2p)−1  16.7  + 1+ dSv mass Sv a Sv A1/3 I + Ze2 /(20RSv )     dL L 2R 3 Ss  + + (1 + 2p)−1  13.7 dSv dipole Sv a 5 Sv A1/3        L Ze2 3 Ss Ss Ze2 −1 dL −1  − + 1 + (1 + 2p) I − dSv skin Sv 7Sv a 10 Sv A1/3 20RSv Sv A1/3  −1.0, (44) where we used L/Sv  3/2, Ss A−1/3 /Sv  0.35 (see Section 5) and R  6.7 fm (i.e., Pb208 ). The general trends are insensitive to the assumed symmetry energy density dependence. Since

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the slopes of these correlations are different, these experiments, taken collectively, offer the hope of constraining symmetry parameters tightly. The neutron skin is especially important, as it is essentially orthogonal to the other two constraints. 3.2. The experimental constraints We first examine the correlation determined by nuclear mass measurements. We initially consider the simple liquid drop model, Eq. (5) in Section 2.1, and ignore shell and pairing effects. 2/3 The net symmetry energy of an isolated nucleus is then EDM,i = Ii2 (Sv Ai − Ss Ai ). The parameters are typically determined by a least-squares fit to measured masses, i.e., minimizing the differences between model predictions and experimentally measured symmetry energies,

2 χ2 = (Eexp,i − EDM,i )2 /σDM , χ¯ 2 ≡ χ 2 /N (45) i

where N is the total number of nuclei and σDM is a nominal error. A χ¯ 2 contour one unit above the minimum value represents the 1σ confidence interval which is an ellipse in this linear example. The properties of the confidence ellipse are determined by the second derivatives of χ¯ 2 at the minimum,

 5/3 4/3  2  61.6, −10.7, 1.87, (46) [χ¯ vv , χ¯ vs , χ¯ ss ]σDM = 2N −1 Ii4 A2i , −Ai , Ai i

∂ 2 χ¯ 2 /∂S

where χ¯ vs = v ∂Ss , etc. The specific values quoted follow from the set of 2336 nuclei with N and Z greater than 40 from Ref. [14]. The confidence ellipse in Ss –Sv space has orientation αDM  = −(1/2) tan−1 |2χ¯ vs /(χ¯ vv − χ¯ ss )| 9.8◦ with respect to the Ss axis, with error widths σv,DM = (χ¯ −1 )vv  2.3σDM and σs,DM = (χ√ ¯ −1 )ss  13.2σDM where (χ¯ −1 ) is the matrix inverse. The correlation coefficient is rDM = χ¯ vs / χ¯ vv χ¯ ss  0.997. In this example, the shape and orientation of the confidence interval depend only on Ai and Ii and not on the binding energies themselves or the location of the best fit. This correlation is therefore largely model-independent and the most valuable of constraints from nuclear experiment. It is straightforward to see that the droplet model energy, Eq. (6), will give a very similar correlation. In this case, however, there will be a small dependence of the second derivatives χ on the position of the minimum as well as the measured energies, in contrast to the liquid drop case. We now summarize correlations between L and Sv based on experiment. This summary is largely taken from Lattimer and Lim [13]. The correlation between L and Sv for measured nuclear masses is taken from Hartree–Fock calculations with the UNEDF0 density functional [17], and the nominal fitting error to be σ = 1 MeV. Ref. [17] chose σ = 2 MeV, but in all likelihood, this value is an overestimate, as the confidence ellipse would then include negative values for L, which imply negative neutron matter pressures. Importantly, the shape and orientation of this confidence ellipse are rather steep in L–Sv space (Fig. 1), roughly in line with the predictions of Eq. (44). Danielewicz [18], on the other hand, also derived an expression for the change in nuclear energy due to polarization, but it has the opposite sign and twice the magnitude of the relevant term in Eq. (38). This accounts for the discrepancy in the position and slope of the correlation between Ss and Sv from Ref. [18] relative to the liquid droplet model and results of Thomas–Fermi and Hartree–Fock simulations. The constraints for rnp for 208 Pb in Fig. 1 were taken Chen et al. [19] who converted the experimental results for Sn isotopes into an equivalent value for 208 Pb: rnp  (0.175 ± 0.020) fm.

