$M1$ resonance in ²⁰⁸Pb within the self-consistent phonon-coupling model

Magnetic dipole ($M1$) excitations in the ²⁰⁸Pb nucleus are the object of numerous experimental and theoretical investigations for several decades. From the theoretical point of view, one of the reasons is the possibility of determining the spin-related parameters of the residual interaction in the calculations of these excitations within the random-phase approximation (RPA) or its extended versions. The calculated energies of the unnatural parity excitations are very sensitive to the values of the underlying model parameters, in particular, of the parameters of the Skyrme energy-density functional. The comparison of these energies with experimental data is the only reliable method to estimate the spin-related parameters of the model. Another reason is the fragmentation (spread) of the isovector $M1$ resonance in ²⁰⁸Pb which is observed in the experiment (see Laszewski et al. (1988)) but which is absent in RPA where the isovector $M1$ strength in this nucleus is concentrated in one state. The description of this fragmentation requires application of more complicated models going beyond the RPA framework (see, e.g., Ref. Kamerdzhiev et al. (2004) for more details).

Most of the early calculations of the $M1$ excitations in ²⁰⁸Pb (see, e.g., Refs. Vergados (1971); Ring and Speth (1973); Tkachev et al. (1976); Speth et al. (1980); Borzov et al. (1984)) were performed within the RPA, the Tamm-Dancoff approximation or within the Migdal’s Theory of Finite Fermi Systems (TFFS, Ref. Migdal (1967)) which in its simplest form used in the applications is equivalent to the RPA with the zero-range residual interaction. In the following, the $M1$ modes were investigated within the generalized models in which the one-particle–one-hole ($1p1h$) RPA configuration space is enlarged by adding $2p2h$, $1p1h\otimes$phonon or two-phonons configurations (see, e.g., Lee and Pittel (1975); Dehesa et al. (1977); Kamerdzhiev and Tkachev (1984); Cha et al. (1984); Khoa et al. (1986); Kamerdzhiev and Tkachev (1989); Tselyaev (1989); Kamerdzhiev and Tselyaev (1991); Kamerdzhiev et al. (1993)). However, the fully self-consistent calculations of the $M1$ excitations in ²⁰⁸Pb have been performed so far only within the RPA (see Refs. Cao et al. (2009); Vesely et al. (2009); Nesterenko et al. (2010); Cao et al. (2011); Wen et al. (2014); Tselyaev et al. (2019)).

In a broad sense, self-consistency means the use of the same energy-density functional (EDF) $E[\rho]$ (where $\rho$ is the single-particle density matrix) for the mean field as well as for the RPA residual interaction. This decreases the number of the free parameters of the theory and, in principle, increases its predictive power. Here we use an EDF of Skyrme type Bender et al. (2003). In a recent paper Tselyaev et al. (2019), we have shown that the adequate description of the low-energy $M1$ excitations in ²⁰⁸Pb within the self-consistent RPA based on the Skyrme EDF is possible only if the spin-related parameters of the known EDF are modified. By re-tuning these parameters we managed to reproduce within the RPA the experimental key quantities: energy and the strength of the $1_{1}^{+}$ state as well as the mean energy and the summed strength of the $M1$ resonance in ²⁰⁸Pb in the interval 6.6-8.1 MeV. However, as mentioned above, the observed fragmentation of the isovector $M1$ resonance and its total width in this model are not yet reproduced.

The aim of the present paper is to study the possibility to describe this fragmentation within the extended self-consistent model including the $1p1h\otimes$phonon configurations on top of the RPA $1p1h$ configurations. This extended model is treated within the time blocking approximation (TBA) which we use actually in its renormalized version (RenTBA, Tselyaev et al. (2018)). Full self-consistency is maintained also for the extended treatment. The method of re-tuning the spin-related parameters of the Skyrme EDF developed in Ref. Tselyaev et al. (2019) is used also for the RenTBA.

The paper is organized as follows. In Section II the formalism of RPA and RenTBA is briefly described. Section III contains the numerical details and the calculation scheme. The main results of the paper are presented in Section IV. In Section V the fine structure of the $M1$ strength distributions in ²⁰⁸Pb and the impact of the single-particle continuum on this structure are analyzed. In Section VI the problem of the fragmentation of the isovector $M1$ resonance in ²⁰⁸Pb is discussed in detail and the necessary condition of the description of this fragmentation is determined. In Section VII we present the results of the RenTBA and RPA calculations of the low-energy electric excitations in ²⁰⁸Pb obtained with the use of the modified parametrizations of the Skyrme EDF. The conclusions are given in the last section.

