FOR GIA IANT RESONANCES M.H. Urin National Research Nuclear - - PowerPoint PPT Presentation

PARTICLE-HOLE DIS ISPERSIVE OPTICAL MODEL FOR GIA IANT RESONANCES

M.H. Urin

National Research Nuclear University “MEPhI” (Moscow Engineering Physics Institute) International Seminar “EMIN-2018” Moscow, Russia, October 8 – 11, 2018

Outline

PARTICLE-HOLE DISPERSIVE OPTICAL MODEL (PHDOM) I. Physical content II. Lines of formulation

• III. Implementations to giant resonances
• IV. Conclusive remarks

I.

• I. Physical content

1.1 Aims and terminology

• The PHDOM is formulated to describe in a semi-microscopic way

commonly the structure and damping of a great variety of high- energy (p-h)-type excitations (including giant resonances (GRs)) in medium-heavy mass “hard” spherical nuclei.

• The term “PHDOM” appears in view of similarity of microscopically-

based formulations of the PHDOM and single-quasiparticle dispersive

• ptical model (SQDOM). Both models are formulated in terms of the

energy-averaged s-q and p-h Fermi-system Green functions (GFs), which obey, respectively, to the Dyson- and Bethe-Goldstone-type equations, having the respective self-energy term (U., PRC`13).

1.2 Relaxation modes

Within the model the main relaxation modes of mentioned excitations are taken into account together. These modes are: i. distribution of the p-h strength, or the Landau damping (a result of shell structure of nuclei); ii. coupling (p-h)-type states to the single-particle (s-p) continuum (nuclei are open Fermi-systems);

• iii. coupling (p-h)-type states to many-quasiparticle (chaotic)

configurations, or the spreading effect (high excitation energy).

1.3 Physical content

• The PHDOM is a semi-microscopic model, in which Landau damping

and coupling (p-h)-type states to the s-p continuum are described microscopically (in terms of a mean field and p-h interaction responsible for long-range correlations), while the spreading effect is treated phenomenologically and in average over the energy (in terms

• f the specific p-h interaction, or the respective p-h self-energy term).
• The PHDOM can be called as “the model of interacting independently

damping quasiparticles”.

1.4 Unique features

The unique features of the PHDOM are concerned with its ability to describe: i. the energy-averaged double transition density and, therefore, various strength functions at arbitrary (but high-enough) excitation energy; ii. direct one-nucleon decays of (p-h)-type states, including the direct + semi-direct (DSD) reactions induced by a s-p external field;

• iii. the spreading (dispersive) shift of the energy of resonance-like

structures related to (p-h)-type states.

II II. . Lines of f formulation (s (schematically)

2.1 Continuum-RPA

• The PHDOM is a microscopically-based extension of the continuum-

RPA (cRPA) versions on taking into account the spreading effect. The standard cRPA version (Shlomo, Bertch, NPA`75) is formulated in terms the p-h Green function (GF), 𝐵𝑑𝑆𝑄𝐵(𝑦, 𝑦′, 𝜕) (𝜕 is the excitation energy), which obeys the Bethe-Goldstone-type equation and determines the strength function, 𝑇𝑊

𝑑𝑆𝑄𝐵 𝜕 , related to a s-p external

field 𝑊

0(𝑦):

𝐵𝑑𝑆𝑄𝐵 = 𝐵0

𝑑𝑆𝑄𝐵 + 𝐵0 𝑑𝑆𝑄𝐵𝐺𝐵𝑑𝑆𝑄𝐵; 𝑇𝑊 𝑑𝑆𝑄𝐵 𝜕 = − 1

𝜌 Im 𝑊

+𝐵𝑑𝑆𝑄𝐵𝑊 0.

Here, 𝐵0

𝑑𝑆𝑄𝐵 is the free propagator, 𝐺(𝑦, 𝑦′) is the p-h interaction

responsible for long-range correlations (e., g. for formation of GRs).

• Formulation of 𝐵0

𝑑𝑆𝑄𝐵 in terms of GFs of the s-p Schrodinger equation

allows one to take exactly the s-p continuum into account. The non- standard cRPA version (U., NPA`08) is formulated in terms of the effective field 𝑊(𝑦, 𝜕), which is defined by the relationship, 𝐵𝑑𝑆𝑄𝐵𝑊

0 = 𝐵0 𝑑𝑆𝑄𝐵𝑊,

and obeys the well-known integral equation: 𝑊 = 𝑊

0 + 𝐺𝐵0 𝑑𝑆𝑄𝐵𝑊.

• In the continuum region, the strength function can be expressed in terms
• f the squared DSD-reaction amplitudes, or partial one-nucleon-escape

strength functions: 𝑇𝑊

𝑑𝑆𝑄𝐵 = − 1

𝜌 Im 𝑊+𝐵𝑑𝑆𝑄𝐵 𝑊 =

𝑑

𝑁𝑊

0,𝑑

𝐸𝑇𝐸,𝑑𝑆𝑄𝐵 2 ≡ 𝑑

𝑇𝑊

0,𝑑

𝑑𝑆𝑄𝐵,↑ .

The amplitudes are proportional to the effective-field matrix elements taken between the bound and continuum s-p states (𝑑 is the set of decay-channel quantum numbers).

2.2 Discrete PHDOM version

• Similarly

to the

• rdinary

(single-quasiparticle)

• ptical

model (formulated by energy averaging the Dyson equation for the single- quasiparticle GF), the PHDOM is formulated by energy averaging the Bethe-Goldstone-type equation for the (generally, non-local) p-h GF 𝐵(𝑦, 𝑦1, 𝑦′, 𝑦′1, 𝜕). Along with the interaction 𝐺(𝑦, 𝑦′), the equation for 𝐵 contains a specific p-h interaction (the energy-averaged p-h self- energy term Π(𝑦, 𝑦1, 𝑦′, 𝑦′1, 𝜕)) responsible for the spreading effect) in the following way: 𝐵 = 𝐵0 + 𝐵0𝐺𝐵; 𝐵0 = 𝐵0

𝑆𝑄𝐵 + 𝐵0 𝑆𝑄𝐵Π𝐵0.

Here, the auxiliary quantity 𝐵0 is the “free” p-h propagator.

The phenomenological quantity Π is properly parameterized to satisfy the statistical assumption: after energy averaging different p-h configurations (with the same quantum numbers) are “decaying” into chaotic states independently of one another. In such a case, the equation for the propagator 𝐵0 (which corresponds to the model non-interacting independently damping quasiparticles) can be approximately solved. Using a mean-field discrete basis (s-p energies 𝜗𝜇, wave functions 𝜚𝜇) one gets the expression for the 𝐵0 expansion elements in a closed form: 𝐵0,𝜇𝜈 𝜕 = 𝑜𝜇 − 𝑜𝜈 𝜗𝜇 − 𝜗𝜈 − 𝜕 + 𝑜𝜇 − 𝑜𝜈 𝑗𝑋 𝜕 − 𝑄 𝜕 𝑔

𝜇𝑔 𝜈

. Here, 𝑜𝜇,𝜈 are the occupation numbers, 𝑋(𝜕) and 𝑄(𝜕) are respectively imaginary and real parts of the strength of the energy-averaged p-h self- energy term, 𝑔

𝜇 is the diagonal matrix element of the Woods-Saxon function

𝑔(𝑦).

Being taken in the local limit, i.e. 𝐵0(𝑦, 𝑦′, 𝜕) = 𝐵0(𝑦 = 𝑦1, 𝑦′ =

2.3 Continuum PHDOM version

The continuum PHDOM version (the basic one) follows from the approximate transformation of the “free” p-h propagator to the form, in which the GFs of the Schrodinger equations, having the addition to the mean field −𝑗𝑋 𝜕 + 𝑄 𝜕 𝑔

𝜉𝑔(𝑦) (𝜉 = 𝜈, 𝜇), are used. (Within

the cRPA, i.e. in neglecting the spreading effect, this transformation is exact). One from these equations determines the continuum-state wave functions 𝜚𝜗 =𝜗𝜈+𝜕 > 0

±

(𝑦). In such a way, there appears an effective optical-model potential, whose imaginary part was found (e. g., from the description of the total width of various GRs) is noticeably less, than the imaginary part of the “ordinary” OM potential taken at the corresponding energy.

