Constant temperature description of the nuclear level densities
CConstant temperature description of the nuclear leveldensities
Mihai Horoi a) and Jayani Dissanayake b) Department of Physics, Central Michigan University, Mount pleasant, MI 48859, USA a) Corresponding author: [email protected] b) [email protected] Abstract.
The spin and parity dependent nuclear level densities (NLD) are calculated for medium-heavy nuclei using shell modeltechniques. The NLD are used to calculate cross sections and reaction rates of interest for nuclear astrophysics and nuclear energyapplications. We investigate a new approach of describing the shell model NLD via a constant temperature parametrization. Thisapproach provides new information about the e ff ects of symmetries on the temperature of the low-lying nuclear states, and it isshown to be more versatile for applications. INTRODUCTION
The nuclear level densities are important quantities that provide interesting information about the nuclear structureproperties of the atomic nuclei [1, 2, 3]. They are very useful ingredients for the calculation of the compound nucleuscross sections [4, 5] of interest for nuclear astrophysics and nuclear engineering applications. In addition, the leveldensities are necessary for the partition functions entering the calculation of the reaction rates for nuclei in hot stellarenvironments [6].Over the years we have developed a methodology [7, 8, 9, 10, 11, 12, 13, 14, 15] for calculating the shell modelspin- and parity-dependent nuclear level densities using methods of statistical spectroscopy [16, 17]. A number ofimportant applications were also reported in Proceedings of di ff erent conferences [18, 19, 6, 20, 21]. Our methodis based on the first two moments, the centroids and the widths, of the nuclear configurations contributing to thelow-energy distributions. The results compare very well with those of full shell model diagonalization of the sameshell model Hamiltonian, which in some cases can be calculated using shell model codes, such as NuShellX [22].We found out that the shell model NLD calculated with nuclear Hamiltonians fine-tuned to describe the low-energynuclear spectroscopy, such as USDA [23] for the sd-shell nuclei or GXPF1A [24, 25] for the fp-shell nuclei describereasonably well the available level counting data for some sd-shell nuclei, such as Al and Si, for which manylow-lying levels are known. For the fp-shell nuclei one also gets good agreement with some of the low-energy neutronresonance data maintained by RIPL [26]. Others approaches to the nuclear level densities based on nuclear shell modelHamiltonians can be found in Refs.[27, 28, 29, 30, 31, 32, 33].The nuclear level densities can be used to understand the atomic nuclei as mesoscopic systems by looking totheir ergodic properties. In Refs. [1, 2, 3] we investigated these properties. For applications in nuclear astrophysicsand nuclear engineering one needs to properly use the NLD as input to nuclear reaction codes, such as TALYS [34]. Inthe past [6] we used our shell model NLD input as tables. This approach is di ffi cult to implement in TALYS and it hasother shortcoming related to the continuation in excitation energy of the NLD beyond 10-12 MeV. Given the recentsuccess of the constant temperature parametrization of the experimental data on nuclear density of states [27, 29, 30]we decided to investigate whether a constant temperature parametrization is appropriate for the shell model spin- andparity-dependent nuclear level densities.The paper is organized as follows. The next section gives a brief description of our moments method approachto calculate the spin- and parity-dependent shell model NLD, which is followed by a description of the constanttemperature parametrization of the NLD. The final two sections are devoted to the results and conclusions. a r X i v : . [ nu c l - t h ] O c t den s i t y ( / M e V ) E (MeV)J=0J=2J=4J=8
FIGURE 1. Fe NLD vs energy for several J -values. The symbols are the results of the moments method, and the lines are thefitting by the constant temperature formula, Eq. (5). MOMENTS METHOD NUCLEAR LEVEL DENSITY
We start following closely the approach proposed in Refs. [14, 8]. For reader‘s convenience, we first repeat the mainequations of the moments method. According to this approach, one can calculate the level density ρ for a given spin J and a given parity π as a function of excitation energy E x in the following way: ρ ( E x , J , π ) = (cid:88) c ∈ con f igs D c ( J , π ) G FR ( E x , E c ( J ) , σ c ( J )) , (1)where c are configurations of protons and neutrons in a set of valance spherical orbitals. Examples of such valencespaces are the sd shell consisting of the 0 d / , 0 d / and 1 s / orbitals, or the f p shell consisting of the 0 f / , 0 f / ,1 p / and 1 p / orbitals. The E c ( J ) and σ c ( J ) are the fixed-J centroids and widths, respectively, corresponding to theconfigurations c of nucleons distributed in the corresponding valence space orbitals. They can be calculated withoutperforming a full diagonalization in the valence space, by calculating traces of the first two powers of the Hamiltonian: E c ( J ) , σ c ( J ) ←− T r
S D c (cid:104) M | H q | M (cid:105) S D c . (2)Detailed expression for these traces can be found in Refs. [14, 8]. A high performance computer implementation ofthe algorithm used to calculate these moments can be found in [7]. Here E x = E − E g . s . , where E g . s . is the groundstate (g.s.) energy. This implies that a good estimation of the g.s. energy is required. This estimation is usually doneby direct shell model diagonalization, or using extrapolation techniques, such as the exponential convergence method[35, 36, 37], or the increased truncation method described in Ref. [8]. CONSTANT TEMPERATURE PARAMETRIZATION
Given the large interest for accurate nuclear level densities, several parametrizations of the density of states wereproposed over the years. Among them, the back-shifted Fermi gas formula [38] and the constant temperature formula[38, 27]. In the von Egidy and Bucurescu [29, 30] approach the constant temperature density of states is given by T ( M e V ) J 64Fe54Fe
FIGURE 2.
