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GROUND-STATE PROPERTIES OF EVEN–EVEN NUCLEI IN THE RELATIVISTIC MEAN-FIELD THEORY
Atomic Data and Nuclear Data Tables 71, 1– 40 (1999) Article ID adnd.1998.0795, available online at http://www.idealibrary.com on
GROUND-STATE PROPERTIES OF EVEN–EVEN NUCLEI IN THE RELATIVISTIC MEAN-FIELD THEORY G. A. LALAZISSIS and S. RAMAN Physics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 and P. RING Fakulta¨t fu¨r Physik, Technische Universita¨t, Mu¨nchen, D-85748 Garching, Germany The ground-state properties of 1315 even– even nuclei with 10 # Z # 98 have been calculated in the framework of the relativistic mean-field (RMF) theory. The Lagrangian parametrization NL3 was used in the calculations. Pairing correlations are accounted within the Bardeen–Cooper–Schrieffer approach. The calculated values for the total binding energy, rms proton radius, rms neutron radius, rms charge radius, neutron quadrupole moment, proton quadrupole moment, charge (proton) hexadecapole moment, quadrupole deformation parameter, and hexadecapole deformation parameter are given in Table I. The RMF predictions of some rare-earth nuclei have been compared with the available experimental information. © 1999 Academic Press
0092-640X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
Nuclear Ground-State Properties
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 4
EXPLANATION OF TABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
TABLE I. RMF Predictions of Ground-State Properties . . . . . . . . . . . . . . . . . . .
11
INTRODUCTION The RMF theory has recently enjoyed considerable success in describing various facets of nuclear-structure properties. With a very limited number of parameters, this theory is able to give a quantitative description of the ground-state properties of spherical and deformed nuclei at and away from the line of b stability [10, 11]. It has been shown that the anomalous kink in the isotope shifts of Pb nuclei and the anomalous isotopic shifts in the Sr and Kr chains can be explained by the RMF microscopic calculations [12, 13]. Such anomalous behavior is a generic feature of almost all isotopic chains in the rare-earth region [14]. Good agreement with experimental data has been found recently also for collective excitations such as giant resonances [15, 16] and for twin bands in rotating superdeformed nuclei [17]. Similarly, good descriptions of the superdeformed rotational bands in the A 5 140–150 region and in the rare-earth region have been provided by the cranked RMF calculations [18 –20]. Very recently it has also been shown that constrained RMF calculations reproduce the excitation energies of superdeformed minima relative to the ground state in 194Hg and 194Pb [21] very well. In the RMF theory, the saturation and the density dependence of the nuclear interaction are obtained by a balance between a large, attractive, scalar s-meson field and a large, repulsive, vector v-meson field. The asymmetry component is provided by the isovector rW meson. The nuclear interaction is, hence, generated by the exchange of various mesons between nucleons in the framework of the mean field. The spin– orbit interaction arises naturally in the RMF theory as a result of the Dirac structure of nucleons.
The development of theoretical models that are able to successfully reproduce and predict the ground-state properties and other properties of finite nuclei throughout the periodic table is of great importance in nuclear-structure studies. A good description of the properties of known nuclei also gives us more confidence in extrapolating to the yet unexplored areas of the nuclear chart. Of course, these extrapolations are based upon the respective ansatz of a model which is usually obtained by fits to the sets of available nuclear properties near the b-stability line. There exist several such theoretical models in the literature. Chief among these are the various mass models [1]. The FiniteRange Droplet Model (FRDM) [2, 3] and the Extended Thomas–Fermi with Strutinsky Integral Model (ETFSI) [4] are two of the currently best-known mass models. In both cases, attempts have been made to obtain the best possible description of nuclear masses and deformation properties. Fully self-consistent calculations within the meanfield theory with suitable relativistic or nonrelativistic effective interactions are a preferred alternative approach. However, the computation time required for such calculations, especially for the number of nuclei approaching the scopes of the above-mentioned mass models, is huge. Therefore, one has to limit these calculations to a smaller number and, at the same time, exploit those symmetries of the mean field that can reduce the computation time. Recently, deformed Hartree–Fock (HF) 1 Bardeen–Cooper– Schrieffer (BCS) calculations with the Skyrme SIII force [5] have been reported for the ground-state properties of 1029 even– even nuclei [6]. This work [6] is the first such extensive calculation within the nonrelativistic mean-field approach. The purpose of the present work is to provide the first systematic study of the ground-state properties of even– even nuclei over a very wide range of isospin within the relativistic mean-field (RMF) theory [7–9].
Relativistic Mean-Field Theory In the RMF theory, the nucleons are described as relativistic particles moving independently in average po2
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
Expressed in terms of the Pauli spin matrices, this Dirac equation reads
tentials determined in a self-consistent way by the exchange of mesons. The relativistic single-particle equation is the Dirac equation. In contrast to the nonrelativistic Schro¨dinger equation, which contains one average potential represented usually by a Saxon–Woods shape and an independently fitted spin– orbit potential represented as the derivative of a Saxon–Woods shape, the Lorentz structure of the Dirac equation allows in principle several types of fields:
S m 1sSp1 V
sp 2m 2 S 1 V
DS gf D 5 e S gf D . i
i
(2)
i
The single-particle wave functions c i are four-dimensional spinors, which describe stationary states of the nucleons with index i and single-particle energy e i . By summing over the occupied orbitals, we can use these wave functions to calculate two types of densities, (i) the usual density
(a) The vector field (V 0 (r), V(r)), which is fourdimensional in space-time and which behaves like a fourvector under Lorentz transformations, is similar in structure to the electromagnetic potentials ( A 0 (r), A(r)) of Maxwell, well known from the Dirac equation in atomic physics. This vector field contains a time-like component V 0 (r), which corresponds to the Coulomb field A 0 (r), and three spacelike components V(r), which are equivalent to the magnetic potential A(r) in electrodynamics. Assuming time-reversal invariance, we can neglect currents and the corresponding space-like parts V(r). We are then left with the time-like part V(r) (for simplicity we neglect the index 0 henceforth). The Lorentz structure of the theory implies that this timelike part of the vector fields is repulsive. As we will see, the essential part of this field is determined by the short-range repulsion of the nucleon–nucleon interaction caused by the exchange of vector mesons. (b) In addition to the vector fields, familiar from atomic physics, the Dirac equation in nuclear physics allows a scalar field S(r), which behaves like the rest mass and stays invariant under Lorentz transformations. The Lorentz structure of the theory requires scalar fields to be attractive. Hence, the scalar field can simulate economically the attractive part of the nucleon–nucleon interaction at intermediate distances. On a more basic level, the attraction is caused in large part by correlated two-pion exchange and by two-pion exchange with a D-particle in the intermediate state. Both processes lead to a parity-conserving mean field, which is simulated by the scalar field. In principle, the Lorentz structure could allow a pseudoscalar field originated by the one-pion exchange. However, this part has to vanish, because it is well known that the nuclear mean field is parity-conserving. Therefore, the pion contributes on the Hartree level only via the two-pion exchange.
Oc A
r ~r! 5
Of A
1 i
ci 5
i51
1 i
~r! f i ~r! 1 g 1 i ~r! g i ~r!,
(3)
i51
which is the zero component of the four-dimensional relativistic current vector, and (ii) the scalar density
O c# c 5 O f A
r s ~r! 5
A
i
i
i51
1 i
~r! f i ~r! 2 g 1 i ~r! g i ~r!.
(4)
i51
In the no-sea approximation [8], these sums do not include the negative-energy solutions of Eq. (2). The fields V(r) and S(r) are obtained by averaging over the interactions induced by the exchange of vector and scalar mesons with the corresponding densities V~r! 5
S~r! 5
E E
v v ~r, r9! r ~r9!d 3 r9,
(5)
v s ~r, r9! r s ~r9!d 3 r9.
(6)
The two-body interactions v v(r, r9) and v s (r, r9) are of the Yukawa type. They correspond to the exchange of scalar mesons s (isoscalar) and vector mesons v (isoscalar) and rW (isovector), where the fields are defined as S~r! 5 g s s ~r!,
(7)
V~r! 5 g v v ~r! 1 g r r 3 ~r! 1 A 0 ~r!,
(8)
in which s(r), v(r), r 3(r) are the classical meson fields and A 0 (r) is the Coulomb field having its origin in the exchange of photons. The equations of motion for the meson fields are the Klein–Gordon equations
Neglecting nuclear magnetism (that is, assuming timereversal invariance of the mean field), we then have a stationary Dirac equation containing only the time-like part of the vector field V and the scalar potential S: $ a p 1 V~r! 1 b @m 1 S~r!#% c i 5 e i c i .
Nuclear Ground-State Properties
(1)
This equation contains the four-dimensional Dirac matrices a and b. The rest mass of the nucleon is denoted by m.
~2D 1 m s2 ! s ~r! 5 gsrs(r),
(9)
~2D 1 m 2v ! v ~r! 5 g v r ~r!,
(10)
~2D 1 m r2 ! r 3 ~r! 5 g r ~ r n ~r! 2 r p ~r!!,
(11)
2DA 0 ~r! 5 e r c ~r!, 3
(12)
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
where r n (r), r p (r), and r c (r) are the neutron, proton, and charge density distributions, respectively, and r c is obtained from the corresponding r p by a convolution with the intrinsic charge structure of the proton. Bearing in mind that the Green’s functions of these equations are of the Yukawa and Coulomb types, we obtain for the interactions v s ~r, r9! 5 2 v v ~r, r9! 5
g s2 e 2m sur2r9u , 4 p ur 2 r9u
If we want to calculate only the ground-state properties of nuclei, why is it necessary to use a relativistic formulation? In fact, the kinetic energies and the Fermi momenta are relatively small compared to the rest mass of the nucleons. Therefore, relativistic kinematics can be neglected. However, the Dirac equation contains more. In contrast to an equivalent Schro¨dinger equation with a potential of depth ;50 MeV, which is also small compared to the nucleon rest mass of 938 MeV, the Dirac equation contains two potentials V(r) and S(r), which are both large (;350 and ;400 MeV), opposite in sign, and nonnegligible compared to the nucleon rest mass. Therefore, one needs relativistic dynamics to describe the interplay of these two strong potentials properly. From Eq. (2), we see that in the upper equation for the large components, only the difference between the absolute values of V and S enters, which is, in fact, small compared to the nucleon rest mass. In the lower equation for the small components, however, the large sum of both potentials, uVu 1 uSu, enters. This term cannot be neglected, and its introduction leads to the strong spin– orbit term in nuclear physics. In fact, it is the advantage of the RMF theory that the strength and the shape of the spin– orbit term are determined in a fully self-consistent way. Because the proper size of the spin– orbit splitting plays a crucial role in understanding the basic properties of nuclei, it follows that a proper treatment of the relativistic dynamics is warranted as is done in the RMF theory. Another example for the importance of relativistic dynamics is the fact that the near equality (but opposite sign) of V and S leads to approximate pseudospin symmetry in nuclear spectra [23]. Finally, we emphasize the fact that the smallness of V 1 S leads to relatively small Fermi momenta and allows, in principle, a nonrelativistic reduction of the Dirac equation to a Schro¨dinger equation with momentum-dependent potentials. Therefore, a nonrelativistic theory with additional spin- and momentum-dependent terms and adjustable parameters can also provide a reliable description of nuclei. However, in general, such a theory requires more parameters, and its predictive power is probably reduced as compared to that of a fully relativistic theory.
(13)
g 2v e 2m vur2r9u g r2 e 2m rur2r9u 1 tW tW 9 4 p ur 2 r9u 4 p ur 2 r9u 1
1 e2 , 4 p ur 2 r9u
(14)
where tW are the isospin matrices. This set of coupled equations for relativistic nucleons moving in classical meson fields are Euler equations obtained from Hamilton’s variational principle based on the relativistic Lagrangian density of the Walecka model [7], + 5 c# ~i g z 2 m! c 1 1
1 1 1 ~ s ! 2 2 m s2 s 2 2 V m v V m v 2 2 4
1 2 2 1 1 1 m v v 2 RW m v RW m v 1 m r2 rW 2 2 F m v F m v 2 4 2 4
2 g s c# sc 2 g v c# g z vc 2 g r c# g z rW tW c 2 e c# g z A
~1 2 t 3 ! c, 2
(15)
where V mv , RW mv , and F mv are field tensors and the dots abbreviate a scalar product in Minkowski space ( g z v 5 g u v m 5 g 0 v 0 2 gW v W ). Using the experimental masses m, m v , and m r for the nucleons and the v and r mesons, we are left with only four parameters, m s , g s , g v , and g r , which are adjusted to the experimental data in a few spherical nuclei. It was recognized quite early that this simple model is not flexible enough to describe quantitatively the properties of real nuclei. An effective density dependence was introduced [22] in replacing the quadratic s-potential 21m s2 s 2 in the Lagrangian by a quartic potential U( s ) including a nonlinear s self-interaction U~ s ! 5
1 2 2 1 1 m ss 1 g 2s 3 1 g 3s 4, 2 3 4
Nuclear Ground-State Properties
Calculational Details In this work, the Dirac equation for nucleons is solved using the method of oscillator expansion, as described in Ref. [24]. For the determination of the basis wave functions, an axially symmetric harmonic-oscillator potential with size parameters
(16)
with the additional two parameters g 2 and g 3 . This change leads to a nonlinear Klein–Gordon equation (9) with a s-dependent mass m s2 ( s ) 5 m s2 1 g 2 s 1 g 3 s 2 , which has to be solved by iteration. More details on the RMF formalism can be found in Refs. [7–10]. 4
b z 5 b z ~b 0 , b 0 ! 5 b 0 exp~ Î5/~16 p ! b 0 !
(17)
b ' 5 b ' ~b 0 , b 0 ! 5 b 0 exp~2 Î5/~64 p ! b 0 !
(18)
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
deformation b 0 is determined for each nucleus in such a way that the resulting mass quadrupole moment Q of the nucleus is given by Q 5 =16 p /5 (3/4 p ) AR 02 b 0 with R 0 5 1.2 A 1/3 . Both proton and neutron pairing correlations have been included using the BCS formalism with constant pairing gaps obtained from the prescription of Ref. [25]. The zero-point energy of the harmonic oscillator has been used for the center-of-mass energy correction. Angular momentum and particle number projection as well as the ground-state correlations induced by the coupling to collective vibrations have been neglected. It is, however, expected that these additional corrections will have only small contributions, and the aim is to describe the ground-state properties of nuclei within the realm of the pure mean field. The effective force NL3 has been adopted for the calculations using a new version of the “axially deformed” code [26]. The parameter set NL3 has been derived recently [27] by fitting ground-state properties of 10 spherical nuclei. Properties predicted with the NL3 effective interaction are found to be in good agreement with experimental data for nuclei at and away from the line of b stability. In Table A, the parameters of the effective force NL3 are given. In the
TABLE A Parameters of the Effective Interaction NL3 in the RMF Theory Together with the Nuclear Matter Properties Obtained with This Effective Force
M ms mv mr g2
5 5 5 5 5
Parameters 939 (MeV) 508.194 (MeV) g s 5 10.217 782.501 (MeV) g v 5 12.868 763.000 (MeV) gr 5 4.474 210.431 (fm 21) g 3 5 228.885
Nuclear matter properties
r 0: (E/A) ` : K: J: m*/m:
Nuclear Ground-State Properties
0.1483 fm 23 16.299 MeV 271.76 MeV 37.4 MeV 0.60
Note. The various quantities are defined in Ref. [27].
is employed. The basis is defined in terms of the oscillator parameter b 0 and the deformation parameter b 0. The oscillator parameter b 0 is chosen as b 0 5 41A 21/3 , and the basis
FIG. 1. Total binding energies for various rare-earth nuclei calculated in the RMF theory (open squares) compared with the experimental values (solid circles) from Ref. [29].
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Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
Nuclear Ground-State Properties
FIG. 2. Isotope shifts for various rare-earth nuclei calculated in the RMF theory (open circles) compared with the experimental values (solid circles) [34]. For Gd, the experimental values are from Ref. [35].
same table, the nuclear matter properties calculated with NL3 are also listed. The number of oscillator shells taken into account is 12 for fermionic and 20 for bosonic wave functions. However, for very heavy nuclei, contributions from higher oscillator shells may not be negligible. Therefore, a larger number of fermionic shells have been used as necessary, checking each time for the correct convergence. The charge radius is calculated using the formula r c 5 Îr 2p 1 0.64
~fm!.
and
H n,p 5 ^r 4 Y 40 ~ u !& n,p 5
9 1 ^8z 4 2 24z 2 ~ x 2 1 y 2 ! 4p 8 1 3~ x 2 1 y 2 ! 2 & n,p .
(21)
The quadrupole deformation parameter b 2 and the hexadecapole deformation parameter b 4 are obtained in such a way that sharp-edged densities with these deformations have the same multipole moments as our selfconsistent solutions. This is accomplished by solving a system of two nonlinear equations, as described in Appendix B of Ref. [28]. In Table I the ground-state properties (total binding energies, nuclear radii, quadrupole and hexadecapole moments, and deformation parameters) of 1315 even– even nuclei are listed. The predictions of the RMF theory are in good agreement with experiment. The total binding
(19)
The factor 0.64 in Eq. (19) accounts for the finite-size effects of the proton. The quadrupole and hexadecapole moments for neutrons (n) and protons ( p) are calculated according to the usual definitions Q n,p 5 ^2r 2 P 2 ~cos u !& n,p 5 ^2z 2 2 x 2 2 y 2 & n,p
Î
(20) 6
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
Nuclear Ground-State Properties
FIG. 3. Quadrupole deformation parameters for various rare-earth nuclei calculated in the RMF theory (open circles) compared with the experimental values (solid circles) [33].
energies are in overall agreement with the empirical data [29], the disagreement amounts to typically 0.5%. (The root-mean-square deviation between calculation and experiment [29] is 2.6 MeV.) Only for some nuclei with N ; Z does the difference appear to be somewhat larger—about 1–2% for some chains. This discrepancy might indicate that for these nuclei additional correlations should be taken into account. In particular, proton– neutron pairing could have a strong influence on the masses [30]. The nuclear radii and deformation parameters also compare well with the available experimental values [31–33]. As an illustration, the RMF predictions for some rare-earth nuclei with 60 # Z # 70 are compared with experiment in Figs. 1–3. The total binding energies are compared in Fig. 1. The isotope shifts d r c2 are compared in Fig. 2. These shifts have been obtained with respect to a reference nucleus (ref.) in each chain as given by d r c2 5 r c2 2 r c2 (ref.). The reference nucleus has N 5 82 for all elements except Dy (Z 5 66), for which the reference nucleus is 156Dy, and Yb (Z 5 70), for which the reference nucleus is 168Yb. Finally, in Fig. 3, the predictions of the RMF theory for the deformation parameters b 2 are com-
pared with the empirical values. It is seen from all three figures that the agreement with the experiment is good. The total binding energies for the three most protonrich isotopes close to the drip line have been compared with the available experimental values in Table I of Ref. [36]. For these nuclei, the root-mean-square deviation between calculation and experiment is slightly larger (3.1 MeV) than the overall deviation (2.6 MeV). Moreover, the calculated binding energies for some light actinide (Z 5 84–92) isotopes exceed the experimental values by more than 5 MeV. These deviations should be attributed to reflection asymmetric shapes of these nuclei. Reflection asymmetry is not included in the current version of the RMF code. In several cases two RMF solutions have been found which differ little in energy. As a result of a complex potentialenergy landscape in the deformation space, the associated shapes are usually of oblate and prolate types. In Table B, the differences in the binding energy of the prolate and oblate minima are listed for those nuclei for which this difference is less than 1 MeV. A negative value implies that the prolate minimum lies lower than the corresponding oblate minimum. The RMF theory predicts that these nuclei are candidates for studying the phenomenon of shape coexistence. 7
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
Nuclear Ground-State Properties
TABLE B Energy Difference (in MeV) and Associated Deformations for Nuclei with Possible Shape Coexistence in the Ground State Nucleus
E pro. 2 E obl.
b 2(pro.)
b 2(obl.)
20.230 20.700 20.360 20.990 20.440 20.390 20.530 0.110 0.150 20.470 0.290 0.200 20.100 20.660 0.250 0.580 20.140 0.250 0.600 20.150 0.010 20.660 0.050 20.080 20.080 20.530 20.070 20.270 20.610 20.390 20.490 0.650 0.300 0.780 20.130 0.270 0.360 0.380 0.120 20.910 0.420 20.170 20.740 20.060 20.330 0.790 0.820 0.530
0.191 0.446 0.296 0.254 0.170 0.230 0.320 0.042 0.301 0.358 0.179 0.113 0.203 0.217 0.227 0.114 0.205 0.230 0.242 0.386 0.099 0.119 0.004 0.132 0.201 0.384 0.375 0.373 0.373 0.374 0.380 0.457 0.338 0.342 0.257 0.276 0.282 0.158 0.137 0.189 0.230 0.181 0.263 0.217 0.255 0.200 0.136 0.094
20.139 20.380 20.261 20.161 20.111 20.165 20.274 20.206 20.374 20.294 20.250 20.242 20.227 20.240 20.261 20.210 20.230 20.265 20.285 20.209 20.107 20.083 20.213 20.128 20.227 20.284 20.341 20.349 20.323 20.299 20.268 20.232 20.264 20.270 20.164 20.175 20.179 20.161 20.138 20.196 20.204 20.174 20.275 20.232 20.220 20.201 20.165 20.119
24
Ne Ne 26 Mg 28 Mg 30 Mg 24 Si 26 Si 30 Si 40 Si 44 S 48 S 50 S 64 Zn 64 Ge 66 Ge 74 Ge 64 Se 66 Se 68 Se 76 Kr 80 Kr 82 Kr 80 Sr 92 Sr 94 Sr 96 Sr 98 Sr 100 Sr 102 Sr 104 Sr 106 Sr 82 Zr 106 Mo 108 Mo 116 Te 118 Te 120 Te 122 Te 124 Te 136 Nd 174 Nd 176 Nd 136 Sm 138 Sm 176 Sm 178 Sm 180 Sm 182 Sm 38
Nucleus 140
Gd Gd 180 Gd 184 Gd 142 Dy 144 Dy 144 Er 184 Er 186 Er 188 Er 190 Er 192 Er 186 Yb 188 Yb 190 Yb 192 Yb 194 Yb 192 Hf 194 Hf 196 W 196 Os 190 Pt 192 Pt 194 Pt 196 Pt 198 Pt 180 Hg 182 Hg 184 Hg 186 Hg 188 Hg 190 Hg 186 Po 188 Po 190 Po 192 Po 194 Po 196 Po 198 Rn 200 Rn 202 Rn 204 Rn 252 Rn 254 Rn 256 Rn 206 Ra 258 Rn 208 Ra 178
E pro. 2 E obl.
b 2(pro.)
b 2(obl.)
20.460 20.260 0.870 0.840 20.100 0.370 20.001 0.400 0.810 0.100 0.840 0.360 20.530 20.080 0.200 0.510 0.460 20.740 0.050 0.010 20.720 20.720 20.280 20.200 0.050 0.720 0.120 0.750 0.210 0.350 0.680 0.180 20.240 20.280 0.200 0.300 0.430 0.910 20.970 20.360 20.470 0.260 0.580 0.910 0.550 20.820 0.920 20.440
0.305 0.277 0.211 0.117 0.306 0.108 0.257 0.207 0.173 0.142 0.106 0.041 0.204 0.177 0.147 0.112 0.047 0.147 0.110 0.097 0.116 0.201 0.161 0.137 0.113 0.068 0.272 0.280 0.285 0.282 0.275 0.257 0.220 0.234 0.204 0.176 0.163 0.140 0.146 0.118 0.104 0.088 0.194 0.168 0.149 0.087 0.127 0.056
20.242 20.230 20.215 20.148 20.245 20.198 20.249 20.229 20.212 20.180 20.133 20.062 20.230 20.214 20.183 20.133 20.070 20.174 20.123 20.106 20.128 20.182 20.170 20.157 20.136 20.105 20.184 20.198 20.216 20.222 20.213 20.197 20.192 20.203 20.207 20.205 20.196 20.186 20.194 20.172 20.109 20.080 20.211 20.201 20.186 20.094 20.169 20.075
Note. A negative value for the energy difference implies that the prolate minimum lies lower than the oblate minimum.
Acknowledgments
is supported by the Bundesministerium fu¨r Forschung und Technologie under Project 06 TM743. The current work was sponsored, in part, by the U.S. Department of Energy under Contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corporation.
G.A.L. is grateful to the Joint Institute for Heavy-Ion Research for arranging his assignment to Oak Ridge. We thank Pertti Tikkanen for his help in producing Table I. P.R. 8
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
G. A. LALAZISSIS, S. RAMAN, and P. RING
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33. S. Raman, C. H. Malarkey, W. T. Milner, C. W. Nestor, Jr., and P. H. Stelson, ATOMIC DATA AND NUCLEAR DATA TABLES 36, 1 (1987)
15. D. Vretenar, H. Berghammer, and P. Ring, Nucl. Phys. A 581, 679 (1995)
34. E. W. Otten, Treatise on Heavy-Ion Science, edited by D. A. Bromley (Plenum, New York, 1989) Vol. 8, p. 517
16. D. Vretenar, G. A. Lalazissis, R. Behnsch, W. Po¨schl, and P. Ring, Nucl. Phys. A 621, 853 (1997)
18. A. Afanasjev, J. Ko¨nig, and P. Ring, Nucl. Phys. A 608, 107 (1997)
35. G. D. Alkhazov, A. E. Barzakh, V. P. Denisov, V. S. Ivanov, I. Ya. Chubukov, N. B. Buyanov, V. S. Letokhov, V. I. Mishin, S. K. Sekatsky, and V. N. Fedoseev, Pisma Zh. Eksp. Teor. Fiz. 48, 373 (1988); JETP Lett. (USSR) 48, 413 (1988)
19. A. Afanasjev, J. Ko¨nig, and P. Ring, Phys. Lett. B 367, 11 (1996)
36. G. A. Lalazissis and S. Raman, Phys. Rev. C 58, 1467 (1998)
17. J. Ko¨nig and P. Ring, Phys. Rev. Lett 71, 3079 (1993)
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EXPLANATION OF TABLE TABLE I.