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Fig. 1. Experimental constraints for symmetry energy parameters, adapted and revised from [13]. See the text for further discussion. G and H refer to the neutron matter studies of Gandolfi et al. [15] and Hebeler et al. [16], respectively.

They also established, from a study of Skyrme Hartree–Fock calculations of 208 Pb in which Sv and L were systematically varied, rnp 7.2028Sv 2.3107L 8.8453Sv2 47.837Sv L 4.003L2 − + . (47)  −0.094669 + + − fm GeV GeV GeV2 GeV2 GeV2 With the aforementioned value for rnp , one obtains d ln L/d ln Sv  −3.75, assuming Sv = 31 MeV and L = 45 MeV. This is steeper than predicted by Eq. (44), but is sensitive to the assumed form of S2 (n). Similarly, the constraint for the electric dipole polarizability αD of 208 Pb is taken from data produced by Tamii et al. [20]: αD  (20.1 ± 0.6) fm3 . Roca-Maza et al. [21] showed, from studies with a series of relativistic and non-relativistic interactions, that the dipole polarizability, bulk symmetry parameter, and the neutron skin thickness for 208 Pb can be constrained by αD Sv  (325 ± 14) + (1799 ± 70)(rnp /fm) MeV fm3 .

(48)

Eq. (47) can be used to convert this into the Sv –L correlation shown in Fig. 1. Although Eq. (48) is not the same as Eq. (41), they predict similar slopes which are smaller than that predicted for nuclear masses.1 The rest of the constraints shown in Fig. 1 are discussed further in Ref. [13]. The white region in Fig. 1 represents the consensus agreement of the five experimental constraints shown in 1 The slope of this correlation is different than shown in Ref. [13], which relied on the erroneous result α ∝ r np D derived in Ref. [22].

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the figure, giving a range 44 MeV < L < 66 MeV.2 Since the model dependencies of these constraints have not been thoroughly explored, this may well be underestimated. If the white region is treated as a 68% confidence interval for the experimental determination of Sv and L, it can be used with Monte Carlo sampling to determine a distribution of values for these parameters. This will be useful to check the neutron matter and astrophysical implications. 4. Neutron matter studies Theoretical studies of neutron matter can also shed light on the symmetry energy parameters. As discussed in Section 1, the neutron matter energy at ns is given by B + Sv if higher than quadratic terms in the symmetry energy isospin expansion are ignored. Furthermore, in the same approximation, the neutron matter pressure at ns is Lns /3. However, until recently, most theoretical calculations of neutron matter, although employing scattering phase shift information to high energies, were unable to be extrapolated with reasonable uncertainty to such a high density. Two recent studies of pure neutron matter using realistic two- and three-nucleon interactions coupled with low-energy scattering phase shift data have changed the picture. The first employed chiral Lagrangian methods [16] and the second quantum Monte Carlo techniques [15]. With the important assumption that higher-than-quadratic isospin terms in Eq. (1) are ignored, the values of the neutron matter energy and pressure at ns from these calculations provide direct estimates of Sv and L. Including their estimated energy error ranges, the allowed ranges of these parameters from fitting to a simple effective interaction [16]     E(n, x) 3  5/3 α α 5/3 2/3 2 (2u) − u = x + (1 − x) + αL − (1 − 2x) T0 5 2 2    η η + ηL − (1 − 2x)2 , + uγ 2 2     1  5/3 α p(n, x) 5/3 5/3 2 α 2 (2u) − u = x + (1 − x) + αL − (1 − 2x) n s T0 5 2 2    η η + γ u1+γ + ηL − (1 − 2x)2 , (49) 2 2 are shown as blue regions in Fig. 1. Symmetry contributions originate from the kinetic energy of non-interacting nucleons and from the interaction potential. In Eq. (49), T0 = (3π 2 ns /2)2/3 h¯ 2 /(2mb ) = 36.84 MeV, u = n/ns , and α, αL , η, ηL and γ are parameters. For the properties of symmetric matter corresponding to E(ns , 1/2) = −16 MeV, p(ns , 1/2) = 0, and K(ns , 1/2) = 236 MeV, where K = 9dp/dn is the incompressibility, one finds γ = 1.32, α = 6.00 and η = 3.93. There are small quartic contributions to the symmetry energy stemming from the kinetic energy, but they are small, amounting to T0 /81 at the saturation density. It is very interesting that the theoretical estimates are mutually consistent, and moreover, apparently consistent with nuclear experiments. Nevertheless, the small displacement of the neutron matter from the consensus experimental region may represent the effects of neglecting quartic or higher-order terms in the effective interaction. 2 The consensus region in Fig. 1 is slightly smaller than given in Ref. [13] because of the additional constraint from isobaric analog states.

