Temperature evolution of the nuclear shell structure and the dynamical nucleon effective mass
TTemperature evolution of the nuclear shell structure and the dynamical nucleoneffective mass
Herlik Wibowo, Elena Litvinova,
1, 2, 3
Yinu Zhang, and Paolo Finelli Department of Physics, Western Michigan University, Kalamazoo, MI 49008, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA GANIL, CEA/DRF-CNRS/IN2P3, F-14076 Caen, France Dipartimento di Fisica e Astronomia, Universit´a degli Studi di Bologna and INFN,Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy (Dated: July 24, 2020)We study the fermionic Matsubara Green functions in medium-mass nuclei at finite temperature.The single-fermion Dyson equation with the dynamical kernel of the particle-vibration-coupling(PVC) origin is formulated and solved in the basis of Dirac spinors, which minimize the grand canon-ical potential with the meson-nucleon covariant energy density functional. The PVC correlationsbeyond mean field are taken into account in the leading approximation for the energy-dependent self-energy, and the full solution of the finite-temperature Dyson equation is obtained for the fermionicpropagators. Within this approach, we investigate the fragmentation of the single-particle statesand its evolution with temperature for the nuclear systems , Ni and Fe relevant for the core-collapse supernova. The energy-dependent, or dynamical, nucleon effective mass is extracted fromthe PVC self-energy at various temperatures.
PACS numbers: 21.10.-k, 21.30.Fe, 21.60.-n, 23.40.-s, 24.10.Cn, 24.30.Cz
I. INTRODUCTION
The problem of predictive description of strongly-correlated many-body systems remains at the frontiersof science for decades. Although its solutions have beenlately boosted by the progress in numerical computation,there is still a need of conceptual and formal advance-ments in the related areas of physics. While finding ex-act solutions is not possible in principle, novel ideas toapproach these solutions without referring to the pertur-bation theory are actively discussed.One of the systematic ways is offered by the equa-tion of motion (EOM) framework. The EOM’s canbe straightforwardly generated for various quantum me-chanical quantities, for instance, the correlation functionsof field operators. The simplest correlation functions areof the propagator type, which are related to the spectralcharacteristics of complex systems. Another advantageof the EOM framework is its general character and thepossibility to accommodate various truncation schemes.For example, the simplest truncation on one-body levelleads to the Hartree-Fock, random phase approximation(RPA), Second RPA (SRPA), the Gorkov theory of thesuperfluidity and the Bardeen-Cooper-Schrieffer (BCS)model. Explicit inclusion of higher-rank propagatorsleads to more complicated sets of coupled equations forpropagators of different ranks. A considerable accuracycan be achieved by cluster expansions of the dynamicalkernels of the fermionic EOM’s in terms of the two-timemany-fermion correlation functions corresponding to therelevant degrees of freedom, as it is discussed, in particu-lar, in Refs. [1, 2]. An attractive feature of the formallyexact EOM’s for these correlation functions, which areknown in condensed matter and quantum chemistry [2–5], is that they have both the static and dynamical ker- nels derived consistently from the same underlying bareinteraction. In nuclear physics, however, the implemen-tations of the analogous methods are still mainly basedon phenomenological interactions.The formal foundation of such methods typically pos-tulate phenomenological Hamiltonians, which imply theexistence of the fermionic quasiparticles and phonons ofbosonic nature. The phonon-exchange interaction be-tween the quasiparticles is added to the pure effectiveresidual interaction between fermions, for instance, inthe nuclear field theory (NFT) [6–10]. Another class ofmodels is based on the phonon degrees of freedom [11–14]. The use of effective phenomenological interactionsallows for simpler calculation schemes, however, more ac-curate and sophisticated versions of the nuclear field the-ory (NFT) were successfully implemented [15–24]. Anal-ogous models operating with mostly the phonon degreesof freedom [11–14], were extended to very complex cor-relations, and there have been a few recent attempts ofusing the bare nucleon-nucleon interaction [25–28].In this work we present a finite-temperature extensionof the many-body model for the nuclear shell structure[29, 30], which combines the relativistic quantum hadro-dynamics (QHD) [31–34] and the quantum field theorytechniques based on the EOM [23, 35–37]. We focuson the single-particle EOM, which is represented by theDyson equation, and elaborate on its dynamical kernel.As it follows from Refs. [23, 36], the leading contributionto the dynamical kernel, also called self-energy, in finitenuclei can be associated with the particle-vibration cou-pling (PVC). Formally similar to the phenomenologicalPVC