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On the relativistic origin of the kink effect in the chain of Pb isotopes

We investigate the origin of the kink effect (KE) in the relativistic mean field theory by transforming the single-particle Dirac equation into a Schrodinger-like equation. It is found that relativistic self-consistent effects as well as contributions from the rho meson determine the actual structure of the KE. However, the spin-orbit force generated by the rho meson has no significant influence on the KE.

The charge radii (r c ) of the Pb isotopes have been measured a few years ago with a very high precision [1], their anomalous kink behavior being the most essential feature of these measurements. The kink effect (KE) means that the experimental data on r c as a function of A in the isotopic chain display a change of the slope at N=126 with a gradual addition of neutrons. This phenomenon has been a matter of a detailed discussion by different theoretical groups during the recent years (see Refs. [2]- [6] and references therein).
At the first stage, it has been shown that the density-dependent Hartree-Fock model with standard parametrizations of the Skyrme (SHF) [2] or Gogny [7] forces fails to reproduce the empirical charge isotopic shifts in the chain of Pb isotopes.
On the other hand, in Refs. [3,4] the anomalous behavior of the charge radii of these isotopes has been studied in the relativistic mean field approach (RMFA). The Pb data are well described by this type of theory. In Ref. [4] it is supposed that the success of the RMFA in reproducing the KE is achieved due to the weak isovector dependence of the spin-orbit force generated by the RMFA.
In Ref. [5] the problem of isotopic shifts has been investigated with exhaustive detail, both in the SHF and RMFA. The authors of Ref. [5] have proposed a new Skyrme functional SkI4 with a more general structure of the spin-orbit force than that of the standard Skyrme functional. They have also succeeded to reproduce the KE, but at the price of introducing an extra parameter for the two body spin-orbit potential.
As the authors of Refs. [3,4] state, the standard RMFA provides an excellent description of the anomalous kink in the isotopic shifts about 208 P b without using extra fitting parameters. In the present paper we shall focus our interest into the problem of the origin of the KE in the RMFA paying special attention to the role of the isovector ρ meson in the kink structure.
In the RMFA [6], the relativistic single-nucleon wave function ψ( r), for a nucleon with the rest mass M, satisfies the Dirac equation where E = M + ε and ε is the binding energy eigenvalue while S( r) and V ( r) are the scalar and vector potentials, respectively, defined as follows: In Eq. (2) σ is the scalar meson field, ω 0 and ρ 0 are the time components of the ω and ρ meson fields, respectively, while V c is the Coulomb potential. g σ , g ω and g ρ are the coupling constants associated with the respective mesons. All meson fields satisfy the corresponding Klein-Gordon equations (see [6], for example).
where the central potential V cent and the spin-orbit potential V SO are given by: By writing the Dirac equation in this form, we are able to study separately the influence of the different terms entering Eq. (4) on the KE (we can analyze, for example, if their respective effects are of relativistic origin or not).
Our calculations have been carried out for the P b chain of isotopes around the 208 P b nucleus in the framework of the standard RMFA, with a linear model (L) [9] and two nonlinear models NL-SH [10] and NL3 [11] containing self-interactions of the scalar field.
For simplicity, we shall mainly concentrate our discussion in the three Pb isotopes 206 P b, 208 P b and 210 P b. The most stable configuration of the 210 P b isotope in the two nonlinear models considered contains two neutrons in the 1i 11/2 states. In the L model, the two configurations in which two neutrons fill up the 1i 11/2 or 2g 9/2 single-particle states are almost degenerate (we shall refer to these two configurations as IC and GC, respectively).
We approach the 210 P b ground state by the IC in both linear and nonlinear models, although the BCS formalism would be more appropriate in the first case. Nevertheless, we expect that our conclusions are also valid for the L model, at least, at a qualitative level.
The self-consistent results for the charge radii corresponding to the exact models are summarized in Table I, case a, and in Fig. 1 (the r c are normalized in this figure to the experimental value of the 208 P b nucleus). The KE appears in the three models considered but it is more strongly pronounced in models with smaller K modulus values. We notice that the KE is quite small for the GC, in spite of the fact that the 2g 9/2 state is less bound than the 1i 11/2 state and that the radius of the 2g 9/2 state is larger than the radius of the 1i 11/2 one. Thus, it is not the radius but the different neutron density associated with these two states that is essential to explain the different KE corresponding to IC and GC.
The KE in the chain of P b isotopes can be only explained if single-particle configurations including 1i 11/2 states significantly contribute to the total nuclear ground state for the isotopes with A > 208.