Let us start with the RPA eigenvalue equation

where $\omega^{\vphantom{*}}_{n}$ is the excitation energy, $Z^{n}_{12}$ is the transition amplitude, and the numerical indices ($1,2,3,\ldots$) stand for the sets of the quantum numbers of some single-particle basis. In what follows the indices $p$ and $h$ are used to label the states of the particles and holes in the basis which diagonalizes the single-particle density matrix $\rho$ and the single-particle Hamiltonian $h$ in the ground state [see Eq. (5) below]. The transition amplitudes are normalized by the condition

where

is the metric matrix in the RPA.

In the self-consistent RPA based on the EDF $E[\rho]$ the RPA matrix $\Omega^{\mbox{\scriptsize RPA}}$ is defined by

where the single-particle Hamiltonian $h$ and the amplitude of the residual interaction $V$ are linked by the relations

In the TBA, the counterpart of Eq. (1) has the form

where

The matrix $\Omega^{\mbox{\scriptsize TBA}}(\omega)$ is energy-dependent due to the matrix $W(\omega)$ which represents the induced interaction generated by the intermediate $1p1h\otimes$phonon configurations. The subtraction of $W(0)$ in Eq. (7b) serves to avoid changing the mean-field ground state Toepffer and Reinhard (1988); Gütter et al. (1993) and to ensure stability of solutions of the TBA eigenvalue equation (see Tselyaev (2013)). The matrix $W(\omega)$ is defined by the equations

where $\sigma=\pm 1$, $\,c=\{p^{\prime},h^{\prime},\nu\}$ is a combined index for the $1p1h\otimes$phonon configurations, $\nu$ is the phonon’s index, $\varepsilon^{\vphantom{*}}_{p^{\prime}}$ and $\varepsilon^{\vphantom{*}}_{h^{\prime}}$ are the particle’s and hole’s energies, $\omega^{\vphantom{*}}_{\nu}$ is the phonon’s energy. The amplitudes ${F}^{c(\sigma)}_{12}$ have only particle-hole matrix elements ${F}^{c(\sigma)}_{ph}$ and ${F}^{c(\sigma)}_{hp}$. They are defined by the equations

where $g^{\nu}_{12}$ is an amplitude of the quasiparticle-phonon interaction.

In the conventional TBA, the phonon’s energies $\omega^{\vphantom{*}}_{\nu}$ in Eq. (8b) and the amplitudes $g^{\nu}_{12}$ in Eq. (9b) are determined within the framework of the RPA. In the non-linear version of the TBA developed in Ref. Tselyaev et al. (2018), the phonon’s energies $\omega^{\vphantom{*}}_{\nu}$ are the solutions of the TBA equation (6), while the amplitudes $g^{\nu}_{12}$ are expressed through the transition amplitudes ${z}^{\nu}_{12}$ which are also the solutions of Eq. (6), namely

The normalization condition for the transition amplitudes ${z}^{\nu}_{12}$ has the form

where

The terms $(z^{\nu})^{2}_{\mbox{\scriptsize RPA}}$ and $(z^{\nu})^{2}_{\mbox{\scriptsize CC}}$ represent the contributions of the $1p1h$ components (RPA) and of the complex configurations (CC) to the norm (11). The model includes only those TBA phonons that satisfy the condition

which together with Eq. (11) means that

The condition (13) confines the phonon space to the RPA-like phonons in agreement with the basic model approximations.

The feedback described above renders the phonon space of RenTBA fully self-consistent. In the present paper we use the version of this non-linear model in which the energies $\omega^{\vphantom{*}}_{\nu}$ and the amplitudes ${z}^{\nu}_{12}$ entering Eqs. (8b) and (10) (and only in these equations) are determined from the solutions of the TBA equation (6) in the diagonal approximation. This model is what we call the renormalized TBA (RenTBA, see Tselyaev et al. (2018) for more details).

The equations of RPA and RenTBA were solved within the fully self-consistent scheme as described in Refs. Lyutorovich et al. (2015, 2016); Tselyaev et al. (2016). Wave functions and fields were represented on a spherical grid in coordinate space. The single-particle basis was discretized by imposing the box boundary condition with the box radius equal to 18 fm. The particle’s energies $\varepsilon^{\vphantom{*}}_{p}$ were limited by the maximum value $\varepsilon^{{}_{\mbox{\scriptsize max}}}_{p}=100$ MeV. The non-linear RenTBA equations were solved by means of the iterative procedure. The phonon space of the first iteration included the RPA phonons with the energies $\omega^{\vphantom{*}}_{n}\leqslant$ 50 MeV and multipolarities $L\leqslant$ 15 of both the electric and magnetic types which have been selected according to the criterion of collectivity

see Tselyaev et al. (2018).

The field operator $Q$ in the case of the $M1$ excitations was taken in the form

where $l$ is the single-particle operator of the angular momentum, $\sigma$ and $\tau_{3}$ are the spin and isospin Pauli matrices, respectively (with positive eigenvalue of $\tau_{3}$ for the neutrons), $\mu^{\vphantom{*}}_{N}=e\hbar/2m_{p}c$ is the nuclear magneton, $\gamma_{p}=2.793$ and $\gamma_{n}=-1.913$ are the spin gyromagnetic ratios, $\xi_{s}$ and $\xi_{\,l}$ are the renormalization constants. The nonzero $\xi_{s}$ and $\xi_{\,l}$ correspond to the effective operator $Q$, however in the present calculations we used $\xi_{\,l}=0$. Thus, the reduced probability of the $M1$ excitations $B(M1)$ is defined as $|\langle\,Z^{n}\,|\,\mbox{\boldmath$Q$}\rangle|^{2}$ in the RPA and as $|\langle\,z^{\nu}\,|\,\mbox{\boldmath$Q$}\rangle|^{2}$ in the RenTBA.

The Skyrme EDF with the basis parametrizations SKXm Brown (1998) and SV-bas Klüpfel et al. (2009) was used both in RPA and RenTBA. The nuclear matter parameters for these parametrizations are listed in Table 1.

There are four experimental characteristics of the $M1$ excitations in ²⁰⁸Pb which serve as a benchmark in our calculations: energy and excitation probability of the isoscalar $1^{+}_{1}$ state ($E_{1}=5.84$ MeV with $B_{1}(M1)=2.0\;\mu^{2}_{N}$, see Shizuma et al. (2008)) and the mean energy and the summed strength of the isovector $M1$ resonance in the interval 6.6–8.1 MeV ($E_{2}=7.4$ MeV with $B_{2}(M1)=\sum B(M1)$ = 15.3 $\mu^{2}_{N}$). The latter two quantities have been deduced by combining the data from Refs. Shizuma et al. (2008); Köhler et al. (1987).

To reproduce these key characteristics, the spin-related EDF parameters $W_{0}$ (spin-orbit strength), $x_{W}$ (proton-neutron balance of the spin-orbit term), $g$ (Landau parameter for isoscalar spin mode), and $g^{\prime}$ (Landau parameter for isovector spin mode) were refitted as explained in Tselyaev et al. (2019) while the remaining spin-related parameters of the functional were switched off. The values of all other parameters of the functional were kept at the values of the original parametrizations. The form of the EDF containing all the parameters mentioned above is given in Ref. Tselyaev et al. (2019). The spin-orbit parameters $x_{W}$ and $W_{0}$ were refitted to reproduce the experimental value of $B_{1}(M1)$. The parameters $g$ and $g^{\prime}$ enter the terms of the modified Skyrme EDF which yield the term $V^{s}$ of the residual interaction $V$ having the form of the Landau-Migdal ansatz

where $C^{\vphantom{*}}_{\mbox{\scriptsize N}}$ is the normalization constant. These parameters allow us to change the calculated energies of the isoscalar and isovector $1^{+}$ states.

Note that the parameter $x_{W}$ was introduced in Refs. Reinhard and Flocard (1995); Sharma et al. (1995) (with the use of slightly different notations in Reinhard and Flocard (1995)) to regulate the isospin dependence of the spin-orbit potential. In the most parametrizations of the Skyrme EDF the value $x_{W}\geqslant 0$ is used (see, e.g., last two lines of Table 2). In particular, the value $x_{W}=1$ (frequently used implicitly) corresponds to the usual Hartree-Fock approximation in the EDF for the two-body spin-orbit zero-range interaction. However, in all these cases the value of $B_{1}(M1)$ in ²⁰⁸Pb calculated within the fully self-consistent RPA is much larger than its experimental value $B_{1}(M1)_{\mbox{\scriptsize exp}}$ = 2.0 $\mu^{2}_{N}$. For instance, $B_{1}(M1)_{\mbox{\scriptsize RPA}}\approx 10\,B_{1}(M1)_{\mbox{\scriptsize exp}}$ for the SLy5 set Chabanat et al. (1998) even with the use of the effective $M1$ operator (16). In Ref. Tselyaev et al. (2019) we have shown that one should use the negative values of $x_{W}$ to decrease the calculated $B_{1}(M1)$ up to $B_{1}(M1)_{\mbox{\scriptsize exp}}$. The values of the parameter $W_{0}$ should be simultaneously increased because from the set of the refitted parameters $x_{W}$, $W_{0}$, $g$, and $g^{\prime}$, only the isoscalar combination of the spin-orbit parameters $C_{0}^{\nabla J}=-{\textstyle\frac{1}{4}}(2+x_{W})W_{0}$ have an impact on the ground-state characteristics of spherical nuclei (see Tselyaev et al. (2019) for more details). This combination remains approximately unchanged in our refitting procedure, so the quality of the description of the ground-state properties with the use of the original and modified parametrizations of the Skyrme EDF is approximately the same.

