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Statistical Properties of 56-57Fe with Multi-Shell Degrees of Freedom

We investigate the statistical properties of 56-57Fe within a model capable of treating all the nucleons as active in an infinite model space including pairing effects. Working within the canonical ensemble, our model is built on single-particle states, thus it includes shell and subshell closures. We include spin correlation parameters to account for nucleon spin multiplicities. Pairing and shell effects are important in the low-temperature region where the quantum effects dominate. The validity of the model is extended to higher temperatures, approaching the classical regime, by including continuum states. We present results for excitation energy, heat capacity and level density of 56-57Fe. The resulting favorable comparisons with experimental data and with other approaches, where applicable, indicate that the methods we develop are useful for statistical properties of medium to heavy nuclei.


I. INTRODUCTION
Iron nuclei, especially 56 Fe, are the foci of extensive studies within various approaches [1][2][3][4][5] since these nuclei play a major role in astrophysics [5]. The iron nuclei are the heaviest nuclei created by fusion of charged particles inside stars, and the starting point of the synthesis of heavier nuclei. Indeed, the statistical properties of 56 Fe are critical for our understanding of the fundamental nucleosynthesis processes in iron rich stars and cataclysmic astrophysical events such as neutron star mergers.
A major thrust of the present work is to develop a statistical method suitable for reaching higher temperatures by allowing the participation of all the nucleons. Another motivation for investigating 56 Fe is to compare our methods with another modern approach also available for this system. Furthermore, a comprehensive experimental list of states exists for 56 Fe [6] as well as experimental level densities for 56,57 Fe [7,8].
Theoretical results for how the excitation energy depends on temperature and how the level density changes with excitation energy for 56 Fe are available from Nakada and Alhassid [5]. Their results are obtained using the auxiliary field Monte Carlo shell model or "Shell Model Monte Carlo" (SMMC) method. Another version of the Monte Carlo shell model has also been used by Rombouts, Heyde, and Jachowicz (RHJ) [2] to obtain 56 Fe statistical properties. Finally, a simple but useful phenomenological formula, the well-known Bethe formula, can be used to compare with various results for the level densities. The parameters of the back-shifted Bethe formula (BBF) for 56 Fe have been determined [9,10].
In this investigation, we utilize the single-particle (SP) spectra and, together with a phenomenological treatment of particle-particle correlations, to develop the properties of the multi-particle system using the canonical ensemble. The SP spectra can be simple, such as the three-dimensional harmonic oscillator, or more realistic, such as states obtained from a phenomenological SP potential [2] or provided by Hartree-Fock (HF) [11]. The SP states are then thermally populated with an approximate treatment of interaction effects to produce the full system's partition function. Observables, such as excitation energy, heat capacity, and level densities, can be directly computed.
We call this method the "Statistical Mechanics using Single Particle States"(SMSPS).
Our approach models the situation of a specific set of nucleons, for example those that comprise the 56 Fe nucleus, embedded in a thermal bath such as the environment of a neutron star or a plasma produced during neutron star mergers. One of our goals is to simulate the physics of exci-tations that extend beyond those of the valence degrees of freedom alone. That is, we incorporate core excitations and excitations into the resonance and continuum regions. Given this span of thermal excitations, our approach must remain elementary in several aspects to make the calculations feasible.
In order to address the nuclear response over a very wide range of temperatures, we adopt a nuclear Hamiltonian that includes a mean field for all nucleons and a residual pairing interaction for nucleons occupying valence states. For states below the valence states we retain only the mean field Hamiltonian. Furthermore, we identify an approximate boundary in excitation energy between resonance states of the mean field and non-resonant continuum states. The resonant states are adopted from the quasi-bound states solutions of the mean field. The non-resonant states are taken to be the single-particle states of a spherical cavity that are smoothly joined by an average energy shift to the resonant state spectrum. The continuum state spectrum naturally depends on the volume (or radius of the spherical cavity) assigned to the nucleus in the thermal medium. In the case of the neutron star, this volume is controlled by external parameters such as gravity and rotation rate of the star.
The SMSPS model has two significant advantages: The first advantage arises, due to the neglect of inter-nucleon interactions, from the exact statistical population of these states which dramatically reduces the computation time. The second advantage is the ability to use a realistic SP spectrum as an input. For example, the SP states will reflect the long-range Coulomb repulsion in the proton degrees of freedom by incorporating the repulsive shift in the proton states relative to the neutron states. This distinct SP spectra, one for protons and one for neutrons, is a dominant aspect of charge dependence in medium and heavy nuclei.
In this investigation, the SP states provided by Ref. [2] are employed within the SMSPS. In section (II) we introduce our approach to the canonical ensemble and present its results. In Section (III) we present an inter-comparison of our results with experimental data and with the Bethe formula. Section (IV) presents our conclusions and outlook to the future.