To understand how the kink is generated by the model we have begun from studying To study the influence of the different components containing the ρ meson field on the charge radii of the nuclei in more detail, it is useful to make the following replacements in eqs. (4) and (5) corresponding to V cent ( r) and V SO ( r), respectively : the ∆V cent term, and g ρ → x 5 g ρ in the the V SO potential. After that, we allow the parameters x 1−5 to vary in a continuous and independent way in the interval [0-1]. We have considered 4 different cases corresponding to the 4 following combinations of the x i parameters: a) The results for these cases are collected in Table I, and can be identified by the corresponding letter a, b, c or d in the last column. We have considered the models L, NL-SH and NL3, and the IC for the ground state. Fig. 2 shows the r c results for the NL-SH model normalized to the r c of the 208 P b nucleus. The case a contains the total ρ contribution and manifests the KE in good agreement with the effect experimentally observed. In the case b, in which the ρ contribution to S + V is almost eliminated, the kink has completely changed its structure in relation to the case a, now the kink seems to be, somehow, the mirror image of the kink in case a. This fact makes evident the crucial role the ρ meson plays in the kink structure through the contribution of the (S + V ) term. Table I shows that r c slightly increases when the parameter x 1 decreases. This fact is related to the decreasing of the proton binding energy with the decreasing of the x 1 value (the opposite is true for the neutron energy). However the qualitative change of the kink structure cannot be attributed to this modification of r c (which has only a small effect on the kink). Notice also that x 1 is taken equal to 0.2 and 0.1 (rather than 0) in the L and NL-SH models, respectively, to maintain the 3s 1/2 proton state bound enough.
The effect of the ρ meson on the kink in the region 208 ≤ A ≤ 210 through the term S + V in the potential is quite small for the more stable IC. However, for the GC, a decreasing of the x 1 parameter produces a sizable turn at the left side of this region of the kink.
With regard to the case c in Fig. 2, the effect displayed is opposite to that shown in the case b and much smaller in magnitude. This result is a consequence of the opposite role the ρ meson field plays in the terms (S + V ) and (S 2 − V 2 )/2M of the central potential. For the GC the effect of the ρ meson on the kink through these two terms remains opposite to the case b, but comparable in magnitude.
Finally, for the case d, i.e., the case without ρ meson contribution, the behavior of r c is intermediate between that corresponding to the cases a and b as it can be appreciated in Table I and Fig. 2.
We have verified that the ρ meson contribution through the ∆V cent and V SO terms has only minor influence on the kink. As we have explained above, this behavior is expected because the relative contributions of the ρ meson to the (S − V ) term, entering V cent and V SO , are very small.
To understand better the nature of the KE it would be useful to examine the partial contributions r ck of different single-particle states "k" to the total charge radii r c of the nuclei under consideration (r c = r 2 c1 + ... + r 2 cA ). Fig. 3 displays these values for the six bound states (closest to the Fermi level) calculated within the NL-SH model for different nuclei in the region 200 ≤ A ≤ 214. We see that the total value of r c for A ≥ 208 is mostly determined by the contribution of the 1h 11/2 , 1g 9/2 and 1g 7/2 proton states, rather than by that of the 2d 3/2 , 2d 5/2 or 3s 1/2 proton states. Let us mention also that in the nuclei considered there are more protons of the first type than those of the second type.
Moreover, the values r ck , for the states of the first type, are increasing functions of A for A ≥ 208 while the respective values for the states of the second type almost remain constant with the increasing of A in this region. One can effectively describe this type of behavior as a result of the repulsive interaction experienced by the 1h 11/2 , 1g 9/2 and 1g 7/2 protons which pushes them outside the central part of the nucleus and increases the value of r c . From Fig. 3 it is also seen that the behavior of the total value of r c for the nuclei in the interval 206 ≤ A ≤ 208 is strongly affected by the r ck value for k = 3s 1/2 . In fact, a similar influence is also shown by the 2s 1/2 state, in spite of its very large binding energy. In cases where the ρ contribution to the (S + V ) term in the central potential is reduced (cases b and d in Fig. 2), the contributions of the two 2s 1/2 and 3s 1/2 states to ∆r − c = r c ( 208 P b) − r c ( 206 P b) are even larger. The two neutrons added to the 3p 1/2 state appear to be very effective to pull on the protons in these two states. Table I  with the reasons given in Ref. [5]. However, the behavior of r c in the region 208 ≤ A ≤ 210 is the opposite. The r c behavior shows that the cases that produce larger variation of ∆r − c are less effective contributing to ∆r + c = r c ( 210 P b) − r c ( 208 P b). This is a reflection of the fact that it is more difficult to increase the r c value by adding extra neutrons when the radius is larger than in the case when it is smaller and when the neutrons previously added are more effective to increase r c .
