Dedicated to Professor Apolodor Aristotel R˘adut¸a˘ ’s 70th Anniversary
COVARIANT DENSITY FUNCTIONAL THEORY AND THE STRUCTURE OF EXOTIC NUCLEI G.A. LALAZISSIS1 , P. RING2 1
Department of Theoretical Physics, Aristotle University, Thessaloniki Gr-54124, Greece E-mail:
[email protected] 2 Physik-Department der Technischen Universit¨at M¨unchen, D-85748 Garching, Germany E-mail:
[email protected] Received June 6, 2013
In recent years covariant density functional theory has developed as a very successful tool to describe nuclear structure phenomena. We discuss in this article, why Lorentz invariance should be taken seriously in this context and show several very successful nuclear applications of relativistic density functional theory. Key words: Density functional theory, nuclear masses, driplines, halo phenomena, exotic resonances, radii, deformations, collective excitations. PACS: 21.60Jz, 21.60Mn, 21.60Jv, 24.10Jv .
1. INTRODUCTION
The possibility to investigate the nuclear chart to the very limits of nuclear binding by new accelerators with radioactive beams has stimulated considerable new efforts on the theoretical side to understand the dynamics of the nuclear many-body problem by microscopic methods. Density functional theories (DFT) play a very important role in this context [1]. One of the underlying symmetries of QCD is Lorentz invariance and therefore covariant density functionals (CDFT) are of particular interest in nuclear physics [2,3] . This symmetry not only allows to describe the spin–orbit coupling in a consistent way, but it also puts stringent restrictions on the number of parameters in the corresponding functionals without reducing the quality of the agreement with experimental data. Therefore, a relatively small number of parameters is necessary, which are adjusted to reproduce a set of bulk properties of spherical nuclei. The resulting energy density functionals (EDF) are considered universal in the sense that they can be used for nuclei all over the chart of nuclides, where mean field is applicable, for ground state properties as well as for excited states. The most successful EDF originate from the Relativistic Hartree Bogoliubov (RHB) model in which p-h and p-p channels are treated simultaneously in a self-consistent manner. A large variety of nuclear phenomena have been described over the years within this model, the equation of state of symmetric nuclear matter, ground state properties of finite spherical and deformed nuclei at and far away from the valley of stability. The RJP 58(Nos. Rom. Journ. Phys., 9-10), Vol. 1038–1047 58, Nos. 9-10,(2013) P. 1038–1047, (c) 2013-2013 Bucharest, 2013
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model is able to describe very successfully exotic phenomena such as halo nuclei, weakening of shell effects away from stability line, super-heavy nuclei. In addition the model can successfully describe excited states, particularly those of collective character. 1.1. COVARIANT DENSITY FUNCTIONAL THEORY
The essential ingredient of density functional theory is the Kohn-Sham equation with its effective single particle potential, the self energy. In a relativistic description it has the form of a Dirac equation f f m + W+ σp = εi . (1) g g σp −m + W− The potentials W+ = V + S and W− = V − S contain an attractive scalar field S and a repulsive vector field V . In the simplest form of the Walecka model [4] for symmetric nuclear matter these two numbers are adjusted to the saturation point (binding energy and density) and it is found that these potentials are very large having nearly the same size, but opposite phase: S ≈ −400 MeV and V ≈ 350 MeV. Therefore the nucleons in the Fermi sea feel a relatively week potential W+ = V + S ≈ −50 MeV. Their Fermi momentum is roughly 250 MeV/c and small as compared to the rest mass of a nucleon m ≈ 940 MeV/c2 . This shows clearly that we do not need relativistic kinematics in order to describe phenomena of low energy nuclear structure. By this reason non-relativistic functionals are very successful, if their parameters are carefully chosen. As compared to the non-relativistic description covariant theories show a number of technical difficulties: The solution of the Dirac equation is definitely more complicated than the Schroedinger equation. In dynamic applications for the investigations of excited states the Dirac sea has to be treated properly [5]. Therefore it is understandable that large parts of the community prefer non-relativistic functionals, where standard computer codes are available and where much experience has been accumulated since more than 40 years. Nonetheless there are important arguments for covariant density functionals. Most of them are based on the simplicity and beauty of the Walecka model [4], which serves as a vehicle to implement relativistic symmetries. The spin degree of freedom is taken into account consistently. Non-relativistic functionals need here additional terms and parameters. An example is the spin-orbit splitting. The elimination of the small components g in Eq. (1) leads to 1 1 1 ∂W− p p+ `s + m + W (2) + f = εf ε + m − W− 2(ε + m − W− )2 r ∂r Particles in the Fermi sea feel a moderate potential W+ ≈ −50 MeV and a very large RJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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spin-orbit term W− ≈ 750 MeV. The size of these fields is determined in nuclear matter by the saturation energy and density [4] and this explains the nuclear shell structure in a very simple and quantitative way without any additional adjustable parameters. In addition there exist strong arguments based on QCD for these large fields. Using in medium QCD-sum rules [6] one can relate the scalar condensate h¯ q qi and † the quark density hq qi to the scalar and vector self-energies of the nucleon in the medium. This yields to an approximate relation S/V ≈ −0.8 ± 0.3 and V ≈ −S ≈ 400 MeV which means that already without any phenomenological adjustment to nuclear data one expects in the nuclear medium large relativistic fields with opposite sign, an attractive scalar S and a repulsive vector field V . It is well known that non-relativistic Brueckner calculations based on twobody forces are not able to reproduce the saturation point in symmetric nuclear matter. Therefore phenomenological three-body forces have been introduced in nonrelativistic ab-initio calculations. The situation is different in relativistic Brueckner theory [7] where one does not need three-body forces in order to reproduce saturation. Although it is still an open question [8], whether there is a connection between three-body forces and relativistic effects, this is a strong hint that relativistic effects should not be neglected in ab-initio calculations. There exists a relativistic saturation mechanism: Non-relativistic Hartree calculations with such simple interactions would lead to a collapse. Therefore one has introduced in such theories a strongly repulsive term depending on the density. In the relativistic model this is not necessary. The source of the attraction is the scalar density, which because of Lorentz invariance is automatically quenched at high densities. This relativistic effect leads always to saturation [4]. As is the case of the spin-orbit force this is a genuine relativistic effect which is taken into account in the present non-relativistic descriptions by introducing additional terms with additional parameters. There are cases where the conventional non-relativistic theories fail. One of those are the famous isotope shifts in the Pb-region. They have been measured with extreme accuracy by atomic beam laser spectroscopy. If plotted as a function of the mass number the experimental radii show a pronounced kink at the magic neutron number N = 126. This kink is neither reproduced by Skyrme [9] nor by Gogny calculations. On the other hand all relativistic investigations show a remarkable agreement with the experimental data [10]. This fact can be traced back to the isospin dependence of the spin orbit term [11], which is rather different in conventional Skyrmeand relativistic models. Usually nuclear density functionals are adjusted to experimental ground state properties or to nuclear matter data. They depend only on time even parts of the potentials and therefore in non-relativistic density functionals the time odd parts show RJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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Bexp. - Bth. (MeV)
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2
0
-2
-4 0
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100
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200
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Fig. 1 – Absolute deviations of theoretical binding energies from the experimental values.
some ambiguity. Covariant functionals have the advantage that the time odd fields described by the currents, i.e. the spatial components of the vector fields, have the same coupling constant as the time-like components. No new parameters are required. This effect is usually called nuclear magnetism [12]. In fact, some of the applications [13] of non-relativistic density functional theory fail to describe the moments of inertia of rotational bands properly. On the other side, relativistic applications to rotational ground state bands [14], to super-deformed bands [15] and magnetic dipole bands [16] show good agreement with experimental data. A proper treatment of nuclear magnetism is crucial. Finally we have pseudospin symmetry. The nearly degeneracy of nuclear single particle orbits with the quantum numbers (n, l, j = l − 21 ) and (n0 = n + 1, l0 = l − 2, j 0 = l0 + 21 ) has lead more than forty years ago to the concept of pseudo-spin (see Ref. [17] and references given therein). This scheme has been used over the years to apply group-theoretical techniques, but it has never been really understood in a nonrelativistic scheme. J. N. Ginocchio [17] could show that this symmetry can be easily understood in the framework of a relativistic description. It can be traced back to the fact that the potentials V and S are nearly equal in size and opposite in sign [17]. 2. APPLICATIONS OF COVARIANT DENSITY FUNCTIONAL THEORY
The Walecka model [4] and its extensions with mesons carrying isospin [18] are not able to provide a quantitative description of nuclei. Its compressibility is much too large and the surface properties are not reproduced well, in particular nuRJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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0.4
β2
Dy 0.3
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0.2
0.2
0.1
0.1
0.0
0.0
5 0.4
Er
Yb 0.3
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RHB/DD-ME2 EXP
-0.1
-0.1 0.1
150
160 A
150
160 A
170
160
170 A
180
Fig. 2 – The calculated and experimental β2 values for Dy, Er and Yb nuclei.
