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The microscopic description of the nuclear equation of state
ProS. P~t
Nm:/.P/W~.,VoL30, pp. 1-15, 1993. Pnntmlin GroatBri~a. Alldgh~ mmtved.
0146--6410/93$24.00 @1993PeqsmcaPre~ Ltd
The Microscopic Description of the Nuclear Equation of State H.MOTHER Institut fllr Theoretische Phy$ik, Auf der Morgensttlle 14, W-7400 Ttlbingen, Germany
ABSTRACT A review is presented on the status of microscopic calculations of the nuclear equation of state derived from realistic nucleon - nucleon interactions. The importance of correlations is demonstrated and the influence of relativistic features on the determination of the saturation properties of nuclear systems are discussed. Special attention is paid to the differences in the behavior of infinite nuclear matter, which may be observed in studies of astrophysical objects, and finite nuclei, which are the object of heavy ion reactions.
KEYWORDS Nuclear equation of state, nuclear matter, finite nuclei, Dirac-Brueckner-Hartree-Fock, realistic nucleonnucleon interactions, hole-line expansion, hadrons in the nuclear medium
INTRODUCTION One of the main motivations for investigating heavy ion reactions at medium and high energies is the hope that such experiments will provide us with information on the equation of state of nuclear matter at high densities. How much energy is required to compress nuclear matter to densities up to two, three or four times the saturation density of normal nuclear matter? Does nuclear matter at such high densities exist in a form similar to the one in normal nuclei or does one observe a phase transition from normal nuclear matter to a quark - gluon plasma? In order to detect signals from such a phase transition in a collision of two heavy ions, one needs very reliable predictions on the properties of conventional nuclear matter compressed to high densities. As an example it should be mentioned that results of computer simulations of heavy ion reactions like the approach of the "Quantum Molecular Dynamic" (see e.g. Aichelin 1991) or the "Boltzmann-UehlingUhienbeck" approximation (see StOker and Greiner 1986) very much depend on the compressibility of nuclear matter. Therefore one typically considers two different equations of state, a soft one with a compressibility as small as K ~ 140 MeV, as advocated by Baron et al.(1985) from their studies of supernova explosions, and a stiff equation of state with a compressibility of K -~ 380 MeV or even higher, which seems to be preferable when one tries to reproduce experimental d a t a of heavy ion reactions (see Keane et.al 1988). Such investigations typically employ information about the equation of state as derived from a phenomenological parameterization of the nuclear barniltonian. Typical examples are the effective nucleon-nucleon (NN) interaction proposed by Skyrme (1956), the Gogny interaction (Gogny, 1975) or the Walecka model which contains relativistic features (Serot and Walecka, 1986). All such effective hamiltonians, however, have been adjusted to describe properties of nuclei at normal densities. Therefore their predictions on the properties of nuclear systems at higher densities are not very reliable. A more microscopic approach is to start from realistic NN interaction. Typical examples
2
H. Mtither
are the Reid soft-core potential (Reid, 1968), the Paris potential (Lacombe et.al. 1980) or the OneBoson-Exchange (OBE) potentials, originally developed in Bonn (Machleidt, 1989). The parameters of such realistic NN interactions are fitted to reproduce the NN scattering data, i.e. to describe the interaction of two nucleons in the vacuum. For these NN interactions one may then try to solve the nuclear many-body problem as rigorous as possible. If such microscopic many-body calculations lead to results for nuclear matter at normal densities, one may then hope that such a theory which predicts the nuclear properties at the saturation density p0 from the NN interaction in the vacuum, will also produce reliable results for higher densities. In this contribution we will focus the attention on a microscopic description of nuclear properties at temperature T = 0. Furthermore we will ignore sub-nucleonic degrees of freedom, like excitations of the nucleons to the A(3,3) resonance in the many-body approach (see e.g. Miither, 1985). In the next section following this introduction the status of non-relativistic many-body calculations for the saturation point of nuclear matter will be reviewed. After the discussion of the relativistic features taken into account in the Dirac-Brueclmer-Hartree-Fock (DBHF) approach, the differences between microscopic calculations for nuclear matter and finite nuclei wiU be discussed, the main conclusions of this contribution axe summarized in the final section.
CORRELATIONS
IN NUCLEAR
MATTER
the simplest approximation to the many-body theory for the description of the ground state of nuclear matter is to perform a Hartree-Fock (HF) calculation. This means that one defines a single-particle hamiltonian h by
< ilhli > = ei = < iltli > + < ilUli > = < iltli > + ~ < ijlVlij > , j_
(1)
where t stands for the operator of the kinetic energy and the single-particle potential U for a nucleon in orbit i is obtained by summing the nntisymmetrized matrix elements of the bare NN potential V for the interaction between a nucleon in orbit i with one in an orbit j below the Fermi surface. For nuclear matter the HF orbits i and j are bound to be plane waves and the summation in eq.(1) must be understood as an integration restricted to all momenta below the Fermi momentum k r , which is related to the density p of the system under consideration by p =
2 kS
(2)
The binding energy can then be calculated as E = 1 ~
[< iltli > +e,] .
(3)
If, however, one calculates the HF energy of nuclear matter at the empirical saturation density (p0 = 0.16 fm -3, which corresponds to a Fermi momentum kF =1.36 fm -1) employing a realistic NN interaction llke the one defined by Reid (1968), one obtains an energy per nucleon of 176.2 MeV instead of the empirical value of -16 MeV as derived from the Beth~Weizsgcker mass formula (see table 1). This result demonstrates quite clearly that a many-body theory employing such realistic interactions must account for correlations beyond the HF approach. As a first step one must account for 2-nucleon corrdations in such a way that the amplitude of the wave function describing the relative motion is small whenever two nucleons are so close to each other that they are exposed to the repulsive core of the NN interaction. This can be achieved either by
The Nuclear Equation of State Table 1: Results for the binding energy per nucleon of Nuclear Matter Results for the energy per nucleon calculated in the Hartree-Fock (HF) the Brueckner - Hartree Fock (BHF) and with inclusion of the 3-nucleon correlations (BHF3) are listed for two densities, characterized by the Fe~Kfi momentum kF=l.2 and 1.36 fm -1 (the empirical saturation density), respectively. Furthermore we give the energy contribution due to two- (AE2) and three-nucleon correlations (AEa). All results have been obtained for the Rdd soft-core potential. kF {fin-z] HF BHF BHF3 AEa AE3
1.2 119.8 -9.1 -12.6 -128.9 -3.5
1.36 176.2 -9.8 -15.2 -186.0 -5.4
considering such correlations in the wave function explicitly or by evaluating an effective interaction which accounts for such correlations. A first approximation for such an effective interaction is the Bruecimer G-matrix which is obtained from the bare NN interaction g by solving the Bethe-Goldstone equation
Q
a ( w ) = v + v w - Qt----~G ( W ) "
(4)
As one can see from this equation, that the G-matrix is very similar to the scattering matrix for 2 free nucleons except for the fact that the Pauli operator Q prevents intermediate two-particle states which are forbidden by the Pauli principle. Also the starting energy W will typically be the energy of two bound nucleons rather than the energy of free particles. The Brueckner-Hartree-Fock approach is obtained by replacing in eq.(1) the matrix elements for the bare potential V by the corresponding ones for G < ijlV[ij >=~< i j [ G ( W = ei + ei)lij > . (5) The BHF approximation yields an energy for nuclear matter which is much closer to the empirical value than the HF approach. This demonstrates that it is inevitable to confider at least for 2 nucleon correlations. Furthermore it is very important account for the quenching of NN correlations in the nuclear medium, i.e. the difference between the G-matrix and the scattering matrix for 2 free nucleons. Ignoring the quenching mechanism would produce an error of the order of 20 MeV per nucleon around saturation density. One now may try to evaluate the binding energy of nuclear matter as a function of density by repeating such a BHF calculation for various densities and determine the saturation point (i.e. energy and density of the stable nuclear matter) as the minimum of the energy per nucleon as a function of density. Such BHF calculations have been performed for various realistic NN interactions, which were all fitted to describe the free NN scattering data. It turned out, however, that all the calculated saturation points were obtained on the so-called Coester band (Coester, 1970 and MSther 1986, see also figure 1) This means that potentials, which reproduce about the correct saturation density predict a binding energy per nucleon which is to small by roughly 5 MeV, and those potentials which lead to the correct predietion for the binding energy yield a saturation density which is twice as large than the empirical one (or results in between). Therefore attempts have been made to improve the solution of the many-body problem and account for three- and more-nucleon correlations. This can be done essentially by solving the three-particle
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Fermi Momentum [fm - 1 ] Figure 1: Binding energy per nucleon and density of the saturation point (expressed in terms of the Fermi momentum) calculated for nuclear matter in the BHF approximation (denoted by asterisks) using various realistic NN interactions. Also displayed are results obtained from BHF with inclusion of pp-hh ring diagrams (triangles) as obtained by Jiang et.al. (1988)
Faddeev equation in the nuclear medium. It turns out that the effects of the three-nucleon correlations are non-negligible and yield in the example of the Reid soft-core potential additional binding energy of about 5 MeV per nucleon (Day 1981, see also table 1). At first sight this seems to improve the agreement with the empirical data. The density-dependence of this three-nucleon correlation effects, however, is such that the resulting saturation point is simply moved along the Coester band. Comparing the effects of two- and three-nucleon correlations one observes that the corrections due to three-nucleon correlations are much smaller. This tends to indicate a fast convergence of the manybody theory. Indeed, estimates for the effects of four-nucleon correlations by Dickhoff et.al. (1982) tend to support such a conclusion. Nevertheless, attempts have been made to improve the non-relativistic many-body calculation such that the saturation point of nuclear matter can be described in terms of a realistic NN interaction. One may argue that the hole line expansion, which treats the particle-particle ladders to all orders but accounts for the hole-hole scattering terms only in a perturbative way is not appropriate for a system with the density of nuclear matter. Rather one may try to use a seE-consistent Greens function approach in which particle and hole states are treated in a symmetric way (for a recent review see Dickhoff and MSther, 1992). Indeed, calculations which include the effects of the so-called particle-particle hole-hole ring diagrams by Jiang et.al. (1988) lead to saturation points of nuclear matter which are in a much better agreement with the empirical point than those obtained for the BHF approach (see also figure 1). However, this success could very well be an accidental one. It is quite possible, that an approach which accounts for collective effects in a more general way than just to include particle-particle hole-hole ring diagrams could very well spoil the agreement again (see also discussion below for finite nuclei).