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Shifting the neutron matter bands up, and in the case of Ref. [15], to the left, would bring them into closer agreement with the experimentally-allowed (white) polygon. In the case of the neutron matter study of [16] ([15]), shifts of ΔSv  +0.1(−1.3) MeV and ΔL  +10.5(+6.0) MeV are indicated. These shifts are much larger than those suggested by quartic contributions to the kinetic energies and could be accommodated with additional quartic terms in the potential part of the effective interaction, e.g., by adding T0 (−βu + δuγ )(1 − 2x)4 to the energy and T0 ns (−βu2 + δγ uγ +1 )(1 − 2x)4 to the pressure in Eq. (49). This does not change the values of α, η, γ , Sv or L since they are fit to symmetric matter, but adds the energy T0 (−β + δ) to the energy and ns T0 (−β + γ δ) to the pressure of pure neutron matter at ns . Thus the neutron matter and experimental regions can be made to overlap in the L–Sv plane if 3γ ΔSv − ΔL = −0.28(−0.24), 3T0 (γ − 1) 3ΔSv − ΔL − 0.27(−0.30). δ= 3T0 (γ − 1)

β=

(50)

5. Neutron-rich matter There is no experimental information that sheds light on the existence of quartic contributions. Nuclei are so symmetric that quartic contributions to the energy, neutron skin, or vibrational frequencies are much smaller than experimental uncertainties. It is therefore of great interest that, recently, the chiral Lagrangian neutron matter studies [16] have been expanded by Drischler, Soma and Schwenk [1] to include matter with small additions of protons, which we will refer to as “neutron-rich” matter. This study computed uncertainty bands for the energy per baryon for densities up to about 1.2ns and for proton fractions in the range 0.0 < x < 0.15. From these calculations, it is possible to directly deduce quartic contributions to the symmetry energy [23]. We first examine how well an effective interaction without quartic contributions to the potential energy, Eq. (49), and with parameters fit to symmetric and pure neutron matter, will fit neutron-rich matter. We establish the parameters α, η and γ from the properties of symmetric matter at ns as above. We fix the values of αL = 1.416 ± 0.015 and ηL = 0.916 ± 0.016 so that the mean (the average of the maximum and minimum) energies of neutron matter are fitted from 0.05ns to 1.3ns using a least squares method. This is straightforward because these parameters enter linearly into the effective interaction. We minimized the χ 2 parameter χ 2 = N −1

