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Finite-nuclei calculations based on relativistic mean-field effective interactions
NUCLEAR PHYSICS A
Nuclear Physics A547 (1992) 447-458 North-Holland
Finite-nuclei calculations based on relativistic mean-field effective interactions Stefan Gmuca
Institute of Physics, Slovak Academy of Sciences CS-842 28 Bratislava, Czechoslovakia Received 26 March 1992 Abstract: The Dirac-Brueckner- Hartree- Fock calculations of nuclear matter are parametrized by the relativistic orw mean-field theory with scalar and vector nonlinear self-interactions . The effective interactions thus obtained are used in the relativistic mean-field studies of the structure of 16 0 and 40Ca nuclei without the introduction of additional free parameters. The calculated binding energies, single-particle spectra and charge radii agree quite well (although not satisfactorily) with experimental data and present an improvement over the nonrelativistic Brueckner-Hartree-Fock approximations .
1. Introduction One of the most ambitious programs in nuclear physics is to understand the structure of finite nuclei starting from the bare nucleon-nucleon (NN) interaction . The nonrelativistic Brueckner-Hartree- Fock (BHF) approach [see e.g. ref. ') and references therein] with various basic forces has been widely and repeatedly applied in the attempts to explain some ofthe nuclear ground-state properties in a parameterfree way. These calculations, ho-Alever, have achieved only a limited success as they were unable to produce reasonable binding energies, single-particle spectra, r.m .s. charge radii and charge distributions for the nuclei considered. Recently these attempts have revived as the relativistic extension of the BHF 2-4 approach [so-called Dirac- Brueckner- Hartree- Fock (DBHF) approach )] has been successfully applied to the nuclear-matter problem. The key point is the use of the relativistic dynamics, governed by the Dirac equation with strong scalar and vector fields, for the single-particle nucleon motion. The explicit treatment of the lower components of the Dirac nucleon wave functions gives rise to strongly density-dependent relativistic effects that move the nuclear-matter saturation point 5 away from the "Coester band" much closer to the empirically acceptable values ). But, unlike in the nonrelativistic BHF case, the DBHF approach, due to its complexity, is much more elaborate when applied to finite systems. In fact, there are still no fully self-consistent Dirac-Brueckner calculations for finite nuclei. The
Correspondence to: Dr. S. Gmuca, Institute of Physics, Slovak Academy of Sciences, Dubravska Cesta 9, CS-842 28 Bratisiava, Czechoslovakia . 0375-9474/92/$05 .00 @ 1992 - Elsevier Science Publishers B.V. All rights reserved
S. Gmucal Finite nuclei calculations
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use of some approximate methods seems to be inevitable . The existing calculations either employ the nuclear-matter results with some kind of local density approximation (LDA) '-"), or rely on various effective interact: - approaches which reproduce " the self-energy terms or G-matrix elements of the DBHF theory 9 o). In the previous paper") we have developed a simple relativistic mean-field parametrization of the effective interactions in nuclear matter which reproduce (rather accurately) the Wnding energy and the scalar and vector (time-like component 3,4) nly) self-energy terms of the recent DBHF calculations of nuclear matter . The model is based on the relativistic aw mean-field theory with scalar and vector self-interactions . The inclusion of the isoscalar vector-meson quartic self-interaction into the effective mean-field lagr.--ngian is essential for obtaining a proper density dependence of the vector potential ") . In the present paper we will use this effective interaction for the calculation of 40 the ground-state properties of the doubly closed-shell nuclei `0 and Ca in the s,imple mean-field approach. In such a way we can bypass the very complicated fully self-consistent DBHF calculations in finite systems . We are interested whether this approach is equally successful in finite nuclei calculations as it is in nuclear matter ones. The arrangment of the paper is as follows . In sect. 2 we will give a short overview of our relativistic mean-field approach (r.m .f.a .) together with parametrizations of recent DBHF results in nuclear matter. Sect. 3 is devoted to the generalization of 40 the model for finite systems. The ground-state properties of the `0 and Ca nuclei are calculated in sect. 4. The comparison with other approaches is presented there as well. Sect. 5 contains a summary and our conclusions. 2.
ean-field approach in nuclear matter
elativistic
2.1. NONLINEAR R.M.F.A. MODEL
We start with the nonlinear r.m .f.a. model which, in addition to the scalar cubic and quartic self-interactions, includes also the vector-meson quartic self-coupling . This extension of the standard r.m.f.a. lagrangian introduces the density dependence also into the vector-meson sector and, in such a way, simulates the density dependence produced in the DBHF approaches. The full discussion of the model is given in refs . 11 J2) . The lagrangian density of the nonlinear aw theory reads L(j p w) = &iya'-(M -g,u) -pSwlee + 1 (hao"a - M 2 a 2) _ l bCrM ( Ig.)3 ff
2
tLV+ I 40ig V&j
1 j
n; 2. &JIJ 01
3
gtr
IA +jCw
4
I
( 0.)4 ra ga
( g2 &J ~L&J IA )2 to
where the symbols have their usual meaning 13). The first line of eq . (1) describes the nucleon (41) part of the lagrangian density including the nucleon-meson interaction terms. The second line is the standard
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449
scalar o,-meson part with cubic and quartic self-interactions. The strengths of these self-couplings are given by the dimensionless constants b,, and c,. The third line contains the vector w-meson part of L including the quartic vector-meson self(g.(d0 )4 interaction term jC. with the dimensionless self-coupling constant c,,,. 4 Restricting ourselves to the mean-field approximation, the scalar and vector fields are replaced by their ground-state expectation values. Consequently the classical equations of meson fields and modified Dirac equation of the nucleon field are easily solved for nuclear matter. They read 11 ) g2 = g,.,Cr
(T 2
M
[pS-b(TM(g(,Cr)2 _ CfjgffG.)3]
(2)
(Y ,,2
9-Ù)O = M"2' [ - MI .
[PB _
CLO
(
gw
WO )3] ~
(3)
+,8M *10 = (E - &,wj 41 .
(4)
Here we utilize the fact that in a uniform system only the time-like component of the vector field survives . The source terms for the scalar and vector fields are the scalar density ps and the baryon (vector) density p B , respectively . Using the standard positive-energy solutions of the Dirac equation (4), they are given by 7 k, m* Y dk 2 + M*2)1/2 PS (5) (27r) 11 (k 9
PB -
6
y . k3 ff 2
F ,)
where we assume that nuclear matter consists of filled nucleon levels up to k F , and y is the spin-isospin degeneracy factor (-/ = 4 for nuclear matter) . The effective nucleon mass M* in eqs. (4), (5) is given by the relation M* = M -
(7)
g"O'. .
The scalar density p s is thus an implicit function of the scalar field. Therefore, the scalar field has to be determined self-consistently at each density by solving eq. (2) and eq. (5) . The energy density of isoscalar nuclear matter is given in the parametric form as 'y k, M2 E,.,,, . dk [k 2 + M*2]1/2 +_ -( g,,Cr )2 +jb(TM(&~0.)3 +1 ( 0.)4 La . -
2
(2 ,w
2 g,
2
3
4 cfr gfr
+
,10)2 1 ( g (1)0)4 . gw 4Cw w 2 gw
The binding energy per nucleon of nuclear matter is then simply Et, = E, .m .f.a ./PB - M-
(9)
450
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TERMINATION OF R.M.F.A. PARAMETERS
he lagrangian (1) given above has to be considered as an effective lagrangian and the meson masses and their couplings are to be understood as effective parameters of the theory. In the previous paper ") we have determined them by reproducing the binding energy and the scalar and time-like component of vector self-energy terms of DBHF calculations of isoscalar nu-lear matter -3,4) . A very good fit was obtained for a broad range of densities (k F < 1 .7 fin-') . The binding energy of nuclear matter was reproduced within 0.3 MeV, and the scalar and vector potentials within 5 MeV in the fitted window. In the present paper we extend these previous results and give more complete r.m.f.a. parametrizations for new recently published DBHF calculations of nuclear matter 14,15) . rockmann and Machleidt 14) presented the DBHF nuclear-matter calculations for three different one-boson-exchange (o.b.e.) potentials (denoted as potential A, and C, respectively) which describe the NN data and are appropriate for the nuclear-structure calculations . These o.b.e. potentials differ mainly by the strengt.h of the tensor force [see ref. 14) for details]. As a result, the corresponding DBHF calculations predict different locations of the nuclear-matter saturation point. They form a new "Coester band" that touches the empirically acceptable area of saturation . To reproduce the DBH F results we used the procedure of ref. "), i.e., we performed a simultaneous fitting of binding energy, scalar and time-like vector components of the self-energy terms. Resulting sets of r.m.f.a. parameters are listed in table 1 . ereafter the r.m .f.a. parameters denoted as BM-A reproduce the DBHF results obtained with the o.b.e. potential A. Analogically, BM-13 denotes the r.m.f.a. parameters obtained by fitting the DBHF calculations with the o.b.e. potential B, and -C denotes those made with the o.b.e. potential C . The qualitty ofparametrizations may be seen from fig. I where the r.m.f.a. fits are compared with the DBHF data for the nuclear-matter binding energy. We see that the DBHF results for all three potentials are reproduced rather accurately (within -0.3 MeV) by our r.m.f.a. model TABLE I Parameters of the r.m .f.a. fits to the DBHF data of Brockmann and Machleidt nuclear matter Parameter 2
gger 2
k, C.C.
BM-A 154.218 226 .887 0.001347 0.007947 0.023870
BM-13 154 .050 235 .109 0.000673 0.009786 0.023605
BM-C 152 .786 238 .691 0 .000201 0.010949 0 .022954
14)
Note in, = 550.0 MeV m.., = 782.6 MeV
in
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451
BM-C
-4 -8
M
LLJ
-12 -16 -20 1 0 .8
1 .0
1 .2 kF
1 .4
1 .6
1 .8
(fM_1)
Fig . 1 . The binding energy per particle E b in isoscalar nuclear matter as a function ofthe Fermi momentum kF for various approaches . The symbols represent the results of the DBHF calculations of Brockmann and Machleidt 14 ) . The curves show the corresponding r.m.f.a. fits. The cross indicates the empirically acceptable area of saturation.
with the scalar and vector self-couplings. Similar conclusions concern also the scalar and vector self-energy terms . In these cases the DBHF results are reproduced within -5-6 MeV. In ref. ") de Jong and Malfliet presented an improved DBHF calculation of nuclear-matter properties as compared to the previous one of the Groningen group The improvement consists ofa more careful numerical treatment ofsome singularities encountered in the DBHF scheme . We have refitted these improved DBHF results by the procedure described above [see also ref. ")]. The resulting set of r.m.f.a. parameters (hereafter denoted as DJM) is listed in table 2. In addition, de Jong and Malfliet ") gave also the results of the calculation of nuclear matter using a so-called relativistic conserving approximation (r.c.a.) which imposes additional constraints on the Dirac-Brueckner scheme . In particular, 4)
16
)
TABLE 2
Parameters Parameter 2'
9 2 . 9 b,, CO-
C.
of
the r.m.fa. fits to the DBHF data Malfliet ") in nuclear matter DJM 134.228 179 .077
0.001469 0.000338 0.005835
DJM-C 137 .485 192 .587 0.000785 0.001330 0.006352
of
deJong and Note
m,, = 571 .0 MeV m_ = 784.0 MeV
-
4S2
S. Ginucal Finite nuclei calculations 0 -4 -8~
0 .8
1 .0
1 .2
1 .4
1 .6
1 .8
Fig. 2. The same as in fig. I but for the DBH F calculations of de Jong and Malfliet '5 ).
the Hugenholtz-van Hove theorem 17), which plays an important role in many-body theories, is almost exactly fulfilled by this approach. The Hugenholtz-van Hove theorem relates the single-particle energy E at the Fermi surface to the binding energy Et, as (10) E = Et,+ Plp,
where p=P2aEjap is the pressure of the system . In a special case, at saturation (P _= 0), the single-particle energy at the Fermi surface should be equal to the binding energy in a conserving approach. The r.c.a. is an improvement of the DBHF approach which is not conserving . This is expressed as the known violation of the Hugenholtz-van Hove theorem in the DBHF. The violation amounts to several MeV at saturation. We note that the r.m.f.a. approach fulfills the Hugenholtz-van Hove theorem exactly. We have fitted the r.c.a. results for nuclear matter by our r.m.f.a. model with scalar and vector nonlinear couplings. The resulting r.m .f.a. parameters are given in table 2. Hereinafter we denote them as DJM-C. The comparison of the DBHF and r.c.a. results of de Jong and Malfliet for the nuclear matter binding energy with the corresponding r.m.f.a. fits is made in fig. 2. 3. Relativistic mean-field approach for finite nuclei The standard relativistic mean-field approach for finite nuclei with only the scalar-meson self-interaction is described in detail for various geometries, e.g. by refs. 13,1 ") . For the sake of completeness we describe here briefly the mean-field procedure when the vector-meson quartic self-coupling is added. Our starting point is the lagrangian density (1) plus the photon field A,, and electromagnetic interaction.
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453
The full lagrangian reads L(4i, o,, w, A)
UCIIA O,. _ M2 Or.2) iy,, (M - g,,a) - g.,y,.,wA]4i + 1(09 2 IA ff jbrrM ( g4'T(7. ) 3 _I C'r'.(g(rO,)4_ I W 1 4
3
+1
2 1£ 2 M w (Oli£O
4
JA
VW
1V
2 +1 ji)2 4Cw(gwwi£w
44'F,,FI" -
2
where the last term describes the electromagnetic interaction. The electromagnetic field tensor is given by FIL"
= a'LA " - a W-1 .
(12)
The equations of motion are obtained from the lagrangian by the standard technique of field variation . In the following we restrict ourselves to the mean field (i .e. meson fields are replaced by their expectation values) and no-sea (ie. we neglect the effects of antiparticles) approximations . In addition, we assume spherical doubly closed-shell nuclei . Using the usual ansatz for the Dirac single-particle spinors 13) tP,, (r) = qjjj ,,, (r) =
1 [
r
] ~
(13)
-Fij(r)Orj,,,(r)
with T=j :E !2 for /=,i ::F'2 and Ojj j r^ ) being the spinor spherical harmonics '9) we obtain the following coupled pair of radial Dirac equations for the upper (G,.,) and lower (F,,) components, + G (r) = [ M - U(r) - V(r) + E ]F , (r) dr r)
(14)
,
d _K F , (r) = [ M - U (r) + V(r) - -- ] Q, (r) . (dr r) The scalar potential U(r) is given simply as U(r) = g,,a(r) ,
(16)
while the vector potential V(r) has a more complicated structure V(r) = g.ojjr) + 2!(I - 73) eAO(r) Here, a represents the spin and angular quantum numbers and quantum number given by K =
The quantity r3 =
+
( - (j+ 21 ) j+(j+j) 2
fori = 1 + I2 forj = I - j2'
1 for neutrons and r3 = -I for protons.
(17) K
is the Dirac (18)
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454
he meson and photon fields obey the radial Laplace e uations d2
-1 (if7
d2
(P
M2,)[ra(r)] = -g,,rjps (r) - b,,M[g,,(Y(r)j2- c,,,[gc
M«I -
Co
)[ rû)~jr)] = _&~rfPB(r) _ Cjg«~~ WO( r)]3j
(19)
~
(20)
d2
(21) à71 [rA,,(r)] - -erp'BP '(r) ,
where the sources are determined by the corresponding densities in the static nucleus. Namely,
(p) PB
Occ Occ 2 ps( r) = I ~~, (r) 41,, (r) = ~ L j' + I [ G 2,, (r) - F,,, (r) ] , . 47rr2 (V
(22)
Occ (r) = lice :-,j,, + I V [G 2 (r) + F2 (r)] , pp(r) - 'E 0 ' (041" 4iTr2
(23)
0',
OL~; C.
147TI'
2 +'F2 Ga (r) Cf
(24)
where w,- adopt the following singie-particle wave function normalization, 1),
I
,(r) + F2 (Y)j dr = 1
(25)
The paii of radial Ditac equations (14), (15) was solved by the four-order Runge-Kutta method while the meson ejuations (,19)-(21) were solved by the standard matrix-inversion scheme which achieves five-point precision by using only three-point formula for the laplacian 20) . An expression for the total energy of the finite system can be derived from the lagrangian (11) in the standard way 13 ). After some manipulations we finally obtain the following formula, 2
a
I
drlg(,o,(r)p s(r)-lb,,M[g,,or(r)] 3
3
1 ]4 _ - -5cjg,,a(r) &,wo(r)Pdr)
2 Cw [gw&j o (r)
+ -1
]4_
eAO (r)p (BP ) (r)j,
(26)
which differs from the standard r.m .f.a. expressio n by the term coming from the vector-meson self-interaction. This energy should be corrected for the spurious centre-of-mass (c.m .) motion . e have estimated this correction from its value in the nonrelativistic harmonic potential which reads Ec .rn. --= - 4 1
hwo = - 4=' X 41 A - 1/3
(27)
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45 5
Finally, the binding energy per nucleon is given by the expression Eb ~ (E,.m.f.a . + Ec.m . - AM)lA .
(28)
In addition to the single-particle properties of each orbital (s.p. energies and wavefunctions), and the binding energy per nucleon of each nucleus, we have also calculated the charge density distributions. These were obtained by folding the proton densities with the proton charge form factor of Chandra and Sauer 21 ) . The correction for c.m. motion was applied as well. 4. Calculation of ground-state properties of 16 0 and 4OCa nuclei In this section we present the results of our r.m.f.a. calculations of ground-state properties of 160 and 40Ca nuclei using the r.m.fa. parametrizations given in tables 1 and 2. As already mentioned, they were obtained by fitting the results of the DBHF calculations of isoscalar nuclear matter and, therefore, they do not contain any isovector mesons. It is expected that due to the isoscalar nature of the nuclei considered the possible contributions of isovector meson fields are small and may be neglected. It has to be noted that the isovector p-meson may be readily included into the r.m.f.a. model 13 ) and its coupling constant may be determined by reproducing the DBHF results for asymmetric nuclear matter 22) or neutron matter. The results of our calculations are summarized in tables 3 and 4. The single-particle separation energies, binding energies per nucleon and r.m.s. charge radii are given there together with the experimental valueS 23'24) with which they can be compared . It appears that these calculations still do not reproduce the experimental data TABLE 3 of 160 Ground-state properties . Results for the single-particle energies of the occupied proton and neutron states, the binding energy per nucleon and the r.m.s. radius of the charge distribution are presented and compared with experimental data 23.24) a).
Proton levels S,/2 (MeV) p3/2 (MeV) PI/2 (MeV) Neutron levels si/2 (MeV) P3/2 (MeV) PI/2 (MeV)
Eb/A (MeV) Rc.h «M) a
BM-A
BM-13
BM-C
DJM
DJM-C
Exp.
-48.12 -22.49 -14.42
-41 .07 -18.77 -12.38
-36.62 -16.43 -11.02
-48 .96 -22.65 -11.66
-52.52 -24.48 -11.83
-39.25 -18.60 -12.98
-53.42 -27.19 -19.09
-46.11 -23.22 -16.81
-41 .47 -20.73 -15.27
-54.18 -27.28 -16.16
-57.83 -29.20 -16.42
-42.20 -22.86 -16.22
-9 .44 2.404
-7 .89 2.528
-6 .89 2.624
-8 .57 2.450
-8.61 2.410
-7 .98 2.730
The calculations were performed with different r.m .f.a. parameters obtained by fitting the DBHF data 14,15) [see text]. The binding energy and charge radius are corrected for c .m . motion . )
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TABLE 4
m1a. calculations of the ground-state properties of 4OCa (see table 3 for details) .
Proton levels ls,12 (MeV) lp3l2 (MeV) ipt/2 (MeV) Idsl, (MeV) 2s, 12 (MeV) ld3/2 (MeV) Neutron levels Is,,, (MeV) IP3/2 (MeV) 113112 (MeV) 1df,~, (MeV) 2s, / :! (MeV) ld3/2 (MeV) Eb/A (MeV) &h (fin)
BM-A
BM-B
BM_C
DJM
DJM-C
Exp.
-53-31 -36-02 -32.04 -17.72 -8.33 -10.40
-45.77 -30.59 -27.39 -14.54 -6.92 -8.66
-40.99 -24.42 -12.57 -5.95 -7.51
-54.05 -36.38 -30.82 -18.02 -7 .20 -8 .45
-58.59 -39.93 -33.67 -20.10 -6.60 -8.79
-53.6 -34.7 -29.8 -15.7 -11.0 -8.5
-63-29 -45.29 -41.42 -26.30 -16-86 -19.03
-55.28 -39.42 -36.30 -22.71 -15.05 -16.85
-50.19 -35.70 -33.02 -20.45 -13.80 -15.40
-63.86 -45.50 -40.05 -26.48 -15.45 -16.83
-68.55 -49.22 -43.10 -28.73 -14.94 -17.37
-61.5 -42.1 -37 .5 -23.6 -18.2 -15.6
-9.20 3.229
-7.72 3.391
1 . 19
-6.75 3.496
-8.44 3.288
-8.63 3.232
-8 .55 3.478
satisfactorily, although a great progress has been reached in comparison with the nonrelativistic BH F calculations') . The improvements consist in the separation and binding energies, and also in the charge radii. Especially, the parameter set BM-B gives results which agree reasonably well with experimental data for both nuclei considered . In the case of the 4 Ca nucleus our r.m.f.a. calculation s with all parameter sets give the crossing of the 2s1/2and Id3/2 single-particle levels; an effect which is in contradiction with the experimental data. However, we are primarily interested in the comparison of our results with other finite nuclei calculations originating from the DBHF approach . 160 6 6ther, Machleidt and Brockmann ) have calculated the properties of the nucleus employing the DBHF approach of ref. 14) in an "effective-density" approxi. 6)] can be compared with ours using the mation . Their results [see table 11 in ref corresponding r.m .f.a. parameters (see table 3, columns BM-A, B, Q. While, in general, the charge radii agree rather well, our r.m.f.a. calculations give significantly more binding (approximately by 2 MeV/nucleon) than the corresponding calculations of ref. 6) . This difference may be due to the "effective-density" approximation used in ref. 6) which combines the nonrelativistic BHF calculations of finite nuclei and the DBHF calculations of nuclear matter. The present r.m .f.a. calculations, on the other side, apply directly the density-dependent effects produced by the DBHF approaches. ur calculatians for 40Ca may be compared only with the results of Marcos et aL'), which use the Dirac-Hartree- Fock approach with the density-dependent coupling constants 25 ) . However, these authors used older DBHF data of ter Haar
S. Gmucal Finite nuclei calculations
45 7
and Malfliet 4) in contrast with our calculations --mploying the improved DBHF results of the Groningen group"). We performed, therefore, also the calculation of the 4'Ca nucleus using the r.m.f.a. parameter set ") which fits the DBHF data of ter Haar and Malfliet 4). In this case, the r.m.f.a. values for the binding energy per nucleon (-7 .74 MeV) and the r.m.s. charge radius (3.34 fm) of 40Ca agree well with the values reported by Marcos et A% -7.9 MeV and 3 .33 fin, respectively. Celenza et A ') have calculated the structure of the "0 and "' Ca nuclei in the DHF approach using their effective interaction ") constructed by adding the pseudomesons to the original o.b.e. interaction. However, since they used a different o.b.e. interaction - HM2 of Holinde and Machleidt 27) - their results cannot be directly compared with ours . Generally speaking, the calculations of Celenza et A give almost correct r.m.s. charge radii, but only with one half of the binding energy. This reflects the saturation properties of the DBHF calculations with the HM2 interaction 2,26) . It is interesting, however, that the calculations of Celenza et aL, which use a different interaction and a different technique, also exhibit the crossing of the 2sl/2 and Id3/2 levels in the 4oCa nucleus; an effect which already appeared in our r.m .f.a. calculations . We were unable to trace to the origin ofthis inconsistency . 5. Summary and conclusions
We have calculated the ground-state properties of the 16 0 and 40Ca nuclei using various relativistic effective interactions . These were constructed in the frame of the relativistic mean-field approach with the scalar and vector nonlinear selfcouplings. The parameters of the effective interactions were determined by reproducing the results of the recent Dirac- Brueckner- Hartree- Fock calculations of nuclear matter. Once the interactions were constructed, there are no additional free parameters in our calculations of finite nuclei . We have found that the effective interactions thus obtained provide reasonable of results for single-particle energies and binding energies '60and 40Ca. In particular, the parameter set BM-B yields a good description of these properties . The calculated charge radii are, however, still smaller than required . This reflects the present state-of-art in the DBHF calculations ofnuclear matter ; the Coester line only touches the high-density side of the empirically acceptable area of saturation . A further improvement is needed which would shift the saturation Fermi momentum to a little lower value (by approximately 0.05-0.1 fm-') while keeping the binding energy almost unchanged. The comparison with other approaches indicates that our r.m.f.a. calculations agree well with the results obtained by the DHF approach with density-dependent coupling constants . These comparisons, however, should not be overestimated, as the finite nuclei calculations originating from the DBHF are still scarce and incomplete. Simply, more calculations are needed to draw meaningful conclusions on various relativistic effective interaction approaches.
4S8
S. Gmucal Finite nuclei calculations
his work was supported in part by the Grant Agency of the Slovak Academy of Sciences under Grant No. GA SAV-517/1992. eferences 1) K.A. Brueckner, J.L. Gammel and H. Weitzner, Phys . Rev. 110 (1958) 431 ; B.D. Day, Rev. Mod . Phys. 39 (1967) 719; K.T.R. Davies, M. Baranger, R.M. Tarbutton and T.T.S. Kuo, Phys. Rev. 177 (1969) 1519; R.L. Becker and M.R. Patterson, Nucl. Phys. A178 (1971) 88 2) M.R. Anastasio, L.S. Celenza, W.S. Pong and C.M. Shakin, Phys. Reports 100 (1983) 327 ; L.S. Celenza and C.M . Shakin, Relativistic nuclear physics : theories of structure and scattering, in : World Scientific Lecture Notes its Physics, vol. 2 (World Scientific, Singapore, 1986) 3) R. Brockmann and R. Machleidt, Phys. Lett. B149 (1984) 283 4) B. ter Haar and R. Malfliet, Phys. Reports 149 (1987) 207 5) B.D. Day, Rev. Mod . Phys. 50 (1978) 495 6) H. Wither, R. Machleidt and R. Brockmann, Phys. Rev. C42 (1990) 1981 ; Phys. Lett. 8202 (1988) 483 7) M. Centelles, X Vifias, M. Barranco, N. Ohtsuka, A. Faessler, Dao Tien Khoa and H. Mifther, J. of Phys. G17 (1991) L193 8) S. Marcos, M. L,6pez-Quelle and Nguyen Van Giai, Phys . Lett. B257 (1991) 5 9) L.S. Celenza, S.-F. Gao and C.M. Shakin, Phys. Rev. C41 (1990) 1768 10) S. Gmuca, J. of Phys. G17 (1991) 1115 11) S. Gmuca, Z. Phys. A, in press 12) A.R. Bodmer, Nucl. Phys. A526 (1991) 703 13) B.D. Serot and J.D. Walecka, Adv . Nucl. Phys. 16 (1986) 1 14) R. Brockmann and R. Machleidt, Phys. Rev. C42 (1990) 1965 15) F. de Jong and R. Malfliet, Phys. Rev. C44 (1991) 998 16) G. Baym and L.P. Kadanoff, Phys. Rev. 124 (1961) 287 17) N.M. Hugenholtz and L. van Hove, Physica 23 (1958) 525 18) Y.K. Gambhir, P. Ring and A. Thimet. Ann. of Phys. 199 (1990) 132 19) A.R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press, Princeton, 1957) 20) F.B. Hildebrand, Introduction to numerical analysis (McGraw-Hill, New York, 1956) 21) H. Chandra and G . Sauer, Phys. Rev. C13 (1976) 245 22) B. ter Haar and R. Malfliet, Phys. Rev. Lett. 59 (1987) 1652 23) S.L. Belostotsky, Yu. V. Dotsenko, N.P. Kuropatkin, O.V. Mikulenko, V.N. Nikulin, O.E. Prokofiev, Yu. A. Scheglov, V.E. Starodubsky, A. Yu. Tsaregorodtsev, A.A. Vorobyov and M.B. Zhalov, Proc. Int. SYmp. on modem developments in nuclear physics, Novosibirsk, 1987, ed. O.P. Sushkov (World Scientific, Singapore, 1988) p. 191 24) H. de Vries, C.W. de Jager and C. de Vries, At. Data Nucl. Data Tables 36 (1987) 495 25) S. Marcos, R. Niembro, M. L6pez-Quelle, Nguyen Van Giai and R. Malfliet, Phys. Rev. C39 (1989) 1134 26) H.B. Ai, L.S. Celenza, A. Harindranath and C.M. Shakin, Phys. Rev. C35 (1987) 2299; H.B. Ai, L.S. Celenza, S.-F. Gao and C.M. Shakin, Phys. Rev. C39 (1989) 236 27) K. Holinde and R. Machleidt, Nucl. Phys. A256 (1976) 479