The Nuclear Equation of State DIRAC-BRUECKNER-HARTREE-FOCK Another step which in recent years has very successfully been made to describe the saturation properties of nuclear matter, has been the consideration of relativistic effects in the so-called Dirac approach (see e.g. Serot and Walecka, 1986, Anastasio et.al., 1983, Brorkmann and Machleldt, 1984). At first sight relativistic effects seem to be negligible since the single-particle potential U (see eq.(1), typically -40 MeV) is very small compared to the rest mass of the nucleons (938 MeV). One must be aware, however, that the single-particle potential U originates from a delicate balance between attractive and repulsive components in the NN interaction. In a OBE model the attractive components are mainly due to the exchange of a scalar meson, the so-called a meson, whereas the short-range repulsion is dominated by the exchange of the w meson which is a vector meson. In the Hartree approximation the exchange of such mesons yield an operator for the self-energy of the nucleons with a scalar component A and a (time-like component of a vector field B fr = A + B 7 ° .
(e)
In the Hartree approximation for nuclear matter A and B are independent of the momentum of the nucleon under consideration and depend on the density of the system, only. Including this self-energy in a Dirac equation for a nucleon with mass m and momentum 1~in the nuclear medium
[,~f+ a (,-. + A)] aCp,.) = (,,
-
B),:Cp,.),
(7)
with 6 and B the usual Dirac matrices, one obtains a solution for the Dirac spinor
(s)
V which is identical to a solution for a free nucleon with an effective mass
rh=m+ A,
~, =
v~
+ f,
(9)
and Xm a Pauli spinor. Since A is large and attractive (-300 MeV) the effective mass rh, which characterizes the ratio of large to small component in the Dirac spinor, is considerably smaller than the bare mass. This change of the Dirac spinors in the nuclear medium leads to different matrix elements for the OBE model of the NN interaction. Due to the change of the nucleon spinors in the medium, the NN interaction in nuclear matter is different from the one in the vacuum. Since the nucleon self-energy ~" is determined from the NN interaction, which itself depends on the solution of the Dirac equation (7) for this self-energy, a new self-consistency problem has to be considered. For a realistic OBE potential one has to go beyond the Hartree approximation for the self-energy and take into account the effects of Fock exchange terms and of two-body correlations by determining the self-energy U from the G-matrix in analogy to the BHF approach outlined in the preceding section. < i j l G ( W = ei + ei)lij > ~
O(i).
(10)
This leads to the so-called Dirac - Bruerkner - Hartree - Fork (DBHF) approximation. The density dependence of the Dirac effects is such that the saturation points resulting from DBHF calculations are effthe Coester baud. Brockmann and Macldeidt (1984) succeeded to find a realistic OBE potential which fits the NN scattering data and reproduces the empirical saturation point of nuclear matter employing the DBHF approach.
H. Miither
Dirac BHF for 160 -3.0
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Inverse Radius [fro**- 1 ] Figure 2: BHF and DBHF calculations for 160. Using OBE potentials A, B and C defined by Machieidt (1989), results for the radius of the charge distribution and the binding energy per nucleon obtained in conventional BHF (stars) and DBHF calculations (circles) are compared to the experimental result.
FINITE NUCLEI The next question to be answered is of course whether the results obtained for the saturation point of nuclear matter are directly related to the features obtained in calculation of ground-state properties for finite nuclei. From a technical point of view the calculation of finite systems is more involved since at each step of the many-body approach one has to evaluate the optimal single-particle wave function which in nuclear matter is determined by the translational invariance of the system. On the other hand, however, the typical densities observed in finite nuclei are below the saturation density. Therefore a many-body approach like the hole-line expansion, which shows a better convergence at low densities, should for a given level for a given order in the expansion provide more reliable results for finite nuclei than for nuclear matter. At first sight the BHF approach for finite nuclei seems to show similar problems as for nuclear matter. As an example we present in figure 2 results obtained for the binding energy per nucleon and the inverse value for the radius of the charge distribution (in order to have a quantity comparable to the Fermi momentum in figure 1) of is O employing various realistic interactions. Also in the case of finite nuclei one obtains a "Coester" band similar to the one of nuclear matter. Figure 2 also displays results of DBHF calculations for this nucleus as obtained by Miither, Machleidt and Brockmann (1990). In this approach the change of the Dirac spinors in the nuclear medium is described in terms of a local density approximation relating the ratio of small to large components in the Dirac spinors for leO to the one obtained in DBHF calculations of nuclear matter at the corresponding densities. The self-consistent solution of the Bethe-Goldstone eq.(4) and the BHF equation (5) are obtained directly for the finite nucleus. Also in this case the inclusion of relativistic
The Nuclear F~luation of Sta~
7
effects improves the agreement with the empirical point, however, a sizable discrepancy remains also for the OBE potential (A) which reproduced the saturation point of nuclear matter. This can be understood as an indication that the Dirac phenomenology alone may not solve the problems of microscopic many-body calculations for nuclear systems. Of course, also for the relativistic calculations one should take into account the effects of many-nucleon correlations and sub-nucleonic degrees of freedom. These effects are non-negligible within the non-relativistic approach to the nuclear matter problem and therefore should also be considered within a relativistic treatment. However, there is a more serious criticism: In the DBHF approach one takes into account that the solution of the Dirac equation for positive energies are modified for the nucleons in the nuclear medium. The presence of the medium, however, also modifies the solutions of the Dirac equation with negative energies. This implies that the Dirac vacuum is modified in the medium as well. Attempts to account for such vacuum renormalization effects demonstrate that such effects are very large and cannot be ignored (Brown et.al., 1990). But also attempts to include the et~ects of pl>-hh ring diagrams, which improved the calculation of the saturation properties of nuclear matter (see above), seems not to produce such a large effect for the finite system (Mavromatis et.al., 1991). Since results obtained in many-body calculations for finite nuclei show remarkable differences from those obtained in nuclear matter (see discussion above) one may suspect that different saturation mechanisms must be considered for finite nuclei than for nuclear matter. To verify this we perform a little theoretical experiment on the compression of nuclear systems. In order to study the compression modulus of the finite nucleus l e o let us consider as a model wave function the Slater determinant appropriate to describe the ground-state of l e o in terms of harmonic oscillator functions for a given oscillator length b. For each oscillator length one can calculate the radial density distribution pb(r) and the average nucleon density
lYb = f p](r) d3r A
(11) '
with A the number of nucleons. For each model wave function or oscillator parameter b one can furthermore calculate the BHF energy by solving the Bethe-Goldstone equation and the BHF equations directly for the finite nucleus under the constraint that the single-particle waves are oscillator functions for the b parameter under consideration. For the discussion of results presented here we have employed the OBE potential "A" defined in table A.1 by Machleidt (1989). The energies per nucleon calculated for v ~ o u s b are displayed by the solid line in figure 4 as a function of fb defined in eq.(11). The minimum of this curve is close to the result for the self-consistent BHF calculation obtained without the constraint for the single-particle wave function. In addition the second derivative of this curve gives an idea about the compression modulus for this finite nucleus. In the next step we now consider the finite nucleus to consist out of slices or shells of nuclear matter and calculate the energy in a very simple local density approximation (LDA) as
ELDA(b)
= / ZNM(Pb(~))d3~,
(12)
where £NM(P) is the energy density for nuclear matter of density p and pb(r) as above stands for the baryonic density calculated for the Slater determinant of oscillator functions with oscillator parameter b. Results for this LDA approximation as a function of density are given by the long-dashed curve in figure 3. One finds that the LDA overestimates the calculated binding energy. At the BHF minimum the difference is about 85 MeV for the total nucleus, which is close to the surface correction contribution in
H. Miither '
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density [fro**-3] Figure 3: Binding energy per nucleon for 160 calculated for various mean values of the density ~ (see eq.(ll)). The results of the BHF calculation directly for finite nuclei (solid line) and the various local density approximations (LDA, see text) were obtained for the OBE potential A defined in table A.1 of Machleidt (1989). the Bethe - Weizslicker mass formula for 160 (m 118 MeV). This suggests that the LDA approximation misses the major contribution to the surface tension and also the major contribution of the surface effects to the compressibility of a finite nucleus. This is supported by the observation that the LDA result for 160 as a function of density is only slightly above the energy versus density curve of nuclear matter. Also one sees that the energy versus density curve calculated directly for the finite nucleus is much stiffer than the corresponding curve in the LDA approach or for nuclear matter. This indicates that due to the surface effects it requires more energy per nucleon to compress a small piece of nuclear matter, as it is done e.g. in a heavy ion collision, than an infinite system or a stellar object. Therefore this surface contribution to the compressibility may to a large extent explain the difference in the equation of state for nuclear matter derived from heavy ion collisions and studies of astrophysical objects. Figure 3 also displays a LDA result (short dashes) which is obtained according to eq.(12) but ignoring the density dependence of the Brueckner G-matrix in ENM(P), i.e. using the G-matrix as calculated for the density p = 0.05 fm -3. Comparing the two curves for the LDA approximation one can see that the density dependence of the G-matrix is a very important ingredient to obtain the saturation point of finite nuclei as it is for nuclear matter. In addition, however, an additional surface effect is needed for finite nuclei, which is absent in nuclear matter or the LDA approach for finite nuclei. LDA AND
FINITE RANGE
OF INTERACTION
In order to find the origin for the discrepancy between the LDA and the explicit calculation for the finite nucleus one can compare the different contributions to this energy. One finds that the LDA
The Nuclear Equation of State
9
approach for the kinetic energy
TLVA(b)= f 3 a2k~-F~ (r)p~Cr)dsr
(13)
with the local Fermi momentum kF(r) determined from the relation
pb(r) = 3-~/c~(r),
(14)
is a very good approximation to the kinetic energy of the corresponding harmonic oscillator configuration. Therefore the deviation must originate from the expectation value for the potential energy. To visualize the difference of the two approaches for the calculation of the interaction energy we assume for the moment the two-body interaction, in the BHF approximation the G-matrix, to be local, which means that it depends on the momentum transfer only. In this case the interaction energy can be calculated as
< V >---- ~_~ / VSMT(q)~SMT(q)dq S,M,T J
(15)
where VSMT(q) is the interaction depending on the momentum transfer q for a given set of spin isoepin quantum numbers S, M, T or a corresponding spin-isospin operator in the interaction V (see e.g. Dickhoff, 1983). The quantity ~(q) represents a two-body density obtained for a given approximation to calculate < V >. For this two-body density all variables have been integrated except the momentum transfer. Therefore this ~b(q) is a measure to which extend the local interaction at momentum transfer q contributes to the total interaction energy. In the BHF approximation this ~(q) can be split into a direct part (related to the Hartree contribution) and an exchange contribution (related to the Fock term) ~SMT(q) SMT SMT = ~di,ect(q) -i- @,zch (q). (16) For isoepin symmetric (Z=N, ignoring Coulomb interaction) and spin saturated systems the direct contributions vmdsh for all spin-isospin channels except the scalar-isoscalar (S=T--0). In nuclear matter the direct term is given by a ~ function / ~
~T~3F
(17)
•000 ,,recttq) = ~ ( q ) and the exchange contributions are evaluated as ~b~fCh (q) =
1
(2~) z 8
q3 qS l
q2 _ ~
+
(18)
for momenta q _< 2kr and zero for larger momentum transfers. As the LDA implies an average over nuclear matter energy densities, the LDA result for the two-body density ~b(q) will be a superposition of nuclear matter functions at various densities. This can be seen from figure 4, where the dashed line exhibits the LDA approach for ~b(q) in the central spin - isovector channel (S--T=I, averaged over the spin projection M). For this channel one does not obtain a direct contribution for 160 and therefore LDA yields a superposition of functions of the form of eq.(18) for various densities. The corresponding two-body density approach for the direct calculation of l e o is given by the solid line and we observe that the LDA obviously yields a very good approximation for the two-body density. The situation is different in the scalar - isoscalar channel (S=T=0) (see figure 5). In this channel one also obtains a direct contribution, which in LDA as in nuclear matter ( see eq.(17)) is given by a ~-function and not made visible in the dashed line of figure 5). As the single-particle wave functions
H. Miithcr
10
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Momentum Transfer [fm-'] Figure 4: The two -body density ~(q) in the central spin isovector channel (S = T = 1, M averaged) used to calculate the interaction energy according eq.(15). Results are presented for the nucleus l e o as obtained in the LDA approach (dashed line) and the exact calculation (solid Line). Also given is a local presentation of a realistic G-matrix as a function of the momentum transfer as introduced by Dickhoff (1983) (dashed-dotted line). Further details in the text.
of a finite system are not eigenstates of the momentum operator, the direct term in ¢(q) calculated directly for the finite system shows a shape different from a 6 function. This can be seen from figure 5, in which the two-body density calculated for the oscillator model of l e o (solid line) exhibits quite a different shape as the LDA.
If the two-body interaction in this spin-isospin channel would be of zero range, i.e. the function V(q) would be a constant as a function of momentum transfer, this difference in the ~(q) would not show up in the calculated energy, since the integral in eq.(15) would yield the same result for the LDA as for the exact calculation. Realistic interactions and also local representations of the G-matrix exhibit a finite range or a dependence on momentum transfer. A typical function Vooo(q) is given by the dashed-dotted llne (referring to the right scale) in figure 5. One can see that such a function is more attractive at q = 0, where the LDA yields the largest contribution, than for those momentum transfers, for which the maximal contribution is obtained in the exact calculation. This explains why LDA overestimates the binding energy. If the nucleus would be compressed, the maximum of ~(q) for the solid line in figure 5 would be shifted to larger momentum transfers, an effect which tends to reduce the calculated energy. On the other hand the LDA still obtain the largest contribution at q = 0. Using the language of momentum representation this explains the difference in the calculated compression moduli shown in figure 3. Our investigation demonstrates that the LDA is valid only for an effective interaction of zero range. For realistic interactions of finite range the LDA underestimates the attraction from the direct interaction in the scalar-isoscalar channel. Therefore we see that the ground-state properties of finite nuclei are much more sensitive to the range of realistic NN interactions than a calculation of the
The Nuclear Equation of State :~o0.0
0.0
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0.0-
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Central-Isoscalar Channel
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--~.0
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-4.0
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Figure 5: The two - body density in the scalar - isoscalar channel ( S = T = 0) used to calculate the interaction energy according eq.(15), Results are presented for the nucleus toO as obtained in the LDA approach (dashed line without the direct contribution 6(q)) and the exact calculation (solid line). Further details see figure 4.
saturation point of nuclear matter.
THE MODEL OF NAMBU
AND JONA-LASINIO
Until now we have taken into account that the effective mass of the nucleons may change in the nuclear medium (see discussion of DBHF above). For the sake of consistency one may also consider a possible change of the meson masses in the nuclear medium. Such a change of the effective meson masses would modify the propagation of the mesons in the medium and give rise to a change of the range of meson exchange interaction between nucleons. As we have seen in our discussion above such a change of the range of the NN interaction would affect in particular the nuclear structure calculation for finite nuclei. Part of this change of the meson propagators can be described within the framework of the conventional many-body theory by considering a coupling of the mesons to the particle-hole excitations of the nuclear system. In this section, however, we would like to account for sub-nucleonic degrees of freedom and try to evaluate the effective masses of baryons and mesons and their medium dependence within a phenomenological model which accounts for quark degrees of freedom. Such an effective theory which reflects the chiral symmetry as one of the basic symmetries of QCD is the one proposed already in 1961 by Nambu and Jona-Lasinio (NJL). In order to recall some features of this conventional NJL model we start writing down the Lagrangian density £ = ¢(i~7 - M ) ¢ + G [ ( 6 1 ¢ ) 2 + ( ¢ i3,s~11¢)2], (19) where ¢ is a (current) quark field with mass M, ¢ the isospin matrices for the SU(2) flavor and I
12
H. Miither
the unit operator for the color part (SU(3)) of the quark fields. The coupling constant G, which is common for the scalar-isoscalar and the pseudoscalar-isovector part of the interaction term has dimension [mass-2]. The Lagrangian of eq. (19) is symmetric under a chiral transformation if we put M -= 0. As a first step we would like to discuss the properties of the quarks in the vacuum. The self-energy E for the quarks can be evaluated in the mean-field or Hartree approximation as E = E" + E p" = 2G1 < ¢ 1 0 > +2Gi7511¥ < ¢i75r-*1¢ > •
(20)
Since the vacuum expectation value < ¢iTsr-*l¢ > vanishes, only the scalar part of the self-energy E" survives and leads to an effective mass for the constituent quarks of the form M* = M -
(21)
2G < ¢ 1 ¢ >
with the scalar density 12
< ¢I¢ >= -~-~-
~A
d3p ~
M* .