proposed quite early by A. Bohr and B. Mottel-son [6, 7] and that of the NFT, it is now understood interms of the EOM derived from ab-initio nucleon-nucleonpotentials. Although in this work we still keep the phe- a r X i v : . [ nu c l - t h ] J u l nomenological effective interaction adjusted in the frame-work of the covariant density functional theory (CDFT)[34] for the static part of the EOM kernel and PVC, it issupposed to pave the way to a fully ab initio descriptionin near future.The approach is designed to clarify the nuclear phe-nomena, which may occur in astrophysical environments,such as neutron stars and supernovae. In such envi-ronments finite temperature becomes an essential fac-tor, which modifies the rates of various nuclear reactions,from the radiative neutron capture to the weak processes[38, 39]. It was pointed out in earlier works, such as Refs.[40, 41], that the nuclear single-particle states underly themechanisms of those processes and impact the statisticalproperties, such as the level density, entropy and specificheat. At the same time, the single-particle states aremodified considerably by the PVC mechanism, that re-sults into an enhancement of the effective mass and of thelevel density around the Fermi surface. The results pre-sented in this work allow for a detailed discussion of thelatter phenomenon and its evolution with temperature. II. DYSON EQUATION FOR THE FERMIONICPROPAGATOR AT FINITE TEMPERATURE
We define the atomic nucleus as a many-body quantumsystem, which consists of protons and neutrons, com-monly called nucleons, in the regimes associated withthe energies below the pion mass. Nucleons are coupledthrough the strong interaction represented by the mesonexchange at such low energies. The Coulomb interactionacts between the positively charged protons. In this workwe will use the concept of effective mesons, whose massesand coupling vertices are adjusted to reproduce the nu-clear masses and radii on the Hartree level. While thelatter constitutes the famous Walecka model for nuclearQHD, it will be used, as in the preceding series of works,only as a starting point and a convenient basis for the de-scription of nucleonic in-medium correlations far beyondthe mean field.One of the most convenient ways to quantitatively ap-proach these correlations is to directly calculate certaincorrelation functions, namely the propagators (also calledGreen functions). We will employ the advantage of thisformalism as it gives a direct access to the excitationspectra and ground state properties of the nuclear sys-tem. Here we are interested in nuclear systems, whichare in thermal equilibrium with the surroundings andcan thus be assigned the temperature. The temperature,or Matsubara, Green function of a fermion is defined as[42–44] G (1 , (cid:48) ) ≡ G k k (cid:48) ( τ − τ (cid:48) ) = −(cid:104) T τ ψ (1) ¯ ψ (1 (cid:48) ) (cid:105) , (1)with the help of the chronological ordering operator T τ ,which acts on the fermionic field operators in the Wick- rotated picture: ψ (1) ≡ ψ k ( τ ) = e H τ ψ k e −H τ , ¯ ψ (1) ≡ ψ † k ( τ ) = e H τ ψ † k e −H τ , (2)where H = H − µN with H being the many-body Hamil-tonian, µ the chemical potential, and N the particle num-ber operator. The subscript k stands for the full set ofthe single-particle quantum numbers in a given represen-tation and the imaginary time variables τ are related tothe real times t as τ = it . The fermionic fields satisfy theusual anticommutation relations, and the angular brack-ets in Eq. (1) stand for the thermal average [43, 44].If the many-body Hamiltonian H is confined by theone-body part, i.e. contains only the free-motion and themean-field contributions, the single-fermion MatsubaraGreen function can be easily calculated and reads (cid:101) G (2 ,
1) = (cid:88) σ = ± (cid:101) G σ (2 , , (cid:101) G σ (2 ,
1) = − σδ k k n ( − σ ( ε k − µ ) , T ) e − ( ε k − µ ) τ θ ( στ )(3)with τ = τ − τ in the basis { k i } of the single-fermion states, which diagonalizes the one-body part ofthe Hamiltonian. In Eq. (3) ε k are the eigenvalues ofthe single-particle part of the Hamiltonian and n ( ε, T )stands for the Fermi-Dirac distribution n ( ε, T ) = 1exp( ε/T ) + 1 (4)at the temperature T . The spectral representation of thethermal mean-field Green function is calculated with theoperation (cid:101) G k k ( ε (cid:96) ) = /T (cid:90) dτ e iε (cid:96) τ (cid:101) G k k ( τ ) , (5)which leads to: (cid:101) G k k ( ε (cid:96) ) = δ k k (cid:101) G k ( ε (cid:96) ) , (cid:101) G k ( ε (cid:96) ) = 1 iε (cid:96) − ε k + µ (6)defined at the discrete Matsubara frequencies ε (cid:96) ε (cid:96) = (2 (cid:96) + 1) πT, (7)where the (cid:96) ’s are integer. The ’ (cid:101) ’ sign in Eqs. (3-6)indicates the mean-field character of the respective Greenfunction.In this work we are interested in non-trivial correla-tions beyond mean field which occur due to the residualinteraction, i.e. in the presence of two-body and higher-rank terms in the many-body Hamiltonian. In this case,the single-fermion propagator