(Et,i − Em,i )2 i

σi2

,

(51)

where i is the index for each density u in the results provided by Ref. [1]. The theoretical energies from Ref. [1] are Et,i and the model energies from Eq. (52) are Em,i , and σi = f u2i (Ehigh,i − Elow,i )−2 , where f is an adjustable parameter and ‘high’ and ‘low’ refer to the upper and lower limits given for the theoretical energies. The choice of σi reflects our finding that it is more important for a global fit to ascribe larger weights to the energies at larger densities than at smaller densities. The value of f , which determines the magnitudes of the uncertainties in the force parameters, is established by requiring that 90% of the model energies chosen from Monte Carlo trials from assumed Gaussian distributions with these uncertainties lie within the black solid curves in the left panel of Fig. 2 for neutron matter (x = 0) which outline the theoretical uncertainty band. For this fit, one finds Sv = 32.0 ± 0.8 MeV and L = 46.1 ± 2.8.

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Fig. 2. Solid lines are minimum and maximum neutron-rich matter energies from x = 0.0 to 0.15 from Ref. [1]. Left: dashed curves are computed from Eq. (49) using parameters derived from neutron matter and symmetric matter properties at ns . Right: dashed curves are computed from Eq. (52) using parameters derived from symmetric, neutron and neutron-rich matter.

The fit to pure neutron matter (x = 0) is adequate, although the low-density energies are slightly overestimated. But the fits to neutron-rich matter, notably for the upper edge of the theoretical energy uncertainty band, are increasingly underestimated with increasing x at ns . The observed linear shifts with x could be due to non-zero quartic contributions to the potential part of the symmetry energy. The effective interaction Eq. (49) does not contain effective nucleon masses, which contributes to the overestimate of the neutron matter energy at low densities [1]. Symmetric matter at saturation density is believed to have effective neutron and proton masses about 0.7 times the bare nucleon mass. We therefore introduce effective masses into the effective interaction in the identical way they are introduced in Skyrme-type potential models. In addition, we also include quartic contributions to the effective potential with the parameters β and δ.     E(n, x) 3 = (2u)2/3 x 5/3 + (1 − x)5/3 (1 + bu) + (a − b)u x 8/3 + (1 − x)8/3 T0 5    α α 2 4 −u + αL − (1 − 2x) + β(1 − 2x) 2 2    η γ η 2 4 +u + ηL − (1 − 2x) + δ(1 − 2x) , 2 2        5 1 5 p(n, x) = (2u)5/3 x 5/3 + (1 − x)5/3 1 + bu + (a − b)u x 8/3 + (1 − x)8/3 n s T0 5 2 2

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  α α + αL − (1 − 2x)2 + β(1 − 2x)4 2 2    η η + ηL − (1 − 2x)2 + δ(1 − 2x)4 . + γ u1+γ 2 2 

− u2

(52)

At ns , the symmetric matter effective mass is m∗ = m/[1 + (a + b)/2] while that of the neutron [proton] in pure neutron matter is m∗n0 = m/(1 + a) [m∗p0 = m/(1 + b)]. We find that fits to low-density energies are improved for small negative values of a. However, as the effective mass term contributes to the energy with the highest power of density, 5/3, negative values of a lead to instabilities in symmetric and neutron matter at high densities. We therefore set a = 0.3 Setting m∗ = m/1.4, with a = 0 and b = 0.8, we obtain γ = 1.207, α = 7.135 and η = 4.586. The remaining parameters in Eq. (52) which best reproduce the energy and pressure of neutron-rich matter with x = 0.05, using the same least squares fitting method used to fit pure neutron matter, are αL = 2.349 ± 0.077, ηL = 1.967 ± 0.090, β = −0.486 ± 0.077 and δ = −0.603 ± 0.090. The model energies predicted by Eq. (52) with effective mass and quartic contributions are plotted in the right panel of Fig. 2. It is interesting that the quartic parameters β and δ determined by this methodology are similar to, but somewhat larger than, those implied by shifting the blue regions in Fig. 1 to overlap with the experimentally-preferred region. These quartic contributions lead to predicted values for Sv and L of about 4 MeV and 20 MeV, respectively, larger than what was obtained by Ref. [16] from fitting the energies of pure neutron matter alone. Nevertheless, the uncertainties in the predicted values of Sv and L, originating from the uncertainties in the predicted energies of pure neutron and neutron-rich matter of Ref. [1], are relatively large. We find Sv = 35.3 ± 4.4 MeV, L = 66.5 ± 14.9 MeV, S4v = −3.9 ± 4.4 MeV and L4 = −25.8 ± 14.6 MeV, where    1 ∂ 2 E(u, x) η 1 + 2a − b α Sv ≡ = T − + − α + η 0 L L , 8 3 2 2 ∂x 2 u=1,x=1/2        η α 2 + 10a − 5b 3 ∂ 3 E(u, x) = T − + 3 − α , + 3γ η L≡ 0 L L 8 3 2 2 ∂u∂x 2 u=1,x=1/2    1 ∂ 4 E(u, x) 1 − a + 2b = T + δ − β , S4v ≡ 0 384 81 ∂x 4 u=1,x=1/2    1 ∂ 5 E(u, x) 2 − 5a + 10b = T0 + 3(γ δ − β) . (53) L4v ≡ 128 81 ∂u∂x 4 u=1,x=1/2 S4v and L4v are the coefficients of the quartic term in the symmetry energy and pressure evaluated at ns . Note that the neutron matter energy and pressure of neutron matter at the saturation density, with quartic contributions, are E(ns , 0)  B + Sv + S4v ,

