Physical properties of hot, dense matter: The general case

Nuclear Physics @ North-Holland

A432 (1985) 646-742 Publishing Company

PHYSICAL

PROPERTIES OF HOT, DENSE The general case+ J.M.

Department

MATTER:

LA’ITIMER*

of Earth and Space Sciences, State University of New York, Stony Brook, NY 11794, USA C.J. PETHICK

Department

of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

1 I10 West Green Street,

and Nordita,

Blegdamsvej

17, DK-2100

D.G. Department

Copenhagen 0, Denmark

RAVENHALL

of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

1110 West Green Street,

and D.Q. Harvard-Smithsonian

Center for Astrophysics, Received

LAMB 60 Garden Street, Cambridge, 14 March

MA 02138, USA

1984

Abstract: The thermodynamic properties of hot, dense matter are examined in the density range 10m5 fmw3 6 n s 0.35 frne3 and the temperature range 0 s Ts 21 MeV, for fixed lepton fractions Yf = 0.4,0.3 and 0.2 and for matter in P-equilibrium with no neutrinos. Three phases of the matter are considered: the nuclei phase is assumed to consist of Wigner-Seitz cells with central nuclei surrounded by a nucleon vapor containing also a-particles; in the bubbles phase the cell contains a central spherical bubble of nucleon vapor surrounded by dense nuclear matter; the third phase is that of uniform nuclear matter. All are immersed in a sea of leptons (electrons and neutrinos) and photons. The nuclei and bubbles are described by a,compressible liquid drop model which is surface, Coulomb energies self-consistent in the sense that all of the constituent properties -bulk, and other minor contributions - are calculated from the same nuclear effective hamiltonian, in this case the Skyrme 1’ interaction. The temperature dependence of all of these energies is included, for bulk and surface energies by direct calculation, for the Coulomb energy by combining in a plausible way the usual electrostatic energy and the numerical results pertaining to a hot Coulomb plasma. Lattice contributions to the Coulomb energy are an essential ingredient, and lattice modifications to the nuclear translational energy are included. A term is constructed to allow also for the reduced density of excited states of light nuclei. All of these modifications incorporate necessary physical effects which modify significantly the matter properties in some regions.

’ Supported in part by US National Science Foundation grants NSF PHY82-01948 and NSF PHYIO25605, by the US Department of Energy grant DOE AC02-80ER-10712, and by the National Aeronautics and Space Administration grant NAGW-246. * Alfred P. Sloan Foundation Fellow. 646

J.M. Lattimer et al. / Hot, dense matter

647

The equilibrium conditions obeyed by the constituents of the model are obtained in analytic form, and the resulting simultaneous equations are solved numerically. Results are presented in tabular form, and also as contour plots in the (n, T) plane. Quantities plotted are the fraction of baryons in nuclei and in a-particles; the electron fraction (for a given lepton fraction): the nuclear mass and charge numbers; the internal energy and the entropy per baryon; the total pressure; the adiabatic index; the chemical potentials of the neutron and of the neutron-proton difference; and the phase boundaries. The adiabatic index along adiabats is also plotted as a function of n. Noteworthy features of the plots are discussed and justified in terms of simple physical models and arguments. A comparison is made with other comparable calculations and some notable differences are revealed; none of the other calculations includes all of the necessary physical effects described here.

1. Introduction

This paper concerns the thermodynamic properties of matter under extreme conditions. Such information is needed in the quite different physical contexts of heavy-ion collisions and stellar collapse. It must be obtained as a theoretical extrapolation from nuclear models based on real nuclei, since the conditions involved are far from those under, which the properties of nuclei are measured in the laboratory. The particular region of densities we wish to address ourselves to here is that in which nuclear properties are modified significantly by the environmental temperature, pressure and the surrounding leptons, but in which nuclear clustering still occurs. We consider temperatures up to the highest critical temperature. A recent review ‘) summarizes a body of previous work related to this problem. Under the conditions of interest, matter may be regarded as a mixture of neutrons, protons, electrons and positrons, neutrinos and antineutrinos, and photons. Neither the temperature nor the density are so high that other degrees of freedom such as quarks can be excited. Electrons interact rather weakly with other electrons and with nuclei, and therefore to a good approximation they may be treated as an ideal Fermi gas. Because of their even weaker interactions, neutrinos and photons may also be treated as ideal gases. The difficult problem is to determine the state of the nucleons, which may be bound in nuclei or may be essentially free in continuum states. At low densities and low temperatures, and provided the matter does not have too large a neutron excess, the relevant nuclei are stable in the laboratory, and in making calculations one can use experimental information directly. The densities of neutrons and protons outside nuclei are then so low that they may be treated as ideal or weakly non-ideal gases, and the equilibrium properties may be calculated by use of the Saha equation to determine abundances. For larger neutron excesses it is necessary to determine nuclear properties by extrapolation from laboratory nuclei. Calculations based on such approaches have been carried out by, among others, Amett ‘), Wilson 3), Bowers and Wilson 4), Lamb 5), Murphy 6), Mazurek, Lattimer and Brown 7), and El Eid and Hillebrandt ‘).

648

J.M. Lattimer et al. / Hot, dense matter

Under

more extreme

conditions

which must be taken into account. outside

nuclei

either to matter

can become having

there are a number

First, at higher densities

comparable

a neutron

of important

excess

to that inside

physical

the density

the nuclei.

so large that neutrons

effects

of nucleons

This can be due “drip”

out of the

nuclei even at zero temperature, or to thermal evaporation of nucleons from nuclei. Under such conditions it is important to treat matter outside nuclei and the matter inside nuclei in a consistent way, as was stressed by Baym, Bethe and Pethick9). A second effect is that at finite temperatures nuclear excited states become populated. Sato lo) was one of the first people to appreciate the importance of this and to incorporate it into his calculations. He realized that the simplest way to do this is to treat nuclei as drops of nuclear matter, whose thermodynamic properties are given by the standard finite-temperature expressions for a Fermi gas. By means of this idea, which was used also by Mazurek, Lattimer and Brown ‘) and by Tubbs and Koonin I’), one is able to give a consistent treatment of nuclear partition functions without introducing arbitrary cut-offs, which lead to difficulties [Fowler, Engelbrecht and Woosley ‘*)I. A third effect is the modification of the nuclear surface by the presence of nucleons . outside. As the matter inside nuclei and that outside become increasingly similar in density, the surface tension decreases, thereby tending to make the nuclear size decrease. In addition, the surface region makes its own contribution to the nuclear density of states. The fact that in dense

matter

the spacing

between

nuclei

may be of the same

order of magnitude as the nuclear size itself leads to a fourth effect, a reduction of the Coulomb energy. Coulomb interaction energies between a nucleus and other nuclei, and between a nucleus and the background electrons, which are typical “condensed matter” effects, can be comparable to the Coulomb energy of an isolated nucleus.

The importance

of this effect in the case of neutron-star

matter was clearly

demonstrated by Volodin and Kirzhnits 13) and by Baym, Bethe and Pethick ‘). The Coulomb energy is reduced by the condensed matter effects, and as a consequence the nuclear size increases. These are the most important

physical

effects. There are others, generally

of lesser

importance, which we shall describe later. One way to take these physical features into account would be to carry out microscopic calculations for unit cells of matter. This has been done at zero temperature by Barkat, Buchler and Ingber 14), by Arponen 15) and by Ogasawara and Sato 16) in the Thomas-Fermi approximation, and by Negele and Vautherin “) in the Hartree-Fock approximation. The extension to finite temperatures of these calculations has been carried out in the Thomas-Fermi approximation by Buchler and Epstein ‘*) (for the case of an isolated nucleus), by Buchler, Barranco and Marcos 19-‘*), by Ogasawara and Sato 23) and by Suraud and Vautherin 24) (for a unit cell) ; and by Bonche and Vautherin 25,26) and Wolff *‘) for a unit cell in the Hartree-Fock case. The fact that a single microscopic hamiltonian is used to describe the nucleons inside and outside nuclei automatically ensures

J.M. L&timer et al. / Hot, dense matter

649

that the properties of matter inside nuclei and that outside are treated consistently. In addition, modifications of the nuclear surface are implicitly allowed for in calculating the properties of the complete unit cell, even though thermodynamic quantities are not separated into bulk and surface contributions. While the microscopic calculations take into account many of the physical effects we wish to incorporate, they are so complicated to perform that it is impractical at present to carry out calculations over the wide range of conditions we wish to survey. Also their integral nature does not permit us to distinguish and scrutinize the separate physical contributions whose interplay we wish to understand. Our aim is to construct a model in which the various physical ingredients are clearly identified, so that their mutual interactions can be examined. If one neglects finite-nucleus effects, a mixture of protons and neutrons can, under some conditions, consist of two coexisting phases, liquid and gas. The conditions under which this is possible give a good guide as to the conditions under which nuclei can exist in dense matter. Such calculations have been carried out by Kupper, Wegman and Hilf *“) for charge-symmetric matter (proton fraction of a half) and by Lattimer and Ravenhall 29), Lamb, Lattimer, Pethick and Ravenhall 30), Barranco and Buchler ‘I), Rayet, Arnould, Tondeur and Paulus 32) and others for arbitrary neutron-proton ratios. These calculations led to the important conclusion that nuclei can exist at temperatures as high as 15-20 MeV, which is the critical temperature. The reason that in earlier calculations based on the semi-empirical mass formula nuclei were found to dissociate at lower temperatures than these is that the effects of nuclear excited states were either neglected or underestimated. They are included implicitly in bulk equilibrium calculations, as we explained earlier. To improve on the bulk fluid approach one includes the effects of finite nuclear size. This will be done here through the compressible liquid drop model. Modifications to the nuclear surface due to the presence of the outside nucleons are taken into account by using the results of microscopic calculations of plane interfaces between two nuclear matter phases. Such calculations have been carried out by a number of authors for symmetric nuclear matter, and for the case of arbitrary proton fraction by Ravenhall, Bennett and Pethick33) for zero temperature, and by Ravenhall, Pethick and Lattimer 34) for finite temperatures. The latter calculations, which are the ones we shall adopt here, were made, as were the earlier ones, using the Skyrme 1’ interaction, which is the hamiltonian we used to evaluate the bulk nuclear properties. This automatically ensures that the bulk and surface properties of nuclei are treated consistently. If one had not done this one would have found, for example, that the surface tension vanished at a different temperature than the critical temperature predicted from bulk equilibrium. Now we turn to the Coulomb energy. This may be evaluated straightforwardly at zero temperature, but at finite temperatures nuclei are no longer localized on sites on a regular lattice. Many calculations of thermodynamic properties of Coulomb plasmas have been carried out [see e.g. Hansen 35)], and we shall incorporate them

650

J.M. Lattimer et al. / Hot, dense matter

in making estimates of the Coulomb energy at finite temperature for the somewhat more complicated problem considered here. Thus we see that the compressible liquid drop model is able to incorporate the physical features important in hot, dense matter. Preliminary results using this model were presented by the present authors 36) some time ago. Apart from its computational simplicity, this model has great advantages when it comes to understanding various phenomena physically. Contributions to thermodynamic quantities arising from, for example, bulk and surface effects, can be clearly separated, whereas in microscopic calculations one knows only the contributions to thermodynamic properties from the complete unit cell. The plan of the rest of this paper is as follows. In sect. 2 we set out the components of the compressible liquid drop, discussing in some detail our treatments of the nuclear surface, the Coulomb energy, translation of the nucleus, and a small correction to allow for the reduced level density of light nuclei. The thermodynamic equilibrium of these components is systematized in sect. 3, for the case of matter with nuclei. A separate discussion of the thermodynamics of the bubbles-matter case is given in sect. 4. In sect. 5 we present contour plots of the density and temperature dependence of the fraction of nucleons in nuclei and in a-particles; the electron fraction and electron chemical potential; the A and 2 of the nuclei; the internal energy; the pressure and the entropy; the adiabatic index; and the nucleon chemical potentials. We also plot the adiabatic index along adiabats. Each of these quantities is given for the cases of p-equilibrium with fixed lepton fractions of 0.4, 0.3 and 0.2, and for p-equilibrium with no neutrinos. The first three cases span the regions of interest for stellar collapse, and the last is relevant for neutron-star matter. We also include in sect. 5 simple analytic and physical discussions of noteworthy features of these graphical results. In sect. 6, comparison is made with some of the results of other workers mentioned earlier in this section. Appendices deal with details of electron screening, of the thermodynamics of the interface (surface) between two-component phases at finite temperature, of the numerical computations, and of thermal fluctuations. 2. The model Under the conditions of interest, hot dense matter consists of a mixture of heavy nuclei, a-particles, neutrons, protons, electrons (and positrons), neutrinos, and photons. The electrons, neutrinos and photons may be treated as perfect gases. Variations in the electron density may be neglected since electron screening lengths are large compared with internuclear separations. As we demonstrate in appendix A, this is a good approximation even when the charge of a nucleus becomes large, as is the case when nuclei fill an appreciable fraction of space. In matter in P-equilibrium the number of muons present is very small under all conditions encountered in stellar collapse at densities of the order of nuclear matter density

J.M. Lattimer et al. / Hoi, dense matter

651

or less, and we shall neglect them. We may write the total free energy per unit volume as the sum of contributions from nucleons both inside and outside heavy nuclei, a-particles, electrons, neutrinos, and photons, plus a baryonic rest-mass term: F=F,,,+F,+F,+F,+F,+n,(m,-m,)c’.

(2.1)

The convention adopted is that the energy of the baryons is measured with respect to the rest-mass energy of an equal number of neutrons. In (2.1) np is the total density of protons, and m, and m, the proton and neutron rest-masses, respectively. For convenience we shall include all Coulomb energies in the nuclear contribution. To ensure charge neutrality the proton and net electron (that is, electron minus positron) densities must be equal. We wish to determine the most favorable thermodynamic state that can be reached by strong and electromagnetic processes, but we shall not impose the condition of p-equilibrium. Since we shall not usually be interested in processes that convert protons into neutrons we adopt the convention that all nuclear energies (and free energies) are measured with respect to the rest mass of the same number of free protons and free neutrons. Under these conditions the proton density is fixed, as are the neutrino and electron densities. We shall assume a single species of heavy nucleus with mass number A and proton number 2 to be present, and take the nuclei to be spherical. We shall treat the a-particle as a rigid sphere of volume v,. The equilibrium state is determined by minimizing the free energy per unit volume with respect to the parameters of the model, keeping fixed the temperature, and the densities of neutrons, protons, electrons and neutrinos. We now discuss the contributions to the free energy and the particle numbers coming from (i) nucleons (both inside and outside nuclei), (ii) cu-particles, (iii) electrons, (iv) neutrinos, and (v) photons. (i) N&eons. We shall treat nuclei using a generalization of the compressible liquid drop model [Baym, Bethe and Pethick ‘)I to finite temperatures. In this model one treats nuclei as spherical droplets of nuclear matter, and writes the (free) energy as a sum of bulk, surface, Coulomb, and translational contributions:

where the f’s are the free energy per nucleus and nN is the number density of nuclei. We now consider the contributions to (2.2) in turn: (a) bulk and surface quantities; (b) Coulomb energy; (c) translational energy. (a) Bulk and surface quantities. One aspect of the physics we wish to discuss in some detail is the division of the free energy of a nucleus and the numbers of nucleons into surface and volume contributions. Just how one chooses to make this separation is to some extent arbitrary, since it is only the total expression for physical quantities that is important. However, some choices are more convenient than others, and we shall now describe the particular choice we make. A detailed discussion of the case of a plane interface is given in appendix B.

J.M. L&timer et al. / Hot, dense matter

652

We begin by considering the total numbers of neutrons and protons. The physical reason for having to look at these quantities carefully is that in neutron-rich nuclei the neutrons and protons do not have the same spatial distribution, and according to most calculations, the neutrons extend beyond the protons. Let us imagine that far inside the surface of the nucleus the neutron and proton densities are uniform, and have the values n,i and npi. Similarly, far outside the interface, the densities have the values nno and npOt. We may define a proton radius by squaring off the proton density profile so that the total number of protons NP is the same in the squared-off distribution as in the original one. This is shown schematically in fig. 1. Algebraically this condition is Np=$Trinri

+( V’-$7rri)npo,

(2.3)

where V’ is the total volume available to nucleons. This is essentially the same procedure as we adopted in our treatment of the plane surface in appendix B. We may also define an analogous radius, r,,, for the neutrons. This will generally not be the same as rP, due to the additional neutrons in the surface region. We shall

oni

h C r 25

Neutrons

Rodiol Actual

rp

rn

Distance System

Surface containing V, =(r,-rphn -nno) neutrons per u$t area

0

rN = rp

rc

Rodiol Distance Model System Fig. 1. Schematic

representation

of the proton and neutron density profiles at the nuclear surface in the actual system, and in the model system.

’ Here and in what follows we shall use the subscripts i and o to denote values inside and outside the nucleus, respectively. We shall use the symbol A to denote the differences between values of the quantities inside and outside the nuclei.

J.M. L&timer et al. / Hot, dense matter

653

find it most convenient to work in terms of the proton radius, since this is the quantity that naturally enters into calculations of the Coulomb energy. We represent a nucleus by a model system consisting of a sphere of uniform nuclear matter Of radius rN = rP and VOhUne vN

=

(2.4)

$r’,

immersed in a less dense sea of nuclear matter and cu-particles. In our description of nuclei we shall write thermodynamic quantities as the sum of the contributions listed in eq. (2.2). The first is that of bulk nuclear matter with neutron and proton densities n”i and npi occupying a volume of radius rN = rP. The second is that of bulk nuclear matter with neutron and proton densities nno and n,,,, occupying the rest of the volume, apart from that excluded by the presence of cY-particles. The final one, which takes into account the remaining contributions, is that from the surface region. To ensure that the model system contains the same number of neutrons and protons as the actual system it is clear that the surface must contain v,=

(%i-

%J(~~;,-

rp)

(2.5)

neutrons per unit area, but no protons. We shall define the nuclear charge by Z = $&nPi,

Q-6)

but we include in the neutron number N the surface neutrons, as well as the neutrons in the bulk, and therefore N is given by N= N’+N,,

(2.7)

N’ = $&n,,i

(2.8)

where

is the number of neutrons in the bulk of the nucleus, and N, = 4mkvs

(2.9)

is the number of neutrons in the surface. The total mass number of the nucleus is given by A= N+Z.

(2.10)

These particular definitions for A and Z have the desirable property of reducing to the usual ones when there are no nucleons outside nuclei. We shall also find it convenient to introduce the total number of nucleons in the bulk of the nucleus, given by A’= N’+Z.

(2.11)

Since n,i + nPi is equal to the total density of baryons ni in the interior of the nucleus,

J.M. Lattimer et al. / Hot, dense matter

654

one has A’ = $rkni

(2.12)

.

Let us now calculate expressions for the total neutron and proton densities. The density of nuclei is nN, and thus the volume per nucleus V, is given by V,=$&=

l/n,.

(2.13)

As we shall discuss in more detail below, we take into account interactions between (Y-particles and the outside nucleons by treating the (Y-particles as hard spheres of volume v,. The space available to nucleons outside nuclei is thus reduced by both the finite size of the nuclei and the finite volume of the a-particles. If we denote the density of a-particles per unit volume of space outside nuclei by n,, the fraction of space excluded by cu-particles is n,u, ( V, - V,)/ V,, and that excluded by nuclei is V,/ V,. Thus the fraction of space available to nucleons outside nuclei, 6, is C=l--nn,u,---

vc-VN

VN

vc

vc (2.14)

=(I-n,v,)(l-u), where u = v,/ vc

(2.15)

is the fraction of space filled by nuclei. The total proton and neutron densities have contributions from nuclei, (Y-particles and nucleons outside nuclei and are given by n,=Zn,+2(1-u)n,+(l-n,u,)(l-u)n,,,

(2.16a)

n,=Nn,+2(1-u)n,+(1-n,u,)(1-u)n,,.

(2.16b)

Likewise the total baryon number density n is given by (2.17)

n=An,+4(1-u)n,+(l-n,u,)(l-u)n,, where n, = nno + nPO.The mean proton number per baryon is YP= n,/n.

(2.18)

Let us now turn to a discussion of the free energy. The bulk contribution

is given

by fbuik=

VNF:(%i,

Q)+(Vc-

Vidl

-wa),)Rnno,

npo),

(2.19)

where $ is the bulk free energy of nuclear matter per unit volume. The surface contribution we take to be

.Lf = nN4~&Ysurf,

(2.20)

where ysurf is the free energy per unit area. As we show in some detail in appendix

J.M. L&timer et al. / Hot, dense matter

655

B, it is convenient to re-express Ysurfin terms of the thermodynamic potential 0; per unit area, which is just the surface tension. The relationship (see eq. (B.21)) is ysurf=

~+PL,Vs

3

(2.21)

where p, is the chemical potential of neutrons in the surface. As is made clear in the next section, this is not the same as the chemical potential of the plane-surface neutrons, nor of the neutrons inside the nucleus. The surface thermodynamic potential u is that appropriate to the surface neutron chemical potential, to be obtained from the plane surface result by the methods described in appendix B. Therefore we have Fsu,f= nN4Tr$(a+&V,).

(2.22)

The fact that there are no pp terms is due to our choosing the volume of the nucleus so that the surface contains no protons. In equilibrium the quantities v, and (T are related by the thermodynamic identity (cf. eq. (B.22)) da us= -7, dp,

(2.23)

where the derivative is to be evaluated along the coexistence curve for two bulk phases, and at constant temperature. (The superscript 0 indicates values for a plane surface; the distinction between p, and &’ is made clear in sect. 3.) The above discussion is quite general, and we now describe the expressions we use for bulk and surface quantities when making the numerical calculations. Under conditions encountered in stellar collapse, matter is more neutron rich than terrestrial matter. Nuclei in neutron-rich matter can have properties vastly ,different from laboratory nuclei, and a mass formula derived from laboratory nuclei cannot be expected to describe their properties. The approach we adopt is a more microscopic one, discussed by Ravenhall, Bennett and Pethick 33), at T = 0, and by Lattimer and Ravenhall 29), at finite T. It is based on the use of a nuclear effective hamiltonian which is ‘a functional of the neutron and proton densities. The properties of the effective hamiltonian have been fitted to data for laboratory nuclei, and to theoretical calculations of the properties of neutron matter. For simplicity, we have used hamiltonians of the Skyrme type 37), which depend only on the local neutron and proton number densities n,(r), n,(r) and their gradients, and the kinetic densities T,(r), 7Jr): H(n,, np, Vn,, Vn,, r,,, rp). The bulk energy we have discussed corresponds to the case that these densities are constant, so that the gradient dependence of H disappears, and the T and n are given by the properties of uniform Fermi fluids at a finite temperature T: 1 %P

* k*.L,Jk)

=7 I0

dk,

(2.24a)

656

J.M. Lattimer

1 Tn,P =

_&p(k)

=

7

et

al / Hot, dense matter

* k4f,,pW

dk

(2.24b)

3

I 0

{ev

[(

&kn,p

-

Tl

L,,,)/

(2.24~)

+ 1I-’ .

Here, &k,,piS the single particle energy arising from H: erst --&;k’+$ f

where t refers to neutron or proton. The particular choice of hamiltonian H =&3t,

- t2)(Vn,+Vnp)~--&3t,+

(2.24d) f

and parameters is as follows: k2 i2)[(vn”)2+(~“r)~,+2m7”+~rp n

+$(t2-f,)(nnT,+~p7p)+~f3[~n~p+~h(nn-np)2](n”+np).

