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AN EXTENDED THOMAS-FERMI CALCULATION OF SUPERNOVA MATTER
X. Viñas, M. Barranco, M. Pí, A. Polls, A. Pérez-Canyellas
To cite this version:
X. Viñas, M. Barranco, M. Pí, A. Polls, A. Pérez-Canyellas. AN EXTENDED THOMAS-FERMI CALCULATION OF SUPERNOVA MATTER. Journal de Physique Colloques, 1984, 45 (C6), pp.C6- 103-C6-110. �10.1051/jphyscol:1984612�. �jpa-00224213�
AN EXTENDED THOMAS-FERMI CALCULATION OF SUPERNOVA MATTER
X. Vinas, M. Barranco, M. Pi, A. Polls and A. Perez-Canyellas*
Facultad de F%sica, Universidad de Barcelona, Diagonal 645, Barcelona-28, Spain
*Fasultad de Fisica, Depavtamento de Fisica Tebvica, Universidad de Valencia, Bwcjasot (Valencia), Spain
Résumé - Nous avons utilisé la méthode "Extended Thomas-Fermi" (ETF) pour calculer les propriétés de la matière stellaire dense et chaude en équilibre bêta avec des électrons et des neutrinos. Nous présentons les résultats concernant l'adiabatique S/A = 0,70k Ye (s Z/A) = 0,35.
Les différentes transitions de phase qui ont lieu dans ce système sont aussi considérées.
Abstract - The extended Thomas-Fermi (ETF) method at finite tem- perature has been used to compute the properties of h o t , dense stellar matter in b e t a equilibrium with electrons and n e u t r i n o s . Calculations along the S/A=0.70 k, Ye ( = Z/A) = 0.35 adiabat are reported and phase transitions considered.
Introduction
In the last few y e a r s , hot dense matter has been the subject of a gro- wing interest. Two are the main reasons for its study. Firstly, high
energy heavy ion reactions 111. Although it is unclear whether equili- brium Thermodynamics can be used in many situations found in this field
, the amount of "hot" papers in the heavy ion -literature has impressi- vely grown.
The other physical situation where one expects to find hot dense matter is in the late evolutive stage of massive stars. H e r e , the situation is far more clear from a thermodynamical point of view. Several groups, using formalisms of different sophystication, have attacked the p r o - blem of computing the equation of state (EQS) of matter at densities around nuclear matter saturation density and temperatures (T) up to 1 0 - 20 MeV / 2 - 9 / . All these calculations but that of ref. 3 were performed in the spirit of the Wigner-Seitz (WS) approximation. That i s , the the£
modynamical properties of a system of n u c l e o n s , electrons and n e u t r i - nos are obtained by minimising the Free energy of the content of a sphe rical cell of radius R e , imposing charge neutrality and eventually, be_
ta equilibrium. Three kinds of nucleon configurations are found in the minimisation. At relative low densities (up to about half the nuclear matter saturation density v a l u e ) , huge nuclei ( A < 1 0 0 0 ) inmersed in a free nucleon sea are the most favoured nuclear configuration. At high- er densities, it is more convenient to think of low density "bubbles"
of nuclear fluid embebbed in an outer, denser nuclear matter sea. F i - nally, at nuclear matter densities and h i g h e r , the nuclear clusters mer ge and an homogeneous system fills the cell.
To describe the nuclear component of the system, hot Hartree-Fock (HF) / 5 , 9 / , Thomas-Fermi (TF) / 4 , 6 - 8 / and a sophysticate liquid-drop models have been set u p , yielding global results in fairly agreement. Never- theless, there are some differences between their results essentially due to the different level of accuracy in the description of the n u - clear surface and to the nuclear force employed.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984612
C6-104 JOURNAL DE PHYSIQUE
Obviously, the non-spherical HF calculations of ref. 9 are the most r e liable ones, together with the spherical HF calculations of ref. 5.
However, they are very much time consuming, specially at high density
( p ) and T. Furthermore, it is rather unfear to use a method based on the nuclear shell structure in a situation where the shell structure is completely whased out.