2.4 Dispersive relationship

The dispersive relationship, which determines 𝑄(𝜕) via 𝑋(𝜕), is

• btained after energy averaging the spectral expansion of the p-h self-

energy term. This expansion is similar to that of the 2p-2h GF. (2p-2h configurations are the doorway-states for the spreading effect). The simplest version of the dispersive relationship 𝑄 𝜕 = 2 𝜌 𝑄. 𝑊.

∞

𝑋 𝜕′ 𝜕′ 𝜕2 − 𝜕′2 + 1 𝜕′ 𝑒𝜕′ is adopted to satisfy the condition: 𝑄 𝜕 → 0 → 0. In current implementations of the PHDOM, a more sophisticated version of the dispersive relationship is used.

2.5 Main PHDOM equations

Most of the main PHDOM equations, namely, the equations for the energy-averaged p-h GF 𝐵0(𝑦, 𝑦′, 𝜕), the strength function 𝑇𝑊

0(𝜕), the

effective field 𝑊(𝑦, 𝜕) looks similar to the respective cRPA equations (p. 2.1) after the substitution: 𝐵0

𝑑𝑆𝑄𝐵 𝑦, 𝑦′, 𝜕 → 𝐵0 𝑦, 𝑦′, 𝜕 .

The difference is concerned with the double transition density 𝜍 𝑦, 𝑦′, 𝜕 = − 1 𝜌 Im𝐵 𝑦, 𝑦′, 𝜕 . Due to taking the spreading effect into account, this quantity can't be factorized in terms of one-body transition density. (The latter can't be defined).

Since existing computer codes for calculation of the inelastic hadron- nucleus scattering accompanied by excitation of a given GR exploit only

• ne-body transition density, we suggest using for these codes the

respective “projected” density: 𝜍𝑕 𝑦, 𝜕 = ∫ 𝜍 𝑦, 𝑦′, 𝜕 𝑊

0,𝑕 𝑦′ 𝑒𝑦′𝑇𝑊

0,𝑕

−1/2, 𝑇𝑊

0,𝑕 = 𝜍𝑕𝑊

0,𝑕 2.

Here, 𝑊

0,𝑕 - specific for excitation of a given GR the external field.

The squared amplitude of the one-nucleon DSD reaction induced by a s-p external field 𝑊

0(𝑦) and accompanied by excitation of one-hole 𝜈−1

state of the product nucleus is expressed in terms of the effective field (U., PRC`13; Gorelik et al., NPA`18) 𝑁𝑊

0,𝑑

𝐸𝑇𝐸(𝜕) 2 = 𝑜𝜈

𝜚𝜗=𝜗𝜈+𝜕

− ∗

𝑊 𝜕 𝜚𝜈 𝜚𝜈

∗𝑊∗ 𝜕 𝜚𝜗=𝜗𝜈+𝜕 +

. Here, 𝑑 is a set of the reaction-channel quantum numbers, which includes the quantum numbers of the one-hole state and considered (p-h)-type excitation, and 𝜗 = 𝜗𝜈 + 𝜕 > 0.

The squared partial amplitudes, which can be called as the partial

• ne-

nucleon escape strength functions 𝑇𝑊

0,𝑑

↑

(𝜕), determine the partial branching ratios for direct one-nucleon decay from a certain excitation-energy interval 𝜀: 𝑐𝑑

↑ 𝜀 =

∫𝜀 𝑇𝑊

0,𝑑

↑

𝜕 𝑒𝜕 ∫𝜀 𝑇𝑊

0 𝜕 𝑒𝜕 .

The total one-nucleon-decay branching ratio 𝑐𝑢𝑝𝑢

↑

𝜀 =

𝑑

𝑐𝑑

↑(𝜀) .

determines the branching ratio for statistical (mainly, neutron) decay: 𝑐↓ = 1 – 𝑐𝑢𝑝𝑢

↑ .

Within the cRPA (i.e. in neglecting the spreading effect), 𝑐↓ = 0

All the main PHDOM relationships are valid at arbitrary (but high- enough) excitation energies. Only in case of using the specific for a given GR external field (probing operator) 𝑊

0,𝑕

to define the “projected” one-body transition density, the energy interval is limited by a vicinity of this GR.

2.6 Unitarity violations

The methods used within the model for describing the spreading effect lead to weak violations of model unitarity. These violations are due to: i. an energy dependence of the averaged p-h self-energy term (takes place for any type of p-h excitations); ii. the use of an approximate spectral expansion for the optical- model-like GF (takes place only for isoscalar monopole (ISM) excitations).

Violation of the first type is approximately eliminated by using the modified “free” p-h propagator 𝐵𝜇𝜈

𝑛 (𝜕):

𝐵0,𝜇𝜈

𝑛

𝜕 =𝐵0,𝜇𝜈 𝜕 1 − 𝑜𝜇 − 𝑜𝜈 𝑒𝑄 𝑒𝜕 𝑔

𝜇𝑔 𝜈

. (the contributions proportional to

𝑒𝑋 𝑒𝜕 2

might be neglected). Then the continuum PHDOM version is formulated using the properly modified optical-model-like GFs and continuum-state wave functions.

In the description of ISM excitations within the PHDOM, the respective probing operator 𝑊

0 𝑠 𝑍 00 should be also modified to avoid spurious

excitations caused by the unit external field: 𝑊

0 𝑠 → 𝑊 0 𝑠 − 〈𝑊 0〉

where averaging 〈… 〉 is performed over the ground-state density. As a result, the unitary version of the PHDOM is formulated (Gorelik et al., NPA`18), and unitarity violations are found to be week. An example is given below.

2.7 Related approaches

Many attempts have been undertaken in past to describe the spreading effect on GR properties. We mention two of them, which seem to us more advanced.

• The approaches, in which the spreading effect is described in terms of

coupling (p-h)-type states to a number of 2p-2h configurations, are related to microscopic approaches (see, e.g., Kamerdjiev et al., Phys.Rep. `05). Being only the doorway-states for the spreading effect, these configurations do not correspond to real nuclear states at high excitation energies, and their level density is much lower than the real one described by the statistical model. For these reasons the mentioned approaches seem not fully adequate to real physical situation. Also, the description of direct-decay properties of GRs is, as a rule, out the scope of these approaches.

• In

past, we actively exploited the so-called semi-microscopic approach, in which the spreading effect is phenomenologically taken into account by means the substitution 𝜕 → 𝜕 + [𝑗𝑋(𝜕) – 𝑄(𝜕)]𝑔(𝑦) directly in the equations of the non-standard cRPA version (U., NPA`08). Except the description of the one-body transition density, the approach can be considered as a “pole” approximation of the PHDOM. However, it is not valid at distant “tails” of GRs. The approach looked as a reasonable prescription, but was not microscopically based.

II III. . Im Implementations

3.1 Input quantities

In first implementations of the PHDOM, a rather simple set of input quantities is used. It includes the Landau-Migdal p-h interaction and a (realistic) partially self-consistent mean field. This field contains a phenomenological isoscalar (Woods-Saxon type) part (including the spin-orbit term), while the symmetry potential and mean Coulomb field are calculated self-consistently. It means that the isovector Landau- Migdal parameter 𝑔′ belongs to the set of mean-field parameters, which are found from the description of observable single-quasiparticle spectra in doubly-closed-shell parent nuclei.

Specific for the model phenomenological quantity, the energy-dependent strength of the imaginary part of the energy-averaged p-h self-energy term 𝑋(𝜕), is parameterized as follows: 2𝑋 𝜕 > Δ = 𝛽

𝜕−Δ 2 1+ 𝜕−Δ 2 𝐶2 , 2𝑋 𝜕 ≤ Δ = 0.

Here, 𝛽~ 0.1 MeV-1 is the adjustable parameter, while the “gap” parameter Δ = 3 MeV and “saturation” parameter 𝐶 = 7 MeV are used, as the universal quantities. The real part, 𝑄(𝜕), is determined by the proper dispersive relationship. The parameter 𝛽 is adjusted to reproduce the

• bserved total width of the given GR in the calculated strength function.

Then, the double transition density, DSD-reaction amplitudes, direct-decay branching ratios are evaluated without the use of any free parameters.

3.2 Properties of ISM excitations

Investigations of ISM excitations are popular due to the possibility of determining the nuclear matter incompressibility coefficient, which depends

• n the mean ISGMR energy.

Within the initial and unitary PHDOM versions, we study the ISM relative energy-weighted strength functions 𝑧𝑕 𝜕 = 𝜕𝑇𝑊

0,𝑕(𝜕)/𝐹𝑋𝑇𝑆𝑊 0,𝑕

related to the probing operators 𝑊

0,1 = 𝑠2𝑍 00 and 𝑊 0,2 = 𝑠2( 𝑠2 − 𝜃) (𝜃 is

an adjustable parameter), which lead to excitation of the ISGMR and its

• vertone (ISGMR2), respectively. The strength functions calculated for 208Pb

within the initial (solid line) and unitary (dotted line) PHDOM versions are shown in Figs.1., 2. (Gorelik et al., NPA`16,`18).

Fig.1a Fig.1b

Usually, the properly normalized (one-body, energy-independent) collective- model transition density of the ISGMR, 𝜍𝑑,1 ~

3 + 𝑠 𝑒

𝑒𝑠 𝑜(𝑠) with 𝑜(𝑠) being

the ground-state density, is used within the DWBA-analysis of 𝛽, 𝛽′ - scattering at small angles. We compare the squared microscopically corrected collective (energy- dependent) transition density defined as follows 𝜍𝑑,1 𝑠, 𝜕 = Λ1

1 2 𝜕 𝜍𝑑,1 𝑠 ; Λ1 𝜕 =

𝑇𝑊

0,1 𝜕

𝜍1𝑊

0,1 2 ,

the squared “projected” transition density, 𝜍𝑕,1(𝑠, 𝜕), and the “diagonal” radial ISM double transition density, 𝜍(𝑠 = 𝑠′, 𝜕), in a vicinity of the ISGMR in 208Pb (Fig. 2). Differences at the ISGMR “tails” are clearly seen.

Fig.3

3.3 Simplest photo-nuclear reactions

The first intensive implementation of the PHDOM was concerned with photo-absorption cross section and DSD (𝛿, 𝑜)- and (𝑜, 𝛿)-reaction cross sections accompanied by excitation of the isovector dipole and quadrupole giant resonances (IVGDR and ISVGQR), respectively) (Tulupov, U., PRC`14). In this consideration, the calculation scheme was extended by inclusion of the isovector velocity-dependent forces taken in a simplest (separable) form (with the dimensionless strengths 𝑙1

′ and

𝑙2

′ ).

Here, we present new results (Tulupov, U., Poster Session) concerned with quantitative estimation of the partial and total branching ratios for direct one-neutron decay of the IVGDR in

• 48Ca. Namely for this

nucleus, the value 𝑐𝑢𝑝𝑢

↑,exp = 39 ± 5% for the excitation-energy interval

𝜀 = 11– 25 MeV is available. As before, we first evaluate within the model the 𝐹1-photo-absorption cross section, which is proportional to the energy-weighted strength function 𝑇1 𝜕 related the external field 𝑊

0,1(𝑦) =

− 1 2 𝜐 3 𝑍

10,

𝜏𝑏,𝐹1 𝜕 = 𝐷𝜕𝑇1 𝜕 where 𝐷 = 16𝜌3𝑓2/ℏ𝑑 , 𝜕 is the 𝛿-quantum energy. From the comparison with the experimental data (Fig. 3), two adjustable parameters 𝛽 (in MeV-1) and 𝑙1

′ are found.

• Fig. 3
• Fig. 4

Then, the partial cross section of the DSD (𝛿, 𝑜)-reaction accompanied by population of the certain one-hole state 𝜈−1 in 47Ca 𝜏𝜈,𝐹1

𝐸𝑇𝐸 𝜕 = 𝐷𝜕 𝜇

𝑁1, 𝜇 ,𝜈

𝐸𝑇𝐸 (𝜕) 2

is evaluated without the use of free parameters (Fig. 5). ((𝜇) = 𝑘, 𝑚 are the quantum numbers of the optical-model-like wave function of escaped neutron, having the energy 𝜗 = 𝜗𝜈 + 𝜕). The calculated total DSD (𝛿, 𝑜)-reaction cross section is shown in Fig. 4. The total branching ratio defined as the ratio 𝑐𝐹1

𝑢𝑝𝑢 𝜀 = 𝜈 𝜀

𝜏𝜈,𝐹1

𝐸𝑇𝐸 𝜕 𝑒𝜕 /𝜕 𝜀

𝜏𝑏,𝐹1 𝜕 𝑒𝜕/𝜕 is then evaluated (𝑐𝐹1

𝑢𝑝𝑢 𝜀 = 35.3%) and found in a reasonable

agreement with the corresponding experimental value.

Fig.5

3.4 The IAR spreading width

The ability to describe distant “tails” of various GRs allows one to get within the PHDOM a quantitative estimation of the (anomalously small) spreading width of the isobaric analog resonances (IARs) (Kolomiytsev et al., arXiv:1712.05146). We incorporate this model into the “Coulomb” description of the isospin-forbidden processes to get expressions for the total and (“single-particle”) proton-escape widths (Γ

𝐵 and Γ 𝐵,𝜉 ↑ , respectively)

in terms of the specific energy-averaged “normal” and proton-escape Coulomb strength functions: Γ

𝐵 = 2𝜌

𝑇𝐵 𝑇𝐷

− 𝜕 = 𝜕𝐵 ; Γ 𝐵,𝜉 ↑

= 2𝜌 𝑇𝐵 𝑇𝐷,𝜉

− ,↑ 𝜕 = 𝜕𝐵 .

Here, 𝑇𝐵 is the IAR Fermi strength (close to the neutron excess), 𝜕𝐵 is the IAR excitation energy, 𝜉 is the set of quantum numbers of a neutron-hole state populated after direct one-proton decay of a given IAR.

The Coulomb strength functions are related to the external field: 𝑊

𝐷 (−) = 𝑉𝐷 𝑠 − 𝜕𝐵 + 𝑗

2 Γ

𝐵 𝜐 −

with 𝑉𝐷(𝑠) being the mean Coulomb field. As functions of 𝜕, the Coulomb strength functions exhibit a maximum, corresponding to the IAR overtone, isovector giant monopole resonance in the 𝛾 − - channel (IVGMR(-)). It means that IAR is located at the distant low-energy “tail”

• f the mentioned GR. An example is given in Fig. 6, where the “normal”

Coulomb strength function is presented in a wide excitation-energy interval, that includes the IAR and IVGMR(-) based on the 208Pb parent- nucleus ground state (the bold line – PHDOM, thin line – cRPA, the arrow marks the IAR energy).

The strength parameter 𝛽 = 0.07 MeV−1 is chosen to reproduce in calculations of the mentioned strength function the experimental total width of the IVGMR(-). Then, the IAR spreading width Γ

𝐵 ↓ = Γ 𝐵 − 𝜉

Γ

𝐵 ,𝜉 ↑

is evaluated without the use of free parameters. The result obtained for the considered IAR, Γ

𝐵 ↓ = 60 keV, is in an

acceptable agreement with the corresponding experimental value, 78 ± 8 keV. The proposed combined approach is used for evaluation of other damping parameters of IAR, and also of the partial branching ratios for direct one-nucleon decay of IVGMR(±). Some results are presented at this Conference (Kolomiytsev et al., Poster Session).

Fig.6

IV

• IV. Conclusive remarks
• In conclusion, the newly developed semi-microscopic model, PHDOM,

is briefly presented. Within the model , the structure and main relaxation modes of high-energy (p-h)-type excitations (including GRs) in medium-heavy mass “hard” spherical nuclei are described

• together. Some implementations of the model are also shown.
• The lines of further development might be new implementations and

the use of more advanced versions of the mean field and p-h interaction.

Many thanks to my colleagues

• M. L. Gorelik, G. V. Kolomiytsev, S. Shlomo, B. A. Tulupov

for fruitful collaboration

Thanks for your attention!