(22)
The two Hartree equations (21) and (22) have to be solved in a self-consistent way and it is clear that for the chiral symmetric case ( M = 0) they have the trivial solution M* -- 0. If, however the interaction strength G is strong enough (for a given cut-off momentum A) the Hartree equation also yield a non-trivial solution with M* > 0. This solution, with the chiral symmetry spontaneously broken, shows up also for the case M > 0 and can be identified from the fact that the constituent quark mass M* obtained in this case is considerably larger than the bare current quark mass M. In order to study a system of finite density one only has to replace the expression for the scalar density given in eq. (22) by 12 ~ A < ¢ 1 ¢ > = -(27r)3
d3pO(ip7- kF)M X / ~
,
(23)
kF
with the usual step function O and the Fermi momentum for the homogeneous system of quark matter. From eq. (23) it is clear that the absolute value for the scalar density tends to reduce with increasing k~,. As a consequence also the effective mass M* is reduced with increasing baryonic density. The scale for this decrease is determined directly by the cut-off parameter A and by the assumption that this cut-off does not depend on the density under consideration. Results for the effective quark mass as a function of the baryonic density are displayed in figure 6 (solid line). The effective mass M* for the constituent quarks can directly be related to the effective mass of the nucleons if we assume that the baryon masses are essentially three times the constituent quark mass. Therefore also the NJL predict a decrease of the effective mass for the nucleon with increasing density as was also obtained in the DBHF approach. In a next step one may then consider the mesons as the collective modes of the quark excitations from the Dirac see. In a nuclear medium these excitations of the Dirac see interfere with the particle-hole excitation of the quarks in the Fermi see This reflects the coupling of the mesons to the medium excitations which are present already in a non-relativistic many-body theory. Therefore one should be very careful if one employs density dependent meson masses as derived e.g. from NJL in a nuclear structure calculation, since these effective masses may already contain effects which would be accounted for again in the nuclear structure calculation. The effective masses of the mesons can be determined from an inspection of the reducible response function (i.e. the excitation spectrum calculated in RPA approximation). Results for the effective mass for the constituent quarks (or baryons) and mesons have been determined by several groups (see
13
The Nuclear Equation of State
Nambu-Jona-Lasinio 700.Sigma uu~-,
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200" t
.
.
.
~jrI
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.
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'
0.0
'
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'
'
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Fermi Momentum [ f m * * - l ] Figure 6: Effective masses as a function of the baryonic density calculated in the NJL model for the quarks (solid line), the Pion (long dashes) and the scalar - isoscalar meson (~r, short dashes). The parameters for the NJL Lagrangian in eq.(19) are M = 5.5 MeV, A -- 630 MeV and G = 0.214 fm -2. See also Henley and MSther (1990).
e.g. Bernard et.al., 1987 and Henley et.al. 1990). As can also be seen from figure 6, one finds that the masses of the scalar ~ meson and the vector to meson are reduced in the medium. This should lead to a larger range of the interaction, which should produce larger radii for the nuclei. Therefore one would expect an improvement of the calculated ground-state properties (see figure 3). In contrast to the medium dependence of these mesons the mass of the pion remains constant for a large range of densities and increases at high densities where the chiral symmetry of the vacuum is restored. This feature is related to the fact that the pion can be considered as the Goldstone boson for the chiral symmetry. It is clear, however, that this medium dependence of the effective masses is a direct consequence of the assumption of a cut-off A which does not depend on the density of the system under consideration. In order to develop a model which is very similar to the NJL model, hut does not require such a strict cut-off, B~ckers et.al. (1991) consider an interaction term between the Dirac epinors in momentum representation of the form V = -G(P)[(~) ~ + (~ i ~ 5 ~ - ~ ) 2 ] . (24) This interaction term is very similar to the original one of the NJL model in eq. (19) and exhibits only two differences: First, the interaction strength is not considered any longer to be a constant, but it should depend on the momentum transfer between the interacting Fermions. Guided by the r , nniug coupling constant of QCD, one assumes that it is reduced for large momentum transfers P. The second difference is that the interaction terms of eq. (24) are of vector-type in color-space. For our present considerations the detailed form of the interaction is not really relevant as long as it is symmetric under chirai transformation. The direct or Hartree contributions to the self-energy are now to be calculated with a "coupling
14
H. Miithcr
constant" G determined at momentum transfer P = 0 (see also discussion in the previous section). For a homogeneous, color symmetric vacuum or baryon system the expectation values for the colorvector terms occurring in (24) vanish and therefore the Hartree terms do not contribute. If, however, one would study a system with an isolated quark of a given color the Hartree terms would diverge and therefore provide a possible mechanism to obtain color-confinement. For the study of the effective mass of quarks in a homogeneous system, however, we have to investigate the exchange or Fock contributions to the self-energy. B~ckers et.al. (1991) show that this yields an effective mass for the quarks which depends on the momentum. This extended NJL model which shows a closer connection to QCD shows similar features as the conventional NJL discussed above.
CONCLUSIONS It has been one aim of this review to demonstrate that conventional microscopic nuclear structure calculations employing realistic NN interactions predict properties of nuclear systems at normal densities which are not really in satisfactory agreement with the empirical data. Improvements beyond the conventional BHF approximation are needed to improve the agreement and therefore establish a microscopic theory which may predict in a reliable way also the main features of nuclear systems at densities well above the normal density. Possible candidates for such improvements are the inclusion of special collective many-body effects like the pp-hh ring diagrams discussed above or the relativistic effects of the DBHF approach. Both of these approaches yield the correct saturation point of nuclear matter but are not yet completely successful to describe the ground-state properties of finite nuclei as well. It has been demonstrated that differences exist in the mechanisms leading to saturation properties of finite nuclei as compared to those in nuclear matter. A local density approximation (LDA) to describe finite nuclei in terms of "slices" of nuclear matter underestimates the surface tension considerably and yields a different compression modulus. Due to the finite range of the interaction it requires more energy to compress finite nuclei as one may estimate from the LDA. This implies that one may deduce a different equations of state from heavy ion reactions (which probes the compression of parts of finite nuclei) than from the analysis of astrophysical objects which might well be described in terms of nuclear matter. Since the structure of finite nuclei is very sensitive to the range of the NN interaction, it may become very important to study a change of this range of the interaction due to a change of the effective meson masses in the nuclear medium. Such a medium dependence might be studied in effective theories related to the QCD (like the NJL model). For more reliable predictions, however, an extended version would be needed which also describes the confinement of quarks in hadrons in the vacuum and inside a nuclear system. The investigations presented here have been performed in collaboration with many colleagues. I would like to list a few of them like R. Brockmann, G.E. Brown, D. Bfickers, P. Czerski, W.H. Dickhoff, P. Ellis, H. Elsenhans, A. Faessler, R. Fritz, E. Henley, M.F. Jiang, T.T.S. Kuo, R. Machleidt, H. Mavromatis, A. Polls, M. Prakash and K.W. Schmid to thank for this collaboration and many fruitful discussions. Also I would like to acknowledge the financial support by the Bundesministerimn fiir Forschung und Technologie (Bonn, project 06 Tfi 714)
The Nuclear Equation of State
15
REFERENCES Aichelin, J. (1991). Phys. Rep. 202, 233. Anastasio, M.R., Celenza, L.S., Pong, W.S., and Shakin, C.M. (1983). Phys. Reports 100, 32T. Baron, E., Cooperstein, J. and Kahanna, S. (1985). Nucl. Phys. A440, 744. Brockmann, R. and Machleidt, R., (1984). Phys. Left. B149, 283. Brown, G.E., MSther, H. and Pr,~b~h, M. (1990). Nucl. Phys. A506, 565. BSckers, D. and M~ther, H. (1991). Nucl. Phys. A523, 629. Coester, F., Cohen, S., Day, B.D. and Vincent, C.1V[.(1970). Phys. Rev. C1,769. Day, B.D. (1981). Phys..~e~. C24, 1203. Dickhoff, W.H., Faessler, A., and Miither, H. (1982). b'ucl. Phys. A389, 492. Dickhoff, W.H. (1983). Nucl. Phys. A399, 287. Gogny, D. (1975). N,~cl. Phys. A237, 399. Henley, E.M. and M~ther, H. (1990). Nucl. Phys. A513, 667. Jiang, M.F., Kuo, T.T.S. and Miither, H. (1988). Phys.Rev. C38, 2408. Keane, D., Chu, S.Y., Fung, S.Y., Liu, Y.M., Qiao, L.J., VanDalen, G., Vient, M., Wang, S., Molitoris, J.J., and StScker, H. (1988) Phys. Bey. C37, 1447. Lacombe, M., Loiseaux, B., Richard J.M., Vinh Mau, R., C6t~ J., Pir~s, and de Tourreil, R. (1980). Phys.Re,J. C21,331. Machleidt, R. (1989). Adv. Nucl. Phys. 19, 189. Mavromatis, H.A., Ellis, P. and M~ther, H. (1991). Nucl. Phys. A530, 251. Miither, H. (1985). Prog. in P,~rt. and IVucl. Phys. 14, 123. Milther, H. (1986). Prog. i,~ P,~rt. a,~d Nucl. Phys. 17, 97. Nambu, Y. and Jona-Lasinio, G. (1961). Phys. Rev. 122, 345 and ibid 124, 246. Reid, R.V. (1968). A,~n.Phy8. 50, 411. Serot, B.D. and Walecka, J.D. (1986). Adv. Nucl. Phys. 16, 1. Skyrme, T.H.R. (1956). Phil. Mag. 1, 1043. St6cker, H. and Greiner, W. (1986). Phys. Rep. 137, 277.
Nm:/.P/W~.,VoL30, pp. 1-15, 1993. Pnntmlin GroatBri~a. Alldgh~ mmtved.