RMF Predictions of Ground-State Properties Z N A BE rn rp rc Qn Qp Hp b2 b4
Atomic number Neutron number Mass number Total binding energy in MeV Root-mean-square neutron radius of the nucleus in fm Root-mean-square proton radius of the nucleus in fm Root-mean-square charge radius of the nucleus (Eq. (19)) Neutron quadrupole moment in b (Eq. (20)) Proton quadrupole moment in b (Eq. (20)) Charge (proton) hexadecapole moment in b 2 (Eq. (21)) Quadrupole deformation parameter Hexadecapole deformation parameter
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Nuclear Ground-State Properties
TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
GROUND-STATE PROPERTIES OF EVEN–EVEN NUCLEI IN THE RELATIVISTIC MEAN-FIELD THEORY G. A. LALAZISSIS and S. RAMAN Physics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37831 and P. RING Fakulta¨t fu¨r Physik, Technische Universita¨t, Mu¨nchen, D-85748 Garching, Germany The ground-state properties of 1315 even– even nuclei with 10 # Z # 98 have been calculated in the framework of the relativistic mean-field (RMF) theory. The Lagrangian parametrization NL3 was used in the calculations. Pairing correlations are accounted within the Bardeen–Cooper–Schrieffer approach. The calculated values for the total binding energy, rms proton radius, rms neutron radius, rms charge radius, neutron quadrupole moment, proton quadrupole moment, charge (proton) hexadecapole moment, quadrupole deformation parameter, and hexadecapole deformation parameter are given in Table I. The RMF predictions of some rare-earth nuclei have been compared with the available experimental information. © 1999 Academic Press
0092-640X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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Nuclear Ground-State Properties
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 4
EXPLANATION OF TABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
TABLE I. RMF Predictions of Ground-State Properties . . . . . . . . . . . . . . . . . . .
11
INTRODUCTION The RMF theory has recently enjoyed considerable success in describing various facets of nuclear-structure properties. With a very limited number of parameters, this theory is able to give a quantitative description of the ground-state properties of spherical and deformed nuclei at and away from the line of b stability [10, 11]. It has been shown that the anomalous kink in the isotope shifts of Pb nuclei and the anomalous isotopic shifts in the Sr and Kr chains can be explained by the RMF microscopic calculations [12, 13]. Such anomalous behavior is a generic feature of almost all isotopic chains in the rare-earth region [14]. Good agreement with experimental data has been found recently also for collective excitations such as giant resonances [15, 16] and for twin bands in rotating superdeformed nuclei [17]. Similarly, good descriptions of the superdeformed rotational bands in the A 5 140–150 region and in the rare-earth region have been provided by the cranked RMF calculations [18 –20]. Very recently it has also been shown that constrained RMF calculations reproduce the excitation energies of superdeformed minima relative to the ground state in 194Hg and 194Pb [21] very well. In the RMF theory, the saturation and the density dependence of the nuclear interaction are obtained by a balance between a large, attractive, scalar s-meson field and a large, repulsive, vector v-meson field. The asymmetry component is provided by the isovector rW meson. The nuclear interaction is, hence, generated by the exchange of various mesons between nucleons in the framework of the mean field. The spin– orbit interaction arises naturally in the RMF theory as a result of the Dirac structure of nucleons.
The development of theoretical models that are able to successfully reproduce and predict the ground-state properties and other properties of finite nuclei throughout the periodic table is of great importance in nuclear-structure studies. A good description of the properties of known nuclei also gives us more confidence in extrapolating to the yet unexplored areas of the nuclear chart. Of course, these extrapolations are based upon the respective ansatz of a model which is usually obtained by fits to the sets of available nuclear properties near the b-stability line. There exist several such theoretical models in the literature. Chief among these are the various mass models [1]. The FiniteRange Droplet Model (FRDM) [2, 3] and the Extended Thomas–Fermi with Strutinsky Integral Model (ETFSI) [4] are two of the currently best-known mass models. In both cases, attempts have been made to obtain the best possible description of nuclear masses and deformation properties. Fully self-consistent calculations within the meanfield theory with suitable relativistic or nonrelativistic effective interactions are a preferred alternative approach. However, the computation time required for such calculations, especially for the number of nuclei approaching the scopes of the above-mentioned mass models, is huge. Therefore, one has to limit these calculations to a smaller number and, at the same time, exploit those symmetries of the mean field that can reduce the computation time. Recently, deformed Hartree–Fock (HF) 1 Bardeen–Cooper– Schrieffer (BCS) calculations with the Skyrme SIII force [5] have been reported for the ground-state properties of 1029 even– even nuclei [6]. This work [6] is the first such extensive calculation within the nonrelativistic mean-field approach. The purpose of the present work is to provide the first systematic study of the ground-state properties of even– even nuclei over a very wide range of isospin within the relativistic mean-field (RMF) theory [7–9].
Relativistic Mean-Field Theory In the RMF theory, the nucleons are described as relativistic particles moving independently in average po2
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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Expressed in terms of the Pauli spin matrices, this Dirac equation reads
tentials determined in a self-consistent way by the exchange of mesons. The relativistic single-particle equation is the Dirac equation. In contrast to the nonrelativistic Schro¨dinger equation, which contains one average potential represented usually by a Saxon–Woods shape and an independently fitted spin– orbit potential represented as the derivative of a Saxon–Woods shape, the Lorentz structure of the Dirac equation allows in principle several types of fields:
S m 1sSp1 V
sp 2m 2 S 1 V
DS gf D 5 e S gf D . i
i
(2)
i
The single-particle wave functions c i are four-dimensional spinors, which describe stationary states of the nucleons with index i and single-particle energy e i . By summing over the occupied orbitals, we can use these wave functions to calculate two types of densities, (i) the usual density
(a) The vector field (V 0 (r), V(r)), which is fourdimensional in space-time and which behaves like a fourvector under Lorentz transformations, is similar in structure to the electromagnetic potentials ( A 0 (r), A(r)) of Maxwell, well known from the Dirac equation in atomic physics. This vector field contains a time-like component V 0 (r), which corresponds to the Coulomb field A 0 (r), and three spacelike components V(r), which are equivalent to the magnetic potential A(r) in electrodynamics. Assuming time-reversal invariance, we can neglect currents and the corresponding space-like parts V(r). We are then left with the time-like part V(r) (for simplicity we neglect the index 0 henceforth). The Lorentz structure of the theory implies that this timelike part of the vector fields is repulsive. As we will see, the essential part of this field is determined by the short-range repulsion of the nucleon–nucleon interaction caused by the exchange of vector mesons. (b) In addition to the vector fields, familiar from atomic physics, the Dirac equation in nuclear physics allows a scalar field S(r), which behaves like the rest mass and stays invariant under Lorentz transformations. The Lorentz structure of the theory requires scalar fields to be attractive. Hence, the scalar field can simulate economically the attractive part of the nucleon–nucleon interaction at intermediate distances. On a more basic level, the attraction is caused in large part by correlated two-pion exchange and by two-pion exchange with a D-particle in the intermediate state. Both processes lead to a parity-conserving mean field, which is simulated by the scalar field. In principle, the Lorentz structure could allow a pseudoscalar field originated by the one-pion exchange. However, this part has to vanish, because it is well known that the nuclear mean field is parity-conserving. Therefore, the pion contributes on the Hartree level only via the two-pion exchange.
Oc A
r ~r! 5
Of A
1 i
ci 5
i51
1 i
~r! f i ~r! 1 g 1 i ~r! g i ~r!,
(3)
i51
which is the zero component of the four-dimensional relativistic current vector, and (ii) the scalar density
O c# c 5 O f A
r s ~r! 5
A
i
i
i51
1 i
~r! f i ~r! 2 g 1 i ~r! g i ~r!.
(4)
i51
In the no-sea approximation [8], these sums do not include the negative-energy solutions of Eq. (2). The fields V(r) and S(r) are obtained by averaging over the interactions induced by the exchange of vector and scalar mesons with the corresponding densities V~r! 5
S~r! 5
E E
v v ~r, r9! r ~r9!d 3 r9,
(5)
v s ~r, r9! r s ~r9!d 3 r9.
(6)
The two-body interactions v v(r, r9) and v s (r, r9) are of the Yukawa type. They correspond to the exchange of scalar mesons s (isoscalar) and vector mesons v (isoscalar) and rW (isovector), where the fields are defined as S~r! 5 g s s ~r!,
(7)
V~r! 5 g v v ~r! 1 g r r 3 ~r! 1 A 0 ~r!,
(8)
in which s(r), v(r), r 3(r) are the classical meson fields and A 0 (r) is the Coulomb field having its origin in the exchange of photons. The equations of motion for the meson fields are the Klein–Gordon equations
Neglecting nuclear magnetism (that is, assuming timereversal invariance of the mean field), we then have a stationary Dirac equation containing only the time-like part of the vector field V and the scalar potential S: $ a p 1 V~r! 1 b @m 1 S~r!#% c i 5 e i c i .
Nuclear Ground-State Properties
(1)
This equation contains the four-dimensional Dirac matrices a and b. The rest mass of the nucleon is denoted by m.
~2D 1 m s2 ! s ~r! 5 gsrs(r),
(9)
~2D 1 m 2v ! v ~r! 5 g v r ~r!,
(10)
~2D 1 m r2 ! r 3 ~r! 5 g r ~ r n ~r! 2 r p ~r!!,
(11)
2DA 0 ~r! 5 e r c ~r!, 3
(12)
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where r n (r), r p (r), and r c (r) are the neutron, proton, and charge density distributions, respectively, and r c is obtained from the corresponding r p by a convolution with the intrinsic charge structure of the proton. Bearing in mind that the Green’s functions of these equations are of the Yukawa and Coulomb types, we obtain for the interactions v s ~r, r9! 5 2 v v ~r, r9! 5
g s2 e 2m sur2r9u , 4 p ur 2 r9u
If we want to calculate only the ground-state properties of nuclei, why is it necessary to use a relativistic formulation? In fact, the kinetic energies and the Fermi momenta are relatively small compared to the rest mass of the nucleons. Therefore, relativistic kinematics can be neglected. However, the Dirac equation contains more. In contrast to an equivalent Schro¨dinger equation with a potential of depth ;50 MeV, which is also small compared to the nucleon rest mass of 938 MeV, the Dirac equation contains two potentials V(r) and S(r), which are both large (;350 and ;400 MeV), opposite in sign, and nonnegligible compared to the nucleon rest mass. Therefore, one needs relativistic dynamics to describe the interplay of these two strong potentials properly. From Eq. (2), we see that in the upper equation for the large components, only the difference between the absolute values of V and S enters, which is, in fact, small compared to the nucleon rest mass. In the lower equation for the small components, however, the large sum of both potentials, uVu 1 uSu, enters. This term cannot be neglected, and its introduction leads to the strong spin– orbit term in nuclear physics. In fact, it is the advantage of the RMF theory that the strength and the shape of the spin– orbit term are determined in a fully self-consistent way. Because the proper size of the spin– orbit splitting plays a crucial role in understanding the basic properties of nuclei, it follows that a proper treatment of the relativistic dynamics is warranted as is done in the RMF theory. Another example for the importance of relativistic dynamics is the fact that the near equality (but opposite sign) of V and S leads to approximate pseudospin symmetry in nuclear spectra [23]. Finally, we emphasize the fact that the smallness of V 1 S leads to relatively small Fermi momenta and allows, in principle, a nonrelativistic reduction of the Dirac equation to a Schro¨dinger equation with momentum-dependent potentials. Therefore, a nonrelativistic theory with additional spin- and momentum-dependent terms and adjustable parameters can also provide a reliable description of nuclei. However, in general, such a theory requires more parameters, and its predictive power is probably reduced as compared to that of a fully relativistic theory.
(13)
g 2v e 2m vur2r9u g r2 e 2m rur2r9u 1 tW tW 9 4 p ur 2 r9u 4 p ur 2 r9u 1
1 e2 , 4 p ur 2 r9u
(14)
where tW are the isospin matrices. This set of coupled equations for relativistic nucleons moving in classical meson fields are Euler equations obtained from Hamilton’s variational principle based on the relativistic Lagrangian density of the Walecka model [7], + 5 c# ~i g z 2 m! c 1 1
1 1 1 ~ s ! 2 2 m s2 s 2 2 V m v V m v 2 2 4
1 2 2 1 1 1 m v v 2 RW m v RW m v 1 m r2 rW 2 2 F m v F m v 2 4 2 4
2 g s c# sc 2 g v c# g z vc 2 g r c# g z rW tW c 2 e c# g z A
~1 2 t 3 ! c, 2
(15)
where V mv , RW mv , and F mv are field tensors and the dots abbreviate a scalar product in Minkowski space ( g z v 5 g u v m 5 g 0 v 0 2 gW v W ). Using the experimental masses m, m v , and m r for the nucleons and the v and r mesons, we are left with only four parameters, m s , g s , g v , and g r , which are adjusted to the experimental data in a few spherical nuclei. It was recognized quite early that this simple model is not flexible enough to describe quantitatively the properties of real nuclei. An effective density dependence was introduced [22] in replacing the quadratic s-potential 21m s2 s 2 in the Lagrangian by a quartic potential U( s ) including a nonlinear s self-interaction U~ s ! 5
1 2 2 1 1 m ss 1 g 2s 3 1 g 3s 4, 2 3 4
Nuclear Ground-State Properties
Calculational Details In this work, the Dirac equation for nucleons is solved using the method of oscillator expansion, as described in Ref. [24]. For the determination of the basis wave functions, an axially symmetric harmonic-oscillator potential with size parameters
(16)
with the additional two parameters g 2 and g 3 . This change leads to a nonlinear Klein–Gordon equation (9) with a s-dependent mass m s2 ( s ) 5 m s2 1 g 2 s 1 g 3 s 2 , which has to be solved by iteration. More details on the RMF formalism can be found in Refs. [7–10]. 4
b z 5 b z ~b 0 , b 0 ! 5 b 0 exp~ Î5/~16 p ! b 0 !
(17)
b ' 5 b ' ~b 0 , b 0 ! 5 b 0 exp~2 Î5/~64 p ! b 0 !
(18)
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deformation b 0 is determined for each nucleus in such a way that the resulting mass quadrupole moment Q of the nucleus is given by Q 5 =16 p /5 (3/4 p ) AR 02 b 0 with R 0 5 1.2 A 1/3 . Both proton and neutron pairing correlations have been included using the BCS formalism with constant pairing gaps obtained from the prescription of Ref. [25]. The zero-point energy of the harmonic oscillator has been used for the center-of-mass energy correction. Angular momentum and particle number projection as well as the ground-state correlations induced by the coupling to collective vibrations have been neglected. It is, however, expected that these additional corrections will have only small contributions, and the aim is to describe the ground-state properties of nuclei within the realm of the pure mean field. The effective force NL3 has been adopted for the calculations using a new version of the “axially deformed” code [26]. The parameter set NL3 has been derived recently [27] by fitting ground-state properties of 10 spherical nuclei. Properties predicted with the NL3 effective interaction are found to be in good agreement with experimental data for nuclei at and away from the line of b stability. In Table A, the parameters of the effective force NL3 are given. In the
TABLE A Parameters of the Effective Interaction NL3 in the RMF Theory Together with the Nuclear Matter Properties Obtained with This Effective Force
M ms mv mr g2
5 5 5 5 5
Parameters 939 (MeV) 508.194 (MeV) g s 5 10.217 782.501 (MeV) g v 5 12.868 763.000 (MeV) gr 5 4.474 210.431 (fm 21) g 3 5 228.885
Nuclear matter properties
r 0: (E/A) ` : K: J: m*/m:
Nuclear Ground-State Properties
0.1483 fm 23 16.299 MeV 271.76 MeV 37.4 MeV 0.60
Note. The various quantities are defined in Ref. [27].
is employed. The basis is defined in terms of the oscillator parameter b 0 and the deformation parameter b 0. The oscillator parameter b 0 is chosen as b 0 5 41A 21/3 , and the basis
FIG. 1. Total binding energies for various rare-earth nuclei calculated in the RMF theory (open squares) compared with the experimental values (solid circles) from Ref. [29].
5
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Nuclear Ground-State Properties
FIG. 2. Isotope shifts for various rare-earth nuclei calculated in the RMF theory (open circles) compared with the experimental values (solid circles) [34]. For Gd, the experimental values are from Ref. [35].
same table, the nuclear matter properties calculated with NL3 are also listed. The number of oscillator shells taken into account is 12 for fermionic and 20 for bosonic wave functions. However, for very heavy nuclei, contributions from higher oscillator shells may not be negligible. Therefore, a larger number of fermionic shells have been used as necessary, checking each time for the correct convergence. The charge radius is calculated using the formula r c 5 Îr 2p 1 0.64
~fm!.
and
H n,p 5 ^r 4 Y 40 ~ u !& n,p 5
9 1 ^8z 4 2 24z 2 ~ x 2 1 y 2 ! 4p 8 1 3~ x 2 1 y 2 ! 2 & n,p .
(21)
The quadrupole deformation parameter b 2 and the hexadecapole deformation parameter b 4 are obtained in such a way that sharp-edged densities with these deformations have the same multipole moments as our selfconsistent solutions. This is accomplished by solving a system of two nonlinear equations, as described in Appendix B of Ref. [28]. In Table I the ground-state properties (total binding energies, nuclear radii, quadrupole and hexadecapole moments, and deformation parameters) of 1315 even– even nuclei are listed. The predictions of the RMF theory are in good agreement with experiment. The total binding
(19)
The factor 0.64 in Eq. (19) accounts for the finite-size effects of the proton. The quadrupole and hexadecapole moments for neutrons (n) and protons ( p) are calculated according to the usual definitions Q n,p 5 ^2r 2 P 2 ~cos u !& n,p 5 ^2z 2 2 x 2 2 y 2 & n,p
Î
(20) 6
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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Nuclear Ground-State Properties
FIG. 3. Quadrupole deformation parameters for various rare-earth nuclei calculated in the RMF theory (open circles) compared with the experimental values (solid circles) [33].
energies are in overall agreement with the empirical data [29], the disagreement amounts to typically 0.5%. (The root-mean-square deviation between calculation and experiment [29] is 2.6 MeV.) Only for some nuclei with N ; Z does the difference appear to be somewhat larger—about 1–2% for some chains. This discrepancy might indicate that for these nuclei additional correlations should be taken into account. In particular, proton– neutron pairing could have a strong influence on the masses [30]. The nuclear radii and deformation parameters also compare well with the available experimental values [31–33]. As an illustration, the RMF predictions for some rare-earth nuclei with 60 # Z # 70 are compared with experiment in Figs. 1–3. The total binding energies are compared in Fig. 1. The isotope shifts d r c2 are compared in Fig. 2. These shifts have been obtained with respect to a reference nucleus (ref.) in each chain as given by d r c2 5 r c2 2 r c2 (ref.). The reference nucleus has N 5 82 for all elements except Dy (Z 5 66), for which the reference nucleus is 156Dy, and Yb (Z 5 70), for which the reference nucleus is 168Yb. Finally, in Fig. 3, the predictions of the RMF theory for the deformation parameters b 2 are com-
pared with the empirical values. It is seen from all three figures that the agreement with the experiment is good. The total binding energies for the three most protonrich isotopes close to the drip line have been compared with the available experimental values in Table I of Ref. [36]. For these nuclei, the root-mean-square deviation between calculation and experiment is slightly larger (3.1 MeV) than the overall deviation (2.6 MeV). Moreover, the calculated binding energies for some light actinide (Z 5 84–92) isotopes exceed the experimental values by more than 5 MeV. These deviations should be attributed to reflection asymmetric shapes of these nuclei. Reflection asymmetry is not included in the current version of the RMF code. In several cases two RMF solutions have been found which differ little in energy. As a result of a complex potentialenergy landscape in the deformation space, the associated shapes are usually of oblate and prolate types. In Table B, the differences in the binding energy of the prolate and oblate minima are listed for those nuclei for which this difference is less than 1 MeV. A negative value implies that the prolate minimum lies lower than the corresponding oblate minimum. The RMF theory predicts that these nuclei are candidates for studying the phenomenon of shape coexistence. 7
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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Nuclear Ground-State Properties
TABLE B Energy Difference (in MeV) and Associated Deformations for Nuclei with Possible Shape Coexistence in the Ground State Nucleus
E pro. 2 E obl.
b 2(pro.)
b 2(obl.)
20.230 20.700 20.360 20.990 20.440 20.390 20.530 0.110 0.150 20.470 0.290 0.200 20.100 20.660 0.250 0.580 20.140 0.250 0.600 20.150 0.010 20.660 0.050 20.080 20.080 20.530 20.070 20.270 20.610 20.390 20.490 0.650 0.300 0.780 20.130 0.270 0.360 0.380 0.120 20.910 0.420 20.170 20.740 20.060 20.330 0.790 0.820 0.530
0.191 0.446 0.296 0.254 0.170 0.230 0.320 0.042 0.301 0.358 0.179 0.113 0.203 0.217 0.227 0.114 0.205 0.230 0.242 0.386 0.099 0.119 0.004 0.132 0.201 0.384 0.375 0.373 0.373 0.374 0.380 0.457 0.338 0.342 0.257 0.276 0.282 0.158 0.137 0.189 0.230 0.181 0.263 0.217 0.255 0.200 0.136 0.094
20.139 20.380 20.261 20.161 20.111 20.165 20.274 20.206 20.374 20.294 20.250 20.242 20.227 20.240 20.261 20.210 20.230 20.265 20.285 20.209 20.107 20.083 20.213 20.128 20.227 20.284 20.341 20.349 20.323 20.299 20.268 20.232 20.264 20.270 20.164 20.175 20.179 20.161 20.138 20.196 20.204 20.174 20.275 20.232 20.220 20.201 20.165 20.119
24
Ne Ne 26 Mg 28 Mg 30 Mg 24 Si 26 Si 30 Si 40 Si 44 S 48 S 50 S 64 Zn 64 Ge 66 Ge 74 Ge 64 Se 66 Se 68 Se 76 Kr 80 Kr 82 Kr 80 Sr 92 Sr 94 Sr 96 Sr 98 Sr 100 Sr 102 Sr 104 Sr 106 Sr 82 Zr 106 Mo 108 Mo 116 Te 118 Te 120 Te 122 Te 124 Te 136 Nd 174 Nd 176 Nd 136 Sm 138 Sm 176 Sm 178 Sm 180 Sm 182 Sm 38
Nucleus 140
Gd Gd 180 Gd 184 Gd 142 Dy 144 Dy 144 Er 184 Er 186 Er 188 Er 190 Er 192 Er 186 Yb 188 Yb 190 Yb 192 Yb 194 Yb 192 Hf 194 Hf 196 W 196 Os 190 Pt 192 Pt 194 Pt 196 Pt 198 Pt 180 Hg 182 Hg 184 Hg 186 Hg 188 Hg 190 Hg 186 Po 188 Po 190 Po 192 Po 194 Po 196 Po 198 Rn 200 Rn 202 Rn 204 Rn 252 Rn 254 Rn 256 Rn 206 Ra 258 Rn 208 Ra 178
E pro. 2 E obl.
b 2(pro.)
b 2(obl.)
20.460 20.260 0.870 0.840 20.100 0.370 20.001 0.400 0.810 0.100 0.840 0.360 20.530 20.080 0.200 0.510 0.460 20.740 0.050 0.010 20.720 20.720 20.280 20.200 0.050 0.720 0.120 0.750 0.210 0.350 0.680 0.180 20.240 20.280 0.200 0.300 0.430 0.910 20.970 20.360 20.470 0.260 0.580 0.910 0.550 20.820 0.920 20.440
0.305 0.277 0.211 0.117 0.306 0.108 0.257 0.207 0.173 0.142 0.106 0.041 0.204 0.177 0.147 0.112 0.047 0.147 0.110 0.097 0.116 0.201 0.161 0.137 0.113 0.068 0.272 0.280 0.285 0.282 0.275 0.257 0.220 0.234 0.204 0.176 0.163 0.140 0.146 0.118 0.104 0.088 0.194 0.168 0.149 0.087 0.127 0.056
20.242 20.230 20.215 20.148 20.245 20.198 20.249 20.229 20.212 20.180 20.133 20.062 20.230 20.214 20.183 20.133 20.070 20.174 20.123 20.106 20.128 20.182 20.170 20.157 20.136 20.105 20.184 20.198 20.216 20.222 20.213 20.197 20.192 20.203 20.207 20.205 20.196 20.186 20.194 20.172 20.109 20.080 20.211 20.201 20.186 20.094 20.169 20.075
Note. A negative value for the energy difference implies that the prolate minimum lies lower than the oblate minimum.
Acknowledgments
is supported by the Bundesministerium fu¨r Forschung und Technologie under Project 06 TM743. The current work was sponsored, in part, by the U.S. Department of Energy under Contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corporation.
G.A.L. is grateful to the Joint Institute for Heavy-Ion Research for arranging his assignment to Oak Ridge. We thank Pertti Tikkanen for his help in producing Table I. P.R. 8
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9
Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999
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Nuclear Ground-State Properties
EXPLANATION OF TABLE TABLE I.
RMF Predictions of Ground-State Properties Z N A BE rn rp rc Qn Qp Hp b2 b4
Atomic number Neutron number Mass number Total binding energy in MeV Root-mean-square neutron radius of the nucleus in fm Root-mean-square proton radius of the nucleus in fm Root-mean-square charge radius of the nucleus (Eq. (19)) Neutron quadrupole moment in b (Eq. (20)) Proton quadrupole moment in b (Eq. (20)) Charge (proton) hexadecapole moment in b 2 (Eq. (21)) Quadrupole deformation parameter Hexadecapole deformation parameter
10
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TABLE I. RMF Predictions of Ground-State Properties See page 10 for Explanation of Table
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