A semiclassical method like TF seems in principle more appropiate for the description of hot dense matter / 4 , 6 - 8 / . However, all the TF EOS calculations performed up to now can be criticized in the sense that they have used the simplest relationship between the kinetic energy d e n s i t y x and the particle density3 . As a result, the nuclear surfa- ce was not correctly described nor the surface Free energy properly e s timated.
It is the aim of this contribution to present a calculation of the EOS using a hot TF method which is a natural extension to finite T of the ETF one /10,11/. The ETF method is known to yield good results for the global properties of nuclei at T=O (i.e., good binding energies and n u clear surfaces at the same time) and gives also results for warm, iso- lated nuclei in agreement with those obtained with the hot HFmodel/l2/
The hot ETF (HETF) method
Since the method is fully described in ref. 11, we just recall that it essentially consistsin introducing the following definitions of the ki netic energy density -5 and particle density 9 - ( f o r each kind of
ticles) : 7 9
where mq is the nucleon effective mass, * is the degeneracy parameter
b
Vq is the single particle potential, J 3 ( ) is the Fermi integral
and 2, and rl( are the gradient corrections to the kinetic energy dey sity due to density inhomogeneities and non local effects (nucleon effective mass and spin-orbit energy)/l3/. We refer the reader to refs 10 and 11 for more details concerning hot TF methods.
The nuclear Free energy is written as
..sed in HF calculations. In our case, we have chosen Tondeur's T6 for- ce/14/ because of its good symmetry properties (a correct descri~tion of the neutron skin and of the neutron gas) and because it has m = m which is a very convenient property for finite temperature calculations ETF calculations are performed within a trial function minimisation procedure. Following the method of ref. 8 we have minimised the nu- clear plus electron Free energy of the WS cell using trial densities of the sort
4
The constant 9% mimicks the external uniform nucleon sea. Electrons are treated as a uniform, very degenerate and relativistic Fermi gas coupled to the nucleons via Coulomb interaction. Neutrinos are comple- tely decoupled from the other particles. Their contribution is deter- mined from beta equilibrium once one knows the nucleon and electron chemical potentials. As the electrons, they turn out to be also extre- mely degenerate.
4
The number of variational parameters per density were two, aq and 5%
The value of 9 was fixed to 1.5 and Rq was obtained from the relation ship existing between 3 , aq and the "nuclear matter radiust1 at the corresponding values of T and Ye /15/. Finally,
d was fixed by nor- malisation.
The trial function procedure has been criticised by several authors /5-7/ because of its supposed lack of flexibility. We want to point out that T=O ETF calculations performed by M. Brack and coworkers using a slightly more flexible density than ours /11/ yield results in excellent agreement with the .HF ones.
The main limitation of function (6) is its unability to reflect the ex pected central depression in the proton density due to Coulomb repul- sion. We have checked that this is a minor drawback because of the small contribution this effect has to the total energy (Note that the presence of electrons reduces considerably the Coulomb energy).
On the contrary, full variational TF calculations cannot deal with c02 plete forces, i.e. including the semiclassical spin-orbit energy, nor correctly describe the nuclear surface. From our experience at T=O,we claim these two effects are much more important than the fact of per- forming a trialanction or a variational calculation. To clarify the situation, a proper: comparison between HF and the two kind of TF cal- culations is in order.
Results
The stellar collapse is isentropic to a good approximation /16/. This is essentially so for neutrinos are trapped at a density around 0.005 fm-3, inhibiting further electron captures,.and because the collapse processes very orderly. Thus, it is convenient to compute the EOS along the adiabats. These adiabats are characterised by Stet, the to- tal entropy per baryon and Y1, the number of leptons per baryon (Y - Y +Yj ) . Actually, it is used to make the calculation at cons- tar& :Y and electron plus nuclear entropy per baryon (S). It turns out
C6-106 JOURNAL DE PHYSIQUE
that the resulting Ye and Stot are almost constant.
We have computed the S=0.7, Ye= 0.35 adiabat. These values are present ly considered as a lower bound of the ones found when the collapse sets off /17/. Our results are presented in figures 1-3 and table 1.
The results shown on the figures include no contribution from the neu- trinos because they are no relevant for the physics we aim to describe with these figures.
I I L L
-7 - 6 -5 - 4 - 3 -2 - 1 0
L O G g
F i g u r e 1
Figure 1 represents in the 3 - T plane the boundary of the region whe- re the homogeneous nuclear system is not the most favoured stable pha- se. It is below the full line that one could expect to find nuclear clusters. The inner, hatched region is the unstability zone. Inside this region the nucleons cannot exist in a single homogeneous phase and nuclei (or bubbles) should be found floating in a free nucleon sea It is worth noting that clusters might be present up to T 2 15 MeV and
Sr.0.14 fm-3 (for Ye=0.35). At both sides of the unstability region there are two metastability zones, the right hand one much narrower than the left hand one. This phase diagram was obtained within the bulk matter approximation, neglecting surface and Coulomb effects. As we shall see later, the EOS so obtained is a surprisingly good zeroth order approximation to the exact one. We refer the reader to refs. 18 and 19 for a through description of this approximation.
Figure 2 shows the enthalpy per baryon versus pressure. Enthalpy is the appropiate thermodynamical potential to describe phase transitions at constant S and p /5/, conditions which are likely to prevail at the nucleus-bubble and bubble-homogeneous matter transitions /20/. From this figure, one may see that a nucleus-bubble phase transition takes place at a pressure p=0.846 MeV fm-3. Within this formalism, it is a first order transition. The density discontinuity is very small; at the transition point we have g =0.062 for nuclei and g =0.065 for bub- bles. These values cannot be considered too seriously because no sphe- r2cal model can deal with this phase transition. Indeed, at this point
an artifact of the imposed spherical symmetry may only be stated by non spherical calculations / g / . Answering this question is beyond the scope of this work and in our spherical model we take the existence of bubbles for granted.
F i g u r e 2
At high densities, bubbles abruptly disappear. Contrarily to the for- mer transition, where two clear metastable regions are observed (da- shed curves near the transitionpoint in figure 2 ) , no metastable re- gion was found for this transition. Actually, our last calculated bub- ble was at Q = 0 . 1 0 fm-3. The single phase nuclear matter starts being stable at $ = 0 . 1 0 3 f r 3 and no bbble configuration was found at
0.105 f ~ n - ~ . P =
In figure 3 we have plotted the EOS (log p vs. log S ) . We remind the reader that neutrinos' contribution is not included.
Another important quantity is the adiabatic index defined as
The value of g is 1 . 3 1 6 in the nucleus regime.It increases up to 1.345 in the bubble region and rises abruptly to 2.098 at the point where bubbles merge. The different pieces of the EOS aredisconnectedbecause the unability of studying the coexistence of a nucleus-bubble or bub-
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ble-homogeneous m a t t e r s y s t e m .
-2.5 - 2 0 -1.5 -1.0 -0.5 0.0
L O G f
F i g u r e 3
A l s o shown on f i g u r e 3 i s t h e b u l k a p p r o x i m a t i o n EOS ( d a s h e d l i n e ) . T h i s a p p r o x i m a t i o n o v e r e s t i m a t e s t h e p r e s s u r e b u t t h e v a l u e o f t h e a d i a b a t i c i n d e x 1 . 3 2 5 i s v e r y c l o s e t o t h e e x a c t a v e r a g e v a l u e .
T a b l e 1 i s t h e t a b u l a t e d EOS a l o n g t h e Ye=0.35, S / A = 0 . 7 a d i a - b a t . Observe t h a t i t c o r r e s p o n d s t o Y 1 r v 0 . 4 5 a n d Sto / A & 0 . 7 0 2 i n t h e h i g h d e n s i t y z o n e . The n e u t r i n o c h e m i c a l potential c a n b e o b t a i n e d from
U n i t s a r e MeV f o r T , c h e m i c a l p o t e n t i a l s a n d F r e e e n e r g y ; MeV fm-3 f o r t h e t o t a l p r e s s u r e ; fm-3 f o r t h e b a r y o n d e n s i t y 9 and fm f o r t h e c e l l r a d i u s R C . A / Z a r e t h e number o f n u c l e o n s / p r o t o n s i n t h e c e l l , w h e r e a s A c l u s / Z c l u s a r e t h e n u c l e o n s c l u s t e r e d i n n u c l e i o r b u b b l e s . Observe t h a t f o r b u b b l e s t h e s e numbers a r e n e g a t i v e meaning t h e n u c l e a r d e - f a u l t i n s i d e t h e c e l l . Z\ i s t h e p l a s m a p a r a m e t e r d e f i n e d a s t h e r a t i o o f t h e Coulomb e n e r g y t o t h e t h e r m a l e n e r g y :
? .005 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 T 2.10 2.45 2.84 3.08 3.28 3.44 3.56 3.67 3.77 3.87 3.97
Rc 19.3 16.5 14.6 14.1 14.0 14.3 14.7 15.5 14.2 13.1 12.2
A/Z 152/ 53 188/ 66 262/ 92 349/122 459/161 608/213 801/280 1082/379 956/335 856/300 756/265 A . /Z , dus olus 145/ 53 181/ 66 254/ 92 340/ 122 449/ 161 598/ 213 788/ 280 -1218/-435 -825/-294 -562/-200 -371/-132
P 100 154 291 495 810 1326 2152 4798 2324 1136 522
r» -3.86 -3.85 -3.85 -3.88 -3.94 -4.00 -4.07 -5.01 -5.00 -5.01 -5.05
rp -21.5 -23.2 -25.5 -27.2 -28.6 -29.9 -31.0 -32.9 -34.1 -35.2 -36.2
N 73.4 92.5 116.5 133.4 146.9 158.2 168.2 177.0 185.0 192.5 199.4 tôt 12.41 18.45 26.11 31.48 35.74 39.31 42.41 45.13 47.44 49.57 51.54
P tot 0.0359 0.0921 0.2354 0.4063 0.5974 0.8042 1.0241 1.2032 1.4423 1.6905 1.9454 S. ./A tôt .728 .728 .727 .727 .726 .726 .725 .725 .724 .724 .723
Y l .422 .432 .441 .445 .448 .450 .451 .452 .452 .453 .453 Table 1
C6-110 30URNAL DE PHYSIQUE
F o r T 5 170 t h e s y s t e m c o n s t i t u t e s a Coulomb l i q u i d w h e r e a s f o r 27 7,
170 we h a v e a Coulomb c r y s t a l / 2 1 / . We c a n s e e from t h e t a b l e t h a t t h e t r a n s i t i o n from Coulomb l i q u i d t o Coulomb s o l i d t a k e s p l a c e a t g d 0 . 0 1 . I n t h e p r e s e n t c a s e t h e s y s t e m r e m a i n s i n t h i s c r y s t a l p h a s e from t h i s p o i n t on b u t i t may happen t h a t f o r a h i g h e r S a n o t h e r p h a s e t r a n s i - t i o n from Coulomb s o l i d t o Coulomb l i q u i d t a k e s p l a c e l a t e r i n t h e bub b l e regime / 4 / .
HETF c a l c u l a t i o n s o f h o t d e n s e m a t t e r p r o p e r t i e s a r e f a s t and r e l i a b l e . P r e s e n t l y , work i s i n p r o g r e s s t o t a b u l a t e a n EOS t h a t c o u l d b e u s e d i n c o l l a p s e c a l c u l a t i o n s .
Acknowledgments
We a r e most i n d e b t e d t o F r a n s o i s Tondeur f o r h i s s u g g e s t i o n s when we s t a r t e d t h i s s t u d y . T h i s work h a s b e e n s u p p o r t e d i n p a r t by t h e C A I C Y T ( S p a i n )
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