- Email: [email protected]
Relativistic Hartree-Bogoliubov calculations with finite range pairing forces
a..__
27June1996
__ BB
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 379 (1996) 13-19
Relativistic Hartree-Bogoliubov calculations with finite range pairing forces T. Gonzalez-Llarenaa,
J.L. Egido a, G.A. Lalazissis b, P. Ring b
a Departamento de Fisica Tedrica C-XI, Universidad Autrinom de Madrid, E-28049, Spain h Physikdeparfment, Technische Universitiit Miinchen, D-85747 Garching, Germany
Received 4 October 1995; revised manuscript received 10 April 1996 Editor: C. Mahaux
Abstract We propose a relativistic extension correlations are taken into account by a fully self-consistent description of investigation of several isotope chains
of Hartree-Bogoliubov theory on the basis of relativistic mean field theory. Pairing an effective interaction of finite range adjusted by Gogny and his group. This allows pairing correlations over the entire periodic table. The method is applied for the of spherical Pb, Sn, and Zr nuclei with magic proton configurations.
PACT: 21.10.Re; 21.60.52; 24.3O.C~
For many years it has been known that ground state properties of nuclei can be well described in the framework of mean field theories with effective forces. Within the non-relativistic scheme the essential breakthrough in this respect was achieved by Vautherin and Brink [ 11, who used an effective force of Skyrme type, i.e. a strongly repulsive density dependent term of zero range in the T = 0 channel in order to guarantee proper saturation of infinite nuclear matter and of finite nuclei. For reasons of numerical simplicity they have also chosen zero range for the attractive part of the effective force as well as for the spin orbit part. Finite range effects were taken into account in an approximate way by a quadratic momentum dependence. This method is extremely successful for Hartree-Fock calculations of spherical doubly magic nuclei, where a sharp Fermi surface guarantees, that the components of this force with large momentum transfer do not play any role. In fact the behavior of the Skyrme force
at large momentum transfer, i.e. at short distances is completely wrong but it has no influence on HartreeFock calculations with a sharp Fermi surface, where only states with small momenta are occupied. In open shell nuclei, where pairing correlations play an essential role, the situation becomes more involved. In fact it is well known that Skyrme forces cannot be used directly for the calculation of pairing properties. It is not the strong density dependent term, which causes problems because it acts only in the T = 0 channel. It is the attractive part of zero range, which is essentially constant in momentum space and leads to a divergence in the gap equation at large momenta. The quadratic momentum dependence does not change this situation. Pairing correlations are therefore taken into account in the Skyrme scheme usually in only a very phenomenological way in a BCS-theory with monopole pairing force adjusted to the experimental odd-even mass difference, i.e. in constant gap approx-
0370-2693/96/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00461-3
14
T. Gonzalez-Llarena
et al./Physics
imation. In practice one just uses occupation factors of BCS-typeuE= i{l-(~k-A)/[(~k-h)*+A*l~/*} with a gap parameter A taken from experimental oddeven mass differences. In all cases, where the only essential effect of pairing correlations is a smearing of the Fermi surface, this procedure works very well. It has only a small problem at large momenta, the corresponding pairing energy Epir = -A ck ukvk diverges. A cut-off has to be introduced in the pairing channel, a so-called pairing window, which takes into account the finite range of the force in a phenomenological way. Several prescriptions are on the market in this context, but since the pairing energy Epaircontributes to the total binding energy only in the percent region this has no essential influence on the success of the model. The limitations of this method show up in regions, where no experimental pairing gap is available, for instance for very exotic nuclei with large neutron or proton excess, or in the fission process, where very elongated shapes can possibly not be described by the same pairing strength as the ground state, and in all cases where the finite range of the pairing force is essential: One example are nuclei with high angular momenta, where alignment processes cannot be taken into account properly by a monopole pairing force only or halo nuclei, where at very small densities the pairing force has to go over into the free nucleon-nucleon force [ 21. There are therefore several modifications of the Skyrme force on the market which use zero range forces also in the pairing channel [3,2,4]. They all need a cut-off, which is adjusted in specific regions, but which reduces the predictive power in other regions considerably. Gogny and his group [ 51 went an essential step forward. They realized that the pairing correlations depend in a very sensitive way on the finite range in the effective force in the pairing channel. Therefore they took from the Skyrme model, in a slightly modified version, the strong density dependent repulsive term in the T = 0 channel as well as the spin orbit term. They replaced, however, the remaining zero-range parts of the Skyrme force by a sum of two Gaussians with finite range and properly chosen spin- and isospinadmixture. For T = l-pairing, which is crucial for all heavy nuclei with open shells, the strong density dependent repulsive term does enter the pairing channel and therefore the finite range of the Gaussian terms which is adjusted to a microscopically determined G-
Letters B 379 (1996) 13-19
matrix guarantees a proper cut off in momentum space. This force can therefore be used for full Hartree-FockBogoliubov calculations in the entire periodic region as well as for rotating nuclei with great success [ 5-71. One no longer depends on the knowledge of experimental mass differences nor on the largely unknown size of quadruple pairing terms and so on. In recent years it has been found that a relativistic mean field (RMF) theory based on effective Lagrangians [ 8,9] can provide a very simple and very elegant parametrization for the description of ground state properties of nuclei. Using only a very small number of mesons describing in a phenomenological way essential properties of the nuclear many-body system, such as the u-meson for the large scalar attraction, the w-meson for the vector repulsion at short distances or the p-meson for the asymmetry properties in nuclei with large neutron or proton excess and fine tuning the surface properties by nonlinear terms in the a-field [ lo] one is able nowadays to describe with only six parameters in an astonishingly accurate way, many essential ground state properties of nuclei all over the periodic table ill]. Recently such investigations have been extended also with great success to excited nuclei such rotating [ 121 and vibrating [ 131 systems. Pairing correlations are usually taken into account in the constant gap approximation [ 141 as in the Skyrme model. In principle a consistent treatment of pairing correlations in the framework of a generalized relativistic mean field approximation has been developed by Kucharek et al. [ 151. However, applications in nuclear matter show clearly that a quantitative description of pairing correlations in the nuclear many-body system cannot be achieved in this way with the presently used parameter sets of relativistic mean field theory. The behavior of the meson exchange forces entering this theory is simply not properly adjusted at large momentum transfer. In principle these forces have finite range and kinematical factors guarantee the convergence of the relativistic gap equation. The large masses of the (T- and the w-mesons, however, do not yield a realistic cut off. Therefore the situation is similar to that of Skyrme forces with zero range: The short range of the relativistic forces produces no problem in the HartreeFock case, where only momenta up to the Fermi surface are involved, but it causes severe problems in the description of pairing.