Nuclear Physics A547 (1992) 447-458 North-Holland
Finite-nuclei calculations based on relativistic mean-field effective interactions Stefan Gmuca
Institute of Physics, Slovak Academy of Sciences CS-842 28 Bratislava, Czechoslovakia Received 26 March 1992 Abstract: The Dirac-Brueckner- Hartree- Fock calculations of nuclear matter are parametrized by the relativistic orw mean-field theory with scalar and vector nonlinear self-interactions . The effective interactions thus obtained are used in the relativistic mean-field studies of the structure of 16 0 and 40Ca nuclei without the introduction of additional free parameters. The calculated binding energies, single-particle spectra and charge radii agree quite well (although not satisfactorily) with experimental data and present an improvement over the nonrelativistic Brueckner-Hartree-Fock approximations .
1. Introduction One of the most ambitious programs in nuclear physics is to understand the structure of finite nuclei starting from the bare nucleon-nucleon (NN) interaction . The nonrelativistic Brueckner-Hartree- Fock (BHF) approach [see e.g. ref. ') and references therein] with various basic forces has been widely and repeatedly applied in the attempts to explain some ofthe nuclear ground-state properties in a parameterfree way. These calculations, ho-Alever, have achieved only a limited success as they were unable to produce reasonable binding energies, single-particle spectra, r.m .s. charge radii and charge distributions for the nuclei considered. Recently these attempts have revived as the relativistic extension of the BHF 2-4 approach [so-called Dirac- Brueckner- Hartree- Fock (DBHF) approach )] has been successfully applied to the nuclear-matter problem. The key point is the use of the relativistic dynamics, governed by the Dirac equation with strong scalar and vector fields, for the single-particle nucleon motion. The explicit treatment of the lower components of the Dirac nucleon wave functions gives rise to strongly density-dependent relativistic effects that move the nuclear-matter saturation point 5 away from the "Coester band" much closer to the empirically acceptable values ). But, unlike in the nonrelativistic BHF case, the DBHF approach, due to its complexity, is much more elaborate when applied to finite systems. In fact, there are still no fully self-consistent Dirac-Brueckner calculations for finite nuclei. The
Correspondence to: Dr. S. Gmuca, Institute of Physics, Slovak Academy of Sciences, Dubravska Cesta 9, CS-842 28 Bratisiava, Czechoslovakia . 0375-9474/92/$05 .00 @ 1992 - Elsevier Science Publishers B.V. All rights reserved
S. Gmucal Finite nuclei calculations
448
use of some approximate methods seems to be inevitable . The existing calculations either employ the nuclear-matter results with some kind of local density approximation (LDA) '-"), or rely on various effective interact: - approaches which reproduce " the self-energy terms or G-matrix elements of the DBHF theory 9 o). In the previous paper") we have developed a simple relativistic mean-field parametrization of the effective interactions in nuclear matter which reproduce (rather accurately) the Wnding energy and the scalar and vector (time-like component 3,4) nly) self-energy terms of the recent DBHF calculations of nuclear matter . The model is based on the relativistic aw mean-field theory with scalar and vector self-interactions . The inclusion of the isoscalar vector-meson quartic self-interaction into the effective mean-field lagr.--ngian is essential for obtaining a proper density dependence of the vector potential ") . In the present paper we will use this effective interaction for the calculation of 40 the ground-state properties of the doubly closed-shell nuclei `0 and Ca in the s,imple mean-field approach. In such a way we can bypass the very complicated fully self-consistent DBHF calculations in finite systems . We are interested whether this approach is equally successful in finite nuclei calculations as it is in nuclear matter ones. The arrangment of the paper is as follows . In sect. 2 we will give a short overview of our relativistic mean-field approach (r.m .f.a .) together with parametrizations of recent DBHF results in nuclear matter. Sect. 3 is devoted to the generalization of 40 the model for finite systems. The ground-state properties of the `0 and Ca nuclei are calculated in sect. 4. The comparison with other approaches is presented there as well. Sect. 5 contains a summary and our conclusions. 2.
ean-field approach in nuclear matter
elativistic
2.1. NONLINEAR R.M.F.A. MODEL
We start with the nonlinear r.m .f.a. model which, in addition to the scalar cubic and quartic self-interactions, includes also the vector-meson quartic self-coupling . This extension of the standard r.m.f.a. lagrangian introduces the density dependence also into the vector-meson sector and, in such a way, simulates the density dependence produced in the DBHF approaches. The full discussion of the model is given in refs . 11 J2) . The lagrangian density of the nonlinear aw theory reads L(j p w) = &iya'-(M -g,u) -pSwlee + 1 (hao"a - M 2 a 2) _ l bCrM ( Ig.)3 ff
2
tLV+ I 40ig V&j
1 j
n; 2. &JIJ 01
3
gtr
IA +jCw
4
I
( 0.)4 ra ga
( g2 &J ~L&J IA )2 to
where the symbols have their usual meaning 13). The first line of eq . (1) describes the nucleon (41) part of the lagrangian density including the nucleon-meson interaction terms. The second line is the standard
S. Gmucal Finite nuclei calculations
449
scalar o,-meson part with cubic and quartic self-interactions. The strengths of these self-couplings are given by the dimensionless constants b,, and c,. The third line contains the vector w-meson part of L including the quartic vector-meson self(g.(d0 )4 interaction term jC. with the dimensionless self-coupling constant c,,,. 4 Restricting ourselves to the mean-field approximation, the scalar and vector fields are replaced by their ground-state expectation values. Consequently the classical equations of meson fields and modified Dirac equation of the nucleon field are easily solved for nuclear matter. They read 11 ) g2 = g,.,Cr
(T 2
M
[pS-b(TM(g(,Cr)2 _ CfjgffG.)3]
(2)
(Y ,,2
9-Ù)O = M"2' [ - MI .