p(ns , 0) 

L + L4v . 3ns

(54)

The symmetry parameters are highly inter-correlated [23]. The optimum values for Sv + Sv4 and L + L4 are approximately equal to those, respectively, of Sv and L when quartic contributions are neglected. 3 This trend was also noticed by Ref. [1], who suggested that m∗ in neutron matter is greater than unity at low densities, n i.e., a < 0.

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Fig. 3. Comparison of correlations predicted between Sv and S4v from liquid droplet model fits to nuclear masses (yellow confidence ellipse) and from neutron matter calculations from Ref. [1] (dots). The red diamonds mark the optimum fits for droplet model fits and for the fits to neutron-rich matter of Ref. [1].

The final range for Sv determined from fitting neutron-rich matter is 30.9 to 39.7 MeV, which overlaps the values obtained without quartic contributions. This large uncertainty can be reduced, however, by considering nuclear binding energies. A quartic contribution to the liquid droplet energy, Sv4 AI 4 , will lead to a significant correlation between Sv and S4v . To estimate the shape and slope of the confidence ellipse for Sv and L, we note that [χ22 , χ24 , χ44 ]σ 2 =

 2 2 4 2 6 Ai Ii Di , Ii Di , Ii8  [34.0, 2.01, 0.126], N

(55)

i

−1/3

where Di = (1 + Ss Ai /Sv )−1 and χij = ∂ 2 χ 2 /∂Siv ∂Sj v (S2v ≡ Sv ). The specific values follow from the set of N = 2336 nuclei with N and Z greater than 40 included in the mass table of Ref. [14] for Ss /Sv = 2.1. The confidence ellipse has the orientation αm = (1/2) tan−1 [2χ24 /(χ44 − χ22 )]  −3.4◦ with respect to the S4v axis, with error widths σi =  (χ −1 )ii , where (χ −1 ) is the inverse matrix of χ . These are σ2  1.04σ and σ4  17.4σ . The √ degree of parameter correlation in this model is rm = χ24 / χ22 χ44  0.973. Nuclear binding energies thus predict that the parameters Sv and S4v are highly correlated, and that the confidence ellipse is nearly vertical in the Sv –S4v plane. Note that the shape and orientation of the confidence ellipse, in this simplified model, largely depend on the masses and charges of nuclei with only a weak dependence on individual binding energies and the location of the energy minimum, just as was found for the properties of the Sv –Ss correlation [13]. The simple droplet model of Eq. (3) is not satisfactory, however, for constraining quadratic and quartic symmetry parameters since its predicted energy minimum occurs far from the minimum location predicted by Hartree–Fock or Thomas–Fermi nuclear model fits. This can be traced to the absence of shell, pairing, and Wigner terms, as well as Coulomb polarization, exchange and diffuseness corrections. When these are included together with a quartic symmetry energy term, the energy minimum occurs at Sv = 32.8 MeV, Ss /Sv = 2.2, and S4v = −13.85 MeV,

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with χ 2 = 1.286σ 2 (Fig. 3).4 But the shape and orientation of the confidence ellipse is nearly identical to that found for the simple droplet model. With no quartic term, the minimum occurs at Sv = 30.7 MeV with χ 2 = 1.482σ 2 . Fig. 3 compares correlations between Sv and S4v from binding energies and neutron-rich matter. The size of the confidence ellipse for the binding energy fit is arbitrary, so we equate it’s Sv error width to that shown in Fig. 1. Although the binding energy correlation itself could be consistent with large quartic contributions to the symmetry energy of either sign, this is incompatible with neutron-rich matter calculations. Mutually consistency between binding energy and neutron-rich matter instead suggests that quartic contributions to S4v nearly vanish. This result, however, does not mean that quartic contributions are negligible; it simply means they nearly cancel in the energy at the saturation density, or β  δ. L4v , however, is substantially negative. More precisely, the parameter values of the points satisfying the neutron-rich correlation and also falling within the binding energy correlation ellipse are Sv = 32.1 ± 0.6 MeV, L = 55.8 ± 2.5, S4v = −0.8 ± 1.0 MeV, L4 = −15.5 ± 3.7 MeV, β = −0.52 ± 0.06, δ = −0.56 ± 0.06, αL = 2.39 ± 0.06, and ηL = 1.92 ± 0.06, where the uncertainties reflect one standard deviation. This shifts the blue regions higher but still overlapping the experimentallyallowed region in Fig. 1. The major effects for astrophysics will be a shift in the core-crust transition, described in the next section. 6. Astrophysics considerations Fixing the symmetry parameters has the greatest significance for the estimation of the neutron star radius. As summarized in Ref. [12], observations of thermal emissions from both bursting and quiescent sources in low-mass X-ray binaries, the neutron star mass–radius relation has been reasonably constrained. Lattimer and Prakash [24] determined that a tight correlation exists between the pressure of neutron star matter and the radii of neutron stars far from the maximum mass. In particular, the radius of a 1.4M star is found to be [13]  1/4 pβ (ns ) R1.4 = (9.52 ± 0.49) km, MeV fm−3

(56)

where pβ is the cold pressure of beta-equilibrium matter. To leading order, and including quartic terms, pβ (ns ) 

  (L + L4v )ns 4Sv + 8S4v 3 3Sv + 6S4v − 4L − 8L4v + ···. + 3 h¯ c 9π 2

(57)

The astrophysical mass and radius constraints without quartic contributions translate into the band shown in Fig. 1. Using our preferred values for S4v and L4v , and adjusting Sv and L accordingly, we find a pressure increase of only 0.07%. Therefore, the concordance among nuclear experiment, neutron matter theory, and astrophysical observations demonstrated, for example by Ref. [13], remains in place. A larger effect will occur for the core-crust phase boundary, which can be estimated from the solution of [6,25] 4 This higher value of S /S compared to the nuclear model prediction from Eq. (37) with p = 1/2, shows that the s v assumed S2 (n) is suspect.