T. Gonzalez-Llarena et al./Physics
The idea of the present work is therefore to combine in a phenomenological way the advantages of a relativistic description in the framework of RMFtheory with the good pairing properties of the Gogny force. From a microscopic theory of pairing correlations [ 161 one knows that one may use different forces in the particle-particle and in the particle-hole channel: the effective force for HF calculations in the phchannel is in Bruckner’s approximation the G-matrix, the sum over all ladder diagrams, whereas the effective force in the pp-channel, the K-matrix, is the sum of all diagrams irreducible in the pp-direction [ 161. We therefore propose to use in the Hartree-channel as usual the conventional RMF-theory and in the pairing channel Gogny’s finite range force. In fact relativistic effects do not play a role in the pp-channel, where we have do deal with matrix elements of a few MeV only. In the ph-channel relativistic effects are very important, because Lorentz scalar and vector fields are each of them several hundreds of MeV in size and the complicated interplay between Lorentz scalar and Lorentz vector quantities is essential for nuclear saturation [ 91. We therefore start with the following Lagrangian:
Letters B 379 (1996) 13-19
15
Using Greens function techniques it is shown in Ref. [ 151 how one can derive a relativistic Hartree-FockBogoliubov theory from such a Lagrangian: After a full quantization of the system the mesonic degrees of freedom are eliminated and the higher order Greens functions are factorized in the sense of Gorkov [ 171, in full analogy to the non-relativistic case. Finally Neglecting retardation effects one ends up with relativistic Dirac-Hartree-Fock-Bogoliubov equations,
(_;*_:*)(;>,=a(F>,T
Ek are qUaSipartiCle energies and the Coefficients uk and vk are four-dimensional Dirac spinors normalized in the following way:
s
u/$&t + v+vk,d3r = t&k!.
- U(a)
- $JP”
+ irn$vP,wP
- $R,,R~” f $msp,pC” - $F~~F~U,
.
(2)
For a realistic description of nuclear properties a nonlinear self-coupling for the scalar mesons has turned out to be crucial [ lo] : U(a)
= irnsu2 + +g2& + $g3u4.
where (+ and w are the meson fields determined consistently from the Klein-Gordon equations:
(6) self-
(1)
where MN is the bare nucleon mass and $ is its Dirac spinor. We have in addition the scalar meson (g), isoscalar vector mesons (0P), isovector vector mesons ( pfi) and the photons (Afi), with the masses mrr, m, and mP and the coupling constants g,,, g,, gP. For simplicity in the following equations we neglect the p-meson and the photon. In the calculations these contributions are, however, taken into account. The field tensors for the vector mesons are given as dz,, = dP~, - a,ti,
(5)
Neglecting the Fock term, as is it mostly done in relativistic mean field theory, we obtain for the average field
h=cup+g,w+P(M+g,u)-A, + @,adPu
(4)
(3)
In the following calculations a center of mass correction for the energy EC,,,= - $4 1A-‘i3 has been taken into account.
(-A
+ rn$-} (T = -g,p,
{-A
+ m;} 0 = &pB ,
- gw2 - gw3 ,
(7) (8)
with the scalar density ps and the baryon density pi (9) k
k
where the sum over k runs only over all the particle states in the no-sea approximation. The pairing potential A in Eq. (4) is given by 1 A ob = 2 c
VP’ nbcd K cd *
(10)
cd
It is obtained from the pairing tensor K = U*VT and the one-meson exchange interaction V:id in the ppchannel. More details are given in Ref. [ 151. As mentioned above, these forces are not able to reproduce even in a semi-quantitative way proper pairing in the
T. Gonzalez-Llarena et al. /Physics Letters B 379 (1996) 13-19
16
Table I The binding energy per particle E/A and the pairing energy I&. of Pb isotopes obtained in the Relativistic Hartree Bogoliubov (RHB) calculations with the parameter set NLl[ 141 and in a non-relativistic HFB-calculation. Details are given in the text. The experimental binding energies are shown for comparison
Table 2 Results for Sn isotopes.
A
112 114 116 118 120 122 124 126 128 130 132
E/A
202 204 206 208 210 212 214
&air
expt.
RHB
Gogny
RHB
Gogny
-7.882 -7.880 -7.876 -7.868 -7.836 -7.805 -7.772
-7.908 -7.903 -7.895 -7.884 -7.841 -7.798 -7.754
-7.811 -7.813 -7.813 -7.815 -7.769 -7.730 -7.692
-14.49 -10.41 -5.57 0.00 -4.79 -8.66 -11.77
-14.41 -10.49 -5.74 0.00 -4.19 -8.64 -12.89
realistic nuclear many-body problem. We therefore replace V$Ed in Eq. (IO) by a two-body force of finite range of Gogny type
V”” ( 1 ) 2) =
c ei=l,2
X
(Wi + BiPa - Hip’ - MiP”P’)
,
(11)
with the parameters pi, Wi, Bi, Hi, and Mi (i = 1,2). In fact this replacement does not violate the variational principle, because we could have obtained identical equations by just subtracting a pairing energy of the form
(12) from the Lagrangian ( 1) and using standard variational techniques for HFB-equations as they are discussed for instance in chapter 7 of Ref. [ 201. The resulting Hartree-Bogoliubov equations (4) and the meson equations (8) are solved by expanding the Dirac spinors Uk ( I) and Vk( r) and the meson fields in a basis of spherical harmonic oscillators up to the major quantum number N = 20 for the large components and N = 21 for the small components. Details are given in Ref. [ 111. The evaluation of the finite range matrix elements of Gaussian shape is carried out using the techniques of Ref. [ 181. In each step of the iteration the chemical potential A is determined by the particle number.
A
The same forces are used as in Table
WA
1
Epair
expt.