P

(2.25)

The parameter values are: to = -1057.3 MeV * fm3, t, = 235.9 MeV * fm5, f2 = -100 MeV . fm5, t3 = 14463 MeV . fm6, x0 = 0.2885 and A = 0.5162. This is the paraIt differs from the original Skyrme 1 parameter meter set used in refs. 29*30~33~34~36*5’~s2). set of Vautherin and Brink 38)in the exchange parameter A,which was adjusted so that for pure neutron matter the results agree with the calculations of Siemens and Pandharipande 39). Lattimer and Ravenhall 29) have described how these equations may be solved to obtain, for a prescribed n,, np and T, the values of p”, pp and all the other relevantthermodynamic quantities. (See also appendix C.) This is how the bulk free energies are obtained. It should be emphasized, however, that the particular hamiltonian we use is not a necessary part of the whole thermodynamic system. That system is applicable quite generally. The surface properties of the interface between two phases of the bulk fluid that are in thermodynamic equilibrium must be calculated using the same functional H( n,, np, Vn,, Vn,, T,,, TV) as is used for the bulk properties. This ensures that the properties of the surface are consistent with those of the bulk fluids. For example, the surface tension vanishes at the critical temperature, but if different hamiltonians are used for calculating the bulk and surface properties, the surface tension will generally vanish at a temperature different from the critical temperature of coexistence of the two bulk phases, which is inconsistent physically. Techniques available for evaluating the surface tension a( T, phase equilibrium), are essentially the same as those employed for isolated physical nuclei. In the geometry of a plane interface between two semi-infinite bulk fluids which are in thermodynamic equilibrium we have used the Hartree-Fock method for T =0, and the Thomas-Fermi approach for general T. The details of these calculations have been reported elsewhere [Ravenhall, Pethick and Lattimer ‘“)I. To obtain complete thermodynamic information we used the surface tension u as a function of the neutron chemical

J.M. Lcrttimer et aL f Hot, dense matter

651

potential and the temperature. For calculating, it is convenient to use the proton concentration of the matter in the dense phase, Xi, as the independent variable, so we shall give fits to u and the chemical potential as functions of Xi and T. The density of excess neutrons in the surface may then be calculated from eq. (2.23), which may be rewritten as vs=-(ao(Xi,

T)/aXi)./(aPf(Xi,

(2.26)

T)IaXi)T*

We wish to incorporate in our fit the known behavior of the surface tension near the critical temperature. The microscopic Hartree-Fock or Thomas-Fermi calculations are difficult to make in this region, and even if they could be carried out accurately, they would yield mean field exponents rather than the true ones. Near the critical temperature, the singular part of the thermodynamic potential per unit volume scales as (T,- T)2-a, where (Yis the specific heat index, and the correlation length scales as (T, - T)-“. The surface thermodynamic potential u scales as the product of these two quantities, that is as (T,- T)2-a-” [Widom “)I. From the scaling relation vd = 2 - (Y,where d is the dimensionality, one sees that u scales as (T,- T)‘? Calculations for systems with a scalar order parameter, which according to universality one would expect to apply to a fluid, and measurements of the surface tension for fluids, give Y% 0.63 [Le Guillou and Zinn-Justin “)I. In our parametrization we have used the value p = 2v = 1.25. [Mean-field theory would give p = 1.5, as shown by Cahn and Hilliard42).] The Xi, T functional dependence of u and Y, has been determined using the Thomas-Fermi numerical values. The Hartree-Fock results, for T = 0, agree well in Xidependence with these, but are bigger by about 20% [ref. “‘)I. We have therefore scaled up our parametric form for u (and hence for vS) to the Hartree-Fock value at Xi= 0.5, T = 0. The fits to these quantities, and to the related critical temperature Tc(Xi), are as follows: Tc(Xi) = Tc(O.S)[l -Cy2-dy4]1'2,

where y = 0.5 -Xi is the neutron excess parameter, d = 7.362; U(Xi,T) =

~(0.5,0)

a (Xi)

=

UC) + U2y2

T,(O.5) = 20.085 MeV, c = 3.3 13, 1 - T2/ Tz(Xi)

16+b l/X’+b+l/(l-Xi)3

1 +U(Xi)T2/Ta(xi) + U4y4

(2.27)

,

1’ ’

(2.28a) (2.28b)

where u(0.5,0)=1.149MeV/fm2, b=24.4, a,=0.935, a,=-5.1, u4=-1.1, andp= 1.25. As in the final part of appendix B we shall denote the neutron chemical potential associated with the surface (in the absence of other finite nucleus effects) by pp to distinguish it from the many other neutron chemical potentials that enter our calculations. In the plane surface calculations it is equal to the bulk neutron chemical potentials in the two coexisting phases. The chemical potential is not

658

J. M. Lattimer et al. 1 Hot, dense matter

expected to exhibit strongly anomalous behavior at T - T,, so a direct double series is the simplest form to use in fitting it. We find 34)

power

(2.29) where moo= -16.01 MeV, m,, = 118.0 MeV, mo2 = -133.0 MeV, mo3= 10.1 MeV, m 04= 93.0 MeV, mzo = -0.06517 MeV-‘, rnzl = -0.03 MeV-‘, rnz2= 0.110 MeV-‘, m30= 5.2 x lop4 MeV-*, m3, = - 4.0 x low5 MeV2, rn3*= - 1.O x 10e3 MeV2, m40= 4.0 X lop6 MeV3,

m4, = -3.0

X

lo-’ MeV3,

and other

mk/ are zero.

Some insight into why the surface tension behaves as x3 for small x may be obtained by considering one-fluid Thomas-Fermi calculations [see e.g. Bethe “‘)I in which one assumes that spatial variations of the neutron and proton densities are proportional to each other. The surface may therefore be described by one density profile, and one finds that the surface tension is proportional to “I

(2.30)

@(n)+P-&“*dn, “0

where P is the pressure in the bulk matter away from the surface, Jo is the chemical potential and ni and n, are the densities far inside and far outside the surface. Because the two phases must be in equilibrium with each other, P(n) must have a common tangent at the densities ni and n,, and therefore the quantity in the square root must vary as (ni - n)“( n - nJ2 for small ni - n,. Consequently from eq. (2.30) one sees that (+ varies as (ni - n,)3. The arguments for a two-component system such as the one considered here are somewhat more complicated, but we expect this qualitative feature derived from the one-component calculation still to hold. At

T =0, when the inside

matter

contains

protons

but the outside

matter

does not,

ni - n, varies roughly as Xi, which accounts for the x: behavior of cr. That the form for a(Xi, 0) based on this argument requires only one other parameter is serendipitous. In the estimates of the surface energy made in ref. ‘) the double tangent condition was not taken into account, and it was assumed that P + P - pn (ni - n)( n - n,), and thus the surface

energy

x2. This high estimate

tension

the nuclear

for the surface

was found

to vary as (ni - no)*, i.e. as

led to unrealistically

large values

for

size.

(b) Coulomb free energy. The Coulomb

free energy includes both the Coulomb energy of an isolated nucleus, and contributions due to interaction with other nucleons, with electrons, and with protons outside nuclei.‘The Coulomb energy of an isolated nucleus, regarded as a uniformly charged sphere of radius rN, is e coul

=

zZ2e2/ rN .

At densities such that the spacing between size, the Coulomb interaction of a nucleus

(2.31)

nuclei is comparable with the nuclear with otner nuclei and with electrons is

J.M. Lattimer et al. / Hot, dense matter

659

of a comparable size and must be taken into account. First let us consider the situation at zero temperature. To calculate the Coulomb energy it is a very-good approximation to use the Wigner-Seitz approximation, in which one considers each nucleus to be at the center of a sphere containing 2 electrons. One then calculates the Coulomb energy by taking the energy of such a cell, and neglects interaction between the cells. The Coulomb energy is given by (2.32) The first term is just the Coulomb energy of an isolated nucleus, eq. (2.31), the second is the lattice energy of a point nucleus, and the final term is a contribution to the lattice energy due to the finite size of the nucleus. This expression has the physically reasonable behavior of going to zero as the nuclei fill all of space, rN + rc. To generalize this result to finite temperatures we wish to take into account a number of features. First of all we consider the protons and (Y-particles outside nuclei. We shall assume that these are uniformly distributed in space. It is then clear that the Coulomb energy is given by the same expression as before, but with the total charge of the nucleus 2 replaced by an effective charge Zefi, which gives the excess charge in the nucleus compared with the charge of the background protons and cu-particles there would be in the same volume. In other words

=z-

V,(a,,(1-&&)+2n,).

(2.33)

The second effect to take into account is thermal displacement of nuclei from their classical equilibrium positions. This obviously does not affect the Coulomb energy of an individual nucleus, nor does it affect the finite size term, which depends only on the background charge density. We may now write the Coulomb free energy in the form (2.34) where 8 is a dimensionless function. The first term is the Coulomb energy of an individual nucleus, the second is due to the finite nuclear size, and the third is the correlation free energy. The first two terms are the same as the first and third terms in (2.32), but with Z replaced by Z,,. In the low-temperature limit the correlation free energy goes over to (2.35) in the Wigner-Seitz approximation, in agreement with (2.32), and differs very little from it if the actual lattice structure is taken into account.

660

J.M. Lattimer et al. / Hot, dense matter

As an interpolation formula that gives the right result for T = 0 and also incorporates the thermal effects on the lattice free energy, one might try (fd0UJ,0, = T4(T) 9 where T4 (r) is the correlation free energy of the one-component ions, with

r = Zzffe2/

rc T.

plasma with point (2.36)

This result is not quite good enough since 4(r) takes into account only the effect of Coulomb correlations, and does not allow for the fact that the finite size of nuclei affects the interaction at distances less than about 2r,. At such distances one should take into account the distortion of nuclei from their spherical shape, and calculate an effective potential in much the same way as one does in calculations of fission barriers. However, such a detailed treatment is beyond the scope of this paper. We construct the interpolation formula by demanding that when the nuclei occupy a small fraction of space, the expression agrees with (2.35), and that when they occupy all of space the Coulomb free energy vanishes, since then the electrostatic potential is spatially uniform. These results must hold at all temperatures. Such an expression is (2.37) For an arbitrary positive value of the index A this reduces to (2.32) for VN< V, and vanishes for V,+ V,, irrespective of the value of the temperature. To get some idea of reasonable values for the parameter A we have investigated results for hard spheres interacting via a Coulomb interaction. As we indicated above, the interaction between two ions at short distances is complicated, but we shall assume that the hard-sphere results give us a qualitative guide to the behavior of the system. The Coulomb internal (rather than free) energy of such a system has been calculated using Monte Carlo techniques by Hansen and Weis 44), and we have fitted their results using an interpolation formula analogous to the correlation part of (2.37): (2.38) where ecoUl(r) is the Coulomb correlation energy for V,/ Vc + 0. For large values of r, ecoUl(r) is close to the value -&Z2e2/rC, and so the parameter A’ is of little importance+. On the other hand, as r decreases, ecOUl( r) becomes increasingly different from -$Z2e2/ rc, and therefore we shall describe the results for r = 20, the lowest value of r for which Hansen and Weis made calculations. Fitting the + Notethat in HansenandWeis’s calculations “) all the charge (except that in the uniform background) resides in the ion, and therefore Z and Z,, are identical.

J. M. L.&timer et al. / Hot, dense matter

661

results for u = V,/ V, = 0 and V,/ V, = 0.343 to (2.38) we find A’ = 0.9 1. If we take the results for V,/ V, = 0 and V,/ V, = 0.4, we find A’= 0.76. The difference between the two values of A’ shows that our interpolation formula is not precise, but is not too bad an approximation. The value of A’ obtained does depend somewhat on I’, tending to decrease with decreasing r. However we shall neglect this effect, and will take A’ to be a constant. We shall further assume that A’ (for the internal energy) is close to A (for the free energy). Our considerations above suggest a value in the vicinity of unity. For the free energy of the one-component plasma we use the fits Hansen 35) made to his Monte Carlo results. The quantity directly calculated is the internal energy per ion, c, which is well fitted by the expression

a2

= r3/2 (b,

a3

:;y2+ b2+r+(b3+r)3/2+(b4+r)2

a4

(2.39)

with the parameter values a, = -0.895 929, b, = 4.666 486 0; a, = 0.113 406 56, b2 = 13.675 411; a3 = -0.908 728 27, b,= 1.890 5603; a4= -0.116 147 73, b4= 1.027 755 4. His e”has the correct small-r behavior given by the Debye-Hiickel formula. It is because of this property that we have chosen Hansen’s approximation over the somewhat more accurate fits, for r > 1, of Slattery, Doolen and Dewitt 45). By integrating the expression for e”Hansen 35) found ~=alYI(U+a2Y2(~)+a,Y3(IT+a4Y4(~),

(2.40)

where Y,(r)=r1/2(bl+r)1/2+fb, Y2(r) = 2P

In 6,-b,

ln[r1/2+(b,+r)L/2],

- 2b:12 arctan (r”2/ b:“) ,

Y,(r) = -2P/(b,+rp2-

In b3+2 In [r1/*+(b3+r)“‘],

Y4(r)=-r”2/(b4+r)+(l/b:/2)arctan(I”/2/b:/2).

(2.41)

There is a further contribution to the Coulomb energy due to electron screening. We shall consider this in appendix A and show that it may be neglected. (c) Translationalfree energy. Now we turn to the translational term. For a classical gas of point particles this is given by

ftrans= T{ln [ ($$)3’2$-] - 1))

(2.42)

where M is the nuclear mass. The translational contribution to the free energy plays an important role at high temperatures, and a number of additional effects must be taken into account. The first of these is that at high densities when nuclei fill an

662

J.M. Lattimer et al. / Hot, dense matter

appreciable fraction of space, the volume available to the nucleus is reduced, hence its entropy is reduced and its free energy is increased. If the nucleus were free to move only within a Wigner-Seitz cell, and if in addition it behaved as a hard sphere, its center would be confined to a sphere of radius rc - rN, and therefore the volume available to it would be $r( rc - rN)3 = ( Vg’ - Vg3)3. More generally, if we go beyond the Wigner-Seitz approximation one might generalize this result to ( Vg’ - /.LV~~)~, where pm3 is the filling factor for close packing, which will depend on the particular lattice structure assumed. This expression would ensure that the volume available to the nucleus vanished at the density corresponding to close packing. For the f.c.c. and h.c.p. lattices, which are the most closely packed ones, one has F = (3&/7r)1’3 = 1.1053, which is the value we used in our calculations. Another physical effect that modifies the result (2.42) is that the effective mass of the nucleus is not equal to the bare mass of the nucleus when the density of the nucleons through which the nucleus moves becomes comparable to that of the nucleus itself. It is not clear to us exactly what expression should be used for the effective mass. One possibility is that the nucleus behaves as a rigid impenetrable sphere and the backflow of nucleons round the sphere leads to an increase in the effective mass of the nucleus M. If the backflow may be calculated using classical hydrodynamics, the increase in the effective mass is equal to half of the mass of fluid displaced, &rn~ V,, where m is the nucleon mass and nb = 4n, +( 1 - nau,)nO is the total density of baryons outside nuclei. Therefore M* = M +$mnkV,

= m(A +$zbVN).

(2.43)

One should note that even if the nuclei behave as hard spheres, the expression (2.43) for the effective mass will not be adequate when nuclei fill an appreciable fraction of space. This expression is appropriate for a sphere moving in an infinite medium, but it must be modified if the typical nuclear spacing is comparable to the nuclear size, and the backflow is impeded. This estimate will also be inadequate if the nuclei do not behave as hard spheres but allow the outside nucleons to penetrate into the nucleus. One could argue that the nucleus effective mass (apart from surface contribution) should vanish when the densities of the inside and outside matter become identical. As a simple estimate one might then take the mass of the surface neutrons plus the magnitude of the density enhancement in the nucleus times the nuclear volume, M*=mV,(ni-nb)+N,.

(2.44)

With v an adjustable parameter, the form M* = (A +ivVNnb)m

(2.45)

incorporates either of the forms (2.43), (2.44). We have used v = 1 in the calculations presented in sect. 5.

J.M. Lattimer et al. / Hot, dense matter

663

There is an additional physical effect which is most conveniently included as part of the translational free energy. In the description of nuclei discussed above, the internal motion of the nucleus has been regarded as possessing A (three-dimensional) degrees of freedom, one for each of the nucleons. However, there are in fact only A- 1 internal degrees of freedom, and one degree of freedom associated with translation of the center of mass, whose contrib.ution to the free energy has been added explicitly. It is therefore necessary to correct for the degree of freedom in the internal motion. We shall not make an explicit correction to the expression for the ground state binding energy, since the mass formula derived from the model described above gives a good account of nuclear masses without it. In some sense the effect of the extra degree of freedom has been mocked up by the other finitenucleus terms. On the other hand, the effect of this additional degree of freedom on the finite temperature properties is large. For example, for a free Fermi gas at low temperatures the finite-temperature contribution to the free energy per nucleon is -&?T*/ TF, where TF is the Fermi temperature (-40 MeV in nuclei). This is of order -&T” MeV, with T in MeV, which is considerable for a temperature of 15 MeV. In light nuclei the surface contributes a similar amount, - - T*/ 10A”3, as one can see from the calculations of the surface free energy. As an estimate for the free energy contribution per degree of freedom we shall use simply the bulk term multiplied by two, -&T* MeV. Subtracting this amount per nucleus from the free energy has the effect of raising the free energy of nuclei, and thereby tends to reduce the number of nuclei present. A second but related effect is that the density of excited states of light nuclei is reduced considerably below the estimates for the same number of nucleons in a bulk Fermi gas. At low excitation energies the effective density of states is small because of the finite spacing between energy levels. At higher energies the effective density of states is also low, in this case because as a consequence of the low binding energy per particle of light nuclei, many excited states correspond to the nucleus fissioning into two or more fragments, and therefore these states should not be treated as excited states of the single original nucleus, but rather as a number of smaller nuclei. To take these effects into account we add a term to the translational free energy which essentially removes all finite temperature contributions to the free energy for A = 4, and which tends to the result for a single degree of freedom for large A. Such an expression is

where the parameters W and A, are chosen to have the values W = 8 MeV and A0 = 12. We shall retain the general expression (2.46) in our calculations since we wish to investigate how sensitive our results are to the choice of parameters. Despite the fact that this term is formally of order l/A per nucleon, and therefore would be neglected compared with the surface terms if one were performing a strict

664

J.M. Lattimer

et al. / Hot, dense matter

expansion in powers of l/A”‘, it is numerically important and has a significant effect on the results of our calculations in the limited regions where A is small. We do not believe that our expression is quantitatively accurate, but feel that it reflects an important physical effect. More fundamental investigations of the density of states and free energies of light nuclei are needed to obtain better estimates of it. These should include also contributions which in a mass-formula approach correspond to surface curvature corrections, which give a free energy per particle varying asA -2/3t . While at low temperatures it is physically reasonable to treat the center of mass motion in a different manner than the other degrees of freedom, this method is not so good when T approaches the critical temperature. First of all the center-of-mass motion is not that of a free particle, due to the fact that the nucleus interacts continually with other particles. Second, many other collective degrees of freedom, such as surface modes, will be excited and we expect that these will be as important as the center of mass motion. As yet there is no consistent treatment of the collective degrees of freedom near the critical temperature. The problem we are considering here is rather different from most critical phenomena problems, since there are two competing interactions. The surface energy, corresponding to gradient energies in a Landau-Ginzburg free energy expansion, tends to favor fluctuations of long wavelength, while the Coulomb energy favors fluctuations of short wavelength. The resulting fluctuations correspond to the well-defined, discrete nuclei that exist at lower temperatures. Only when the correlation length becomes greater than the electron screening length l/km (eq. (2.51)), does one expect the behavior to be the same as that of an uncharged fluid. Because the electron screening length is large, this is the case only extremely close to the critical temperature. On the basis of this discussion, we argue that as T approaches the critical temperature, treating the center-of-mass motion as that of a free particle overemphasizes its importance. Lacking any fundamental treatment of the collective degrees of freedom, we therefore reduce it by a factor h(T) which vanishes as T + T,. We apply a similar factor to the finite nucleus effects, eq. (2.46), since these have their origin in similar sorts of physics. Thus our final expression for the translational contribution to the free energy is

It seems physically reasonable for the factor h to be close to one for T c ;T,, and to tend to zero for T = T,. Such a function is zs

’ Surface curvature corrections the density of the nuclei-bubbles

1;

h(z)=O,

z, 1,

(2.48)

can play an important role in some situations, such as determining phase transition [Pethick, Ravenhall and Lattimer ‘“)I.

J. M. httimer

et al. / Hot, dense matter

665

where z = T/ T,, and 7, a numerical factor, is chosen to be 4, so that h(f) = 0.91. This expression varies as (T,- T)2 near T,, and the leading finite temperature contributions for T + 0 vary as T2. These properties ensure that ah/aT vanishes as T + 0 and T + T,, and therefore the total translational contribution to the entropy will vanish as T + 0 and as T + T,. The remaining contributions to the free energy are relatively simple compared with the nuclear terms, and we now turn to them. (ii) Alpha particles. Alpha particles are included as a separate nuclear species. Interactions between nuclei and cr-particles are taken into account by confining cu-particles to regions outside nuclei. Interactions between the outside nucleons and a-particles are also taken into account in a similar way, in that we attribute a volume V, to an a-particle and exclude the outside nucleons from that region. For this effective volume of the a-particle we have used the value V, = 24 fm3, which is obtained by calculating the scattering length aPOlfor the pa optical potential given by Satchler et al. 47). Since upu is defined in the p(~ relative-coordinate system, the excluded volume in space is then V, =$&u,~)‘. The a-particle has relatively few excited states, so we shall not include their contribution to the free energy. Treated as a classical gas, the free energy of the cY-particles is given by F.=n.(l-~)[T{ln[($$)3’2n.]+l}-B~],

(2.49)

where B, is the a-particle binding energy (relative to the rest mass of two free neutrons and two free protons). Under some conditions we shall find that the most favorable heavy nucleus has A - 4, and is therefore rather like an alpha particle. One might then ask whether it is consistent to include both a-particles and heavy nuclei in the calculation. We shall find that when A-4, a-particles are far more favorable than the nuclei and therefore the inclusion of them both does not lead to appreciable overcounting difficulties: (iii) Electrons. Electrons and positrons are treated as ideal Fermi gases, since for relativistic electrons, electron-electron interactions give contributions to the free energy of order (Y= e2/hc compared with those of an ideal gas. A more important contribution to the energy comes from electrons screening the Coulomb interaction between ions. For point ions and at zero temperature one finds in the Wigner-Seitz approximation [Salpeter ‘“)I W screening =‘k2 35

where kFT is the Fermi-Thomas

FT

r2C Wlattice,

(2.50)

screening wave-number, given by k’, = 4re2 an,/ap, .

