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STATICS AND DYNAMICS OF COMPRESSED NUCLEI
W. Stocker
To cite this version:
W. Stocker. STATICS AND DYNAMICS OF COMPRESSED NUCLEI. Journal de Physique Collo-
ques, 1984, 45 (C6), pp.C6-289-C6-295. �10.1051/jphyscol:1984635�. �jpa-00224237�
STATICS AND DYNAMICS OF COMPRESSED NUCLEI
W. Stocker
Sektion Physik, Universitat Miinchen, 0-8046 Garching, F. R. G.
R6sum6 - On Btudie en particulier la tension superficielle des noyaux comprim6s ainsi que les effets de compression sur la barriPre de fission. Les cons6quences de ce calcul statique sur la dynamique d'un noyau vibrant sont analysges, en utilisant une mgthode hydrody- namique. On pr6sente en outre les caract6ristiques de compression de la matisre nuclgaire homogsne 5 partir d'un modsle de gaz de Fermi.
Abstract - We especially study the surface tension of compressed nuclei and compression effects on the fission barrier. We analyse the consequences of this statical calculation on the dynamics of a vibrating nucleus using a h~drodynamical approach. In addition we present compression properties of excited homogeneous nuclear matter starting from a Fermi gas model.
Nuclear experiments now are performed in an energy range where nuclei can be compressed and heated up to states which are far off from the well known region of saturation where pressure p and temperature T are vanishing. The structure of compressed - eventually also heated
-nuclei is also of interest in astrophysics. The line of P-stability, nuclear decay properties, nuclear reactions,the single-particle level scheme etc. might be appreciably affected by an external pressure.
In the present contribution we report on calculations of static com- pression properties of nuclei taking into account nuclear surface effects. These results are then used as input into a hydrodynamical model in order to study the dynamical behaviour of a nucleus in the O+-monopole state. In the last section we start from a Fermi gas model in order to derive compression properties of heated homoqeneous nuclear matter.
I - STATIC COMPRESSION PROPERTIES OF FINITE NUCLEI
A value of
K,- 230 MeV for the incompressibility of infinite symmet- ric nuclear matter is now well established from calculations of static properties of finite nuclei as well as from analyses of the nuclear breathing mode /I/.- In the case of finite nuclei this value is appre- ciably altered by surface effects /I/. The surface contribution to the incompressibility depends on the compression mode, i. e. on the way how the surface region is compressed.
In order to study the surface incompressibility in the case of a plane surface we have carried out an analytical model calculation /2/ and also performed Hartree-Fock calculations usinq realistic Skyrme inter- actions
/ 3 / .For the simulation of the external pressure a density- dependent constraint
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984635
C6-290 JOURNAL DE PHYSIQUE
has to be included which also depends on the compression ratio q
=nc/n of the bulk density nc and the bulk saturation density no and on tRe surface compression mode characterized by a parameter B. In the case of the model energy density of ref. /2/ we obtain the follow-
ing closed expression for the surface tension
o,ofq,P)
=3 1 oor4q 132
-3q 1+3
+ql-Pj -
( 2 1oo is the surface tension at saturation. This expression reflects the essential features of the surface tension of compressed nuclear matter which are also found by constrained Hartree-Pock calculations /3/. The surface tension is found to be stationary at the saturation density no independently of the compression mode 8. Thus
This theorem was first found by Myers and Swiatecki /4/ for the
special case of a density multiplied by an overall factor, correspond- ing to our case
I3 = 0.It is found that the surface tension for rea- sonable values of p is a maximum at the saturation density. Around no the surface tension turns out to be lowest for B
=1 which is the case where the density is antiscaled, i. e. where a compression in the bulk is combined with a more diffuse surface. The so-called scalinq of the density, B
=- 1/3, leads to a surface tension which is larger than for the case B
=1.
We used this static result on the dependence of the surface tension on the bulk compression ratio q for a given surface compression mode p in order to calculate the dependence of the fission barriers of finite nuclei on compression. In a simple liquid drop description the fission barrier is determined by the interplay between Coulomb and surface effects. Since the surface tension is stationary at saturation the effect of compression on the fission barrier in first order comes only from Coulomb effects. Using the parameterization of Nix /5/ for the description of the nuclear deformation we obtain for the deformation energy Edef the following expression
7 '7
L L
- -.
3
- 3
2 2Ed,f(~;qrB)
=asurfA F(qrB)q {(BsUrf(~)-1)+Zm*
( B CO &)-1) 1 (41 where
F(q,P)
=3
1[4q J3+2-3qB+3+ql-P1
•asurf is the surface energy coefficient asurf
%18 MeV, A the mass number, and the shape dependent quantities Bsulf and Bc are func- tions of the deformation
--parameter y as given In ref. /!?y! We present the results for Ede£ in fig. 1 for the nuclei 2 0 8 ~ b and
2 3 8 ~ .One sees that the fission barriers are lowered by compression and increased with
Ye-compression. For a system vibrating between q
= 1 - Eand q
= 1+
Efor realistic values of the surface effect in second order will lead
to a decrease of the time-averaged flssion barrier since Coulombeffects
should average to zero and the surface effect in second order would
lead to a decrease of the surface tension. Experiments /6/ indicate the
opening of the fission channel during the monopole vibration of 2 0 8 ~ b .