The parametrizations obtained in the result of the refitting procedure described above are SKXm_-0.54 and SV-bas_-0.50 for the RPA and SKXm_-0.49 and SV-bas_-0.44 for the RenTBA (here and in the following the numerical subindex of the modified parametrization indicates the value of the parameter $x_{W}$). The values of the refitted parameters for these sets are shown in Table 2 along with several sets discussed in Sec. VI. In addition, we used the renormalization constant $\xi_{s}$ in the field operator of the $M1$ excitations (16) to fit the isovector $M1$ strength. The values of this constant for the RPA and RenTBA are also shown in Table 2.

Note that the set of the phonons in the RenTBA after the renormalization procedure with the use of the condition (14) included 123 electric and 83 magnetic phonons for the parametrization SKXm_-0.49 and 121 electric and 85 magnetic phonons for the parametrization SV-bas_-0.44.

Most of the calculations presented below have been performed within the discrete versions of RPA and RenTBA that means that the model equations are solved in the discrete basis representation with the use of the box boundary conditions for all functions entering these equations. It is convenient to present these results as well as the experimental data in the form of the strength functions $S(E)$ obtained by folding the discrete spectra with a Lorentzian of half-width $\Delta$:

The results for the modified SKXm parametrizations SKXm_-0.49 (RenTBA) and SKXm_-0.54 (RPA) obtained with $\Delta$ = 20 keV are shown on the upper panel of Fig. 1. The experimental spectra were taken from Refs. Shizuma et al. (2008) [²⁰⁸Pb$\,(\gamma,\gamma^{\prime})$ reaction, data below the neutron separation energy $S(n)=$ 7.37 MeV] and Köhler et al. (1987) [²⁰⁷Pb$\,(n,\gamma)$ reaction, data above $S(n)$].

The RenTBA, in contrast to the RPA, reproduces the experimental splitting of the $M1$ resonance into two components separated by the dip near 7.4 MeV. The quantitative characteristics of this splitting are given in Table 3 in comparison with the experiment.

The experimental summed $M1$ strength in the energy interval below the neutron threshold $\sum B(M1)_{<}$ is greater than the strength above the threshold $\sum B(M1)_{>}$ by about 50%, while the respective theoretical values are approximately equal to each other. Nevertheless, the total theoretical summed $M1$ strength in the interval 6.6–8.1 MeV is equal to the experimental one according to the conditions of construction of our modified parametrizations. The absolute values of the calculated mean energies $\bar{E}_{<}$ and $\bar{E}_{>}$ are close to the experimental values, however the differences $\Delta\bar{E}=\bar{E}_{>}-\bar{E}_{<}$ are different: the theoretical value $\Delta\bar{E}_{{}_{\mbox{\scriptsize theor}}}$ = 0.14 MeV is less than the experimental one $\Delta\bar{E}_{{}_{\mbox{\scriptsize exp}}}$ = 0.31 MeV by a factor of two. To estimate the fragmentation of the $M1$ resonance we have also calculated the equivalent Gaussian width $\Gamma$ in the interval 6.6–8.1 MeV both for the experimental and for the theoretical strength distributions. The results presented in last column of Table 3 show that the total width of the resonance is still underestimated.