(2.51)

666

J.M. Lattimer et al. / Hot, dense matter

An exact calculation

by Dyson 49) taking

the lattice structure

into account

gives for

approximation,

Wscreening/Wlattice = O.l1578k*~r~, but for our purposes the Wigner-Seitz whose numerical factor & is 0.11429, is quite adequate. When all

the protons

reside

a bee lattice

in nuclei

it is easy to show

(see appendix

A) that

/CANTS =

= (Z/4100)2’3. An interesting prob3( 12/7~)“~2”~ (Y,and therefore Wscreening/Wlattice lem arises when one takes the finite size of the nucleus into account. When nuclei begin to fill a significant fraction of space the lattice energy largely cancels the Coulomb energy of an individual nucleus. It is then conceivable that the screening free energy could be small compared with the lattice energy, but still significant compared with the total Coulomb energy. We calculate the screening contribution to the energy in appendix A, and show that it is reduced significantly when nuclei fill a substantial fraction of space, and that it is never of importance for the situations we encounter. One remark in passing is that the screening corrections are generally more important for matter having the relatively high proton fractions encountered in stellar collapse than in cold catalysed matter, which is very neutron rich. This is because Z can be significantly larger in the former case, as we shall explain in sect. 5. The thermodynamic

properties

of electrons

and

positrons

may

be calculated

simply when particles are very relativistic and the effects of interactions can be neglected [Bludman and Van Riper ‘“)I. This is the case if pY, > 10’ g cme3 or Ts 1 MeV, which is satisfied under all conditions of interest to us. The free energy per unit volume of electrons and positrons is (2.52) The total electron of positrons)

number

density

(that is, the number

of electrons

minus the number

is n, = ne- - n,+ = q&d

+ n-*T*PJ

,

(2.53)

where (2.54)

qe = g,/6m2(fic)3, and g,(=2) is the spin degeneracy factor. solves the cubic equation (2.53) for we:

To calculate

Fe as a function

of n, one

(2.55) where a = n,/2q, and b = [a’ +&rr6T6]“‘, and inserts the result into (2.52). From the condition of charge neutrality, the proton fraction YP = nP/ n is related to the electron and positron fractions by the equation nY,=n,=n,_-n,+=nY,_-nY,+

(2.56)

J.M. Luttimer et al. f Hot, dense matter (iv)

667

Neutrinos. If the proton fraction of the matter is known, it is generally rather

unimportant how one treats the neutrinos in calculating thermodynamic properties of the matter since they contribute little. We shall treat the electron neutrinos as ideal two-component fermions, and when it is desirable to include their effects, we shall assume them to be in thermal equilibrium. The free energy density and number density are given by the equations corresponding to those of the previous section, but now involving neutrino variables, with the spin weight factor g, = 1. (u) Photons. The photons, which are always close to thermai equilibrium, give a contribution to the free energy and the pressure. The photon energies are shifted from those of free photons by plasma effects, but these have a negligible effect on the~odynamic properties of the matter since plasma effects are important only when photons contribute little to the thermodynamic properties. We therefore treat the photons as fully relativistic free particles. 3. Equilibrium conditions and physical quantities The equilibrium state is determined by minimizing the free energy per unit volume, keeping the total baryon number and, of course, the temperature fixed. Since we do not wish in general to impose the condition of @equilibrium we consider only variations in the parameters of the model that also keep the number density of protons fixed. As independent variables we may use A, Z, N,, nN, VN, npo, nno, and n,. Two constraints on these are the conditions that the numbers of protons and baryons given by eqs. (216a) and (2.17) be fixed, and thus six further conditions may be obtained by minimizing the free energy. There is a large amount of arbitrariness in the manner in which independent variations are chosen. One way to do this would be to take six of the eight independent variables and vary each of them in turn, keeping the seven other variables fixed. This procedure leads to rather lengthy algebraic expressions, which may, however, be simplified by evaluating various linear combinations. We shall adopt the altemative approach of choosing our variations of the variables so that they immediately yield simple equilib~um conditions with straightforward physical interpretations. For the first three of these we consider variations analogous to those made in determining the equilib~um conditions for coexistence of two bulk phases, as was done in our earlier calculations for nuclear matter 30). The first of these is that it cost no free energy to transfer a neutron from the outside of a nucleus to the inside, that is, the neutron chemical potentials pn in the two phases must be equal. The second condition is the analogous one for protons, that the proton chemical potentials, tip, in the two phases be equal. In practice it will be more convenient to use the condition that pn-pp be the same in the two phases. The third condition is that the pressures of the two phases be equal. In the presence of heavy nuclei we take as the first variation the transfer of a neutron to the nucleus from outside, keeping fixed the proton number of nuclei, 2,

668

J.M. Lattimeret al. / Hot, dense matter

the number of surface neutrons per nucleus, N,, the nuclear volume, V,, the density of nuclei, nN, the density of cu-particles, nu, the density of protons outside nuclei, npO,and the total number density of nucleons, n. In this change the density of matter increases due to the addition of a neutron, while that outside decreases. Note that the condition that the total proton number is fixed is satisfied by this variation. Mathematically the condition is (3.1) Because the total number density of protons is given in terms of their outside number density nPOby eq. (2.16a), we may write the variation in (3.1) as being carried out at constant nP instead of npo. The second condition is that the difference of the proton and neutron chemical potentials be the same in both phases. The particular variation we consider is the same as in the first case, in that we transfer a neutron from the outside nucleons to the nucleus, but in addition a proton is transferred from the bulk interior of the nucleus to the outside nucleons. We again keep N,, nN, VN, and n, fixed. The total number of nucleons, A, in the nucleus is unchanged in this operation, as is the density of nucleons outside. We may therefore write the condition (3.2) i.e. the free energy to change a proton into a neutron must be the same in a nucleus as in the nucleon liquid outside. The pressure equilibrium condition expresses stationarity of the free energy under variations of the nuclear volume. We choose to make the variation keeping fixed 2, A, N,, the number of neutrons outside nuclei, the number of protons outside nuclei, the number of rr-particles, and the number density of nuclei. The average number density of ar-particles is n,( 1- $, VN), since they have a density n, in the region outside nuclei. Because of the constraints on the total neutron and proton numbers and the other constraints we have imposed in making this variation, holding the numbers of neutrons and protons outside nuclei fixed is equivalent to holding nP and n, fixed, and therefore we may write the condition as =AP=O.

(3.3)

Three additional conditions determine the number of surface neutrons, the size of the nuclei (and hence their number density), and the density of cr-particles. The first is the condition that it cost no free energy to transfer a neutron to the surface of the nucleus from another part of the system. We choose to consider the addition of a neutron to the surface region from the interior of a nucleus, keeping fixed the number of neutrons outside nuclei, and the number of cY-particles. We also keep

J. M. Lather

et al. / Hot, dense matter

669

fixed the nuclear volume, the proton number 2, the density of protons outside nuclei nPO,which, together with the condition that n, be fixed, ensures that nP be fixed. Since the neutron number N is fixed in the variation, the condition that the total density of neutrons n, is fixed is sufficient to ensure that nno is fixed. The condition may therefore be written (3.4) The next condition determines the nuclear size. We demand stationarity under the variation of the free energy with respect to the nuclear volume, keeping fixed the fraction of space occupied by nuclei, nN VN.In addition the densities of neutrons and protons both inside and outside nuclei are held fixed, as are the total number of surface neutrons per unit volume of matter and the density of cY-particles. One way to express this condition is as follows: =o. n~V~,z/A’,n~N,,n,,,n,,,n,,n

(3.5)

Clearly any variable specifying the scale of the nuclei (A, 2, V, = l/ nN) could equally well have been used as the independent variable. The great advantage of holding constant the particular quantities specified is that properties of bulk matter inside and outside nuclei do not enter into the equilibrium condition, which expresses the fact that the equilibrium size of nuclei is determined by competition between various finite nucleus effects, such as those due to Coulomb interactions, the presence of the nuclear surface and the translation of nuclei. The final condition is that the free energy be a minimum with respect to converting two protons and two neutrons into an a-particle. The particular variation we consider is to convert two neutrons and two protons outside nuclei into an a-particle, keeping nN, VN, N,, A, 2, np and n, constant. Thus we have = 0. n~,h,+4,A,Z~,&

(3.6)

The variations we have chosen to perform may at first sight appear rather complicated. However, the reason for using these variations rather than simpler ones is that they lead to especially compact expressions for the equilibrium conditions. Once the configuration that minimizes the free energy has been determined, the thermodynamic properties of the system may be evaluated in terms of the free energy and its derivatives. The total proton and baryon number densities are given by (2.16a) and (2.17). The neutron and proton chemical potentials may be determined by evaluating the change in free energy when a single neutron or proton is added, and the total

J.M. Lattimer et al. / Hot, dense matter

670

pressure

P may be calculated

from the thermodynamic n2

P=

(@F/n)> an

where the Yi = nil n are the fractional entropy per unit volume is given by

yp,

relationship (3.7)

9

Y”,

compositions

ye.Y” of the various

components.

The

(3.8) from which one can evaluate

the internal

energy:

E=F+TS. We now evaluate contributions properties. First let us consider potentials

inside and outside

to the equilibrium conditions and thermodynamic the condition of equality of neutron chemical

nuclei,

Apu, = (Apn)b”‘k +(ApJurf The bulk term, obtained

(3.9)

eq. (3.1). In an obvious

notation

+(A,un)=‘” +(Apn)fra”s +(Ap,,)*

by differentiating

the bulk free energy

one may write = 0.

(3.10)

(2.19), is (3.11)

(‘+‘L,)bu,k = P!-?k - CL::lk , where

(3.12) and similarly and i and

for pzilk. The tilde indicates

o denote

quantities

the bulk

to be evaluated

free energy

for neutron

inside and outside the nucleus, respectively. The surface term vanishes, as one can see from the following the number

of neutrons

in the surface

per unit volume,

and proton

is being held fixed, according

densities

argument.

Since

to the variation

(3.1) that we have chosen, the only possible source of variation of Fsurf, eq. (2.22), is due to variations of E.L,and of cr. The quantity u is a function of only two independent variables, ps and the temperature, and since it satisfies the equality (3.13) (see eq. (B.22)), we find that Fsurf remains unaltered by the variation. This invariance and fixed T will of Fsurf with respect to pu, at fixed number of surface neutrons simplify considerably the discussion of a number of the later equilibrium conditions. The effective charge Z,, (eq. (2.33)) and the nuclear radius and nuclear separation are unaltered by the variation, and therefore the Coulomb energy is not altered, term, the effective mass of the nucleus and ( Ap,JcO” vanishes. As for the translation is given by eq. (2.45), and transferring one neutron to the nucleus from outside

J.M. Lattimer et al. / Hot, dense matter

671

changes M* by an amount SM*=

(3.14)

The variation offfin (eq. (2.46)) is easily evaluated, and we must also take into account the dependence of the suppression factor h(T) on the proton concentration Xi of the nucleus, through the dependence of T, on this quantity, which will be altered by the variation considered. We find

(4drans = -%

A+f& o

VN

l-+-

v

v,-v,

N

From eq. (2.49) it is easy to see that (ApJ simply bulk Pni

&lk

+

h&T’ xi ah ftrans 1 (3 15)

7GiTF~T

*

vanishes. Thus the condition (3.10) is

(ApJranS

=0,

(3.16)

with ( Ap,Jtrans given by eq. (3.15). We now turn to the condition (3.2) for the difference between neutron and proton chemical potentials inside the nucleus compared with that outside, which we write as A(~u,-~p)=A(~,-~~)b”‘k+A(~,-~~)s”rf+A(~,-~~)CoU’+A(~,-~p)franS

(3.17)

+A(~n-pp)m=O.

The bulk terms are straightforward to evaluate, and the surface terms give no contribution. The Coulomb term leads to a contribution due to the change in Z,,. We may write eq. (2.33) as vNzout

z,,=z--

v,-

(3.18)

v,’

where Z,,, = [ nPo(l- n,u,) +2n_]( V,- VN) is the total number of protons per nucleus, outside nuclei. The variation we are considering involves removing one proton from inside the nucleus and adding one outside, and therefore dZ,,, = -dZ, whence (3.19)

dZ..=dZ(l+&)=&dZ. From eqs. (3.2) and (2.34) we therefore find

A bu, - up)-‘” =

=

(3.20)

---

VC

vc-v,

2ecd

z,,

a

(3.21)

612

J.M. Lattimer

et al. / Hoi, dense matter

Here = T$

e

is the Coulomb energy per nucleus, with 0 =fcoUl/ T (eq. (2.34)). The translational term has a contribution only from the dependence of the suppression factor h(T) on proton concentration, since in the variation considered the total number of nucleons within a nucleus and the total number outside both remain unchanged. We find (3.22) Finally, A(pn - pp)O vanishes, and therefore the condition (3.17) is

1 ah _Ls=o

bulk _~~~lk_~~b:lr_~~~Ik)-l_:N,~~~---Pni

A’ axi

h

*

(3.23)

The pressure balance condition (3.3) may be written Ap = Ap”“lk + ApsUrf+ &Jc”“’ + Apt”“” + &=” = 0 .

(3.24)

The bulk term is just the difference of the bulk pressure inside and that outside: Apburk= p;urk _ Pzrk ,

(3.25)

where P”‘k( n,, n,) = j$Iknn +j$5$

- S( n,, rIP) .

(3.26)

The surface term is easily evaluated from eq. (2.22) for the surface free energy, since we have chosen to make the variation keeping the total number of neutrons in the surface fixed. Also any changes in (+ due to variations of cc, cancel the explicit CL,variations, and therefore we find APsUrf= -2a/r,,

(3.27)

which is the well-known Laplace formula expressed in our notation. The variations of the Coulomb free energy (2.34) come from two sources, the explicit dependence on V,/ V,, and the dependence of Z,, on V, as indicated by (3.18). One finds ar AP Coul_ - __

ah

afcoul -_-

ar

afcoul ah ze,

(3.28)

673

J. M. L&timer et al. / Hot, dense matter

The pressure due to translational Apt’““”

=

while the a-particle

h(T)

(

&T

motion is easily evaluated, and is found to be

p T&y3

VC nb -A+vnbVN V,-V,

contribution

’

Vg3-pVg3

is AP” = -n,T.

(3.30)

The pressure balance condition therefore reads

,

-GutVc 2ecoul +(v,-VN)=

z’,

+h

-- VC

tvTA+Ij;lby,

V,-VN

p TV;;“’ V~3_pV~3

The condition (3.4) has only bulk, surface and translational reduces simply to bulk Pni

xi _ -cL”-A’ax,h.

ah

1

-naT=O*

contributions,

ftrans

(3.31) and

(3.32)

This is the condition for equality of the neutron chemical potentials in the surface and the interior of the nucleus. From it one can see that the chemical potential to be used in evaluating surface quantities differs slightly from the bulk chemical potential in the interior of the nucleus. The equilibrium conditions for the differences of neutron and proton chemical potentials and pressures across the nuclear surface produce values of these quantities which are all modified from the plane-surface results, of course. Let us write eq. (3.5) determining the nuclear size as --V, nN

aF (

avN

~A~~‘k+Asu~+Acou’+Atra”s+A~ )

~0.

(3.33)

~N.VN,Z/A’,~N,N,,~,,,~~.~,,~

The contribution from bulk terms, Abulk, vanishes, since the proportion of the bulk phases and the densities of neutrons and protons in them remain unaltered by the variation considered. This is one of the great advantages of making the variation keeping the total number of neutrons in the surface fixed. The surface term may be evaluated easily; again terms from CL,variations cancel, and we find simply

A s”rf= -f(4m&)

The contribution

A”‘“’ is best obtained indirectly,

.

(3.34)

as follows. From the Coulomb

674

free energy

J. M. Lattimer

(2.34) (which

et al. / Hot, dense matter

as given is per unit cell), we find A

Coul

= -

= The Coulomb contribution may be written

TB(T) +$TF

aW)

-fed +2ecoul .

to the entropy

per nucleus,

(3.35) another

needed

quantity,

scou1=

Wr)

=-e(r)+r~.

(3.36)

Thus we find A

Cod = 2 &ou,

The translational contribution is easily evaluated to V, for the variation we consider. We find A The a-particles

trans

=

give no contribution

(3.37)

+$fcoul .

since M* and A are proportional

-f;,,,-thT-$g. to A, and therefore

the equilibrium

condition

(3.33) is (3.39) This is the generalization to finite temperatures of the result obtained by Baym, Bethe and Pethick9) that in equilibrium the surface energy is twice the Coulomb energy. Note that when the neutron skin is taken into account it is the surface thermodynamic

potential

a, and not the surface

free energy,

ysurf, that is important.

Under some conditions the extra T-dependent terms occurring here, especially&,,, and the last term, modify dramatically the equilibrium state, as we shall describe in sect. 6. One advantage of the particular variation we consider in (3.6) for determining the number of cY-particles is that there are no Coulomb, surface or translational contributions. The Coulomb contribution vanishes because of our assumption that the densities of protons and cy-particles are uniform outside nuclei, and therefore the Coulomb energy of an a-particle is equal to that of two protons outside nuclei. By straightforward differentiation we find 2(piIlk

+piU,lk) - T In

(3.40)

J.M. L&timer et al. 1 Hot, dense matter

67.5

The interaction between cy-particles and the nucleon vapor is contained in the last term of (3.40). The equilibrium state is determined by solving simultaneously the six conditions (3.16), (3.23), (3.31), (3.32), (3.39), and (3.40), together with baryon and charge conservation. We now describe how thermodynamic quantities are evaluated. First of all we consider the neutron and proton chemical potentials, which are the changes in the free energy when a neutron or a proton is added to the system. In equilibrium the neutron and proton chemical potentials are the same everywhere (although in each region they differ from their bulk values), so we may choose to add the particles in such a way as to produce simple expressions for the chemical potentials. The choice we make is to add neutrons and protons outside the nuclei, keeping V,, nN, A, Z, N,, and n, fixed: 1 (3.41) I&= nN( vc - v, - vol) V,.n,,A,Z *N I? n u. n p ’ 1 i+=

nN(VC-

VW

(3.42)

V,)

V,,n,,.%z,N,,n,.n,

(The prefactor is needed because npOand nno are the densities of the protons and neutrons outside. Part of the volume V, = (V,- V,)n,v, is filled with a-particles, and is not available to them.) The neutron chemical potential has only bulk and translational contributions, and we find

The proton chemical potential has in addition Coulomb and rest-mass contributions+: 1 VIV -~ 2v, eooUi pp = pLbpfk- ;hTv -+(m,-m,)c2. (3.44) A+fvnbVN

V,-

V,

V,-

V,

Z,,

The Coulomb contribution is just the same as that part of the contribution to A(p,,-pp) due to variations of Z,,,, (see eqs. (3.18)-(3.21)). Having obtained the chemical potentials we can now evaluate the pressure from eq. (3.7). Because the free energy is stationary with respect to many variations of the configuration of nucleons there is a large amount of arbitrariness about how one evaluates eq. (3.7). If one does this by varying V,, keeping V,, A, Z, and N, fixed, one finds T 1-U 1 %“, u P=p~‘k+nJ+~ ---2 -fr- n,,3 +AuA(M+4) 3 -Gr (l-u) vc I C +n,WT)

_$,,3-1

’ We caution proton-neutron

3 SM* 1 2 M* l-u

the reader that the proton mass difference term.

chemical

1 [ 1 +p,+p,.

potential

defined

(3.45) in ref. 30) does not contain

the

J. M. L&timer et al. / Hot, dense matter

676

This result can also be derived from the chemical potentials and the free energy, using the thermodynamic identity P= -0 = p”n,+p.,n,+pen,+p,n,-

(3.46)

F.

To evaluate the entropy, given by eq. (3.8), we may keep A, 3 N,, nN, VN, npo, nno and n, fixed since in equilibrium the free energy is stationary with respect to variations of the latter quantities. We therefore find s =

+ &f

nN(hulk

+ koul

+ had

+ &

+ &

The entropy per nucleus in bulk matter is I

sbulk

=

-

(3.47)

+ &.

_

vN~(n.i,n,i)+(vC-vN-v,,~(n,,,n,,)

1.

(3.48)

The quantity sSurfis to be evaluated at phase equilibrium, with p-L,held constant:

and the Coulomb contribution

is given by (3.36). The translational

term is (3.50)

Strans = -

and that from cr-particles is (3.51) The electron and neutrino entropies are the values for free particles: S,,, = qJ(&

+&r2T2) ,

(3.52)

where qe,” are the factors (2.54) related to the spin degeneracy. The analysis of this section, by minimizing the free energy with respect to appropriate variables, has determined the properties of the most probable WignerSeitz cell in the matter. The expressions we have just given assume that matter is composed entirely of such cells, all identical. The presence of thermal fluctuations means that there is in fact a distribution of cells, and there are therefore fluctuation corrections to these thermodynamic properties. In appendix D we summarize a discussion of fluctuations appropriate to this problem, and give expressions for the corrections. 4. Bubbles Up to now we have assumed that matter consists of blobs of nuclear matter (nuclei) immersed in less dense nuclear matter. When the denser phase occupies

AM. L&timer ef aL f Hot, dense matter

677

more than about half of space it is favorable for nuclei to turn inside out and form a state consisting of a denser phase with isolated regions of less dense nuclear matter (bubbles), since such a rearrangement reduces the surface and Coulomb free energies. his possibiiity was first suggested in refs. 9*36).The physical situation is analogous to that for nuclei, and we denote the bubble volume by V, and the total volume per bubble by V, = l/nu, where na is the density of bubbles. We again allow for the possibility of cu-particles being present in the less dense phase, which in this case is the phase inside the bubble, whereas in the nuclear case it is the phase outside the nucleus. If the number density of a-particles inside bubbles is denoted by n,, and the number densities of neutrons and protons inside bubbles (in the regions not occupied by cY-particles) by a,i and hpi, the total number of protons associated with the bubble is given by Z = VahLi

(4.1)

and the total number of neutrons (including those in the surface) by N=N’+N*.

(4.2)

Here N’= V,n~i

(4.3)

is the number of neutrons in the bulk, and N,=4?r&v,

(4.4)

is the number of neutrons in the surface. The average densities of neutrons and protons inside the bubble are given by nbi = ?Zpi(1 - ??aUa)+2h, 3

(4.5)

?Z~i=&i(l-?8*et,)+282~*

(4.6)

The bubble radius, rB, is related to the bubble volume by VB=+;.

(4.7)

The total proton and neutron densities are given by ?lp=Znu-t(l-q)n,,

(4-g)

n,=Nrl~-t”(l-q)n,,

(4.9)

(cf. (2.16a) and (2.16b)), where 4” Va/Vc

(4.10)

is the fraction of space occupied by bubbles. The total baryon density is given by n=An,+(l-q)n,, where A = 2 +N, as in the case of nuclei (2.10).

(4.11)

J.M. Lattimer et al / Hot, dense matter

678

The various contributions and the nuclear contribution

to the free energy may be expressed may be written as

as in eq. (2.1),

(4.12)

.&UC= %(fb”lk +Lurf +fcou, +_L,,,) (cf. eq. (2.2)). The bulk term is given by j&k=

Va(l

-%ua)E(Gi9

npi)

which differs from (2.19) only because of outside

the nuclei.

The surface

+tvC-

VB)F(nno,

$0)

3

the cY-particles are inside the bubbles

(4.13) instead

term is given by

_Lf= 4diYsurf,

(4.14)

where ysurf is given by (2.21). Note that the surface energy is to be referenced in terms of the proton concentration of the denser phase, in this case the “outside” one, but in the nuclear case the “inside” one. Also a correction is to be applied because of the difference between the outside pu, and the plane-surface value. The Coulomb free energy of a collection of charged spheres depends only on their effective charge and the fraction of space occupied by them, and is independent of whether they are bubbles or nuclei. For bubbles the effective charge is given by (4.15)

GE = 2 - VB$o 2 which differs from (2.33) only because there Thus the Coulomb free energy per bubble is

are no cY-particles

outside

fcou,= mT, WV,), where

r is related

(4.16)

to Z,, by eq. (2.36), and 13is defined

For the translational

free energy

a possible

procedure

by eq. (2.34). would

be to take

.LS=~(~/7.(x,)){~(ln[(~)3’z(v~,3_~v~,3)3]-l)}. This differs from the corresponding

result

First of all the finite size correction

has been

bubbles.

(4.17)

for nuclei

(2.47) in a number

neglected.

term for bubbles analogous to (2.46) for nuclei it nucleons outside bubbles (not including the surface) values of A,,, we find, it would have little effect on that the suppression factor h is to be evaluated for sponding to the denser phase, that is, the “outside” The effective mass is given by M* = (A +fvV,n,)m,

of ways.