RHB
-8.513 -8.523 -8.523 -8.517 -8.505 -8.489 -8.467 -8.444 -8.418 -8.388 -8.355
-8.558 -8.564 -8.563 -8.554 -8.538 -8.515 -8.487 -8.453 -8.414 -8.369 -8.319
Gogny
RHB
Gogny
-8.419 -8.433 -8.437 -8.43 -8.417 -8.400 -8.378 -8.353 -8.326 -8.300 -8.283
-22.84 -22.93 -22.75 -22.40 -21.89 -21.06 -19.68 -17.44 -13.97 -8.72 0.00
-19.04 -19.29 -19.39 -19.15 -17.92 -16.76 -14.94 -12.50 -9.45 -5.48 0.00
I
In the following we discuss several applications of this theory for the isotopic chains 202Pb-214Pb, ’ 12Sn‘32Sn and the exotic chain *‘8Zr-‘26Zr. We compare relativistic Hartree-Bogoliubov calculations (RHB) in an oscillator basis of N = 20 major shells with nonrelativistic HFB-calculations using the Gogny force DlS [ 191 in a basis of N = 14 major shells. In the relativistic case we used in the pairing channel for all these calculations the parameter set DlS renormalized by an overall factor 1.15, such that the pairing energy of the Pb isotopes obtained in the relativistic calculations is more or less in agreement with the nonrelativistic results. In Table 1 we compare the binding energies per particle E/A of spherical Pb isotopes obtained in the relativistic Hartree-Bogoliubov theory with the experiment and with the values obtained in a non-relativistic Hartree-Fock-Bogoliubov calculation with the Gogny force. We observe in both calculations excellent agreement between theoretical and experimental results. It is somewhat better for the relativistic calculations, which produce in general a slight over-binding of roughly 0.2% in contrast to the non-relativistic calculations, where we find a small under-binding of roughly 1%. This small difference could be connected with the fact that the non-relativistic calculations are carried out with N = 14 oscillator shells only as compared to N = 20 shells for the relativistic calculations or it could be a minor weakness in the parametrization of the Gogny force. For the isotopes with the mass numbers A = 212 and A = 214, both theories give an
T. Gonzalez-Llarena et al. /Physics Letters B 379 (1996) 13-19
Q
Isotope Shifts: Pb Nuclei
I
“g “2
0.0 -
4
P-qDl
-0.4 200
204
208
212
216
A
Fig. 1. Isotope shifts of Ph nuclei obtained in the relativistic Hartree-Bolgoliubov calculations with the parameter sets NLl and NL-SH. Details are given in the text. These results are compared with the experiment and with non-relativistic HFB calculations based on Gogny’s force Dl and DlS and with a HPBCS calculation based on the Skyrme force SkM* from Ref. [4].
under-binding of less then 1%. This might be understood by the fact that these nuclei belong already to the transitional region where fluctuations in deformation space become increasingly important. We also show in Table 1 the pairing energy [ 201 of neutrons E+r = - iTrAK*. This is not an experimentally accessible quantity, but it is a measure for the size of the pairing correlations in the theoretical calculations. In both theories we find as expected, vanishing pairing correlations for the double magic nucleus 208Pb and steeply increasing pairing correlations if we go away from the magic neutron number N = 126. In Table 2 we show similar results obtained with the same parameter set for a large chain of isotopes in the Sn region. Again we find very good agreement with experimental binding energies in the 1% range for the non-relativistic theory and in the 0.5% range for the relativistic theory. Again there is a slight over-binding in the relativistic theory and a slight under-binding for the non-relativistic theory in most cases. The pairing energy is strongest in the half filled shell at A e 114116 and it decreases to zero if we reach the double magic configuration ‘32Sn. In Fig. 1 we show the isotope shifts in the Pb region with the famous kink at the magic nucleus
17
208Pb, which has been investigated in great detail with non-relativistic Skyrme forces in Ref. [ 41 and within relativistic mean field theory in Ref. [ 211. In Refs. [ 22,231 it could be shown that the isospin dependence in conventional Skyrme forces which is different from that in relativistic theories is the origin of this discrepancy. In Fig. 1 we show non-relativistic HF + BCS calculations with the Skyrme force SKM* [ 251 and non-relativistic HFB calculations with two Gogny forces Dl [ 51 and DlS [ 191 together with relativistic RHB calculations using two parameter sets NLl [ 141 and NL-SH [24] in both cases with the finite range pairing part of the DlS force modified by a factor 1.15 as discussed above. In earlier calculations [ 211 pairing was treated only in the constant gap approximations and it was shown in Ref. [ 41 that modifications of the pairing force can have influence on the kink behavior of the isotope shifts. It is satisfying to see that the present improved treatment of pairing with a finite range Gogny force does not change the good agreement with experimental data. Since nonrelativistic Gogny calculations use the same two-body spin orbit force as the Skyrme calculations, this could be the reason why these calculations also fail to reproduce the kink in the experimental isotope shifts properly. Finally we turn to exotic nuclei with a very large neutron excess close to the neutron drip line, the chain of Zr isotopes with Neutron numbers 78-86, which play a role in the understanding of the r-process in astrophysics [ 261. The question whether shell effects are quenched at the drip line is still much debated: In non-relativistic HFB-calculations in the continuum based on the Skyrme force SkP [ 271 shell effects are washed out at the drip line, whereas in relativistic calculations [28] one observes a clear shell effect at the magic number N = 82 for ‘**Zr. There have been two discrepancies between both calculations: (a) the isospin dependence of the spin orbit term is rather different in both theories, as we have seen before. In the relativistic theory this leads to clear a gap in the single particle spectrum at the magic number N = 82, in the Skyrme calculations this is not the case and pairing stays important, (b) the relativistic calculations have so far been carried out only in the constant gap, i.e. BCS approximation. This is certainly a deficiency, because the continuum is not treated properly in such a theory. In fact several levels in the continuum are
T. Gonzalez-Llarena et al./Physics
18
I
I
I
,
,
Zr Isotopes about Drip line
-940
1
m
118
I 120
122 A
124
126
Fig. 2. Binding energies of Zr isotopes in the neighborhood of the magic neutron number N = 82. Relativistic Hartree-Bogoliubov calculations with the parameter sets NLI and NL-SH based on a finite range pairing force ( 11) are compared wifh the predictions of the finite-range droplet model of Ref. [29]. The parameters of the pairing force are those of Gogny’s force DIS multiplied with an overall strength factor 1.15.