As explained above, although the ρ meson contribution to the total energy is rather small in comparison with that of the σ and ω mesons, it is just the ρ meson that is mainly responsible in the RMFA for the kink structure. However, even without ρ meson contribution (g ρ = 0) there is a residual kink, case d in Fig. 2. In this case, we have investigated the role of the different terms entering V cent and of V SO in producing the residual kink. To do that, we have artificially changed the contribution of these terms in a similar way as we did it for the different components of the ρ meson, but readjusting now the strength of the (S + V ) term, if necessary, to keep the r c values close to the experimental ones for the P b isotopes studied, (the same conditions are used for the three isotopes). These studies reveal that the terms ε V M , ∆V cent and V SO contribute in a slight and positive way to the kink structure, whereas the term (S 2 − V 2 )/2M has an opposite effect that approximately cancels the effects mentioned above. We notice that taking about the 70% of the strength of the (S + V ) term entering Eq. (3) as the total potential, the KE (obtained in the P b isotopes studied taking g ρ = 0) is mainly reproduced as an effect of the self-consistent procedure. For example, if the S and V potentials are approached by the Woods-Saxon functions with the radius proportional to A 1/3 the KE disappears, although different diffuseness parameters for the potentials are used.
We have studied above the role that V cent and V SO play in generating and forming the structure of the kink. Another crucial relativistic effect comes from the fact that the source term in the scalar field equation, the scalar density (ρ S ), is smaller than the actual baryon density (ρ B ). This difference of two densities is just the mechanism that brings about the saturation in the RMFA, and essentially distinguishes relativistic models from nonrelativistic ones. Thus it would be interesting to check also the role of this mechanism in the KE. One naive way would be to replace ρ S in the scalar field equation by ρ B . Since the nucleus would become unstable in this case, we add to V cent in the single-particle Schrödinger equation (1) a repulsive density-dependent term formally identical to the term proportional to t 3 appearing in the corresponding equation for the Skyrme-Hartree-Fock model. The coefficient t 3 is chosen to fit the r c ( 208 P b). Under these conditions we observe that in two cases, with and without the ρ meson (cases a and d, respectively), the quantities r + c and r − c are reduced and so is the ratio r + c /r − c . Thus, the KE is reduced about 15% in the case a and 60% in the case d. The residual KE without the ρ meson almost disappears.
Our conclusions can be summarized in the following points: 1. The KE in Pb isotopes chain appears to be a general feature of the RMFA. It is observed in both linear and nonlinear models, being more strongly manifested in the models with smaller values of K. We have found that the ρ meson plays an essential role in forming the KE. In the case without the ρ meson a small kink remains that can be attributed to relativistic as well as to self-consistent effects.
2. The KE mainly comes from the (partially) destructive interference of contributions of the ρ meson to the (S + V ) and (S 2 − V 2 )/2M components of the single-particle potential. The (S +V ) term leads to a KE of positive sign while the (S 2 −V 2 )/2M component produces a KE of negative sign. The total effect has a positive sign and corresponds to the trend observed empirically (see cases b and c in Fig. 2).
3. Excluding the ρ contribution from the two components of V cent mentioned in the previous point, i.e., taking x 1 = x 2 = 0, one gets a strong reduction of the KE. This situation is quite similar to that in which the ρ meson contribution is completely excluded (g ρ =0, case d in Fig. 2). The residual KE in this case is related to the self-consistent procedure. This effect completely disappears if one uses a Woods-Saxon potential in the Dirac equation rather than a self-consistent calculation. 4. The contribution of the ρ meson to V SO has no influence on the KE. difference between the scalar density and the baryon density, also contributes to a certain increase of the KE with respect to the nonrelativistic case in which saturation is produced by a repulsive density-dependent term as in the Skyrme-Hartree-Fock model.
In a forthcoming publication we shall investigate the problem of the KE in the relativistic Hartree-Fock framework.
One of the authors (L.N.S.) is very grateful to the University of Cantabria and to the Technical University of Munich for hospitality. This work has been supported in part by the DGESIC grant PB97-0360. Fig. 1. The charge radii of P b isotopes obtained with the L [9], NL-SH [10] and NL3 [11] models, normalized to the experimental value of the 208 P b nucleus. The empirical values (EXP) are taking from Ref. [10]. Fig. 2. The same as in Fig. 1 for the NL-SH [10] model, and the four cases considered in Table I: a: x 1−5 = 1 (the exact model); b: x 1 = 0.1, x 2−5 = 1; c: x 1,3−5 = 1, x 2 = 0; d: