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APPLICATIONS OF A FULLY VARIATIONAL
METHOD FOR SOLVING ZERO ORDER
THOMAS-FERMI EQUATIONS
Eric Suraud
To cite this version:
Colloque C6, supplkment au n06, T o m e 45, juin 1984 page C6-161
APPLI CAT1
ONS OF A FULLY VARIATIONAL METHOD FOR SOLVING ZERO ORDER
THOMAS-FERMI EQUATIONS
E . S u r a u d
Division de Physique The'orique*, I n s t i t u t de Physique Nucle'aire,
91 406 Orsay Cedex, France
~ 6 s u m 6
-
Nous pr6sentons quelques r6sultats obtenus dans un calcul Thomas-Fermi effectu6 sans resteindre l'espace variationnel et nous le comparons $ ceux obtenus avec des fonctions d'essais et h ceux de la m6thode du champ moyen, dans le cas de llkquation dl&tat de la matikre dense.Abstract - We present some results obtained in a Thomas-Fermi calculation without restriction of the variational space. We compare them to those obtained with trial functions and to the results of the mean field approximation, in the case of the equation of state of hot dense matter.
In the Thomas-Fermi type calculations, the nucleus is described with the help of the densities pntp (r), by eliminating the kinetic and spin-orbit energy density terms in the energy functional. Various methods have been developed for this purpose. In most cases the variational parameters pntp(r) are approxima- ted by trial functions / I / which destroy the self-consistency of the calcula- tion. We have constructed a method of solution of the Thomas-Fermi equations without restriction of the variational space of the density profiles. We apply this method to calculations of the hot dense matter which occurs in the super- novae explosions 121. We show that the zero-order Thomas-Fermi method is a very good approximation to the Hartree-Fock results at high temperature, while, in contrast, the use of trial Fermi functions gives a rather poor agreement with Hartree-Fock calculations.
Method
We minimize the free energy functional F(N,Z,T,R ) of a system containing
N
neutrons and Z protons inside a spherical box of rsdius Rc and at a temperature T ; the nucleon-nucleon interaction being a Skyrme type one. For this purpose we construct a series of profiles
~p)(r)
(a labelling neutrons or protons) defined bvwhere F&(") is the functional derivative oTn1the free energy with respect to the density pa(r), calculated for p (r) G p (r), and X Z is a parameter of the
n7P n7P method, similar to the imaginary time step 131.
The recursive relation (1) leads to the minimization of the free energy by means of successive first order variations. The technical details concerning this method have been described in previous publications /4,5/.
In figure 1 we present a comparison between Hartree-Fock and Thomas-Fermi density profiles. The agreement is very good for the bulk parts while the description
*Laboratoire associg a u C.N.R.S.
C6-162 JOURNAL DE PHYSIQUE
of the surface is slightly different, due to the lowest order Thomas-Fermi approximation we use. 10 E + 0- 0.05 0.01
Fig.1 - Proton and neutron distributions obtained in lead 208 with interaction
5111 in complete Thomas-Fermi and in Hartree-Fock calculations. The box size is 14 fm and the temperature 6 MeV.
I I I 2 0 8 ~ b protons -
-
-
Thomos Fermi - - - tiorfree. Fock- . -. -. - . - HFlno spin orbit)
-
*7
Applications to the Equation of State
0 5
l o R ( f r n ) 14
I I I
The equation of state, which gives the pressure P as a function of density p and temperature T, is an important ingredient in the description of stellar collapse and supernovae explosions. For densities up to about nuclear density, the matter is a mixture of protons, neutrons, nuclei, electrons and neutrinos.
neutrinos being included as free gases.
Various methods have been used to describe this exotic system, ranging from the bulk matter approximation / 6 / and the Liquid drop model / 7 / to more micros- copic approaches as Thomas-Fermi /8/ and Hartree-Fock calculations /9/. The mean field approximation gives the most reliable results but the numerical work involved becomes untractable at high temperature. The Thomas-Fermi method which is much simpler seems to be very adequate because the shell effects are washed out at the relevant temperatures (T % 5 MeV) and because we are
only interested in average quantities such as pressures and entropies.
In figure 2 we show the density profiles pn(r) and p (r) calculated for a
P
particular configuration of the equation of state labelled by the entropy per baryon s = 1 and the lepton fraction YE = 0.35 /lo/. The results obtained with trial Fermi functions /11/ are in rather bad agreement with those of the mean field approximation 191. On the contrary our complete Thomas-Fermi calculation givesa satisfactory description of both the surface and the bulk parts, which shows the limitations of using an arbitrary restricted subspace for the density profiles pn (r).
9 P
Fig.2- Proton and neutron distributions in the particular configurationp= 0.02 fm
'
and T =3.89 MeV of the equation of state labelled by s=l and Yq, = 0.35, for Hartree-Fock (full line (HF)), complete Thomas-Fermi (dashed line (TFI)) and Thomas-Fermi with trial functions (dashed-dotted line (TF2)) calculations. The interaction is SkM.C6-164 JOURNAL
DE
PHYSIQUEpressure) to more than 513 (nucleons pressure). This products pressure waves which generate the shock and the possible explosion of the outer shells of the star. The success or failure of the shock results from a tenuous energy balance and a crucial parameter appears to be the deviation of the mean adiaba- tic index y from 413, for the densities up to p
0' P M B o
M B B
v BBCWI
*
B V 8 2 x S V I IFig.3
-
Comparison of the equations of state P( p ) obtained in bulk matter approximation (PMB /14/), Liquid drop model (BBCW /15/),Thomas-Fermi with trial functions (MBB /a/), Hartree-Fock (BV 82 191) and our complete Thomas- Fermi calculations (SV). The scales are logarithmic.The a u t h o r s e s t i m a t e t h e energy o f t h e shock i n f u n c t i o n o f t h e a d i a b a t i c i n d e x y , t h e e l e c t r o n f r a c t i o n Ye and t h e d e n s i t y o f t r a n s i t i o n ptrans between " n u c l e i " and homogeneous m a t t e r
Eshock DC f(463 - y ) Ye 1013 'trans 113
where f i s r e l a t e d t o t h e mass o f t h e i n n e r c o r e o f t h e supernova.
I n t a b l e 1 we p r e s e n t t h e term f ( 4 / 3 - y ) c a l c u l a t e d f o r t h e e q u a t i o n s o f s t a t e o f f i g u r e 3. The d i f f e r e n c e s between t h e v a r i o u s approaches a r e now v e r y obvious. I n p a r t i c u l a r t h e 30 % i n c e r c i t u d e on t h e v a l u e o f f o b t a i n e d i n t h e e q u a t i o n s o f s t a t e o f r e f e r e n c e s / 8 / and 1151 i s t o o b i g t o g i v e r e l i a b l e e s t i m a t i o n s o f t h e shock energy.
- -
E q u a t i o n PMB BBCW MBB
Table 1 - I n f l u e n c e o f t h e a d i a b a t i c index y on t h e shock energy. The t e r m f ( 4 / 3 - y ) (eq.3) c a l c u l a t e d f o r v a r i o u s equations o f s t a t e ( b u l k m a t t e r appro- x i m a t i o n (PMB / 1 4 / ) , L i q u i d drop model (BBCW / 1 5 / ) , Thomas-Fermi w i t h t r i a l f u n c t i o n s (MBB / 8 / ) , and our complete Thomas-Fermi (SV) i s compared t o t h e sa- me term o b t a i n e d i n Hartree-Fock c a l c u l a t i o n s (BV 191). The mean a d i a b a t i c i n d e x y i s determined f o r 0.02 f m - ' Sp 5 0.07 fm-'.
Conclusion
We have shown t h a t t h e use o f t r i a l Fermi f u n c t i o n s i n z e r o o r d e r Thomas-Fermi c a l c u l a t i o n s may l e a d t o bad approximations. On t h e c o n t r a r y , t h e f u l l y v a r i a t i o - nal. Thomas-Fermi method, we have developed, g i v e s r e s u l t s i n v e r y good agreement w i t h those o f Hartree-Fock c a l c u l a t i o n s . I n p a r t i c u l a r , t h i s method i s v e r y s u i t a b l e f o r t h e sta~dy o f h o t dense m a t t e r and i t would be w o r t h w h i l e t o develop a p p l i c a t i o n s i n t h i s d i r e c t i o n .
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151 SURAUD E., Proceedings o f t h e Workshop on C o l l a p s e and Numerical R e l a t i - v i t y , Toulouse, France, November 7-11, 1983.
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