If we were to include

a

would

involve the number of instead of A, and for the the results. A second point is the critical temperature correphase in the case of bubbles. A,,,

(4.18)

which differs from (2.43) only due to the substitution of V, by V,, and of n; by n,. The latter replacement follows from the fact that for the bubble case there are no cY-particles outside nuclei. Because, however, of the relative uncertainty of

J.M. Lattimer et al. / Hot, dense matter

619

these conjectures, in the calculations reported in this paper we have taken f,,,, = 0 for bubbles matter. The equilibrium conditions may be evaluated in the same manner as for matter containing nuclei. The condition for equality of neutron chemical potentials inside and outside bubbles is

Pni -ccm -2 bulk

bulk

3h

T

VB

1 -fV---

A+$vn,VB

Vc-

; xo WxJ .Ls 1

VB

--=o, 8x0

Am,

(4.19)

hbo)

compared with eqs. (3.15) and (3.16) for the nuclear case. The condition for the equality of pn-pLp is bulk _p~_(CL~~lk--pb$)_~ Pni

l_i,v

(4.20)

+k?!$dk=o, B

C

Cl

0

instead of (3.23). The pressure balance condition is ppU’k _ p:lk

2+3)

_ -

rB

ZO”,vc

2%xll

+ z,,

+${fy(l-~)+“($*[$$+Tt$(r)]} pTvg213

-_ vc

no

vB)2+h 3vT A

(V,-

A+Vil,V,

Vg3-pVz3

v,-vB

1

+n,T=O,

(4.21)

compared with (3.31) for nuclei, and the surface chemical potential condition is (4.22)

compared with (3.32) for nuclei. Finally the bubble size equation is $4rr&(xJ

-

($Wou~

+$Ld

+_A,,,

+%(x0)

T=

0,

(4.23)

which differs from (3.39) for nuclei only due to the neglect of the subtraction term. For the thermodynamic quantities we find

/l,=CL~U.lk-th(X.)TvA+?~n $-y,

(4.24)

v

2

p-cp=

b”‘k-$h(x,)Tv

PPO

oBC

1 A+$JTt,V,

B

-_--

vB

v,-

2eCoul v,

z,,

% v,-

(4.25) v,’

1 (4.26)

s = nB(&lk

+

brf+

sCoul+ %?.d+ &

+ se+SW3

(4.27)

680

J.M. Lattimer et al. 1 Hot, dense matter

where q = V,/ V,. These equations should be compared with eqs. (3.43)-(3.47) for nuclei. The, various contributions to the entropy are given by

(4.30)

(4.3 1) (cf. eqs. (3.48)-(3.51) for nuclei). The quantities scoUl, S, and S, are given by eqs. (3.36) and (3.52), as in the nuclear case. 5. Results

In the previous sections we have considered the problem of determining the thermodynamically most favorable state of matter containing a given density n, of neutrons and a given density nP of protons, that is, for an electron fraction Y,= n,/( n, + nP). In the later stages of stellar collapse, neutrinos are trapped in matter, and therefore the total lepton fraction YE is fixed, but not the separate electron and neutrino fractions. In this paper we shall present results for fixed lepton fractions. (Results for fixed Y, will be presented in a later paper.) Here we shall further assume that the neutrinos are in thermal equilibrium, and that the matter is in P-equilibrium. The equation expressing the latter condition is 42=P”-PrT=Y,-&.

(5.1)

We remind the reader that pn and pP are both measured from the neutron rest-mass energy, while pu, contains the rest-mass contribution: consequently no rest masses occur explicitly in (5.1). The actual lepton fractions we consider are Y&=0.4, 0.3 and 0.2, which span the range of values of interest in stellar collapse. Another case we shall consider is that of P-equilibrium in the absence of neutrinos. The proton fraction is then determined by (5.2) k=cLe. The properties of matter under these conditions are of interest in investigating the behavior of hot neutron stars. The properties of matter depend rather sensitively on the proportion of matter in nuclei. In the phase with nuclei, the fraction of nucleons in heavy nuclei is given by XH=n,A/n,

(5.3)

J.M. L&timer et al. / Hot, dense matter

681

while the fraction of nucleons in a-particles is

The total fraction of matter in nuclei, X,,,,

is given by

X Nuc=XI-I+Xor.

(5.5)

In the bubbles phase the fraction of matter “in heavy nuclei” is defined as the fraction of nucleons in the bulk matter outside the bubbles plus the fraction in cY-particles, which occur only in very small numbers: X” =

[n,(l - 9) ++?&21/~*

(5.6)

Numerical values of some of the significant physical quantities are given in table 1. They have been taken from the fine grid (see appendix C) used to prepare the contour plots we describe later. They are included to provide precise benchmark values, but they are for rather widely spaced points in (n, T). To show details and to exhibit trends we have made contour plots, and it is to these that we shall make reference in the rest of this section. In figs. 2a-d we show contours of XNuc and X, in the (n, T) plane. The treatment of the a-particle as a separate component of the matter is necessary because of its special role in astrophysical processes. We can observe later, however, that in the region where X, becomes large the A and 2 of the ‘heavy nucleus’ component tend to small values -4 and -2 respectively. This and the smoothness of the X,,, contours, which sum both components, is additional justification for our allowance for the reduced level densities of light nuclei by introducing the factor&,, eq. (2.46). For these reasons we plot X,,, contours rather than those of X,. By comparison with our earlier bulk-equilibrium results 30) it may be seen that the regions in which an appreciable fraction of the nucleons reside in nuclei have the same qualitative shape as the boundaries of the two-phase regions in the bulk-equilibrium approximation. As a consequence of finite-nucleus effects, however, nuclei are seen to dissociate at somewhat lower temperature than that approximation suggested. At lower densities, the fractions of matter in cu-particles is seen to increase rapidly as the temperature rises. This reflects the dissociation of heavy iron-peak elements into cr-particles. As the temperature increases further, the X, and X,,, contours become essentially coincident since there are very few heavy nuclei present, and X, (and therefore also XNuc) then decrease with increasing temperature as n-particles are dissociated into their constituent protons and neutrons. In order to understand the behavior of X,,, and other variables at low temperatures and low densities we need to consider the behavior of b. In the region where most of the nucleons are in nuclei, fi must be a function mainly of xi, depending little on density. Some arguments can be made on the basis of this

J.M. Lattimer et al. 1 Hot, dense matter

682

TABLE

Numerical

values

T [Meal

-4.5

.75

Y, log

-3.704

-4.91

-3.09

-3.5

-3

.374 -3.051 -.43

-2.5

T and log,,, (n(fmm3)),

-2

.347

.361

.334

-2.396

-1.739

-1.082

3.38

8.88

16.81

.383

.242

.307

-1.5

-0.5

.323 -.425 28.27

.187

.316

.307

.196

1.550

44.36

.I44

.I18

14.39

17.28

20.61

24.45

29.16

36.91

hl

-5.40

-4.81

-4.11

-3.29

-2.48

-1.77

-1.28

-2.47

.365 P

.363

-4.221

-3.659

e

-I .62

-1.60

s

3.177

1.668

;I

6.19

8.62

-7.31

Y,

,324 P

-6.41 .319

.360

.353

.343

.333

-3.031

-2.386

-1.734

-1.079

.43

3.94

9.25

17.07

1.098

,564

.780

-5.28

-4.13

-3.05

-2.13

-1.49

-2.59

108.80

.322

.327

.329

.327

9.19

8.52

7.91

7.75

11.10

18.20

s

8.403

6.362

4.343

2.373

t

3.55

4.27

6.44

e

.320

-8.67 .312

-3.42

-2.25

-3.04

108.80

.312

.318

-1.034

14.51

14.14

14.98

15.09

15.60

20.31

4.383

L

4.40

4.74

5.33

7.56

.315

-9.96 .309

2.530

-3.36

-3.76

108.70

.3ll

24.65

18.93

19.31

20.79

21.97

11.740

8.843

7.104

5.482

;1

4.80

5.67

6.11

7.14

.324

-14.68 .314

3.759

108.50

38.20

28.67

28.90

31.27

11.360

11

4.12

6.65

U”

-59 .Ol

log

P

e

-2.242 520.00

.436 -2.165 182.00

-37.02 .370

-17.75 .316 -1.145

77.81

50.16

46.92

22.900

12.310

;

2.13

5.16

9.35

-92.98

10.47

-1.617

54.190

-111.60

.328

5.024

-1.975

s

h

-26.86

-74.16

8.344 12.31 -56.28

.513

-4.73

18.900

,486

93.03

-4.73

79.70

Y,

1.003

-6.86

.308

s

-47.71

47.31

-9.63

e

8.76

1.350

1.554

43.22

-1.340

8.00

32.40

.307

.2lO

36.74

-1.852

6.587

2.121

.316

25.88

-2.350

8.350

23.84

.317 -.388

17.31

-2.760

P

.316 -.983

10.64

-3.005

log

.386

-5.10

s

-21.01

92.37

-7.15

e

.356

.753

43.24

-1.492

-27.90

46.01

.307 1.553

36.76

-2.003

.422

.993

.204

26.84

-2.522

-35.07

30.52

.316

19.05

-3.025

U”

1.457

.319 -.406

12.28

-3.444

Y,

.321

-1.611

.326

.258

-5.08

-2.120

.358

91.89

-6.78

6.266

P

.499

43.26

-2.632

-14.04

45.07

36.83

-3.150

-18.96

.648

.307 1.551

27.92

-3.641

-24.20

29.22

.I99

21.44

7.932

Y,

.903

.316

15.36

9.824

log

1.357

-321 -.417

10.11

s

%

.I29 43.27

e

.334

91.60

36.90

-1.064

P

.245

28.86

-1.700

Y,

1.550

23.75

-2.294

log

.307

.197 44.50

.309

108.80

.316

19.38

-2.813

-11.28

28.45

.064 43.27

15.42

-3.327

-14.31

-.424

91.52

II.80

-3.626

U”

.323

.412

for Yt = 0.4

-1

i2.13

log

15.00

on a grid of temperature

10.34

%

9.00

quantities,

II

log

6.00

.440

.544

Y,

4.50

.386

-4.355

s

3.00

-4

.395 P

e

1.50

of physical

la

6.354 13.99 -39.80

.310 -.845 33.52 3.428 15.51 -11.26 .312 -.651 49.67 4.810 17.04 -25.70

.315 -.330 38.14 2.109

.317

.307

.227

1.560

50.93 1.488

94.90 .762

24.61

36.93

43.17

-7.92

-7.46

108.20

.313 -.180 52.45 3.332 24.59 -16.99

.318 .328 60.05 2.269 38.86 -12.95

.307 1.574 100.50 1.227 43.14 106.80

J.M. L&timer et al. / Hot, dense matter TABLE

683

lb

The same as for table I a for Yr = 0.3

T NW

-4.5

.75

Y, log

.305 P

e s

1.50

-3.00

-1.21

1.143

.a33

1.48

5.36

10.94

18.94

.611

-3.10

-2.31

-1.58

-.78

P

-4.285

-3.739

-1.56

-1.67

s

3.599

t

8.06 -6.55

Y,

.257

2.241

.2aa -3.132

.283

.436

109.10

-.28

2.13

5 .go

11.23

I .608

1.158

.809

20.51

25.80

-5.23

-3.77

-2.30

-.a4 .264

.544 31.92 .58 .262 -1.194

e

7.93

7.13

6.30

5.40

7.56

12.37

s

8.199

6.305

4.443

2.566

1.569

;

4.55

5.44

8.14

.273 P

e

.253

.245

.246

.252

.256

13.26

12.19

12.50

12.10

11.60

14.34

5.71

6.23

6.93

9.69 -9.13 .243

2.619

1.556

e

23.87

16.95

16.48

17.17

17.38

17.64

s

11.620

8.618

6.896

5.355

;

6.11

7.51

8.12

9.37

log

-13.95

.249

3.734

25.74

24.68

25.81

11.250

;

5.17

8.58

log

P

e

-2.242 520.20

.414 -2.165 182.10

-36.21 .329

-16.61 .252 -1.188

76.67

46.15

40.57

22.900

12.240

;1

2.68

6.52

12.06

-92.40

13.96

-1.642

54.200

-111.30

.270

4.880

-I .980

*

hl

-25.97

-73.21

8.143 16.58 -54.98

-.515 22.96 1.376

1.503

31.72 .743 49.90 .91

70.34 .376 58.04 109 .oo

.253

.243

.060

I .505

32.98 .904

70.99 .499

108.80

37.05

11.80

2.176

.254

.243

.048

-.03

18.890

6.376

1.038

.252

-I .47

79.66

-47.07

.251

-4.69

.243

s

.476

69.00

-a.44

e

-58.59

.497

58.02

-I .397

%

30.81

49.81

-1.898

Y,

1.502

35.11

-2.382

10.74

.243

.040

22.99

-2.771

8.126

.253

13.79

-3.006

P

21.18

-.06

.251 -1.081

.262

.255 -.542

-2.73

.245 -1.564

-20.43

.702

-5.70

-2.061

.308

19.91

36.47

-2.568

-27.36

.257 -.562

25.53

-3.050

.394

109.10

15.98

-3.455

-34.63

2.05

109.10

II

hl

1.89

1.61

4.377

Y,

58.06

1.09

6.128

P

50.28

-.89

7.713

log

.I26

39.32

-3.24

-1.149

.249

69.60

-5.61

-1.700

-13.53

.247

58.05

-2.186

.263

30.25

50.08

-2.686

-18.53

.361

1.501

37.95

-3.193

.309

19.14

.243

.035

28.71

-3.669

-23.80

-.576

.253

20.30

9.615

hl

1.040

.259

13.08

s

Y,

.063

2.17

-1.810

Y,

69.53

2.13

.268

-2.382

log

.121

1.09

-1.223

-7.97

30.11

.I1 .276

15.81

-10.85

1.501

58.07

-2 A78

-14 .oo

.243

.033

50.37

-3.380

“n

.253

39.61

-1.867

.260

-0.5

33.24

-2.505

.255

.I83

-1

28.39

11.60

.251

.290

-3.870

P

.260 -.579

“n

.290

.270

-1.5

-1.231

24.62

.289

.281

-2

-1.882

21.58

log

15.00

-4.20

-291

-2.5

-2.531

19.05

“n

9.00

-3.179

16.56

e

6.00

.290

-3.825

-3

i

log

4.50

.300

-3.5

-4.455

1.486

Y,

3.00

-4

6.131 19.13 -38.26

.246 -.921 26.43 3.362

.250 -.433 28.66 2.125

.252 .060 36.77 1.486

.243 1.510 72.80 .740

20.53

32.83

49.81

57.99

-9.42

-4.70

-3.41

108.40

.247 -.706 41.35 4.639 23.11 -23.71

.249 -.258 41.52 3.216 33.26 -13.89

.254

.243

.225

1.526

44.98 2.178

78.14 1.187

52.67

58.07

-7.66

106.90

J. M. Lattimer et al. / Hot, dense matter

684

TABLE

lc

The same as for table la for Yp = 0,2

-4.5

T [Meal -75

.224

Ye log

P e s

1.50

.I84

-2.058

-1.413

-.768

.61

3.01

6.56

.199 -3.330

-2.94

-2.05

-.99

2.079

1.537

21.33

24.33

un

-2.46

-1.52

-.65

,199

,199

.19a

1.048 27.79 .27 .IPb

-2.5 .I95

,653

-2

”370

4.76

6,52

107 .so

.I89

-.7f

-*91

-*13

1.29

3.52

6.91

s

4.334

3.052

2.278

l.h36

1.os9

;

9.12

13.91

18.69

24.05

-5 .a9

-4.34

-2.72

-1.07

e

6.82

s 7. ”

‘1.961 5.85

un

-13.71

Y, log

.212 P

e

‘ iao

.64 .I87

.f86

5.17

4.16

5.22

8.09

6.241

4.558

2.8SO

1.841

I.204

b.87

9.96

.182

-7.32 -174

-4.52

-1.65

,174

.178

.182

IO.31

9.69

3.m

9.96

4.365

2.159

1.695

-23.44 .x3

-18.11 .f98

-13.02 .178

-8.33

-4.29

.I72

.I73

23.51

fS.38

i3.90

13.99

13.61

13.14

11 .S3O

8.397

7.48

9.73

un 9.00

-34.24

Y,

.366

-26 .a4 .263

-19.85 ,199

-13.22 .178

-2.59

1.61

-1.944

-1 A55

e

79.93

36.54

23.44

20.87

20.99

$

18.890

IL.170

s

6.22

10.59

-XI.19

un Y, log

,467 P

c

-2.242 520.60

-46.49

.392 -2.164 182.60

14 -02 -35.44 -290

.2X3

-15.47

.I82 -I .228

16.37

43.35

34.95

22.910

12.190

ii

3.24

7 -89

14.81

-91.85

-2s .oa

la.33

-1.660

54.213

-111.10

15.77

4.701

-1.981

s

h

6.125

-12.35

7.968 21.42 -53.78

I A25

-7.29

-2.408

7‘909

15.72

44 .a4

-2.775

e

-.650

28.74

-3 .a05

log

2.288

.Iai

17.30

.I73

5.878 25.72 -36.71

,733

52.31 .3b3

107.30

s

12.16

20.42

5.44

d

10.73

1.o97

.173 t.444

2.95

.I78 -t.172

CJ

13.99

.181 -.13S

-.SS

-1.634

3.698

‘182 -.694

75.24

-2.11s

5.195

.243

64.70

-2.615

6.647

SL.86

46.36

-3.086

P

.49S

31.98

-3.461

log

19.53

.173 1.443

107.40

to.44

19.69

~181 -.15i

107.50

6.06

12.36

12.10

6.43

.I22 75.25

3.98

-1.263

a.96

64 “$4

51.59

1.25

-I -779

a.15

.24a

75.24

-2.250

7.21

.764

18.99

.173 1.442

64.77

-2.740

;

12.14

-.I62

47.73

-3.237

5.955

.I83 -.731

.181

35.37

-3.b90

7.4b0

,404

4.61

6 .O?

-10.43

11.93

2.52

-1.334

24 .a0

-.759

48.67

-I.918

IS.91

.I84

38.22

-2.461

9.428

Q

30.34

.682

-2.940

s

Y,

.184

,061

2.85

.193

e

.tn

51.52

1.38

-1.392

-3.430

.I22

75.25

-2.0!8

.I84

18.84

.I73 2.441

64.86

-2.631

-3.9I4

.I205

*tat -.I65

-0.5

48.89

-3.227

e

II.70

-1

39.01

-3.801

Y,

-1.5

32.35

-4.323

P

log

15.00

.189

-2.698

.203 -3.944

18.34

hl

6.00

-3

;

log

4.50

-3.5

-4.510

2.434

Y,

3.00

-4

,174 -.994 20.52 3.274

.178

30.83 -21.67

64.611 4.57 ,180

24 -92

2.138

-1 .a2

4.424

,962

2t*i3

-7.58

33.93

21.61

-.071

41.61

-.7bO

-.liS

-.536

26.32

.I76

“18L

-178 -.332 32.34 3.015 43.76 -10.65

1.407

,173 1.646 52.93 .482 75.24 107.20 -173 I.452 54.66 -713

64.89

75.31

2.05

106.70

“182 -130 32.65 2.059 69.30 -2.06

.I73

1.468 59.15 1.139 75.77 105.10

TABLE Id The same as for table la for &equilibrium with no neutrinos

T NW

-4.5

.75

Y, log

.363 P

e s

1.50

-1 .oo

%I

-4.16

-2.35

.I?4

.155

-4.330

-3.820

e

-.kl

-.39

6

4.543

3.431

P

.I12

1.528

.123

1.65

3.30

5.81

9.58

.087

.a09

.410

.204

92.70

121 .oo

2.53

4.98

a.31

14.26

97.53

.059

,043

.034

.45

1.37

2.33

3.73

6.03

9.71

2.808

2.150

1.420

-2.14

-.I2

.053

,043

.789

.402

91.81

121.00

2.09

4.79

8.22

14.11

97.51

*Ok5

.Okl

.033

-2.521

-2.024

-1 .soa

-.9a5

e

6.02

5.05

4.48

4.24

4.25

5.13

7.00

8

7.705

6.075

4.671

3.417

2.284

1.418

L

6.98

9.60

Y, log

.230 P

.130

-6.44 .07

-2.99

1

,044

-3.707

-3.272

-2.809

-2.333

11.61

30.82 .39 .034 -1 .a51

.823

7.79

13.83

97.45

,036 -1.373

.035 -.897

7.20

6.97

7.18

8.36

5.607

4.227

3.012

1.965

1.188

;

6.74

9.93

%

-23.54 ,318

-17.82 .i96

-12.38 ,109

-7.18

-2.13

.061

‘040

13.48

97.35

.03a -.809

e

20.91

14.36

11.68

10.33

9.60

9.28

9.93

s

10.950

8.172

3.515

2.363

1.485

t

5.87

9 .a4

“n

-34.70

Y, log

.430 P

-26.82 .316

-19.34 ,197

-12.24 .112

26.76 -5.40 .065

41.17 -60 .046

-3.084

-2.813

-2.428

-I .9a7

-1.527

-1.075

e

64.22

32.28

21 .a2

17.63

15.59

14.28

8

16.610

10.540

;

3.80

a.28

hl

-59.15

Y, log

,494 P

e

-2.351 400.00

-47.16 -446 -2.254 144.10

7.672 14.15 -35 .k2

.349

.230

-13.89 .137 -1.260

65.13

40.22

30.93

19.540

11.210

c

1.60

4.53

10.73

-93.25

-24.31

28.69

-1.682

43.530

-111.90

20.74

4.320

-2.033

8

h

5 .a04

-73.66

7.678 19.94 -54.12

5.607 31.02 -35.77

.316

7.03

.037 -1.260

19.19

35.48

2.57

-1.721

1.4.16

.638

121.70

-2.195

4.825

10.92

.025 1.301

95.04

-2.661

6.322

-.265

62.02

-3.105

P

-039

41.96

-3.496

log

.213

3.98

7.94

Y,

35.10

121.20

7.180

26.85

.45 1 91.53

9.15

18.53

10.20

.025 1.299

61.60

9.192

13.44

.034 -.302

44.56

8

e

.108

62.20

-3.009

19.97

34.86

45.80

-3.488

13.71

.232

.025

f .29a

34.73

-3.94’1

-9.92

-034 -.323

-4.03

.054

62.09

-1.054

-5.74

34.80

45.98

-1.648

VII

.025

i .29a

35.77

-2.224

26.78

-13.52

.I12

-2.762

20.30

.076

.035

.46

14.73

P

-0.5

-.322

10.09

.133

-033

-1

-1.075

.82

-3.289

.043

-1.5

-1.704

29.37

-.66

.064

-2

-2.329

1.337

24.30

-2.5

-2.868

Y

%

15.00

-3.02

18.94

.Y,

9.00

-4.82

13.63

log

6.00

-3.328

1.112

-3

.201

-3.821

.a04

Y,

4.50

,303

-3.5

-4.397

;

log

3.00

-4

3.031

45.02 -19.23

.41?

5.93

12.96

97.20

‘044 -.652 13.59 1.970

2.90

4.067

36.01

122.30

-4.58

26.58

.831 96.47

64.50

,084

11.93

,026

1.304

62.93

41.33

-.BlB

-04 1 -.221

.062 -.399 23.63 2.797

.Ok? -.I23 14.60 1.187

.028 1.312 37.48 .613

99 .k9

123.90

11.45

96.76

.061

,033

.068

1.333

21.82 1.794

41.90 .9a2

66.72

104.90

127.20

-5.72

6.60

95.19

Listed at each grid point are the electron fraction Y,, the log,, of the pressure P in MeV. fm-“, the intemai energy per baryon e in MeV, the entropy per baryon s in ka, the neutron-proton chemical

J.M. Lattimer et al. / Hot, dense matter

686

0 10'5

10-I

Fig.2s. Contours of constant X,,,

lO-3

IO+ 10-l n ffl8) (full lines) and X, (dashed lines) in the n - T plane for a lepton fraction Yc = 0.4.

10 T (MN 5

to-"

10+

10-3 n (fme3)

Fig.2b. The same as fig. 2a, but for a lepton fraction Yr = 0.3.

10-l

J.M. L&timer et al / Hot, dense matter

687

10 T WV) 5

_ 0 10 -5

ZP7_

.d,\b.~

10-4

I

I111111

1

1 I I 1111

I

10-l

10-z

1o-3

n (frnm3) .95 Fig. 2c. The same as fig. 2a, but for a lepton fraction

1

I1IIIII

Yf = 0.2.

WeV) 5

0

10-5

lo-’

10-j

1o-2

10-l

n (fmq3) Fig. 2d. The same as fig. 2a, but for &equilibrium

in the absence of neutrinos.

1

688

J.M. Larrimer et aL / Hot, dense matter

information alone. Here, however, it is necessary to have a more specific expression, and we draw on the semi-empirical mass formula. In this approximation, valid at low densities and low temperatures, we may write the energy per baryon of the optimal nucleus as e = eo+4S(X,-XJ2,

(5.7)

where e,, = -9 MeV, s = 20 MeV, and x, = 0.485. From this expression we obtain p”= e,+4s(x,2-xf) ,

(5.8)

b = BS(X,-Xi) .

(5.9)

Our parameters were chosen to make neutron drip, /.L,., = 0, occur at Xi = Xd = 0.35. If the electrons and neutrinos are assumed to be completely degenerate, their chemical potentials are given by /_Le = hC(3T2nY,)“3 )

p,, = hc(61r2nY,)“3,

(5.10)

where the one third powers are to be taken as the real solutions. These expressions allow us to write the P-equilibrium condition as A Y;/3-2’/3

Y;”

= (3T2n;,,3hc

(5.11)

.