A. The Single Particle Partition Function
The first basic ingredient of our model is the Hamiltonian which can be written aŝ where H m is the mean field Hamiltonian, and H P is a Hamiltonian that accounts for the residual or "effective" interactions that induce correlations beyond those accommodated by the mean field.
For open-shell nuclear systems, such as 56 Fe, the nuclear pairing Hamiltonian [12] is particularly important. For our specific application, we therefore include pairing interaction contributions to states above the Fermi level. The second ingredient of our approach is the SP partition function Z 1 defined as Here, the index α denotes the set of quantum numbers of each, possibly degenerate, state and k labels the degenerate substates. E max denotes an energy cutoff defining the maximum resonance state of the mean field Hamiltonian, V denotes the volume of continuum state. The first term in the RHS of Eq.(2) represents the sum of the discrete single-particle states (SPS) generated by the mean field Hamiltonian plus the effect of the residual interaction. The 2 nd term in the RHS of Eq.(2) represents the sum of continuum states and the integral can be computed to yield Assuming the adopted mean field has spherical symmetry (which we adopt here) there is degeneracy with respect to angular momentum projection, m Jα , the sum over discrete states can be written as where is the degeneracy for the α th state. The partition function in Eq. (4) represents the partition function of SPS plus pairing. One common description that links pairing correlations to thermal effects is implemented within the Bardeen-Cooper-Schrieffer theory. In BCS the pairing Hamiltonian is expressed bŷ where α andᾱ denote a state and its time-reversed state, respectively. To have this model effective for hot nuclei, a thermal population of particles among the states is used, mainly by implementing a Monte-Carlo technique [1][2][3]13]. This technique, however, requires significant computational resources, even for limited orbits in the proximity of the valence shell. Another model for including pairing effects is based on Hartree-Fock-Bogoliubov (HFB) [14,15] theory. However, the HFB model is limited by particle number fluctuation and quasi-particle parity mixing issues [16,17]. In our approach, as commonly done in BCS calculations at zero temperature, the pairing interaction is considered acting only on energy levels above the Fermi level. Phenomenologically we can express this effect on the energy levels as where θ(ε f ) is the step function Thus the SP partition function in Eq.(4) becomes The effect of the pairing is to increase the energy gap of the Fermi level. The gap shift ∆ is set to be temperature dependent, expected to be maximum near zero temperature and to decrease as temperature increases to reach zero at a higher temperature. The calculation of ∆ will be discussed in sec.III.
Within the framework of the SMSPS the effects of pairing and spin-orbit splitting are averaged by arranging the nucleons using a technique, which we call configuration-restricted recursion. We implement this technique for a fixed species of protons or neutrons by constructing the partition function for a subgroup of that species that, for example, limits the values of the magnetic projection of the total angular momentum quantum number to positive values, signified by |m J |. Thus, we obtain the SPS partition function of that restricted range, denoted as Z SPS 1 |m J | , and given by where