[PB _
CLO
(
gw
WO )3] ~
(3)
+,8M *10 = (E - &,wj 41 .
(4)
Here we utilize the fact that in a uniform system only the time-like component of the vector field survives . The source terms for the scalar and vector fields are the scalar density ps and the baryon (vector) density p B , respectively . Using the standard positive-energy solutions of the Dirac equation (4), they are given by 7 k, m* Y dk 2 + M*2)1/2 PS (5) (27r) 11 (k 9
PB -
6
y . k3 ff 2
F ,)
where we assume that nuclear matter consists of filled nucleon levels up to k F , and y is the spin-isospin degeneracy factor (-/ = 4 for nuclear matter) . The effective nucleon mass M* in eqs. (4), (5) is given by the relation M* = M -
(7)
g"O'. .
The scalar density p s is thus an implicit function of the scalar field. Therefore, the scalar field has to be determined self-consistently at each density by solving eq. (2) and eq. (5) . The energy density of isoscalar nuclear matter is given in the parametric form as 'y k, M2 E,.,,, . dk [k 2 + M*2]1/2 +_ -( g,,Cr )2 +jb(TM(&~0.)3 +1 ( 0.)4 La . -
2
(2 ,w
2 g,
2
3
4 cfr gfr
+
,10)2 1 ( g (1)0)4 . gw 4Cw w 2 gw
The binding energy per nucleon of nuclear matter is then simply Et, = E, .m .f.a ./PB - M-
(9)
450
S. Gmucal Finite nuclei calculations
TERMINATION OF R.M.F.A. PARAMETERS
he lagrangian (1) given above has to be considered as an effective lagrangian and the meson masses and their couplings are to be understood as effective parameters of the theory. In the previous paper ") we have determined them by reproducing the binding energy and the scalar and time-like component of vector self-energy terms of DBHF calculations of isoscalar nu-lear matter -3,4) . A very good fit was obtained for a broad range of densities (k F < 1 .7 fin-') . The binding energy of nuclear matter was reproduced within 0.3 MeV, and the scalar and vector potentials within 5 MeV in the fitted window. In the present paper we extend these previous results and give more complete r.m.f.a. parametrizations for new recently published DBHF calculations of nuclear matter 14,15) . rockmann and Machleidt 14) presented the DBHF nuclear-matter calculations for three different one-boson-exchange (o.b.e.) potentials (denoted as potential A, and C, respectively) which describe the NN data and are appropriate for the nuclear-structure calculations . These o.b.e. potentials differ mainly by the strengt.h of the tensor force [see ref. 14) for details]. As a result, the corresponding DBHF calculations predict different locations of the nuclear-matter saturation point. They form a new "Coester band" that touches the empirically acceptable area of saturation . To reproduce the DBH F results we used the procedure of ref. "), i.e., we performed a simultaneous fitting of binding energy, scalar and time-like vector components of the self-energy terms. Resulting sets of r.m.f.a. parameters are listed in table 1 . ereafter the r.m .f.a. parameters denoted as BM-A reproduce the DBHF results obtained with the o.b.e. potential A. Analogically, BM-13 denotes the r.m.f.a. parameters obtained by fitting the DBHF calculations with the o.b.e. potential B, and -C denotes those made with the o.b.e. potential C . The qualitty ofparametrizations may be seen from fig. I where the r.m.f.a. fits are compared with the DBHF data for the nuclear-matter binding energy. We see that the DBHF results for all three potentials are reproduced rather accurately (within -0.3 MeV) by our r.m.f.a. model TABLE I Parameters of the r.m .f.a. fits to the DBHF data of Brockmann and Machleidt nuclear matter Parameter 2
gger 2
k, C.C.
BM-A 154.218 226 .887 0.001347 0.007947 0.023870
BM-13 154 .050 235 .109 0.000673 0.009786 0.023605
BM-C 152 .786 238 .691 0 .000201 0.010949 0 .022954
14)
Note in, = 550.0 MeV m.., = 782.6 MeV
in
S. Gmucal Finite nuclei calculations
0
451
BM-C
-4 -8
M
LLJ
-12 -16 -20 1 0 .8
1 .0
1 .2 kF
1 .4
1 .6
1 .8
(fM_1)
Fig . 1 . The binding energy per particle E b in isoscalar nuclear matter as a function ofthe Fermi momentum kF for various approaches . The symbols represent the results of the DBHF calculations of Brockmann and Machleidt 14 ) . The curves show the corresponding r.m.f.a. fits. The cross indicates the empirically acceptable area of saturation.
with the scalar and vector self-couplings. Similar conclusions concern also the scalar and vector self-energy terms . In these cases the DBHF results are reproduced within -5-6 MeV. In ref. ") de Jong and Malfliet presented an improved DBHF calculation of nuclear-matter properties as compared to the previous one of the Groningen group The improvement consists ofa more careful numerical treatment ofsome singularities encountered in the DBHF scheme . We have refitted these improved DBHF results by the procedure described above [see also ref. ")]. The resulting set of r.m.f.a. parameters (hereafter denoted as DJM) is listed in table 2. In addition, de Jong and Malfliet ") gave also the results of the calculation of nuclear matter using a so-called relativistic conserving approximation (r.c.a.) which imposes additional constraints on the Dirac-Brueckner scheme . In particular, 4)
16
)
TABLE 2
Parameters Parameter 2'
9 2 . 9 b,, CO-
C.
of
the r.m.fa. fits to the DBHF data Malfliet ") in nuclear matter DJM 134.228 179 .077
0.001469 0.000338 0.005835
DJM-C 137 .485 192 .587 0.000785 0.001330 0.006352
of
deJong and Note
m,, = 571 .0 MeV m_ = 784.0 MeV
-
4S2
S. Ginucal Finite nuclei calculations 0 -4 -8~
0 .8
1 .0
1 .2
1 .4
1 .6
1 .8
Fig. 2. The same as in fig. I but for the DBH F calculations of de Jong and Malfliet '5 ).