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μpp −

μ2np μnn

  1/3 + 4 πe2 β0 − 4α0 β0 9πn2p = 0,

where μij = ∂μi /∂nj , α0 = 1/137 is the fine structure constant, and    μnp μnp 2 β0  100 1 − 4 + MeV fm5 . μnn μnn

19

(58)

(59)

It is found that the transition density and pressure are nt /ns = 0.435 (0.52) MeV fm−3 with (without) quartic contributions. Therefore, the predicted quartic contributions will moderately lessen the fractions of mass and moment of inertia within the crust of a neutron star. 7. Concluding remarks In retrospect, the nuclear equation of state employed by BBAL is remarkably similar to the best estimates now available, despite the fact that 35 years ago the uncertainties in the relevant parameters were significant. It is therefore not surprising that its general description of collapsing matter preliminary to a supernova explosion is still applicable. Acknowledgements This research was supported in part by the U.S. DOE under grant DE-AC02-87ER40317 and by the Yukawa Institute of Theoretical Physics, and I thank its staff and faculty for their hospitality. I wish to express my deepest appreciation to Gerry for his many insights, for providing me with the opportunity to work with Hans Bethe, and for convincing me to come to Stony Brook. But it is our friendship that I will remember most of all. His influence will remain among us for eons and eons.5 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

C. Drischler, V. Somá, A. Schwenk, Phys. Rev. C 89 (2014) 025806. C.F. von Weizsäcker, Zeit. für Phys. 96 (1935) 431. W. Myers, W.J. Swiatecki, Nucl. Phys. 81 (1966) 1. W.M. Myers, W.J. Swiatecki, Ann. Phys. 55 (1969) 395. H.A. Bethe, G.E. Brown, J. Applegate, J.M. Lattimer, Nucl. Phys. A 324 (1979) 487. G.A. Baym, H.A. Bethe, C.J. Pethick, Nucl. Phys. A 175 (1971) 225. A. Heger, K. Langanke, G. Martinez-Pinedo, S.E. Woosley, Phys. Rev. Lett. 86 (2001) 1678. K. Langanke, et al., Phys. Rev. Lett. 90 (2003) 241102. H.-Th. Janka, K. Langanke, A. Marek, G. Martinez-Pinedo, B. Müller, Phys. Rep. 442 (2007) 38. J.M. Lattimer, C.J. Pethick, D.G. Ravenhall, D.Q. Lamb, Nucl. Phys. A 432 (1985) 646. E. Lipparini, S. Stringari, Phys. Rep. 103 (1989) L175. J.M. Lattimer, A.W. Steiner, Astrophys. J. 784 (2014) 123. J.M. Lattimer, Y. Lim, Astrophys. J. 771 (2013) 51. G. Audi, A.H. Wapstra, C. Thibault, Nucl. Phys. A 729 (2003) 337.

5 The opening line in the original draft of BBAL was “Massive stars live for eons and eons, developing peacefully as heavier and heavier nuclear fuels are burned.” The phrase eons and eons was eventually replaced, at my insistence, by ∼ 107 yr. A geological or astronomical eon is, of course, 1 Gyr. And according to the dictionary, the more common usage of eon is for any long, indefinite, period. Perhaps this is a reason I was hesitant to become associated with the manuscript after an initial reading.

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[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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S. Gandolfi, J. Carlson, S. Reddy, Phys. Rev. C 85 (2012) 032801. K. Hebeler, J.M. Lattimer, C.J. Pethick, A. Schwenk, Phys. Rev. Lett. 105 (2010) 161102. M. Kortelainen, et al., Phys. Rev. C 82 (2010) 024313. P. Danielewicz, Nucl. Phys. A 727 (2003) 233. L.-W. Chen, C.M. Ko, B.-A. Li, J. Xu, Phys. Rev. C 82 (2010) 024321. A. Tamii, et al., Phys. Rev. Lett. 107 (2011) 062502. X. Roca-Maza, et al., Phys. Rev. C 88 (2013) 4316. P.-G. Reinhard, W. Nazarewicz, Phys. Rev. C 81 (2010) 051303. J.M. Lattimer, 2014, in preparation. J.M. Lattimer, M. Prakash, Astrophys. J. 550 (2001) 426. K. Hebeler, J.M. Lattimer, C.J. Pethick, A. Schwenk, Astrophys. J. 773 (2013) 11.