G obeys the Dyson equa-tion G k k ( ε (cid:96) ) = G k k ( ε (cid:96) ) + (cid:88) k k G k k ( ε (cid:96) )Σ k k ( ε (cid:96) ) G k k ( ε (cid:96) ) , (8)where G is the free propagator and Σ is the self-energy,or the mass operator. As it can be shown within the equa-tion of motion (EOM) framework, the exact self-energy isdecomposed into the energy-independent (static) (cid:101) Σ andthe energy-dependent (dynamical) Σ e parts:Σ k k ( ε (cid:96) ) = (cid:101) Σ k k + Σ ek k ( ε (cid:96) ) , (9)that is also valid at finite temperature. In ’ab-initio’calculations based on the Hamiltonians with bare inter-actions the static part of the self-energy is given by thecontraction of the matrix element of the bare interactionwith the exact one-fermion density, while its dynamicalpart is represented by the three-fermion correlated prop-agator contracted with two matrix elements of the bareinteraction [23, 36, 45].Using Eq. (9), it is convenient to eliminate the un-perturbed propagator G from Eq. (8), and work withthe thermal mean-field propagator (cid:101) G which satisfies theequation: (cid:101) G k k ( ε (cid:96) ) = G k k ( ε (cid:96) ) + (cid:88) k k G k k ( ε (cid:96) ) (cid:101) Σ k k (cid:101) G k k ( ε (cid:96) ) . (10)Then, the Dyson equation for the full propagator takesthe form: G k k ( ε (cid:96) ) = (cid:101) G k k ( ε (cid:96) ) + (cid:88) k k (cid:101) G k k ( ε (cid:96) )Σ ek k ( ε (cid:96) ) G k k ( ε (cid:96) ) . (11)The energy-dependent part of the mass operator Σ e describes the coupling between single fermions and in-medium emergent degrees of freedom. In this work, weemploy the particle-vibration coupling model, which ap-proximates the exact energy-dependent part of the self-energy Σ e by a cluster expansion truncated at the two-body level [23]. When retaining only the coupling tonormal phonons, the analytical form of this self-energy,in the leading approximation, readsΣ ek k ( ε (cid:96) ) = − T (cid:88) k ,m (cid:88) (cid:96) (cid:48) (cid:88) σ = ± (cid:101) G k ( ε (cid:96) (cid:48) ) σg m ( σ ) k k g m ( σ ) ∗ k k iε (cid:96) − iε (cid:96) (cid:48) − σω m (12)where g m are the phonon vertices and ω m their frequen-cies, which can be found from the EOM for the two-fermion correlation functions. The phonon vertices aredetermined via g mk k = (cid:88) k k (cid:101) U k k ,k k ρ mk k , (13) g m ( σ ) k k = δ σ, +1 g mk k + δ σ, − g m ∗ k k (14)where ρ mk k are the transition densities of the phononsand (cid:101) U k k ,k k are the matrix elements of the nucleon-nucleon interaction. In principle, the relationship (13)is model-independent and, ideally, the transition densi-ties are the exact ones, while the interaction (cid:101) U is the bare interaction. However, numerous models employ-ing effective interactions and the random phase approx-imation based on these interactions for the computationof the phonon vertices and frequencies typically providequite realistic approaches to the dynamical self-energy.In this work, we use the effective interaction of the co-variant energy density functional (CEDF) [33, 34] withthe NL3 parametrization [46] and the concept of the ’no-sea’ relativistic random phase approximation (RRPA)[47] adopted to finite temperature in our previous de-velopments [38, 48, 49].The summation over (cid:96) (cid:48) in Eq. (12) can be transformedinto a contour integral by the standard technique [44].After the analytical continuation to complex energies, wethen obtain the final expression for the mass operator Σ e of the form:Σ ek k ( ε ) = (cid:88) k ,m (cid:40) g mk k g m ∗ k k N ( ω m , T ) + 1 − n ( ε k − µ, T ) ε − ε k + µ − ω m + iδ + g m ∗ k k g mk k n ( ε k − µ, T ) + N ( ω m , T ) ε − ε k + µ + ω m − iδ (cid:41) , (15)where N ( ω m , T ) = 1exp( ω m /T ) − ω m . It is easy to see that in the limit T → ek ,k ( ε ) = (cid:88) k ,mε k >ε F g mk k g m ∗ k k ε − ε k + ε F − ω m + iδ + (cid:88) k ,mε k ≤ ε F g m ∗ k k g mk k ε − ε k + ε F + ω m − iδ , δ → +0 , (17)where ε F is the Fermi energy, i.e. the energy of the lastoccupied single-particle state. Indeed, in the limit T → N ( ω m , T ) →
0, and thefermion occupation number n ( ε k − µ, T ) takes the value1 (0) for ε k ≤ ε F ( ε k > ε F ). III. NUMERICAL SOLUTION OF THEFINITE-TEMPERATURE DYSON EQUATION:RESULTS AND DISCUSSION
We have selected the atomic nuclei , Ni and Feto illustrate the performance of the developed approach.This choice was determined, in particular, by the as-trophysical relevance of these nuclear systems. Indeed,nickel and iron isotopes with the mass number A = 56play a very important role in current understanding ofstellar evolution. For instance, they are associated withthe final stage of formation of massive evolved stars be-fore core collapse. Characteristics of all the three nucleiare important ingredients for understanding presuper-nova and neutron stars. Also, comparison between Nand Ni can be informative for evaluating the range ofthe relevant characteristics within a single isotopic chain.The calculation scheme consists of the following steps.First, we solve the closed set of the relativistic meanfield (RMF) equations with the NL3 parametrization[46] of the non-linear sigma-model with the thermalfermionic occupation numbers (4). This leads to a setof temperature-dependent single-particle Dirac spinorsand the corresponding single-nucleon energies, whichform the basis for subsequent calculations. Second,the finite-temperature relativistic random phase approx-imation (FT-RRPA) equations are solved to obtain thephonon vertices g m and their frequencies ω m . The setof the obtained FT-RRPA phonons, together with theRMF single-nucleon basis, forms the pp ⊗ phonon and ph ⊗ phonon configurations for the particle-phonon cou-pling self-energy Σ e ( ε ). Third, Eq. (11) is solved inthe truncated configuration space, as described in [29]for the T = 0 case, in the diagonal approximation, i.e.Σ ek k ( ε ) = δ k k Σ ek ( ε ).The particle-hole basis for the FT-RRPA calculationsof the phonons was limited by the particle-hole ( ph )configurations with the energies ε ph ≤
100 MeV andthe antiparticle-hole ( αh ) ones with ε αh ≥ − J π = 2 + , − , + , − , + below the energy cutoff, whichamounts to 20 MeV for the considered nuclei. This cut-off is justified by our previous calculations. A truncationof the phonon space was applied according to the valuesof the reduced transition probabilities of the correspond-ing electromagnetic transitions: we included the modeswith the reduced transition probabilities B ( EL ) equal ormore than 5% of the maximal one (for each J π ). Thesame truncation criteria are applied to the phonon en-ergy, J π and the reduced transition probability for alltemperature regimes in order to make a fair comparisonof the calculated single-particle strength distributions atdifferent temperatures. As in our previous calculations[38, 48, 49], at high temperatures we see the appearanceof many additional phonon modes as a consequence ofthe significant thermal unblocking, that may cause slowersaturation of the results with respect to the B ( EL ) cut-off. This happened typically at the temperatures of 5-6MeV, which we do not consider here because of their lit-tle relevance to the astrophysical applications. Anothertruncation was made on the single-particle intermediatestates k in the summation of Eq. (15): only the stateswith the energy differences | ε k − ε k | ≤
50 MeV wereincluded in the summation.We investigated the neutron and proton states in the -25-20-15-10-505 E [ M e V ] Neutrons -30-25-20-15-10-50 Protons T = 0 T = 1T = 2 T = 3 T = 4MeV MeV MeV MeV MeV 2p RMF T = 0 T = 1 T = 2 T = 3 T = 4MeV MeV MeV MeV MeV1d Ni FIG. 1. Single-particle states in Ni at zero and finite tem-peratures calculated in the RMF approximation. -25-20-15-10-505 E [ M e V ] Neutrons -30-25-20-15-10-50 Protons T = 0 T =1 T = 2 T = 3 T = 4MeV MeV MeV MeV MeV 1d T = 0T = 1T = 2 T = 3 T = 4MeV MeV MeV MeV MeV
RMF+PVC Ni FIG. 2. The dominant fragments of the single-particle statesin Ni at zero and finite temperatures calculated in theRMF+PVC approximation. approximately 20 MeV energy window around the re-spective Fermi energies of , Ni and Fe nuclei. While Ni is a doubly-magic, or closed-shell, nucleus, the pro-ton subsystem of Ni is of the closed-shell nature andthe neutron subsystem is open-shell. The Fe is, inturn, an open-shell nucleus for both neutrons and pro-tons. Thus, the superfluid character of the respectivesubsystems, because of the presence of the pairing corre-lations, can importantly affect the single-particle spectra.Within our formalism, this phenomenon was discussedin Refs. [30, 50]. On the mean-field level, the Bardeen-Cooper-Schrieffer (BCS) or the Bogoliubov’s approxima-tions typically give a reasonable description of pairingcorrelations, which manifest themselves through consid-erable redistributions of the single-particle states in thevicinity of the Fermi energy and their fractional occu-pancies in the superfluid subsystems, along with someminor rearrangements of the single-particle states in theirnon-superfluid counterparts. In the calculations beyondthe mean field, such as RMF+PVC, pairing correlationsmay have stronger impacts. The main underlying reasonfor these impacts is that taking pairing correlations intoaccount leads to the appearance of the phonon modes,first of all, the quadrupole modes, at significantly lowerenergies. Such phonons play the major role in the dy-namical self-energy. As it has been shown in Ref. [30],the modification of the phonon spectrum due to the pair-ing correlations further affects the single-particle struc-ture, as compared to the mean-field approach. The PVCfragmentation effects become stronger in general and, inparticular, the shell structure of the non-superfluid coun-terparts are modified considerably toward much highersingle-particle level densities.As the superfluidity in the Bogoliubov’s or BCS sensevanishes at the critical temperature T c , which has a well-established relation to the pairing gap ∆ p at zero temper-ature: T c ≈ . p ( T = 0), the role of pairing correlationsdiminishes quickly with the temperature growth. For thenuclei considered in this work, whose pairing gaps wereadjusted to the odd-even mass differences using the three-point formula and the data on nuclear binding energiesfrom Ref. [51], the values of pairing gaps deduced by thisprocedure are ∆ ( n ) p = 1.6 MeV for neutrons in Ni, and∆ ( n ) p = 1.8 MeV and ∆ ( p ) p = 2.1 MeV for neutrons andprotons in Fe, respectively. For these cases the criticaltemperatures have the values in the 0 ≤ T ≤ T = 0 and neglectedfor T ≥ T = 0 and T = 1 MeVtakes place in Ni. In Fe, where the neutron pair-ing gap ∆ ( n ) p vanishes at approximately 1.1 MeV and theproton pairing gap ∆ ( p ) p at approximately 1.3 MeV, weassume that the role of pairing correlations is negligiblealready at T = 1 MeV, too. The T = 0 calculations withpairing correlations are performed within the superfluidRMF+PVC approach developed originally in Ref. [30].The obtained single-particle shell structure for Niis shown in Figs. 1 and 2. Fig. 1 displays the statescomputed within the thermal RMF approach, while Fig.2 presents the dominant states, which are the outcomeof calculations within the finite-temperature particle-vibration coupling (RMF+PVC). It is quite commonin the literature to assume that the nuclear mean fieldremains almost unchanged with temperature in a rela-tively broad temperature range 0 ≤ T ≤ T = 1 MeV, the overall trend in theneutron subsystem is the relatively minor densifying ofthe spectrum with the temperature increase, while theproton mean-field states move up nearly uniformly by1-2 MeV in the 0 ≤ T ≤ Ni,for the each state close to the Fermi energy there is onefragment with the spectroscopic factor of 0.8-0.9, whilethe rest of the fragments are characterized by very little,mostly less than one percent spectroscopic factors. Thefragment with the largest spectroscopic factor is com-monly called dominant, and the states with this frag-mentation pattern are called good single-particle states.Typically, for such states the energy of the dominant frag-ment is rather close to the energy of the original mean-field state. In other words, in non-superfluid systemsthe single-particle states in the vicinity of the Fermi en-ergy are not very much affected by the PVC. In contrast,the states far away from the Fermi energy are stronglyfragmented. For many states it is still possible to iden-tify the dominant fragment, although with a considerablyquenched spectroscopic factor, and this fragment can bequite far from the original mean-field state. The alge-braic reasons behind this qualitatively outlined pictureare discussed, for instance, in Ref. [29].For the second type of states, i.e. the states remotefrom the Fermi energy, often two or more fragments ex-hibit comparable spectroscopic factors. These are thecases, for instance, for the 1f / state in the neutron sub-system and 2s / state in the proton subsystem, whichshow 0.28/0.17 and 0.39/0.37 shares of the spectroscopicfactors between two dominant fragments, respectively, at T = 0. Our calculations reveal that the general patterndescribed above persists with the temperature increasewithin the interval 0 ≤ T ≤ ν / , ν / , π / and π / show splitting into two major fragments withcomparable strengths. The gaps between these fragments -30 -25 -20 -15 -10 -500.10.20.30.4 S ( ν f / ) -30 -25 -20 -15 -10 -500.10.20.30.4-30 -25 -20 -15 -10 -5E [MeV]00.10.20.30.4 S ( ν f / ) -30 -25 -20 -15 -10 -5E [MeV]00.10.20.30.4 ν T = 0 MeV ν T = 2 MeV ν T = 1 MeV ν T = 3 MeV Ni FIG. 3. Temperature evolution of the neutron 1f / statein Ni. Blue bars represent the spectral strength distribu-tions (spectroscopic factors) of the fragmented state calcu-lated within the thermal ’RMF+PVC’ approach. The red barcorresponds to the pure RMF state, and the dashed green lineindicates the chemical potential. -8 -6 -4 -2 0 2 4 6 800.10.20.30.40.5 S ( ν / ) -8 -6 -4 -2 0 2 4 6 800.10.20.30.40.5-8 -6 -4 -2 0 2 4 6 8E [MeV]00.10.20.30.40.5 S ( ν / ) -8 -6 -4 -2 0 2 4 6 8E [MeV]00.10.20.30.40.5 ν T = 0 MeV ν T = 2 MeV ν T = 1 MeV ν T = 3 MeV Ni FIG. 4. Same as in Fig. 3, but for the neutron 3d / state in Ni. as well as the ratios of their spectroscopic factors changewith the temperature increase.In order to illustrate and better understand the tem-perature evolution of the fragmentation mechanism, wedisplay four examples of strongly fragmented single-particle states in Ni at the temperatures of T = 0 , , / in Niis shown in Fig. 3. At T = 0 it consists predominantly ofthe two fragments approximately 6 MeV apart with thespectroscopic factors of 0.28 and 0.17 located on the op-posite sides of the uncorrelated, or mean-field, hole state(below the Fermi energy). The phase transition, whichoccurs around T = 1 MeV, together with the beginningthermal unblocking, slightly changes the strength distri- -50 -40 -30 -20 -10 000.10.20.30.40.5 S ( π / ) -50 -40 -30 -20 -10 000.10.20.30.40.5-50 -40 -30 -20 -10 0E [MeV]00.10.20.30.40.5 S ( π / ) -50 -40 -30 -20 -10 0E [MeV]00.10.20.30.40.5 π T = 0 MeV π T = 2 MeV π T = 1 MeV π T = 3 MeV Ni FIG. 5. Same as in Fig. 3, but for the proton 1d / state in Ni. bution preserving, however, the general two-peak struc-ture. With further temperature increase, the two-peakstructure persists, while each of the two peaks undergofragmentation. At temperatures T = 3 MeV and T =4 MeV we find that the lower-energy major fragmentdominates, although its spectroscopic factor continues toquench. The evolution of the state 3d / in the neutronsubsystem of Ni is illustrated in Fig. 4. This is the par-ticle state (well above the Fermi energy), which appearsat T = 0 as a structure with a single dominant peakand where at T = 1 another major fragment appears tocompete with the share of 0.21/0.39 between the spectro-scopic factors. In contrast to the case of the neutron 1f / state, these fragments are only about 1 MeV apart in en-ergy and rather close to the mean-field 3d / state, fromwhich they originate. With the temperature increase oneof the two major fragments undergoes further fragmen-tation, and at T = 3 MeV the other fragment, whichis closer to the original mean-field 3d / state, becomesabsolutely dominant.The examples from the proton subsystem are repre-sented by the states 1d / and 1g / shown in Figs. 5 and6, respectively. The case of the proton 1d / state is sim-ilar to the one of the neutron 3d / state discussed above.At T = 0 the dominant state appears surrounded by themultitude of weaker fragments and a relatively strongsecond fragments with 0.23/0.45 share of the spectro-scopic factors. With the temperature increase, the entirespectrum undergoes gradual fragmentation with strongquenching of the dominant fragment. In all temperatureregimes the major fragment is located very close to theoriginal mean-field state. The behavior of the proton1g / state is again different. At T = 0 the dominantfragment splits out from a number of weaker fragments,which are grouped around the original mean-field state.After the phase transition, rearranging the distributioninto a competing two-peak structure, the temperature -16 -12 -8 -4 0 4 800.10.20.30.40.50.6 S ( π / ) -16 -12 -8 -4 0 4 800.10.20.30.40.50.6-16 -12 -8 -4 0 4 8E [MeV]00.10.20.30.40.50.6 S ( π / ) -16 -12 -8 -4 0 4 8E [MeV]00.10.20.30.40.5 π T = 0 MeV π T = 2 MeV π T = 1 MeV π T = 3 MeV Ni FIG. 6. Same as in Fig. 3, but for the proton 1g / state in Ni. growth causes a redistribution and eventually a strongerfragmentation of this group of fragments, which never-theless remain around the mean-field state. The domi-nant fragment retains its dominance, although its spec-troscopic factor gets gradually quenched as the temper-ature increases.Thereby, while the evolution of the good single-particlestates is quite similar to the evolution of the mean-fieldstates, the strongly fragmented states may exhibit vari-ous scenarios. The latter is determined by the spin andparity of the state, its closeness to the Fermi energy aswell as by the magnitudes of the most relevant PVCmatrix elements and the associated phonon frequencies.These factors define the interplay of coupling to vari-ous phonon modes [55]. As we have shown explicitlyfor the case of the most important quadrupole modes inRef. [49], the non-trivial temperature evolution of thephonon spectrum gives the corresponding feedback onthe PVC and the related fragmentation of the nuclearexcitation modes. On the large scale of temperatures,the fragmentation is enhanced with the temperature in-crease because of the general trend of the thermal un-blocking. However, we also observed a counter trend atmoderate temperatures, when the thermal unblocking isnot yet developed sufficiently to generate new strong low-energy phonon modes, but weakens those modes whichplay the major role at T = 0. The feedback of this effecton the nuclear excited states was observed as a minorweakening of the fragmentation at small and moderatetemperatures of about 1-2 MeV, while with further tem-perature increase the fragmentation became reinforcedagain. Now we can see that some of the single-particlestates show a similar behavior: the examples of those arethe neutron 3d / and the proton 1g / states.An example of a nucleus with strong pairing correla-tions in both neutron and proton subsystems is repre-sented by Fe. Indeed, the first excited state in this -30-25-20-15-10-505 E [ M e V ] Neutrons -35-30-25-20-15-10-5051015 Protons T = 0 T = 1T = 2 T = 3T = 4MeV MeV MeV MeV MeV T = 0 T = 1T = 2T = 3 T = 4MeV MeV MeV MeV MeV
RMF Fe FIG. 7. The single-particle states in Fe at zero and finitetemperatures calculated in the RMF approximation. -30-25-20-15-10-505 E [ M e V ] Neutrons -35-30-25-20-15-10-5051015 Protons T = 0MeV T = 1MeV T = 2MeV T = 3MeV T = 4MeV 2s T = 0MeV T = 1MeV T = 2MeV T = 3MeV T = 4MeV