Existence of the dip near the neutron separation energy in the experimental distribution of the $M1$ strength in ²⁰⁸Pb is generally an uncertain point because the reliability of the experimental data Shizuma et al. (2008); Köhler et al. (1987) goes down in this region. To some extent, the possible existence of this dip is supported by the more recent data of the ²⁰⁸Pb$\,(p,p^{\prime})$ experiment Poltoratska et al. (2012). The partial $M1$ cross section $\sigma_{M1}$ of this reaction is shown on the lower panel of Fig. 1. The dip in energy dependence of $\sigma_{M1}$ near 7.4 MeV exists though it is less pronounced than for the strength function obtained from the data Shizuma et al. (2008); Köhler et al. (1987). Note, however, the following. First, the direct comparison of the $M1$ strength functions $S(E)$ and the cross section $\sigma_{M1}(E)$ is hindered by the fact that they are determined by the different reaction mechanisms. The distribution of the $B(M1)$ values can be obtained from the cross section of the $(p,p^{\prime})$ reaction only within the framework of some model assumptions, see, e.g., Ref. Birkhan et al. (2016). Second, the dip near 7.4 MeV is absent in the distribution of $dB(M1)/dE$ deduced in Birkhan et al. (2016) from the data of Ref. Poltoratska et al. (2012) and shown in Fig. 3(b) of Ref. Birkhan et al. (2016). But this fact can be explained by the different (and quite large) widths of the used energy bins that corresponds to the large and energy-dependent values of the smearing parameter $\Delta$ of the strength function (18).

To show the fine structure of the theoretical and experimental strength distributions and to study the role of the single-particle continuum (which in principle can manifest itself above the neutron separation energy) we have calculated the $M1$ strength functions in ²⁰⁸Pb within the continuum RenTBA with $\Delta$ = 1 keV and 0.1 keV. The single-particle continuum was included within the response function formalism according to the method developed in Ref. Tselyaev et al. (2016). In this approach the strength function $S(E)$ is expressed through the response function, and the right-hand side of Eq. (18) is supplemented with the contribution of the continuum part of the spectrum. The parametrizations SKXm_-0.49 and SV-bas_-0.44 (the latter is discussed in more detail in Sec. VI) were used.

The results for $\Delta$ = 1 keV are shown on the upper panel of Fig. 2 in terms of the function $\tilde{B}_{M1}(E)$ defined as

Here we use this function because, as follows from Eq. (18),

So, if the $\Delta$ is small, the peak values of the function $\tilde{B}_{M1}(E)$ are close to the excitation probabilities at the peak energies. Note that Eq. (20) makes sense only for the states of the discrete spectrum. However, if the $\Delta$ is greater than the escape width of the quasidiscrete state in the continuum, the peak value of the function $\tilde{B}_{M1}(E)$ allows us to estimate the integrated strength of the single resonance.

In the RenTBA calculation with the SKXm_-0.49 set and $\Delta$ = 1 keV, the fragmentation of two main peaks shown in Fig. 1 for the strength distributions with $\Delta$ = 20 keV is very small. This picture does not match the detailed fragmentation structure of the experimental distribution composed from data of Refs. Shizuma et al. (2008); Köhler et al. (1987) and shown on the lower panel of Fig. 2. The $M1$ strength in the interval 7–8 MeV obtained in the RenTBA with the parametrization SV-bas_-0.44 is concentrated in one state without visible fragmentation, as in the case of the RPA.

The lack of fragmentation in the presented RenTBA calculations can be explained by the limited (though extended as compared to the RPA) kinds of the correlations included in the model. There are two natural generalizations of the RenTBA which enable one to include the additional correlations. First is the model taking into account the so-called ground-state correlations beyond the RPA. In Refs. Kamerdzhiev and Tselyaev (1991); Kamerdzhiev et al. (1993), it was shown that the inclusion of the correlations of this type increases the fragmentation of the $M1$ resonance in ²⁰⁸Pb. The second generalization is the replacement of the intermediate $1p1h\otimes$phonon configurations by two-phonon configurations according to the scheme suggested in Tselyaev (2007) and in analogy with the first versions of the quasiparticle-phonon model Soloviev (1992). Note that the relative importance of these additional correlations increases at low energies due to the low level densities as compared to higher energies.

To analyze the effect of the single-particle continuum we first note that the theoretical neutron separation energies are equal to 7.30 MeV for the parametrization SKXm_-0.49 and 7.64 MeV for the parametrization SV-bas_-0.44. So, the single peak of the RenTBA strength distribution for the SV-bas_-0.44 set shown on the upper panel of Fig. 2 (blue dashed line) is in the discrete spectrum, while the main strength of the distribution for the SKXm_-0.49 set (red solid line) lies in the continuum.

The effect of the continuum is determined by the values of the escape widths of the resonances. The full width at half maximum (FWHM) of the single peak of the strength distribution corresponding to the one or several overlapping resonances is formed by the escape and spreading widths and by the artificial width of $2\Delta$ introduced by the smearing parameter. Thus, the FWHM can serve as an upper bound of the escape width. The distribution for the parametrization SKXm_-0.49 shown on the upper panel of Fig. 2 contains three main peaks with the energies 7.313 MeV, 7.325 MeV, and 7.457 MeV. These peaks correspond to four states of the discrete RenTBA spectrum with the energies 7.313 MeV, 7.326 MeV, 7.457 MeV, and 7.459 MeV which exhaust 92% of the summed strength of the $M1$ resonance in the interval 6.6–8.1 MeV. So, we can confine ourselves to analyzing the widths of only these peaks. The respective values of the FWHM are equal to 2.1 keV for the quasidiscrete states with $E=$ 7.313 MeV and 7.325 MeV and to 3.4 keV for the resonance with $E=$ 7.457 MeV. The last FWHM value is appreciably greater than $2\Delta$. This is explained by the fact that the peak with $E=$ 7.457 MeV is formed by two overlapping resonances which correspond to two states of the discrete spectrum mentioned above.

In the calculation with $\Delta$ = 0.1 keV, the widths of the main peaks decrease. The values of the FWHM for the quasidiscrete states with $E=$ 7.313 MeV and 7.325 MeV become less than 0.3 keV. The peak with $E=$ 7.457 MeV is split into two peaks separated by the small interval of 2 keV and having the widths which are less than 1 keV. Thus the escape widths of the main peaks of the distribution for the SKXm_-0.49 set are safely less than 1 keV. These results show that the inclusion of the single-particle continuum has no appreciable impact in the calculations with $\Delta=$ 20 keV presented in the paper.

The splitting of the isovector $M1$ resonance in ²⁰⁸Pb into two main peaks obtained in RenTBA with the use of the parametrization SKXm_-0.49 is not a common result for the self-consistent calculations in our approach. In the typical case, if the EDF parameters $g$ and $g^{\prime}$ are fitted to reproduce the experimental energy of the $1_{1}^{+}$ state and the mean energy of the $M1$ resonance in ²⁰⁸Pb, the fragmentation of the isovector $M1$ resonance is reduced to the quenching of the main peak without appreciable broadening. This quenching is compensated by decreasing the renormalization constant $\xi_{s}$ after which the forms of the RenTBA and RPA $M1$ distributions become close to each other. This is illustrated in Fig. 3 where we show results for the modified SV-bas parametrizations.

To clarify the problem, we note that the effects of the fragmentation of the RPA states in TBA and RenTBA are determined by the energy-dependent induced interaction ${W}(\omega)$, Eq. (8a). The fragmentation of the RPA state with the energy $\omega_{{}_{\mbox{\scriptsize RPA}}}$ is strong if (i) one or more energies $\Omega^{\vphantom{*}}_{c}$ of the $1p1h\otimes$phonon configurations in Eqs. (8) are close to the shifted energy $\tilde{\omega}_{{}_{\mbox{\scriptsize RPA}}}$ (shifted due to the regular contribution of the remaining $1p1h\otimes$phonon configurations) and (ii) the respective amplitudes ${F}^{c(+)}_{ph}$ are non-negligible. In the case of the nucleus ²⁰⁸Pb, the isovector $M1$ strength in the RPA is concentrated as a rule in one state with the energy $\omega_{{}_{\mbox{\scriptsize RPA}}}(1^{+}_{2})$ (the $1^{+}_{1}$ RPA state is isoscalar) which is formed by two $1p1h$ configurations: $\pi(1h_{9/2}\otimes 1h^{-1}_{11/2})$ and $\nu(1i_{11/2}\otimes 1i^{-1}_{13/2})$. So, the $ph$ indices of the amplitudes ${F}^{c(+)}_{ph}$ producing appreciable fragmentation of the $1^{+}_{2}$ RPA state should be one of these two combinations. Under this condition and according to the selection rules for the $M1$ excitations, the minimum value of $\Omega^{\vphantom{*}}_{c}$ in ²⁰⁸Pb is determined by the configuration $c=\{\pi(1h_{9/2}\otimes 3s^{-1}_{1/2})\otimes 5^{-}_{1}\}$, that is