0146--6410/93$24.00 @1993PeqsmcaPre~ Ltd
The Microscopic Description of the Nuclear Equation of State H.MOTHER Institut fllr Theoretische Phy$ik, Auf der Morgensttlle 14, W-7400 Ttlbingen, Germany
ABSTRACT A review is presented on the status of microscopic calculations of the nuclear equation of state derived from realistic nucleon - nucleon interactions. The importance of correlations is demonstrated and the influence of relativistic features on the determination of the saturation properties of nuclear systems are discussed. Special attention is paid to the differences in the behavior of infinite nuclear matter, which may be observed in studies of astrophysical objects, and finite nuclei, which are the object of heavy ion reactions.
KEYWORDS Nuclear equation of state, nuclear matter, finite nuclei, Dirac-Brueckner-Hartree-Fock, realistic nucleonnucleon interactions, hole-line expansion, hadrons in the nuclear medium
INTRODUCTION One of the main motivations for investigating heavy ion reactions at medium and high energies is the hope that such experiments will provide us with information on the equation of state of nuclear matter at high densities. How much energy is required to compress nuclear matter to densities up to two, three or four times the saturation density of normal nuclear matter? Does nuclear matter at such high densities exist in a form similar to the one in normal nuclei or does one observe a phase transition from normal nuclear matter to a quark - gluon plasma? In order to detect signals from such a phase transition in a collision of two heavy ions, one needs very reliable predictions on the properties of conventional nuclear matter compressed to high densities. As an example it should be mentioned that results of computer simulations of heavy ion reactions like the approach of the "Quantum Molecular Dynamic" (see e.g. Aichelin 1991) or the "Boltzmann-UehlingUhienbeck" approximation (see StOker and Greiner 1986) very much depend on the compressibility of nuclear matter. Therefore one typically considers two different equations of state, a soft one with a compressibility as small as K ~ 140 MeV, as advocated by Baron et al.(1985) from their studies of supernova explosions, and a stiff equation of state with a compressibility of K -~ 380 MeV or even higher, which seems to be preferable when one tries to reproduce experimental d a t a of heavy ion reactions (see Keane et.al 1988). Such investigations typically employ information about the equation of state as derived from a phenomenological parameterization of the nuclear barniltonian. Typical examples are the effective nucleon-nucleon (NN) interaction proposed by Skyrme (1956), the Gogny interaction (Gogny, 1975) or the Walecka model which contains relativistic features (Serot and Walecka, 1986). All such effective hamiltonians, however, have been adjusted to describe properties of nuclei at normal densities. Therefore their predictions on the properties of nuclear systems at higher densities are not very reliable. A more microscopic approach is to start from realistic NN interaction. Typical examples
2
H. Mtither
are the Reid soft-core potential (Reid, 1968), the Paris potential (Lacombe et.al. 1980) or the OneBoson-Exchange (OBE) potentials, originally developed in Bonn (Machleidt, 1989). The parameters of such realistic NN interactions are fitted to reproduce the NN scattering data, i.e. to describe the interaction of two nucleons in the vacuum. For these NN interactions one may then try to solve the nuclear many-body problem as rigorous as possible. If such microscopic many-body calculations lead to results for nuclear matter at normal densities, one may then hope that such a theory which predicts the nuclear properties at the saturation density p0 from the NN interaction in the vacuum, will also produce reliable results for higher densities. In this contribution we will focus the attention on a microscopic description of nuclear properties at temperature T = 0. Furthermore we will ignore sub-nucleonic degrees of freedom, like excitations of the nucleons to the A(3,3) resonance in the many-body approach (see e.g. Miither, 1985). In the next section following this introduction the status of non-relativistic many-body calculations for the saturation point of nuclear matter will be reviewed. After the discussion of the relativistic features taken into account in the Dirac-Brueclmer-Hartree-Fock (DBHF) approach, the differences between microscopic calculations for nuclear matter and finite nuclei wiU be discussed, the main conclusions of this contribution axe summarized in the final section.
CORRELATIONS
IN NUCLEAR
MATTER
the simplest approximation to the many-body theory for the description of the ground state of nuclear matter is to perform a Hartree-Fock (HF) calculation. This means that one defines a single-particle hamiltonian h by
< ilhli > = ei = < iltli > + < ilUli > = < iltli > + ~ < ijlVlij > , j_
(1)
where t stands for the operator of the kinetic energy and the single-particle potential U for a nucleon in orbit i is obtained by summing the nntisymmetrized matrix elements of the bare NN potential V for the interaction between a nucleon in orbit i with one in an orbit j below the Fermi surface. For nuclear matter the HF orbits i and j are bound to be plane waves and the summation in eq.(1) must be understood as an integration restricted to all momenta below the Fermi momentum k r , which is related to the density p of the system under consideration by p =
2 kS
(2)
The binding energy can then be calculated as E = 1 ~
[< iltli > +e,] .
(3)
If, however, one calculates the HF energy of nuclear matter at the empirical saturation density (p0 = 0.16 fm -3, which corresponds to a Fermi momentum kF =1.36 fm -1) employing a realistic NN interaction llke the one defined by Reid (1968), one obtains an energy per nucleon of 176.2 MeV instead of the empirical value of -16 MeV as derived from the Beth~Weizsgcker mass formula (see table 1). This result demonstrates quite clearly that a many-body theory employing such realistic interactions must account for correlations beyond the HF approach. As a first step one must account for 2-nucleon corrdations in such a way that the amplitude of the wave function describing the relative motion is small whenever two nucleons are so close to each other that they are exposed to the repulsive core of the NN interaction. This can be achieved either by
The Nuclear Equation of State Table 1: Results for the binding energy per nucleon of Nuclear Matter Results for the energy per nucleon calculated in the Hartree-Fock (HF) the Brueckner - Hartree Fock (BHF) and with inclusion of the 3-nucleon correlations (BHF3) are listed for two densities, characterized by the Fe~Kfi momentum kF=l.2 and 1.36 fm -1 (the empirical saturation density), respectively. Furthermore we give the energy contribution due to two- (AE2) and three-nucleon correlations (AEa). All results have been obtained for the Rdd soft-core potential. kF {fin-z] HF BHF BHF3 AEa AE3
1.2 119.8 -9.1 -12.6 -128.9 -3.5
1.36 176.2 -9.8 -15.2 -186.0 -5.4
considering such correlations in the wave function explicitly or by evaluating an effective interaction which accounts for such correlations. A first approximation for such an effective interaction is the Bruecimer G-matrix which is obtained from the bare NN interaction g by solving the Bethe-Goldstone equation
Q
a ( w ) = v + v w - Qt----~G ( W ) "
(4)
As one can see from this equation, that the G-matrix is very similar to the scattering matrix for 2 free nucleons except for the fact that the Pauli operator Q prevents intermediate two-particle states which are forbidden by the Pauli principle. Also the starting energy W will typically be the energy of two bound nucleons rather than the energy of free particles. The Brueckner-Hartree-Fock approach is obtained by replacing in eq.(1) the matrix elements for the bare potential V by the corresponding ones for G < ijlV[ij >=~< i j [ G ( W = ei + ei)lij > . (5) The BHF approximation yields an energy for nuclear matter which is much closer to the empirical value than the HF approach. This demonstrates that it is inevitable to confider at least for 2 nucleon correlations. Furthermore it is very important account for the quenching of NN correlations in the nuclear medium, i.e. the difference between the G-matrix and the scattering matrix for 2 free nucleons. Ignoring the quenching mechanism would produce an error of the order of 20 MeV per nucleon around saturation density. One now may try to evaluate the binding energy of nuclear matter as a function of density by repeating such a BHF calculation for various densities and determine the saturation point (i.e. energy and density of the stable nuclear matter) as the minimum of the energy per nucleon as a function of density. Such BHF calculations have been performed for various realistic NN interactions, which were all fitted to describe the free NN scattering data. It turned out, however, that all the calculated saturation points were obtained on the so-called Coester band (Coester, 1970 and MSther 1986, see also figure 1) This means that potentials, which reproduce about the correct saturation density predict a binding energy per nucleon which is to small by roughly 5 MeV, and those potentials which lead to the correct predietion for the binding energy yield a saturation density which is twice as large than the empirical one (or results in between). Therefore attempts have been made to improve the solution of the many-body problem and account for three- and more-nucleon correlations. This can be done essentially by solving the three-particle
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Faddeev equation in the nuclear medium. It turns out that the effects of the three-nucleon correlations are non-negligible and yield in the example of the Reid soft-core potential additional binding energy of about 5 MeV per nucleon (Day 1981, see also table 1). At first sight this seems to improve the agreement with the empirical data. The density-dependence of this three-nucleon correlation effects, however, is such that the resulting saturation point is simply moved along the Coester band. Comparing the effects of two- and three-nucleon correlations one observes that the corrections due to three-nucleon correlations are much smaller. This tends to indicate a fast convergence of the manybody theory. Indeed, estimates for the effects of four-nucleon correlations by Dickhoff et.al. (1982) tend to support such a conclusion. Nevertheless, attempts have been made to improve the non-relativistic many-body calculation such that the saturation point of nuclear matter can be described in terms of a realistic NN interaction. One may argue that the hole line expansion, which treats the particle-particle ladders to all orders but accounts for the hole-hole scattering terms only in a perturbative way is not appropriate for a system with the density of nuclear matter. Rather one may try to use a seE-consistent Greens function approach in which particle and hole states are treated in a symmetric way (for a recent review see Dickhoff and MSther, 1992). Indeed, calculations which include the effects of the so-called particle-particle hole-hole ring diagrams by Jiang et.al. (1988) lead to saturation points of nuclear matter which are in a much better agreement with the empirical point than those obtained for the BHF approach (see also figure 1). However, this success could very well be an accidental one. It is quite possible, that an approach which accounts for collective effects in a more general way than just to include particle-particle hole-hole ring diagrams could very well spoil the agreement again (see also discussion below for finite nuclei).
The Nuclear Equation of State DIRAC-BRUECKNER-HARTREE-FOCK Another step which in recent years has very successfully been made to describe the saturation properties of nuclear matter, has been the consideration of relativistic effects in the so-called Dirac approach (see e.g. Serot and Walecka, 1986, Anastasio et.al., 1983, Brorkmann and Machleldt, 1984). At first sight relativistic effects seem to be negligible since the single-particle potential U (see eq.(1), typically -40 MeV) is very small compared to the rest mass of the nucleons (938 MeV). One must be aware, however, that the single-particle potential U originates from a delicate balance between attractive and repulsive components in the NN interaction. In a OBE model the attractive components are mainly due to the exchange of a scalar meson, the so-called a meson, whereas the short-range repulsion is dominated by the exchange of the w meson which is a vector meson. In the Hartree approximation the exchange of such mesons yield an operator for the self-energy of the nucleons with a scalar component A and a (time-like component of a vector field B fr = A + B 7 ° .
(e)
In the Hartree approximation for nuclear matter A and B are independent of the momentum of the nucleon under consideration and depend on the density of the system, only. Including this self-energy in a Dirac equation for a nucleon with mass m and momentum 1~in the nuclear medium
[,~f+ a (,-. + A)] aCp,.) = (,,
-
B),:Cp,.),
(7)
with 6 and B the usual Dirac matrices, one obtains a solution for the Dirac spinor
(s)
V which is identical to a solution for a free nucleon with an effective mass
rh=m+ A,
~, =
v~
+ f,
(9)
and Xm a Pauli spinor. Since A is large and attractive (-300 MeV) the effective mass rh, which characterizes the ratio of large to small component in the Dirac spinor, is considerably smaller than the bare mass. This change of the Dirac spinors in the nuclear medium leads to different matrix elements for the OBE model of the NN interaction. Due to the change of the nucleon spinors in the medium, the NN interaction in nuclear matter is different from the one in the vacuum. Since the nucleon self-energy ~" is determined from the NN interaction, which itself depends on the solution of the Dirac equation (7) for this self-energy, a new self-consistency problem has to be considered. For a realistic OBE potential one has to go beyond the Hartree approximation for the self-energy and take into account the effects of Fock exchange terms and of two-body correlations by determining the self-energy U from the G-matrix in analogy to the BHF approach outlined in the preceding section. < i j l G ( W = ei + ei)lij > ~
O(i).
(10)
This leads to the so-called Dirac - Bruerkner - Hartree - Fork (DBHF) approximation. The density dependence of the Dirac effects is such that the saturation points resulting from DBHF calculations are effthe Coester baud. Brockmann and Macldeidt (1984) succeeded to find a realistic OBE potential which fits the NN scattering data and reproduces the empirical saturation point of nuclear matter employing the DBHF approach.
H. Miither
Dirac BHF for 160 -3.0
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,
,
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Inverse Radius [fro**- 1 ] Figure 2: BHF and DBHF calculations for 160. Using OBE potentials A, B and C defined by Machieidt (1989), results for the radius of the charge distribution and the binding energy per nucleon obtained in conventional BHF (stars) and DBHF calculations (circles) are compared to the experimental result.
FINITE NUCLEI The next question to be answered is of course whether the results obtained for the saturation point of nuclear matter are directly related to the features obtained in calculation of ground-state properties for finite nuclei. From a technical point of view the calculation of finite systems is more involved since at each step of the many-body approach one has to evaluate the optimal single-particle wave function which in nuclear matter is determined by the translational invariance of the system. On the other hand, however, the typical densities observed in finite nuclei are below the saturation density. Therefore a many-body approach like the hole-line expansion, which shows a better convergence at low densities, should for a given level for a given order in the expansion provide more reliable results for finite nuclei than for nuclear matter. At first sight the BHF approach for finite nuclei seems to show similar problems as for nuclear matter. As an example we present in figure 2 results obtained for the binding energy per nucleon and the inverse value for the radius of the charge distribution (in order to have a quantity comparable to the Fermi momentum in figure 1) of is O employing various realistic interactions. Also in the case of finite nuclei one obtains a "Coester" band similar to the one of nuclear matter. Figure 2 also displays results of DBHF calculations for this nucleus as obtained by Miither, Machleidt and Brockmann (1990). In this approach the change of the Dirac spinors in the nuclear medium is described in terms of a local density approximation relating the ratio of small to large components in the Dirac spinors for leO to the one obtained in DBHF calculations of nuclear matter at the corresponding densities. The self-consistent solution of the Bethe-Goldstone eq.(4) and the BHF equation (5) are obtained directly for the finite nucleus. Also in this case the inclusion of relativistic
The Nuclear F~luation of Sta~
7
effects improves the agreement with the empirical point, however, a sizable discrepancy remains also for the OBE potential (A) which reproduced the saturation point of nuclear matter. This can be understood as an indication that the Dirac phenomenology alone may not solve the problems of microscopic many-body calculations for nuclear systems. Of course, also for the relativistic calculations one should take into account the effects of many-nucleon correlations and sub-nucleonic degrees of freedom. These effects are non-negligible within the non-relativistic approach to the nuclear matter problem and therefore should also be considered within a relativistic treatment. However, there is a more serious criticism: In the DBHF approach one takes into account that the solution of the Dirac equation for positive energies are modified for the nucleons in the nuclear medium. The presence of the medium, however, also modifies the solutions of the Dirac equation with negative energies. This implies that the Dirac vacuum is modified in the medium as well. Attempts to account for such vacuum renormalization effects demonstrate that such effects are very large and cannot be ignored (Brown et.al., 1990). But also attempts to include the et~ects of pl>-hh ring diagrams, which improved the calculation of the saturation properties of nuclear matter (see above), seems not to produce such a large effect for the finite system (Mavromatis et.al., 1991). Since results obtained in many-body calculations for finite nuclei show remarkable differences from those obtained in nuclear matter (see discussion above) one may suspect that different saturation mechanisms must be considered for finite nuclei than for nuclear matter. To verify this we perform a little theoretical experiment on the compression of nuclear systems. In order to study the compression modulus of the finite nucleus l e o let us consider as a model wave function the Slater determinant appropriate to describe the ground-state of l e o in terms of harmonic oscillator functions for a given oscillator length b. For each oscillator length one can calculate the radial density distribution pb(r) and the average nucleon density
lYb = f p](r) d3r A
(11) '
with A the number of nucleons. For each model wave function or oscillator parameter b one can furthermore calculate the BHF energy by solving the Bethe-Goldstone equation and the BHF equations directly for the finite nucleus under the constraint that the single-particle waves are oscillator functions for the b parameter under consideration. For the discussion of results presented here we have employed the OBE potential "A" defined in table A.1 by Machleidt (1989). The energies per nucleon calculated for v ~ o u s b are displayed by the solid line in figure 4 as a function of fb defined in eq.(11). The minimum of this curve is close to the result for the self-consistent BHF calculation obtained without the constraint for the single-particle wave function. In addition the second derivative of this curve gives an idea about the compression modulus for this finite nucleus. In the next step we now consider the finite nucleus to consist out of slices or shells of nuclear matter and calculate the energy in a very simple local density approximation (LDA) as
ELDA(b)
= / ZNM(Pb(~))d3~,
(12)
where £NM(P) is the energy density for nuclear matter of density p and pb(r) as above stands for the baryonic density calculated for the Slater determinant of oscillator functions with oscillator parameter b. Results for this LDA approximation as a function of density are given by the long-dashed curve in figure 3. One finds that the LDA overestimates the calculated binding energy. At the BHF minimum the difference is about 85 MeV for the total nucleus, which is close to the surface correction contribution in
H. Miither '
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,
0.3