We can use this equation to show that at sufficiently low densities all nucleons will be in nuclei (Xu = 1) for all Ye. For this to occur, we must have Xi = Y, > xd, so that p,, is negative. Since Y,, = Ye- Y,, the baryon density at which neutron drip first occurs is given by eq. (5.11) as 1 nd=3

8s 6

3 xs-xd x;/3_21/3( y,_xd)‘/3

> ’

(5.12)

For Ye = 0.4, 0.3 and 0.2 we obtain nd = 3.2 X 10d3 fm-3, 2.8 X lo-* fm-‘, and 1.7 X lo-’ fme3 respectively, while for the no-neutrino case nd = 1.3 X 10m4fmw3. The positions of the X,,, =0.99 contours in figs. 2b, c and d are quite consistent with these limiting densities. For the case Ye= 0.4, however, our crude approximation for pn is not accurate enough at 3.2 x 10e3 fm-‘, as may be seen from fig. 9a, and in this case neutrons at zero temperature do not drip at any density. From eq. (5.11) we also obtain the limiting behavior of matter at densities far below nd, and at temperatures such that the leptons are completely degenerate. Since the left side of the equation is finite, as n tends to zero Xi must tend to xs, the proton fraction of stable isolated nuclei with b = 0. All nucleons are in nuclei, so that Y, = x,, and the neutrinos and antineutrinos provide the correct Yr, with Y, = Yp -x,. Thus the nuclear part of the matter consists of the familiar iron-peak elements, unaffected by the surrounding leptons, whose Fermi energies are too low to disturb /?-equilibrium.

J.M.

Lattimer

et

689

al. / Hot, dense matter

From fig. 2d it is seen that for the no-neutrino case, nuclei are present in appreciable numbers at densities close to nuclear densities even at temperatures as high as 10-15 MeV. The quantity which determines the critical temperature is xi, and in this region fig. 4d shows Xi to be -0.2. It is this relatively large inside-proton fraction that enables nuclei to survive to such higher temperatures. While our treatment of cu-particles does include their interaction with the surrounding medium, their internal properties (e.g. B,) are assumed to be unaffected. This is expected to give reliable results at densities lower than the nuclear saturation density n,, where the dripped matter surrounding them has a much lower density than their internal density. The X, contours at n, and above indicate the presence of very small numbers of a-particles also in this region, but the actual values are not to be taken seriously, since the density of the surrounding matter there is comparable with or greater than the a-particle’s internal density. Since, however, there are so few a-particles, the thermodynamic properties in these regions are essentially unaffected by them. The other general compositional variable is the fraction of electrons, Y,, whose contours are shown in figs. 3a-d. Y, is controlled by the P-equilibrium conditions (5.1) and (5.2), which we have exploited in eq. (5.11). By looking at variations of this equation, one can easily see why, at T = 0, Ye should decrease with increasing

20 I I I I

I

150

200

I

I I

I

I

I I

I

10 T WV) 5

1o-5

lo-’

10-s

1O-2

10-l

n (fm-‘) Fig. 3a. Contours

of constant electron fraction Y. (full lines) and electron chemical lines) for a lepton fraction Yr = 0.4. Units of CL, are MeV.

potential

p. (dashed

690

J.M. L&timer et al. / Hoi, dense matter

MN I I

I I

I

I

0 1o-5

lo-'

10-2

1o-3

10-l

n (frnw3) Fig. 3b. The same as fig. 3a, but for a lepton fraction

Yp = 0.3.

I I I

I I I I I

10

I I

T

I I I

WV)

I I

5

I I I

1o-5

lo-'

1o-3

lo-*

10-l

n (frnm3) Fig. 3c. The same as fig. 3a, kut for a lepton fraction

Yp = 0.2.

J. M. l&timer et al. / Hot, dense matter

691

15

10 1 (W 5

n ;0-5

lo-'

10-3

1o-2

10-l

n (ftnw3) Fig. 3d. The same as fig. 3a, but for matter in p-equilibrium

with no neutrinos.

density in the density range where nuclei are present. From the fact that S Y, = -8 Y, at fixed lepton number one finds that the density variation and the Y, variation are related by the equation (5.13) if one assumes that fi is a function only of Xi,as for example in the simple expression (5.9). (The result (5.13) applies to the fixed Ye case, but is valid also for P-equilibrium with no neutrinos if one drops the pu, terms.) For fixed Ye, eq. (5.13) gives

*ye=

-sn n 1-4 Ye+kI

l-b-i&

Yv-3(axilaYe)(aiVaxi)*

(5.14)

CL,is greater than py, and therefore, since k decreases with increasing proton richness (for XiCO.5) and Xi increases with Y,, one sees from (5.14) that dY,/dn must be negative, so that Y, decreases with increasing density. When nuclei are no longer present the argument breaks down since then the density dependence of & must be taken into account. The value of Y, generally falls also with increasing temperature. This is due to the evaporation of neutrons, which at fixed Y, would lead to an increase in Z/A

692

J.M. Laftimeret

al. / Hot, dense matter

and a decrease in b. As a result p,, (and Yy) rise to compensate, forcing a decrease in Y,. This phenomenon continues until nuclear dissociation is complete and neutron evaporation ceases. At constant Yp, Y, is remarkably independent of temperature just below nuclear densities. To understand this behavior we again start from the P-equilibrium condition. Both electrons and neutrinos are very degenerate and we may neglect the temperature dependence of their chemical potentials. If one denotes by SY, and Sfi the finite temperature contributions to Y, and b for fixed density, one finds from (5.13) that (5.15) or (5.16) k depends rather weakly on temperature at densities near that of nuclear matter, as may be seen in the plots of fig. 9, which show that at fixed density b varies by at most -5 MeV. Thus SY, is very small, since pu, and py are typically between 100 and 200 MeV. To take a specific case of a density of 0.1 frnm3and Ye = 0.4, one has Y, = 0.3 and Y, = 0.1, p, = 190 MeV and pv = 160 MeV. From these quantities eq. (5.16) yields SY, = a;/750 MeV, which is less than 1%. The temperature dependence of k is small because it is the difference between the neutron and the proton chemical potentials, and since these have very similar temperature dependences there is significant cancellation in &. To see this explicitly, we assume that the temperaturedependent contributions to the neutron and proton chemical potentials are those of a free Fermi gas: one has ?T*P( E, - E&J

(5.17)

4EpEn

for the leading corrections, where E, and E, are the kinetic contributions to the neutron and proton Fermi energies. If we take a proton concentration Xi= 4 and assume the Fermi energies to be those appropriate for nuclear density, we find E,- 25 MeV, E, 2: 44 MeV, and Sfi = 0.04T2 MeV, when T is measured in MeV. Thus at T = 10 MeV, finite temperature contributions to & would be only -4 MeV, in order of magnitude agreement with what our detailed calculations show. At densities above nuclear densities, Skyrme-type effective interactions produce a rapid increase in L (see later discussion). This leads to an increase in Y,, and consequently there is a broad minimum in Y, at densities somewhat below nuclear densities. The composition in the region of high temperatures and low densities may easily be understood on the basis of the P-equilibrium condition (5.1). At high temperatures

693

J.M. Lattimer et al. / Hot, dense matter

the densities of neutrons and protons are related to their chemical potentials by the usual Boltzmann result which, because n,/ nP = ( 1 - Y,)/ Y,, produces the relationship 1-Y ;=Tlny+(m,-m,)c2. e

(5.18)

In this region the electron and neutrinb chemical potentials are small compared with the temperature, and therefore, from (2.53) and its analogue for neutrinos, we find (5.19) n, = q,r2 T2p,, (5.20)

n, = qym2T2pv. These expressions are valid to better than - 10% so long as pe d T, or when

qe n<--2T3z10-4

fmW3.

Ye

(5.21)

(Under these conditions the number of neutrinos and the number of antineutrinos are both of order T3, and much greater than the net neutrino number n,) Using these equations to determine pe and y, one finds -=L(Y,-2Y”) (5.22) When (5.22) and (5.18) are substituted into (5.1) we obtain the following equation for the Y, contours: Tln -+(m,lmye Y,

(5.23)

m,)c2=~ 112;2q (3Y,-2~0. e

At very high temperatures and low densities, the right-hand side may be neglected with respect to the rest-mass term, and therefore Y, tends to 0.5 from above as T + co. As the density is increased or the temperature lowered Y, has a maximum, and then will again pass through Y, = 0.5. The logarithmic term in (5.23) vanishes there, and therefore the Y,=O.5 contour is given approximately by fme3.

(5.24)

For n = 10e5 frne3 this expression gives T = 11.2 MeV for Yp = 0.4 and T = 12.7 MeV for Yp = 0.3, in excellent agreement with the Y, = 0.5 contours shown in figs. 3a and b. At lower temperatures, but ones still sufficiently high for (5.23) to be valid, the rest-mass term is less important and the contours of constant Y, are given approximately by n - T3.

694

J.M. L&timer

et al. j Hot, dense mutt@

Contours of constant A and constant Z are shown in figs. 4a-d. The main features of these contours may be understood from eq. (3.39). At low temperatures the translational contribution may be neglected, as may scoUl_One then finds 4&a

= 2ec,,,:

(5.25)

the surface thermodynamic potential per nucleus is twice the Coulomb energy. When there are no protons outside nuclei the Coulomb energy at T = 0 is given by (2.32), and therefore the number of nucleons in the bulk of a nucleus, A’, is given by D

(5.26)

As the density increases, nuclei occupy an increasing fraction of space, and therefore rN/rC increases. The Coulomb self-energy and the lattice energy partially cancel, thereby lowering the Coulomb energy coefficient. This shifts the equilibrium nuclear size to larger values of A’, and therefore also to larger values of A. At finite temperatures the. surface tension cr decreases below its T = 0 value, which accounts for the decrease of A with increasing temperature. This decrease is particularly rapid at low lepton fractions and when there are no neutrinos, since,

Fig. 4a. Contours of constant mass number A (f&I lines) and atomic number Z (dashed lines) for a lepton fraction Ye = 0.4.

J.M. L&timer et al. / Hot, dense matter

695

- AZ

1

0

10-5

lo-’

I III,,,

10-3

1o-2

10-l

n (frnm3) Fig. 4b. The same as fig. 4a, but for a lepton

fraction

Yf = 0.3.

- AZ

0

10-s

I I1111111

lo-’

.-I

1o-3

1o-2

10-l

n (frnm3) Fig. 4c. The same as fig. 4a, but for a lepton

fraction

Yr = 0.2.

J.M. Lattimer et

696

I I I I , I I, 20 -

01. /

Hot, dense matter

1 I 1 I I I 1I

I

11,1111

I

f9equ no v

I111111

I

I

10 T bw

5-

I 1 6 ,,lcI 0 1o-5 10-'

10-j

1O-2

10-l

n (frnW3) Fig. 4d. The same as fig. 4a, but for matter

in p-equilibrium

with no neutrinos.

because of the lowered critical temperature of matter at low proton fractions, the surface tension varies more rapidly with temperature there. Further reduction in A is brought about by the translational term, which for reasons of phase space favors splitting up nuclei into smaller pieces. At low temperatures and up to densities - 1O-3 fmm3, the values of A and 2 appear to be remarkably insensitive to lepton number. Below the density for neutron drip . . nd given m eq. (5.12), this can be related to the fact that Y, is also insensitive to Y,. A and 2 are primarily

functions

of Xi = Y, through

eq. (5.26).

In this region

P-equilibrium is described by eq. (5.11). If we take as a convenient fiducial curve that Ye0 for which Y,, = 0 at a given density nP, eq. (5.11) yields the relationship for Y,, = YeO: (5.27) where we have used eq. (5.9) for I;. At higher densities matter will contain a net number of neutrinos (n, > 0) while at lower densities it will contain a net number of antineutrinos (n, < 0). By similar methods to those used in obtaining eq. (5.13) we can relate the variations of Y, and Yp from their fiducial values by SY,-6Y,=

Y,”

(5.28)

J.&f. L&timer et al. / Hot, dense mutter

697

where we have used na = 10e5 fme3, corresponding to Yp,= 0.423. Thus for Ye = 0.4, 0.3 and 0.2 the SY, that occur at the density na are -0.008, -0.016 and -0.020, or Y,=O.415, 0.407 and 0.403. Thus Y, at this density, and thus also A and 2, show little variation with Yp. That this insensitivity extends also into the drip regime can be understood as follows. If one adds additional neutrons to the matter at the density nd for neutron drip, these will be accommodated in continuum states, and therefore, with increasing neutron density, the neutron chemical potential will rise only rather slowly compared with the rate with which it would rise if the neutrons were in states bound in nuclei. For example, if neutrons with a density of 10e5 fmm3 are added outside nuclei, the neutron chemical potential rises by only about 0.1 MeV. Thus to a. first approximation, after the neutrons have been added the p-equilibrium condition is still satisfied, and the numbers of electrons and neutrinos is the same as before, as is the Xi= Z/A ratio in nuclei. However,_the electron and neutrino fractions Y, and Y, both scale as (1 + no/rid)-‘‘’ where n, is the density of neutrons outside nuclei, and the total density is n,+n,. This gives an approximate scaling relationship with density for the properties of matter at different lepton fractions. Therefore Z/A varies rather slowly with density near neutron drip, and by our previous argument 2 and A are also insensitive to variations of Yf. If one substitutes into (5.26) the value Xi= 0.35 appropriate for neutron drip one finds A= 90, which is consistent with what is shown on the contour plots. For the case of &equilibrium with no neutrinos, A and Z have rather different behaviors with increasing density since the proton fraction of matter in nuclei decreases more rapidly with increasing density than it does in the case of fixed Y<. For 0.1 G Xi~ 0.3 one finds a(xi) - ~3, and therefore from eq. (5.26) A- Xi. The lower values of A for this case are therefore accounted for by the lower Xi*Heavy nuclei disappear rapidly with increasing temperature due to the lowered critical temperature for matter with a low proton concentration. We now consider the contours for constant energy per nucleon e, shown in figs. Sa-d. This quantity is needed for calculating the mass density in terms of the baryon density, and in many hydrodynamical calculations it is one of the independent variables. At low temperatures and low densities e is of the same order of magnitude as the energy per nucleon of the iron-peak nuclei, e. - -9 MeV. The actual energies are somewhat higher than this for a number of reasons: first, matter is neutron rich, which implies that nuclei are neutron rich and therefore have a smaller binding energy. Second, when the lepton fraction is low there are significant numbers of dripped neutrons outside nuclei, and at low densities their energies will be significant. Third, the energy includes a contribution from electrons and neutrinos. This latter terms becomes increasingly important as the density increases, an effect seen clearly in the contour plots. As the temperature increases the energy increases by terms varying as T2, the result of heating up the degenerate nucleons, electrons and neutrinos. At high

J.M. Lattimer et al. / Hot, dense matter

698

20 t

e s

15

10 T (MeV) 5

0 lo-’

10-5

1o-3

10-l

10-2 n (frnm3)

Fig. 5a. Contours

of constant energy per baryon e (full lines) and entropy per baryon for a lepton fraction Yp = 0.4. The units of e are MeV.

s (dashed

20

15

10 T (MeV) \

I

5

0 ,o-5

‘-5

lo-’

1o-2

10-3

10-l

n (frnm3) Fig. 5b. The same as fig. 5a, but for a lepton

fraction

Yp = 0.3.

lines)

699

10

T

WV) 5

_ +=-=~- -

-=-__=- =_ _0 -5 10-s lo-'

F

10-3

1o-2

10-l

n (fm’31 Fig. SC. The same as fig. 5a, but for a lept&

fraction

Yc = 0.2.

10

T

@VI 5

10-s

lo-'

1o-3

1o-2

10-l

n (frnm3) Fig. Sd. Ke

same as fig. 5a, but for matter in ~-equilib~um

with no neutrinos.

700

J.M. Lattimer et al. / Hot, dense matter

temperatures and low densities e increases with increasing n and is the lepton number, since the equation of state is dominated in electron-positron and neut~no-antineutrino pairs and by photons case of no neutrinos, which of course lacks neutrino pairs). The (2.52) and the T3 term in eq. (3.51) dominate, and the energy per case of fixed Ye is 29 e =40

1 37?(hC)3

p 7

independent this region (except in T4 term in nucleon in

of by the eq. the

(5.29)

(In the case of no neutrinos, one replaces g by $$.) This expression is valid so long as eq. (5.21) is satisfied. At somewhat higher densities or lower temperatures the electron fraction Y, decreases and the neutrino fraction Y, increases as n increases at fixed T and Ye (see fig. 3). Whether the energy increases or decreases at fixed T depends on a delicate balance between Y, and Y,, and on the introduction of semi-degenerate terms in eq. (5. lo), which depend more strongly on the- density. The result is that in some regions e may be nearly independent of n. The baryon contribution, being of order nT, is not impo~ant here. In the case of ~-equilib~um with no neutrinos this balance does not occur, and a gradual transition to the degenerate state ensues. At still higher densities and lower temperatures, when the leptons become degenerate, e increases with density, since under these conditions the lepton energy varies as s”~( Yz” +21’3 Yz”). The decrease of the factor in parentheses coupled with the decrease in the baryon energy with increasing density is not sufficient to overcome the increase from the n”3 factor. Finally we remark that in this region of T and n, e decreases with decreasing Yp, due mainly to the reduced number of leptons. We now turn to adiabats, which are also displayed in figs. 5a-d. Their general form is rather similar to that of the adiabats for the bulk-equilib~um case. In the region where there are appreciable numbers of heavy nuclei, the temperature remains low since the entropy resides in thermally populated nuclear excited states. For Ye = 0.4, the contour in which half of the matter is in nuclei coincides roughly with the s = 3 adiabat, while for Yp = 0.2 it corresponds to the adiabat with s = 2.5. Except at the lowest temperatures, adiabats are remarkably insensitive to the lepton fraction for 0.2 G Y86 0.4. The low entropy adiabats for Y,=O.3 and 0.2, and especially for matter in /3equilibrium with no neutrinos, lie rather close to the n-axis for densities -5 X 10m4fm-‘. Physically this corresponds to the region in which neutron drip is occurring. Matter here has a high specific heat, since raising the temperature leads to the evaporation of large numbers of neutrons from nuclei. This in turns leads to the compression of the adiabats.

J.M. Lutiimer et al. / Hot, dense matter

701

At high temperatures and low densities, where pairs and photons dominate, the total entropy may be written in

Y,‘;(l-

Y,)‘-x

1.

(5.30)

At densities of 10m3fmd3 and T - 20 MeV, the baryons contribute about 6.9 units to s. Next we consider the pressure P, contours of which are shown in figs. 6a-d. For reference, adiabats have been superimposed on the plots. As a rule the pressure increases with increasing temperature and increasing density. At low densities and high temperatures the pressure is independent of composition (i.e. Y,) since it is dominated by lepton pairs and photons, and is obtained by multiplying eq. (5.29) by fn. Thus the pressure becomes independent of density. When the density reaches nuclear matter density, the pressure begins to rise rapidly with increasing density, due to the pressure of the degenerate nucleons and to the increasingly repulsive nucleon-nucleon interaction at smaller separations. At subnuclear densities, even at low temperatures, the pressure is mostly leptonic; hence it is the pressure of fully degenerate leptons with a T2 correction. The pressure contours begin to curve I

1

I

llll,

I

20 - Y,=O.4

I

- Ps

7.; I I

: (a>

I

15 -

10

(MeV) 5

n lo-5

lo-’

1O-3

lo-* n

10-l

(fmq3)

Fig. 6a. Contours of constant pressure P (full lines) and entropy per particle s (dashed lines) for a lepton fraction Yf = 0.4. The contours are labeled with log,,, P, with the pressure measured in units of MeV . fme3.

J.M. Lattimer et al. / Hot, dense matter

702

20

15

10 T WV) 5

0 1o-5

lo-’

10-l

10-2

1o-3 n

(fin-‘)

Fig. 6b. The same as fig. 6a, but for a lepton

fraction

Yr = 0.3.

20

10 T (MeV) 5

1o-5

lo-’

10-l

10-2

1o-3 n (fd)

Fig. 6c. The same as fig. 6a, but for a lepton

fraction

Ye = 0.2.

J. M. L&timer et al. f Hot, dense matter 20 - fiequ no v

1o-5

lo-’

1o-2

1O-3

10-l

n (ffnm3) Fig. 6d. The same as fig. 6a, but for matter in p-equilibrium

with no neutrinos.

downwards where the electrons become degenerate and show a kink where nuclei dissociate. There are a few conditions, at low temperature and low density, where the pressure decreases with increasing temperature. The explanation for this behavior is that under the conditions where the phenomenon occurs, the neutron chemical potential is close to zero. As the temperature rises pn drops rapidly, which leads to a reduction pLp.Since the p-equilibrium condition (5.1) must be satisfied, as well as in b=pnthe condition of constant lepton number, this implies a reduction in pe and an increase in CL,,.The reduction in pressure is due to the reduced number of electrons. Under other conditions this effect is swamped by other ones which tend to increase the pressure as the temperature increases. An unusual feature visible in figs. 6b-d is that at densities near n = 10e4 fmP3, the temperature on the lowest adiabats decreases as the density increases. The effect is more pronounced for the small lepton fractions and for the no-neutrino case, and is due to the onset of neutron drip, as we now discuss. As a consequence of the thermodynamic identity

fi=($)o=-(~),=nz(~)s, avas

(5.31)

704

J.M. Latiimer et al. / Hot, dense matter

where u = l/n, the pressure at fixed density decreases with increasing entropy if the temperature on an adiabat decreases with increasing density. This phenomenon is well known in the case of water just above its freezing point. The fact that P decreases with increasing entropy in this region is clearly seen from the plots. The physical reason for (aT/an), being negative is that the proportion of nucleons outside nuclei increases as the density increases. Nucleons outside nuclei have a much lower Fermi energy than those inside, and their specific heat, which at low temperatures is inversely proportional to the Fermi energy, is much higher. Consequently a given entropy is reached at lower temperatures at the higher densities. Another consequence of (t3P/ds), being negative in some region is that adiabats will cross in that region on a P - n or a P - V plot. Of particular interest for applications to stellar collapse is the adiabatic index r=

aln P t alnn > S’ -

(5.32)

whose contours are shown in figs. 7a-d. Evident in all of the plots is a certain amount of jitter in the contours where nuclei are evaporating rapidly. The derivative calculations needed in the r-contour plots have encountered at these places a residual error due to the finite gridding of the arrays of P- and s-values (see appendix

20

15

10 T (MeV) 5

1o-4

1o-3

lo-*

10-l

n (fmm3) Fig. 7a. Contours

of constant

adiabatic index r (full lines) and entropy per particle s (dashed for a lepton fraction Ye = 0.4.

lines)

J.M. Lattimer et al. / Hot, dense matter

705

20

15

10 T

WV) 5

0 1 n (fin-‘) Fig. 7b. The same as fig. 7a, but for a lepton fraction Yr = 0.3.

T

WV)

10-5

lo-’

10-j

1O-2 n (fm-‘)

Fig. 7c. The same as fig. 7a, but for a lepton fraction Y/ = 0.2.

I”

J.M. Luttimer et al. / Hot, dense matter

106

15

10 T (MeV) 5

0

10-s

IO-’

10-2

l0-3

10-l

n (fmm3) Fig. 7d. The same as fig. 7a, but for matter in p-equilibrium

with no neutrinos.