It turns out that for most Skyrme EDF parametrizations the value of $\Omega^{\mbox{\scriptsize min}}_{c}$ is substantially greater than the mean energy of the isovector $M1$ resonance in ²⁰⁸Pb, that is $\Omega^{\mbox{\scriptsize min}}_{c}>7.4$ MeV. Thus, if the parameters of the EDF are fitted to reproduce this mean energy, the fragmentation of the isovector $M1$ resonance is reduced to its quenching as mentioned above. The parametrization SKXm_-0.49 is an exception because the value of $\omega_{{}_{\mbox{\scriptsize RenTBA}}}(5^{-}_{1})$ comes close the experimental value which, in turn, yields an $\Omega^{\mbox{\scriptsize min}}_{c}$ close to 7.4 MeV. This is shown in Table 4 in comparison with the case of the SV-bas_-0.44 parametrization.

Note that the splitting of the isovector $M1$ resonance shown in Fig. 1 is achieved only in the RenTBA. In conventional TBA, the energies of the phonons in Eqs. (8) are calculated within the RPA. In the case of the parametrization SKXm_-0.49, the energy $\omega_{{}_{\mbox{\scriptsize RPA}}}(5^{-}_{1})$ = 3.64 MeV that increases the energy $\Omega^{\mbox{\scriptsize min}}_{c}$ and leads to the RPA-like result in the TBA similar to shown in Fig. 3 by the red solid line.

On the other hand, the fragmentation of the isovector $M1$ resonance in ²⁰⁸Pb in itself can be obtained also in the case $\Omega^{\mbox{\scriptsize min}}_{c}>7.4$ MeV if the isovector $M1$ strength is shifted to higher energies by increasing the EDF parameter $g^{\prime}$. This is shown in Fig. 3 for the parametrization SV-bas${}^{\prime}_{-0.44}$ which is constructed from the set SV-bas_-0.44 by changing the parameter $g^{\prime}$ from 1.03 for SV-bas_-0.44 to 1.46 for SV-bas${}^{\prime}_{-0.44}$ (however, the set of the phonons in this illustrative RenTBA calculation for SV-bas${}^{\prime}_{-0.44}$ was used the same as for SV-bas_-0.44). Thus, the simultaneous description of the mean energy of the isovector $M1$ resonance in ²⁰⁸Pb and of the fragmentation of this resonance in the self-consistent calculation is seemingly possible only in rare circumstances as, e.g., in case of the parametrization SKXm_-0.49.

Note that the fragmentation of the isovector $M1$ resonance in ²⁰⁸Pb was obtained in the early calculations within the shell model in the $1p1h+2p2h$ space Lee and Pittel (1975) and within the models based on the TFFS Migdal (1967) and including the particle-phonon interaction on top of the RPA (see, e.g., Dehesa et al. (1977); Kamerdzhiev and Tkachev (1984, 1989); Kamerdzhiev and Tselyaev (1991); Kamerdzhiev et al. (1993)). This result is explained by two reasons. First, the mean energy of the isovector $M1$ resonance in these calculations was greater than the experimental value. The shift to higher energies increases the spreading of the $M1$ strength as was noted in Ref. Lee and Pittel (1975) and is demonstrated in Fig. 3. Second, the phonon energies in the calculations of Refs. Kamerdzhiev and Tkachev (1984, 1989); Kamerdzhiev and Tselyaev (1991); Kamerdzhiev et al. (1993) were fitted to their experimental values that makes the value of $\Omega^{\mbox{\scriptsize min}}_{c}$ more close to the mean energy of the isovector $M1$ resonance, see Table 4.

To demonstrate the role of the intermediate $1p1h\otimes$phonon configuration $\pi(1h_{9/2}\otimes 3s^{-1}_{1/2})\otimes 5^{-}_{1}$ in the effect of the fragmentation under discussion we show in Fig. 4 the results of three RenTBA calculations with the use of parametrizations SKXm_-0.49, SKXm${}^{\prime}_{-0.49}$, and SKXm${}^{\prime\prime}_{-0.49}$ (see Table 2).