clear deformations are much too small [19]. In order to obtain a realistic density functional one needs an additional density dependence. Over the years three types of models have been developed, non-linear meson coupling models [20–25], density dependent meson-coupling models [23, 26–29] and point-coupling models with density dependent vertices [30–32]. 2.1. GROUND STATE PROPERTIES
We show here results of calculations for ground states properties of spherical and deformed nuclei using the effective interaction DD-ME2 [28]. Pairing correlations have been included in the relativistic Hartree Bogoliubov (RHB) model [2] with the finite range force of Gogny D1S [33] in pairing channel. The binding energies energies of approximately 400 nuclei are compared with experimental values in Fig. 1. The calculated binding energies are generally in very good agreement with experimental data. The rms error including all the masses shown in Fig. 1 is less than 970 KeV. This is a significant improvement compared with the previous RMF calculations with non linear-forces NL3 [22], where the rms error was around 2.5 MeV [34]. The theoretical values of the quadrupole deformation parameters are displayed in Fig. 2, in comparison with the empirical data from Ref. [35].We notice that the RHB results reproduce not only the global trend of the data, but also the saturation of quadrupole deformation for heavier isotopes. The RHB model with density dependent meson-nucleon coupling constants is highly successful in describing the ground state properties ( e.g binding energies, nuclear radii, the deformations, isotope shifts etc). RJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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20 15 208
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Sn
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ISGM R9fm /MeV)
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R(fm /MeV)
ISGM
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Sn
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Sn
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R(e fm )
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10 5 0
4 2
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10 20 E (MeV)
0
10
20
Fig. 3 – The isoscalar monopole (upper panels) , and the isovector dipole strength distributions in 208 Pb and in 116,118,120 Sn nuclei calculated with the effective interaction DD-ME2. 2.2. GIANT RESONANCES
In time dependent systems, the Runge Gross (RG) theorem [36] provides the formal foundation of time-dependent density functional theory. It shows that the density can be used as the fundamental variable in describing quantum many-body systems in place of the wave function, and that all properties of the system are functionals of the density. Starting from RG theorem one is able to derive the time dependent KS equations and then using the adiabatic approximation one can describe excited states such as rotational bands in normal deformed and superdeformed nuclei [3] and collective vibrations. Rotations are treated in the cranking approximation, which provides a quasi-static description of the nuclear dynamics in a rotating frame. For the description of vibrations, a time-dependent mean field approximation is used by assuming independent particle motion in time dependent average fields. In the small amplitude limit, one obtains the relativistic Random Phase Approximation (RRPA) [5] and for open shell nuclei the corresponding relativistic quasi-particle random phase approximation (QRPA) [37]. This method provides a natural framework to investigate collective and non-collective excitations of ph character. It is successful, in particular, for understanding the position of giant resonances and spin-or/and isospin excitations such as the Gamow-Teller Resonance (GTR) or the isobaric Analog Resonances (IAR) [38]. The corresponding eigenmodes can be determined eiRJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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ther by diagonalizing the RRPA / RQRPA equations in an appropriate basis [5] or by solving the linear response equations in time-dependent external field [39]. The two methods lead, in principle, to exactly identical results. It should be noted however, that the second method provides a more consistent treatment of the coupling to the continuum. The fully self-consistent RRPA [40] and RQRPA [37] have been used to calculate excitation energies of giant resonances in doubly-closed and open-shell nuclei, respectively. The RQRPA is formulated in the canonical basis of the RHB model and, both in the ph and pp channels, the same interactions are used in the RHB equations that determine the canonical quasiparticle basis, and in the matrix equations of the RQRPA. For 208 Pb the RRPA results for the monopole and isovector dipole response are displayed in the left part of Fig. 3. In Fig. 3 we also compare the RQRPA results for the Sn isotopes with experimental data on IVGDR excitation energies [41] and ISGMR energies [42]. In contrast to the case of 208 Pb, the strength distributions in the region of giant resonances exhibit fragmentation and the energy of the resonance EGDR (lower right panels) is ¯ = m1 /m0 , calculated in the same energy window defined as the centroid energy E as the one used in the experimental analysis. The arrows denote the location of the empirical values. In Table 1 the predictions of the CDFT for the (m3 /m1 )1/2 ratios of several nuclei are compared with very recent experimental information [42]. It is quite clear that the RHB+RQRPA calculation with the DD-ME2 interaction reproduces in detail the experimental excitation energies and the isotopic dependence of both the IVGDR and ISGMR. Table 1 RHB results for the giant isoscalar monopole (m3 /m1 )1/2 ratios of several Sn isotopes.