This, however, could also been an inertia effect. Results similar to
ours have been obtained earlier in ref. /15/.
pressionratiosq; a
=0.
I1 - DYNAMICS OF THE MONOPOLE MODE
Neither experimental analyses /7/ nor microscopic calculations /I/ for the monopole breathing mode exhibit an antiscaling like behaviour of the transition density during monopole density vibrations as one would naively expect from the static calculations of sect. I. In order to study the effects of dynamics and inertia on the behaviour of the transition density we used the static results on the compressibility as input for the restoring force parameters in a hydrodynamical approach to the breathing mode /a/. We also included Coulomb contri- butions. Imposing the dynamics on the system by giving the (q,P)- dependence of the density the velocity field can be obtained from an integration of the continuity equation /a/. Thus, the inertial para- meters are found without the dynamical Euler equation.
Starting from the results of sect. I we were able to treat a aeneral class of dynamical couplings of the surface to the bulk of a spherical nucleus. The two ei~enmodes, found by diagonalizing the hamiltonian corresponding to the quantities describing bulk and surface compres- sion as independent collective variables, coincide with the two extrema of the energy of the mode as a function of the coupling para- meter (Rayleigh's principle). The lower of the two eigenmodes is found to be close in energy to the pure bulk vibration and - starting from the realistic SkM*-interaction - in excellent agreement with the experimental monopole energy. Its transition density is of scaling type. The upper mode which has a pronounced antiscaling behaviour is appreciably higher in energy than the pure surface mode.
I11 - COMPRESSION PROPERTIES OF EXCITED NUCLEI
Nuclear experiments now can be performed in an enerSy ranae where the nuclear equation of state far from the saturation point can beprobed.
Especially phase transitions and critical phenomena in nuclei, which might manifest themselves in e. g. special fragmentation processes, are of actual interest
/ 9 / .The theoretical approach to the nuclear equation of state concentrates
on Hartree-Fock (HF) and Thomas-Fermi (TF) like procedures. In the
KF-method /lo/, which has to be based on a finite temperature single-
C6-292 JOURNAL
DE PHYSIQUEparticle distribution, the free energy of an isolated nucleus at a given temperature is minimized. Also semi-classical methods have been used to describe excited nuclei /11-14/. The earlier calcula- tions assume potential energy density functionals which do not depend explicitly on temperature. Semi-classical calculations usually assume the nuclear liquid to be in phase equilibrium with its own vapour.
This phase equilibrium is characterized by equal pressures as well as equal chemical potentials for both phases. Speaking in terms of standard thermodynamics, a Maxwell construction forthe determination of the equilibrium pressure is needed. Such a phase equilibrium might be realized very well in neutron stars; it might, however, not be the appropriate picture for an isolated nucleus in a nuclear laboratory.
As an alternative to the phase equilibrium description of an excited nucleus we have therefore proposed /11/ to describe the heatednucleus as a metastable system (represented by a point lying on the metastable part of the isotherm) with vanishing external pressure. Such a system in standard thermodynamical terms will not be stable and decays after some time. The same holds for an excited nucleus. It might be that this description is another extreme compared to the phase equilibrium picture, the excited equilibrated nucleus bein? a system in between the two extremal conditions, especially if there
isadditional dyna- mica1 internal motion.
In order toinvestigate compression properties of bulk nuclear matter we start from a model energy density introduced in ref. /11/ with parameters guaranteeing to reproduce all well known nuclear ground state properties. The kinetic energy density nkin is that of a free Permi gas with the given density n. The potential part of the energy density is parametrized as a function of the density n,
Choosing
a =-322.7 MeV fm3, b
=-17.47 MeV fm, c
=491.9 MeV fm5 the well known ground state enerqy per nucleon eo
=-15.9 MeV, the satura- tion density no
=0.17 fm-3 and an incompressibility K
=230 MeV are reproduced. Since from general thermodynamics the entropy s per
nucleon and the internal energy e per nucleon are related by
the model entropy per nucleon has to be taken for consistency reasons as that of a free Fermi gas with density n.
The thermodynamics of the model system is now fully defined and it is more or less straightforward to calculate the isothermal and adiabatic compressibility moduli, KT and KS. We additionally relate them to the specific heats Cp and Cv and calculate also the surface tension o(T).
Of special interest for us are the critical points for the two cases of (i) a metastable system with vanishing external pressure and (ii) a gas-liquid phase equilibrium system. The critical temperature in the p
=0-system is defined by the isotherm which touches the n-axis in a(n,T)-diagram since above that temperature there is no point on the isotherms with p
=0, i. e. no such system can exist. The isothermal incompressibility KT is generally obtained fromthe freeenergy f=f(n,T) per nucleon
2 a2f a f K~
=9[n
(--23 +2n(-) I .
an
Tan T
( 7 )The equilibrium states in case (i) are found as zeroes of the isotherm
in a (n,T)-diagramm; in case (ii) a Maxwell construction is needed. The
isentropic (adiabatic) incompressibility
KSis found from
The e q u i l i b r i u m d e n s i t i e s n = n ( T ) a r e t h e more d i f f e r e n t f o r t h e
c a s e s ( i ) and ( i i ) t h e more t h e c r i t i c a l t e m p e r a t u r e TErit f o r c a s e ( i ) pproached. The c r i t i c a l t e m p e r a t u r e i n . c a s e ( i ) i s ound t o b e
;%ift
5 12.2 MeV, i n c a s e ( i i ) we g e t T ('+& 5 16 !lev. The r e l e v a n t c r l t l c a l d e n s i t i e s a r e n< 5 0.09 fmwSr:nd n 6 i f t 5 0.06 fm-3. I n f i g s . 2 and 3 t h e i n c o m p r e s s i b i l i t i e s KT and KS o r t h e two c a s e s a r e d i s p l a y e d . I t i s s e e n t h a t KT v a n i s h e s i n b o t h c a s e s a t t h e c o r r e - sponding c r i t i c a l t e m p e r a t u r e . I n c a s e ( i i ) t h a t i s a w e l l known r e s u l t . I t i s a l s o expected i n t u i t i v e l y t h a t KS i s somewhat l a r g e r t h a n KT i n b o t h c a s e s . For KS no s p e c i a l b e h a v l o u r c a n be s e e n a t t h e c r i t i c a l t e m p e r a t u r e s , n e i t h e r i n c a s e ( i ) n o r i n c a s e ( i i ) .T h e r e i s a fundamental r e l a t i o n s h i p between KS and KT and t h e s p e c i f i c h e a t s C p and Cv ( p e r n u c l e o n ) ,
'dl TEMPERATURE (MeV)
FIG. 2 - I s o t h e r m a l incompressi- FIG. 3
-
Same q u a n t i t i e s a s i n f i g . 2--
b i l i t y KT and a d i a b a t i c incom- f o r t h e system i n phase e q u i l i b r i u m . p r e s s i b i l i t y Ks, s p e c i f i c h e a tp e r nucleon f o r c o n s t a n t volume, Cv, and f o r c o n s t a n t p r e s s u r e , C p , a s f u n c t i o n s o f t e m p e r a t u r e f o r t h e m e t a s t a b l e system w i t h v a n i s h i n g e x t e r n a l p r e s s u r e .
T h e r e f o r e we have a l s o c a l c u l a t e d t h e s p e c i f i c h e a t s C and Cv f o r c o n s t a n t p r e s s u r e and c o n s t a n t volume, r e s p . They a r e ghown i n f i g s . 2 and 3 f o r t h e two c a s e s . The s i n g u l a r b e h a v i o u r o f C a t Tcrit i s an i n d i c a t i o n t h a t a phase t r a n s i t i o n w i t h a l a t e n t h e a r o c c u r s .
I t i s a w e l l known f a c t t h a t t h e s u r f a c e t e n s i o n a, i. e . t h e s p e c i f i c f r e e energy p e r u n i t a r e a , v a n i s h e s a t T = TCrit. The same d o e s n o t h o l d f o r t h e i n t e r n a l s u r f a c e e n e r g y . To g e t a s i m p l e model e n e r g y f o r a non-homogeneous system we make a l o c a l d e n s i t y approximation and complete t h e bulk i n t e r n a l e n e r g y d e n s i t y by a g r a d i e n t t e r m
JOURNAL DE PHYSIQUE
To obtain the realistic ground state
7 I I Isurface tension o
=1.12 MeV fm-2
-
Eone has to take a2
=65 MeV fm5. The
determination of the density n (z) of a semi-infinite liquid with -T
>0 goes via the solution of an Euler-
z Lagrange equation. In fig. 4 we give
the resulting surface tensions. It
-should be pointed out that the re-
sulting densities in case (i) show a characteristic behaviour. They end
LL
at a finite value with a horizontal
tangent abruptly. In case (ii) the
5 10 15density of the liquid phase asympto-
T E M P E W R E [MeV)
tically reaches the bulk liquid
value exponentially and ends qua- dratically at the qas value. The FIG. 4 - The surface tension a0 strange behaviour of the density in for case (i) and oliquid-gas for case (i) reflects the fact that ame- case (ii) . tastable statehas tobe represented.
Due to the vanishing of the isothermal incompressibility KT as well as the vanishing of the surface tension at Tcr.t the nucleus might break into pieces of any size since the (isothermaf) fragmentation does not need any expenditure of energy, even creating a surface is nothindered energetically. It is interesting to note that the recent analysis of ref. /9/ gives a gritical temperature of about 12 MeV which could
in-dicate that case (i) is a better model for the experimental situation than case (ii) .
Most of the results of sects. I and I1 were obtained in collaboration with M. Brack (cf. refs. /2/ and /8/) and J. M. Pearson and M. Farine
(cf. ref. /3/). For valuable discussions concerning sect. I11 the author is indebted to M. Brack.
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