The RenTBA calculation with the use of parametrization SKXm_-0.49 coincides with one shown in Fig. 1. In the calculation with the use of SKXm${}^{\prime}_{-0.49}$, the $5^{-}_{1}$ phonon was excluded and the EDF parameter $g^{\prime}$ was slightly changed to fit the mean energy of the isovector $M1$ resonance to the experiment. The calculation with SKXm${}^{\prime\prime}_{-0.49}$ represents the opposite case: only the $5^{-}_{1}$ phonon was included in the phonon basis of the RenTBA and the EDF parameters $g$ and $g^{\prime}$ were changed to fit the energy of the $1^{+}_{1}$ state and the mean energy of the isovector $M1$ resonance to the experiment. The renormalization constant $\xi_{s}$ was also changed to compensate decreasing of the quenching of the $M1$ strength. However, the characteristics of the same phonons (energies, etc.) were the same in all three calculations. These results show that the splitting of the isovector $M1$ resonance in ²⁰⁸Pb is determined in the considered model practically exclusively by the configuration $\pi(1h_{9/2}\otimes 3s^{-1}_{1/2})\otimes 5^{-}_{1}$. The other $1p1h\otimes$phonon configurations produce only the shift of the $M1$ resonance and the quenching of the $M1$ strength.

In section VI, we have shown that the RenTBA using the modified Skyrme EDF SKXm_-0.49 gives an energy of the first $5^{-}$ state in ²⁰⁸Pb close to its experimental value. Here we consider the results of the RenTBA and RPA calculations for the first excited states of natural parity in ²⁰⁸Pb with the multipolarity $L$ from 2 to 6 both for the SKXm_-0.49 and the SV-bas_-0.44 parametrizations. The results are presented in Tables 5 and 6.

Note that the RenTBA results have been obtained without use of the diagonal approximation which is used in the model only for the phonons entering the intermediate $1p1h\otimes$phonon configurations. It explains the small difference between the energies of the $5^{-}_{1}$ state listed in Tables 4 (where the diagonal approximation is used) and 5.

The RenTBA energies calculated with the parametrization SKXm_-0.49 agree fairly well with the experiment. The deviations for SV-bas_-0.44 are slightly greater (except for the $4^{+}_{1}$ state). The RPA gives too large energies for both parametrizations. The energy shift $\omega(\mbox{RPA})-\omega(\mbox{RenTBA})$ is between 0.2 MeV for the $3^{-}_{1}$ state and 0.6 MeV for the $6^{+}_{1}$ state.

The situation is the opposite for the excitation probabilities shown in Table 6.

The RPA results are closer to the experiment as compared to the RenTBA results (and are in a good agreement with the experiment for $2^{+}_{1}$, $3^{-}_{1}$, and $4^{+}_{1}$ states). The decrease of the $B(EL)$ values in RenTBA is caused by the quenching as in the case of the $M1$ excitations.

By construction, the modified parametrizations SKXm_-0.49 and SV-bas_-0.44 describe the nuclear ground-state properties within the Skyrme EDF approach (with approximately the same accuracy as the original parametrizations SKXm and SV-bas) and reproduce the basic experimental characteristics of the $M1$ excitations in ²⁰⁸Pb within the RenTBA. The results of this section show that the RenTBA with the use of these modified parametrizations is applicable also to the description of the low-energy electric excitations in this nucleus.

The present paper is a continuation of our recent work Tselyaev et al. (2019) in which we investigated the low-energy $M1$ excitations in ²⁰⁸Pb within the self-consistent RPA based on the Skyrme energy-density functionals (EDF). Here we use the extended self-consistent model including the particle-phonon coupling within the renormalized time blocking approximation (RenTBA, Tselyaev et al. (2018)). As in the case of the self-consistent RPA, the description of the basic experimental characteristics of the $M1$ excitations in ²⁰⁸Pb (energy and strength of the $1_{1}^{+}$ state as well as mean energy and summed strength of the isovector $M1$ resonance) requires refitting some of the spin-related parameters of the Skyrme EDF within the self-consistent RenTBA. We have determined several sets of these parameters from this condition. It has been shown that the observed fragmentation of the isovector $M1$ resonance in ²⁰⁸Pb which is absent in all the RPA calculations can be to a certain extent described within the self-consistent RenTBA. However, this description is not fully quantitative and is attained only in some cases of the modified functionals of the Skyrme type. We have found that the necessary condition to obtain this fragmentation in our model is the proximity of the energy of the intermediate $1p1h\otimes$phonon configuration $\pi(1h_{9/2}\otimes 3s^{-1}_{1/2})\otimes 5^{-}_{1}$ to the mean energy of the isovector $M1$ resonance in ²⁰⁸Pb, i.e. the proximity of the energy of $5^{-}_{1}$ phonon to the experimental excitation energy of the $5^{-}_{1}$ state in ²⁰⁸Pb. We have also shown that the modified parametrizations of the Skyrme EDF presented in the paper can be used in the description of the low-energy electric excitations within the RenTBA.