Structure and decay properties of spin-dipole giant resonances within a semimicroscopical approach

A semimicroscopical approach is applied to calculate: (i) strength functions for the charge-exchange spin-dipole giant resonances in the 208Pb parent nucleus; (ii) partial and total branching ratios for the direct proton decay of the resonance in 208Bi. The approach is based on continuum-RPA calculations of corresponding reaction-amplitudes and phenomenological description of the doorway-state coupling to many-quasiparticle configurations. The only adjustable parameter needed for the description is found by comparison of the calculated and experimental total widths of the resonance. Other model parameters used in calculations are taken from independent data. The calculated total branching ratio is found to be in reasonable agreement with the experimental value.


Introduction
The intensive experimental and theoretical studies of the direct nucleon decay of various giant resonances have been undertaken in recent years in an attempt to understand better the interplay of single-quasiparticle, collective and manyquasiparticle modes of nuclear motion. This work was stimulated by the appearance of experimental data on (total) branching ratio for the proton decay of the spin-dipole giant resonance of the (pn −1 ) -type (SDR (−) ) in 208 Bi [1]. The branching ratio was deduced from the analysis of the 208 Pb( 3 He,t) 208 Bi and 208 Pb( 3 He,tp) 207 Pb reactions cross sections at E( 3 He)=450 MeV. Being related to the spin-flip giant resonance, this branching ratio is a valuable addition to the experimental data on partial proton widths of the Gamow-Teller resonance (GT R), which were also obtained in Ref. [1].
The main aim of this work is to calculate the above-mentioned branching ratio within the approach proposed previously [2,3] and called, for brevity sake, the semimicroscopical approach. Another aim is to analyze the strength functions of the inverse resonances (SDR (+) ) in 208 Tl within the same approach. The basic points of the semimicroscopical approach are the following: (i) continuum-RPA (CRPA) calculation of the reaction amplitudes corresponding to the excitation of the considered GR by an external single-particle field; (ii) Breit-Wigner parameterization of the calculated amplitudes to deduce the parameters of those particle-hole-type doorway states, which form the GR; (iii) a phenomenological description (with averaging over the energy) of the doorway-state coupling to many-quasiparticle configurations. In the considered case the only adjustable parameter needed for the description is found by comparison of the calculated and experimental total widths of the SDR (−) . The phenomenological mean field and the isovector part of the Landau-Migdal particle-hole interaction are used in the calculations. The calculation results are compared with both the experimental data taken from Ref. [1] and results of some previous calculations of the GT R partial proton widths [4].
2 Calculation scheme CRPA equations. All CRPA-equations used in this work are given in the form accepted within the finite Fermi-system theory [5]. LetV JLSM (x a ) = V (r)T JLSM ( n)τ (∓) be an external single-particle field acting upon the nucleus in the process of GR excitation. Here, T JLSM ( n) is the irreducible spinangular tensor operator of rank J, τ (∓) , τ (3) are the Pauli matrices acting in the isospin space. Bearing in mind the SDR (∓) excitation we put below J = 0, 1, 2; L = S = 1 (for the GT R J = S = 1, L = 0). Within the CRPA the strength function ("inclusive reaction" cross section) is defined by the following expression: where P (∓) V,J (ω) is the nuclear polarizability ("forward scattering" amplitude) corresponding to the given external field, (rr ′ ) −2 A (∓) J (r, r ′ ; ω) is the radial part of free particle-hole propagator carrying quantum numbers J, L and S, ω is the excitation energy measured from the parent-nucleus ground-state energy andṼ (∓) J are the so-called effective fields. They satisfy the integral equations: where G ′ is the intensity of the spin-isospin part of Landau-Migdal particle- can be expressed in terms of: (i) occupation numbers n α µ (α = n, p), (ii) radial boundstate single-particle wave functions r −1 χ α µ (r) (µ = ε µ , j µ , l µ ) and Green functions g α (ν) (r, r ′ ; ε) of the radial single-particle Schrodinger equations ((ν) = j ν , l ν ). The expressions for A (∓) J are well-known and given, for instance, in Ref. [2] for the case of the GT R description.

Doorway-state parameters.
In the vicinity of the GR with not-too-large excitation energy the reaction amplitudes calculated within the CRPA exhibit narrow, as a rule nonoverlapping, resonances. These resonances correspond to the particle-hole-type doorway states forming the GR. Breit-Wigner parameterization of the amplitudes P V,J and M c allows one to deduce the doorway-state parameters: energy ω g , partial strength R g , partial and total escape widths Γ ↑ gc and Γ ↑ g , respectively. The possibility to use the above parameterization can be checked by satisfying the equal- Eq. (5) are smooth functions of energy. Note that amplitudes R 1/2 g , (Γ ↑ gc ) 1/2 are not sign-fixed quantities and only their products found with the help of parameterization (5) are used below.
V,J (ω) dω calculated with the use of Eq. (3) for a rather wide excitation energy interval δ 12 = ω 2 − ω 1 define the GR total strength, so that ratios x g = R g /R V,J are the relative partial strengths. Calculations of the above strength functions can be checked by a comparison of difference R V,J calculated for a long-wave external field V (r) are expected to be small for nuclei with large neutron excess due to Pauli blocking.
Doorway-state coupling to many-quasiparticle configurations. This coupling leads to doorway-state spreading and formation of the GR as a single resonance in the energy dependence of energy-averaged reaction cross sections. We take this coupling into consideration phenomenologically by independently spreading each doorway-state resonance [2,3]. It means that the transition to the energy-averaged reaction amplitudesP V,J andM c can be realized by the following substitution in Eqs. (5): The doorway-state spreading width Γ ↓ is considered as the only adjustable parameter of the semimicroscopical approach. It can be found by equating the total width Γ (dependent on Γ ↓ ) of the calculated energy-averaged strength function of the SDR (−) to the total width Γ exp of the SDR (−) in the experimental inclusive reaction cross section. Because this cross section is parameterized by a single-level formula [1], we approximate calculated strength function (6) by the same formula:S where R and ω m are, respectively, the calculated total strength and mean excitation energy of the SDR (−) .
Because each doorway-state resonance in the energy dependence of ampli-tudesM c (ω) becomes rather broad, it is necessary to take also into account changing the penetrability of the potential barrier for escaping protons over the resonance. It can be done as follows [4]: Here, ε gµ = ω g + ε n µ ,P g(ν) is the penetrability averaged over the resonance: Thus, the energy-averaged partial cross sectionsσ µ (ω) = (ν) |M c (ω)| 2 (the fluctuational part of these cross sections is neglected) can be calculated without the use of any free parameters. Summation in the above equation is performed over the quantum numbers of the escaping proton, which are compatible with the selection rules for the spin-dipole transitions. Cross section σ µ (ω) corresponds to population of single-hole state µ −1 in the product nucleus after the SDR (−) proton decay.
The SDR (−) branching ratios and partial widths for the direct proton decay are defined as follows: whereS is defined by Eqs. (6). Note that this strength function, its singlelevel approximation (7) (parameters R When only one doorway state corresponds to the considered GR (for instance, in case of the GT R) one can calculate the average partial escape widths of this GR directly with the help of Eq. (8): Such a procedure was realized in Ref. [4]. To take the averaged potential-barrier penetrability more accurately into consideration in accordance with Eqs. (8), (9) we use the experimental one-hole state energies ε n exp µ instead of calculated ones. For comparison with experimental data the calculated quantities b µ orΓ ↑ µ should be also multiplied by spectroscopic factors S µ of the corresponding single-hole states in the product nucleus 207 Pb.

Calculation results
Choice of model parameters. The nuclear mean field and particle-hole interaction are the input data for any RPA calculations. In the following the isoscalar part of the phenomenological nuclear mean field (including the spin-orbit interaction) is chosen in accordance with Ref. [2]. Only mean field amplitude U 0 is slightly increased (54.1 MeV instead of 53.3 MeV) to describe better the nucleon separation energies for 208 Pb. The strengths (F ′ , G ′ ) = (f ′ , g ′ ) · 300 MeV · fm 3 of the isovector part of the Landau-Migdal particle-hole interaction are chosen as follows: f ′ = 1.0 (to describe the experimental difference of the neutron and proton separation energies in 208 Pb); g ′ = 0.76 (to describe the experimental energy of the GT R in 208 Bi [1]). The above strengths are close to those used in Refs. [2,4]. The isovector part 1 2 τ (3) v(r) of the mean field is calculated in a self-consistent way (see e.g. Ref. [6]): v(r) = 2F ′ ̺ (−) . The nuclear Coulomb field is calculated in Hartree approximation via the proton density ̺ p .
Calculation results for the 208 Pb parent nucleus. The nuclear mean field chosen above allows one to describe satisfactorily the single-quasiparticle spectrum of the 208 Pb parent nucleus. The calculated neutron single-hole spectrum (energies ε n µ ) is given in the Table 1 in comparison with the experimental one (energies ε n exp µ ). All experimental quantities (except for the spectroscopic factors S µ and neutron separation energy S n ) in this Table are taken from Ref. [1]. The values of S µ and S n are taken from Refs. [7] and [8], respectively.
In CRPA calculations of the SDR (−) and the GT R strength functions the radial dependence of the external field V (r) is chosen as r/R and 1, re-spectively (R is the nuclear radius). The GT R strength function is similar to that given in [2] and is not shown here. The SDR (−) strength functions S In the case of J π = 0 − and J π = 1 − the main part of the total strength is exhausted by one doorway state (85% and 81%, respectively). In the case of J π = 2 − the calculated strength function exhibits an essential gross structure: eight doorway-state resonances exhaust 91% of the total strength (the relative strength of most resonances x g (in %) is shown in Figs. 1a-1c). The ratios x J and B J are also given in these figures.
The energy-averaged strength functionsS  Table 2. The strength function calculated for each interval is approximated by single-level formula (7) so that adjustable parameter Γ ↓ (doorway-state spreading width) and calculated SDR (−) energy ω (−) m are determined by fitting to the SDR (−) experimental total width Γ exp = 8.4 MeV [1]. These parameters are also given in Table 2.
Partial and total branching ratios for the SDR (−) proton decay are calculated according to equations (8)- (10). Calculated total branching ratios b depending on considered interval δ 12 are given in Table 2. The use of ω (−) 1 = 17 MeV seems to be reasonable for description of experimental data [1]. The reasons are the following: (i) calculated strength functionS (−) V (ω (−) ) is satisfactorily described by single-level formula (see Fig. 2) used for approximation of the experimental inclusive reaction cross section [1]; (ii) the most part of the calculated SDR (−) strength (x δ = 83%) is exhausted within this interval δ 12 ; (iii) doorway-state spreading width Γ ↓ = 4.7 MeV found with the use of the experimental SDR (−) total width is reasonably greater than the experimental GT R spreading width Γ ↓ GT R = 3.54 MeV [1]. The partial SDR (−) proton branching ratios b µ (b = 16.1%) calculated for the chosen excitation-energy interval are given in Table 1.
The averaged proton partial widths of the GT RΓ µ calculated according to Eqs. (8), (9) are also given in Table 1 in comparison with the experimental data. In above calculations the following expression for the penetrability is used [9]: P (kR) = kR [F 2 (kR) + G 2 (kR)] −1 , where k 2 = 2mε/h 2 , F and G are well-known Coulomb functions.

Discussion of results.
The quality of the performed CRPA calculations of the SDR (−) spin-dipole strength functions is satisfactory, because calculated ratios x V,J are close to unity (Figs. 1a-1c). As expected, Pauli blocking leads to suppression of the SDR (+) spin-dipole strengths R for 208 Pb), because calculated ratios B J are rather small (Figs. 1a-1c).
Calculated proton partial widths of the GT R are in a reasonable agreement with both the experimental data and the results of previous calculations performed within the same approach. The difference between two sets of Γ

Conclusion and acknowledgments
In this work the semimicroscopical approach is applied to calculate the branching ratios for the direct proton decay of the SDR (−) in 208 Bi. A reasonable description of the experimental data on the proton total branching ratio is obtained. Previous calculations on the partial proton widths of the GT R in 208 Bi are refined. The energy position of the second SDR (+) in 208 Tl is predicted.
It will be possible to perform more detailed comparison of the results obtained within the semimicroscopical approach and experimental data provided that the experimental partial proton branching ratios b µ will become available and different SDR (−) spin components will be separated [10]. In this connection further experimental and theoretical studies of proton and γ -decays of the SDR (−) seem to be very promising. financial support from the "Nederlandse organisatie voor wetenschappelijk onderzoek" NWO during his stay at the KVI.     Table 1 Calculated escape widths and branching ratios for proton decay of the GT R and the SDR (−) in 208 Bi. Rather small contribution of deep-hole states to the calculated b value ( µ ′ b µ ′ = 1.94%) is not shown. Experimental data contain also Γ ↑ tot = 1.18 ± 0.35 MeV and b = 14.1 ± 4.2% for the SDR (−) . Table should be put after the first mention in the text. Table 2 Dependence of the SDR (−) parameters on the choice of excitation-energy interval δ 12 = ω  Table should be put after the first mention in the text.