partially occupied, which leads to a unstable situation. As has been shown by Dobaczewski et al. [3] HFBtheory treat the continuum properly. Because of the essential difference between the canonical basis and the quasiparticle basis in such a theory a nucleus can be localized, although its Fermi level is close to the continuum limit and the occupations in the canonical basis are distributed on both sides of this level. In order to investigate this question more carefully we have carried out full relativistic Hartree-Bogoliubov calculations in the region of the nucleus 122Zr. In fact, as shown in Fig. 2, we observe again a shell effect (a kink in the binding energy) at the magic number N = 82 in full agreement with the old relativistic calculations using a BCS-treatment and in agreement with finite range droplet model of Nix et al. [ 291. As described above, our calculations are carried out in a finite oscillator basis of 20 major shells and comparing it with calculations in 18 and 19 shells we found convergence. Therefore we think that we have found a localized solution of the full Hartree-Bogoliubov equations with a proper treatment of the continuum. We can conclude, that the gap in the single particle spectrum caused by the spin orbit term is crucial for the occurrence of shell effects. Since this gap is large, reasonable pair-
Letters B 379 (1996) 13-19
ing correlations as those of the Gogny force do not change this result considerably, even if the coupling to the continuum is taken into account properly. Let us summarize: As long as a full microscopic theory of pairing forces is still missing, we propose to use the very successful phenomenological pairing force of Gogny in connection with relativistic mean field calculations. In this case we have the advantages of the relativistic scheme, which turned out to be so fruitful in recent years, as well as the proper treatment of pairing correlations where finite range forces are important. In this work we have applied this method to several problems recently under investigation, such as pairing correlations in single magic isotope chains, the isotope shifts in the Pb-region and the question of shell effects for very neutron rich nuclei. In the first two cases we could reproduce and sometimes even slightly improve the results of older relativistic calculations, which could only be obtained in the framework of the constant gap approximation using experimental values for the gap parameters. Having reproduced these results without the help of experimental gap parameters we can hope to have now a theory with a high predictive power also in regions, where experimental masses are not available, and in cases where the finite range of the effective force in the pairing channel is crucial. This work has been supported in part by DGICyT, Spain under project PB94-0164. P.R. wishes to express his gratitude to the Spanish Ministry for Education for support of his work at the Universidad Autbnoma de Madrid in the framework of the A. von Humboldt - J.C. Mutis program for the Scientific Cooperation between Spain and Germany. G.A.L. acknowledges support by the European Union under the contract HCM-EG/ERB CHBICT-9305 1. In addition we acknowledge partial support of this work by the BMFT, Germany under contract number 06 TM 743 (6). References [ 11 D. Vautherin and D.M. Brink, Phys. Rev. C 5 ( 1972) 626. [2] G.E Bertsch and H. Esbensen, Ann. Phys. 209 ( 1991) 327. [3] J. Dobaczewski, H. Placard and J. Treiner, Nucl. Phys. A 422 (1984) 103. [4] N. Tajima, l? Bonche, H. Flocard, P.-H. Heenen and MS. Weiss, Nucl. Phys. A 551 (1993) 434.
T. Gonzalez-Llarena
et al./Physics
] 5 ] J. Decharge and D. Gogny, Phys. Rev. C 21 ( 1980) 1568. ]6] J.L. Egido and L. Robledo, Phys. Rev. Lett. 70 ( 1993) 2876. [7] M. Girod, J.P. Delaroche and J.F. Berger, Phys. L&t. B 325 (1994) 1. [S] J.D. Walecka, Ann. Phys. (N.Y.) 83 (1974) 491. [9 I B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [ IO] J. Boguta and A.R. Bodmer, Nucl. Phys. A 292 (1977) 413. [ 111 Y.K. Gambhir, P Ring and A. Thimet, Ann. Phys. (N.Y.) 198 (1990) 132. 1121 J. Konig and P. Ring, Phys. Rev. Lett. 71 (1993) 3079. ] 13 ] D. Vretenar, H. Berghammer and l? Ring, Nucl. Phys. A 581 ( 1994) 679. [ 141 P.G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A 323 (1986) 13. [ 15 1 H. Kucharek and P Ring, Z. Phys. A 339 (1991) 23. ] 16 1 A.B. Migdal, Theory of Finite Fermi Systems, Applications to Atomic Nuclei (Wiley Interscience, New York, 1967). [ 17 1 L.P. Gorkov, Sov. Phys. JETP 7 (1958) 505. [ 18 I J.D. Talman, Nucl. Phys. A 141 ( 1970) 273. I 19 1 J.F. Berger, M. Girod and D. Gogny, Nucl. Phys. A 428 ( 1984) 32~.
Letters B 379 (1996) 13-19 [20] [21] [22] [23] 1241 [25] [26] [27] [28]
[29]
19
P Ring and P. Schuck, The Nuclear Many-body Problem (Springer Verlag, Heidelberg, 1980). M.M. Sharma, G.A. Lalazissis and P Ring, Phys. Lett. B 317 (1993) 9. M.M. Sharma, G.A. Lalazissis, J. Konig and P Ring, Phys. Rev. Lett. 74 (1995) 3744. P-G. Reinhard and H. Flocard, Nucl. Phys. A 584 (1995) 467. M.M. Sharma, M.A. Nagarajan and P Ring, Phys. Len. B 312 (1993) 377. M. Brack, C. Guet and H.B. Hakansson, Phys. Rep. 123 (1985) 275. K.-L. Kratz, Rev. Mod. A&on. 1 (1988) 184. J. Dobaczewski, I. Hamamoto, W. Nazarewicz and and J.A. Sheikh, Phys. Rev. Len. 72 (1994) 981. M.M. Sharma, G.A. Lalazissis, W. Hillebrandt and I? Ring, Phys. Rev. Lett. 72 (1994) 1431; see also Phys. Rev. Lett. 73 (1994) 1869. P Moller, J.R. Nix, W.D. Myers and W.J. Swiatecki, Atomic Data and Nuclear Data Tables 59 ( 1995) 185.