the Hugenholtz-van Hove theorem 17), which plays an important role in many-body theories, is almost exactly fulfilled by this approach. The Hugenholtz-van Hove theorem relates the single-particle energy E at the Fermi surface to the binding energy Et, as (10) E = Et,+ Plp,
where p=P2aEjap is the pressure of the system . In a special case, at saturation (P _= 0), the single-particle energy at the Fermi surface should be equal to the binding energy in a conserving approach. The r.c.a. is an improvement of the DBHF approach which is not conserving . This is expressed as the known violation of the Hugenholtz-van Hove theorem in the DBHF. The violation amounts to several MeV at saturation. We note that the r.m.f.a. approach fulfills the Hugenholtz-van Hove theorem exactly. We have fitted the r.c.a. results for nuclear matter by our r.m.f.a. model with scalar and vector nonlinear couplings. The resulting r.m .f.a. parameters are given in table 2. Hereinafter we denote them as DJM-C. The comparison of the DBHF and r.c.a. results of de Jong and Malfliet for the nuclear matter binding energy with the corresponding r.m.f.a. fits is made in fig. 2. 3. Relativistic mean-field approach for finite nuclei The standard relativistic mean-field approach for finite nuclei with only the scalar-meson self-interaction is described in detail for various geometries, e.g. by refs. 13,1 ") . For the sake of completeness we describe here briefly the mean-field procedure when the vector-meson quartic self-coupling is added. Our starting point is the lagrangian density (1) plus the photon field A,, and electromagnetic interaction.
S. Gmucal Finite nuclei calculations
453
The full lagrangian reads L(4i, o,, w, A)
UCIIA O,. _ M2 Or.2) iy,, (M - g,,a) - g.,y,.,wA]4i + 1(09 2 IA ff jbrrM ( g4'T(7. ) 3 _I C'r'.(g(rO,)4_ I W 1 4
3
+1
2 1£ 2 M w (Oli£O
4
JA
VW
1V
2 +1 ji)2 4Cw(gwwi£w
44'F,,FI" -
2
where the last term describes the electromagnetic interaction. The electromagnetic field tensor is given by FIL"
= a'LA " - a W-1 .
(12)
The equations of motion are obtained from the lagrangian by the standard technique of field variation . In the following we restrict ourselves to the mean field (i .e. meson fields are replaced by their expectation values) and no-sea (ie. we neglect the effects of antiparticles) approximations . In addition, we assume spherical doubly closed-shell nuclei . Using the usual ansatz for the Dirac single-particle spinors 13) tP,, (r) = qjjj ,,, (r) =
1 [
r
] ~
(13)
-Fij(r)Orj,,,(r)
with T=j :E !2 for /=,i ::F'2 and Ojj j r^ ) being the spinor spherical harmonics '9) we obtain the following coupled pair of radial Dirac equations for the upper (G,.,) and lower (F,,) components, + G (r) = [ M - U(r) - V(r) + E ]F , (r) dr r)
(14)
,
d _K F , (r) = [ M - U (r) + V(r) - -- ] Q, (r) . (dr r) The scalar potential U(r) is given simply as U(r) = g,,a(r) ,
(16)
while the vector potential V(r) has a more complicated structure V(r) = g.ojjr) + 2!(I - 73) eAO(r) Here, a represents the spin and angular quantum numbers and quantum number given by K =
The quantity r3 =
+
( - (j+ 21 ) j+(j+j) 2
fori = 1 + I2 forj = I - j2'
1 for neutrons and r3 = -I for protons.
(17) K
is the Dirac (18)
S. Gmucal Finite nuclei calculations
454
he meson and photon fields obey the radial Laplace e uations d2
-1 (if7
d2
(P
M2,)[ra(r)] = -g,,rjps (r) - b,,M[g,,(Y(r)j2- c,,,[gc
M«I -
Co
)[ rû)~jr)] = _&~rfPB(r) _ Cjg«~~ WO( r)]3j
(19)
~
(20)
d2
(21) à71 [rA,,(r)] - -erp'BP '(r) ,
where the sources are determined by the corresponding densities in the static nucleus. Namely,
(p) PB
Occ Occ 2 ps( r) = I ~~, (r) 41,, (r) = ~ L j' + I [ G 2,, (r) - F,,, (r) ] , . 47rr2 (V
(22)
Occ (r) = lice :-,j,, + I V [G 2 (r) + F2 (r)] , pp(r) - 'E 0 ' (041" 4iTr2
(23)
0',
OL~; C.
147TI'
2 +'F2 Ga (r) Cf
(24)
where w,- adopt the following singie-particle wave function normalization, 1),
I
,(r) + F2 (Y)j dr = 1
(25)
The paii of radial Ditac equations (14), (15) was solved by the four-order Runge-Kutta method while the meson ejuations (,19)-(21) were solved by the standard matrix-inversion scheme which achieves five-point precision by using only three-point formula for the laplacian 20) . An expression for the total energy of the finite system can be derived from the lagrangian (11) in the standard way 13 ). After some manipulations we finally obtain the following formula, 2
a
I
drlg(,o,(r)p s(r)-lb,,M[g,,or(r)] 3
3
1 ]4 _ - -5cjg,,a(r) &,wo(r)Pdr)
2 Cw [gw&j o (r)
+ -1
]4_
eAO (r)p (BP ) (r)j,
(26)
which differs from the standard r.m .f.a. expressio n by the term coming from the vector-meson self-interaction. This energy should be corrected for the spurious centre-of-mass (c.m .) motion . e have estimated this correction from its value in the nonrelativistic harmonic potential which reads Ec .rn. --= - 4 1
hwo = - 4=' X 41 A - 1/3
(27)
S. Gmucal Finite nuclei calculations
45 5
Finally, the binding energy per nucleon is given by the expression Eb ~ (E,.m.f.a . + Ec.m . - AM)lA .
(28)
In addition to the single-particle properties of each orbital (s.p. energies and wavefunctions), and the binding energy per nucleon of each nucleus, we have also calculated the charge density distributions. These were obtained by folding the proton densities with the proton charge form factor of Chandra and Sauer 21 ) . The correction for c.m. motion was applied as well. 4. Calculation of ground-state properties of 16 0 and 4OCa nuclei In this section we present the results of our r.m.f.a. calculations of ground-state properties of 160 and 40Ca nuclei using the r.m.fa. parametrizations given in tables 1 and 2. As already mentioned, they were obtained by fitting the results of the DBHF calculations of isoscalar nuclear matter and, therefore, they do not contain any isovector mesons. It is expected that due to the isoscalar nature of the nuclei considered the possible contributions of isovector meson fields are small and may be neglected. It has to be noted that the isovector p-meson may be readily included into the r.m.f.a. model 13 ) and its coupling constant may be determined by reproducing the DBHF results for asymmetric nuclear matter 22) or neutron matter. The results of our calculations are summarized in tables 3 and 4. The single-particle separation energies, binding energies per nucleon and r.m.s. charge radii are given there together with the experimental valueS 23'24) with which they can be compared . It appears that these calculations still do not reproduce the experimental data TABLE 3 of 160 Ground-state properties . Results for the single-particle energies of the occupied proton and neutron states, the binding energy per nucleon and the r.m.s. radius of the charge distribution are presented and compared with experimental data 23.24) a).