RMF+PVC Fe FIG. 8. The dominant fragments of the single-particle statesin Fe at zero and finite temperatures calculated in theRMF+PVC approximation. nucleus is the 2 + state at 846.78 keV followed by the4 + state at 2085.1 keV, according to the experimentalmeasurements [56]. Although the relativistic quasipar-ticle RPA (RQRPA) can not reproduce accurately theexcitation spectrum, it nevertheless returns the energies E (2 +1 ) = 1.12 MeV and E (4 +1 ) = 4.07 MeV, which stip-ulate strong PVC effects. As a result of the RMF+PVCcalculations with pairing correlations for this nucleus at T = 0, we obtain strong fragmentation of the single-particle states even around the Fermi surface. This isreflected in Fig. 8, where one can see quite a number ofsuch states represented by pairs of their competing ma-jor fragments. A significant compression of both neutronand proton single-particle spectra relative to the RMFcalculations shown in Fig. 7 is another consequence ofthe strong PVC in the broad energy region around theFermi energy. As it can be seen in Fig. 8, the first step -30 -20 -10 0 10 E [MeV] M ( E ) / M -30 -20 -10 0 10 E [MeV] neutron, ∆ = 2 MeVneutron, ∆ = 5 MeVproton, ∆ = 2 MeVproton, ∆ = 5 MeV Ni T = 0 Ni FIG. 9. Neutron (blue) and proton (red) dynamical effec-tive masses in Ni (left) and Ni (right) computed with theimaginary parts of the energy variable ∆ = 2 MeV (dashedcurves) and ∆ = 5 MeV (solid curves) at T = 0. The dy-namical effective masses are averaged over the single-particlestates within 40 MeV energy window around the neutron andproton Fermi surfaces, respectively. of the temperature evolution of Fe is the phase tran-sition from the superfluid to the non-superfluid state,which is indicated by the drastic decrease of the den-sity of the single-particle spectra in both neutron andproton subsystems, when the temperature raises from T = 0 to T = 1 MeV. Further temperature evolutionconsists of a rather smooth redistribution of the spectro-scopic strength between the major fragments.It is difficult, however, to assess the global evolutionof the shell structure by looking only at the major frag-ments. A very important characteristic of the strongly-coupled fermionic systems, namely the effective mass, islinked to both the single-particle level density and thenucleonic self-energy and, thus, can help evaluate thegeneral trends. For relativistic systems, the authors ofRef. [57] introduced the non-relativistic type effectivemass, whose energy dependence at low energies is similarto that of non-relativistic systems [58, 59]:¯ M ( k ) ( ε ) M = 1 − ddε R e Σ e ( kk ) ( ε ) , (18)where the energy argument is a complex number and M is the mass of the bare nucleon. We can call the quan-tity ¯ M ( k ) ( ε ) /M dynamical, or energy-dependent, effec-tive mass. The indices in the brackets indicate the re-duced matrix elements: k = { ( k ) , m k } , where m k is theprojection of the total angular momentum on the quanti-zation axis, which is commonly called magnetic quantumnumber in spherical symmetry. The effective mass aver-aged over the single-particle levels reads [59]: (cid:104) ¯ M ( ε ) M (cid:105) = (cid:88) ( k ) (2 j ( k ) + 1) ¯ M ( k ) ( ε ) M / (cid:88) ( k ) (2 j ( k ) + 1) . (19)In order to evaluate the nucleonic effective mass in nu-clei, a finite imaginary part of the energy variable is used in Eq. (18): ε = E + i ∆. The role of the imaginarypart is to soften the singularities of the self-energy byaveraging over the discrete single-particle spectrum. Inthis way, the self-energy acquires both real and significantimaginary parts. While the real part contributes to theeffective mass as in Eq. (18), the imaginary part playsthe role of the optical potential. The value of ∆ shouldbe thus associated with the average distance between thesingle-particle states. In our approach, for the medium-mass nuclei in the iron and nickel mass region the valueof ∆ = 5 MeV satisfies this criterion. However, in or-der to have an idea about the sensitivity of the effectivemass to this parameter, we have calculated it with ∆ = 2MeV and ∆ = 5 MeV for two isotopes of nickel, Ni and Ni, at T=0, where we have neglected the pairing corre-lations in the latter nucleus. The calculations and aver-aging were performed within symmetric 40 MeV intervalsaround the Fermi energies for both neutrons and protons.The results for the neutron and proton effective massesare displayed in Fig. 9. One can see that, indeed, theeffective mass as a function of energy is sensitive to theaveraging parameter, which reflects the non-continuityof the single-particle spectrum. Independently of that,the neutron and proton effective masses can be different,although both of them exhibit broad peaks around thecorresponding Fermi energies, when a sufficiently largeaveraging is used. In the N = Z nucleus Ni the neutronand proton effective masses show very similar behavior,although the neutron effective mass varies in a slightlylarger range, than the proton one. The situation is dif-ferent in the neutron-rich nucleus Ni, where the protoneffective mass demonstrates a remarkably stronger vari-ation between its central and peripheral values. Thisresult is consistent with the general trend, according towhich the effective mass increases with the density ofstates: indeed, one can conclude from Figs. 1 and 2 thatthe overall density of the proton single-particle states in Ni is more affected by the PVC effects than the densityof states in the neutron subsystem.Fig. 10 illustrates the sensitivity of the dynamical ef-fective mass to pairing correlations by displaying the neu-tron and proton effective masses in Fe at T = 0. Asthe neutron excess in this nucleus is small, the effectivemasses of neutrons and protons are represented by simi-lar functions of energy, although the variation of the pro-ton effective mass is somewhat stronger. Both effectivemasses show peaks in broad energy intervals around theFermi surfaces. The widths of the peaks diminish whenthe pairing correlations are taken into account. The pair-ing correlations also cause stronger variations of the effec-tive masses: the higher peak values and the lower valuesin the peripheral areas. Overall, this result is consistentwith the well-established proportionality of the dynami-cal effective mass to the density of states, which is higheraround the Fermi surface, when pairing is included.Finally, Fig. 11 illustrates the temperature evolutionof the dynamical effective masses in , Ni. As it wasmentioned above, it is difficult to make definite conclu- -20 -10 0 10
E [MeV] M ( E ) / M ∆ p = 0 ∆ p > 0 -20 -10 0 10 E [MeV] ∆ p = 0 ∆ p > 0 Fe T = 0
Neutrons Protons
FIG. 10. The dynamical effective masses of neutrons andprotons in Fe calculated with (solid curves) and without(dashed curves) pairing correlations at zero temperature. Thevalue ∆ = 5 MeV was used for the imaginary part of theenergy variable. The symbol ∆ p stands for the superfluidpairing gap. -30 -20 -10 00.850.90.9511.051.1 M ( E ) / M -20 -10 0 10 T = 0T = 1 MeVT = 2 MeVT = 3 MeVT = 4 MeV
Neutrons Protons Ni -20 -10 0 10 E [MeV] M ( E ) / M -30 -20 -10 0 10 E [MeV]
T = 0T = 1 MeVT = 2 MeVT = 3 MeVT = 4 MeV
Neutrons Protons Ni FIG. 11. Temperature evolution of the nucleon dynamicaleffective masses in , Ni calculated with ∆ = 5 MeV. sions on the evolution of the overall density of states bylooking at the fragmented states themselves. The calcula-tions of the effective masses, however, offer this opportu-nity. The results of calculations at various temperaturesshow a smooth evolution of the effective masses towarda more uniform energy dependence. With the temper-ature increase, the effective masses in the peak regionbecome less pronounced, while in the peripheral areastheir values grow. In all the cases, when the temperaturegrows to higher values T ≥ Fe shows similartrends, that are consistent with the study of Ref. [40],although we have obtained overall smaller effective massvalues at the Fermi surfaces.
IV. SUMMARY AND OUTLOOK
We have developed a many-body approach to describefragmentation of the single-particle states in strongly-coupled fermionic systems at finite temperature. Thedynamical finite-temperature self-energy is detailed forapplications to atomic nuclei, where the leading fragmen-tation mechanism is the coupling to correlated particle-hole pairs, which represent emergent phonons of pre-dominantly vibrational character. The Dyson equationwith the particle-vibration self-energy has been solvednumerically in the basis of the relativistic mean field formedium-mass nuclei, such as , Ni and Fe, which playimportant roles in understanding stellar evolution. Thetemperature evolution of the fragmentation mechanismhas been analyzed in detail by extracting the completefragmented single-particle spectra in the 40 MeV windowaround the Fermi energies of the considered nuclei. Var-ious scenarios realized for different types of states havebeen discussed.To characterize the spectra globally, we have com-puted the averaged dynamical neutron and proton ef-fective masses as functions of energy at various temper-atures. In cold nuclear systems, i.e. at zero tempera-ture, the dynamical effective masses show a clear bell-shaped behavior with maxima around the Fermi energy.We found that, depending on the interpretation and onthe value of the averaging parameter, the variation ofthe dynamical effective mass can reach 10-20% betweenthe central and peripheral values, with respect to unity.Pairing correlations were found to sharpen the dynamicaleffective mass as a function of energy, while the temper-ature increase reduces the variation between the centraland peripheral values.The obtained results may have some significance forastrophysical modeling of various stages of stellar evo-lution. As the nucleon effective mass is directly relatedto the density of states, entropy and symmetry energy,its correct temperature dependence can be important,in particular, for the core-collapse supernova modeling[40]. The existing strategies to accommodate this depen-dence suggest simple parametrizations of the dynamicaleffective mass, however, some sensitivity studies wouldbe helpful to establish whether its non-smooth oscillat-ing behavior can be neglected on the global scale.
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