density [fro**-3] Figure 3: Binding energy per nucleon for 160 calculated for various mean values of the density ~ (see eq.(ll)). The results of the BHF calculation directly for finite nuclei (solid line) and the various local density approximations (LDA, see text) were obtained for the OBE potential A defined in table A.1 of Machleidt (1989). the Bethe - Weizslicker mass formula for 160 (m 118 MeV). This suggests that the LDA approximation misses the major contribution to the surface tension and also the major contribution of the surface effects to the compressibility of a finite nucleus. This is supported by the observation that the LDA result for 160 as a function of density is only slightly above the energy versus density curve of nuclear matter. Also one sees that the energy versus density curve calculated directly for the finite nucleus is much stiffer than the corresponding curve in the LDA approach or for nuclear matter. This indicates that due to the surface effects it requires more energy per nucleon to compress a small piece of nuclear matter, as it is done e.g. in a heavy ion collision, than an infinite system or a stellar object. Therefore this surface contribution to the compressibility may to a large extent explain the difference in the equation of state for nuclear matter derived from heavy ion collisions and studies of astrophysical objects. Figure 3 also displays a LDA result (short dashes) which is obtained according to eq.(12) but ignoring the density dependence of the Brueckner G-matrix in ENM(P), i.e. using the G-matrix as calculated for the density p = 0.05 fm -3. Comparing the two curves for the LDA approximation one can see that the density dependence of the G-matrix is a very important ingredient to obtain the saturation point of finite nuclei as it is for nuclear matter. In addition, however, an additional surface effect is needed for finite nuclei, which is absent in nuclear matter or the LDA approach for finite nuclei. LDA AND
FINITE RANGE
OF INTERACTION
In order to find the origin for the discrepancy between the LDA and the explicit calculation for the finite nucleus one can compare the different contributions to this energy. One finds that the LDA
The Nuclear Equation of State
9
approach for the kinetic energy
TLVA(b)= f 3 a2k~-F~ (r)p~Cr)dsr
(13)
with the local Fermi momentum kF(r) determined from the relation
pb(r) = 3-~/c~(r),
(14)
is a very good approximation to the kinetic energy of the corresponding harmonic oscillator configuration. Therefore the deviation must originate from the expectation value for the potential energy. To visualize the difference of the two approaches for the calculation of the interaction energy we assume for the moment the two-body interaction, in the BHF approximation the G-matrix, to be local, which means that it depends on the momentum transfer only. In this case the interaction energy can be calculated as
< V >---- ~_~ / VSMT(q)~SMT(q)dq S,M,T J
(15)
where VSMT(q) is the interaction depending on the momentum transfer q for a given set of spin isoepin quantum numbers S, M, T or a corresponding spin-isospin operator in the interaction V (see e.g. Dickhoff, 1983). The quantity ~(q) represents a two-body density obtained for a given approximation to calculate < V >. For this two-body density all variables have been integrated except the momentum transfer. Therefore this ~b(q) is a measure to which extend the local interaction at momentum transfer q contributes to the total interaction energy. In the BHF approximation this ~(q) can be split into a direct part (related to the Hartree contribution) and an exchange contribution (related to the Fock term) ~SMT(q) SMT SMT = ~di,ect(q) -i- @,zch (q). (16) For isoepin symmetric (Z=N, ignoring Coulomb interaction) and spin saturated systems the direct contributions vmdsh for all spin-isospin channels except the scalar-isoscalar (S=T--0). In nuclear matter the direct term is given by a ~ function / ~
~T~3F
(17)
•000 ,,recttq) = ~ ( q ) and the exchange contributions are evaluated as ~b~fCh (q) =
1
(2~) z 8
q3 qS l
q2 _ ~
+
(18)
for momenta q _< 2kr and zero for larger momentum transfers. As the LDA implies an average over nuclear matter energy densities, the LDA result for the two-body density ~b(q) will be a superposition of nuclear matter functions at various densities. This can be seen from figure 4, where the dashed line exhibits the LDA approach for ~b(q) in the central spin - isovector channel (S--T=I, averaged over the spin projection M). For this channel one does not obtain a direct contribution for 160 and therefore LDA yields a superposition of functions of the form of eq.(18) for various densities. The corresponding two-body density approach for the direct calculation of l e o is given by the solid line and we observe that the LDA obviously yields a very good approximation for the two-body density. The situation is different in the scalar - isoscalar channel (S=T=0) (see figure 5). In this channel one also obtains a direct contribution, which in LDA as in nuclear matter ( see eq.(17)) is given by a ~-function and not made visible in the dashed line of figure 5). As the single-particle wave functions
H. Miithcr
10
Central ~pin-Isovector Channel
0.0 -
~
-lO0.O-
_
\ \.\ ~, ,..
"'-...
- oo.o
-250.0
~ / ~
0.4
_
0.0
De=Ity
-o2 /I
. . . .
-
-0~1
-300.0 -350.0 0
~'"'''-.... r
,
1
2
-LO -12
3
4
Momentum Transfer [fm-'] Figure 4: The two -body density ~(q) in the central spin isovector channel (S = T = 1, M averaged) used to calculate the interaction energy according eq.(15). Results are presented for the nucleus l e o as obtained in the LDA approach (dashed line) and the exact calculation (solid Line). Also given is a local presentation of a realistic G-matrix as a function of the momentum transfer as introduced by Dickhoff (1983) (dashed-dotted line). Further details in the text.
of a finite system are not eigenstates of the momentum operator, the direct term in ¢(q) calculated directly for the finite system shows a shape different from a 6 function. This can be seen from figure 5, in which the two-body density calculated for the oscillator model of l e o (solid line) exhibits quite a different shape as the LDA.
If the two-body interaction in this spin-isospin channel would be of zero range, i.e. the function V(q) would be a constant as a function of momentum transfer, this difference in the ~(q) would not show up in the calculated energy, since the integral in eq.(15) would yield the same result for the LDA as for the exact calculation. Realistic interactions and also local representations of the G-matrix exhibit a finite range or a dependence on momentum transfer. A typical function Vooo(q) is given by the dashed-dotted llne (referring to the right scale) in figure 5. One can see that such a function is more attractive at q = 0, where the LDA yields the largest contribution, than for those momentum transfers, for which the maximal contribution is obtained in the exact calculation. This explains why LDA overestimates the binding energy. If the nucleus would be compressed, the maximum of ~(q) for the solid line in figure 5 would be shifted to larger momentum transfers, an effect which tends to reduce the calculated energy. On the other hand the LDA still obtain the largest contribution at q = 0. Using the language of momentum representation this explains the difference in the calculated compression moduli shown in figure 3. Our investigation demonstrates that the LDA is valid only for an effective interaction of zero range. For realistic interactions of finite range the LDA underestimates the attraction from the direct interaction in the scalar-isoscalar channel. Therefore we see that the ground-state properties of finite nuclei are much more sensitive to the range of realistic NN interactions than a calculation of the
The Nuclear Equation of State :~o0.0
0.0
~}0.0
'aT is°'°-
0.0-
11
Central-Isoscalar Channel
/
• -.
~
/ /
.......
\ \
/
.~
Potential
-to
-----Density
/"
__
--~.0
-50.0 0
+ 1
i
l
g
3
MornenLuli1 Transfer [fro-1]
-4.0
4
Figure 5: The two - body density in the scalar - isoscalar channel ( S = T = 0) used to calculate the interaction energy according eq.(15), Results are presented for the nucleus toO as obtained in the LDA approach (dashed line without the direct contribution 6(q)) and the exact calculation (solid line). Further details see figure 4.
saturation point of nuclear matter.
THE MODEL OF NAMBU
AND JONA-LASINIO
Until now we have taken into account that the effective mass of the nucleons may change in the nuclear medium (see discussion of DBHF above). For the sake of consistency one may also consider a possible change of the meson masses in the nuclear medium. Such a change of the effective meson masses would modify the propagation of the mesons in the medium and give rise to a change of the range of meson exchange interaction between nucleons. As we have seen in our discussion above such a change of the range of the NN interaction would affect in particular the nuclear structure calculation for finite nuclei. Part of this change of the meson propagators can be described within the framework of the conventional many-body theory by considering a coupling of the mesons to the particle-hole excitations of the nuclear system. In this section, however, we would like to account for sub-nucleonic degrees of freedom and try to evaluate the effective masses of baryons and mesons and their medium dependence within a phenomenological model which accounts for quark degrees of freedom. Such an effective theory which reflects the chiral symmetry as one of the basic symmetries of QCD is the one proposed already in 1961 by Nambu and Jona-Lasinio (NJL). In order to recall some features of this conventional NJL model we start writing down the Lagrangian density £ = ¢(i~7 - M ) ¢ + G [ ( 6 1 ¢ ) 2 + ( ¢ i3,s~11¢)2], (19) where ¢ is a (current) quark field with mass M, ¢ the isospin matrices for the SU(2) flavor and I
12
H. Miither
the unit operator for the color part (SU(3)) of the quark fields. The coupling constant G, which is common for the scalar-isoscalar and the pseudoscalar-isovector part of the interaction term has dimension [mass-2]. The Lagrangian of eq. (19) is symmetric under a chiral transformation if we put M -= 0. As a first step we would like to discuss the properties of the quarks in the vacuum. The self-energy E for the quarks can be evaluated in the mean-field or Hartree approximation as E = E" + E p" = 2G1 < ¢ 1 0 > +2Gi7511¥ < ¢i75r-*1¢ > •
(20)
Since the vacuum expectation value < ¢iTsr-*l¢ > vanishes, only the scalar part of the self-energy E" survives and leads to an effective mass for the constituent quarks of the form M* = M -
(21)
2G < ¢ 1 ¢ >
with the scalar density 12
< ¢I¢ >= -~-~-
~A
d3p ~
M* .