C). (When these derivatives are evaluated with more accuracy, as in figs. 8, the f-values in these regions are seen to be continuous but with possible rapid variation.) We have not manually smoothed the r-contours so as to avoid removing genuine features together with the noise. In the regions of the phase transitions (n 10-l fmm3), r changes too rapidly and discontinuously to be approximated by derivatives evaluated from our grid. The behavior of I’ in the region of the phase transitions has been discussed in refs. 5’P52). One of the salient features of the contour plots is that r is close to : in the high-temperature low-density region, where electron-positron and neutrino-antineutrino pairs, which behave as relativistic perfect gases, are the dominant contributors to the pressure. At T = 0 for the fixed- Yp plots the pressure is provided chiefly by relativistic electrons, and r is again close to $ except at the highest and the lowest densities exhibited. r increases rapidly with increasing density above nuclear density due to the large bulk modulus of uniform nuclear matter. The drop in r at the lowest densities and temperatures in the Ye = 0.2 and 0.3 plots is due to the presence of a net number of antineutrinos, as we shall explain in more detail below. For densities less than -IO-* fmm3, r generally first of all increases with increasing temperature, due to the increasing number of free nucleons, which have an adiabatic index of 2. At higher temperatures the leptons become nondegenerate and their pressure (- T4) overwhelms that of the baryons (-T), and r therefore

J.M. L&timer et al / Hot, dense matter

707

has a maximum, before falling to the value -! characteristic of a relativistic gas. To obtain a crude semi-quantitative estimate of r at the maximum we treat the pressure as being made up of electron and baryon components. If one includes the leading finite temperature contributions to the electron pressure one finds P,=+n,pz

(1 +wT2/(/d2j,

(5.33)

where pz is the electron chemical potential at T = 0. We assume the nucleons to be an ideal gas, since in the vicinity of the maximum they are all in continuum states, and therefore their pressure is P,,= nT.

We estimate r by taking a pressure-weighted components: $P&+$P,

r==

pe+pb

4

=j+T/&+;y,(l

(5.34)

average of the T’s for the two 4Tld +$r’T’/(p;)“)’

(5.35)

This approximation does not take into account possible transfer of entropy between the two components when the density is varied, or changes in Y, with density, but we expect them to have little effect. The maximum value occurs for T = 6”2/ v&!, and its value if r,,,,, = $ +$/( 1 + 7rY,/&. If one puts Ye = Y,, the estimate for r,,,,, is 1.55 for Y, = 0.4, 1.57 for Y, = 0.3 and 1.60 for Y, = 0.2. The values show the same trend with Y, as the more detailed calculations, and the actual values are in reasonable agreement given the simplicity of the above estimate. The temperatures for the maximum predicted by the simple model are about 50% higher than those given by the detailed results on the contour plots, and we attribute this to the fact that the low temperature result (5.33) for the electron pressure underestimates it at temperatures in the vicinity of the maximum and higher. The reduction in r at low temperatures and at densities somewhat below nuclear matter density is due to the Coulomb lattice energy. This gives a negative pressure of order Z2’3e2/hc times the electronic pressure, which leads to a reduction in r, since Z increases with increasing density. In the bubbles phase r remains relatively low, again as a consequence of the Coulomb energy. For a more detailed discussion of these effects we refer to refs. 51S52). In the infall stage of stellar collapse, the entropy is conserved to a good approximation after neutrino trapping. The behavior of r on adiabats, shown in figs. 8a-d, is therefore of special interest. The behavior of r when nuclei disappear depends on how the nuclei disappear. At low entropies nuclei merge, in some cases passing through the bubbles state, into uniform nuclear matter. For such cases r falls with increasing density, and shows discontinuities at the various phase transitions. A more detailed description of these features has been given in refs. s’*52).At higher entropies (typically s = 3 or more) nuclei disappear by dissociation, which gives rise to a marked dip in r,

708

J.M. Lattimer et al. / Hoi, dense matter

Fig. 8a. Adiabatic index r as a function of density on a number of adiabats for a lepton fraction Yp = 0.4. The value r = $ appropriate to an ideal relativistic gas is also shown. The curves are discontinuous at the phase boundaries (see fig. 10). In many cases the curves for the uniform phase lie outside the graph.

lO-5

lO-4

lO-3

IO-*

10-l

n (fmm3) Fig. 8b. The same as fig. 8a, but for a lepton

fraction

Yr = 0.3.

J. M. L&timer et al. / Hot, dense matter

n (fmT3) Fig. 8c. The same as fig. 8a, but for a lepton fraction Yp = 0.2.

1.6-

I .4 -

4 3

0.8 -

0.6 (d)

1 I I Ill,, 10-5

I 1 I f Illll 10-4

I I , I11111

10-s

I I I I,,,, I

10-2

10-I

n (fmm3) Fig. 8d. The same as fig. 8a, but for matter in p-equilibrium

-

I I I I ,,,J_

with no neutrinos.

710

J. M. Lattimer et al. / Hot, dense matter

qualitatively similar to what occurs in most ionization or dissociation processes. In this case r shows no discontinuity, but smoothly approaches its value for uniform matter. Earlier we remarked on the region in the Ye contour plots at low densities and low temperatures where r falls below :. This behavior shows up clearly on the plots of r on low-entropy adiabats. To understand it, consider matter at T = 0 with a fixed electron fraction Y,. There will generally exist some density, which we have denoted by np in eq. (5.27), at which the matter is in /3-equilibrium with no neutrinos. At higher densities, matter will contain net numbers of neutrinos (py > 0), while at lower densities matter will contain net numbers of antineutrinos (pV ~0). The simplest situation to consider is one in which the electron fraction is so high that all protons reside in nuclei. If we further neglect the lattice contribution to the pressure, the total pressure is the sum of the electron and neutrino contributions: (5.36)

P=+nYepu,+&Yvp,,

where CL,and py are the electron and neutrino chemical potentials, given by eq. (5.10). Note that eq. (5.36) holds for both n, = nY, > 0 and n, < 0, since if n, < 0, both CL,,and Y, are negative, and therefore their contribution to the pressure is positive. The adiabatic index is therefore given by (5.37)

In deriving the last result we used the fact that Yp = Y, + Y, is constant and therefore To evaluate (aY,/an), we consider the /3-equilibrium @Y&n),,= -@Y&n),,. condition (5.1), with fi having a character such as we have incorporated in eq. (5.9). It is a good approximation so long as dripped particles and Coulomb lattice energies can be neglected. If we denote the electron chemical potential at density ns by pep, the P-equilibrium condition there is (5.38)

b ( YP) = Pep * Subtracting this result from the general P-equilibrium /.2(YJ-;(Y,)=(P~-/-Q)-P”.

condition one finds (5.39)

If one expands b and cc, in powers of Sn = n - np and Y,, = Yp- Y, and neglects all but the first-order terms one finds 1 Sn Y (5.40) -CL”, ( P> e

-yvg=pepj ,_g

or (5.41)

J. M. Lattimer et al. / Hot, dense matter

The leading contribution

711

to p,, for small &t/n and therefore small Y’ is wCLt$~;,

1 Sn

(5.42)

or 3

Y&,=&Y, Er 0n

.

(5.43)

Thus (5.44) and therefore the leading contributions

to r for small &n/n are

r=4 -_- 2 -Sn ’ ’ 3 27 0 n

(5.45)

This shows that r is exactly $ when Sn = 0, that is when there are no neutrinos, and that it decreases for non-zero 6n of both signs. Our numerical calculation for Yp = 0.3 and 0.2, where the numbers of dripped nucleons are sufhciently small that our simple model should be reasonable, shows that r on the lowest adiabats (see figs. 8b and c) reaches a maximum extremely close to G, and that the neutrino chemical potential there is very close to zero. Generally, Y, is approximately zero at the local maximum in r for all the fixed-lepton-number cases. The plots show that I’ decreases much more rapidly below na than above. This too may be understood in terms of our modei. First of all we note that eq. (5.41) shows that Y, is an odd function of &I, and therefore aY,/an is an even function of 6n and cannot be of help. However, the factor (Yp- Yy)“3-(2Yy)“3 1-f&n/n Ye Ye/he + Yv& = ( Y, - Y,y + (2 Y4y)“3= tie--P”

(5.46)

in (5.37) does have a term linear in &n/n, and will make r vary more rapidly for n < np (i.e. Y, ~0) than for n > np (Y, > 0). When n > np, both pe and 1” are positive, while for n < na, JL, is positive and py is negative. The large dip in the r versus n curves for P-equilibrium and the secondary dip on the s = 0 adiabat of the Yp = 0.3 case at n = 9 x 1O-’ fmT3 have a different origin. They are due to the onset of neutron drip. As was shown in ref. 9), at T =0 r is non-analytic at the neutron drip density nd, and it drops as (n - nd)‘12 for n > n& With increasing temperature, or entropy, the square-root singularity gets considerably smoothed out. On the s = 0.5, 1.O and 2.0 adiabats for ~-equilib~um with no neutrinos there are rather pronounced dips at densities of 0.05 fm-‘, 0.02 fm-’ and 0.007 fmV3, respectively, as well as a kink in the s = 3 adiabat at 0.0015 fme3. These appear in the

J.M. Lather

712

contour

et a/. / Hot, dense matter

plot in fig. 7d as a valley around

T = 3 MeV. This valley is associated with

a rapid decrease in the nuclear size from intermediate masses (A b 40) to small masses (A- 10). This rapid decrease in A at approximately 3 MeV can be noted from fig. 4d. One sees that on the s = 2.0 or 3.0 adiabats the abundance of nuclei passes rather smoothly to zero since they pass above the critical temperature. Lower adiabats pass through a phase transition directly to uniform matter. (Y, is too small in the case with no neutrinos to permit the existence of a bubbles phase, as may be seen from fig. 10d.) The final set of contours (fig. 9) we show are those for $ = CL,- pP and for p,,. Their general shapes are very similar to those for bulk equilibrium for fixed proton fraction 30),except at high temperatures and low densities. At fixed Y, the &contours are horizontal in the high-temperature, low-density region, but at fixed Ye the electron number Y, increases with increasing temperature (cf. fig. 3). This leads to a larger number of protons relative to neutrons, and therefore to a lower value of k compared with the value for fixed Y,. In this part of the (n, T) plane (5.47)

I_~=)(L,+--~p=(m,-mp)c2+Tln

10 T NeV) 5

3 J

0

10-5

lo-’

1o-2

1o-3 n (frnb3)

Fig. 9a. Contours of the difference lines) and CL, (dashed

lo-’

(a)

between the neutron and proton chemical potentials I; = pL, - pP (full lines) for a lepton fraction Ye = 0.4. The units are MeV.

J. M. Luttimer et 01. / Hot, dense matter

713

20

15

10

T

NV) 5

n 10-Z

10-3

n (frnq3) Fig. 9b. The same as lig. 9a, but for a lepton fraction Yr = 0.3.

lo-’

(b)

i i I I I I I I

10

l&C IO1 I I I I

T &eV) . . 5

0 1o-5

lo-’

1o-3

lo-* n (frnm3)

Fig. 9c. The same as fig. 9a, but for a lepton fraction Yr = 0.2.

10-l

714

J.M. Lattimeret

al. / Hot, dense matter

n 10-3

10-2

10-l W

n (frnm3) Fig. 9d. The same as fig. 9a, but for matter

in P-equilibrium

with no neutrinos.

and therefore on contours of constant Y,, ii is a linear function of T. Consequently the fi-contours have a steeper slope than the Y, contours. & increases with increasing density, due mainly to the decrease in Y, with increasing density. When most of the matter is in nuclei, $ is proportional to x,-Z/A’ = x, - Y,. The highest density points on the r-2 contours at relatively low temperatures occur where nuclei are breaking up. The general structure of the contours coexistence

in this region line found

is a smoothed in the bulk

out version

equilibrium

of the sharp

calculation

features

on the

30). This structure

has

also already been commented on in the discussion of Y, contours in fig. 3. The neutron chemical potential increases with increasing density and decreases with increasing temperature, as one would expect, except in the vicinity of phase transitions. At high T and low n, CL,,is just given by the classical Boltzmann formula. One unusual feature of the results for Y,=O.4 is that at low temperatures and low densities k first of all increases with temperature and then, at a temperature -0.75 MeV, suddenly begins to decrease. To understand this we recall that at T = 0 and for this lepton fraction, all nucleons are in nuclei, with no dripped neutrons outside. Imagine now what happens if matter is warmed up at fixed composition. All particles are degenerate fermions, and their chemical potentials therefore decrease. The largest decrease will be for the neutrinos, which have the lowest Fermi

J. M. Lattimer et al. 1 Hot, dense matter

715

energy. To restore P-equilibrium at fixed lepton number it is then necessary to decrease Y,, which increases @ and increases py until condition (5.1) is satisfied. Since & - (x, - Y,) and we - Y"3 e ,the small decrease in Y, is mostly reflected in a k, rather than a pe, change. However, at higher temperatures neutrons evaporate from nuclei more readily than do protons. The evaporation proceeds exponentially as T increases -the free neutron abundance reaches 7% at 0.75 MeV - which leads to a higher proton concentration of matter in nuclei, and therefore to a lower b. The interplay of these two opposite trends gives rise to the cuspy appearance of the k-contours at this temperature. At higher densities, neutrino Fermi energies are more comparable to electron ones, and therefore the neutrino contribution to the chemical potential balance condition (5.1) is no longer the dominant one. Therefore, the cusp in the &-contours disappears at higher densities. A general feature of the &contours at low densities and temperatures is their linear T-dependence on the low-T side of the v-shaped valley, above the region of characteristically T* dependence and the cusp just mentioned. This behavior can be understood qualitatively in terms of the behavior of the evaporated neutrons as a Maxwell-Boltzmann gas. If we neglect cu-particles the fraction of nucleons evaporated is related to the fraction in nuclei by X0 = 1 -X,. Charge neutrality requires that Y, = xix,. Combining these relationships, we obtain (5.48) With the approximation (5.9) for &, and the assumption that the electrons and neutrinos are completely degenerate, /3-equilibrium results in eq. (5.11). Small excursions from the fiducial contour Y,= Yp can be treated by the same methods as were used to obtain eq. (5.13). From (5.11) we obtain at fixed Yp and n the result that (5.49)

and since Yy=O in the vicinity of the Y,= Ye contour, SY,-0. From (5.48), and using the approximate expression (5.8) for jam,we can then obtain (5.50) showing that Sxi depends linearly on ST,sothat from (5.9) fi depends linearly on T As mentioned at the end of sect. 3, the results already presented assume that all Wigner-Seitz cells are identical. The A, 2 and derived quantities we have discussed are those pertaining to the most likely cell. Because of thermal fluctuations there is a distribution of A and 2 about those values. Appendix D summarizes a discussion of the fluctuations expected in the nucleon content of a cell and the resulting

J.M. Lattimeret at! / Hot, dense matter

716

modifications in thermodynamic quantities. The reader is also referred to more detailed discussions by the authors [Burrows and Lattimer 53), Pethick and Ravenhall ‘“)I. The two methods discussed in appendix D estimate lower and upper limits on the fluctuations. To give an idea of the orders of magnitude involved, we show in table 2 some results for the s = 1 and s = 2 adiabats, at a lepton fraction Yp = 0.3, TABLE 2 The effect of thermal

n [fm-‘1

tluctuations

on the number a cell

of nucleons

A’ and protons

K

T [Meal

A

(A&,

lo-4 10-3 10-z

0.300 0.286 0.263

0.640 1.285 2.881

94.2 112.3 178.9

9.69 15.9 34.4

Y,=O.3, s=2 lo-4 10-3 lo-2

0.293 0.265 0.252

1.389 2.615 5.715

84.3 94.8 98.7

14.5 24.3 52

n [fm-‘1

z

(A.%,

(AZ),

.? in

(A&

Yr = 0.3, s = 1 9.86 17.5 39.7

15.9 30.0 -170

A,,

Amin

Yp=0.3, s= 1 IO+ 10-3 10-r

28.3 32.1 47.0

2.62 4.18 8.70

2.71 4.74 10.3

10.00 16.4 35.4

0.89 1.13 1.69

Y( = 0.3, s = 2 lo-4 10-r 10-r

24.7 25.2 24.8

3.84 6.01 12.7

4.49 7.96

14.9 25.0 53

1.16 1.38 1.6

-44

n [fm-‘1

Af IMeVl

AP WeVJ

AP/P

AS

Yp=0.3, s=l lo-‘+ lo-’ 10-r

-0.027 -0.054 -0.096

-0.026 -0.047 -0.052

0.0011 0.0028 0.0070

0.043 0.042 0.033

Y,=O.3, s=2 lo-4 1OP lo-2

-0.077 -0.148 -0.36

-0.07 1 -0.121 -0.1 I

0.0038 0.008 1 0.033

0.056 0.057 0.064

The guctuations (AA) and (A.?) with subscripts 1 and 2 are respectively the lower and the upper estimates, as described in appendix D. The quantities A with subscripts maj and min give the major and minor axes of the correlation ellipses. The last four columns are the thermal fluctuation corrections to the free energy per baryon, the nucleon chemical potentials, the pressure and the entropy per baryon. Results are given for Yr = 0.3 and for the adiabats s = 1 and s = 2.

J.M. httimer

et at! f Hot, dense matter

111

at densities of 10-4, lop3 and 10-q fmv3. The quantities A and z are the number of nucleons and of protons (including those in the vapor), and f(AA),,2 and *(A.& are the rrns thermal fluctuations in those quantities according to the two methods. They thus represent lower and upper bounds, in the approximation that the populations have gaussian distributions. It can be observed that A and Z increase as the densities and temperatures increase, so that on the s = 2 adiabat at n = lo-’ fm-‘, AA/A1. Clearly the A and 2 of a cell (and thus the A and 2 of its nucleus) are not precisely fixed quantities under these conditions. Two features of the results presented in table 2 are especially noteworthy. First, the fluctuations in x and 2 are highly correlated. The columns A,, and Amin give the semi-major and semi-minor axes of the A, 2 distribution. (For brevity we give here the results only for method 1, the lower limit, since it is evident that the AA and AZ of the two methods do not differ greatly under most conditions.) The correlation ellipse is always very narrow. Physically, the major axis represents the variation in cell size with a fixed proton ratio for the matter. It is mediated by the surface and Coulomb energies. The minor axis corresponds to fluctuations in the charge ratio Z/A, which the nuclear symmetry energy constrains much more tightly. The second feature is that the effect on the~odynamic quantities of the thermal fluctuations is very small, even when the fluctuations in 2 and 2 are comparable to the quantities themselves. On the free energy per baryon and on the nucleon chemical potentials it is usually less than 0.1 MeV, although it increases with temperature. The relative effect on the total pressure is usually a few tenths of a percent, and on the entropy per baryon it is of order 0.05 (kB). Thus, although A and 2 can have large fluctuations, the thermodynamic quantities are not affected in an important way. All of the quantities quoted in the previous tables and in the figures do not contain these fluctuations. There are occasional irregularities, or even discontinuities, in some of the contours at densities around n - 0.08 fmm3. These phenomena are due to first-order phase transitions involving the three different phases that may occur: matter with nuclei, matter with bubbles, and uniform matter. The phase boundaries for each lepton fraction are shown in figs. 1Oa-d. Inside the double lines, matter consists of a mixture of the adjacent pure phases. The bubbles phase does not occur for the no-neutrino case, and for the fixed lepton fractions it can exist only for T up to about 10 MeV (for Yp = 0.2) or 13 MeV (for Ye = 0.4). Above these temperatures the matter makes a transition directly from nuclei to uniform matter. The various features of these boundaries and their dependence on T and on Yp have been discussed elsewhere 5’*52).It has recently been discovered 55*56)that in the region from n - 0.1 n, to the upper phase boundary and at zero temperature, aspherical nuclei will have a lower energy than the spherical objects we consider in this paper. In fact, it appears that at some densities cylindrical or planar “nuclei” are energetically favored. The actual structure at finite temperatures has yet to be explored. One must therefore

718

J.M. L&timer et al. / Hot, dense matter

0 1o-2

10-l n (fmw3)

Fig. 10a. Boundaries between the nuclei phase, the bubbles phase and the uniform phase in the n - T plane for a lepton fraction Yc = 0.4. Inside the double lines matter exists as a mixture of the adjacent phases. The scales of this figure are identical to those of the contour plots.

I I ,,,!,I 20 -

I I

YcO.3

- phase boundaries 15 -

10 -

T

-

(MM 5-

0

- (b) 111111.11

10-2

I

I

10-l n (fmS3)

Fig. lob. The same as fig. lOa, but for a lepton fraction

Y,=O.3.

J.M. Lather

et al. / Hot, dense matter

phase boundaries

t

719

1

10

(Ml") 5

1o-2

10-l n (fmm3)

Fig. 10~. The same as fig. lOa, but for a lepton fraction Yt = 0.2.

I

20 -

I

1 1

I11111

1 equ no v

’

- phase boundaries 15 -

10 -

(ML)

: 5-

: (4 0, 1O-2

I 111111' 10-l

"

n (fmm3)

Fig. 10d. The same as fig. lOa, but for P-equilibrium with no neutrinos. There is no bubbles phase in this case.

720

J.M. Luttimer et al. / Hot, dense matter

bear in mind that the results we present that the constituents most affected are affected

are constrained

here were all obtained to be spherical.

in this region by the future removal most by our presently

calculated

under the assumption

The variables

of this constraint

phase transitions,

which

will be

are those which i.e. s, P and 1’.

All of the plots shown in this section have been computer drawn to the same scale. Comparison of results for different Ye, for example, or of the values of quantities on the phase boundaries, may be made easily by taking a set of transparencies of the figures and overlaying them.

6. Discussion In this paper we have described a unified approach to calculating the properties of matter at subnuclear densities. Within this framework it is possible to include many important aspects of the physics in a relatively straightforward way. Among these one may mention the following: (i) a consistent treatment of the matter inside and outside nuclei; (ii) automatic inclusion of effects of nuclear excited states through the finite temperature contributions to thermodynamic quantities; (iii) the temperature and proton-concentration dependence of the properties of the nuclear surface; (iv) the Coulomb energy, including “lattice” contributions and finite temperature effects; (v) nuclear translation. One further effect that can readily be taken into account is thermal fluctuations in the nuclear distribution. At low temperatures fluctuations are negligible, but they become increasingly important as the temperature increases. Throughout most of the paper we have neglected them, but we show in appendix D how one may readily extend the model to take them into account, and we have given estimates of their effects in sect. 5. We shall give more detailed results of including fluctuation effects elsewhere 53*54).Fluctuations are expected to be especially important near the critical temperature, where, in addition, our description of nuclei as spherical droplets (or, in the case of bubbles, spherical rarefactions) is an oversimplification. Even at low temperatures it is now known 55356)that nuclear matter will adopt aspherical shapes at densities above about one tenth of nuclear saturation density. We now turn to a discussion of our results, and a comparison of them with other calculations. The results given in sect. 5 survey the thermodynamic properties of matter at densities through nuclear saturation density and at temperatures through the highest critical temperature. Prominent features and trends have been identified and examined by means of simple physical arguments and relationships. We shall make comparison with other calculations only at this level, and only in cases where there are differences of a very basic nature. Some of the features we have examined occur in regions dominated by leptons, and these are results which any other computation must be able to reproduce. They represent the simple physics of leptons and free nucleons, however, and are implicitly contained in most treatments of the problem. Other features of the plots reflect the presence of nuclei and of the surrounding nucleon vapor. To the extent that the

J. M. Lurtimerei al / Hot, dense matter

721

nuclear model used represents the properties of physical nuclei, the low-temperature, low-density features for sufficiently large Yr that neutron drip does not occur should still be independent of the nuclear model used. For Yr small enough that neutron drip occurs the situation is less clear cut, and the nature of the nuclear and many-body physics used begins to reveal itself. At the outset, the calculations which include finite nuclei divide into two groups: those which, like ours, treat the nucleon vapor by the same techniques as they treat the nuclei, with the systems in mutual thermodynamic equilibrium; and those which extrapolate the properties of terrestrial nuclei, with or without a different and separate nucleon vapor. As a representative of the latter method we consider the calculation of El Eid and Hillebrandt *). The quantity most sensitive to the nuclear physics is T’, and we therefore compare fig. 3 of ref. *), where r along adiabats is shown for Yp =0.25, with our figs. 8b and c, which are for Y,=O.3 and 0.2 respectively. In the neighborhood of the lowest density given there, n = 4.8 x 10e4 fmd3, the T’s on their s = 1 and s = 2 adiabats are coincident and have r = 1.35. We get at that density r(s = 1) = 1.338, r( s = 2) = 1.352. At larger densities, where nuclear physics begins to show, their s = 2 curve lies considerably below their s = 1 curve. Our results indicate the opposite order-in fact, r increases steadily with s in the region just below nuclear saturation density. Our curves of r for all Yp descend well below $ in that region, a consequence of the nuclear attraction and the phase transitions to bubbles and to uniform matter. The complicated physics which produces the detailed structure in our curves near n = 0.1 fme3 is presumably not present in the model of ref. *), since there is no sign in their fig. 3 of the details that it produces in r at that density. In view of the critical role played by the value r = $ in stellar-collapse calculations, these are significant differences. We note important differences also at smaller densities: the deep minimum in their s = 3 curve at n = 6 x 10S3fmm3, and the subsequent rise, are to be compared with the small kinks in our curves at about that density. The physical origin of their minimum in r on the s = 3 adiabat is unclear, since none of their tabulated parameters, such as the fraction of matter in heavy nuclei, appear to vary rapidly with density. By contrast, the sharp dips in I’ at densities a little below 0.1 fme3 on the s = 3 adiabats in our calculations are associated with the rapid disappearance of heavy nuclei near the critical point of the neutron-proton mixture. As noted above, such differences in r could have major consequences in collapse calculations, since in them it is the difference of r from the value $ that matters. At higher entropies, some of the differences between the calculations may be due to the fact that fluctuations were allowed for in ref. *), whereas the results we have presented here were calculated, for the most part, neglecting fluctuations. However, we suspect strongly that the major source of the differences between our results and ones like ref. “) is that in the latter calcuiations matter inside nuclei and matter outside are described in different ways, lattice contributions to the Coulomb energy are neglected, and the effects of finite temperature on surface properties are not taken into account.

722

J.M. Lattimer et al. / Hot, dense matter

We now turn to a comparison with models in which all of the nuclear properties are obtained consistently from one effective hamiltonian. The fitting of the hamil-’ tonian to properties of physical nuclei is a strong constraint on its properties, although because of the relatively few adjustable parameters contained therein it involves compromises, so that there remains a certain limited variability in the models employed ‘). Neutron drip represents a considerable extrapolation of the properties of interacting nucleons and nuclei. The fitting of the effective hamiltonian to the properties of neutron matter [as calculated in ref. 39) or in the more recent work of Friedman and Pandharipande 57)], a generally accepted procedure, is a very necessary “lever arm” in pinning down the dependence of the interaction on the proton fraction at and beyond neutron drip. Models with differing charge symmetry properties will predict somewhat different drip values of the proton fraction, but so long as they ascribe a reasonable behavior to neutron matter, the thermodynamic quantities may be expected to show the same qualitative features. It is therefore to be expected that at zero temperature all calculations which use one fitted effective hamiltonian for all parts of the calculation should agree at the level we are discussing. [Note, however, that differences between results calculated with different hamiltonians may be important in stellar collapse through their influence on electron capture and neutrino generation [Cooperstein and Wambach “), Fuller 59)], and on supernova energetics [Yahil and Lattimer6’), Bethe, Brown, Applegate and Lattimer “‘)I. The temperature dependence obtained with a given model is related to the level density the model predicts for isolated nuclei, and at higher temperatures to the nucleon effective mass. At low temperatures and at energies close to the Fermi surface the nucleon effective mass m* is about unity (in units of the bare nucleon mass), while at an energy of more than about 10 MeV from the Fermi surface the effective mass is about 0.7. [For a review of these effects, see ref. “‘).I Realistic calculations of the effective mass “) show also a temperature dependence. Such energy dependent effects are not included in the Skyrme hamiltonian, whose effective masses depend only on proton and neutron densities. Another consequence of the simplified nature of the Skyrme interactions is that they do not reproduce in detail the level spacing and level ordering of physical nuclei. For the Skyrme I’ interaction that we use, m* is about 0.85 at nuclear saturation density, which is intermediate between the low temperature and high temperature values. Most thermodynamic quantities depend rather weakly on the nucleon effective mass: at low temperatures m* enters finite temperature contributions to the free energy, but these are small since they vary as T2. However, the nucleon contribution to the entropy varies as m* T, and therefore the temperature at a given entropy and density will be sensitive to the value of m*. Thus at low entropies, the temperatures on adiabats in our calculations will be slightly higher than is realistic, while at higher temperatures they will be slightly lower. Simple estimates suggest that the characteristic temperature scale for variations of the effective mass is of order 10 MeV, and therefore

J. M. Lattimer et al. / Hot, dense matter

723

our use of an average effective mass -0.85 is probably quite reasonable for conditions in the infall stages of stellar collapse. At temperatures greater than a few MeV pairing effects and collective states will not be important, and therefore, although these features are not present in our model our results are unlikely to be in serious error thereby. At lower temperatures these effects will alter the expression for the temperature as a function of the entropy, but will have little influence on other thermodynamic functions. In calculations of the kind we are discussing, the characteristic temperature determining the scale of gross variations of the thermodynamic properties is the critical temperature, T,, of the neutron-proton mixture. We expect that all models whose interactions are based on real nuclei will show similar behavior at densities less than nuclear saturation density, apart from a scaling of the temperature according to T,. Various models for the nucleon-nucleon interaction give critical temperatures for symmetric nuclear matter in the range 15-20 MeV, a rather modest spread. We intend to carry out detailed calculations to investigate how thermodynamic properties are affected by using a different hamiltonian. Besides the temperature dependence implicit in the effective hamiltonian there are other specifically many-body finite-temperature effects which we have described in earlier sections, and they must be considered. They include the finite-T contributions to the free energy arising from the thermal motion of the nuclei, and the effect this motion has on Coulomb energy. In addition, a necessary property is that these effects disappear at T, (cf. our factor h(T), eq. (2.48)). That procedure is reasonable in view of the fact that close to T, one expects the critical behavior to be given by the classical indices, which do not depend in any way on the translational degrees of freedom, since, as is well known, in classical calculations the translational degrees of freedom may be integrated out trivially. To illustrate the importance of these explicitly temperature-dependent effects we consider the s = 1 adiabat for Ye = 0.4. The range of densities from n = 0.01 fmw3 up to the bubbles phase transition might be thought of as a region where the nuclei are big enough that thermal motion effects could be neglected. Yet on this adiabat the Coulomb free energy is about 10% higher than the zero-temperature Coulomb energy used in refs. 24-27),and the nuclear translation energy is about -10% of the surface energy. The entropy per baryon contributed by Coulomb temperature dependence and translation on this adiabat can be as much as -0.016 and 0.067 respectively. This adiabat is typical of what is expected on collapse paths, and the omission of -5% of the entropy may have important consequences. Other parts of the n - T plane can suffer much more from such omissions. On the s = 2 adiabat for Ye = 0.4 at T = 8 MeV, where the density is 0.0275 fmm3 and nuclei have A = 102, 2 = 37, the Coulomb free energy is 30% larger than the zero-temperature version; the translational free energy is even somewhat larger than the Coulomb energy (and negative). The entropy per nucleon contributions of Coulomb and translational effects are respectively -0.01 and 0.066. The pressure contributed by the translation is 7% of the nuclear pressure. The most

J. M. Laitimer et al. / Hot, dense matter

724

dramatic

effect of neglecting

increased

by large factors

we expect

differences

than the differences

nuclear

(sometimes

translational

energy

is that A and Z may be

by a factor -2 or more). Thus in these regions

due to the omission

of these effects to be considerably

due to the way in which the nuclear

dynamics

is handled

bigger or to

the precise hamiltonian used. If such drastic modifications result from contributions which can be added only a posteriori even in the more precise treatments of the nuclear part of the problem such as Hartree-Fock and Thomas-Fermi, the advantage of such treatments over the liquid drop model is not so evident. Barranco and Buchler 22), with a Thomas-Fermi approach, give some results which include the temperature dependence of the Coulomb energy, and nuclear translational energy, both of them in rather less intricate forms than we have developed in sect. 2. At T = 4 MeV, at the four density points given in ref. 22), there is quite close agreement with our results so far as thermodynamic properties are concerned. The A- and Z-values they give are smaller than ours by -20%. (We have allowed for their slightly different definition of A and Z, in terms of the excess of matter inside the nuclei.) The difference is small, however, compared to the effect of neglecting these temperature-dependent contributions entirely, which they find makes differences in A and Z of a factor -2. Nonetheless, most of the results given in refs. 19-**) do not include these effects. One difficulty described in ref. 20) was also encountered initially in our work. This was an inability to find realistic solutions of the equilibrium conditions near the critical point, in the region where nuclei dissolve into a uniform fluid. In this region of the (n, T) diagram uniform matter is not stable, i.e. it lies within the two-phase region of bulk equilibrium. The difficulty is partly that the effect of translation is overemphasized by eq. (2.42), and partly that the densities of states of light nuclei are overestimated by the compressible liquid-drop bulk energy. We have resolved these the subtraction

problems by including the cut-off factor h(T), eq. (2.48), and term ffin, eq. (2.46). We believe that any model of the equation is to be used in this (n, T) region must take these effects into

of state which account. Near the critical point, nuclei will not be well defined entities, will be important. It may well be preferable to describe properties

and fluctuations of matter under

those conditions in terms of the density fluctuation spectrum, as one does in the case of uncharged fluids, but making allowance for the long range Coulomb interaction which will inhibit fluctuations on length scales larger than the electron screening length. Under conditions where nuclei become small, our estimates for their energies using liquid drop ideas are only a first approximation. Better calculations could be made by employing more realistic estimates of properties of small clusters, such as could also give the ones described by RGpke and coworkers 63364).Such calculations more reliable information about the density of excited states in small clusters, and thus enable us to improve on the simple cut-off procedure we employed.

J.M. Lattimer et al. / Hot, dense matter

725

The

Hartree-Fock calculations of Bonche and Vautherin 25*26)omit from the free energy the contributions from nuclear translation and from the thermal modification of the Coulomb energy. The former is claimed to be negligible, but from results we have quoted, in the region T < 10 MeV for which those calculations are apparently intended that is by no means the case. So far as the Coulomb energy is concerned, it is perhaps necessary to point out that the nuclear thermal motion which causes the modification in it is not contained in the dynamics of the interior of one unit cell, no matter how accurately that is calculated. Despite such omissions in refs. 25,26),there is general agreement between those results and ours even though a different method and a different interaction are employed. In the region around the nuclei-bubbles transition on the s = 1 adiabat, the density dependence of the temperature, pressure and composition of the matter appear to be very similar to those we obtain. Attention has been paid to the bubbles phase-in both calculations, and in the (n, 1”) plane, both their and our adiabats are smooth through this region. The Hartree-Fock calculation of Wolff *‘) shows a very different and puzzling s = 1 adiabat: it lies below ours by a factor - 1.6 at a density of 0.06 fmm3, while at a density of 0.15 fme3 it rises abruptly to be a factor - 1.6 above our adiabat. The latter behavior may be ascribed to the small nucleon effective mass used in the interaction of ref. *‘) (rn$Jm&. - 3). But then we would have expected that adiabat to lie above ours at lower densities also. A comparison of the adiabatic indices reveals a similar situation: our results for r, as given in figs. 8, are quantitatively quite similar to those of refs. 25726),but those of ref. “> are quite different. The latter curves show T(s = 1) increasing slightly in the region of subnuclear densities, and T(s = 2,3) increasing steadily beyond n - 6 x 10m3fmm3. For instance, at n = 0.02 fme3 their graphical results are r( s = 2) = 1.36 and r(s = 3) = 1.5 1. Our values for these quantities are respectively 1.283 and 1.314. Remembering the significance of r-$, these differences are huge. The calculations of Wolff *‘) seem to be modelled on those of Bonche and Vautherin 25,26),and like those, they omit the effect of thermal motion on the Coulomb energy. In addition, they deliberately avoid the bubbles phase, even where it is favored energetically over the nuclei phase, because.of a supposedly undesirable appearance of its density profile at r = R, [ref. *‘)I. The transition from nuclei to uniform matter is reported to have the appearance of a second-order phase transition’. Good reasons have been given elsewhere for why the bubbles phase should exist, in the liquid drop model 5’), in the Hat-tree-Fock treatment 26) and in the Thomas-Fermi treatment 24), and why the transition between it and its neighboring phases should be of first order 51). We suspect that the abrupt behavior of the s = 1 adiabat of ref. *‘) just below nuclear saturation density may be due to the omission of this phase, and may in fact be a computer-smoothing of a physical discontinuity .’ W. Hillebrandt, private communication. Professor Hillebrandt.

We acknowledge a lively discussion of these points with

726

J.M. Lattimer et al. / Hot, dense matter

between

the nuclei and the uniform

in stellar-collapse

calculations,

standing of the adiabats In a future publication of different effective nuclear properties.

phases.

it would

In view of the importance

be very helpful

of ref. *‘). we shall address

nuclear

the question

interactions.

of this feature

to have a physical

under-

of the effect on our results

We shall also give more

details

of the

We wish to thank A. Burrows for helpful reading of the manuscript, and for interesting discussions. We are grateful to Michael Fisher for helpful remarks about the behavior of the translational term at higher temperatures, and to Gordon Baym for many valuable discussions during the course of this work. We thank consultants of the University of Illinois Computer Services Office for patient help, particularly A. Tuchman for invaluable instruction in the use of various graphics programs, and S. Lathrop for advice and assistance with the communications system. We thank also Bjiirn Nilsson for introducing us painlessly to the Univac computer at the University of Copenhagen. One of us (D.Q.L.) gratefully acknowledges the support of a John Simon Guggenheim Foundation Memorial Fellowship and the hospitality of the Aspen Center for Physics during part of this work. Most of us (D.Q.L., J.M.L., D.G.R.) have enjoyed the generous hospitality of Nordita at various times during the course of this research.

Appendix A ELECTRON

SCREENING

In this appendix electrons, allowing

we calculate the correction to the energy due to screening by for the finite size of the nucleus. We consider first the case of

zero temperature, and we shall employ the Wigner-Seitz approximation. If we denote the deviation of the electron density at point r from its average value by h,(r), the change in the energy is given by the sum of the compressional energy of the electrons and the change theory,

in the Coulomb

the compressional

energy

energy.

Within

the framework

of linear

response

is

(‘4.1)

$(8n,(r))2d3r e

while the change

in the Coulomb

energy

Wpot=-e

is

6V(r) I

h,(r)

d3r,

(A.21

where 8V(r) is the deviation of the electrostatic potential from uniformity, which may be calculated assuming the electron distribution to be uniform. Note that since the total charge remains unchanged under the distortion of the charge distribution,

721

J.M. Lattimer et al. / Hot, dense matter

we have the condition j &r,(r) d3r = 0, and therefore the uniform component

of the electrostatic potential does not enter. The electrostatic potential varies slowly over distances of order the interelectronic spacing, and therefore it is a good approximation to take the response of the electron density to the potential to be local: h,(r)

=

e?

6V(r).

(A.3)

e

Combining (A.l), (A.2) and (A.3) we find for the total change in the energy due to the electron response W screening= Wcomp + Wpot =-_-

1 ape 8nf(r) d3( r) = -fez

M”(r)

2 an,

d3r.

(A-4)

Now we calculate the electrostatic potential. For simplicity we assume that all the protons reside in nuclei, as our calculations indicate is the case at zero temperature. The electrostatic potential within the nucleus is given by r
(A.5)

where the first two terms come from the protons and the last two from the electrons. Outside the nucleus we have contributions Ze/r from the nucleus, and the same as in (A.5) for the electrons:

64.6) Performing the average in (A.4). one finds W screening

=

--

:;5

~/&r$(l

-y)‘(l

-f-y-$“)

(A.7)

C

per nucleus, where y = rN/ r,. For screening to have a small influence on the Coulomb correlations this energy must be small compared with the lattice energy, which at zero temperature is -&Z2e2/rc. From (A.7) we find W screening =&k2mrc( 1 -~)~(l -+Y-$~)

Wlattice.

w-0

When the nuclei fill an appreciable fraction of space the lattice energy largely cancels the other contributions to the Coulomb energy. If screening is to have a negligible effect it is then important that the screening energy be small compared with the total Coulomb energy $(Z’e’/rN)( 1 -& which can be considerably

+$‘)

=$(z’e’/+,)(

1-

y)‘( 1 +$y) ,

smaller than the lattice energy. We find

W screening = -Ak2,rcy

1-fy-fy2W 1 +;y

Coulomb

(A.9)

728 and

J. M. Latiimer et al. / Hot, dense mutter the

ratio

Wscreeningl ~~~~~~~~ is therefore finite even for y = 1, since both the

energies vanish as (I- y)* for y + 1. Thus, we see that the screening energy is small compared with both the lattice energy and the total Coulomb energy provided k,r,< 1, whatever the value of y. To see what this means in terms of the nuclear size, we make use of the fact that for relativistic electrons k2,r~=3(12/#3a,

(A.lO)

where CY= e’,/hc is the fine structure constant and the electron separation parameter r, is related to the electron density, n,, by the equation n, = ($rr~)-‘. Also if all protons are in nuclei, rc = Z”3re and therefore k2 r* -3 FT c-

.!? “3zz/3a 1 o ‘zl

(A.1 1)

From (A.9) we then find W screening W Coulomb

(A.12) The largest values of 2 are encountered when the nuclei fill roughly half the available volume (y’ = f), and when y is larger than this there is a transition to the bubbles state. Even for 2 = 1000, which is what may be found for y3 = f at the higher proton concentrations (x 2 0.3), Wscreeningf WCoulombis only - -0.16, which can cause some quantitative changes but does not alter the basic picture. Generally W screening/ WCouhnb will be much less than this value and will be completely negligible. Had we found Wscreening ,/ WCoulombof order unity or more, this would have signaled a tendency of the system to divide up into droplets of nuclear matter with linear dimensions large compared with the electron screening length k$. In that case the matter would be to a good approximation electrically neutral locally, and not only on average. The system would then consist of two electrically neutral bulk phases in equilibrium. Two of the equilibrium conditions would be the same as for the case of mixtures of neutrons and protons considered earlier [Lattimer and Ravenhall 29)], namely that the pressures and neutron chemical potentials in the two phases should be the same. The third condition would be that the sum of the electron and proton chemical potentials in the two phases should be the same, which expresses the physical condition that it cost no free energy to transfer a proton and its attendant screening electron from one phase to the other. At finite temperatures we expect the effects of screening to be less than at T = 0 since 2 decreases with increasing temperature. We therefore condude that it is a good approximation to neglect screening by electrons, and assume the electron density to be uniform.

729

J.M. Lattimer et al. / Hot, dense matter

Appendix B PROPERTIES

OF A PLANE

INTERFACE

Here we discuss various aspects of the thermodynamics of surfaces, paying particular attention to the neutron skin. Consider the plane interface between two coexisting bulk phases of nuclear matter which we label by 1 and 2. For simplicity we neglect the effects of the Coulomb interaction. The neutron and proton density profiles are shown schematically in fig. 1. We denote the total length of the system perpendicular to the interface by L, and its cross sectional area by A. The neutron and proton densities far inside the interface are denoted by nnl and nPl, and those far outside the interface by nn2 and nP2. We wish to express the number of neutrons, the number of protons, and the free energy as sums of two bulk contributions plus a surface contribution: N, = N,, + N,z + Nn,su,r , Np = Np, f

Np2

+

(B.1)

Np,surf ,

(B.2)

F=F,+F2+Fsuti.

(B.3)

While the total quantities are well defined, there is some arbitra~ness in how one makes the separation into bulk and surface contributions. The bulk contributions are defined in terms of the free energy densities. and number densities attributed to the bulk phases and the volumes occupied by the two phases. For the bulk properties, a natural choice, although not a necessary one, is to take the properties of bulk matter far from the interface on the appropriate side. This is the one we shall make. We shall choose the volumes for the two bulk phases in such a way that the total volume occupied by the two bulk phases is equal to the volume of the system. Thus, if we take the length of the denser phase to be z,, we have F, +F2 = zok+z,,,

~~,)+(L-zo)~~(~~*,

n,J t

(B.4)

where fi(n,, n,,) is the free energy of bulk matter per unit volume, and therefore we find F surf =F-zo/i8(n,,,

n,,)-(L-z,)ii&n,,,

npz).

(B.5)

We therefore see that the definition of Fsurf depends on the choice of zo, and we find

1 aF,urf -A - az, = %“2, $2) =

&pl,

n,2>

I.Ln(%l, - nn2)+Pclp(npl - np2)-

The second line follows from the first by noting that Rn,,

np) = &I% +&Jr,-

p,

(B.6) (B-7)

J.M. Lattimer et al. / Hot, dense matter

730

where

P is the pressure,

and using the fact that pn, pp and P are the same on the

two sides of the interface. Two possible choices of z0 are the position

of the effective

proton

and neutron

surfaces zp and z,. They are defined so that if, in the case of protons, say, protons at density npl occupied a length zp and protons at density npZ occupied the rest of the volume, the total number of protons per unit area would be the same as that in the actual system, i.e. n,,z,+n,,(L-z,)= Similarly,

for the neutrons

one would

&/ii.

(W

have

n,,z,+n,,(L-z,)=N,/a.

(B.9)

The lengths zp and z, obtained by so squaring off the particle distributions are shown schematically in fig. 1. For many purposes it is convenient to work, not with the free energy associated with the surface, but rather with the thermodynamic potential 0. If we denote the total number of neutrons and protons by N, and Np, then R is given by R=F-/_L,,N,,-~~N~. We may split R up into volume

and surface

contributions

(B.lO) in the same way we split

up F: R = 0, +c&+nsurf.

(B.ll)

If we again choose the total volume occupied by the two bulk phases to be equal to the total volume of the actual system, and if we further take the thermodynamic potential densities used in evaluating the bulk contributions to be the values for the actual conditions far away from the surface, then it is clear that -Pv,

n, +n,= where

V= La is the volume

of the system.

choice of z,. Thus the thermodynamic

(B.12)

Note that this is independent

potential

associated

with the surface

of the is given

by n

surf =

0

+

pv,

which does not depend on the choice of zO, in contrast to what is the case for the free energy. In our calculations we shall find that the surface thermodynamic potential is the quantity of primary importance. This point is stressed particularly by Landau and Lifshitz 65), and is implicit in the calculations of nuclear surface energies by Ravenhall, Bennett, and Pethick 33). We shall denote the thermodynamic potential per unit area of the surface by u.

J.M. Luttimer et

731

al. / Hot, dense matter

Let us now turn to the question of how many particles one associates with the surface. From (B.l), (B.2), (B.8), and (B.9) it follows that N n,surf= (n”i - n”z)(z” - G) ,

(B.13)

N p,surf= (n,l - n,d(z, - 4 .

(B.14)

From this we see that N”,surf and Np,+,“r cannot vanish for a common value of z. if z,# z”, and we must therefore allow for particles associated with the surface. If we knew the thermodynamic potential R as a function of the two independent variables CL”and c(.~ we could calculate N”,Surf and Np,surf directly, using the thermodynamic identities N n,surf= -

N p,surf= -

afL”rf

( )pJ’ (app) .

(B.15)

aP”

aaurr

(B.16)

P”.T

However, we known onsurfonly for chemical potentials consistent with coexistence of the two bulk phases. Along the coexistence curve one has equality of pressure of the two phases, which implies that possible variations of pp and p” are related by the conditions dP = uPI dpp + n”, dp” = np2dpp + nn2dp” ,

(B.17)

and therefore (B.18)

(n”,-n”2)d~“+(np1-np2)d~p=0. Variations of onsurfmay be written donsurf= -Np,surf Gp - Nn,surf h,

,

(B.19)

which along the coexistence curve may be written dp”

Nn,surf--Np,surfnPl

=

-

nP2 >

-(n,, - nn2)(zn - zp)dpn .

(B.20)

The physical significance of this result is that while N”,surf and Np,surf depend on the choice of zo, the value of the combination occurring in eq. (B.20) does not, and is equal to (n”, - n,,)(z, - zp). In other words, we are free to choose some linear combination of N”,surf and Np,surf arbitrarily, provided the combination occurring in the second line of eq. (B.20) is fixed. The choice we make is to take Np,surf= 0, that is, zp = zo, so the surface gives a contribution to the neutron number, but not to the proton number. We stress, however, that the physical results do not depend on the choice of z,. When we alter zo, the bulk and surface contributions to the

J.M. Lattimer et al. / Hot, dense matter

732

thermodynamic potential and the neutron and proton numbers both change, but in such a way as to keep the totals fixed. To summarize, we describe the plane interface as two bulk phases, occupying volumes z,A and (L- .z,),& and having the properties of the bulk matter in coexistence, plus a surface phase. The surface has a thermodynamic potential onsurf and contains Nn,surf neutrons and no protons. Its contribution to the free energy is therefore given by F surf = Gurf + pnNn.surf.