Nucleus 112
Sn Sn 116 Sn 118 Sn 120 Sn 122 Sn 124 Sn 114
1/2
1/2
(m3 /m1 )(th.)
(m3 /m1 )(exp.)
16.9 16.7 16.7 16.6 16.5 16.4 16.2
16.7±0.2 16.5±0.2 16.3±0.2 16.3±0.1 16.2±0.2 15.9±0.2 15.8±0.1
2.3. PYGMY RESONANCES
Using the relativistic QRPA scheme CDFT can also be used to study nuclear excitations and new modes have been found. On the neutron-rich side, in particular, RJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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Fe
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E[MeV] 46
9.44 MeV
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E [MeV]
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Fe
0.1 0.0 -0.1
neutrons protons
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2
4
6
r[fm]
8
0
2
4
6
8
r [fm]
0
2
4
6
8
10
r[fm]
Fig. 4 – The RHB+RQRPA isovector dipole strength distributions in the N=20 isotones, calculated with the DD-ME2 effective interaction. For 44 Cr and 46 Fe the proton and neutron transition densities for the main peak in the low-energy region are displayed in the lower panel and, for 46 Fe, the transition densities for the main GDR peak.
the possible occurrence of the pygmy dipole resonance (PDR), i.e. the resonant oscillation of the weakly-bound neutron skin against the isospin saturated proton-neutron core, has been investigated. On the theory side, various models have been employed in the investigation of the nature of the low-lying dipole strength [43], in particular also the relativistic quasiparticle RPA (RQRPA) [44]. In the right panel of Fig. 3 the pygmy resonance peak in 208 P b is seen. The calculated value (7.13 MeV) is in excellent agreement with recent experimental information. We can also use the relativistic QRPA to study the evolution of low-lying dipole strength in proton-rich nuclei [44]. In Fig. 4 we display the RQRPA dipole strength distributions in the N=20 isotones: 40 Ca, 42 Ti, 44 Cr, and 46 Fe. They are dominated by the isovector giant dipole resonances (GDR) at 20 MeV excitation energy. With the increase of the number of protons, low-lying dipole strength appears in the region below the GDR. In the lower panel of Fig. 4 we plot the proton and neutron transition densities for the peaks at 10.15 MeV in 44 Cr and 9.44 MeV in 46 Fe, and compare them with the transition densities of the GDR state at 18.78 MeV in 46 Fe. Obviously the dynamics of the two low-energy peaks is very different from that of the isovector GDR. The low-lying state represents a fundamental mode of excitation: the proton electric pygmy dipole resonance (PDR). RJP 58(Nos. 9-10), 1038–1047 (2013) (c) 2013-2013
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2.4. BEYOND MEAN FIELD THEORY
It should be noted, however, that nuclear DFT based on the mean field framework cannot provide an exact treatment of the full nuclear dynamics. It is known to break down in transitional nuclei, where one has to include correlations going beyond the mean field approximation by treating quantum fluctuations through a superposition of several mean field solutions as for instance in the generator coordinate method (GCM) [45]. It is also known that even in ideal shell model nuclei such as in 208 Pb, with closed proton and neutron shells one finds in self-consistent mean field calculations usually a single particle spectrum with a considerable enhanced Hartree-Fock gap in the spectrum and a reduced level density at the Fermi surface as compared with experiment. The situation is considerable improved by taking into account the energy dependent part of the self-energy and treating it in terms of the particle-vibration coupling model [46–48]. Moreover, we get accurate information not only for the position of the giant resonances but also for their widths in agreement with the experimental observations [49] 3. SUMMARY
In conclusion, covariant DFT provides a very successful and microscopic description for ground states and excited states in nuclei. Most of the successful functionals are at present phenomenological. On the mean field level there is no energy dependence of the self-energy, no fluctuations are taken into account and symmetry violations are not treated properly. The energy dependence of the self energy can be treated by means of the particle vibration model, while the symmetries and fluctuations can be considered by projections and the GCM method. Many things should be done in future. On the static part, we are still far from a microscopic derivation of the model. We have to improve the functionals both in ph- and pp-channels. On the dynamical part: the PVC model is so far restricted to spherical systems and GCM is so far restricted to very few degrees of freedom. Acknowledgements. This work has been supported in part by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de).
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