B. The Nuclear Partition function and Observable Calculations
We then recur the partition function in Eq.(8) for the given m-projected configurations of Z 1 protons (or N 1 neutrons) using the recursion formula for spinless identical fermions, given in Ref. [18].
This formula for n identical spinless (or polarized) fermions is where n has to be replaced by Z 1 for the proton case (N 1 for the neutron case) in the |m J | states and by Z 2 for the proton case (N 2 for the neutron case) in the −|m J | states. We denote Z = Z 1 + Z 2 as the total number of protons, N = N 1 + N 2 as the total number of neutrons, and A = Z + N as the total number of nucleons. Note that for even Z (or N ) then For odd Z (or N ) we choose Z 1 = (Z + 1)/2 and Z 2 = (Z − 1)/2 (or N 1 = (N + 1)/2 and The recursion formula (9) gives exact values of the partition functions at specific values of n and β. Up to this point we obtain Z and Z |m J | N 1 for Z 1 protons and N 1 neutrons, respectively, which thermally populate the |m J |'s states. Following the corresponding procedure, we now obtain for Z 2 protons and N 2 neutrons that populate the −|m J |'s states.
The total nuclear partition function is constructed by With this product in Eq.(10) of restricted sub-configuration partition functions, the total magnetic projection quantum number is restricted, on average, to zero for each nucleon species (protons or neutrons). Thus, the fact that each nucleon species occupies SP states predominantly in spin-paired configurations is taken into account, on average.
Eq.(10) also implies that, at sufficiently high β, due to the degeneracy of time-reversed single particle states, for each proton (or neutron) in state J with the same statistical weight. Then, as β decreases, excitations enter that break this symmetry while retaining only a total M = 0 constraint for each nucleon species. We believe that this feature of our total partition function is reasonable since we expect that polarization effects are minimal at zero temperature and those polarization effects beyond simple thermal excursions that are accommodated here are unlikely to contribute as temperature increases.
In this way, we approximate average pairing of total spin projections suitable for the low to moderate excitation region. The additional coherent pairing energy associated with the ground state and lowest-lying states is very important when we calculate the level density for an open-shell even-even nucleus. This will be addressed phenomenologically when calculating level densities via the parameter ∆ in Eq. (5). The configuration-restricted recursion technique and the coherent pairing energy shift defines our proton-proton and neutron-neutron correlation approach in our SMSPS theory.
Once the nuclear partition function is computed for the desired system at a given temperature, the observables such as the average thermal energy E A , the heat capacity C A , and the number of levels per unit energy g A can be computed, respectively, in the canonical ensemble as [19] and The factor Z A (β) exp (βE A (β)) is the number of microstates Ω A (E). We also notice that the heat capacity and the level densities are independent of the lowest-state energy E gs as expected. We can easily prove in the canonical ensemble that [20] ∆E Therefore Eq. (12) can be written as As β → ∞, ∆E → 0 and the statistics become very low. Thus, at T = 0 we switch over to the micro-canonical ensemble to obtain level densities [20].  state to the0g 9/2 state, leaving a core of 8 protons and 8 neutrons. We refer to this model space as 11 orbits. The final model space is the no-core case which includes all available orbits (18 orbits) and continuum to attain a no core system. The volume of the continuum state is considered to be a spherical cavity with a radius equal to 2.3 times the mean nuclear radius. We adopt a mean nuclear radius R = r 0 A 1/3 where r 0 = 1.25 fm. Thus R = 4.78 fm for 56 Fe and R = 4.81 fm for 57 Fe. Support for this choice of volume for the continuum states will be presented when we discuss the heat capacity. Fig.(2) shows the excitation energy of 56,57 Fe versus temperature computed with SMSPS using 4orbit, 11-orbit, and no-core model spaces. Here, the excitation energies of the SMSPS are compared with the excitation energy obtained for 56 Fe using the Shell-Model Monte Carlo (SMMC) method of Nakada and Alhassid [5]. The SMMC is a standard technique since it includes a microscopic Hamiltonian that accounts for important residual interactions.

III. RESULTS AND DISCUSSION
The general trend for the excitation energy is that it is zero at and near zero temperature and it does not exhibit any significant increase for T < 0.2 MeV. This indicates the thermal energy is insufficient to excite nucleons to higher valence states. As temperature increases above T = 0.2 MeV, internal energy increases rapidly with temperature. The rate of rising excitation energy with temperature tapers off for 4 orbits and then for 11 orbits model spaces, as the model space eventually cuts off and yields no additional higher-energy states for the excited nucleons to occupy. That is, the internal energy saturates towards the maximum energy offered by the quantum configurations of these limited model spaces. Thus, above a certain temperature a larger model space is needed to provide more particles and states to contribute to the excitation energy.
The SMSPS 4 orbits results agree well with those of the SMMC up to T ≤ 1 MeV. The results of the SMSPS for T ≤ 1 MeV are controlled by phenomenological SP spectra that are in good agreement with experimental information. Beyond the 1 MeV temperature, the SMSPS results lie below those of SMMC even for higher model spaces. Note that the SMMC results are obtained with an additional orbital, the 0g 9/2 , included within its valence space. We observe that when we include this orbital and additional orbitals in the 11 orbit space, we extend the temperature range over which our SMSPS results agree with the SMMC results. We also observe that the SMSPS internal energy is close to the internal energy predicted in Ref. [2] using the shell-model quantum becomes lengthy. Thus we truncate the minimum temperature limit for larger model spaces such that the results of larger model spaces join smoothly with the results in the smaller model spaces.
In the case of 57 Fe, there are 11 neutrons in the f p shell at T = 0. Therefore, this isotope has a leading unperturbed configuration which is one neutron less than needed to fill the 1p 3/2 subshell.
We found that the addition of one neutron to 56 Fe does not have a significant contribution to the excitation energy as a function of T as seen by comparing results in the two panels of Fig.(2).
Although it is not apparent from Fig.(2), for E  higher lying orbits. Below the TVM temperature, however, all model spaces produce exactly the same behavior. This is understandable since at sufficiently low temperatures the valence nucleons dominate. We conclude that valence shell contributions to the excitation energy and heat capacity are dominant for T < 1.0 MeV. Fig.(4) shows the SMSPS heat capacity for 56,57 Fe computed at no core model space including the continuum for an extended temperature range. There is another maximum for heat capacities at T = 8.8 MeV, which marks the transition from resonance states to the continuum. The value and the position of this maximum depend on the continuum state volume: the larger the volumethe larger the maximum. As we vary the volume, the position of the maximum versus temperature can shift either towards higher temperature or towards lower temperature. We fix the volume such that the maximum is located at a temperature corresponding to the binding energy per nucleon.
In case of 56,57 Fe BE/A ≈ 8.8 MeV we find there is a unique value for the spherical volume of the continuum which satisfies this condition -when the radius of the volume equals to 2.3 mean nuclear radius of 56,57 Fe.
The rise of the value of the maximum when the volume increases, implies that the system finds it more favorable, as expected, to excite nucleons from resonance states to continuum states. We note that this maximum corresponds to the beginning of a phase transition from quantum gas to classical gas regime since the values of the heat capacities C/k B are trending towards the classical is computed for valence shells only. As we can understand from E x vs T and C vs T graphs the valence contribution is sufficient for T < 1.0 MeV or E x < 6.0 MeV. The experimental level density is histogrammed into 1 MeV bins using the state list in Ref. [6]. The experimental spectrum appears incomplete for E x > 4.0 MeV. The circular points in the figure represent the experimental level densities extracted from primary γ spectra for 56,57 Fe nuclei obtained with ( 3 He, αγ) and ( 3 He, 3 Heγ) reactions on 56,57 Fe targets using the Oslo method (OM) [7,8].
The Bethe backshifted formula values (BBF) displayed in Fig.(5) are computed using [9,10] where g 0 = 2/π and σ is the spin cut-off parameter given by [24]  MeV the effect of the continuum is imperative and has to be incorporated into the calculation for reliable results.

B. The Pairing Effect
We have commented that shifting the excitation energies by 1.   The agreement of the SMSPS with experimental data and BBF is gratifying. This agreement with BBF for 56,57 Fe extends to higher excitation energies, as Fig.(10) shows, beyond energies we used to evaluate ∆ as a function of T .

IV. CONCLUSIONS AND OUTLOOK
We have shown that the SMSPS is a flexible tool to compute the thermal properties of isotopes in the Iron region. The SMSPS has the ability to produce results in both near zero and high temperature at large model spaces. Two primary variables needed to be adjusted in order to refine the SMSPS approach: (1) The volume of the continuum states which needs to be chosen such that the heat capacity exhibits a phase transition between quantum to classical gas at temperatures approximating the binding energy per nucleon, (2) The pairing parameter ∆ which has to be calculated to achieve the correct backshift energy of the level density. The volume variable is crucial in the high-temperature region, whereas the pairing variable is important for even-even open-shell nuclei in the low temperature region. According to the SMSPS approach, A = 56 and 57 Iron isotopes have the following properties 1. For 0 < T < 0.9 MeV these nuclei behave like a superfluid with significant pairing effects.
2. For 0.9 < T < 8.8 MeV these nuclei exhibit properties reflective of a mixture of fluid (in the core) and quantum gas (in the valence and resonance states).
3. For T > 8.8 MeV these nuclei exhibit the transition to a non-interacting classical gas.
The main advantages of the SMSPS are: 1. Depends on the chosen SP states which means that the SMSPS accommodates physically realistic shell structures, including shell closure effects.
2. The theory is capable of treating all nucleons as active in a model space that extends to continuum.
3. The theory can admit pairing correlations in a phenomenological manner.

The SMSPS theory is computationally inexpensive.
The disadvantages of the SMSPS are its sensitivity to the mean field SP states, and its lack of a microscopic Hamiltonian. However, if the energy gap function due to the paring correlation of such SPS is obtained and the volume parameter of the continuum state is determined, the SMSPS is found to yield excellent values of level densities. In the future, we will investigate alternative models for the SPS such as those from mean field theory. The SMSPS model may be further improved by refining the configuration-restricted recursion technique with input from a fully microscopic NN interaction. However, this requires significant additional developments that we plan to report in the near future.