27June1996
__ BB
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 379 (1996) 13-19
Relativistic Hartree-Bogoliubov calculations with finite range pairing forces T. Gonzalez-Llarenaa,
J.L. Egido a, G.A. Lalazissis b, P. Ring b
a Departamento de Fisica Tedrica C-XI, Universidad Autrinom de Madrid, E-28049, Spain h Physikdeparfment, Technische Universitiit Miinchen, D-85747 Garching, Germany
Received 4 October 1995; revised manuscript received 10 April 1996 Editor: C. Mahaux
Abstract We propose a relativistic extension correlations are taken into account by a fully self-consistent description of investigation of several isotope chains
of Hartree-Bogoliubov theory on the basis of relativistic mean field theory. Pairing an effective interaction of finite range adjusted by Gogny and his group. This allows pairing correlations over the entire periodic table. The method is applied for the of spherical Pb, Sn, and Zr nuclei with magic proton configurations.
PACT: 21.10.Re; 21.60.52; 24.3O.C~
For many years it has been known that ground state properties of nuclei can be well described in the framework of mean field theories with effective forces. Within the non-relativistic scheme the essential breakthrough in this respect was achieved by Vautherin and Brink [ 11, who used an effective force of Skyrme type, i.e. a strongly repulsive density dependent term of zero range in the T = 0 channel in order to guarantee proper saturation of infinite nuclear matter and of finite nuclei. For reasons of numerical simplicity they have also chosen zero range for the attractive part of the effective force as well as for the spin orbit part. Finite range effects were taken into account in an approximate way by a quadratic momentum dependence. This method is extremely successful for Hartree-Fock calculations of spherical doubly magic nuclei, where a sharp Fermi surface guarantees, that the components of this force with large momentum transfer do not play any role. In fact the behavior of the Skyrme force
at large momentum transfer, i.e. at short distances is completely wrong but it has no influence on HartreeFock calculations with a sharp Fermi surface, where only states with small momenta are occupied. In open shell nuclei, where pairing correlations play an essential role, the situation becomes more involved. In fact it is well known that Skyrme forces cannot be used directly for the calculation of pairing properties. It is not the strong density dependent term, which causes problems because it acts only in the T = 0 channel. It is the attractive part of zero range, which is essentially constant in momentum space and leads to a divergence in the gap equation at large momenta. The quadratic momentum dependence does not change this situation. Pairing correlations are therefore taken into account in the Skyrme scheme usually in only a very phenomenological way in a BCS-theory with monopole pairing force adjusted to the experimental odd-even mass difference, i.e. in constant gap approx-
0370-2693/96/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00461-3
14
T. Gonzalez-Llarena
et al./Physics
imation. In practice one just uses occupation factors of BCS-typeuE= i{l-(~k-A)/[(~k-h)*+A*l~/*} with a gap parameter A taken from experimental oddeven mass differences. In all cases, where the only essential effect of pairing correlations is a smearing of the Fermi surface, this procedure works very well. It has only a small problem at large momenta, the corresponding pairing energy Epir = -A ck ukvk diverges. A cut-off has to be introduced in the pairing channel, a so-called pairing window, which takes into account the finite range of the force in a phenomenological way. Several prescriptions are on the market in this context, but since the pairing energy Epaircontributes to the total binding energy only in the percent region this has no essential influence on the success of the model. The limitations of this method show up in regions, where no experimental pairing gap is available, for instance for very exotic nuclei with large neutron or proton excess, or in the fission process, where very elongated shapes can possibly not be described by the same pairing strength as the ground state, and in all cases where the finite range of the pairing force is essential: One example are nuclei with high angular momenta, where alignment processes cannot be taken into account properly by a monopole pairing force only or halo nuclei, where at very small densities the pairing force has to go over into the free nucleon-nucleon force [ 21. There are therefore several modifications of the Skyrme force on the market which use zero range forces also in the pairing channel [3,2,4]. They all need a cut-off, which is adjusted in specific regions, but which reduces the predictive power in other regions considerably. Gogny and his group [ 51 went an essential step forward. They realized that the pairing correlations depend in a very sensitive way on the finite range in the effective force in the pairing channel. Therefore they took from the Skyrme model, in a slightly modified version, the strong density dependent repulsive term in the T = 0 channel as well as the spin orbit term. They replaced, however, the remaining zero-range parts of the Skyrme force by a sum of two Gaussians with finite range and properly chosen spin- and isospinadmixture. For T = l-pairing, which is crucial for all heavy nuclei with open shells, the strong density dependent repulsive term does enter the pairing channel and therefore the finite range of the Gaussian terms which is adjusted to a microscopically determined G-
Letters B 379 (1996) 13-19
matrix guarantees a proper cut off in momentum space. This force can therefore be used for full Hartree-FockBogoliubov calculations in the entire periodic region as well as for rotating nuclei with great success [ 5-71. One no longer depends on the knowledge of experimental mass differences nor on the largely unknown size of quadruple pairing terms and so on. In recent years it has been found that a relativistic mean field (RMF) theory based on effective Lagrangians [ 8,9] can provide a very simple and very elegant parametrization for the description of ground state properties of nuclei. Using only a very small number of mesons describing in a phenomenological way essential properties of the nuclear many-body system, such as the u-meson for the large scalar attraction, the w-meson for the vector repulsion at short distances or the p-meson for the asymmetry properties in nuclei with large neutron or proton excess and fine tuning the surface properties by nonlinear terms in the a-field [ lo] one is able nowadays to describe with only six parameters in an astonishingly accurate way, many essential ground state properties of nuclei all over the periodic table ill]. Recently such investigations have been extended also with great success to excited nuclei such rotating [ 121 and vibrating [ 131 systems. Pairing correlations are usually taken into account in the constant gap approximation [ 141 as in the Skyrme model. In principle a consistent treatment of pairing correlations in the framework of a generalized relativistic mean field approximation has been developed by Kucharek et al. [ 151. However, applications in nuclear matter show clearly that a quantitative description of pairing correlations in the nuclear many-body system cannot be achieved in this way with the presently used parameter sets of relativistic mean field theory. The behavior of the meson exchange forces entering this theory is simply not properly adjusted at large momentum transfer. In principle these forces have finite range and kinematical factors guarantee the convergence of the relativistic gap equation. The large masses of the (T- and the w-mesons, however, do not yield a realistic cut off. Therefore the situation is similar to that of Skyrme forces with zero range: The short range of the relativistic forces produces no problem in the HartreeFock case, where only momenta up to the Fermi surface are involved, but it causes severe problems in the description of pairing.
T. Gonzalez-Llarena et al./Physics
The idea of the present work is therefore to combine in a phenomenological way the advantages of a relativistic description in the framework of RMFtheory with the good pairing properties of the Gogny force. From a microscopic theory of pairing correlations [ 161 one knows that one may use different forces in the particle-particle and in the particle-hole channel: the effective force for HF calculations in the phchannel is in Bruckner’s approximation the G-matrix, the sum over all ladder diagrams, whereas the effective force in the pp-channel, the K-matrix, is the sum of all diagrams irreducible in the pp-direction [ 161. We therefore propose to use in the Hartree-channel as usual the conventional RMF-theory and in the pairing channel Gogny’s finite range force. In fact relativistic effects do not play a role in the pp-channel, where we have do deal with matrix elements of a few MeV only. In the ph-channel relativistic effects are very important, because Lorentz scalar and vector fields are each of them several hundreds of MeV in size and the complicated interplay between Lorentz scalar and Lorentz vector quantities is essential for nuclear saturation [ 91. We therefore start with the following Lagrangian:
Letters B 379 (1996) 13-19
15
Using Greens function techniques it is shown in Ref. [ 151 how one can derive a relativistic Hartree-FockBogoliubov theory from such a Lagrangian: After a full quantization of the system the mesonic degrees of freedom are eliminated and the higher order Greens functions are factorized in the sense of Gorkov [ 171, in full analogy to the non-relativistic case. Finally Neglecting retardation effects one ends up with relativistic Dirac-Hartree-Fock-Bogoliubov equations,
(_;*_:*)(;>,=a(F>,T
Ek are qUaSipartiCle energies and the Coefficients uk and vk are four-dimensional Dirac spinors normalized in the following way:
s
u/$&t + v+vk,d3r = t&k!.
- U(a)
- $JP”
+ irn$vP,wP
- $R,,R~” f $msp,pC” - $F~~F~U,
.
(2)
For a realistic description of nuclear properties a nonlinear self-coupling for the scalar mesons has turned out to be crucial [ lo] : U(a)
= irnsu2 + +g2& + $g3u4.
where (+ and w are the meson fields determined consistently from the Klein-Gordon equations:
(6) self-
(1)
where MN is the bare nucleon mass and $ is its Dirac spinor. We have in addition the scalar meson (g), isoscalar vector mesons (0P), isovector vector mesons ( pfi) and the photons (Afi), with the masses mrr, m, and mP and the coupling constants g,,, g,, gP. For simplicity in the following equations we neglect the p-meson and the photon. In the calculations these contributions are, however, taken into account. The field tensors for the vector mesons are given as dz,, = dP~, - a,ti,
(5)
Neglecting the Fock term, as is it mostly done in relativistic mean field theory, we obtain for the average field
h=cup+g,w+P(M+g,u)-A, + @,adPu
(4)
(3)
In the following calculations a center of mass correction for the energy EC,,,= - $4 1A-‘i3 has been taken into account.
(-A
+ rn$-} (T = -g,p,
{-A
+ m;} 0 = &pB ,
- gw2 - gw3 ,
(7) (8)
with the scalar density ps and the baryon density pi (9) k
k
where the sum over k runs only over all the particle states in the no-sea approximation. The pairing potential A in Eq. (4) is given by 1 A ob = 2 c
VP’ nbcd K cd *
(10)
cd
It is obtained from the pairing tensor K = U*VT and the one-meson exchange interaction V:id in the ppchannel. More details are given in Ref. [ 151. As mentioned above, these forces are not able to reproduce even in a semi-quantitative way proper pairing in the
T. Gonzalez-Llarena et al. /Physics Letters B 379 (1996) 13-19
16
Table I The binding energy per particle E/A and the pairing energy I&. of Pb isotopes obtained in the Relativistic Hartree Bogoliubov (RHB) calculations with the parameter set NLl[ 141 and in a non-relativistic HFB-calculation. Details are given in the text. The experimental binding energies are shown for comparison
Table 2 Results for Sn isotopes.
A
112 114 116 118 120 122 124 126 128 130 132
E/A
202 204 206 208 210 212 214
&air
expt.
RHB
Gogny
RHB
Gogny
-7.882 -7.880 -7.876 -7.868 -7.836 -7.805 -7.772
-7.908 -7.903 -7.895 -7.884 -7.841 -7.798 -7.754
-7.811 -7.813 -7.813 -7.815 -7.769 -7.730 -7.692
-14.49 -10.41 -5.57 0.00 -4.79 -8.66 -11.77
-14.41 -10.49 -5.74 0.00 -4.19 -8.64 -12.89
realistic nuclear many-body problem. We therefore replace V$Ed in Eq. (IO) by a two-body force of finite range of Gogny type
V”” ( 1 ) 2) =
c ei=l,2
X
(Wi + BiPa - Hip’ - MiP”P’)
,
(11)
with the parameters pi, Wi, Bi, Hi, and Mi (i = 1,2). In fact this replacement does not violate the variational principle, because we could have obtained identical equations by just subtracting a pairing energy of the form
(12) from the Lagrangian ( 1) and using standard variational techniques for HFB-equations as they are discussed for instance in chapter 7 of Ref. [ 201. The resulting Hartree-Bogoliubov equations (4) and the meson equations (8) are solved by expanding the Dirac spinors Uk ( I) and Vk( r) and the meson fields in a basis of spherical harmonic oscillators up to the major quantum number N = 20 for the large components and N = 21 for the small components. Details are given in Ref. [ 111. The evaluation of the finite range matrix elements of Gaussian shape is carried out using the techniques of Ref. [ 181. In each step of the iteration the chemical potential A is determined by the particle number.
A
The same forces are used as in Table
WA
1
Epair
expt.
RHB
-8.513 -8.523 -8.523 -8.517 -8.505 -8.489 -8.467 -8.444 -8.418 -8.388 -8.355
-8.558 -8.564 -8.563 -8.554 -8.538 -8.515 -8.487 -8.453 -8.414 -8.369 -8.319
Gogny
RHB
Gogny
-8.419 -8.433 -8.437 -8.43 -8.417 -8.400 -8.378 -8.353 -8.326 -8.300 -8.283
-22.84 -22.93 -22.75 -22.40 -21.89 -21.06 -19.68 -17.44 -13.97 -8.72 0.00
-19.04 -19.29 -19.39 -19.15 -17.92 -16.76 -14.94 -12.50 -9.45 -5.48 0.00
I
In the following we discuss several applications of this theory for the isotopic chains 202Pb-214Pb, ’ 12Sn‘32Sn and the exotic chain *‘8Zr-‘26Zr. We compare relativistic Hartree-Bogoliubov calculations (RHB) in an oscillator basis of N = 20 major shells with nonrelativistic HFB-calculations using the Gogny force DlS [ 191 in a basis of N = 14 major shells. In the relativistic case we used in the pairing channel for all these calculations the parameter set DlS renormalized by an overall factor 1.15, such that the pairing energy of the Pb isotopes obtained in the relativistic calculations is more or less in agreement with the nonrelativistic results. In Table 1 we compare the binding energies per particle E/A of spherical Pb isotopes obtained in the relativistic Hartree-Bogoliubov theory with the experiment and with the values obtained in a non-relativistic Hartree-Fock-Bogoliubov calculation with the Gogny force. We observe in both calculations excellent agreement between theoretical and experimental results. It is somewhat better for the relativistic calculations, which produce in general a slight over-binding of roughly 0.2% in contrast to the non-relativistic calculations, where we find a small under-binding of roughly 1%. This small difference could be connected with the fact that the non-relativistic calculations are carried out with N = 14 oscillator shells only as compared to N = 20 shells for the relativistic calculations or it could be a minor weakness in the parametrization of the Gogny force. For the isotopes with the mass numbers A = 212 and A = 214, both theories give an
T. Gonzalez-Llarena et al. /Physics Letters B 379 (1996) 13-19
Q
Isotope Shifts: Pb Nuclei
I
“g “2
0.0 -
4
P-qDl
-0.4 200
204
208
212
216
A
Fig. 1. Isotope shifts of Ph nuclei obtained in the relativistic Hartree-Bolgoliubov calculations with the parameter sets NLl and NL-SH. Details are given in the text. These results are compared with the experiment and with non-relativistic HFB calculations based on Gogny’s force Dl and DlS and with a HPBCS calculation based on the Skyrme force SkM* from Ref. [4].
under-binding of less then 1%. This might be understood by the fact that these nuclei belong already to the transitional region where fluctuations in deformation space become increasingly important. We also show in Table 1 the pairing energy [ 201 of neutrons E+r = - iTrAK*. This is not an experimentally accessible quantity, but it is a measure for the size of the pairing correlations in the theoretical calculations. In both theories we find as expected, vanishing pairing correlations for the double magic nucleus 208Pb and steeply increasing pairing correlations if we go away from the magic neutron number N = 126. In Table 2 we show similar results obtained with the same parameter set for a large chain of isotopes in the Sn region. Again we find very good agreement with experimental binding energies in the 1% range for the non-relativistic theory and in the 0.5% range for the relativistic theory. Again there is a slight over-binding in the relativistic theory and a slight under-binding for the non-relativistic theory in most cases. The pairing energy is strongest in the half filled shell at A e 114116 and it decreases to zero if we reach the double magic configuration ‘32Sn. In Fig. 1 we show the isotope shifts in the Pb region with the famous kink at the magic nucleus
17
208Pb, which has been investigated in great detail with non-relativistic Skyrme forces in Ref. [ 41 and within relativistic mean field theory in Ref. [ 211. In Refs. [ 22,231 it could be shown that the isospin dependence in conventional Skyrme forces which is different from that in relativistic theories is the origin of this discrepancy. In Fig. 1 we show non-relativistic HF + BCS calculations with the Skyrme force SKM* [ 251 and non-relativistic HFB calculations with two Gogny forces Dl [ 51 and DlS [ 191 together with relativistic RHB calculations using two parameter sets NLl [ 141 and NL-SH [24] in both cases with the finite range pairing part of the DlS force modified by a factor 1.15 as discussed above. In earlier calculations [ 211 pairing was treated only in the constant gap approximations and it was shown in Ref. [ 41 that modifications of the pairing force can have influence on the kink behavior of the isotope shifts. It is satisfying to see that the present improved treatment of pairing with a finite range Gogny force does not change the good agreement with experimental data. Since nonrelativistic Gogny calculations use the same two-body spin orbit force as the Skyrme calculations, this could be the reason why these calculations also fail to reproduce the kink in the experimental isotope shifts properly. Finally we turn to exotic nuclei with a very large neutron excess close to the neutron drip line, the chain of Zr isotopes with Neutron numbers 78-86, which play a role in the understanding of the r-process in astrophysics [ 261. The question whether shell effects are quenched at the drip line is still much debated: In non-relativistic HFB-calculations in the continuum based on the Skyrme force SkP [ 271 shell effects are washed out at the drip line, whereas in relativistic calculations [28] one observes a clear shell effect at the magic number N = 82 for ‘**Zr. There have been two discrepancies between both calculations: (a) the isospin dependence of the spin orbit term is rather different in both theories, as we have seen before. In the relativistic theory this leads to clear a gap in the single particle spectrum at the magic number N = 82, in the Skyrme calculations this is not the case and pairing stays important, (b) the relativistic calculations have so far been carried out only in the constant gap, i.e. BCS approximation. This is certainly a deficiency, because the continuum is not treated properly in such a theory. In fact several levels in the continuum are
T. Gonzalez-Llarena et al./Physics
18
I
I
I
,
,
Zr Isotopes about Drip line
-940
1
m
118
I 120
122 A
124
126
Fig. 2. Binding energies of Zr isotopes in the neighborhood of the magic neutron number N = 82. Relativistic Hartree-Bogoliubov calculations with the parameter sets NLI and NL-SH based on a finite range pairing force ( 11) are compared wifh the predictions of the finite-range droplet model of Ref. [29]. The parameters of the pairing force are those of Gogny’s force DIS multiplied with an overall strength factor 1.15.
partially occupied, which leads to a unstable situation. As has been shown by Dobaczewski et al. [3] HFBtheory treat the continuum properly. Because of the essential difference between the canonical basis and the quasiparticle basis in such a theory a nucleus can be localized, although its Fermi level is close to the continuum limit and the occupations in the canonical basis are distributed on both sides of this level. In order to investigate this question more carefully we have carried out full relativistic Hartree-Bogoliubov calculations in the region of the nucleus 122Zr. In fact, as shown in Fig. 2, we observe again a shell effect (a kink in the binding energy) at the magic number N = 82 in full agreement with the old relativistic calculations using a BCS-treatment and in agreement with finite range droplet model of Nix et al. [ 291. As described above, our calculations are carried out in a finite oscillator basis of 20 major shells and comparing it with calculations in 18 and 19 shells we found convergence. Therefore we think that we have found a localized solution of the full Hartree-Bogoliubov equations with a proper treatment of the continuum. We can conclude, that the gap in the single particle spectrum caused by the spin orbit term is crucial for the occurrence of shell effects. Since this gap is large, reasonable pair-
Letters B 379 (1996) 13-19
ing correlations as those of the Gogny force do not change this result considerably, even if the coupling to the continuum is taken into account properly. Let us summarize: As long as a full microscopic theory of pairing forces is still missing, we propose to use the very successful phenomenological pairing force of Gogny in connection with relativistic mean field calculations. In this case we have the advantages of the relativistic scheme, which turned out to be so fruitful in recent years, as well as the proper treatment of pairing correlations where finite range forces are important. In this work we have applied this method to several problems recently under investigation, such as pairing correlations in single magic isotope chains, the isotope shifts in the Pb-region and the question of shell effects for very neutron rich nuclei. In the first two cases we could reproduce and sometimes even slightly improve the results of older relativistic calculations, which could only be obtained in the framework of the constant gap approximation using experimental values for the gap parameters. Having reproduced these results without the help of experimental gap parameters we can hope to have now a theory with a high predictive power also in regions, where experimental masses are not available, and in cases where the finite range of the effective force in the pairing channel is crucial. This work has been supported in part by DGICyT, Spain under project PB94-0164. P.R. wishes to express his gratitude to the Spanish Ministry for Education for support of his work at the Universidad Autbnoma de Madrid in the framework of the A. von Humboldt - J.C. Mutis program for the Scientific Cooperation between Spain and Germany. G.A.L. acknowledges support by the European Union under the contract HCM-EG/ERB CHBICT-9305 1. In addition we acknowledge partial support of this work by the BMFT, Germany under contract number 06 TM 743 (6). References [ 11 D. Vautherin and D.M. Brink, Phys. Rev. C 5 ( 1972) 626. [2] G.E Bertsch and H. Esbensen, Ann. Phys. 209 ( 1991) 327. [3] J. Dobaczewski, H. Placard and J. Treiner, Nucl. Phys. A 422 (1984) 103. [4] N. Tajima, l? Bonche, H. Flocard, P.-H. Heenen and MS. Weiss, Nucl. Phys. A 551 (1993) 434.
T. Gonzalez-Llarena
et al./Physics
] 5 ] J. Decharge and D. Gogny, Phys. Rev. C 21 ( 1980) 1568. ]6] J.L. Egido and L. Robledo, Phys. Rev. Lett. 70 ( 1993) 2876. [7] M. Girod, J.P. Delaroche and J.F. Berger, Phys. L&t. B 325 (1994) 1. [S] J.D. Walecka, Ann. Phys. (N.Y.) 83 (1974) 491. [9 I B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [ IO] J. Boguta and A.R. Bodmer, Nucl. Phys. A 292 (1977) 413. [ 111 Y.K. Gambhir, P Ring and A. Thimet, Ann. Phys. (N.Y.) 198 (1990) 132. 1121 J. Konig and P. Ring, Phys. Rev. Lett. 71 (1993) 3079. ] 13 ] D. Vretenar, H. Berghammer and l? Ring, Nucl. Phys. A 581 ( 1994) 679. [ 141 P.G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A 323 (1986) 13. [ 15 1 H. Kucharek and P Ring, Z. Phys. A 339 (1991) 23. ] 16 1 A.B. Migdal, Theory of Finite Fermi Systems, Applications to Atomic Nuclei (Wiley Interscience, New York, 1967). [ 17 1 L.P. Gorkov, Sov. Phys. JETP 7 (1958) 505. [ 18 I J.D. Talman, Nucl. Phys. A 141 ( 1970) 273. I 19 1 J.F. Berger, M. Girod and D. Gogny, Nucl. Phys. A 428 ( 1984) 32~.
Letters B 379 (1996) 13-19 [20] [21] [22] [23] 1241 [25] [26] [27] [28]
[29]
19
P Ring and P. Schuck, The Nuclear Many-body Problem (Springer Verlag, Heidelberg, 1980). M.M. Sharma, G.A. Lalazissis and P Ring, Phys. Lett. B 317 (1993) 9. M.M. Sharma, G.A. Lalazissis, J. Konig and P Ring, Phys. Rev. Lett. 74 (1995) 3744. P-G. Reinhard and H. Flocard, Nucl. Phys. A 584 (1995) 467. M.M. Sharma, M.A. Nagarajan and P Ring, Phys. Len. B 312 (1993) 377. M. Brack, C. Guet and H.B. Hakansson, Phys. Rep. 123 (1985) 275. K.-L. Kratz, Rev. Mod. A&on. 1 (1988) 184. J. Dobaczewski, I. Hamamoto, W. Nazarewicz and and J.A. Sheikh, Phys. Rev. Len. 72 (1994) 981. M.M. Sharma, G.A. Lalazissis, W. Hillebrandt and I? Ring, Phys. Rev. Lett. 72 (1994) 1431; see also Phys. Rev. Lett. 73 (1994) 1869. P Moller, J.R. Nix, W.D. Myers and W.J. Swiatecki, Atomic Data and Nuclear Data Tables 59 ( 1995) 185.