(B.21)

N n,surfis calculated from N l&surf=

--dfinsurf Gu, ’

(B.22)

where the full derivative means that it is to be evaluated along the coexistence curve. It would have been possible to avoid having neutrons in the surface of the model system if we had added the surface neutrons to the bulk phases of the model system. One could then work with bulk phases in the model system whose properties are not those of the two bulk phases in equilibrium. While this approach has its advantages in some applications, it has the disadvantage that the properties of the model system are not simply related to those of the actual system. In addition it makes discussion of equilibrium conditions very unwieldy. In applying the plane-interface results to our spherical nuclei and bubbles, where other effects modify the two-phase equilibrium conditions, the quantities transferred are the thermodynamic potential CT,the surface neutron density Nn,surf, and the plane-surface neutron chemical potential p”nS.For numerical convenience these are expressed as functions of T and of the proton fraction x of the denser phase. Under the conditions pertaining in spherical nuclei the surface neutron and proton chemical potentials differ from those of the corresponding plane-surface situation. We allow for the effect of these modifications to first order by inverting eqs. (B.15) and (B.16):

=

@“,f

-

bns- /&I Nn.surf 9

(B.23)

where the subscript s on the chemical potentials reminds us that the surface values may be different from those inside the nuclei. The surface free energy under the bulk conditions for spherical nuclei may also be calculated from this asurf:

@u,+ d,Nn,surf ,

Fsu,f= Gurf + pUnsNn.surf =

(B.24)

the second equality following from (B.23). Thus the surface free energy is not

J.M. L&timer et al. ,f Hot, dense mutter

733

OnsUrf is affected, however, and it is this affected to this order by the modifi~tion in pLnS. quantity which enters into the equilibrium equations of sect. 3. In earlier treatments of the model, such as that of Baym, Bethe and Pethick9), where ysurf was parametrized in terms of density or free-energy differences across the interface, some of the equilibrium conditions appear to introduce derivatives of ySUrfbecause of its apparent dependence on these quantities. The necessity for phase equilibrium means that such derivatives are incomplete and misleading. The presence of Nn,Surfin the counting of particles introduces modifications which are, in effect, the correct way to include the dependence of surface properties on the bulk properties. Appendix C COMPUTATIONAL

DETAILS

The core of the computations is the determination of the thermodynamic properties of the bulk phases of neutrons and protons, eqs. (2.24). In terms ofthe Fermi integrals

(C-1) the nucleon number densities and kinetic densities are

%P

=~(r,,p)-‘/‘~~,*(Y,,) ,

T”,P =

~(r,,p)-‘~2F,12(Y”,p) ,

(C.2) (C.3)

where (C-4)

Given r~,.,~and T, eq. (C.2) may be inverted to yield

(C.6) As discussed in ref. 29), the argument in eq. (C.6) depends only on the densities nn,p, and therefore the functions F;f2(z) and F3&) are all that are needed. We have approximated these functions, to an accuracy of about six digits, by the

J.M. L&timer et al. / Hot, dense matter

734

following

expansions: -x

3

(

1 - 1 cnx-2n , II=, >

x = (;p,

z> z,

z*> z> z3

-n~04(ln(~))n,

(C.7)

F3/2(_d=

fexp(-&hfj9

l-

shoe-Y I

YZ
(

i h,eeny , n=l >

y>y,.

(C.8)

The coefficients and the limits for F$ and F3,* are given in tables Cl and C2 respectively. For both functions, linear interpolation is used inside the intervals not specified in these expressions. The limiting values to be used in the interpolation are F$(z,)

F3,*( y,) = 18.5363404 ,

= -5.00011045,

TABLE

z, = 7.83797 c, = 0.822467032 c,= 1.21761363 cj =9.16138616

.z, = 7.6200 1, z, = 0.126063 d,=O.513636531 d, = 1.38707911 d,=O.187698933 dx = 0.0587095388 d,=O.O131648069 d, =2.14356866 x 1O-3 ds = 1.33227675 x 1O-4 d, = -2.51095876 x lo-’

TABLE

y, = -5.0 f, = 6.16850274 fi= - 1.77568655 f3= -6.92965606

Cl z, = e,,= e, = ez = e, =

0.114588 1.12837917 0.39894228 1 0.07327482 16 8.28 106450 x 1O-3

C2

yz = -4.9, ys = 1.5 g,, = 0.263252146 g, = 0.882378266 g, = 0.0405880718 g, = -5.78471772 x 1O-3 g, = -5.54437791 x 1o-4 g,=2.17431259x10-4 g, = 4.72987303 x 1O-5 g, = 2.83997426 x 1O-6

y4= h, = h, = h, = h, =

1.6 1.32934039 0.176776695 -0.0641500299 0.03 125

et al. / Hot, dense matter

J. M. Ldtimer

F$( ZJ = -4.89997247 ,

F3,&)

= 17.7620763 ,

F;QZj) = 1.90029415,

F&J

= 0.190509951 ,

F&L,)

= 0.172962007 .

F$(

ZJ = 1.9999985 ,

735

The thermodynamics of sect. 3 imposes six equilibrium conditions on the matter. Thus if, at a given T and lepton fraction Ye, matter is required to have density n, then mutually compatible values must be obtained for rN, rN/ rc = u”~, $9 Xi, no and xoso as to satisfy these conditions. These six quantities are obtained iteratively by Newton’s method. The sequence of iterations differs somewhat between two independent versions of the program, one made by J.M.L. and the other by D.G.R., but the results of the programs are in complete agreement to within the accuracy with which the equilibrium conditions are met. The extra requirements of the present problem, compared with that of the twophase region of bulk equilibrium described in ref. 29), involve rN and u”~. In the uniform-fluid region above the phase boundary, and at lower densities corresponding to the boundary for two-phase equilibrium in the bulk approximation, when X, falls below a small value the programs revert to a one-fluid nuclear phase plus a-particles. To prepare the text figures, arrays of the illustrated quantities, and some others, were prepared on a mesh of densities n over the range -5.0 s lnlo n (fmp3) s -0.45 with an interval of 0.05, and temperatures T over the range 0.15 MeVC T c 21 .OMeV with an interval of 0.15 MeV.

Appendix D FLUCTUATIONS

IN NUCLEAR

SIZE

Throughout the paper we have assumed that the nuclei are all identical. This is a good approximation at low temperatures, but becomes poorer as the temperature increases. Here we estimate the fluctuations in the sizes of nuclei. A more detailed account of this work will be given elsewhere [Pethick and Ravenhall ‘“)I. A somewhat different approach has yielded very similar results [Burrows and Lattimer “)I. The problem of evaluating the distribution of nuclear sizes is a complicated one, since it necessitates the determination of the free energy of very inhomogeneous and disordered distributions of nucleons. One is therefore forced to make approximations. In the first case we consider, we assume the electron distribution to be spatially uniform. This should be a reasonable approximation since electron screening lengths are large compared with typical nuclear spacings. We further assume that matter may be regarded as a number of electrically neutral cells each with a nucleus in the middle and a dilute nucleon vapor in the region outside. We denote the number of nucleons in the cell by A, and the number of protons (and also of electrons) by 2.

736

J.M. Lattimer et al. / Hot, dense matter

From the charge neutrality condition, the volume of the cell, v,J~, is VA,2= 2/n,

)

CD.11

where m, is the density of background electrons, which is for this case related to the proton fraction of the matter and the density by n,=nY,.

CD.21

We shall assume that the free energy of a configuration having nA,z cells with the specified numbers of nucleons and protons may be written as

F{n,qil= C

A.2

nn,if(k Z n,i,i)

,

(D.3)

where f is the free energy of a cell. Because the free energy of a cell contains a Coulomb “lattice” contribution, f will depend on the cell volume Z/n,. Since, however, n, is assumed to be constant for all cells, this dependence is contained in the g-dependence and need not be indicated explicitly. The free energy per cell depends on the density of nuclei also because of the translational contribution; otherwise the free energy is completely determined by 2 and A. It is convenient to express our present calculations in terms of the free energy of cells calculated under the assumption that all cells are identical, since that is what is used in the body of this paper. It is the quantity f(A, 2, l/v,& evaluated when there is one cell for every volume occupied by 2 electrons, i.e. for nA.2 = 1/(2/n,) = l/vn,z. We denote this quantity by

and it is given by F/q.,, where F is given by eq. (2.1). The neutrinos and photons play no role in the physics of this appendix, and we shall neglect their contributions to the free energy. When the fraction of cells of a given type, nA,ivA,i, is less than unity, the free energy has an additional negative term T In (n~v~,~), which reflects the larger volume available to each nucleus. This leads to a larger entropy per nucleus and a reduced free energy: _ _ _ I CD.3 f(A, 2) =f’(A, 2) + T ln (nA.2, VA.~). The most probable distribution of nucleon numbers is obtained by minimizing the free energy (D.3) with respect to n ~,i subject to the constraints that the total number of neutrons and the total number of protons be fixed: & finA. = n(lJ2.i%Ai=nYP.

YP),

(D-6)

0.7)

Since we have imposed the constraint that the proton density of every cell is fixed,

J. M. Lattimer et al. / HOI, dense matter

131

the condition that the total volume occupied by cells be unity, namely

c

?IA,iUri,i

=

A.2

1,

(D.8)

is equivalent to the proton conservation condition (D.7), because of (D.l). By minimizing the free energy one finds that the most probable distribution is given by nA,iuA;i=exp{-(fO(A,Z)-#-&/T},

(D-9)

where 5 and 5 are Lagrange multipliers. The value of A and 2 for which the fraction of space occupied by that species has a maximum is obtained by maximizing (D.9). Denoting those values by A0 and $ one finds af” - (D.lO) ;;i5(Ao, 20) = 5, (D.11) The quantities 5 and 5 are essentially chemical potentials for the neutron and proton respectively. However, they are not the usual chemical potentials: firstly, the derivatives in (D.lO) and (D. 11) are not evaluated keeping the volume of the cell fixed, and 5 and t therefore contain pressure terms in addition to the usual chemical potential contributions. Secondly, 5 contains a contribution from the electron chemical potential, since in our electrically neutral cells an electron is always added when a proton is introduced into the cell. This point will become clearer when we consider a second model for evaluating fluctuations later. If fluctuations are small, we may expand f” as

f”(A,Z)-&4-Z)-~Z=fo(Ao,Zo)-&io-$)-~~o +$,&AA’

+f,&AAAZ +&f&AZ*,

(D.12)

where AA = A - 2, and AZ = ,? - z,, and the fz are second derivatives off ‘. The fluctuations of i and 2 may be calculated straightforwardly, and one finds AA’=f&T/detf’, AA AZ = -f& T/det f" , A.?*=f&T/detf’,

(D.13) (D.14) (D.15)

where det f” = det (fi), and the bar denotes a thermal average. Results similar to (D.13)-(D.15) have been derived previously by, among others, Mackie 66). Our results are more general in that they allow for a more complex physical situation, including the finite pressure of nucleons outside nuclei.

738

J. M. Lattimer et al. / Hot, dense matter

To calculate the change substitution of (D.5) reads

in the free energy

F{n.z,A= 2

na,i(f”(A,

we return

to eq. (D.3),

.

Z) + T In (n~,iu~,i))

which

after

(D.16)

A.2

The leading correction term to the identical-cell is that due to the logarithm, and one finds

approximation

at low temperatures

(D.17) The quantity

n~~~~u~~i, is fixed by any of the conditions

(D.6)-(D.8),

and is (D.18)

Since we are interested in the leading corrections at low temperatures we have replaced all quantities by their values for A = A0 and 2 = go, except in the exponent. Then the Helmholtz

free energy

is given by (D.19)

where F” is the free energy calculated neglecting fluctuations. From this expression the fluctuation contributions to the chemical potentials, the pressure and the entropy may be found

directly: a(F-F”) AP,, =

an, (D.20) d(F-

A/l, = Ap=

F”)

(D.21)

= A/-G,, an, @(F-F’)ln) an

As= _L a(F-F”) aT n

=$~ln((de~~~li2)

=kln

( (de:;;“‘)

7

*

(D.22)

(D.23)

The expression (D.23) has a simple physical interpretation: 2rT/(detf’)“’ is the effective number of nuclear species present in significant numbers, and therefore the additional entropy per cell is just the logarithm of it. One point to notice is that in evaluating fz,only nucleon contributions need to be included. This follows because the electronic contribution to the free energy is linear in the total number of electrons, since by assumption n, and therefore the

J.M. Laitimer et al. f Hot, dense matter

739

electron chemical potential are fixed, so that second derivatives of the electronic free energy with respect to electron number vanish. The approximation described above clearly understimates the size of the fluctuations because the electron density has been held fixed. A second approximation is to assume matter to be made up of electrically neutral cells, but to allow the electron density to vary from cell to cell. This will underestimate the energy of a fluctuation since we neglect contributions to the energy due to the spatial variation of the electron density. Such cont,ributions are likely to be considerable since electron screening lengths are large compared with internuclear spacings. However, the calculation is an instructive one, since it gives a plausible estimate for an upper limit to the fluctuations. One then proceeds exactly as before, except that the free energy of a cell has an additional dependence on the cell volume u. The arguments given above go through exactly as before, with the exception that the constraint on the volume (D.8) is now independent of the proton number condition since the electron density in a cell can differ from its average value nY,,. It is therefore necessary to introduce another Lagrange multiplier, and one finds n~iiuqi

=

exp (-(fO(A, Z, u) - [fi - AZ - vu)/ T) .

(D.24)

The Lagrange multipliers 6, A and v are the neutron chemical potential, the sum of the proton and the electron potentials, and the pressure, respectively. We denote them by symbols different from the usual ones since it is important to remember when performing expansions of quantities about most probable values that they are the parameters to be held fixed. To make contact with the earlier treatment, where u = Z/n, with n, independent of 2, we observe that the Lagrange multiplier 5 in (D.9) corresponds to A + v/n, in the present case. In principle we could treat the fluctuations in the cell size on the same footing as those of A and .?. However, these fluctuations correspond to the excitation of breathing modes of cells, and in calculating their effects one should also add kinetic energy terms associated with this collective motion. This would take us far beyond the rather simple picture in this paper, and we shall merely assume that the cell size t, for given A and 2 is that which maximizes the exponent in (D.25). This gives as the implicit equation for v (D.26) where p. is the pressure when fluctuations are absent. To bring out clearly the difference between this treatment and the earlier one where the electron density is held fixed it is convenient to expand quantities about the volume u4~ = Z/n,, where n, is the mean electron density. We assume the pressure condition (D.26) to hold for A = Ao, 2 = .?, for a volume &,,& = zo/ n,. The proton fraction of the matter is Z/A. Expanding the pressure condition about

J.M. L&timer et al. / Hot, dense matter

740 _

_

w

.

_

A = A,,, Z = 2, and VA,~ = vt.2 = Z/n, @O zAA

-

one finds to first order

ape ape - +;Avai

+zAZ

= 0.

(D.27)

All derivatives are to be evaluated for A = Ao, Z = 2, and VA,~ = V~,KXTo second order, the compressional free energy of the nucleus A,-?, due to changing the volume from vi.2 to VA,~ is f”( A, Z, VA,~) - VV,Q = -: One finds that the distribution

of nuclei

1 C+po - * ---&A, Z, v:,i)A&

.

(D.28)

is given by

1

ni,ia-

VS,i (D.29) to A and 2, the volume

of the cell is to

be held equal to z/n,, and therefore the electron contribution proportional to Zfe( n,), has no second-order derivatives. Thus the in (D.29) may be replaced by derivatives off”. The result (D.28) from the earlier one with cells of constant electron density only compressional term, which lowers the energy of a fluctuation.

In evaluating

of y, which is derivatives off0 therefore differs by virtue of the Substituting the

expression

the derivatives

with respect

for AvA,~ from (D.27) into (D.29) n,z,z- a-

1

one finds the distribution

exp [-$h,AXiAXj/

T] ,

is given by (D.30)

f&i

where

x1=/I ,

x2=2,

(D.31)

/‘$!r_apoapo ap,_ axi axj

axi axj

I

av

(D.32)

All of the earlier results (D. 18)-(D.23) now go through, provided one replaces det f” by det h. While a2fo/axi axjcontains no electronic contribution as we remarked earlier, the electronic contributions to the derivatives of p. are very important, and give the dominant contributions under most circumstances. In this appendix we have couched our arguments in terms of the numbers of nucleons and of protons in a cell. Of course, related results can be derived for fluctuations in the numbers of nucleons and of protons in the nucleus. The numerical evaluation of AA and AZ quoted in sect. 5 has been performed by disabling or modifying those routines of the iterative equilibrium search which re ate to the l&b-tear radius, the baryon density and, in the second case, the electron density. The resultr. 7 changes off’ from its equilibrium value may be identified

J.M. Lattimer et al. / Hot, dense matter

741

with a power series expansion in the changes AA and AZ and, in the second method, the local electron density, thus providing an evaluation of fiA, etc. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45)

J.M. Lattimer, Ann. Rev. Nucl. Part. Sci. 31 (1981) 337 W.D. Arnett, Ap. J. 218 (1977) 815; Ap. J. Lett. 263 (1983) L55 J.R. Wilson, Ann. NY Acad. Sci. 336 (1980) 358 R.L. Bowers and J.R. Wilson, Ap. J. Suppl. 50 (1982) 115; Ap. J. 263 (1982) 366 P. Lamb, M.N. Roy. Astron. Sot. 188 (1979) 565 M.J. Murphy, thesis, LJ. of Chicago (1980); Ap. J. Supp. 42 (1980) 385 T.J. Mazurek, J.M. Lattimer and G.E. Brown, Ap. J. 229 (1979) 713 M.F. El Eid and W. Hillebrandt, Astron. Astrophys. Suppl. Ser. 42 (1980) 215 G. Baym, H.A. Bethe and C.J. Pethick, Nucl. Phys. Al75 (1971) 225 K. Sato, Prog. Theor. Phys. !I4 (1975) 1325 D.L. Tubbs and SE. Koonin, Ap. J. 232 (1979) 59 W.A. Fowler, C.A. Engelbrecht and S.E. Woosley, Ap. J. 226 (1978) 984 V.A. Volodin and D.A. Kirzhnits, ZhETF Pisma 13 (1971) 450: JETP Lett. 13 (1971) 320 Z. Barkat, J.-R. Buchler and L. Ingber, Ap. J. 176 (1972) 723 J. Arponen, Nucl. Phys. A191 (1972) 257 R. Ogasawara and K. Sato, Prog. Theor. Phys. 68 (1982) 222 J.W. Negele and D.Vautherin, Nucl. Phys. A207 (1973) 298 J.-R. Buchler and R.I. Epstein, Ap. J. Lett. 235 (1980) L91 J.-R. Buchler and M. Barranco, J. de Phys. COB. C2 (1980) 31 M. Barranco and J.-R Buchler, Ap. J. Lett. 245 (1981) L109 M. Barranco and J.-R. Buchler, Phys. Rev. C24 (1981) 1191 S. Marcos, M. Barranco and J.-R. Buchler, Nucl. Phys. A381 (1982) 507 R. Ogasawara and K. Sato, Prog. Theor. Phys. 70 (1983) 1569 E. Suraud and D. Vautherin, Proc. Second Workshop on nuclear astrophysics, Ringberg Castle, Tegernsee, Max-Planck-Institut fiir Physik und Astrophysik, Munich 1983, p. 50; to be published P. Bonche and D. Vautherin, Nucl. Phys. A372 (1981) 496 P. Bonche and D. Vautherin, Astron. Astrophys. 112 (1982) 268 R.G. Wolff, thesis, Technische Universitat, Miinchen (1983); reported in W. Hillebrandt, K. Nomoto and R.G. Wolff, Astron. Astrophys. 133 (1984) 175 W.A. Kiipper, G. Wegman and E.R. Hilf, Ann. of Phys. 88 (1974) 454 J.M. Lattimer and D.G. Ravenhall, Ap. J. 223 (1978) 223 D.Q. Lamb, J.M. Lattimer, C.J. Pethick and D.G. Ravenhall, Nucl. Phys. A360 (1981) 459 M. Barranco and J.-R. Buchler, Phys. Rev. C22 (1980) 1729 M. Rayet, M. Arnould, F. Tondeur and G. Paulus, Astron. Astrophys. 116 (1982) 183 D.G. Ravenhall, C.D. Bennett and C.J. Pethick, Phys. Rev. Lett. 28 (1972) 978 D.G. Ravenhall, C.J. Pethick and J.M. Lattimer, Nucl. Phys. A407 (1983) 571 J.P. Hansen, Phys. Rev. A8 (1973) 3096 D.Q. Lamb, J.M. Lattimer, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett. 41 (1978) 1623 T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615 D. Vautherin and D.M. Brink, Phys. Lett. 32B (1970) 149; Phys. Rev. C5 (1972) 626 P.J. Siemens and V.R. Pandharipande, Nucl. Phys. A73 (1971) 561 B. Widom, in Phase transitions and critical phenomena, vol. II, ed. C. Domb and MS. Green (Academic, New York, 1972) p. 79 J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. B21 (1980) 3976 J.W. Cahn and J.E. Hilliard, J. Chem. Phys. Zs (1958) 258 H.A. Bethe, Phys. Rev. 167 (1968) 879 J.P. Hansen and J.J. Weis, Molec. Phys. 33 (1977) 1379 W.L. Slattery, G.D. Doolen and H.E. De Witt, Phys. Rev. A21 (1980) 2087

142

J.M. Laftimer et al. / Hot, dense matter

46) C.J. Pethick, 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66)

D.G. Ravenhall and J.M. Lattimer, Phys. Lett. 128B (1983) 137 G.R. Satchler, L.W. Owen, A.J. Elwyn, G.L. Morgan and R.L. Walter, Nucl. Phys. All2 (1968) 1 E.E. Salpeter, Ap. J. 134 (1961) 669 F.J. Dyson, Ann. of Phys. 63 (1971) 1 S.A. Bludman and K.A. Van Riper, Ap. J. 224 (1978) 631 D.Q. Lamb, J.M. Lattimer, C.J. Pethick and D.G. Ravenhall, Nucl. Phys. A411 (1983) 449 C.J. Pethick, D.G. Ravenhall and J.M. Lattimer, Nucl. Phys. A414 (1984) 513 A. Burrows and J.M. Lattimer, Ap. J. (1984), in press C.J. Pethick and D.G. Ravenhall (1984), in preparation D.G. Ravenhall, C.J. Pethick and J.R. Wilson, Phys. Rev. Lett. 50 (1983) 2066 C.J. Pethick and D.G. Ravenhall, Numerical astrophysics, ed. J. Centrella and R. L. Bowers (Jones and Bartlet, Portola Valley, CA, 1984) B. Friedman and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 J. Cooperstein and J. Wambach, Nucl. Phys. A420 (1984), 591 G. Fuller, Ap. J. 252 (1982) 741 A. Yahil and J.M. Lattimer, in Supernovae: a survey of current research, ed. M.J. Rees and R.J. Stoneham (D. Reidel, Dordrecht, Holland, 1982) p. 53 H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A324 (1979) 487 C. Mahaux, in Nuclear physics, ed. C.H. Dasso, R.A. Broglia and A. Winther (North-Holland, Amsterdam, 1982) p. 319 G. Riipke, L. Miinchow and H. Schulz, Nucl. Phys. A379 (1982) 536 G. Ropke, M. Schmidt, L. Miinchow and H. Schulz, Nucl. Phys. A399 (1983) 587 L.D. Landau and E.M. Lifshitz, Statistical physics, pt. I, 3rd ed. (Pergamon, Oxford, 1980) ch. XV F.D. Mackie, Ph.D. thesis, U. of Illinois at Urbana-Champaign (1976)