Proton levels S,/2 (MeV) p3/2 (MeV) PI/2 (MeV) Neutron levels si/2 (MeV) P3/2 (MeV) PI/2 (MeV)
Eb/A (MeV) Rc.h «M) a
BM-A
BM-13
BM-C
DJM
DJM-C
Exp.
-48.12 -22.49 -14.42
-41 .07 -18.77 -12.38
-36.62 -16.43 -11.02
-48 .96 -22.65 -11.66
-52.52 -24.48 -11.83
-39.25 -18.60 -12.98
-53.42 -27.19 -19.09
-46.11 -23.22 -16.81
-41 .47 -20.73 -15.27
-54.18 -27.28 -16.16
-57.83 -29.20 -16.42
-42.20 -22.86 -16.22
-9 .44 2.404
-7 .89 2.528
-6 .89 2.624
-8 .57 2.450
-8.61 2.410
-7 .98 2.730
The calculations were performed with different r.m .f.a. parameters obtained by fitting the DBHF data 14,15) [see text]. The binding energy and charge radius are corrected for c .m . motion . )
S. Gmucal Finite nuclei calculations
4S6
TABLE 4
m1a. calculations of the ground-state properties of 4OCa (see table 3 for details) .
Proton levels ls,12 (MeV) lp3l2 (MeV) ipt/2 (MeV) Idsl, (MeV) 2s, 12 (MeV) ld3/2 (MeV) Neutron levels Is,,, (MeV) IP3/2 (MeV) 113112 (MeV) 1df,~, (MeV) 2s, / :! (MeV) ld3/2 (MeV) Eb/A (MeV) &h (fin)
BM-A
BM-B
BM_C
DJM
DJM-C
Exp.
-53-31 -36-02 -32.04 -17.72 -8.33 -10.40
-45.77 -30.59 -27.39 -14.54 -6.92 -8.66
-40.99 -24.42 -12.57 -5.95 -7.51
-54.05 -36.38 -30.82 -18.02 -7 .20 -8 .45
-58.59 -39.93 -33.67 -20.10 -6.60 -8.79
-53.6 -34.7 -29.8 -15.7 -11.0 -8.5
-63-29 -45.29 -41.42 -26.30 -16-86 -19.03
-55.28 -39.42 -36.30 -22.71 -15.05 -16.85
-50.19 -35.70 -33.02 -20.45 -13.80 -15.40
-63.86 -45.50 -40.05 -26.48 -15.45 -16.83
-68.55 -49.22 -43.10 -28.73 -14.94 -17.37
-61.5 -42.1 -37 .5 -23.6 -18.2 -15.6
-9.20 3.229
-7.72 3.391
1 . 19
-6.75 3.496
-8.44 3.288
-8.63 3.232
-8 .55 3.478
satisfactorily, although a great progress has been reached in comparison with the nonrelativistic BH F calculations') . The improvements consist in the separation and binding energies, and also in the charge radii. Especially, the parameter set BM-B gives results which agree reasonably well with experimental data for both nuclei considered . In the case of the 4 Ca nucleus our r.m.f.a. calculation s with all parameter sets give the crossing of the 2s1/2and Id3/2 single-particle levels; an effect which is in contradiction with the experimental data. However, we are primarily interested in the comparison of our results with other finite nuclei calculations originating from the DBHF approach . 160 6 6ther, Machleidt and Brockmann ) have calculated the properties of the nucleus employing the DBHF approach of ref. 14) in an "effective-density" approxi. 6)] can be compared with ours using the mation . Their results [see table 11 in ref corresponding r.m .f.a. parameters (see table 3, columns BM-A, B, Q. While, in general, the charge radii agree rather well, our r.m.f.a. calculations give significantly more binding (approximately by 2 MeV/nucleon) than the corresponding calculations of ref. 6) . This difference may be due to the "effective-density" approximation used in ref. 6) which combines the nonrelativistic BHF calculations of finite nuclei and the DBHF calculations of nuclear matter. The present r.m .f.a. calculations, on the other side, apply directly the density-dependent effects produced by the DBHF approaches. ur calculatians for 40Ca may be compared only with the results of Marcos et aL'), which use the Dirac-Hartree- Fock approach with the density-dependent coupling constants 25 ) . However, these authors used older DBHF data of ter Haar
S. Gmucal Finite nuclei calculations
45 7
and Malfliet 4) in contrast with our calculations --mploying the improved DBHF results of the Groningen group"). We performed, therefore, also the calculation of the 4'Ca nucleus using the r.m.f.a. parameter set ") which fits the DBHF data of ter Haar and Malfliet 4). In this case, the r.m.f.a. values for the binding energy per nucleon (-7 .74 MeV) and the r.m.s. charge radius (3.34 fm) of 40Ca agree well with the values reported by Marcos et A% -7.9 MeV and 3 .33 fin, respectively. Celenza et A ') have calculated the structure of the "0 and "' Ca nuclei in the DHF approach using their effective interaction ") constructed by adding the pseudomesons to the original o.b.e. interaction. However, since they used a different o.b.e. interaction - HM2 of Holinde and Machleidt 27) - their results cannot be directly compared with ours . Generally speaking, the calculations of Celenza et A give almost correct r.m.s. charge radii, but only with one half of the binding energy. This reflects the saturation properties of the DBHF calculations with the HM2 interaction 2,26) . It is interesting, however, that the calculations of Celenza et aL, which use a different interaction and a different technique, also exhibit the crossing of the 2sl/2 and Id3/2 levels in the 4oCa nucleus; an effect which already appeared in our r.m .f.a. calculations . We were unable to trace to the origin ofthis inconsistency . 5. Summary and conclusions
We have calculated the ground-state properties of the 16 0 and 40Ca nuclei using various relativistic effective interactions . These were constructed in the frame of the relativistic mean-field approach with the scalar and vector nonlinear selfcouplings. The parameters of the effective interactions were determined by reproducing the results of the recent Dirac- Brueckner- Hartree- Fock calculations of nuclear matter. Once the interactions were constructed, there are no additional free parameters in our calculations of finite nuclei . We have found that the effective interactions thus obtained provide reasonable of results for single-particle energies and binding energies '60and 40Ca. In particular, the parameter set BM-B yields a good description of these properties . The calculated charge radii are, however, still smaller than required . This reflects the present state-of-art in the DBHF calculations ofnuclear matter ; the Coester line only touches the high-density side of the empirically acceptable area of saturation . A further improvement is needed which would shift the saturation Fermi momentum to a little lower value (by approximately 0.05-0.1 fm-') while keeping the binding energy almost unchanged. The comparison with other approaches indicates that our r.m.f.a. calculations agree well with the results obtained by the DHF approach with density-dependent coupling constants . These comparisons, however, should not be overestimated, as the finite nuclei calculations originating from the DBHF are still scarce and incomplete. Simply, more calculations are needed to draw meaningful conclusions on various relativistic effective interaction approaches.
4S8
S. Gmucal Finite nuclei calculations
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