(22)
The two Hartree equations (21) and (22) have to be solved in a self-consistent way and it is clear that for the chiral symmetric case ( M = 0) they have the trivial solution M* -- 0. If, however the interaction strength G is strong enough (for a given cut-off momentum A) the Hartree equation also yield a non-trivial solution with M* > 0. This solution, with the chiral symmetry spontaneously broken, shows up also for the case M > 0 and can be identified from the fact that the constituent quark mass M* obtained in this case is considerably larger than the bare current quark mass M. In order to study a system of finite density one only has to replace the expression for the scalar density given in eq. (22) by 12 ~ A < ¢ 1 ¢ > = -(27r)3
d3pO(ip7- kF)M X / ~
,
(23)
kF
with the usual step function O and the Fermi momentum for the homogeneous system of quark matter. From eq. (23) it is clear that the absolute value for the scalar density tends to reduce with increasing k~,. As a consequence also the effective mass M* is reduced with increasing baryonic density. The scale for this decrease is determined directly by the cut-off parameter A and by the assumption that this cut-off does not depend on the density under consideration. Results for the effective quark mass as a function of the baryonic density are displayed in figure 6 (solid line). The effective mass M* for the constituent quarks can directly be related to the effective mass of the nucleons if we assume that the baryon masses are essentially three times the constituent quark mass. Therefore also the NJL predict a decrease of the effective mass for the nucleon with increasing density as was also obtained in the DBHF approach. In a next step one may then consider the mesons as the collective modes of the quark excitations from the Dirac see. In a nuclear medium these excitations of the Dirac see interfere with the particle-hole excitation of the quarks in the Fermi see This reflects the coupling of the mesons to the medium excitations which are present already in a non-relativistic many-body theory. Therefore one should be very careful if one employs density dependent meson masses as derived e.g. from NJL in a nuclear structure calculation, since these effective masses may already contain effects which would be accounted for again in the nuclear structure calculation. The effective masses of the mesons can be determined from an inspection of the reducible response function (i.e. the excitation spectrum calculated in RPA approximation). Results for the effective mass for the constituent quarks (or baryons) and mesons have been determined by several groups (see
13
The Nuclear Equation of State
Nambu-Jona-Lasinio 700.Sigma uu~-,
600.
",
500. q)
•~
/
400. ~t
• Quark Mass 300"i
200" t
.
.
.
~jrI
,,
~
I
/-~/
.
100. i Pion
O.
'
0.0
'
'
'
l
0.5
'
'
'
'
I
1.0
'
'
'
'
I
1.5
'
'
'
'
I
'
'
2.0
Fermi Momentum [ f m * * - l ] Figure 6: Effective masses as a function of the baryonic density calculated in the NJL model for the quarks (solid line), the Pion (long dashes) and the scalar - isoscalar meson (~r, short dashes). The parameters for the NJL Lagrangian in eq.(19) are M = 5.5 MeV, A -- 630 MeV and G = 0.214 fm -2. See also Henley and MSther (1990).
e.g. Bernard et.al., 1987 and Henley et.al. 1990). As can also be seen from figure 6, one finds that the masses of the scalar ~ meson and the vector to meson are reduced in the medium. This should lead to a larger range of the interaction, which should produce larger radii for the nuclei. Therefore one would expect an improvement of the calculated ground-state properties (see figure 3). In contrast to the medium dependence of these mesons the mass of the pion remains constant for a large range of densities and increases at high densities where the chiral symmetry of the vacuum is restored. This feature is related to the fact that the pion can be considered as the Goldstone boson for the chiral symmetry. It is clear, however, that this medium dependence of the effective masses is a direct consequence of the assumption of a cut-off A which does not depend on the density of the system under consideration. In order to develop a model which is very similar to the NJL model, hut does not require such a strict cut-off, B~ckers et.al. (1991) consider an interaction term between the Dirac epinors in momentum representation of the form V = -G(P)[(~) ~ + (~ i ~ 5 ~ - ~ ) 2 ] . (24) This interaction term is very similar to the original one of the NJL model in eq. (19) and exhibits only two differences: First, the interaction strength is not considered any longer to be a constant, but it should depend on the momentum transfer between the interacting Fermions. Guided by the r , nniug coupling constant of QCD, one assumes that it is reduced for large momentum transfers P. The second difference is that the interaction terms of eq. (24) are of vector-type in color-space. For our present considerations the detailed form of the interaction is not really relevant as long as it is symmetric under chirai transformation. The direct or Hartree contributions to the self-energy are now to be calculated with a "coupling
14
H. Miithcr
constant" G determined at momentum transfer P = 0 (see also discussion in the previous section). For a homogeneous, color symmetric vacuum or baryon system the expectation values for the colorvector terms occurring in (24) vanish and therefore the Hartree terms do not contribute. If, however, one would study a system with an isolated quark of a given color the Hartree terms would diverge and therefore provide a possible mechanism to obtain color-confinement. For the study of the effective mass of quarks in a homogeneous system, however, we have to investigate the exchange or Fock contributions to the self-energy. B~ckers et.al. (1991) show that this yields an effective mass for the quarks which depends on the momentum. This extended NJL model which shows a closer connection to QCD shows similar features as the conventional NJL discussed above.
CONCLUSIONS It has been one aim of this review to demonstrate that conventional microscopic nuclear structure calculations employing realistic NN interactions predict properties of nuclear systems at normal densities which are not really in satisfactory agreement with the empirical data. Improvements beyond the conventional BHF approximation are needed to improve the agreement and therefore establish a microscopic theory which may predict in a reliable way also the main features of nuclear systems at densities well above the normal density. Possible candidates for such improvements are the inclusion of special collective many-body effects like the pp-hh ring diagrams discussed above or the relativistic effects of the DBHF approach. Both of these approaches yield the correct saturation point of nuclear matter but are not yet completely successful to describe the ground-state properties of finite nuclei as well. It has been demonstrated that differences exist in the mechanisms leading to saturation properties of finite nuclei as compared to those in nuclear matter. A local density approximation (LDA) to describe finite nuclei in terms of "slices" of nuclear matter underestimates the surface tension considerably and yields a different compression modulus. Due to the finite range of the interaction it requires more energy to compress finite nuclei as one may estimate from the LDA. This implies that one may deduce a different equations of state from heavy ion reactions (which probes the compression of parts of finite nuclei) than from the analysis of astrophysical objects which might well be described in terms of nuclear matter. Since the structure of finite nuclei is very sensitive to the range of the NN interaction, it may become very important to study a change of this range of the interaction due to a change of the effective meson masses in the nuclear medium. Such a medium dependence might be studied in effective theories related to the QCD (like the NJL model). For more reliable predictions, however, an extended version would be needed which also describes the confinement of quarks in hadrons in the vacuum and inside a nuclear system. The investigations presented here have been performed in collaboration with many colleagues. I would like to list a few of them like R. Brockmann, G.E. Brown, D. Bfickers, P. Czerski, W.H. Dickhoff, P. Ellis, H. Elsenhans, A. Faessler, R. Fritz, E. Henley, M.F. Jiang, T.T.S. Kuo, R. Machleidt, H. Mavromatis, A. Polls, M. Prakash and K.W. Schmid to thank for this collaboration and many fruitful discussions. Also I would like to acknowledge the financial support by the Bundesministerimn fiir Forschung und Technologie (Bonn, project 06 Tfi 714)
The Nuclear Equation of State
15
REFERENCES Aichelin, J. (1991). Phys. Rep. 202, 233. Anastasio, M.R., Celenza, L.S., Pong, W.S., and Shakin, C.M. (1983). Phys. Reports 100, 32T. Baron, E., Cooperstein, J. and Kahanna, S. (1985). Nucl. Phys. A440, 744. Brockmann, R. and Machleidt, R., (1984). Phys. Left. B149, 283. Brown, G.E., MSther, H. and Pr,~b~h, M. (1990). Nucl. Phys. A506, 565. BSckers, D. and M~ther, H. (1991). Nucl. Phys. A523, 629. Coester, F., Cohen, S., Day, B.D. and Vincent, C.1V[.(1970). Phys. Rev. C1,769. Day, B.D. (1981). Phys..~e~. C24, 1203. Dickhoff, W.H., Faessler, A., and Miither, H. (1982). b'ucl. Phys. A389, 492. Dickhoff, W.H. (1983). Nucl. Phys. A399, 287. Gogny, D. (1975). N,~cl. Phys. A237, 399. Henley, E.M. and M~ther, H. (1990). Nucl. Phys. A513, 667. Jiang, M.F., Kuo, T.T.S. and Miither, H. (1988). Phys.Rev. C38, 2408. Keane, D., Chu, S.Y., Fung, S.Y., Liu, Y.M., Qiao, L.J., VanDalen, G., Vient, M., Wang, S., Molitoris, J.J., and StScker, H. (1988) Phys. Bey. C37, 1447. Lacombe, M., Loiseaux, B., Richard J.M., Vinh Mau, R., C6t~ J., Pir~s, and de Tourreil, R. (1980). Phys.Re,J. C21,331. Machleidt, R. (1989). Adv. Nucl. Phys. 19, 189. Mavromatis, H.A., Ellis, P. and M~ther, H. (1991). Nucl. Phys. A530, 251. Miither, H. (1985). Prog. in P,~rt. and IVucl. Phys. 14, 123. Milther, H. (1986). Prog. i,~ P,~rt. a,~d Nucl. Phys. 17, 97. Nambu, Y. and Jona-Lasinio, G. (1961). Phys. Rev. 122, 345 and ibid 124, 246. Reid, R.V. (1968). A,~n.Phy8. 50, 411. Serot, B.D. and Walecka, J.D. (1986). Adv. Nucl. Phys. 16, 1. Skyrme, T.H.R. (1956). Phil. Mag. 1, 1043. St6cker, H. and Greiner, W. (1986). Phys. Rep. 137, 277.