Temperature vs J for two iron isotopes. ρ CT ( E x ) = T e ( E x − E ) / T (3)Then, the J-dependent nuclear level density can be calculated by the approximate formula ρ CT ( E x , J ) = J + σ e − J ( J + / / (2 σ ) ρ CT ( E x ) . (4)where σ is an appropriate spin cuto ff parameter that depends on the excitation energy and other isotope specific factors.Here, the assumption is that both parities have the same nuclear level density. This approach has the disadvantage thatconstant temperature applies only to the density of states, and therefore the temperature is not constant for each J (orparity).The constant temperature approach success can be understood as due to a pairing first order phase transition.The semiclassical BCS description of the pairing is consistent with a second order phase transition. However, highernonlinearities induced in many body e ff ective Hamiltonians could lead to first order pairing phase transitions [39, 40,41, 42]. These many body interactions can be originating from full shell model Hamiltonian projected on restrictedmodel spaces, such as those included in the BCS approach. A detailed connection between these points of view is notyet available, but seems quite plausible. Additional arguments in favor of an exponential (rather than a Gaussian) tailof the distributions described by shell model Hamiltonian can be extracted from Figs. 9-10 of Ref. [3].Although in the shell model the phase transitions are not sharp, one can see that they are likely to depend on thesymmetries left [39]. Therefore, here we consider a constant temperature formula for each angular momentum J andeach parity π : ρ CT ( E x , J , π ) = T J π e ( E x − E J π ) / T J π (5)This parametrization of the spin- and parity-dependent NLD is similar to that of Eq. (3), except that the temperatures T J π and the energy shifts E J π are expected to be di ff erent for each parity and spin. It is expected to better describe anergodic system without mixed symmetries. T ( M e V ) A J = 2J = 5/2
FIGURE 3.
Temperature vs the mass number A for some sd - (left) and f p -shell (right) nuclei. Results for sd- and fp-shell nuclei
We calculated the moments method NLD for a large number of sd-shell and fp-shell nuclei using the USDA andthe GXPF1A Hamiltonians, respectively. In addition, we fitted the moments NLD to the spin and parity constanttemperature formula, Eq. (5). The results are analyzed below.Figure 1 shows the NLD of Fe calculated for J = , ,
4, and 8. The symbols are the results of the momentsmethod, and the lines are the fitting by the constant temperature formula, Eq. (5). The calculations were done in the f p valence space using the GXPF1A Hamiltonian. The slope of di ff erent curves indicates di ff erent temperatures fordi ff erent symmetries (total angular momenta), while Eqs. (3) - (4) are using the same temperature for all J s.Figure 2 shows the temperature for two isotopes as a function of the total angular momentum. This results clearlyindicate that for the same nucleus the temperature is dependent on the total angular momentum. In addition, one cansee that di ff erent isotopes of the same element having di ff erent masses could exhibit significantly di ff erent temperaturecharacterizing the low-lying spectra. In addition, one can observe the odd-even J staggering typical for the pairinge ff ects in even-even nuclei (see e.g. Figs. (5)-(6) of Ref. [39]).Figure 3 shows the spin and parity temperature as a function of the mass number A for series of even and oddmass sd-shell and fp-shell isotopes. For the even mass isotopes the selected total angular momentum is 2 (and positiveparity), and for the odd mass isotopes the selected total angular momentum was 5 / =
16 and A =
40, was also observed in the data [29, 30]. The general decrease of thetemperature with the mass number observed in [29, 30], is also present in our results. The temperatures tend to stay innarrow bands for a large number of isotopes of di ff erent masses, indicating that a parametrization and extrapolationof the quantities to other nearby isotopes could be possible.Figure 4 shows the spin and parity energy shift E J π as a function of the mass number A for the same isotopes,angular momenta, and parities as in Fig. 3. The energy shifts tend to be much more scattered than the temperature inFig. 3, suggesting that a better understanding of their larger range is needed before considering any extrapolations. E ( M e V ) A J=2J=5/2
FIGURE 4.
Energy shifts E vs the mass number A for some sd - (left) and f p -shell (right) nuclei. CONCLUSIONS AND OUTLOOK
In conclusion, we studied the moments method fixed spin and parity nuclear level densities for a large number of sd - and f p -shell nuclei using realistic shell model Hamiltonians. We also investigated the adequacy of the constanttemperature approximation for these shell model densities, and we argued that a constant temperature approach tothese densities may be better justified than for the density of states, which includes subsets from di ff erent symmetryrepresentations (e.g. di ff erent spins).We found that indeed, the temperatures corresponding to di ff erent total angular momenta of the same isotopecould be quite di ff erent. We also showed that these temperatures exhibit and odd-even spin e ff ect similar to that foundin the pairing strength of the low-lying states of nuclei. This feature supports the conjecture that the constant temper-ature description of the low excitation energy nuclear level density could be explained by a pairing phase transition atalmost constant temperature. However, more investigations need to be done to fully establish this connection.Finally, we investigated the behavior of the parameters entering the constant temperature formula Eq. (5), namelythe temperature and the energy shift, and we found that one could potentially extrapolate these parameters to otherneighboring isotopes of interest. This outcome could be very useful for a simple integration of the results of Eq. (5)within reaction codes, such as TALYS. ACKNOWLEDGMENTS
Support from the U.S. NSF Grant No. PHY-1404442 and the NUCLEI SciDAC Collaboration under U.S. Departmentof Energy Grant No. DE-SC0008529 is acknowledged.
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